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NBS SPECIAL PUBLICATION 567

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

Accuracy in

Powder Diffraction
-

NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU

OF STANDARDS

LIBRARY

Accuracy in Powder Diffraction

APR 2 1 198D

QC/OO

/9BO

Proceedings of a Symposium on Accuracy in Powder Diffraction

Held at the National Bureau of Standards

Gaithersburg, Maryland

June 11-15, 1979

Edited by:

S. Block

C. R. Hubbard

Center for Materials Science

National Measurement Laboratory

National Bureau of Standards

Washington, D.C. 20234

Sponsored by:

International Union of Crystallography

5 Abbey Sqare

Chester, CHI 2HU, England

Chemistry Division National Research Council of Canada

Ottawa, Canada K1AOR6

National Measurement Laboratory

National Bureau of Standards Washington, D.C. 20234

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

6 NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

f?('Mi\ (1U

Issued February 1980
Library of Congress Catalog Card Number: 80-600010

National Bureau of Standards Special Publication 567

Nat. Bur. Stand. (U.S.), Spec. Publ. 567, 553 pages (Feb. 1980) CODEN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1980

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Stock No. O03-OO3-O2153-9 (Add 25 percent additional for other than U.S. mailing).
Preface

The symposium on Accuracy in Powder Diffraction was held to meet the deep and

widespread need to assess the current position of powder diffraction including both the

fundamentals underlying the accuracy attainable in the entire measurement and analysis

I system as well as new developments in techniques and applications. The last international

I meeting on powder diffraction was in Stockholm, Sweden, in June 1959. In the past two

decades there have been great developments in materials science. Parallel with this has

been a renewed and enlarged interest in x-ray powder diffraction both for materials

analysis and characterization. The applications range from stress determinations to quanj titative analysis to structural determination from line shape.

The symposium was cosponsored by the International Union of Crystallography, the

I Chemistry Division of the National Research Council of Canada, and the National Bureau of

Standards. The organizing committee had an advisory board representing seven countries.

There were 115 attendees from 17 different countries.

These proceedings include most of the invited talks (two were not available at the time

| of publication), extended abstracts or papers from selected contributions and abstracts of

I all remaining papers. Pertinent discussion of the papers is also included. It is hoped

that these proceedings will serve not only as a record of the symposium, but as a source

book for the latest knowledge in this increasingly important field.

S. Block and C. Hubbard

Editors

It

Disclaimer:

Certain trade names and company products are identified in order to adequately specify the experimental procedure. In no case does such

identification imply recommendation or endorsement by the National

Bureau of Standards, nor does it imply that the products are necessarily

the best available for the purpose. Views expressed by the various

authors are their own and do not necessarily represent those of the

National Bureau of Standards.

ii i
Acknowledgments

The editors wish to thank the advisory committee and other members of the organizi

committee as well as the participants in the symposium. Special thanks are due to the

rangements committee for their assistance and the publication committee for producing

proceedings.

Committees

ORGANIZING

S. Block, National Bureau of Standards, Chairman

L. Calvert, National Research Council of Canada

J. B. Cohen, Northwestern University

C. R. Hubbard, National Bureau of Standards

P. Suortti , University of Helsinki, Finland

ADVISORY

W. G. Bronger, Inst. f. Anorganische Chemie, West Germany

M. Cernohorsky, J. E. Purkyne University, Czechoslovakia

A. J. Guinier, Universite Paris SUD, France

S. Hosoya, University of Tokyo, Japan

G. B. Mitra, Indian Inst, of Techn. , India

W. Parrish, IBM, San Jose, CA, USA

T. M. Sabine, New South Wales Inst, of Techn., Australia

ARRANGEMENTS

D. Appleman, Smithsonian Institution

R. B. Johnson, NBS

J. A. Lorden, NBS

R. F. Martin, NBS

B. A. Myers, NBS

K. Stang, NBS

PROCEEDINGS PUBLICATION

J. Bartlett, NBS

B. Myers, NBS

E. Ring, NBS

iv
Table of Contents

preface

[Acknowledgments

[Welcome

i

I. TOTAL PATTERN

Chairperson, L. Calvert

COMPONENTS OF THE TOTAL X-RAY SCATTERING (Invited Paper) 1

P. Suortti

NEUTRON DIFFRACTION — THE TOTAL POWDER PATTERN

(Invited Paper) 21

T. M. Sabine

SYNCHROTRON RADIATION AND ENERGY-DISPERSIVE DIFFRACTION

(Invited Paper) 33

B. Buras

ACCURACY IN X-RAY WAVELENGTHS (Invited Paper) 55

R. Deslattes, A. Hening, and E. G. Kessler, Jr. NEGLECTED CONSIDERATIONS FOR INTENSITY MEASUREMENT

(Invited Paper) 73

L. D. Jennings

II. INSTRUMENTATION AND AUTOMATION

Chairperson, Floyd Mauer

POSITION-SENSITIVE DETECTORS FOR POWDER DIFFRACTOMETRY

(Invited, Abstract only) 85

R. W. Hendricks, M. K. Kopp, and A. H. Narten

DETERMINATION OF THE SPECTRAL INTENSITY OF THE INCIDENT

BEAM IN ENERGY DISPERSIVE X-RAY DIFFRACTION (Abstract) 87

R. Uno and J. Ishigaki

TIME-OF-FLIGHT NEUTRON POWDER DIFFRACTION AT PRESSURES TO

35 KI LOBARS (Abstract) 89

J. D. Jorgensen

POWDER— A COMPUTING SYSTEM FOR X-RAY POWDER DIFFRACTION

CALCULATIONS (Abstract) 91

B. C. Osgood and R. L. Snyder

THRESHOLD LEVEL DETERMINATIONS FROM DIGITAL X-RAY POWDER

DIFFRACTION PATTERNS (Abstract) 93

C. Mai lory and R. L. Snyder

v
III. PROFILE FITTING

Chairperson, J. Rush

ACCURACY OF THE PROFILE FITTING METHOD FOR X-RAY POLYCRYSTALLINE

DIFFRACTOMETRY (Invited Paper) 95

W. Parrish and T. C. Huang

PROFILE REFINEMENT OF NEUTRON POWDER DIFFRACTION

PATTERNS (Invited Paper) Ill

A. W. Hewat

STRUCTURAL ANALYSIS FROM X-RAY POWDER DIFFRACTION PATTERNS

WITH THE RIETVELD METHOD (Invited Paper) 143

R. A. Young

A NEW PATTERN FITTING STRUCTURE REFINEMENT PROGRAM FOR X-RAY POWDER DATA (Abstract) 165

Ch. Baerlocher and A. Hepp

THE DETERMINATION OF STRUCTURAL PARAMETERS AND THEIR STANDARD

DEVIATIONS FROM POWDER DIFFRACTION PATTERNS (Paper) 167

M. J. Cooper, M. Sakata, and K. D. Rouse

STRUCTURAL REFINEMENT OF NEUTRON AND X-RAY DATA BY THE RIETVELD

METHOD: APPLICATION TO A1

203 and BiV04 (Paper) 189

D. E. Cox, A. R. Moodenbaugh, A. W. Sleight and H.-Y. Chen

TEMPERATURE DEPENDENCE OF THE ATOMIC THERMAL DISPLACEMENTS IN U0

2

: A TEST CASE FOR THE RIETVELD PROFILE REFINEMENT METHOD

(Paper) 203

A. Albinati, M. J. Cooper, K. D. Rouse, M. W. Thomas, and

B. T. M. Willis

STUDIES OF THERMAL MOTION USING CONSTRAINED PROFILE

ANALYSIS (Abstract) 211

E. Prince, C. S. Choi and S. F. Trevino

IV. ANALYSIS OF PEAK SHAPE

Chairperson, P. Suortti

ACCURACY OF CRYSTALLITE SIZE AND STRAIN VALUES FROM X-RAY DIFFRACTION LINE PROFILES USING FOURIER SERIES (Invited Paper) 213

R. Delhez, Th.H. de Keijser, and E. J. Mittemei jer

ACCURACY OF CRYSTALLITE SIZE AND STRAIN DETERMINED FROM THE INTEGRAL BREADTH OF POWDER DIFFRACTION LINES (Invited Paper) 255

J. I. Langford

DETERMINATION OF COMPOSITIONAL VARIATIONS BY X-RAY DIFFRACTION

LINE PROFILE ANALYSIS (Invited Paper) 271

E. J. Mittemei jer and R. Delhez

vi
STRUCTURES FROM POWDER DATA: DATA SAMPLING, REFINEMENT AND ACCURACY (Paper) 315

W. J. Mortier

V. LATTICE PARAMETERS AND INDEXING

Chairperson, Marlene Morris

ACCURACY IN METHODS OF LATTICE-PARAMETER MEASUREMENT

(Invited Paper) 325

A. J. C. Wilson

SOME STATISTICAL ASPECTS OF LATTICE PARAMETER EVALUATION

(Invited Paper) ..... 353

J. Mandel

DATA ACCURACY FOR POWDER INDEXING (Invited Paper) 361

R. Shirley

SUCCESSIVE DICHOTOMY METHOD FOR INDEXING POWDER PATTERNS

(Abstract) 383

D. Louer

THE DETERMINATION OF THE PRECISE LATTICE PARAMETER OF A DIFFUSE

MINOR PHASE IN A DILUTE BINARY URANIUM ALLOY (Abstract) 385

D. A. Carpenter and C. M. Davenport

PRECISION GUINIER X-RAY POWDER DIFFRACTION DATA (Abstract) 387

J. W. Edmonds

A SIMPLE GRAPHICAL METHOD FOR OBTAINING REASONABLY ACCURATE CELL

DIMENSIONS FROM X-RAY POWDER PHOTOGRAPHS OF HEXAGONAL AND TETRAGONAL MINERALS (Abstract) 391

E. E. Fejer

THE RELIABILITY OF POWDER INDEXING PROCEDURES (Paper) 393

A. D. Mighell and J. K. Stalick

X-RAY POWDER DIFFRACTION IDENTIFICATION OF CRYSTAL PHASES

WITH SUPERIMPOSED LINES BY THEIR SELECTIVE CRYSTALLIZATION

(Abstract) 405

I. Mayer, I. Gedalia and B. Laufer

COMPARISON BETWEEN DEBYE-SCHERRER, TRANSMISSION AND REFLECTION MEASURING MODES (Abstract) 407

E. Woelfel

vii
VI . APPLICATIONS

SESSION I. Chairperson, M. Kuriyama

STATISTICAL ANALYSIS OF THE MEASUREMENT OF GRAIN AND

PARTICLE SIZE WITH X-RAYS (Invited, Abstract only) 409

J. Hilliard

ANALYSIS AND TOPOGRAPHY OF LATTICE DEFECTS IN POWDER

DIFFRACTION PATTERNS (Invited Paper) 411

S. Weissmann

DETERMINATION OF PREFRACTURE DAMAGE IN FATIGUED AND STRESS-CORRODED

MATERIALS BY X-RAY DOUBLE CRYSTAL DIFFRACTOMETRY (Paper) 433

R. N. Pangborn, R. Yazici, T. Tsakalokos, S. Weissman and

I. R. Kramer

HIGH RESOLUTION POWDER DIFFRACTION AT ARGONNE'S ZING-P' PROTOTYPE

PULSED NEUTRON SOURCE (Abstract) 451

J. D. Jorgensen, F. J. Rotella, and M. H. Mueller

SESSION II. Chairperson, L Mordfin

STRESS ANALYSIS FROM POWDER DIFFRACTION PATTERNS (Invited Paper) .... 453

J. B. Cohen, H. Dolle, and M. R. James

X-RAY RESIDUAL STRESS EVALUATION BY AN ENERGY DISPERSIVE

SYSTEM (Paper) 479

M. Kuriyama, W. J. Boettinger, and H. E. Burdette

SESSION III. Chairperson, J. B. Cohen

STANDARD REFERENCE MATERIALS FOR QUANTITATIVE ANALYSIS AND

d-SPACING MEASUREMENT (Invited Paper) 489

C. R. Hubbard

STRUCTURAL ANALYSIS FROM GUINIER FILM DATA (Paper) 503

P. -E. Werner

OPTIMAL DEGREE OF AUTOMATION IN QUANTITATIVE X-RAY DIFFRACTION PHASE

ANALYSIS (Abstract) 511

A. Griger

VII. FURTURE OPPORTUNITIES IN POWDER DIFFRACTION

Chairperson, S. Block

STANDARDS FOR THE PUBLICATION OF POWDER PATTERNS: THE AMERICAN CRYSTALLOGRAPHIC ASSOCIATION SUBCOMMITTEE'S

FINAL REPORT (Invited Paper) 513

L. D. Calvert, J. L. Fl ippen-Anderson, C. R. Hubbard,

Q. C. Johnson, P. G. Lenhert, M. C. Nickols, W. Parrish,

D. K. Smith, G. S. Smith, R. L. Snyder, and R. A. Young

viii
SUGGESTIONS FOR A QUANTITATIVE EVALUATION OF POWDER

PATTERNS (Paper) 537

G . Donnay

X-RAY POWDER DIFFRACTION 543

L. D. Calvert

FUTURE OF POWDER NEUTRON DIFFRACTION 547

M. H. Mueller

JCPDS - INTERNATIONAL CENTER FOR DIFFRACTION DATA: PRESENT

AND FUTURE ACTIVITIES 551

G. McCarthy and D. K. Smith

POSTERS

DESIGN OF A HIGH-ACCURACY GONIOMETER FOR X-RAY POWDER

DIFFRACTOMETRY AT CONTROLLED TEMPERATURE 557

J. F. Berar, G. Calvarin, J. Chevreul, M. Gramond and

D. Weigel

THE EFFECT OF TWINNING ON PARTICLE SIZE BROADENING IN SOME OXIDES DERIVED BY DEHYDRATION REACTION 559

F. Watari

LIST OF PARTICIPANTS 565

ABSTRACT

The proceeding of the Symposium on Accuracy in Powder Diffraction presents the

papers, abstracts and discussions of the symposium held at the NBS, Gaithersburg,

Maryland on June 11-15, 1979. The symposium was jointly sponsored by the NBS, the

National Research Council of Canada and the International Union of Crystallography.

These proceedings contain a total of 24 invited and contributed abstracts. Many

papers are followed by an edited discussion. The proceedings are divided into the

following topics: Total Pattern, Instrumentation and Automation, Profile Fitting,

Analysis of Peak Shape, Lattice Parameters and Indexing, Applications, and Future

Opportunities in Powder Diffraction.

Key words: Applications; lattice parameters; peak shape; powder diffraction;

profile fitting; x-ray neutron.

ix
WELCOME

John B. Wachtman, Jr. Director, Center for Materials Science

National Bureau of Standards

Good morning and welcome to the National Bureau of Standards. Many of us here have

long and useful contacts with individuals and organizations in the field of powder

diffraction. I suspect all of you know of our participation in the Joint Committee for

Powder Diffraction and the role of some of our staff members in the publication of CRYST/ DATA and the Powder Data File. We have a deep concern for accuracy in powder diffractioi

Our work includes experimental and theoretical improvements in means for obtaining and

analyzing both x-ray and neutron diffraction data. I hope many of you will be able to tlj the tours on Wednesday of our neutron and x-ray facilities. We also seek to contribute

accuracy in powder diffraction through supplying standard powder for x-ray diffraction.

Because of our measurements and standards mission and our deep involvement in powde

diffraction we are very pleased to co-sponsor this Symposium together with the National

Research Council of Canada and the International Union of Crystallography.

This Symposium is very timely. There is a resurgence of interest in powder diffraci

as evidenced by recent formation by the American Crystal lographic Association of a speci

group in applied crystallography with powder diffraction as one of the major components.

Several developments contribute to this resurgence. New experimental developments, incl

synchrotron radiation and improved detectors, are opening exciting new possibilities.

Advances in theory coupled with powerful, on-line computing ability are greatly improvin

feedback-controlled experiments and analysis of patterns. Availability of microprocess

reasonable prices is making many of these techniques available for general use. Concer

precise control of phase composition and microstructure is enhancing the role of powder

diffraction in quality control. Concern with toxic substances and pollution control pi

a^ new demands and opens new opportunities for powder diffraction.

As I indicated, we at NBS are very pleased to be co-sponsoring so timely a Symposiu

If there is anything which you would like to recommend for the future role of NBS, we wc welcome your suggestions.

x
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

COMPONENTS OF THE TOTAL X-RAY SCATTERING

P. Suortti

University of Helsinki

Si ltavuorenpenger 20 D SF-00170 Helsinki 17

Finland

Separation of the non-Bragg components of scattering is discussed

in analytic and synthetic approaches. Various interactions of the

x-ray photons with the electrons are considered, and the radiative

processes are studied in more detail. The spectrum of the radiation

is smeared by the detection system, and the possibilities of separating

the elastic or nearly elastic scattering on the basis of energy is

discussed. Calculations of TDS and multiple scattering are treated

briefly, and the use of the approximative sum rules of kinematical

diffraction in connection of an absolute intensity measurement is

introduced. The role of the background in the diffraction pattern

synthesis by profile fitting is also considered. A few examples of

extrapolation of the background level from the separable parts of

the Bragg reflections are given with the Voigtian as the profile

function. The evaluation of the thermal parameters from the powder

diffraction data is considered as a concrete example. The results

demonstrate the importance of a correct background separation and

a realistic profile function.

1. Introduction

The total scattering of (monoenergetic) x-rays from a crystalline sample is a very

"oad subject, and we have to limit the scope right from the beginning. Our point of view

> very practical: we want to separate the elastic part of the total scattering and resolve

: to individual reflections. However, it has been demonstrated that this separation is far

rom trivial, and it cannot be performed accurately without a careful consideration of all

he components of scattering.

Absorption and scattering of photons in medium is summarized by the Kramers-Kronig

Rations. The significance of these relations to our practical goal is that the dispersion

i i
terms of the atomic scattering factors, f and f " , can be evaluated from the attenuation

coefficient. The scattering can be divided into resonant and non-resonant parts, and we

will summarize briefly the various processes and study how they can be separated from eac

other.

It is clear that an unambiguous treatment of the total scattering must be based on t analytical approach outlined above. Some parts of this analysis are based on theoretical

calculations or independent data which may not be available for the compound being studie

In such a case, the results are synthetized within a phenomenological model of scattering

diffraction, and the results are ultimately justified by internal consistency. The con sistency may be artificial, however, and the parameters of the model do not necessarily

correspond to the physical quantities they are purported to represent. Test cases, where

the analytic and synthetic approaches can be compared, are therefore crucial for a critic

appraisal of the reliability of the fitting procedures. We will demonstrate the problems

involved by a simple example and suggest improvements to the present day methods.

2. Scattering Cross Section

The interaction Hamiltonian between an atomic electron and the radiation field is gi

by

2

H' = " A-p + -5-

2 A-A , (1) mc " ~ 2mc^ ~~

i \

where p and A are operators with the classical analogues of the electron momentum and thj

vector potential of the radiation field, respectively. The A-p term makes no contributij

in first order, but taken twice it gives rise to the resonant scattering, which may be m^ larger than the contribution of the A-A term. In the resonant scattering, the atomic st

A absorbs the incident photon of energy fiu^ and becomes an intermediate state I and simu

taneously or subsequently state I emits the outgoing photon of energy and changes in; state B. In the A-A interaction, there is no intermediate state. These processes are illustrated by the space-time diagrams of figure 1. The resonant scattering consists of fluorescence lines of Lorentzian shape or of re onant Raman scattering, which has a low energy tail due to sharing of energy between the!] I outgoing electron and photon [1-4] 1

. Both components are isotropic and give only a smooth background to the powder diffraction pattern, or can be eliminated altogether by

energy discrimination.

1 Figures in brackets indicate the literature references at the end of this paper.
tM

tfhere rg is the electron scattering length, uu = tu^-u^,

Figure 1. Space-time diagrams for scattering of photons by atomic electrons. In the

resonant scattering (a and b) an intermediate state prevails between t-^ and and

c represents non-resonant scattering. The wave-vector and the (unit) polarization

vector of the incident photon are given by k^ and e^, and those of the scattered

photon by and e^.

The non-resonant part can be written as

d

2a

du^dQ (?)

(?)

<!i-!2>"

II <B e AB I I ^'I:|a>| 2 6(E

B-EA

-1iu)) (2)

(e-j/^r S(k,u>)

k = k, - k9

, and S(k,io) is the

Jynamic scattering factor [5]. In the energy region of fib = 10 keV, the various scattering processes are characterized by the entity accepting the momentum and energy transfers,

k and Ae = fiu). For Bragg and disorder scattering k is absorbed by the average lattice

and Ae = 0; in the thermal diffuse scattering (TDS) the interaction is with the vibrating

atoms, and Ae is from 0 to 0.1 eV; in the plasmon scattering collectives of electrons are

involved and Ae = 10 eV; and finally, in the Compton scattering k is carried by one elec2

tron with an energy change of order 10 to 10 eV. The energy spectrum of the various

processes is illustrated in figure 2. For completeness, also the resonant contributions

3re included.

3. Spectral Analysis

I In most cases, the subject of interest is the unmodified (Bragg + disorder) part of the

Scattering, and the foremost practical problem of powder diffraction is the separation of

3
Coh.

fia>,-(Q L+EF ) ^^-(Q^Ep)

I

-1500 -1000 -50 0 E

K/3

Coh.

II

-10000 -9000

b)

X/2

-1000 0

Ar

17440

Figure 2. Spectra of the radiation scattered by a mono-atomic sample: a corresponds approximately to scattering of CuKa from Ni at 28 = 20° and b to scattering of MoKa from Ni at 26 = 150°. Coherent scattering (Bragg, disorder,

TDS) is denoted by Coh, Compton by C, plasmon by P, RRSa and RRSB signify the

(ls,2p), (ls,3p) contributions to the resonant Raman scattering, respectively,

\/2 the coherent scattering of the half-wavelength, and Ka plus KB the specimen

fluorescence. The energies are in eV.

this part from the rest, from the background. We will start with considering the possibilities of making this separation on the basis of the energy shifts of the background

components.

It is possible to separate the elastic scattering from the TDS and other inelastic

scattering by means of the MSssbauer effect [6-8]. The available intensity from the y-ray

57 sources (usually the 14.4 keV line of Co) is rather low, and the technique has been

applied only to measurements of single crystal reflections. The background level (TDS + Compton) in the recent measurements has been typically of order 1 count/s, which would

probably be sufficient for an absolute determination of the background intensity over the

whole measuring range. The spectra of figure 2 are convoluted by the detector resolution function, and the cases of the scintillation counter and the semiconductor detector (SCD) are illustrated in

figure 3. The scintillation counter is barely able to reject the A/2 component or low

energy fluorescence, but it has other advantages such as low price, reliability, and large

effective area. At large incident energies and scattering angles, the SCD's separate the

4
RRSd

RRS/3 + P+OCoh.

Figure 3. The spectra of figure 2 as convoluted by the detector resolution function;

in a by that of an SCD and in b by that of a scintillation counter. The intensity

ratios correspond to background scattering.

Compton contribution completely from the elastic peak, and this is utilized in the Compton

profile measurements. Also at the crystal lographic energies of order 10 keV some degree of

separation is possible at large scattering angles, where the energy shift becomes a few

hundreds of eV [9]. This is very important in studies of liquid or disordered structures.

The resolving power of a crystal monochromator placed in the diffracted beam is typically a few tens of eV, comparable to the energy difference between Ka^ and Ka^ of the

commonly used radiations. This is not sufficient for separation of the TDS and plasmon or Compton scattering at low angles, but the rest of the background is effectively eliminated.

Accordingly, this construction has become increasingly popular in the recent years. The

drawback of the diffracted beam monochromators is their non-uniform response. Most of the

commonly used monochromators are mosaic crystals, and the reflectivity may vary much from

point to point, see figure 4. On the other hand, elastic bending of nearly perfect crystals

(Ge, Si, quartz) results in monochromators with large areas of uniform sensitivity. The

reflecting range of a perfect crystal is very narrow but the peak intensity high, and with

proper dimensions very effective constructions are possible [10,11]. Much of the renewed

interest in monochromators is due to the increasing use of synchrotron radiation, which will

open up new avenues also for the powder method [12,13].

5
1—

• •

• • oo°o<fc°Oo< '°oo 0

o 0o o

• •

• • < •

• ••

•

•

• • •. •• • v

1 1 1

Figure 4. Reflectivity of bent crystal monochromators along a line of the crystal

face in the diffracting plane: graphite (00.2) (•) and quartz (10.1) (o). The

diameter of the pinhole beam has been of order 0.1 mm.

Elastic and Nearly Elastic Scattering

4.1. Kinematical theory

The preceding discussion implied that in practice the coherent part of the scatterin;

Bragg, disorder and thermal diffuse, is detected together. Their separation is based on differences in the angular behavior, which, in turn, is derived from a model for diffracti

from the powder sample. In the following, we will consider the general properties of thei

components and the practical possibilities of separating them.

The coherent scattering from a modulated structure can be divided into two parts: f

from the average structure, I,(k), and that from the distortions, l^W- In the kinematicj

approximation of diffraction [14], in electron units,

I x(k) = |F(k) £

m=0

i k*r

1,(10 = £ 4> m (k)V(r) e

i k- r

m=0

(3a)

(3b)
where r is the separation between two unit cells, V is the volume of the crystal, and V

c that of the average unit cell. The structure factor of the average crystal, F(k), and

that of the modulation, <t> m

( k )> are defined by

F (k) = F(k) + <|> (k) ; 4 (k) = $ 4 (4)

n ~ ~ Yn ~ ' Ym ~ Yn Y n+m

v 1

V(r) is the autocorrelation function of s(r), the shape function of the crystal, and S(k)

is the Fourier transform of s(r). The Bragg reflections are centered at k^ , and their

profile is given by S(k). It is worth noting that the diffuse scattering from an infinite crystal, I^k), is convoluted by the same profile function S(k). Also, the

effects of varying k^ and (beam divergences) can be included in S(k).

In general, the coefficients §mof the Fourier series expansion of IJ,(k) are functions

of k

00 ik*r

= £ <L (k) e ~ ~m

» (5)

m=0

which means that I^(k) is only quasi-periodic in k. Suppose, however, that we can find a function f(k) such that I£(k) = I^kV^k) is periodic in k. If there are essentially

different contributions to I^Ck), division to a few terms may be useful; I£ .(k)/,P .(k) , so that the functions *j(k) would be smooth. For instance, these could account for the

contributions of different vibration modes to the total TDS. The Fourier coefficients

mjj' which are independent of k are

V r - i k • r - -

c J

l»(k) e ~

~m

d

3

k , (6)

'm , .3 J

x 2 (2n)

v

*

c

and

ra (k) = V(k) ^ , (7)

3 where the integration is over the reciprocal unit cell V* = (2n) N^. The first term in

the series (6) is

f

I 2

( b) *\

V*

c

(8)
and if the average value of I£(k) is normalized to unity,

VIP = F n

( !P

F n

( !P " F(

!P

F(

!P = t(k)

• (9)

In the following, we will exclude substitutional disorder, and accordingly for thermal motior

and homogeneous strains

<t» 0(k) = |F(k) 2

|- |F(k)|

2

. (10)

In this case, the diffuse intensity from a disordered structure consists of a periodic

function I£(k) modulated by ^(k), which covers the variation of the structure factor from

one reciprocal unit cell to another. For the TDS, the relevant parameters of <l>Q(k) are the

amplitude ratios of the various vibration modes: for a specific example see [15].

The integral value of a Bragg reflection hkl is

i3 A<« d Vl kwl 2 (2n)

• TM v

*

c

An approximation for the diffuse scattering is obtained by taking (}>q( k) at k^

(2n) w *

(12)

V

c

This indicates that the Bragg plus diffuse scattering approximately equals to the scattering from the undistorted structure (cf. [16]).

4.2. Powder TDS and multiple scattering

It is rather difficult to describe the scattering from static distortions without some specific model, whereas the features of the TDS are common to all crystalline solids. In

v -2

the harmonic approximation, the first order term of I£(k) is proportional to Ej(q)u) j(Q)i

where E.(q) is the energy of the mode of angular frequency w.(q) and wave-vector q. For a

J ~ J ~ crystal with n atoms in the primitive unit cell, three of the branches j are acoustic with

ui^(q=0) = 0, and 3(n-l) are optic with u)g(q=0) i- 0. The acoustic vibrations make the TDS

to peak at the reciprocal lattice points, while the contribution from the optic modes varies

slowly with q, as well as that from the processes where two or more phonons are involved.

The spherical averaging which takes place in powder diffraction is illustrated in

figure 5. This smooths I£(k) appreciably, but the peaks at k^ persist. The number of the

reciprocal lattice points that contribute to I£(k) increases with k, and the peaking part of' I£(k) becomes less pronounced; also the increase of the multi-phonon processes has a similar^

effect. Figure 6 shows the result of a calculation for a mono-atomic fee crystal. In the I

8
general case of many atoms per cell, this would represent only the fluctuating part of I^k),

which lies on the top of the smooth contributions to I£(k). The average value, I£(k) = 1,

is also shown in the figure.

Figure 5. (Oil) plane in the reciprocal lattice of the fee crystal. The Brillouin

zones are marked with thin lines. Powder averaging is indicated by the circles of

radi i and k r :i "2 contribution at k„.

Vector q shows one-phonon TDS at k^ and q^ + q 2

a two-phonon

The insert shows schematically the TDS functions for an NaCltype crystal along the broken line.

For the general behavior of I^k) also the spherical average of <J> 0 (k) is to be calculated. The result is the Debye equation for the difference of scattering of one distorted

unit cell from that of one perfect cell,

<t» 0 (k)

n n I I

j

i

-M M fj(k)f.(k) {1

i , si nkr

1 kr (13)
where f is the atomic scattering factor, M the Debye-Waller factor, and r = r. - r. . At

~J ~ 1 large values of k, the terms i ^ j become small, and

-2M

> Q (k) -> 1 f

2Ak) {1 - e

J

} , (14)

which approximates the average value of I^k).

2a sinS/X

Figure 6 The powder TDS, l£(k) from acoustic phonons for the fee structure. The in^ parameters are those of Ni. The broken line indicates the average value

singular at the reciprocal points, but the peaks are rounded off by the convolution

with the instrument function.

I£(k) is

Now we will turn to the practical problem of separating the background from the Bragg

reflections. Suppose that all other components but the TDS have been subtracted by energy

analysis or by calculation from independently known scattering cross sections. This latte

approach requires an absolute measurement of the scattered intensity, and this is supposedj to be the case also in the following analysis.

The Bragg reflections lay on the top of an intensity distribution exemplified in

figure 6, although the sharp peaks are rounded off due to limited resolution. It is clear

that some fraction of the TDS peaks is counted with the Bragg intensity, if a straight

background line is assumed. On the other hand, the tails of the reflections may extend a few degrees to both sides of the peaks and they begin to overlap even at moderate values o

10
hkl. Therefore, the real problem of background subtraction is often that of finding the

correct level of the background, and to this end the average I£(k) suffices.

Utilization of eq. (13) or (14) requires a knowledge of the thermal vibration para2 meters, = Mj/(sin0/A) , but in a typical case they are among the parameters to be

determined from the powder pattern. 4>q( k) can be determined at small or intermediate

angles and extrapolated to the large angles, but the problem is that f j(k)exp(-Mj) is a function of both k and j. At high enough temperatures Ej(q) = kgT, and the average B is proportional to T in the harmonic approximation. Accordingly, measurements at two tem- peratures would yield the correct separation between the TDS and Bragg scattering. This

method has been used for determination of the B-values with good results [17]. However,

many important cases are excluded because of large anharmonic effects or phase transitions. Perhaps, the best experimental method for finding the background level is a measurement on a well crystallized powder sample. Here, the limiting factor is the

instrumental broadening of the Bragg reflections.

The correction for the included TDS is based on a calculation of the scattering from

the acoustic phonons. The minimum information for the calculation is an approximate

knowledge of the elastic constants. The sharp maxima at the Bragg peak positions are rounded off by the convolution with the profile function. An estimate for the fractional

correction to the integrated Bragg reflection is given by the Chipman-Paskin formula [18],

which is valid for well separated reflections. A formula which may be applied to crystals

with more than one atom in the primitive unit cell is given by [19];

_

8" k

BT A(28)cos6 sin 6 n ,. a

p ,2 ,2

UDJ

where A(29) is the width of the scan in rad, p is the density of the diffracting material,

and v. the average velocity of the transverse long-wavelength acoustic phonons. To give

. . -23

an idea of the magnitude of the correction we substitute typical values as k^T = 4.10 J,

p = 4.103 kgm"

3

, v. s 2.10 3 ms"

1

, A(26)cos6 = 0.05, X = 1.5 A. This results in a = 0.2 A

2 2 (sine/A) . This is of the same form as the temperature factor, M, and if the TDS correction

p; °2 . is ignored, the average B-value would be too small by about 0.1 A in the present example.

The above calculation was for the case where the reflections could be separated

completely from the background, and the included TDS was the peak above the straight background line. The use of the average TDS as the background may compensate the correction

for the TDS to a large extent. This is illustrated in figure 7. This compensating effect

occurs when the reflections are wider than the TDS peaks above the average value. Evidently,

this balancing becomes exact when the overlap of the peaks is large enough to make I-^(k) + ^(k) everywhere larger than the average TDS. However, sharing of intensity between the

reflections is not usually correct, particularly when strong and weak reflections are next

to each other. The neutron case is more favorable, as phonons which are faster than the

neutrons do not contribute to the TDS peak.

11
I Figure 7. Separation of the background and correction for the included TDS. Total

scattering is marked by the solid line, TDS by the broken line, and resolving to individual reflections by the dotted line. The partial compensation of the include*

TDS by the tails of the reflection below the average background line (-.-.-) is

indicated by shading.

The x-ray photons that are scattered more than once in the sample cause another con- tribution to the observed radiation. In case that most of the scattering is only slightly

modified, the calculation is reduced to a geometrical problem, where the single scattering

is used as input [20]. Typically, there are small peaks plus a smooth contribution, which

can be ignored in most cases. However, if the absorption coefficient is small and the

sample large, the contribution to the total scattering may be large enough to deserve a detailed calculation.

The analytical approach to the total x-ray scattering is used very seldom, although

in many cases data is available for a complete calculation. It has been demonstrated

that the x-ray scattering can be resolved to identified components without a remainder

through an energy analysis and with sufficient knowledge of the lattice dynamics of the

crystal [21]. From the practical point of view, an important result of this analysis is that the actual widths of the reflections are very large, typically of the order of 5°

in 26. More fundamentally, comparison of the reflection profiles and the integrated

intensities reveals deviations from the predictions of the kinematical theory.

The analysis of the background even in the simplest form where only the average TDS

is calculated requires a knowledge of the absolute scale of the intensity. It can be

estimated on the basis of the kinematical theory, but it is far more preferable to determine the scale experimentally. The primary flux of the x-ray photons can be measured

with a set of calibrated foils [22], or with a reference scatterer, such as the standard

Ni sample for CuKa radiation [21].

5. Powder Pattern Synthesis

In the synthetic methods, the profiles of the Bragg reflections are "known," and the

reflections are separated from the background and from each other by the least squares

12
fitting of the superposition of these profiles to the experimental pattern. This method

was first applied to the neutron diffraction patterns [23,24], and in recent years several

studies of x-ray case have been published [25-27]. Typically, the output includes cell

dimensions, atomic positions, and anisotropic thermal vibration parameters. The main problems of the method are separation of the background and finding the correct profile function,

and some of the results are very sensitive to the choices made.

5.1. Separation of background

Profile fitting presumes that the Bragg peaks are superimposed on a constant or slowly

varying background. We have seen that the actual background is not smooth, and so the fitted

pattern is always affected by the TDS to some extent. The peaking part of the TDS is included

in the profile function, and part of the correction for the TDS is taken care by the use of

the average TDS as the background level. However, the integrated intensity of an individual

reflection may be in error by an amount comparable with a from eq. (15).

In the actual fitting procedures, the background is determined at a few scattering angles

on the basis of visual examination of the pattern, and linear interpolation is used between

these points. The selected background level depends on the assumed profile of the powder

reflections; for instance, if a Gaussian profile is assumed, the background level should be

found at a distance of 1.5 to 2 times the FWHM from the peak. The relation between the

background level and the profile function can be made explicit by including the background

level in the parameters to be refined [28].

The correct background level can be found if the behavior of the tails of the reflections

is known. The particle size effects are specific to the sample, but in many cases even the

tails of the reflections are determined by the instrumental factors. In such a case, the

profile function for a given instrument can be extrapolated from a few carefully studied

test cases.

5.2. Profile function

The actual profile can be found only through the analytic approach described earlier.

Various functions have been compared with the experimental powder profiles [29-32], but

sufficient attention to the separation of the tails from the background has not been paid.

Scattering of CuKa radiation from Ni powder was analyzed very carefully in order to give

an example of the possibilities of the powder method [21]. The various components were

subtracted from the total one by one, and at the end only the Bragg reflections were left

on a small residual background of amorphous scattering. This data can be used for an analysis of the profile functions without the ambiguity arising from the unknown backo ground level. Also the scattering of 1.51 A neutrons from the same powder was measured

and the background was determined on the basis of the x-ray profiles. Furthermore, the

x-ray profiles were deconvoluted by the instrument function in order to reveal the

effects of particle size and strain. A few examples are shown in figure 8.

13
Figure 8. Neutron and x-ray reflections from Ni powder on arbitrary scales. In the

neutron case reflections 311 and 222 are shown in the logarithmic scale, the full

line is the Voigtian fit, where the individual contributions are shown by the dotted

lines, and the dash-dotted line is the Gaussian fit. The symmetrized x-ray reflects

111 from a sample pressed at 480 MPa is shown in linear and logarithmic scales. The

full line is the Voigtian fit and the broken line the Lorentzian fitted to the

-1 -2

L $ tail of the reflection.
The peak part of the reflection profile is broadened by the lattice strains, but this

broadening may be masked by the instrumental factors. In practice, convolution of the

various effects results in a shape that can be approximated by a Gaussian. According to the

-1 -2

kinematical theory the tails of the reflections fall off as L , where L is a characteristic dimension of the crystallites and = A(20) is the angular distance from the center of

jthe reflection. Also the effects of the other factors may extend to this region. Inclusion

Jof the TDS above a straight background line affects the tails, and at least in neutron

diffractometry the instrumental effects are seen in the tails, too. This gradual change

from the Gaussian peak to a long, Lorentzian type tail is covered by the Voigtian, which is

the convolution of the Gaussian and the Lorentzian [33,34]. The examples of figure 8

I . .

' "2-3

demonstrate that a very close fit is achieved down to an intensity level of 10 to 10

times the peak intensity, and that the improvement over the customarily used Gaussian is

clear in the tails of the reflections. The deviations in the x-ray case show, however,

that strain and particle size effects cannot be simultaneously parametrized by the

-2-2 ° Voigtian. The tail is of form L § with L = 1000 A, and the shape between 0.1° and

o 0.5° is mostly due to strains. The crystallites are understood to be about 1000 A diameter but appreciably strained at the same time.

The good description of the neutron diffraction profiles of Ni encouraged us to study a number of other cases, although a similar analysis of the background was not possible. The

parameters that determine the Voigtian, the FWHM and the integral width, turned out to be

typical to the instrument used (fig. 9). At least in this case, it looks feasible to use

the Voigtian as the profile function and even to include the background level in the parameters being refined, although the convergence of such a procedure must be studied carefully.

In any case, it is obvious that the improved profile function will make the sharing of

intensity between the reflections much more correct, particularly when strong and weak

reflections are closely spaced.

0.50 100 150 26(°)

ted

cti<

Figure 9. The ratio of the FWHM to the integral width, 2w/p, of powder reflections

o o measured with 1.38 A to 1.91 A neutrons from various compounds. The broken lines

indicated by G and C show the values of 2w/p for a pure Gaussian and a pure Cauchy

(Lorentzian) profile, respectively. Ag/\ 1s tne ratl ° °f the area under the

Gaussian to that under the Voigtian, when the peaks and FWHM's coincide. The solid

2 2 line is a fit of form A /A = exp(-2ABsin d/K ) to the filled dots measured for Ni. g v

K y 15
6. Thermal Motion Parameters

The reduction of the Bragg intensity with the increasing temperature can be related

to the vibration amplitudes of the atoms through the harmonic approximation. The integrate^ counts of a reflection in the ideal powder approximation is

n - n J- ^poAkl . I" ,2 n ,, Vl - n 0

r

e -..2 . . ~r Ahkl l F hkl I ' (16)

8nV

c sin6

hkl sin 6^

where n^ is the incident photon (neutron) flux, tjj the planar angle intercepted by the slit

height, K -j the polarization factor, and p^ the multiplicity. The absorption factor

A^

k i depends on the geometry and

F hkl = 1 ( f

j

+f

j

+if?e^'1

V,

hu M '

where h is used for the Miller indices h,k,l. In the neutron case r is substituted by

u ' ' e

J unity, and f . by b., the nuclear scattering length. In the following, we will study a cubi

J J o metal with one atom per r the primitive r unit cell, and so 8 = B-6 = B/4a , where a is th ' "pv pv 1 lattice constant. The thermal parameter B can be determined from an absolute measurement of n^ provide^

that f (or b) is known, or B can be evaluated from a comparison of the relative values of j of many integrated reflections. In the case of Ni , we can compare the results obtained by

different methods. These include absolute x-ray measurements on ten different samples and

p three types of analysis of relative neutron measurements from the same powder.

The x-ray intensities are affected by the specimen granularity, preferred orientation

and primary extinction. The effects of the first two could be corrected experimentally,

and extinction-free values could be extrapolated from the measurements on different samplesThe background was separated through the analysis described earlier. Although large, the ; E correction for anomalous dispersion, f

1 , is known accurately from the interferometric

measurements [35]. The specimen effects are negligible in the neutron case, and the only

problem is the separation of the background. There is no need for the TDS correction

as almost all phonons are faster than the 1.51 A neutrons. The reflections were integrated point by point and from the fitted profile functions. The actual fit was made only

with the Gaussian and the difference to the Voigtian was estimated from the ratio of the

integral values, A

g/A

v

» taken from figure 9. The results of table 1 show that a careful analysis of the background leads to a good

consistency of the x-ray and neutron values. Also the agreement with the theoretical and j most of the other experimental values is close. The remarkable features are the failure of|

the Gaussian fit and the success of the Voigtian. This difference is due to the tails of

the reflections, because the same background was assumed.

16
Table 1. The harmonic thermal motion parameter, B, for Ni.

hod B (A

2

)

powder, 1.542 A x-rays

o

1. 51 A neutrons

calculation, force constant model

o powder, 0.709 A x-rays

o powder, 0.709 A x-rays

o powder, 0.8 to 1.5 A neutrons

powder, neutrons, T0F

0.34 ± 0.02

0.34 ± 0.02

0.46

0.32

0.381

0.386

0.37

0.426

0.34

present, absolute f-values

present, relative b-values

present, Gaussian fit

present, Voigtian fit

Barron and Smith [36], from

phonon dispersion data

Paakkari [17], two temperatures

Inkinen and Suortti [37],

relative f-values

Cooper and Taylor [38]

Windsor and Sinclair [39]

Discussion

The preceding chapters have centered on the problem of separating the Bragg scattering

from the other radiation recorded by the detector. An exhaustive analysis is feasible

only in the simplest cases, but these may serve as indispensable reference points for

studies of complicated compounds. It has been found that the actual widths of the powder

reflections are typically several degrees, and so the real problem is finding the correct

background level. Quite little information is needed for an approximate calculation of

this level if the absolute scale of intensity is determined. It is emphasized that it is

very little additional effort to obtain absolute values rather than accurate relative

values. From the synthetic point of view, it may be possible to find profile functions

which are correct enough to allow separation of the tails of the reflections from the

background. Again, the results of this approach must be tested in cases where also the

analytic approach is possible.

When the reliability of the integrated Bragg reflections improves, the problem becomes

that of conversion of the scattering power to physically meaningful parameters. The Bragg

scattering is affected by departures from the conditions of the ideal powder sample, which

is assumed in the derivation of eq. (16). The powder particles of the actual sample are

not independent and randomly oriented, and they do not consist of small perfect crystallites.

There are correction methods for the effects of granularity and preferred orientation,

plthough they require tedious subsidiary measurements. On the other hand, there is no adequate diffraction theory for a sample where the crystallites are large enough to cause

>rimary extinction and are strained at the same time, and which contains also amorphous

Jnaterial between the crystallites. The effects of these deviations are seen in the x-ray

profile of figure 8, and they have been analyzed more thoroughly elsewhere [21]. In a 17
typical case, the F-values obtained by a straightforward application of eq. (16) may be in error by several percent. Considering the other limitations of the powder method, this

may not be much, but even at the moment powder data is used for evaluating values of

physical parameters which are sensitive to errors in F. It is hoped that some of the

present activity with the practical aspects of powder diffractometry will be directed to the fundamental problems of the method.

The author wishes to thank Dr. L. 0. Jennings for many years of collaboration and for

putting forward many of the ideas presented in this paper.

[1

[2

[3

[4

[5

[6

[7

[8

[9

[10

[11

[12

[13

[14

[15

[16

[17

[18

[19

[20

[21

[22

[23

References

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Eisenberger, P., Platzman,* P. M. , and Winick, H. , Phys. Rev. Letters, 36, 623-626

(1976) . Bannett, Y. B. , Rapaport, D. C. , and Freund, I., Phys. Rev. A16, 2011-2021 (1977).

Suortti, P., Phys. Stat. Solidi. (b) 91, 657-666 (1979).

Eisenberger, P. and Platzman, P. M. , Phys. Rev. A2, 415-423 (1970).

O'Connor, D. A. and Butt, N. M. , Phys. Lett. 7, 233-235 (1963).

Albanese, G. and Ghezzi , C. , Phys. Rev. B8, 1315-1323 (1973).

Barnighausen, E. , J. Appl. Cryst. 11, 221-228 (1978).

Paakkari, T. and Suortti, P., Phys. Rev. B9, 1756-1761 (1974).

Berreman, D. W. , Rev. Sci. Instrum. 26, 1048-1054 (1955).

Berreman, D. W. , Stamatoff, J., and Kennedy, S. J., Appl. Optics, 16, 2081-2085

(1977) . Hastings, J. B. , Kincaid, B. M. , and Eisenberger, P., Nucl. Instr. Methods, 152, 167:3

(1978) . Lemonnier, M. , Fourme, R. , Rousseaux, F. , and Kahn, R. , Nucl. Instr. Methods, 152 , 173-177 (1978).

Guinier, A., X-ray Diffraction (W. H. Freeman, San Francisco, 1963).

Smith, T. , in Phonons jji Perfect Lattices and in Lattices with Point Imperfections ,

R. W. H. Stevenson, ed. , chap. 6, (Oliver and Boyd, Edinburgh, 1966), pp. 161-169.

LaFleur, P. L. G. M. , Acta Cryst. A26, 674-678 (1970).

Paakkari, T. , Acta Cryst. A30, 83-86 (1974).

Chipman, D. R. and Paskin, A., J. Appl. Phys. 30, 1998-2001 (1959).

Haav, A. A., Pel jo, E. , and Surotti , P., Phys. Stat. Solidi. (b) 80, 255-264 (1977).

Suortti, P., Phys. Fenn. 10, 161-166 (1975).

Suortti, P. and Jennings, L. D. , Acta Cryst. A33, 1012-1027 (1977).

Chipman, D. R. , Acta Cryst. A25, 209-214 (1969).

Rietveld, H. M. , J. Appl. Cryst. 2, 65-71 (1969).

18
[24] Hewat, A. W. , J. Phys. C 6, 2559-2572 (1973).

[25] Werner, P.-E. , Ark. Kern. 31, 505-511 (1970).

[26] Young, R. A., Mackie, P. E. , and Von Dreele, R. B. , J. Appl. Cryst. 10, 262-269 (1977).

[27] Ayers, G. L. , Huang, T. C. , and Parrish, W. , J. Appl. Cryst. 11, 229-233 (1978).

[28] Sabine, T. M. and Clarke, P. J., J. Appl. Cryst. 10, 277-280 (1977).

[29] Malmros, G. and Thomas, J. 0., J. Appl. Cryst. 10, 7-11 (1977).

[30] Hall, M. M. , Veeraraghavan, V. G. , Rubin, H. , and Winchell, P. G. , J. Appl. Cryst. 10,

66-68 (1977).

[31] Rondot, D. and Mignot, J., J. Appl. Cryst. 10, 84-90 (1977).

[32] Khattak, C. P. and Cox, D. E. , J. Appl. Cryst. 10, 405-411 (1977).

[33] Langford, J. I., J. Appl. Cryst. 11, 10-14 (1978).

[34] Suortti, P., Ahtee, M. , and Unonius, L. , J. Appl. Cryst. 12, in press (1979).

[35] Bonse, U. and Materlik, G. , Z. Physik, B24, 189-191 (1976).

[36] Barron, T. H. and Smith, T. , J. Phys. Chem. Solids, 27, 1951-1952 (1966).

[37] Inkinen, 0. and Suortti, P., Ann. Acad. Sci. Fenn. A VI, 147 (1964).

[38] Cooper, M. J. and Taylor, R. I., Acta Cryst. A25, 714-715 (1969).

[39] Windsor, C. G. and Sinclair, R. N. , Acta Cryst. A32, 395-409 (1976).

Discussion

Question (Kuriyama): Regarding the flux conservation within a reciprocal unit cell

when diffuse scattering exists, I believe that statement holds only for small momentum

transfers within a reciprocal unit cell.

Response (Suortti): Yes, this is consistent with replacing the structure factor, of

the disorder scattering by its value at the center of the Brillouin zone in the

approximate formula of the conservation of scattering.

Comment (Langford, Mittemeijer, and DeKeijser): For Dr. Suortti ' s paper it is shown that

a Voigt function may provide a significant improvement in profile fitting, as compared to the functions formerly used. Also this function incidently does not provide a complete

description of the tails of the profile in case of pure size broadening. Thus from the

"Voigt" tail an erroneous value for the crystallite size is obtained. We would like to remark that if a quick estimate of size and strain values is to be obtained from line

breadths, then the method using Voigt functions (J. I. Langford, this conference) can

give valuable results, since the size-strain parameters are not solely obtained from the

tails of the profiles. However, if the detailed shape of the profile is taken into

account (e.g. as in the Fourier and Variance methods) any assumption on the shape of the

profile is avoided.

Response (Suortti): The author agrees that the bulk of the information of size and strain

of the powder crystallites is contained in the intense part of the reflection. However,

19
the use of the tails of the reflections has some advantages. First, that part is not critically affected by the inaccuracies of the deconvol ution procedures. Second, the

size effect can be isolated from the tails of the reflection without any assumptions of

the type of the strain, and this is a very concrete measure of the mosaic crystal model,

namely the average thickness of the diffracting crystallites. A fuller discussion of

this aspect is given in reference [21].

Comment (Kuriyama): In your characterization of the background scattering, you have

identified x-ray Raman scattering as resonant Raman. I would like to remind you, however,

that there is a Raman scattering which does not require the resonant conditions: that is

2 the Raman scattering caused by the A interaction term in the Hamiltonian.

Response (Suortti): It is true that there is also a Raman scattering contribution to

the non-resonant scattering. However, this is very small, and has been neglected in the

present study. The resonant Raman scattering arises from the A • p interaction taken

twice, and it increases rapidly when the resonant condition is approached from below

going over to fluorescence at the absorption edge.

20
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg , MD, June 11-15, 1979.

(Issued February 1980)

NEUTRON DIFFRACTION—THE TOTAL POWDER PATTERN

T. M. Sabine

Institute of New South Wales of Technology

P. 0. Box 123

Broadway, New South Wales

Australia 2007

1. Introduction

Neutrons are of particular use in the study of solids, liquids and gases because of

the following basic properties:

(i) The thermal neutron wavelength is comparable to the interatomic spacing. Hence,

there will be pronounced interference effects when neutrons are scattered by

condensed atomic systems;

(ii) The change in neutron energy from inelastic processes involving the creation and

annihilation of excitations in condensed matter are of the order of the incident

neutron energy;

(iii) The neutron has a magnetic moment. This gives it a unique value in probing the

magnetic structure and magnetic dynamics of atomic systems;

(iv) The neutron is a neutral particle. Because of this it can penetrate deeply into

matter;

(v) Neutrons are a gentle wind, compared with the strong breeze of x-rays and the

raging gale of electrons, in causing radiation damage in the systems under

investigation.

As an experimental science, neutron diffraction started seriously in 1950 when neutron

12 _ o -l beam fluxes of about 10 n cm sec became available. The theoretical basis had been

laid by Halpern and Johnson [l] 1

, Weinstock [2], and Cassels [3].

The difficulty of obtaining quantitative information from single crystals due to the

Dreakdown of the kinematical approximation in crystals sufficiently large for appreciable

scattering to occur was noticed by the earliest workers [4]. To overcome these extinction

effects Wollan and Shull [5] constructed a powder di f f ractometer. Powder methods gave

(accurate data for simple compounds, however the problem of overlap between closely spaced

Reflections precluded study of materials with large unit cells or low symmetry.

21
Many workers interested in the application of neutron diffraction to chemical com- pounds tried to reduce extinction effects or to correct for them. Early work was carried

out by Bacon and Lowde [6] and Petersen and Levy [7].

These studies showed that the effects of extinction could be reduced by destroying

the perfection of the crystal. Quenching or irradiation damage were favoured techniques

[8]. The unfortunate result of these treatments is the introduction of dislocations,

point defects and defect clusters, all of which produce appreciable diffraction effects

and require explanation in a proper discussion of the structure. Zachariasen [9] treated extinction in terms of a crystal containing perfect crystal

blocks of a certain size and separated by a certain degree of misorientation. His equations were soon incorporated into the least squares refinement procedure with provision

for anisotropic block sizes and anisotropic mosaic distribution [10]. Coppens and Sabine

[11] used these equations in a refinement of the structure of oxalic acid dihydrate,

however, in a direct examination of the same material by Lang topography [12], the dislocation density was found to be orders of magnitude less than that predicted by the results

of the Zachariasen treatment. While these discrepancies exist there must be doubt about

this solution to the extinction problem.

The powder method continued to be developed slowly, but its use was largely limited

to magnetic systems. Studies of the optics of powder diff ractometers were made by

Cagliotti, Paoletti and Ricci [13] and by Sabine and Browne [14].

Rietveld [15] made a significant improvement in methods of analysis by realising that

powder profiles closely followed a Gaussian form. He calculated the ordi nates of the

Bragg profiles and refined them against the observed pattern. This method, particularly

after the development of high resolution neutron diffractometers [16], led to the satisfactory refinement of data from a large number of inorganic compounds [17] with accuracies!

in positional parameters comparable with those found from single crystal work.

Agreement for thermal parameters was far less satisfactory [18] and a treatment in

which all the neutrons scattered by the specimen could be included in the refinement was

given by Sabine and Clarke [19].

Development of the appropriate expressions for calculation of the ordinate of a powder pattern is the subject of this paper.

In this analysis it will be assumed that a conventional two-axis powder diffractomete

is being used. The monochromatic neutron beam is of sufficient size to completely bathe

the specimen. The pattern is scanned by a counter which is fitted with a narrow slit.

Typically, this slit subtends ten to twenty minutes of arc at the specimen position. The

diffractometer arm moves in small steps of 26 under the control of a monitor counter

placed in the incident beam. The time spent at each 26 position is t.

2. The Total Neutron Powder Pattern

2.1. Bragg scattering

22
j The total number of neutrons diffracted into a length h of the Debye-Scherrer ring,

der these experimental conditions, is [20,21]

- ,3... .. M 2,-2 -2M $ A hVp jN F e

p _ _y «- 8rrRp sin 0 sin 28

(a glossary of terms is given in table 1) .

To determine the ordinate of the profile at any point we will assume, following

etveld [15], that the profile of a reflection is obtained by convoluting the above

oression- with a Gaussian broadening function of area unity. Because of the Maxwell ian

tribution of neutron velocities in the reactor moderator and the Gaussian distribution

mosaic blocks in the monochromator and the specimen, the profile is accurately Gaussian.

> full width at half maximum intensity is given by

B = U tan

2

6 + V tan 0 + W

] where U, V, W are constants for a given experimental arrangement. The profile for

igg scattering is then

1(261 - y3hVp

* jN

c F2e

~ 2M

2 (WU 4 £n 2 1(26 3 _ 8nRp sin 0 sin 20 B \/T/ exp ~^2~ (26

i 28)

is the Bragg position for the ith reflection. It is assumed for the moment that there

no overlap between reflection profiles.

In an experiment the ordinate of the profile is measured by recording the number of

trons received by the counter through a slit of width w in time t.

Then the Bragg component is given by

/*20H

/n (26) = t|

*20+W/2R

yn (26) = tl I (20) d(20)

'20-W/2R

iugh the approximation

»tei x+^ 2 2 2

a e"Px dx = a Ax e"Px rx

2

23
Table 1. Glossary of Terms

F Structure factor per unit cell

-2M

e Debye-Waller factor for intensities

b Coherent scattering length

j Multipl icity

A Wavelength

V Specimen volume

p' Measured density

P Theoretical density

V

c Unit-cell volume

N

c Number of unit cells per unit volume

*o

— 9 — 1 Incident flux (neutrons m s ) h Height of counter slit w Width of counter slit

s Step size in 26

R Specimen-to-counter distance

t Time spent for each step

B Full width at one-half maximum intensity

y(2e) The number of neutrons recorded in the counter in

C

i Site occupation factor for ith site

f

i The magnetic form factor of the ith atom

S

i The spin of the ith atom

I Total cross section per unit cell

we obtain

* A^hwVp 'jN^F^e ^a[exp(-px^)]t

yi(2e) = -2 ±

8rcR p sin 9 sin 26

where

2.2. Diffuse scattering

If we assume 1^ is the diffuse scattering cross-section per unit cell the number

neutrons received by the counter in time t will be

24
* hwtN Vp'

y? (26) = £

* 4nRp

he total neutron count is then

3 2 -2M * hwtN p'V (N \ if e 2 ^ y (26) = ° / JlJ. a e"?* + I 4nR p ( 2 sin 6 sin 29

u

j

jperimposed on this count is the experimental background.

Measurement of all the quantities in front of the bracket will enable the data

) be put on an absolute scale. In general we will put

* hwtN p'V

K= ° / . 4nR^p

Hi I

2.3. Magnetic Bragg scattering

Scattering from ordered magnetic ions in the crystal is included through the magnetic

ructure factor which is given by [22]

F(k) = I p. (k) exp 2nik-r. [tx(n.xt)]

j

j „ „ ~j „ ~j „

P,(k) = ^4 <S,>f,(k) . J mc J J

<S.> is the average magnetic moment on the jth site, f .(k) is the magnetic form factor

J J TM

' this atom, n. is a unit vector parallel to the direction of the jth spin, t is a unit

[tor parallel to the scattering vector. When unpolarized neutrons are used there are no

|ss terms between nuclear and magnetic scattering and the intensities add. For polarized

| trons the structure factors add.

3. Evaluation of 1^

I In a system exhibiting long range order, the diffuse scattering arises because of

jiations from the average structure. These deviations can be compositional fluctuations

both chemical and isotopic composition) or random displacements from lattice points, or

25
3.1. Atomic short range order

The intensity scattered in the reciprocal lattice director k (k = -

s^n 9

) is given y

N-l N-l

I(k) =1 lb. b.,* exp 2nik-r.-r.,

0 0

J J " I

J

where N is the total number of atoms in the crystal.

Following Wilson [23] we put

j' = j + m hence

N-l

I(k) = I (N- m ) J exp 27tik-r

-(N-l) m ~ ~

where J mis the average value of the structure factor product for sites separated by th(

vector r . m

To evaluate this average we consider a disordered AB alloy with concentration C. o AA AB BA

A and C D of B. Set P as the probability that r joins two A sites, P (= P ) the

B m

y J BB

' m

v my probability that it joins an A site and a B site, and Pm the probability that it joins

two B sites then

I

i

, _ .2 DAA 9h . DAB . .2 D BB

J m - b. P + 2b. bp P_ + b R mAmADmBmP_ • f

In terms of the Cowley [24] short range order parameters, c< m

, these probabilities

become

C

= CA<

CA

+ amV>

Pm

B = C B(

C B + a

m

CA)

hence

J m =

( CA

bA

+ C B

b

B )2 + CA

C

B

( bA

" b

B

) 2 am

26
Substituting back gives

I(k) = (CA

bA

+ C B

b

B

) 2 I (N- m) exp 2nikT

m -(N-l)

+ WbA

" b B )2 (N" m } "» 6XP 27ti ~* r5

For large values of N the first term becomes Bragg scattering from the average lattice,

s am

will be non zero only for small values of m, the second term represents diffuse

attering and can be written

I(k) = NC

A

C

B (bA

- b

B

) 2 1 a

m exp arik-r.

m

ir a powder pattern we must average over all directions of k and obtain

9

oo s in 2n kr

I n = NC.CD (b« - b D r lea - m-1 m m

In krm

ere c is the number of atoms in the mth shell and r is its radius. Hence

m m

I = 4nl

SRO

u

;re N is the number of atoms in the unit cell.

The cross-sections for several components of the pattern can be deduced from this

ression.

3.2. Incoherent scattering

If b-^ is the scattering length associated with the 1th isotope and spin state, then

a =C mm>0

contribution from incoherent scattering becomes (a =1, a =0) a o ' m

[|<b

2

> - <b

1 I. > , = 4n{<b> - <b,>

2 incoh

27
3.3. Partial site occupancy

For systems with incomplete site occupancy

C A = C. b A = b; C

B = (1-C); b

g = 0

where is the site occupancy factor chosen so that for C. = 1 all sites in all unit cell are occupied. Then

I = 4n I C.(l-C.) b. disorder i=l

where n is the number of atoms in the unit cell.

This expression assumes that there is no correlation between vacant sites. If there

is clustering of either vacancies or occupied sites modulated diffuse scattering of the f

9 sin 2n kr I = 4ntTc(l-c) lac —= - ... v m m In kr disorder m m

will appear for each atomic species. The cross sections will simply add if the ordering

each species is independent but, in general, cross terms could be expected.

3.4. Magnetic diffuse scattering

The treatment for systems in which the magnetic moment varies from site to site foll'/sj

the treatment of composition fluctuations.

Assuming that the magnetic system is col linear and parallel to the z axis

I(k) = (1 - xh I 1 p,(k)p.-(k) exp 2nik-r.-r..

j j

~ ~ ~

" )

Defining an AB alloy with atoms of magnetic scattering factor p. and Cn with

scattering factor p

g a Bragg term containing the average scattering factor is obtained

and the diffuse scattering is given by

diffuse = W ~ ^ CaWIP " magnetic J

1

sin 2n kr

x I c a —^—j ,| mm Zn krm

1

28
The first term reduces to the expression used by Shull and Wilkinson [25]. Marshall

[26] has introduced short range order parameters to describe the perturbation of the

imoments on B atoms which are close neighbours of atoms of type A. These parameters are

equivalent to c< m

.

3.5. Paramagnetic scattering

The appropriate cross-section is

- 87t/e^\2 2 S.(S. + 1) f

2Ak)

z " 3 L2 j

j=1

para

[j labels the magnetically active atoms in the unit cell.

I

3.6. Displacement disorder

The formalism of this problem is similar to that of the composition fluctuation problem.

|n atom can be displaced from a lattice site by a defect in the unit cell or by a large

Jiscrepancy in atomic size. Detailed calculations have been given by Warren, Averbach

nd Roberts [27] and by Borie [28,29].

These calculations lead to the expression

I = 4n CAC

B (bA

- b

B

) 2 sin 2n kr

T m 2 a

m

cm 2n kr m m

(sin 2n kr —r - cos In kr

2n kr m

m

lere a is the short range order coefficient and 6 the size effect coefficient.

3.7. Thermal diffuse scattering

At any temperature the intensity removed from the Bragg peaks by the Debye-Waller

ictor reappears as in elastic scattering in the diffuse background.

The simplest approximation to this term is based on the assumption that the atoms

brate independently. With this assumption

-2M.

unit

cell

29
I For simple structures it is possible to find expressions which recognize the correlati;

between atomic vibrations. Treatments have been given of correction to integrated intensity

for the component of the TDS which peaks at the position of the Bragg peak [30].

r Marshall and Lovesey [22] have given expressions in which the elastic and one-phonon

terms are treated exactly and the multi-phonon terms approximated. These are not, at prese|

convenient for calculation, but could form the basis for a detailed treatment of the TDS j contribution to the powder pattern.

4. Conclusion

Expressions have been given for scattering mechanisms which contribute to the total \ powder pattern. Inclusion of these removes the necessity for a subjective estimate of

the background. |j Those quantities which are the result of local deviations from the average structure

are included in the refinement process through their contributions to both the Bragg ji scattering and the diffuse scattering. A constraint that the total scattering is

conserved can then be applied.

References \

[1] Halpern, 0. and Johnson, M. H. , On the magnetic scattering of neutrons, Phys. Rev. 35, j 890-923 (1939).

[2] Weinstock, R. , The inelastic scattering of slow neutrons, Phys. Rev. 65, 1-20 (1944). \ [3] Cassels, J. M. , The scattering of neutrons by crystals, Prog. Nucl. Phys. 1, 185-215

(1950).

[4] Fermi, E. and Marshall, L. , Interference phenomena of slow neutrons, Phys. Rev. 71,

666-677 (1947).

[5] Wollan, E. 0. and Shull , C. G. , The diffraction of neutrons by crystalline powders, |i Phys. Rev. 73, 830-841 (1948).

[6] Bacon, G. E. and Lowde, R. D. , Secondary extinction and neutron crystallography,

Acta. Cryst. 1, 303-314 (1948).

[7] Peterson, S. W. and Levy, H. A., The use of single-crystal neutron-diffraction data

for crystal -structure determination, J. Chem. Phys. 19, 1416-1418 (1951).

[8] Sanger, P. L. , An experimental determination of the ionic state of MgO by singlecrystal methods, Acta. Cryst. A 25, 694-702 (1969).

[9] Zachariasen, W. H. , A general theory of x-ray diffraction in crystals, Acta. Cryst. 23,

558-564 (1967).

[10] Coppens, P. and Hamilton, W. C. , Anisotropic extinction corrections in the Zachariasen

approximation, Acta. Cryst. A 26, 71-83 (1970). [ [11] Coppens, P. and Sabine, T. M. , Neutron diffraction study of hydrogen bonding and therma v motion in deuterated a and p oxalic acid dihydrate, Acta. Cryst. B 25, 2442-2451 (1969)

30
f'[12] Michell, D. , Smith, A. P., and Sabine, T. M. , The dislocation arrangement in oroxalic

[13

14

I

acid di hydrate, Acta. Cryst. B 25, 2458-2460 (1969).

Caglioti, G. , Paoletti, A., and Ricci, F. P., On resolution and luminosity of a neutron

powder spectrometer, Nucl . Instr. 3, 223 (1958).

Sabine, T. M. and Browne, J. D. , A geometrical consideration in the design of neutron

powder spectrometers, Acta. Cryst. 16, 435-436 (1963).

15] Rietveld, H. M. , A profile refinement method for nuclear and magnetic structures,

J. Appl. Cryst. 2, 65-71 (1969).

16] Hewat, A. W. , Design for a conventional high resolution neutron powder dif fractometer,

Nucl. Instr. Methods, 127, 361-379 (1975).

17] Cheetham, A. K. and Taylor, J. C. , Profile analysis of neutron powder data, J. Solid

State Chem. 21, 253-275 (1977).

18] Andresen, A. F. , Hewat, A. W. , and Sabine, T. M. , The reliability of neutron powder

profile refinement, J. Appl. Cryst. (in press).

19] Sabine, T. M. and Clarke, P. J., Powder neutron diffraction - refinement of the total

pattern, J. Appl. Cryst. 10, 277-280 (1977).

20] Bacon, G. E. , Neutron Diffraction (Oxford University Press, 1962).

21] Warren, B. E. , X-ray Diffraction (Addison-Wesley , Mass., 1969).

£2] Marshall, W. and Lovesey, S. W. , Theory of Thermal Neutron Scattering (Oxford

University Press, 1971).

!3] Wilson, A. J. C. , Imperfections in the structure of cobalt. II Mathematical treatment

of proposed structure, Proc. Roy. Soc. A 180, 277-285 (1942).

!4] Cowley, J. M. , An approximate theory of order in alloys, Phys. Rev. 77, 669-675 (1950).

!5] Shull , C. G. and Wilkinson, M. K. , Neutron diffraction studies of the magnetic

structures of alloys of transition elements, Phys. Rev. 97, 304-310 (1955).

Marshall, W. , Neutron elastic diffuse scattering from mixed magnetic systems,

J. Phys. C. 1, 88-101 (1968).

Warren, B. E. , Averbach, B. L. , and Roberts, B. W. , Atomic size effects in the x-ray

scattering by alloys, J. Appl. Phys. 22, 1493-1496 (1951).

|8] Borie, B. , X-ray diffraction effects of atomic size in alloys, Acta. Cryst. 10,

89-96 (1957).

! ] Borie, B. , X-ray diffraction effects of atomic size in alloys II, Acta. Cryst. 12,

280-282 (1959).

Willis, B. T. M. and Pryor, A. W. , Thermal Vibrations in Crystal lography (Cambridge

University Press, 1975).

Discussion

istion (Thomas): Have you included the effect of TDS which peaks under the Bragg peaks

23

isen

hero your MgO data?

31
Response (Sabine): No, in the approximation I used at present TDS is a smooth function

sin 6A. I am preparing an expression for TDS which contains components peaking under

Bragg positions.

Comment (Ladell): The shape of a diffraction profile, whether Gaussian or Lorentzien,

can be determined experimentally by using samples which have high preferred orientation.

In this case, it is possible to see Lorentzien character in the profile stretching over

many degrees.

Response (Sabine): This may be true in the x-ray case in which the Lorentzien profile o the spectral line, which is a consequence of the nature of x-ray production, dominates tf distribution. However, for neutrons, we have a Maxwellian distribution in the moderator

followed by Gaussian mosaic block distributions in the monochromator and specimen. From

central limit theorem, the final profile is fairly accurately Gaussian.

32
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

SYNCHROTRON RADIATION AND ENERGY-DISPERSIVE DIFFRACTION

B. Buras

Physics Laboratory II, University of Copenhagen

Uni versitetsparken 5, DK-2100 Copenhagen, and

Ris*J National Laboratory, DK-4000 Roskilde

Denmark

X-ray energy-dispersive diffraction utilizes white radiation;

in this method the scattering angle is fixed during a measurement, a semiconductor detector performs an energy (wavelength) analysis of the

diffracted x-rays and all reflections are recorded simultaneously. As was demonstrated at DESY (Hamburg, F.R.G.), NINA (Daresbury, U.K.), VEPP-3

(Novosibirsk, U.S.S.R.) and recently also at the storage ring DORIS

(Hamburg, F.R.G.), synchrotron radiation with its special features

particularly a continuous and smooth spectrum extending to short

wavelengths, excellent collimation, high intensity, and well-defined

polarization—is ideal for this method. In the case of a storage ring,

the spectral distribution of the incident radiation can also be calculated

from the machine parameters, essential in making quantitative structural

analyses.

The paper discusses briefly the following problems, important for

structural studies by means of synchrotron radiation and energy-dispersive

diffraction: incident spectrum, polarization, large scattering vectors,

profile fitting, momentum resolution and precision of interplanar distance

measurements, studies of samples in special environments, counting time

and counting rate, time-resolved studies, corrections and limitations.

1. Introduction

X-ray energy-dispersive diffractometry (XED) invented in the late sixties [l,2] 1 and

nee then developed and utilized at conventional x-ray sources [3-50], requires a llimated incident beam of a smooth "white" spectrum. Thus, synchrotron

jigures in brackets indicate the literature references at the end of this paper.

33
radiation is the proper x-ray source for this method. Other qualities of synchrotron

radiation, such as high intensity and known polarization, make it an almost ideal source,

as was recently demonstrated at DESY [51-54], NINA [55-58], VEPP-3 [59] and DORIS [60-

61].

This review gives a brief description of the XED method, recalls the characteristic

features of synchrotron radiation and discusses the following problems important for

powder diffraction by means of synchrotron radiation: incident spectrum, polarization,

large scattering vectors, profile fitting, momentum resolution and precision of

interplanar distance measurements, studies of samples in special environments, counting

time and count-rate, time-resolved studies, corrections and limitations.

The discussion is rather brief and is intended to give only a general picture of the

present status of energy-dispersive diffractometry. More details may be found in the

quoted papers.

The use of XED for single crystal studies is beyond the scope of this paper.

However, references to single crystal work are included in the list of references.

2. The Method

Figure 1 shows the principle of the method. A white, collimated x-ray beam is

scattered by the powdered sample through a fixed angle 2Q

Q

and the wavelength (energy)

distribution of the scattered photons is analysed by a semiconductor detector connected to

a multichannel pulse height analyser. Each set of crystal lattice planes of spacing d^ (H

indicates indices of reflection HKL) selects from the incident white spectrum photons of

energy E H (wavelength X H

) fulfilling the Bragg equation

2 d

H .^0 .^).j£. «-4

iff*) 0,

(h is Planck's constant, c the velocity of light), and reflects them to the detector.

Figure 2 shows an example of a pattern measured at DORIS.

The integrated intensity I H

is given by the formula

I„- Cr1 (E)i

o (E)jH dH

2 fJ 2

. A(E,0

Q

)C (kin)/sine

Q

, (2) I

where C is a constant, n(E) the detector efficiency, i 0

(E) the incident beam intensity per

unit energy range, j H the multiplicity factor, F^ the structure factor including atomic

factors, A(E,6

Q

) the absorption factor, and C

p the polarization factor. It should be

noticed that in order to derive the structure factors from the measured integrated

intensities one must know i 0

(E) and C

p

. We return to this point later.

34
<

Q. oQ

CD

O Z

l- o

III « £ O

a. cr

LU<

crO

i- S o ^ o < z

lu >. cr m to o z

cr

cr

LU

_j

X CL

cr

LUI

LU (J

Q- z < D

I— 0.

cr

LU\-

LUzoo

—1 (—

LU I

NN

g ER < LU to I I > Ob

LUCO 1

AL

_l z D D < 2 a.

LU

CO

LU

cr

LU

0.X

LU

*5

4->i +J

a>10

+•> c

a>E

•i—sOJCLX

OJ

a.

ra ac

ro>

•r-10

s-CL

to

>>

ai

s-

<ue

a;

a)

+-> M—O

<DCL

•r—o

QEU

i-cn

35
36
The main characteristic features of the method, as compared with the x-ray

'monochromatic beam (angular scan) method, are the following:

(a) it uses a white beam,

(b) the scattering angle is fixed during a measurement but can be optimized for each

particular experiment,

(c) the detector is an energy-dispersive one,

(d) all reflections are recorded simultaneously,

(e) the exposure time is relatively short (this point is discussed in some detail in Section 4.7).

A white x-ray beam--essential for energy-dispersive diffractometry--can be obtained

either from an x-ray tube (Bremsstrahl ung) or from a synchrotron radiation source. In the

following section the characteristic features of synchrotron radiation are briefly

jrecalled and compared with those of Bremsstrahl ung from an x-ray tube.

3. Synchrotron Radiation Versus Bremsstrahl ung from an X-ray Tube

i Synchrotron radiation is emitted by electrons in circular motion with velocities

close to that of light. The emitted synchrotron radiation is continuous . Figure 3 shows

the spectral distribution of radiation from electrons passing the bending magnets of the

(National Synchrotron Light Source (NSLS) now under construction in Brookhaven National

laboratory. It extends from infrared (not shown in the figure) to the x-ray region. It can be characterized by the so-called critical wavelength x c dividing the spectrum into

two halves of equal radiation power. This wavelength is a bit shorter than the one for

o which the intensity of the spectrum reaches its maximum. In the NSLS A c

- 2.5 A. By

applying a stronger magnetic field locally the radius of curvature becomes smaller and the

emitted radiation is shifted toward shorter wavelengths; thus harder x-rays are emitted.

This is accomplished by the so-called wigglers. The proposed NSLS wiggler will give X_ =

3.6 A. As the rule of thumb, useful intensity can be obtained down to 1/5 of \ r and thus

in the case of the NSLS to * 0.5 A (^ 25 keV) from bending magnets and ^ 0.12 A (^ 100

keV) from the wiggler. The latter wavelengths are shorter than those from conventional

sources. Synchrotron radiation is emitted tangentially to the orbit with a very small mean

angular spread ay in the plane perpendicular to the orbital plane2 (fig. 4). This spread

is inversely proportional to the electron energy. For 2.5 GeV electrons of the NSLS AS' Equals 0.2 millirad « 42 seconds of arc for x c

. AY decreases slowly with decreasing

(wavelength.

phe spectral brilliance is also much larger, especially at dedicated synchrotron

padiation source.

37
CO CN o o o

(euD|d |doi^j9a ui pepj6e4ui)

L _(Lj;pjMpuDq o/o |/o) ^pdjuj • t _s • SNOlOHd

T3

sajQ- T3Coo

CU

c

o -—. $_ JZ +->

+-> o

C7> S- c

o

c

I/Oc

<a

CUo.o

S-3

s- „—

cu

0) o J=

Q_ ai +-> cn

</) its -a C s- e

o 10 +j >

o to c

.c o

Q_ CU •i—

V E +-> u

-t->

• CO T3 +J

+-> C 00

ai to c

c o

cn CU o

ns C

E to S1 cu cn Q- -a C c

•ii— 3

-a (O c o ^—

ai •i

.Q +-> 00

s- r) E ai O > •» i. c

<4- cu cu j= >

T3 4-> CD x:

-t-> C

+-> o

'i

o

•a s- a> aj

+-> -

^ c 10 o s- m

•r" _i

+-> 0) CO

10 +j —

c • -o •rTM ffl fO cn o

s_ •1

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s- C 1 3

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to (J

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sz UJ

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s- s-

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10 CL n3 *I

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o c_> •1— CO N Q •r* (13 cu S- E

5- o o t> 3 x: s- cn 4- a:

38
At|> =

£

MeV

(E -electron energy)

For E = 5 GeV

= 10" 4 rod =0.1 mrad = 21

n

1

AO)

T

Figure 4. Geometry of emission of synchrotron radiation by electrons in circular

motion with velocities close to that of light. B-bending magnetic field perpendicular to the electron orbit.

39
Radiation is emitted from the whole orbit within a flat pancake in the plane of the

electron orbit. Using a beam definer one cuts from this pancake a small slice and is able

to obtain a beam of synchrotron radiation very well collimated both in the vertical and

horizontal planes. The radiation is in a very high degree plane-polarized in the plane of

the orbit and elliptically out of this plane. For an ideal orbit the radiation would be

completely plane-polarized in its plane.

Figure 5 shows a comparison between x-rays from a conventional x-ray generator and

those produced by a storage ring . The radiation from the focal spot of the anode is emitted into a half sphere while synchrotron radiation emerges within a small vertical

angle, as already discussed (fig. 5a). The intensity of synchrotron radiation from large

machines, discussed here, is several orders of magnitude larger than Bremsstrahl ung from

the most powerful rotating anode tubes (fig. 5b). The comparison between the

monochromatic characteristic radiation from an anode and the monochromatic radiation which

can be obtained from the continuous synchrotron spectrum (by means of a monochromator) is

beyond the scope of this paper. It can be said, however, that in general synchrotron

radiation is superior.

The intensity of radiation from an anode is constant in time while from a storage

ring it comes out in short pulses , typically of 0.1 - 1 ns length, separated by 2 ns - 1 ys (fig. 5c). This is a result of the fact that electrons in a storage ring travel in

short bunches few centimeters long. The separation time depends on the number of bunches

travelling in the storage ring.

4. The Energy-Dispersive Method at a Storage Ring

4.1. Polarization

As can be seen from eqs. (2) a quantitative structure analysis requires knowledge of

the polarization factor; this in turn requires knowledge of the polarization of the

incident beam. In the case of conventional sources Bremsstrahl ung polarization is dependent on the photon energy and it cannot be calculated. The measurements are difficult, time consuming and rather inaccurate [46], In the case of synchrotron

radiation the polarization of the incident beam is well known and easy to take into

account . Because the radiation is in a very high degree plane polarized in the plane of

the orbit (which is horizontal), a vertical scattering plane is advantageous. In the case

of a horizontal scattering plane the polarization factor decreases from 2Q

Q

= 0 or 180° to

2e Q

= 90° and becomes very small at 2Q

Q

= 90. However, for scattering angles smaller than

30° or larger than 150° the loss is smaller than 25 percent and can be in most cases easily tolerated taking into account the high incident intensity.

4.2. The incident spectrum and crystal structure analysis

As already mentioned in Section 2, in order to perform a quantitative structural

analysis one must know besides the polarization also the incident spectrum i 0 (E). In the

40
a

SYNCHROTRON

X-RAY TUBE

1

ANGLE

CHARACTERISTIC

RADIATION

BREMSSTRAHLUNG

WAVELENGTH

INTENSIT -*-0.1 ns -1 ns

2ns - 1 [is

TIME

Figure 5. Comparison between x-ray synchrotron radiation and Bremsstrahlung from

an x-ray tube.

41
case of conventional x-ray sources i Q (E) cannot be calculated. The measurements of i Q (E) are time consuming and present a source of errors. Nevertheless, it was shown that the

determination of structure factors using XED with conventional sources is in principle

possible [34]. It was also demonstrated that in the case of a synchrotron (NINA)

operating in a dedicated mode a Rietveld type profile fitting procedure (in which both the

incident spectrum and the crystal parameters are fitted) can be successfully applied

[58].

In the case of a storage ring the situation is much more favourable. It is well

known that the spectral distribution of the emitted radiation can be calculated from the

machine parameters and it was recently shown that the calculated spectrum (in this case at DORIS in Hamburg) can be used for structural XED studies [61]. For several substances the

structural parameters were obtained from the measured integrated intensities or from a Rietveld type profile fitting procedure. The reliability factors were around 0.03 for

simple substances (Silicon, Iron) and R prof was equal to 0.10 and 0.14 in the case of Urea

and Naphthalene, respectively (R-j nt was equal to 0.05 and 0.06, respectively). Figure 6 shows, as an example, the results of refinement for Urea. T T T

UREA(H2 N-CO-NH2

) 29 0 = 16.33°

300 400 500 600

CHANNEL NUMBER

Figure 6. Diffraction pattern of Urea (after background subtraction). Crosses

denote the experimental points and the full line the refined profile.

42
4.3. Large scattering vectors

-> The modulus of the scattering vector Q is given by

q( A-1) -41. sine. E(keV) -sine. (3)

and thus it increases with the photon energy. For the maximum photon energies to be

obtained at the NSLS and quoted in Section 3 one should be able to use scattering vectors

O -I O-l

as large as ^_ 25 A~ x and 100 A with radiation from bending magnets and wigglers , O I

j respectively . This should be compared with about 17 A from a conventional source

(MoKa). The importance of measurements at large scattering vector does not need to be

I advocated, however, one should remember that because of thermal vibrations and the

decreasing atomic form factor the scattered intensity decreases with increasing scattering

vectors; thus even for intense synchrotron radiation in each case a certain limit exists

beyond which large scattering vectors are useless for structural studies.

4.4. Momentum resolution and precision of interplanar distances measurements

The momentum resolution is limited by the energy resolution of the detector system

and by the collimation used. There is little hope that the energy resolution can be

improved beyond 100 eV at 5.9 keV (at present the best detectors have a resolution of ca. 140 eV at 5.9 keV). A major advantage in working with synchrotron radiation is the

excellent collimation of the incident beam and its high intensity, which enables small

samples to be used as well as fine slit systems and large distances between sample and

detector (or fine Soller collimators for large samples). In addition the high intensity

secures good statistics.

As a result of the very good collimation, the best resolution is obtained not, as one might think, in backscattering but at small (but not too small) scattering angles [53].

Figure 7 shows the dependence of the FWHM of a diffraction peak (5dp W^M-measured in the dinterplanar spacing scale) on the scattering angle for different d-spacing and different

col 1 imations, Ae Q

. One notes a pronounced minimum for A6

Q

= 10"^, obtainable with

synchrotron radiation. However, small scattering angles require high photon energies and

this points to the need for hard x-rays. Nevertheless, using the formulas of reference

[53] one can in each case find the scattering angle for optimum resolution . It follows

also from references [26] and [53] that with XED it is possible to measure interplanar

I distances with a_ precision of 0.01 percent .

43
Figure 7. The dependence of <$dpWHM

(the FWHM of a diffraction peak measured in the

interplanar spacing scale) on the Bragg angle e o for Ae

Q = 10"4 rad (full lines)

and Ae

Q

= 5 • 10" 3 rad (dotted lines) for different interplanar spacings d using

a germanium detector with a resolution of 150 eV and 485 eV at 5.9 keV and 122 keV,

respectively [53].

44
4.5. Fixed scattering angle and high energy photons

These two features taken together facilitate studies of samples in special

environments (e.g., high pressure, high and/or low temperatures, etc.) where special

windows must be used. The loss of intensity due to absorption is smaller for high-energy

photons, and the fixed scattering angle enables the use of no more than two (or even one,

j if backscattering is used) small windows.

These types of measurements have been already made with XED at conventional sources

! [4,15,4.0,50] and some demonstration experiments were performed as well with synchrotron

radiation [52,54], For high-pressure measurements in diamond anvils the sample volume is very small (0.01 mm and less), so that it is essential that the incident intensity be

very high.

4.6. Simultaneous recording of all reflections

I This feature of XED makes the method independent of the decrease of the incident beam

intensity (it affects only the statistics), due to the decay of the electron current in

the storage ring. Thus, no monitor is necessary . The simultaneous recording of reflections greatly facil ities also, among others,

j studies of phase transitions .

4.7. Counting time and count-rate

4.7.1. High count-rate

Because of the simultaneous recording of all reflections and the high incident

intensity of synchrotron radiation the counting time is relatively small. Counting times

of Is from powdered samples of a volume of 0.2 mnr measured at DORIS (3.7 GeV, 14 mA, x r = 1.34 A) working in a single bunch mode were reported [61] (e.g., see fig. 2). A further

substantial decrease of the counting time is limited by the present maximum count-rate of

semiconductor detectors 10 counts/s). However, even with the present maximum count

rate for a single detector energy-dispersive diffractometry offers a possibility of

, improving the counting statistics and thus decreasing the counting time. Because the

1

scattering angle is fixed one can simultaneously use several detectors placed in a circle

and simply sum up the data (fig. 8). Such a system is at present expensive but some simplifications lowering the cost can be already envisaged (e.g., a joint cooling system

and/or a multichips detector instead of several detectors). The future may bring some development in this field.

45
46
4.7.2. Low count-rate

For very low count-rates one detector (or several detectors in parallel) covering a

part or the whole of the Debye-Scherrer ring (cone) can be used (fig. 8). In the case of

very small samples (e.g., in a diamond cell) one detector of several centimeters in

diameter equipped with a conical collimator might be sufficient [50].

4.8. Time-resolved structural studies

The relatively short counting time, simultaneous recording of all reflections and the

possibility of quantitative structure analysis make it possible to collect and analyse

data in a few seconds and thus allow structural parameters to be obtained very rapidly.

Thus , it seems feasible , for instance , to carry this out semicontinuously with changing

temperature and/or pressure and construct a motion picture of the change in crystal

structure during a phase transition . This type of time-resolved studies is possible in

the time range of few seconds and above. Faster repetitive processes (relaxation times

between 10 ns and 1 s depending on the available electronics) can be studied by means of a stroboscopic method. This is illustrated in figure 9 with an example of a repetitively

pulsed external electrical field applied synchronously with the x-ray pulse. Thus, XED

seems to be a suitable method for studies of transient phenomena in a large range of

relaxation times . 4.9. Some other applications

As was already demonstrated at conventional sources, XED can be used for

determinative mineralogy [7], Debye-Waller temperature dependence studies [33,44],

attenuation coefficients measurements [45], on-stream measurements [17,18], particle size

measurements [41] and texture studies [9,23,28,36,49]. In some of these cases the

possibility of time-resolved studies, discussed in Section 4.8 is of great importance

(e.g., development of texture, annealing of texture, creation of strain). This applies to

studies of chemical reactions as well.

4.10. Choice of detectors and bunch mode

The detectors available on the market and tested with synchrotron radiation and XED

[51,52,54-58,60,61] are Si(Li) and extra-pure Ge. For low energy photons (below ^ 30 keV)

Si(Li) detectors are preferred (with only negligible escape peaks [27] and a low energy

absorption edge of Si at 1.84 keV). For higher photon energies Ge detectors are much

better because of the higher efficiency at energies above 30 keV. However, escape peaks

'Complicate the diffraction pattern [22], and the presence of the absorption edge at 11.1

keV disturbs the efficiency in this region. In both cases pile-up rejectors should be

applied. The pile-up problem deserves a special discussion.

47
Time -resolved studies

'External

field

0

n Incident X-rays intensity

/ V.

JUL. JlAAjl

Eph Eph Eph

6t- variable phase shift

Figure 9. Time- resolved studies by means of a stroboscopic method: by using

f one (or several counters, as shown in the figure) and changing the phase

shift one is able to follow the structure changes induced by the application):

(or removal) of an external field.

48
Because of the already mentioned large dead-time of detector systems (ca. 10 ys) they

record either 0 or 1 photon from each bunch and are "dead" for the next 10 ys. Thus, a multibunch mode is better than a single one only when the probability of recording 1 photon from 1 bunch is smaller than unity (as with small samples, fine collimation, high

absorption, etc.)*

5. Summary and Discussion

XE.D at a storage ring can be used for fast quantitative structure analysis and seems

to be especially suitable for phase transition and chemical reaction studies, fast

identification of known structures, studies of samples in special environments, liquid and

amorphous material studies, determination of textures, and time-resolved studies. Some of

these studies could not be made with XED at all or in any case with great difficulty and

poor accuracy using conventional sources; some others would need a much longer measuring

time if the same precision is to be attained.

In the early days of XED it was thought that some corrections might be much more

difficult to apply in the case of a white beam than for a monochromatic one. However, it was shown later that thermal diffuse [33], Compton [35], and absorption

[31,34,35,57,58,61] corrections can be applied without special difficulties.

The main disadvantages of XED at present are:

(a) the fluorescence induced by the white beam obscures the diffraction pattern in some cases, although by a proper choice of the scattering angle the diffraction

lines can be shifted to the most convenient position in the pattern;

(b) the white beam impinges on the sample and this may result in heating it

excessively or even in damaging it by radiation;

(c) the momentum resolution is limited by the energy resolution of the detector

system, however, the availability of hard x-rays partly improves the resolution;

(d) the dead-time of the detector system (of the order of 10 ps) limits the counting

rate to * 10^ counts/s for one detector.

As concerns the characteristics of synchrotron radiation required by energydispersive diffractometry, the importance of hard x-rays should be stressed . It is

necessary for obtaining larger scattering vectors, smaller attenuation factors and better

momentum resolution.

To sum up: the XED method takes full advantage of the white beam and also utilizes

other characteristic features of synchrotron radiation . It is simple to use and

relatively fast . It does not provide the accuracy obtainable with sophisticated

monochromatic beam methods ; however , it has a large number of applications in research and

development studies , and in selected areas it has advantages over the monochromatic beam

methods.

49
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Di scussion

Question (Hewat): Is it fair to conclude that the major limitation of energy dispersive

techniques is likely to be counter technology? Firstly, because of the limited resolution

possible (Ad/d ~ 3 x 10 ), and secondly, because of the limited count rate?

Comment (Buras): At present it is a fair statement. However, as mentioned in Section 4.7

of the paper, there exist some possibilities of improving the count rate.

Comment (Parrish): It would have been interesting to see a comparison for a moderately

complicated pattern of a synchrotron-SSD and a scanning diffractometer with

monochromator.

Comment (Buras): We are planning to perform this summer such a comparison at DORIS using

our triple axis crystal spectrometer.

Question (Uno): How accurate is the absorption factor which you can get in the

synchrotron radiation measurement?

Comment (Buras): I am not sure whether I understood your question correctly. I think

that accuracy should not depend on whether one uses synchrotron radiation or Bremsstrahl ung from a conventional x-ray generator. However, when one applies a profilefitting procedure one can treat the packing fraction as a fitting parameter.

54
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg , MD, June 11-15, 1979.

(Issued February 1980)

ACCURACY IN X-RAY WAVELENGTHS

R. D. Deslattes, A. Henins and E. G. Kessler, Jr. Center for Absolute Physical Quantities

National Bureau of Standards

Washington, DC 20234

The prominent Ka^ lines most frequently used in precision diffractometry are well known to be afflicted by the proximity of Ka^ radiation,

they suffer quite appreciable line widths (none is narrower than 300 ppm)

and have substantially asymmetric profiles. On the other hand, these

vagaries have not, for most of x-ray history, had as noticeable an effect

on final results as has uncertainty in the x-ray scale itself. This

situation led very early to introduction of arbitrary "standards" in the

x-ray region according to several different conventions leading to a situation where the dominant component of uncertainty for diffraction

measurements lay in the conversion factor, A. Only with the greatest

difficulty did the classical ruled grating scheme yield uncertainties in

A below 15-20 ppm. The past few years have seen a reversal of this situation since,

using x-ray and optical interferometry, we can now reckon single crystal

periods to near 0.1 ppm. Using other crystal periods derived from these

and accurate goniometers, the wavelengths of narrow lines (i.e., y-ray

transitions) can nowadays be determined to approximately the same accuracy

(0.1 ppm). Wavelength values assigned to x-ray lines in this way are thus

no longer limited in accuracy by the scale used, but by the characteristics

of the lines themselves. This is a subject on which recent work has shed

somewhat unexpected light and which carries with it some limitations for

crystal measurements (single crystal and powder methods) to which a measure

of attention needs to be drawn.

1. Introduction

For some years now, we have worked toward an improved knowledge of crystal spacings,

ay wavelengths and y-ray wavelengths. These efforts have borne fruit in the fundamenconstants area, in tests of theory for normal and exotic atoms and in determination of masses of elementary particles. For the most part the changes in numerical estimates

55
of x-ray wavelengths and the increased refinement of our results have not had much signi

icance for crystal structure determination. The recent discovery of appreciable fine

detail within strong characteristic x-ray lines is probably of even less concern, given

effectively low equivalent spectral resolution with which especially powder procedures a

;

carried out. Nevertheless, it is possibly useful, at times, to attend to the foundations of one business to be sure they are tidy and robust. It is with this end in mind that we have

gathered here a brief summary of the fundamental problems involved together with the new

results on profiles and the wavelength scale. There are, in fact, two cases in which thl

new results may have some practical significance, if one imagines that technical progresr

continues in structure studies. First, in the comparisons between observed and calculat!

densities, cell edge dimensions need to be known in terms of the cube root of volume meai sure (most usually this is expressed as a length and derived from the customary standard

but see below). Second, in the area of detailed profile analysis, although a smooth (as

metric) model suffices at present, it may someday be important to know that this approxi

mation is, in detail, wrong.

2. X-ray Wavelength Scales - A Little History

Crystal diffraction measurements, whether carried out for structural or spectroscop

purposes, yield only angles, twice the sines of which are A/d ratios as was known early

this century from Sir Lawrence Bragg and from Laue's equations. The first good estimate

a macroscopic measure of "d" came in 1912 from Sir William Bragg [l] 1 who equated the ma contained in a unit cell as calculated from atomic measure and from density and (a priori

fi m

unknown) macroscopic volume. For a hypothetical cubic crystal with cell edge a

Q

, n molef per unit cell of average molar mass (molecular weight), M, Bragg 1 s equation would be: ^ TMJ

PV - pa^ = CIS

where p is the macroscopic density and N^ is the Avogadro constant. For an estimate of

Bragg made use of the then recently available oil -drop value of the electron's charge, e and the already well-known electrochemical Faraday, F, according to

NA = F/e . (2)>

• .1 This approach gave an estimate for the cell edge dimension of rock salt a = 2d = 5.62 x 10 cm. Not long after, Mosely [3] computed a somewhat more accurate value d = 2.814 x|

10

8 cm (~ 1700 ppm). (Unfortunately, the oil-drop experiment contained a serious syste

••• J|

atic error which was to remain undetected for many years and an unknown source of confli

for many more years. An adequate digression on this point would render this report much

overlong but it is planned to treat it elsewhere.)

a

«1

figures in brackets indicate the literature references at the end of this paper.

56
! As the second decade of this century passed, it became clear that quite accurate A./d

|ios (and evidently A'/A. ratios) were emerging from careful angular measurements of symric reflections as suggested in figure 1. Indeed the accuracy of these "tube-spectrometer"

surements was far better than that of Sir William's estimate of the rock-salt cell dimenI. The automatic response of a scientific sub-discipline to such a problem, then as now,

to introduce a local unit.

Figure 1. Plan of a "Tube" spectrometer of the sort used in early precision

spectroscopic and crystal lographic measurements.

In the case at hand of the x-ray spectroscopy/crystal structure business, it was Manne

[jlbahn [4] (who died, by the way, only last year) who began to use a unit for wavelength

lominally 10 ^ cm and denoted by xu. He found calcite to be superior to NaC£ and by the

of his 1925 book [5] was advocating fixing the 18 °C grating space of calcite, d = 3029.04

Such a local standard permits succinct exchange of precise information provided only that

local standard is independently reproducible. Unfortunately, as now we know so well, latparameters of chemically similar specimens do vary appreciably and such variations have

ceable effects even in the case of such a well behaved natural crystal as calcite.

Clearly, instead of defining a local unit by a crystal dimension, the scale could have

i fixed by assigning an exact numerical value to a prominent x-ray line as was, naturally,

lested many times. Resistance to this alternative (at least in principle) seemed to cenon the fact that the resolution (or sharpness of definition) of a crystal repeat distance

superior to the fractional widths of x-ray lines; x-ray lines are also troubled by asymy and, for the strongest ones, a ubiquitous partner. (As will be seen in the following

ion, the sharpness of definition possible for the repeating distance of certain crystal

jimens is indeed remarkable while later it will be shown that x-ray lines are rather more

insive than had been imagined in the past.)

| As happens in many areas, standards of practice for the x-ray scale evolved somewhat

Ipendently of the "crystal definitions". For convenience and economy of effort, for meaments not of limiting accuracy, conventional values came into use for the wavelengths of

57
Cu Kc^ and Mo Ka^. Spectral tables and papers especially in Europe tended to be based on I

A.(Mo Ka-^) = 707.831 xu while papers in the U.S. tended to favor reporting data using

\(Cu Ka-^) = 1537.400 xu. That the units so-defined were, in fact, inconsistent was "discovered" by the spectroscopists only in 1964 [6] but was evidently already well-known to crystal lographers [7]. To escape the x-unit ambiguity and take the opportunity to elimi-Hl

o nate the 0.2 percent offset between the x-unit and the milli Angstrom, Bearden proposed a| o*

new, and more consistent, scale [8]. The unit for this scale was denoted by A which was

o*

chosen in such a way that A.(W Ka-^) = 0.2090100 A . In reference [8], considerable pains

were taken to express previously available data on this more consistent scale. Unfortuna^

ly, as will appear below, this scale seems also to suffer a measure of inconsistency. Sin

crystal lographers seldom use W Ka-^ radiation it was necessary to wait for a while in ordaj

to reach this conclusion in a direct way. [»

To return to the historical line, it is noted that, shortly after the demonstration a

x-ray diffraction by a ruled grating [9], a refined measurement was reported by E. Backliijj

in 1928 [10]. The output datum from this exercise A = ^gAg represented the ratio of a wavelength as measured on the "grating" scale to the same wavelength as measured on the

"Siegbahn" or x-unit scale. By the time of Backlin's measurement, a considerable and

accurate body of information on x-ray wavelengths and cell dimensions was available. Witflrf a value of A, this could be then turned to use in determining fundamental physical consteM»i For example, one could rewrite eq. (1) as a determination of in terms of a cell volume^

derived from edge dimensions a

x (in xu) and A as

NA =

nM

3 A 3 pa/

(3)

Other constants were also available in this way, as, for example, h/e from the high freqiU

limit (HFL) of the continuous spectrum where

hv = hc/A = hcA

xA eV (4)!

for electrons of energy eV, with V the voltage across the x-ray source and A. = \

xA the w< length corresponding to the HFL. Unfortunately, the value of "e" derived in this way is

variance with the final result of Millikan's oil drop studies [11]. The Millikan result

widely accepted and strongly held so that Backlin's thesis was to begin a debate lasting

almost two decades.

Because of its great difficulty and its pivotal role in the controversy, great attei

tion was given to the ruled grating measurement of x-ray lines. J. A. Bearden [12], in I particular, made gr^at contributions to the plane grating measurement technology involvei

58

1

ft
Jen

rile H. A. Kirkpatrick and J. W. M. DuMond [13] attempted to produce a large, focusing

istrument in which the important work of Tyren [14] (who compared calculable hydrogenic

lectra with M Ka^ 2) night be repeated. In all cases, the plan of measurement was to

impare A£ Ka-^ or the peak of the partly resolved doublet with an optical wavelength standard.

the culminating work of the plane grating school, A. Henins was able to report a measure

198

Cu Ka^ in terms of the Hg green line to a claimed accuracy of approximately 10 ppm [15].

At an earlier stage, the weight of cumulative evidence against the oil-drop "e" had be-

|me large and the origin of the systematic error (in rotating cylinder air viscosity meairement) had been discovered [16]. At that point, there came a brief "golden age" during

ich x-ray measurements contributed important and, in some cases, dominant information

garding the fundamental constants. By the sixties, this influence had passed as other

j'thodologies improved toward the few ppm level while x-ray measurements remained burdened

io|) a which stubbornly refused to reach even 10 ppm. (The 10 ppm result noted above came

ter, at a time when its impact was not large.)

That at the present time, an x-ray method (though not a spectroscopic one) is the domint source of information on (at the 1 ppm level), suggests that some new technologies

ve intervened as indeed they have. These technologies have permitted optical determination

Y

- ray lines (which are very sharply defined) with accuracies approaching 0.1 ppm while

tarohowing location of characteristic x-ray features to the lesser accuracy (~ 1 ppm) their

lejljrger widths entai 1

.

3. New Crystals - New Technology

As one looks at a diagram such as that of figure 1, it appears that an x-ray line can

used to measure a crystal spacing or vice versa but neither can be better known than the

ler. Since the minimum width of x-ray lines exceeds 300 ppm, there would appear to be a

jpm floor on d-measurements and on y-ray to visible comparisons if they use x-ray interJiate steps. It was clear by the very early sixties that, in response to the needs of

niconductor device technology, crystals of increasing perfection were coming to be available

Jecially in the case of Si. How could one extract the full scientific benefit from these

j/elopments in spite of poorly structured characteristic x-ray lines? This question was

tier active debate in the few remaining x-ray spectroscopy laboratories in the early sixties

sn the answer became clear in the first beautiful demonstrations of x-ray i nterferometry

U. Bonse and M. Hart [17].

Early interferometers were cut from single crystal silicon blocks such as indicated in

jure 2 for the most primitive LLL (Laue, Laue, Laue) case [18]. These blocks showed atomictie Moire patterns sometimes with appreciable areas of uniform intensity. Best of all,

ey are substantially achromatic, i.e., the picture seen is rather weakly dependent on x-ray

/elengths. A procedure to determine directly a lattice period (independent of x-ray wave-

|igth values) was evident to all but not necessarily easy to carry out. What had to be done

I to provide for motion of the third crystal lamella relative to the first two and deterie the motion by optical i nterferometry with a (visible) wavelength standard. Several

59

tie'
approaches to this problem have been made, three of which were described at a conference

held at NBS in 1970 [19]. Numerical results have, thus far, been reported for one of the

efforts although progress is evident in the case of two others. In any case, the optical

interferometric measurement of the repeat distance in a particular (Si) specimen marks

only the first step in a process leading to one or another invariant output. Any assumpt

that the numerical result can be attributed to crystals other than the object measured

involves one in a "generic" approximation which has obviously limited validity and an er structure not easily quantified. The one study so far carried out shows a range of varia

of 1 ppm for float zoned samples and 1 ppm for Czochralski material (with a 3 ppm offset)

from a small number of distinct sources [20].

Figure 2. Monolithic symmetric LLL (Laue, Laue, Laue) x-ray Moire interferometer after U. Bonse and M. Hart [17]. A lattice measurement can be

secured by freeing one of the wafers, equipping both it and the pair left

with optical mirrors and scanning a common baseline.

The measurement carried out at NBS is shown schematically in figure 3. The optical

j

interferometer is of the hemispherical Fabry-Perot type and the visible standard is a 63j

3 22

nm He Ne laser stabilized with respect to a saturated absorption feature in molecular I 129 iodine, namely the "B" peak in I 9 [21]. This particular laser source was measured wii; 86 respect to both the currently defined wavelength standard ( Kr, 606 nm) and the Cs freq^

standard. Its numerical value, 632990.079 pm (4 ppb), is consistent with a possible futij

redefinition of the meter such that c = 299792458 ms . It is also closely related to t Balmer a transition in hydrogen (which is immediately the Rydberg constant, R ) hence

j offering a suitable basis for comparisons with atomic theory, quantum electrodynamics an() for fixing masses of elementary particles. \

60
heste

a]

Figure 3. Schematic of combined x-ray and optical interferometers used to determine 220 repeat distance in a Si sample.

| The measurement proceeded by successive refinements toward fixing a value for the ratio

1where \ is the optical wavelength and d is the 220 repeat distance in our Si specimen

roximately 0.2 nm). This ratio has an integer part and a fractional part which are

rately determined. The integer part was determined by scanning to and fro over one or

(300 nm) optical periods while recording the more rapidly varying signal (0.2 nm period)

ie x-ray interference channel. The result (which was, of course, known ahead of time)

1648. The first approximation to the fractional part of the ratio, f, was obtained by

tronically "locking" the system to successive optical fringe maxima and noting the x-ray

isity. With due attention to the circular progression, these intensities could be

iged to give values for the cumulative x-ray phase as a function of optical order number.

i values from such curves, averaged for two scan directions, were used as initial approxijns to f. More precise estimates came from runs with larger "steps" covering two or

(optical orders between lock points. Finally, we chose pairs of integers, m, n, such

f m/n. In this case, lock was established only every n optical orders and the slow

a change due to the difference, f-m/n, measured. This procedure gave rather precise

Its in short times which was helpful in dealing with drifts encountered on the subic scale being measured.

Data from approximately 160 measurements were reasonably well fit by a Gaussian distention having a = 0.5 ppm which implies for the mean an estimated a = 0.04 ppm. Several

actions are needed for systematic effects, namely: 2.50 ± 0.01 ppm due to diffraction

1 shift of the visible light; -0.27 ± 0.03 ppm because of path curvature of 1.17 ± 0.06

| arc sec/200 optical fringes; 0.34 ± 0.01 ppm for specimen compression due to the fact

the interferometer measurement was carried out in vacuum while subsequent wavelengths

done at atmospheric pressure. Results for each measurement were corrected back to a

61
common reference temperature of 25 °C from the measurement temperatures (23.8 °C to 24.4 °C) using a = 2.56 ± 0.03 ppm K . When these corrections are applied and their

estimated uncertainties (intended to reflect a 67 percent confidence interval) combined

in quadrature with the statistical measurement uncertainty, a , the result is d(Si 220) =

192.01707 pm (0.1 ppm) at 25 °C or a

Q = 543.10628 pm (0.1 ppm). This result pertains onlj

to the particular piece of Si which was measured.

To progress from one specimen to other specimens and other species, we need procedur

by which this single piece of information can be disseminated to other crystals both of t same chemical identity and of different species. Some of these crystals may be useful in

themselves, as for example in the case of an Avogadro constant measurement while, for the

majority of applications, we require yet another measurement (of diffraction angle) to go

to an x-ray or y-ray wavelength. These processes, namely crystal -to-crystal transfer mea

surements and diffraction profile measurements are treated in the next two sections.

^

4. Crystal -to-Crystal Comparisons

\

To date all of our crystal -to-crystal comparisons have included an implicit transfer

in which it is assumed that two specimens from adjacent parts of the same boule have equa

unit cells and that the variously available reflections have spacings which are related t

2 2 2 -1/2

the cell edge dimension, a

Q

, by d^ = a

Q

(h + k + £ ) for the cubic crystals. How

good are these assumptions? We have cumulative arguments to the effect that they are not very bad at least down to a few parts in 10^. As regards uniformity within a boule, we h some worst case limits from the survey mentioned above [20]. In particular, where refere

[20] compared crystals obtained from the same source with similar specifications, 6d/d ap bounded by 2 in 10

7

. Where measurements have been made of the density of sections taken

along a single boule, a maximum density change of 0.1 ppm/cm is noted [22]. It would be

extraordinarily worst case hypothesis that the density variation be due entirely to changil

in a

Q along the boule. We have demonstrated that the density change between samples is

predominantly due to isotopic fractionation and that only a somewhat uncertain but small

residue can be due to lattice changes. A reasonable guess is that 6d/d likely does not exceed 0.01 ppm/cm.

2 2 2 -1/2 Questions regarding the reliability of using d

hkJi = aQ

(h + k + SL ) seem addre:

able in two ways which, though not logically independent, seem to make separate contributions to one's confidence in this matter. The first experimental test asks to what extenare we sure that d^ = d^

hk = d^? More generally, do all triplets, hk£, for which a

Q

(| + k^ + $}) ^ ^ = d

Q

/n give equal values for the primitive spacing? As one test of this p< degeneracy of the 511 and 333 reflections has been verified and we have a substantial bodj

of evidence that different orders of a particular primitive reflection are related by appi

priate integers. These results are near the 0.1 ppm level in some cases. In addition, x- "polygons" were constructed using four-fold and pseudo six-fold "standards" in a goniomet*

calibration exercise [23]. The results of these exercises were consistent with "perfect"

62

?
our-fold and six-fold symmetry within 0.02 arc sec which, at the Bragg angles used (near 45°)

mplies effective lattice equality within 0.1 ppm. It is asserted on the basis of the above

rguments that equating the results obtained from adjacent parts of the same boule and with

quivalent reflections is permissible at the 0.1 ppm level.

Given admissibility of the implicit transfers, how can we make accurate explicit transers between different crystals of the same species without being limited by the imperfection of x-ray lines? Several modalities are available to this end. Those we have used are

'Related to ones discussed in one form or another by Hart and collaborators as indicated

\elow. One of the schemes applicable to comparison of highly degenerate specimens and

! 1st able to handle, as well, cases of rather approximate degeneracy is shown in figure 4.

0

i jhe long crystal shown at A has a grating spacing approximately equal to one of the specia lpns being compared. When a crystal specimen already characterized as having a repeat disknce, d

g

, is placed at the position B, diffraction maxima will be observed as separated by

ti angle B Q. When the specimen to be calibrated which has a spacing d

u is placed at B, Effraction maxima, for the symmetric ray-paths shown, will be separated by B. If, with

je attention to algebraic sign, we let 6 = B - p, then

^ = cos(6/2) - sin(6/2) cote (5)

ere 0 is the Bragg angle for the first crystal. Evidently knowledge of the x-ray wavength used is required only for evaluating the cote term. Since this term appears multiied by sin(6/2), wavelength information need not be terribly precise provided 6 is quite

all. When such a procedure is applied to equivalent reflections in the same species, the

velength dependent term is always so small that an extremely crude estimate of the waveinge "9th suffices.

dref

bu

;te

is pi

bod) 0 1

ect

1

Figure 4. Quasi-non-dispersive arrangement for transfer of crystal calibration.

Unknown sample and standard crystals are alternately placed in position B and the difference, B, between the left and right diffraction paths are measured. ^
On the other hand, when it is required to compare cell edge dimensions for different

crystal species, one must look for near coincidences in order to remain relatively insens'

tive to x-ray wavelength values. In the main case of interest here, we have made use of 1 near degeneracy of the 800 reflection in Ge with the 355 reflection in Si, as suggested

Baker and Hart [24]. Since these spacings differ, nominally, by 240 ppm, a wavelength val good even to 100 ppm would already overwhelm any potential need when due account is taken

of the intrinsic widths of x-ray lines.

Altogether there are three regimes in which these "quasi-non-dispersive" transfer me surements can be carried out. In the first, and obviously most desirable, spacing differ

are so small, e.g., a few to a few hundred ppm, that the observed profile appears simple

symmetric. In this case one does not need either a good estimate of line shape or a good

algorithm for establishing a wavelength scale. In the second case, mismatch is sufficienl

large that and begin to appear partly resolved. This is a difficult region in whid 1

to work because of the required corrections for overlap. The problem can surely be hand! 1 by a detailed modeling of the spectrum; up until the present, we have avoided this region!

worked, instead in a third regime. For this third mode, mismatch of A and B is sufficien

that a, and are clearly resolved though, of course, they appear more closely spaced th

in typical single- or double-crystal profiles because of the near cancellation of disperse

This regime does make some more demands on angle measurement and wavelength information ti the others, however it still offers a distinct gain over the fully dispersive case. For example, when one makes a comparison using 1 percent mismatch, there is a 100-fold reduci

in the accuracy of angle measurement and of wavelength input data required to generate af

targeted output accuracy. One case where this regime was tried involved comparing (440)

spacings in Si and Ge where the mismatch is 4 percent. Here the accuracy gain was not qi sufficient to compensate for a relatively poor angle calibration then available for the

goniometer in use and limitation of knowledge of the Ag Ka, radiation used began to becod

perceptible as we were seeking 0.1 ppm in that particular exercise. Any less stringent ? requirement would have been easily met. » To summarize this part of the measurement chain, we have transferred spacing informiii from the interferometrical ly calibrated Si specimen to all other crystals used in subseqf

precision measurements of x-ray and y-ray lines. These include Si crystals cut for symm

Laue diffraction using the 220, 400 and 111 reflection and Ge crystals oriented for 400 f reflection. As described in the next section, these have been used in measurement of x-ji and y-ray diffraction profiles having scale errors below the 1 ppm level.

f 5. Double-Crystal Diffraction Measurements

Crystals calibrated as described above can be used for accurate wavelength measured;

provided angle measurement facilities are adequate. At accuracy levels suited to work b< 1 ppm, the required angle scale needs to be established by an i_n situ first-principles c bration. Also the scale refinement needed (0.1 arc sec at large angles to well below O.f arc sec at small angles) strongly suggests use of i nterferometric angle measurements. [ 64

!
(a) DETECTOR

(b)

Figure 5. Diagrams of large angle (a) and small angle (b) double crystal

diff ractometers.

id; Geometric arrangements of the two instruments used thus far are shown in figure 5.

j ure 5a shows the typical arrangement of a Bearden-Ross instrument [25] which we have used

the Cu Ka and Mo Ka measurements. This instrument has a hybrid angle measuring system

qu|i h a ± 3° i nterferometric goniometer surmounted by a large angle indexer with fairly

roducible 1° stations. The large angle generator was calibrated with respect to a 12-

coijed optical polygon which, in turn, had been calibrated from first principles. The angle

it erferometer (resolution limit near 0.01 arc sec) was then calibrated using the indexer to

erate up to 5° increments at several points in its range. Because of irreproducibility in

i indexer and the several stages required in calibration, this instrumental arrangement was

sefjilj/ marginally adequate even for relatively large angle reflections (~ 30° - 50°) with Cu

I yimt

»j 2 anc* M° 2 rac'iation.

j The situation of the small -angle instrument [26] indicated in figure 5b was more satistory from the point of view of calibration, but it had a very limited angular range of

5°. (This limitation was determined by certain, by now irrelevant, choices made 15 years

ier.) The small angular range could, of course, be entirely covered by i nterferometric

iurement permitting a very satisfactory calibration procedure. In this procedure, all

irfacial angles of a 72-sided optical polygon were successively measured by the interureiif))meter; the closure condition, namely that the sum is 360° suffices for the calibration

^([pendent of polygon errors. The angle interferometers are, in this case, quite sensitive

i e5 (:|ig easily readable to 0.05 milli arc sec. With the aid of a sensitive autocol 1 imator , such

bration exercises have been carried out over the past four years with individual measureerrors of a few parts in 10 . The calibration constant changes slowly with time (y 0.7

65
ppm/year) requiring that the value be read from the curve of growth. This appears to have

present a limiting accuracy near 0.1 ppm which is entirely satisfactory for the x-ray measu

ments but limits slightly the y-ray determinations. Unfortunately, the limited angular rar

has restricted this instrument's application to only the case of W Kc^ ^ among the x-ray

lines measured. (Still a third instrument with an interferometric range of ± 15° has been

built [27] and is available to obtain slight improvements in the results to be presented

below should these be required.)

Results available thus far for Cu Ka^, Mo Ka-p and W Ka-^ are as follows: \(Cu Ka-^) =

154.05974 pm (1.0 ppm); A(Mo Ko^) = 70.93184 pm (0.6 ppm); A(W Kc^) = 20.901349 pm (0.9 ppH

Numerical values given for Cu Ka^ and Mo Ka^ differ slightly (0.44 ppm) from those we

published in 1974 [28] due to improvements in the lattice parameter measurement and small

systematic corrections. If we compare these above data with the three x-ray scale definio tions, values for A are: A(Cu Kc^ = 1537.400 xu) = 1.0020797 A/kxu (1.0 ppm); A(Mo Ko^ = 707.831 xu) = 1.0021013 A/kxu (0.6 ppm); A(W Kou = 0.2090100 A*) = 1.0000167 A/A* (0.9 ppm

Such conversion factors may be of some utility in interpreting past work, however, we sugg

that they are likely to be less needed in the future.

These procedures have also been applied to almost two dozen y-ray secondary standard

198 192

lines in the range 0.06 < E < 1.2 MeV. Initial results on Au and Ir y-lines [29] ha

been followed by new measurements on lines from "^Yb and ^^Tm [30]. All these results h accuracy claims near 0.3 ppm; there are several cases in which satisfactory closure tests

have been made among three-level transition groups where E-^ = E-^2 + E23.

6. Line Shape Refinement

1 There is one final issue, namely, what are line shapes in detail? Many studies in thj

past have reported widths and asymmetry indices but there were no really detailed profiles I published by the mid- seventies with the high quality crystals by then available. Over 20 I years ago Lyman Parratt had developed some qualitative pictures suggesting that all x-ray I lines should conceal a detailed infrastructure [31]. There had also been, from time to til

communications reporting observations of some fine scale detail.

In 1976, Sauder, et al . [32] using a tunable arrangement of a high resolution monolilj

monochromator [33], studied in detail the profile of Cu Ka^ ^ wi tn resulting data as showr

in figure 6. Calculations on the positions and line strengths for the double vacancy trai

sition arrays corresponding to ls2p

6 3d9 1,3

D » ls

2 2p

5 3d9 1,3

PDF and ls2p

6 3p5 1,3

P -> ls

2 2p! 13' SPD are also shown in figure 6 along with the result of subtracting Lorentzian doublet

components from the raw data. The association between at least the general clumping of tl predicted satellite lines and experimental residues appear quite suggestive but some skepi

cism remains in order. There was, however, other work underway, at the time not known to

the U.S. workers which gives striking confirmation of the general picture and offers, in

the end, stern warnings regarding the limitation of detailed profile descriptions.

66
n r

(a)

(b)

j I Hi I.! L J , I

(c)

0 -10 20 30

RELATIVE PHOTON ENERGY (eV)

Figure 6. The Cu Ka, „ region as observed, a; spectra resulting from subtracting model distribution, b and c. 67
i Using synchrotron radiation from the electron storage ring at Orsay, Briand, Travennie

and Chevallier were studying widths of transition metal lines using near threshold fluores-f cent excitation [34]. Their instrumentation had far less resolution than that used to obtain figure 6; it was, however, quite adequate to see the remarkable changes in width

shown in figure 7. What is happening here is that quite clearly when the incoming photons

have energy insufficient for producing a particular double-vacancy initial configuration, t associated satellite array simply disappears. Since these extra transitions are scattered

about in some way such as that suggested in figure 6, one sees a narrowing for near threshold excitation. Not only does one have to expect a narrowing but, since the distribution

of the satellites has no requirement to be symmetric, shifts must be expected in any smooth!

fit property of the profile, e.g., peak location, centroid, etc. Instrumentation used in

reference [34] was not quite adequate to show these shifts but their presence seems inevit^

ble. Subsequently, further confirmation [35] of this model has emerged in a detailed stud^

of Cu KB.

i

li

4000

spectrum A FWHM=2 7eV

spectrum B FWHM = 2 9eV

fi

480

Figure 7. Profile of Fe Ka^ as obtained very close to single vacancy

threshold, A, and with excitation of 500 eV above the single vacancy

threshold, B.

I

r

i

What then are the implications of these results for crystallography? As was suggested

in the Introduction, nothing in this entire report can be disturbing to crystallography as

it is now practiced since there are other, more dominating, sources of error in all currenl

methodologies, especially those using powdered samples. On the other hand, there is at lei an aesthetic point in understanding basic limitations and potential sources of confusion.

Clearly, no one uses tunable photon excitation of his or her diffraction tube so it remain*

68
) be considered what can be expected from electron bombardment sources. Crudely, the elec-

'on excitation behavior with respect to anode voltage represents a complex kinematic averagig of the photon excitation function with necessary allowance for energy loss processes.

IIth constant potential excitation at high tube voltages there is little doubt that, for a ean metal, gross smearing has covered most traces of the above detail. On the other hand,

th unfiltered quasi-dc or ac excitation it is clear that the line is changing shape during

ie time that voltage is near threshold. Surely, this is not a large effect and it occupies

small part of the excitation waveform, but it is there nonetheless. Finally, there is one

her aspect of this problem which may prove to be a more troublesome one in the end. It is

ill known that x-ray emission satellites, when they are clearly resolved from their parent

ansitions, display far greater chemical sensitivity (shifts and intensity changes) than do

ie diagram lines. It would surely be imprudent to assume that the close-in satellites which

e important here have some immunity from this problem. We have, therefore, a mechanism

rough which aging of the anode in the residual gases of the envelope or its alloying with

thode and shielding materials may lead, over time, to changes in the shapes and positions

the characteristic lines.

8. Summary and Conclusions

An attempt has been made to indicate the state of current knowledge of some of the x-ray

nes frequently used in structure studies. The data indicate that a particular profile has atures such as peak location which are established at or near the 1 ppm level. It is most

sirable that such first-principles determinations be undertaken in at least one or two

her laboratories so that the presently available results could be confirmed or challenged.

Certain recently discovered complexities have been recapitulated which tend to make a mple parametrization of the line profiles somewhat suspect. Indeed these complexities are

own to point up the existence of a certain level of i reproducibility in the smoothed feares of characteristic x-ray lines obtained under different excitation conditions.

At the present time there appears to be no structural methodology that fully utilizes

e available accuracy and certainly none is presently troubled by the last mentioned subtle

obi em areas.

References

1] Bragg, W. H. , Proc. Roy. Soc. (London) 88A, 428 (1913); ibid. 89A, 246 (1914); 89A,

430 (1914).

2] Mill i kan, R. A., Phys. Rev. 26, 198 (1908); Phil. Mag. 19, 209 (1910).

3] Mosely, H. G. S. , Phil. Mag. 26, 1024 (1913).

4] Siegbahn, M. and Leide, A., Phil. Mag. 38, 647 (1919); Siegbahn, M. , Arkiv. Matt. Astron.

Fys. 14, Nr. 9 (1919).

5] Siegbahn, M. , The Spectroscopy of X-rays (Oxford Univ. Press, London, 1925), p. 84.

6] Bearden, J. A., Henins, A., Marzolf, J. G. , Sauder, W. C. , and Thomsen, J. S. , Phys.

Rev. 135, A899 (1964).

69
[7] One of us (R. Deslattes) was given this impression while trying to tell of the then

"new" results.

Bearden, J. A., Rev. Mod. Phys. 39, 78 (1967).

Compton, A. H. and Doan, R. L. , Proc. Natl. Acad. Sci. U.S. 11, 598 (1925).

Backlin, E. , Inaugural dissertation, Uppsala Univ. Arsskrift, 1928; later work is

described in Z. Physik 93, 450 (1935).

Millikan, R. A., Phys. Rev. 7, 355 (1916).

Bearden, J. A., Phys. Rev. 37, 1210 (1931); 48, 385 (1935).

I

1 Kirkpatrick, H. A. , DuMond, S. W. M. , and Cohen, E. R. , Proceedings of the Third Int national Conference on Atomic Masses , R. C. Barber, ed. (Univ. of Manitoba Press,

Winnpeg, 1965), p. 347. Tyren, F. , Zeits. Physik 109, 722 (1938).

Henins, A. , in Precision Measurement and Fundamental Constants (D. N. Langenberg

and B. N. Taylor, eds.), National Bureau of Standards Special Publication No. 343 (U

G.P.O., Washington, D.C., 1971), p. 255.

This origin for the discrepancy was apparently first suggested by K. Shiba, see discussion by J. S. Thomsen and A. F. Burr, Am. Jour. Phys. 36, 803 (1968).

Bonse, U. and Hart, M. , Appl. Phys. Lett. 6, 155 (1965); Z. Phys. 188, 154 (1965).

For a review, see U. Bonse and W. Graeff, in X-ray Optics , H.-J. Quesser, ed. (Sprin

Verlag, Heidelberg, 1977), p. 93. Precision Measurement and Fundamental Constants , D. N. Langenberg and B. N. Taylor,

eds., National Bureau of Standards Special Publication No. 343 (U.S.G.P.O.,

Washington, D.C. , 1971).

Ando, M. , Bailey, D. , and Hart, M. , Acta. Cryst. A34, 484 (1978).

Schweitzer, W. G. , Jr., Kessler, E. G. , Jr., Deslattes, R. 0., Layer, H. P., and Whe

stone, J. R. , Appl. Opt. 12, 2927 (1973).

Bowman, H. A., Schoonover, R. M. , and Carroll, C. L. , J. Res. NBS 78A, 13 (1974), sej

especially p. 39, fig. 6. This work (unpublished) was carried out with the help of B. Paretzkin.

Baker, J. F. C. and Hart, M. , Acta. Cryst. A31, 364 (1975).

Deslattes, R. D. , Rev. Sci. Inst. 38, 815 (1967).

A description of this instrument together with a more thorough description of the

entire measurement chain is being prepared.

This instrument which may, over the next few years cover the range 0.05 < E < 5 MeV, was designed and built in collaboration with two scientific visitors, W. Schwitz,

University of Fribourg and Swiss National Science Foundation, and L. Jacobs, Assoc.,

IKS Leuven University and SCK, Mol , Belgium.

Deslattes, R. D. and Henins, A., Phys. Rev. Lett. 31, 972 (1973).

Kessler, E. G. , Deslattes, R. D. , Henins, A., and Sauder, W. C. , Phys. Rev. Lett. 40

171 (1978). Background and some details can be found elsewhere; see also,

R. D. Deslattes in, Proceedings of the International School of Physics "Enrico Fermi'

Course LXVIII 12-14 July (1976) (to be published).

70

it

Si

I

i

ti

x
ltd

0] Kessler, E. G. , Jr., Jacobs, L. , Schwitz, W. , and Deslattes, R. D. , Nucl. Instr. and

Meth. 160, 435 (1979).

|l] Parratt, L. G. , Rev. Mod. Phys. 31, 616 (1959).

2] Sauder, W. C. , Huddle, J. R. , Wilson, J. D. , and LaVilla, R. E. , Phys. Lett. 63A, 313

(1977).

3] Deslattes, R. D. , Appl. Phys. Lett. 12, 133 (1968).

4] Chevallier, P., Travennier, M. , and Briand, J. P., J. Phys. B 11, L171 (1978).

5] LaVilla, R. E. , Phys. Rev. A 19, 717 (1979).

Discussion

estion (Shirley): Can you please advise us what criterion of line position you used, con- dering that for example the CuKa, line does not have a very regular shape?

Can you state the wavelength in the form of, for example, medium position at 80 percent

ximum intensity, so that the result may be more readily applied?

sponse (Deslattes): This is a very good question to which there is not a simple answer

pecially below 1 ppm. Firstly, the measurements we made on CuKa-^ and MoKa^ were carried

(t before observation of the internal fine structures mentioned in the text and with insuficient resolution and stability to have seen these features clearly. (In the case of W Ka^

s linewidth itself may preclude observation of such structures although they are certainly

jsent. The line position criterion was, in each of the above cases, peak location which

; given (approximately equi valently) by the asymptotic bisector of the media of chords or

formal differentiation of a fitted profile. Discovery of the internal fine structure

rtainly means that a carefully observed profile would exhibit significant jitter in mid3rd location as a function of chord height thereby precluding the realization of a wellined asymptote. Similarly, no fitting to simple model functions or polynomials of

Jerate order could exhibit statistical control. Therefore, the formal differentiation

jorithm fails also. Thus, all that can be said is that: if one smears the "true" proe adequately, then a convolved result emerges which exhibits the needed properties and

(is fold has a peak position which is well-defined to a certain extent and is possibly

inewhat independent of the details of the smoothing function. This is clearly not a :isfying outcome but is nonetheless what appears to emerge from our best efforts to date,

could, possibly we should, publish some horribly detailed profile (lumps and all) with

absolute energy, wavelength or frequency scale and leave it to the user to smooth this as

i or her application requires. Remembering that the texture is chemically sensitive and

)endent on excitation conditions, we are counseled to moderation in thinking to undertake

|h an enterprise.

71
If

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l«l

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ii
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

NEGLECTED CONSIDERATIONS FOR INTENSITY MEASUREMENT

L. D. Jennings

Army Research Center

Watertown, MA 02172

Although the primary thrust of powder diffraction is the measure- ment of the position and shape of the peaks, some applications require

accurate intensity measurements. Some considerations and techniques

which seem not to be generally appreciated are discussed: (1) the

technique of evaluating dead time by measuring the apparent absorption of a single foil as a function of count rate; (2) the role of

small angle scattering, for example within an absorbing foil, in this

and other connections; (3) the recognition that a typical monochromating

crystal displays extreme secondary extinction, thus yielding polarization

ratios far from the mosaic value; (4) simple techniques for measuring

this polarization ratio using amorphous scattering; and (5) the value of

using a wide receiving slit to measure integrated intensities rapidly and

with high resolution.

1. Introduction

Although most of the discussion at this conference concerns the positions and shapes

of powder diffraction peaks, there are also analytical, characterization and structural

problems for which intensity information is valuable. With modern photon counting apparatus,

j such intensities are often available with an apparent accuracy of a fraction of one percent.

i The actual accuracy is much harder to determine. It was therefore appropriate to conduct an International Powder Intensity Project (under IUCr auspices) [l] 1 some dozen years ago. A few caveats and techniques were developed at that time which may not be generally recognized,

and it seems desirable to repeat them here.

Good accuracy requires both reasonable resolution and small and well defined background. Thus, first preference must be given to crystal monochromated radiation. Ayers,

Huang, and Parrish [2] have also suggested that standard wavelength dispersive diffractometry,

when properly carried out, is more rapid than is energy dispersive or position sensitive

detection. I, therefore, will couch the discussion in terms of an ordinary diffractometer

with a crystal monochromator. It is often convenient to place the monochromator after the

Figures in brackets indicate the literature references at the end of this paper.

73
sample, to remove fluorescence for example, but, for the highest accuracy, the monochromator

must be placed between the source and the sample. I will therefore explicitly consider the I latter configuration, though most of the remarks will also apply to other situations.

Rather than merely highlighting a large number of possible considerations, I would

prefer to restrict the discussion to just a few points. I hope in that way to give a real

appreciation for each one.

I

i

2. Dead Time

All apparatus which counts individual photons must fail to distinguish pulses which

arrive nearly simultaneously. The fraction of the counts lost will depend on the count rate,

so some correction must be made, either automatically by the apparatus or in the data reduc- 1

tion step. In any case, the careful experimenter will wish to verify that the correction is

accurately made. The usual technique is to compare the expected and measured effect of the

insertion of a number of presumably identical absorbing foils. The neglected technique is to make use of a single foil at a number of different count rates. The usual technique is illustrated in figure 1. An intense beam is attenuated by the

successive insertion of nearly identical foils. The apparent count rate is plotted against

the number of foils inserted using a semilogarithmic scale. A straight line is fitted to the

data at low count rate and a dead time is determined which will bring the points at high coun

rate up to the line. The procedure has some disadvantages, however. The plot of figure 1 does not readily reveal modest errors or inconsistencies; as shown, a 5 percent effect would

scarcely be noticed. Furthermore, there is usually no stated criterion as to how to choose

which points are to be fitted with the straight line. Most importantly, the technique does

not explicitly take into account the possibility that the foils are not identical.

The neglected technique is much more satisfying and graphic. A master foil is chosen

which can conveniently (perhaps automatically) be inserted and removed from the beam. Thus,

if the beam is attenuated by foils which are adequately uniform, but whose attenuation factorj

is otherwise arbitrary, it is possible to plot the apparent absorption of the master foil

(R) vs the count rate N

Q

(with the foil out). Figure 2 shows that such a plot may be represented very closely by the equation R = Rt

- N

Q

T(R

t

- 1). A little algebra confirms that

this result corresponds to the usual dead time formula, N. = N/(l - Nx), where N. is the true count rate corresponding to the observed count rate N. Note that the entire derivation

relies solely on the assumption that the same foil replaced in the beam in the same way

always has the same absorption. There is no question about which points are to be used in

the fitting of the line and any discrepancy is clearly visible; 0.1 percent effects can now

be seen.

74
0 1 8 12 16 20 24 APPARENT COUNT RATE (10

3 PER SECOND)

Figure 1. A typical plot of observed count rate vs number of

absorbers placed in the beam.

0 2 4 6 8 10

NUMBER OF ADDITIONAL ABSORBERS

Figure 2. Measured values of apparent absorption of a master foil vs the apparent

count rate with the foil removed. The intercept on the y axis is Rt and the slope

is t(l - R+ ), where t is the dead time.

1

75
There are many other issues which need to be considered and also benefits to be derived

from having a readily available set of calibrated foils. The choice of foil material is of

importance. One has to consider possible beam hardening (the small curvature at low count

rate in figure 2 arises partly from beam hardening). For a typical scintillation detector, i< the entire results depend on the analyzer settings and also perhaps on the beam distribution- over the crystal area. Dark counts must of course be taken into consideration. On the othe

hand, if genuine curvature is found in the plot, it can easily be taken into account by adding a second order term to the dead time formula. In any case, a very accurate value (R

t

) i obtained for the absorption of the master foil and absorption of other similar foils can be

easily obtained by comparison with the master at high count rate. Such values may be used ti make measurements on an absolute basis, or help with relative measurements at widely

differing count rates. These and other considerations are discussed in more detail in

Chipman's paper [3].

In any case, the important thing to remember is that the master foil method is more convenient and more reliable than a method which relies on several foils being identical.

3. Small Angle Scattering

There is one aspect of absorbing foils which is especially illustrative. For concreteness, consider the case of a monochromator followed by one or more absorbing foils

followed by a small sample. In such a case, one might replace the small sample by an aperture of the same size so as to study the portion of the beam of interest. In this

situation, we have often found the apparent absorption of a foil to have an unexpected value

The situation is most marked in the case that the foil is composed of ground powder embedded

in plastic (such as is available commercially). The explanation is simple: the mono- chromator gives rise to a non-uniform beam and the ground powder gives rise to a small angle

scattering. Thus, for example, if the sample is in a portion of the beam that is especially

intense, the foil will appear to have unusually large absorption. f Although such circumstances are perhaps rather specialized, they have actually arisen in! our own work. However, the principle illustrated must always be considered: small angle

scattering may sometimes best be thought of as slight broadening of the direct beam; in otherj

situations it is appropriate to think of the small angle scattering as part of the

scattering. In this latter case, there is the additional consideration of whether or not the]

scattering is part of the absorption. This point comes up, for example, if the absorption

coefficient is needed in an absolute measurement of a structure factor on a powder. Any

measurement of this coefficient which does not have a discussion of angular acceptance of the

apparatus is essentially worthless (at least for light elements in powder form where the

scattering is appreciable).

In summary, for an accurate experiment, always reflect on whether or not small angle

scattering may be a troublesome side effect.

incit

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BSS

76
4. Values of the Polarization Ratio

The x-ray reflection from a sample is naturally divided into two components, n with the

polarization in the plane of incidence and a with the polarization normal to the plane of

incidence. The remainder of the system may have differing efficiencies for the production or

heif! transmission of these two components. The ratio of these two efficiencies is called the

polarization ratio, K [4]. The two most important contributions to polarization ratios difi£| fering from unity are polarization arising in the x-ray source and from reflection by a crystal monochromator.

In the typical case where an experiment is performed using line radiation from a tube,

account would have to be taken of whether the adjacent bremstralung wavelengths are con- sidered to be part of the background or to be part of the desired beam. In either case, our experience is that line radiation may be considered unpolarized to moderate accuracy. On the

other hand, in cases where a crystal monochromator is used, it will be shown that it is

essential that the polarization ratio be measured and this measurement will, of course,

include any effects arising from the x-ray source.

If the source of x-rays, as viewed from any one point on a monochromating crystal, is

substantially uniform over an angular range greater than the mosaic spread of the crystal,

then the diffracted power will be proportional to the integrated reflection of that crystal.

In this case, there are again two principal polarization directions, and there will be a different integrated reflection for each of these directions. The ratio of these integrated

reflections has been called the polarization ratio, a, by Chandrasekaran [5]. If, in

addition to the uniformity condition just mentioned, the source of x-rays is unpolarized,

then the polarization ratio defined in this paragraph is identical to that defined in the

above paragraph. In the cases that the assumptions are not met, the context makes it clear

whether it is a or K which is being considered.

It has been assumed by many workers, including Chandrasekaran, that the possible range

2 of a l s from cos 28 to cos 26, where 6 is the Bragg angle. It has also been assumed that the

possible range for K is the same, but that for a highly imperfect monochromating crystal the

2

K value would be close to the ideally mosaic value, cos 28. All these assumptions are incorrect. The implications of and reasons for this are discussed below.

Of especial interest in connection with this symposium is the observed range of K, which

enters in an essential way into the polarization factor required to adjust experimental data. For the two most common geometries, an error of A in K leads to a fractional error of A/(l +

K) or A/[K(1 + K)] in the polarization factor at 90° in comparison to the low angle value.

Taking a Li F monochromator at CuKa as an example, there are published values of K ranging

from 0.62 to 0.78. There would thus be errors ranging from about 8 percent to about 25

percent if the ideally mosaic value of 0.50 were used instead of the measured value. Note

also that some of the measured values are greater than cos 26 = 0.71.

77
Because this wide range of K values is not generally appreciated, the Apparatus Commission of the International Union of Crystallography is conducting a survey [6] of measure

values. Further information may be obtained from the author who would be very happy if some of the symposium participants were encouraged to perform the necessary experiments.

It is not necessary to understand the physical basis for K values differing so greatly I 2 • t from cos 28 in order to make use of the correct polarization factor. On the other hand, sue an understanding may serve as encouragement for making the necessary subsidiary measurements' Consider, then, a very imperfect monochromating crystal with small absorption. Consider als

the case that the rocking curve of the crystal is relatively narrow, yet whose width is

greater than the angle intercepted by the source of x-rays as viewed from a point on the

i monochromator. If, in addition, the structure factor is sufficiently large, the kinematic

approximation would give far more than total reflection. This is, of course, impossible.

The shortcoming in the kinematic approximation is that it does not take account of the

diffraction contribution to reduction of the incident beam, an effect referred to as secondary extinction in the present case. In actual fact, for suitably prepared materials,

it is possible to achieve the condition that there is very strong secondary extinction (i.e.

the power reflected is very much less than that calculated on the kinematic approximation)

and yet the a component of the incident beam is almost totally reflected . In this case, for

2 the n component, the kinematic approximation would give a value smaller by a factor cos 28^

with 8^ referring to the monochromator. This means that the n component would penetrate

farther into the crystal and there would be somewhat more photoelectric absorption of this

component. But this is still small in comparison to the portion diffracted so the n

component is also almost total ly reflected . Therefore, the polarization ratio, under these

conditions, is nearly unity . In actual cases both the geometry and the crystal preparation may not conform to the

idealizations discussed above. The general principles do apply, however, and so the thrust

of this section is that the polarization ratio may assume a wide range of values, none of

2 which (for a high efficiency monochromator) is close to the kinematic value cos 28. It is

therefore essential that the polarization ratio be measured for work requiring moderate to high accuracy.

5. Measurement of the Polarization Ratio

The original impetus for measuring the polarization ratios was the Powder Intensity

Project [1] which aspired to the very highest accuracy. Some of the methods used are described in an IUCr sponsored report [7] on accuracy in powder diffraction. For ordinary work

at about 1 percent accuracy, much simpler techniques will suffice. These are described in

the appendices to the IUCr polarization ratio survey announcement [6] (available on request

from the author) and in the work of LePage, Gabe, and Calvert [8].

78
The idea is to measure amorphous scattering at 90° in each of the two principal planes.

The ratio of these two results gives the polarization ratio directly. The only problem is to ensure that identical solid angles are intercepted by the detector in the two cases. A jig to do this is sketched in figure 3. If the construction permits a full circle rotation

of the detector, it is only necessary to put the axis of the jig near the center of the beam

and average counts taken at diametrically opposite positions in each of the two principal

planes. If such a full circle rotation is not possible, it is then necessary to determine

the center of gravity of the beam in each of the two planes so that the effective solid

angle can be made the same during the two measurements. This determination may be made by

scanning through the beam in each of the planes. Alternatively, the median of the beam

power, determined by cutting off half the beam with a stop, may be an adequate approximation

to the mean. The thrust of this section is that it is relatively easy to measure a polarization

ratio. I should like to encourage all workers to make such measurements and also submit

the results to me as a part of the IUCr survey of polarization ratios.

Figure 4a shows a typical region of a powder scan made with a narrow receiving slit.

If the slit were replaced with a somewhat wider slit the count rate would be increased so

it would be possible to take the scan of figure 4b somewhat more rapidly and still get the

same statistical accuracy. However, it is clear that the scan of figure 4b has poorer resolution. Thus, using the slit of figure 4b, it would be more difficult to separate the intensity belonging to the two different peaks. Because of this situation, it is easy to fall

into the error of assuming that a wide slit is inferior to a narrow slit when high accuracy

is required. This is not at all the case, as shown by the following example.

Figure 3. A jig used for measuring the

polarization ratio. An amorphous

scatterer is designated by S, the beam

catcher by BC, and the receiving aperture

by RA.

6. Use of a Wide Receiving Slit

79
ANGLE

Figure 4. Two typical neighboring peaks in a powder scan. Figure 4a shows the

situation with a narrow slit where the received power has dropped to the background

level at points A, B, and C. Figure 4b shows the decreased resolution of a somewhatJ

wider slit. A very wide slit, spanning the angular range from point A to point B integrates the peak while giving very well resolved definition of the end points.

Suppose that the problem is to find the relative area of the two peaks in figure 4a

as accurately as possible. This might be important in a study of preferred orientation or

of relative abundance of two components, for example. The straightforward technique would ) be to use the same slit that yielded figure 4a and, using either step scanning or continuou

scanning techniques, to determine the area between points A and B and also between points B and C. If the wider slit implied by figure 4b were used so as to reduce the time required '

get adequate statistical accuracy, it would not be clear as to where to stop scanning the

first peak and start scanning the second peak. Suppose, however, that a slit were availabli

whose angular width was just the separation of point A and B. Then, in the not unusual

situation that the positions of the peaks were known in advance, it would be possible to place the slit so as to integrate the area from A to B without scanning. Then the slit wou

be moved so as to integrate the area from B to C, again without scanning. The required

information would thus be available directly from a comparison of the two count rates.

It is important to note that in the case described the ends of the "scan" are precisely

fixed by the edges of the slit. Thus, the use of the wide slit implies the best possible

resolution rather than any loss of resolution. Furthermore, in addition to having the possi1 bility of achieving an accurate value in a short time, the constant count rate allows for a

80

jcfl
convenient correction for dead time. Finally, the effect of aberrations (on the integrated

reflection) is reduced using the wide slit scheme, as discussed in detail by Suortti and

Jennings [9].

Summarizing this section, when intensities rather than peak shapes are required, it is

well to consider the use of a slit wide enough to encompass the entire angular range of

interest.

7. Concluding Remarks

Everything discussed in this paper has been known to me for over a decade and, no doubt,

to other workers long before that. I would like to thank the organizers of this symposium

for giving me the opportunity to present this old material in the hope that it will give

encouragement, even in this day of sophisticated and titillating computer programs, to imaginative consideration of the pedestrian aspects of dif fractometry.

References

[1] Jennings, L. D. , Current status of the I. U. Cr. powder intensity project, Acta. Cryst.

A25 [1], 217-222 (1969).

[2] Ayers, G. L. , Huang, T. C. , and Parrish, W. , High-speed x-ray analysis, J. Appl. Cryst.

11 [4], 229-233 (1978).

[3] Chipman, David R. , Conversion of relative intensities to an absolute scale, Acta. Cryst.

A25 [1], 209-214 (1969).

[4] Jennings, L. D. , Polarization of crystal monochromated x-rays, Acta Cryst. A24 [4],

472-474 (1968).

[5] Chandrasekaran, K. S. , Study of perfection of crystals using polarized x-rays, Acta

Cryst. 12, 961-922 (1959).

[6] International Union of Crystallography, Polarization ratio for x-rays--a survey by

the Commission on Crystal lographic Apparatus, Acta Cryst. A34 [1], 159-160 (1978).

[7] Suortti, P. and Jennings, L. D. , Accuracy of structure factors from x-ray powder

intensity measurements, Acta Cryst. A33 [6], 1012-1027 (1977).

[8] LePage, Y. , Gabe, E. J., and Calvert, L. D. , X-ray beam polarization measurements,

J. Appl. Cryst. 12 [1], 25-26 (1979).

[9] Suortti, P. and Jennings, L. D. , Effects of geometrical aberrations on intensities

in powder dif fractometry, J. Appl. Cryst. 4 [1], 37-43 (1971).

Discussion

Comment (Parrish): I don't believe your statement that the best resolution is obtained with

a wide receiving slit is correct. Resolution refers to the width of profiles and ability to resolve them. You seem to have confused the meaning of the term in the context you used.

81
Response (Jennings): To the extent that resolution refers to the width of a scanned profile

you are certainly correct. If we extend the definition to refer to the possibility of

assigning received powder to a well defined portion of the diffraction cone, then a fixed

wide receiving slit gives the best possible definition of the region, in addition to the

primary advantage of a high count rate. »

f Comment (Parrish): I am not certain of the correctness of your statement that the best measure of the integrated intensity is with a receiving slit as wide as the full profile. I the past, the wider the r.s. the wider the profile and the lower the peak-to-background rati

the increased background may be considerable. It also implies the r.s. width would have to change with 20 which may be impractical and introduce errors due to changing background

levels. In the case of a diffracted beam monochromatic changing r.s.w. has little effect orj the profile width. !-

Response (Jennings): Yes, there is no point to a wide receiving slit with a diffracted bean

monochromator; this is one of the disadvantages of such a monochromator. Furthermore, in til coarse of an unknown sample or one with numerous peaks, the usual scanning strategy is best.

Nevertheless, I assert that, in some cases where the problem is well defined and/or the peat are widely separated and/or the integrated intensity is required with high accuracy in the

j minimum counting time, the use of a fixed wide receiving slit is the optimum strategy. In this case it is inappropriate to discuss the peak to background ratio (or the apparent width

of a profile that might be obtained by scanning the wide slit. Instead one should focus on the required quantity: the difference between the power in some interesting region of the

j diffraction cone (the "peak area") and a nearby region (the "background"). Ideally, the

fixed slit should be exactly as wide as the interesting region. We have found it helpful ti use subsidiary slits (with known area relative to the standard slit) to help determine back 1

ground in the case of closely spaced peaks. Thus, I agree that the optimum configuration

would depend on 29 (and other parameters) and a decision would therefore be required for ea^ problem as to whether the much higher count rate obtained with the fixed slit warrants the

overhead of getting organized.

Question (Shirley): I'm not too happy about your use of the term "resolution" when referri! to intensity measurement using a wide slit. Might this not be better described as aiming a

the highest intensity signal/noise ratio (assuming the background level to be known) consis

tent with maintaining separation from other liner?

Comment (Jennings): Although a wide slit does give high count rates, there are other advan

tages which I wished to emphasize. In particular, the modest sized slit of figure 4b fails

to resolve the peaks and thus a deconvol ution would be required to get the individual

integrated intensities. One might naively assume that a wider slit could only be worse.

82
However, a fixed wide slit spanning A to B and B to C in figure 4a completely "resolves" the

two peaks. Thus, in this sense, the fixed wide slit has better "resolution" than the scanned

slit of moderate width. The former cannot, of course, conveniently show the shapes of the

peaks.

Comment (Ladell): The use of multiple foils for evaluating dead-time can be self-defeating

if proper account is not made of the fluorescent x-rays generated by the foils, and also if

selective absorption which modifies the transmitted spectral distribution is neglected.

Response (Jennings): These and other considerations (such as small angle scattering) must be

taken into account of course, but they apply with equal force to both the master foil method

and the multiple foil method. Beam hardening is considered in some detail in reference [3].

Using 6 filter material (e.g., Ni for CuKa) for the foils and crystal monochromated

radiation, we found that the effects of fluorscence were negligible, but this should be

checked in each case.

to c

ni i

83

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

POSITION-SENSITIVE DETECTORS FOR

POWDER DIFFRACTOMETRY 1

R. W. Hendricks, M. K. Kopp, and A. H. Narten

Oak Ridge National Laboratory

Oak Ridge, TN 37830

We have applied both one- and two-dimensional position-sensitive

detectors to powder diffractometry. The advantages of such detectors

are (1) they are fast and have a wide dynamic range, (2) the acquired

data are digital, and (3) in the case of neutrons they can also be used

for time-of-f1 ight (TOF) spectroscopy. The use of straight one- dimensional or flat two-dimensional detectors may introduce parallax

errors if the detected angular range is too large. We will report on our experiences in neutron diffractometry with a 50 cm straight detector

with 0.5° resolution and our evaluation of a prototype 130° two theta

curved linear detector. We will also discuss a flat area x-ray detector

(20x20 cm) which has been in use for several years with the 10 meter

small-angle x-ray scattering camera. With this detector, powder patterns from structures with large d-spacings have been obtained. Finally,

methods for improving the performance of such detectors both in spatial

resolution and in count- rate capability will be outlined.

Research sponsored by the Division of Materials Science, Department of

Energy under contract W-7405-eng-26 with the Union Carbide Corporation

85
I
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

I (Issued February 1980)

DETERMINATION OF THE SPECTRAL INTENSITY OF THE INCIDENT

BEAM IN ENERGY DISPERSIVE X-RAY DIFFRACTION

R. Uno and J. Ishigaki

Department of Physics

College of Humanities and Sciences

3-25-40 Sakurajosui , Setagaya-Ku,

Tokyo, Japan

To determine the spectral intensity of the incident continuous x-ray

in the energy dispersive x-ray diffraction, it was the best way to measure

the integrated intensity of a well-defined Bragg reflection at several

energies (J. Appl . Cryst. 8, 578 (1975)). The spectral intensity was obtained from the calculated structure factors of the reflection at relevant energies. However, this method requires measurements at several

Bragg angles, so that one of the merits of the energy dispersive method is

lost.As a different method, the spectral intensity was directly measured

by the SSD, when the x-ray tube current was reduced to several microampere.

In this method we found two problems. The one was that the focus of tube

current changed and sometimes another focus appeared, and the other was that the obtained spectral intensity depended on the intensity of the

incident beam, when the spectrum was measured by the use of a usual multi

channel analyzer. It seems better that the x-ray tube is new and the

tube current is as low as possible if the x-ray due to the cold emission

is very weak.

In the energy dispersive method, the structure factors are usually

estimated under the assumption of unpolarized incident beam. The spectral

intensity obtained from the diffraction measurement is modified by the

polarization of the incident beam near the high energy limit, but this is preferable for the determination of the structure factors.

An irregular behavior of our apparent spectrum against x-ray

intensity was found to be mainly due to the anomalous spotty x-ray source

87
which appeared at low tube-current. At higher tube-current the anomalous

source became weaker than the regular one and negligible at the normal tubecurrent (see fig. 1). The pile-up effect of the amplifier was not so much

when x-ray total intensity was less than 2000 cps.

Change of X-ray Source by the Tube Current

a New W-Tube at 40 kV

2mA 25sec 27 pA Ahrs

58 pA 30min 405 jjA 3min

Cold Emission 26

::y:

it

ijin

ice

inly

sol

Question (Wilson):

spectrum?

Figure 1.

Discussion

How accurately does Kramer's law describe the high-energy end of tht,

Response (Uno and Ishigaki): Our results on the spectrum of the continuous x-rays fromij ^ the tungsten tube was modified by the quantum efficiency of the Si(Li) solid state detei Its quantum efficiency becomes lower and lower at energy higher than about 15 keV. As tube was operated at 40 kV, the high-energy end of the spectrum was modified very much.

I did not mention it, because we could get the structure factor without any knowledge o the quantum efficiency of the detector since the integrated intensity of reflection was, also measured with the same detector.

88

icie

:tua

upli
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg , MD, June 11-15, 1979.

(Issued February 1980)

i

TIME-OF-FLIGHT NEUTRON POWDER DIFFRACTION AT PRESSURES TO 35 KILOBARS

i J. D. Jorgensen

Solid State Science Division

Argonne National Laboratory

Argonne, IL 60439

Neutron diffraction has proved to be particularly useful for obtaining structural infornation for samples under high pressure conditions because of the excellent penetrating power

bf neutrons. The time-of-f 1 ight technique offers the further advantages of being able to obtain complete diffraction data at a single scattering angle and markedly increasing the

:ount rate by using large time-focussed detector arrays. The resolution function of such

in instrument can be accurately characterized and remains constant over long periods of time

fince a typical instrument has no moving parts or variable physical parameters. Profile

jmalysis techniques can be used to obtain atomic positions from the data. Moreover, the

•esolution function is sufficiently well known that peak broadening due to small symmetry

listortions can be seen and analyzed.

The time-of-f1 ight neutron powder diffractometer presently operating at Argonne 1 s CP-5

eactor consists of a chopper 0.3 meters in front of the sample and a detector array 3.4

3 eters from the sample at 20 = 90°. The detector array contains 14 He counters 2.5 cm iameter x 46 cm long. Samples are contained in a piston-cylinder pressure cell capable of

eaching 35 ki lobars. The powder sample along with a suitable hydrostatic liquid is con- ained in a sealed teflon capsule inserted into the bore of an A^Og cylinder. The A^Og

s supported radially by a hardened steel binding ring which has slits at appropriate angles

or the incident and scattered neutrons. The attenuation of neutrons in the A^O^ is small. A recent study of the pressure-induced strain transition in Ni F2 illustrates the utility of this technique for measuring small lattice distortions [l] 1

. Ni F2 has a tetragonal

froi jutile structure at atmospheric pressure and undergoes an apparently continuous transition ^ d a closely related orthorhombic structure at 18.3 kbar. The orthorhombic strain is sufA$ iciently small (reaching about 1.3 percent at 32 kbar) that orthorhomic doublets are not

uch. ctually resolved even at the highest pressure. However, the use of profile analysis,

96 1 Jupled with an accurate knowledge of the instrument's resolution function, allows the

1* etermi nation of the orthorhombic lattice parameters and the atomic displacements associated

ith the transition.

igures in brackets indicate the literature references at the end of this paper.

89
I

I

A study of compression mechanisms in the orquartz structures SiO^ and GeC^ further

illustrates the ability of this method to measure small changes in atomic positions [2],

By determining bonds lengths and angles versus pressure it was possible to show that SiC

compresses by a nearly rigid rotation of its corner-linked SiO^ tetrahedra while isostri,

tural GeC^ compresses by distortion of bond angles within its GeO^ tetrahedra.

References

[1] Jorgensen, J. D. , Worlton, T. G. , and Jamieson, J. C. , Phys. Rev. B 17, 2212 (1978]

Jorgensen, J. D. , Worlton, T. G. , and Jamieson, J. C. , High Pressure Science and Tj

nology , Vol. 1, 152; K. D. Timmerhaus and M. S. Barber, eds. (Plenum Publ . Corp., j [2] Jorgensen, J. D. , J. Appl. Phys. 49, 5473 (1978).

i

90
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg , MD, June 11-15, 1979.

(Issued February 1980)

POWDER— A COMPUTING SYSTEM FOR X-RAY POWDER

DIFFRACTION CALCULATIONS

B. C. Osgood and R. L. Snyder

NYS College of Ceramics,

Alfred University

Alfred, NY 14802

A system has been designed and written to maintain all pertinent

information and carry out the calculations associated with x-ray

powder diffraction. The system has incorporated the following design

features:

(1) All computations may be carried out interactively.

(2) All component programs may be executed independently, or as overlay segments of a large program, either interactively or

in batch mode.

(3) The system control program and all component programs are written almost exclusively in ANSI machine independent

FORTRAN.

(4) Language and notation used is designed to be understood by

researchers and students outside the area of crystallography.

(5) All program options default to reasonable values.

(6) Incorporation of existing or newly developed programs is

easily accomplished.

The system control program, called POWDER, creates, updates, and

maintains a data file containing all of the needed information for a sample. Data may be input directly from automated diffractometer

files or from films or strip charts. The most commonly used programs

for phase identification, lattice parameter refinement, indexing and

pattern calculation have been incorporated. The input sequence in the

existing programs has not been altered, allowing for easy insertion of

a new version of the code. This feature is accomplished by having the

system control program read its data file and set up an input file for

any of the application programs.

The purpose of the system is to allow the many powerful computational procedures developed in recent years to be routinely available to workers both in and outside the area of crystallography.

91

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979

(Issued February 1980)

THRESHOLD LEVEL DETERMINATIONS FROM DIGITAL X-RAY POWDER

DIFFRACTION PATTERNS

C. Mai lory and R. L. Snyder

NYS College of Ceramics

Alfred University

Alfred, NY 14802

A major difficulty in the reduction of digital x-ray powder diffraction data is the determination of an appropriate threshold level.

Data above that threshold level may be considered as being due to a real diffraction peak. Traditional methods involve a mean value (u)

for the background and determine the threshold based upon the standard

deviation (a), i.e. threshold = u + Na. The value chosen for N introduces an undesirable degree of arbitrariness into the process. The

result is an approximation of the threshold that often leaves a number

of true background points present while, in other regions of the

spectrum, small peaks may be rejected.

A procedure has been developed for determining a threshold level

based directly on the maximum count rates found in the raw digital

data. In this procedure the maximum intensity point in each 20

segment, of a given width, in the pattern are analyzed for statistically

significant increases from the previous maximum point. The test is

based upon the standard deviation of the count rate of the individual

maximum points. Data not passing this criterion are rejected as being

part of a significant peak and a polynomial is fit to the remaining

data. A spline fit is used to eliminate all points below the threshold.

The threshold level is then defined as the intensity value of the

polynomial evaluated at each 26 step within each given 29 segment.

This procedure produces a threshold line which passes through the

maximum point in each 26 interval determined to contain only background

data and passes smoothly between these intervals under peaks. The

subtraction of this threshold from the observed data, while giving all

negative values, produces a series of clean peaks or peak clusters

whose statistical significance is determined by the adjustable 26

interval rejection criteria.

93
1

Itl
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Raithersburg , MD, June 11-15, 1979.

(Issued February 1980)

ACCURACY OF THE PROFILE FITTING METHOD FOR X-RAY POLYCRYSTALLINE DIFFRACTOMETRY

W. Parrish and T. C. Huang

Research Laboratory

International Business Machines Corporation

San Jose, CA 95193

This paper presents the results of an experimental study of the

accuracy of the profile fitting method in determining profile shapes,

diffraction angles, peak and integrated intensities in x-ray powder

diff ractometry. The precision R in determining the standard profile

shapes (W*G) of a number of powder profiles free of line broadening, is

calculated from the differences between the fitted profile and the experimental points. Collecting about 30,000 counts on the peak of the profiles

with a step size 0.03° gave an average better than R = 2 percent for 35

profiles covering the range 15° to 165°; RI (integrated intensity) =0.3

percent for the same set of data. The precision of reflection angles and

intensities of three profiles were each measured 10 times in 0.01° steps

collecting 40,000 counts at the peaks and calculating up to 0.05° steps.

Fifty-six percent of the 150 peak angle determinations agree to ±0.0001°

and 97 percent to ±0.0004°. The precision of the peak and integrated

intensities was 0.2 percent for steps 0.01° to 0.05° and 0.5 percent for

steps 0.06° to 0.09°. Data for weak peaks and overlapped peaks resolved

by profile fitting are described. The method applied to a typical powder

pattern (garnet) routinely recorded at moderate- (25 min) and high-speed

(3 min) gave R = 3.2 percent and 6.6 percent, respectively, and average

agreement exceeding ±0.01° (26) and ±1 percent intensities. This precision

exceeds the curve fitting results of others, e.g., the Rietveld method as currently applied to x-ray powder diffraction. A method for handling systematic errors in lattice parameters using profile fitted angles is presented.

1. Introduction

The advantages of the profile fitting method are becoming more apparent and it is

jly that it will soon be widely used in a variety of applications. In this paper, we

[jate the precision of our profile fitting method [1.2] 1 from measurements of powder

jres in brackets indicate the literature references at the end of this paper.

95
diffractometer profiles, reflection angles, and intensities. The emphasis will be on precision or reproducibility rather than accuracy because of uncertainties in absolute

values available for comparison.

The precision achieved in the present state of development of the method already far

exceeds most of the data in the standard Powder Diffraction File. The method also greatly

reduces the time and labor in collecting and reducing powder data. It was recently shown

that even a complicated pattern could be automatically recorded and a precise set of d's

and 1

1 s obtained in a few minutes [3].

The use of computers is essential in applying the method making it possible to auto- matically control the diffractometer in collecting large sets of data and to do the

involved calculations rapidly and accurately. Modern powerful minicomputers can now be

used for this method [4].

Our method has greater precision and is quite different from the Rietveld method [5]

applied to x-ray powder diffractometry. No prior crystal lographic information on the

material to be analyzed is required. It was not designed specifically for structure

determination but rather for general diffractometer analysis. It determines integrated

and peak intensities and diffraction angles of complete patterns or individual profiles

with high precision. The experimental data are used without smoothing. Accurate fitting

of the entire profile is achieved and we believe this to be essential to produce precise

results. The ability to resolve overlapping peaks with a resolution far exceeding the

original diffractometer recording makes it possible to derive good data from clusters of

peaks and complex patterns. The method has been developed to the point that after the

operator selects the experimental parameters, the entire run from data collection to reduction is automatic. It has also been used for qualitative analysis of mixtures [6]

and variations of the method for energy dispersive spectral analysis [7].

2. Basis of the Profile Fitting Method

Our method is based on the well-known relation

P(8) = W * G * S + background (1)

where P(6) = Profile measured with the diffractometer,

W = Observed x-ray Ka spectral line profile,

G = Convolution of all aberrations of the diffractometer

and the diffraction process, and

S = True diffraction effects of the specimen.

In our earlier papers, we showed that W must be the actual distribution observed in

| the apparatus being used rather than the double crystal profile, and that it was closely

approximated by a high angle reflection virtually free of aberrations such as silicon

(444) with CuKa radiation. In practice, there is no need to determine W separately. The | convolution W*G is determined by accurate measurements of a series of profiles of a 96
)er of carefully prepared "standard" powder specimens free of line broadening effects.

> is equivalent to determining the instrument function in terms of the profile shapes

various reflection angles as described in Section 4. When using CuKa radiation the Ka-^ and Ka^ peaks are each fitted with three Lorentzian

res and the weak Ka^ satellite group with one. The number of Lorentzians required to

lin an accurate fit was determined empirically. Two Lorentzians gave good accuracy

the symmetrical higher angle reflections, but the more asymmetric lower angle reflects required three. Four did not significantly improve the accuracy but greatly

"eased the computing time.

Each Lorentzian is represented by three parameters, 26, I, and w, defining the peak

tion, intensity, and width, respectively. The W*G program adjusts the 21 parameters

hat the sum of the seven Lorentzians makes the best fit to the experimental data points.

The W*G profiles are normalized and interpolated between reflections to provide the

c profile shapes at all 28's. The data (i.e., the parameters) are stored in the comr for use in later computations of S. Once W*G is known, the true diffraction effect S of the specimen can be precisely

rmined. The diffraction profile s^ of a reflection without asymmetric broadening can epresented by a single Lorentzian with 20^., I., and w^ as its parameters. If the

imen profiles are asymmetrically broadened, an asymmetric Lorentzian with different

'.-widths on each side of the peak is needed to obtain a good fit. The contribution S

he specimen to the profile is obtained by adjusting the parameters 26, I, and w of

reflections recorded so that its convolution with W*G, i.e., (W*G)*S matches the

rimental data points as closely as possible. The algorithm used to determine W*G and

essentially the nonlinear simplex method [8,9,10].

Irregular backgrounds such as those of a thin film on an amorphous or crystalline

trate can be handled routinely.

3. Instrumentation

The experiments were carried out with a Norelco vertical scanning-plane diffracer, radius 185 mm and reflection specimen [11]. Various improvements that evolved

the years have been incorporated, although they are not essential for applying the

le fitting method. The principal changes were the addition of a diffracted beam,

graphite monochromator and rebalanced worm gear, elimination of the parallel slit

acted beam collimator, a vacuum path constructed in four sections and a rotating

men device. The stepper motor is mounted at the rear and coupled directly to the

drive. We used the Norelco long fine focus x-ray tube with 12° take-off angle,

ited at 50 kV, 20 mA. The alignment and calibration was made with methods previously

hbed [11].

|The computer system is an on-line IBM System/7 for di f f ractometer control and data

fction and an off-line host IBM System 370/168 for data reduction (reference [2],

1 and 2). We have recently developed the use of the IBM Series/1 minicomputer to

97
perform all the tasks of the large two computer system with reasonable turn-around time.

This stand-alone computer has EDX and FORTRAN capabilities, and a description is being

prepared [4].

4. Determination of W*G

A set of W*G standard profiles covering the angular range to be used is required for

each set of experimental conditions. Any change in instrument geometry or x-ray wavelengths that modifies the profile requires a separate set of standard profiles. Readjustment of the monochromator may modify W and the angular calibration, changing slit

widths changes the profile widths and asymmetry, and different di f f ractometer geometries

and x-ray tube targets will require a new set of standard profiles. The recording con- ditions such as the diff ractometer step-angle A28, and counting time do not change the

shape. Remeasuring the W*G dataset a few times a year is a good practice to assure that

the profiles are accurately represented. It is essential that the adjustments, alignment

and calibration be carefully done prior to starting the computer automation method.

The standard specimens were carefully prepared powders with average crystallite

sizes 5 to 20 urn, mounted with 1:10 col lodion: amyl acetate on a flat 3/4 inch square

single crystal silicon wafer cut parallel to (510). The silicon was mounted on an aluminum cylinder which was continuously rotated during data collection. We used all the

reflections of silicon and tungsten in a silicon + tungsten mixture, and some of the

reflection of quartz, Gd^Ga^O^ garnet, PbCNO^^. and a steroid for small 26' s. All

reflections used had high intensity and P/B, no overlaps and showed no line broadening.

The goodness of fit between the observed and calculated profiles relating the

individual step-scanned points ^.(obs) to the same 20-angle points of the fitted profile

I.(PF) is defined as

£ [Ii (obs)-Ii (PF)]

2 /^I^obs)2

i=l / i=l

R(%) = x 100

and the match of the integrated intensity of the entire profile is defined as

,2

(2)

RI(%) =

n / n Z [I

i (obs)-Ii (PF)] 2 / £ ^.(obs)

i=l / i=l

x 100 (3)

We learned from many previous runs that the precision given in table 1 is typical

and satisfactory for determining the W*G profiles. The overall averages of the 23 profiles listed (35 were measured) were better than R = 2 percent and RI = 0.5 percent when i accumulating about 30,000 counts at the Ka^ peaks and using the step-scan angles shown.

Four of these profiles (fig. 1) were selected to illustrate typical cases of low R (a)

and (c), and higher (though still good) R (b) and (d). The differences between the exper

mental and fitted points is shown below each profile. It should be noted that the fitted

profiles (a) and (c) have a high accuracy as shown by the curve passing through all the \

98
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99
Figure 1. Four typical profiles used for W*G determination, x = experimental

measurements, solid line = fitted profile. A26 = 0.01°, alternate x's omitted in (a) and (b). Angular aperture = 1° for (a) and (b) and 4° for

(c) and (d). (a) and (c) have the lowest R, and (b) and (d) the highest

R in table 1.

100
xperimental points including the tails that (b) and (d) have sufficient accuracy to

jepresent the true profiles. Profile fitting of this accuracy is required to properly

lepresent the experimental data and to achieve the precision in measuring reflection

ingles and intensities as described in the following sections.

5. Precision of Peak Angle Measurements

It was explained above that the calculations for determining the reflection angles

jnd intensities require only one Lorentzian for each reflection. The computer program

etermines the angle corresponding to the peak of the reflection. Other measures such as

he mid-point of chords at various heights, the centroid, etc., could be programmed if

esired.

The following experiments were planned to determine the inherent precision of the

jrofile fitting method by measuring individual reflections with good counting statistics,

lie time required to collect data for a single profile varied from 12 to 24 minutes

ing on the scanning range required to record the entire profile and the background

ith 0.01° steps. It will be shown that the step sizes can be increased to 0.05° and

ounting time decreased without significant loss of precision, and hence, the experiental time can be greatly reduced.

Three reflections of a specimen containing a mixture of silicon and tungsten powder

ere used: silicon (111) to illustrate an unresolved doublet, silicon (311) for a partially

esolved doublet, and tungsten (321) which is broader and has nearly complete resolution,

en consecutive runs were made on each reflection with A26 = 0.01° steps and counting times

ufficient to collect about 40,000 counts at the Ka-^ peaks. To determine the effect of

he step size and to avoid rerunning with the different A20's, we used the same data for

26 = 0.02° by using alternate steps; this was repeated in the calculations up to

26 = 0.05° by using every fifth step.

The average peak angle was calculated from the five different A20 sets of each of

he ten runs. The differences between these averages and the angles calculated from each

26 in each run is used as a measure of the accuracy of the profile fitting program

ecause systematic and drift errors are avoided. The average absolute difference is .0001° or about 1/3 second of arc for tungsten (321) as shown in table 2. Figure 2 is a istogram of the 150 peak determinations of the three reflections; 56 percent lie between

0.0001° and 97 percent between ±0.0004°. Although we have not included them, the calcuations were continued to A26 = 0.09° with virtually the same results obtained with the

mailer A26's.

These data demonstrate the high precision that can be achieved by using the profile

itting method. However, there are well-known factors which may introduce systematic

rrors that are considerably larger than the inherent precision of the fitting procedure.

Ihese include the zero angle calibration of the di f f ractometer , specimen surface dislacement, the geometrical aberrations such as flat specimen and transparency, the specien temperature, slight shifts of the position of the x-ray tube focal line due to changes

101
Table 2. Precision of Peak Angle Determination Tungsten (321), CuKa

Average

Run

q 01° to 0.05° A26= 0.01° 0. 02° 0.03° 0.04° 0.05°

1 131.1521 +0.0001 +1 -2 0 0

2 .1521 -2 0 -1 0 +2

3 .1515 -1 0 .0

" +2 0 4 .1511 -1 0 +1 0 0 5 .1508 0 0 +1 +2 -4

6 .1504 -1 0 -1 0 +4

7 .1498 0 -1 0 -1 +2

8 .1502 -1 +1 +1 +1 -4

9 .1498 -1 +1 0 -1 +1

10 .1497 +1 +1 +2 -4 +1

Average 131.1508 0.00009 05 09 11 18

1

10

33

25

26

19

16

1 -7 -6 -5 -3 -2 -1 0 +1 +2 +3 +4 +5

0.000 x °2d

«-56%-*

97%

Figure 2. Precision peak angle determination by profile fitting. Fifty runs with A26 = 0.01° to 0.05° for each of Si (111), Si(311) and W(321). Each horizontal unit = 0.0001°(28). Number of runs shown at top of bars.

102
j the cooling water temperature, and others. The reduction of these sources of errors

the small errors achieved in profile fitting is a formidable task.

This problem is illustrated by the data in the second column of table 2 which shows

gradual decrease in the average reflection angle in the ten consecutive runs. No temrature control of the specimen was used although the specimen was in a vacuum chamber

d air conditioned room. The runs were made in a four-hour period following a one-hour

rm-up of the x-ray generator (Philips PW 1310) at full power. The temperature was

nitored at the edge of the rotating specimen device with an electronic thermocouple

junted through the specimen vacuum chamber. The temperature rose from 24.2 to 26.2 °C

ring these runs. This rise was caused by a slow increase in room temperature and heat

nerated by the synchronous motor and bevel gears of the rotating specimen device. The

crease of 0.0024° corresponds to the thermal coefficient of expansion 0. OO11°(20)/°C

r the tungsten (321) reflection.

At a later time, the same specimen was remounted and another set of ten runs was made

thout specimen rotation. This time the temperature increase was 1 °C, the angle decreased

001°(26) which checks well with the previous runs. However, the average angle was 0.003°

gher. This corresponds to a difference of specimen surface position of only 6 urn, which

curred when the specimen was remounted. It is probably impossible to avoid such small

splacements of powder specimens.

6. Precision of Intensity Measurements

The profile fitting method provides precision measurements of the peak and inteated intensities even when overlapping occurs. The data listed in table 3 summarize

e profile fitted intensity data of the same set of three profiles with the same experintal conditions used in table 2. The data were extended to 28 = 0.09° to show that

curate values can be obtained even when using large steps. No systematic drifts in the

tensities was observed in these runs.

Each number in table 3 is the ratio (multiplied by 100) of the average of 10 runs

r that A26 to the average of all 90 determinations (10 runs, A20 = 0.01° to 0.09°);

ch A is the ratio of the average of the differences of each of the 10 runs of that A28

taset. For example, the average peak intensity of 10 runs of silicon (111), A29 = 0.02°

js 37,543 counts, the overall average for the 90 runs was 37,506 and the ratio 1.00098

listed as 100.1; the average of the differences regardless of sign was 58 and the ratio

0015 is listed as 0.2. The precision is thus about 0.2 percent for peak and integrated

tensities with small A26 and increases to about 0.4 to 0.5 percent with large steps.

In Section 5, it was shown that errors from several sources may be larger than the presion with which the reflection angles can be measured with profile fitting. The same difculty occurs in measuring intensities. Preferred orientation, crystallite size statistics

'd inhomogeneous specimen preparation may cause large errors. The use of transmission specen or Seeman-Bohlin diffractometers to supplement the reflection specimen geometry data [12],

rotating specimen device and careful specimen preparation will greatly reduce the errors.

103
Table 3. Precision of Intensity Measurements

A20°

Integrated Peak

Si (111) OX loJLXJ

0.01 100.1 ± 0.2 100.3 99.7 100.1 + 0.1 100.0 100.0

0.02 99.9 0.2 100.4 99.8 100.1 0.2 99.9 100.0

0.03 99.9 0.2 100.7 100.3 100.0 0.2 99.7 100.1

0.04 99.9 0.3 100.4 100.4 99.9 0.2 98.9 100.1

0.05 100.3 0.2 99.3 100.2 99.8 0.3 100.6 100.2

0.06 99.8 0.3 101.9 100.1 100.7 0.3 101.4 100.2

0.07 100.8 0.5 99.4 100.1 100.4 0.4 98.9 100.4

0.08 99.9 0.4 98.0 99.8 99.5 0.2 98.4 99.4

0.09 99.3 0.5 99.7 99.7 99.6 0.2 102.1 99.8

Average Peak Counts 37506 36320 42163

7. Weak and Overlapped Peaks

In this section, we will present some results obtained with low intensity, low P/B

and overlapped peaks. For the case of an isolated peak, we used quartz (12.0) with

P/B = 1.25. Ten runs were made with A28 = 0.01° and t = 10 seconds per step, and these

were repeated with t = 1 second. The profile fitting results are summarized in table 4,

which includes data for A26 = 0.02° to 0.05° obtained by skipping steps in the calculation as was done before. Although the large reduction of counts resulted in lower

precision, the standard deviation was 0.001° with 1300 peak counts and 0.008° with 130

counts, and the average peak intensities were better than ±1% for both.

A more difficult case is the measurement of a weak peak occurring on the tail of a strong peak such as quartz (00.3) on (11.2). The Ko^ peaks are separated 0.48°, the Ka doublet 0.13°, and the ratio of the intensities and P/B of (11.2) = 45. Using

A26 = 0.01°, collecting 741 counts on the (00.3) Kc^ peak, R = 2.4. The results of the

profile fitting are shown as a plot in figure 3(a).

The ability to resolve overlapping reflections and to derive precise values of the

26's, integrated and peak intensities of the resolved peaks is an important advantage of the J method. When it is used in an automatic routine analysis, the ability to resolve overlaps

depends on the separation of the peaks and the quality of the data. If the peaks are separated by at least the HWHM regardless of the Ka doublet separation, they can be resolved with

104
Table 4. Precision Determination of Weak Quartz Peak (12.0),

P/B = 1.25, 10 Runs for Each Counting Time

Counting

Time/Step A20 26

Standard

Deviation

Integrated 1

(Normalized)

Peak I (Counts)

10 sec. 0.01° 57.2075 0.0009 103 1292

0.02 72 06 101 8

0.03 75 23 99 0

0.04 66 14 98 6

0.05 87 19 99 0

Average 57.2075 0.0014 100 1293

1 sec. 0.01 57. 2088 0.0057 108 130

0.02 83 72 101 129

0.03 59 76 101 129

0. o o OJ C £ OD

0.05 68 60 96 126

Average 57.2076 0.0066 100 129

I (counts)

x 103

100

REL

30.1a.

Figure 3. Profile fitting of quartz powder profiles. Dots = experimental

measurements, solid line = fitted profile, dashed line = resolved profiles,

(a) Resolution of (00.3) peak on tail (11.2); (b) Resolution of three

overlapped CuKa doublets.

105
good precision. If the indices and lattice parameter are known, peaks can be resolved with

even smaller separations.

When several peaks are closely overlapped, it is difficult to determine the correct

relative intensities and reflection angles with the usual recording methods. The often-used

quartz cluster shown in figure 3(b) illustrates this. The CuKa^ peaks of (12.2) and (20.3)

are separated 0.40°, (20.3) and (30.1) 0.17°, and the doublets 0.19°. The intensity of the

(30.1) peak read from a strip chart would be in error by about 40 percent. The correct

relation of the three reflections is shown by the dashed curves obtained with profile fitting

8. Profile Fitting Powder Patterns

An example of the profile fitting method applied to a typical powder pattern used for

identification is shown in figure 4; these are Calcomp plots of the experimental data of a garnet collected at 0.05° steps. The powder was mounted on 0.125 mm thick Be foil and run with the transmission specimen di f f ractometer using a diffracted beam logarithmic spiral

quartz monochromator [11,12,13]. The upper pattern has 29 reflections (plus 3 Be peaks),

was recorded with 1 second counting time per step, and completed in 25 minutes. The profile

fitted data were R = 3.2 percent, RI = 0.1 percent. The lower pattern was recorded with 0.1

second counting times and required only 3 minutes. To avoid computing difficulties arising

from poor counting statistics on the weak peaks, the program used only peaks whose intensity

exceeded 3(B) for the computer fitting. The 11 reflections dropped (marked x) were all less than 3 percent of the strongest peak. The results for the remaining 18 reflections

were R = 6.6 percent, RI = 0.6 percent, average precision 0.008°(29) and peak intensities

0.8 percent. These results are similar to those we recently reported on high-speed x-ray

analysis of a more complicated powder pattern [3].

The accuracy reported here exceeds recently published results obtained with the

Rietveld method for fitting x-ray powder patterns as shown by the following few examples.

Malmros and Thomas [14] used a modified asymmetric Lorentz function to fit automatic

microdensitometer recordings of Guinier-Hagg camera films of aBi^O^; their diagrams show

large discrepancies between observed points and the fitted profiles. Young, Mackie and

von Dreele [15] reported that neither Gaussian nor Cauchy (Lorentz) functions have "adequate fittings" of powder diffractometer patterns and that better representations were

needed to lower the R values in structure refinement. Khattak and Cox [16] used various

functions to fit a rhombohedral powder diffractometer pattern containing 41 reflections

in a 76°(28) range. Even though over 63 hours were used to record the pattern using

0.02° steps, 60-second counting times and CuKp radiation (to avoid the Ka doublet "problem"), the matches between fitted and observed patterns were "distinctly inferior to that

obtained in a typical neutron profile analysis" and their best value of R = 13.7 percent

was obtained using the Lorentz function.

106
4860 Counts

465 Counts

J\>~*Jiz—Lj—Ijl.

Gd3 Ga

5 012 on Be Substrate

(a) Step Scan: A20 = 0.05°, At = 1 sec Total Exp. Time: 25 minutes R = 3.2%, RI = 0.1%

(b) Step Scan: A20 = 0.05°, At = 0.1 sec Total Exp. Time: 3 minutes R = 6.6%, RI = 0.6%

20 30 40 50 7

60 70 80

°20

I 90

Figure 4. Transmission powder diffractometer patterns of garnet GdgGaj-O-^ powder

sample mounted on thin Be foil, CuKa radiation. Eleven reflections marked x in

(b) had peak intensities less than 3(B 2

) and were dropped from the profile fitting calculations; average of 18 peaks 0.008°(26) and 0.8 peak intensities for

3 minute recording time.

9. Lattice Parameter Determination

It is well known that systematic errors limit the accuracy of lattice parameter

determinations. It was shown in Section 4 that a good practical method of handling these

errors is needed to take full advantage of the accuracy of the profile fitted values. A least-squares type of refinement of the profile fitted angle data may be useful for this

purpose as shown in the following example.

A powder sample of a Czochral ski-grown garnet boule, Gd^a^O^, was prepared on a silicon (510) substrate. Twenty-five patterns were run in the front-reflection region

90° to 16° which includes 27 reflections. The specimen was removed and replaced between

runs to include specimen surface displacement errors. Each run was made with 0.05° steps

and 1-second counting time.

The 26 's obtained by profile fitting were used to determine the lattice parameter by

refinement of the coefficients defining the zero angle calibration, and the trigonometric

107
dependence of the specimen surface displacement and transparency corrections. The refinement was reached when the weighted sum of the absolute differences between the calculated

26 's obtained from these coefficients and the profile fitted 28 's of all reflections in a run was a minimum. The weighting factor of each reflection was made proportional to tan8

and the square root of the intensity.

o o Using Bearden's value for CuKa-. = 1.540562 A, the average a = 12.3827 A and the

-5

average Aa/a = 2.3x10 . This accuracy of about 1:50,000 was obtained from 25 minute

routine runs using only the front-reflection peaks. Because of the tan8 dependency of Aa,

it is likely that the back-reflection region would have yielded a much higher accuracy.

10. Conclusions

A series of experimental tests of our profile fitting method applied to powder diffractometry shows it has high precision in determining full profile shapes, peak reflection

angles, peak and integrated intensities. R is better than 2 percent on the average in

fitting profiles when about 30 K counts are collected at the peak, peak angles can be

determined to better than ±0. 0004°(28) , and peak and integrated intensities to 0.5 percent.

The precision for weak peaks (130 counts) is better than 0.01° and a few percent in peak

intensities. Overlapped peaks can be resolved if their Ka^ peaks are separated by at least

HWHM, thus avoiding gross errors in reading ratemeter strip charts. A possible method is

introduced for handling the inherent systematic diffractometer/specimen errors with a least-squares type of refinement using the profile fitted angles for precision lattice

parameter determination.

The method applied to a typical powder pattern routinely recorded at moderate- and

high-speeds for identification purposes gave data better than 0.01°(28) and 1 percent peak

intensities. The Rietveld method as currently applied to x-ray powder di f f ractometer

does not yet approach this precision.

References

[1] Huang, T. C. and Parrish, W. , Accurate and r Appl. Phys. Letters, 27, 123-124 (1975).

[2] Parrish, W. , Huang, T. C. , and Ayers, G. L. , computer x-ray instrumentation and analysis,

(1976).

[3] Ayers, G. L. , Huang, T. C. , and Parrish, W.

,

Cryst. 11, 229-233 (1978).

[4] Parrish, W. , Ayers, G. L. , and Huang, T. C.

,

x-ray analysis, 28th Denver Conf. on Applic.

[5] Rietveld, H. M. , A profile refinement method

Appl. Cryst. 2, 65-71 (1969).

ipid reduction of experimental x-ray data,

Profile fitting: a powerful method of

Trans. Am. Cryst. Assoc. 12, 55-74

High-speed x-ray analysis, J. Appl.

A minicomputer and methodology for

of X-ray Anal. (August 1-3, 1979).

for nuclear and magnetic structures, J.

108
[6] Huang, T. C. and Parrish, W. , Qualitative analysis of complicated mixtures by profile

fitting x-ray diff ractometer patterns, in Advances in X-Ray Analysis , Vol. 21 (Plenum

Press, New York, 1978), pp. 275-288.

[7] Huang, T. C. , Rapid and precise method for analysis of energy dispersive x-ray

spectra, IBM Jour. Res. & Dev. 29, 206-213 (1979).

[8] Nelder, J. A. and Mead, R. , A simplex method for function minimizations, Computer

Jour. 7, 308-312 (1965).

[9] Taupin, D. , Automatic peak determinations in x-ray powder patterns, J. Appl. Cryst.

6-, 266-273 (1973).

[10] Robaux, 0., Analysis de problems de "deconvolution," Rev. du Cathedec, NS74-2 (1974).

[11] Parrish, W. , X-ray Analysis Papers (Centrex Publ. Co., Eindhoven, 1965).

[12] Parrish, W. , Role of diffractometer geometry in the standardization of polycrystalline data, in Advances in X-Ray Analysis , Vol. 21 (Plenum Press, New

York, 1974), 97-105.

[13] Huang, T. C. and Parrish, W. , The transmission-specimen powder diffractometry and

the profile fitting method, Am. Cryst. Assoc. Meeting, Univ. of Hawaii (March 21,

1979).

[14] Malmros, G. and Thomas, J. 0., Least-squares refinement based on profile and analysis

of powder film intensity data measured on an automatic densitometer, J. Appl. Cryst.

10, 7-11 (1977).

[15] Young, R. A., Mackie, P. E. and Von Dreele, R. B. , Applications of the patternfitting structure- refinement method to x-ray powder diff ractometer patterns, J. Appl. Cryst. 10, 262-269 (1977).

[16] Khattak, C. P. and Cox, D. E. , Profile analysis of x-ray powder diffractometer data:

structural refinement of La

Q 75

SR

Q 2 5Cr03' J - APP ] - Cryst. 405-411 (1977).

Discussion

Comment (Edmonds): The R factor's are the same mathematically in form. However, a straight

comparison is misleading since Malmorse and Thomas applied the neutron diffraction Rietveld

structure refinement technique to digitized Guinier x-ray diffraction film data for which

they had no accurate estimation of the true form and variation of the profile, whereas

Dr. Parrish has accurately calibrated his instrument over a wide 26 range. The profile

shape for Guinier film data is much more complex than the single modified Lorentzien Mal- morse and Thomas assumed, and the variation with 26 is not the straightforward 3 parameter

relationships used in the Rietveld technique.

Comment (Sabine): You state that your R factor is very much better than that given by the

Rietveld method. You are measuring very different things. His Y calc is constrained by the

structure. Yours is not; hence, it must be lower.

109
Response (Parrish): No direct comparisons were intended, but it is evident that the modifiec

Lorentzian and other forms used by Mai morse and Thomas did not give as accurate a fitting as our model using the form of three Lorentzian for CuKL, and the same for CuKa.

Comment (Jorgensen): You have apparently developed a very accurate representation of the

x-ray line shape. Your resolution function should now be used in a Rietveld-like analysis

before you can give a meaningful comparison of which method is better, since previous x-ray : Rietveld analysis programs have probably not used a resolution function as accurate as yours.

At ANL, we have tried using direct deconvolution methods similar to yours on time-of-f 1 ight i neutron data where the peak shape is well known. We found it difficult to find peaks whenev^

three or more peaks recurred in a cluster. Moreover, the Rietveld approach gives far superiq

structural information, in our case.

Response (Parrish): We do not use deconvolution; we synthesize profiles which very

closely match the x-ray di f f ractometer profiles. As you saw in my slides, we were suc- cessful in resolving overlaps as for example the three quartz reflections at 68°.

Comment (Jorgensen): The R value you calculate is vastly different than the Rietveld R value and it is meaningless to compare them. The difference results because your Ycalc

is simple a scale factor times the appropriate element of your compound Lorentzian reso- lution function while in the Rietveld program, Ycalc includes all of the structural

parameters (i.e. Ycalc must obey the constraints of a structural model and is not free

to independently scale to Yobs). The result of this is that your R value is only a measure of how well your resolution function fits the observed peak shapes while the

Rietveld R is primarily dependent upon how well the structural model fits the data.

Response (Parrish): The R we use is given in the same mathematical form by Malmorse

and Thomas who used a modified Lorentzian. I showed that we obtain accurate fitting

of the entire profile at all 20 's and that we believe this is essential to obtaining

accurate values of reflection angles and intensities. I stated at the beginning of

my paper that our method was for general use in powder patterns; of course the precision

integrated intensities obtained with the method could be used for structure determinations.

Comment (Prince): In the discussion of these papers, there seems to be a general misconception that there is a conflict between the profile fitting of Parrish and the total

pattern analysis of Rietveld. The Rietveld technique takes a diffraction pattern and an assumed peak shape and determines crystal lographic parameters. The Parrish work is a good representation of x-ray peak shape. What is needed now is the marriage of these

techniques for x-ray refinement.

Response (Parrish): No comment made or necessary.

110
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg , MO, June 11-15, 1979.

(Issued February 1980)

PROFILE REFINEMENT OF NEUTRON POWDER DIFFRACTION PATTERNS

A. W. Hewat

Institut Max von Laue - Paul Langevin

156X Centre de Tri, 38042 Grenoble

France

The refinement of crystal structures directly from neutron

powder diffraction patterns, without first extracting the structure

factors, has become an important crystal lographic technique in the

ten years since Rietveld first introduced it. This profile

refinement technique is related to, but different from, the line

profile analysis techniques developed even earlier for x-ray powder

diffraction. In this paper we first follow the contributions made

by Rietveld and others by reviewing the applications of profile

refinement to specific types of problems in solid state physics.

Starting with magnetic structures, profile refinement was quickly

adopted for studies of uranium halides, phase transitions, hydrogen

bonding and disordered structures, especially ionic conductors. It has removed many of the limits imposed on classical crystallography

by the availability of suitable crystals, extinction, twinning and

different sample environments. In fact, it is equally easy to study

the solid structure of materials which are normally gases or liquids. With this historical and applications background, we then

examine the technique itself in more detail. The correct choice of

weights, the use of molecular constraints and Fourier synthesis, the

best description of thermal motion for individual atoms and rigid

molecules, are all subjects still being debated, although a working

consensus does exist. Improvements are also being worked out for

the description of the line shape, the background contribution and

the effect of absorption. We then turn to the existing experimental

techniques, emphasizing the need for high resolution diffractometers, efficient multidetectors and focussing monochromators to give a wide choice of neutron wavelengths. The sample preparation

and environment are seen to be simpler than for x-ray powder

diffraction, large cryostats, furnaces, and pressure cells can be

accommodated in a routine way and since the sample itself is much

111
larger, preferred orientation is a very much less serious problem.

Finally, we mention some of the new experimental techniques being

developed, especially the use of high efficiency multidetectors, and

intense pulsed neutron sources for time-of-fl ight diffractometers.

We do not believe that these new developments will produce the same

kind of revolution that profile refinement did. Rather, we expect

the continued improvement in resolution, intensity, and computing

techniques to extend the application of neutron powder diffraction

to even more fields in solid state chemistry and physics. Such work

will be increasingly concentrated at a few large national or international centers open to a wide community of users.

Interest in powder diffraction profiles is not new. Long before neutron powder

diffraction became important, x-ray line profiles were being studied in detail [l] 1. Similar techniques of line profile analysis, though less widely known, have been describedi

for single crystal data [2], When computers became available, they were applied to x-ray . line profile analysis, and this technique can now be used automatically for on-line

experiments with a cycle time of the order of seconds [3]. Indeed, cycle times of the

order of milliseconds may soon be expected for the study of the chemical kinetics of

catalytic surfaces, the electrolysis of battery electrodes etc. [4],

What then is different about the Rietveld Type of profile refinement? With profile [ line analysis, the emphasis is still on treating single lines or groups of lines to

extract the positions and intensities of their components. With the Rietveld technique,

j

1

the crystal structure itself is refined to fit directly the complete diffraction

pattern. There is no intermediate step of extracting intensities or structure factors. ji This does not mean that the standard crystallographic techniques, such as Fourier

synthesis of the crystal structure, are not available [5]. However, it does mean that a i starting model for the structure is necessary, and the Rietveld technique is of no use foHl direct methods of structure solution. An intermediate type of profile refinement is

needed if such a starting model is not known.

To understand the differences between profile line analysis and Rietveld profile

refinement, it is first necessary to understand some of the difficulties with the

former. The Rietveld technique is an effort to circumvent these difficulties. Both

techniques are of most value when the powder diffraction lines are not all completely

resolved; otherwise integrated intensities can be used for the structure analysis, and

profile refinement is really only useful for improving the precision of the measurements

of peak positions and hence of the crystal cell dimensions.

Figures in brackets indicate the literature references at the end of this paper.

1. Line Profile Analysis and Rietveld Refinement

112
With profile line analysis, one attempts to fit a group of overlapping lines with a function which is the sum of a number of individual lines. Obvious parameters are the

positions and intensities of each of the individual lines, but one might also need

additional parameters describing the background level and line shape as a function of the

scattering angle. With x-ray powder diffractometers, the line shape in particular, may be

a very complicated function (e.g., see [1]). Then, we need 2-4 parameters for each line

in the diffraction pattern, which might contain up to lfP lines for the complete scan. If

| we are not to have an impossibly large number of parameters, we must identify distinct

parts of the scan separated by regions where no lines contribute: these parts can then be

treated independently, at least for a given cycle of refinement. The problem of finding

such independent groups of lines is less serious for high resolution x-ray studies of

inorganic compounds, but becomes virtually impossible for more complex organic structures,

especially with relatively low resolution neutron diffractometers.

Apart from the number of parameters, and the need for regions where no lines

contribute, the line profile analysis technique suffers from the problem of correlation

'between parameters. For example, for two closely spaced lines, there will be a strong

correlation between the widths and positions (separation) of the lines: there are similar

problems if one attempts to refine the background for a complex pattern containing only

small regions where no lines contribute. For these reasons, the background levels are

usually fixed at "reasonable" values, and an attempt is made to find a function which will

also fix the shape of all the lines. This is not easy, because both background and line

shape are both sample dependent, and cannot be measured once and for all for a given

diffractometer. Even if this were possible, there would remain twice as many parameters

as there are diffraction lines, and correlation would still exist between the different

line positions and intensities.

The Rietveld technique drastically reduces the number of parameters, and new

parameters are directly related to physical quantities. All of the line positions are

determined by the unit cell dimensions (at most six parameters) and the line intensities

are no longer independent, but are determined by the crystal structure itself. There is

no longer any need to break the pattern into groups of lines and groups of parameters;

indeed, the best results are obtained when all of the structural parameters are simultaneously refined to fit the complete diffraction pattern (profile). There are then

only about 102 parameters for the 103 lines of the complete scan.

2. Applications of Neutron Powder Profile Refinement

2.1. Magnetic structures

Neutron profile refinement, in the Rietveld [6,7] sense, predated its application to

] x-rays by a decade. For some years, most other laboratories neglected the work at Petten,

iwhich was concerned largely with a rather special field of study, magnetic structures,

which have little interest for x-ray crystal 1 ographers. Even neutron diffractionists

113
tended to discredit the use of powder diffraction for magnetic structures, since it was

well known [8] that for high symmetry structures, as most commonly studied, some information about the nuclear spin orientation is irretrievably lost when the crystal is

powdered. As well, powerful single crystal techniques for magnetic structures depend on applying magnetic fields in a specific crystal lographic direction, and in using beams of

polarized neutrons and measuring the difference in scattering for different relative

orientations of polarization and crystal axes [9],

Such techniques lose much of their force with the random orientation of crystal axes

in a powder. Powder diffraction, though most important in the early days of neutron

crystallography, had come to be regarded as a poor man's alternative to these new single

crystal techniques which had been made possible with the advent of high flux reactors.

However, much valuable work was accomplished in these early years at Petten on materials for which single crystal work was difficult for various reasons [7].

Similar early studies were taken up at Kjeller (Norway) with work by Andresen and van Laar [10] on the magnetic structure of Fe^Se^, and by Holseth, Kjekshus, and Andresen

[11], on FeSb2 and the magnetic transition in CrSb2« In the following years, Andresen and

his colleagues remained an important group of users of the Rietveld technique for mainly

magnetic structures.

2.2. Uranium hal ides

It was not, however, until 1973 that most other neutron diffraction laboratories took

an interest in profile refinement. Taylor and Wilson [12,13] using their own profile

refinement program, adapted from the ORFLS single crystal program, extended the work of

Loopstra and Rietveld [14] on uranium oxides to hal ides such UCI4 and UClg. The interest

in uranium compounds was of course mainly due to their importance in the nuclear research

projects at Petten (The Netherlands) and Lucas Heights (Australia). Such hal ides are often volatile, corrosive, hygroscopic and unstable, and thus very difficult to study

using normal crystal lographic techniques. The early achievements of Zachariasen [15], who

succeeded in obtaining the correct structure for UClg and many other such hal ides with

milligram samples, must be regarded as an uncommon "tour de force." With powder profile

refinement, Taylor and Wilson not only achieved more precise structures for many of these

materials at low temperature, but demonstrated the transformation of some, such as MoFg

and WFg to plastically crystalline phases, with rotations of the halide octahedra before

the crystal actually melts.

2.3. Phase transitions in perovskites

Phase transitions were also being studied at Harwell at this time, using neutron

profile refinement. Perovskite ferroelectrics, such as KNbC^ and BaTiC^ undergo a series

of phase transitions to lower symmetry structures as the temperature is lowered. It is

114
\ry difficult to obtain single crystals of such materials, since the crystal usually

jatters or becomes multiply twinned on passing into the lower symmetry phase. Even

lough such transitions involve very small displacements of the atoms 0.05 A), the

jutron powder diffraction patterns look quite different for the different phases (fig. 1)

the structures can be readily obtained from profile refinement [16a]. In this case,

|ere were stringent checks on the correctness of the results. Not only could the higher

mmetry phases be compared with careful single crystal measurements, but the low symmetry

ructures could be used to calculate the spontaneous electrical polarization of these

rroelectric materials; these calculations compared exceptionally well with macroscopic

lasurements. Many such ferroelectric and antiferroelectric phase transitions have been

udied since [16b]. With the ILL high flux reactor, complete data sets can be obtained

r many different temperatures in a short time. The temperature dependence of the order

rameter in the antiferrodistorti ve material KCaF

3 has thus been studied [17]. The

fi suits are in excellent agreement with NMR measurements in the tetratragonal phase, and a lantitative relation has been found between the temperature dependence of the order

i rameter, the lattice dimensions and the anisotropic Debye-Waller factors.

i 2.4. Phase transitions in inorganic materials

i

Structural phase transitions are really quite common in many "ordinary" materials.

( e "crystal structure" usually reported at standard temperature and pressure may not then

present the true equilibrium structure, but only an approximation. The real crystal

ructure must be determined near absolute zero temperature and although this presents

fficulties for conventional techniques, there is no special problem with neutron powder

thods. Temperatures of millikelvins are even possible, but the boiling point of liquid

lium is usually sufficiently low for most structural studies.

For example, Pb30^ undergoes a subtle phase transition at 170 K in which the

) lifting of the diffraction lines is only visible on a high resolution diffractometer.

varri, Weigel and Hewat [18] have shown, by profile refinement of neutron powder data,

lat the transition occurs because of the opposition between Pb ions, the Pd* and 0

fins playing no part. The structure simply contracts when the temperature is lowered, and

I 170 K, the Pb^+

ions come into contact: there is then a shearing of the structure to

'oid a too close contact.

Such phase transitions represent very small changes of the structure: atomic

splacements are a fraction of an angstrom unit. However, relatively large changes

:sult in the diffraction pattern, which is clearly very sensitive to any splitting of the

ines. With a line resolution of Ad/d ^ 10 , even changes ten times smaller have a kiceable effect.

Bi VO^ is an example of a large group of so called "ferroel astic" materials, where the

(rection of polarization can be changed by some (often small) external stress.

115
I

Figure 1. Successive phase transitions occur in most perovskite ferroelectrics. Neutron powder profile refinement has been used to study many

such materials.

David, Glazer and Hewat [19] have shown how the bismuth atoms and the rigid VO4 tetrahedra

move along the c-axis to produce this polarization; as the temperature is lowered, the

crystal enters the ferroelastic phase at 255 °C. The data is sufficient to resolve even

the position of the vanadium atom, which has an unusually weak scattering length for

neutrons.

i, A number of other ferroelastic materials have been studied by Glazer, and also by

a Hidaka using profile refinement (e.g., see [20]).

L 2.5. Hydrogen bonds and phase transitions

I

Phase transitions in hydrogen bonded materials have also been studied for some time

using neutron powder diffraction. For example, in NH^H^PO^ [21] single crystals are

i

116
iestroyed on passing into the low temperature phase, which is anti ferroelectric in pntrast to the ferroelectric structure of the potassium isomorph K^PO^ [22]. This

h'fference was shown to arise because the new hydrogen bonds between the NH4 and PO4

Iroups in NH4H2PO4 force the H2 atoms to take up an alternative hydrogen bonding

Irrangement, which is apparently less favorable in KH2PO4. More recent measurements on

IH4H2PO4 [23] are sufficiently precise to allow a detailed interpretation of the

jnisotropic temperature factors obtained from profile refinement (fig. 2). We notice

mmediately that above the phase transition both the NH4 and PO4 groups are librating

trongly.. It is remarkable that the major axes of the thermal ellipsoids, although not

onstrained to lie in any particular direction, refine to be perpendicular to the bonds,

s required for rigid body librations. The disordered 0..D..0 bonds appear to be

ignificantly nonlinear, with the two half deuterium positions displaced from the line

oining the oxygen atoms. This is not the correct interpretation; the two PO4 groups are

0 doubt librating 180° out of phase, synchronized with the motion of the D atom in its

[ouble well. The 0-D..0 or 0..D-0 bond can be linear at any given instant: the 1/2 D Positions do lie on the line joining the opposite extremities of the 0 ellipsoids. The

D4 ellipsoids show strong libration plus vibration along y, so as to make and break the

-D..0 hydrogen bonds: this motion must also be synchronized with that of the PO4 and D roups. Then, instead of the simplified picture of hydrogen bonds tunneling between two

inima of a double well potential, which is the classical picture of H-bonded

erroelectrics, we can see that the true "soft-mode" involves all of the atoms, coupled in more complex collective oscillation. The transition mechanism can be more easily

isualized by viewing the 5 K structure on top of that at 300 K. This example illustrates the utility of using a complete description of the thermal

otion, and indicates the potential of profile refinement for studying structures as a unction of temperature.

2.6. Disordered structures

After 1973, an explosion of papers on neutron profile refinement occurred, as ummarized by Cheetham and Taylor [24]. In one of the first, Cheethan and Norman [25]

efined the structures of YF3 and Bi F3 from data collected at Harwell. The group of

ender in Oxford applied the technique to many of the problems in chemical crystallography

hat they had already pioneered using classical neutron powder techniques [26,27,28].

In particular, the technique was applied to disorder configurations in non- tochiometric oxides and hydrides. For example, Titcombe, Cheetham and Fender [29]

epeated and extended their earlier conventional powder diffraction work on hydrogen

rdering in CeD2 + x [30].

1 Also at Oxford, Von Dreele and others applied profile refinement to large disordered

jtructures made from simple blocks of titanium, niobium and germanium oxides [31], These

Rock structures are best known from the x-ray work of Wadsley [32] and especially the

117
PHASE TRANSITION IN ND4 D 2 P0

4

300K I42d

1/2

7.53 A c = 7.55 A a = 7.52 A b = 7.55 A c = 7.44 Figure 2. Ammonium di hydrogen phosphate is an example or a more complex type

of phase transition studied by profile refinement.

electron diffraction lattice imaging technique developed by Allpress [33] and others (see

Cowley and Iijima [34] for references). The characteristic feature of such sructures is

usually the regular array of defects joining large blocks of perfect order. Anderson,

Bevan, Cheetham, Von Dreele, Hutchison, and Strahle [35] therefore had recourse to these

latter techniques to interpret their profile refinement results for germanium niobium

oxide.

In all of these non-stochiometric problems, one attempts to refine the occupancy of

the various sites, and the danger is the correlation to be expected with the Debye-Waller

factor. One needs very good data, and even then a knowledge of the average occupation may

be difficult to interpret in terms of the detailed crystal structure.

fit

tri

i

2.7. Fast ion conductors

Structures showing partial occupancy are, however, of increasing importance for the

study of ionic conductors, and powder diffraction can make an important contribution here,

provided the proper precautions are taken. Wright and Fender [36] studied the phase

transition in which Agl becomes a conductor for Ag+ ions. England, Jacobsen and Tofield

[37] showed that the hydrogen atoms in deuterated g-alumina, which are bonded to the

oxygen atoms at 4.5 K, become available for ionic conduction at 823 K. A more detailed

118
D;:

iunt, in which Fourier techniques are used to examine the hydrogen distribution at high

eratures, is given by Tofield, Jacobsen, England, Clarke, and Thomas [5].

Bogacz, Bros, Gaune-Escard, Hewat, and Taylor [38] showed that in the ionic conductor

Brg, the sodium ions are freed at two successive phase transitions. Hewat [39] has

ured the complete structure of Bi^C^ at six temperatures from 300 K through the

er-ionic" phase transition at 1000 K, where the oxygen ions become distributed over

jmpletely occupied holes. The intermediate phase previously reported on lowering the

erature, was not found when the sample was prevented from taking up additional

Ispheric oxygen.

2.8. Location of hydrogen atoms

Just as neutron diffraction is of value for studying heavy atom structures, such as of uranium, it is invaluable for the location of very light atoms such as

pgen. The scattering power for neutrons is of the same order for most elements, and

/Proportional to the atomic number, as it is for x-rays. In addition to those quoted

| earlier headings, the following example may be noted.

Some materials absorb so much hydrogen that they are known as "hydrogen sponges", and

p this has possible applications for energy storage, work is currently under way to

| the mechanism of such absorption using neutron powder diffraction. Earlier studies

'drogen in metals using neutron profile refinement include the work of Kuijpers and

tra [40].

Sodium per carbonate (^003 -1.5 H2O2) is a well known industrial bleach, which is

usually stable vehicle for hydrogen peroxide, or active oxygen. Adams, Pritchard,

lewat [41] have determined the precise hydroxide configuration in this material, which

sordered at room temperature but transforms to an ordered structure at low

ratures. Other perhydrates, such as those of the oxalates of potassium and rubidium

s, Ramdas and Hewat, [42] have also been studied in the same way. In such materials,

much of the structure is already known from x-ray methods, the difference Fourier

ique applied to neutron powder data immediately reveals the hydrogen atom

ions.

2.9. Solid liquids and gases

The crystal structure of materials which are normally liquids or gases can most

y be studied using powder diffraction. Sandor has studied the phase transitions in

e molecular solids in this way, (e.g. see ref. [43]). Solid ammonia is another

le which is ideal for studying the dynamics of such materials, and one may ask if the

119
ammonia molecules in the solid undergo hindered rotation as they do in many salts such as

NH4 N03 [44],

As an aid to the study of the molecular dynamics of ammonia, the solid was examined

by neutron profile refinement as a function of temperature between 2 K and 180 K [45]. No phase transition was found, but even near absolute zero temperature, the ammonia molecule

is librating strongly. The amplitude of libration does not increase with temperature as quickly as required for a purely harmonic model, and this suggests a possible quantum

mechanical tunneling between the hydrogen sites.

Solid oxalic acid [46] and acetic acid [39] are being studied by profile refinement

at low temperature to investigate the possibility of "resonance" structures in which the

C-0 and C = 0 bonds become equivalent above a certain transition temperature, with

disordering of the hydrogen bonds.

3. Profile Refinement Theory

3.1. Basic principals

All types of profile refinement can be justified by the following simple arguments

[47]. Suppose that the count y^obs) at the point i in the diffraction pattern is a sample of some model function y^calc). In the case of line profile analysis, this model

function is the sum of the profiles of all of the individual lines; with the Rietveld

method, these individual line profiles are calculated directly from the crystal structure

model. Then the probability p^ that a given sample count y^obs) will differ from

yn-(calc) is given by

P

II!

if]

H

ll

Pi a 4. eX P

) - 2

i I a.

h [y^obs) - y^calc)]^

since each sample count comes from a normal (expone ntial) distribution centered on

y.j(calc), with standard deviation = J y"J (cal c) . The probability P that all of .the counts are samples of the population y(calc) is the product p^p^ of the individual

probabilities. Then

P = n.p. « I exp

since the product of exponentials p n

- is the exponential of the sum of the separate

exponents. The probability P is maximized if the exponent factor

at

120

I
x

2 = s. to. [y^obs) - y^calc)] 2

inimized using the weighting scheme to. = —=• - —7—^—y = —, t \ .

1 a

2 y^calc) y^obs)

j It should be noted that the statistical weight is of course calculated from the raw

I before subtracting the background. Since the background is a smooth function

jined from the average of many points, the statistical error introduced by the

ground subtraction is usually neglected, as in single crystal measurements. However,

ematic errors may occur; for example, an underestimation of the contribution of

dial diffuse scattering will lead to an underestimation of the overall temperature

pr for the crystal structure.

j The "R-factor" x

2 is well known in statistics. Indeed, the "chi-squared test" can be

I to test the significance of the initial hypothesis that the y^ (obs) are samples of

yj(calc) i.e., that the observations are consistent with the model. Given x

2

, the

I o

;r of observations i, and the number of parameters used to minimize x » then one can

jit a table to determine the probability that the model is correct. In practice, we

ilate the x expected if the model were exact, and if the fluctuations in y^(obs) were

ly statistical. The x obtained by refinement will always be larger than this, but

approach the ideal value as systematic errors in the data or the model are

inated. The R-factor for integrated intensities is also calculated at the end of the

lement, after dividing each yn- (obs) between the contributing lines according to the

ilated intensities. This R-factor is twice the R-factor for structure factors usually

id by crystal lographers.

3.2. The weighting scheme

The weighting scheme used for profile refinement, derived purely from statistical

lents, is to be contrasted with the different weighting schemes devised for

mtional single crystal structure analysis. In the latter case, an attempt is usually

Sto give lower weights to those observations most affected by systematic errors such

Itinction. With neutron powder measurements, such systematic errors are usually much

important. However, if special care is taken to eliminate such systematic errors, a y statistical scheme can also be used for single crystal measurements. Sakata and

r [48] propose that the weight for an integrated Bragg reflection should then be

sely proportional to the integrated intensity, just as it is inversely proportional

e intensity for profile refinement. With this assumption, they then go on to show

[the calculated standard errors are different for the same data treated by profile

121
refinement or by refinement on the integrated intensities. They conclude that the errors calculated by profile refinement must therefore be wrong. The details of this argument

are correct, but the basic assumption and conclusion are not.

In fact, each count y 7-(obs) is an independent estimate of the integrated Bragg

intensity; simply adding together these counts does not give the best possible estimate o the integrated intensity, but only an approximation which is not even very good for weak

peaks on a high background. The idea of making the weight inversely proportional to this

integrated intensity is again an approximation which takes no account of the fact that a number of independent, but not equally good, estimates (a step scan) was made of a function which we know to be a peak. Nor is it possible to argue that the observations

are not independent estimates, because they all come from the same Bragg peak, nor to argue even that the Bragg peaks are not independent because they come from the same structure. Here we are concerned only with the notion of statistical independence.

Sakata and Cooper have then correctly demonstrated that, given these approximations,

the calculated errors are not the same for the integrated intensity refinement as for

profile refinement. They correctly attribute this to the different weighting scheme used

especially for weak reflections. However, the correct conclusion should be that profile

refinement must be used for single crystal data, as was shown long ago by Diamond [2],

3.3. Constrained profile refinement

mi

re

The principals developed above apply to all types of profile refinement: only the

parameters are different with different methods. With line profile analysis, we assume

nothing at all about the structure; in fact this technique is especially useful for a preliminary analysis of unknown structures or even mixtures of unknown materials.

The positions and strengths of the strong lines can be obtained automatically and

compared with a computer file to identify the components. However, if a model structure

is already known, it is better to use the Rietveld technique to reduce the number of

parameters to those few that are physically meaningful, thereby reducing the correlation

between parameters and improving the result of the structure refinement. Intermediate

types of refinement can also be envisaged. For example, the positions of the first few

lines might be obtained by line profile analysis, these lines indexed using one of the

computer programs now available [49], and then a new profile refinement attempted in whic

the parameters were just the lattice constants and the line intensities. This should hav' some of the advantages of the Rietveld technique in removing parameter correlation withoi|ttu

the need to know the details of the structure, which could then be obtained from standard

crystallographic techniques and perhaps even direct methods.

This intermediate profile refinement and the Rietveld refinement, can be regarded a? in constraining the basic line profile refinement to give results consistent with some additional knowledge—first of the unit cell, and then of the structure itself. We can f even further; if we assume that some components of the structure are rigid units, we need

Sil

122
only refine the positions and orientations of these units as a whole. A simple

iple, which is already possible with the Rietveld program, is the well known rigidity

he oxygen octahedra in most perovskite oxides [47], These additional constraints

n reduce the number of parameters and the correlations between them.

Pawley, Mackenzie and Dietrich [50] have written a new profile refinement program

gned to exploit the full power of such constraints, which Pawley [51] has already

floped for single crystal refinements. He has shown that in many molecular crystals,

molecules, or at least parts of them, can be constrained to move as rigid units. This

tly simplifies the refinement, which can then give useful results for quite complex

ctures, using only the relatively small amount of information contained in a powder

ern. These developments will no doubt increase the upper limit on the size of the

cture that can be treated with powder diffraction, now estimated to be a monoclinic

of volume up to 3500 A

J [52].

Such constrained refinement techniques are also of value in studying the behavior of

rials under different conditions of temperature and pressure. Often, it is only

ain features of the structure which are changed; for example, one molecule sliding

another, or a rigid unit rotating to accommodate new crystal stresses. Once such

ures are identified, only the appropriate parameters need to be refined at each

jerature or pressure.

3.4. Thermal motion and anisotropic Debye-Waller factors

Rietveld [7] worked with normally stable structures, often at low temperatures to

O y magnetic ordering. He used rather long wavelength neutrons (2.4 A) for reasons of

lution and intensity [53] and therefore obtained data of limited range in sine/x.

r these conditions, isotropic Debye-Waller factors, or even an overall temperature

or, was sufficient. When the profile refinement technique was applied to other

lems, especially those involving structural phase transitions, these approximations

no longer sufficient. A phase transition occurs because a structure is basically

able, and precurser effects are seen as characteristically strong vibrations of groups

toms. Such effects are described as "soft modes", and are far from isotropic. The

e-Waller tensor may be four times larger in the soft direction than in the other

ctions and this makes a big difference to the intensities of high angle lines,

ite this difficulty, powder diffraction may often be the only way of refining such

ctures where the crystal itself may be destroyed by the phase transition. Anisotropic

fe-Waller factors were therefore soon introduced into profile refinement [47], Apart

permitting fits to the high angle data which could not be refined with the normal

Jram, they offered physical insight into how soft modes produced phase transitions,

low resolution diffractometers of limited sine/x range, the absolute values of the

e-Waller factors may not be well determined because of systematic errors in the

ee|ground measurement. However, even then, the relative values often showed the

123
characteristic librations or vibrations of groups of atoms. These effects are most convincing when the results of profile refinement are plotted with the ORTEP program, as

in figure 2.

3.5. Cumulant expansion of thermal motion

However, even anisotropic Debye-Waller factors are not adequate for describing

certain types of motion found in crystals, especially at high temperatures. These

problems do not occur so often for single crystal measurements simply because such

measurements are so often limited to standard temperatures and pressures. For example, in

MoFg at 263 K, the octahedra, which are very rigid units, begin to rotate [54], Such

dissociative phase transitions may be precursors to melting itself, or may for example be

associated with the onset of superionic conductivity, as in ^UBrg [38], Levy et al.

[54] have shown how the cumulant expansion developed by Seymour and Pryor [55] can be used to describe such structures with a few physically meaningful parameters suitable for

profile refinement. The dissociation of the structure can then be followed as a function

of temperature, since many complete data sets can be collected in a short period.

A similar solution to a related problem has been developed independently by KurkiSuonio. In NH4NO3 [44] this method has been used to find the preferred orientation of the

NH4 and NOg" groups in the high temperature phase. These groups are rotating, but there

is a definite correlation between their orientations. This type of hindered rotation must

in fact be quite common, and many more examples will no doubt be discovered now that

j. structures can be precisely measured at many different temperatures with profile

refinement.

A related effect occurs in dynamically disordered materials. In CsPbC^ [56] the

anions occupy holes which are too large for them. At high temperature, they therefore

move around these potential wells, but nevertheless, have preferred positions which can be|

found using the cumulant expansion technique. This effect can be called "anharmonic"

thermal motion. At lower temperatures in CsPbClg, the ions no longer have the energy to move around, and become frozen in particular positions after a series of phase

transitions. Profile refinement, using powerful on-line computers and interactive

programs will be increasingly used to explore such effects, if only because life is not long enough to survey such structures as a function of temperature and chemical

composition using traditional single crystal structural studies.

3.6. Fourier techniques

The Rietveld program refines the structure directly to fit the diffraction pattern,

without first extracting the structure factors which are needed for Fourier synthesis.

Fourier techniques cannot then be immediately applied. However, if a large part of the

structure is already known, for example from x-ray work, it is possible to use this part

124
ind the remainder from a difference Fourier synthesis. The refinement is set up in

usual way, and just the scale factor and cell dimensions are refined using the partial

cture. At the end of the refinement, even though the agreement between calculated and

srved profiles will not be very good, the experimental structure factors are extracted

dividing up intensities due to overlapping peaks according to the ratio of the

:ulated structure factors. The phases are also calculated from the model of the

tial structure. A difference Fourier is then constructed and searched for features

eh might be identified with the unknown parts of the structure.

Some of the unknown atoms can then be placed, and the whole procedure repeated to

d any remaining unknown atom positions. Such an approximate procedure does not of

rse produce precise atom coordinates, but only starting positions which can later be

ined in the usual way. It works best when no more than about one quarter of the atom

itions are unknown, and of course when the structure is centrosymmetric so that the

pes are simply 0° or 180°.

Fourier syntheses have often been used to locate hydrogen atoms in structures where

'heavy atoms have already been found with x-rays (Sections 2.7 and 2.8). Such

ihniques should also be of value for disordered structures.

3.7. The line shape

The classical neutron powder diffractometer has an advantage over x-ray powder

ras and diffractometers for profile refinement: the line shape is much simpler. This

ses for a number of reasons.

(i) A monochromated white source of neutrons does not of course contain sharp

characteristic lines, such as the and compounds of x-rays,

(ii) The resolution must usually be relaxed for neutrons, since the radiation source

is much less bright. Then the effects of particle size and strain are not so important and the line shape is largely determined by the diffractometer, rather

than the sample.

(iii) The sample is much larger cm ) for neutrons, so that preferred orientation,

surface texture effects etc. are not usually important,

(iv) The geometry is simpler for neutron diffractometers. The sample is cylindrical

and concentrated at the center of the counter bank. Sbller collimators or position sensitive detectors are used to define the angular divergence: both

have triangular acceptance functions, so that when several are convoluted

together, an almost perfect Gaussian results.

Even for high resolution powder diffractometers (Ad/d <\> 10" ) the line shape is

ribed very well by a simple Gaussian [57]. The only concession made by Rietveld [7]

he diffractometer geometry was to allow the line width to vary in a simple way

125
with scattering angle, and to correct the small asymmetry observed at low angles due to the finite vertical divergence. These approximations are the only ones used for almost

all neutron powder work at present, but a number of authors have shown how they can be

improved to describe more rigorously the experimental profiles. L Cooper and Sayer [58] and Thomas [59] have derived more complete corrections for the

effect of vertical divergence. They have shown that in addition to a more complex form o asymmetry of the line shape, the centers of the lines are shifted slightly, and the

integrated intensities must also be corrected. Of course, these effects are really only

important for low (or high) angle lines on high resolution machines with rather large

vertical divergences 5°). However, since new diffractometers do aim for fairly high

resolution, with the vertical divergence increased to make up for the resulting intensity

losses, these more precise corrections should be used more regularly.

Again, with higher resolution neutron diffractometers, the Gaussian line shape

approximation which has up to now served so well, needs revision. Suorrti [60] has shown

that with high resolution, tails can be seen on each side of the line; these tails cannot

of course be described by the Gaussian approximation. He has introduced a Lorentz

component to account for these tails and has shown that a very much better fit is obtaine

when the line shape is described by the convolution of a Gaussian and a Lorentzian, known! as a Voigt function.

Since the effect of the tails becomes systematically more important at high

scattering angles, the use of the usual Gaussian leads to an overestimation of the overal

Debye-Waller factor, although the refined structural parameters are presumably not much

affected.

Similar conclusions have been reached for x-ray line profiles. Langford [61]

considers a Voigt function represented by the convolution of m Lorentzians and n Gaussian

to represent the lattice strain, crystal imperfections and the shape and distribution of

the crystallites in addition to the instrumental profile. Both Suorrti and Langford

indicate how these more exact line shape functions might be incorporated into profile

refinement programs.

li

3.8. Background subtraction

The existence of long tails on each line emphasizes a rather basic problem with

profile refinement: what to do about the background. Profile refinement is needed to resolve overlapping lines for more complex structures, but in that case it becomes

difficult to find regions where there are no lines and where the background can be

measured. Sabine and Clarke [62] have proposed that just as the structure is refined to

fit the peaks, other features such as defect distributions might be refined from the

background. They see the background as a valuable source of additional information about

the crystal, rather than a problem to be overcome. Windsor [63] has argued that the , Debye-Waller factors should be refined to account for the angular dependence of the

126
;rmal diffuse scattering as well as that of the Bragg peaks. Fender [64] has used

lulations in the background, in addition to the Bragg peaks, to obtain the defect

;tribution. Boysen and Hewat [65] have shown that such background modulations in

>nClg are due to a dynamic disorder at room temperature which tends to disappear as the

fstal undergoes successive phase transitions at lower temperatures. In many such

imples, the background can supply new information about the powdered crystal, but to set

a model of the background for profile refinement, which contains only a few physically

wingful parameters, is not something that can yet be done routinely. Some physical

ight into each special case is usually needed. At present, the background is measured

positions where no lines appear to contribute, and is thus not part of the profile

inement.

There are also experimental approaches to the background problem which will be

cussed in Section 4.3.

3.9. Absorption corrections

Except for a few isotopes (for example, the most common isotopes of H, B, Cd, Gd, Eu other transuranics) , the effective absorption for neutrons is very much smaller than

is for x-rays. Samples at least 1 cm thick are common. This means that a large

iindrical sample can be used, greatly simplifying the geometry, and providing a true

k average of the crystal properties. However, even this small absorption (yR ^ 1) can

se systematic errors in the temperature factors, which will then be underestimated,

se et al. [66] and others have published tables of absorption corrections for such

pies, but in fact the effect on the Debye-Waller factors can be calculated directly

,68]. Such corrections are then best left to the end of the refinement.

In the few cases mentioned above where absorption is strong, a nonabsorbing isotope

be used in place of the natural element. Such isotopes are quite inexpensive for the

iter elements due to the advances made with isotope separation in the nuclear

jstry. For the transuranic elements, the absorption can in some cases be greatly

o jced by using a shorter neutron wavelength, below the absorption edge (e.g., 0.5 A

:rons for Gd).

4. Experimental Techniques

4.1. The classical neutron diffractometer

The classical neutron powder diffractometer, such as that used at Petten or Harwell

many of the early measurements using profile refinement, consists of three main parts

j). 3): a large monochromating crystal, a counter turning around the sample, and Sbller

jimators cc^, and 013. In some cases, such as at Lucas Heights and Brookhaven, an

tional analyzing crystal is used in front of the counter to eliminate elastically

tered neutrons contributing to the background [69].

127
Figure 3. A conventional neutron powder diffractometer with a bank of Soller

collimators and counters [57].

128
The transmission function for a Soller collimator is triangular, and the convolution

two such triangular functions is very nearly a Gaussian [70]. The mosaic spread of the

mochromator (&) is also a Gaussian if an average is taken over a volume large compared

th the mosaic block size, as is true in this case. The intrinsic line shape from the

rticle size and strain in the sample does not enter for the typical medium resolution

chine, where the instrument itself accounts for most of the line width A-^ at half

ight (typically 0.5° in 2e). Caglioti et al . [71] have derived the dependence of

scattering angle 2e as

^

A

l/2^

2 = U tan ^

+ VtanS + W

^re U, V, W are strictly functions of the divergence angles c^, c^, 013 and e, but are

^ially treated as parameters in the refinement.

Together with an elementary correction for line asymmetry at small scattering angles,

lis simple description of the line shape continues to serve for almost all neutron

pfile refinements, and is in fact a very good approximation. It is only in the case of

gh resolution diffractometers that more sophisticated treatments become important

ection 3.7. )

.

I

4.2. Requirements for high resolution

The line width can of course be reduced simply by reducing the divergences 0^, 0^, C13 the collimators. However, since the integrated line intensity is proportional to the

duct otj, c^, 03, 8 [71] such a simplistic appproach is not usually possible: the

egrated intensity would be reduced by a factor of 16 for a reduction of a factor of 2 each of a^, c^, C13 and $. A one day experiment would then take two weeks. In ctice, the situation is not quite so bad, since it is the peak intensity and not the

egrated intensity that is important for profile refinement: the reduction in peak

ensity would be "only" a factor of four. As well, we will see that because of

iissing considerations, it is only necessary to reduce and a^: the peak intensity

1 then be simply proportional to the resolution.

Loopstra [53] has proposed another means of improving the resolution. If a longer

elength is used, the low angle peaks are spread across the entire angular range, and

therefore better resolved. As well, the same total scattering power is now

tributed over fewer reflections, so that the line intensity will actually increase for

129
longer wavelengths. This is offset somewhat because the peak intensity from the reactor

o o comes at about 1 A, but nevertheless, there is a net gain up to wavelengths of about 2.5 A

The diffractometer of Petten is therefore designed to use such a relatively long

wavelength, and a special graphite filter is used after the monochromator to remove

shorter wavelength harmonics.

This solution is especially interesting for work on magnetic structures, where the

magnetic form factors make it impossible in any case to collect data at large values of

sine/x. For crystal structures it is interesting to work with shorter wavelengths to

better resolve the structure. For example, on the diffractometer D1A at the ILL, the

o o usual wavelength is 1.9 A, which gives data to sine/x = 0.5 A, the usual copper limit for

o x-rays. However, wavelengths as long as 5.7 A are used to index complex diffraction

patterns, where the positions of the first few lines must be determined most precisely

[72]. Longer wavelengths are also useful for very large molecular structures, where

constrained molecular refinement can be used to limit the number of parameters, and hence

reduce the amount of information needed from the diffraction pattern [50],

Fortunately, it is not necessary to have equally good resolution for all values of

scattering angle. The density of lines reaches a maximum at 2e ^ 90°, and there are fewer

lines at very low or very high angles. The best resolution is obtained when the

scattering angle is about equal to the monochromator angle; the different wavelengths ax from the monochromator are then scattered back by the sample along the same direction as the incident beam. The line width is then essentially given by the convolution of aj and

a-j; the collimator a2 between the monochromator and sample, and the mosaic spread g may

have any values. The monochromator angle should then be about 90° to coincide with the

region of maximum line density. The objection to using such large monochromator angles e M

is that the reflectivity is then lower. This is partly due to the Debye-Waller factor for

high order reflections from the monochromator, but mainly due to the reduced AX of

wavelengths reflected

—— = v cot e M

.

The solution is to increase the monochromator mosaic spread b, which does not affect

the focussing resolution, and eventually to cool the monochromator to reduce its DebyeWaller factor. With new hot pressing techniques for monochromators [73], the first

solution is now easy, and the second is feasible but not usual. In practice, the

monochromator angle is made even larger than 90° (on D1A it is 122°), to compensate for

the rapid increase in line width at high angles (with sin2e). A more complete account of

these considerations is given by Hewat [52].

130
4.3. Background

The experimental determination of the background level is difficult, and this sets

pits on the precision with which site occupation numbers and temperature factors can be

tained. We need to have regions throughout the diffraction pattern where the Bragg peak

itribution is negligible in order to measure the background level and interpolate to the

ner points. In particular, it is necessary to find such regions at high scattering

pies. On a conventional diffractometer, this is still possible if the take-off angle is

jiater than 90°, because of the fall off in peak density again in this high angle

gion. Oh a time-of-fl ight machine this is more difficult, because the resolution is

istant while the line density continually increases with sine/x.

4.4. Multicounters

The simplest way of improving the statistics for a conventional diffractometer is to

^ilace the single counter by a bank of counters and Soller collimators. However, until

I cently, it has been very difficult and expensive to make a bank of identical

llimators. If the collimators are not identical, the profiles will not be identical

ther, and it will not be possible to simply add them together. Banks of counters have

^n used since at least 1966 for integrated intensity measurements (CURRAN at Harwell),

. p because of intercal ibration difficulties the idea was dropped on later machines (PANDA

Harwell), and only taken up with recent advances in collimator design.

The best Soller collimators for neutrons are now made from stretched mylar foil,

ited with neutron absorbing paint [74]. Such plastic collimators, first suggested by

lister et al. [70] have efficiencies of the order of 95 percent, but more importantly,

or llimators of 5' or 10' divergence can be made to be virtually identical at relatively

* cost. Such collimators were first used routinely on the D1A diffractometer in

;noble, where the results of a bank of ten collimators and counters are automatically

led together to give a single composite profile for refinement. The computer program

ces the remaining small corrections for relative efficiency and alignment, and each

int in the profile is weighted according to the number of counters averaged to obtain it n.

i

i

:t

4.5. Position sensitive detectors

The DIB diffractometer at Grenoble also uses a multiple counter, but there are no ilimators. Instead the counter consists of 400 independent elements, spaced at 0.2°

of

j mm) to cover a total angle of 80° [75]. The angular resolution is determined by the

ze of the sample, the counting element, and the distance between them (1.5 m). This is

O n a medium resolution machine operating at rather long wavelengths (2.5 A), which is of

l

131
particular interest for very high rates of data collection. For certain problems, useful 1

diffraction patterns can be collected in times of the order of one second, so that for

example, the crystalline products of chemical reactions can be monitored as a function of t time. For such "kinetic" reaction experiments, even faster cycle times will be achieved j. on new instruments at the high flux reactor.

Collimators are still needed however, for high resolution diffractometers, since

otherwise the sample would have to be very small.

4.6. Focussing monochromators

k

Here we are talking about focussing a broad beam of radiation, up to 20 cm high, ont<j (ks

a small sample, 2 cm high for example. This is quite different to focussing a range of

wavelengths into the same counter direction, which we have already discussed in Section

4.2. The monochromator is curved about a horizontal axis, so there is little effect on the wavelength composition of the beam.

Large gains in intensity are thus obtained for the usual small samples without much

affecting the resolution. The reason is that the beam divergence in the vertical

direction may be 5° or more for even a high resolution diffractometer, while the vertical

divergence available from the reactor is usually 1° or less because of the extensive

radiation shielding. Vertically focussing monochromators can therefore be used to increase this divergence at the expense of beam area, which is normally much larger than

available samples. For the same reason, vertically extended counters are used to match

the vertical divergences between counter/sample and sample/monochromator. The only

disadvantage of divergences of 5° or more is an increased asymmetry of the low and high

angle lines, but the corrections for these asymmetry effects [58,59] is a small price to

pay for a large increase in intensity. The gains are of particular interest for

diffractometers such as D1A and DIB on neutron guide tubes far from the reactor.

The 15 cm high focussing monochromator on D1A is composed of 30 strips of hot-pressed

germanium, each 5 x 5 x 40 mm, glued onto a flexible aluminum plate. A relative alignment

of the strips to within 3' of arc is assured because they are cut from much larger strips

of germanium already glued to a rigid alignment plate. The alignment of the original

single crystal is thereby conserved throughout the cutting and transference to the

aluminum focussing plate. The orientation of the parent crystal is chosen so that the

mosaic spread is large in the horizontal plane, to ensure the selection of a large

wavelength band AX, and small in the vertical plane to ensure the smallest focal spot size

(15 mm). On D1A an order to magnitude is gained in the intensity using this focussing

monochromator.

i

tir

132
sed

4.7. Wavelengths between 1.1 A and 5.7 A

A supplementary advantage of using a large monochromator angle, is that a wide choice

wavelengths is then available simply by selecting different reflection planes. The D1A

o lochromator was designed to use the [117] plane in germanium to give 1.38 A, but many

ler [hhl] planes can be obtained by simple rotation about the vertical axis. Thus

0 ° :ussing is still very good for [115] giving 1.9 A and [119] giving 1.1 A. Other useful

^lengths are 1.2 A, 1.3 A, 1.5 A, 2.3 A, 3.0 A and 3.7 A.

j Longer wavelength harmonics are in all cases forbidden for such [odd, odd, odd]

flections, and of course x/2 is forbidden for the germanium structure. X/3 and higher

o 'monies are filtered out by the guide tube for X < 2.6 A, but a graphite filter should

o o used for 3 A, and a cooled beryllium filter for 5.7 A.

o o The most common wavelengths used in practice are 1.9 A for new work, 1.38 A for

o o Pining well determined structures and 3 A or 5.7 A for indexing unknown structures. Any

relength can be obtained within seconds, and under computer control if required.

4.8. Cryostats, furnaces and pressure cells

Closed circuit refrigerators are available down to 10 K, but a helium cryostat is

ill preferred for most work. Such a cryostat works anywhere between 1.5 K and 310 K,

i the ILL design has the advantage that the sample chamber is quite independent of the

nainder of the cryostat, so that the sample may be changed within a few minutes.

The standard vacuum furnace works between 300 K and 1300 K with the sample usually

sled in a quartz glass tube. Special furnaces are available for the ranges of 20 °C to

) °C, 1000 °C to 1600 °C and 1000 °C to 2400 °C. Experiments at the highest

nperatures must be completed in the shortest possible time because of possible crystal

)wth [76]; the sample may be sealed for example in an electron beam welded tungsten

i. Various types of pressure cells for neutron scattering have been developed at the

jnoble CNRS laboratories under the direction of D. Block. The simplest type works with

npressed helium gas, up to at least 7 kbar and in a cryostat at lower pressures,

pending on the temperature. The sample size is of the order of 0.4 cm . For pressures

to 10 kbar, 0.1 enr samples are contained in a 5 mm hole in a single crystal of

jpphire supported by an aluminum alloy cell. The pressure is applied by opposed tungsten

rbide pistons acting on a special organic pressure transmitting liquid. Pressures of up

30 kbar can be obtained by replacing the sapphire crystal by sintered alumina, but then

ditional lines due to the AI2O3 powder are obtained in the diffraction pattern. Such a

11 has been used, for example, to study the behavior of the hydrogen bonds in rroelectric KH2PO4 under pressures [77].

Pressures of up to 45 kbar can be obtained by using a special black alumina, which

s additives to improve its mechanical properties. Unfortunately, these additives

133
contribute yet more lines to the diffraction pattern, and this type of cell has so far

only been used for studying compressibilities, and the behavior of isolated lines, not profile refinement.

4.9. Sample preparation

Because such a large sample (1-5 cm ) is used, preparation is not so critical as for . x-ray powder diffraction. The material is crushed very finely in a pestle and mortar. A dry glove bag is used if the sample is at all sensitive to moisture, because water

scatters neutrons strongly, increasing the apparent absorption and background.

Even materials which are normally gases or liquids are best treated this way. In this case, the mortar is immersed in a bath of liquid nitrogen inside the glove bag. The

boiling nitrogen readily excludes atmospheric moisture, which would otherwise condense.

The pestle, sample can and spatula used to transfer the sample are likewise immersed in

the nitrogen bath. The full sample can is left immersed in the liquid nitrogen until it

is transferred to the cryostat (a few seconds exposure to the atmosphere), and then any

liquid nitrogen which may have leaked into the sample is boiled off by briefly raising the

temperature to 80 K. This procedure is surprisingly easy, and has been used even for

poisonous gases such as ammonia after alternative methods, such as sublimation or shock

freezing produced samples showing preferred orientation [45]. Even materials which have

melting points above ambient temperature may be too soft to grind, and may benefit from

such treatment.

The sample is pressed into the sample can, never shaken, and provided it has been

thoroughly ground, preferred orientation cannot usually be detected. If it is, the

preparation should be repeated more carefully. There are, however, three other techniques

which may be needed occasionally. Firstly, since the sample is a cylinder, it can be

rotated at say 1 revolution per second throughout the measurement: some workers routinely

use this technique, which reduces the effect, but does little if the preferred axis is the

axis of rotation. Riekel, Convert, Gobert and Lazaro [78], have used a spherical sample

container, randomly spun on a jet of gas, so that a "powder pattern" may be obtained even

from a single crystal.

The second technique is to dilute the sample with a large amount of some other powder

which does not show preferred orientation. Materials with low absorption and simple

diffraction patterns are needed, since their effect must later be subtracted. Powdered

aluminium has been used, but there is a danger of reaction with finely divided metal

powders, especially when used with oxidizing samples in furnaces. Carbon fibre or quartz

wool has been used to prevent extensive crystal growth when a liquid is frozen, [50] and

powdered quartz wool would seem to be the most generally useful material; it can be

removed later by dissolving the sample.

Finally, if all else fails, a preferred orientation correction can be made during the

refinement, as introduced by Rietveld [7]. Then something must be known a priori about

134
the

tely

ider

-tz

the nature of the preferred orientation. None of these three techniques, however,

•eplaces a thorough grinding of the sample, at liquid nitrogen temperatures if necessary

followed by annealing at high temperatures, and a careful filling of the sample can.

The can itself is normally a thin vanadium tube of between 5 and 15 mm diameter,

lany such tubes have been manufactured at Harwell from electron beam welded vanadium

sheet, with titanium ends made vacuum tight by indium wire seals. Because of the very

bmall coherent scattering length of vanadium, no extra lines are introduced into the

Effraction pattern. Such cans (without the indium) are also used in vacuum furnaces up lb about .800 °C. For higher temperatures (up to 1100 °C) the sample is sealed under an tppropriate atmosphere in a quartz glass tube, which has much lower absorption than

ordinary borated glass. The diffuse scattering from the glass then adds to the

background, but again there are no Bragg peaks. For still higher temperatures, or for

1

1

^active samples, other materials such as electron beam welded tungsten have been used,

[ieutron absorbing materials, such as cadmium in a cryostat, or boron nitride in a furnace,

ire used to shield parts which otherwise might scatter neutrons. With Soller collimators

the furnace or cryostat dimensions can be chosen so that they are not seen by the

ounters. Normally then, there are no additional lines in the diffraction pattern, and it

s not necessary to make a separate background scan with the sample removed.

4.10. Data collection and treatment

Although the operation of a powder diffractometer is relatively simple, step scanning

ith one motor and storage of the results of the different counting elements, a small

edicated computer, such as the LSI-II based system used on D1A is of great value. Not

nly does it permit the storage of the results of a large bank of counters, and the

utomatic transmission of the data to the central computer, but it allows on-line

xamination and treatment of the data in the background mode even while controlling the

iffractometer in the priority foreground mode (RT-II operating system). As well, with a ultiple counter, rapid scans can be programmed at different temperatures and the results

ompared in a search for phase transitions. More detailed treatment of the data, and in articular, profile refinement, is best done by transmitting the results directly to a

jentral computer, which can be used on-line in a time sharing mode. Eventually, such a ystem might lead to an automatic structure refinement "machine", which could be used to tudy the effect of external conditions, such as temperature and pressure, on a given

rystal structure. Such a machine could reveal much more about a structure, its stability

nd the balance of interatomic force than do the present "static" crystal 1 ographic

echniques.

the

135

!
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spectrometer for neutron diffraction, Nucl. Inst. Methods, 3 223-228 (1958).

139
[72] Hewat, A. W. , Shirley, R., and Pawley, G. S., Automatic indexing of neutron powder

diffraction patterns, J. Appl. Cryst. (1979), to be published.

[73] Freund, A., Proceedings of the conference on neutron scattering, R. M. Moon, ed.

ORNC Report CONF-760601-02, pp. 1143-1150, Oak Ridge (1976); see also, Freund, A. and

Forsyth, J. B., A treatise on materials and technology, H. Herman, ed., 15, chap. 10 (Academic Press, NY, 1979).

[74] Carlile, C. J., Hey, P. D., and Mack, B., High efficiency Soller slit collimators for

thermal neutrons, J. Phys. E _10 , 543-546 (1977).

[75] Allemand, R., Bourdel , J., Roudaut, E., Convert, P., Ibel , K. Jacobe, J., Cotton, J. P., and Farnoux, B., Position sensitive detectors for neutron diffraction, Nucl.

Inst. Methods 126, 29-42 (1975).

[76] Aldebert, P. and Traverse, J. P., Etude par diffraction neutronique des structures de

haute temperature de La

2 03 and Nd2 03

, Mat. Res. Bull. U_, 303-323 (1979).

[77] Nelmes, R. J., Meyer, G. M., Baharie, E., Hutton, J., and Hewat, A. W., The pressure

and temperature dependence of the structure of para-electric (tetragonal) KH2PO4 and

KD

2 P04

, XI Int. Conf. Cryst. Warsaw (August, 1978).

[78] Riekel, C, Convert, P., Gobert, G., and Lazaro, C, A spherical sample container for

neutron powder diffractometry, Nucl. Inst. Methods 151 , 201-203 (1978).

Discussion

Comment (Ladell): The difference between Parrish et al . line profile analysis approach

and the Rietveld method is that the former isolates the instrumental, spectral and

abberational line broadening factors from the crystal structure factors. By using the

Rietveld approach, the least squares refinement may account for the instrumental effects

at the expense of the crystal lographic parameters since the minimization of errors to determine position and thermal parameters are clearly influenced by the approximation used

to define the profile shape. If the models used on the Parrish procedure are correct, and

his work seems to confirm that these models are correct at least for the x-ray case; the

correct procedure should be to use these models to isolate the variables before using a Rietveld procedure.

Response (Hewat): There are only four "instrument parameters", U, V, W, and F used by

Rietveld, and these are not at all strongly correlated with the crystal structure

parameters. However, there are small errors in Rietveld' s description of the line shape

which should be corrected in the manner advocated by Parrish and others.

Comment (Edmonds): (Directed to question of difference between Parrish' s profile fitting

method and Rietveld analysis, and added after Ted Prince's (NBS) comment:)

140
Rietveld' s technique applies crystal structure determining and data reduction

parameters in a least squares fit to the raw data (entire pattern) while Parrish's

suggestion would first apply data reduction techniques then least square crystal structure

parameters to the derived data (discrete hk& and I). If the same data reduction parameter

were available to both formalisms, and the entirety of the powder pattern were used, there

would be no differences in the results. However, using discrete data poses the problem of

what to do with the weak or below threshold data when multiple hkS-'s may be present, where

1 the Rietveld technique will deal with the intensity sums on a dynamic basis. In any case,

j we are able to fit structural models only by comparing calculated data to observed data,

be the latter derived or raw in nature. Forgetting this point will only magnify

artificial or superficial differences between techniques and limitations.

Response (Hewat): The techniques would not be equivalent when applied to a complete

diffraction pattern, because there would be many more parameters to determine with

I Parrish's technique. The Rietveld technique greatly reduces the correlation problems that

one would have in this case. However, the Parrish technique is valuable for improving the

: description of single lines.

I Comment (Prince): In the discussion of these papers, there seems to be a general

| misconception that there is a conflict between the profile fitting of Parrish and the

total pattern analysis of Rietveld. The Rietveld technique takes a diffraction pattern

and an assumed peak shape and determines crystal lographic parameters. The Parrish work is

a good representation of x-ray peak shape. What is needed now is the marriage of these

\ techniques for x-ray refinement.

Response (Parrish): No comment made or necessary.

Question (Rush): I have several questions related to your high resolution diffraction

: procedures at Grenoble. How do you determine and subtract background? Do you change

collimation or resolution conditions depending on the complexity of the structure? Do you

have to worry about subtle changes in line shape when you vary the incident wavelength?

Response (Hewat): The background is measured in regions where no peaks are thought to

, contribute, using the cursor on a Tektronix screen. The wavelength is chosen according to

o o the complexity expected for the pattern: usual choices are 1.91 A and 1.38 A, but

o o wavelengths as long as 3 A or 5.7 A are used for indexing unknown patterns. No changes

are possible in the collimation. The peak widths change with changes in the wavelength;

three parameters are refined to describe the angular dependence of peak width.

I

141

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

STRUCTURAL ANALYSIS FROM X-RAY POWDER DIFFRACTION PATTERNS

WITH THE RIETVELD METHOD

R. A. Young

Georgia Institute of Technology

Atlanta, GA 30332

The Rietveld method of whole-pattern fitting has now been

successfully applied to x-ray data by several groups in Europe

and the USA and others are known to be working with the method.

Two types of angle-dispersive data (diffractometer and GuinierHa'gg camera) and energy-dispersive data (synchrotron source)

have been used. Structures have been successfully refined for ~30

different materials in 15 space groups ranging from inorganic

synthetics and an organic through several minerals to human tooth

enamel. In the few cases wherein both x-ray and neutron data were

analyzed, comparable results were obtained for the atom position

coordinates. Standard deviations (a) tend to be an order of magnitude larger than with good single crystal refinements, partly

because (sin 6)A is often more limited. The R values tend to be

higher with angle dispersive x-ray data (e.g. R = 12-28 percent

P with average near 20 percent) than with neutron data (e.g. R = 5-17

percent with average near 10 percent). The extra size of the x-ray

R 's is, probably, largely a consequence of the inadequacy of the

reflection profile functions so far used (Gaussian, Lorentzian, two forms of modified Lorentzian). Voigt functions, the Pearson VII

form, an Edgeworth series, and a polynomial form are now being

tried by various groups.

The computer programs used with x-ray data have generally

been adaptations of Rietveld' s programs written for the neu- tron case, but recently at least three new programs for the

angle dispersive case have been written. Added features include

single pass operation, modular construction, multicomponent

refinement capability, and generalized coordinates to deal with

constraints.

Two general approaches have been followed by the various

workers. In one, the lattice parameters are first determined in a separate step and then fixed while structure refinement is

143
carried out, often with the goal of obtaining "observed" Bragg

intensities which are then used for the final structure analysis.

In the second approach, lattice parameters are refined simultaneously with the structure parameters and the "best" structure

results are taken to be those obtained directly from the Rietveld

refinements. Position a's are comparable. Physically significant individual temperature parameters have so far received

little attention.

Flat and false minima seem to be a particular problem,

perhaps because the model is generally inadequate, at least

in the reflection profile functions used. In initial refinements with many natural materials, there may be sufficient

substitutions at unknown sites so that the structural model

will be inadequate to the extent that false minima and the true minimum can not be distinguished by values, especially when R > ~ 20 percent.

M. J. Cooper (priv. comm.) has pointed out that the

standards deviations are ordinarily calculated incorrectly,

in principle. Comparison of results from different data

sets on the same material suggest that the true a's may be

2 to 3 times those calculated.

The various difficulties of the method are receiving

attention, however. Highly useful results are being obtained,

the Rietveld method constitutes a major breakthrough for the

usefulness of x-ray powder patterns, and activity in the field

is growing rapidly.

i

1. Reported Results

1.1. Scope

The Rietveld [1.2] 1 method of crystal structural model refinement with powder diffraction data, widely used with neutron data for some years, has now been applied with

x-ray data by several groups. In this method, the parameters in a structural model, plus

necessary instrumental parameters, are adjusted in a computer calculation until the best

fit, in a least squares sense, is obtained between the entire calculated and observed powd<

patterns. The diffraction data are obtained in digitized form through a step-scanning, or equivalent, procedure. Thus, the whole-pattern fitting is actually a fitting of calculate^ f to observed intensity at each and every point, simultaneously, in the step scan. No

a priori assumptions are made about how overlapped reflections should be resolved into i i separate Bragg reflections, and the absence of intensity also constitutes useful, and

1 Figures in brackets indicate the literature references at the end of this paper.

144

I
used, information. Figure 1 gives, as an example, a visual comparison of observed and

calculated x-ray diffraction patterns for fluorapatite after Rietveld structure

refinement.

i . More than 30 different structures, with 15 different-space groups, have now been suc- cessfully refined from x-ray data with the Rietveld method. This number is small compared

to that for neutrons (>200), but it is only two years since the first x-ray refinement was published [3]. The approximately thirty structures which have been Rietveld-ref ined from

• x-ray data, and for which we have been able to obtain information, are listed in table 1

J along with some of the key information about each. Reference to this table is implied in

the ensuing discussions of what the various research groups have done.

1.2. R factors

The various R factors listed are defined as follows, where Ig is the intensity assigned

to a particular Bragg reflection and y. is the intensity observed at the ith step in the

step-scanned pattern.

R

F = l|Vl

B ("obs") - VlB (calc)

I Vl

B ("obs")

I I B ("obs") - I(calc)

RB =

I I B ("obs")

I y^obs) - - yi (calc)

TyTTobsT

wp

1 w.(y.(obs) - \ yi (calc) )2

1 w.(y.(obs))'

In the foregoing, c is a scale factor and w^ is the weight.

In the definition of R c and R D the "obs"--for observed—has been written in quotation

r ts

,

marks because the Bragg intensities, IgC'obs"), are not actually observed but are derived

from a parceling out of the observed intensity, in a given 26 range, in proportion to the

calculated Bragg intensities, Ig(calc). Some workers appear to favor citing Rp or Rg

because they most closely resemble the "conventional" R factors still widely used in single

crystal structure analysis. Rp has the advantage that it is based on actual observations

rather than partially calculated ones. The most significant R factor, however, is Rw

because the numerator of R is the quantity that is actually minimized in the least

WP

145
146
1 squares refinement procedure. To be objective, then, and to be able to tell whether one

|i refinement model or result is better than another, one must cite R even though it is,

wp

discomfortingly, the largest of the four. Unfortunately not all authors have listed Rwp

,

! so we must look to other R's in table 1.

1.3. Different approaches

The approaches used by the various groups who have published data differ somewhat.

The Stockholm/Upsala groups have used Guinier-Hagg film data digitized with a micro-

! densitometer [4]. The monochromating crystal was set to allow only the a-^ lines of the

a doublet to pass. Typically, though not always, they refine the lattice parameters in

a separate operation and keep them fixed in the Rietveld refinements. This association

of groups is responsible for the major portion of the x-ray Rietveld refinements published to date. These groups have frequently used the Reitveld refinement as a way to

i parcel out the observed intensity to individual Bragg reflections in order to use the

| resulting structure amplitudes (|F|) in further refinements and in construction of

electron density maps.

Khattak and Cox [5] used step-scanning diffractometer data and avoided the problem

of having to handle the two wavelengths in the Ka doublet by using KB radiation. They

j also provided an interesting direct comparison of x-ray and neutron results by refining

the same structure with both kinds of data. They included lattice parameters as a part

of the Rietveld refinement.

Glazer, Hidaka and Bordas [6] took advantage of some of the newest instrumentation,

primarily as a demonstration. They used a synchrotron source and an energy dispersive

detector. In this case there was no a doublet to deal with, only one wavelength at a time, but special computer programming was, of course, necessary to deal with this type

of data. The Rp values reported are notably lower than those reported for other methods,

probably because the instrumental profile is well fit by Gaussian in this case. Both the Atlanta group and Immirzi have used step-scan diffractometer data collected

i with the use of crystal monochromatized Ka radiation in which both the and the com- ponents were present. Their refinement programs are able to handle the two wavelengths.

Both also refine the lattice parameters as a part of the Rietveld refinement.

1.4. Generalizations

In addition to the lattice and crystal structural parameters, there are, usually, an overall scale factor, an overall "temperature" factor, a 20-zero correction, and two or three reflection-profile breadth parameters to be refined. The total number of refined

parameters indicated by table 1 varies from 7 to 51. Site occupancies have been refined with seeming validity, as judged from the fractional standard deviations shown for the Atlanta group in table 1, but never simultaneously with individual temperature factors. No one seems yet to have tried to refine

anisotropic thermal parameters with x-ray data. Some individual isotropic temperature

I 147
Comment Initial

structure

from

Patterson

with

resolved

reflections,

then

Rietveld.

then

Fourier

analysis

with

assigned

F(obs).

Initial

structure

solution

Jrom

Patterson

based

on

resolved

reflections.

Impurity

lines

removed.

Four

specimens

were

analyzed.

HL

clearly

best

in

tails

but

no

profile

function

great.

Convergence

incomplete

but

main

structural

featured

det'd.

Fourier

methods

were

used.

V=1441A

3

.

s

Budapest/Stockholm Budapest/Stockholm Budapest/Stockholm

Werner

(1977)

Stockholm Stockholm Stockholm

&

Uppsala

Stockholm

&

Uppsala

Stockholm

&

Uppsala

Stockholm Stockholm Stockholm

s

Farkas,

Gado &

Werner

(1977)

Farkas,

Gado &

Werner

(1977)

Farkas,

Gado &

Waltersson,

Werner

&

Wilhelmi

(1977a)

Waltersson,

Werner

i

Wilhelmi

(1977b)

Nord

(1977)

Malmros

&

Thomas

(1977)

Halmros

i,

Thomas

(1977)

Halmros

&

Thomas

(1977)

Berg

&

Werner

(1977)

Werner,

Harinder

&

Hagneli

(1978)

Werner,

Harinder

&

Hagneli

(1979)

d

1

(J) 554S S

(5)9 6

16-20 10-30 13-30 12- >

100

UOLTM)

'pjooo

0.6-1

0.4-0. "too large' 0.5-9 0.2-4 1-4

0.5-8 0.6-9 0.6-9

1 ! (3i6u

;» •

1

s

(1,D

I £ S ? \

s>|e3d B6ejg o|j*\ 1 s

Thermal

parameters

refined

I till* i * t II

r. as

1

23.9 22.1 19.5 27.5 25.6

s = s

12.1 12.7 9.4 « £ S "

s d ;

(E 3 lqn BBS) i i i £ i * o -, £ * i i lilt I ! II! I ! ! ll-= ajn^anjis . . - B . a s s s s a

t/(9l^S) -xbm 8 5 3 S S S 3

o o o o d o d

I III 1 II

adij X2) r f f r § £ § f i

dnoj6 aaeds

(X) ^Mnd

s

I ill 1 !

. 1

- 1 fff 1 ?! J

148
I ill

I i ?

s s

5 £

i i

1 «

C S3

iiH

llij

2 5 2 S

![

,J 1 !HI ! I

i i ! is i i s I j film iiii i

1 s iiiilil i fi ..

liinii i ii in

j

sIf

1

5 J I I I I 5 ~ | ^

J I J I J l jj I i 1 f J f J

A 2_^.

§1

§

3

1-2 i

3 s „ „ S h 2

0.002 0.001 0.001 0.001 0.004

d d d d J d d

a a a a 5 a a sasassiSRgssa

i t

16.8 R ft a ti 2 a a

25.5 12.4 17.9 7.2 8.1 10

14.2 22.3 13.7 15.7 14.

7.8

12.1 21.4

d m <f m oi d

i I I I i 1 1 1 \\:n

s a s a a s „ s § SSS! 8

a a 8 Si 3

d

s

d s

d

5

d d

s

d

0.56 0.

56

0.56 0.56 0.4 0.4

1 I I 1 II 1111 1

1 f § f r

I ->s i § I I * * is 1

?!

4

<lib

I I

f i

d d d d d *

149
I I I I I I.

I I ill I fill

I I :

II I? 1! ill! f! is if i. i

f I fi I I ii fl i! if I

s 1

S 5

S 3 !!!!!! T

a—

s~ 11111 rrm i i §

* > * > *• > > *

ill

H

s i i II I I i . s j j i f

i I

j t_j__£_jj_|_i t

150

ill

IP

11!

Ill

ill!
factors were refined satisfactorily in the Khattak and Cox [5] work and in the synIhrotron experiment [6]. However, in most of the refinements based on Guinier-Hagg film

iata these temperature factors tended to become negative. The Stockholm/Upsala authors

ave often mentioned this in their papers and suggest it is due to uncorrected absorption.

Standard deviations in the coordinate parameters for the various atoms in the refine-

-4 -2 -3

ents listed in table 1 range from 10 to 10 of the unit cell dimension with 10 being

verage. For single crystal x-ray work, one would probably expect an average of about

ine-tenth of this, i.e. a standard deviation of about 0.0001 of the cell dimension.

j Except for the unusual case of the synchrotron experiment, the Rp values run from

jbout 12 percent to 27 percent with an average around 20 percent. There is little or

So correlation between Rp values and the number of parameters adjusted. This average

1 value is about twice that for the neutron cases, which range from 5 percent to 17

prcent with an average near 10 percent for the 170 structures reviewed by Cheetham

pd Taylor [7].

I 2. Computer Programs for Rietveld Analysis of X-ray Data

Most of the results indicated in table 1 have been obtained with one or another modification of Rietveld' s [2] program written for the neutron case. This fact is indicated

n table 1 by the letter R as the first symbol in the "Program" column. By arranging

xperi mentally to have only one wavelength present, both Malmros and Thomas [3] and Khattak

id Cox [5] avoided the need to make the program handle two wavelengths. The "RHVM" [8], on

le other hand, does handle explicitly the two wavelengths of the a doublet. The ratio of

densities of the two wavelengths (via scale factors) is one of the parameters refined,

lewatt modified Rietveld 1 s program to handle anisotropic thermal parameters and von Dreele

id Mackie then modified that program to handle the x-ray case, hence the designation RHVM.)

)th the RMT and the RHVM programs have been distributed to more than a dozen groups in many

juntries.

Recently, several programs have been written specifically for the angle-dispersive

(ray case, quite independently of the Rietveld program. Table 2 gives some comparison

features for those program about which we have enough information to say anything,

i the table, a blank at an intersection means that we have no information about that

Mature for that program. A dash means that the implication from information we do

five is that a "No" belongs there, but our information is not explicit. A question

irk means that the implied answer is "Yes" but, again, we do not have explicit

liformation. Although a multiphase capability is roundly desired and many workers intend to

iplement one, the RWSMT program [9] modified from the original Rietveld one, seems

j> be the only reported success to date.

' The DBW(II) program (D.B. Wiles, version II, see Wiles & Young [10]) has been

^signed for user convenience and for easy use by groups having access to any of the

iandard large computers. It is written to ANSI code standards for portability, is of

151
Table 2. Some Computer Programs for Rietveld Analysis of Angle Dispersive X-ray Dat

FEATURE PROGRAM

RMT RHVM RKC RWSMT AI DBW(II) BH

All space groups "as is" - No No -

? Yes Single pass operation - No No -

? Yes -

Modul ar - No No Yes

ANSI - No No -? Yes

a doublet No Yes No No Yes Yes No Multiphase capability No No No Yes No No ~ Constraint functions Yes Yes Yes Ye Generalized coordinates No No No No Yes No No Profile functions ML G,L G,ML,L,IL ML PVII G,L PC Refines preferred

orientation parameter Yes Yes Yes Ye

1

Refines asymmetry parameter Yes Yes Yes y|

Refines lattice parameters - Yes Yes - Yes Yes Ye R-factors in various shells No No Yes No N( Spt unU yJ list IIOU for 1 \J \ anothpr Ql IU Ul IC 1 run 1 UN No(?) Yes

1 Plotter output Yes Yes Data type GH 6-26 6-26 GH 6-26 8-26 a- Used by other groups Yes Yes No Anomalous dispersion Yes No Yes l

if

modular construction to facilitate changes, and handles all space groups without additional programming. (Only a space group code is required in the input.) It incorporates a large data base of atomic scattering factors and anomalous dispersion correction

The AI (A. Immirzi [11,12]) program has been written particularly with complex

structures in mind, e.g. molecular crystals, polymers, fibrous structures, in which a great deal of information may already be available about multiple-atom structural units

In such cases, the total number of parameters can be greatly reduced if the a priori

information is entered into the programs as constraints. Thus, Immirzi has designed

his program to allow treatment of any kind of constraint with the writing of "only a very small ad hoc routine." In an application to a molecular complex of pyrene and

tetracyanoqui nodi methane he used D2^ molecular models and refined three rotation angles

per model. He reports a convergence neighborhood of^at least 15° and good agreement wi

single crystal results.

Another unusual feature of the AI program is the use of the Pearson VII profile

function, to which we return later in this paper.

j 152
The BH (Barlocher and Hepp [13]) program is designed to make use of weighted con- straints, both holonomic and non-holonomic, and the basic reflection profile to be used

I programmatical ly determined by least-squares fitting of a resolved reflection by a articular polynomial expression (see the following section).

Four of the seven programs in table 2, including all of the three which are indesndent of the Rietveld program, have been completed within the last year. One hears

f others being written. Clearly, the field of x-ray Rietveld analysis will not long be

anting for a selection of good programs.

3. Profile Functions

While Gaussian functions fit neutron powder diffraction profiles well, such is not

le case for x-ray powder diffraction profiles (Bragg reflections). As was cited in

;ction 1 on the basis of table 1 plus the Cheetham and Taylor [7] review, Rp values typiiv

tiont

illy run nearly twice as large for the x-ray cases as for the neutron cases. The reason

| primarily the inadequacy of the profile functions used. The actual reflection profiles

e neither Gaussian (G) nor Lorentzian (L). In fact, as can be seen from the Atlanta

oup's quartz work cited in table 1, how these two compare for poorness of fit depends

i the broadening of the diffraction profiles. Further, the different R factors increase

id decrease together in the Khattak and Cox [5] work and the Malmros and Thomas [3] work

i table 1, but this is not true for the Young, Mackie and von Dreele work [8] on quartz,

i which two specimens were studied in which differing degrees of crystallite size and

;rain broadening were present. By the criterion that R^ is the really significant R facr, the ML (modified Lorentzian) function is better in the Malmros and Thomas [3] work but

t as good as either the IL (intermediate Lorentzian) or L function in the Khattak and Cox

] work. X-ray emission spectra are Lorentzians but in the angle-dispersive x-ray methods,

ie actual instrumental resolution functions and intrinsic diffraction broadening functions

e not. It is easily realized that, until better profile functions can be implemented for

ray Rietveld analysis, much of the structural detail, that which makes only small differces in R , will stay out of actual reach of the method; an important part of the method's

wp

tential will not be realized.

Several pragmatic suggestions which have been made for replacement of the Gaussian and

rentz functions are listed in table 3. Malmros and Thomas [3] and followers have used the

dified Lorentzian (ML), while Khattak and Cox [5] found the intermediate Lorentzian (IL)

be preferable in their work. Immirzi [11,12] has suggested the "Pearson VII" function

d has incorporated it in his computer program (table 2) because, with appropriate choices

m, it varies from Lorentzian (m = 1) to Gaussian (m = <»). In principle, one could refine

e parameter m as a part of the Rietveld refinement. In fact, Immirzi 's program provides

r this. But he notes (priv. comm.) that, even in a test case with simulated error-free

ta, m underwent very large oscillations in the process of convergence. With actual data,

en, convergence might be denied. Clearly, this is an interesting function to explore,

th by refining m and by fixing it at different values, and we will no doubt be seeing

re of it in future work.

Igg
Table 3. Profile Functions

Symbol Name Function: I(X) = Reference

G

L

ML

IL

Gaussian

Lorentzian

Modified Lorentzian

A e (-x

2 /k

2

)

A (l+k2x

2

) _1

A (l+k

2 x

2 )" 2

2 2-15

Intermediate Lorentzian A(l+k x )

'

Pearson VII

Voigt

Poly Polynomial

iViiiV^i i

A / L(x') G(x-x')

1+4 (2 -1) -m

Malmros & Thomas [3]

Khattak & Cox [5]

Immirzi [ll][Hall

et al, 1977, J. Appl

Cryst. 10, 66.]

(OS

Hi

If

jie

iff

dx'

P and Q are polynomials with

even exponents, only.

Langford [14]; Suortlr icii Ahtee and Unonius [11

Barlocher and Hepp [

NOTE: In the above expressions:

x is distance from Bragg peak position, e.g. 29.-26^

A is the required normalization factor

k is a constant related to profile breadth

Another function that has been suggested [14] because it is intermediate to Gaussi

and Lorentzian functions is the Voigt function. Langford [14] has carried out the indi

convolution, via Fourier transforms, to obtain the two parameters (essentially the inte1

breadths of the component Gaussian and Lorentzian function) to be adjusted in the refin

process. Suortti , Ahtee, and Unonius [15] have tested the function by applying it to b x-ray and neutron data on Ni powders. They found it was a substantial improvement over Gaussian function for the neutron case but not so satisfactory for the x-ray case. It better than either Gaussian or Lorentzian, but still did not satisfactorily lead to det

nation of the average crystallite size and strain values, as had been hoped. Nonethele

apparently would lead to lower R values in other cases, and therefore better detail i wp

structure refinement, and hence should be explored by other workers.

The most recent suggestion for a new profile function is that by Barlocher and Hep

[13], the polynomial expression in table 3. They do a least squares fitting of this

function to a resolved profile, then store the result and modify it as need be to take

account of the dependence of breadth on 26. They report good results but no quantitati;

details were available at this writing.

154
I Another potentially interesting profile function, not yet tried in Rietveld analyses,

combination of Lorentzians such as has been successfully used in pattern decomposiTaupin [16] represented the x-ray reflection profile as a series of Lorentzians,

- |lly 3 or 4, in a markedly successful automatic method for peak determination and

[ration. Huang [19], Parrish [17], Parrish, Huang and Ayers [18], and Parrish have

ja composite of seven Lorentzians (three for a^, three for a^, and one for a^) in

^posing patterns into component (overlapping) reflection peaks. Probably the threehtzian representation could accommodate diffraction broadening better than a single

9 ML, or- IL function if the breadths, relative heights, and separations of the three

itzians were made adjustable variables for each new pattern.

I A function with a better theoretical basis, but one which has not yet been tried

jn'mental ly, has been suggested by Wilson [20]. Noting the variety of contributions to

[Jiffraction profile broadening, and drawing on the central limit theorem, he suggested

| some complex of Gaussian functions with skewness and kurtosis might be physically

bnable and appropriate. He has, therefore, suggested the Edgeworth series, in which the

Ificients have the added advantage of being related directly to the moments, m, of the

le. The Edgeworth series may be written as

: Kx) = <Kx) - |r v

x ^"(x) + |r v

2 + If i\ <t> vi U) + .

,( X) = -Jl_ e"*

2/2

V-i = kg a - coefficient of skewness

-4

^2 = k^ a = coefficient of excess (curtosis)

k^ = m^ = m

2 2 ^2 ~ m

2

" m

l - °

3 ^3 ~ m

3

~ ^m

im

2 + 2m

i

2 2 2 k^ = m^ - 3ni2 - 4m

1m3 + 12m

1m2

- 6m^

is expressed in units of a, the square root of the variance.

\ny profile whose moments are all finite can be represented by such an Edgeworth series

In using it with Rietveld analyses, one would hope that very few terms would

peded.

In conclusion to this section, one may say that while it is true that better profile

Lions are needed, especially more fundamentally based ones, several already exist which

j not been thoroughly evaluated.

155
4. Characteristics of the Refinements

4.1. Convergence range

More than a hundred Rietveld refinements have now been carried out in our laboratory

with x-ray data. Our general experience has been that, compared to single crystal analy)

the starting model has to be more nearly correct or the Rietveld refinement will not "go

Other workers have also remarked orally to us upon this characteristic. One normally

proceeds in steps, first refining only one or two parameters (e.g. overall scale factor)

and then gradually letting more and more of the parameters be varied in the successive

least-squares refinement cycles. That is to say, the range of convergence is not as great with Rietveld refinements as it is with single crystal structure refinements done

with, for example, the widely used Busing, Johnson, Ellison, Thiesen and Levy [22] program. This is, of course, related to the prevalence of false minima discussed in the next section. Rather than being surprised at the limitation on the convergence range,

perhaps one ought to be surprised that the refinements converge to meaningful results at all--the F(hk£) data are badly superimposed by multiplicity, reflection overlap produces

more superimposition and partial reduction of the effective (sin 8/A.) range (Young,

Mackie and Wiles [23]) the general signal-to-noise level is poor compared to that for

most single crystal data, and background intensities cannot be well determined independently of the refinement (except in such simple cases that Rietveld refinement is not needed anyway). The use of two wavelengths at once (e.g., the a doublet) may make

matters worse by superimposing two complete patterns— although the additional data shoul

improve the statistics and the simultaneous fitting of the calculated patterns to two

sets of observed data (one with each wavelength) can also be regarded as a valuable

internal consistency requirement.

4.2. "Flat" and false minima

"Flatness" of the minima in Rietveld refinements is another characteristic about

which many users comment privately. "Flatness" is indicated by relative insensitivity f

R (for example) to parameter shifts greater, or much greater, than the standard deviawp

tions in them. Such insensitivity is often noted. Since the R^ is held rather high

(compared to good neutron cases) because of model inadequacies such as in the profile

functions, because of the background assessment problems, and because multiplicity and

reflection overlap cause the number of effectively independently determinable Bragg

2~ to be much smaller than it is for the single crystal case1 F hk£

reflection intensities

"flatness" of the minima must be expected. Further, even when better profile functions

are used, some "flatness" must be still accepted as an intrinsic characteristic of struc ture refinement with powder diffraction patterns.

False minima also seem to be more prevalent with Rietveld refinements than with

single crystal refinements. Table 4 gives an example with fluorapatite (Ca^PO^gF).

Refinements A and B were carried out with the same data and program; the only difference

156
Table 4. Fluorapatite (P6

3 /M)

(All Parameters x 10

3

)

itor

lalj

Ca(l) Ca(2) P 0(1) 0(2) 0(3)

Z ' Y XY XY XY X V

t A 014 246 992 402 356 416 524 640 484 363 276 032

t B 001 246 992 399 369 328 485 587 465 344 258 070

A 0^ 245

Q

991

Q

403

Q

363

Q

398-,^ 513

1 652 1 490 1

342-,^ 262

Q 04^

B 002

Q 243Q 992Q 398Q 367 Q 326Q 488

Q 589-L 467Q 341Q 260Q 066 Q

Site-Occupancy Parameters

Ca(l) Ca(2) P 0(1) 0(2) 0(3) F a(A) C(A) Rwp(%)

t A 263 500 483 440 294 940 250 9.3653 6.8826

ft B 270 477 492 280 580 940 255 9.3653 6.8826

A 353

2 5943 4573 504

g 461. 940- 1592 9.3664 6.8816 23.0

B 299

x 454

x 429 £ 4182 425 2 940- 147

1 9.3667 6.8821 15.5

(It!

id B used same data and program,

ycles; all final A < a.

in the starting parameters used. The initial A and B locations for 0(1) differ by 0.72

o o

id the refined positions by 0.59 A. For 0(2) the differences are 0.43 A initially and

p o A finally. Thus, choices of atom starting positions that differ by less than 1 A have,

i, led to refined ("minima") positions differing by about one-half angstrom, which is

| than 50 times the calculated standard deviations for the atom positions.

In this fluorapatite case, the false minima could be distinguished from the true

|mum on the basis of R (23 percent vs 15.5 percent), but that is not always possible.

>rapatite is a very nearly stoichiometric, well crystallized material and its structure

cnown in great detail. [The final position parameters for refinement B in table 4 do

ie with the single crystal results [24] within 3 a and mostly within 2 a except for

i(l)), for which the A result happens to be the single crystal result.] Most materials

which one would wish to use the Rietveld method will not be stoichiometric ones with

Jctures well known in detail. The structural model will, at first at least, have signifit deficiencies and as a consequence the R factors will tend to be higher than they were

the B refinement of fluorapatite. Table 5 gives an example of a false minimum in the

157
Rietveld refinement of human tooth enamel [25]. The final positions for 0(1) and 0(2)

o o differ by 0.51 A and 0.43 A, respectively, in the two cases (A and B). In this case, it not possible to tell from the R values which, if either, of the minima is the true one. wp

Because both R values are as high as that for the false minimum with fluorapatite, one wp

might try to assume that both A and B refinements represent false minima. But that is no permissible, either. At this stage, the TE structure refinement is still preliminary ano;. some moieties known to be present, e.g. a few wt% of C0

3 and H2

0, are not included in the. model because their locations are not known. One would like to know which of the refineir A or B, is the better one to take as a starting point for adding in these moieties in tri|

locations in the structural model to be refined. The difference in R between case A an wp

is not obviously large enough to permit such a judgment and the familiar Hamilton [26] R- test is not applicable because the observations are not directly of independent Bragg int. ties. Thus, one can only conclude that, at least when the R^ values are still as high a

Table 5. Human Tooth Enamel on Hydroxyapatite Model (P6,/M)

3 (All Parameters x 10 )

Ca(l) Ca(2) P 0(1) 0(2) 0(3)

\ Z X Y X Y X Y X Y X Y

Start A 014 249 989 402 356 416 524 640 484 363 276

| 03

Start B 009 250 985 406 362 323 487 613 476 357 282 05

End A 009

Q

247

Q

988^^ 400

1 357

1

406^^ 517

1 6422 4722 366

1 2781

[ 04

End B 003

1 250Q

994

1

402^^ 359

1 3321 484

2 5891 4S01 3641 277

Q

os|

End C 002

1

249-,^ 993

1 4031 3581 326

1 483

1 5931

451-j^ 353.,^ 264

1 06

»5

"it

Site-Occupancy Parameters

T

ia

sa

sa

)K

ir] Ca(l) Ca(2) 0(1) 0(2) 0(3) X-I0N at Z = c(A)

0.05 0.15 0.25

Rwp(%

Start A 263

Start B 280

500-

365

483

404

440

307

294

416

976 169

1000- 125

423

125

012

125

9.4415

9.4415

6.8964

6.8964

c

End A 302

2 500-

End B 239

2 3892 End C 298

3 419

3

A and B: same data.

494

4 592

g 281g 12002 412

3 2575 4695 1000-

413

4 364? 433 y 1000-

199

6 40013

048

8

9. 4406 6 8761 271 162

4

345

10 034g 9. 4417 6. 8770 25

091

6 18614

066

9

9. 4451 6 8813 25

C: Separate data, same material, same initial parameters as B. Eight cycles, all A < a. t 53
25 percent, false minima can occur relatively nearby true ones and cannot be effectively

ecognized on the basis of R values.

It may be that, if the second order derivative term in the shifts were implemented

the computer program, which it is not2 in those we have used, the convergence neighborhood might be significantly increased and false minima would be less prevalent. This

ems to be a possibility that should be explored in order that the effectiveness of the

iietveld method can be maximized.

4.3. Standard deviations

The calculated standard deviations seem to be physically reasonable in some cases,

it in others they are too small.

Two examples of reasonableness are these:

(1) Rietveld results and single crystal results frequently agree within 3 a or less,

g. for fluorapatite (cf. table 4 with Sudarsanan, Mackie, and Young [24]) and for quartz

oung, Mackie and von Dreele [8]).

(2) Rietveld refinements were carried out with 29 sets of data collected at different

mperatures (25-500 °C) on the same TE specimen. With few exceptions, all 29 values of

ich positional parameter were the same within 2 o [23]. (Experimentally, this was someing of a disappointment because we were hoping to see some dramatic changes produced by

ating.

)

In the fluorapatite example, the refinements were done with the DBW program (table 2)

d the others were done with the RHVM program. Because of an internal difference the RHVM

ogram calculates larger standard deviations. (Factors of 1 to 3 have been observed.)

There are also cases in which the calculated standard deviations seem to be too small,

finements B and C in table 5 illustrate one. The data came from specimens taken from

e same large and presumably homogenized supply of dense-fraction human tooth enamel,

e same starting parameters and program (DBW) were used for both refinements. But the

fined positional parameters differ by more than 3a in 5 instances and nearly 10 in one. early, if the calculated a's were 2 or 3 times as large, we would probably conclude that

e positional parameters did not differ significantly between the two refinements.

Dr. Martyn Cooper [27,28] has pointed out that the calculations of standard deviations

jne in most or all current Rietveld refinement programs are wrong in principle. A central

rt of this point is easily grasped. The usual way to calculate standard deviations is as

he calculated parameter- shift term formally given by

I 2w

i tyoi c yci J 8Xjax

k \dx.

J

I ax

k J

^re Xj, x^ are the jth and kth parameters being varied, is approximated by dropping

I second order derivative terms.

159
2 -l

1 w-j (y^ Cobs) - -12y^calc))

CTj = A

jj FP

where y. is the intensity observed at the ith step in the scan, A. \ is the (diagonal)

J J element of the inverse normal equations matrix, P is the number of parameters adjusted,

and N is the number of independent observations. Most of the parameters are crystallographic and depend on F(hki>) but not on the reflection profile shape or position. The

value used for N is usually the number of steps in the scan, which far exceeds the num-

| ber of independent F(hk£)'s. Thus, the denominator is made too large, and a too small,

by this usual procedure. Is there any reasonably practical way to treat separately the

two kinds of parameters and observations, Bragg intensity and profile shape and position

in order to calculate a's correctly in principle? Perhaps the work of Dr. Cooper or others will tell us. If not, we will have to rely on the accumulating experience of man workers to learn some kind of recipe for the factor by which the calculated a's are too

small under what conditions.

5. Conclusions

I

Application of Rietveld analyses with x-ray data lagged by several years the appli- ! cation with neutron data. Probably because the x-ray reflection profiles are less well

represented by simple functions, x-ray results to date for Rp are typically about twice,

those for neutron results although standard deviations on positional parameters are

comparable. Several all-new computer programs with various special virtues have

recently been written for x-ray Rietveld analysis. The problem of appropriate repref sentation of the reflection profiles is receiving attention. Flat and false minima

problems do limit the straightforward application of the method somewhat, but there

is reason to hope that these problems can be reduced with the use of improved profile

functions and algorithms. The calculated standard deviations are often too small,

perhaps by a factor of 2 or 3. This problem is receiving attention.

Nonetheless, x-ray Rietveld analyses have already produced highly useful structural results for a number of materials which are simply not available, or for other

reasons cannot be studied, in single crystal form. Human tooth enamel is one of them.

The pace of activity in x-ray Rietveld analysis is picking up rapidly now and, because

x-ray beams are much more widely available than neutron beams, one can anticipate that

the x-ray activity will soon exceed the very substantial neutron activity.

1,

It is a pleasure to thank Dr. J. 0. Thomas for keeping me informed of the work

of the Stockholm/Upsala group at the manuscript stage, Dr. A. M. Glazer for keeping me similarly informed about the work of the Oxford and associated groups, Dr. A. Immirzi

160
ice

irzi

or supplying extensive information on his still unpublished program and test work with

, Dr. Martyn Cooper for alerting me to the in-principle problem with standard deviaons via a draft manuscript and "discussions" of unpublished work, Professor A. J. C. lson and many other colleagues here and abroad for discussions and correspondence,

. Barlocher for an advance copy of his abstract for this conference, my colleagues

. P. E. Mackie and D. B. Wiles for assistance and permission to quote unpublished

ork, and the USPHS for financial support through NIH-NIDR Grant DE-01912.

References

1] Rietveld, H. M. , Line profiles of neutron powder-diffraction peaks for structure

refinement, Acta Cryst. 22, 151 (1967).

2] Rietveld, H. M. , A profile refinement method for nuclear and magnetic structures,

J. Appl. Cryst. 2, 65-71 (1969).

3] Malmros, G. and Thomas, J. 0., Least-squares structure refinement based on profile analysis of powder film intensity data measured on an automatic microdensitometer, J. Appl. Cryst. 10, 7-11 (1977).

4] Malmros, G. and Werner, P. E. , Automatic densitometer measurements of powder

diffraction photographs, Acta Chemica Scandinavica, 27, 493-502 (1973).

5] Khattak, C. P. and Cox, D. E. , Profile analysis of x-ray powder diff ractometer

data: Structural refinement of La

Q y^SrQ 25

Cr®3' J ' APP^- Cryst. I9_» 404-411

(1977).

6] Glazer, A. M. , Hidaka, M. , and Bordas, J., Energy-dispersive powder profile refinement using synchrotron radiation, J. Appl. Cryst. 11, 165-172 (1978).

7] Cheetham, A. K. and Taylor, J. C. , Profile analysis of neutron diffraction data:

Its scope, limitations, and applications in solid state chemistry, J. Solid State

Chem. 21, 253-275 (1977).

8] Young, Mackie and von Dreele (1977).

9] Werner, P. E. , Salmone, S. , Malmros, G. , and Thomas, J. 0., Quantitative analysis

of multi component powders by full-profile refinement of Guinier-Hagg x-ray film

data, J. Appl. Cryst., in press (1979).

0] Wiles, D. B. and Young, R. A., Improved computer programs for Rietveld analysis

of x-ray powder diffraction patterns, American Crystallographic Association Program

and Abstracts, 6, 56 (1979).

1] Immirzi, A., Profile fitting refinements using generalized coordinates, Acta

Cryst. A34, S348-S349 (1978).

2] Immirzi, A., (unpublished work, 1979).

|3] Barlocher, Ch. and Hepp, A. , A new pattern fitting structure refinement program

for x-ray powder data (1979). This conference.

4] Langford, J. I. , A rapid method for analyzing the breadths of diffraction and

spectral lines using the Voigt function, J. Appl. Cryst. 11, 10-14 (1978).

! 161
[15] Suortti , P., Ahtee, M. and Unonius, L. , Voigt function fit of x-ray and neutron

powder diffraction profiles, Report series in Physics HU-P-163, University of

Helsinki, 16 pp. (1978).

[16] Taupin, D. , Automatic peak determination in x-ray powder patterns, J. Appl

Cryst. 6, 266-273 (1973)

[17] Huang, T. C. and Parrish, W. , Accurate and rapid reduction of experimental x-ray

data, Appl. Phys. Lett. 27, 123-125 (1975).

[18] Parrish, W. , Huang, T. C. , and Ayers, G. L. , Profile fitting: A powerful method

of computer x-ray instrumentation and analysis, Trans Am. Cryst. Assoc., 12,

55-75 (1976).

[19] Parrish, W. and Huang, T. C. , Accuracy of the Profile Fitting Method for X-Ray

Polycrystal 1 ine Diffractometry, this volume.

[20] Wilson, A. J. C. , Note on the aberrations of a fixed-angle energy-dispersive

powder dif fractometer , J. Appl. Cryst. 6, 230-237 (1973).

[21] Cramer, H. , Mathematical Methods of Statistics (Almquist and Wiksells, Uppsala,

1945), pp. 228-229.

[22] Busing, W. R. , Johnson, C. K. , Ellison, R. D. , Thiesen, W. E. , and Levy, H. A.,

World list of crystallographic computer programs, 0RXFLS3, Accession No. 84,

J. Appl. Cryst. 6, 309-346 (1973).

[23] Young, R. A., Mackie, P. E. , and Wiles, D. B. , Rietveld pattern-fitting

structure-refinement ('Profile Refinement') with x-ray powder diffraction

data, Acta Physica Polonica, in press (1979).

[24] Sudarsanan, K. , Mackie, P. E. , and Young, R. A., Comparison of synthetic and

mineral f 1 uorapatite, Ca^PO^^F, in crystallographic detail, Mat. Res. Bull.

7, 1331-1338 (1972).

[25] Young, R. A. and Mackie, P. E. , Crystallographic studies of human tooth enamel:

Initial structure refinement, submitted for publication (1979).

[26] Hamilton, W. C. , Significance tests on the crystallographic R factor, Acta Cryst.

18, 502-510 (1965).

[27] Cooper, M. (private communication).

[28] Sakata, M. and Cooper, M. , An analysis of the profile refinement method, Report

MPD/NBS Bibliography/95, AERE, Harwell (1978).

Bibl iography

Kajzar, F. , A profile refinement method for determination of the contribution of

rhombohedral structure in graphite, Rap. - Inst. Fiz. Tech. Jad. AGH, Issue

INT 112/PS, 32 pp. (1976) (Abstract in CA).

Buras, B. , Gerward, L. , Glazer, A. M. , Hidaka, M. , and Staun Olsen, J., Quantitative structural studies by means of the energy dispersive method with x-rays

from a storage ring, to be published (1979).

162
Discussion

omment (Sabine): I think that you should not state that the Rietveld method understimates standard deviation. All that is known is that the two methods give different

easurements in some cases.

jesponse (Young): The full text of this presentation cites examples in which the e.s.d.'s

Ippear to be physically reasonably larger as well as examples in which they are smaller

han the actual observed lack of reproducibility. However, the RHVM program has an

nternal feature which causes it to produce e.s.d.'s larger by a factor of ~2 than does

he DBW II program. Unfortunately, the RHVM was used for the examples in which the

.s.d.'s were realistically large and the DBW II program was used in the examples in

jhich the e.s.d.'s could be seen to be "too small." Thus, I am afraid, we do not yet

|ave unambiguous examples of the e.s.d.'s being as large as intercomparisons suggest

|hey should be.

uestion (Snyder): Given the high degree of precision obtainable from this method, what

s your estimate of the degree of accuracy of the lattice parameter and how are the

any systematic errors allowed for?

i

esponse (Young): We have made no effort to assess accuracy, only precision. However,

he lattice parameter results we have obtained in Rietveld analysis of fluoroapatite,

or example, compare well (within 3o) with those obtained with single crystals and with

ther powder methods. The only systematic error currently allowed for is a 20 zero

rror. It should not be difficult to incorporate a specimen displacement error correcion also, but the small estimated standard deviation we are getting suggests that such

correction, if incorporated, would usually be small for the rather strongly absorbing

t aterials we have been studying.

163

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

A NEW PATTERN FITTING STRUCTURE REFINEMENT

PROGRAM FOR X-RAY POWDER DATA

Ch. Baerlocher and A. Hepp

Institut fur Kristal lographis und Petrographie

ETH Zurich, Switzerland

An entirely new program package has been written for the profile

fitting method, designed specifically for x-ray powder data. It uses

a novel function for the peak shape which gives excellent agreement

between observed and calculated profiles. Additional information such

as known interatomic distances and angles can be readily included as weighted constraints and/or their expected ranges imposed as boundary

conditions. This improves greatly the convergence and it becomes also

possible to refine structures with up to 100 or more positional parameters. To achieve the necessary resolution, diff ractometer data are

used with strictly monochromatic a x radiation.

In a first step the true background is determined and 20 and Lp

corrections are applied. All possible reflections (excluding extinctions) are generated from the lattice parameters and assigned to the

measured profile steps. Next, the function which approximates best

the form of a single, non-overlapping peak is determined with a special

least squares procedure. This function has essentially the form

,- const. r , A20-,

F = —p

• [1 " -Q-] ,

where P and Q are polynomials having only even exponents. All starting

parameters are estimated by the program. The final function is normalized, its values stored and then used as a standard in the subsequent

profile fitting structure refinement program. This employs a variable

metric algorithm, which efficiently handles the above mentioned con- straints and boundary conditions by a projection method. The algorithm

is also very stable towards high correlation. Apart from structural

parameters, the program refines lattice constants, linear 26 correction,

half width as a function of 28 and hk£ (if necessary), preferred

orientation, and profile asymmetry. The input for the programs is kept to a minimum and is mostly in free format. Examples will be

shown.

165

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

THE DETERMINATION OF STRUCTURAL PARAMETERS AND THEIR

STANDARD DEVIATIONS FROM POWDER DIFFRACTION PATTERNS

in

M. J. Cooper, M. Sakata 1 and K. D. Rouse

Materials Physics Division

AERE Harwell, 0X11 ORA, England

An analysis of the Rietveld Profile Refinement Method has shown

that statistical correlation, which must occur for points on the same Bragg peak, introduces bias into the estimates of the structural parameters and also results in the standard deviations being determined

incorrectly. Because of this correlation it is not valid to derive the

standard deviations for the profile and structural parameters in the same way, since the structural parameters depend only on the intensity

of the Bragg peaks and not their shape. This has been confirmed by the

analysis of a number of simple neutron diffraction patterns using both

Profile Refinement and integrated intensity methods, and in some cases

the standard deviations were found to be underestimated by the Profile

Refinement Method by a factor of at least 2, compared to those given by

an integrated intensity method, although overestimation is also possible.

Our analysis indicates that all standard deviation values which have

been derived by the Rietveld Profile Refinement Method are wrong and no simple modification of the method is possible to overcome this problem. An alternative approach has therefore been formulated in order to provide

reliable estimates of the structural parameters and their standard deviaI I tions, and results of its use are presented.

1. Introduction

The Rietveld Profile Refinement Method [l] 2 is now extensively used for the analysis of

wder diffraction data. The method was first developed some ten years ago but, until recent-

> , no adequate theoretical understanding existed of the results which it gives. It is the

rpose of this paper to outline the results of a recent analysis [2] aimed at providing such

n leave from Nagoya University, Nagoya, Japan.

igures in brackets indicate the literature references at the end of this paper.

167
a theoretical understanding, at least for simple cases, and to indicate how the analysis o powder diffraction data might be developed in order to provide more reliable results.

2. Structural Parameter Values

The Rietveld method involves the minimization of the function:

M

P =

f w

i - z

k y i,k>

2 <w

where y. is the observed background-corrected intensity at a single point on the diffracti

profile and w. is the statistical weight associated with that observation, yi

k is the co tribution from a single Bragg reflection to the theoretical intensity at that point and is

given by:

y; >k =Ki k (k,p

c )Gi)k (i,k,p

p

) (2)

where K is a scale factor, I k is the calculated Bragg intensity and G. ^ is a normalized p shape function, which for neutron diffraction is usually taken to be a Gaussian. The

refinable parameters can be divided into structural and profile parameters and it is usua" assumed that I k is a function of the structural parameters (p

c

) only and G.

k is a functic

of the profile parameters (p ) only, as indicated in equation [2]. As we shall see, the s P ration of the two types of parameters in this way has an extremely important consequence 1 the results given by the method.

The easiest way to visualize the significance of the results given by the method is 1 compare them with those which would be obtained from a conventional least-squares analysi;

the integrated Bragg intensities. For a diffraction pattern with overlapping peaks the

minimization function M becomes:

P

m

p =! w

t (yi " K ?

r

k Gi,k>

2 (3)

whereas that for an integrated intensity analysis would be:

M

i = I

k \^i,k- K v

2

< 4)

9M

p 9Mj

The values of and , where x is a structural parameter, are thus:

oX dX

9M 31. "85 =

* ^k <*i - ^

! k ^k> "ax"

< 5 >

168
3M 31

is =_2KI

k w

k

( f

y M~ KI K^ ^

In general, therefore, the two methods will not give identical values for the structural

parameters, although it is not clear how they will differ.

In order to clarify the situation we have therefore considered the more simple case when the peaks are resolved. In this case the minimization functions can be written as:

K J

where j represents a point on a single Braggs peak and the summation over j can be carried

out before the summation over k. 3M

p 3Mj

The values for -s- and —r— are then: dX dx

3M

p 31. -^ = -2KIIw.

)k G.)k (y.k -KI k Gj>k)^ (g)

k j

3M 31 ^ = " 2K ' W

k (Zyj,k- KI

k k>^

^ j

These again show that the values of the structural parameters will not, in general, be

identical. However, it is now possible to derive fairly simple relations between the weighting schemes which would result in the derivatives being equal and thus give identical values

for the structural parameters. Two particularly simple cases are:

i "j,k = VGa,k < n>

w

j,k

= 1/y j,k andu

k =1/^j,k < 12 >

The latter is that appropriate to a case and zero background (B..=0) which is, of course,

rather unrealistic. That given by equation [11] shows that the weights would have to be

increased for points on the tails of the Bragg peaks. This is indicative of the different

statistical bias introduced in the two methods, but does infer that the values given by the

profile refinement method may possibly be more appropriate estimates of the structural

parameters, although they may well not differ significantly from those given by an integrated

intensity analysis.

169
3. Standard Deviations

We must also consider the standard deviation values given by the profile refinement

method. These are derived from the minimization function through the standard equation:

2 -1 M

CTi = A

il 1TP < 13 >

where is the appropriate diagonal element of the inverse normal matrix and N-P is the

number of degrees of freedom (N is the number of statistically independent observations an<;

P is the number of independently varied parameters).

Again, comparing the results with those for a conventional integrated intensity analysis we see that the elements of the normal matrix, which are given by:

9 8(K I. ) 3(K I. )

8(K I ) 8(K I ) KM k p q

and hence the values of A.., will not usually be the same. We can define an agreement factor, given by

A-C^Tp) 2 (16)

the values of which are:

A

P = { f

Iwj,k Aj,k^N p- p p- p c) < 17 > K J

A

B = & Wk Ak* /(NB

" PC> (18)

where A. . = y. . - y'. . , A = I A. . , N D is the number of points in the profile, ND is th JjKJjKJjKKjJjKr

number of Bragg reflections and P

p and P c are the number of profile and structural paramet

respectively. Again, these will be different and it is not clear how the values of the

estimated standard deviations will differ.

However, it is at this point that we must consider the significance of the fact that

we can separate the observed intensity into two factors, each of which depends on one type

of parameter only. In particular 1^. has an identical value for each point on the same Bra

peak, so that the values of A. , are in fact correlated over each peak. This is illustrat

J > K in figure 1 for a peak for which A^ is positive, giving a positive mean value for the i

170
II

individual A. . values. This correlation between a sequence of A. . values in fact invaliJ > K J , K dates the use of the agreement factor Ap in the derivation of the standard deviations.

GJ.k\

Observed

Gaussian

Calculated

Gaussian

Figure 1. Illustration of the correlation between A. . values for the same Bragg peak. The observed points are distributed randomly about the observed

Gaussian rather than the calculated Gaussian, so that the differences which

must be considered statistically are A. and A. . -G. .A. rather than A. . . K J>KJ,KK J,K

The observed points are distributed about an observed Gaussian curve rather than the

calculated curve, so that the differences which must be considered statistically are A^ and

Aj,k" Gj,k V rather than A

j,kThis correlation is illustrated for some experimental data in figure 2(a). This shows

| the differences observed for a number of peaks for a UG^ sample. The solid lines represent

the mean difference values for each peak, with the shaded areas corresponding to A. . This

is also illustrated in figure 2(b) in which the solid curve represents the difference

is (

171

between the observed and calculated Gaussians (G . . A.)

J > K K

We must therefore consider two agreement factors relating to the fitting of the peak

shapes and the peak intensities respectively, given by:

A* = {I w

k a£}/(Nb

- P c

) (20)

The ratio of the estimated standard deviation given by the two methods is then given

by:

7 -1 7 f

1 9 ^ 1

which can also be written in the form:

—o— —ri— + e j (22)

(af)j (A^j A* N

p

- (P

c * Pp

)

£ and e

1 are both functions of the weighting scheme, having values in the range 1 to 2, given

by:

"k"*

4*'? j"!.**!.* 4*

<23>

£i 1 w- = n "j> 6ja • <24)

Equation [21] enables us to predict the ratio from the agreement factors given by the

two methods, whilst equation [22] shows how the ratio is related to the agreement factors

appropriate to the shape and intensity fitting. Since the number of degrees of freedom is

normally much larger for the profile data than for the integrated intensity data, the

second term will be small and the standard deviation ratio will usually be determined by the

ratio of these agreement factors, rather than the different number of degrees of freedom.

4. Experimental Results

We have analyzed a number of experimental data sets using both methods in order to provide a quantitative illustration of these results. Table 1 summarizes the results

obtained for the UC^ data which include those shown in figure 2. The values of the structural

parameters are not significantly different, but the agreement factors are, and consequently

173
the standard deviation ratio is less than 0.5, i.e. the profile refinement method underestimates these standard deviations by more than 2. The magnitude of this factor is well

predicted by our analysis.

Table 2 shows some similar results for data from orAgl, which was studied by Wright

Fender [3] using the Rietveld method. We have therefore reanalyzed their data using an integrated intensity analysis and compared the two sets of results for two different stru^

tural models. Again, the most significant differences are in the values of the standard

deviations, the profile refinement method underestimating them by a factor which varies

between 2 and 5, thus giving values which are particularly misleading. For example, the

difference between the values given for the positioned parameter x. appears to be 4^ timtf

fly its standard deviation instead of just over 1.

Table 1. Results of Profile and Integrated Intensity Refinements of UO^ Data

Profile Intensity

nQ(rel.occ.) 1.0C

B

u (A

2

) 0.2E

B„ (A

2

) 0.54

oR(F

t )(%) 4.03

(14) 0 977 (32)

(32) 0 246 (75)

(38) 0 522 (82)

-2,

R(I) (%) - 3.92

A 1.60 -

A

B

- 2.92

Scale (rel) 1.0000 1.0139

Table 2. Results of Profile and Integrated Intensity Refinements for a - A^I

Model 1 Model 2

Ag site

xAg

Bj (A*)

R(D(%)

Profile Intensity Profile Intensity

12 (d) 12 (d) 24 (g) 24 (g)

0.308 (2) 0.299 (7)

11.7 (4) 12.4 (2.1) 9.7 (8) 11.5 (1.4)

7.5 (3) 7.8 (9) 7.1 (3) 7.8 (5)

8.5 8.3 4.5 4.7

174
1

true

We have recently analyzed several further sets of data from UC^, at five different

j temperatures [4]. Table 3 summarizes some of the relevant results. This gives the values

I of e

1 and the agreement factors for the two methods of analysis. From these we can predict

a value for the mean standard deviation ratio, which can then be compared with the observed

value. It can be seen that, although the ratio varies between 0.6 and 1.7, we can in all cases predict its value to within 10 percent. These results also show that the profile

refinement method can overestimate, as well as underestimate, the standard deviations. However, the agreement factor Ap is somewhat larger for these data than is usual and it is our

opinion that, at least for neutron data when the peak shape is usually quite well predicted,

it is more likely that the standard deviations will be underestimated.

Table 3. Standard Deviation Ratios for uo

2

Temp

O-p/Oj Calc

(°C)

e' A

p AI Calc Obs Obs

20 1.37 3.07 4.23 0.62 0.60 1.03

800 1.46 2.47 2.01 1.02 0.99 1.03

1100 1.51 2.51 1.10 1.86 1.71 1.09

1400 1.40 2.85 3.24 0.74 0.68 1.09

1460 1.39 2.48 1.94 1.08 1.00 1.08

Mean 1.43 2.68 2.50 1.06 1.00 1.06

5. An Alternative Approach

The situation is clearly more complicated when appreciable peak overlap occurs and the

results given by the Rietveld method may well become more unreliable. Moreover there is no simple way in which the method can be modified to give reliable standard deviation estimates.

We have therefore formulated an alternative approach, which is illustrated in figure 3. This

lis a prototype system using existing programs as far as possible.

Firstly KDRREF generates a list of reflections so that any unseparable peaks, i.e. peaks

separated by less than the step size, can be identified. Profile data are generated as before by the Rietveld PREPR0F program and initial values for the Bragg intensities are generated from the structural model by the TAILS program. The SCRAP program then refines the

profile parameters and the values of the Bragg intensities, which are then analyzed by the

TAILS program to give the structural parameters. This method thus allows any uncertainty

associated with the magnitude of the Bragg intensities to be reflected in the standard

deviations estimated for the structural parameters.

175
KDRREF

hkl

list PREPROF

Profile

Data

Initial

I. values

k

3» SCRAP >

Profile

Parameters

i r Refined

I. values

k TAILS

Structural

Parameters

Figure 3. Analysis of powder data.

We have now used this program to analyze a number of sets of data. The results for H

room temperature UC^ data, referred to earlier, are illustrated in table 4 and figure 4. Table 4 gives the final values for the refined parameters and Bragg intensities, with theiri estimated standard deviations. The standard deviations of the intensities are larger than

those calculated by the Rietveld program from the specified weights by about 50 percent.

Figure 4 shows the fitting of a number of the peaks, which can be seen to be quite good,

fitted values for the Bragg intensities were then analyzed by the TAILS program, giving th

values for the structural parameters given in the last column of table 5. The SCRAP analysis leads to values of the agreement factor and the R-factor which are lower than those for the other two methods indicating that, for these data, the SCRAP progr provides the most reasonable estimates of the Bragg intensities and their standard devia

«

tions. The standard deviation values are also appreciably less than those given by the

integrated intensity analysis. This is therefore an example in which the peak shape is qui

'"Wt well approximated by a Gaussian, as illustrated by the data shown in figure 4, so that the

use of a profile fitting technique has advantages over a simple summation of the Bragg

i ntensity.

176

i> l!tl

iff

ft

p

Hi

f

id

f

lid
Table 4

V(I)

•0.58010977E+04

0.18237734E+04

S.D. OF V(I)

0.81595856E+02

0.31276672E+02

leii

the

are

n

1 AMRRA n i nno.QT cnp+m U. XUUOOXOUCTUX U. HUUjjODDt

Urroc

n 9nfiftiA"?RFTNT in 1 MDNU.

X 791 / 2- Xoft

. L9 1 R"? R

oc

A7ft 7 79ICO

0 9RA

At K9nn ft i An 7

r nXXOr 4.

. H RQ ft c

D

R0.R7 7 1 0.0. 9 7 119 XX£ ft

. O

oo XXI 9 DD . 0 Q lA9ftn ft 90A n

ixun 31 ft9 1 OXOc. . X

110. XXO. 9L XX R9RA ft ~\X1 "\ XJ / . 0

1 9 •3ft94 0 1 9°. A

1 3. 1 Q7 K 79 ft

1

Xf ftO.90. 0 XDO . 3

1XOR iXofrap; i O .

QQ A

16 149.3 77.8

17 2432.4 112.6

18 2859.9 120.1

19 172.2 84.7

20 12146.5 202.0

21 3725.8 136.9

22 1295.3 112.7

23 1293.5 116.6

24 89.3 103.2

25 7962.2 181.8

RESIDUAL SUM OF SQUARES = 0. 75901294E+03

DEGREES OF FREEDOM = 301

VARIANCE RATIO = 0. 25216370E+01

PROBABILITY =0.0

The same is not true, however, when the peak shape is not well approximated, as llustrated in the following example. This is the U0£ data set obtained at 1100 °C [4], which

e have also analyzed using the SCRAP program. The results are compared with those for the

netveld and integrated intensity methods in table 6. In this case the estimated standard

eviation values given by the Rietveld method are nearly twice those obtained from the

ntegrated intensity method, with the latter giving agreement factor and R-factor values less

han half those given by the Rietveld method. In contrast to the previous example the SCRAP

rogram leads to values of the standard deviations and the R-factor which are larger than

hose given by the integrated intensity analysis, and the agreement factor is also further

rom 1. These results are therefore consistent with an inadequacy of the peak shape fitting,

hich is illustrated in figure 5 and arise from deviations from a Gaussian shape function

hich are present at all temperatures. In these circumstances profile fitting therefore

eads to poorer estimates of the Bragg intensities and their standard deviations.

177
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181
Table 5. Results of Different Refinements of U0

2 Data at R.T.

Proii I

Intensi ty SCRAP

n Q (rel. occ; l.UUb (.14 .) u.y// 1.001 (21)

Op

d n 9R/i /"jo's B

u (A ; U.^D4 U. c.'va v. '3 ) 0.262 (46)

B

o

(.A ; U.041 {io) n coo /oo\ 0.520 (50)

K{r ) (.A ) 4.Uo -

R(D (%) 3.92 3.33

A

p 1.60

A

B 2.92 1.36

Scale (rel) 1.0000 1.0139 0.9977

Table 6. Results of Different Refinements for UOg data at 1100 °C

Profile Intensity SCRAP

B

u (A

2

) 1.31 (9) 1.40 (5) 1.30 (7)

B Q(A

2

) 2.19 (11) 2.20 (6) 2.22 (10)

R(F

2

) (%) 2.57 1.13

R(D (%) 1.76

A

p 2.51

A

B 0.95 0.89

Finally, we have also analyzed a set of data from A^O^, for which a certain amount j peak overlap occurs, as illustrated in figure 6. (Beyond 28 = 92° peak overlap becomes mi severe and so further background points are observed.) The results given by the SCRAP pn

gram are shown in table 7. In this case a few pairs of reflections occur within 0.1 or 0 of each other and the intensities of these cannot therefore be estimated accurately, as si by the large standard deviation values and correlation factors. Examples of the latter a; shown in the section of the correlation matrix is given in table 8. Results from the leaf squares analysis of these intensities are compared with those given by Rietveld and integ

intensity analyses in table 9. Because of overlap 40 Bragg reflections were included in

Integrated intensity, but as only 18 separate observations, 7 of which are multiple leaks

containing up to 8 Braqq reflections. In this case the integrated intensity and SCRAP an. gave similar results, but as would be expected the standard deviations are appreciably la

for the integrated intensity analysis. We have not been able to obtain a stable refineme

using the Rietveld method, even with the starting values for the parameters set very clos> 182
those given by the other methods. This is illustrated by the results given in table 7 following one cycle of refinement only. Although the standard deviation values are already comparable to those given by the SCRAP analysis the value of Bq is negative and differs signifiantly from the values given by the other methods. What is perhaps more significant is the

act that when we keep the temperature factors fixed at their starting values the Rietveld

lethod still gives a solution with identical R factor values. This is indicative of an [xtremely flat minimum of the type commented on by Young [5]. In contrast, the SCRAP program

js well behaved and converges quickly to a good solution.

Table 7. Results of Different Refinements of A^O^ Data

PROFILE INTENSITY SCRAP

N

I 43 18 (40) 43

X

A1

0 3524 (4) 0. 3528 (5) 0.3529 (3)

X

0

0 5571 (4) 0. 5559 (7) 0.5559 (4)

BA1

Oo

(A^) 0 38 (9) 0. 23 (15) 0.12 (8)

B o

°2

(A ) -o 13 (5) 0. 27 (7) 0.22 (5)

R(D (%) 8 89 2. 16 5.34

A

p

3. 82

A

B

4. 32 1.38

6. Conclusions

Our analysis indicates that all standard deviation values which have been derived by the

ietveld Profile Refinement Method are wrong and that no simple modification of the method is

ossible to overcome this problem. If the peak shape is well determined it is possible that

he standard deviations are underestimated, particularly if appreciable peak overlap occurs. The SCRAP method, outlined in this paper, provides reliable estimates of the structural

jarameters and their standard deviations, which are superior to those given by an integrated

ntensity method if the peak shape is well determined. Comparison with the Rietveld method

ndicates that the profile fitting is both faster and more stable with the SCRAP method.

183
4000

2000

62 64 66

2-THETA (IN DEGREES)

t—

68 70 72

4000

2000

1 — I 1 \ ^ -ruu^.

1 n

""^ —

1 1

1—r ... _ 1 72 74 76

2-THETA (IN DEGREES)

78 80 82

4000

2000

0 L 1

'1 1 1 8l " 84 86

' 88 90 92

2-THETA (IN DEGREES)

Figure 6,

184

!
Table 8

U 0.52210477E+05 0. 17139536E+04

V -0.32285629E+05 0. 15905103E+04

W 0 . 65861289E+04 0 . 35240942E+03

OFFSET -0.17595169E+02 0. 96862018E-01

INT NO.

1 544.2 65.6

2 7653.5 168.9

3 69147.7 482.1

4 135.8 53.6

5 8919.1 182.0

6 34440.0 344.4

7 3775.6 127.8

8 0.0 234.6

9 1975.0 247.4

10 8428.5 182.1

11 38477.1 366.0

12 2918.0 121.0

13 42.0 69.7

14 4298.2 446.5

15 24755.3 514.7

16 2453.6 279.6

17 199.8 270.3

18 12473.2 268.2

19 19045.7 417.5

20 1281.9 345.9

21 2067.4 1082.6

22 1732.6 1011.5

23 733.6 108.6

24 5382.0 292.6

25 246.4 259.9

26 1633.4 500.9

27 15267.3 545.2

28 4198.8 156.8

29 2550.8 173.6

30 4379.2 536.8

31 417.5 562.3

32 676.1 189.5

33 3750.1 2584.3

34 4.7 3950.9

35 0.0 8655.4

36 4990.5 7372.4

37 13566.7 446.1

38 9498.8 2242.4

39 851.9 2554.3

40 0.0 801.9

41 2755.1 605.6

42 22670.9 922.4

43 4101.4 2073.8

RESIDUAL SUM OF SQUARES = 0. 16787471E+04

DEGREES OF FREEDOM = 509

VARIANCE RATIO = 0. 32981272E+01

PROBABILITY =0.0

185
31 32 33 34

Table 9

35 36 37 38 39

31 -1 000 -0 025 -0. 008 0. 009 -0. 002 -0. 006 0. 000 -0. 005 0. 024

32 -0 025 1 000 -0. 082 0. 039 -0. 032 0. 015 -0. 000 -0. 001 0. 006

33 -0 008 -0 082 1. 000 -0. 473 0. 390 -0. 210 0. 002 -0. 000 -0. 008

34 o 009 o 039 -0. 473 1, 000 -0. 933 0. 578 -0. 006 0. 004 0. 006

35 -0 002 -o 032 0. 390 -0. 933 1. 000 -0. 707 0. 007 -o. 006 0. 002

-0 006 o 015 -0 210 o 578 -0. 707 1. 000 -o Oil n\J U _L C -n U 1J 37 o 000 -o 000 o 002 -o 006 o 007 -o Oil 1 000 -0 989 o 846

38 -o 005 -o 001 -o 000 0 004 -o 006 0 012 -0 989 1 000 -o 905

39 0 024 0 006 -o 008 0 006 0 002 -o 015 0 846 -o 905 1 000

40 -o 027 -0 007 0 009 -o 007 -o 001 0 015 -o 817 0 880 -o 998

41 0 021 0 006 -o 009 0 008 -0 002 -o 009 0 299 -o 339 0 470

42 -o 003 -o 001 0 001 -0 001 -o 000 0 003 -o 172 0 194 -0 268

43 0 006 0 002 -o 002 0 002 -o 000 -0 003 0 154 -0 175 0 244

44 -0 Oil -o 003 0 005 -o 005 0 002 0 004 -0 112 0 128 -o 185

45 -o 040 -o 010 0 016 -o 017 0 007 0 009 0 057 -o 057 0 049

46 0 051 0 014 -0 022 0 023 -o 010 -0 013 -0 034 0 028 -o 000

47 -o 019 -o 005 0 009 -o 009 0 004 0 005 0 025 -o 024 0 018

1!

•o.i

-o.c

:!

00o.c

0OA

0

0

1

-0.5

•0

0.; •O.f

0.

-o. o.

•o.

We are grateful to Mr. G. G. Proudfoot, Dr. M. W. Thomas and Dr. B. T. M. Willis

providing us with unpublished experimental data.

H

References

' :

[1] Rietveld, H. M. , Acta Cryst. 22, 151-152 (1967); and J. Appl. Cryst. 2, 65-71 (19

[2] Sakata, M. and Cooper, M. J., J. Appl. Cryst. (1979), in the press.

[3] Wright, A. F. and Fender, B. E. F. , J. Phys. C. 10, 2261-2267 (1977).

[4] Albinati, A., Cooper, M. J., Rouse, K. D. , Thomas, M. W. and Willis, B. T. M. Si to Acta Cryst. (1979).

Discussion

Comment (Sabine): I do not think that you can state that the Rietveld method under- I estimates e.s.d.'s. All you can say is that the two methods give different results i: cases. Response (Cooper, Sakata, and Rouse): I can state that the Rietveld method provides

liable estimates for the standard deviations of the structural parameters. These may

underestimates or overestimates of the best values, depending on the relative goodnes

fit of the peak shape function and the Bragg intensities and also on the degree of pe overlap. It is probable that these factors will lead to underestimation of the stand

deviations for most neutron diffraction data.

186

i
1

se the residuals obtained from their comparison with calculated data are. This means that

e Rietveld method is in fact not statistically correct. Since the expression for calcuted intensity is separable into two terms, each of which depends on one type of parameter

ly, correlation between the two types of parameter and between different Bragg peaks can

and is, taken into account correctly in the SCRAP method. It is in fact the Rietveld

thod which fails to do this, which is why no one has been able to give a valid justificaon of the method.

I

jmment (Prince): The individual data points within the same Bragg peak are not statstically

rrelated. The Rietveld procedure is in fact mathematically basically correct. A procedure

ich refines integrated intensities, particularly if profile and crystallographic parameters

e treated separately, neglects various cross correlations between the intensities of over- pping peaks and between profile and structural parameters. If these are correctly included,

e procedure becomes substantially equivalent to the Rietveld procedure.

sponse (Cooper et al.): Individual observed data are of course not correlated, but in this

S I

la]

tand

187
ll
il National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

STRUCTURAL REFINEMENT OF NEUTRON AND X-RAY DATA BY THE RIETVELD METHOD:

APPLICATION TO Al^ AND BiV0

4

j

D. E. Cox and A. R. Moodenbaugh

Brookhaven National Laboratory 1

Upton, New York 11973

A. W. Sleight and H.-Y. Chen

E. I. du Pont de Nemours and Company

Wilmington, Delaware 19198

The application of the Rietveld refinement technique to neutron

and x-ray powder data from standard samples of A^O^ is described. The

results are compared with those obtained by conventional refinements of

the corresponding integrated intensities, and also with x-ray single

crystal data. The Rietveld technique has also been used to analyze

neutron and x-ray powder data in a structural study of BiVO^. Since V has a very small neutron scattering amplitude, the x-ray data allow a more accurate determination of the V position to be made.

The use of the Rietveld technique for refinement of powder diffraction data has been

'eviewed in these proceedings by Hewat [l] 2 and Young [2] for the case of neutrons and

:-rays, respectively. The neutron technique is now firmly established and has been used

in hundreds of structural investigations, but the x-ray technique is still in its infancy,

ind a number of fundamental problems remain. Chief among these are the lack of a suitable

function to describe the peak shape, and where to truncate the peak tails. It may also be

lecessary to make allowance for preferred orientation.

The possibility of combining the powerful Rietveld technique with the better resolution

)f x-ray diffractometers provides a powerful stimulus for overcoming these limitations. In addition, the availability of monochromated radiation from synchrotron sources in the next

few years offers the prospect of obtaining very high resolution powder data with peak profiles which are much closer to Gaussian.

pThis work is supported in part by the Department of Energy under contract number EY-76-

C-02-0016.

2 Figures in brackets indicate the literature references at the end of this paper.

189
One approach to the analysis of x-ray powder data is described in these proceedings

by Parrish and Huang [3]. In this, integrated intensities can be obtained by deconvolution of powder data with the aid of measured values of the full profiles of a number of

standard specimens. As far as is known, this technique has not yet been applied in

structural refinement. A more common current approach, as described by Young [2], is to modify the Rietveld program for neutron refinement. In the present work, this has

been done as simply as possible. A comparison is made between the results obtained in

refinements of neutron and x-ray data for A^Og, which is a commonly used standard, and

BiVO^, which undergoes an interesting ferroelastic phase transition at 255 °C [4].

The neutron data from A^Og were collected for an intercomparison project of the

Neutron Diffraction Commission of the International Union of Crystallography [5].

The sample was a sintered pellet made from Linde A grade A^Og [6] about 1.4 cm in

diameter and 1.8 cm in length weighing 10.5 g. Pyrolytic graphite was used as monoo chromator in the (002) reflection setting to give neutrons of wavelength 1.424 A, and

as an analyzer in the (004) setting. Higher order components of wavelength were removed

with a graphite filter. Collimation was 20' in-pile, 40' monochromator-to-sample, 40' sample-to-analyzer, and 10' analyzer-to-counter. Data were collected at 0.05° steps over a 26 range of 22.0 - 124.4° with a counting time of 38 seconds per step. Refinement was carried out with a modified version [7] of the Rietveld program [8] with individual isotropic temperature factors. Scattering amplitudes were taken as 0.345 and

-12

0.580 x 10 cm for Al and 0, respectively. The results are summarized in table 1, and

the fitted profile is shown in figure 1. Agreement is very good, as seen by comparison i of R^p, the weighted profile R factor, with R^, the expected, or statistical, value (for [ definitions, see reference 7).

The x-ray data were collected on a General Electric XRD-5 diffractometer with a fine-focus Cu target (take-off angle 6°) and a bent graphite monochromator of normal

mosaic spread 0.4° in the diffracted beam tuned for Cu KB radiation. The sample con- sisted of a pressed pellet of NBS 0.3 urn A^Og powder which had been sintered overnight

j at 1000 °C. Collimation consisted of a standard 1° horizontal divergence collimator

with medium resolution Soller slits in the incident beam, and medium resolution Soller

slits and an 0.05° receiving slit in the diffracted beam. Data were collected at 0.02° , steps over selected ranges of 20 between 22° and 136° with counting times ranging

between 20 and 80 seconds according to the intensity of the peak. A scan of the (116) peak is shown in figure 2a, and for comparison, the corresponding neutron scan in figure 2b. A Gaussian peak gives a good fit to the latter as shown

by the solid line, with Rwp = 0.038 (R

£ = 0.031). The x-ray data in figure 2a were

fitted to three Lorentzian functions of the type A/[l+C(A20) 2

] n

; a simple Lorentzian

L(n =1), an intermediate IL(n = 1.5), and a modified ML(n = 2). The results are sum- marized in table 2, and it can be seen that the quality of the fit is not as good as

in the neutron case. The intermediate function gives the best fit over this range of

data, which is about five full-widths at half-maximum (FWHM), and is shown as a solid

line in figure 2a. 190
Table 1. Results of structure refinements for A^O^. N(hk£) is the number of

reflections used in the refinement. Other quantities are defined in text or

ref. [7]. Rj and Rwpare based on point-by-point or integrated intensities

as appropriate. A value of 0.04 A

2 has been added to the neutron B's to allow for the effects of absorption as described by Hewat [9].

Neutron (1.424 A)

6 X-ray (1.392 A) X-ray

Parameter Profile Integrated Prof i le Integrated Crystal [12]

Al:z 0. 35221(8) 0.3521(2) 0 35200(7) 0.3522(2) 0.35219(1)

°2

A1:B(A ) 0. 22(2) 0.26(5) 0 40(1) 0.24(6) _ _ _ 0:x 0. 30636(9) 0.3060(2) 0 3064(3) 0.3066(8) 0.30633(5)

0:B(A^) 0. 23(1) 0.25(3) 0 51(2) 0.15(8) - - - — -2.

G(rad c

) o 0 195(9) 0.19(3)

a V n / 4. 7640(1) 4 75855(2)

c(A) 13. 0091(3) 12 9906(1)

U(deg

2

) 1. 42(11) 0 0084(12)

V(deg

2

) -1. 62(14) 0 0117(19)

W(deg2

) 0. 55(4) 0 0073(6)

R

I 0. 018 0.019 0 038 0.053

0. 071 0.024 0 149 0.082

R

E

0. 060 0 111

N(hkl) 60 60 69 71

Table 2. Results of least-squares fits to the data in figure 2. Background

was determined well away from the peak, and was not a variable in the

refinement. NO is the number of observations.

X-ray

(1.392 A)

Neutron

(1.424 A)

L IL ML G

Peak amplitude 2224(76) 2054(53) 1958(70) 4359(40)

Peak position (deg) 51.494(2) 51.494(2) 51.493(2) 52.77(1)

FWHM (deg) 0.100(3) 0.121(3) 0.132(4) 0.312(2)

Background -21.3 35.3

RWP 0.106 0.086 0.124 0.038

R e — 0.041 0.031

NO - 31 32

191
1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 1 1 1 J

m 0

—

O

P- O O O O O O OOOOOOtM tO LO CO CM — O^- CM O CM ^

I'SlNflOD N0ain3N 2 ot x IV

I

192
Figure 2a. Diffraction data for the [116] reflection of Al^: x-rays Cu Kp.

Solid lines are fitted curves as described in text.

Figure 2b. Diffraction data for the [116] reflection of Al^:

Neutrons. Solid lines are fitted as described in text.

193
p = 0.165, R^ = 0.066), but the intensities showed unmistakable evidence of (001)

A modified version of the Rietveld program described previously [10] was used to refine the x-ray data, which were fitted to the intermediate Lorentz function over a 3+ 2- range of two FWHM's on each side. Scattering factors for Al [11] and 0 [12] corrected

for the real dispersion term were used. At present the program has not been modified

to incorporate the imaginary dispersion term. The initial fit was quite encouraging

preferred orientation. In Rietveld' s original program [8], provision is made for a

2 correction of the type I = I . exp(-G<{> ) where <J> is the acute angle between the

C O IT* T* ODS scattering vector and the normal to the crystallites. G is the preferred orientation

parameter and is a measure of the half-width of the assumed Gaussian distribution of

the normals about the preferred orientation direction, which is along the axis of the

cylindrical samples usually used in powder neutron diffraction. This expression is not appropriate for flat-plate x-ray samples, but inspection of the observed and calculated

intensities showed as exponential -type dependence which could be approximated by a

2 slightly modified expression of the type I corr

= I Qj)s exp[-G(n/2 -<|>) ]. With this correc

tion, most of the systematic discrepancies disappeared and Rwpshowed a significant

decrease to 0.149. This is illustrated in figure 3, which shows the ratio of the

observed and calculated intensities for several of the stronger peaks with and without

the correction.

0.5

Al 2 0

3

I c

' =I C exp{G ( it/2- 4>) 2

}

f - o 2d* = °

CM O

3 K>TM

0 07 O

O ,n - 10— ro

40

o G = O

• G=O.I9

<tf CC CXJ o — zzro

0 0 VS,

80

Figure 3. Ratio of observed and calculated integrated x-ray intensities s> for for A^Og as a function of <(>, the angle between the normal to the

plane hkl and the [001] axis. Open circles--no correction. Closed

circles—preferred orientation correction applied.

194

i

I

1
rrec

The results of the refinement are summarized in table 1, and the profile fit is

shown in figure 4, with all intensities normalized to a counting time of 40 seconds.

Integrated neutron and x-ray intensities were determined and refined with a program which allows overlapping peaks to be included. The final results are also summarized in table 1, together with some recent x-ray single crystal results [13].

There is agreement among the various sets of positional parameters to within 1-2

jesd's. However, the esd's derived from the integrated intensities are a factor of 2-3

higher than those from the profile refinements. This subject is discussed in more

jdetail by Cooper et al. [14] elsewhere in the proceedings.

The x-ray profile temperature factors are clearly systematically high by a factor

Df two or so. Since the integrated values are in reasonable agreement with the neutron

values, it is likely that the problem lies in the peak shape function rather than in

other possible sources of error such as surface roughness, choice of scattering factors

of the Lorentz-polarization correction.

The difference between the neutron and x-ray lattice parameters determined in the

•profile refinements can be largely accounted for by a small error in the neutron wavelength of about 0.13 percent. The apparent precision of the x-ray values is rather

unexpected, since no attempt was made to allow for systematic deviations in peak positions

as a function of 28 or aberrations in peak shape. This precision could be quite valuable

in investigations of phase transitions or thermal expansion behavior.

The observed and fitted values of peak FWHM's are shown as a function of 26 in

figure 5 for the neutron and x-ray profile refinements. The usual expression

2 2 (FWHM) = U tan 6 + V tan 8 + W has been employed in both cases. The x-ray Rietveld technique has been used in conjunction with the neutron technique in a structural study of BiVO^ [15]. This has monoclinic symmetry at room temperature and undergoes a ferroelastic structural transition at 255 °C to a tetragonal

scheel ite-type structure. The neutron scattering amplitude of V is small, about -0.05

I -12

x 10 cm, and the V shifts from the ideal scheelite positions are rather poorly detero o mined (^0.06 A, a = 0.03^ A) in the neutron refinement. An x-ray refinement was accordingly

undertaken in order to obtain a more accurate value. A comparison of the neutron and x-ray

results is given in table 3, and the fitted profiles are shown in figures 6 and 7. Once

again, in the initial stages of the x-ray refinement systematic discrepancies were observed

which indicate (001) preferred orientation. Inclusion of a correction of the form described

earlier gave a much-improved fit, with R^p decreasing from 0.223 to 0.178.

From table 3 it can be seen that the Bi positional parameters are equally well determined in the two refinements, while those of vanadium and oxygen are a factor of about

five better determined in the x-ray and neutron refinements, respectively, as is to be

expected from the relative scattering amplitudes. Agreement is generally within 2-3 esd's.

Although these may be underestimated as discussed by Cooper et al. [14], it seems likely

that part of the disagreement lies in systematic errors in the x-ray analysis. This cer- tainly appears to be the case for the temperature factors, where agreement is very poor.

It is to be noted that the neutron results, including temperature factors, are in better

agreement (1-2 esd's) with those of an independent neutron study by David et al . [16].

195

I———I 1 I I I I I 0 20 40 60 80 100 120 140 COUNTER ANGLE, 28 (deg)

Figure 5a. Variation of FWHM of A^Og diffraction peaks as a function of

scattering angle 28 for the experimental conditions listed: x-rays,

Cu Kp.

Figure 5b. Variation of FWHM of AlpOg diffraction peaks as a function

of scattering angle 20 for the experimental conditions listed:

o neutrons, K = 1.424 A. 197


Table 3. Structural parameters for BiVO^ at 295 K from Rietveld

refinement of neutron and x-ray powder data. N(hk£) is the

number of reflections used in the refinement.

Neutron X-ray

(1.302 A) (1.392 A)

Bi : z 0

/•Tor / O \ 6335(2) 0. 6341(2)

°2

Bi:B(A ) u on/ o \ 80(3; 0. 36(2)

V: z AU 0. 1315(6)

v:B(A ) U

•3 /"/I A 3(4) 0. 13(6)

0(l):x au

1 A C C ^ /I \ I4bb(4 J 0. "I ^ o / o \ 143(3 J A ( 1 \ . w 0(l):y u

C A~7~7 f A \ 50/7(4; 0. 512(3)

0(l):z 0 2082(2; 0 211(1)

°2

0(1):B(A ) u 85(4; 1 9(2)

0(^).X U dbl)b(.o ) 0 £0£(o J ou;:y U 0 0(2;.

U 0

A AC f ~\ \ 44b (1)

°2

0(2):B(A ) 0 85(4) 2. A / O \ 0(2)

v\i au j o nu ,/«\

o 5 1935(3; 5 1956(i;

b(A) 5 0898(3) 5 0935(1)

c(A) 11 6972(8) 11 7044(2)

y(deg) 90 387(4) 90. 383(1)

U(deg

2

) 2 53(6) 0 049(7)

V(deg

2

) -2 03(6) -0. 020(6)

W(deg2

) 0 495(13) 0. 0165(11)

R

I 0 028 0. 077

RWP

0 123 0. 178

R

E 0 117 0. 097

N(hkJL) 391 154

There is once again a small systematic difference between the neutron and x-ray lattice parameters which reflects an 0.05 percent error in the neutron wavelength. The two values of the monoclinic angle y, which is independent of wavelength, are in excellent

agreement. The precision of the x-ray values is again quite striking.

From the above comparisons, one can conclude that the refinement of x-ray data by

the Rietveld technique can give useful results for the positional parameters in spite of

the problems of peak shape, truncation, and aberrations. However, further work in this area is obviously necessary. One useful approach is suggested by the work of Wertheim

200
et al. [17], in which a Voigt line shape is approximated by a linear combination of

Gaussian and Lorentzian components. Such an approximation gives a significantly better

fit to the data in figure 2a, Rwp

decreasing to 0.070. It is also necessary to obtain

more rigorous expressions for the angular dependence of the FWHM, and for preferred

orientation.

We would like to thank L. D. Calvert for sending us single crystal data on Al^

prior to publication.

References

[1] Hewat, A. W. , Symposium on Accuracy in Powder Diffraction, National Bureau of Standards, Gaithersburg, Maryland, June 11-15, 1979.

[2] Young, R. A., Symposium on Accuracy in Powder Diffraction, National Bureau of Standards, Gaithersburg, Maryland, June 11-15, 1979.

[3] Panrish, W. and Huang, T. C. , Symposium on Accuracy in Powder Diffraction, National

Bureau of Standards, Gaithersburg, Maryland, June 11-15, 1979.

[4] Bierlein, J. D. and Sleight, A. W. , Solid State Commun. 16, 69-70 (1975).

[5] Andresen, A. F. , Hewat, A. W. , and Sabine, T. M. , Conference on Diffraction Profile

analysis, Cracow, Poland August 1978. J. Appl. Cryst. (to be published).

[6] Andresen, A. F. and Sabine, T. M. , J. Appl. Cryst. 10, 497-501 (1977).

[7] Hewat, A. W. , U.K. Atomic Energy Authority Research Group Report RRL 73/897 (1973).

[8] Rietveld, H. M. , J. Appl. Cryst. 2, 65-71 (1969).

[9] Hewat, A. W. , Acta Cryst A35, 248 (1979)

[10] Khattak, C. P. and Cox, D. E. , J. Appl. Cryst 10, 405-411 (1977).

[11] International Tables for X-ray Crystal! ography , Vol. Iv (The Kynoch Press, Birmingham, 1974).

[12] Tokonami, M. , Acta Cryst. 19, 486 (1965).

[13] LePage, Y. , Gabe, E. J., and Calvert, L. D. (to be published).

[14] Cooper, M. J., Sakata, M. , and Rouse, K. D. , Symposium on Accuracy in Powder Diffraction, National Bureau of Standards, Gaithersburg, Maryland, June 11-15, 1979.

[15] Sleight, A. W. , Chen, H.-Y. , Ferretti , A., and Cox, D. E. , Mater. Res. Bull, (to be

publ ished).

[16] David, W. I. F. , Glazer, A. M. , and Hewat, A. W. , J. Phys. C. (to be published).

[17] Wertheim, G. K. , Butler, M. A., West, K. W. , and Buchanan, D. N. E. , Rev. Sci. Instrum. 45, 1369-1371 (1974).

201

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

TEMPERATURE DEPENDENCE OF THE ATOMIC THERMAL

DISPLACEMENTS IN U0

2

: A TEST CASE FOR THE RIETVELD PROFILE REFINEMENT METHOD

A. Albinati

Institute of Chemistry

The Polytechnic

Milan, Italy

M. J. Cooper, K. D. Rouse, M. W. Thomas, and B. T. M. Willis

Materials Physics Division

Atomic Energy Research Establishment

Harwell , 0X11 ORA, England

The neutron diffraction powder pattern of UO2 has been mea- sured at five temperatures between 20 °C and 1460 °C, and the data

have been analyzed by two methods: the Rietveld profile refinement

method and the more conventional procedure based on integrated

intensities. The thermal displacements of the uranium and oxygen

atoms, as derived by the two methods, are in agreement within one standard deviation at each temperature. The standard deviations

from the Rietveld analysis are lower at some temperatures than

those from the conventional analysis, but at 1100 °C the e.s.d's

from the Rietveld method exceed those from the conventional procedure.

These differences are readily explained in terms of the different

ways of calculating the e.s.d's in the two methods. The observed displacements are a little higher than those calculated by lattice dynamics, but the discrepancies are probably caused by the neglect of

anharmonic effects in the calculations.

We shall also discuss the system, hexamethylene tetramine,

which has been analyzed in a similar way at a number of temperatures. Here, both positional and thermal parameters are obtained

from the powder data.

203
1. Introduction

For the past ten years, the profile refinement method of Rietveld [1.2] 1 has been

widely used for the interpretation of neutron diffraction powder patterns; more recently,

the procedure has been extended to the analysis of x-ray diffraction powder data too (see

references [3] and [4]). Intensity points at small intervals in the diffraction

pattern are treated as individual observations which may arise from contributions

from a number of Bragg reflections, so that the profile refinement method can be used

even when there is a large number of overlapping peaks. On the other hand, in the more

conventional refinement procedure the integrated intensities are derived by assuming

that the Bragg reflections are separated from one another; this approach cannot be used

where there is severe overlapping of adjacent peaks.

Sakata and Cooper [5] have carried out a statistical analysis of the Rietveld method

and have raised a number of questions regarding its validity. In particular, they

show that the crystal lographic parameters derived by the two types of refinement are not necessarily the same, and that the estimated standard deviations of the parameters are different.

In the present paper we test these theoretical predictions by applying both refinement procedures to the same experimental measurements. The system we have chosen is polycrystalline uranium dioxide, examined by neutron diffraction over a wide range of temperature from 20 °C to 1460 °C. By using a sufficiently long neutron wavelength and a high

take-off angle at the monochromator, most of the Bragg peaks were well separated in the

diffraction pattern. The scattering data could be analyzed satisfactorily, therefore, by

both methods.

2. Conventional Refinement

The neutron powder diffraction pattern of U0^ was recorded with the PANDA diffractometer installed at the PLUTO reactor, Harwell. The polycrystalline sample, of composition

close to UO2 qq, was enclosed in a cylindrical can of tantalum. With a wavelength of

1.334 A and a range of scattering angles of 18° ^ 26 ^ 114°, all reflections up to 533 were

observed.

Relative values of the observed structure factors F(obs) were obtained at the five

temperatures 20, 800, 1100, 1400 and 1460 °C. A least-squares refinement was then

carried out for each temperature, where the quantity

hkl hk1 (F(obs)) 2

- s(F(hkl))

2 (1)

was minimized to yield the scale factor s and the thermal parameters By and BQ

. F(hkl) is the calculated structure factor and is of the form

figures in brackets refer to the literature references at the end of this paper.

204
F(hkl) = 4b

u exp(-B

u sin

28/\2

) + 8b

Q

exp(-B

Q

sin

26/\2

)

h+k+1 = 4n

= 4by expC-By sin

26A2

) .... h+k+1 = toil (2)

= to

y expC-By sin

26/A

2

) - 8b

Q exp(-B

Q

sin

26A2

)

h+k+1 = 4n+2

h'th b n and bn representing the neutron scattering amplitudes of uranium and oxygen

U U

2 fespectively. w

hkl in eq. (1) is the weight of (F(obs)) and was calculated from

I 1 f C

, . .^2 (I + E)"2 W

hkl = — ' °hkl = (F(0bs))

(I - B) (3) a

hkl

j/here B is the background count and I is the integrated intensity count plus the background

fount. The computer program used for the refinement was ORXFLS 3 [6].

Table 1 lists the thermal parameters of the two atoms, together with their estimated

tandard deviations. We also give the agreement factor Ag, defined by

A

B

2 = Xw|(F(obs)) 2

- s(F(hkl))

2

|

2 - (Nj - P s

) (4)

/here Nj is the number of independent integrated intensities and Ps(=3) is the number of

tructural parameters.

Table 1. Results given by Conventional Refinement Method

(using Integrated Intensities)

Temp. B

U

B0

N

I AB (°C)

Oo

(A ) Op

(A^)

20 0.18 0.66 9 4.23

(10) (12)

800 0.91 1.68 9 2.01

(7) (9)

1100 1.40 2.20 9 1.10

(5) (6)

1400 1.68 2.79 8 3.24

(17) (21)

1460 1.73 3.03 8 1.94

(10) (14)

205
'it

3. Profile Refinement

The same set of observations was then analyzed by the Rietveld method. Each observation y.j of the background-corrected intensity at the scattering angle 28^ was assigned a weight

w = I— (5)

1 Y

i + Bi

where B. is the background and (= y^ + ) is the total intensity. B. was estimated by

interpolation between regions with no Bragg intensity. The number of observations y. was restricted to cover the same range of 0.. 's as was used in the conventional refinement

(Section 2). The computer program used for the profile refinement has been described by

Hewat [7], and has minor improvements made later by one of the present authors (M.W.T. ). The refinement yielded the three structural parameters (s,B^,Bq), four instrumental

parameters and the lattice parameter a

Q

. The four instrumental parameters are U, V, W,

governing the 8-dependence of the half-width of the reflections (see ref. [2]), and the *"e

zero-20 position of the diff ractometer. Table 2 lists the thermal parameters and the

lattice parameters, together with their e.s.d's. Also given are: the number of experimental points Np, separated by 0.1° in 20, at which the profile was sampled; and the agree- ment factor Ap defined by

2 I w.(y.(obs) - s y.(calc))

2 _ i A

P ~ N Z~P r_ y N P ' ^ p

r pr $

where Pp(=5) is the number of profile parameters (i.e. U, V, W, zero-20 and a

Q

), and P^ is

the number of structural parameters.

Table 2. Results given by Profile Refinement Method

c is Temp. B

U

B0

N

p Ap ao (°C)

°2

(A*)

°2

(A )

o(A)

20 0.12 0.51 99 3.07 5.4710

(6) (7) (2)

800 0.84 1.70 105 2.47 5.5135

(7) (9) (3)

1100 1.31 2.19 107 2.51 5.5333

(9) (11) (3)

1400 1.59 2.70 93 2.85 5.5538

(11) (15) (4)

1460 1.67 2.92 89 2.48 5.5580

(10) (14) (1)

206

In

I!

k
The B values from tables 1 and 2 are in close agreement, at each temperature, for the

two methods of analysis. In principle, the structural parameters given by the two treatments are not identical [5], but in the present case they do not differ significantly.

The ratio of (o"

n-)p, the e.s.d. of the i*" parameter as determined by the Rietveld

method, to (o"

n-)j, the corresponding quantity for the conventional treatment, is given by:

<°i

2

>p . Op a

p 2 <Vl <Ol*B

where A.. ^ is the diagonal element of the inverse normal matrix corresponding to the \^

11 -1 -1

parameter. Sakata and Cooper have shown that the value of (A.. )p/(A.. )j can be estimated from the approximate expression:

Op. i

where the factor e

1 is determined by the weights given to the observations in the two

methods of refinement. The equation defining e

1 is:

2

e lw, = 2 I w. , G. , k

k kj j,kj,k

the summation on the left-hand side is over the k reflections, each with weight w^, and

the summation on the right-hand side over j refers to the profile points for an individual

reflection with each point weighted by w. . . G. . represents a normalized Gaussian centered

at 28

k

, with variance 2-/ln2/H

k where H

k is the F.W.H.H. of the reflection. Equation (7) can

be replaced, therefore, by

(°i>P. i a

p ...

The standard deviation ratios predicted by equation (8) are compared with those

observed, in table 3. Columns 2 to 5 in the table list the e.s.d. values given by the two

janalyses and the ratios of these are given in columns 7 and 8. Column 6 gives the standard

(deviation ratio derived from equation (8), using e' =1.5 and the agreement factors listed

in tables 1 and 2. (None of the individual values of e' differ from 1.5 by more than 10

(percent.) The ratios in column 6 are in reasonable agreement with the observed ratios in

(the last two columns. We conclude that equation (8), derived by Sakata and Cooper,

jaccounts satisfactorily for the differences in the estimated standard deviations of the

tructural parameters obtained by the two methods of analysis.

207
Table 3. Estimated Standard Deviation Ratios

for Profile and Conventional Refinements

Temp.

(°C)

(o-.y

prof i le

<*i>r

conventional

1 ^

v i^p (a.) T

i = B.. °2

(A )

i = b„

0

°2

(A*)

i = B,. °2

(A )

i = B„

0 °2

(A^)

J? A

B B

U

B0

20 0.062 0.072 0.102 0.120 0.59 0.61 0.60

oUU 0.071 0.091 0.074 0.120 1. UI U . 30 1. Ul

1100 0.086 0.107 0.052 0.061 1.87 1.65 1.75

1400 0.113 0.145 0.166 0.211 0.71 0.68 0.69

1460 0.104 0.137 0.104 0.138 1.04 1.00 0.99

.1

it

lit

r.

ita

4. Hexamethylenetetramine (HMT)

We h)ave also examined fully deuterated hexamethylenetetramine, CgN^D^. powder

neutron diffraction, using the Rietveld method to analyse the data. DHMT is especially

suitable for study, as it crystallizes in a body-centered cubic lattice, so that the

peaks are well separated from one another in the powder pattern and are uniformly spaced.

j Here, we shall present some preliminary results only. I Choosing the origin of the unit cell at the center of the molecule, the atomic coord" nates are

u 0 0 for carbon

v v v for nitrogen

and x x z for hydrogen (deuterium).

Thus, there are four positional coordinates: u, v, x and z. A powder diffraction pattern at 1 i quid- hel i urn temperature was analyzed by the Rietve"

procedure to give

u = 0.238 (1), v = 0.126 (1), x = 0.084 (1), z = -0.332 (1).

208

ill

':

in

v
These estimates can be compared with

u = 0.2391 (8), v = 0.1243 (4),

x = 0.0918 (8), z = -0.3278 (12)

which were derived by Duckworth, Willis and Pawley [8] in a room-temperature neutrondiffraction study of a single crystal of HMT. The atomic coordinates of the carbon and

nitrogen atoms for the powder and the single crystal are within one or two standard deviations of one another. The agreement is less satisfactory for the hydrogen atom, but this

may be due to the replacement of H by D in the polycrystal 1 ine sample.

5. Conclusions

Sakata and Cooper [5] have analyzed several sets of powder diffraction data obtained

Iwith neutrons, and have found that for most of these the standard deviations were underestimated by the Rietveld procedure by a factor of at least two. The inference that this

factor could be as large as this for most neutron diffraction data is not supported by the

present results. Indeed, in two of the five data sets (800 °C and 1460 °C) for U0

2 the

standard deviations given by the conventional and Rietveld refinements are about the same,

jand in one data set (1100 °C) the Rietveld method leads to standard deviations which exceed

those from the conventional refinement by a factor of nearly two. However, the differences between the e.s.d's obtained at each of the five temperatures

for the two types of refinement can be accounted for satisfactorily by the analysis of

Sakata and Cooper. The agreement factor A

R (see eq. (4)) in the conventional refinement

reflects the degree of matching between the observed and calculated integrated intensities,

whereas the agreement factor Ap (see eq. (6)) for the Rietveld refinement reflects not only

the degree of matching between integrated intensities but also that between the shape

functions describing the profile of the individual reflections. The structural parameters

are determined by the integrated intensities and the profile parameters by the shape

functions: it is not surprising that the e.s.d's from the Rietveld refinement are different from those from the conventional refinement. Equation (8) with e

1 =1.5 accounts

satisfactorily for the e.s.d. ratios in U09

. The larger values obtained for (a.

)

D/(a.

)

T are due to the unusually small agreement factors Ap for these data sets. The numerical estimates of the thermal parameters of U^, derived from either the profile or the integrated intensity method, are in fairly close agreement with one another over

the temperature range 20-1460 °C. There is similar agreement in the estimates of the

positional parameters of all the atoms (except hydrogen) in hexamethylenetetrami ne. In

(the latter case the comparison was made between a single-crystal, room-temperature analysis of HMT and a powder, liquid-helium profile analysis of DHMT.

209
References

[1] Rietveld, H. M. , Line profiles of neutron powder-diffraction peaks for structure

refinement, Acta Cryst. 22, 151-152 (1967).

[2] Rietveld, H. M. , A profile refinement method for nuclear and magnetic structures,

J. Appl. Cryst. 2, 65-71 (1969).

[3] Young, R. A., Mackie, P. E. , and Von Dreele, R. B. , Application of the patternfitting structure-refinement method to x-ray powder diffractometer patterns, J. Appl.

Cryst. 10, 262-269 (1977).

[4] Malmros, G. and Thomas, J. 0., Least-squares refinement based on profile analysis

of powder film intensity data measured on an automatic microdensitometer, J. Appl.

Cryst. 10, 7-11 (1977).

[5] Sakata, M. and Cooper, M. J., An analysis of the Rietveld profile refinement method,

J. Appl. Cryst. (1979), in the press.

[6] Busing, W. R. , Martin, K. 0., Levy, H. A., Ellison, R. D. , Hamilton, W. C.

,

Ibers, J. A., Johnson, C. K. , and Thiessen, W. E. , 0RXFLS3, Oak Ridge National

Laboratory Report 0RNL-59-4-37 (1972).

[7] Hewat, A. W. , The Rietveld computer program for the profile refinement of neutron

diffraction powder patterns modified for anisotropic thermal vibrations, Report No. RRL 73/239, Atomic Energy Research Establishment, Harwell, England (1973).

[8] Duckworth, J. A. K. , Willis, B. T. M. , and Pawley, G. S. , Neutron diffraction study

of the atomic and molecular motion in hexamethylenetetramine, Act Cryst. A26,

263-271 (1970).

Discussion

Comment (Cooper): The existence of very large residuals for these data is illustrated in

figure 5 of the paper by Cooper, Sakata, and Rouse. These large residuals occur systematically at all temperatures and lead to unusually large values of the Agreement Factor Ap.

Since the Gaussian function is not a good approximation to the observed peak shapes, these

results cannot be considered to be typical neutron diffraction results, but are nevertheless

entirely consistent with the conclusions of Cooper, Sakata, and Rouse.

210
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

STUDIES OF THERMAL MOTION USING CONSTRAINED PROFILE ANALYSIS

E. Prince

National Measurement Laboratory

National Bureau of Standards

Washington, DC 20234

C. S. Choi and S. F. Trevino

Energetic Materials Division

LCWSL, ARRADCOM

Dover, NJ 07801

A constrained refinement procedure which treats atomic groups as rigid bodies and uses the elements of the T, ^L, and £ tensors as the

independent variables has been successfully applied to single crystal

diffraction data. In order to determine if similar procedures can be

applied to powder data, in spite of the loss of information due to over- lapping peaks and the difficulty of determining background, we have

modified the Rietveld-Hewat profile analysis program to incorporate

rigid-body constraints. Subroutines have been added which evaluate

anisotropic temperature factor coefficients, and also third cumulant

coefficients, and their derivatives in terms of the T L S tensor elements,

and the treatment of constraints has been changed so that relationships

are applied directly so as to reduce the size of the Hessian matrix. The

program has been applied to an analysis of data taken from potassium azide,

with results that are generally consistent with single crystal results,

although they are considerably less precise. Constrained refinement has

also been applied to powder data from nitromethane and a high temperature

phase of ammonium nitrate, with reasonably satisfactory results. It appears that, although profile analysis is much less powerful than cor- responding single crystal techniques, it can in favorable circumstances

yield meaningful thermal motion information.

211
Discussion

Comment (Edmonds): One normally considers the low angle three dimensional x-ray diffraction

data as determining the positional parameters and the high angle data defining the thermal i motion parameters. Since the Rietveld method uses the collapsed one dimensional data, there!

is a significant loss of information necessary to define thermal anisotropy. One must be

very careful in choosing a constrained model which may neglect information which is present

in order to allow access to information which may be only marginally present. At this point

I would still feel quite cautious on this matter for the x-ray case.

Response (Prince): I agree that the loss of information due to collapsing the three

dimensions of reciprocal space to one dimension is particularly detrimental to the determination of thermal parameters. This makes it especially important to constrain the model

to conform to the laws of physics. For x-rays, the problem of line shape, among others,

will have to be solved before thermal parameters can be considered meaningful.

212
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

ACCURACY OF CRYSTALLITE SIZE AND STRAIN VALUES FROM X-RAY DIFFRACTION

LINE PROFILES USING FOURIER SERIES

R. Delhez, Th.H. deKeijser and E. J. Mittemeijer

Laboratory of Metallurgy

Delft University of Technology

Rotterdamseweg 137, 2628 AL Delft

The Netherlands

The accuracy of size-strain analysis using Fourier series is

determined by: (i) the methods and experimental procedures used in

obtaining the Fourier coefficients of the pure, only structurally

broadened, line profile; (ii) the model used to describe the arrangement of the atoms in the specimen and the approximations used in relating Fourier coefficients to the structural parameters of the model; and

(iii) the availability of corrections for systematic errors. Attention is paid to procedures for deconvolution, elimination,

correction for the angle dependence of Lorentz and polarization factors,

assignment of origins and the choice of background level. The errors in

the Fourier coefficients of the measured profiles and of the pure, only

structurally broadened profile are considered.

Methods for the separation of the order-dependent and orderindependent parts of the broadening in the Warren-Averbach theory are examined (including single-line techniques). The best way of separating these parts depends on a priori knowledge about the imperfections

(e.g., distribution functions), but in general a plot of the cosine

Fourier coefficient versus the square of the order of the reflection

is most appropriate.

Correction methods for the errors accumulated in values for size

and strain are discussed for the case of a non- ideal standard line

profile. Especially the mean square strain values are affected heavily

by application of a standard specimen with a spacing only a few percent

different from that of the ideal one. Nevertheless, without these

corrections, relative determinations of size and strain can be made

accurately.

213
The determination of the crystallite size distribution is con- sidered separately in view of the high sensitivity to error propagation.

1. Introduction

Properties of crystalline materials are determined by the imperfectness of their

structure. The x-ray powder diffraction method is a non-destructive technique and it

averages over a reasonable volume. From the analysis of x-ray diffraction line profiles

measures for deviations from the perfect crystal structure can be obtained. Principal

imperfections are crystallite size and lattice distortions.

For the analysis of the peak shape several methods have been proposed. Much work has

been done using the integral breadth and full width at half maximum. This technique has

gotten an impulse recently r.1,2,3] 1

. For a more detailed analysis a variance method [4]

and a Fourier series method [5] are available.

In this paper, we will confine our attention to size-strain analysis using Fourier

series representations of the line profiles measured. From a minimum of assumptions very

detailed information can be obtained.

The accuracy of size-strain analysis using Fourier series is determined by:

(i) the methods and experimental procedures used in obtaining the Fourier

coefficients of the pure, only structurally broadened, line profile;

(ii) the model used to describe the arrangement of the atoms in the specimen

and the approximations used in relating the measured Fourier coefficients

to the structural parameters of the model ; and

(iii) the availability of corrections for systematic errors.

2. Errors in Line Profiles

In this paper, it is assumed that all line profiles are recorded by a step-scanning

technique. Most aspects of the choice of step size, angular range sampled, radiation and

monochromatization, slit widths, counting time and dead time of the counter can be found

in handbooks (e.g. reference [6]). Specimen preparation should be carefully dealt with:

effects of surface roughness, specimen transparency, inhomogeneous density and grain size

may be overlooked easily.

2.1. Elimination of background

Estimation of the background level is still more an art than a science, although metho

have been developed to find a justified estimate [7,8,9]. Usually it is assumed that the

background is linear, which is often not true, because of the presence of e.g. thermal

Figures in brackets indicate the literature references at the end of this paper.

214
ffuse scattering2 or the white spectrum (including a K-edge of the K0 filter). Since one

p

s become accustomed to the use of computers in powder diffraction work the experimentalist

uld calculate these effects before estimating a remaining linear background with one of the

thods referred to. Because all physical quantities are calculated from profiles minus background and

cause the background has to be taken as a smooth curve all statistical fluctuation in the

(tal profiles are found in the net profile. Therefore, it is advantageous to keep the

ckground as low as possible. This can be cone by using (i) a vacuum chamber on the

Tfractometer to reduce air scatter, (ii) an x-ray wavelength such that fluorescence is

rtimal, and (iii) a monochromator (which also avoids the step in the background due to

a K-edge of the K

p filter).

2.2. Elimination of the broadening due to instrumental aberrations and

the x-ray spectrum used

A line profile h measured on a specimen containing structural imperfections is deterled by:

(i) The structural imperfections to be investigated which cause line broadening

effects contained in the structurally broadened profile f. (ii) The non-ideal geometry of the diff ractometer and the specimen,

(iii) The wavelength distribution,

(iv) Angle dependent factors (Section 2.4.).

Usually (ii) and (iii) are taken together into one function g called the standard line

ifile, which can be observed with good approximation on a suitable standard specimen (cf.

:tion 5.1.). In most experiments (ii) and (iii) constitute the major part of the profile

at low Bragg angles (ii) is predominant and at high angles it becomes negligible, while

I (iii) the reverse holds [11].

The line profile h can be described by a convolution

* /

h(x) " fl f(y) g(x-y)dy (2.1)

+00

re A = j

g(z)dz. It is advantageous to take x, y, and z on a sine-scale [12].

i example of a significant contribution of thermal diffuse scattering to the background

found during the evaluation of the concentration-penetration curve from line profiles

diffused copper-nickel "single crystals" [10]: the analysis would have been impossible

a linear background had been chosen.

215
This equation implies that the shape of g should be constant within the range of the

profile. If this condition is not met satisfactorily, f can only be evaluated using a se of linear equations analogous to the convolution integral (cf. e.g. reference [13]).

2.2.1. Deconvolution

Many investigators in many disciplines have noticed difficulties in the deconvolutio

(i) arbitrarily small variations of h may correspond arbitrarily to great changes in the

deconvolute (unstable solution), and (ii) eq. (2.1) does not have a solution for every h. For a short discussion on seven methods used in x-ray powder diffraction see referen

[13]. Here we will restrict ourselves to the solution using Fourier series, firstly appl to powder diffraction by Stokes [14]. Since convolution of functions is equivalent to mi plication of the complex Fourier coefficients of these functions, deconvolution can be

accomplished using

\

F(n) = H(n)/G(n) , (2.

a

where F(n), G(n), and H(n) are the Fourier coefficients of the profiles f, g and h respec tively, normalized such that F(o) = G(o) = H(o) = 1 and n is the harmonic number3

. Other procedures for deconvolution used in powder diffraction may be regarded as simple deconvolution by use of eq. (2.2) after smoothing of the profiles [16]. Thereformost conclusions given here will also hold for these other methods.

Croche and Gatineau [13] state the thumbrule that deconvolution should be applied oij if the ratio of the widths of the profiles h and g is between 2 and 6: below 2 any exisj

deconvolution procedure becomes too inaccurate4 and above six deconvolution seems to hav^ no effect but enlarging the statistical fluctuations. Further, they propose a procedure

choose optimal experimental conditions such that the ratio mentioned is as close as possf to 6 with yet acceptable counting accuracy.

In some cases, the f profile cannot be recovered from the h profile with the g prof:

belonging to it, namely if the Fourier coefficients of the f profile are not vanishing bi

i those of the g profile. An example is a line profile of a diffused copper-nickel single

crystal already referred to in Section 2.1.: the f profile contains two "discontinuitie

at the Bragg positions of the pure metals: between these positions a continuously varyi

intensity is present, outside there is no intensity. For such a case see Section 2.2.2.

3A good, and also pictorial, description of the convolution equation and the use of Four series in connection with this is given by Brigham [15].

4 Schwartz and Cohen [17, p. 391] state that h should be at least 20 percent broader thanl

which implies an upper limit of about 100 nm for the determination of effective crystalTM

size.

216
pe

' Every deconvolution procedure leaves some "residual smearing" in the resulting profile,

can be regarded as a residual instrumental function [18], which is obtained, when the

rumental profile g is deconvoluted with itself using the same number and values of paramas in the deconvolution of h with g. In the case of the Fourier method discussed in

paper, the f profile is synthesized from a limited number of its Fourier coefficients:

ourier transform of the f profile is truncated. Since the Fourier transform of a rec- lar truncation function is of the form (sin ax)/x, this is also the shape of the

Jual instrumental function for the Stokes method. It is this function that leaves the

ier ripple," as it is sometimes referred to, in the synthesized f profile 5. It is

>us, that details corresponding to Fourier coefficients above the cut-off value will

>e found in the f profile, although they are present in the h profile.

2.2.2. Elimination of the component6

Deconvolution is impossible in cases as mentioned in Section 2.2.1. or if no appropritandard specimen is available. In these cases an elimination should be performed at

, since the doublet causes the greater part of the broadening especially at not too

ragg angles. The spectral part of the resulting profile may be removed to a large

it using a Cauchy or Voigt profile having a width in agreement with known data (see

note Section 5.2.1.). Further an elimination is helpful to separate partially overng peaks.

The assumptions on which elimination is based, are: (i) the component has the

shape as the a, component, (ii) the intensity ratio R [= I (max)/I (max)] is known

x a£ a^

(iii) the doublet separation 6 is known. In practice the first assumption is much

r fulfilled on a A-scale (equivalent to a sinG-scale) than on a 20-scale [12]. The

et separation 6 can be calculated from literature data of wavelength values and the

R is about 0.5 in most cases, but can be significantly different if a monochromator

ed. R can then be found by optimization [11].

If I^, Ip and denote the total, the a, and ct^ profiles respectively, then the

nation can be written using the basic assumption mention:

10ji

itii

!

I 1(x) = I t (x) - RI^x-6) . (2.3)

he expression for I^(x-6) obtained by replacing x by x-6 in eq. (2.3) is substituted

(2.3) and the substitution is made repeatedly eq. (2.3) becomes

thiff' f profile would have been obtained too, if the h profile prior to deconvolution

/sta ip have been convolved (smoothed) using the appropriate function (sin ax)/x.

main field of application of the a„ elimination is its use in line profile analysis

ds on the basis of line widths.

217
I^x) = ^ ( " R)m

• Vx-"1 " 6 ) • (2-4 m=o

If H t (n) is the Fourier coefficient of 1^, normalized such that Ht

(o) = 1, then using eq. (2.4) the normalized Fourier coefficient H^(n) of 1^ is obtained as (cf. reference [19])

H^n) = H t (n)(l+R/{l+Rexp2nin6/a} = Ht (n)/P(n) (2.5

where a is the period on a sine scale.

2.3. Counting statistics

When a line profile is sampled, e.g. using the fixed time counting method, each

observed number of counts is subject to a statistical error; the variance is equal to the

number of counts of the observation concerned. The observations are statistically independent. Statistical errors can be handled in three ways:

(i) By smoothing. As stated in Section 2.2. a number of deconvolution procedures contain an implicit smoothing process [16]. The experimentalist

should always be aware of the fact that smoothing may introduce periodicities which were not present initially (cf. the residual instrumental

function in Section 2.2.) or that it may eliminate meaningful periodicities,

(ii) By curve fitting. Then the mathematical form of the line profile has to be

known to a good approximation. Additional advantages are that uncertain

regions (peak overlap) may be ignored and that the tails of a profile may

be estimated more accurately. [ (iii) By regarding these errors as an intrinsic part of the profile. This

implies that the statistical fluctuations are deconvolved along with

the exact h profile resulting in much larger fluctuations in the resulting

profile and its Fourier transform (see also footnote to eq. (5.12)).

The approach (iii) will be followed in this paper. For most deconvolution procedure

the sensitivity to these errors obliges to measure the profiles with less than 1 percent,

statistical error [13]. This emphasizes once more the importance of a low background le\|

(cf . Section 2.1. ). \

2.4. Corrections for the angle dependence of Lorentz, polarization and atomic

scattering factors

In the interpretation of the broadening of x-ray diffraction line profiles it shoulc

be recognized that the Lorentz (L), polarization (P) and atomic scattering factors are

218
the

iurf

ant

lei

|e dependent within a given profile. If accurate line profile analysis is demanded this

je dependence should be accounted for. With respect to values for crystallite size,

| experimental line profiles it was shown that errors of 10 percent may occur when this

e dependence is ignored [20].

Rules are given which, in our opinion, should be followed when correcting for the

e dependence of L, P and atomic scattering factors.

2.4.1. Formulae for the Lorentz-polarization factor

In textbooks normally the LP factor is given for the integrated intensity of a reflec-

\. However, in structurally broadened line profiles the LP factor relevant to the

ribution of the intensity over 26 has to be considered. Warren and Averbach [21] have

pulated this LP factor for a powder specimen. Following the same procedure one can

ulate the corresponding LP factor for a single crystal.

It can be useful to perform line profile analysis on a sine-scale [12]. Then the LP ors for the integrated intensity and for the distribution of the intensity over sine

the same. This LP factor (omitting constant factors) for a single crystal is

2 1 + Bcos 26 , 0 c , —sTH2l— (2 - 6)

for a powder specimen

1 + Bcos

2 26 (2.7)

sin 6cos6

an unpolarized incident beam and without monochromator B = 1, whereas with an ideally

2 srfect monochromator crystal B = cos 26 , where 6 is the Bragg angle for the reflecting

m m

les in the monochromator. For a particular monochromator crystal B can be easily

iured [22].

In case one has to deal with (even strongly) texturized specimens it can be shown that

s allowed to use the LP factor for a powder specimen [23].

2.4.2. Correction for the angle dependence of the Lorentz-polarization factor

Here only line profiles on a sin6-scale are considered. The analogous correction of

» profiles on a 26-scale is described elsewhere [24]. Two cases have to be considered

practice.

I (i) The broadening due to instrumental aberrations is dominant in the g profile. This occurs at the smaller 26 values. If we assume that the

wavelength distribution is a delta function, each f value is diffracted

at a sharply defined diffraction angle; however, the intensity is recorded

over a range of measured 28 values for which the L and P factors are

219
determined by that one sharply defined diffraction angle. Therefore, both

the g and h profile should not be corrected for the angle dependence of the

L and P factors. After the deconvolution procedure the f profile should be

corrected for the angle dependence of the L and P factors,

(ii) The spectral broadening dominates in the g profile. This occurs at the

larger 26 values. With only spectral broadening present, each f value is

diffracted by the whole wavelength distribution, leading to recorded intensities over a range of measured 26 values. The recorded intensity at each

26 value is dependent upon the L and P factors for that 26 value. Therefore,

both the g and the h profile should be corrected for the angle dependence of

the L and P factors before performing a deconvolution procedure.

2.4.3. Correction for the angle dependence of the atomic scattering factor

Within the g profile taken from an ideal standard specimen (no structural broadening

the atomic scattering factor is not angle dependent at all. It is clear that a correcti

for the angle dependence of the atomic scattering factor should be performed after the

deconvolution procedure.

3. Errors in Fourier Coefficients of Line Profiles

3.1. Sampling and truncation

In practice no continuous line profile is recorded. Instead, the profile is sampled!'

i at certain equidistant locations on the s-axis. This can be considered as a multiplicati

of the profile f(s) with the sampling function A(s), where A(s) denotes an infinite serie

of equidistant impulses separated by the sample interval T. Then, according to the convo

lution theorem, in Fourier space (v domain) the effect of sampling implies a convolution

the continuous Fourier transform F(v) of f(s) with the Fourier transform A(v) of A(s), wh

A(v) denotes an infinite series of equidistant impulses separated by 1/T. As a result, t Fourier transform of the sampled profile has become periodic with period 1/T (the first

period is confined to -1/2T < v £ 1/2T). It will be clear that in case the true Fourier > transform has no vanishing values at v

c = 1/2T, sampling will result in overlapping:

"aliasing." Aliasing can be reduced by enlarging the period 1/T of A(v), i.e. reducing

the sample interval T on the s-axis. It is concluded that there is no loss of information if sampling occurs such that F(v) = 0 for v > v^, where v^ijvj. This can physically

be interpreted as that no significant changes occur in the line profile within the sampli

distance T. Further, it has to be recognized that the Fourier transform cannot be computed as

a continuous function; only sample values can be determined. As sampling of the profile

leads to periodicity in Fourier space (see above), the reverse is also true. Hence the

discrete Fourier transform pair amenable to machine computation (by for example using a F Fourier Transform technique [15]) requires that both the line profile and its transform a modified such that they become periodic functions.

220
atii

A second, normally more serious, distortion of the true Fourier transform arises from

lie limitation that intensity measurements have to be restricted to a certain range in

i-space. This inevitable truncation can be considered as a multiplication of the profile

(s) with the function r(s), where

r(s) = 1 if -a/2 < s <; a/2 and elsewhere r(s) = 0 . (3.1)

hen, in Fourier space the affected transform F'(v) is obtained from the convolution

F'OO-/ F(p) dp . (3.2)

\n important effect of truncation may be its contribution to the so-called "hook" effect:

tioiijrtie curvature of F'(v) near v = 0 will be smaller than the corresponding one of F(v). In the

field of x-ray diffraction this effect has been recognized first by Bertaut [25]. Model

:alculations by Young, Gerdes, and Wilson [26] showed that a truncation of the line profile

at positions corresponding to intensity values of the order of 1 percent of the peak maximum

bay introduce a detectable "hook" already (see also Section 4.4.4.).

Further, the Fourier transform becomes broader (smeared) and a ripple may appear on it

due to the oscillatory nature of the Fourier transform of the rectangular truncation

function.

Without intervention the rectangular truncation function is implied by experiment. The

rietjripple on F'(v) thus introduced, can be suppressed by multiplying the profile by a non- rectangular artificial truncation function with a corresponding Fourier transform which

possesses oscillations of smaller magnitude (see for example the Hanning function [15,

up. 141]). The effect of such a truncation function is to reduce the effects of the discontinuities of the rectangular truncation function. However, the more the ripple is

reduced in this way, the broader the F'(v) curve will become. Although never applied in

the analysis of x-ray diffraction line profiles such a procedure may be of some value in

calculating size distributions (cf. Section 6.), since this calculation is highly sensitive

to fluctuations on the F'(v) curve. Finally, it can be remarked that the transform of the g profile is usually much broader

than the one of the h profile and thus the error in the Fourier transform of the f profile

induced by truncation originates from the h profile [26].

3.2. Background errors

In practice the background level is more readily over- than underestimated, owing to overlapping tails of neighboring reflections. Then, a zero level is assigned unjustly to

the line profile at its extremities. Such a line profile can be considered as truncated at

221
positions where the intensity apparently falls to zero. Suppose that these positions art, found at -b/2 and b/2 (b/2 < a/2, where a is the period in s-space; cf. Section 3.1.) anc that the background is overestimated with an amount <{>. It follows for the background

affected Fourier transform F'(v) [26]:

F'(v) -SlSfcL (3,1

/IV

The discrete Fourier transform is sampled at v = n/a (n = 0, 1, 2, ...; cf. Section 3.1.1

~1 ^ Normally, [F(o) - F(l/a)]/F(o) < [<|)b - <t»(sinnb/a)(n/a ]/())b and as a consequence a "hook'

J

effect will be introduced. For this case the effects on size-strain values as obtained

from the Warren-Averbach analysis (cf. Section 4.) have been assessed by model calculatic

from simulated line profiles [27]. Although the Fourier coefficients of a single line

J profile may be affected appreciably, it was shown that compensation for the errors in tht size-strain values, arising from analogous errors on the first and second order reflectic can occur. This effect may be more pronounced for the mean square strain values than for the average crystallite size values (see also reference [28]).

If the background is constant it may be advantageous to underestimate the backgroun

with an amount <j) as has been proposed by Young, Gerdes and Wilson [26]. Then it follows:

«

ft

Now F'(v) = F(v) for v = n/a (n = 1, 2, ), which are exactly those values of v where

the Fourier transform is sampled except for n = 0, (cf. Section 3.1.). This implies that

an underestimated constant background need not be corrected, except for F'(o). Then, in

the Warren-Averbach analysis (cf. Section 4.2.) no errors are introduced when the size an strain values are obtained from the other coefficients.

3.3. Statistical variance

The errors due to counting statistics in Fourier coefficients will be estimated by \ calculating the statistical variances. Only line profiles recorded using the fixed time

counting method will be considered. Calculations for the fixed count timing method are analogous, but of less interest because the variances are generally larger than for fixed,

time counting. The minimum variance counting method will not be considered here [9].

Variances a due to counting statistics can be estimated using the formulae (cf. [29

|

a

2 (W) =y^fg f£ cov (x., Xj ) i,j = 1, 2, 3,.... (3.5

1J 222

!
aMith

andl

hile

1,

>d

itii

thi

:tio i

for

2 cov(Xj, Xj) = a (Xj)

cov(u, v)= w.^ , (3 6)

J

here W is a linear function of the variables x^, x^, x^ and U and V are linear

unctions of the variables y^, y^, y^, •••• and where cov(z^ , z^.) means covariance of the

unctions of the independent variables z. and z. and cov(z., z.) = 0, if z. and z. are

' p " ' J

1 J tatistically independent and cov(z.j , z^) = a (z.), the variance of z.. If W is a non- inear function of x^, x^, x^, then eq. (3.5) may yield a good approximation if the

ariance of each x^ is small.

On the basis of eqs. (3.5) and 3.6) Wilson [29] calculated the variances of quantities

seful in x-ray powder diffraction. His formulae (slightly rearranged) concerning the

ormalized cosine (A) and sine (B) Fourier coefficients of a measured profile are adopted

a

2 [A(n)] = a{L.+L

b+Li A(2n) + 2(Lb -L.)A

2 (n)}/2L2 t (3.7)

a

2 [B(n)] = a{L

i +Lb -Li A(2n) + 2(Lb-L.)B

2 (n) + 2(-l)

n a(G

R-G

L )B(n)/nn}/2L2 t (3.8)

cov[A(n),B(n)]=a{L

i B(2n)+2(L

b-L.)A(n)B(n)+a(G

R-GL )[(-l)

nA(n)-l/4]/7tn}/2L2 t (3.9)

that lere:

e an

A(n) = U

1 £ (I.-G.cos(2nnj/a); B(n) = lT 1 £ (I .-G ,)sih(2itnj/a);

j j L

1 £ dj-Gj) ; Lb = (GR+GL )a/2

) these formulae the period a is taken here as the (odd) number of observations; t is the

)tal measuring time; I. and G. are the observed intensity and the interpolated background

J J itensity at the j-th position; G

L and GR are the background intensities measured just

itside (left and right) the range a of the profile (to make the background statistically

idependent of the profile) and n is the harmonic number.

Jnly normalized Fourier coefficients are of interest in size-strain analysis (Section 4.).

jor not-normalized Fourier coefficients eq. (3.5) holds strictly. In the evaluation of

qs. (3.8) and (3.9) slight approximations were made.

223
As can be seen from the limiting forms of eqs. (3.7) and (3.8) as given in table 1 the general trend for the variance of the cosine Fourier coefficient is an increase with

n if L. > 2L

b and a decrease with n if < 2l_ b

, while the general trend for the variance

of the sine Fourier coefficient is an increase with n. All variances will be the smaller

the lower the background.

3.3.1. Deconvolution

On the basis of eqs. (3.5) and (3.6) cumbersome expressions are obtained for the

variances of the cosine (A^) and sine (B^) Fourier coefficients of the pure line profile

f, which expressions are approximations because A

f and are not linearly dependent on the

cosine (A^ and Ag) and the sine (B^ and Bg) Fourier coefficients of the h and g profiles

(cf. eq. (2.2)). Three limiting cases are considered: (i) n very close to zero, (ii)

vanishing values of B^ and Bgj this is usually the case for small values of n if origins

are chosen on centroids in the Fourier series evaluation, and (iii) vanishing values of A^

and B

h (and nonvanishing values of Ag and Bg); this is the case for large values of n. In these cases the covariances are neglected. The resulting expressions are listed in table

The expression for the mean square value of the modulus of the error as calculated, for all

n, by Stokes [14] is analogous to the result for n small given here.

During deconvolution there is the problem of choosing the cut-off harmonic number.

This choice may be based on the formulae in table 1 for large n. Evidently, only the

statistical errors in the h profile determine such a criterion.

3.3.2. elimination I

On the basis of eq. (3.5) expressions are obtained for the variances of the

cosine (A.^) and the sine (B^) Fourier coefficients of the line profile. Since A^ and

B^ are linearly dependent on the cosine (A

t

) and sine (B^) Fourier coefficients of the

total line profile the calculation of variances from eq. (2.5) holds exactly. Results

are given in table 1. It is seen that the variances are periodic with n (cf. reference

[19], where also the effect of errors in R and 6 is considered).

4. Separation of Size and Strain Broadening

4.1. Elements of the Warren- Averbach theory

Both small crystallite sizes and lattice distortions in (poly)crystal 1 ine materials

are known to bring along x-ray diffraction line broadening. Other causes of line broadening may be the presence of deformation and twin faults (cf. reference [5]) and concentration variations [30].

224

!
i— cr> i/> -i- <o cu cu </) CU -t-> ss- c s. -o

r- O C - CD U ID

— *- CD

cu <o -i- u

E -=C CD l/>

+J i—

•r- (0 •!-

•r- 4->

<:

+J i— co

3 CD

s- </l

-<-> > CD -o s01 I— H3 u.

jz

•)-> +J

s- c (/) o o 4-> <f- M C\J <u to

ra c ' CJ

c/> =J E -Q O" E cu

S- </) o cu c Ll. CU CM 3 +J

o

<=> l\

... o

w cn w oi + +

CM Cl +

CM -r- C\J *t—

I—

Nlb

+

a:

i—

M

b

—- +

cm a w a CM CL CXI Q_

XIb

XIb

> i—

<

Nlb

225
The presently most widely applied theory to describe size-strain broadening was developed by Warren and Averbach [21] 8

. In this theory, regions in the sample which

scatter the x-rays coherently are designated as domains. The different domains are thought to diffract essentially incoherently.

According to the kinematical diffraction theory the diffracted intensity of a domai

corresponds to

1 * E E. e*P 2niH.(R. " R ) (4.

where H and Rm are the reciprocal lattice vector and the position vector of the m-th uni

cell respectively according to

H = h^ + h2b

2 + h 3

b

3

R = (m, + X )a, + (m„ + Y )a_ + (m_ + Z )a_ -m

v 1 m -1 2 m -2 v 3 nr-3

where a^, a^, §3 and b^, b^, b^ are the unit vectors in real and reciprocal space respec

tively and X , Y , Z are the displacements with respect to the reference lattice in uni m m m

a^, a^. Each (hkl) reflection from a crystal of any symmetry can be considered as a (00£) reflection from a crystal having orthorhombic axes [33,34]. Thus, the length of t diffraction vector can be approximated by ^jj^j

Further, the domain is thought to consist of columns of unit cells, perpendicular t the (00£) planes. Then it can be shown (cf. reference [5]) that the diffracted powder p unit length of the intersection of the diffraction cone with the receiving surface can b expressed as the Fourier series

P(h

3

) ^ 23 F(n) exp 2ninh

3

(4. n=-»

with

F(n) = A

S (n)FD (n,£) (4.

The size Fourier coefficient A (n) is real and independent of the order & of the reflect

in contrast with the distortion Fourier coefficient F^(n,£):

8 0ther theories are given by Krivoglaz [31] and Ergun [32].

226
A

S (n) = ^ = ±-

I

{ i n )p(i)di (4 4)

1= n

F D (n,£) = A

D (n,£) + iBD (n,£) = <exp2ni£Z(n)> (4.5)

s, averaging over all columns in the sample, N(n) is the average number per column of

s of unit cells a distance of n cells apart, is the average domain size in unit cells

?ndicular to the reflecting planes, p(i)di is the fraction of columns of lengths between

i i+di cells and Z(n) is the difference between the displacements (in units a^ ) of two

I cells n cells apart in a column. Usually Z(n) is replaced by the strain e(n) = Z(n)/n.

I eq. (4.4) it is obtained

dA

S (n)

dn

n4-o

joi

i a

ft

Now the problem arises of what a domain represents actually. The theory was developed

inally for cold-worked materials. According to Warren [35,5] cold work can produce

ocation arrays, such as small angle boundaries, which subdivide the material into the

ins as mentioned above. Then the theory implies that the change in orientation over

boundary is abrupt enough to cause the domains to scatter incoherently. With respect

he latter point some doubts have been raised [38,28]: A small angle boundary does not

rrupt coherence for very small n-values (corresponding to an averaging distance for the

e factor (eq. (4.5)) of a magnitude equal to the range of the strain field of the

ocation array), since the orientation changes gradually and not abruptly over the

dary. Also, according to Ergun [32], if the lattice is highly defective, the domain

ept may fail and then the notion of defect broadening can be introduced, which gives

to Cauchy-shaped profiles. It was demonstrated that the experimental size Fourier coefents for cold-worked tungsten and aluminum might be described by such Cauchy functions

ed. Thus, for the case of cold-worked materials care should be taken in identifying the

in with a physical entity. 9

the case of deformation and twin faults are present, the initial slope of the size

rier coefficient curve (cf. eq. (4.6)) does not correspond to the true column size N^,

its value is denoted as an "effective" size. Therefore, the methods to be presented

the following sections, which provide sizes on the basis of eq. (4.6), are equally well

licable to obtain effective sizes in case deformation and twin faults are present,

icedures exist to evaluate the true column size from the effective size (cf. [5]).

227
On the other hand, for many specimens the interpretation is less ambiguous: In cas< the powder specimen to be investigated consists of a system of loosely bound particles

(e.g. catalysts, colloids) each particle may be monocrystal 1 ine and hence can be identif

as a domain. Also, the domain concept is very useful in case abrupt changes in oriental

occur, as is, for example, the case with high angle boundaries.

4.2. Classical method

The original analysis for separating the effects of size and strain is due to Warrei

and Averbach [37]. Normally, in the Fourier analysis of line profiles the origins are confined to the respective centroids and consequently the imaginary parts of the Fourier

coefficients are small. Then the analysis is restricted to the real part of the Fourier

coefficients:

A(n,£) = <cos27i£ne(n)> N(n)/N

3

. (4.7

D S Since A (n,A) is dependent on & and A (n) is not, size and strain effects can be separat

by taking the logarithm of eq. (4.7):

In A(n,£) = ln[N(n)/N

3

] + In <cos27t£ne(n)> . (4.8 v

Two power series expansions are applied:

<cos2rt£ne(n)> = 1 - <[^ne(n)]2

> + Rcos

J « 1 - 2n

2£2

n2 <e

2 (n)> (4.9

ln[l-2n2£

2

n2 <e

2 (n)>] = -27t2£2

n 2 <e

2 (n)> + rJ"

* -27tVn2 <e

2 (n)> , (4.1

where R^

os and R^

n are the remainders of the two series expansions. From eqs. (4.8),

(4.9), and (4.10), it follows that

In A(n,£) ~ ln[N(n)/N,] - 2nVn2 <e

2 (n)> (4.1;. O II

2

" If at least two orders of a reflection are available then values for N(n)/N- and <e (n)>

2 can be obtained from the slope and intercept of a plot of In A(n,£) versus £ . 228

<
cas v

irre

'e

iei

•ier

irat

,1

2 2 Now the errors arising from both series expansions will be investigated. If <e (n)>

i COS i y be approximated by <e (n)>, a maximal estimate for the remainder |R^

J and a minimal

timate for R^

n yields (cf. reference [38]:

Jn

mm

= 3

Since the cosine is underestimated with

j?^ n

| it follows from eq. (4.12) that the logarithm series expansion introduces an error

i the final result at least twice that of the cosine series expansion.

Let us also consider the ratio R of the relative errors caused by truncation of the

2 2 2 2 2 ;ries after the terms with £ . Defining q = 2n £ n <e (n)>, it follows from eqs. (4.9)

2 h

id (4.10) with the additional assumption e(n) = <e (n)> that:

,cos

.cos

max

(4.12)

and the logarithm is over-estimated with

R =

Rj

n/ln(l-q)

Rjos/cosV2q

[ln(l-q) - (-q)]/ln(l-q)

[cosV2q-(l-q)]/cosV2q~ (4.13)

he relative errors in the cosine and logarithm series expansions and their ratio R, as

(efined by eq. (4.31), are plotted in figure 1. It is clear that at small values of

i the neglect of higher-order terms in the logarithm series is much more important than in

e cosine series; at larger values of q the difference becomes less pronounced.

4.3. Modified method

The errors introduced by the series expansion for the logarithm (eq. (4.10)) can

e avoided [39]. The direct substitution of the power series expansion for <cos2n£ne(n)>

ccording to eq. (4.9) into eq. (4.7) yields:

A(n,£) s* N(n) N(n)

No

? ? ? 2 2n £ n <e (n)> (4.14)

ow the influence of size and strain are not separated as in eq. (4.11). However, if

2

two orders of a reflection are available the values of N(n)/N^ and <e (n)> are readily

2 btained from a plot of A(n,£) versus £ . The results derived from both methods of separation will be compared. As can be

S 2 hown [39] values of A (n) and <e (n)> obtained with the modified method (eq. (4.14))

ire always smaller than the values deduced with the classical method (eq. (4.11)) and

hese differences increase with increasing values of n. This also implies that the average

:rystallite size as determined from eq. (4.6) is smaller for the modified method. This is Illustrated for some practical examples in table 2. In practice the second order of a reflection is usually measured less accurately than

phe first order. Therefore, spurious variations may be present on the Fourier coefficients

229
R^

os Figure 1. The relative error in the series expansion for the cosine: /V2q;

the relative error in the series expansion for the logarithm: R. /ln(l-q) and

their ratio R as a function of q = Zn £ n

of the second order mainly. This will produce spurious ripples on the A (n) curve. These

S ripples will be enhanced in the A (n)-curve obtained from eq. (4.11) as compared to the

S 2 A (n)-curve obtained from eq. (4.14). Also <e (n)> obtained from eq. (4.11) is more sensitive to spurious variations in the Fourier coefficients of the second order as compared to

2 <e (n)> obtained from eq. (4.14) [39].

Seldom more than two orders of a reflection are available. But for such a case one more term in the series for <cos2n£ne(n)> can be considered: If terms in £ and higher

orders are neglected, the Fourier coefficient of the total line profile can be expressed

as:

!

1

A(n,£)

2 4 a-b£ +c£ (4.15)

4 4 4 a2n n <e (n)>/3. As in eq. (4.14), again the

2 2 2 where a = N(n)/N

3

; b = a2n n <e (n)> and c influences of size and strain are not separated, but the three coefficients a, b, and c are readily obtained if three orders of a reflection are available. After substitution of

2 4

a [=N(n)/N

3

] in b and c, values for <e (n)> and <e (n)> are found.

230
Table 2. Effective [see footnote to eq. (4.6)] crystallite sizes (D ) 2 %

e and root mean square strains (<e (L)> ) in some cold-worked metals

and alloys, as obtained by the classical method [eq. (4.11)] and the

modified method [eq. (4.14)] (data from Misra and Ghosh [47]).

Metal/Alloy (hkl) Type Classical Modified

Method Method

Ti 001 D

e (nm) 38.0 37.3

<e

2 (10 nnO^xlO3 1.58 i. DO

1 001 D

p (nm)

e 28.0 21.0

<e

2 (10 nm)>\l03 2.10 1. 66

Ag-15%In 111 D

e (nm) 8.0 7.5

<e

2

(D )>\l03

e 3.50 2.60

100 D

e (nm) 4.5 4.0

<e

2 (D )>\l03

e 5.50 J. JU

Cu-12.46%Ge 001 D e (nm) 9.2 8.8

<e

2 (5 nm)>\l03 3.30 3.00

Cu-17.72%Ge 001 D

e (nm) 17.6 17.0

<e

2 (5 nm)>\l03 2.65 2.42

4.4. Evaluation of results

4.4.1. Mean square strain values

The x-ray diffraction technique is one of the few experimental tools which provide

quantitative information about non-uniform microstrains averaged over moderately large

volumes. Therefore, a discussion will be dedicated to the mean square strain val ues

found.

231
2 Normally, the <e (n)> values found will increase as n decrease 10

, and the largest

value might be found at a distance close to the interplanar spacing (n = 1), or, as an approximation, zero averaging distance. There, the true strain component is found. This

extrapolation is difficult to perform in practice (due to experimental uncertainties

such as the "hook" effect (Section 4.4.4.)). An extrapolation of the logarithm of the

strain value versus n to n = 0 might be carried out easier [40]. If an extrapolation

is omitted, one often tabulates the (root) mean square strain values at a fixed distance

L = n |ag |, such as 5 or 10 nm, or L = aN^

(see table 2).

, where a can be 1/2 or 1, for example,

Rothman and Cohen [41,42] suggested that, if dislocations are the principal cause of

2 v microstrains a reasonable approximation for <e (n)> may be

<e

2 (n)> = c^n (4.16)

where c^ is a constant for the reflection considered. From eq. (4.16) with eqs. (4.5) and

(4.9) it is found:

A

D (n,£) ~ 1 - 2n2A2 nc

1 ~ exp {^nVc^)

Eastabrook and Wilson [43] showed, on another basis, that for larger values of n the

following approximation holds

A

D

(n,i>) = exp {-c n} . (4.17)

The above suggests that the approximation (4.16) is especially useful for larger values of n. Also, Gangulee [44] concluded that eq. (4.16) will generally not hold for small

values of n. Hence, a method [45] for the separation of size and strain effects by use of

the initial slope (n4-o of the A(n,£)-curve, based on eq. (4.16), may be less reliable 11 . For arbitrary dislocation configurations the mean square strain curves may be calcu2 lated numerically. Comparison of experimental and simulated <e (n)> curves may give

some insight into the imperfect structure. Such a computer program has been developed

for elastically isotropic materials by Turunen [46].

2 In general for different (hkl) types of reflections, different <e (n)> values will

be found. This may, for example, be due to oriented defects (such as precipitate particles

growing preferentially on specific (hkl) planes) and/or the elastic anisotropy of the material. One might rationalize (e.g. for the case of cold-work deformation) the (hkl) dependem

of the microstrain in the same way as can be done for the macrostrain: For macrostrain it

j

10 Instead of n, the variable L = n|a

3

|, a real distance perpendicular to the reflecting

planes, is often used.

xl In case the approximation (eq. (4.16)) is used, nevertheless, a single-line method

(with n > 0) should be preferred (cf. Section 4.5.).

232
has been found that the Voigt-Reuss-mean (also designated as Voigt-Reuss-Hill) model affords

a sufficient description [48,41]. Models with a better theoretical basis [49] do not give

essentially different results [50,51]. If e

h and ep are the strain in the specimen surface

and in a direction perpendicular to the surface respectively, a denotes the stress parallel

to the surface and s^. are the elastic coefficients in Voigt notation, it follows for cubic

crystals

jwhere

and

phtt. n ,u, q

. , 45(S

11

+25

12

)+5S44(S 11

+5 12>

e

h - (1/2) {s n

s12

s.S + } a ^ - U + c <; + 2S(2

11

+2S

12 )+5S

12

5 44

e , p -{h2 + s - S + 6s+5s

44

* a < 4 - 18>

s = h

2 k2 + hV + k

2l2

(h

2 + k2 + £

2

) 2

s s ll~

s 12s 44/^

These formulae can be derived on the basis of reference [52] and are valid only if no

2 h

texture in the specimen exists. If <e (n)> is identified with e , e. and a can be

p n calculated while the (hk£) dependence is provided by S.

4.4.2. Strain distribution

4

V

•

' f

Introducing a strain distribution g[e(n)] such that

J g[e(n)]de(n) = 1, it follows

•'-oo

A (n,£) = <cos2n£ne(n)> = I g[e(n)] cos{2n£ne(n)}de(n) . (4.19)

2 2 2 2 -1

For a Gaussian distribution g[e(n)] = aA/n) exp{-a e (n)} with <e (n)> = (2a ) it is

obtained:

A

D (n,£) = exp {-7tVn2 /a

2

} = exp {-2n2£

2

n 2<e

2 (n)>} . (4.20)

Hence, for this special case the classical method of separating size and strain (eq. 4.11))

yields the exact results. However, other distribution functions may be nearer to practice

(for example, a Cauchy function with cut off tails as suggested by Williamson and Smallman

[53]; see also reference [35]). Also, there is no a priori need that the shape of the distribution function is independent of n. Therefore, in general, the modified method (eq.

(4.14)) is expected to give more correct results.

233
A consequence of eq. (4.19) is that the strain distribution function can be determined if sufficient orders of a reflection are available, since by Fourier transformation

oo g[e(n)] = 2n j A

D (n,£) cos {2n£ne(n)}d£ . (4.21)

In general, however, not enough data will be available to determine the dependence of the

distortion Fourier coefficient on £. Only for cold-worked elastically isotropic materials,

where all peaks available can be combined (cf. reference [5]), it is possible to determine

strain distributions for not too small values of n [54].

4.4.3. Determination of the average crystallite size

Usually, the average crystallite size is determined from the initial slope of the

size Fourier coefficient curve (eq. (4.6)). In principle, it is possible to obtain

the average crystallite size before size and strain are separated. If it is considered

2 that d[<e (n)>]/dn (n-t-o) = finite, it immediately follows from eq. (4.14)

dA(n,£)

dn = dA

S (n)

dn

n4-o

fl- (4-22)

n-i-o

2 In case <e (n)> is described by eq. (4.16) an additional term will be introduced at

the right-hand side of eq. (4.22) (cf. reference [45]). However, since finite values

2 of <e (n)> for n+o are more realistic (cf. references [54,40,36]) eq. (4.16) is not suitable for n-t-o. In practice the determination of the initial slope may be problematic

because of the "hook" effect (see next section).

4.4.4. "Hook" effect

S In the experimental A(n,£) and A (n) curves often a negative curvature near n = 0 is

observed, whereas the Warren-Averbach theory prescribes a positive curvature (eq. (6.1)).

Two experimental errors have been indicated for this effect: truncation of the line profile (Section 3.1.) and a background estimated too high (Section 3.2.).

The usual method of correction for the "hook" effect consists of extrapolating the

straight portion of the Fourier coefficient curve at small values of n to n = 0. However, it was shown [28,36] that there may be also a theoretical background to a

c "hook" effect in the A (n) curve. For wide strain distributions the truncation of the

series expansion in eq. (4.9) (and in eq. (4.10)) if the classical method of size-strain

2 separation is used) after the term with <e (n)> may introduce errors such that a plot of

2 2 A(n,£) versus £ (classical: In A(n,£) versus £ ) is no longer linear in fact. Linear

234
'extrapolation, if performed, may introduce a "hook" in the A (n) curve. In such a case,

an average crystallite size determination according to eq. (4.22) may be advised before

size and strain effects are separated.

4.5. Single-line methods

There are many practical cases where crystallite size and strain are anisotropic and

where higher orders of a reflection cannot be measured reliably or only one reflection is

available (e.g. for supported catalysts, oriented thin films, multiphase composites and

heavily deformed materials). Hence, one is interested in obtaining size-strain parameters

from a single line profile. Then additional assumptions are required.

All single-line methods proposed approximately the size Fourier coefficient (eq.

! (4.4)) by

A

S (n) = l-n/N

3

. (4.23)

This implies that all columns in the range of n-values considered are assumed to be of equal

length. If e(n) is approximated with its mean <e>, it follows

A(n,£) = (l-n/N

3

) cos2n£n<e> . (4.24)

Then the parameters and <e> can be solved from eq. (4.24) for any two Fourier coefficients A(n1} £) and A^.A). In general, different (if plausible) values will be obtained

from different (n^,n

2

) pairs. Then some kind of curve fitting will provide more realistic

estimates [55]. In case it is assumed in addition that the strain distribution is

Gaussian and independent of n [56], it is obtained

In A(n,£) = ln(l-n/N

3

)-27t2£

2

n 2 <e

2

> (4.25)

which can also be considered as obtained by series expansion from eq. (4.24) with

2 2 <e> <e >. The average crystallite size can be determined according to eq. (4.22)

p and according to eq. (4.25), <e > can be obtained from the slope of the straight line

2 in the plot of ln{A(n,£)/(l-n/N

3

} versus n . The methods discussed above characterize the microstrains present in the crystallites

by a single parameter. In our opinion these single-line Fourier methods are less practicable

than a recently proposed rapid single-line integral breadth method [3]. In this method, it was assumed that the crystallite size effect gave rise to a Cauchy profile (corresponding to

S A (n) = exp{-n/N

3

}) and that the microstrain effect gave rise to a Gaussian profile (corresponding to A

D (n,£) = exp{-2rt2£

2

n

2 <e2>}).

On the other hand single-line Fourier methods may provide additional information if a 2 certain dependence of <e (n)> on n, say f(n), is assumed. It follows

A(n,£) = (l-n/N

3 )[l-2rtVn2 f(n)]

235

(4.26)
For example, f-^(n) = c^/n (c^o) (cf. eq. 4.16)) or f

2 (n) = exp{c3 n+c4

}(c

3

,c4

<o) may be

applied. With f-^(n) from eq. (4.26) N

3 and c.^ can be solved for any two Fourier coefficients ACn^A) and A(n2> £) and with f2 (n) from eq. (4.26) N

3 , c 3 and c^ can be solved for1^

three Fourier coefficients A(n

1

,£), A(n

2 ,£) and A(n

3 ,£).

Applying f-^(n) Gangulee [44] found valid and invalid solutions (e.g. negative size and^ strain values). Especially for very small values of (n^,n

2

) invalid solutions were obtaine<

which may be attributed to the invalidity of the approximation f-^(n) for very small values | of n (see the discussion below eq. (4.17)) and experimental errors (e.g. the "hook" effect). For this case, Mignot and Rondot [57] tried to overcome these difficulties by fitting a pol}j

nomial of second degree (cf. eq. (4.26) after substitution of f^(n)) to the experimental

Fourier coefficient curve using a least squares method for different ranges of n-values and

selecting the most appropriate solution.

No experience exists with f

9 (n). It may give better results for small values of n. 2

f 9 (n) was applied in reference [40] to determine <e (o)> by extrapolation from the values

2 of <e (n)> obtained by a multiple line method.

5. Line Profile Analysis Using a Non-Ideal Standard Specimen

Because of unavoidable alignment errors [6] the standard line profile g cannot be

calculated, but has to be determined experimentally. However, the ideal standard specimen

is seldom available. In many cases a standard line profile is recorded from a standard

specimen different in composition [58] or even made of another material (e.g. reference

[59]). Furthermore, the standard specimen may contain some structural defects (e.g. ref- erence [60]). This leads to non-ideal, approximate standard line profiles at an incorrect

sine range with an incorrect broadening. Large errors may result from such defects (cf.

reference [61,62]).

It is the purpose of this section to present some rules for the assignment of origins

to the profiles in the deconvol ution procedure and to propose correction factors for the

elimination of errors due to incorrect broadening of the standard line profile.

5.1. The ideal standard line profile

The ideal standard line profile g comprises exactly the broadening by the instrumental

conditions and the wavelength distribution present in the h profile. The ideal standard

line profile can be described by the convolution g^g5

, where g

1 and gs represent the broadening by the instrumental conditions and the wavelength distribution respectively. The origins for g

1 and g are taken at the same value of sine and the centroid of gs is at the origin

The ideal standard line profile is not really constant over the sine range of the f profile. In practice, applying Stokes 1 method (eq. (2.2)) some position in the sine range

of the h profile should be chosen as a reference for the ideal standard line profile. For that purpose the centroid of the h profile is used in this paper. Thus, the ideal standard

line profile should be recorded from an ideal standard specimen with an interplanar spacing

236
for

ues

ect)

poiy\

I

| c such that the centroids of the h and g profiles coincide (d

c is determined after

;orrection for the centroid shift due to instrumental aberrations). By choosing the origins

f the g and h profiles at the same value of sin0 the centroid of the resulting f profile

ill be at the origin (since <h> = <g> +<f> and <h> = <g>; the brackets denote the centroids

f the profiles on the sin6 scale).

For the subsequent interpretation of the structural broadening the centroid of the

profile has to be identified with sine = <\

c>/2dc

, where <X

c

> is the centroid wavelength

f the x-ray spectrum used. The strain values e(n) obtained from the Warren-Averbach

nalysis are then related to the reference spacing d

c> A macrostrain follows from the

ifference d

c

- d

Q

, where d

Q

is the spacing of the material under investigation when the

laterial is ideally crystallized.

It is remarked that an ideally crystallized specimen does not necessarily produce

jm ideal standard line profile because its centroid is not necessarily at the same sine

jfalue as the centroid of the h profile (cf. references [5,62]).

5.2. The non-ideal standard line profile and its relation to the ideal standard line profile

In the non-ideal standard line profile the following types of deviations are distinguished:

(i) the standard profile is measured at a position on the sine scale different

from that of the desired exact standard profile g;

(ii) the standard profile is measured from a standard with a specimen transparency effect different from that of the specimen to be investigated; and,

(iii) the physical profile f of the standard specimen is not a delta function,

but it is (slightly) broadened by strain and finite crystallite size.

In general, the measured non-ideal standard line profile gm can be described by the

convolution g

1 * g

s * f . It may be displaced at the sine scale relative to the ideal 3m

3m oJ r standard line profile and contain an erroneous broadening. With Gm

(n) being the Fourier

coefficient of the measured line profile g and F (n) the Fourier coefficient of the pro- r 3m o file f such that F (o) = G(o) = G (o) = 1, we introduce a correction factor

o o m

C<n>

" ^' -W* " gW<h)

(51)

where F m

(n) = H(n)/G

m

(n) is the approximate, "measured" Fourier coefficient of the pure

profile and the superscripts refer to the corresponding profiles.

In general, the correction factor C(n) (eq. (5.1)) is the product of the correction factors for displacement and incorrect broadening. Provided the origins for the

h and g profiles are chosen at the same value of sine, the correction for the displacement

237
can be accomplished by

C(n) = exp

2ninA

(5.2)

where A = <g > - <g> = <g > - <h>. m m

It is not necessary to calculate the centroids, since

A can be found from the initial slope of the sine Fourier coefficients B (n) of the f m m profile obtained by using the gm profile in the deconvolution. It follows that

Fm<°>

dB (n) m

v J dn

n4-o

2nA

a (5.3)

Alternatively, in case the centroids are determined the correction implied by eq. (5.2) may

be performed by choosing the origins at the centroids of the h and gm profiles.

Correction factors for the errors due to the incorrect broadening of gm

, after choosin

the origins at the centroids of the profiles, will be discussed below.

5.2.1. A standard specimen with an improper spacing

Assuming the same effects due to the instrumental aberrations in the g and gm profiles

(see also below eq. (5.5)) the displacement A (eq. (5.2)) takes the form

According to Bragg' s law, a non-ideal standard specimen with a spacing dm^d

c "projects

the wavelength distribution on the sin8 scale at a different position and with a different

broadening as compared to the ideal standard line profile.

A solution to this problem may be obtained by applying the profile fitting method [63]

In this method, an instrumental line profile is approximated by a small number of Cauchyfunctions. Several instrumental line profiles taken over a large range of 9 values are

measured. The parameters for the Cauchy-f unctions are obtained by fitting. Normally, an instrumental line profile is needed at an unmatched 8 value. The parameters necessary for

the evaluation in terms of Cauchy-functions of that profile are obtained by interpolation.

However, in case of size-strain analysis on the basis of the method developed by Warren and

Averbach, the Fourier coefficients of the pure profile are required. Therefore, a direct

description in terms of Fourier coefficients is preferable. For the determination of

Fourier coefficients of instrumental profiles at unmatched 0-values an analogous interpolation procedure can be followed as for the determination of the parameters of the

Cauchy-functions.

Although the above mentioned interpolation procedures are possible methods for sizestrain analysis, it should be recognized that these procedures are laborious and still do not give exact results. Moreover, any change in the experimental conditions (e.g. slit

238
losingb

iies

ects"

width, x-ray spectrum 12

, absorption in the specimen) obliges to measure again the instrumental profiles.

An alternative, analytical and quick method is possible to correct for standard pro- files measured at incorrect 0 values [61,62]. The method is based on an analytical

description of the Ka doublet [11]: The standard line profile g(x), with x defined on a sine scale, is taken as the sum of the and components. The shape and position of

the components is related to a standard profile g

c (x) intermediate between that of the

and a

?c components, g

c (x) is considered as the convolution of

. the spectral profile gtW c of the intermediate wavelength distribution and the profile g (x) due to the broadening

c C caused by the instrumental conditions. For g

c (x) the generally accepted Cauchy wavelength

distribution is adopted [6,64]. This model has been experimentally verified [62,65].

Applying eq. (5.1) and using the model for the Ka doublet the correction for the

incorrect broadening of gmcan be accomplished by

r ( > Gc,m

(n)

(1+R) expI-inR^} * Rmexp{ina

m

} 1 G%) (1+R

m

) exP{ _1'nRa} + Rexp{ina} A

c

X exp {- «g (1- - !_)} (5.5) m c

where G

n (n) is the Fourier coefficient of g

1

; R = g (max)/g (max); a = n (<k9>

-

Of 2 ^

1 <\^>/d(l+R)a, where <A-j> and <k^> are the centroids of the and wavelength distributions respectively [<^

c

> = (<\-j> + R<A.2> V(1+R)] and w 1S tne weighted average (cf. <A-C

> ) of the widths at half height of the spectral and components. From experimental

results on the determination of the broadening by the instrumental aberrations [11] it was concluded that for practical cases G

1 (n)/G1 (n) = 1, especially for the small values

of n, which are of interest in size-strain analysis. Then the correction factor C^(n)

can be calculated since all other factors are known 13 . R should be taken from the h profile since Rm

is not necessarily equal to R [66]. A method to determine R-values is

given in reference [11].

in

' 12 Assuming that a high angle reflection of a standard specimen is broadened only by the

and x-ray spectrum used [63] a high angle reflection has to be measured again with change

of x-ray spectrum.

13 In addition it is noted that if a deconvol ution procedure cannot be applied [10,30] the

Fourier coefficient of the pure f profile can be approximated by dividing the Fourier

coefficients of the h profile by (1+R)

1 [exp{-inRa} + Rexp{ina}] exp{-nnw/2da} . This

is a better correction than a single elimination, since not only the component is removed from the line profile, but also the broadening by the spectral component (see

also Section 2.2.2. ).

239
5.2.2. Standard specimen with an improper transparency

Specimen transparency causes an asymmetric line broadening: an extension of the line

profile to lower sine values. As compared to a specimen with no transparency at all, it ce be shown that for a specimen with an effective linear adsorption coefficient u the centroicj

of the line profile is shifted on the sine scale by 6

t = -sin6c cos

2

6

c/2ur, where r is the

radius of the goniometer and 6

C corresponds to the centroid on a sine scale (cf.[67]).

A standard specimen with a linear absorption coefficient Mm different from the linear

absorption coefficient u of the specimen to be investigated diffracts in a different sine

range with a different broadening as compared to the ideal standard specimen.

The displacement A (eq. (5.2)) of gm with respect to g is

2 sine cos e , , A, = - §- (-^ - —) (5.6)

2 2r v u u M m

M

Using eq. (5.1) and a model of Keating and Warren [68] for the broadening due to

^ specimen transparency it follows that the correction for the incorrect broadening of gm

, in case "infinitely thick" specimens are considered, is performed by

(l-2nin6

t/a)exp{2nin6

t/a}

C

2 (n) = (l-2nin6

t m/a)exp { 2nin6

t m

/a} (5 " 7)

j?

a It is remarked that the model of Keating and Warren is less accurate when the angular

aperture of the diverging slit is not smaller than that of the receiving slit and the

absorption coefficient is very low [69].

5.2.3. Standard specimen containing structural defects

Many standard specimens give rise to a slight structural broadening (cf. reference

[60]). Applying the Warren-Averbach method (see Section 4) from eq. (5.1) the following correction factor for the incorrect broadening of gm

can be obtained:

C

3 (n) = F Q

(n) = A^(n)A^(n 5 £) = l-n/N30)(l-2n

2£2

n2<e^(n)>) (5.8)

where the subscripts "o" refer to the non-ideal standard specimen. The distortion sine

coefficients of the non-ideal standard specimen were assumed to be zero, which is exact

for symmetrical distortion broadening. The crystallite size distribution in the non-ideal

standard specimen was neglected, which is justified for a correction factor.

240
5.2.4. Combined effects

In case the effects of different spacing, different specimen transparency and structural

broadening in the standard specimen are superimposed it follows that

A = A

x

+ A

2 and C(n) = C1 (n)C2 (n)C3 (n) . (5.9)

5.3. Effect of an approximate standard profile in size-strain analysis

5.3.1. Consequences of the displacement A of the standard line profile

Clearly, values calculated for the strain will be heavi-ly affected because the fractional displacement A/a can easily be of the same order of magnitude as the strain e(n).

S Since the average crystallite size follows from dA (n)/dn for (n-i-o) (eq. (4.22)) it can be shown from eqs. (5.1) and 5.2) that in principle the crystallite size is not affected by a displacement. However, because of the "hook" effect (Section 4.4.4.), the

first cosine Fourier coefficients are normally determined too small. The usual method of

correction for the "hook" effect consists of extrapolating the straight portion of the

cosine Fourier coefficient curve at small values of n to n = 0. Therefore, in practice

the crystallite size determined will be affected by the displacement too. Errors due to the displacement are avoided when the origins for the Fourier series

evaluation of the profiles are chosen at their centroids (see discussion to eqs. (5.2)

and (5.3)).

5.3.2. Consequences of the incorrect broadening of the standard line profile

By power series expansion of the exponentials in eqs. (5.5) and (5.7) the product

C^(n)C

2 (n) can be approximated by

1_n2Rmam/2 Wnu, i i l+2n

2

n

262/a2

f - ft . r fr*\r fr>\ - m m r 7tnW f 1 lx, w t . (D.1U)

Cl<n>C2(n> " ^ ' ' "= % V] * 1+2*W /a

2

t ,m

Thus, within the approximations used, this factor is real. Since all quantities in eq. (5.10) are known a corrected cosine Fourier coefficient can be defined (cf. eqs. (5.1)

and (5.8)):

(n,£) = A

m (n,A) C 1 (n)C

2 (n) = V"'^ (5 - 11} ^corr

A^(n) A^(n

241
cow

With respect to the separation of size and strain effects, a plot of In A (n,£) versus

2 corr 2 m £ and a plot of A

m (n,£) versus £ (classical and modified methods; Sections 4.2 and

4.3) both yield

and

A*(n) = A

S (n)/A^(n) (5.12) 14

<e*(n)> = <e

2 (n)> - <e^(n)> (5.13)

The relation between the measured crystallite size

m

, the true crystallite size

and the crystallite size q of the standard specimen follows from eqs. (5.12) and (4.6)

(N

3 ( m>

_1 =

" ^.(P"

1

• < 5 " 14 >

Note that this relation also holds when B^(n,£) ± 0 (see Section 5.2.3.). In an x-ray

diffraction study of deformation in Ag-Si alloys [70] formulae related to eqs. (5.13)

and (5.14) were used. These equations were applied for relative determinations. From

the treatment given here it may be concluded that accurate relative determinations using

these equations are possible even in the case that the line profile under investigation

and the line profile of the reference specimen are at different sin6 values, provided the

correction implied by eq. (5.11) has been performed. Furthermore, from these relations

the accuracy of size-strain analysis can be estimated.

From eqs. (5.1), (5.5), and (5.7), it follows directly that

dA

m (n,£)

dn =

TtW /_1 _ _l\

(5 15)

N- 2a d

A 1 (o.io)

3,m l m

which means that can be determined directly from the A (n,£) values. Omittance of the

3,m m spectral term in eq. (5.15) produces errors of a few percent in the value calculated for

the crystallite size.

Finally, because determinations of crystallite size and strain can be performed independently of distortion (eq. (5.14)) and strain (eq. (5.13)) in the standard specimen, it

might be suggested that, for example, with a view to extinction problems, for the determination of crystallite size a slightly deformed standard specimen should be used and for

the determination of distortion colloids as a standard specimen should be used.

14rr — -j. 1 1 aS For an ideal standard specimen A (n) = 1 (infinite particle size) for all n. In practice,

0

S the standard specimen has a large, however finite, particle size q. Hence A

Q

(n) tends

to zero at large n. Combined with the effect of counting statistics this may be responsible (cf. eq. (5.12)) for the large spurious ripples in the Fourier coefficients of a deconvoluted line profile at large n (n > N

3 Q

) (see also Section 3.3.1.).

242
6. Determination of Column Length Distribution

In many areas of applied science knowledge of the particle size distribution is desired.

I For example, in the field of catalysis size distributions influence the activity and selecj tivity of the catalyst.

It has been first shown by Bertaut [71,72] that the column length distribution can be

c

i obtained from the second derivative of the size Fourier coefficient A (n) (cf. [5,73]:

d

2AS (n) = Ps (n) = Py (n) (6.1)

dn

2 <N

3,s

> n

where p

g (n) and Pv (n) denote the area weighted and volume-weighted distributions respectively. Obviously, the distribution functions can also be obtained directly from a purely

S size-broadened line profile f (x) according to

+a/2

d

2 AS (n) _ _ 4n2 f x2

f

S (x)exp{ 2^^} dx (6.2)

dn

2 a3 -a/2

The mean values of the p

g and p

v distributions are the area-weighted and volumeI weighted average crystallite sizes (mean column lengths) <N^

s > 15 and <N

3 v

> respectively,

measured in unit cells along the (00£) direction. The following relation holds

<N 3,v

> = <N

3,s

> 7 <N

3,s

>

" (6 ' 3)

<N~ > can be obtained from the integral breadth of the line profile (by application of the

•5 , V Scherrer-equation <I*L > = A./6cos0, where p is the integral breadth on a 20-scale, 0 can be

o , V obtained from the sum of the real parts of all Fourier coefficients), whereas <N

3 s > follows

from the initial slope of the Fourier coefficient curve (Section 4.4.3.). Since

2 2 <N~ > > <N- > , as obtained from the Schwarz's inequality, it follows

J , 5 O y S

<N3v> I

<N 3s> (6.4)

2 Thus, for a monodisperse system of spheres with diameter N^, it is obtained <N^

s > =3

N d and <N

3 v> = |

N d. The differences between <N

3 s > and <N

3 v

> will be the larger the

broader the distribution, as the small crystallites have a relatively large contribution

in the area-weighted average.

When the relation with the surface area available in the specimen is desired (e.g.,

as with catalysts), only the area-weighted size and distribution are of interest. For

15 It should be recognized that the symbol N

3 as used elsewhere in this paper, actually

denotes <N 0 >.

3 , s

243
example for a polydisperse system of spheres the surface area S is given by the same relation as for a monodisperse system [74]:

S = 4M/p <N

3 s > (6.5)

where M and p are mass and density of a sphere, respectively. It is concluded that for the

relation between x-ray diffraction line broadening and surface area the Fourier methods

discussed in this paper should be preferred over integral breadth methods.

Since for the calculation of the column length distribution a twofold differentiation

of the Fourier coefficient curve is required, the resulting distribution function is highly

sensitive to error propagation. Erroneous oscillations on the distribution may be obtained,

which can easily be of the same order of magnitude as the actual p values 16. However, it

is interesting to note that experiment shows that from the error affected volume-weighted

distribution functions mean crystallite sizes are obtained which are close to the values

derived from integral breadth measurements [75,76].

The spurious features may be caused, for example, by low accuracy in the tails of the

line profile and truncation effects. The consequence of the "hook" effect (Section 4.4.4.)

should be noted: negative p values for very small n. Furthermore, in case size and strain broadening occur simultaneously, both influences

have to be separated, which in practice can be done only reliably for small values of n (cf. Sections 4.2. and 4.3.). Therefore, in our opinion, a trustworthy column length distribution can only be obtained if the line profile to be investigated is purely sizebroadened 17 . With this restriction in mind, some of the methods to remove or to avoid

the spurious features on the size distribution functions will be discussed shortly.

6.1. Elimination of the effect of truncation on the

size distribution function

Using Bertaut's result [25] truncation of the profile (cf. Section 3.1) leads to

2 S d Vn) - 1_ r- sin7tn [ P-(n') sin n(n-n')

dn

2 "<N3,s

>

C TM

+ J

n < n

- n,) J

1 r sinnn . , , _ Ps,r/ n ^ (6.6)

3,s 3,s

16This situation resembles the problems to be dealt with in calculating the radial distribution function from diffraction data of amorphous materials.

17 In case both size and strain broadening occur, size distribution functions may, for

example, be obtained by small-angle x-ray scattering (e.g. [77]).

244
where Am

s is the size ... Fourier coefficient obtained from the truncated profile and p is m

K K s,m

the apparent distribution function. From the first term within the brackets it is seen

that p is negative for n = o, while for the other n values p' = p s,m "

r

s r s,m

The true distribution p

g should be obtained from p^ by deconvolution. Dealing with

the form (sin 7tn)/nn, standard deconvolution procedures are impossible. This problem has

been tackled by Popescu and Benes [78]. A modified version of the successive convolution

unfolding method [79] was proposed: the convolution product should only be calculated over

the range where the distribution function has no vanishing values. Outside this range the

convolution has no physical bearing. This "finite-range" successive folding method gave

reasonable results for a theoretical experiment. It was shown that a qualitatively correct

distribution function might be obtained from a line profile taken over a range of two or

three times the line width, instead of the normal range of five to six times the line width.

6.2. Smoothing and iterative methods

The fluctuations due to counting statistics in the tails of the line profile may lead

to serious errors in the distribution function. Smoothing of the tails of the profile may

2 lead to improvement. For example, the tails may be approximated with a/x + b [80] or

aexp{-bx} [81] where a and b are found by curve fitting.

Very often negative values are observed in the distribution function obtained. These

are physically unrealistic. Then the following iterative method may be applied to eliminate

the spurious features on the distribution function [81]:

(i) Cut off all negative regions in the distribution function.

S Recalculate f (x) by Fourier tran< "corrected" distribution function.

S (ii) Recalculate f (x) by Fourier transformation (cf. eq. (6.2)) from this

S . . (iii) Replace, except for its tails, this f (x) by the experimental function.

(iv) Calculate the new distribution function from this modified line profile.

(v) Repeat steps (i) - (iv) until a sufficient convergence is obtained.

The Fourier method can be avoided. In case of pure size broadening the line profile

equals

2

,S, v 1 V""* , n sin nnx , c f (x) = ^TTT 2^ P

s (n) ~2— (6 - 7)

3,s n=l sin nx

This result can be considered as a set of linear equations which may be solved by leastsquares methods. It can be shown that the resulting P

s (n) has no essential differences

with the result obtained from the second derivative of the Fourier coefficient (eq. (6.1))

[76]. The same spurious features are present. Hence, as compared to the Fourier

method, no specific advantages are expected from methods based on eq. (6.7). Nevertheless, in case one desires to apply eq. (6.7), of course, methods can be developed to suppress the oscillations [75,76], as for the Fourier method.

245
6.3. The true particle size distribution

From x-ray diffraction line broadening only the mean column length and the column

length distribution can be obtained, whereas one wishes to know the average particle

size and the particle size distribution.

A first limitation arises from the recognition that each particle need not be mono- crystalline. In that case, the centroid of the distribution function as determined by a technique as small-angle x-ray scattering will be larger than the one obtained from the

x-ray diffraction line broadening method (for an example see reference [75]).

A second limitation is introduced by the notion of column length (cf. Section 4.1.)).

Particle size and particle size distribution are intuitively expressed in terms of a para- j meter A such as the diameter of a sphere or the edge of a cube. In general, these measures

1

are not compatible with the mean column length and the column length distribution. Thus

2 I

for a system of spheres with diameter it follows <N^

s > =

g and although the system

is monodisperse a column length distribution does exist. If the particle size distribution

in terms of the parameter A is denoted by p(A), then for the total number of columns with

lengths between n and n + dn it follows

where g(n,A)dn corresponds to the number of columns with lengths between n and n + dn presj

ent in a crystallite of length A and A

Q

(n) is the minimum value of A for a certain n (for aj

sphere as well as for a cube, viewed along the edge, A

Q

(n) = n). Now the question is to derive p(A) from the P

s (n) data. Smith [82] has considered this problem for some simple

shapes and orders of reflection for crystals of the cubic system. One of the interesting | results is that irrespective of the nature of the true particle size distribution a maximum

in the column length distribution will always occur from any reflection from a system of

spheres and will never occur from any reflection from a system of tetrahedra, while a system of cubes takes an intermediate position (hoo reflection maximum; hho and hhh

reflections: no maximum).

In general, it can be said that information on the true particle size distribution can only be obtained with difficulty from the column length distribution (a third or higher

differentiation of the original Fourier coefficients may be necessary; cf. eq. (6.8)).

We are indebted to Professor B. Okkerse for his guidance and to Dr. F. W. Schapink

for his interest, with respect to our research in this field during the last years.

00

g(n,A) p(A) dA dn (6.8)

246
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analysis of x-ray diffraction profiles, Indian J. Pure Appl. Phys. 16, 620-622 (1978).

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messung, Mitt. K. -Wi In. -Inst. Eisenforschg. 21, 261-269 (1939).

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pretation of x-ray line broadening from cold-worked iron and molybdenum, Acta Cry;

7, 574-581 (1954).

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J. Appl. Phys. 24, 52-56 (1953).

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[56] Mitra, G. B. and Misra, N. K. , Line shape analysis of cold-worked magnesium, Acta"

Cryst. 22, 454-456 (1967).

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microdeformations a parti r des coefficients de Fourier d'un seul profil de raie di diffraction X, Acta Met. 23, 1321-1324 (1975).

[58] O'Holleran, T. P., McKinstry, H. A., and Stubican, V. S. , Microstrain in some oxi

crystalline solutions, Phil. Mag. 35, 485-492 (1977).

[59] Haider, N. C. and Johnston, E. E. , X-ray diffraction study of HCP metals II. Lini' broadening in polycrystal 1 ine Zn and Mg powder, Z. Naturforsch. 30a , 825-830 (197

[60] Urban, J. P., X-ray measurements of strain and mosaic particle size in annealed

tungsten powder, J. Appl. Cryst. 8, 459-464 (1975).

[61] Keijser, Th.H. de and Mittemeijer, E. J., Correction for errors in microstrain vaf from x-ray diffraction line profiles, Phil. Mag. 36, 1261-1264 (1977).

[62] Keijser, Th.H. de and Mittemeijer, E. J., Analysis of x-ray diffraction line broa

ening using a non-ideal standard specimen, Z. Naturforsch., 33a 316-320 (1978). j [63] Huang, T. C. and Parrish, W. , Qualitative analysis of complicated mixtures by pro

fitting x-ray diffractometer patterns, Adv. X-ray Analysis, C. S. Barrett, D. E. !

J. B. Newkirk, and C. 0. Ruud, eds. , 21, 275-288 (1978).

[64] Compton, A. H. and Allison, S. K. , X-rays in theory and experiment (McMillan, Lon 1935).

[65] Delhez, R. , Keijser, Th.H. de, Mittemeijer, E. J., and Pers, N. M. , van der, An experimental determination of the x-ray diffraction line broadening due to the no ideal geometry of the diffractometer; consequences for size-strain analysis,

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press.

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coefficient of specimen on position and shape of powder diffraction maxima, J. Scf Instrum. 27, 321-325 (1950). f

\

:

250

I
iKeating, D. T. and Warren, B. E. , The effect of a low absorption coefficient on x-ray

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Dineen, C. , On the correction of the shapes of x-ray diffraction line profiles for

absorption broadening, J. Appl. Cryst. 6, 474-477 (1973).

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deformation of Ag-Si alloys, Scripta Met. 6, 875-882 (1972).

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dans une poudre cristalline, C. R. Acad. Sci. Paris, 228, 492-494 (1949).

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de Bragg dans les poudres polycristal 1 ines , Acta Cryst. 3, 14-18 (1950).

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127-130 (1972).

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Cryst. 10, 191-196 (1977).

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from x-ray line-profile analysis, J. Appl. Cryst. 11, 50-55 (1978).

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by means of small-angle x-ray scattering, J. Appl. Cryst. 9, 433-440 (1976).

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to x-ray scattering intensities, J. Appl. Cryst. 1, 19-23 (1968).

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antiphase dans un alliage ordonne, J. Appl. Cryst. 7, 493-497 (1974).

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J. Appl. Cryst. 9, 187-189 (1976).

Discussion

*ion (Edmonds): Have you based the determination of the unknown line broadening

85 15

in unusual sample (e.g. Cu' Ge' ) only upon a standard which is very close in

jparency to the sample (e.g. Cu), or have you tried using standards which are very

rent in transparency, for instance aAL^?

(Onse (Delhez, Keijser, and Mittemei jer): In case one is obliged to use a non-ideal

[lard specimen, correction for an improper spacing (e.g. (5.5) in this paper) and an

t>per transparency (e.g. (5.7) in this paper) are possible (see also e.g. (5.10)). The

action factors are based on models that are approximations. Therefore, the differences

251
in spacing and transparency between the standard specimen used and the ideal standard speci

( men should be kept as small as possible. We have not investigated the effect of the cor- rections in case of large differences.

Question (Cohen): (1) Can you tell us, for example, if the standard peak is 1° or 2° or

3° from the peak to be analyzed, by what percent are the cosine Fourier coefficients

affected, in your opinion? It would be useful to have such information, as to when a correction is really needed. (2) Can you make any comments on the recent work by

M. Wilkens, where he shows that a hook effect can arise due to strains from dislocations?

2 2 2 He also has questioned the usual two order analysis, In A^ vs h + k +SL .

Response (Delhez, Keijser, and Mittemei jer): (1) Especially strain values are affected

when standard specimens with an improper spacing and transparency are used. When the

centroids of the standard peak and the peak to be analyzed are just a few degrees apart

(x = 0 is confined to the centroid in the Fourier series evaluation) already errors of more than 10 percent occur in the mean square strain values for small harmonic number

(see refs. [61-62] in this paper); the error in the crystallite size is a few percent.

(2) We have not yet studied Wilkens paper (ref. [36] in this paper) in detain, however,

a few comments are made in this paper. It seems that the domain concept (see Section 4.1.

of this paper) may introduce difficulties. The two-order analysis of the Fourier coefficients is indeed not always allowed because of the truncation of the series expansions

(see Section 4.4.4. of this paper).

i

i Question (Laddell): In deconvolutioning by means of Stokes method, what do you do when

the denominator Fourier coefficient is zero or close to zero? This could happen for

real profiles which are not strictly monotonic decreasing in both sides.

Response (Delhez, Keijser, and Mittemei jer) (See comments made by J. Cohen.) In addition

to the comment of Professor Cohen on this question, it is remarked that the profile h to

be investigated (with the coefficients H) is the convolution of the structurally broadened

profile f (with coefficients F) and the instrumental profile g (with coefficients G).

Therefore, H = F x G. When G is zero, H should be zero too, and, in principle, there are ! no problems in the deconvolution. However, because of experimental errors, difficulties

can arise when H and G are near to each other for the same harmonic number. As pointed

out by Cheng, Williams, and Cooper (ref. [16] in this paper, see also Section 2.2.1.) otheV

procedures for deconvolution may be regarded as a Stokes deconvolution with additional

smoothing (see Section 2.3. of this paper). The best way out for obtaining reliable

results is to improve the quality of the measurement.

I Comment (Cohen): Dr. Laddell has asked what happens when the Stoke's correction diverges,

because a Fourier coefficient of the standard becomes zero. Actually, in doing this

analysis, because of the various approximations, useful information is obtained only from

the Fourier coefficients of low order. This divergence does not occur in this region.

252
r Also, to obtain some idea of the accuracy of these techniques, they have been

iared to particle sizes determined by hydrogen chemisorption, in a paper by S. Sashital,

SI., S. of Catalysis, 1978. Let me add a comment to my reply to Dr. Laddell's question concerning the possible

o o irgence of An. For particles sizes 1000 A - 2000 A or less there is generally no

llem, as I mentioned above. However, when there is a broadening less than about 20

tent greater than the standard, i.e. for larger sizes, then such a problem can arise,

jrould be interesting to follow up on Dr. Laddell's suggestion that in such a case

; other more recent method than the Stoke 1 s correction could prove interesting.

253
;
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

ACCURACY OF CRYSTALLITE SIZE AND STRAIN DETERMINED

FROM THE INTEGRAL BREADTH OF POWDER DIFFRACTION LINES

J. I. Langford

Department of Physics

University of Birmingham

Birmingham B15 2TT

England

i Although the integral breadth has been used extensively as a measure of diffraction broadening for the past fifty years, there have

been few serious attempts to assess its accuracy or estimate the uncer- tainty in any crystallite size or strain derived from it. Some errors,

such as the effect of counting statistics, for example, are calculable,

but others, including the correction for instrumental aberrations, cannot

readily be treated analytically. Statistical methods can be used to estimate the uncertainty in the apparent size, but the reliability of any

estimate of the true size is influenced by the shape of the crystallites

and the distribution of size. Similarly, analysis of the strain will

depend on the validity of any assumed stress or strain distribution.

The application of the integral breadth requires careful collection

and systematic interpretation of the data, due consideration being given

to counting statistics. The broadening of standard data used to correct

for instrumental effects should be inappreciable and any assumed functions

used in the analysis must approximate closely to experimental profiles.

If these precautions are observed, the integral breadth provides a method

for determining crystallite size and strain which is both reliable and, in 1 general, easy to apply; any remaining uncertainties should then be mainly

attributable to the properties of the specimen under investigation.

1. Introduction

The integral breadth was introduced as a measure of diffraction broadening in 1926 by

|aue [l] 1 and it has since been used extensively in the study of imperfections in powder

amples. A comprehensive account of the method and its applications has been given by Klug

nd Alexander

I

Figures in brackets indicate the literature references at the end of this paper.

255
[2] and only the main features will be reviewed here. Recent developments are, however,

considered in greater detail, since these should result in an improvement in the accuracy

of strain and crystallite-size measurements.

The integral breadth is defined as the width of a rectangle having the same area and

height as the diffraction line after subtraction of the background2

, or

S P = • (1)

I - B -o -o

It is thus relatively easy to obtain, compared with the calculation of the variance of a diffraction line [3] or deconvolution procedures, such as the Stokes correction [4] or the LWL method [5]. Furthermore, the integral breadth is less prone to subjective errors

than the full width at half maximum intensity (FWHM), particularly if a least-squares

parabola is fitted to the peak in order to find 1 [6]. In any detailed study of imperfection broadening where reliability and accuracy are of importance, it is usually desirat

and often necessary, to use two or more measures of breadth, as is shown below. However,

the purpose of this review is to discuss the accuracy of the integral -breadth method and

the reliability of other measures of line broadening are not considered.

The principal disadvantage of the integral -breadth method, as with the FWHM, is that

the instrumental, broadened and diffraction profiles are assumed to approximate to analyt

functions, usually Cauchy (Lorentzian) or Gaussian. In the Cauchy case the integral breac

(or FWHM's) of the constituent profiles are additive, or

ph =

Pf + V

(2)

and for the Gaussian the squares of the breadths are additive, or

There is evidence that the Gaussian is a better approximation in many cases, particularly

strain broadening is dominant for example [7-9], and hitherto this function has been widelij

used in practice. Equally, it has been shown that the profile generated by a distribution

crystallite sizes is nearer Cauchy in form [10,11], as is the instrumental function with K} radiation [12]. It is clear, therefore, that in general the observed line will be of inte

mediate form, as will be the majority of diffraction profiles. Although equations (2) and

(3) greatly facilitate the separation of the constituent breadths, they impose a considers1

limitation on the reliability of any derived parameters and should only be used where orde

of-magnitude calculations are acceptable. A better approximation is the Voigt function, o convolution of Cauchy and Gaussian curves [12-16] and its use will be assumed in the folio

ing discussion. (See also reference [17].) j ^

2 Principal symbols are listed in the appendix.

256

ft
The integral breadth of a diffraction profile is also proportional to the inverse sum

f the Fourier cosine coefficients [4]:

p

f = A(26)/I A

n

. (4) — n —

Thus, if the h-profile is corrected for instrumental effects by means of the Stokes method,

the resulting estimate of p^ does not depend on ascribing a closed function to the constituent profiles. However, the problem of separating the contributions from various imperfections remains. The accuracy of p^ can be evaluated from the expressions given by Wilson

18-20] for the standard deviation in A

n> (See also Delhez, de Keijser and Mittemei jer:

his symposium.

)

2. Crystall ite-size and Strain Determination

The importance of careful and systematic collection and interpretation of data in any

line-broadening application cannot be overemphasized. Indeed, failure to observe certain

basic requirements can lead to serious and often unpredictable inaccuracies in derived

parameters. The recommended procedure in the integral -breadth method, assuming that the

jdata are collected by step-scanning each profile, is as follows. The technique is essen- tially the same for continuous scanning, but the analysis of errors will differ somewhat.

2.1. Data collection

Whenever practicable, data should be recorded for as many reflections as possible,

30th for the sample under investigation and for the "standard." The method can be applied

to a single line [21,22], but only at the expense of accuracy and the amount of information

tfhich can be obtained about any imperfections present. The step length should not be too

large and, though the optimum value varies from sample to sample, as a rule the step length

should not be greater than about 5 percent of the FWHM for accurate work. A counting strategy should be chosen to reduce random errors to an acceptable level, as is indicated below.

£See also reference [18].) It is customary to increase the counting time (or to time a larger

count) in the tails of a profile to reduce uncertainties in the estimated background and it

jis sometimes advantageous to reduce the standard error of the peak count in the same way.

hKfrhe usual precautions must be taken to ensure that intrinsic broadening of the standard data

is negligible and that instrumental broadening is the same as that of the specimen under

'investigation.

2.2. Correction for Ko^ - Ka^ doublet

lm") Ideally the data should be obtained with monochromatic radiation, though this is less

limportant if the integral breadth is derived from the Fourier coefficients. Otherwise, if

(a x-rays are used, a correction for the a-^"^ doublet must be applied in the majority of

jpplications. Of the various methods available, that of Rachinger [23] has been widely

257
used in the past, though the correction curves of Klug and Alexander [2] are easier to app

A more systematic approach to the problem of removing the component is that devised by

Delhez and Mittemeijer [24], or the least-squares method of Narayan [25].

2.3. Correction for instrumental broadening

3

In common with most other measures of line broadening, the integral breadth of the hf unction must be corrected for instrumental effects and the normal procedure is to obtain

(j-profile from a "standard" sample. In the present method the correction requires the pea

height, total intensity, integral breadth and FWHM of each profile. Particular care shoul

be taken in estimating the background level, but the slope is less important, provided it not too large. The variance method automatically gives an estimate of the true background)

level, but the slope is less important, provided it is not too large. The variance methoc

automatically gives an estimate of the true background [26], as does the Fourier method in

principle, though this is less sensitive to small changes in the background level [27]. F haps the most appropriate method of obtaining the background is that described by Lundy an Eanes [28]. This approach is based on the same assumption as is inherent in the variance

procedure, namely, that there is an approximate inverse-square variation of intensity in 1 tails of a profile, and it avoids the use of more elaborate methods. A useful technique i obtaining the integrated intensity, particularly when the peak-to-background ratio is smal has been devised by Rigault [29].

The next step is to see if the c[- and h-profiles are approximately Cauchy or Gaussiar) jit or whether the Voigt is a better approximation, when the Cauchy and Gaussian components mi be obtained. Langford [12 (22)] has derived the following explicit equation for a Voigt

function in terms of the parameters defining the constituent curves:

I(x) = Re jpcI oCI oGuj[7tVp

G

+ ijOj , (5)

where u)[u + iv] is the complex error function3 and

k = P

CM\, (6)

and has shown that the curve may be characterised by the ratio 2w/p. This ratio has limi

values of 2/n = 0.63662 (Cauchy) and 2(log

g2) Vn 2 = 0.93949 (Gaussian). Examination of 1

ratio for the experimental data thus gives an indication of the nature of the cp and h- functions. If 2w/0 lies outside this range, and the precautions given above have been

observed, then the Voigt function is not a good approximation to the profile. The FWHM a integral breadth are related by the equations (see ref. [12 (27)])

Re ju)[7t

i5 (2w)/2p

G +

ik]J = \

u)[ik] = p

fl/2p, (7)

3 An algorithm for evaluating the complex error function has been devised by Gautschi [30]

258
hich can be solved to give p^ and pg. Thus, provided that the lines are reasonably

Symmetrical , p^, p^g and p^, p^. can be found from equation (7) and p

f ^, p f

g are then

btained from equations (2) and (3). The method has recently been extended by de Keijser,

littemeijer and Vogels [22] to take into account an asymmetry of the g-function.

The solution of eq. (7) can be carried out by graphical methods or by interpolation

rom tabulated data, but de Keijser, et al . have also devised the following approximate

equations which are more amenable to calculation:

p

c/p = aQ + a^ + a

2

<t. 2 (8)

i ind

2 PG/P = b Q + b^U - 2/n) 2 + b^ + b

2

<|> , (9)

where <|> = 2w/p, a

Q = 2.0207, = -0.4803, a2 = -1.7756, bQ = 0.6420,

L = 1.4187, b ± = -2.2043 and b

£ = 1.8706. (10)

iThe errors arising from the use of eqs. (8) and (9) do not exceed 1 percent.

2.4. Determination of strain and crystallite size

p.^ and p^g are then separated into the contributions from size effects (Pq c> Pgc

) and

strain (Pq s

» Pqs

)» if these are the dominant source of broadening. From the Scherrer equation,

the apparent crystallite size is

e = p/K„ — = \/p cose, (11)

p £

where P

c

is the size component of p

f in units of 29, and Stokes and Wilson [31] defined an apparent strain as

r| = p

s cote 5.0 < e

2 >

2

, (12)

where p

g is the strain component of p^ and <e2 > is the mean-square strain. Thus, for the

Cauchy components

pfC cose = p Cc + p Cs sine (13)

and, for the Gaussian part,

PfGcos2e = & + Pes 51 " 20

- (14)

259
^Cc' ^Cs are then the s1ope and intercept of equation (13), B^ cos6 being plotted as a function of sine for several reflections, and Bq

c , Bqs are obtained from the slope and intj

cept of equation (14). If the intercept of (14) is zero, then there is no Gaussian size

component and the apparent size is given by the substitution of B^ for B

c in (11).

Similarly, Bq

s is substituted for B

g in (12) if the slope of (13)-

fs zero? However, in

general B^, Bg

c and Bgg are then recombined by the method given by Langford [12] to give

B

c and BsT~or alternatively the following approximate equation of de Keijser, et al. can

be used: -

P x = B^/C-W5 + h(nk

2 + 4)^ - 0.234kexp(-2.176 k)], (15)

where x = c or s. If the data are for two or more orders of a reflection, the crystallite size and

strain relate to one particular crystal lographic direction. Otherwise, the values obtained

are averaged for all reflections.

2.5. Analysis of crystallite size and strain

Equations (11) and (12) yield the average apparent size and strain. Furthermore, the

2 2 deviation of B.^ cos6 and B-q cos 0 in (13) and (14) contains three components: a random

contribution arising from counting statistics, which is calculable, and systematic contribi

tions due to the variation of the apparent size with hkl and to anisotropy of strain. If * crystallite size is the dominant source of broadening, a comparison of the apparent sizes

for several reflections often yields information about the shape and true size of the cryst

lites [32]. If, on the other hand, size effects are negligible or can be separated, the

variation of the apparent strain can be interpreted in terms of elastic anisotropy and possibly stress variations within the crystallites [31,33]. The extent to which intermediate

cases can be analysed depends on numerous factors, such as the number of orders of reflecti

which can be recorded, or the relative magnitudes of the Cauchy and Gaussian components of

strain and size breadths. In this instance it is usually better to compare parameters deri

from several measures of breadth, combined with information obtained from peak displacement

3. Sources of Error

For many years diffractionists have been conscious of the limitations of the integral

breadth method and various procedures for reducing uncertainties in the derived parameters

have been summarised by Klug and Alexander [2]. However, there do not appear to have been

any serious attempts to estimate the magnitude of the errors in crystallite-size or strain

measurements, though Wilson [18-20] evaluated the standard deviation in the integral

breadth due to counting statistics.

Inaccuracies arise from systematic errors in the collection and analysis of data, whicli in general cannot be treated analytically, and from random errors due to counting statistic?

260

nbr

inn
ledi

*hich are calculable. A major potential source of error is the use of standard data for

intt^hich diffraction broadening is not negligible. This can be avoided if a check is made to

snsure that the breadths of the cj-profiles are not appreciably greater than that of the

wavelength distribution, perhaps by measuring the a^""^ separation, if the doublet is present.

The breadths should also increase monotonical ly with angle, aside from the scatter due to counting statistics.

A second source of systematic error can arise from the removal of the line. This

Jshould be inappreciable if the correct procedure is adopted (see Section 2.2.) and is

eliminated if the component is removed by means of a monochromator.

An incorrect choice of background level can also introduce serious inaccuracies. For example, if an incorrect background results in the integrated intensity being underestimated

by 5 percent, then the component breadths of an assumed Voigt function are overestimated by

about 15 percent. Again, background errors are not easy to quantify and if possible the

level should be obtained by one of the methods given in Section 2.3. In any event background data should be collected at points well away from the profile tails if they are not overlapped by neighbouring lines. The background under the peak is usually assumed to vary

linearly with angle. This is a possible source of error, since it makes no allowance for the

increase in thermal diffuse scattering under the Bragg peak and similar effects.

A further cause of appreciable uncertainty could arise from the assumption that the

h- and f-profiles all approximate to Cauchy or Gaussian functions. The resulting error

could perhaps be evaluated, but in many cases it is reduced substantially if the Voigt

method is used to correct for instrumental effects and to analyse the breadths of the

diffraction profile. The errors arising from counting statistics with this technique are derived in Section 4. The standard deviation in the observed and derived parameters can

be calculated and, if desired, the procedure can be used in reverse to design an experiment

which will yield results having a predetermined accuracy.

If the foregoing points are observed, it should be possible to evaluate the error in

the apparent size and strain for a particular crystal lographic direction. Uncertainties

in the estimated "true" size and strain are, in general, difficult to quantify, since they

depend on the shapes of the crystallites, the distribution of size, any assumed models of stress or strain distribution, and so on. It may, however, be possible to separate these

effects from uncertainties due to data collection and analysis and to represent them as "apparent errors."

4. Errors Due to Counting Statistics

4.1. Integral breadth and FWHM

The statistical variances of the integral breadth and FWHM, together with other lineprofile parameters, have been evaluated by Wilson [18-20] for various counting strategies,

but only fixed-time counting will be considered here, since this is the method commonly

l| used in practice. For fixed-count timing and other strategies, the original paper should be

consulted.

261
Wilson assumed that observations are made at an odd number of equally spaced points

with the range adjusted until the mid point is within a step of the peak or centroid, that

the effect of variations in the step length is negligible, and that the background varies

linearly under the peak. He showed that the variance of the integral breadth is (Wilson's

equation 64)

B [Rp_ - (R - B) ] + 2 ' Rio (16)

provided the peak-to-background ratio is not too small. (For weak lines, Wilson's equation

63 should be used.

)

The points of inflection of a line profile occur at roughly half the maximum height, si the variation in intensity is approximately linear in the region where the profile is

intersected by a chord drawn at this height and parallel to the background. If the peak

is reasonably symmetrical and the linear region on one side has an intercept a and slope b,

the FWHM is

2w s hi + B - 2a), (17)

b -o -o -

and its variance (Wilson's equation 58) is

o 1 I Bn 2a

ai2*^-/f + i

+ P> (18)

where the linear region on each side of the peak is averaged over s points.

4.2. Form factor (<|> = 2w/B)

The FWHM and integral breadth are statistically dependent, since both are functions

of I Q

and B

Q

4

. The variance of <|) is therefore given by

°

2 ^-

($fcy)

2 »

2 ^' + <H>

2 °2 <« + (sflwy if) • < 19 >

where

cov /3(2w) \

9 /3(2w) \ ^ ? (—If) —o^ + (-5B- If ) o^ • ^ 2 °)

\ -o -o / \ -o -o /

4 Strictly, they are also functions of I. (i ^ 0), since a few intensities on each side of

the peak are used to obtain a and b. The resulting correlations are neglected in the

above analysis. 262
From eqs. (1) and (17)

8(2w) 8(2w)

1 anH 8p_ _ 9§_ - 3L BZ n b ana

31 ~ 8B ~ "

. 2 -o -o - -o -o L (21)

and

cov(2w,p) = - £

(22)

Hence

soo ct2

(<|>) = 4> 2

2 „2 (2wT p' b(2w) (23)

4.3. Cauchy and Gaussian components of the integral breadth

The approximations to the components of the Voigt-function breadth given by eqs. (8)

and (9) can be used to estimate the variances of p^ and p^. That of pc

is

'8p

c 8(3^ ° ( V = [w] aW + a w + [W W cov(M) ' (24)

where

cov "•> - (if ffc)

o2<I°} * ft k) °

2(5o>

+ (S §)

°2(1) (25)

with a similar expression for a (p~). From eqs. (1) and (17),

bS

(I

o + Bo

- 2a)(I - -o- -o

B J (26)

Hence,

|f - 2(I0-a)/bS, |f - 2(rB0 )/bs, | - " *'% and ff = 1/L„ -o -o

(27)

From Wilson ([18] eqs. (23) and (145)), provided that the background slope is small,

a (S) = S + RB

Q (1 + R/p_) . (28)

Hence,

2

8 C a

2 (B

c

) = a

2 (B) + p^a-^a^)2 a

2

(<|)) + p^a^a^) cov(p,<|>) P 263

(29)
where

-0 ^ cov(M) = { £ (a-Io)In + I (r§n )B - I [ P L + RB„d+R/£)] • (30) -0-0 b 0 -0 3 LM-o —

to VSimilarly,

P

2

a

2 (PG

) = °

2 (P) + P2 L"r (t>- d/") + b

2

+ 2b

2

<|>] a

2

(<t>)

+ p G l-f (-2/ic) + + 2b

2

4>] cov(p,<|>) . (31)

4.4. Integral breadth of the size or strain function

After the size and strain components of the constituent functions have been separated

by the method described in Section 2.4, PGc , pGc , or P Cg , PGg

in general are recombined to give the breadths P

c , pg of the size or strain profiles. The variance of these parameters

is

(3Bx \2 /ap

x

\ 2 /ap

x apx \

^

j °v + (

)

^+ ( 4*e j

cov(

^ • (32)

where x = c or s and, from the approximation of de Keijser, et al. given by eq. (15),

9PX

P 2

_! 8PZ

= " ZT- ^Cx^Cx + 4P

2Gx) ^

- 1 - 0. 264(1-1. 228pCx/pGx)

j str

J Cx 2p Gx

2 , „„2 *

exp(-1.228p

Cx/pGx )] - 2pGx

(p^

Cx

+ 4p^Gx )

and

2 2 a Px

Px

2pGx p Cx _^

ap=-

= "i- + 0.162 -2= exp(-1.228p

Cx/pGw

)-2pGx (p

2Cx

+ 4p

2

fix

) ] — P Cx * P Gx ____ _

(33)

264
2 2 and a (PCx ), o (PGx ) and cov(P

Cx , pfix

) are obtained in the usual way from the variances of

the constituent parameters. For example, the variance of P

c when the contribution from size

broadening is entirely Cauchy is ~

| a

2 (p£

) = a

2 (p^) = a

2 ^) * a

2 (^) , (34)

2 2 where a (P^), o (PhC ) are given by eq. (29). Similarly, the variance of p

$

when strain

broadening is entirely Gaussian is

a

2 Os

) = a

2 0^) = -|- [P^a2 (^) + P^O^)] , (35)

P fG

2 2 where a (P^q)> ct (P

n(;) are stained from ecl- (31)

From eqs. (11) and (12), the standard deviation in the apparent crystallite size and

strain, arising from random fluctuations in the count rate, are then

a(e) = f- <j(p ) (36)

and

o-(n) = 9- cr(B ) (37)

P

s i

5. Summary

The accuracy of the crystallite size and strain obtained from the integral breadths of

powder diffraction lines can be affected by numerous factors. Sources of systematic error

include the use of standard data, to correct for instrumental effects, the removal of the

Ka^ line, background corrections and the choice of analytic function to describe the line

profiles. Additionally, there are calculable random errors arising from statistical

fluctuations in the count rate.

In general, systematic errors cannot readily be calculated, but can be minimised

by means of careful experimental procedure (Section 2) and by applying various tests to ensure that their effect is inappreciable (Section 3). The Voigt function is considered

to be an adequate approximation to line profiles in the majority of applications (Sections

2.3. and 2.4.), and this is used to estimate the error in the apparent crystallite size and

strain due to counting statistics.

It is thus possible to estimate the accuracy of the apparent size and strain. There

will, however, be additional uncertainties in the "true" values which cannot readily be

calculated, since they depend on the shapes of the crystallites, the distribution of size

and strain and other factors.

265
Appendix

List of Principal Symbols

A Fourier cosine coefficient of order n -n

a,b Intercept and slope of approximate linear region of peak

B

Q

Mean background level

C (Suffix): Cauchy function

c (Suffix): Crystallite-size function

2 h

<e > r. m. s. strai

f (Suffix): Pure diffraction profile, including contributions due to crystallite and strain

2 (Suffix): Instrumental profile, including wavelength distribution

and contribution from physical factors

G (Suffix): Gaussian function

h (Suffix): Experimental (broadened) profile

I. Observed intensity at ith step

I Observed peak intensity

Kp Integral -breadth Scherrer constant

k = p£/n\

L

Q Peak intensity of 1 ine- less-background

£ True crystallite size; factor by which counting time is increased to determine background

£ Factor by which counting time is increased to determine peak

R Total number of steps

5 Integrated intensity of line-less-background

s Number of intensities on each side of peak used to estimate FWHM; (Suffix): Strain

2w Full width at half maximum intensity (FWHM)

p Integral breadth

A(26) Angular range of profile for which Fourier coefficients are calculated

e Apparent crystallite size

q Apparent strain

6 Bragg angle

\ Wavelength

((> Form factor (= 2w/p)

266
References

[1] Laue, M. von, Lorentz-faktor und Intensitatsverteilung in Debye-Scherrer-Ringen,

Z. Krist. 64, 115-142 (1926).

[2] Klug, H. P. and Alexander, L. E. , X-ray Diffraction Procedures for Polycrystal 1 ine

and Amorphous Materials , 2nd edition (Wiley, New York, 1974).

[3] Wilson, A. J. C. , Mathematical Theory of X-ray Dif fractometry (Centrex, Eindhoven,

1963) . [4] Stokes, A. R. , A numerical Fourier-analysis method for the correction of widths

and shapes of lines on x-ray powder photographs, Proc. Phys. Soc. Lond. 61 , 382-391 (1948).

[5] Louer, D. , Weigel, D. , and Louboutin, R. , Methode directe de correction des profils

de raies de diffraction des rayons X. I. Methode numerque de deconvolution, Acta

Cryst. A25, 335-338 (1969).

[6] Wilson, A. J. C. , The location of peaks, Brit. J. Appl. Phys. 16, 665-674 (1965).

[7] Murdock, C. C. , The form of the x-ray diffraction bands for regular crystals

of colloidal size, Phys. Rev. 35, 8-23 (1930).

[8] Warren, B. E. , X-ray studies of deformed metals, in Prog. j_n Met. Phys. 8, 147-202

(Pergamon Press, London, 1959).

[9] Wagner, C. N. J. and Aqua, E. N. , Analysis of powder pattern peaks from cold-worked

face-centred cubic metals, in Adv. in X-ray Anal

7, 46-65 (Plenum Press, New York,

1964) . [10] Jones, F. W. , The measurement of particle size by the x-ray method, Proc. Phys. Soc. Lond. A166, 16-43 (1938).

[11] Alexander, L. E. , Geometrical factors affecting the contours of x-ray spectrometer

maxima. I. Factors causing asymmetry, J. Appl. Phys. 19, 1068-1071 (1948).

[12] Langford, J. I. , A rapid method for analysing the breadths of diffraction and spectral

lines using the Voigt function, J. Appl. Cryst. 11, 10-14 (1978).

[13] Ruland, W. , The integral width of the convolution of a Gaussian and a Cauchy function,

Acta Cryst. 18, 581 (1965).

[14] Ruland, W. , The separation of line-broadening effects by means of line-width relations,

J. Appl. Cryst. 1, 90-101 (1968).

[15] Schoening, F. R. L. , Strain and praticle size values from x-ray line breadths.

Acta Cryst. 18, 975-976 (1965).

[16] Haider, N. C. and Wagner, C. N. J. , Separation of particle size and lattice strain in

integral breadth measurements, Acta Cryst. 20, 312-313 (1966).

[17] Hulst, H. C. van de and Reesinck, J. J. M. , Line breadths and Voigt profiles,

Astrophys J. 106, 121-127 (1947).

[18] Wilson, A. J. C. , Statistical variance of line-profile parameters, Measures of

intensity, location and dispersion, Acta Cryst. 23, 888-898 (1967).

[19] Wilson, A. J. C. , Statistical variance of line-profile parameters, Measures of

intensity, location and dispersion: Corrigenda, Acta Cryst. A24, 478 (1968).

267
[20] Wilson, A. J. C. , Statistical variance of line-profile parameters: Addendum,

[21] Gupta, R. K. and Anantharaman, T. R. , Shape analysis of x-ray diffraction broadening

from deformed tungsten, Z. Metal lk. 62, 732-735 (1971).

[22] De Keijser, Th. H. , Mittemeijer, E. J., and Vogels, A. B. P., Experiences with the

Voigt function for a single-line technique in the analysis of x-ray diffraction line

broadening, submitted for publication (1979).

[23] Rachinger, W. A. , A correction for the doublet in the measurement of widths of

x-ray diffraction lines, J. Sci. Instrum. 25, 254-255 (1948).

[24] Delhez, R. and Mittemeijer, E. J., An improved elimination, J. Appl. Cryst. 8,

609-611 (1975).

[25] Narayan, R. , The separation of Ka^ and Ka^ components in x-ray line profile analysis,

J. Appl. Cryst. 10, 351-352 (1977).

[26] Langford, J. I. and Wilson, A. J. C. , On variance as a measure of line broadening in

diffractometry: some preliminary measurements on annealed aluminum and nickel and

on cold-worked nickel, in Crystal lography and Crystal Perfection , G. N. Ramachandran,

ed. (Academic Press, London, 1963), pp. 207-222.

[27] Langford, J. I., The variance and other measures of line broadening in powder

diffractometry. I. Practical considerations, J. Appl. Cryst. 1, 48-59 (1968).

[28] Lundy, D. R. and Eanes, E. D. , A computer program for the separation of linearly

sloping backgrounds in x-ray line broadening analysis, J. Appl. Cryst. 5,

380-381 (1972).

[29] Rigault, J., Determination de l'intensite integree en step-scanning, J. Appl. Cryst.

12, 116-118 (1979).

[30] Gautschi, W. , Algorithm 63: complex error function, Commun. A. CM. 12, 635 (1969).

[31] Stokes, A. R. and Wilson, A. J. C. , The diffraction of x-rays by distorted crystal

aggregates. I. Proc. Phys. Soc. Lond. 56, 174-181 (1944).

[32] Langford, J. I. and Wilson, A. J. C. , Scherrer after sixty years: a survey and some new results in the determination of crystallite size, J. Appl. Cryst. 11, 102-113

(1978).

[33] Langford, J. I, X-ray powder diffraction studies of Vitromet samples, J. Appl.

Cryst. 4, 164-168 (1971).

Question (Jennings): If an experimental peak can be fitted within experimental accuracy by

a 3 (or 4 or 5) parameter function, clearly these parameters (or quantities derived from theti represent the most that can be said. How many parameters are required in a typical case? Dc

2 you agree with me that the coefficient of the 1/x behavior of the wings is one of the important parameters (where x is the deviation of sin 6 A from the Bragg condition)?

Acta Cryst. A25, 584 (1969).

Discussion

Response (Langford): Four parameters are required if an experimental peak is assumed to be

Voigtian--the peak height, integrated intensity, FWHM and mean background. These parameters

268
are sufficient to give the lattice strain and the apparent crystallite size, based on a volume average. A fifth parameter could be included to represent skewness if the profile

is asymmetric. Provision for background slope could also be included, but this is not important in the method described.

*. 2 The coefficient of the 1/x behavior in the profile tails is an important parameter,

since it provides a size estimate, based on an average. If this is also obtained, a com- parison of and Eg may well give an indication of the "average" shape of the crystallites.

Question (Suortti): When subtracting the background, the non- linear behavior of the TDS

was not taken into account. What effect would this have?

Response (Langford): The integral -breadth method does not produce results of high accuracy

and it is doubtful if the omission of a TDS correction leads to a measurable increase in

the overall error. However, an appropriate correction could be made if the Voigt function

is used in structure determination from powder data.

269
I

:

i
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

DETERMINATION OF COMPOSITIONAL VARIATIONS

BY X-RAY DIFFRACTION LINE PROFILE ANALYSIS

E. J. Mittemeijer and R. Delhez

Laboratory of Metallurgy

Delft University of Technology

Rotterdamseweg 137, 2628 Al Delft

The Netherlands

In the analysis of the x-ray diffraction line broadening due to

concentration variations three types of specimens are distinguished:

(a) powder samples where the "same" concentration variation occurs within each individual crystallite; (b) powder blends with concen- tration variations across the contact places of the particles (each

particle consists of a number of crystallites or domains); (c) mono- crystals where the concentration variation extends over many coherently

diffracting domains.

For the first type of specimens the concentration profile can be

obtained from the sine Fourier coefficients of a single line profile.

This analysis appears to be especially suited for the investigation of

concentration inhomogeneities occurring within the crystallites of alloy

catalysts.

For the description of diffusional homogenization of compacted

powder blends-- the second type of specimens--a method for the calculation of concentration-effective penetration curves was proposed by

Rudman. A fast and simple method is possible utilizing the position of

the maximum of an intensity band. Applying such methods best results

are obtained by modeling the powder blend as a conglomerate of concentric spheres. For the third type of specimens Houska and co-workers calculated

the concentration profile by: (i) an iterative method using two or more

orders of the reflection and (ii) computer simulation of a single profile.

Further, using a single profile, a direct method without iterations or simulations is possible. The background is dominated by thermal diffuse

scattering (TDS) and thus a linear background profile is a poor approximation to the real background. The TDS background can be calculated itera-
1. Introduction

From a technological point of view the study of concentration variations is very important, in particular in those cases where the concentration variations occur over small

(about 1 urn) and very small (about 1 nm) distances. This is illustrated by the following

examples which are closely related to this paper: (a) The numerous applications of microelectronic devices and metal coatings stimulate the interest in the diffusion-induced con- centration variations in thin metal films [l] 1

. (b) The increasing importance of alloy

catalysts necessitates the study of local concentration inhomogeneities occurring in the

small catalyst particles [2]. (c) Several technologically important binary metal alloys ar, produced by diffusional homogenization of a compacted powder blend of the constituting metal

components [3].

Lattice parameter variations broaden x-ray diffraction line profiles. Therefore the

analysis of x-ray diffraction line broadening may afford a powerful nondestructive quantitat

method to study compositional variations (cf. fig. 1).

26 -26 -26 -26

U i

Figure 1. 220 line profiles recorded with CuKa radiation from an RbCl/KCl

powder specimen interdiffused at various temperatures.

Until now there has been a very limited interest in the application of x-ray diffractio

methods to study concentration variations (for a review of the older literature see referenc

[4]). In this paper recent methods for the analysis of x-ray diffraction line broadening du

1 Figures in brackets indicate the literature references at the end of this paper.

272
to concentration variations are presented for three types of specimens: (i) powder samples

where the concentration variation occurs within coherently diffracting crystallites or domains; (ii) powder blends where the concentration variations occur across the contact

places of the particles (each particle consists of a number of crystallites or domains);

(iii) monocrystals where the concentration variation extends over many coherently diffracting

domains. Results obtained and difficulties encountered in practice are discussed.

This survey is preceeded by a development of the theory of x-ray diffraction line

broadening of a domain containing a gradient of spacing.

2. Theory of Diffraction of a Domain Containing a Gradient of Spacing

For a coherently diffracting domain the diffracted intensity can be expressed as the

Fourier series [5,6],

N

3 -l

P(h

3

) = C ^

F(n) exp27tinh

3

, (2.1)

n=-(N

3 -l)

where

N

3 -l-|n

F(n) = ]T] exp27ri£S

c (m

3

,n) (2.2) m

3=0

with

S„(m,,n) = Am . I n rAm for n > 0

c 3 m

3+|n| m3 —

= A -A .

1 1 for n < 0. m

3 m3+ |n| =

where A

m <a

3

> is the displacement of the m

3 ~th lattice plane from its reference position

3 m

3

<i3> * * n tne following, only positive values of the harmonic number n will be considered

Orthorhombic axes have been used, which transform reflections of (hkl) type into (00£) type

[7,8]. The reciprocal lattice vector is defined by h

3

b_ 3

, where |b

3

| = l/|<a

3>|. The number

of diffracting lattice planes is Hy The quantity C comprises the usual angle-dependent

factors [5]. If a

3 denotes the first interplanar spacing, the m

3 ~th lattice plane is found

at

273
m

3

Rm

3 =

m

3 ^3

+ jZ) A

k[

k=l

or in terms of the reference spacing <a

3

>

R = m, <a,> + Am <a,> (2.3) —

3

5 -3 —5

where

m_ N 0 ~l ,

, 3 3 a, siE^irf-E^KH (2 - 4)

k=l

6 k=l6

If it is assumed for our model calculations that the spacing varies linearly such that

Aj^ = (k-l)6 (2.5)

then substituting eq. (2.5) into eq. (2.4), using the approximation

J a_3 |/|<a

3

>| = 1, and

summing the arithmetic series it is obtained

A

m

3

= \

6m

3 (m3" N

3+1) (2 ' 6)

Substitution of eq. (2.6) into eq. (2.2) yields

N

3-l-n

F(n) = exp27tiA[6m

3

n + h 6n(n-N

3+l)]

m

3=0

sinnfin£(N

3 -n) ^ 7)

sinn6n£

after summing the geometric series. In this case "long-range" displacements exist, unlike in

the case of Warren's strain broadening. This Fourier coefficient is analogous to expressions

obtained by Borie [9] and Houska [10].

In practice the replacement of the sine in the denominator of F(n) by its argument and

the replacement of the summation in eq. (2.1) by an integration is justified. Hence it

follows for the diffracted intensity (N--1SN,):

274
N

3 /sirvt6n£(N--n)

exp(2ninh,)dn . (2.8)

n6n£

2 After definition of the factor S = 6£N

3 and replacement of variable n by x=n/N

3

eq. (2.8) can

be rewritten

1

P(h,)=CN*/ sinnSx(l-x) exp[2nix(N h )] dx . (2 .9)

J V, ttSx 6 6

We now introduce the notion of comparable graphs for the shape of the intensity function of a particle with lattice spacing variations [11]. As is well known, the intensity function of a

2 2 particle without lattice spacing variation is given by C sin 7tN 3

h 3 /sin nh

3 having an absolute

maximum proportional' to N

3 and a width proportional to tiy Let us define a function which

characterizes —independent of the total amount of diffracting material —the shape of the

intensity function of a particle without lattice spacing variation. This can be realized by .... 2 dividing the intensity values by CN

3 and multiplying the variable h 3 by N

3

- In a similar way

the intensity function of a particle with lattice spacing variations is normalized. Then it

is obtained

1

p

n (hn

}

J sinnSx(l-x) exp(2nixhn )dx (2 10)

-1

where

P n (h

3

) = P(h

3 )/CN

3 with hlj = h 3

N

3

.

It is concluded that the shape of the diffraction profile is adequately described by the

2 combination of 6, £ and N

3 in the factor S=6£N

3

. Characteristic diffraction profiles are presented in figure 2. For small S-values the

profile resembles the profile due to pure size broadening, while for large S-values the

profile has a rectangular shape (see also ref. [10]). This can be understood as follows: For small S

sin nSx(l-x) „ = ~

nSx 3

which is exactly the Fourier transform for crystallite size broadening.

275
For large values of S, for no vanishing values of the Fourier transform x«l, and

sinnSx(l-x) sinnSx

nSx nSx

which is the well-known (e.g. ref. [12]) Fourier transform of a rectangular function in real space.The changes in the diffraction profile can be illustrated clearly by plotting T=l-P

n (h

3 = £Nj), which represents the decrease in top intensity, as a function of S (fig. 3). For

small values of S (S<1.5), T is a parabolic function of S [a series expansion of eq. (2.10)

2 2 results in T^tc S /180]. T is small in this region of S and the shape of the profile is

approximately the ideal one. For larger values of S the subsidiary maxima rise at the cost

of the originally absolute maximum at h^= $> (cf. fig. 2). If S>6 the profile has

approximately a rectangular shape and the following formulation can be given:

Defining a^ and a

3 as the two outermost mterplanar spacmgs present, it follows for th<

difference Aa

3=a

3

- a

3 that

Aa

3 = (N

3 -2)6a3 ^N36a3 (2.11)

The distance |AH|=Ah

3

|b

3

| along the reciprocal lattice vector, where in the case of a (00£)

reflection intensity can be found, conforms to

| AH | = Ah

3 |b3

| = A(l/|a

3 ( - l/|a

3

|)- < 2 - 12 )

Using eqs. (2.11) and (2.12) it is found for the shape factor S=6H N

3 that

S = Ah

3

N

3

a3 |/|<a

3

>| (2.13)

Because the diffraction profile is of rectangular shape and the integrated intensity equals

j the constant CN

3

, it follows for the top intensity of the normalized diffraction profile that

i

P n (h

3=£N3

) = CN

3/Ah3 CN3 = l/Ah

3

N

3 (2.14) |

I Applying eqs. (2.13) and (2.14) it is finally found that

T = l-P

n (h

3=£N

3

) = 1- a

3

|/ <a

3

>| S = 1-1/S (2.15)

For those cases where the concentration profile extends over many coherently diffractinc

domains, the methods used in sections 4 and 5 divide the concentration profile into segments

involving that on the 20-axis the diffracted intensity from a specific segment is restricted

to a narrow range around the 26-value corresponding to the average composition of that

segment according to Bragg 1 s law. Each segment may consist of one or more domains. Then, ir this segmenting procedure it is always assumed that the effect of domain size broadening is 276
small, which will normally be the case. It is implied also that the additional broadening

due to the change of spacing should be small. Otherwise in the integrated intensity ascribed

to a segment intensity of its own will be missed and intensity from neighboring segments

will be present; in general these effects will not compensate each other.

(c)

S = 5.6 0.008 Ni-i

1-2

N 3 — h1 + 3N 3

Figure 2. Diffraction profiles for some values of the shape factor S. Figures 2a-c

have been normalized [see eq. (2.10)]. In figure 2d no normalization has been car- ried out to demonstrate the rectangular shape for large values of S.

277
Figure 3. The decrease in top intensity T versus the shape factor S.

Approximating the change of spacing in the domain as linear, the above discussion lecto the requirement that S should be small, say S<1.5 (dominant size broadening; see above

From the definition of S [below eq. (2.8)] it then follows for the domain size Lq(=N

3

I <a

3

>

1.5 d

I Aa t / 1

(2. If

where 6 has been substituted according to eq. (2.11) with |Aa|/|a|=|Aa

3 |/|a

3

| correspondir)

to the fractional change of lattice parameter in the segment and d=|a

3 |/£ is the lattice

spacing of the diffracting (00£) planes.

For the Cu/Ni and RbCl/KCl systems investigated in Sections 4 and 5 total fractional

changes of spacing (|Aa|/|a|)

t of 2.5 percent and 4.5 percent, respectively, occur over diffusion zones Z of the order of 5 urn. Then the criterion (2.16) for the domain size car

278

lire
rewritten as (see also ref. [13])

l d

<_ j-rHrf- (2 - 17)

which leads to Ln Cu/Nl(nm) < 173 {d(nm)}

3 * and L^bC1/KC1

(nm) 4 129{d(m)} h

. From measurements

[14] an upper estimate for the domain size in the Cu/Ni system may be taken as 100 nm.

2 It

is concluded that in general the criterion (2.17) will be fulfilled for practical cases where

the concentration profile extends over several micrometers.

However, if the entire concentration profile is restricted to a very small range in the

specimen (e.g. in the early stages of interdiffusion), a 1 percent change in spacing may be

taken as a maximum before coherence is disrupted anyhow. Then from the inequality (2.16)

LD 1 150 d . (2.18)

For these cases this criterion may lead to a critical domain size which can be smaller than

the domains diffracting actually coherently; i.e. the actual domain size corresponding to a 1 percent change in spacing may be larger than the critical domain size prescribed by (2.18).

3. Concentration Variations Within Small Crystallites

In this section we deal with the x-ray diffraction investigation of concentration

variations occurring within the individual crystallites (coherently diffracting domains). Up

to the present the x-ray diffraction method was applied to non-systematical deviations in

lattice parameter only (e.g. resulting from cold-work) by using the Warren-Averbach analysis

[5,6] based on the cosine Fourier coefficients of the pure line profile. Here, an x-ray

diffraction method will be presented for the investigation of long-range displacements (cf.

Section 2) by employing the sine Fourier coefficients of the pure line profile.

This method may be especially useful in the field of alloy catalysis [18]. Alloy

catalysts are developed to obtain a higher selectivity, a higher activity and a higher

stability (i.e. resistance against poisoning).

It is well known that deviations from the average composition can occur in alloy

catalysts. The concentration i nhomogeneity is localized within the crystallites because

significant variations of composition on a macroscale are improbable usually [2].

The line broadening phenomena to be analysed are illustrated in fig. 4, where the 400 cr^ line profiles are shown of a Au standard powder sample, a sintered AuPt catalyst (9 at% Pt)

and an unsintered AuPt catalyst (14 at% Pt). It is seen that both AuPt specimens give rise

I to considerable line broadening. In the sintered specimen, no large concentration variations

within the crystallites are expected. The line profile should then be mainly symmetrically

2 The domain boundaries may be grain boundaries which developed during interdiffusion:

diffusion-induced substructure" (cf. refs. [14-17]).

279
broadened due to small particle size, and, accordingly, the peak position should practical^

correspond with the average composition, as is observed. The line profile of the unsinter

specimen is substantially asymmetrically broadened which is indicative (see Section 3.2) q considerable concentration variations. For this case, the peak position, in general, will

correspond with the average composition, as is observed.

3.1. Method of profile analysis [19]

Crystallite size, cold work, and concentration variations contribute to the x-ray

diffraction line broadening. Here, only spacing changes due to concentration variations a considered. The presence of strain from another origin is not considered, which is justif

for alloy catalysts. Further, the composition dependence of the atomic scattering factor:

not taken into account. This is not a serious limitation, since in many cases the scatter factors are approximately equal and/or the concentration variation usually covers only a p of the possible composition range.

In Section 2 the kinematical theory of x-ray diffraction from a crystallite with a linear concentration profile has been developed. Nonlinear concentration profiles will be dealt with below.

i Analogous to the Warren-Averbach treatment [5,6], the diffracted intensity of a syste

of unequally sized binary A/B crystallites in which a lattice spacing variation due to a concentration profile is present, can be expressed as the Fourier series (see also Sectior

P(h

3

) = C y ^

F(n) exp2ninh

3

. (3.lj

n=-oo

As before, the quantity C comprises the usual angle-dependent factors and orthorhombic axe, have been used which transform reflections of (hkl) type into (00.2.) type. Here, the

reciprocal lattice vector is defined by h-^bg where ig'bg = l(a^ and ag are the 00£ spacinc

of the pure components A and B, respectively).

The Fourier coefficient F(n) = A(n)+iB(n) can be written as the product

where

and

F(n) = A

S (n)F

C (n,£), (3.2;

A

S (n) = N

n/N

3 (3.3;

F C (n,£) = <exp27ti£S

c (m3 ,n)> (3.4]

S A (n), the "size" Fourier coefficient, is real and independent of the order SL of the

c c c ! reflection in contrast to the "concentration" Fourier coefficient F (n,£)=A (n,£)+iB (n,m

N

n denotes the average number per column of pairs of unit cells a distance n cells apart.

280

P hr
The average in eq. (3.4) is the average over such pairs] and N

3 is the average crystallite

domain) size in unit cells perpendicular to the reflecting (00£) planes. The displacement

f lattice plane m

3 from its reference position m

3 aB is Am^ag and thus (n>o) [cf. eq. (2.2)]

S (mOJ n) = A . -A

c

v 3' 1 m^+n m^

(a)

101 103 26 (degrees)

(3.5)

(b)

101 103 — 26 (degrees)

Figure 4. 400 line profiles of the Au standard powder sample (a), the

sintered AuPt (9 at% Pt) powder specimen (b), and the unsintered AuPt

(14 at% Pt) powder specimen (c).

jy denoting c

ft k as the atomic fraction of component A in the k-th lattice plane and using

(egard's law which is valid for most binary systems 3, it follows that

More complicated composition—lattice parameter dependencies can be dealt with analogously.

281
m

3 +n

S

c (m3 ,n) = 6a ^ CA,k < 3 - 6 >

k=m

3+l

where

5

a = (|aA |-|aB |)/|aB

| .

r

F (n,£) is approximated for small values of n [cf. eq. (3.6)] by

V

•jfl

and thus

FC (n,£) = l+2ni£<S

c (m3 ,n)>-2nV<S

c (m3

,n)

2

> (3.7)

A(n,£) = A

S (n)-AS (n)2n

2£2±<S

c (m3 ,n)

2

> (3.8)

and

B(n,£) = A

S (n)27t£<S

c (m3 ,n)> . (3.9)i

Equation (3.9) makes evident that the presence of concentration variations within fi Fj crystallites gives rise to non-zero sine Fourier coefficients indicating asymmetrical line

broadening.

2 Obviously investigation of <S

c (m3 ,n)> is more easily done than of <Sc (m

3 ,n) >. Therefore, in the following, it is attempted to obtain information on the concentration

profile from the sine Fourier coefficients B(n,£) of the line profile.

Use of eq. (3.9) necessitates knowledge of the "crystallite-size" Fourier coefficient

S S A (n): If two orders, and £„, of a reflection are available, values of A (n) [and

2 2 <S (nu,n) >] are obtained by plotting A(n,£) versus S, [cf. eq. (3.8) and see also ref. [2

S -1

The particle size N 0 follows from {-[dA (n)/dn] , } . <S (m.,,n)> can be rewritten

o n*o c o

N

3 -l-n

<S

c (m3 ,n)> = [l/(N3 -n)]£ \+n~%^

m

3=0

n-1

=[l/(N

3 -n)]^ (A^.^ -A^) . (3.10

m

3=0

We shall restrict ourselves to symmetrical concentration profiles, which is a condition

usually met in practice. Then,
A

N

3 -l- m3 = 6

a

N

3

c " A

m

3 -1 • < 3 - n>

/here c is the known average composition of the crystallite. Substitution of eq. (3.11) into

;q. (3.10) leads to

<S

c (m3 ,n)> = [l/(N

3 -n)][n6

a

N

3 c-f(n)L (3.12)

/here

n-2

f(n)=A

n

. 1+2^ V (3.13)

k=0

tombination of eqs. (3.9) and (3.12) finally results in

f(n) = n6

a

N

3 c-[B(n,£)/AS (n)][(N

3 -n)/2n£]. (3.14)

Ml factors at the right-hand side of eq. (3.14) are known. Hence, f(n) can be calculated

or successive values of n(n=l,2, . .

. ). Then, as follows from eq. (3.13), the A

n values

n=0,l,2,...) can be calculated from the f(n) values by applying the recursive equation

A

n = f(n+l)-f(n)-An

_ 1

. (3.15)

n n iecause A

n=6a ^ c^k [cf. eq. (3.6)], ^ c^

k can also be calculated, implying the detern'nation of the concentration profile. In the above derivation no assumption is made on

he shape of the particles, but it only holds for not too large values of n [cf. eq. (3.7)].

3.2. Results from AuPt alloy catalysts

S According to eq. (3.8) the crystallite size Fourier coefficients A (n) can be determined

S rom two orders of a reflection4

. After substitution of the A (n) coefficients into eq. 3.14), the f(n) values are obtained. On these f(n) values a correction for the "hook"

iffect can be performed [19]. The resulting concentration profiles as calculated from the 200

ine profiles of a sintered AuPt catalyst (9 at% Pt) and of an unsintered AuPt catalyst (14

tt% Pt) are shown in figures 5a and b, respectively.

The result for the sintered specimen (fig. 5a) shows that in the interior of the

crystallite the concentration maintains a constant value of 9 at% Pt which agrees very well

c In case only a single line profile is available A (n) can be approximated with l-n/N

3

, where N, can be determined from the initial slope of the A(n)-curve [21].

283
with the average composition. In the surface region, enrichment by gold is observed. Th

in agreement with Auger electron spectroscopical [22] and hydrogen adsorption [23] data,

enrichment of the surface region by the component with the lower heat of sublimation (i.ej

gold) was also theoretically predicted [24,25].

From figure 5a it is seen that after the surface region where enrichment of gold is

j observed, an inversion region occurs with platinum enrichment, as compared to the average

composition. Until now this effect has not been explained theoretically by thermodynamic

models for endothermic systems as AuPt (cf. ref. [25]). After submission of the above res for publication [19], comparable results for CuNi alloys were published [26]; applying

ultraviolet photoemission spectroscopy, it was shown that the surface region was enriched

with Cu and that the composition as a function of depth below the surface did not approadj

the bulk value monotonical ly, but at least one oscillation occurred. For both endothermic

systems, AuPt and CuNi, neither the quasichemical approach nor the regular solution model

explain such behaviour. These theories explain surface enrichment on the basis of a driv

force due to a difference in the pure metal surface energies. A more general theory [27]

also includes the influence due to a difference in atomic size and the contribution of th< heat of solution. But for the systems considered the latter effects are expected to be

negligible. It is suggested that spinodal decomposition in the surface region may be

responsible for the oscillation observed.

The result for the unsintered specimen (fig. 5b) shows that the concentration variat^

are much larger than in the sintered specimen. For those concentration variations, neithe,

f other experimental data, nor theoretical predictions, exist. Since the specimens are prej

by chemical reduction it may be expected that the nucleus of the crystallite should be

platinum-rich, surrounded by a gold-rich shell. This corresponds with the experimental

observation. The platinum enrichment of the surface, as compared to the average composit

is not understood at present.

Surface compositions of alloy catalysts are determined usually by Auger electron

spectroscopy and selective chemisorption [18]. The validity of the latter technique has tL questioned [28] since the method itself may influence the surface composition. With Augev( electron spectroscopy it is possible to obtain an impression of the concentration variatic

below the surface using low- and high-energy transitions, which allow to distinguish betwe

the compositions of the first and deeper atomic layers [29]. Analogously qualitative rest can be obtained by ultraviolet photoemission spectroscopy, referred to above [26]. For m

methods it is difficult to assign a certain concentration to a specific location below the surface. Further, the destructive sputter depth profiling techniques are difficult to quantify [30], for example preferential sputtering of one of the components might occur [2, Therefore, in the field of alloy catalysts, the non-destructive x-ray diffraction method

| presented above can be useful, the more since, in contrast with the other methods mentione,

it enables to investigate the supported alloy catalysts as used in practice.

The method of profile analysis presented in this section may also be used profitable

investigate spinodal decomposition, clustering phenomena and mechanical deformation and

interdif fusion in very thin solid films which are transparent for x-rays.

284
— 16

20 . 30

L(A)

20 . 30

• L(A)

Figure 5. Atomic fraction of platinum versus the distance L perpendicular

to the reflecting planes for the sintered AuPt catalyst (9 at% Pt) (a), and

the unsintered AuPt catalyst (14 at% Pt) (b).

4. Diffusional Homogenization in Powder Blends

The fabrication of alloys by diffusional homogenization of compacted powder blends is

technologically important [3]. To control such processes characterization of the interdiffusion occurring in the powder blend is necessary. On diffusion-annealing numerous diffusion couples operate simultaneously in a powder blend. One wishes to characterize the

progress of homogenization in terms of average parameters simplifying the powder blend as a conglomerate of model systems.

X-ray diffractometry yields a description of the imperfect structure as an average over

a moderately large diffracting volume. Therefore x-ray diffraction line profile analysis

appears to be a powerful quantitative nondestructive method for the study of diffusional

homogenization in powder blends (Note the prominent changes in the line profiles on diffusion-annealing in fig. 1.)

tt

' 4.1. A model of interdiffusion: the concentric sphere model

Metallographic evidence [31-33] indicated that the geometry in binary powder specimens

may be described by the concentric sphere geometry. A perfect distribution of the minor

component A in the matrix of the major component B is assumed, or in case there are equal

portions of both components it is generally assumed that the material with the lower melting

point envelopes the particles of the material with the higher melting point at the start of

homogenization. Then, a concentric sphere with A and B constituting the sphere and the shell,

respectively, may be a reasonable approximation of reality.

A general solution to Fick's second law with a concentration independent diffusion

coefficient for diffusion in a sphere with impermeable surface was given by Crank (eq. (6.47)

285
in ref. [34]). In our case the concentration distribution f(r) at t=0 is given by

0<r<a f(r) = 1 and a<r<b f(r) = 0 (case 1)

where a and b are the radii of the nucleus and the concentric sphere respectively.

It can be shown experimentally (cf. Section 4.3.2) that at the start of interdiffu:

(very fast) surface diffusion occurs at the original interfaces of both components. Th< more realistic description of homogenization may be obtained if a pre-alloyed shell at A/B interface is thought to be present at t=0. Then, for the initial concentration

distribution f(r) it follows

0<r<a-A

1 f(r) = 1; a-^ <r<a+A2 f(r) = c

L

and

a+A

2 <r<b f(r) = 0, (case 2)

3 3 where A-j+A

2 is the radial dimension of the pre-alloyed shell and c^ = {a - (a-A-^) }/{(<

A

2

)^ - (a-A-^}. Evaluation of the integrals in Crank's formula yields

.3 o K V

2

"*

i o) sina r c(r) = *j + f Z^exp {-Dt a

2 \

E

n (4

b

J n=l 1 n

' sin a b

n

n

with E„ - ~ cos a a + 0 0 sin a a for case 1

n „ ^2 n 2, 2 n a b a b

n n

E n — cos or (a-A,) + -o—o— sin a (a-A-,) (1-c.) +

a b a b 1 n n i

"(a+A

2

) + £ cos «

n (a+A2

) + sin a

n

( a+A2^ lab a b

* n n

c^ for case 2

Ifi

m

where c(r) is the volume fraction of material A and a is the n-th positive root of the

transcendental equation ba cot ba =1.

n n

5 For the concentration profile as obtained from the x-ray diffraction experiment, the

concentration is expressed normally in terms of the mole fraction of one of the componen

Before the results can be related with eqs. (4.1), (4.2) and (4.3), more fractions shoul

be converted into volume fractions.

^qq
us;

[(a i

The quantity of material m

t which has crossed the interface after a given diffusion time

ind the quantity of material m which will cross in infinite time follow from

a/b

U-c)(£)

2 d(£) + 471

x

a/b

> 2 d(i)

3 3 m _ 8na a , 3b* b*

The degree of interdiff usion F as defined by the ratio m./m^ (cf. Section 4.3.1) can be

culated from (0 < F < 1)

F = 1- exp {-Dta„}

^ (1- ^) n=l

b* b*

a bsin a b

n n

, a 1 . c (- r cos a a + —r sin a a) E b n ab n'n (4.2)

If it is assumed that the position of the maximum of the line profile observed

responds to the concentration of which the maximum amount is present, then this position--

ressed as the "relative peak position" p^ [cf. Section 4.2.2. and eq. (4.9)]--can also be

ulated from eq. (4.1). It follows that this position usually corresponds to the

position at r=b. Taking the peak position as the reciprocal of the lattice spacing of the

lecting planes it is obtained for cubic substances (0 < P r

< 1)

P =

r 3

1

2

3 exp{-Dta } — r

-. Kl

n

J sina b n=l n -1 +

-1

(4.3)

"e a^ and ag correspond to the lattice parameters of nucleus and shell material

>ecti vely.

4.2. Methods of profile analysis

4.2.1. Rudman's method [35]

The concentration profile is divided into a number of segments. Each segment may

ist of one or more coherently diffracting domains. It is implicitly assumed that the

lin size is smaller than the critical domain size as prescribed by eqs. (2. 16)-(2. 18).

287
The present version of this method differs from the previous ones [35,36], where

concentration profile was described as a continuous function, although a stepped functi

implicitly assumed by the diffraction equations used. According to the kinematical

diffraction theory the integrated intensity Pm

of a segment corresponds to

KP =

m 16nR c (Lp ) m

Ff d/vf) A mm m m pr mm

(4

where P Q is the power in the primary beam, R is the radius of the goniometer and C = r with r and A being the classical radius of the electron and the x-ray wavelength

6 2 . respectively; LPm comprises the Lorentz and polarization factors, Fm

is the squared mod

of the structure factor (including the Debi je-Wal ler factor) and vm

is the unit cell vo A is the absorption factor which is approximately constant: A =1/2 u ,,, where u „ i m m "eft eff

effective linear absorption coefficient in the sample; pm denotes the volume fraction

compositions contained within the segment.

Previous investigators [3,31,32,35] did not consider the concentration dependence

unit cell volume. [This implies that 9 percent and 15 percent intensity lapses in the

profiles of Cu/Ni and RbCl/KCl powder blends (cf. Section 4.3) are not taken into acco Applying eq. (4.4) the volume fraction p can be obtained from experimentally dete

intensity bands (cf. fig. 1). Subsequently an effective penetration depth of segment

be calculated according to

y(x

m

} =

P

i °=y=

1

i=l

where x

m

is the average composition of segment m and then a plot of x versus y is anal to normal concentration—penetration curves. Rudman [35] proposed the degree of interdiffusion F as a parameter to character!' z< stage of homogenization. F is defined by the ratio of the quantity of material which

crossed the Matano interface after a given diffusion time to the quantity of material v will cross in infinite time:

rx(y=yM

)

J x=o

•x=l

(yM-y)dx + j

(y-y

M

>

ci

f =

x (y=y

M

)

x 2l

-'x=o (yM-y)dx

*yM + (i-yM Xi-x) 2xy,

where y^ represents the Matano interface and x the average composition of the sample

concentric sphere model the ratio F is calculated analytically in Section 4.1. The

2 dependence of F on Dt/a , where D is the interdiffusion coefficient, a is the radius o nucleus of the concentric sphere and t is the annealing time, is shown in figure 6. [ previous paper [36] a numerical error was made for the curve based on the modified con

sphere model. The correct one is presented here.]

288
ofl

)UN

0.5

(4>

original model

modified model

(Dt/a2

)

Figure 6. The dependence of the degree of interdiff usion F and the relative peak

position p

r on Dt, where D is the diffusion coefficient and t the annealing

time, according to the original concentric sphere model and the modified one

(cf. Section 4.2), for a 30A/70B powder blend.

4.2.2. Line profile simulation method [36]

Consider the segmented concentration profile. Each segment m will diffract at a certain

s(=2sin0A) value, say s m

. We assume that all the segments give rise to a diffracted profile

of the same shape. This implies that the structural broadening of a segment owing to its

concentration variation is neglected as compared to the spectral and instrumental broadening.

The intensity of the profile of segment m will be proportional to the amount of material of

segment m. Then, according to the concentric sphere geometry the profile i' m

(s) of segment m

|can be written as

of

vm

< R s-Vi>- (Rv -r y s nr (4.7)

289
where C is a constant (including the absorption factor which is approximately constant)

is the average unit cell volume of segment m,R

g is the radius of the concentric sphere,

radii r , and r enclose segment m(m=l,2,... and r =R ) and f describes the shape of t\* m I m os

diffracted profile such that f(o)=l.

The total line profile of the powder specimen will be the sum of the profiles of a" segments. Dividing the intensities by the peak intensity I q of the major component befc' interdiffusion, the total line profile is given by

v 2 = , B

5— )

' 4 f(R -r -,) 3

" ( R -r ) 3

) f ( s

" s )

'< 4

.3 /n . x 3, / j 2

1 s m-1 s m5 nr I„ t«--(R

s-rB

)-) V <

lliil

tu Mostly the spectral broadening dominates over the broadening due to the instruments rati conditions [37]. It is generally accepted that the spectral component can be described

Cauchy-f unction [38]. Therefore, this function was used as a reasonable approximation

shape function f. The full width at half maximum of this function should be determinedit i experimentally from the component of the line profile recorded from a standard speci|i§ifi

(cf. refs. [36,39]).

Results of simulations of the 220 line profile of an interdiffused Cu/Ni powder sp on the basis of eq. (4.8) are shown in figure 7 for various stages of interdiffusion (t

corresponding concentration profiles were calculated according to the concentric sphere

model ).It is proposed to characterize homogenization by the relative peak position p defined as

p

r = (p-po )/( Poo- Po ) (4

where p is the actual peak position and p

Q

and p^ correspond to the peak positions at t start and at the end of homogenization respectively, p^ can be calculated from the mea

composition. In figure 7 the behaviour of the copper-side peak maximum on annealing is indicated (dashed curve). On the basis of the line profile simulations according to th 2 concentric sphere model the relative peak position is plotted as a function of Dt/a in

J figure 6. This plot plays the same role for the relative peak position as the correspc!; one for the degree of interdiffusion; once the value of p^ is established the value of

is obtained according to the concentric sphere model. It is clear that the relative p^

position is much easier as well as much faster to determine than the degree of

interdiffusion.

290
the:

nil

4.3. Practice

4.3.1. Determination of concentration profiles and degrees of interdiffusion;

deconvol ution versus elimination

Deconvolution potentially is the best method for purification of experimentally

etermined line profiles, but often only elimination is or can be employed. If

onsiderable amounts of pure component material are still present after annealing,

econvolution is inappropriate because the Fourier series of the deconvolved line profile

i verges due to the presence of 6-functions. The elimination does not remove all

nstrumental and spectral broadening. Therefore, the behaviour of the corresponding concenration profile in the neighbourhood of y=0 and y=l is not physically significant: the range

f compositions seems to be wider than in reality. Examples of concentration--ef fective

tafenetration curves as calculated according to Section 4.2.1. from the line profiles of

i be u/Ni powder blends (0.3 atomic fraction nickel, [36]) are shown in figure 8. For experiments with equiatomic RbCl/KCl powder blends at 500, 575 and 650 °C [39] the

oncentration-ef fective penetration curves could be obtained from the deconvolved line

rofiles, except for the experiment of 0.25 h at 500 °C where the concentration profile was btained from the eliminated line profile. The corresponding degrees of interdiffusion F, btained from the concentration-effective penetration curves according to eq. (4.6), are slathered in table 1. The incomplete removal of instrumental and spectral broadening by

limination leads to an apparent enhancement of interdiffusion at short diffusion times and

n apparent retardation at long times. If F^ and F^ denote the degrees of interdiffusion

btained after elimination and after deconvolution respectively, we expect F^ >F

d for

mall F and F^ <F

d for large F. This corresponds with the experimental finding (table 1).

4.3.2. Surface diffusion

From the degrees of interdiffusion or the relative peak positions Dt-values can be

btained (cf. fig. 6). A typical result is given in figure 9. If the concentric sphere model

ould be ideally suited to describe the interdiffusion occurring, straight lines of Dt versus

I should have been obtained, with slopes corresponding to the respective diffusion

oefficients. Clearly, this is not the case. Experiments with powder blends at relatively low temperatures show that (very fast)

urface diffusion contributes significantly to the homogenization observed at the start of

omogenization [36,39]. This process is demonstrated by the 220 line profile of the Cu/Ni

owder specimen annealed at 634 °C for 15 min (fig. 10). Two maxima between the nickel and

opper peak can be observed, which show that homogenization occurred locally not affecting

he bulk of the copper and nickel particles.

291
Figure 7. Computer simulations of 220 a-^ line profiles of 70 Cu/30 Ni powder

specimens for various stages of i nterdi f fusion according to the concentric

sphere model. The behaviour of the copper-side peak maximum on annealing isiil indicated by the dashed curve. The full width at half maximum of the shape 1 -3 °-l function was taken as 0.8 x 10 A .

0.0

0.5 -

Figure 8. Concentration-effective penetration curves from Cu/Ni

powder specimens (0.3 atomic fraction nickel) as calculated

according to the method outlined in Section 4.3.1. x is the atomic

fraction nickel and y is the effective penetration distance

[eq. (4.4)]. 292
Table 1. Degrees of interdif fusion as calculated from c^-el iminated line

profiles (F

Q

) and deconvoluted line profiles (F^) and the relative peak

position p . T is the annealing temperature and t is the annealing time.

T 500 °C 575 °C 650 °C

too Fa

2

F d Pr Fa

2

F d Pr Fa

2

F d Pr

0.25 0.32 0.29

a 0.03 0.55 0.54 0.75 0.78 0.79 0.93

1.0 0.45 0.44 0.19 0.71 0.72 0.91 0.86 0.88 0.96

2.0 0.53 0.52 0.40 0.78 0.79 0.93 0.89 0.91 0.97

4.0 0.60 0.60 0.78 0.83 0.84 0.94 0.89 0.92 0.97

j Unreliable value (see text).

4.3.3. Activation energy and diffusion mechanism

It is possible to determine the activation energy of the diffusion process without

knowing the actual diffusion geometry in the powder specimens employed. Assuming one diffusion mechanism to be dominant, it is recognized that equal values of the relative peak

! position or the degree of interdiffusion at different temperatures imply identical stages of

homogenization in the powder specimen. Hence the ratios of the diffusion coefficients at these temperatures are given by the ratios of the corresponding annealing times.

From Arrhenius plots the activation energies for Cu/Ni and RbCl/KCl powder blends are obtained

I I Qcu/Ni = 7 - 6 kJ/mo1 Wl/KCl = 10 kJ/m01

As compared with data on volume diffusion [40,41] these activation energies indicate a dominant grain boundary diffusion for Cu/Ni (cf. ref. [42]) and a dominant volume diffusion

for RbCl/KCl. This difference between Cu/Ni and RbCl/KCl may be understood since it has been

experimentally shown [43] that grain boundaries do not enhance the cation dif fusi vities of

j the salts.

Figure 9 shows that the apparent diffusion coefficients become small at large annealing

times. This can be attributed to deviations in the powder specimen from the diffusion

geometry adopted: larger diffusion distances (e.g. contiguous Ni and KC1 particles,

respectively, for the powder blends considered) become dominant at longer diffusion times

causing a progressive decrease of the rate of homogenization. This problem can be dealt with

| using two theses: (i) at one temperature the diffusion coefficient D should be constant and

1 (ii) equal values of p

r or F at different temperatures indicate equal stages of

homogenization. From (i) follows that the apparent change found in D can be ascribed to a change in radius a of the nucleus of the concentric sphere since for a concentration

j independent D, Fick's second law can be solved for concentric sphere geometries in terms of

293
Dt/a . The change of a with annealing time is described by a change in the effective

particle size a =a/a , where a is a reference particle size (cf. ref. [32]). From (ii)i r e r r > follows that equal p

r or F values observed at different temperatures imply that the effe^

particle sizes are the same. In this way effective particle sizes at different temperatt

can be brought into accordance (see fig. 11). It is seen that all effective particle si

observed lie approximately on a single curve. This illustrates that one single diffusioi I mechanism is dominant in the temperature range applied, as was assumed for the determinai

of the activation energy.

1

I

4.3.4. Deficiencies of the concentric sphere model

To investigate the validity of the concentric sphere model results obtained from the relative peak position and the degree of interdiffusion can be interrelated . The values

I P

r and F calculated for the concentric sphere model are plotted in figure 12 together wi1

experimental data: theory does not agree very well with experiment. Recalling the fast

surface diffusion process at the start of homogenization (cf. Section 4.3.2.) a better

description of the diffusion behaviour may be obtained if a prealloyed layer at the inter

of both components is assumed to be present at t=0. Although a better fit of theory witf

experiment is obtained at the first stages of homogenization no improvement is observed i the later stages [39]. Other models proposed [44,45] are less general than the concentri!

sphere model and do not afford a better correspondence of theory with experiment.

For an overall description of homogenization the concentric sphere geometry remains

attractive because it is applicable to powder blends of any average composition and the

experimentally determinable particle size of one of the components enters into the

equations.

4.3.5. Comparison of both methods of profile analysis

With the Rudman method (Section 4.2.1.) a Dt-value can be obtained after the degree I interdiffusion had been determined from the concentration profile calculated from the lifi profile. With the line profile simulation method (Section 4.2.2.) a Dt-value can be obta

after the relative peak position has been determined from the line profile. It has been i shown that applying both methods the same conclusions are obtained from the experiments

[36,39]. However the relative peak position is much more rapidly obtained, and therefore

this parameter may be applied more efficiently in practice to characterize the stage of

homogenization in a powder blend.

5. Concentration Profiles in Small Epitaxial Binary Diffusion Couples

The accessibility of small diffusion zones is of great technological importance in v of the many applications of microelectronic devices. X-ray diffraction may afford a powe

tool for the investigation of interdiffusion in thin films where the concentration variat

occurs over distances of one to several micrometers.

294
O 800 C

v 900 °C

1000 °C

t(h)

Figure 9. Graphs of Dt versus t for Cu/Ni powder blends (0.3 atomic fraction

nickel); Dt values are calculated according to the concentric sphere model.

Figure 10. 220 line profile recorded with CuKa radiation from a Cu/Ni powder specimen annealed at 634 °C for 15 min.

295
8

(TJ

o 800 °C

v 900 °C

1000°C

(a)

0 0.1 0.2

Dt /a2

10

(0

500 °C

575 °C

650°C

0.05 0.10 0.15

Dt/a:

Figure 11. The effective particle sizes a

g as a function of Dt/a , where a is the

radius of the nucleus, for Cu/Ni (a) and RbCl/KCl (b) powder blends. The effect

particle size was arbitrarily set equal to 1 for the experiment at 800 °C with 1! annealing time for Cu/Ni and for the experiment at 500 °C with 15 min. annealing

for RbCl/KCl. 296
Figure 12. The relative peak position p

r as a function of the degree of

interdiffusion F according to the concentric sphere model for Cu/Ni

(a) and RbCl/KCl (b) powder blends. The experimental data points are also

shown. 297
In the following three methods for the analysis of x-ray diffraction line profiles frf small epitaxial binary diffusion couples will be discussed.

A few notes should be made in advance: (i) The penetration of x-rays is restricted

depths of several times sin0/p where u is the linear absorption coefficient of the materia

This is at the same time the limitation and the power of x-ray diffraction analysis, (ii]

With respect to the determination of the concentration profile, the spatial resolution

obtainable is directly proportional to the difference [A(2si nQ/A.)] between the reciprocal

spacings 1/d^ and 1/dg of the reflecting planes of both components, (iii) The x-ray

diffraction method is non-destructive and may be performed j_n situ , (iv) The methods to discussed in this section assume that the concentration varies monotonically with the

distance below the surface (in contrast with the method discussed in Section 3).

5.1. Basis

Consider the geometry of figure 13: the specimen may be thought to consist of a serif

segments parallel to the free surface by adopting some fixed change in concentration; the

composition of each segment may be considered constant. In this way the gradual change ii composition is approximated by a series of steps. Each segment may consist of one or morj

coherently diffracting domains. It is implicitly assumed that the domain size is smaller

the critical domain size as prescribed by eqs. (2. 16)-(2. 18)

Figure 13. Schematic representation of specimen and diffraction geometry.

Applying the kinematical diffraction theory [6], the following expression is obtains

for the integrated intensity P of segment m extending from the penetration distance y J ym [46]

P = P C(LP) F 2 (l/v

2 )A V , m o m m m m m'

2,3

(5.1

where P q is the power in the primary beam and C=r A , with rg and A being the classical

radius of the electron and the x-ray wavelength respectively; LPm comprises the Lorentz (|

298
2

fr|

1 ari zati on factors; Fm

is the squared modulus of the structure factor (including the

bije-Waller factor), and vm

is the unit-cell volume. The incident beam is absorbed

djrtially in reaching segment m, and the intensity diffracted by segment m is absorbed

j a|,rtially on leaving (m>l). The absorption by the nondiffracting segments 1 to m-1 is

scribed by

m-1

A

m = exp{-2^u

jAyj /sin0} ) m=2,3..., (5.2)

j=l

d for m=l, Aj=l. The absorption in the diffracting segment m is included in

V = g [l-exp(-2u Ay /sinG )] (2u )~ 1

1 (5.3) m m ^M mJ m mM m

'

iere 0 is the angle of incidence, u is the linear absorption coefficient, and Ay =y -y ,

, m

3 ' r m

v ' •'m 'm •'m-l

ie orientation factor gm (0<g

m<l) is introduced, since in the specimens employed in practice,

ten only a fraction of the material is in epitaxial position [47-49].

5.2. Methods of profile analysis

5.2.1. Multiple-line method

Houska and collaborators [47,50] approximated the product A

m

Vm with

y

g Ay

V. - 5TST «P -I Tjsr : Mfyioyj

m f m [ M(y)dy>

J n

iere y^ = \^m-i

+ym

)- Tnen after substitution of eq. (5.4) into eq. (5.1) and by taking the

)garithm at both sides of the latter equation it follows

y'

£nP

m

= lna " iW / ^ (y)dy (5 - 5) m

n n,i r,\ r2 1 ^(l/^m a =

P o C(LP)

m

F m T iffiT

v m

m

299
For each segment m, / u(y)dy can De determined from measurements of P from at least two

m

W 1 Jo ' in f

Ky)d>;

Jo

orders of a reflection. Then the concentration profile can be obtained from the

data using an iterative procedure starting with [47]

u(y)dy = J-2 [M(o)+u(y

m ] y^

It should be noted that approximation (5.4) is only valid for small values of the " M

tt i thickness Ay

m of the segment. In case a segment extends over several microns as can be the case for the end regions of the concentration profile, a procedure based on eq. (5.4) cann

be used. Further the need for at least two orders of a reflection and the iterative nature

1 '

of the procedure may be considered as disadvantages of this method.

5.2.2. Line profile simulation method6

When a concentration profile is assumed and segmented subsequently the integrated

intensity of each segment can be calculated by application of eq. (5.1). Then a certain

distribution function for this integrated intensity around 0

m has to be assumed. A Cauchy-shaped profile and a Gaussian-shaped profile may provide suitable descriptions at first and final stages of interdif fusion respectively [51]. After summing the intensity

values for all segments, a simulated intensity band is obtained. This method was applied

successfully [14,48,49,51,52] although the approximation (5.4) was used again. It is

suggested that a higher accuracy may be obtained by using the more exact eqs. (5.2) and

(5.3). Further, the concentration profile assumed has to be adopted by trial and error, u satisfactory agreement exists between simulation and experiment.

5.2.3. Direct single- line method

A direct method can be applied for the determination of the concentration profile [46]

J

yJ =y ,+Ay (m=l,2,...), (5.6a)

m •'m-l °m v

'

' "

where according to eqs. (5.1)-(5.3)

6 Note the parallel with the line profile simulation method for the characterization of

diffusional homogenization in powder blends (Section 4.2.2.).

300
this way the concentration profile can be calculated recursively, starting from the free

face with yQ

=0 and A-^l [cf. eq. (5.2)].

The price for this single-line method is that P should be known: P can be obtained

o o her by direct measurement [53,54] or by indirect calculation [For example, P Q

(as well as plating thickness) may be obtained from integrated intensity measurements before

erdiffusion. ] The direct method has the advantages that neither iterative nor trial and error prolures are required and that only a single order of a reflection is required. The latter

iect may be of great practical importance; since the penetration depth depends on the Bragg

lie, the reflection to be measured can be chosen in relation to the plating thickness

j>lied to provide optimal information (thin plating: low order; thick plating: high order).

5.3. Elimination of the background from an intensity band

j Commonly the intensity bands measured are corrected for background radiation by applying

linear interpolation between both extremeties of the intensity bands [47]. The background

jions of a Cu/Ni bicrystal before diffusion annealing are shown in figure 14. Note that

i signs of the slopes of the background profile at both sides of each Bragg reflection are

josite, and that no constant background level is reached. Therefore, a linear background

a poor approximation to the actual background profile. 7 The type of background observed

ig. 14) is characteristic for thermal diffuse scattering (TDS).

TDS contributes significantly to the integrated intensities measured, as is shown by the

lio a(=integrated TDS intensity/integrated kinematical intensity) for copper and nickel in

I 222 to 555 MoKa line profiles at room temperature (see table 2). These calculations are

;ed on the approximate analytical expression for first-order TDS derived by Nilsson for

Die single crystals [55-58].

In the following calculation [46] of the background of an intensity band from a ffusion zone, it is assumed that the background consists dominantly of TDS. Since the TDS

of i 1 e from each segment extends over the whole 20 range investigated, overlap should be

ken into account. The background for segment i is determined by the total TDS contribution

om all M segments, so we have

t^ZN/j (5-7)

J j=l

specially at the early stages of interdiffusion a linear background tends to overestimate

e real one at 0 positions between those of the pure components. This may result in

parently negative intensity values, although noticeable interdiffusion has occurred.

301
Figure 14. The 444 MoKa intensity band recorded from a Cu/Ni bicrystal before

diffusion annealing. The intensity scale for the background regions is 50x

larger than for the Bragg reflections. The TDS character of the background

is less pronounced for Ni , since the integrated kinematical Bragg intensity

and a are smaller (cf. table 2).

where a... denotes the ratio of the integrated TDS intensity of segment j at the 20 range o segment i to the integrated intensity P. (without background b^) of segment j. The a^.

values can be calculated from the Nilsson formula [55]. Because the P. values are unknown

the b. values cannot be determined directly.

302
Table 2. The ratio a (= integrated TDS/integrated kinematical intensity)

for Cu and Ni at room temperature in the 222, 333, 444, and 555 MoKa line

profiles according to Nilsson's formula (see ref. [55]). The elastic con- stants needed are taken from ref. [63]. The 20 integration ranges given are 1/2 and 1/4, respectively, of the separation of the Cu and Ni Bragg positions.

The uj integration range is 10°.

Integra- Integration

Cu Ni tion

Cu °Ni

range 20

(%) (%)

range 20

(%) m

222 1.08° 9.4 5.9 0.54° 5.6 3.5

333 1.77° 21.0 13.1 0.885° 12.6 7.8

444 2.79° 37.6 23.2 1.395° 22.5 13.9

555 4.97° 59.5 35.4 2.485° 35.6 21.1

By assuming that the main contribution to TDS in segment j originates from segment j,

a first estimate of background is obtained from

, M a . . . bf =£ -U- Pi, (5-8)

1 , 1+a.. J J=l JJ

where is the observed intensity (including background b.) of segment j. Since the TDS

J

1 • profiles of the different segments overlap bj is an overestimate. Then, the following

iterative method can be applied to determine the integrated intensities:

P

k = - b^ (5.9a)

l l i

M

b

k+l = \"\..p

k (5 9b)

j=l

The correct values of b. and are approached alternatingly by underestimates and

overestimates (k=l,2,...). Calculations should be continued until all segments give rise to

P. or b. values which differ less than an arbitrarily fixed fraction from the previous set obtained in the iterative procedure. It was found that normally 3-10 iterations (depending

on the reflection considered) suffice if for any segment a 5 percent difference between the

last two iterations is allowed.

The calculation of the background in the manner proposed above gives a physically more

sound base for the calculation of concentration profiles from x-ray diffraction line profiles

than a linear interpolation procedure.

303
5.4. Practice

In the diffractometer recording of the intensity bands care should be taken to collect !!

all diffracted intensity. Because of the mosaic structure of the samples at each value of \ angle 20, intensity measurements should be performed over a range of angles w and x» the ui axis being perpendicular to the diffractometer plane and the x axis being in the plane of 1 diffraction perpendicular to the uj axis. Integration over x can be established by removing;

the Soller slits, thus allowing for ± 4° axial divergence. Integration over uj is performed

by rocking the crystal.

Purification of the measured profiles by deconvol ution will be impossible usually since

the presence of pure components (pure component B will normally be visible in the line

profile, cf. fig. 13) should lead to Dirac-functions. Therefore, the component should b eliminated experimentally by using a high resolution monochromator or by computation [59]. For the latter technique the ratio R[=I (max)/I (max)] can be determined according to j

ref. [37]. Corrections for the angle dependence of Lorentz and polarization factors should

be applied [21], while the angle dependence of the scattering factor can usually be taken a? f=xfg+(l-x)f^, where x is the atomic fraction B. In figure 15 the concentration profile is shown of a Cu/Ni bicrystal diffusion-annealec

for 1 h at 925 °C. The bicrystal was prepared by electrodeposition of 14 urn Cu onto a Ni monocrystal with a 111 surface. From the integrated intensities measured before

interdiffusion, it was found that for this specimen the orientation factor g=l, indicates

perfect epitaxy. The concentration profiles were calculated according to the direct

single-line method from the 222, 333, 444, and 555 intensity bands after elimination of the;

respective TDS background profiles [46]. For this relatively thick specimen it should be

noted that the lower the order of the reflection, the smaller the part of the concentration'

jf profile which can be obtained reliably, due to decreasing penetration depths of the x-rays \ (this stresses the need for a method where the concentration profile is obtained from a single order of the reflection). It is seen from figure 15 that the results of the intensi^

bands coincide very well as far as penetration depths of the x-rays allow comparison.

In figure 16 the concentration profile is shown of a Cu/Ni bicrystal diffusion-annealec

for 16 h at 550 °C [46]. The bicrystal was prepared by vapour deposition of 2 urn Cu onto ai

Ni monocrystal with a 111 surface. From the integrated intensities measured before interj diffusion, it was found that the orientation factor g=0.70 for the vapour-deposited copper

layer. For the calculation of the concentration profile, it was assumed that g3 in eq. m (5.6b) was constant (0.70) for the segments in the region originally occupied by the pure

to

304
Figure 15. The concentration profiles as calculated from the 222 (+), 333 (x),

444 (a) and 555 (o) MoKo^ intensity bands for the "thick" Cu/Ni bicrystal

diffusion annealed for 1 h at 925 °C. It is remarked that the lower

the order of the reflection, the smaller the part of the concentration

profile which can be obtained reliably, due to decreasing penetration

depths of the x-rays.

copper layer (in the nickel region g=l), because previous experiments with the Cu/Ni system

did not show a concentration dependence of g [60]. This need not be true for other cases

(cf. ref. [48]). The concentration profiles were calculated according to the direct

single-line method from the 222 and 333 intensity bands after elimination of the respective

305
Figure 16. The concentration profiles as calculated from the 222 (+) and 333

iiii (x) MoKa, intensity bands for the "thin" Cu/Ni bicrystal diffusion annealed

1 |l if for 16 h at 550 °C.

i

TDS background profiles. The results for both orders coincide satisfactorily especially i! the surface region; at relatively large penetration depths some disagreement is observed.

The result from the 222 intensity band, however, is considered to be the most reliable, sij

. t ier the TDS calculations revealed that in the 20 region corresponding to the larger penetratic

p My depths about 60 percent of the diffracted intensity in the 333 intensity band is TDS, wher

it is 25 percent for the 222 intensity band. With respect to the TDS background, the 111

intensity band would be even more useful, but the effects of instrumental braodening are t large. The 222 intensity band is the best compromise.

It is concluded that a single order of the reflection may be selected which can be us most profitably for the calculation of the concentration profile [thick plating: high orde

(cf. fig. 15); thin plating: low order (cf. fig. 16)]. Hence, in practice the direct sine

line method may be advised provided P Q can be determined (cf. Section 5.2.3.). Otherwise

simulation method may be used.

306
I The methods of profile analysis discussed in this section open perspectives not only to ^investigations of interdiffusion (or evaporation or depletion of a component from an alloy)

put also to the investigation of mechanical deformation (e.g. by rolling, or enclosure

jaccompanying the electrocrystal 1 ization of metals) as a function of the distance to the

|surface.

We are indebted to Prof. B. Okkerse for his guidance and to Dr. F. W. Schapink for his

interest, with respect to our research in this field during the last years.

Appendix

Suggestion for a Rapid Method for the Characterization of the Stage of Interdiffusion.

In case one desires to estimate very quickly the stage of interdiffusion in a binary

system accessible to x-ray diffraction, the more accurate methods discussed in this paper may

be less appropriate.

Here a rapid method is suggested which may be useful for the characterization of the

stage of interdiffusion directly from the measured line profiles. The method requires only

the determination of the peak position in a line profile and can be used for specimens where

the concentration variation extends over many coherently diffracting domains: types (ii) and

(iii) mentioned in Section 1 and dealt with in Sections 4 and 5. The method is based on a simple model applied successfully in electron diffraction investigations on diffusion-anneal ing [61,62,16].

A.l. A model

Before diffusion occurs the specimen is described by a binary system consisting of

crystal A (N

A moles) with spacing dA and of crystal B (Ng moles) with spacing dg. After a certain time of diffusion the average spacings <d^> and <d

g

> in crystals A and B, respectively,

may be given by

<d

A

> = (1" X B,A

)dA+XB,Ad

B (A ' la)

<d

B

> = (l-xM)dB+

x MdA

(A. lb)

307
where Xg ^

is the mole fraction of component B present in crystal A after a certain time

annealing, and x^ g is defined analogously. From eqs. (A. la, and b) it follows

1

- <d

A

>

' <d

B

>

dA"d

B

XA,B+XB,A

(A.

It is assumed that the diffusion coefficient is constant and hence the quantity of materia

that has crossed the original interface is the same in both directions. The degree of int diffusion F is defined as the ratio of the quantity of material m

t which has crossed the

interface in both directions at time t to the quantity of material m^ which will cross in

infinite time [cf. eq. (4.6)]. It follows

m

t = 2X

B,A

N A = 2X

A,B

N

B

(A. 3a

and since xA g(at t=») = N /\/( N

/\ +N g)

m = 2

NA

N

B VN

B

(A. 3b;

From eqs. (A. 3a and b) it is obtained

XA,B+XB,A m ~

F (A.4)

and thus [eqs. (A. 2) and (A.4)]

<d

A> <d

B

>

dA"d

B = F (A.5)

If the peak positions after a time of annealing t are identified with <d^> and <dg>,

respectively, the approximate degree of interdiffusion (o<F<l) can be calculated using eq

(A.5). This implies that this method can be used only as long as the peaks of the two components are observed separately. It should be remarked too that this procedure does not take into account the effects of specimen absorption on the incident and diffracted x-rays

This is justified for powder blends (Section 4). For specimens with a plane geometry (cf.

fig. 13; Section 5) absorption has major effects on the line shape, but the effects on the

[ peak position are much less. Further, substrate B (cf. fig. 13) may often be considered as

j semi-infinite medium and the x-rays will "see" so much of pure component B that peak "B" hardly moves at all on diffusion-annealing. Then only peak "A", corresponding to diffraction

from the surface layers, moves significantly and thus dominates for the changes in the left

hand side of eq. (A.5).
in a

,3a)

I

It may be valuable to continue this analysis one step further: specific expression

for the dependence of F on Dt, where D is the diffusion coefficient, can be obtained if the

appropriate diffusion geometry is considered. For the concentric sphere geometry F is

calculated in Section 4.1 [eq. (4.2)]. For the plane geometry (cf. fig. 13) the derivation

is given below.

If and z B denote the thicknesses of layers A and B, respectively, it follows for the

int|| concentration profile in the specimen as a function of the depth z below the surface [34]

16

00

C A^=

z%" + sin 7~7"t~t exp |- 5^-,cos T^f^l (A.6)

a b n ^f n <vv I (zA+

z B

) 2 <vy)

where o<c

A

il. Since (dm

t/dt)=-2D(3c/8z)(at z=zft ) it is obtained from eq. (A.6) for the

degree of interdiffusion F^^/m^

p _ 2 (zA+Z

B )2 Vl - 2 nnzA

n

£ ZA

Z

B (z

fl +zR

) n=l

A B

1-exp

2 2 n

. -n n Dt

< z aV

(A.7)

In case no late stages of homogenization are considered, then in solving Fick's second

law reflection at the surfaces need no be considered (double-infinite case) and eq. (A.7) is

replaced by

(A. 8)

From the expressions for F [eqs. (4.2), (A.7) and (A.8)] 8 the dependence of F on Dt can

be calculated. Once F is determined according to eq. (A. 5) a value of Dt is found from the curve F(Dt).

This simple model with the resulting eqs. (A.7) and (A.8) for the plane geometry was applied successfully in an electron diffraction investigation of interdiffusion in Cu/Ni

foils [61,62,16]. In that case (<d

A>-<d

R >)/(d

A-d

R

) [cf. eq. (A. 5)] corresponds to the ratio

M

Q

/M where M

Q

and M are the moire spacings observed at the start of interdiffusion and after

a given annealing time respectively.

8 In fact these equations for F only hold if the concentration is expressed in terms of

volume fractions, while F in eq. (A. 4) is based on mole fractions. Conversions of mole

into volume fractions and vice versa can be performed easily, but, in view of the simplicity of the model used here, these will not be necessary normally.

309
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[18] Ponec, V., Surface composition and catalysis on alloys, Surf. Sci. 80, 352-366 (1979)

310
[19] Mittemei jer , E. J. and Delhez, R. , Concentration variations within small crystallites

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[22] Kuijers, F. J., private communication.

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(1966).

[25] Williams, F. L. and Nason, D. , Binary alloy surface compositions from bulk alloy

thermodynamic data, Surf. Sci. 45, 377-408 (1974).

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the compositional depth profile of CuNi alloys: a study by UPS, Surf. Sci. 74,

612-620 (1978).

[27] Miedema, A. R. , Surface segregation in alloys of transition metals, Z. Metallkde. 69,

455-461 (1978).

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Cu-Pt alloys; a comparison of low-energy ion-scattering results with theory, Surf. Sci.

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[29] Losch, W. , and Kirschner, J., Surface composition of polycrystal 1 ine Au-Cu alloys as a function of temperature, J. Vac. Sci. Technol. 15, 1591-1598 (1978).

[30] Lee, W. Y. , Lee, M. H. , and Eldridge, J. M. , Alloy compositional profiles by AES, ESCA,

and ion sputtering: air-exposed Fe-^_ xPd

x films, J. Vac. Sci. Technol. 15, 1549-1553

(1978).

[31] Fisher, B. and Rudman, P. S. , X-ray diffraction study of interdiffusion in Cu-Ni powder

compacts, J. Appl. Phys. 32, 1604-1611 (1961).

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concentric-sphere diffusion model, Trans. ASM, 57, 443-463 (1964).

[33] Holloway, P. H. and Mohanty, G. 0. , A diffraction study of diffusion in the Mo-Cr

system, J. Phys. Chem. Solids, 32, 2656-2659 (1973).

[34] Crank, J., The Mathematics of Diffusion (At the Clarendon Press, Oxford, 1956).

[35] Rudman, P. S. , An x-ray diffraction method for the determination of composition

distribution in inhomogeneous binary solid solutions, Acta Cryst. 13, 905-909 (1960).

[36] Delhez, R. , Mittemeijer, E. J., and Bergen, E. A. van den, X-ray diffraction line

profile analysis of diffusional homogenization in powder blends, J. Mater. Sci. 13,

1671-1679 (1978).

[37] Delhez, R. , Keijser, Th.H. de and Mittemeijer, E. J., The x-ray diffraction line

broadening to the di f f ractometer condition as a function of 20, J. Phys. E: Sci. Instrum. 11, 649-652 (1978).
.r. I'!

[38] Klug, H. P. and Alexander, L. E. , X-ray Diffraction Procedures for Polycrystall ine an Amorphous Materials, 2nd edition, p. 297 (John Wiley, New York, 1974).

[39] Bergen, E. A. van den, Delhez, R. , and Mittemeijer, E. J., An x-ray diffraction study

diffusional homogenization in RbCl/KCl powder blends, Phys. Stat. Sol. (a), 44, 517-5

(1977).

[40] Butrymowicz, D. B. , Manning, J. R. , and Read, M. E. , Diffusion in copper and copper | alloys, part IV. Diffusion in systems involving elements of group VIII, J. Phys. Chem

Ref. Data, 5, 103-200 (1976).

[41] Kantola, M. and Lindstrb'm, R. , Chemical diffusion coefficient to the KCL-RbCl couple,

Ann. Acad. Sci. Fenn. , AVI 222 , 1-11 (1967).

[42] Askill, J., Tracer Diffusion Data for Metals, Alloys and Simple Oxides, p. 11 (Plenum

Press, New York, 1970).

[43] Benard, J. and Cabane, J., Diffusion in the alkali halides, Sintering and Related

Phenomena, G. C. Kuczynski , N. A. Hooton and C. F. Gibbon, eds. , Proc. Intern. Conf. on Sintering and Related Phenomena, June 1965, pp. 1-28 (Gordon and Breach, New York,

1967).

[44] Duwez, P. and Jordon, C. B. , Application of the theory of diffusion to the formation f| alloys in powder metallurgy, Trans. ASM, 41, 194-212 (1949).

[45] Weinbaum, S. , Alloying of metal powders by diffusion, J. Appl. Phys. 19, 897-900

(1948).

[46] Delhez, R. and Mittemeijer, E. J., On the analysis of x-ray diffraction line profiles:

from small epitaxial binary diffusion couples: Determination of concentration profiles

and influence of TDS, J. Appl. Phys. 49, 4770-4775 (1978).

[47] Tenney, D. R. , Carpenter, J. A. , and Houska, C. R. , X-ray diffraction technique for

investigation of small diffusion zones, J. Appl. Phys. 41 4485-4492 (1970).

[48] Murakami, M. , Fontaine, D. de and Fodor, J., X-ray diffraction study of interdiff usicj

bimetallic Au/Pd thin films, J. Appl. Phys. 47, 2850-2856 (1976).

[49] Murakami, M. and Fontaine, D. de, X-ray diffraction study of i nterdi ff usion in

bimetallic Ag/Cu thin films, J. Appl. Phys. 47, 2857-2861 (1976).

[50] Carpenter, J. A., Tenney, D. R. , and Houska, C. R. , Method for determining compositid

profiles and diffusion-generated substructure in small diffusion zones, J. Appl. Phys

42, 4305-4312 (1971).

[51] Unnam, J. and Houska, C. R. , An X-ray study of diffusion in the Cu-Ag system, J. Appl

Phys. 47, 4336-4342 (1976).

[52] Unnam, J., Carpenter, J. A., and Houska, C. R. , X-ray diffraction approach to grain

boundary and volume diffusion, J. Appl. Phys. 44, 1957-1967 (1973).

[53] Chipman, D. R. , Conversion of relative intensities to an absolute scale, Acta Cryst.

A25 , 209-214 (1969).

[54] Inkinen, 0., Assessment of accuracy in powder intensity measurement, Acta Cryst. A25

214-217 (1969).

[55] Nilsson, N. , On the corrections of the measured integrated Bragg reflections due to r thermal diffuse scattering, Ark. Fys. 12, 247-257 (1957).

312
52;

ion

;56] Cooper, M. J. and Rouse, K. D. , The correction of measured integrated Bragg intensities

for first order thermal diffuse scattering, Acta Cryst. A24, 405-410 (1968).

57] Willis, B. T. M. , Lattice vibrations and the accurate determination of structure factors

for the elastic scattering of x-rays and neutrons, Acta Cryst., A25, 277-300 (1969).

;58] Renninger, M. , Experimental determination of the integrated contribution of temperature

diffuse scattering in x-ray reflections, Adv. X-ray Analysis, J. B. Newkirk and G. R. Mallet, eds., 10, 42-45 (1967).

j[59] Delhez R. and Mittemeijer, E. J., An improved elimination, J. Appl. Cryst. 8,

609-611 (1975).

]60] Houska, C. R. , personal communication.

61] Dijk, T. van and Mittemeijer, E. J., The effect of interdiffusion on moire patterns of

thin bimetallic films, Thin Solid Films, 41, 173-178 (1977').

62] Mittemeijer, E. J. and Beers, A. M. , Recrystal 1 i zation and interdiffusion in thin

bimetallic films, to be published.

]63] K. H. Hellwege (ed.), Landolt-Bornstein Numerical Data and Functional Relationships in

Science and Technology, New Series, Group III, Vol. 1, 6-7 (Springer Verlag, Berlin,

1966) and Vol. 2, 2-6 (Springer Verlag, Berlin, 1969).

Discussion

Question (Snyder): It seems to me that producing a quantitative concentration profile is

:onsiderably more difficult than you have shown. In general, the problems arising from the

iifferences in the mass absorption coefficients of the various crystallites present, and the

iifferences in the diffracting abilities of the end members, are very serious and certainly

:annot be neglected. Has any effort been put into allowing for these problems?

Response (Delhez and Mittemeijer): I presume your question deals with the powder blends.

rom the powder blends a concentration-effective penetration curve is obtained (cf. Section

1.3.1. of this paper). This "average" concentration profile represents in a useful way the

lighly complicated processes occurring on diffusion-annealing in powder blends. Differences

in the intensity profiles of the different segments are not important; only the integrated

(intensities are used. Problems associated with differences in mass absorption coefficients

)etween the components are small, because the incident and diffracted beams pass through a lumber of grains in which material of all possible compositions is present. Thus, for the

"ecorded intensity of a specific segment, an effective absorption coefficient can be

introduced which is constant for all segments. Allowance is made for the differences in scattering factor (cf. refs. [36, 3g]). (Corresponding remarks for the other types of

ppecimens are made in this paper).

Question (Suoritti): Referring to your concentric sphere model for interdiffusion in powder

)lends, have you studied the actual shape of the powder grains by scanning electron mi- roscopy?

313
Response (Delhez and Mittemei jer): Yes, we did. Also, optical microscopy and microprobe

analysis were applied. It was found that after the start of homogenization the geometry in

the powder blend can be approximated indeed by the concentric sphere geometry (cf. Sections

4.1 and 4.3.2. of this paper and see also the optical micrographs published in refs.

[31-33]).

Comment (Parrish): It is advisable in practice to consider the use of other analytical

methods which may be more appropriate than x-ray powder analysis. For example, in material

characterization in solid state electronics, wavelength-and/or energy-dispersive methods wi

the electron microprobe and scanning electron microscope gives far better spartial resoluti

and chemical composition for segregated and nonhomogeneous specimens than the broad area analysis of x-ray powder diff ractometry. Changing electron beam voltage and/or using Auger

electron spectroscopy permits measuring the variations in composition with respect to depth

which we use in thin films such as permalloy. Direct SEM observation provides size and

morphology of small particles, even if only a few occur as is often the case. Single- and

double- crystal methods are necessary for measuring the precise lattice parameter and Aa/a

epitaxial films. We cannot become parochial in selecting analytical methods to obtain the

best possible results.

Comment (Ladell): In the care of the epitaxial single crystal example, how do you account

j for the change in diffraction profile due to entering first a perfect crystal then

proceeding, through a region which is imperfect and then coming again to a perfect region? | Near the surface, the diffraction is dynamic, in the immediate interior kinematic. This | means shifted peaks and enhanced intensity which influence the assessment of diffusion

concentration.

i Response (Delhez and Mittemei jer): As a result of specimen preparation (e.g., vapour

deposition or electrocrystal 1 ization) the order layer (cf. fig. 13) is not a perfect single

cyrstal . Further, on diffusion-annealing a diffusion-induced substructure develops (dc. re [14-17]). Then, the kinematical diffraction equations can be applied to the diffraction frf

the material covered by the concentration variation

ids

jar

314
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

STRUCTURES FROM POWDER DATA: DATA SAMPLING, REFINEMENT AND ACCURACY

j9 W. J. Mortier

Katholieke Universiteit Leuven

Centrum voor Oppervlakteschei kunde en ColloTdale Scheikunde

De Croylaan 42

B-3030 Heverlee, Belgium

Some important features of the procedure followed to obtain accurate

intensity data for powder samples and for the determination of crystal

structures from these data are presented. One extension of an earlier

described method for the deconvolution of overlapping peaks using an experimental profile is described, with the refinement of a width function

and the c^/a^ ratio. Topics of the least-squares refinement for powders

and the statistical evaluation of the parameter sets for two isotypic

structures are discussed. A general program package for the determination of structures from powder samples is presented.

1. Introduction

There are several reasons why one has to rely on crystal structures determined from

powder samples for the research of some materials. The most obvious reason is that single

crystals might not be available. Studying the physicochemical properties of materials such

as zeolites, severe pretreatment conditions are required, resulting more often than not in

the destruction of single crystals.

Moreover, parameters of critical importance such as temperature, dehydration and

adsorption of gases are more easy to control for powders. The influence of a single

parameter can be evaluated from the gradual changes in a series of structures with this

parameter as a variable. Finite conclusions could rarely be derived from a single study

based on powder data.

This type of study requires a highly accurate data sampling with a minimum number of

variables, a most careful refinement of the structural parameters, (some of which are highly

correlated), a refinement which is based on independent observations such as the intensities

of combined peaks and the statistical comparison of isotypic structures for the evaluation of

the significance of differences between such isotypic structures.

During several years we have been able to develop the data sampling, refinement procedure

and the statistical evaluation for powder samples. The recent availability of automated

315
powder diffractometers has substantially increased the accuracy and has considerably extende

the possibilities. A package of programs, allowing a flexible data collection, crystallographic computations and statistical evaluations is presented here. All of these

have proved to be useful for the determination of structures from "poorly" crystalline

zeolite powders.

2. Data Sampling

2.1. Experimental

E if

The recordings reported here were made with an automated SEIFERT-SCINTAG PAD II powder

diffractometer using Ni-filtered Cu radiation and a proportional counter. Technical specifications of the diffractometer slit system are given in figure 1. A high resolution was ensured by the use of a large goniometer measuring circle (R = 220 mm), a low take-off angle

(3.5°), a fine focus (0.4 x 8 mm) Phillips x-ray tube, the use of a supplementary scatter

slit (S) in connection with the divergence slit (D) (width of the scatter slit S being twice

the width of the divergence slit (D)), and the use of long parallel slits (P) (50 mm long,

1 mm between the foils) in both the incident and diffracted beam. A receiving slit R of

0.2 mm was used for all samples. An angular divergence <|> of the incident beam of 0.4° was

used for the recording of the Si powder standard and for the lower 20 angles of the zeolite

powders reported. For higher angles of the zeolite powder diffraction profile, an aperture

of 0.8° was used. Fixed counting times per step of 0.01° varied between 10 s (Si) and 18 or] 36 s (zeolite). A supplementary Ni filter was placed in the diffracted beam for all samples

Figure 1. PAD II diffractometer slit system. F: x-ray linefocus;

D: divergence slit; S: scatter slit (aperture S = 2 x aperture D);

R: receiving slit; <}> divergence angle; P: parallel slit assembly.

All dimensions are given in mm.

316
2.2. Deconvolution of overlapping peaks

In an earlier study [l] 1 a method has been presented for the deconvolution of over- lapping peaks in x-ray powder patterns with the use of an experimental profile. This

profile is the a-^ profile as it can be derived from the Fourier coefficients of a combined

diffracted peak as described by Gangulee [2]. The Fourier series being unique for

a particular peak, a transformation of the abscissa accounts for the angular dependence

of the actual profile and the height is adjusted by a constant. The intensity function

for a combined profile was given as [1]

I (x) = 1

i=l

h

1 a; (1 + R) / 2

+ * ( AnPn,i " B nVi } C0S n (X " 3

i + b)

n-l

+

n=i

(B

nP M

+ A

n q n,i } sin n (x " ai + b)

j

where s = number of o-j^ comDined peaks

pr

. = 1 + R cos n A.

n, i l

q . = R sin n A.

A. = the separation of the CuKa-^ and CuKa^ components

R = intensity ratio c^/a^

a.. = abscissa values of the maximum of the a components

b = abscissa value of the maximum of the standard profile.

H. = the height of the peak

A' and B

1 = the Fourier coefficients of the standard profile.

The arguments of the sine and the cosine function must be within the range [0, 2n] and

all abscissa values are expressed in the same units. It was reported before [1] that the

width parameter, adjusting the abscissa increments such that at each diffracted angle the

same half width as the standard profile was obtained, could be derived from the width of the

component of three separate peaks in the range of the data collection. It was also

1 Figures in brackets indicate the literature references at the end of this paper.

317
Titei

mentioned that a linear variation of the width with 26 was observed in the same range. How- ever, this was found to cause problems in that it was sometimes difficult to find intense

well resolved peaks at higher 26 angles. The method was therefore extended to refine on a width parameter. A supplementary parameter (w) can be introduced on the abscissa values

modifying the arguments of the sine and cosine functions as n(x - a^) w + b. Accordingly, |:>

:f,s

A. must also be multiplied by w. Furthermore, R can also be taken as a variable and refined

A combined profile can therefore be fitted using a very limited number of parameters, i.e. H one single parameter b since the relative position of the peaks can be calculated from the

lattice parameters, one width parameter w for the entire profile and one parameter R. A wid

function can however be established by fitting a number of intense profiles refining on H.

,

b and w. R can be kept constant once determined. Therefore, usually only s + 1 parameters

are fitted for a profile of s peaks and at most s + 3 parameters. During our data sampling

all diffracted peaks were fitted giving the advantage of a more reliable intensity estimatio

for small peaks.

2.3. The standard profile

An analysis of the diffraction profiles of Si powder was made using two different

standards: the a x profile of the N (=h

2 + k2 + £2

) = 8 peak (26 % 47° for CuKa radiation)

and N = 35 (26 « 114°). It was observed that there was a difference in line shape between

the lower order and higher order diffractions. Therefore, two standard profiles had to be

used, i.e. N = 8 up to 26 80° and N = 35 for higher 26 values in order to obtain a sufficiently good agreement between the observed and calculated profiles (i.e. 1 a 2 percent

difference in observed and calculated intensity). The difference in profile is illustrated

for the lowest diffracted peak, i.e. N = 3 (26 = 28.47°) in figure 2. A good fit was obtained using the N = 8 standard profile (full line, 1). When using

the N = 35 standard, deviations in the profile are important in the tails and the top (dotted

line, 3). The two different fits with the N = 35 profile were refinements including the

a2^a

l ratl ° and Wltn fixed R ratio. It is important to emphasise that when refining with

an incorrect standard profile, the calculated a^/a^ ratio was considerably in error. Indeed, the better fit (intermittent line, 2) on the component for the N = 35 standard

was obtained with an a^a

i

ratio of 0.353 ± 0.025. Using the N = 8 profile, the a^/ct-^ ratio

was 0.513 ± 0.024. There is no such important deviation however when the and cip components are better resolved as for the higher order diffractions.

2.4. Intensities

In the study of powder samples such as zeolites, peaks above a diffraction angle of 80°

(26) are almost not present or they do not contain sufficient information (too many coin- heas ciding reflections). The same standard profile for N = 8 of Si powder can be used through

out to separate overlapping peaks as it is shown in figure 3 for peaks diffracted at 26 « 16°

and 70° of a dehydrated CaAgA zeolite and in figure 4 for a series of peaks at a diffraction

318
I

jl e of « 68° for a dehydrated Ru exchanged NH^Y zeolite. The highest peak of figure 4 is

ly three times the background. The relative peak heights and widths together with the

jndard deviation and correlation matrix are given in the caption of this figure. Converge is usually obtained after three cycles. If only the peak heights and the positional

ameter is refined, two cycles are sufficient.

Figure 2. Si-powder 111 peak (29 « 28.5°) in steps of 0.01°(«), counting time 10 s, fitted with (1) standard profile derived from the silicon 220 peak (20 « 47°)

(full line), and (2), (3) fitted with the a-^ standard derived from the silicon

531 peak (20 « 114°). Intermittent line 2 is profile fitted when refining on the a^/oi-^ intensity ratio. The dotted line 3 is fitted with constant a„/a^

ratio of 0.5.

Errors in the intensity of the individual peaks include counting errors and deconvol ution

)rs. The number of counts per step and per minute being given by e. {= accumulated counts

'measurements time (t)}, and a

c =Vc/t. The total number of counts in a profile is obtained

)U 5' fp = Ic. and = (Za^ ) . Since the intensity I. of a single peak in the profile is

iiortional to the height H. (I. 1 H.

l

l

C)

319
1

(o

H

• C)

2 + (a

c

• H^2

i

1/2

au being obtained by the least squares fit and IH. being treated as a constant.

H . 1

1

3. The Least-Squares Refinement

Unlike single crystal studies, powder samples do not always allow the determination

structure factors from diffracted peaks because of coinciding reflections. An example i 2 2 2 ... cubic system where all reflections with the same N = h + k + £ are coinciding. To de

the structure factors, one can only rely on the ratio of the calculated intensities to determine the contribution of the individual h k SL reflections to the intensity of syste

atically coinciding reflections. In this way, the model is already included in the obse

intensities which results in a biased parameter estimation. It is possible by this howe

o c 2 1

by minimizing I w^ (1^ - 1^) in the least-squares refinement. Divergence and oscillati

can easily be Avoided by the use of damping factors.

It is also difficult to estimate the scale factor. This is more so if partial occu pancies are to be refined as it is the case for zeolite-type structures. This l^solved

the use of a modified Wilson- type plot [3]. Since

it t / on sin 0.

k ^bs

= W' exp ( " 26

'

where k = scale factor and B = overall temperature factor, we have

£n <WW> = £n(1/k) " 2B(^).

2 2 A plot of £.n(I obs

/I ca -j c

) vs (sin 6/A ) for every 10 reflections, e.g. allows to determir

scale factor and temperature factor corrections.

4. Accuracy

For zeolites, the structural studies parallel the research of the physicochemical

catalytic properties. The influence on the structure of some parameters can be foil owe: structure analysis varying a single parameter. Considering the reduced amount of inforj

the statistical significance of the differences observed in the structural parameters 1

be determined. A statistical comparison might however be limited because of the correl

problems (scale factor, occupancy factors and temperature factors) which do not allow t determination of the entire variance-covariance matrix. If no full matrix is available

tests of significance of the parameters can only be based on the conditional distributi

320
6 6 3

7 4 4

8 4 1 8 3 3

9 1 0

Figure 3. Selected peaks of a dehydrated CaAgA zeolite powder (Pm3m, a

Q =

12.34 A) fitted with the silicon a., standard derived from the Silicon 220

peak: (a) 210 peak (20 «16°), counting time per step: 18 s, (b) 663,

744, 841, 900 and 833, 910 diffracted peaks, counting time per step: 18 s,

26 « 68.5°.

321
• RuNH4 Y

/* \

16 8 2

12 12 6

14 8 8 i

1* A.

f \*

/ 18 2 0

' 16 6 6 NX 15 9 5

13 9 9

\ * *\

^ 1 ^

Figure 4. Part of profile fitted with the silicon standard (220 reflection)

for a dehydrated RuNH^Y type zeolite (Fd3m, a

Q = 24.78) at 26 «68.5°,

counting time per step: 36 s. Peak heights (H. , arbitrary units):

0.0075(5), 0.0770(12), 0.0175(11); width parameter w: 0.380(6);

Correlation between and Hj 1, 2: 0.098; 2, 3: 0.094; 1, 3: 0.030;

Correlation between width and of peaks 1, 2 and 3: 0.191, 0.786 and

0.327, respectively.

the parameter set. It has been shown [4] that the evaluation of Q

(Pj -p

Q

)

1 s1

(p 1

- pQ

) _

on

til

n be

I IS a

It

i to

ienst

Q =

2 qs'

q, n-p (a)

can be used to test the significance of the parameter differences (p-^ - Pq). [n = number

of observations, p = number of parameters; q = number of parameters compared, is the

matrix of the coefficient of the normal equations for the q compared parameters; and

2 2

s = Iw (AF) / (n-p)]. This F test is then based on the conditional density function

of the subset of q parameters. If the entire matrix of normal equations (S) is available

the test can also be performed on a subset using the marginal distribution if the rows -1 -1 2 and columns are crossed from (S ) giving before evaluating Q. Notice that t

=

c 1, n-p

I

!

and that the expression can be used for the comparison of two single parameters

(u.^ by (Q. -

inverse of S.

u})/s(2S:j) 1/2

t where S-. is the diagonal element of the matrix

n-p n M

322
I

5. The Programs

I The programs for the above calculations are available from the authors. ALPHA1 calculates

le fourier coefficients of a standard profile starting from a combined profile.

|AKFIT performs the deconvol ution of the peaks on the basis of this experimental profile.

|e program POWLS is a general least-squares program for centric spacegroups (a largely

jdified version of Hamilton's [5] program) and allows the refinement based on structure

Ictors, on individual intensities 1^ or on coinciding or quasi-coinciding relections.

her programs are the same as for single crystal studies such as JIMDAP for the calcution of the electron densities, using an intermediate file written by POWLS; ORFFE [6]

r the calculation of interatomic distances and bond angles and LIST for the listing

j observed and calculated structure factors for publication. The statistical F-test

n be performed by a program VMATVP, using the matrix of normal equations punched by

(WIS as an input file. All programs are written in FORTRAN IV.

')

| The author wishes to acknowledge the "Belgisch Nationaal Fonds voor Wetenschappel i jk

jderzoek" for a research grant as "aangesteld navorser" and the Belgian Government

jliensten voor programmatie van het wetenschapsbeleid") for financial aid.

References

.] Mortier, W. J. and Costenoble, M. L. , J. Appl. Cryst. 6, 448 (1973).

!] Gangulee, A., J. Appl. Cryst. 3, 272 (1970).

I] Lipson, H. and Cochran, W. , The Determination of Crystal Structures , (G. Bell

& Sons Ltd, London, 1968), p. 72.

\] Mortier, W. J., Acta Crystallogr. A29, 473 (1973).

i] Hamilton, W. C. , P0W0W, Brookhaven National Laboratory, Brookhaven, New York

(1962).

I] Busing, W. R. , Martin, K. 0., and Levy, M. A., ORFFE, Report 0RNL-TM-306, Oak Ridge

I

i National Laboratory, Oak Ridge, Tenn.

Discussion

ie bmment (Edmonds): I would like to clarify the use of the term "deconvol ution" . Deconblution should be reserved to describe the separation of physical phenomena that interact

jultipl icatively to give the observed profile. Peak resolution, or another appropriate term,

hould be used to describe the separation of individual peak shapes that add to yield the

final observed profile.

323
Question (unidentifiable source): Did you determine the accuracy of your fitting method b calculating an R-Factor?

! Response (Mortier): We calculate the sum of the squared errors of fit as a means of

following the convergence, although we could as well have used a reliability index. The

latter, however, would not teach us much more for low intensity data with a large scatter '

in the measured points. Therefore, a visual inspection is our criterion for correction

of the background or the standard profile.

324

..sta

rang

I' k

iffi

'H,

'

: Cll[
National Bureau of Standards Special Publication 567 Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

ACCURACY IN METHODS OF LATTICE- PARAMETER MEASUREMENT

A. J. C. Wilson

Department of Physics

University of Birmingham

Birmingham B15 2TT, England

The geometrical and physical aberrations relevant in powder

diffractometry are discussed, particularly in connection with the

determination of lattice parameters by means of a Bragg-Brentano

(Parrish) parafocusing powder diffractometer. Other diffractometers

and photographic methods are mentioned more briefly. The displacements of the centroid of the line profile that depend only on the

specimen can be eliminated simply by extrapolation methods. Some

of those, depending on the radiation and the apparatus, can be

eliminated in the same way, but others require more detailed

treatment. Attention is given also to peak methods, to profilefitting methods, to energy-dispersive diffractometers, and to

statistical assessment of the parameters obtained.

An extensive, but not exhaustive, bibliography contains some

papers not explicitly cited in the main text.

1. Historical Background

Since about 1930, it has been claimed that the lattice parameters of cubic and other

gh-symmetry substances could be measured within one part in about 50,000 by powder

thods (see, for example, reference [l] 1 ). It is unquestionable that measurements on

e same material by the same investigators were reproducible within this limit, even

en different cameras, different radiations, and different film mountings were used

ee, for example, reference [2]). However, different investigators often did not

hieve this degree of agreement with one another, and in April 1956 the Commissi on

ystallographic Apparatus of the International Union of Crystallography decided to

range an inter- laboratory comparison of measurements of the lattice parameters of three

ibic substances (diamond, silicon, and tungsten) with widely different linear absorption

(efficients! Powders of the three substances, of the highest purity readily available,

re homogenized and samples distributed by sixteen laboratories in nine countries,

igures in brackets indicate the literature references at the end of this paper.

325
though not all laboratories reported on all three. The results [3] were illuminating,

though perhaps disappointing. For each substance there was one maverick determination,

differing from the mean of the rest by one or two parts in 10,000. The remaining deter

mi nations agreed reasonably well with one another, the standard deviation from the mean'

1

being about one part in 30,000-- twice the expected amount. No correlation could be four

between the deviation from the mean and the method, wavelength, or type of apparatus usr There were, however, few dif fractometer measurements, and it would be interesting to 1 ! '

repeat the comparison now, as di f f ractometers are generally believed to be less subject

than cameras to systematic error.

2. Ideal and Real Diffractometers

An ideal angle-dispersive di f fractometer would consist of a point source emitting

strictly monochromatic x-rays of wavelength A, a specimen of negligible physical

dimensions and without strain or other intrinsic defects, and a point detector. An idee

energy-dispersive di f fractometer would be similar, except that the source would emit a wide range of wavelengths (photon energies) and the detector would have perfect wavelength (energy) discrimination. In either case, the lattice spacing d of the diffractir

planes would then be related to the wavelength A. and the diffraction angle 26 by Bragg 1 s

1 aw

2dsin6 , (1)

Ml

lit

«1

ill

2i

the only deviations from it being random errors in the measurement of the angle (wavelength, energy). If measurements of several different interplanar spacings were made tt lattice parameters could be obtained from the known relations between interplanar

spacing, Miller indices, and lattice parameters (International Tables, Vol. II, 1959,

pp. 106-158) and the values obtained from different sets of spacings would agree within |

limits of the random errors in measuring the 6's (A's).

In practice, in order to obtain sufficient intensity for any measurements to be

made, the source, specimen, and detector must all be of finite size and a finite range o wavelengths must be used. The parameters now show a systematic drift with angle—there I are systematic as well as random errors affecting them. The systematic errors are usually called aberrations, and can be divided into two main classes: (i) geometrical

aberrations, and (ii) physical aberrations. The geometrical aberrations are those that ! depend on the dimensions of the source, specimen and detector (or of the slits that 1 imi

their effective dimensions), the physical aberrations are those that depend on the

intensity distribution in the range of wavelengths used (in the angle-dispersive case; on the response characteristics of the detector and associated circuits in the energydispersive case).

There are two errors that in ordinary circumstances would not reveal themselves

through inconsistencies or systematic variations in the values obtained for the lattice I parameters. The first is that the wavelength to be inserted in eq. (1) is not known wi til 326
i "8

litd

gh accuracy. The wavelengths given by spectroscopi sts (the exact feature to which they

fer is not known, but it is probably nearer to the peak than to the centroid) are subject

uncertainties of one part in 50,000 (see, for example, reference [4], especially, p. 157),

ough this uncertainty is reduced by a factor approaching ten for more recent measurements

lown to refer to the peak defined by, say, the extrapolated mid-point of chords [5]. Energyspersive systems are usually calibrated by reference to such x-ray wavelengths, so the

^solute energy scale is uncertain to at least the same extent. The wavelength distributions in the emission spectra of the elements ordinarily used in

;ystallography are not noticeably affected by the methods used for preparing the targets,

lough Lipson and Rogers [6] have reported some variation for manganese. There will be a |ight dependence, at about the limit of detectabi 1 ity , on operating voltage, take-off angle,

!id degree of filtration [7-11]. Effective monochromators will produce large variations

b]. However, eq. (1) depends only on the ratio of d to A, so that relative spacings can be

kermined without regard to the accuracy of A, provided that nothing is done that alters the

ivelength distribution between measurements, and that the same identifiable feature of the

stribution is used throughout.

The other error of this type is that arising from refraction. X-rays, unless incident

irmally, are refracted away from the normal on entering matter, and while inside matter have

Monger wavelength than i_n vacuo . Both effects are small, but the former leads to a ?asurable geometrical aberration for solid specimens (that is, without voids or binder) with

at surfaces (single crystals or polished metal blocks). For the usual powder compacts, it

jads to a broadening rather than a displacement Wilson, [13,14]; but compare Wilkens, [15],

le greater wavelength within the crystal leads to a pseudo-aberration; the actual wavelength

jght to be used in eq. (1), and if the in-vacuo wavelength is used the lattice spacing

)tained will be too small by a fraction equal to the amount by which the refractive index

ffers from unity.

3. Centroid and Peak Displacements, Line Profiles

The diffractometer aberrations shift and distort the diffraction maxima. The study of

leir effects can be divided into four stages, corresponding to four levels of mathematical

jfficulty, and the stage to which it is necessary to carry the calculation depends on the

jrpose in view and the 'identifiable feature' of the wavelength distribution that it is itended to adopt as a measure of the position of the line profile. The three usual features

e: (i) the centroid (centre of gravity, mean, average) of the wavelength distribution;

i) the peak (mode, maximum); and (iii) the best overall fit between the observed and the

/nthesized line profile. The first of these, the centroid, requires only the first stage of

alculation for the geometrical aberrations and the first and second stages for the physical,

lie third, best overall fit, requires all four stages; the second, the peak, logically

pquires all four, but approximations can be obtained at the second stage.

The first stage of the calculation is the determination of the effect of the

ijjerration on the centroid of the diffraction maximum, and ordinarily this presents

327
I little, or at least no insurmountable, difficulty, and is all that is required for the.

correction of centroid poistions of line profiles for geometrical aberrations [16-23].

The additivity of centroid displacements should be exact for geometrical aberratij

and other functions with finite variance, but is subject to some limitations for funct

like the wavelength distribution and the diffraction profile, and depends to some exte

the truncation procedures [24]. This limitation of the use of centroids has been disc

by Wilson [25]; it depends on the initial slopes and curvatures of the Fourier compone

the profiles that are folded together. The initial slopes vanish for the geometrical

rations, so that non-additi vity should not affect the determination of the lattice par

of well -crystallized materials.

The second stage is the calculation of the mean-square broadening (variance). Tlj

is not necessary for parameter determination from centroid measurements, but it can be,

used to obtain a reasonable approximation to the correction to peak positions over a vj

range of Bragg angles [26-28]. To this approximation, the position of the observed pe

is given by

talc

( 26 ) 0 bs = (26) true

+ <A(26)> + WT"/2!"' \

where <A(28)> is the centroid and W is the variance of the geometrical aberration andj

I"' are the second and third derivatives of the observed line profile evaluated at it;

maximum. The physical aberrations of the centroid depend on the variance of the partj

the wavelength distribution used in determining the centroid [7,11,29]. Those of the

depend on the ratio of the peak intensity I to its second derivative I" [11,26,30].

The third stage is the calculation of the line profile corresponding to each geo

aberration. These aberration profiles can be combined by convolution (folding), eith

directly or by Fourier methods, and, in the fourth stage, the combined aberration prof

be convoluted with the profile of the wavelength distribution of the emission profile

x-ray source (or of the emission profile as trimmed by a monochromator , pulse-height

j filter, etc.) and with the diffraction profile corresponding to the state of strain,

| tallite size, etc. of the specimen. This calculation of the composite line profile i)

necessary preliminary to an exact use of peak positions or overall profile-fitting in'

lattice-parameter determination. Such calculations were proposed many years ago, forj

example, by Alexander [31-34], and have been used by Beu and his collaborators (for

references, see Section 6), and Boom [35]; Boom and Smits, [36]).

Until recently the use of exact-peak or overall -fitting methods of lattice-param

determination could only be regarded as an occasional tour-de-force. With the develo

and greater availability of computers, they are now within striking distance of beconj

routine. The most common approach, based on the proposal of Rietveld [37,38], is emp

rather than the fundamental one outlined above. In outline, a function resembling tlj

profile and containing several functions of angle with adjustable parameters is chose

an adjustable multiple of it is placed near the expected position of each diffraction

The adjustable parameters (multiple, centroid shift, breadth, skewness, excess, etc.)*

then varied until the best least-squares fit between the total observed pattern and

1||

328

k
jcalculated pattern is obtained [37-44]; (reference [44] contains references to many other

papers). In much of this work the emphasis has been on either the closeness of the fit

jbetween the observed and calculated patterns of individual lines, or on obtaining structural

parameters by making the 'multiple' depend explicitly on them, so that deducing lattice

parameters from the centroid shifts has received little attention, and it is not clear to

l/hat extent the results are dependent on the choice of the approximation function.

j 4. Aberrations

The symmetrical Bragg-Brentano (Parrish) and the Seemann-Bohl i n angle-dispersive diffractometers are so well known that no description is needed here. The centroid displacements and the variance of the aberrations of the Parrish type have been collected by Wilson

[11,30] and displayed in tabular form ([45], pp. 34-35, 143). A similar table for the peak

jdisplacements has been given by Wilson [26]. For the Seemann-Bohl i n type they are collected

jin table 1, mainly from Wilson [46] and the references cited there. The aberrations are expressed in inverse powers of the source-specimen distance S and the specimen-detector distance R, and tend to be larger for the Seemann-Bohl i n than for the symmetrical arrangement.

For the latter, S and R are constant and equal to the radius, say R , of the diff ractometer,

hereas for the former

S = 2R si ni|> (3)

and " -°

R = 2R

Q

sin(2())-(Ji), (4)

offwhere i|> is the constant angle that the incident x-rays make with the specimen surface. In

the Seemann-Bohl i n case S will be constant at a value depending on the choice of the angle i\>,

a;3ut probably less than Rq

, and R will vary with 2<J>, approaching zero as approaches ty.

cThere will thus be a range of 2<)> for which the Seemann-Bohl i n aberrations containing R become

isf/ery large. Mack and Parrish [47] have confirmed experimentally the expected differences in

favour of the Bragg-Brentano arrangement for general use, even though the effective equatorial divergence ('flat-specimen error') can be greatly reduced by appropriately curving the

specimen. The aberrations in table 1 reduce to those for the symmetrical case for R = S =

* 0

> «l» = <t>- An energy-dispersive di ff ractometer can be made from an angle-dispersive one simply by

|i) replacing the usual counter by a solid-state detector, (ii) replacing the usual electronic

ircuits by a multi-channel pulse-height analyser, and (iii) keeping the specimen and detector

stationary while the counts are accumulated. When so used, the geometrical aberrations are

thl^ssentially the same as those of an angle-dispersive di ff ractometer , though the greater penes enpration of the higher-energy x-rays means that greater attention must be paid to the irradiated

/olume and the specimen transparency [41,48-51]. As Sparks and Gedcke [52] emphasize, spacing

measurements made with such a di ff ractometer are subject to large specimen-surface displacement

|t;fnd transparency aberrations, and the corrections required to allow for them are difficult

329
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0 make. [There seems to be either a typographical or a mathematical muddle in their

quation (5), which carries over into equations derived from it.] Fukamachi , Hosoya, and

erasaki [53]; and Nakajima, Fukamachi, Terasaki and Hosoya [54] show that this difficulty

an be avoided if the Soller slits are rotated about the beam directions by 90°, so that they

imit the equatorial divergence instead of the axial; this was, of course, the orientation

sed by Soller himself [58]. Any effect of specimen-surface displacement and transparency is

hen negligible if the ordinary care in adjustment is used, and the specimen may be placed in

ither the reflection, the symmetrical transmission, or the unsymmetrical transmission

position [59]. The aberrations of an energy-dispersive di f f ractometer are collected in table

1 mostly from the paper of Wilson [59] just cited. The geometrical aberrations apply to the

jiriginal Soller orientation of the Soller slits; as already mentioned, the usual expressions

jipply for the other orientation. Except where otherwise stated, the physical aberrations are

(.he same for both orientations. The most difficult one is probably the correction for the

fcnergy distribution in the incident x-ray beam; aspects of this have been discussed by

jSourdillon, Glazer, Hidaka and Bordas [60]; Buras, Staun Olsen, Gerward, Will, and Hinze

[61]; Fukamachi, Hosoya and Terasaki [53]; Laguitton and Parrish [62]; and Wilson [59]; only

r he last is directly relevant to the lattice-spacing problem. The best results so far

'eported seem to be those of Fukamachi, Hosoya and Terasaki (0.01 percent in the lattice

>arameter).

Okazaki and Kawami nami [63] have suggested the use of a stationary specimen followed by

analysis of the diffracted x-rays with a single-crystal spectrometer. This would give some

)f the advantages of energy-dispersive dif f ractometry (easy control of temperature, etc.,

)ecause only small windows would be needed), but there would be no reduction in the time

"equired for recording a pattern.

Statistical fluctuations in the numbers of counts recorded are not exactly aberrations,

)ut influence the precision with which angles of diffraction, and hence lattice parameters,

:an be determined. Space does not permit a detailed discussion, and a paper on statistical

aspects is scheduled elsewhere in this Symposium. Among the many papers relevant to the

problem are Mack and Spielberg [64], Pike and Wilson [23], Thomsen and Yap [65,66], Wilson

[30,67-72], Wilson, Thomsen and Yap [73], and Zevin, Umanskij, Khejker and Pancenko [74].

The formulae are complicated, and depend on the measure of location that is adopted for the

diffraction profile. In general, however, the standard deviation of the angle is inversely

proportional to the square root of the number of counts accumulated.

From eq. (1) it follows immediately that in angle-dispersive di ff ractometry the

fractional error in a lattice spacing is related to the error in the measurement of the Bragg

angle by

5. Extrapolation Methods

Ad/d = -cot9A6. (5)

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335
The effect of random errors in measuring 8, therefore, becomes very small as the Bragg

angle approaches 90°. The same is true for all aberrations that remain finite in the

back-reflection region. Reference to table 1 (or more conveniently, tables 3-3 and 3-4 of

Wilson, [25,45] shows that this includes all the geometrical aberrations except part of the

axial divergence not dependent on the specimen size (Wilson, [11], p. 39). If the lattice

parameter is plotted against 8, therefore, and a smooth curve through the experimental point

is extrapolated to 8 = 90°, the extrapolated value should be free of most geometrical

aberrations. Refraction and the part of the axial aberration dependent on the source and th

detector (Wilson, [11], p. 39), extrapolate to a constant fractional error in the spacing,

for which a correction can be applied when it is large enough to be significant.

A plot of lattice parameter against 8 is usually strongly curved, so that considerable

time and ingenuity have been expended in attempts to find a function of 8 such that a plot

against it would give a straight-line extrapolation. Perhaps the best simple function for

diffractometry is cos8cot8, the correct form for the specimen-surface-displacement and the

specimen-transparency aberrations. These are ordinarily the largest aberrations in the

low-and moderate-angle range, though axial divergence may take over in the back-reflection

region. Functions for camera use have been discussed by Bradley and Jay [1], Nelson and

2 -1

Riley [75], Taylor and Sinclair [76], and others. The function ^cos 8(csc8 +8 ) is often

very satisfactory; 8 must, of course, be expressed in radians.

The preceding paragraph has been written with the implicit assumption that the extra- polation is done graphically. This procedure is quick and easy for cubic substances with

only one lattice parameter to be determined, and is possible for hexagonal, rhombohedral and

tetragonal [2,77], but it is not elegant, and becomes practically impossible for less

symmetric substances. Graphical methods, even in the one-parameter case, are somewhat

subjective, and Cohen [78] proposed that an extrapolation line (in the cubic case) should be

fitted by least-squares methods. Such analytical extrapolation is readily extended to crystals of low symmetry; in the triclinic case there are six lattice parameters (a, b, c,

B, y), plus one coefficient of the extrapolation function, to be found. The least-squares

method is thus used to fit a hyper-plane, and was very tedious before computers came into common use. The program described by Mueller, Heaton and Miller [79] allows up to three

trigonometric extrapolation functions to be used simultaneously, so that, for example,

specimen-surface displacement, axial divergence, and a physical aberration could be allowed

for simultaneously; other programs have been described by Vogel and Kempter [80] and later

i workers. The number of well-resolved and accurately measured lines required for the reliabl

estimation of lattice parameters goes up with the number of parameters and coefficients of

extrapolation functions that have to be estimated from the data. Weighting schemes have bee

discussed by Hess [81]; Langford and Wilson [82]; Langford [83]; Warczowski and de Wolff

[84]; Warczowski [85], and others. Fitting the full pattern (see Section 3) can be regarde

as an advanced method of analytical extrapolation.

The effect of geometrical aberrations either vanishes on extrapolation or, at worst,

leads to a small fractional error in the lattice spacings that does not interfere with the

extrapolation process. Some physical aberrations for the centroid, on the other hand, becorr

i 336

deri
th

3 very large at high angles, varying approximately as tan 6, and thus producing a curvature

when the usual extrapolation functions are used. Gillham and King [57] have confirmed this

theoretical prediction experimentally, using a series of alloys that shifted the lines

progressively to higher angles. The net effect on the peaks is considerably smaller,

increasing only as tan 8, and thus producing a fractional error in the derived parameters

that does not interfere with the extrapolation process--they are approximately equivalent to

a wavelength error (see Section 2) for peak measurements (Wilson, [11], pp. 79-81; [30], pp. 87-90). Analytical extrapolation procedures, as mentioned in the preceding paragraph, could,

in principle, cope with this behaviour, even for the centroid case, but progress has been

made with correcting the measurement of each diffraction maximum for all known aberrations

before the lattice spacing is calculated, and thus obtaining lattice parameters that do not show a systematic variation with angle. A few examples are given in Section 6.

6. Elimination of Aberrations

In the absence of aberrations, all lines in the pattern of a cubic material would give

the same value of the lattice parameter a. Similarly, for a non-cubic material, any

selection of lines would give the same values of the two, three, four or six lattice

parameters required to characterize its unit cell. Aberrations lead to a dependence of the

derived lattice parameter(s) on the particular choice of line(s). Graphical or analytical

Sextrapolation methods are an attempt to remove the effect of aberrations, making use of an approximate knowledge of their functional dependence on Bragg angle, but without making any

attempt to assess their magnitude a priori . Obviously, it would be more satisfactory if it were possible to assess the actual magnitude of each aberration for each line, and correct

each measured Bragg angle for all aberrations, thus obtaining estimates of the lattice

parameter(s) that show no drift with angle. Straumanis and levins' [86] attempted this for

the geometrical aberrations by reducing the dimensions of the collimating apertures of their

camera and the diameter of the specimen to the point where the aberrations became negligible,

but more recent work (for example, Delf, [87]; Boom and Smits, [36]; and Boom, [35]) has

relied on careful measurement of the di f f ractometer dimensions and of the properties of the

specimen, so that each aberration of the type listed in table 1 can be individually calculated

and applied to the measured Bragg angle. Delf used the centoid (see Section 3) as the measure of line position, and his measurements have been used for illustration by, for

example, Beu [88] and Wilson [25,45], pp. 46-47). Boom and Smits used the best overall fit

between observed and synthesized line profiles (see Section 3), and Bond [89] used peak

positions (extrapolated mid-points of chords) in a study of a silicon single crystal. All

achieved remarkable constancy of the lattice parameter (of cubic substances) derived from

lines at a number of Bragg angles. The question arises: is the observed variation more than

would reasonably be expected on statistical grounds?

Karl Beu and his collaborators have published an extensive series of papers on accuracy

and precision in the determination of lattice parameters [88,90-94]. They had photographic

methods chiefly in mind, and the full program was frustrated by Beu's death; but the papers

by Beu, Musil and Whitney [90,91] and Beu [88] go a long way toward answering the question at

337
the end of the preceding paragraph. They proposed a particular method of estimating the 1 al' tice parameters, the maximum-likelihood method, and a test for deciding whether the remanent differences between observed and calculated Bragg angles were too great to be reasonably

attributed to statistical fluctuations, and thus indicative of uncorrected systematic error]

The modern use of maximum-likelihood estimates seems to go back to Fisher ([95], where

the name is not used explicitly; [96,97]). Their merits relative to other estimation metho

have been the subject of some debate (see, for example, Edwards, [98], who cites an historii

example from 1760). They are not, however, very familiar to the average crystallographer,

though Beu et aJL_ have been followed in their use by Boom and Smits [36], Boom [35], and Ki and Caulfield [99]. More recently, Price [100] has proposed a maximum-likelihood method of

estimating structural parameters, assuming a Poisson distribution of counting fluctuations

Maximum-likelihood estimates depend fundamentally on a knowledge of, or on an assumption

about, the exact form of the distribution function of the statistical fluctuations, and are thus more model -dependent than the method of least squares, which requires little more than

that the variance of the fluctuations be finite. Both maximum-likelihood and least-squares

estimates may be biased when based on a small number of observations. Beu, Musil and Whitn

[90] assumed that the random errors in the measured Bragg angles were normally distributed

with known variance, possibly without realizing, and certainly without emphasizing, that

under these assumptions the maximum-likelihood and the least-squares estimate are equivalen

(see, for example, Hamilton, [101], pp. 37-42, or Bard, [102], p. 63). Since determination

of parameters by least squares is much more familiar, and avoids detailed assumptions abou

distribution functions, it seems worthwhile to rewrite the argument of Beu et al

in the more familiar form.

In the present application, the quantity to be minimized is the sum of the squares of I the differences between the observed Bragg angles and those calculated from the lattice pa-j

rameter (in the cubic case: lattice parameters in systems of lower symmetry) with approJ

1 priate weights; in other words the residual to be minimized is

R = Ew.($. - e.)

2 (6)

where is the observed Bragg angle and 0. is the Bragg angle calculated f rom

\ = 2d.sin0. ; (7) ~1 1

0.j is thus a function of the lattice parameters entering into d^ , the Hh interplanar

spacing. In the cubic case

d. = a(h.

2 + k.2 + 1 . Z )' h

, (8) _1 - -T ~1 ~1

I formulae for the less symmetrical crystal systems are to be found in most elementary texts. The residual R is thus a function of the single parameter a for cubic crystals and up to a

parameters (a, b, c, a, B, y) for crystals of lower symmetry. Provided that the number,

338
say n, of lines measured is greater than the number, say m, of parameters to be determined,

the residual can be minimized by standard methods and least-squares estimates of the pa- rameters found.

Under the hypothesis that the observed angles 0. have been corrected for all systematic

errors, so that the differences between the iji.'s and the 0^'s are due only to the random

errors of measurement, and with the weight w. chosen as the reciprocal of the variance of

2 -

the corresponding 0. , say a. , eq. (6) for the residual becomes

E = 2c V" el

)2/a

i

2

' (9)

I I I i=l

the sum of n variables each having the mean value unity. In theory, of course, the sum could have any value between zero and infinity, but very large values are unlikely. The

expected value is

n

<R> 1 = n , (10)

1=1

--or it would be if the n terms entering into R were entirely independent. In fact, when

the lattice parameters are chosen so as to minimize R, they are slightly influenced by

the actual errors in determining the in such a direction as to make the expected

value of each term somewhat less than unity, and a little calculation shows that the

reduction is 1 for each parameter determined. The expected value of £_.•_ is thus n - 1 for cubic crystals, n - 2 for tetragonal or hexagonal (including rhombohedral )

, n - 3 for

orthorhombic, n - 4 for monoclinic, and n - 6 for triclinic, say n - m in general. The

variance of R . depends on the exact distribution of the errors of measurement, but for

-mi n K

a normal distribution (as assumed by Beu, Musil and Whitney), it is easily found to be

aD 2 = 2(n - m). (11)

Other reasonable distributions would change the numerical coefficients of n and m. The probability of getting a random deviation of 2a

R or more from the expected value is a few percent for any reasonable distribution, so that values of R . up to

(R . ) ... = <R> + 2cr v D -min y critical - R = n - m + 2[2(n - m)]*5 (12)

are not unlikely to arise by chance, but larger values are progressively less likely, and

if large values are found they indicate that some systematic error has not been accounted

339
for—if the remanent systematic error in <|>. is 5^ there will be additional component of

about

E6i

2 /a

i

2 (13)

i - -

in R . . The contribution of the systematic errors to R . is positive, whatever the

-mi n _mi

sign of the actual error 6-. Some critical values of R . calculated from eq. (12) for various values of (n - m) ar —mm — —

given in the second column of table 3; the meaning of the third column is given in below.

Beu, Musil and Whitney argue that values of R . as large as those given in table 3 can happen quite frequently through random error, so that (though we may harbour our private

suspicions) if Rm -j n is less than that given in the appropriate line of the table, we cannot

reject the hypothesis that the corrections for systematic error have, in fact, been done

correctly. On the other hand, values very much larger are more and more improbable, so that if R . much exceeds the appropriate value we can be practically certain that systemat

errors are still present.

Table 3. Some values of R corresponding to a deviation of Zo^ greater

than chance expectation; for the third column, see -below.

n - m n - m + 2[2(n-m)]

J'

2 5% critical value of

chi-squared

1 3.85 3.84

2 6.00 5.99

3 7.90 7.82

4 9.66 9.49

5 11.32 11.07

6 12.93 12.59

7 14.48 14.07

8 16.00 15.51

9 17.49 16.92

10 18.94 18.31

Beu applied the likelihood-ratio method to Bond's [89,103] measurements on silicon

[90] and to Delf's [87] measurements on tungsten [88]. In the latter case, it was found

that the six corrections applied by Delf were necessary and sufficient (R . = 2.5; critica

value for n - m = 3 is 7.9). In the former case, in contrast, it was found that only one (refraction at the crystal surface) was necessary; those for axial divergence and Lorentzpolarization factor were not, and, in fact, caused a small increase in R . (from 0.3 to 0.( -mm

either value is small compared with the critical value of 6.0 for n - m = 2). Possibly the

correct inference would have been that the corrections were incorrect or incomplete rather

than unnecessary. There are at least six papers dealing with specimen- and beam-tilt error

in Bond's method (Burke and Tomkeieff, [104,105]; Gruber and Black, [106]; Halliwell, [107]

Hubbard and Mauer, [108]; Walder and Burke, [109]). All seem to be incomplete in the sense

340

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that they assume a parallel beam at a fixed angle of incidence, whereas in reality the angle

of incidence varies over a range determined by the size of the x-ray source and the slit

system. Bond [89] himself attempted to deal with this aberration, but his treatment is presumably in need of the same type of improvement. Some preliminary results have appeared

[110,111]. Bond applied no correction for dispersion, stating that none is necessary if the

tabulated wavelength values refer to peak positions. The physical aberrations resulting from

dispersion and Lorentz and polarization factors are closely connected, and their combined

effect on peak positions has been discussed by Pike and Wilson [26,30,112]; the treatment

in Wilson [11] is less complete).

There is some evidence [3,113,114] that the lattice parameter of powdered silicon is a

5 few parts in 10 smaller than that of single-crystal silicon, even when both specimens are taken from the same boule. It is not known whether this phenomenon is specific to silicon

(resulting, according to one suggestion, from stress generated by formation of a layer of

amorphous Si^ on the surface of the powder grains), or is a function of the surface/volume

ratio and would be shown by other substances if investigated with the same care, or whether

there is an as-yet undetected systematic error in powder di f f ractometry. Variations in the

lattice parameters of metals after coid-working are, of course, well-known (see, for example,

[115,116]. There are many explanations of this behaviour, and differences generally become

imperceptible after suitable annealing.

The method of Beu, Musil and Whitney, as set out by them, is complex in notation and

somewhat difficult to follow. It is hoped that the above description is easier to grasp and

that it does not differ in any essential. The function actually minimized by them is

n

W =^ni log

e [1 + - e.)

2/^.2

] , (14)

i=l

where n. (not to be confused with n) is the number of measurements made of the position of

the ith~line and s. is the estimated standard deviation of a single measurement (not to be — 2 2 confused with a.; roughly n.a. = s. , but a. is the standard deviation of the 'population'

of all possible~measurementi of <j>.,~and is never known exactly, being estimated from the

scatter of the actual observations). The function W has approximately the same distribution

as the chi-squared function (see, for example, Cramer, [117], p. 233) critical values of

which are given in the third column of table 3. It will be seen that, at the 5 percent level

used by Beu et al . , they do not differ appreciably from those given by the 2a

R criterion.

The second term within the square brackets in eq. (14) is of the order of r^. , so that

W reduces to R if n. is great enough to allow the approximation log

e (l + x) =~x. There is a procedural difficulty if n. is not large: the variances s.. as well as the lattice parameters are estimated from the angle observations, so that the value of m to be used in testing for remanent systematic error becomes indefinite without a more detailed statistical study of the whole process.

341
It must be pointed out that certain systematic errors would not be detected by the

likelihood-ratio test proposed by Beu, Musil and Whitney or the practically equivalent

procedure outlined above. As mentioned in Section 2, errors in the assumed wavelength do

j not reveal themselves through any inconsistency between values of lattice parameters

determined at different Bragg angles. Consideration of eq. (5) indicates that the same would be true of any aberration producing a systematic error in 9 proportional to tan0 or! approximately so. For the symmetrical diffractometer, the summaries given by Wilson

[11,45,46] suggest that this undetectable category would include part of the axial diver-:

gence; absorption in the specimen, filters, etc.; variation of quantum-counting efficiency

refraction; and, for peaks but not for centroids, di spersion-pl us-Lorentz- factor.

7. Summary

I Analysis of the systematic errors affecting angle-dispersive powder dif fractometry

and the agreement between measurements made with different techniques within the same laboratory indicate that the accuracy of lattice-parameter measurement by powder methods

ought to approach one part in 100,000.

A comparison made twenty years ago showed that the accuracy actually achieved in

inter- laboratory comparisons was considerably lower, being hardly better than one part in

10,000. The development of dif fractometry in the intervening years makes it probable that

better accuracy could be achieved now, but there has been no recent comparison on the same|

scale.

Lattice-parameter measurement by energy-dispersive di f fractometry has not been invest

gated as thoroughly; the best accuracy so far claimed is one part in 10,000.

There may be a systematic discrepancy between lattice parameters determined by single

crystal and by powder methods (see Section 6), but, on the whole, the difference seems to I more probably related to imperfections introduced in the process of powdering.

Statistical tests exist for the effectiveness of the correction of certain systematic

errors (see Section 6).

I am indebted to Professor B. Buras, Professor Michael Hart, Dr. L. D. Jennings,

Dr. John Mandel , Dr. William Parrish and Professor R. A. Young for correspondence

and offprints. Travel assistance from the International Centre for Diffraction

Data is gratefully acknowledged.

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Birmingham, 1962), pp. 144-156.

Parrish, W. and Mack, M. , Seemann-Bohl i n x-ray di f f ractometry. I. Instrumentation,

Acta Crystallographica, 23, 687-692 (1967).

349
I

Parrish, W. and Wilson, A. J. C. , Precision measurement of lattice parameters of

polycrystal 1 ine specimens, International Tables for X-ray Crystal lography , Vol. II (The Kynoch Press, Birmingham, 1959), pp. 216-234.

Parrish, W. and Wilson, A. J. C. , Precision measurement of lattice parameters of

polycrystal line specimens: Erratum in the International Tables for X-ray

Crystallography , Acta Crystallographica, 14, 439 (1961).

Pike, E. R. , Lattice parameter determination from broad diffraction lines, J. Sci. Instrum. 35, 34-35 (1958).

Pike, E. R. , Counter diff ractometer - the effect of vertical divergence on the

displacement and breadth of powder diffraction lines. Errata, J. Sci. Instrum. 36,

52-53 (1959a).

Pike, E. R. , Counter di ff ractometer - the effect of dispersion, Lorentz and

polarisation factors on the positions of x-ray powder diffraction lines, Acta

Crystallographica, 12, 87-92 (1959b).

Pike, E. R. and Ladell, J., The Lorentz factor in powder diffraction, Acta

Crystallographica, 14, 53-54 (1961).

Taylor, J. , Mack, M. , and Parrish, W. , A method for the determination of the

centroid of an x-ray distribution, Acta Crystallographica, 16, 1179-1180 (1963).

Tempest, P. A., The influence of specimen absorption and beam divergence on the

accurate determination of lattice parameters by the Debye-Scherrer method, J. Appl

Cryst. 10, 238-246 (1977).

Wassermann, G. and Wiewiorowsky , J., Uber ein Geiger-Zahl rohr-Goniometer nach dem

Seemann-Bohlin-Prinzip, Z. Metal lkunde, 44, 567-570 (1953).

Weyerer, H. , Verfei nerung der Auswerteverfarhren fiir die Prazisionsbestimmung von Gitterkonstanten kubischer Kristalle nach der Pul vermethode , Z. Kristal logr. 109 , 338-353 (1957a).

Weyerer, H. , Extrapolationsmethoden fur Pul verauf nahmen von nichtkubischen

Kristal len, Z. Kristal logr. 109, 354-366 (1957b).

Willis, K. E. , An explicit expression for the lattice parameters of cubic crystals

for the ratio method, Acta Crystallographica, 14, 1090-1091 (1961).

Wilson, A. J. C. , Straumanis 1 method of film-shrinkage correction modified for use without high-angle lines, Rev. Sci. Instrum. 20, 831-832 (1949).

Wilson, A. J. C. , X-ray emission spectra and lattice-parameter determination,

Bulletin of the National Institute of Sciences of India, No. 14, pp. 91-93 (1959a).

Wilson, A. J. C. Some problems in the definition of wavelengths in x-ray crystallography, Z. Kristallogr. Ill, 471-476 (1959b).

Wilson, A. J. C. , On variance as a measure of line broadening in diff ractometry.

I. General theory and small particle size, Proc. Phys. Soc. Lond. 80, 286-294

(1962b).

Wilson, A. J. C. , Aberrations and line broadening in x-ray powder di ff ractometry

,

in Advanced Methods of Crystal lography , G. N. Ramachandran , ed. (Academic Press,

London, 1964), pp. 221-250.

350
de Wolff, P. M. , Reliability of unit cells derived from powder diffraction

patterns, Acta Crystal lographica, 14, 579-582 (1961).

Young, R. A., Gerdes, R. J., and Wilson, A. J. C. , Propagation of some systematic

errors in x-ray line-profile analysis, Acta Crystallographica, 22, 155-162 (1967).

Zevin, L. S. , Umanskij, M. M. , and Khejker, D. M. , Geometric aberrations and

optimum conditions for diffractometric measurements of polycrystal s , Soviet

Physics-Crystallography, 8, 663-673 (1963).

Discission

lestion (Calvert): Do you have any comments on reports of accuracy of 1:10^ by Baker

d co-workers at Harwell with a di f f ractometer (Publ. in Nature)?

sponse (Wilson): The reference is T. W. Baker, J. D. George, B. A. Ballamy and

Causer, Nature, 210, 720-721 (1966). The method is essentially that of Bond, though

|e accuracy of angle measurement, temperature control etc. is unusually high. The

ecision of 1 in 10^ is thus credible.

351
on

li

lit

»i

fe

iiii

«

lit

N

I

::
j National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

SOME STATISTICAL ASPECTS OF LATTICE PARAMETER EVALUATION

John Mandel

National Measurement Laboratory

National Bureau of Standards

Washington, DC 20234

1. Introduction

Translated into statistical terminology the problem of estimating lattice parameters

y be regarded as that of estimating the unknown parameters of a distribution function,

ven a statistical model and a set of experimental data. In this formulation, the nature

the distribution function of the observed data is assumed to be known. In practice, a ussian distribution is generally assumed to hold. On the other hand, the method of least

bares does not require that such a specific assumption be made, unless it is desired to

ke tests of significance.

In Chapter 10, Kaelble's Handbook of X-Rays [l] 1

, entitled "The Precise and Accurate

termination of Lattice Parameters", by Karl E. Beu, a special statistical technique,

own as the Likelihood Ratio Method (LRM), is proposed for the statistical treatment of

ttice parameter measurements. The objective of this paper is to discuss this technique

d to compare it with the more commonly used least squares estimation procedure.

2. The Likelihood Ratio Method

As illustration, we will use the data discussed by Beu in his chapter: Delf's centroid

ta on tungsten. The data are summarized in table 1. The objective is to determine the rameter a

Q

in the equation

i

i a sin 6. = K. (1)

o 1 1

ere K. is a known function of wavelength for each diffraction line. The statistical model

Iderlying the Likelihood Ratio Method is best understood by rewriting the model as shown in

pie 2. It is assumed that a set of (systematic) "corrections" have been applied to the

Iserved Bragg angles, yielding the corrected values where the index j refers to the

f

1 replicate measurement made for the diffraction line i_. A random measurement error, 6...

j standard deviation a., affects the corrected value. Furthermore, the possibility is

jnsidered that the systematic corrections that were made may be inadequate, leaving an

igures in brackets indicate the literature references at the end of this paper.

353
Table 1. Delf's Centroid Data on Tungsten

i = (hkl) n. ib. degrees

S.

i degrees

(110) 7 19.75061

-4

8.42 xlO

(211) 7 36.23581 5.91 "

(310) 9 49.96961 9.23 "

(321) 8 65.30461 26.23 "

Table 2. Statistical Model

a sin 6. = K. O 11

i = (hkl)

a sin[(i|>. + 6.). + e.] = K. o LVS

j j'l 1 J

1

*J = corrected measurement of Bragg angle

6.

J = random error

e

i = remaining systematic error

unknown residual systematic error e. that affects all measurements made for the diffrac

line j_. This yields the equation

a sin[(ib.+6.)-+e.] = K. . (2)

o J ] 1 1 1 i

For the Delf date, i 90es from 1 to 4. Therefore the number of parameters is 9: one f a

Q

, four for the a. and four for the e^. We can visualize the values of these paramete

as a point in a nine-dimensional space, and the objective of the statistical analysis p posed by Beu is to find the "best" point in this space, and to estimate its adequacy,

this effect, a two-step analysis is made.

The first step, which uses the Likelihood Ratio, is essentially a statistical test whether the point in question lies in a specified subspace uj, (see fig. 1) given that i lies in the larger subspace Q. This requires that both subspaces uj and Q be completely

specified. Beu specifies them through the statements shown in figure 2. Thus, the que tion to be answered by the Likelihood Ratio Test is whether it is reasonable to conclud

354
Likelihood Ratio Method, I

Parameters

e

= 1 to 4 = 1 to 4

9 parameters

Nine-Dimensional Parameter Space

cj.

e

Figure 1.

Likelihood Ratio Method ,n

Q defined by X e = = o

W defined by e, = o for al

Figure 2.

355
that no systematic errors e. remain, given that the algebraic sum of the four systematic

errors e. is zero. The Likelihood Ratio Test consists in first finding the point in Q fo which the probability of the observed data is maximum, and calculating this probability,

denoted by P^x^)- Next the same procedure is followed for u>, yielding the probability

P (u)). Finally, the "likelihood ratio" k is calculated by the equation (see fig. 3):

p _ Max(rn)

A large value of A indicates, roughly stated, that the probability that the true par

meter point is in uj is almost as large as the probability that it is in Q, thus giving

credibility to the hypothesis that the point is in w. The statistical procedure for test

K is through the use of the chi-square distribution [2].

If the outcome of the Likelihood Ratio Test points to the acceptability of w, then t second step of the analysis is carried out. This consists in adopting the point in uj fof

which the maximum probability was realized, as the best estimate point of the unknown par

meters, and to calculate the standard errors of these parameter estimates.

The results of Beu's analysis are shown in table 3, for six different sets of correc

tions (see Beu's chapter for a description of these corrections). For purposes of compar

son, the table also shows the results of the weighted least squares analysis.

For our problem, the least squares analysis reduces to a simple weighted straight li regression [3,4], by writing eq. (1) in the form:

sin 6. = — K. + error (4)

o

The 8. are estimated by the corresponding (see table 1). Equation (4) shows that — 1 1 a

o the slope of a straight line going through the origin. The error term can be specified b

calculating the standard error of sin 6. , for each i_, using the estimates of the standarcr

deviations s. of table 1, and the law of propagation of errors [4], to convert an error q an angle to the corresponding error of its sine. The calculations are those of a weighte

straight line regression through the origin. The quantity denoted as "Weighted Root Mearj_

Square Residual" is discussed below.

Discussion

In

The Likelihood Ratio method, as used by Beu, raises a number of important points.

1. The specification for the 0 region is highly artificial. There appears to exis

no logical basis for defining Q by the constraint that the algebraic sum of the remaining

systematic errors must be zero. Against this it may be argued that the test is probably fl

not too much affected by the exact nature of the constraint used to define Q. Be that as it may, the logic of the procedure appears to be less than saf isfactory.

356

' free

tie

;»3
Likelihood Ratio Method,IH

1. In 0,find point for which the probability

of the observed data is maximum

P

Max (O)

2. In w , find point for which the probability

of the observed data is maximum:

P Max (")

3. Test criterion forw:

P

Max < w > A =

P

Max < n >

Figure 3

Table 3. Results for Delf's Data

Maximum Likelihood

rrections X

2 Test

Weighted Least Squares

sa

j;me 3. 183657 14 X

10" 6 significant 3 187413 6000 X10" 6 61 67

+ 3 3. 165020 14 X

10" 6 signi f icant 3 165175 207 X io"

6 21 68

+ 2 + 3 3. 164937 14 X 10" 6 signi f icant 3. 165061 163 X io"

6 17 04

+ 2 + 3 + 4 3. 164934 12 X

10" 6 signi f icant 3. 164966 48 X10" 6 4 99

+ 2 + 3 + 4 + 6 3. 164934 10 X

10" 6 significant 3. 164930 18 X io"

6

1 89

il 6 3. 164942 10 X

10" 6 not significant 3. 164942 9 X io"

6 90

357
2. Beu uses nine parameters to describe the parameter space. Four of these are thi standard deviations a.. But our interest is predominantly in a

Q

and the adequacy of thefilith corrections (i.e. the equality of all e

i to zero). To include the a. in the estimation 5 no process appears to be undesirable. By contrast, the least squares procedure uses the estjiisfa mates of the a. only as weighting factors. It is well known that the results of a least

squares analysis are not greatly affected by moderate changes in the weights. If attenti

is at all directed to the a., through examination of their estimates s., it should be in terms of detecting possible outliers in the data, by comparing the s. with corresponding

values obtained in previous experiments.

3. The Likelihood Ratio Method requires the assumption of a specific underlying er distribution for the measurements. It is here assumed that the random errors in measurin

Bragg angles are Gaussian. To give credibility to this assumption, it should be subjecte

to experimental verification. The method of least squares, on the other hand, can be jus

fied without recourse to assumptions of normality and is indeed less vulnerable than the

Likelihood Method to departures from Gaussian behavior.

4. The use of the chi-square distribution to test the likelihood ratio k is valid

asymptotically when the number of degrees of freedom is large [2]. In the present case,

it is used for 3 degrees of freedom, raising questions of validity of the test.

5. Finally, there appears to be an advantage in the use of the Likelihood Ratio

Method, over the least squares procedure in that the former provides a statistical test f the adequacy of the corrections. But it has already pointed out that the validity of thi test is questionable for small numbers of degrees of freedom. Furthermore, the weighted

least squares method too provides means to judge the adequacy of the corrections. Indeed

an examination of the weighted residuals, and their mean-square, can throw considerable

light on the adequacy of the data. This may be illustrated by comparing, for example,

third and the sixth sets in table 3.

11 at

l( ber

Let us first define the "weighted residuals." If w. is the weight (reciprocal of tf

th

1 error variance) for the i point in the equation

y. = 8 x. + e.

I then the weighted residual is defined by

d- = (y-"P x.) VwT

where B is the least squares estimate of B. In our case, x. = K. and y^ = sin tjj^ . It a

be shown that if the w. are the correct weights (or reasonably close to the correct weigf

and if the straight line assumption is valid, then the sum of the squares of the d^ , divi

by one less than the number of points, has an expected value of unity. Furthermore, thes

residuals should be randomly distributed above and below the fitted line.

358
The weighted residuals for the two sets in question are shown in table 4, together

h the weighted root mean square residuals. It is clear that the correction set (1+2+3)

s not yield a satisfactory fit, whereas the correction set (1+2+3+4+5+6) is entirely

isfactory.

Table 4. Weighted Residuals

Corrections (1+2+3) Corrections (1+2+3+4+5+6)

-21.66 -1.38

-12.25 .52

8.65 -.28

13.31 .40

WRMSR

3

17.04 .90

(a) weighted root mean square residual

4. Conclusion

The Likelihood Ratio method presents an interesting approach to the problem of lattice

ameter estimation. It lends itself to the simultaneous estimation of any number of

ameters, provided sufficient data are available. However, this method suffers from a ber of drawbacks not shared by the least squares estimation procedure. There seems to

no convincing reasons for adopting this technique without further study and refinements.

References

Kaelble, E. F. , Handbook of X-Rays (McGraw-Hill Book Company, N.Y. , 1967).

! Larson, H. J. , Introduction to Probabil ity Theory and Statistical Inference

j (John Wiley & Sons, N.Y. , 1974).

I Draper, N. R. and Smith, H. , Applied Regression Analysis (John Wiley & Sons, N.Y.

| 1966).

j Mandel , J. , The Statistical Analysis of Experimental Data (John Wiley & Sons, N.Y.

r 1964).

359
Discussion

Comments (Wilson): Since submitting my typescript I have looked into the problem o estimatinq the standard deviation. an , of R . , . . K .. - . i y R -mm when the distribution of the random error measured angles <)>. is not normal. The expected value of R^-p remains n-m, but equation

is replaced by

o\ = (2+Y2 )n - 2(1+y

2

)!!! , (15)

plus a further complicated term depending on the fourth-order products of the derivative

of Q. with respect to the parameters to be estimated; ^ 1S a measure, called the excess

or the kurtosis , of the extent to which the error distribution is taller and slimmer tha

the normal (shorter and fatter for negative y

2

)- Examination of the raw data used by

Langford (1973) indicated that for actual film measurements some observers had appreciab

non-normal error distributions, with values of ^ lyi ng in the range -1 to +1. For diff ractometer measurements a theoretical value of y

2 could be obtained by an extension i of the methods used for the variance (Wilson, 1968), but I have not yet tried to do so. In the simple case of m = 1 the additional complicated term reduces to

i

vanishing, as it should, for ^ ~ 0> and being in any case of the order of ^/.D.-

i

V

iI

360
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15 1979

(Issued February 1980)

DATA ACCURACY FOR POWDER INDEXING

Robin Shirley

Chemical Physics Department, University of Surrey

Guildford, Surrey GU2 5XH, England

Powder indexing involves taking the one-dimensional spacing information present in a powder diffraction pattern, and hence reconstructing

the previously unknown three-dimensional lattice that produced it. We need to be able to do this economically and reliably for any single

solid phase, irrespective of symmetry. For most materials of orthorhombic symmetry or lower, this is only practicable by computer methods,

and care taken over the resolution and accuracy of the diffraction mea- surements will be amply rewarded in terms of reliable cells and short

computing times.

The crucial requirement is resolution, which must be to at least

0.25° in 20 (CuKa), so that use of a dif fractometer or focussing camera

is essential. Effective resolution can be improved more than tenfold

by use of a profile analysis and peak-stripping program on carefully

measured profiles.

Only the first 20-30 lines are needed, but must be measured with

great care. In this low-angle range, the systematic errors can for

the most part be combined into a single 26 zero-error term, which

must be corrected to better than 0.02° in 26, preferably by calibration

against an internal standard. It is shown here that self-calibration

using second-order diffraction lines is often also possible. The problem of spurious lines is discussed, and the need emphasised for these

to be identified as such and removed before indexing commences. Data supplied for indexing purposes as a list of d-spacings ought

not to be rounded to less than four decimal places, and it is urged that

measured 26's should always be reported in literature patterns.

An outline of the nature of indexing solution space is given. A powder pattern does not usually lead to a single obvious indexing solution, which can be verified as correct by visual inspection alone. Apart

from alternative settings within the same lattice, there may be geometrical ambiguities, and there will usually be a number of multiple solutions

which have quite distinct Niggli reduced cells, especially when a dominant

361
r

>

rfer

It in

; <iei

ill

zone is present. These solutions can be assessed by figures of merit, and

it is shown that N^q is more suitable for this purpose than F^, which is however a good measure of data quality. Some further criteria for

recognising a correct solution are given.

Available indexing programs are classified both according to their

search methods and their sensitivity to data quality, and a guide provided to their typical success rates and run times. A list of data

problems with advice concerning their remedies is provided, and an attempt made to estimate what effects these will have on computing time,

both singly and in combination.

1. Importance of Accurate Data

There can be few fields of endeavour in which careful data preparation is as dramat

ally rewarded as in computer-based powder indexing. A properly measured and calibrated

pattern is usually solved almost immediately, while mediocre data can run forever!

By powder indexing, I mean taking the one-dimensional spacing information present i

a powder-diffraction pattern, and reconstructing the previously unknown three-dimensiona

lattice that produced it, and doing this reliably and economically for any single solid

phase of any symmetry.

Some of you already have a wide experience of indexing, and will be familiar with

much of what I shall say, but I assume that there are also many who have very little

acquaintance with modern indexing practice and probably regard the subject with some scepticism, because manual powder indexing used to have a deservedly poor reputation,

order to clarify what these methods can offer the practical powder diffraction scientist

we shall start by outlining same basic features of the subject.

However, the emphasis will be on accuracy of data, because that is the subject of

this symposium, and most appropriately so, for this is a field in which a few minutes

extra care from the experimenter can save literally hours of computing. Lack of that c^ ^ can turn a rapid interactive computation into a long trail of abortive and expensive bat'i tura( runs that may never reach a satisfactory conclusion. Until, that is, the experimenter i |, what he should have done in the first place, and goes back to remeasure the pattern (wit

little more feeling this time).

Powder indexing is not like structure analysis, which works well on good data, and'<|f{

| usually get by on poor data given a little more time and attention. Powder indexing wo*

beautifully on good data, but with poor data it usually will not work at all. fi ^

^ ted 2. Introduction to Powder Indexing -

:,

( 2.1. Powder constants and zones

If a lattice is described by its six direct cell constants, the general d-spacing

equation needs implicitly to include transformations between direct and reciprocal spacfc

362

E)

d ca
I becomes complex and unwieldy. For indexing purposes, it is much simpler to convert

king data to Q = 1/d , and the reciprocal cell to its quadratic form, giving the six

vder constants:

QA = a*

2 %

- 2b*c*cosa*

Q

B = b*

2 = 2c*a*cosp*

Q

c = c*

2 ^F = 2a*b*cosy*

The various calculated are then qiven by the equation:

Q

hlcJ> = h \ + k\ + Z\ + + £h(

^E + hk(V

•

I Reciprocal lattice points lying in a plane through the origin constitute a central

rie in reciprocal space, and the corresponding powder lines are said to belong to the same

fder zone . Assuming a provisional assignment of axes, any such zone may be described as

), and needs only three powder constants:

^hkO

= h \ + k\ + hk

^F

•

I Q-values tend to be small fractions, and to give more convenient numbers, it has become

4 stomary to multiply them by 10 . Throughout this paper we shall adopt these Q-units of

[ x 10" 4 = 10

16

m"

2 (abbreviated to QU).

2.2. Figures of merit

Experience has often shown that a list giving apparently good agreement between

served and calculated d-values is an untrustworthy guide to the correctness of a proved cell. This is because d-spacings are so non-linear with respect to the measurement

-curacy of 26, and because an increase in the volume of the proposed cell can "improve"

ijch a comparison to any arbitrary degree.

:

l To correct these deficiencies, we need to relate the discrepancies between observed

d calculated lines to the maximum value they could have had, and to do so using the

re linear Q-values in place of d's. The di screpancy 6^ for the i-th observed line is fined as its absolute difference in QU from the nearest calculated line. This difference

nnot exceed a target e . which is half the separation in QU between the two calcuted lines which bracket the observed one. The ratio <e>:<6> between mean target and

jan discrepancy resists spurious "improvement" with increasing volume, and forms the

sis of a well-behaved family of figures of merit [l] 1

.

'igures in brackets indicate the literature references at the end of this paper.

363
The most useful of these for comparison purposes is the simplest, Ml:

It

iw

not I

Ctl

(i:

mi =

<£> - calcmax

<6> " 2 N calc<6>

N , is the number of distinct and potentially observable calculated lines up to ^ rre ctr C3. I C the last observed line used (N^), and should not include additional symmetry-equivalent

lines, nor space group absences, as far as they are known. It is thus effectively the hire same as N Qss

of Smith and Snyder [2] and of de Wolff [3,4] when = 20.

If N ^ = 20, and all 20 observed lines are counted as indexed, we arrive broadly at

the original de Wolff [3] figure of merit, M^q, which has become the generally adopted ^

indicator of the reliability of proposed powder cells. Using the nomenclature of this

paper, it is defined as:

(3 Q obsmax

M

20 2 N -. <6>

calc

The question of impurities and the treatment of unindexed lines is one that we shall

be returning to later. Note that the formula above will also work correctly for values

of N (i.e., N

o |3s

) other than 20, to give a figure of merit M^, as used, for example, by

Marriner, Langford and Tarney [5]. A different form MJj has been used by Snyder, et al

cause

\

ngs t

[6] and Smith and Snyder [2], but this is not really a generalisation of M^q because

it has acquired an extra factor of N ^ in the numerator, and so is strongly dependent on line number.

i Smith and Snyder [2] have also proposed a quite different figure of merit based

on 26 differences, and claimed that it should replace I^q. We shall be returning to this

point later, where it will be argued that while F^ is more suitable for assessing measure- ment accuracy, f^Q is still superior for evaluating indexing solutions. Using the nomen- clature of this paper, F^ is defined as:

F

N obs

N <626> N , calc

In practice it is hard to give an exact and definitive value of M^q (or F^) for a particular indexing solution, because of sensitivity to slight cell parameter changes, etc which cause disproportionate changes to the residuals in the denominator of the figure of

merit formula. Slightly different refinement conditions may well yield figures of merit

differing by more than 10 percent for the same solution, and only differences between solu,

tions that are clearly larger than this should be considered significant. Further work irij|]

this area leading to a firmer concensus would be useful. Heavy-handed rounding of the

d-spacings has particularly disastrous effects here, a point that we shall be coming back s later.

364

pat

ace [

the

MIPS

ipsi

ter
2.3. Recognising a correct solution

It will be shown later that, because of zone dominance, it is ultimately only the

lative value of I^q compared with other possible solutions that matters, and so we mot lay down any absolute value of M

2Q (or any other figure of merit) that guarantees

rrectness. Nevertheless, one can still give some very helpful guidelines:

(1) False solutions generally have M^q < 10, and resist attempts to improve them by

11 refinement or adjusting the calibration constants, because the information simply is

It there.

(2) A solution that indexes all the first 20-30 lines and has M

2Q

> 20 is nearly

tfays correct, especially if this is twice that of the next best solution and the list

I 20 differences contains no unexplained excursions greater than three times the

jpected a(26).

J (3) The correct solution also gives itself away in less quantitative ways to the

Idicious eye. It tends to recur with different indexing methods and from different start- g points. It is also readily improved by fine adjustments to the calibration constants,

cause its internal consistency takes over as a stronger indicator of the correct setngs than the original calibration method.

! (4) Finally, because it contains genuine information, the correct solution will

lually start to make crystallographic and structural sense. A level-by-level listing of

e pattern will usually suggest systematic absences that will lead to a few possible

ace groups. The formula weight and a density estimate will lead to a clear indication

the number of formula units per cell that will be consistent with the possible spaceoups and molecular symmetry. Finally, the cell geometry will often indicate relationips with other materials and perhaps suggest a packing scheme and structure, or do so ter transformation to a more suitable setting. (However, one should not be too insistent

this, as one of the attractions of research is that the truth is sometimes unexpected.)

3. Nature of Indexing Solution Space

The attention we have been giving to figures of merit and choosing the correct

ilution, implies that there may be a number of solutions to choose between, and this is ideed the case, as we shall see.

3.1. Alternative settings

j A lattice can be described by an infinite number

.....

of choices of unit cell within it, and

rery high-symmetry lattice yields numerous, less convenient cells in lower crystal systems.

L pwever, these alternative settings are easily recognised as the same solution, because they

f |l lead to the same reduced (Niggli) cell [7]. If sub- or superl attices are obtained, for

(cample by halving or centering the true cell, again the reduced cells will indicate that

lese are not independent solutions. In all such cases, the cell volumes will be either

365
equal or in a simple ratio; this is a useful guide, but not as conclusive as reduced cell

compari sons.

3.2. Geometrical ambiguities

Mighell and Santoro [8] have shown that certain distinct lattices, with different,

though related reduced cells, can give geometrically identical powder patterns. This

geometrical ambiguity is only found when special relationships exist between the parameters of the two lattices, and cannot occur between general cells of orthorhombic or lower symmetry. However, it always occurs when the metrical symmetry of one of the

lattices is higher than orthorhombic. For example, if one examines the numerous lowersymmetry cells that can account for a hexagonal pattern, it will be found that some of

them are true geometrical ambiguities.

In practice this is easily detected, because as far as is known, the volumes of

such cells will always be in a simple relationship (though not necessarily an integer one --factors of -{2 can occur, for example). The higher symmetry solution is usually the cor rect one, although one cannot be sure of this without studying the line multiplicities or intensities, and exceptions have occurred. For example a-Li^E^O^ [3] can be indexed exact

as hexagonal, but has been shown by single crystal work actually to be orthorhombic.

3.3. Multiple solutions

If we look at the figure of merit surface for a particular powder pattern, possible

3 solutions will appear as peaks. Figure 1 shows the sharpened figure of merit (WM3 ) sur- face for hexamethyl benzene , in the plane through the correct solution with only and

varied (generated by the grid-search method using the POWDER 49 program). This is equivalent to varying only the reciprocal angles a* and p* while keeping the remaining

cell constants fixed. It can be seen that there is only a single prominent peak, and

so in this case a unique solution. We are usually not so fortunate.

A quite different situation is shown in figure 2, the Ml surface for the blue pigment

orCu phthalocyanine. In this case there are numerous peaks, and hence many plausible

false solutions (pseudo-solutions) in addition to the correct one. The material is

actually triclinic, though pseudo-monocl i nic with angles of 90.8° and 90.3°, and was sufficiently colloidal to exhibit appreciable line-broadening. The best five solutions

index all the first 20 lines with figures of merit of 12.7, 9.7, 9.6, 8.6, and 8.5

respectively, so this was not an easy case to solve.

Fortunately the first solution (M^q = 12.7) is indeed the correct one (indicated

by an arrow on the diagram), but all the first 20 solutions look acceptable in the form

of the traditional dobs" 0^-^ list, although in fact they are quite distinct, with different volumes and densities. a-Cu phthalocyanine was one of the first instances of multiple solutions to be studied in detail, but many more striking examples have since

been discovered.

366
The first three solutions for monoammonium urate have I^q = 21, 15, and 13 while for

monosodium urate monohydrate (the crystals which cause gout) the best five cells give 30,

21, 20, 19, and 18, and no less than 31 cells index all the first 20 lines with a figure of

merit better than 10. There are, in fact, some 200 tolerably plausible solutions for this

pattern, although only the best 36 have been examined in detail. More extreme examples are

jnot uncommon: using the L0SH and LZ0N programs, the search for solutions that index the

'first 20 lines within their 26 error limits is often terminated at 500 cells, the present

storage maximum. These cases generally arise through the mechanism of zone dominance.

Figure 1. Hexamethyl benzene ** WM3 .

I
Figure 2. LPHA CU-Phthalocyanine HK1 > WW W/0 001 Theta < 17.5 ** Ml.

3.4. Dominant zones

The powder zone with the largest direct space constants, and hence whose reciprocal

net cell has smallest area S, will generate most of the calculated lines near the origin,

and, unless it is also a glide plane, will index most of the low angle observed lines.

2/3 When S approaches half the geometrical mean value (V*) , this dominant zone will prc^ ably index about 10 out of the first 20 lines, leaving the remaining three powder constant:

to explain only about 10 upper- level lines. This is not very difficult to do by chance, a1 least approximately, so such materials will tend to give many plausible, multiple solution'

as occurs with the three examples above. In each case the solution with highest figure of

merit is correct, but can only be identified by comparison with the entire set. Dominant zones can in principle occur in any crystal system, but their effects are moi

noticeable among triclinics. In such cases no level of figure of merit can, by itself, gu< antee correctness, since false solutions can occur with arbitrarily high values. It is the. better to work with more than 20 lines so as to include a higher proportion of upper- level

r ones, and to use the fact that, for sufficiently accurate data, the correct solution will

always fit appreciably better than any pseudo-solution.

368

iti

I to

0 "ore

i *rac

III!

Nt

if

P uiir
Given good data, materials with dominant zones no longer present any serious difficulty, since this is the type of problem that grid-search programs (such as LOSH, LZON

land POWDER 49) like best.

3.5. Problem of immensity

] It would be reassuring to be able to locate all the multiple solutions for each patjtern by an exhaustive search, but as we go to lower symmetry and so have more variable

dimensions, the combinatorial explosion takes over. Assuming reasonable search limits and

14 Working on a 5 QU grid, it turns out that more than 10 different lattices are possible.

'ht the rate of one per 3 ms (just about achievable on a CDC7600), they would take 12000

jyears to search by direct exhaustive trial, and at least 10 years by the optimal algorithm

Jin parameter space (successive dichotomy, a form of binary search). It is thus not likely

jthat an exhaustive search will ever be feasible in the general case.

It is, however, a reasonable method for high-symmetry systems down to orthorhombic

,

if a suitable algorithm is used (programs L0UV and POWDER = TAUP). In lower symmetry

systems, semi -exhaustive methods predominate, in which the search is restricted to two or three dimensions by carefully chosen simplifying assumptions.

Similar conclusions can be drawn if one approaches the problem by assigning Miller

indices to each of N observed lines, where N need not exceed the number of unknown parameters

so that solutions are located in a 3N-dimensional index-space in which only integer

coordinates are possible.

The obvious moral is that we cannot hope to tackle more than one unknown phase at

a time in lower symmetry systems (up to six more parameters per impurity), and are likely

to have great difficulty with inadequately calibrated patterns (at least one or two

nore parameters).

4. Data Requirements

For indexing work it is not necessary, nor even desirable, to measure the entire diffraction pattern. Information about the correctness of the unit cell is concentrated at the low angle end, where the lines are relatively well separated and are resistant to displacement by small cell parameter changes. It is generally sufficient to measure the

'[positions of the first 20-30 lines, provided that this is done thoroughly and accurately.

If powder measurements are progressively degraded by errors, indexing becomes difficult and more time-consuming, and eventually impossible, even in principle. In general,

it is a waste of time to attempt to index poor data, unless the solution is trivially

simple.

| If data errors were negligibly small, the correct solution would have an enormous figure of merit compared with the much smaller ones of the pseudo-solutions. Data errors degrade the true solution, but have relatively little effect on the field of pseudosolutions produced by accidental superimpositions. For every peak that declines,

another is as likely to improve, so that the general character of the group of false

369
solutions is resistant to change. In the limit, the true solution (the signal) is no longer distinguishable from the pseudo-solutions (the background noise), and the data

become unsolvable.

4.1. Resolution

Probably the most exacting requirement for successful indexing is the need to observe

most of the first 20-30 lines (that are not too weak) as distinct individuals, rather than

merged groups. Typically, this means the need to distinguish adjacent lines separated by

no more than 0.25° of 26 at CuKa wavelengths, which effectively rules out Debye-Scherrer

data. Resolution in this sense can be taken as being numerically equivalent to the line

width, expressed as full-width at half maximum height (f.w.h.m.). Inadequate resolution

nearly always makes the data impossible to index.

The paramount importance of resolution for indexing work explains the high success

rate for focussing camera data, especially from Guinier-Hagg instruments, whose resolution

can only be described as superb. It is rather less common (and considerably more expensive) to obtain as good resolution with di f f ractometer data.

The effective resolution of any instrument can be increased without theoretical

limit (other than signal/noise ratio considerations) if the full profile is measured and

a profile analysis and peak stripping program used.

Resolution can also generally be improved by increasing the wavelength, which is especially relevant for neutron work. Some very recent experiments on the D1A neutron

diff ractometer at the I.L.L., Grenoble, demonstrated that its effective resolution, which

o o was insufficient for most indexing work at 1.9 A, became quite reasonable at 3 A and very

o satisfactory by 5.7 A. We can say in general that different kinds of work have different minimum resolution

requirements. For phase identification and quantitative analysis, it is often sufficient

to work at quite low 26 resolution (about 1°, for example), if this matches that of the

standards used. For indexing, adjacent lines must be resolved, so a higher resolution

(<0.25°) is required. This is also more than sufficient for many structure refinements

using the Rietveld profile-fitting method, for which <0.5° will often suffice. It is,

however, not nearly enough for ab initio structure solution, for here we need to observe

not only a large number of lines but also the gaps between them, to identify accidental

absences and establish background levels in order to build up the 50-200 terms needed

for Pattersons and AF syntheses. Hence, the highest available resolution (<0.1°) is desirable.

4.2. Accuracy

Measurement accuracy comprises precision (random errors) and correction (systematic

errors), and powder indexing responds very differently to these two contributions.

370
Indexing is surprisingly tolerant of imprecision. Random variation with a(26) up

o 0.03° (absolute error limits of ±0.08°) will increase the computation time, but

irobably not prevent the solution from being found. This is easily met, since

(26) < 0.02° is common in routine measurements.

Systematic error is much more serious. In the low angle region that is of interest

;o powder indexing, the important effects can be summarized as a zero error in 26. Quite

small increases will cause computing time to increase rapidly and depress the figure of

herit of the correct solution so much that it becomes hard to recognize with any confidence

Zero error particularly affects powder dif fractometers with the usual parafocussing

eometry, because of sensitivity to errors of less than 0.1 mm in the effective sample

josition, varying from sample to sample with absorption and surface roughness. The

>nly completely satisfactory answer is to do a calibration run using an internal standard

;o obtain unbiased 26' s for the major peaks, from which a run using the pure sample may

>e corrected.

Line profiles in the low 26 range of interest are often erratic and sharply skew

lear the intensity maximum, and so appreciable error can result if peak position is taken

is the criterion of line 26. An equally rapid and much stabler procedure is to use the

:enter of the chord at about 80 percent of maximum height.

4.3. Self-calibration

In principle, calibration knowledge for a pattern is present in the measured powder

lines themselves, but cannot be accessed until the pattern has been indexed (or at least

individual zones). However, one special case exists for which indices are not required —

that of any observed line 26-^, and its second order 26

2 (other higher orders are seldom

available).

The required zero correction Z 2Q can be obtained from the iterative formula:

Z 26 = Z 26

+ 2

< 2arcsin |

sin(6

2 corr) -26^ corr'

A/here 26 = 26 + Z 0 „. From an initial estimate of zero, this expression converges

corr raw 26

Irapidly in about five cycles, and is easily implemented on a programmable pocket calculator

A quick search through the line listing will usually reveal three or four possible

(second order relations, of which the one yielding smallest Z 2Q

would be first choice for

self-calibration. If two such relations produce the same Z 2Q , it may be accepted with

considerable confidence. For high-precision data, this simple procedure can produce

reliable zero calibrations to better than 0.01°, and I have found it invaluable both for

refining an existing calibration and for salvaging precise but poorly calibrated data

from other sources.

Z 20

determination from internal consistency of the data offers further interesting

possibilities. If Z 2Q were included in zone indexing programs as a variable parameter

during the refinement of trial zones, correct zones would tend to reveal themselves by

371
all giving the same Z2Q

value. Thus, we might be able to overcome one of the major

obstacles to routine fast indexing--the lack of a reliable criterion for identifying

correct zones.

4.4. Spurious lines

It is very desirable to eliminate spurious lines which do not belong to the phase

under investigation, because, as we have seen, they introduce additional unknown parameters, as well as serious methodological difficulties.

If we permit observed lines to be discarded as "not indexed", we are then changing

our data to fit the proposed model--a dubious scientific practice in itself, and very

hazardous when multiple solutions are to be expected. The only safe course is to remove

all doubtful lines on a priori grounds.

For a start, an impurity level of <1 percent should be sought, and preferably verified

in a density gradient column. Known impurities need not cause serious trouble because thei'

lines can simply be omitted, although they may obscure some of the unknown pattern. Differences in line widths, or variations in intensity ratios in different samples, may offer jl some useful clues here, but direct examination by microscopy or electron probe microanalysis is the surest test. Finally, it is nearly always best to omit all very weak

lines at this stage, and only return to them after the cell has been found. They will

rarely be precisely measured, and should be suspected as possible contaminants until

their indexings have been confirmed.

Spurious lines are particularly damaging to exhaustive search procedures. Although

some programs permit a specified number (NSPURI) of poor indexings to be omitted, this is

liable to cause an exponential increase in run times, and so should not be used without

good cause.

4.5. Number of digits specified

A surprisingly common source of data degradation is simply heavy-handed rounding of

d-values when a measured pattern is reported.

Werner [9] has shown that rounding errors alone will place a ceiling on the figure

of merit for a particular cell volume V, according to:

3md

2Q Monmax «

~ J '20",aA ~ 8nV<6>

where d 2Q

is the d-spacing of the 20th observed line, m is the multiplicity factor and

<5> is the estimation precision in 1/d^, in this case arising from rounding. A simple

estimator for V is given by Smith [10].

It can be shown that two decimal places are virtually never enough, three are often

insufficient, and even four may not be overgenerous. This point is also made by Smith

and Snyder [2], who join the overwhelming majority of those engaged in indexing work in

372
urging that the list of observed 26 's should be published, as is done for National Bureau

of Standards data. It is to be hoped that this will soon become the accepted convention,

as has been recommended by the A.C.A. subcommittee on this subject, whose report we shall

be hearing later in the symposium (Calvert, et al. [11]).

5. Indexing Programs

I

19 Details of the methods used by particular indexing programs are discussed in a recent [1], and we shall confine ourselves here to a classification and overview of

the various programs, with particular reference to their sensitivity to data quality.

5.1. General classification

Powder indexing requires either that the correct values of the cell constants (or

['powder constants, etc.) are found in a continuous parameter- space of up to six dimensions,

or that the appropriate values of the Miller indices for each line are located in a 3N-dimensional integer- valued index- space , where N is the number of observed lines to be

indexed. These two objectives are equivalent, and are both fulfilled when the correct

I solution is found.

I Index- and parameter- space methods are considerably different in approach, and can act

as useful complements (or checks) for each other. In general, parameter-space methods lend

themselves more to the inclusion of cell information and constraints, while index-space

programs are the stronger in the presence of unhelpful accidental or systematic absences.

Programs may also be classified according to whether they adopt a mainly deductive

or exhaustive approach. Broadly, deductive methods try to infer the values of lattice

parameters from coincidences and relations between the observed lines, to achieve speed

at the cost of rigour. By contrast, exhaustive methods systematically search the rele- vant solution space, gaining rigour at the expense of speed. Deductive methods try to work out quickly what the solution ought to be, but may fail with poor data or a tricky

problem. Exhaustive methods eliminate everywhere that the solution cannot be a procedure

which must succeed but is slow in low-symmetry cases. The classification is not rigid, because programs often incorporate some aspect of

both approaches. For example, deductive programs usually try some kind of systematic

combination of the deduced possibilities, while most of the new programs are semi-exhaustive , making judicious deductions to limit the solution field to be searched,

in order to gain speed.

373
Parameter-space Index-space

Deductive:

Semi-exhaustive:

Visser (ITO=FZON)

Shirley (P0WDER49)

Louer & Shirley (LOSH)

Shirley, Louer & Visser (LZON)

Smith & Kahara (QTEST)

Werner (TREOR=TRER)

Kohlbeck (TM0=K0HL)

Exhaustive: Louer (LOUV, LOUM, DICVOL) Taupin (POWDER=TAUP)

Modern indexing programs tend to be large and complex, especially the semiexhaustive ones, because they are effectively experiments in artificial intelligence.

They apply carefully evolved heuristics to try to quickly locate solutions that are wholly out of reach by the best exhaustive algorithms.

A summary of the characteristics of the major programs is shown in table 1. More

details are given in Shirley [1]. This library is available from me, on receipt of a magnetic tape and information about preferred tape formats.

In general, exhaustive methods are excellent down to orthorhombic, but run away from

us somewhere between three and five unknown parameters. For monoclinic, and certainly for

triclinic, we should first try deductive programs (although these are less tolerant of data

errors), and then semi-exhaustive ones.

5.2. Tolerance of data errors

The table which follows should be regarded as an approximate guide only:

Tolerance of spurious lines

Sensitive Insensitive

Sensitive LZON.TAUP FZON.TRER, (QTEST)

Insensitive LOUV, LOUM KOHL

DICVOL, LOSH

5.3. Standardization of data formats

It is obviously desirable that all indexing programs should accept data in the same

format, so as to eliminate the labour and introduction of errors involved in reformatting

and retyping for each program tried, and to simplify the exchange of data sets between

374

Tolerance of

Inaccuracy
O

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laboratories. A Standard Powder Data Interchange Format (SPDIF - see fig. 3) has been

designed for this purpose, in consultation with most of the program authors and other

interested parties [12]. In due course, all the major indexing programs, as well as those for many related calculations, will be able to accept SPDIF as an alternative to their own input formats. In the interim I have written two short programs, PDAT and

QDAT, which accept data in an easily typed form, and then generate suitably formatted

data decks for any or all of the major indexing programs, as well as an SPDIF deck.

SPDIF 5 TITLE RUBIDIUM CARBONATE, RB2C03.H20, AFTER J.W.VISSER

REMARK TEST DATA FOR ZONE-INDEXING PROGRAMS (40 LINES) REMARK MONOCLINIC I-CENTERED CELL, GIVING M20=54.2

DEFPAR 71 0.03 1.54050

LINE 139.0

LINE 1868.9

END TITLE ALPHA CU-PHTHALOCYANINE: ALL DATA>VVW, R.SHIRLEY

REMARK DATA MEASURED BY R. SHIRLEY FROM PHOTOGRAPH SUPPLIED BY C.J.BROWN

REMARK TRICLINIC, PSEUDO-MONOCLINIC. FIRST 8 LINES IN DOMINANT HKO ZONE DEFPAR 11 REMARK NONIUS GUINIER-DE WOLFF CAMERA: XMM GIVES 4THETA IN DEGREES

REMARK CALIBRATED AGAINST BETA CU-PHTHALOCYANINE TRACK ON SAME FILM

40

XCALIB 0.06 0.5

CELMAX 1000. 576.084 1.6

LINE 31.5 VVS 6 1

• • • • • •

• *

* LINE 85*25 M 24

END FINISH

Figure 3. Example of SPDIF Deck, Containing Two Data Sets

6. Comparison of Figures of Merit

Figures of merit for powder data are needed in two rather different circumstances,

j and to some extent the requirements are conflicting. \ For phase identification, we need to know the quality of measurement of the reference

patterns in the literature. For this purpose, the published unit cells can be assumed to

be correct, because if not, the real figure of merit will be higher, and we shall merely

376
have erred on the side of caution. Thus only the accuracy and completeness of the line

listings are relevant, and any dependence on cell volume, symmetry, etc., will only be a disadvantage.

Clearly meets these criteria better than I1,q, and would do so more if its wavelength dependence could be eliminated, as we shall see.

In powder indexing the situation is quite different—it is not the quality of the

{data but the reliability of proposed unit cells that we wish primarily to assess.

In this case the fall in V\^q with lower crystal system and with increasing cell

volume becomes an advantage, because it demands a higher standard of agreement for proosed large, low-symmetry cells (those with more degrees of freedom, and hence overdeterined by a smaller margin), and this is supported by both theory and practical experience.

Hence, to give an example, we might find I^q = 50 given both by an excellent set of

bata for a large monoclinic or triclinic cell, and by a relatively poor cubic pattern.

|n both cases the reliabilities of their cells would be about the same (i.e. very good),

although the data qualities of the two are very different. By Occam's razor, we require

stricter evidence to support the more complicated model.

If we compare M^q and F^ for N ^ ~ 20:

F 20 =

40 I6Q

M

20 " Qobsmax

' 1626

_ An I(sin29-626)

qx

2

n

' 1626

9A Q obsmax

The first term is constant for a particular data set, while the second term indicates

:he slightly different weighting of the residuals for the two measures and approximates to sin28>, depending on the particular 26 1 s present. Comparing data at different wavelengths

: 2 or the same sample, we see the strong 1/K dependence of F M , as at present defined. Com-

-2/3

laring different materials at the same wavelength, a variation of M

2Q

with V occurs via

the term in Q . A more subtle effect of this term is to introduce the greater sym- ^obsmax.

lietry dependence of M

2Q , because of its relative increase with increasing symmetry as more

•eciprocal lattice points are needed on average to produce a new observed line. The

1/3

(sin26'626) term introduces a further approximate dependence on V , so that the overall

-1/3

olume dependence of Mon is roughly as V . It is argued here that these features of

20 are beneficial to its purpose, but that the A dependence of F^ is not.

As presently defined, F N measures the quality of the experimenter— how well he or she

jieasured the actual 26 's at the wavelength used. But it would be more useful to remove the

'avelength dependence and know how good the data are. This could be done by converting the

377
observed differences in 26 to differences in d, then back to differences in 26 at a standan

wavelength, say CuKa-^. The standardized would then reflect the improvement in accuracy

that can be obtained by measuring the expanded pattern at a longer wavelength.

An instance of this arose last month while collecting neutron powder data at the

I.L.L. , Grenoble, where it was possible to more than double the accuracy in d by changing

( to a longer wavelength, but this was not reflected in F^, which was in fact halved because p of a drop in the uncorrected 26 resolution with increasing wavelength! » Ideally, three assessment measures should be given, because there are three distinct

needs to be satisfied. These are: (1) a(26), the estimated standard deviation in the

original 26 measurements, so that we may judge the accuracy of experimental measurement , assuming the cell to be correct; (2) the standardized figure of merit F^, with 26 's ,i referred to a standard wavelength (CuKa^), so that we may gauge the quality of the resultf ing data set , again assuming the cell to be correct; (3) the properly calculated de Wolff

figure of merit I^q or M^, so that we may assess the reliability of the proposed unit cell , or, if this is not in question, to add to the accumulated experience through which we make

this judgement in other cases. >

The two intolerable problems for indexing work are insufficient resolution or not enough digits reported for the data. The presence of either of these will ensure that a study is abortive. However, there are a number of other problems, which may be tolerable

in isolation, though probably not in combination:

(1) imprecision (lengthens computing time);

(2) poor calibration, particularly zero error (self-calibration is often possible);

(3) impurity contamination (exponential rise in computing time if using NSPURI

facility);

(4) awkward accidental absences, such as from a strong subcell (index space methods

J preferable here);

(5) awkward cell shape, producing a dominant zone and numerous multiple solutions

(grid-search programs needed);

(6) high cell volume (requires correspondingly high accuracy).

Low symmetry is not a problem in itself, but will make the other problems much worse,'

since there may be a need to use exhaustive methods, which are much slower in low symmetry

As a rough guide, we may relate the number of problems to probable computing time

needed (CDC 7600) as follows:

7. Summary of Data Problems for Indexing Work

Number of problems 7600 time

0 under 1 minute

1 5-30 minutes

2 1-3 hours

3 get better data

378
8. Conclusion

Modern computer-based powder indexing is sufficiently easy and well-proven for it to

e worth the while of serious powder diffraction workers to include it in their repertoire

jf techniques.

We now have available a number of powerful and mature programs, all of which usually

fucceed in at least 50 percent of cases, and many achieve better than 80 percent. Because

they use quite distinct and complementary methods, their limitations tend to affect different problems, so that with several of them available one will seldom be unable to index

i properly measured pattern. By knowing the right order to apply them, which broadly means

trying the speedy deductive methods first, then working down from high symmetry with the

lore powerful exhaustive and semi-exhaustive methods, computing cost should not be a problem. But that takes us back to where we started--the quality of the data. Although

phere remain many fascinating ideas to be followed up in the field of indexing programs,

ve now understand that aspect of the problem reasonably well, and the ball is back in

;he court of the experimentalists.

The equipment required to record patterns with excellent resolution and accuracy

is quite inexpensive (under $8000 for a Guinier-Hagg camera), and in any case, as the

1CPDS Associateship at NBS have shown, if you keep in mind the important requirements,

it is very likely that existing equipment can be persuaded to produce data of sufficient

Until a pattern is indexed, it is difficult to demonstrate its quality, or to show

that it does not contain spurious lines. Without indexing, it is impossible to proceed

to the powerful and elegant structural investigations that we have been hearing about.

[1] Shirley, R. , Indexing powder diagrams, Crystal lographic Computing , H. Schenk,

ed. (Delft University Press, 1978).

[2] Smith, G. S. and Snyder, R. L. , F^: A criterion for rating powder di '/fraction

patterns and evaluating the reliability of powder-pattern indexing, J. Appl.

Cryst. 12, 60-65 (1979).

[3] de Wolff, P. M. , A simplified criterion for the reliability of a powder pattern

indexing, J. Appl. Cryst. 1, 108-113 (1968).

[4] de Wolff, P. M. , The definition of the indexing figure of merit M

20 , J. Appl.

Cryst. 5, 243 (1972).

[5] Marriner, Langford and Tarney, J. Appl. Cryst. (1979).

[6] Snyder, R. L. , Johnson, A.C., Kahara, E. , Smith, G. S. , and Nichols, M. C. , An analysis of the powder diffraction file, Report UCRL-52505 (Lawrence Livermore

Lab., University of California, 1978).

[7] Mighell, A. D. , The reduced cell: Its use in the identification of crystalline

materials, J. Appl. Cryst. 9, 491-498 (1976).

}ual ity.

References

379
[8] Mighell, A. D. and Santoro, A., Geometrical ambiguities in the indexing of powder

patterns, J. Appl. Cryst. 8, 372-374 (1975).

[9] Werner, P.-E., On the determination of unit cell dimensions from inaccurate

powder data, J. Appl. Cryst. 9, 216-219 (1976).

[10] Smith, G. S. , Estimating the unit-cell volume from oneline in a powder diffraction pattern: the triclinic case, J. Appl. Cryst. 10, 252-255 (1977).

[11] Calvert, L. D. , Fl ippen-Anderson, J. L. , Hubbard, C. R. , Johnson, Q. C.

,

Lenhert, P. G. , Nichols, M. C. , Parrish, W. , Smith, D. K. , Smith, G. S. Snyder, R. L. , and Young, R. A., The standard data form for powder diffraction

data, Proceedings of Symposium on Accuracy in Powder Diffraction, Washington,

D.C. , June 1979.

[12] Shirley, R. , SPDIF - A standard interchange format for powder diffraction data,

Proceedings of Applied Crystallography Conference , Kozubnik, Poland, 1978 - in

press (1979).

Bibl iography

Commission on Crystal lographic Data, Powder data, J. Appl. Cryst. 4, 81-86

(1971).

Ishida, T. and Watanabe, Y. , Probability computer method of determining the

lattice parameters from powder diffraction data, J. Phys. Soc. Japan, 23,

556-565 (1967).

Ishida, T. and Watanabe, Y. , Analysis of powder diffraction patterns of

monoclinic and triclinic crystals, J. Appl. Cryst. 4, 311-316 (1971).

Ito, T. , A general powder x-ray photography, Nature, 164, 755-756 (1949).

Ito, T. , X-ray Studies on Polymorphism , 187-228 (Maruzen, Tokyo, 1950).

Kohlbeck, F. and Horl, E. M. , Indexing program for powder patterns especially

suitable for triclinic, monoclinic and orthorhombic lattices, J. Appl. Cryst. 9,

28-33 (1976)

Kohlbeck, F. and Horl, E. M. , Trial and error indexing program for powder

patterns of monoclinic substances, J. Appl. Cryst. 11, 60-61 (1978).

Lipson, H. , Indexing powder photographs of orthorhombic crystals, Acta Cryst. 2,

43-45 (1949).

Louer, D. and Louer, M. , Methode d'Essais et Erreurs pour l'Indexation Automatique

des Diagrammes des Poudre, J. Appl. Cryst. 5, 271-275 (1972).

Marriner, G. F. , Langford, J. I. and Tarney, J., Crystal data for CaF

2 «Si0

2

, J. Appl. Cryst. 12, 131-132 (1979).

Rouse, J. E. , M.Sc. Dissertation, University of Surrey, England (1973).

Runge, C. , Analysis of Crystal Structure by x-rays, Z. Zeits. 18, 509-515 (1917).

Shirley, R. , Recent advances in determining unknown unit cells from powder diffraction data, Acta Cryst. A31, S197 (abstract) (1975).

380
Shirley, R. and Louer, D. , New powder indexing programs for any symmetry which

combine grid-search with successive dichotomy, Acta Cryst. A34, S382 (abstract)

(1978).

Smith, G. S. and Kahara, E. , Automated computer indexing of powder patterns: the

monoclinic case, J. Appl. Cryst. 8, 681-683 (1975).

Smith, G. S. , Estimating unit cell volumes from powder diffraction data: the

triclinic case, J. Appl. Cryst. 9, 424-428 (1976).

Taupin, D. , Une methode generale pour l'indexation des diagrammes des poudres,

J. Appl. Cryst. 1, 178-181 (1968).

Taupin, D. , A powder-diagram automatic- indexing routine, J. Appl. Cryst. 6,

380-385 (1973).

Visser, J. W. , A fully automatic program for finding the unit cell from powder

data, J. Appl. Cryst. 2, 89-95 (1969).

Werner, P.-E., Trial and error computer methods for the indexing of unknown

powder patterns, Z. Kristal logr. 120, 375-387 (1964).

de Wolff, P. M. , On the determination of unit-cell dimensions from powder diffraction patterns, Acta Cryst. 10, 590-595 (1957).

de Wolff, P. M. , Detection of simultaneous zone relations among powder diffraction

lines, Acta Cryst. 11, 664-665 (1958).

Wolff, P. M. de, Indexing of powder diffraction patterns, Adv. X-Ray Anal. 6,

1-17 (1963).

juestion (Snyder): Assuming a constant measuring error A29 it is clear that AQ will

increase with line number and, in turn, Q and will be functions of line number.

/ill also show functional dependence on cell volume and symmetry. F^, however, is more

learly independent of cell volume symmetry and line number. Thus, will have a :onstant set of boundaries determining the corrections or incorrections of indexing. Why

io you prefer which requires a different set of criteria for each crystal class?

Response (Shirley): It does not seem to be in dispute that large volume and/or lower

symmetry make it harder to establish the correctness of a proposed cell. Therefore, a reliability measure for indexings ought to vary with volume and with symmetry to the

appropriate degree so as to correct for these influences. N^q does this to approximately

the right extent, because it was intended for this purpose, but F^ was specifically

designed to be independent of volume and crystal system, so as to be a more suitable

fneasure of data quality. F^ tests the quality of the data , M^q the reliability

jjof the model . requires approximately the same set of criteria for each crystal class,

because it has already taken into account the different level of agreement (from random

Discussion

381
scatter) that each class requires to achieve a particular level of confidence. The

1/3

increase of AQ with line number contributes a factor of VI which is included in the

1/3

overall volume dependence of V exhibited by M2Q , which is appropriate for its function

of evaluating proposed indexings.

Comment (Girgis): (1) When indexing powder patterns, if I get a primitive cell, I take

care and I am a little bit suspicious. If I get extinctions and can find a space group,

then I feel at ease. (2) The percentage of the indexed lines to the calculated lines:

the higher percentage gives an indication of the right solution.

Response (Shirley): (1) If you can see systematic absences that come together to make

a valid space group, or short list of possible space groups, then this is a good confirmation that the indexing is correct. I would not be worried by a primitive unit cell,

because most materials do indeed have primitive lattices, as long as the figure of merit

was over 20 and this did not conceal particular lines with unacceptable A20's. With

triclinic cells, of course, I should not expect to see any systematic absences, and

would feel that more investigation was needed if any seemed to be present.

(2) I agree that good coverage of the calculated pattern by observed lines is a com- forting situation, although I would only pay attention to the region near the origin of tht reciprocal lattice, because the general picture is already taken care of by the Nobs/Ncalc

term contained in all the continually used figures of merit. Low angle unobserved lines

that are not accounted for by systematic absences should not be present in quantity, although nearly every pattern seems to have two or three. Absent reflections like 200, 020,

002 are definitely disquieting and deserve further investigation, but correctly indexed pa terns sometimes exhibit such absences, so it is best to keep a reasonably open mind.

Comment (Girgis): We found it very helpful for indexing using besides the indexing programs, other physical measurements: e.g., (1) optical measurements to find out if the

crystal is uniaxial, (2) electron diffraction, and (3) density measurements.

Response (Shirley): I fully agree--this kind of supplementary information is very

valuable, especially electron diffraction which can give definite confirmation of the

corrections and symmetry of a number of zones. Polarizing microscopy is useful,

especially to confirm a high symmetry cell, but does usually require that the material

is transparent, and can be seriously impeded by twinning. Density measurements can be

extremely helpful, as Dr. Louer's talk will show, but one must be careful that the

chemical formula of the material is correct and that it is stoichiometric.

382
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

SUCCESSIVE DICHOTOMY METHOD FOR INDEXING POWDER PATTERNS

D. Louer

Universite de Rennes

Laboratoire de Cri stall ochimie

Campus de Beaulieu

35042 Rennes-Cedex - France

The successive dichotomy method [1] is an efficient procedure for

indexing accurate powder patterns. Such a procedure was used in

computer programs for indexing orthorhombic of higher symmetry systems

or, more recently, by incorporating if in a grid search program [2]

for lower symmetry systems.

The successive dichotomy method is a form of binary search with

an exhaustive strategy in parameter-space. Each variable parameter is divided into sections. Combination of these sections forms domains.

Using rigorous inequalities based on the experimental error of the

observed lines, each domain is tested. When a possible solution is

contained in a domain, a dichotomy process is used for reducing the

domain boundaries.

The introduction of a partition of volume-space, or domains of

volumes defined from density and error on density, decreases the cpu

time, and allows an extension of the dichotomy method to monoclinic

system. With this strategy two previous experimental versions of

programs (LOUV and LOUM) [3] were realized, and described in a paper

by Shirley [4]. They have been incorporated in DICVOL, which allows a search for systems down to monoclinic. The realization of a definitive

program based on the successive dichotomy method is in progress.

References:

[1] Louer, D. and Louer, M. , J. Appl. Cryst. 51 271-275 (1972).

[2] Shirley, R. and Louer, D. , Acta Cryst. A34, S 382 (1978).

[3] Louer, D. , unpubl i shed work.

[4] Shirley, R. , Crystal lographic Computing (Delft University Press Oosthoeks, 1978).

383
I

;

j

|

i

i

I

i

i

I

f
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS , Gaithersburq, MD, June 11-15, 1979

(Issued February 1980)

THE DETERMINATION OF THE PRECISE LATTICE PARAMETER

OF A DIFFUSE MINOR PHASE IN A DILUTE BINARY URANIUM ALLOY

D. A. Carpenter and C. M. Davenport

Oak Ridge Y-12 Plant 1 Oak Ridge, TN 37830

When a uranium-5.9 atomic percent niobium rod is extruded in the

alpha-phase region, a metastable two-phase microstructure results.

The bcc y^ phase, ranging in composition from 18 at. percent niobium

to about 34 at. percent niobium, constitutes approximately 15-20

percent of the volume of the extruded material, with the remaining

volume occupied by the orthorhombic alpha phase. The x-ray diffraction

pattern consists of intense, broadened alpha-phase peaks and one weak,

broad, low-angle [(110) at 37° 20, CuKa radiation] phase peak

partially overlapping the alpha-phase (002) peak. The (110) peak

has a FWHM value of 0.4-0.5° 20 and a peak height of 15-30 c/s.

The lattice parameter is very sensitive to the niobium concen- tration which varies with changes in the metastable a + thermodynamic state. The analysis of the diffraction pattern requires the

precise determination of the (110) peak position as an aid to understanding the physical metallurgy of the extrusion process. This is carried out by first resolving the (110) peak from the alpha (002)

peak by a nonlinear least-squares fit to a modified Lorentz function.

The resulting 26 value is then corrected for sample displacement by

using the drift correction of the alpha (002) peak calculated from a lattice parameter determination of the alpha phase. Repetitive mea- surements on different samples gave average standard deviations of

-5 - 0.006° 26 for the y-^ (100) peak position and 6 x 10 nm for the y1 lattice parameter.

Examples of this analysis carried out on alpha-phase extruded

U-5.9 at. percent Nb rods as a function of extrusion variables will be

presented.

Operated for the Department of Energy by Union Carbide CorporationNuclear Division under Contract W-7405-eng-26.

385
L

!'
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg , MD, June 11-15, 1979.

(Issued February 1980)

PRECISION GUINIER X-RAY POWDER DIFFRACTION DATA

J. W. Edmonds

Analytical Laboratories

The Dow Chemical Company

Midland, MI 48640

The opportunity to employ microcomputers to aid in x-ray powder diffraction data

quisition/reduction by microdensitometer reading of high resolution Guinier films not only

ovides readily available high quality data for phase characterization [l] 1 and reliable

ttern indexing, but indicates the accessibility of a much higher degree of precision for

ese systems than has previously been acknowledged.

The applications of least square fitting of profiles to diffraction data [2,3] raised

'O questions with respect to microphotometered Guinier film data: (1) would accuracy of the

termination of 26 ^5 be improved by the fit of an analytical profile and (2) would the type

I profile yield information concerning camera and/or sample aberrations affecting peak

ppe.The analytical profile chosen was the Pearson Type VII function due to the computaonal ease with which it can be varied in form from Cauchy to Gaussian [4]. Allowance for

ak asymmetry was made to using two split profiles for each peak with a common position

d height but differing breadths. Using an internal standard of size selected silicon [5], own sample preparation and film scanning techniques [1], fitting analytical profiles to gitized data from the entire Ewald sphere for CuK , radiation (front and back reflection

tterns from the same sample preparation), and determining cell constants by least squares

t of only those 26

Qbs

which lie between known Si lines, the following results were

tained:

Table 1

o-Al 00

Zn

2U3 0 a

Cu

0 b

o a,

4.75939 (5)

2.66474 (7)

3.6152 (3)

o c,A

12.9919 (1)

4.9480 (1)

0.004°

0.005°

0.013°

A28 Nobs/Nposs F.. max

K N

0.015°

0.010°

0.045°

37/57 199

19/19 194

8/8 77

a ° Crystallite size >10,000 A.

D ° Crystallite size ~300 A.

igures in brackets indicate the literature references at the end of this paper.

387
Two back reflection lines for Cu^ (400 and 420) are extremely weak and one (the 40. buried under a strong sample support line (Al foil). Peak resolution is difficult in tl two extreme cases, and the corresponding A20 values are -0.037° and 0.045°, respective!

0 o Deleting these values yields for Cu : a = 3.6152 [1] A; |A26| = 0.004°, A26 = O.t

F c = 185.

6 0 These results represent precision of 1 to 2 parts per 100,000 except for Cu (1 pa

in 35,000), and are within la of the cell constants reported by the National Bureau of

Standards.

These results give strong indications for the following conclusions:

1. Extremely high quality data is available from Guinier Cameras using high qua! i| internal standards.

2. The inherent high quality can be extracted with simple profile fitting techniq

to determine observed peak position.

3. For the fitted asymmetrical profiles, the best peak position at all values of [ is the profile center, a better estimation of the observed peak maximum. Instrumental

rations which shift the observed peak maximum from the true position are negligible or least very smal 1

4. Substantial peak asymmetry may be present but appears to be determinated by sa rather than instrumental aberration. The asymmetry is not a simple function of 26 as h been assumed in some applications of the Rietveld method to x-ray film data [6].

5. Profile variations in one film are a function both of background selection and

line intensity due to the effects of film sensitivity in addition to the effects noted '

The profile will not easily be "quantitated" as it can be for the dif fractometer [2].

6. Various effects on lattice parameters can be pronounced, especially temperaturi

variation during data collection, and changes in chemical composition and physical stat|

(strain, imperfections) of the sample. Even though it is possible to measure lattice

parameters with this level of precision, the same level of reproducible accuracy will b more difficult to achieve due to the variation of the above parameters from laboratory

j laboratory, especially during the simplest of sample preparations.

References

[1] Edmonds, J. W. and Henslee, W. W. , in Advances in X-ray Analysis , Vol. 22, 143-150

(Plenum, New York, 1979).

[2] Parrish, W. , Huang, T. C. and Ayers, G. L. , Trans. Am. Cryst. Assoc. 12, 55-73 (19

See also these proceedings.

[3] See papers by Hewat, A. E. , Young, R. A., Baerlocher, Ch. , Cox, D. E. , Albinati, a! and Prince, E. , in these proceedings.

[4] Hall, M. M. , Jr., Verraraghavan, V. G. , Rubin, H. andWinchell, P. G. , J. Appl. Cr

10, 66-68 (1977).

388
5] NBS SRM 640. Particle size reduced by

particles which dropped out within the

using.

6] Malmros, G. and Thomas, J. 0., J. Appl.

suspending in isopropyl alcohol and regrinding

first half hour. Fines were vacuum dried before

Cryst. 10, 7-11 (1977).

389

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

A SIMPLE GRAPHICAL METHOD FOR OBTAINING

REASONABLY ACCURATE CELL DIMENSIONS FROM X-RAY

POWDER PHOTOGRAPHS OF HEXAGONAL AND TETRAGONAL MINERALS

E. E. Fejer

British Museum, Natural History

London, S. W. 7 Cromwell Road

ENGLAND

This is essentially a graphical, high-angle extrapolation method

using the Nelson-Riley function of obtaining cell dimensions from

11.46 cm diameter powder photographs of cubic, tetragonal, hexagonal,

2 and occasionally even of orthorhombic compounds. Sin 6 corrected to

AKa, is obtained by direct measurement of 2s = 360° - 46 of suitable

lines on the film and divided by a+pR where a is

.22h + k for tetragonal

2 2 2 crystals, p is £ , and R is a /c . A (the ratio obtained) is plotted

against corresponding values of the Nelson-Riley function. A straight

line drawn through the points thus obtained is extended to where it

meets the axis. From the intercept at this point a and c can be

calculated. By slight and judicious alteration of R the graph can usually be improved considerably. Erroneous indexing will also become

obvious. From a final A-value, and taking into account possible

errors, reproducible cell dimensions can be calculated, and these were

found to be correct to about 3-4 parts in ten thousand.

391

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

THE RELIABILITY OF POWDER INDEXING PROCEDURES

A. D. Mighell and J. K. Stalick

Center for Materials Science

National Bureau of Standards

Washington, DC 20234

Recent work has proved that identification/registration of molecular

materials by cell parameters is extremely effective. The NBS Crystal

Data File contains unit-cell data for 60,000 substances; of these, 30,000

are molecular materials which crystallize predominantly in the monoclinic, triclinic, and orthorhombic crystal systems. If one can obtain

a unit cell of a crystalline compound from a powder sample, the basis for

a convenient identification procedure is established. Consequently, it

is essential to determine the reliability of indexing procedures for the

low- symmetry crystal systems.

In our analysis, we used a standard indexing program on fifty powder

patterns measured by the JCPDS Associateship at the National Bureau of

Standards. The indexing program determined the correct cell in about

50 percent of the cases and a derivative lattice in many of the remaining

cases. As the quality of data decreases, the probability of obtaining either the correct lattice or a derivative lattice continually

decreases. We have noted that unindexed lines often occur, not from

a small amount of a second phase, but because the program has found a derivative lattice. This study has shown that the existence of

geometrical ambiguities constitutes a major problem to the indexing

program.

1. Introduction

A solid-state material can conveniently be characterized by its unit cell. Recent

^search with large data bases has proved that the lattice of a crystalline material

highly diagnostic; this is particularly true for molecular compounds, which usually

ystallize in such lower-symmetry crystal systems as monoclinic and triclinic. Thus, a

'terial can be registered or identified once any primitive cell which defines the lattice

393
has been determined. The procedure [l] 1 consists of the following steps (figure 1): M

determine a primitive cell in reciprocal space; (2) convert this cell to a primitive j»] in direct space; (3) reduce the cell; (4) check against the file of known materials, jjh NBS Crystal Data File2 contains unit-cell information for about 60,000 substances and ^

growing at the rate of about 5,000 entries per year. This represents one of the largit

evaluated numerical data bases, roughly twice the size of the Powder Diffraction Filel 1

The Crystal Data File is soon to be an active component of the NIH/EPA Chemical

Information System. To automate the entire registration/identification procedure, itif

important to develop simple, inexpensive, and rapid methods for unit cell determi nati; l.

UNKNOWN

CRYSTAL

PRIMITIVE

CELL RECIPROCAL SPACE

PRIMITIVE

CELL DIRECT SPACE

REDUCED

CELL

IDENTIFICATION CHECK REDUCED

CELL FILE OF KNOWN

COMPOUNDS

DERIVATIVE CELLS

REDUCED

DERIVATIVE CELLS

Figure 1. Outline of a crystal lochemical identification procedure based

on the NBS Crystal Data File reduced cells. Identification is still

possible if the unknown cell defines a derivative lattice (e.g. a sub-,

super-, or composite lattice).

i

1 Figures in brackets indicate the literature references at the end of this paper.

2 This file contains data that have been published in the Crystal Data Determinative Jb}

[2], as well as more recent data.

3 JCPDS--International Centre for Diffraction Data. r

394
If a single crystal of suitable size can be obtained, one can determine the unit cell

(flii'th an automated diffractometer or by film techniques. However, this is not always

possible owing to the lack of necessary equipment or to the nature of the sample. Often

fi highly crystalline sample may be available but with crystals too small for the

1 In'ngle-crystal method. This is frequently true for molecular compounds such as organics

|Sl|and organometall ics. The organic component of the file has more than 30,000 entries

|md contains many compounds of industrial importance. For these materials, one potential

(approach is to take a powder pattern, and to index the pattern to obtain a unit cell.

j)nce a cell has been obtained, the Crystal Data File can be checked for identification.

o<! If powder indexing procedures are to be successful for molecular substances, they

Lust work well in the lower- symmetry crystal systems. Figure 2 shows the space-group

distribution for organic and organometall ic compounds. The most common space groups are

5 2^/c and PI. In fact, 75 percent of the materials fall into five space groups, the

first four of which are primitive. Figure 3 shows a classification of these materials by

reduced form. A few reduced forms dominate; they correspond to the primitive orthorhombic, monoclinic, and triclinic Bravais lattices. Consequently, in our analysis of

[oowder indexing procedures we have concentrated on the indexing of powder patterns for

substances that crystallize in the lower- symmetry crystal systems. An earlier analysis by

3. G. Johnson and J. W. Visser [3] has shown that Visser's indexing program [4] is indeed

a valuable tool for obtaining unit cells from powder data. In our study special emphasis

[has been placed on analyzing the geometrical problems associated with the indexing of

ipowder data.

2. Experimental

We used a sample of fifty powder patterns in the indexing study. The data were

selected from "Standard X-ray Diffraction Patterns" taken by the JCPDS Associateship at

J the National Bureau of Standards [5]. These patterns are of high quality with an average

error of slightly more than 0.01° in 28 for the lines in the powder pattern. An automatic

[program for finding the unit cell from powder data [4] was used. The following five types

jpf test problems were used as input to the indexing program: (1) the first 20 observed

.powder lines with the observed 26 values; (2) the first 20 observed powder lines with

Ijcalculated 28 values; (3) the first 20 calculated lines (whether observed or not) with

ijcalculated 28 values; (4) 20 lines obtained by selecting every other calculated line with

i[calculated 28 values; (5) 20 lines obtained by selecting every third calculated line with

[calculated 28 values. The first option gave us a test of the indexing program on actual

jldata. The other options, all of which used exact data, gave us insight on the interaction

jOf the indexing program with geometrical problems associated with indexing.

When using actual data, the program found a cell defining the correct lattice in 50

[percent of the cases. In the remaining cases, the program found either a cell defining a [^derivative lattice or a cell defining a lattice that bore no simple relationship to the

icorrect lattice. Test problems (2)... (5) repeatedly illustrated the troublesome nature of

^geometrical problems associated with the indexing of powder data. No matter how reliable

395
Space-Group Frequency

(Organic)

Space group Symbol and no. DO irI 111

r 1

(19)

it/

r*o//* MO\ 10/

rDCa (D If Pnma IA91 Pna21 (33) Pbcn (60)

P1 (1) C2 (5) Cc (9)

Pca21 (29)

P21 /m (11) C2/m (12) P21 21 2 (18)

P2/c (13)

Fdd2 (43)

Pc (7) R3 (148)

I 1 1 1 1 1 1 1—^ 1

1 2 3 4 5 6 7 12

Percent of the total compounds

Figure 2. Relative frequency of space group occurrence for 21,051 organic

and organometal 1 ic crystalline compounds. The data are from Volumes 1 and

3 of Crystal Data Determinative Tables and from the structural entries in

the Cambridge Data Bank.

<m the observed 26 values are, one often does not obtain the true lattice. Furthermore,

found the indexing program extremely sensitive to the quality of the data. Thus, wher

using the first 20 observed powder lines with calculated 26 values, we obtained the

correct solution in 75 percent of the cases. In the remaining cases, we usually obtai

a derivative lattice as one of the solutions.

396

I
REDUCED FORM FREQUENCY

(21051 ORGANIC COMPOUNDS)

BRAVAIS LATTICE REDUCED _ FORM , SYSTEM^CENTERING NO.

"

C

RC

R

C

T

T0

RM

T

H0M

T

0M

T

0M

T

H

0

RM

0M

M

M

M

A

0M

M

M

0M

0M

0M

0M

A

FHR

P

HR

I

I

I

I HR

C OR I P

PC

C OR I

IF C OR I

I

I C OR I P

PCHR

C OR I F COR I C

CC

P

P

P

P

PC

C OR I C

C OR I C

C OR I

I

IP

12

34

5

6

7

89

10

1112

1314

15

161718

1920

2122

2324

25

26

2728

2930

31323334

35

36

37383940

41 h

42

4344

PERCENT OF TOTAL SAMPLE -+~ 2_3 4 5 6 7 8 9 10 11 12 13 14 15 1 6 17 18 19 20 21 22 23 24 25

t i i i i i ii—i—i—

n—i—i—i—i—i—i—

•8.25% TRICLINIC P 24.6% ORTHORHOMBIC P

43.7% MONOCLINIC P

6.8% TRICLINIC P

' '

i i ' i I i I I I I l I Ii i i i

i i i i

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 PERCENT OF TOTAL SAMPLE "KEY: A/M/O/R/T/H/C FOR TRICLINIC/MONOCLINIC/ORTHORHOMBIC/

RHOMBOHEDRAL/TETRAGONAL/HEXAGONAL/CUBIC.

Figure 3. Classification of organic materials by reduced form number (see

reference [6]). The figure shows the relative frequency of occurrence for

each of the 44 reduced forms. The compounds are the same as those in

figure 2. These statistics were obtained in collaboration with John Rodgers

of the Cambridge Crystal lographic Data Centre.

397
3.1.

3. Discussion

Theoretical background

Because we have data that correspond only to the lengths of reciprocal lattice

vectors, two or more lattices are often consistent with large regions of the powder

pattern. Indexing problems of a geometrical nature can best be understood in the

framework of derivative lattices. In a paper by Santoro and Mighell [7] derivative

lattices were defined and divided into three types that are specified by the nature of

transformation matrix S that relates the original lattice and the derivative lattice,

derivative lattice is defined in [7] as follows:

b. = *f..a. (i,j = 1,2,3)

The original lattice is specified by a., (three noncoplanar translations); the derivativ<

lattice is specified by b^ (three noncoplanar translations that can be regarded as the

edges of a primitive cell); and S is the transformation matrix relating the two lattice^

A derivative lattice is a superlattice if the elements S^. of matrix S are integers and]

the determinant |S| is greater than one. A derivative lattice is a sublattice if the

elements T.j of matrix T (where T = S ) are integers and if the determinant )T| is greater than one. Finally, a derivative lattice is a composite lattice if one or more

the elements S.. of matrix S and one or more of the elements T. . of matrix T are

lj ~ i J ~

fractional

Derivative lattices have features in common. Thus all nodes of a superlattice are nodes of the original lattice, but the reverse is not so; likewise all nodes of the oricj nal lattice are nodes of the sublattice but not all nodes of the sublattice are in the

original lattice. A composite lattice is neither a sublattice nor a superlattice but it partakes of the two. Of particular interest to the indexing problem is the fact that

derivative lattices have a certain number of d-spacings in common. With the indexing

program we commonly obtained superl attices of twice the volume of the true lattice,

sublattices of one-half the volume, and composite lattices of the same volume. Also, we obtained more complex derivative lattices with nonintegral volume relationships to the

correct lattice.

3.2. Total geometrical ambiguities

A total geometrical ambiguity occurs if two or more different lattices have exactly

the same set of d-spacings. These ambiguities have been discussed in detail in a paper

Mighell and Santoro [8]. Figure 4 shows a schematic example of a four-fold total

geometrical ambiguity. The same set of d-spacings is consistent with four different

398
GEOMETRICAL AMBIGUITY

20

Cubic I Lattice 1

Tetragonal P

Lattice 2 Orthorhombic F Lattice 3 Orthorhombic P

Lattice 4

V/2 V/3 V/4

Figure 4. Total geometrical ambiguities. From any cubic I lattice, one can calculate a tetragonal P, an orthorhombic F and an orthorhombic P lattice

all of which have exactly the same d-spacings (ref. [8]) as depicted

schematically. The transformation matrices required to produce this fourfold ambiguity are:

cubic I » tetragonal P

cubic I * orthorhombic F

cubic I * orthorhombic P

(0 \\ I 0 \ \ / \

0 0)

(||o/ooT/iTo)

(JJo/ooJ/fJo)

All four lattices are represented by different reduced cells with primitive

cell volumes in the ratios: ^ : ^ as shown.

lattices, characterized by four different reduced cells with four different volumes.

\lthough total geometrical ambiguities are probably restricted principally to the higher

symmetry systems, their existence is proof that indexing is by no means simple and

ometimes more than one lattice is consistent with a given set of d-spacings.

3.3. Partial geometrical ambiguities

A partial geometrical ambiguity occurs if two or more different lattices have some

)ut not all d-spacings in common. Sometimes two different lattices are consistent with

399
large regions of the same powder pattern (hence the importance of using all the observedj

data in the indexing program). The indexing program is always confronted by_ partial

geometrical ambiguities because every lattice has associated derivative lattices and

derivative lattices have d-spacings in common. A schematic example of this is shown in ; figure 5. In this figure, lattices (1) and (2) have many d-spacings in common. These

two lattices are in a composite relationship with each other as they have a common superlattice (they may be likened to siblings). If one allows for accidental absences, U it is quite jDossibl e that more than one consistent solution exists (hence the importances

of using all possible weak lines as well as strong lines). I;

29

PARTIAL AMBIGUITIES

True lattice Sublattice (1) Sublattice (2) Sublattice (7)

X1

V/2 V/2 V/2

Figure 5. Partial geometrical ambiguities. Composite lattices often have many

d-spacings in common and sometimes the d-spacings can be identical over a large

region in 20. For example, the seven sublattices of V/2 (i.e. those with defining primitive unit cells of 1/2 the volume of the unit cell defining the

original lattice) that are associated with any given original lattice are in a composite relationship to one another. For a procedure to generate such derivative lattices see Santoro and Mighell [7,9], and for a listing of the transformation matrices see Mighell [1]. All d-spacings in each of the sublattices are

in the original lattice. Also each sublattice has d-spacings in common with the

other sublattices. Here sublattices (1) and (2) have d-spacings in common except|

Hj

for line 'A' which is not in sublattice 2, and 'B' which is not in sublattice (1)

Usually such a close match does not occur over the entire powder pattern but it

is not uncommon for such close agreement to occur over large regions of 26. Acci

dental absences will compound the problem.

400
3.4. Misref inements

The concept of the existence of partial geometrical ambiguities allows us to explain

certain deceptive types of solutions that occur in the indexing of powder patterns. The

indexing program often finds a solution that indexes the powder pattern except for a few

lines. In many cases unindexed lines may result not from a second phase but rather from

the fact that one has determined a derivative lattice. This is illustrated in figure 6.

Here, cells (1) and (2) of the same volume define lattices that are in a composite relationship with one another. Let us suppose that the indexing program finds cell (2) and then

refines- this cell with the data that really are from cell (1). Since cell (1) and cell (2)

have a common superlattice, many d-spacings are in common. Two things can happen: first,

if one is fortunate, such a refinement will use only the lines in common to both lattices

REFINEMENT OF CELL (2) WITH DATA OF CELL (1)

2 9 Superlattice Lattice (1) Lattice (2)

v

Figure 6. Schematic example of how a misref inement can occur. Here the true lattice

(1) and lattice (2) of volume V (V = volume of the primitive unit cell defining the

lattice) have common superlattices of 2V. Hence all lines in lattice (1) and lattice (2) are in the superlattice. Lattices (1) and (2) are in a composite relationship with each other and have many lines in common. The indexing program mistakenly

tries to refine cell (2) with the data for cell (1). The line marked 'A' will not cause a problem as it will be considered an accidental absence. Line 'B' will cause

a problem because it is observed but is not present in lattice (2). As 'B' is very

close to line 'C, the program will use line 'B

1 for 'C which will result in a misref inement. Such misrefinements are not uncommon especially with data of poor

qual ity. 401

1 1 rue lattice)

2 V
and several lines will remain unindexed. Such cases occur and it is theoretically

possible to transform cell ^2) to cell (1) with a transformation matrix S as defined

earlier for a composite lattice. Thus, although one has not obtained the correct lattice,

one has obtained a correct derivative lattice. Second, if one is "unlucky" the refinement!

will use lines common to both lattices and some of the lines that do not belong (but are close) to allowed d-spacings for lattice (2). This results in a misref inement and one obtains an answer that is neither the correct lattice nor a derivative lattice.

Misref inements become more and more common as the quality of the data decreases. Thus,

the chance of a misrefinement can be minimized by taking good data and by adjusting

parameters of the indexing program so that it will use only observed data in the

refinement that are close to calculated values. The data must be good and the program

must be told so.

i

4. Conclusions

The major conclusion of this study is that automated indexing programs have great

potential for finding cells suitable for identification and for characterization.

However, the data must be of high quality. If single crystals of appropriate size cannot

be obtained, an attractive alternative is to measure a powder pattern and to employ an indexing program to obtain a unit cell. As indicated earlier, if one has a correct

primitive cell of the lattice, identification against the Crystal Data File is almost 100

percent certain, especially if one has some chemical information. Also, recent work with

the Crystal Data File and with new lattice identification techniques indicates that

identification is still likely if one has determined a simple derivative lattice.

Our study has revealed several considerations in using an indexing program: (1) as many lines as possible should be used, including small peaks; (2) the observed data must

be of high quality (see also [10]); (3) peaks in the low 26 region of the powder pattern

are especially important; (4) unindexed peaks can occur either from a second phase or from;

a derivative lattice; (5) figures of merit should be interpreted with caution because in

certain cases derivative lattices can give high figures of merit; (6) sometimes more than I one solution exists (i.e. two or more lattices that are consistent with the same set of

d-spacings). Finally, the study has shown that the power of the indexing program can be

significantly augmented by devising new algorithms that take into account the existence ofj

derivative lattices and will find the transformation matrix S that relates a derivative

lattice to the true lattice.

II

We wish to thank Antonio Santoro for many helpful discussions in connection with this! work and Timothy Blizzard for his assistance with the many computer runs required for the

survey.

402
References

I [1] Mighell, A. D. , J. Appl. Cryst. 9, 491-498 (1976).

[2] Crystal Data Determinative Tables , Third Edition: Volumes 1 and 2, J. D. H. Donnay,

H. M. Ondik, S. Samson, Q. Johnson, M. H. Mueller, E. R. Ryba, G. M. Wolten,

M. E. Mrose, 0. Kennard, D. G. Watson, and M. V. King, eds. (1972,1973);

Volume 3, 0. Kennard, D. G. Watson, J. R. Rodgers, S. M. Weeds, A. Doubleday, and

B. Hummel ink-Peters, eds. (1978); Volume 4, H. M. Ondik, A. D. Mighell, M. E. Mrose,

C. R. Robbins, and J. K. Stalick, eds. (1978). U.S. Department of Commerce, National

Bureau of Standards and the JCPDS-International Centre for Diffraction Data, Swarthmore, PA.

[3] Johnson, G. G. and Visser, J. W. , Paper F10, American Crystallographic Association Spring Meeting, University of Virginia, Charlottesville, Virginia (1975).

[4] Visser, J. W. , J. Appl. Cryst. 2, 89-95 (1969).

[5] Standard X-ray Diffraction Patterns published in Powder Diffraction Data from the

Joint Committee on Powder Diffraction Standards Associateship at the National

Bureau of Standards, JCPDS--International Centre for Diffraction Data, 1601 Park

Lane, Swarthmore, PA, 19801 (1976).

[6] Internationa l Table s for X-ray Crystal! ography , Vol. 1 (Kynoch Press, Birmingham,

1969), pp. 530-535.

[7] Santoro, A. and Mighell, A. D. , Acta Cryst. A28, 284-287 (1972).

[8] Mighell, A. D. and Santoro, A., J. Appl. Cryst. 8, 372-374 (1975).

[9] Santoro, A. and Mighell, A. D. , Acta Cryst. A29, 171-175 (1973) (See appendix).

[10] Shirley, R. , "Data Accuracy for Powder Indexing," Symposium on Accuracy

in Powder Diffraction, National Bureau of Standards, Gaithersburg, Maryland (1979).

Discussion

Comment (Girgis): You showed a slide with four "correct" solutions of indexing. One of

them is cubic, others are orthorhombic P, orthorhombic F, and tetragonal P. I would use optical measurements which would give me a clear differentiation between these crystal

systems giving the right solution.

Response (Mighell and Stalick): In our study, we analyzed the reliability of indexing

procedures assuming that only a set of d-spacings is available. In actual practice,

however, optical measurements should routinely be carried out. We completely agree with

your statement that such measurements are often invaluable in distinguishing between

several possible cells that could index a powder pattern.

403

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

X-RAY POWDER DIFFRACTION IDENTIFICATION OR CRYSTAL

PHASES WITH SUPERIMPOSED LINES BY THEIR SELECTIVE CRYSTALLIZATION

I . Mayer

Department of Inorganic and Analytical Chemistry

I. Gedalia and B. Laufer

Preventive Dentistry

Dental Morphology

Hadassah School of Dental Medicine

Hebrew University, Jerusalem, Israel

The identification of chemical compounds by x-ray powder diffraction is based on the d values of the three strongest lines of

their x-ray powder pattern. This method is seriously handicapped when

two phases coexist in the system and the strongest lines of one of

them are superimposed by the lines of the other. This problem and its

possible solution by selective crystallization is demonstrated in the

following example:

The fluoridation of Ca^PO^OH, the main mineral constituant of

dentin, was studied. When Ca^PO^^OH is treated by a NaF solution in

acid media, CaF2 can form. CaF2 has its three strongest lines at d values of 1.93^, 3.15g and 1.65^, and the first two were found to be

superimposed by the (222) and (102) reflections of Ca

5 (P0

4

) 30H. As a result of this, CaF2 could not be detected in the system, because even

if present, its relatively small amounts cannot change the intensity

of the (222) and (102) reflections. These difficulties have been overcome when the samples were heated to about 600 °C. After this

heat treatment, the intensity of the lines at d values 1.93 and 3.15

relative to the intensity of other reflections of the pattern has

increased to such an extent that leaves no doubt that the origin of

the increased intensity has to be attributed to the presence of CaF2>

The above results seem to indicate that at 600 °C the crystallization

of CaF2 in the hydroxyapatite is significantly improved. Consequently,

sharp increase in the intensity of the reflections of CaF2 takes place

relative to that of Ca^POJ^OH.

405

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

COMPARISON BETWEEN DEBYE-SCHERRER, TRANSMISSION

AND REFLECTION MEASURING MODES

E. Woelfel

Stoe

Darmstadt, W. Germany

An automated powder counter dif fractometer for Debye Scherrer

capillaries, powder plates and thin powder layers has been developed

on which transmission and reflection measuring modes can be utilized.

The instrument is also suitable for low angle scattering experiments.

Curved graphite or silicon monochromators are used in SeemannBohlin geometry. For reduction of fluorescence radiation a secondary

monochromator between specimen and counter is available.

The instrument will be described in details and examples will be

discussed which show advantages and disadvantages of the different

measuring modes.

407
!
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

STATISTICAL ANALYSIS OF THE MEASUREMENT OF GRAIN

AND PARTICLE SIZE WITH X-RAYS

J. Hilliard

Department of Materials Science and Engineering

Northwestern University

Evanston, IL 60201

In collaboration with J. B. Cohen and C. Rinik, a comparison has

been made of the determination of grain size in commercial steel

samples by optical microscopy and by the variation in the intensity of

the x-ray beam diffracted from different areas of the specimen. By

using a ratio of two of the moments of the optically measured interceptlength distribution, an exact comparison could be made of the average

grain size yielded by the two methods. Good agreement was found, provided the grains were reasonably equiaxed.

The x-ray method is not only non-destructive, but it is also sufficiently rapid that it could be used for the continuous monitoring of

grain size during the rolling of sheet steel. The application of the

procedure to the sizing of loose powders will also be discussed.

409

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

ANALYSIS AND TOPOGRAPHY OF LATTICE DEFECTS IN POWDER DIFFRACTION PATTERNS

S. Weissmann

College of Engineering

Rutgers University

Piscataway, NJ 08854

An x-ray method is described which enables the investigator to analyze quantitatively the defect structure of individual crystallites in a polycrystal 1 ine specimen, and which provides him also

with x-ray topographic images of the analyzed crystallites. The

analysis of the defect structure is based on the principle of the

double-crystal di f f ractometer. The specimen is irradiated by a crystal -monochromated beam and each reflecting crystallite is considered to function independently as the test crystal of a double-crystal diff ractometer. By carrying out discrete, angular

specimen rotations and appropriate film shifts, to separate the

spot reflections corresponding to successive rotation positions,

one obtains an array of spots for each reflecting crystallite.

These spot arrays, with their intensity dependence on the angular

position, represent rocking curves. On the basis of the rocking- curve parameters, the excess dislocation density of the crystallite

can be determined. If a resolvable substructure exists within the

crystallite, the rocking curve exhibits a multimodal intensity

distribution, the analysis of which gives the excess dislocation

density in the subgrain boundary and in the subgrain. The topographic imaging of the reflecting crystallites is obtained by

taking reflection topographs at the specimen surface. The

correlation of these surface images and the analyzed spot reflections along the Debye arcs on the cylindrical film is accomplished

by outward tracing of the reflection images. Using an exposure

method which employs a superposition technique of unfiltered and

crystal -monochromated radiation, the location of the imaged

crystallites on a flat specimen is established. In this way, the

morphology of the crystallites studied by the x-ray methods can be

411
linked to that analyzed by other imaging methods, such as light

microscopy, transmission and scanning electron microscopy.

Applications of the x-ray method to important technological

problems in metallurgy are given.

Key Words: Double-crystal di f f ractometer ; excess dislocation

density; lattice defects; rocking curves; subgrain structure; x-ray

topography.

1. Introduction

Research work in materials science during the last decades has clearly shown that j many physical properties of solid materials, such as the self-diffusion, strength and j plastic properties of metals and alloys, are structure-sensitive and are, therefore,

intricately linked to faults and lattice inhomogeneities in crystal structure. Consequently, it is not surprising that an increased effort has been made in recent years to elucidate the interrelationship of the crystalline fine-structure of a solid with the

observed structure-sensitive properties.

Metal lographic and x-ray methods are usually employed for the study of lattice

inhomogeneities. If single crystals are studied, no particular difficulty is encounter?

in correlating the metal lographic and x-ray observations, although the correlation in

fine-structural details, as revealed by both methods, may sometimes pose a serious

problem. The problem becomes much more difficult if one wishes to investigate the

substructure characteristics of fine-grained polycrystal 1 i ne materials, viz. metals,

alloys, or ceramic materials. In such instances, the observations made by metal lographij

and x-ray methods cannot usually be directly correlated to the individual grain or crystallite and one has to rely on statistical sampling procedures. A correlation,

however, between metal lographic and x-ray observations with respect to the identical

crystallites of the specimen becomes increasingly more important, particularly if

structural changes that result from chemical or physical processes are to be studied.

Problems of this sort arise in the study of deformation, creep, fatigue, recovery,

recrystal 1 ization, precipitation and solution-hardening, to mention only a few. Such

correlation of observations has been accomplished by a method combining light microscopy

with x-ray reflection microscopy and diffraction analysis and will presently be describe

An unfiltered, divergent x-ray beam P passing through the collimator C is made to impinge at a small grazing angle on the surface of the polycrystal 1 ine test specimen S located at the center of a Debye-Scherrer camera E (fig. 1). A fine-grained photographi

2. Diffraction Method

2.1. X-ray reflection microscopy

412
plate D is placed at a distance of about 0.5 mm from the specimen surface. Owing to the

divergence and spread of wavelengths of the beam, a great many crystallites, on the

specimen surface satisfy the Bragg conditions of reflection and are, therefore, recorded

on the photographic plate in the form of an x-ray reflection micrograph (Berg-Barrett

topograph). This x-ray micrograph reveals the topographical relationship of the

crystallites and can, therefore, be directly compared to the corresponding light

micrograph. A correlation of the observations made of each crystallite by light and

x-ray reflection microscopy is thus achieved.

Figure 1. Experimental arrangement for taking reflection x-ray

micrographs and tracing of reflecting images. Solid lines with

crystal A retracted indicate arrangement for use of heterogeneous

radiation. Dashed lines with crystal A in reflecting position

indicate arrangement for use of crystal -monochromated radiation.

The successive positions of the films for the outward tracing of

reflections are given by Roman numerals.

413
For the diffraction analysis of lattice imperfections of the crystallites, additional experimental steps have to be carried out. First of all, there are usually too many reflection images recorded and the number of reflecting crystallites has to be

drastically restricted. This is conveniently accomplished by interposing a single

crystal A into the path of the beam P and adjusting the crystal so as to reflect the

incident beam. The (111) reflection of a silicon or germanium crystal, possessing a hi degree of lattice perfection, is preferred. The reflected, monochromated beam M thus

subtends an angle of 26 with the primary beam P, where 6 is the Bragg angle of the

reflecting (hk£) planes of the monochromator. By rotating the entire assembly, con- sisting of specimen S, photographic plate D, cylindrical camera E, radiation detector R and slit system B, by the exact angle of 26 around an axis 0 coincident with the vertice|

axis of rotation of the monochromati ng crystal, the identical area of the specimen S is now irradiated with crystal -monochromated radiation M. This experimental arrangement i^

schematically represented in figure 1 by dashed lines. Because of the stringent

reflecting condition imposed by the monochromatization of the beam, fewer crystallites

will now be recorded on the x-ray micrograph. The interposition of the monochromator

may, therefore, be likened to the use of a high-power objective in a light microscope,

except that in the x-ray analogue the selectivity of the image is achieved by restrictTM

the orientation rather than the size of the crystallites in the field of view [1.2] 1

. \ the specimen, however, consists of well-annealed grains and has an appropriate size

distribution, the x-ray reflection micrograph, taken with a monochromated beam, may

frequently suffice to establish the identity of the grains with those on the corres- ponding light micrograph. Figures 2a and 2b show the metallograph and x-ray reflection!

micrograph of an annealed 304 stainless steel sample, respectively. The identity of a few grains is given by numbers.

2.2. Tracing of reflection images

I

The individual reflection images can be traced outward in space by recording the 1

images photographically at increasing distances from the specimen surface. The successive positions of the photographic plates are schematically indicated in figure 1 by Rorj numerals, whereby V represents the circumference of the Debye-Scherrer camera. By mean; of this tracing technique, there is established a direct correlation between the spot

reflections on the Debye-Scherrer lines and the crystallites on the specimen surface

giving rise to these reflections [1]. Figures 3a and 3b show the spatial tracing of tht grain reflections of figure 2b. Selected reflection micrographs at distances of 2.5 ani 27.5 mm from the specimen surface are shown. For the sake of clarity only three image

reflections, marked by numbers 1, 2, and 3, are shown. These images were selected not only for the purpose of illustrating the image tracing technique, but also to demonstraj

a special versatility of the combination method, which will be described in a unique

application of this method to stress-corrosion studies (Section 5). It is perhaps

Figures in brackets indicate the literature references at the end of this paper.

414
arthwhile noting that, due to different crystal orientation, images that appeared in

jpographic proximity at the surface ended up at different (hk£) reflections on the

ircumference of the Debye-Scherrer camera (compare figure 2b to figure 4a).

Figure 2. Identification of grains of 304 stainless steel, (a) Metal 1 ograph.

(b) X-Ray reflection micrograph. Crystal-monochromated OK radiation.

a

l

415

I
Figure 3. Outward tracing of reflection images of Fig. 2b. (a) 2.5 mm

from specimen surface, (b) 27.5 mm from specimen surface.

2.3. Quantitative diffraction analysis of lattice defects

After the spot reflections along the Debye-Scherrer arcs have been correlated b\ spatial image tracing technique to the topography of the grains at the surface, and

j via x-ray topography to light metallography, a quantitative analysis of the spot

reflections is undertaken. This analysis is based on the principle of the double- i crystal di f f ractometer. Each reflecting crystallite of the polycrystal 1 ine test spei, functions independently as the second crystal (test crystal) of a double-crystal

416
I diffractometer. Consequently, by carrying out discrete, angular specimen rotations and

I appropriate film shifts between each specimen rotation to separate the spot reflections

corresponding to successive rotation positions, one obtains arrays of spots for each

! reflecting crystallite, such as those shown in figures 4a and 4b. These spot arrays

represent rocking curves, or rather angular samplings of the rocking curves, of the

crystallites, since the intensity variation as a function of discrete specimen rotation

is being recorded. The angular specimen rotation of the 304 stainless steel sample of

figures 4a and 4b was carried out in intervals of 3 minutes of arc. The type of diagram

obtained is conveniently referred to as multiple-exposure diagram.

3. Photometric Transformation of Crystallite Rocking Curves [3,4]

The arrays of spots at the equatorial reflection of the cylindrical camera, with

their intensity dependence on specimen rotation, are analogous to the rocking curves of

the conventional double-crystal diffractometer, because for equatorial reflections the

prystallites have their plane normals parallel to the plane normal of the monochromating

crystal, regardless whether they reflect in the parallel (m, -n) or antiparallel (m, +n)

irrangement relative to the monochromating crystal. Because the reflecting crystallites

:ontain lattice defects, viz. dislocations, one may conceive the picture that the

reciprocal lattice point of a (hk£) reflection of a perfect crystal is now replaced by a Reciprocal lattice volume. The change of shape and intensity of the diffracted spots in

he sequence of the array depends now on the extension of the contact of this reciprocal

attice volume with the Ewald sphere. For equatorial reflections, the Ewald sphere in

he double-crystal diffractomer arrangement is sharply defined by virtue of the monohromating property of the first crystal. The horizontal convergence of the monochroated beam irradiating the crystallites is controlled by the rocking curve half-width of

he monochromatizer and amounts typically to only about 10-15 seconds of arc. The

ertical convergence for equatorial reflections is zero.

If one analyzes, however, the rocking curves of perfect crystallites for nonquatorial reflections, one will observe a systematic increase of the rocking curves with

jfncrease of azimuthal elevation t|i. The apparent increase is due to two factors, namely

I me velocity factor associated with the rotation of the crystallites, and the increased

pntribution of the vertical convergence of the beam, which remains unaltered after

Hi

i eflection from the monochromating, first crystal. For crystallites containing lattice

* pfects, the non-equatorial reflections exhibit still larger increases of the rocking

I jrves, and represent the transform of the convolution of lattice defects with the

"Srizontal and vertical convergence of the beam. Although for many experiments there may

"1 i a sufficiently large number of equatorial reflections available to extract a statiscal half-width parameter of the analyzed rocking curves, p, which can be regarded as tpresentative of the defect structure of the entire crystallite population, it is wjivertheless desirable to utilize, also, non-equatorial reflections so as to increase the

nfidence level of statistical sampling. What follows now is an analysis of the

417
photometric transformation of the rocking curves of the crystallites, with the aim of;

utilizing all reflections on the Debye-Scherrer arcs and extracting from them statisti,

parameters representative of the defect structure of the entire crystallite population

Figure 4. Detail of multiple-exposure diagram of 304 stainless

steel. Array of spots represents samplings of the rocking curve

of each reflecting grain. Discrete, angular specimen- rotation

3 minutes of arc. CrK radiation, (a) Annealed, (b) Subjected

to stress-corrosion, (c) Enlarged detail of rocking curve of

annealed grain 3. (d) Enlarged detail of rocking curve of grain

3 after SC. (Note increase of rocking curves after SC.)

418
Figure 5 represents the unit sphere with center at the specimen S. The vector r ^represents a ray of the pencil that converges to the specimen. With this ray and the

Bragg angle 6 is associated the Debye-Scherrer cone of which a typical element is the ray

L. With the converging pencil is associated a two-parameter family of conjugate cones

with vertices at S that intersect the unit sphere in a zone B, which will be called the

reflecting zone. A configuration such as this will be associated with each point of the

small, but finite, irradiated area. For the purpose of the present investigation,

however, all such points may be considered to be concentrated at S, since only the

integrated photographic intensities will be required.

I

Figure 5. Reflecting zone and coordinate system.

419
The region A represents an area on the surface of the unit sphere pierced by the|

misaligned normals of a crystallite, one of which is marked n. When the specimen is

rotated about the vertical axis, reflection from this facet will occur when n coincicj;

with one of the typical vectors n^. That is, reflection occurs when the area A inter?, sects the reflecting zone B, and the intensity of the resulting photographic spot wi

depend, among other things, on the amount of this intersection.

Let U(|), V(n) be the normalized x-ray intensity distributions in the horizontal!

vertical directions of the converging beam. Since the maximum angle subtended by the1

' 2 2 beam is small, there will be values of (|,n) say (a,b) such that a ,b ~Q and U(|),

V(n) « 0, when [|| > a,

| n,

| > b. With every element dA of the area A will be associated a reflecting power w(A)dAin|

which w(A) is a two-dimensional point function. Let x be the angular measure along a great circle passing through the centroid of the distribution and making the angle y th the circle of longitude through this centroid. If y is the geodetic measure in the

direction perpendicular to the x-circle and

W(x) = S w(A)dy,

I

then W(x)dx is the one-dimensional reflecting power of the strip of width dx shown in

figure 6, and W(x) is the true rocking curve of the crystallite in the direction of x

In terms of a rotation of the specimen, x may be written

x = ((» cos t|i sin y

as shown in figure 7. It should be noted that, if <|> is the general coordinate shown in figure 5, a rotation of the specimen is given by the difference " $j °^ two ^ ocatlons of tne

normal n. The origin of $ is immaterial. For general relations on the surface of th

sphere, <|> is taken to be the angle shown in figure 5, but when referring to rotations

the specimen, is measured as shown in figures 7 and 8. In what follows, the origin

<J> is always specified.

In particular, let the origin of § be the point of intersection of the circle of

latitude and the conjugate cone determined by the ray (1,0,0), and let be the

coordinate of the intersection of this circle and the cone determined by the ray (1,4,

as shown in figure 8. Using eq. (2), we have

W(x)dx = cos i|j sin y W[(<|> - ^cos i|i sin y]d<|>.

420
The intensity of the x-rays associated with a point of the reflecting zone is the

is ;5um of the intensities associated with the conjugate cones that intersect at this point

ijknd is, therefore, Jc U(4)V(n)dc where c is the arc of the circle with center at the point

tjiand geodetic radius equal to n/(2 - 6). Since the extremes of (4,n) at which the

irl|rradiation is measurable are extremely small, the maximum angles of intersection of

[Relevant conjugate cones are very small except for 6 nearly equal to 90°. Hence, locally

the irradiation may be considered constant in the direction of the circle determined by

ljthe cone (1,0,0), the gradient of x-ray intensities being perpendicular to this direction

e^except, possibly, in cases of extreme back-reflection).

Figure 6. Distribution of misaligned normals.

421
Figure 7. Great circle measure X. 422
Figure 8. Photometric transformation.

423
Letting y be the angle the circle (1,0,0) makes with the circle of latitude, m

total intensity that reaches the photographic film from the misaligned region at thi

setting is

I(<))) = cos iji sin y

x If u(4)v(n)w@((t> - 4> 1 )cos * sin ^]d|dn [4

where the double integration extends over all values of (|,n)- This is shown scheme

ically in figure 8. We may obtain <J>^ and sin y in terms of |, n,» and 6 in the following manner: it

<t> be the general longitudinal coordinate of n as shown in figure 5. The equation olthi conjugate cone associated with the ray (1,0,0) is

cos t|/ cos <J) + sin e = 0 (5

and that of the cone associated with the ray (l,|,n) is

cos (J) cos((J> + <t>-^) + n cos <J) sin(<f> + <|>^) + 4 sin tp + sin 6 = 0.

2 2 2 From these two equations and the relations | ,n, ,<)) w 0, we find

Also,

4»t = n + i sin <\> sin <)> cos i|j 4 sin u> (cos t|i - sin 9)

tan y tan (}> cos 4* dtp sin i|>

dt|< (from eq. (5))

Therefore

,

(cos 4* - sin 6)'

sin 9 sin tjj

(cos t|< - sin 9) cos 9 cos 4< sin y

424

i
Let 0^,0^,

2 be the variances of U(|), V(n), W(x). If X

2 is the variance of the

hotometric intensities, then

2 2 I = J" <j) I(<)))d<t> = cos (J< sin y

X JJJ <t> 2 U(4)V(n)W[(<)) -

<t. 1 )cos q> sin Y]d|dn.d<|>

2 . 2

2

-

~ a 2.2 + a + 1 ; n 2 - 2 cos i|> sin y cos ij) - sin 6 (9)

his relation between the four variances is independent of the analytical forms of U,V,W.

enerally, the half-width is a more convenient parameter than the variance, but a relation

etween half-widths may be obtained from eq. (9) only when the analytical forms of the

istribution functions are given.

A reasonable hypothesis is

U(|) = 1/K,

I I

I < K/2

= 0, |4| > K/2

V(n) = r- exp[- n

2/2a2

] (10)

(2n)\

n

H

W(x) = exp[-x2/2a2

] . (2nf2a

r these forms of the distributions, it is readily found that

o>

2 = K

2 /12, L = 2.36a , B = 2.36a

ere L is the half-width of the horizontal intensities of the irradiating beam and 6 is

e half-width of the true rocking curve of the crystallite in the x-direction. Substiting these values in eq. (9), we obtain

B = (5.57I

2

- L2- 0.46A<)) 2

2

)

!5 cos 41 sin y (11)

:re

A<t> 2 = K sin i|;/(cos 2

>\> - sin

2 Q)'z

. (12)

425
4. Determination of Specific Lattice Defects from Rocki ng-Curve Parameters

4.1. Determination of excess dislocation density of crystallite

Using eqs. (11) and (12), the half-width of the true rocking curve of the reflec

crystallite is obtained. In order to apply these equations, one has to determine the

rocking-curve half-width of the monochromating crystal, L, and the vertical convergenc

of the beam, K. Typical values for the experimental arrangement employed, using the

(111) reflection of silicon, are: L = 15 seconds of arc, K = 10.8'. These values are then used for all calculations based on eqs. (11) and (12) to obtain the corrected

half-width value of the crystallites, B. Next, the azimuthal elevation of the spot

reflection, t|>, is determined by the relation

. . _ Y/R

sin t|i = 5-r

2 sin 6[1 + (Y/R) ]

2

where Y is the vertical distance of the spot reflection measured from the equator and

is the camera radius. Using the i|i and 8 values for the reflection, y 1S calculated ft eq. (8) and Aty^ from eq. (12). Thus, when the half-width of the uncorrected rocking

curve, 2.361, is experimentally determined, all parameters are now given to compute b} eq. (11) the half-width of the corrected rocking curve, 6. It will be seen that for

equatorial reflections, where 4> and y are 0 and 90°, respectively, eq. (11) reduces tc 2 2 2 B = 5.571 - L , which is the difference between the square of the half-width of the

observed rocking curve of the crystallite and that of the monochromating crystal.

The 6- values obtained do not give information about groupings of adjacent disloq

tions of opposite sign. The specific species of lattice defects measured by 6, howevt

is the excess dislocation density, D, which can be obtained from the relation given bv Kirsch [5]:

? 7 D = 6 /9b

where b is the magnitude of the Burgers vector. It is the excess dislocation density,

which has been shown to be a most significant indicator of microplasticity [6]. If the angular lattice misalignment of the crystallite is large, the total exces: dislocation density can also be assessed from the vertical spread of the spot reflect is

AiJj. Indeed, for gross lattice defects, the measurements of the A4> spread, in conjunct

with the horizontal A<}> measurements , can be very effective in characterizing the over; I excess dislocation density of the crystallite.

i.

426
4.2. Determination of substructure characteristics

If the crystallites have a substructure, then the rocking curves are no longer

inimodal but multimodal; i.e., more than one peak is recorded in an array of spot of a iiulti-exposure diagram [7]. From the intensity data and known specimen settings,

[tatistical parameters can be computed which are a measure of the angular tilt between

LLwo adjacent subgrains and also the angular misalignment within the subgrain. If the

|ieans <j>^ and ^ °f two neighboring individual components of 'the multipeaked rocking curve

ire determined, then the angle, A(j), subtended by the means is the projected tilt of two

jdjacent subgrains. Figure 9 may serve to illustrate this point. From the tilt or

Sisorientation angle, A(|>, the excess dislocation density in the subgrain boundary, D^g,

[an be computed. Since the dislocation arrangement in the subgrain boundaries is not

andom but exhibits a preferred alignment, the following relation given by Hirsch [5] is

ised to calculate D CD :

(15)

i? gure 9. Comparison of the experimental data and theoretical Gaussian intensity distributions for equatorial reflection of Al 96% cold-rolled, annealed 1 hr at 300 °C, (222)

| line, (1, +1) position. Circles: experimental data; full line: theoretical curve [7].

427
where t is the subgrain size, which can be determined from the corresponding image of th

reflection micrograph.

In order to determine the angular lattice misalignment existing within a subgrain,

the half-width values of the individual components of the multipeaked rocking curves, p^ are measured. If detailed experimental data can be obtained, such as those shown in

figure 9, the standard deviation, o, of the individual components of the distribution

2 2-2 curve can be determined, using the relation a = §

-

<t> • For the illustrative array,

the computation of a was particularly simple, since reflections occurred near the equato

and the contribution of the vertical convergence of the beam to the rocking curve of the

grain reflection was negligible.

Assuming the dislocation density within the subgrain to be random (viz. cell

j, structure), the excess dislocation density in the subgrain, D^, is computed again by eq.w (14), substituting p^ for p. U

r 5. Applications and Versatility of the Combination Method

Perhaps the versatility of the combination method can best be demonstrated by citinl: some of the successful applications of the method to problems in materials science. The;

method was applied to such diverse problems as the determination of the deformation

substructure of nickel and its correlation to mechanical properties [8], growth processes

in the recrystal 1 ization of aluminum [9,10], substructure formation in iron during creep

[11], and residual stress and grain deformation in BeO ceramics [12]. I The combination method also proved to be very useful in a recent study of a precipitation-hardening alloy of the Ti-Al-Mo system. High- temperature strength and

ductility increases were obtained for this alloy by applying a step-aging program based ji on principles of particle-dislocation interactions [13]. Thus, precipitation of ordered

TigAl particles in the high- temperature beta titanium phase field induced the moving

dislocations to bypass the particles. Consequently, the slip became dispersed in the

hard p-phase and in this way increased the ductility of the alloy. Small alpha titanium

and Ti^Al particles which were precipitated in the low- temperature phase field during the

step-aging program were cut by the moving dislocations, thereby increasing the yield

strength of the alloy. The materials science problem involving the x-ray analysis was toj elucidate and verify the effect of TigAl particles as efficient agents of slip dispersal

in the high- temperature p-field. To achieve this objective, the x-ray combination method

was employed [14]. Since the p and TigAl reflections were clearly separated on the

Debye-Scherrer film, the rocking curves of the grains of the respective phases could be

identified easily. It was noted that after deformation of 2 percent, the angular range

of reflection for the Ti^Al particles did not exceed 12 minutes of arc, while that for

| the p-grains exceeded more than 1°. Hence, the lattice misalignment in the p-matrix

became at least five times as large as that in the TigAl particles. It could be

concluded, therefore, that room-temperature deformation of the p + Ti^Al phase caused the

p-matrix to undergo preferentially plastic deformation by homogeneous slip, while the

undeformed, ordered TigAl particles functioned as hard particles controlling the slip

distribution.

^28
In a recent application of the diffraction method to determine the prefracture

| damage in fatigued and stress-corroded materials [15], the analysis of the excess dislocation density in surface and bulk led to the prediction of the lifetime of the

S':(fatigued and stress-corroded materials. Such predictions are of great importance to

technology. It should be pointed out that single-crystal studies of stress-corroded

jnetals and alloys have been notoriously unsuccessful in the past. This led many

investigators to believe that the phenomenon of stress corrosion is not applicable to

3f 'single crystals. By applying, however, the combination diffraction method to stresscorroded polycrystal 1 ine materials, one analyzes and characterizes in essence the defect

structure of small single crystals in a polycrystal line aggregate. In the stressijcorrosion (SC) study of 304 stainless steel (figs. 2-4), a very low stress, i.e., 55

percent of the yield stress, was applied and the stressed alloy was subjected to a corrosive medium of MgC^-^O solution at 154 °C. Neither the applied stress nor the

^olution alone had any deleterious effect on the alloy; only the combined effect caused

the SC attack. Because the applied stress was very low, grain (crystallite) rotation did

hot occur and it was possible, therefore, to analyze the identical grains for the induced

'S Refect structure as a function of stress-corrosion time. Comparison of figures 4a and 4b

innll show that as a result of SC attack only specific grain reflections showed an

increase in the rocking-curve half-width, such as those marked by the numbers 1, 2, and

ml, while the rocking curves of other grains remained invariant. For the grains 1, 2, and

I J, p was about 14' in the annealed state and increased to about 20' after the alloy was

subjected to SC for 75 percent of its lifetime. Compare the enlarged rocking-curve

details of annealed grain 3 in figure 4c with those after SC in figure 4d. Thus, in

analyzing the defect structure of the grains in slow, creeping SC attack, the method

Proved to be capable of specific selectivity.

6. Advantages and Limitations of the Diffraction Method

Although the essential features of the combination method of reflection topography

! sand rocking-curve analysis were developed by the author some 25 years ago, and although

I the method was applied successfully to a number of problems of technological importance,

the execution of the method was tedious and time-consuming. The drawback, however, did

Iiot lie in the method per se, but was principally due to the inefficient x-ray tubes

!;:kiich were available at that time. However, with the great efficiency of today's x-ray

Itubes, the availability of powerful rotating anodes and the advent of Synchrotron

radiation, the method has experienced a new renaissance and has assumed new significance.

Experiments that used to take weeks of exposure can now be performed in a few hours'

1 Itime. The great advantages which the method offers can be stated as follows:

(a) It combines the topographic and analytical analyses of the defect structure of

the individual crystallites of a polycrystal 1 ine specimen without surrendering

the ability of statistical sampling and group representation which the study of

a polycrystall ine material usually offers.

429
(b) If the defect structure of crystallites becomes pronounced in certain crysta'

lographic directions, as a result of physical operations, viz. magnetization

plastic deformation or diffusion, there will be pronounced changes in the

rocking-curve characteristics of corresponding (hk£) reflections. Such chan

frequently become self-evident by mere inspection of the spot arrays in a multi-exposure diagram. Such was the case, for example, when the deformatio

characteristics of a silicon powder was investigated and the rocking curves

(111) reflections corresponding to the active slip planes exhibited conspicu

ously larger values [3].

(c) By correlating the quantitative analysis of the defect structure to surface

topography; a bridge is formed which permits subsequent detailed structural

analysis and characterization of the identical crystallites by light microsc

scanning and transmission microscopy.

The limitations of the method are rather obvious. If the crystallite size is sma

i.e., of the order of a few micrometers, the spatial tracing of the reflections from

specimen surface to the circumference of the Debye-Scherrer camera is difficult to car'

out. Consequently, the topographic relationship between the analyzed rocking curves ai the surface sites of the crystallites cannot be established unequivocally. Finally, 1' either as a result of a special growth or severe deformation process, the crystallites

are so small and numerous that even by application of a microfocus tube and use of

nuclear track plates the reflecting crystallites cannot be resolved (size limit about

1.5 urn) , then neither topographic mapping nor rocking-curve analysis of the crystallit*

can be performed.

References

[1] Weissmann, S. , Method for the study of lattice inhomogeneities combining x-ray

microscopy and diffraction analysis, J. Appl. Phys. 27, 389-395 (1956).

[2] Weissmann, S. , Substructure characteristics of fine-grained metals and alloys

disclosed by x-ray reflection microscopy and diffraction analysis, Trans. ASM, 52,; 599-614 (1960).

[3] Reis, A. J., Slade, J. J., Jr., and Weissmann, S. , A new x-ray diffraction method

for studying imperfections of crystal structure in polycrystal 1 ine specimens, J. j Appl. Phys. 22, 665-672 (1951).

[4] Slade, J. J., Jr. and Weissmann, S. , Study of imperfections of crystal structure i! polycrystal line materials, J. Appl. Phys. 23, 323-329 (1952).

[5] Hirsch, P. B. , Mosaic structure, in Progress Met . Phys . 6, 283 (Pergamon Press,

Ltd. , New York, 1956).

[6] Tsunekawa, Y. and Weissmann, S. , Importance of microplasticity in the fracture of

silicon, Met. Trans. 5, 1583-1593 (1974).

[7] Weissmann, S. and Evans, D. L. , An x-ray study of the substructure of fine-grained

aluminum, Acta Cryst. 7, 733-737 (1954).

430
[8] Weissmann, S. , Quantitative study of substructure characteristics and correlation

to tensile property of nickel and nickel alloy, J. Appl. Phys. 27, 1335-1344

(1956).

.[9] Weissmann, S. , Growth process in recrystal 1 ization of aluminum, Trans. ASM, 53,

265-281 (1961).

[10] Weissmann, S. , Imura, T. , and Hosokawa, N. , Recrystal 1 ization and grain growth of

aluminum, in Recovery and Recrystal 1 ization of Metals , L. Himmel , ed. (Interscience,

New York, 1963), pp. 241-267.

j[ll] Garofalo, F. , Zwell , L. , Keh, A. S. , and Weissmann, S. , Substructure formation in

iron during creep at 600°C, Acta Met. 9, 721-729 (1961).

|[12] Smith, D. K. and Weissmann, S. , Residual stress and grain deformation in extruded

polycrystall ine BeO ceramics, J. Am. Ceram. Soc. 51, 330-336 (1968).

,[13] Hida, M. and Weissmann, S. , High-temperature strength and ductility increases of

Ti-Mo-Al alloys by step-aging, Met. Trans. 6A, 1541-1546 (1975).

14] Hamajima, T. and Weissmann, S. , Thermal equilibria and mechanical stability of

Ti

3Al phase in Ti-Mo-Al alloys, Met. Trans. 6A, 1535-1539 (1975).

15] Pangborn, R. N. , Yazici, R. , Tsakalakos, T. , Weissmann, S. , and Kramer, I. R.

,

Determination of prefracture damage in fatigued and stress-corroded materials by

x-ray double crystal diffractometry, this volume .

431

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

DETERMINATION OF PREFRACTURE DAMAGE IN FATIGUED AND STRESS-CORRODED MATERIALS BY X-RAY DOUBLE CRYSTAL DIFFRACTOMETRY

R. N. Pangborn, R. Yazici, T. Tsakalak'os, S. Weissmann

College of Engineering

Rutgers University

Piscataway, NJ 08854

and

I. R. Kramer

David W. Taylor Naval Ship R&D Center

Annapolis, MD 21402

The lattice defects induced in fatigued aluminum alloys and in

austenitic stainless steel and titanium-aluminum alloys subjected

to stress corrosion were investigated by a method based on x-ray

double crystal diffractometry combined with x-ray topography.

X-ray rocking curves of the grain reflections from the surface

layer and bulk material of the specimens were obtained, and a statistical halfwidth value for the rocking curves, p, was determined from the population of grain reflections. This

parameter characterized the excess dislocation density of the

analyzed grains. It was shown that a defect structure was rapidly

produced in the surface layer for both the fatigue cycled and

stress corroded specimens. In addition, for both deformation

processes, gross mechanical instability and fracture occurred when

a critical value, (}*, was attained. Investigation of the induced

excess dislocations in depth by incremental surface removal showed

that there exists a dynamical interplay between the buildup of excess dislocations in the surface and bulk. By measuring the

progressive buildup of excess dislocations in depth as a function

of the number of cycles for fatigue, or the elapsed exposure time

for stress corrosion, the induced damage to the material could be

assessed. Consequently, the remaining lifetime of the materials

could be predicted by the nondestructive x-ray method at various

stages of their lives.

Key Words: Excess dislocation density; failure prediction;

fatigue; stress corrosion; x-ray double crystal diffractometer;

x-ray topography.

433
1. Introduction

The ability to predict the failure of materials subjected to static stressing in

aggressive environments, or to dynamic or repeated loading, is of great technological

importance, both for mechanical and structural applications. Since the physical

properties of solid materials, such as their strength and plastic response to deformation

are structure sensitive, the prefracture damage may also be related to changes in the

defect structure and lattice misalignment. For deformation processes which are known to

be highly surface sensitive, the evaluation of these structural changes with respect to depth from the surface is equally important to the comprehensive characterization of

deformation induced damage. Because x-ray diffraction methods are nondestructive, they

represent an ideal tool with which to elucidate structural changes, and the degradation

in mechanical performance they represent.

1.1. Stress corrosion

Stress corrosion cracking is the phenomenon of metal disintegration under the

combined action of chemical corrosion and mechanical stress. Although a number of

theories have been proposed to explain the stress corrosion (SC) process, no single

theory has received general acceptance. Indeed, some investigators [l] 1 maintain that no general mechanism is applicable to SC, but rather that different mechanisms may be

operative for different metals and environments. Thus, some studies [2,3] evoked a "dissolution" theory, which interpreted SC as a preferential dissolution process at the

crack tip. Other investigators [4,5] attributed SC to a decrease in the surface energy

by absorption. A "brittle film" model was first proposed by Logan [6], and this concept

was pursued further by a number of investigators [7,8]. This latter proposal suggested

that the crack propagation rate could be attributed to the dissolution of the substrate

at cracked sites in the film, and was controlled by a repassi vation process which

prevents dissolution. Several investigators [9,10] also gathered evidence in the

particular case of austenitic stainless steel, that hydrogen is evolved at the tip of the

advancing crack during SC, causing local embrittlement.

Kramer [11,12] has ascribed the formation of a surface debris layer or film to the

surface stresses set up in deformed metals and alloys. His preliminary studies [13,14]

of uniaxially strained metals revealed that the surface layer is work hardened preferentially compared to the bulk material. Other scientists [15-17] have made etch pit

density measurements, TEM analyses, and x-ray diffraction studies of tensile deformed

metals to provide evidence that the dislocation density is higher at the surface than in the bulk. It would be expected that the application of static loads less than that

required to produce macroscopic yield or creep will, under the influence of a corrosive

environment, produce substantial multiplication and accumulation of dislocations in the

1 Figures in brackets indicate the literature references at the end of this paper. 434
surface layer. Indeed, Fourier analysis of the x-ray diffraction line broadening in

austenitic stainless steels revealed an increase in dislocation density due to SC,

despite the small amount of plastic deformation observed to result from the testing [18].

in addition, crack initiation was associated with a critical value of the stored energy,

estimated from the broadening of the diffraction lines, which was approximately the same

for all specimens tested.

1.2. Fatigue cycling

Like stress corrosion cracking, failure under repeated stressing, employing loads in

the elastic range, is both difficult to explain and to predict. Many studies [19-22]

fiave been made in the past to relate the defect structure induced by fatigue cycling to

^he slip morphology exhibited on the surface. The topological features of the surface,

buch as persistent slip bands, intrusions and extrusions, developed during cycling have

also been related to subsequent crack initiation and eventual failure [23-26]. Despite

the considerable effort expended to understand the fundamental mechanisms governing crack

'initiation and propagation, no real consensus on what constitutes fatigue damage prior to

these phenomena has yet emerged. Previous investigations employing x-ray diffraction

analysis techniques have been unsuccessful in forecasting ultimate failure on the basis

)f changes in the patterns. Increased asterism or x-ray line broadening occurred either

very early stages of the cycling, remaining invariant thereafter [27,28], or were

restricted to that period which follows the initiation of catastrophic, macroscopic

failure [29,30]. Consequently, these parameters could not be associated with a marked,

linear accruement of fatigue "damage" or the amount of reduction in the lifetime.

The propensity for surface layer work hardening during uniaxial straining, noted

previously, may be extrapolated to fatigue by viewing a simple tensile test as the first

naif-cycle of push-pull fatigue. Kramer [31-33] has obtained evidence that such a surface layer is, in fact, formed during cycling of single crystals and commercial

illoys. The preferential work hardening, disclosed by the enhancement of the surface

layer flow stress, increased linearly during the life to a critical value at fracture

rtiich was independent of the stress amplitude [34]. It was proposed that the surface

fayer represents a barrier to dislocation migration from the bulk, and that fracture is

III jr' V initiated if the accumulation or pileup of excess dislocations at the surface produces

focal stresses in excess of the fracture strength.

1

' 1.3. Direct examination of the induced defect structure

The first objective of this study was to evaluate the structural changes in stress

:orroded and fatigue cycled metals and alloys as a function of their lifetimes, measured

in terms of the exposure time, or number of cycles, respectively. Particular attention

[/as focused on elucidating the distribution of structural defects from the surface to

j)ulk induced by these deformation processes. X-ray double crystal diffractometry and

435

jL
reflection topography represented sensitive, yet powerful research tools by which to analyze the prefracture damage. Thus, through a better understanding of the fundamental

mechanisms leading to material failure by SC or fatigue, a new approach for predicting

the onset of failure was sought.

2. Experimental Procedure

2.1. Material selection and specimen preparation

Three model materials were chosen for the preliminary investigation of tensile

deformed single crystals: silicon, a low stacking fault energy, brittle material which

is ductile when deformed at elevated temperature, and for which dislocations are virtually immobile after cooling to ambient temperature; aluminum, a high staking fault

| energy material with microstructure typical of ductile fee metals; and gold which L exhibits little propensity for oxide layer formation. Flat test specimens were prepared

|. from the crystals with tensile axis and surface orientations which would insure single orjj multiple glide and symmetric x-ray diffraction conditions, respectively.

Two types of alloys were selected for the stress corrosion investigation: austeni- a tic stainless steel of 304 commercial grade with composition Fe-18Cr-8Ni-2Mn-lSi-0. 8C; jg and alpha-titanium with nominal composition Ti-lOAl. Flat, pin-loading specimens were U cut from sheet stock with tensile axis parallel to the rolling direction.

The fatigue studies were carried out for aluminum single crystals of <100> tensile

| axis orientation, and Al 2024 alloy with nominal composition Al-4Cu-l.4Mg-0.5Mn.

Cycl indrical , thread-mounting specimens were fabricated from rod stock.

All the specimens were heat treated in vacuum or in argon atomo sphere, and water

quenched. Suitable annealing times were determined to stress-relieve the specimens and

to obtain grain sizes appropriate for the x-ray diffraction analysis. Prior to testing, j, the specimens were chemically or electrolytical ly polished to insure damage- and

contamination-free surfaces.

2.2. Mechanical testing

The tensile tests of single crystals were performed on an Instron universal testing

machine under conditions indicated in the results section. The SC tests were carried out: at constant tensile loads employing the Instron equipment and a cantilever- type, dead

load machine. The corrosive medium for austenitic stainless steel was a boiling MgC^-^O

solution, maintained at 154 °C by controlled heating and water flow. The orTi alloy was tested at room temperature in two types of corrosive media: a 0.5 molar, aqueous

solution of NaCl , and a solution consisting of 0.5 percent HC1 , 1 percent H

20 and 98.5

percent CH

30H. A wide range of cell potentials was applied during the testing.

The fatigue cycling was performed in the push-pull mode with zero mean stress, using

a Tatnel 1-Krause apparatus and Instron servohydraul ic equipment (Model 1350). The

constant applied loads for SC, and applied stress amplitudes for fatigue cycling,

represented chosen fractions of the yield stress or proportional limit, respectively,

predetermined from static tensile tests. 436
2.3. X-ray diffraction analysis

The principal research tool for structural characterization was the x-ray double

crystal diffractometer method, in combination with Berg-Barrett topography to afford a visualization of the lattice inhomogeneity [35,36]. The application of this technique

and a detailed discussion of the information it yields, both qualitative and quantitative, are presented in an invited contribution to this symposium [37]. Briefly, however,

[the lattice (Disorientation and configuration of crystal defects can be evaluated from the

jx-ray microscopy. The density of excess dislocations of one sign is calculated from the

breadth of the diffraction peak, or rocking curve halfwidth. Since practical interest in

SC required investigation of commercial alloys, an adaptation of double crystal diffractometry to polycrystal 1 ine materials was employed in this study. Debye patterns of the

'jreflections from favorably oriented grains in the sample are recorded photographically.

|By carrying out incremental specimen rotations and corresponding film shifts, each

grain's rocking curve can be analyzed as if it functions independently as a small single

crystal. The most remarkable aspect of this special x-ray method is that the progressive

change in selected grains can be followed during the deformation process. This is

especially useful in the study of SC, since only particular grains are affected by the

deformation, while others remain virtually unchanged. Thus, the double crystal diffractometer technique facilitated identification and analysis of only those grains which were

critically affected by SC, so that by these direct measurements the computational

averaging procedures, applied in conventional x-ray line broadening analysis to characterize the defect density, were avoided. For the fatigued alloy, on the other hand, the

microplasticity was more uniformly distributed among the grains, and could therefore be

treated statistically. The defect concentration in this case was characterized simply by

•taking the average rocking curve halfwidth for the analyzed grain population.

3. Results and Discussion

3.1. Change in the surface layer defect structure induced by SC or fatique

SC experiments were performed on austenitic stainless steel at three constant stress

levels corresponding to 95, 75 and 55 percent of the static yield stress at 154 °C. From

the dependence of fracture time on the applied stress, shown in figure la, it can be seen

that the fracture time increased rapidly when the applied stress dropped below the proportional

! limit, equal to about 75 percent of the yield stress. Scanning electron microscopy revealed a transcrystalline fracture mode for all the specimens tested. The broken curve in figure 2 [illustrates the time dependence of the strain for the specimen loaded to 55 percent of

the yield stress. A small initial increase was observed due to the elastic loading, and

jvirtually no subsequent deformation, caused either by corrosion or creep, was observed for

J19.5 hours, or about 80 percent of the total lifetime, t^. After this critical

lexposure time, t , however, development of a large central crack and concommitant

437

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t 100

CO Q_ CO O

3*

FAILURE TIME, t f (minutes)

NUMBER OF CYCLES, N

f

Figure 1. Dependence of specimen lifetime on stress condition: (a)

failure time for SC as a function of applied static stress; (b)

cycles to failure as a function of stress amplitude (S-N curve) for

fatigure.

reduction of the cross-sectional area produced obvious mechanical instability. The

rocking curve halfwidths for the reflecting grains, measured during SC, varied in a similar fashion with the exposure time. The solid curves in figure 2 depict the

dependence of the ratio of measured-to-intrinsic (undeformed) halfwidths during the

lifetime. The lower curve, denoted A, refers to the average B/B value for all the

grains analyzed, while the upper curve, labeled B, represents the change in halfwidth for specific grains which showed a preferential or persistent broadening as a result of SC.

| (The selectivity of grain attack by SC is shown by the details of the multi-exposure

diagram in figures 4a and 4b of reference [37]). Both curves ascend to maximum values [l during the first 67 percent of t^, after which they decline conspicuously to lower valueiat

the critical time t . Finally, the macroscopic mechanical instability occurring between..

and t^ is reflected on a microscopic scale by the rapid rise of p/B

Q

from 80 to 100

percent of t,.

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for all grains CURVE B

: Half width dependence

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Strain dependence

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FRACTION OF FAILURE TIME, t/t

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Figure 2. Changes in the strain and x-ray rocking curve halfwidths

during progressive exposure to SC.

Fatigue tests were carried out on AT 2024 specimens to generate an S-N curve typical

3r tension-compression cycling. As shown by figure lb, there exists a distinct

milarity between the dependence of on stress amplitude and the dependence of t on

bplied stress for SC in figure la. Three stress amplitudes corresponding to 50, 75 and

)0 percent of the proportional limit for static tension were chosen for the x-ray

lvestigation. Figure 3 shows the cyclic dependence of the halfwidths, p, corrected for

ie average intrinsic, or undeformed halfwidth, p . pis computed according to the

li - 2 2 h

Ration: p = (p ~P0

)

2

, where p is the average halfwidth measured after deformation,

lie curve, generated by using shallow-penetrating Cu radiation to analyze only the

|irface layer, reveals many similarities to the progressive changes in the rocking curves

'nduced by SC, depicted in figure 2. After an initial period of rapid halfwidth

ipansion comprising the first 20 percent of the life (Stage I), a markedly decreased

jope was obtained for the long cycling duration extending to 95 percent of the total num-

|r of cycles to failure, N f

. As in the case of SC, the macroscopic mechanical

^stability during the final Stage III caused another rapid enhancement of the rocking

>j<rve breadths. Remarkably, the progressive changes in the halfwidths, as well as the

fitical halfwidth at failure, p*, were nearly identical for cycling at all three stress

llvels.

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25 50 75

FRACTION OF FATIGUE LIFE, N/Nf (%)

100

Figure 3. Change in the rocking curve halfwidths at the specimen suface

during progressive fatigue cycling at selected stress amplitudes.

3.2. Depth distribution of the lattice defects

A preliminary study involved tensile deformed Si, Al and Au monocrystal s. The

distribution of excess dislocations in depth was determined by rocking curve measurement

[37] after discrete surface layer removals, effected by chemical or electrolytic

polishing. As shown in figure 4, a decreasing gradient in the excess dislocation density

was obtained from surface to bulk for all three specimens. Significant declines from a high surface density to a constant density at about 150 urn into the interior were

measured, even for the gold crystal deformed to only 3 percent plastic strain. Thus

oxide formation is presumably not a necessary prerequisite for preferential accumulation

of excess dislocations in the surface layer.

Using a similar procedure, the depth dependence of the rocking curve breadths after

SC and fatigue was analyzed. Figure 5 shows profiles of p7PQ as a function of depth

distance from the surface, x, for austenitic stainless steel, subjected to SC at 55

percent of the yield stress. After an exposure time corresponding to only one-tenth of t

a notable decrease from surface to bulk for the halfwidths of grains selectively

affected by the SC was obtained. These high surface halfwidths, analyzed with Cu radiation which penetrates only 1 urn into depth, indicate that the plastic deformation

occurs principally in the surface layer. Like the profiles for tensile deformed single

440
ystals, x-ray diffraction analysis of SC specimens after surface layer removals

Sealed that a constant minimum value for the halfwidths is established in depth. By

ntrast, when p/PQ was measured in depth after SC to the critical exposure time, t , the

lk p/pQ

values were equivalent to those of the surface layer An identical behavior

s' obtained for the orTi for SC at 60 percent of the yield stress and an impressed cell

tential of -200 mV (rest potential = -230 mV).

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o

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ICL.

CO

oo_JCO

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DEPTH FROM SURFACE, x(p)

300

Figure 4. Experimental gradients

with depth from the surface for

[Aluminum: <100> tensile axis,

tensile axis, (112) reflection;

reflection.

]

of the excess dislocation density

tensile deformed monocrystals.

(100) reflection; Silicon: <110>

Gold: <123> tensile axis, (113)

Al 2024 specimens subjected to fatigue cycling also exhibited surface layer

tjfwidth enhancements analogous to those for the tensile and SC tests. Figure 6 shows

| profiles obtained for alloy specimens fatigued to various stages of the fatigue life.

>Ajer only 0.15 percent of the life a simple decreasing gradient typical of that for

mlotonic deformation was obtained. After cycling to 25 percent of N^, however, the

Pjfile conformation exhibits an additional salient feature. After declining to a mjimum value at 100 urn in depth, the halfwidths increase again further into depth to a Pjteau level at about 250 urn from the surface. The figure also shows that the halfwidth

Mansion at the surface occurs early in the life while the halfwidths for the bulk

i prease more gradually and never exceed the surface value.

441
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1 ft \ 4/4 - A 1 I A J T / I- - U. 1

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0 25 50 100 125 DISTANCE FROM SURFACE, x(/tm)

Figure 5. Depth profiles, showing the halfwidth dependence on distance from the surface of austenitic stainless steel, for

various stages of SC.

DISTANCE FROM SURFACE , x ( )

Figure 6. Depth profiles for Al 2024 specimens after various amounts

of push-pull fatigue. ...
For both the stress corroded and fatigue cycled specimens, x-ray reflection

jtopography [37] disclosed significant differences between the defect structures of the

surface layer and bulk material. The spot reflections from the surface grains displayed

substantial breakup in the azimuthal direction, along the tangent to the Debye arcs. In

i (the radial direction, incremental specimen rotations in the parallel incident radiation

jbroduced a broad range of reflection, featuring sequential intensity maxima and minima.

Both of these observations are indicative of the introduction of a deformation substructure [37], increases in the defect density, and concomitant work hardening of the

surface layer. The reflections from grains located in the bulk, on the other hand, were

well- resolved and unimodal. The reduced plastic deformation in the core region of the

specimens was manifested by the short angular duration of the reflection range of the

jrains.

3.3. Influence of the surface layer on prefracture mechanical behavior

The surface layer of stress corroded or fatigue cycled specimens clearly plays a similar role in the two respective deformation processes. In both cases, the rocking

:urve halfwidths, representative of the induced excess dislocation densities, increased

preferential ly in the surface layer early in the lifetime. A long period of rocking

:urve invariance was exhibited during intermediate life fractions, before a final

Expansion stage occurred just prior to failure. The rapid saturation of the surface

ayer halfwidths and decreasing gradients in depth suggest that a barrier effect is

:reated at the surface, which restricts the egression of excess dislocations generated in

;he bulk. As a result, a more gradual but steady accumulation of defects occurs in the

nterior throughout the lifetime.

In SC, the electrochemical attack was initiated at the surface by preferential

leformation of the surface layer between t = 0 and t = 0.1 t

c

. The surface corrosion was

ustained during the period from 0.1 t to t by progresssive deformation of the surface

ayer with exposure time. The corrosion process may be conceived in terms of a galvanic

ell, with the plastically deformed, film-free metal grains functioning as an anode, and

he undistorted grains serving as the cathode. The diffusion of the chemical species is

resumably assisted by the dislocation activity in the surface layer. At the same time,

Ihis effective work hardening of the surface layer confers to it the property of a (arrier to dislocation egression, and induces a dynamical interplay between surface and

julk. Thus, the effects of corrosive embrittlement of the surface layer and dislocation

Accumulation due to deformation under the applied load are superimposed, so that

licrocracks advance by a combination of electrochemical and mechanical action. The

esistance to failure is therefore substantially reduced, compared to that of a material

tressed in a non-corrosive medium. The approach toward a critical value, (p/0

Q )*, at

jhich macroscopic mechanical instability at t

c

, and then brittle failure at t^, occur is 'ccelerated in the aggressive environment.

I

443
An analogous barrier aspect of the surface layer is illustrated by the results for

j fatigue cycling. Preferential work hardening of the surface layer occurs during the

first 20 to 25 percent of the fatigue life and impedes the outward migration of internally

generated defects. The mutual interplay between the defect structure in the surface

layer and the plastic deformation introduced to the bulk sustains the strengthening of

this barrier. According to the model proposed by Kramer [33], microcracks are initiated

at the surface when the local stress fields associated with dislocation pileups exceed

the fracture strength. The rate of inward propagation of the cracks is controlled by the I dislocation accumulation in depth, produced by continued cycling.

To test the surface barrier model for fatigue, a further experimental sequence was devised and carried out, as depicted by the composite diagram in figure 7. Part A of thei

figure presents typical depth profiles for Al 2024 specimens cycled to 75 and 95 percent

of their fatigue lives. These curves were obtained by incrementally removing the surface

layers by electropol ishing and performing x-ray rocking curve analyses at each depth.

After a total thickness of 400 urn was removed from the original surfaces of the speci- ! mens, the fatiguing was continued at the same stress amplitude. The curves in Part B, obtained by analysis after short cycling increments, demonstrate a conspicuous decline in! the B/B

Q

values on the initial recycling. The dislocation structure and arrangement in

the bulk is obviously very unstable during cycling when the blocking effect inherent to the deformed surface layer is absent. After reaching a minimum at 1 to 2 percent of the

life, the halfwidths increased again since a new, work hardened surface layer was being

formed. When the recycling process was interrupted after 5 percent to obtain a second

depth profile, as exhibited in Part C, the B/B values throughout the cross section of

o the specimen given 75 percent prior cycling were significantly less than that induced by j the original fatigue, as shown by reference to the plateau level in Part A. Indeed the

halfwidths were comparable to the value for an undeformed, virgin specimen. Furthermore,

uninterrupted recycling of the specimen previously fatigued to 95 percent of its life and

polished to 400 urn in depth produced a 75 percent net increase in the normal fatigue

life. These experiments not only provide a dramatic demonstration of the blocking effect

of the work hardened surface layer, but also explain the remarkable extensions of fatigue

life afforded by judicious, periodic surface layer removal [32,38]. Though the improvement in the lifetime has been ascribed in the past to the elimination of microcracks,

persistent slip bands, and other surface relief, it may actually derive primarily from

the instability of the bulk defect structure in the absence of the work hardened surface

layer.

3.4. Nondestructive prediction of failure

The results of the x-ray diffraction studies imply that the critical accumulation of excess dislocations in the surface layer, experimentally determined by (p/PQ

)*, governs

the fracture of the material. For SC of both austenitic stainless steel and alphatitanium, mechanical instability set in when the critical halfwidth value was approached,

regardless of the particular conditions employed in the tests; routes to failure

444
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I TT

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i 0 100 200 300 400

DISTANCE FROM SURFACE , x

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1.8

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FRACTION of LIFE DISTANCE FROM SURFACE , x N/Nf(%) ( pm )

Figure 7. Composite diagram for AT 2024 specimens given prior fatigue

cycling to 75 and 95 percent of their fatigue lives at ±200 MPa,

followed by surface removal and recycling treatment (A & B), and

either continued recycling or a second depth profile analysis (C).

solving application of high stress and short SC exposure time or low stress and long SC

sure time produced a consistent value for (B/P

Q

)*. Similarly, fatigue failure

curred at a critical corrected halfwidth value, B*, which was independent of the stress

jplitude. The recognition of the importance of the critical value of the halfwidth to

jese modes of mechanical deformation opens up an exciting vista with regard to predicpn of a material's lifetime.

I For SC, it may be seen from figure 2 that the dependence of the measured halfwidths

<l exposure time is much steeper when only selected grains are considered (Curve B), than

| an average for all the grains is taken (Curve A). The enhanced potential for

(timating the failure time from any intermediate point along Curve B derives from the

(jique capability of double crystal diffractometry to distinguish those grains exhibiting

oronounced susceptibility to SC. An alternative method for predicting the remaining

jfe of a stress corroded material resides with the disparity in the rates of change of

1j5 surface and bulk halfwidth values. As shown in figure 5, about half of the total

(ange in the surface B value occurred during the first 10 percent of the critical

•posure time. The plastic deformation in the bulk during this period was very minimal,

fl/ertheless , at the critical time, t

c

, the surface layer and bulk material halfwidths

445
exhibit near parity. Thus, the rate of halfwidth expansion of bulk grains must be

greater than that for grains at the surface during intermediate stage of the life. The

broken curves shown in the figure indicate speculative conformations of the profiles for I times t * 0.4 t and 0.7 t which would provide a nearly linear dependence of the bulk

c c halfwidths on exposure time.

Reference to figure 6 shows that such a linear dependence, in fact, occurs for

fatigue. While the surface halfwidths again exhibit an early-life saturation behavior

(as shown by figure 3, and by the nearly equivalent values for p/p at x = 0 for 25 and

95 percent of the life in figure 6), the bulk plateau values show nearly equal increases

for each 25 percent increment in the expended life. The in-depth study demonstrated why

in the past the x-ray patterns, which analyzed only the surface grains of cycled

specimens, were inconclusive in predicting fatigue failure. Because the work hardening

at the surface occurs so early in the life, it cannot be used to predict the onset of

failure, nor can it even be interpreted as a true indication of fatigue damage. This

latter conclusion, first suggested by Barrett [39], was also demonstrated by corres- ponding studies of fatigued Al monocrystal s. The profile in figure 8, obtained for a

r 5 crystal fatigued in the high cycle range (±1.03 MPa, 2x10 cycles), shows that despite

the surface work hardening, at sufficiently low stress amplitudes, no change in the bulk

halfwidths occurred. It may be seen that the interior plateau level remained at essentially the same value as for the undeformed crystal, P

Q

, and the subsurface region

extending from 50 to 150 urn actually displayed p values lower than the intrinsic

halfwidth of the virgin crystal.

OX

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crz>o

o

z

o

o

uwo

¥>

O 50 IOO 150 200 250 300

DEPTH FROM SURFACE , x ( urn )

Figure 8. Profile of the measured halfwidth dependence on depth from

the surface for a fatigued Al single crystal. [<100> axis, (100)

reflection.

]

446
The Al 2024 depth profiles revealed how a more accurate determination of the failure

e could be achieved. Figure 9 shows the dependence of the rocking curve halfwidth on fraction of life for analysis with copper, molybdenum, and chromium radiation. The

iation produced by these three target materials provides a spectrum of penetration

ths in the Al alloy of about two orders of magnitude. Chromium radiation which

atrates a maximum of 12 urn from the surface produces very little change in p from 5 to

jercent of the life. Copper radiation which penetrates up to 37 pm, or about one

in diameter, exhibits little more of an incline during this period (the absolute

jes of p differ from those in figure 3 due to the smaller average grain size of the

by stock used in the experiment). By contrast, molybdenum radiation which penetrates

| depth of up to 350 pm ascends steeply from about 5 minutes of arc at about 5 percent

the life, to 35 minutes of arc at 95 percent of the life, coinciding with the critical

width value measured using copper radiation. The steep, nearly single-stage curve

lted from the added contribution of the defect structure in the bulk to the x-ray

ysis. A new criterion for fatigue life and failure prediction has therefore emerged:

jre-establ ishing the critical halfwidth value, fj*, and comparing it to the 0 value

! lured in depth at any stage in the fatigue process, the remaining fraction of the life

i|be estimated. By determining the degree of prefracture fatigue damage in this way,

rfatigue life, and probably the life for other deformation processes as well, may be

; ssed easily, accurately, and nondestructi vely.

owos

•

e

J,

IQQ.

O

Ul

OO

CuK« radiation MoKaradiation O Cf Ka radiation

25 50 75

FRACTION OF FATIGUE LIFE, N/Nf (%)

100

Figure 9. Comparative dependences of the rocking curve halfwidths on

the fraction of fatigue life for analyses employing incident

radiation from chromium, copper and molybdenum x-ray targets.

447
4. Conclusions

1. Austenitic stainless steel and alpha-titanium were subjected to stress corrosion, and Al 2024 was cycled in the push-pull fatigue mode. Selected grains at the

surface of SC specimens, and the grains located in the surface layer of fatigued

specimens exhibited a higher density of excess dislocations than the bulk material, as determined by x-ray diffraction analysis. It was proposed, therefore, that the surface

layer functions as a barrier to the egression of dislocations generated in the bulk, anc that prefracture damage is accrued as a result of a dynamical interplay between the work

hardened surface layer and the induced defect structure of the bulk material.

2. For both SC and fatigue, the x-ray rocking curve halfwidths for the surface

layer increased rapidly early in the life. By contrast, the bulk material revealed a steep, linear enhancement of the measured halfwidths throughout the life, to a critical

value at the onset of failure. By application of deeply-penetrating radiation to sample

x-ray rocking curves in depth at various stages of the life, the measured halfwidths

could be compared to this critical value. Thus, for fatigue, the remaining fraction of

life prior to failure could be accurately and nondestructi vely estimated by this

experimental technique. A similar approach for prediction of failure due to SC is currently under investigation.

The authors wish to express their gratitude to the Office of Naval Research

(Metallurgy Program) for the support of the stress corrosion studies, and to the David V- Taylor Naval Ship R&D Center for their sponsorship of the fatigue investigation.

References

[1] Pugh, E. N. , On the mechanism(s) of stress-corrosion cracking, in Envi ronment-SenS 'i Mechanical Behavior , A. R. C. Westwood and N. S. Stoloff, eds. (Gordon Breach, New! 1965), pp. 351-401.

[2] Hines, J. G. and Hoar, T. P., The stress corrosion cracking of austenitic stainles;

steels. II. Fully softened, strain hardened and refrigerated materials, J. Iron

Steel Inst. (London) 184, 166-172 (1956); ibid. 182, 124-143 (1955); ibid. 177, 24! (1954).

[3] Sanderson, G. and Scully, J. C. , Stress corrosion of titanium alloys in methanolic

solutions, Corrosion Sci. 8 [7], 541-548 (1968).

[4] Coleman, E. G. , Weinstein, D. , and Rostocker, W. , On the surface energy mechanism jl for stress-corrosion cracking, Acta Met. 9 [5], 491-496 (1961).

[5] Powell, D. T. and Scully, J. C. , Fractographic observations of the stress corrosion

cracking of titanium alloys in methanolic environments, Corrosion, 25 [12], 483-49;'

(1969).

448
Logan, H. L. , Fi lm- rupture mechanism of stress-corrosion, J. Res. NBS, 48 [2],

99-105 (1952).

] Forty, A. J. and Humble, P., The influence of surface tarnish on the stresscorrosion of crbrass, Phil. Mag. 8 [86], 247-264 (1963).

] McEvely, A. J. , Jr. and Bond, A. P. , Initiation and growth of stress-corrosion

cracks in tarnished brass, J. Electrochem. Soc. 112 [2], 131-138 (1965).

Mehta, M. L. and Burke, J., Role of hydrogen in stress corrosion cracking of

austenitic stainless steels, Corrosion, 31 [3], 108-110 (1975).

Sudarshan, T. S. , Louthan, M. R. , Jr., and McNitt, R. P., Hydrogen induced

suppression of yield point in A- 106 steel, Scripta Met. 12 [9], 799-803 (1978).

Kramer, I. R. and Podlasek, S. , Stress-strain behavior of aluminum crystals at low

pressures, Acta Met. 11 [1], 70-71 (1963).

Kramer, I. R. , Correlation of the surface layer stress coefficient and stress

ij corrosion cracking, Corrosion, 31 [11], 383-391 (1975), ibid. 391-394.

Kramer, I. R. and Demer, L. J., Effect of surface removal on the plastic behavior

of aluminum single crystals, TSM-AIME 221 [4], 780-786 (1961).

Kramer, I. R. , Influence of the surface layer on the plastic flow deformation of

aluminum single crystals, TMS-AIME 233 [7] 1462-1467 (1965).

Kitajima, S. , Tanaka, H. , and Kaieda, H. , Dislocation distribution near the surface

of weakly deformed copper crystals, Trans. Jap. Inst. Met. 10 [1], 12-16 (1969).

Tabata, T. and Fujita, H. , An electron microscope study of dislocation arrangement

in the surface layer of aluminum single crystals, J. Phys. Soc. Jap. 32 [6],

1536-1544 (1972).

Sumino, K. and Yamamoto, M. , Preferential plastic deformation in the surface region

of Al and orFe single crystals, J. Phys. Soc. Jap. 16 [1], 131 (1961).

Kamachi , K. , Otsu, T. , and Obayashi , S. , A study of mechanical behavior of

austenitic stainless steels in the process of corrosion testing, J. Jap. Inst. Met.

35 [1], 64-70 (1971).

Segall, R. L. , Partridge, P. G. , and Hirsch, P. B. , The dislocation distribution of

face-centered cubic metals after fatigue, Phil. Mag. 6 [72], 1493-1513 (1961).

McGrath, J. T. and Waldron, G. W. J., An electron microscope study of dislocation

Ljarrangements in fatigued Al-1% Mg crystals, Phil Mag. 9 [98], 249-259 (1964).

IlLaufer, E. E. and Roberts, W. N. , Dislocations and persistent slip bands in

fatigued copper, Phil. Mag. 14 [127], 65-78 (1966).

BlLevine, E. and Weissmann, S. , The structure and development of fatigue striations

lin Al, Trans. Quar. ASM 61 [1], 128-138 (1968).

B'Thompson, N. and Wadsworth, N. J., Metal fatigue, Adv. in Physics (Phil. Mag.

i||Suppl.) 7 [25], 72-170 (1958).

W|Forsyth, P. J. E. and Stubbington, C. A., The mechanism of fatigue failure in some

jlkiinary and ternary aluminum alloys, J. Inst. Met. 85, 339-343 (1956-7).

*ftCottrel 1 , A. H. and Hull, D. , Extrusion and intrusion by cyclic slip in copper,

llproc. Roy. Soc. (London) A242 [1229], 211-213 (1957).

ii ii
[26] Alden, T. H. and Backofen, W. A. , The formation of fatigue cracks in aluminum

single crystals, Acta Met. 9 [4], 352-366 (1961).

[27] Bennett, J. A. , A study of fatigue in metals by means of x-ray strain measurement,

J. Res. NBS 46 [6], 457-461 (1951).

[28] Taira, S. and Hayashi , K. , X-ray investigation on fatigue fracture of notched steel

specimen, Bull. JSME 9 [36], 627-637 (1966).

[29] Gough, H. J. and Wood, W. A. , A new attack upon the problem of fatigue of metals,

using x-ray methods of precision, Proc. Roy. Soc. (London) A154 [883], 510-539

(1936).

[30] Spencer, R. G. and Marshall, J. W. , An x-ray study of changes that occur in

aluminum during the process of fatiguing, J. Appl. Phys. 12 [3], 191-196 (1941).

[31] Kramer, I. R. , Work hardening and creep of aluminum and copper during alternate

stressing, Trans. ASM 62 [1], 521-536 (1969).

[32] Kramer, I. R. and Kumar, A., Relaxation and cyclic hardening of the surface layer

of copper, Met. Trans. 3 [5], 1223-1227 (1972).

[33] Kramer, I. R. , Prediction of fatigue damage, 2nd Int. Conf. on Mech. Beh. Mat.,

Boston, Mass., 812-816 (Aug. 1976).

[34] Kramer, I. R. , A mechanism of fatigue failure, Met Trans. 5 [8], 1735-1742 (1974).

[35] Intrater, J. and Weissmann, S. , An x-ray diffraction method for the study of substructure of crystals, Acta Cryst. 7 [11], 729-732 (1954).

[36] Weissmann, S. and Evans, D. L , An x-ray study of the substructure of fine-grained,

aluminum, Acta Cryst. 7, [11], 733-737 (1954).

[37] Weissmann, S. , Analysis and topography of lattice defects in powder diffraction

patterns, this volume.

[38] Thompson, N. , Wadsworth, N. and Louat, N. , The origin of fatigue fracture in coppel

Phil. Mag. 1 [2], 113-126 (1956).

[39] Barrett, C. S. , The structure of cold-worked metal, Chap. XVII, Structure of

Metals; Crystal logr. Methods, Principles and Data , 1st ed. (McGraw Hill, New York,

1943), pp. 377-380.

450
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

HIGH RESOLUTION POWDER DIFFRACTION AT ARGONNE'S

ZING-P' PROTOTYPE PULSED NEUTRON SOURCE

J. D. Jorgensen, F. J. Rotella and M. H. Mueller

Argonne National Laboratory

Argonne, IL 60439

Argonne National Laboratory has actively pursued the development of pulsed spallation

jtron sources as a viable alternative to reactors since 1974 when the first prototype spal-

? pi on source, called ZING-P, was operated [l] 1. The basic principles of spallation neutron

i|jrces for slow neutron scattering have recently been reviewed by Carpenter [2]. A full

l^le pulsed neutron source, called IPNS-I, has now been funded for construction at Argonne and

'expected to be operational in 1981. The present prototype source, called ZING-P', began

I:>ration in November 1977 and will continue until IPNS-I comes on line.

The ZING-P 1 facility consists of an H source, 50 MeV linear accelerator, electron

sp'pper, and a 500 MeV proton synchrotron. The 500 MeV protons are projected in short

12

Bses of up to 10 protons per pulse at a rate of 10-30 Hz onto a heavy metal target where

r iitrons are produced by spallation. Both tungsten and uranium have been used as target

Iierials. The present uranium target yields 20-25 neutrons per proton. The high energy

r itrons are then slowed to thermal energies in hydrogenous moderators of nominal dimension

]'i ;10x5 cm. A high resolution time-of-f1 ight powder diffractometer has been constructed on an 18.3

it er flight path viewing one of these moderators. Large time-focused [3] detector banks

i' positioned 1 meter from the sample at 26 = ± 160° and 1.5 meters from the sample at 26 =

J 3 2 t0°. The time-averaged flux at the sample position is presently about 3x10 n/cm -sec

| a beam size of 1x5 cm. In spite of this relatively low flux, useful count rates are

a ieved because of the large detector areas. The 160° banks cover 0.08 sterradians and

a|ieve Ad/d = 0.003 (FWHM), while the 90° banks cover 0.03 sterradians and achieve Ad/d =

o'o5. The resolution of a pulsed source diffractometer remains nominally constant over a He range of d because of the moderator characteristics.

Profile analysis techniques have been used to analyze several sets of data from this

i trument. The resolution function used to fit the asymmetric, non-Gaussian peaks is a tee-part stepwise-continuous function made up of a rising Gaussian, decaying Gaussian and

daying exponential. The wavelength dependence of the four peak shape parameters has been

a urately determined by fitting individual peaks in a set of Fe powder data. A standard

flures in brackets indicate the literature references at the end of this paper.

451

J ,

1
A^Og sample has been studied in order to make comparisons with conventional two-axis

j fractometers. The profile analysis of this data in a hexagonal lattice setting includi

o o 235 reflections where 0.5 A < d < 2.2 A. The effectiveness of a pulsed-source diffrac

for collection of low-d data, stemming from the enhanced epithermal neutron flux, is d matically illustrated by this data. For a total counting time of 160 hours the 113 rej

i tion has an integrated intensity of 55,000 counts and a signal to noise ratio of 90.

References

[1] Carpenter, J. M. , Mueller, M. H. , Beyerlein, R. A., Worlton, T. G. , Jorgensen, J. Burn, T. 0., Skold, K. , Pelizzari, C. A., Peterson, S. W. , Watanabe, N. , Kimura,

and Gunning, J. E. , Proc. Neutron Diffraction Conf. , Petten, August 5-6, 1975, Re Centrum Nederland, RCN-234, 192 (1975)

[2] Carpenter, J. M. , Nucl. Instrum. Methods 145, 91 (1977).

[3] Carpenter, J. M. , Nucl. Instrum. Methods 47, 179 (1967).

452
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

STRESS ANALYSIS FROM POWDER DIFFRACTION PATTERNS

J. B. Cohen and H. Db'lle 1 Department of Materials Science and Engineering

The Technological Institute

Northwestern University

Evanston, IL 60201

and

M. R. James

Science Center

Rockwell International

Thousand Oaks, CA 91360

A brief review of the method of measuring residual stresses in

polycrystalline materials with x-rays is given. The effect of

counting statistics on precision is discussed as well as factors

that affect accuracy. Beam penetration, stress gradients, and the

form of the stress tensor can each seriously affect accuracy, if

traditional methods are employed, and practical procedures to deal

with such situations are outlined.

1. Introduction

Knowledge of stresses (applied or residual) is quite important in an industrial

ironment. Crack initiation and propagation in static or fatigue loading, or in stress

rosion, can be impeded by compressive stresses normal to the crack, and greatly

elerated by tensile stresses. Residual stresses can be produced during quenching

cause of the different cooling rates of the surface and the interior and any volume

mges due to transformations), during manufacturing (due to the difference in deforma-

|n of the surface and the interior—as in straightening, machining, rolling or shot

ning), or due to the different response of particles and matrix in a multiphase

erial. During welding, the resistance of the cold base metal to the solidifying (and

i

I B. Cohen is Frank C. Engelhart, Professor of Materials Science and Engineering, and

DSlle is a Post-Doctoral Visiting Scholar.

453
shrinking weld pool will produce stresses in both regions. These stress patterns can a do change in service, and there is increased interest in monitoring these. For a recert overview, see reference [l] 2

. Apart from deliberately induced residual stresses, the measured stresses can devel

due to applied loads, or even in a quasi-uniform stress field, from differences in resp

of the surface and the interior, one grain with respect to another, or one phase with

respect to another. In the case of residual stresses, the tensile and compressive comp nents must balance across a section, but to see this it may be necessary to examine

values in different phases. This is possible with x-rays, but it has not often been of

sufficient interest; only the stresses in the major constituent are usually examined.

Also, even in a single phase material there may be situations where it is not possible

to remove material to examine gradients because of the geometry of the piece.

Actually, a variety of non-destructive techniques, based on the acoustic and magne

response of a material to stresses, is being evaluated for this purpose, but so far the

have proved to be too sensitive to microstructural variations such as changes in textur

[2]. The x-ray method, which was first proposed in the 1920 's by Lester and Aborn [3],

has withstood the test of time in industry; it is the subject of this review, in which

we shall examine not only the method, but its limitations, accuracy and precision, as well as the latest developments which have made it of considerable use in field situati

While there are a number of ways of examining strains with x-rays, we shall concentrate

only on those that are widely employed in industry. To prove their validity, these

approaches have been compared to measurements of distortion after mechanical dissection

[1].

The basic principle that is employed is quite simple: An interplanar spacing (d)

in the material serves as an internal strain gage. As shown in figure la, in a polycrystalline specimen in the normal position on a diff ractometer, only those grains

with planes nearly parallel to the surface form a diffraction peak. If there are com- j pressive stresses in the near-surface regions, the "d" spacing is increased for these

planes due to the effect of Poisson's ratio. When the specimen is tilted with respect

to the x-ray beam (or the beam to the specimen), figure lb, the diffracting grains have,

their planes more nearly perpendicular to the stresses in the surface and their increas in spacing is less, or even decreased. (Components of the surface stress parallel and per

ndicular to the planes now affect this spacing.) The 26 positions of diffracting peaks ion) such planes and Bragg' s law are all that are required to examine the strain tensor, froi which the surface stresses may be determined.

2 Figures in brackets refer to the literature references listed at the end of this paper,

454

I
Figure 1. Top: Schematic of a di f f ractometer. The incident beam

diffracts from planes that satisfy Bragg' s law, in grains with

these planes parallel to the sample's surface. If regions near the

surface are in compression, with the stress parallel to the

surface, because of Poisson's ratio, these planes are further apart

than in the stress-free state. The "d" spacing is obtained from

the peak in intensity vs. scattering angle 28 (insert), and Bragg'

law. Bottom: after the specimen is tilted, diffraction occurs

from other grains, but from the same planes, and these are less

separated than in (a), because there is a component of the residual

stress normal to the planes.

2. The Basic Equation

In what follows, we shall employ the axial system, L. , for the measurements (a

iratory" system) and axes P^ for the specimen, as shown in figure 2. Primed

•ities will refer to measurements in the laboratory system, unprimed in the sample

on. With d^, the interplanar spacing at tilts <t>, tj>, and d

Q

, the value for the strain

material, the strain along L~ can be written [4]:

455
d ~ d

£ 33 ~ ~ = e ll

COs2<,)S ^ + £ x2sin2*sin^ + £ 1 3

cos<t)s "i n 2»|< + e22sin^<l>sin^

+ £22 s i n*sl

24" + £ 33cos ^ (1

Figure 2. The axial system. The L. are the laboratory system and

measurements of Ad/d are made along L^. The P. describe the specimen.

Once the strains are determined (by the procedures to be described below), the

stresses, a.., can be calculated from:

1

S 1 (hk£) a

ij " ^S

2(hk£) [£

ij " 6ij 35 S 2 (hkl)+3S

1

(hkil) + £ 22

+ £ 33 )] 1

The term 6.. is the Kronecker delta function, and since 6. . = 0, i^ j, the shear strain

alone determine the shear stresses. The S. are x-ray elastic constants for a particular hk£ reflection. For an elasti

cally isotropic solid,

h S 2 (hk£) = , S

1 (hk£) = (- g),

where p is Poisson's ratio and E is Young's modulus. Theory exists to calculate the

constants (for each hk£ reflection) for an anisotropic crystal coupled to an isotropic

matrix, and this theory shows that the values are within a few percent of the average c

456
e well-known Reuss and Voight formulae, which assume, respectively, constant average

ress and constant average strain for all grains in the material. In addition, theory

developing for an elastically anisotropic material with severe texture. (This entire

ea has recently been reviewed by one of us, reference [5].) In the latter case, the

a'stic constants associated with a particular hk£ reflection may vary with $ and i|i tilt.

has also been well established that the x-ray elastic constants can vary appreciably

th amount of plastic deformation; changes as large as 20-40 pet. have been reported,

lis phenomenon is not understood at the moment and can obviously severely affect the

jcuracy of any stress measurement if calculated values are assumed to be independent of

or strain. Fortunately, it is possible to test for this problem, and, in many cases, measure the effective constants. We discuss this further below. In addition, it is

;en the relative change in stress with some change in condition that is important, in

ch case the constants employed are not as important.

We turn now to the various methods, after which we will discuss other errors that

ect accuracy, and also precision.

3. Surface Stresses Only

As a free surface cannot support a stress normal to it (a^ = = = 0), the

less tensor in the sample system shown in figure 2 is:

ano

12

0

a

12

CT220

^ 3a ^

purse, this tensor may be written in terms of principal axes in a rotated axial

suem:

o

1

0 0

0 a

2 0 J C3b)

0 0 0

, it can be shown that:

d. - d

e 33 =

d = % s 2

( hk£ )C a

iC0S * + a

2 sin^]sin »]> + S

1 (hk£)[a1 + a^. (4)

that at t(i = 0°, a comparison of d^ to d

Q

immediately indicates whether the sum of

irincipal stresses is positive or negative.

457
Subtracting the value for ip = 0°:

d. -d

( d

o

d

4>,t|)=90° " do ^ % " d

»,t|)=0

d

o

' ~ d*,^0

2

(5)'!

The stress in any direction, <t>, on the surface can be determined simply from the f slope of d vs sin The stress-free value, d

Q

, is not needed. A typical plot is shownj

in figure 3. The principal stresses and their directions can be obtained [6], although I their * directions are often known from the nature of the applied stress system. By

application of known elastic stresses on an identical specimen, the effective elastic

constants can be obtained from eqs. (4) or (5). |

1.1703;

1.1702

1. 1701 -

.1700

1.1699-

Figure 3. Interplanar spacing, "d", vs. sin normalized AISI-1010

steel, deformed in tension along to a true strain of 8 pet. 211

peak, CrK

a radiation; o^_qO=-107. 1 MPa (-15,303 psi). From

reference [22].

It cannot be emphasized too strongly that the linearity of "d" vs sin^t|; must be co firmed in each situation, but i_f this is the case, the equation can be further simplified, with the aid of Bragg' s law, to involve only two tilts, ij) = 0° and, typically, t|i 45° or 60°:

0^ = KA20 (in degrees),

with:

K =

180

S 2 (hk£) sin t|> 27

cot ^Vu +

(6,

(61

458
In this "two exposure method," a high angle peak is chosen, and the iji tilt is made

is large as possible, to minimize K and hence maximize the sensitivity of the peak shift

|.o the stress. For example, for steel, CrK^ radiation, the 211 reflection at «156° 26,

Ind a i|rtilt of 45°, a + 0.1° 26 shift corresponds to a stress of «-60 MPa (-8600 psi).

Shifts as small as 0.01 - 0.02° 26 can be detected. If it has been verified that d vs

19' 2

Jin i)j is indeed linear, it is possible to develop simple portable equipment (for use in

.he field or in a factory environment) which can measure the surface stress in seconds.

,n example of this equipment is shown in figure 4 [7]. It is even possible to use only

me incident beam direction and examine, simultaneously with two detectors, two points on

'.he diffraction cone which come from grains tilted at different ^ values [8]. However,

fit has been shown [1] that in this case the stress constant, K, is much larger, which

mplies a larger error in stress from a given uncertainty in 26. (Film equipment

mploying this "single exposure" technique is still sometimes used for very large

n'eces.

)

Figure 4. The PARS system [7]--a portable (x-ray) analyzer for residual stresses.

(A) linear position sensitive detector to record the entire diffraction profile

without detector motion; (B) air-cooled miniature x-ray tube (50 kV, 2 mA);

(C) exit soller slits; (D) rods to position unit with respect to a specimen;

(E) one (of two) handles.

459
Because of the simple form of eq. (6a), it is particularly easy to measure K by

applying a series of known elastic loads to a specimen. Tensile or bending loads are commonly employed [9,10].

4. The General Stress State

Because the measured stresses are averages over a finite depth of beam penetratior we should consider the general stress tensor:

G

ll a12

CT 13v

CT12 °22 a

23

) (7

a

13

CT23CT 33^ •

2 Referring to eq. 1, the presence of e^^ and ^3 ~^ ea^ s to curvature in the d vs sin i|) relationship, as their contribution depends on sin2t|j. Such terms can be important in

machining, or in wear; that is, when there are strong components of the deforming force

tangential to their surface, and strong gradients (since the strains £-^» £23, £33 mus vanish at the surface). Examples of this effect (from reference [11]) are shown in

2

* figures 5 and 6. The principal effect of the shear stresses is to cause "d" vs sin <|> be non-linear in opposite senses for positive and negative \\t tilts. This i|i splitting a large curvature near i)j = 0 are distinguishing features of this phenomenon. The actual stress tensors are given in the captions to the figures, and the small value of the she stress to cause the "4< splitting" is particularly noteworthy. Also, the curves for

positive and negative i|j (or by changing <}> by 180°) reverse when the sign of the shear

stress reverses (fig. 6). This curvature and its cause were first reported by Walburge1

[12], and a simple method of analysis of the data was suggested by Dolle and Hauk [12]. It was employed in an extensive study for the first time by Db'lle and Cohen [11],

introducing the terms a^ and a2» employing eq. (1) and using carats to indicate averagej over the depth of penetration:

+ e^J = <£

33

> + [(<£ n

>cos

2* + <£

12

>sin20. + <£

22

>sin2<i>

- <£

33>]sin (8

2 ~ ^%\\>+ " £

<tHjr^ = [<£ 13

>cos* + <£

23

>sin4>] sin

| a 2i|>| . 0

2 Thus, < £33> can be obtained from the intercept of a^ vs sin t|<; this value is

independent of <t>, and so it can be verified by measurements at different <i> tilts. The

2 components £.^, e-^* £22 can be obtained from 3a^/8 sin tji. At <t> = 0 , - £33, and

hence, £-^, is obtained, whereas for * = 90°, £22 can be measured. The £^ component i then determined from

3a,

[ 1 -

1 3sin2

4) *=45°

460
Figure 5. Normalized plain carbon steel strip (0.6 wt pet carbon) ground

under flowing water to remove 5 urn per pass [11]. Various * tilts shown.

The curves are fits to the data points with eq. (1). The stress tensor

in MPa is: 14 63

306 -1

-1 92

211 peak, CrK

a radiation.

8a,

8s in 2t|>

e 23for * = 90c is calculated for * = 0°, The curves in figures 5 and 6 are the fit

the data points with eq. (1).

The stresses are then obtained from eq. (2).

In this procedure, it is necessary to know "d . " For steels, measurements generally

0 o ;ld a value for the lattice parameter of 2.8665(1) A. The uncertainty leads to a

inge of ss ± 30 MPa in the normal stresses and hardly any variation in the shear

jresses. Measurements with a well annealed powder of Cr indicate that the precision is

5 a for a^, a^. and better than 8 MPa for o^g. This uncertainty is

ten less than the variation of the actual stresses from point to point in a specimen.

Another way of examining the stress in three dimensions is with high-energy x-rays

neutrons, isolating the volume element to be examined by suitably narrowing the slits

fining the incident and diffracted beams. The feasibility of such a procedure is

nsidered in the next paper.

461
2 Figure 6. <e 33

> vs - S1

' n *\>- Normalized plain carbon steel strip (0.6 wt pet

carbon) ground to remove 5 urn per pass under flowing water [11]. <t> = 0°. -3

Strain in units of 10 . For specimen C5, the direction of grinding was reversed each pass with the final pass opposite to that for C2 (in the -P^

direction). The stress tensor for C5 is given in figure 5. For C2 (in

MPa) it is: 199 -10 -63

-10 86 5 -63 5 84

Note that the curvature for <J> > 0, t|i < 0 reverses when reverses. The curves are the fits to the data points with the stress tensors and eq. (1).

0: t|i < 0; I ip > 0. 211 peak, CrK

a radiation.

Except for materials with appreciable content of a second phase (such as a steel

with more than « 0.4 wt pet C), some kinds of deformation may produce strong texture at

2 modest strains. In such cases, oscillations have been observed in d vs sin ijj as shown n

figure 7. The precise interpretation for this effect is still uncertain; it may be due

to elastic or plastic anisotropy [1,5]. Tests are currently underway to decide which

factor is most important, after which procedures may be possible to minimize this

problem. For example, any contributions from elastic anisotropy can be minimized (for

any material) by employing hoo or hhh reflections [5].

462
t r

~2 Z A t

SIN

2

<i>

Figure 7. Interplanar spacing, "d", vs. sin for a Cu^Au specimen

pulled in tension along to a true strain of 35 pet [22], 420

peak, CuK

a radiation. The term f(0,i|0 is the area under the Bragg

peak at each ijj position.

5. Gradients

The ability to detect stresses normal to the surface, as was discussed in the

previous section, is based on the presence of gradients, and the fact that the x-ray beam

averages over its penetration (D). Therefore, it is appropriate to examine the effects

of such gradients because, even in the absence of stresses normal to the surface, there

will be effects on <o^> , <0

22

> ' <G

12

> ' Corrections are available for this [1,5], and

also for stress relief due to layer removal [1], if the gradients are examined in this

fashion. But the presence of steep gradients can also be detected and examined from the

I 21 form of "d" vs sin i|i without layer removal, as illustrated in figure 8. For this figure,

the stresses were obtained from an assumed stress profile (in the z-direction) [5]:

D

<cr,.> = o..(z=0) + J" exp(-z/t)g,,(z)dz . (9)

Here, with p the linear absorption coefficient:

.2 .2

_ sin 9 - sin j> , nn » T " 2psin9 cost|j

uua;

463
for t|< tilts around the 6 axis, and:

sine costl) , T 2u (10b

for tilts around an axis parallel to the plane of the diff ractometer. The term 9jj(z)

describes the variation of stress with depth (z) below the surface. (Once these average

stresses are evaluated for a particular strain distribution, < £

3 3

> can be calculated at each iJj and d vs. sin^ obtained as will be shown below.)

Note particularly the difference in curvature near t|< = 0 between this effect (if

shear stress <^3

> , <a

23

> = 0) ar| d for t|< splitting (compare figure 8 with figures 5 or

6). Also, the values of d at ±<f< should be identical in this case, so the effect is

readily distinguished from 41 splitting.

Type I

o

o

o

Type 2 Type 3 Type 4

<G„> < 0

<%> <o

<o^> <0

<gfl > >o

«rfl > >0

<9n> <0

<an> >0

<%> >o

<q«> >0

<Q33> >0

>o

<o

«T33> <0 «T33> <o

:

sin

2 T

2 Figure 8. The effect of various gradients on "d" vs sin »|». The gradient

is assumed to be linear over the x-ray beam's depth of penetration, with

glope g.j. Typical values were assumed (for a-^, 200-600 MPa over 20 um, for a

33 , 100 MPa per um (zero at the surface)).

This curvature can be employed to obtain information about gradients, and we now present a concrete example of how this can be done, with data from a SAE 1040 steel,

normalized and shot peened. We make the following assumptions (all of which could be

checked in a more thorough investigation):

a)

b)

the x-ray elastic constants and S£ do not vary with depth, z: a., varies with z only:

c) * = 0° (by definition)

464
d) a

]j( z ) = CT22^ Z ^ ( c' ue to snot P eeninQ normal to the surface);

e) a^.jCz), i f j = 0 (no appreciable shear deformation).

Then:

2 9 <e

33

>

4>=0 = J5S 2^ nkA ^sl

' n »l<<CTii

> + C 1 " sin i(j) <o-33>] + SL (hkJi)[2<a

;L1> + <o" 33

>]. (H)

From this equation:

<e22

>*,tb=0 = 2S

1Chk£)<an

> + <o^> \h S 2 (hk£) + S^hkA)] , (12a)

3<eJn

>

(b=n a

. 2 = «2 (hk£)[<au

> - <a

33

>]

9si n ij) (12b)

With eqs. (12a, b), first estimate of <o"-q> and <0" 33

> can be obtained, fitting a straight line to the data. Typical data is shown in figure 9. From this figure, these

first estimates are: <<3y\? ~ <a

33

> = Comparison of figure 9 with

figure 8 assuming the gradients are linear indicates that either g.^ < 0 or g 33

> 0, or

both.

.908

sin

2 V

Figure 9. Measured "d" vs. sin shot peened (normalized) AISI 1040

steel; data averaged over nine specimens. Error bars from observed

variation in 26 in repeated measurements. C° Ka

> 310 reflection.

The measured macroscopic gradient is illustrated in figure 10a. It can be seen that

the stresses vanish at « 500 urn. A simple representation of this figure was taken to be

figure 10b. The terms z

9 and g. . were varied to give the best fit to the data: <o,-.>

,

<033

> ' and tne measured slopes of "d" versus sin i|j at various angles.

465
Figure 10. (a) The gradient measured by layer removal for one of the

specimens in figure 9. (Etchant: 100 parts 30 pet H

20, 10 parts

48 pet HF). Data uncorrected for layer removal and depth of

penetration. 310 peak, CoK

a radiation, (b) Assumed gradients for

calculations described in text to fit data in (a).

From the average <a0 ,> = 140 MPa and the penetration depth, t = 10 urn, g33 was the

order of 15 MPa/um. From the curvature of "d" vs. sin <|<, it could be concluded that gn \ -10 to -15 MPa/pm. These values were actually obtained simply by choosing various z2 values and surface stresses. Of course, more sophisticated least-squares approaches are

possible. This procedure might be particularly fruitful when many similar samples have

to be examined for their gradients, and the gradient in one specimen has been measured by

layer removal.

f!

466
Finally, it is worth noting that peak shapes, as well as positions have been

employed by Murakami [14] to obtain information on gradients in thin foils. While his

assumptions as to the stress tensor are too restrictive for general use, his procedure

could be expanded.

6. Factors Affecting Accuracy

We have already mentioned one possible source of inaccuracy, the elastic constants.

The actual error in stress from the effects to be discussed below can be evaluated

through the stress constant K and eqs. (6a, b), or by employing experimental x-ray elastic

constants and A particular error in 26, when multiplied by K for the chosen

material, gives a first estimate of the error in stress.

All data should be corrected for the variation in scattering factor and Debye-Waller

factor if the peak is very broad (> «5° in half breadth). Corrections for the Lorentz

polarization and absorption factors (LPA) can be written for each position, 26, across a peak. Assuming filtered radiation:

2 LPA =

1 + c ° s 26

(1 - tanijj cote), (13a)

sin 6

for tilts around around the 26 axis, and:

LPA =

1 + CO/26

, (13b)

sin 6

for tilts around an axis parallel to the goniometer. The latter method [15,16] is increasingly employed in Europe. It is not limited to|i|>|< 6, as is a tilt around the 6 axis. Furthermore, the path lengths of the incident and diffracted beams within the

specimen are equal, so absorption corrections are not involved (eq. (13b) vs. eq. (13a).

It is a helpful technique for measurements in difficult locations such as on gear teeth,

where the incident or diffracted beam can be blocked by the gear itself.

Because these corrections are small and straightforward, there is no need to avoid

them, and therefore no need to estimate errors from these sources. Locating the specimen over the center of rotation can be particularly troublesome

especially when the geometry of the specimen is complicated, and this is one of the

largest sources of inaccuracy. There are two ways of achieving this location. The

lattice parameter for two or more different reflections (widely separated in 26) can be

o plotted vs. cos 8/sin6. Then (for i|i=0°), with "a" the lattice parameter [17]:

2 Aa _ _ 26 cos 6 . a R sine ' u

467
with 6 the sample displacement and R the goniometer radius. The displacement can therefore be calculated from the slope of such a plot, and the sample displaced appropriately.

Another way, particularly important when *|< splitting is suspected, is to examine th, peak position of a stress free powder (painted in a thin layer on the specimen in an acetone solution) at a fixed 20 (close to the reflection angle of the specimen), and thej

varying An example of such a method is shown in figure 11.

S =-l mm

Figure 11. The effect of sample displacement on "d" vs. sin

Measurements made with an annealed Cr powder painted on a steel

specimen in an acetone solution. 211 reflection, CrK^ radiation.

In Japan, a parallel beam method is employed [18], to eliminate the sensitivity to sample position. In figure 12, a comparison of the peak shift due to sample displacement

is made between this method, the parafocusing method (for which, the receiving slits are moved to the correct focus at each i|0 , and a stationary slit, divergent beam. The

parallel beam technique is obviously least sensitive. However, the peak is broadest in

this case and the time for data analysis is « 1/3 longer for the same statistical error (Section 7) than parafocusing, and the reproducibility twice as poor for sharp profiles

[19]. The stationary slit method appears to be the best compromise. However, in this case (but not in parafocusing), the shape of the peak may vary with i|< tilt. How this

alters various methods for peak location is illustrated in figure 13. Bias corrections

may be taken from this plot.
KSI

50

Figure 12. Measured effects of sample displacement, for various

focussing techniques employed in stress measurements. Normalized

1045 steel, 211 reflection, CrK radiation.

3

half-width of doublet

in degrees

Figure 13. Calculated peak shift due to K c<2 doublet for various

definitions of the peak, (a) Special slits to symmetrize peak (Wolfstieg, U.

,

Die Symmetrisierung unsymmetrischer Interferenzl inien mit Hilfevon, Spezial

Henden, Harterei-Techn. Mit. 31, 23-26 (1978)). (b) Center of gravity,

(c) Middle of half-width, (d) Parabolic fit to top 15 pet. The same Gaussian

functions were assumed for the a-^, components. 211 reflection, CrK

a radiation

469
(If parafocussing is employed, and the sample has a strong texture, it is important to introduce a permanently fixed slit, defining the beam along the 8 axis; it is not moved when the scanning slit is moved to the correct focus. This assures that the same portion of the diffraction cone is seen at all t|> tilts [20].)

An excellent presentation of the bias from various instrumental misalignments can ji found in reference [2]; while the equations in this reference are based on the center

gravity of a peak, they can be used as a guide for any definition of the peak. Another source of analytical approximations to the errors is reference [22]. For this paper, wj

have made computer calculations of the peak shifts by a ray-tracing method. The

intensity of the beam was assumed to be uniform across its cross section. This over- emphasizes the error and the resultant shifts were multiplied by 1/3, in agreement with)

the findings in similar comparisons by Zantopoulos and Jatczak [13]. The errors summarized in table 1 are the shift in 26 for a i|> tilt from +45° to -45°. If only i employed, the errors are reduced by an additional factor of 1/2. This table can serve

a quick guide to the important aspects in alignment of the diffractometer for stress determinations. Table 2 gives- some typical errors for the two- tilt method and parafocussing, with tilt around the 6 axis. Clearly, the bias due to geometric errors can held to quite small values, with good alignment and a judicious choice of slits.

Table 1. The influence of various mis-alignments on the accuracy of stress analys

+ = large error (0.07° or more)

- = small error (0.01° or less)

medium error

type of error

i|> tilt

around

0 axis

i|i tilt

around

axis in

diffraction

plane

centers of 8, t|) rotation

not coincident (A = 0.1 mm)

4° horizontal divergence

4° vertical divergence

1° horizontal tilt of

beam to the zero direction

1° vertical tilt of beam

to true zero direction

tube mis-alignment

(Ax, Ay, Az 1 mm)

displacement of sample

from the tilt axis

(0.2 mm)

sample curvature (beam

much smaller)

+ -

+

to m

m to +

to m

470
Table 2. Typical bias in peak location for 26 = 156°

(approximately the 211 CrK^ peak from iron).

cause peak shift between

4» = 0° and i|> = 45°

peak shift between

4< = 0° and i|) = 60

(1

°26 MPa °26 MPa

jeak location ± 0.01°26

11

+ .02 11.9 ±.02 ±1700

[otal horizontal beam

liveregence of 1° -.0006° -0.35 -.0025 -215

ertical beam divergence

'assuming strong texture

ting divergent Soller slit;

jo receiving Soller slit) ±.002 ±1.20 ±.002 ±170

ample displacement,

V = ±.025 mm ±.0034 ±2.1 ±.0088 ±760

,-axis displacement,

K

1 = ±.025 mm ±.002 ±1.2 ±.0068 ±585

kximum total errors

3 '*3 1 in -20 direction

) in +28 direction

-.008

+.0068

-4.8

+4.1

-.0201

+.0149

-1150

+860

tote: Maximum error is either on of these but not the total range.

Calculated for steel from aA

= K. (20-26,) where K.r = 602 MPa/°26 and Kcn = 408.1 MPa

4> i|> i|r ' 45 60

ii/°26. Does not include error in peak location as it is dependent on time of data

hi lection.

7. Factors Affecting Precision

!

Automation of stress measurements has been achieved in several locations. One such

Kigram, for example, includes sample alignment over the tilt axis, and preliminary scans

| locate the peak and to determine the time to achieve an operator-specified precision

i ii.9,24]. Such on-line controls allow studies to be made of factors affecting reproducijity.

R As shown in table 2, the error in determining a peak's location is one of the

largest sources of uncertainty in the results. A number of methods have been presented

I define the angle of diffraction, 26, from the diffraction profile, including the apex

f a parabola or cubic polynomial fit to the upper portion of the profile [25], the

Imtroid [26] and the half value breadth [18]. Comparisons between these methods

Jdicate that the apex of a least squares parabola fit to that portion of the profile

Love 85 percent of the maximum intensity usually provides the most reproducible
procedure for defining the diffraction profile when determining peak shifts [19,24].

This is shown in table 3.

Table 3. Precision of various measures of profile position

(10 measurements)

Sample Time FWHM

(sec) (°26)

Half-Value Breadth

(°20)

Centroid

(°28)

Parabo,

(°26)

AISI 1090-1 50 .45 156.149 (± .021)'

AISI 1045-2 100 3.45 155.336 (± .064)

156.096 (± .011) 156.186 (± 01

155.396 (± .085) 155.413 (± 02

FWHM is the full width at half of the maximum intensity.

bThe term in () represents one standard deviation from the average position over the

10 measurements.

If the intensity 1^. accumulated at each jth interval in 26 is taken at 2n+l observ

tion points (the center point being defined as the working origin 20

Q

) in equal incre

ments, 6, of 28, the peak location in a parabolic fit is given by [24].

where:

20

p = 26

o

- (n

2

2 -n

o

n4 )6

2 M

1/2n2 (n26

2M

o -n

oM2

) ;

n . , n n. = I J

1 and M. = 6

1 I J I. . 1 j=-n

1 j=-n J

(1

(Angular intensity corrections such as the Lorentz polarization and absorption factors

must be performed on the raw data before determining the peak position.)

Scatter in the peak position arises from statistical fluctuations in the x-ray

intensities and the precision of the measured value is properly evaluated by its standaf deviation due to counting statistics. The standard deviation in peak location (eq. (15

due to random counting errors is [19,24]:

S(26

p

)

4 2 2 6 (n 9 -n nA ) n

, 0 0 9

<

iJ ?

I (n

o8\f + (n

26\-noM

2 )6^j

4n

2 (n 0 62 M -n M„)" J=-n 11 o o I

-n

2d\)2

S

2 (I

j

)

2 where the variance in the power, S (I.) depends on the method of data accumulation and : given by:

472
s ft<v y* for fixed time, t,

2 2

S pr (I.) = I./c for fixed counts, c.

J J

(17a)

(17b)

The error, S(28p), introduces an error in the calculated interplanar spacing given by:

S(d) = [ Acose -. S(29

p

)

, n , ,_2„ J ,„ 480 J 2sin 6 V2

(18)

where X represents the characteristic wavelength and S(2r) is in degrees 26. The subsequent statistical counting error in measured residual stress is:

I (sin24)-sin

2

i)j) 2-S

2

(dij>) f

7 7 7 7 [I (sinVsin \\>) ]

[d -S /

(hki°] L o 2

7 2

J ' (19)

for the sin 4> technique (eq. (5)) and:

S^) = K[S

2 (26

o

) + S

Z (2Q^)t

2

, (20)

for the two-tilt method (eq. (6)).

Figure 14a shows the distribution of errors in a large number of actual measurements

when 0.01° 26 was chosen as the desired statistical precision for each peak in a control

program employing the above equations. The half-width of the distribution is only

-3°

2.4-10 26. But when the distribution of errors in repeated measurements on the same specimens was examined (without moving it), the results in figure 14b were obtained.

Clearly, the scatter is greater than the statistical error. In a series of experiments

[19,24], the factors required to achieve reproducibility equivalent to the statistical

lerror were examined. The results can be summarized as follows:

S (a) Unless the peak is very sharp, at least 7 points should be employed in a parabolic fit, if the two-tilt method is employed- not the three points usually taken.

!j(The total measuring time can be kept fixed.)

(b) If the peak-to-background ratio is low (2:1) or the peak is broad (2° or more),

background subtraction is required, at least in the first scan to determine the region of

[fit. (This was not done for the data in figure 14, and this is the main reason for the

(poorer than expected precision.)

Even with (a) and (b) the scatter may be twice that predicted by the above statisical analysis for the two-tilt procedure. But:

473
c) With the total time kept fixed, the sin i|j method, with at least a three point

parabolic fit and background subtraction will achieve the expected error.

d) An alternative to (c) is to employ a position sensitive detector (PSD) and 30 or more points in the region of fit. In this case, background subtraction is not necessary.

E

3

I

2 40

standard deviation in IO~ degrees

in 2e

0 12 3 4

standard deviation in I0~

2 degrees in 2e

Figure 14. (a) Precision of peak location (parabolic fit, top 15

pet). 572 peaks; actual standard deviations when the indicated

precision was requested, (b) 440 peaks each examined 4 times;

indicating the spread in each of these 4 repeated measurements.

Armco iron and steels, 211 reflection, CrK

a radiation. 7 point

parabolic fit for peak positions.

474
Automation alone will reduce the time of measurement by a 2. A PSD will result in

me saving of a factor of 5 or more [24,27]. Indeed, if an error of 34 MPa (5 ksi) is

tceptable, it has been shown that in some cases, measurements can be completed with the

[o tilt method with 2 seconds for each tilt [7] with the instrument in figure 4. Finally, it is instructive to consider the total error possible in a residual stress

:asurement due to all the factors discussed. Samples circulated to many investigators

in give perhaps the best view of the combined accuracy and precision to be expected,

he following data is taken from reference [28]. The two tilt method was employed, with

teel specimens.

I 1. SAE Round Robin No. 3 (11 laboratories reported, 1959)

a. flat specimen (broad peak)

stress level: 14 MPa

standard deviation among laboratories: ±10.3 MPa

b. 1010 annealed flat steel (sharp peak)

stress level: -5 MPa

standard deviation: ±16.5 MPa

2. SAE Round Robin No. 4 (25 laboratories, including many with little or no experience

in residual stress measurements)

a. flat, shot-peened (R

c 63)

stress level: 593 MPa

standard deviation: ±41 MPa

b. 1045 Round bar, 6.22 cm diameter (R

c 61/62)

stress level: 910 MPa

standard deviation (axial direction): ±56.5 MPa

standard deviation (longitudinal direction): ±72 MPa

We are grateful to 0NR for their continued support of our efforts in this area,

ny of the experiments were carried out in Northwestern University's Long-Term X-ray

cility supported (in part) by the NSF-MRL program through grant No. DMR 76-80847. We

lank M. McClinton for the data on his shot-peened steels employed by us to examine

adients. One of us (H.D.) also thanks the Council for the International Exchange of

[holars for a travel grant from the Fulbright-Hays Program.

References

1] James, M. R. and Cohen, J. B. , The measurement of residual stresses by x-ray

diffraction techniques, in Experimental Methods i_n Material s Science , Vol. 1, H. Herman, ed. (Academic Press, in press).

2] James, M. R. and Buck, 0., Quantitative nondestructive measurements of residual

stresses, in Critical Reviews in Sol id State and Material s Science (in press).

475
[3] Lester, H. H. and Aborn, R. H. , Behavior under stress of the iron crystals in

steel, Army Ordnance, 6, 120-127, 200-207, 283-287, 364-369 (1925).

[4] Evenschor, P. D. and Hauk, V., Uber nichtlineare Netalbenenabstandsverteilungen bei

rontgenographischen Dehnungsmessungen, Z. Metallkde. 66 [3], 167-168 (1975).

[5] Dolle, H. , Influence of multiaxial stress states, stress gradients and elastic

anisotropy on the evaluation of (residual) stresses by x-rays, J. Appl. Cryst. , in

press.

[6] Nye, J. F. , Physical Properties of Crystals (Clarendon Press, Oxford, 1975).

[7] James, M. R. and Cohen, J. B. , PARS - a portable x-ray analyzer for residual

stresses, J. of Testing and Evaluation, 6 [2], 91-97 (1978).

[8] Steffen, D. A. and Ruud, C. 0. , A versatile position sensitive xray detector, in

Advances in X-ray Analysis , Vol. 21, H. F. McMurdie, C. S. Barrett, J. B. Newkirk,

and C. 0. Ruud, eds. (Plenum Press, New York, 1978), pp. 309-315.

[9] Marion, R. H. and Cohen, J. B. , The need for experimentally determined x-ray

elastic constants, in Advances j_n X-ray Analysis , Vol. 20, H. F. McMurdie, C. S. Barrett, J. B. Newkirk, and C. 0. Ruud, eds. (Plenum Press, New York, 1977), pp. 355-367.

[10] Prevey, P. S. , A method of determining the elastic properties of alloys in selected

crystal lographic directions for x-ray diffraction residual stress measurements, in

Advances in X-ray Analysis , Vol. 20, H. F. McMurdie, C. S. , Barrett, J. B. Newkirk,

and C. 0. Ruud, eds. (Plenum Press, New York, 1977), pp. 345-354.

[11] Dolle, H. and Cohen, J. B. , Residual stresses in ground steels, submitted for

publ i cation.

[12] Walburger, H. , Conference on Residual Stress Analysis, Plattig, Germany, May 3-4

(1973).

[13] Dolle, H. and Hauk, V., Rontgenographische Spannungsermittlung fur Eigenspannungsysteme allegemeiner Orientierung, Harterei-Techn. Mitt. 31, [3], 165-169 (1976).

[14] Murakami, M. , Residual strains of Pb thin films deposited onto Si substrates, Acta

Met. 26, 175-183 (1978).

[15] Wolfsteig, U. , Rontgenographische Spannungsmessungen mit breiten Linien, Arch.

Eisenhlittenwes, 30, 447-450 (1959).

[16] Macherauch, E. and Wolfstieg, U. , A modified dif fractometer for x-ray stress measurements, in Advances j_n X-ray Analysis , Vol. 20, H. F. McMurdie, C. S. Barrett, J. B. Newkirk, and C. 0. Ruud, eds. (Plenum Press, New York 1977), pp. 369-378.

[17] Schwartz, L. H. and Cohen, J. B. , Diffraction from Materials (Academic Press, New

York, 1977).

[18] Chrenko, R. H. , X-ray residual stress measurements using parallel beam optics, in

Advances in X-ray Analysis , Vol. 20, H. F. McMurdie, C. S. Barrett, J. B. Newkirk,

and C. 0. Ruud, eds. (Plenum Press, New York, 1977), pp. 393-402.

[19] James, M. R. , An Examination of Experimental Techniques in X-ray Residual Stress

Analysis, Ph.D. Thesis (Northwestern University, Evanston, IL, 1977).

476
0] Marion, R. H. and Cohen, J. B. , Anomalies in measurement of residual stress by

x-ray diffraction, in Advances j_n X-ray Analysis , Vol. 18, W. L. Pickles, C. S. Barrett, J. B. Newkirk, and C. 0. Ruud, eds. (Plenum Publ. Co., New York, 1975),

pp. 446-501.

1] Lecroisey, F. , Miege, B. and Saint-Etienne, A., La Mesure de Constraints Residuelles

Method de Determination par Rayons X (Centre Techniques de Industries Mechaniques,

B.P. 67, 60304 Senlis, France, 1978).

2] Marion, R. H. , X-ray Stress Analysis of Plastically Deformed Metals, Ph.D. Thesis

(Northwestern University, Evanston, IL, 1972).

3] Zantopulos, H. and Jatczak, C. F. , Systematic errors in x-ray diffractometer stress

measurements due to specimen geometry and beam divergence; in Advances in X-ray

Analysis , Vol. 14, C. S. Barrett, J. B. Newkirk, and C. 0. Ruud, eds. (Plenum

Press, New York, 1971), pp. 360-376.

I] James M. R. and Cohen, J. B. , Study of the precision of x-ray stress analysis, in

Advances j_n X-ray Analysis , Vol. 20, H. F. McMurdie, C. S. Barrett, J. B. Newkirk,

and C. 0. Ruud, eds. (Plenum Press, New York, 1977), pp. 291-309.

|5] Koistinen, D. P. and Marburger, R. E. , A simplified procedure for calculating peak

position in x-ray residual stress measurements on hardened steel, Trans, of the

ASM, 51, 537-555 (1959).

5] Singh, A. K. and Balasingh, C. , Effect of x-ray diffractometer geometrical factors

| on the centroid shift of a diffraction line for stress measurement, J. Appl. Phys.

42 [13], 5254-5260 (1971).

H James, M. R. and Cohen, J. B. , The application of a position sensitive x-ray

detector to the measurement of residual stresses, in Advances in X-ray Analysis , Vo. 19, R. W. Gould, C. S. Barrett, J. B. Newkirk, and C. 0. Ruud, eds. (Kendall/

Hunt Publ. Co., Dubuque, Iowa, 1976), pp. 695-708.

i] Residual Stress Measurement by X-ray Diffraction - SAEJ784a, M. E. Hi 1 ley, J. A. Larson, C. F. Jatczak, and R. E. Ricklefs, eds. (Society of Automotive Engineers,

Inc., Warrendale, PA., 1971).

477

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

X-RAY RESIDUAL STRESS EVALUATION BY AN ENERGY DISPERSIVE SYSTEM

Masao Kuriyama, William J. Boettinger and Harold E. Burdette

National Bureau of Standards

Washington, DC 20234

In place of traditional Bragg dif fractometers , energy dispersive

systems are now employed more increasingly as a tool for crystallography. These systems are not only suitable for rapid data collection

and processing, but also are simple and compact, since no elaborate

x-ray optical alignments, nor delicate moving parts are required.

In addition, the energy of x-ray photons can be extended to considerably higher values, hence, permitting better penetration into materials.

Currently, improved quality control of industrial components

demands quantitative information concerning the stress distribution

near cracks and residual stress distributions after different types

of cold working and heat treating or under various conditions of

load. These demands naturally lead to the necessity of measuring

residual strains i_n the interior of materials. With a little extra

precaution on the divergence of an incoming x-ray beam and the

opening of the detector window, energy dispersive systems will

respond to these demands. Furthermore, the curve fitting of each

diffraction peak will provide sufficient precision for the determination of strains in materials. The use of these systems makes

it possible to determine the strain tensor in predetermined volumes

in the interior of materials. Mapping of strains will become

possible throughout the material.

1. Introduction

One of the nondestructive methods currently used for measurements of residual

sses in industrial materials involves the application of x-ray analysis for stress

urement. X-ray diffraction phenomena are used to determine macroscopic residual

sses in engineering components. The methodology of analysis and its basic principles

well known. Briefly, when materials are under stress, x-rays are diffracted with a angle which is slightly different from the Bragg angle expected in unstressed

'a trials. The change in the Bragg angle is related to the alteration of atomic interplanar

479

1
spacings when the crystal lattice of the materials is strained. As a measurement systil

the essential part of this x-ray technique is the accurate measurement of lattice consll (or their changes) in materials by use of well-refined Bragg di ff ractometry. The surfilie residual stress can be evaluated from the strains measured by x-ray diff ractometry usir set of assumptions regarding material elasticity and homogeniety. Although the accura: obtained by this technique is, in principle, superior to any other methods, an obvious^

shortcoming lies in its incapability of detecting strains (or stresses) in the interiojo

bulk materials.

Currently, improved quality control of industrial components demands quantitative

information concerning the stress distribution near cracks and residual stress distrib i after different types of cold working and heat treating. These demands naturally leadio the necessity of measuring residual strains in the interior of materials. Ordinary Brfig diff ractometry will not be sufficient to respond to these demands. There is an entire

different approach to the x-ray evaluation of residual strains although it uses the

principles of x-ray diffraction. This approach involves the use of energy dispersive

state detectors with high energy x-ray photons. The use of solid state detectors has

introduced very effectively in the fields of scanning electron microscopy (microanalys

and x-ray fluorescence analysis of materials in the past ten years. It is called ener. dispersive spectroscopy. Unlike these successful applications, this technique has not applied to the residual stress (strain) measurements of materials. Extremely high

resolution and accuracy are required for strain measurements in comparison with the ot applications. If one can improve the resolution sufficiently and overcome other impor.

technical problems, this new system would be ideal for residual stress measurement int. industry; the equipment will be compact, results will be displayed visually and almosl

instantaneously, and analysis of the data will be made on site by a computer. Also tl system, in general, works better for high energy photons. High energy photons also

practical the use of the transmission geometry, since the x-ray absorption coefficient

materials is approximately inversely proportional to the third power of the energy ofj- x-rays.

The use of an energy dispersive system for possible residual stress evaluation w<, tested in 1973 by Leonard [l] 1

, who concluded, however, that this system did not have! sufficient accuracy for this purpose. In this paper, we will prove that his conclusit w premature and it is still too soon to dismiss the possible industrial use of this tecliq

for the evaluation of residual stresses. In Section 2, we will propose [2] a method ]

determining a strain tensor in predetermined volumes in the interior of a material, ai discuss the principles involved, although they are trivial. In Section 3, we will pp

some data showing the attained accuracy in this system, and discuss what kind of pre-'

cautions should be taken to make this system workable in practical use. Although thi; system should be used ultimately with high energy photons up to 300 keV, the data shoj

this paper have been obtained with an ordinary laboratory x-ray source up to 50 keV. ! should be noted, however, that the use of low energy photons for the test of resoluti 1

1 Figures in brackets indicate the literature references at the end of this paper. 480
:uracy does not negate the conclusion of this paper for high energy photons, since the

polution usually improves for those high energy regions. In other words, the

Inonstration of the sufficient accuracy for low energy photons certainly guarantees the

jfectiveness and accuracy of this technique when one uses more penetrating high energy

jtons for residual stress evaluation.

2. Method

1 In energy dispersive spectroscopy, one is concerned with the energy spectral profiles

| peak positions at different energy values, instead of Bragg angles which play a key role

Bragg diffractometry. Lattice constants will be measured in energy dispersive spectropy by the energies of diffracted photons:

d(A) = . v 1 v , (1) sm6 ' E(kev) ' v '

Ire d is an atomic interplanar spacing related to lattice constants, E is the energy of

fracted photons measured from peak positions of the spectra, and the coefficient including

f the scattering angle 6 is constant for a given scattering geometry. Figure 1 shows a ematic diagram of an energy dispersive diffractometry system. Usually, each Soller slit

jsists of a vertical and a horizontal slit. Often Soller slits can be replaced by a sycomb slit originally used as medical filters. For the present purpose, these slits

1 be replaced by a special slit system which improves the spatial resolution to a desired

"ee as described later. A solid state detector is mounted at a fixed scattering angle,

our purpose, the scattering angle should be smaller than 30° to ensure complete transition geometry. The geometrical orientation of a sample is indicated by the unit vector

fine role of which will become obvious later.

For a given sample geometry, one obtains an energy spectrum which contains many Bragg

I s obtained from hk£ planes which are perpendicular to ji. From a set of interplanar

l|/|ings for different (hk£), one can define the "stretch" in the ji direction as I

, _ ,Ad^ _ hk£

Q

hkJ>

inje d°^ is the interplanar spacing for hk£ diffraction obtained either from a reference

injjrained material, or from an arbitrarily chosen reference part of the sample. This

tity is related to the strain tensor by

\. = V e. . n.n

n L, ij i j ij

(3)

481
SOLLER OR SPECIAL SLITS

\Ge(Li) WHITE RADIATION (Ge(Intrin B io) (HIGH ENERGY)

Figure 1. A schematic diagram of an energy dispersive system for powder

diffractometry. The geometrical orientation of a sample is indicated

by vector n, the role of which is significant for residual stress determination.

where e^. is the i,j component of the strain tensor and n.. is the > component of the \ in a frame imbedded in the sample. Since the strain tensor has, in general, six inc components £-^, £-^> £ ^3> e £2' £ 23' anc* £ 33' one mus * nave a se^ °^ S1X simultaneous t, which can be obtained using (3) for independent measurements performed in six differed

orientations.

There are many ways to perform six independent experiments. Here we mention onejji

that the same predetermined volume in the interior of the sample is viewed by the det«:oi for all these six measurements. As shown in figure 2, the sample is rotated around tMl

dent x-ray beam three times using two detectors to obtain six independent measurement^

the incident beam direction is used as a rotational axis, one cannot obtain six indepiJei

j, sets if the same scattering angle is used.) As the sample is translated up and down ti ways, one can essentially map the entire volume of the sample. The incident beam mus'se prepared to have a very small cross section by a slit system and also to be extremely ?W (no angular divergence). In addition, the detector must receive the scattered beam wjn' 1

extreme narrow angular range so that the scattered beam should be originated from a p' termined volume inside the sample. This simple geometrical argument on the scattering

is justified as long as the scattering (Bragg diffraction) is kinematical.

482
SAMPLE

DETECTOR

ROTATION

TRANSLATION

Figure 2. An example of the geometrical arrangements for the evaluation of residual

strains in a predetermined volume inside materials. The sample is rotated three

times around an x-ray beam to give six independent sets of measurements with two detectors. The mapping of strains throughout the volume of the sample is achieved

by translations.

3. Accuracy

-4

In the application to residual stress measurements, a strain (Ad/d) of less than 10

10 percent) should be detected; in other words, the effective resolution of the energy

J I . -2

jersive system is required to be smaller than 10 percent. In 1967, Giessen and Gordon

jnstrated a resolution of 0.18 percent with the use of a lithium drifted Si detector [3].

le, Laheenmaki and Kantola claimed in 1971 the resolution of 0.05 percent [4]. A year

r, Fukamachi , Hosoya and Terasaki used a lithium drifted Ge detector with Soller slits

their x-ray optical scattering system and estimated the resolution of slightly larger

l 0.01 percent [5].

Bragg' s law states that a measure for resolution is given by

= - ^| - cot 6 A9, (4)

Ipe AE is the energy resolution of a detector and AG is the angular divergence created by

Hi incident beam divergence and the receiving angle of the detector. For intermediate

!ljhiy photons, the first term is of order of 10 for the resolution of two energies. This

ll becomes smaller as the photon energy increases. The second term is purely controlled

>S:he x-ray optical arrangement. Therefore, it should be kept smaller than the first term

>jjntroduci ng appropriate slit systems before and after a sample. As described before, it

iilecessary to limit the physical spatial size of the incident x-ray beam in order to

iw[ure the spatial variation of strains. The reduction of the second term should, thereIlk be achieved anyway,

i 483
One may claim then that the energy resolution of current solid state detectors is It small enough to reduce the value in (4) for the evaluation of residual strains. It has:ie known fortunately that each diffraction profile in the energy spectrum obtained from a :>li state detector is very close to a Gaussian. If the mathematical shape is known for a \ spectral profile, then the claim mentioned above can be circumvented: mathematically r fitting the observed peaks with Gaussian functions, one can determine the centroids (on

peaks) of the profiles far more accurately than the AE/E value stated above would indie As shown below, one can improve by a factor of 100 in the accuracy of determining the

centroid positions.

Figure 3 shows the energy spectra obtained from a commercial material, 304 Stainle

Steel. The scattering angle was set at 20 = 30.00°, with the incident beam divergence

-4 -3

4.6 x 10 rad (0.026° = 1.6') and the receiving divergence much less than 10 rad.

2 The spatial size of the beam on the sample is less than 0.5 x 0.5 mm . Under the prese receiving slit condition, the predetermined volume is roughly 0.5 x 0.5 x 0.6 mm^. It

^ be noted in the figure that the spectrum shows reflections of quite high order in the

well -control led background scattering. In principle, the higher order reflections shoul

be used for the evaluation of strains, as the relative error decreases in the high ener

range.

Figure 3. A Bragg diffraction energy spectrum obtained from a commercial 304

stainless steel sample. The scattering angle is 30.00°, ensuring a trans-

-4

mission geometry. The incident beam divergence is less than 5 x 10 rad

-3

(0.027° = 1.6'); and the receiving divergence is much less than 10 rad. 2 The spatial size of the beam on the sample is less than 0.5 x 0.5 mm .

484
Figure 4 shows as example of the observed spectral profile which has been fitted by

Gaussian curve with a linear background. The accuracy of determining the centroid

r peak) positions has been found to depend simply upon the counting statistics.

producibi 1 ity of those positions has also been tested; as long as the same predetermined

lume is viewed, the centroid position remains identical within the statistical error,

shown in table 1. As a reference to possible instrumental instability, Ag Ka line

jectrum in each run has been fitted with a Gaussian curve. (White radiation was

itained using a Ag target tube.) The Ag Ka line result indicates that the instability,

any, is not significant to jeopardize the accuracy in the determination of diffraction

-5

ak positions. Table 1 demonstrates that strains of 5 x 10 can be detected with

liable reproducibility, if sufficient counting statistics is established, and strains of

-4

x 10 can be detected even with moderate counting statistics.

Table 1

220 200 AgKa (22.2 kev)

Peak Peak Peak Peak Peak Peak

Height Position FWHM Height Position FWHM Height Position FWHM

1307.66 840.07 12.9 174.08 592.31 12.3 6046.04 986.870 13.56

±7.88 ±0.04 ±4.51 ±0.18 ±35.14 ±0.047

1526.20 840.02 12.9 485.71 593.51 11.8 6267.04 986.900 13.61

±8.67 ±0.04 ±6.59 ±0.09 ±33.70 ±0.044

410.79 840.70 13.2 100.03 592.13 11.9 1868.17 986.935 13.95

±6.02 ±0.010 ±3.36 ±0.21 ±19.34 ±0.086

In conclusion, an energy dispersive diffraction system has demonstrated its capability

a useful tool for the determination of residual stresses inside materials, when the

"ve fitting technique is used simultaneously. The present demonstration has been carried

t with low energy photons. The use of high energy photons certainly makes more practical

use of these energy dispersive systems for residual stress evaluation. More penetration

b increased energy resolution of detectors will improve the resultant accuracy and

|tectability of strains. Since the energy dispersive system can be made simple and compact,

^hout any delicate moving parts, the system is quite ideal for an industrial use on

Site, particularly when data are handled by a mini-computer. By no means, an energy

: tapersive system such as used here has reached the ultimate resolution; further

provements on x-ray optics and detectors should be made, in particular, with high

l?rgy photons. To demonstrate the x-ray optical quality of the energy dispersive

stem used in this paper, a diffraction spectrum obtained from a commercial NiP amorphous

I* jtal is shown in figure 5.

i

485
2000

391 Stainless Steel

(220)

800 820 840 8S0 88o 900

CHANNEL NUP1BER DATA.4C 5/29/79

igure 4. (a) An example of the observed diffraction profile fitted with a

Gaussian curve with linear background.

100

-50 -

-100 840

CHANNEL NUP1BER DATA.4C 5/29/79

900

Figure 4. (b) The residuals from the Gaussian fit which are well within

limits expected from counting statistics.

486
I

tli-P Glass

26=22.00'

0 10 20 30 40 50

Figure 5. A diffraction energy spectrum obtained from a commercial NiP amorphous

alloy, demonstrating the x-ray optical quality of the energy dispersive system

used in this paper.

Leonard, L. , Franklin Institute Research Lab (Philadelphia) Report F-C3454 (July 1973).

2 Kuriyama, M. , Boettinger, W. J., and Burdette, H. E. , ASNT National Fall Conference,

, October, Denver, CO, p. 49 (1978).

3; Giessen, B. C. and Gordon, G. E. , Science, 159, 973 (1968).

4 Laine, E. , Lahteenmaki , I., and Kantola, M. , X-ray Spectrometry, 1, 93 (1972).

5 Fukamachi , T. , Hosoya, S. , and Terasaki , D. , J. Appl. Cryst. 6, 117 (1973).

References

487

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

STANDARD REFERENCE MATERIALS FOR QUANTITATIVE ANALYSIS AND d-SPACING MEASUREMENT

Camden R. Hubbard

Ceramics, Glass, and Solid State Science Division

National Bureau of Standards

Washington, DC 20234

The National Bureau of Standards has an on going program in x-ray

powder diffraction to provide Standard Reference Materials (SRMs) to the

diffraction community. The first product of this program was SRM-640 Si Powder, certified in 1974 as a d-spacing standard. While this material

has become widely used it is not a universal reference material. In

o particular, the large d-spacing region (>4 A), accessible to most diffraction apparatus is not covered. To fill this need we are reviewing

materials as candidates for a large d-spacing SRM. Candidate materials

currently include molybdenum disulfide, arsenic tri oxide, sodalite,

fluorophlogopite, muscovite, and thallous acid phthalate.

Intensity SRMs for quantitative analysis will soon be certified.

The five materials (orA^Og, T^-rutile, ZnO, C^Og and CeO^ were

selected to cover the range of linear absorption coefficients encoun- tered in practical samples. By matching approximately the linear absorption

coefficients, errors in analysis due to microabsorption can be minimized.

The relative intensities and the reference intensity ratio for each material

are being measured on Bragg-Brentano focusing di f f Tactometers. The results

are compared with values obtained from calculated powder patterns. The use

of the reference intensity ratio as a basis for tabulation of the calibration

constants needed in internal standard methods of quantitative analysis will

be reviewed.

1. Introduction

Standard Reference Materials (SRM's) have been produced, certified and issued by NBS

nee 1905. NBS now issues in 70 major categories, approximately 37,000 SRM units each

sar to over 12,000 customers. Nearly 900 different SRM's are currently available [l] 1

.

i general SRM's are well-characterized, stable materials (or artifacts) which are proiced in quantity and have one or more physical or chemical properties measured and certied by NBS. SRM's are used to: calibrate instruments and measurement system; facilitate

igures in brackets indicate the literature references at the end of this paper.

489
the exchange of goods; maintain quality control in industry; assure the development and

enforcement of equitable regulations; and assure the long term accuracy and compatibilifl

of measurements.

In the field of x-ray powder diffraction there are a wide range of possible classe

of SRM's including standards for instrument calibration and quantitative analysis. A standard for either purpose can be mixed with the sample to be studied (an internal sta a or used independently (an external standard). It is possible that the same material ca|bi

used for a variety of purposes. However, as we will see below, the requirements for ea class of standards are somewhat different.

2. External Instrument Standards fM

Materials that are used to aid and/or check instrument alignment, to evaluate reso' lution, to determine an instrument systems stability and to intercompare various instruments are usually a stable briquette or a disk. Some instrument manufactures have prov

powdered silicon as a bonded disk or fine grain quartz slabs. Materials that have been'isi to test resolution include BaS0

4 (l% Sr) [2], tungsten [3] and quartz [4].

The use of an external instrument standard to calibrate the 20 scale, however, has limited accuracy. The accuracy is limited ultimately by the knowledge concerning the

5

'3 mm.

standard itself to about 1 part in 10 . However, errors due to systematic displacement

of the Bragg peaks usually dominate. The displacements arise from geometrical properti

such as axial divergence, physical factors such as sample position, and pentration of t

1

beam into the sample. The peak displacement effects for the standard, and separately f

a sample, combine to limit accuracy in cell parameter determination to between 1 and 1C 4 . parts in 10 in routine work.

3. Internal Instruments Standards

One of the easiest methods to obtain improved accuracy in lattice parameter detem

nations is to initimately mix a powdered internal instrument standard (also known as a d-spacing or 20 standard) with the powdered phase to be characterized. In this case, displacement of Bragg profiles due to sample displacement and beam penetration affect

the standard and sample reflections almost identically. The location of the standard

lines are then used to generate a calibration curve (see section "Calibration Methods"

1

The historical development of internal instrument standards is not well documentei

Their use likely began as early as the beginning of x-ray diffraction itself. During

the 1940's, and probably earlier, NaCl and quartz [5] were used as internal standards.

It became apparent that reference sample preparation methods and lack of homogeneity

limited the usefulness of these materials. This was further accentuated by the fact 1

that no single source or institution provided well characterized reference

materials. The first major improvements in internal standards came in the 1960's.

The IUCr Commission on Apparatus and Standards headed by W. Parrish organized a

"

round robin on cell parameter measurement. Three materials (diamond, silicon and

490

2
ingsten) were carefully selected and distributed to a number of laboratories. While

e discrepancy between the results of the various laboratories was larger than expected

4 -5

j| part in 10 ), mean lattice parameters with a/a =;3 x 10 were obtained [6]. The

stributed samples and "best" lattice parameters became de facto standards. They were

ten used to calibrate secondary standards for routine laboratory use [7]. At NBS, the

ill parameters of high purity tungsten, silver and cadmium oxide [8] were determined to

5 itter than one part in 10 . These standards have been used for nearly 15 years by the

iS in collaboration with the Research Associateship of the International Centre for

ffraction Data - JCPDS. Approximately 1000 reference patterns have been carefully

[corded using one or the other of these internal instrument standards. However, these

tference materials were not made publically available. To overcome this deficiency, NBS

ioduced SRM 640 Si powder [9]. The lattice parameters of the high purity Si were meajred relative to that of NBS tungsten by powder dif f ractometry and measured directly by

Lie 1.5. single crystal Bond technique. The absolute uncertainty in a is reported to be within

Jparts in 10 . Since its issuance in 1974, over 550 units have been sold to laboratories

^roughout the world.

A few comments on the limitations of this material are required. First, the grinding

the single crystal boules probably introduced some strain. Thus, the sample should not

| used directly to define the true instrumental profile. Second, the SRM sample contains

Lime particles up to 20 urn in length. These small single crystals can produce undesirable

jiikes on the side of a recorded powder diffraction line. The effects of the spikes can

|| minimized by careful interpretation of the strip chart. However, for automated systems

iploying computer analysis of the digital data a spike can cause a serious error in deter-

| ning the peak position. To overcome this problem some laboratories have further sieved

le Si powder (<400 mesh). It is possible, though highly unlikely, that such a procedure

mid produce a powder with lattice parameters shifted by as much as 10 ppm. No definitive

; ists have been performed at NBS. The third, and probably far more important, limitation

Li the use of SRM 640 is that for CuKa x-radiation the lowest Si diffraction peak occurs at ^443° 28. This leaves the region between 28° and 0°28 calibrated only through extrapojtion.

I

The value of using a d-spacing internal instrument standard was demonstrated in a ;kent Round Robin on Search/Match Procedures, [10]. One of the test problems was a mixlire containing three phases and the Si standard. One-half of the participants were told

sjiat the sample contained SRM 640 Si powder, and they were provided with the expected 26 's

i>r CuKcv-^ radiation for the Si. The other half of the participants did not receive this

^formation. All respondents measured the d-spacings and attempted to identify the phases

resent in the mixture. To ascertain the quality of the observed d-spacings six lines with

o •values between 4.4 and 1.54 A (20° and 60° 26 for CuKo^) were selected and the average of

le magnitude of the relative error in d-spacing (d-quality) was calculated. Results

fovided by the respondents fell into three categories as shown in table 1. The accuracy

I measured by d-quality improved significantly when the standard was used for calibration

Vposes. The success rate (rating) in phase identification (5 points for each correct

11 491
phase, 20 maximum) also improved as the quality of data improved. However, the d-quali,

3 of 1 part in 10 is a factor of 10 below that expected. It must be remembered, however

that the purpose of the exercise was phase identification and not accurate d-spacing

measurements. Significant improvement of the success in identifying phases in multipha

mixtures due to high quality data have been reported by Edmonds and Henslee [2].

Table 1. Round robin d-quality.

|

Use of

Standard # of Response d-Qual ity Rating

None

Visual check on 26 values

Corrected 26 values

26

6

7

0.0018

0.0016

0.0009

16.1

17.0

18.0

4. Large d-Spacing Standards

To improve on the calibration of the low 26 range NBS is currently examining sever candidates for use as a large d-spacing reference material. The criteria for a large

d-spacing standard are: (1) that the powdered material is stable in the atmosphere, in x-rays and in

mixtures with other powders;

(2) that it is available in homogeneous quantities of 1 to 20 kg;

(3) that the diffraction pattern exhibits a few sharp well spaced strong lines

o with at least one d-value >6 A. Besides receiving suggestions from co-workers and colleagues, we have searched the Hanav, Index of the Frequently Encountered Phases Search Manual [11] for possible materials. i\ candidate materials are given in table 2 along with the value of the largest d-spacing,

the relative intensity of that line and pertinent notes. The materials described as "ply

exhibit extreme preferred orientation. This can be used to advantage with diffractomete

where only the 00£ reflections will appear (fig. 1). However, the preferred orientation

is a disadvantage to their use as a standard for Guinier cameras. Both sodalite [12] arji thallous acid phthalate [13] have desirable characteristics except that above 30° 26 (Cu ) there are many interfering reflections and a second sample mixed with Si would likely be required to calibrate the 26 scale above 30° 26. We intend to accurately measure the

d-spacing of one or two of the candidate materials by both powder diffractometry and the,! Bond single crystal method, as was done for Si SRM 640.

492
Table 2. Candidate large d-spacing standards

Phase dmax r

e1 Notes

MoS

2 6.15 A 100 platy

Sodal ite

v 8 6 6 24 2 6.27 40 cubi c

As

203 6.39 60 Used at Dow

Chemical Corp.

f 1 uoro-phlogopite

(KMg

3 (Si3A1010

)F

2

) 9.96 100 platy

muscovite

(KM

2

Si

3A101Q

)F

2

) 10.0 100 platy

thallous hydrogen

phthalate (T£C

g

H

504

) 12.9 100

• mono, xtal

2000 —

1800 —

1600 —

1400 —

2. 1200 —

c 1000-

0

c

800 —

600 —

400 —

200 —

001

d = 10?

003

d=3.33

004 d = 2.50

005 d= 2.0

006 Jim-'

J I I

I I I 12 16 20 24 28 32 36 40 44 48 52 56 26 (degrees)

Figure 1. X-ray powder diffraction pattern of highly oriented phlogopite measured

on a di f f ractometer with CuKa radiation and 6-compensati ng slits. The peaks are labeled by their Miller indices and d-spacings.

493
5. Calibration Methods

There are several ways to use a d-spacing standard to correct the observed 26 va The simplest method which is generally used for strip chart recorded patterns is to c. culate the correction factors (^=

20^ rue-2e

o ^ s

) and then for each 26 of the sample to perform a linear interpolation between adjacent standard lines. In table 3, seven caj

bration points were obtained using Si and W internal standards in an a-A^O^ sample.

The mid-cord position at 80 percent height of each peak was determined from a strip clrt recording measured to ±0.01° (26). The plotted data (fig. 2) shows that linear inter|-

lation will provide corrections which are probably as accurate as the measurements. i|

plying corrections with a o- ^

= 0.01° to data measured with a o

Q ^ s = 0.01° leads to ji 1 corrected data with 0" cQr = -f~2 x 0.01°. Fitting a low order polynomial to the indica d corrections reduces the uncertainty in the correction factor and hence reduces a

cor

- Guinier films, Sonneveld and Visser recommend using a second order polynomial [14]. low in his paper on alignment and calibration of Guinier cameras, uses a smooth curve thr< gh the plotted points [15]. A more fundamentally sound method is to remove the aberratiis

due to line shape and instrumental aberrations and then perform a least-squares refinisn

of values for 26-zero, sample displacement and sample transparancy [16].

Table 3. 28 correction data.

Sample

Standard

a-Al

2

0

3 Si and W

Radiation: Cu Ka (Ni f i 1 tered)

Standard 26.true obsA

Si 28.443° 28.46° -0 02°

W 40.262 40.26 0 00

Si 47.303 47.30 0 00

Si 56.123 56.11 +0 01

W 58.251 58.22 +0 03

Si 69.131 69.11 +0 02

W 73.184 73.14 +0. 04

494
I,

0.10 c •

o «

5S (0 o a) V 9

g I

-0.10 30 40 50 20 (degrees)

60 70

Figure 2. Correction curve based on Si and W internal Standards in a-A^O^.

6. Intensity Standards

An external instrument standard for calibrating or checking an instrument's intensity

response should be a powder formed into a disk or briquette. Two examples are pressed Ni disks [17] and a fine grained quartz mineral sample such as novacculite. Generally, for

highest accuracy, each sample must be independently calibrated because of sample inhomogeneity and variable preferred orientation. Another more common use of intensity standards

is for quantitative analysis. Before discussing standards for quantitative analysis it

is necessary to review the basis of quantitative phase analysis [18]. The fundamental

equation used throughout all methods of quantitative analysis is:

iJ

K., X,

i J J Pi M*T (1)

tfhere I., is the intensity of the i t '1 line of the J-th phase, K., is an instrument sensi-

:ivity constant, Xj is the weight fraction of phase J in the mixture pj is the density of

)hase J, and is the mass absorption coefficient of the mixture.

The absolute intensity method of quantitative analysis works particularly well for

picrogram quantities such as quartz on a filter. The basic assumption involved is that

495
a monolayer or less of sample is deposited. In this case, u*y for the quartz grains is u* of quartz and is independent of the matrix or amount of quartz. With these assumptions,

eq. (1) simplifies to I.j = k Xj. A method which requires calculation of u*y is the

multisample - multiphase method of Zevin [19]. For this method, all phases in the mixture

(including amorphous phases) need to be included in the analysis since the constraint

2 Xj = 1 is imposed. Zevin 1 s method requires that one set of phases (m) be present in all samples (n) and that m < n. Furthermore, for reasonable precision the weight fraction of

each phase should vary significantly throughout the set of samples. More general methods

for quantitative analysis are based on the addition of a phase to a mixture in order to eliminate u*j. The three possible methods are dilution [20], doping with the analyte

[21,22,23] and addition of an internal standard [24]. The dilution method has low precision of analysis and will not be considered further. Doping with the analyte is most

sensitive when the analyte weight fraction is low. The internal standard method yields

the greatest precision in most other cases. This method is also the most generally

applicable since for a preselected standard a table of constants can be prepared (discussed below). NBS is preparing intensity SRM's for the internal standard method.

Before we can discuss the internal intensity standards we need to examine the internal

standard method in detail. The basic equation is derived directly from eq. (1) by

evaluating the ratio of intensities of the i* line of the analyte J to the k^ line of th( standard:

I., /K.JA /p u*\ k' 1C X,

i J / lJ J \ / Ks M T \ JS J

I

fcS " T ) I K kS

X

S ] "

X

S (2)

VQ 1 Since K. , = c I.,, where c is a combination of a instrument sensitivity factor and numer-

~\%J 1

ical constants, the factor k'j<. is independent of the instrument. Simply, k'j^ is the

ratio of the intensity I.j to 1^ for a 1:1 mixture by weight. Provided that sample

factors such as strain or degree or extinction are not important, the factor k'j^ may be

measured at one laboratory and used at any other laboratory.

In cases when all phases in a mixture are crystalline their weight fractions can be

determined without addition of the standard. Applying the assumed constraint that

Z Xj = 1, eq. (2) can be transformed into [25,26]

k'

1

T

2^ £

! L/. , (3)

L J L=lK LS

Only ratios of intensities of all the phases and the constants k'.<. are required. An example of application of this equation is given in table 4 for a 50-50 rutile-anatase

mixture.

Table 4 phase hk£ k' I(cps)

X calc

X cnown Ti0

2-rutile 110 3.4 200 0.50

4

0 .50

Ti0

2-anatase 101 4.3 156

496

0.49

6

0 .50
is example is undoubtedly optimum for analysis by this method since, (1) p*j is indeIndent of the weight fractions, and (2) as a laboratory example the weight fraction

nstraint is likely valid. The accuracy of analysis is limited by the accuracy of the

tensity measurements and that of the quantitative constants k

1 . When IX , = 1 does not

jld, ratios of weight fractions can still be determined without adding a standard. For

lases J and L in a multiphase mixture if k'j^ is known and I.j and 1^ are measured then

(2) yields xyx

L

. To avoid the necessity of determining the constants k' for each analysis, de Wolff

1 Visser [27] proposed tabulating k' for the strongest line of the sample and a fixed

andard. In addition to de Wolff and Visser, Swanson [28] and Scott [29] recommended

iH^Og (corundum) as the reference standard. This particular ratio has become known as

> Reference Intensity Ratio and is abbreviated as I/Ic. Values of I/Ic for about 1000

[ises are published in the Powder Diffraction File [30]. The NBS Monograph 25 series [31]

ibo gives I/Ic values. The majority of these I/Ic values were determined by direct

I T*G 1 ,'plication of the definition. That is, the ratio of the strongest line (I = 100) of

re 1 sample and of a-AlgOg (I]j3 = 100) were measured for a 1:1 mixture by weight. Generally,

fbios of peak height intensities are used as an approximation to ratios of integrated

yensities. Providing that the two phases have approximately equal diffraction line

fiadths and that the two measured lines are within 10° 26 this approximation introduces

< error of probably less than 5 percent in I/Ic. Errors due to preferred orientation,

ntroabsorption and counting statistics further limit the accuracy of I/Ic values

insured to between 5 and 10 percent. Unfortunately, error estimates were not reported

;il recently.

A new procedure has been adopted at NBS to reduce the errors due to preferred ori enion and counting statistics and an error estimate is now given. A mixture of known

ght fraction ratio X/X

c is prepared which yields approximately equal intensities of

I strongest line of each phase within the constraint that 0.20 < X/X

c

< 5 in order to

fejn'd errors due to particle statistics. Multiple lines of the sample and corundum are

insured. Lines are paired which are close in 20 and the I/Ic value is calculated for

e h pair using:

i x T yy

t n - xx c vJ I/Ic = w

* r • px

I c

|jre is the observed intensity of a sample line with relative intensity yy and I*

x

the observed intensity of a corundum line with relative intensity xx. The relative

ensities yy and xx must have been determined independently in order that errors due

preferred orientation effects are reduced. An example of application of this approach

shown in table 5. In this case, the intensity of the 113 line of orA^Og could not be

d due to an interfering line overlap. The average I/Ic is 5.18 with a rms deviation

fjm the average of 0.12(^2 percent). Repeating these measurements at another value of X/X

c

Wjjld then be performed to further reduce systematic errors due to preferred orientation

497
in a given mounting and to reduce random errors due to counting statistics. Typical

measurements yield an internal consistency ranging between approximately 2 and 10 percent. We expect to further reduce this by application of automation and calibration

of secondary reference standards.

I

I)

Table 5. Determination of I/Ic for ZnO. (X/Xc = 4.57)

ZnO a-Al

203 I/Ic

hk£

jrel I(cps) hki> jrel I(cps)

100 59 249 012 61 235 4.99

002 46 191 104 92 333.5 5.24

101 100 413 104 92 333.5 5.21

110 31 128 116 83 296 5.29

Because of the increasing accuracy in Reference Intensity Ratios, we need to be

clearer as to what sample volume and experimental conditions are employed in the mea- surements. First, relative intensities and I/Ic values should be representative of a constant diffracting volume of the sample. If variable slits are employed in the

measurements which lead to changing sample volume, then conversion of the observed data

is required. Second, the intensities should be recorded with CuKa radiation. Hubbard

and Smith [32] have discussed conversion to alternate wavelengths and the effect of

anamolous scattering. Finally, I/Ic values should be corrected for polarization effect

of a monochromator if one was used [32]. (Reference Intensity Ratios should not be

published unless they meet these criteria).

Although the reference standard has been chosen to be a-A^O.^ this does not imply

that a-Al^O^ must be used as the internal standard in application. For example, if

another reference standard eliminated peak overlap, the quantitative constant k

1 of

eq. (2) for analyte A relative to standard S is given by:

)\ te

! If

( a

k'K = — AS I r

Thus, tabulating accurate Reference Intensity Ratios will enable rapid quantitative

analysis by the internal standard method even when crAKO^ is not the appropriate stand;! Further, if all the phases in the mixture are known and if 1 Xj = 1 is a good approxima- 1

tion, then a standard need not be added. In both cases all that is required is accuratf fe knowledge of Reference Intensity Ratios. BP

498
Besides derivation through experimental measurements, the Reference Intensity Ratio may

I be calculated along with a simulated powder pattern from knowledge of the atomic positions

and thermal motion [33]. These simulated patterns are free from extinction and preferred

orientation. The accuracy of calculated Reference Intensity Ratios is limited by the

accuracy of the structural information of the analyte and of corundum. Errors in atomic

positions and thermal parameters can easily introduce significant errors. As a general

rule, a single crystal R-factor below 0.08 appears adequate. Typographical errors in structural data must be detected by careful analysis of the published structure. The

atomic parameters of orA^O^ are well known but the aspherical charge density is not yet

well known [34] limiting the accuracy in calculated Reference Intensity Ratios to about

5 percent. If the powder sample of the analyte exhibits extinction or microabsorption

the error in the calculated I/Ic can be as large as 100 percent. Tests on the accuracy

of analyses using I/Ic values from various sources have been reported previously [26,33].

When I/Ic values have been determind from the same sample from which the test mixtures

have been prepared, one can expect errors to be less than ±1 weight percent. When values

from the PDF or from calculations are used, errors of 1 to 10 percent by weight are not

uncommon. This indicates that preferred orientation, extinction and/or microabsorption

were present in one or more of the measurements.

To improve the accuracy of routine quantitative analysis, NBS is in the process of

certifying five intensity standards. The five materials (table 6) were selected to cover

the range of linear absorption coefficients likely to be encountered in analysis. By

approximately matching u of the standard to that of the analyte as well as grinding the

sample to < 10 urn, microabsorption can be minimized [35].

Table 6. Proposed x-ray diffraction intensity standards.

Phase u(cm

1

) I/Ic(calc)

a-Al

203 124.9 1.00

ZnO 288.0 5.43

Ti0

2 (rutile) 549.5 3.44

Cr

203 952.1 2.16

Ce0

2 2082.9 14.1

These five materials will be certified for relative intensities and Reference Intensity

Ratios. Currently, measurements have yielded relative intensities reproducible to 2-4

percent without rotating the samples. When a sample rotator is employed, we expect to lower the uncertainty in the relative intensity of the strongest lines to 1-2 percent.

These future SRM's will generally be used for rapid, fairly accurate quantitative analysis.

499
They can also be used to check instrumental response and to compare the performance of

various instruments. The cell parameters of these materials have been determined with

5 uncertainty of approximately ±5 parts in 10 .

7. Summary

The x-ray powder diffraction Standard Reference Materials program at NBS will soon certify five intensity standards for use as internal standards for quantitative

analysis. Coupled with tables of Reference Intensity Ratios they can provide quantitative information with minimal experimental effort. Following the intensity standards; we plan to certify one or two large d-spacing standards to complement SRM 640 Si powder

Possibilities for other SRMs exist including a particle size standard and respirable

quartz, Cristobal ite and tridymite. Which future standards will be prepared, if any, depends on input from the user community. Letters recommending needed standards are

welcome.

References

[1] NBS Standard Reference Materials Catalog, NBS Special Publication 260, 1979-80

Edition, National Bureau of Standards, Washington, D.C. 20234.

[2] Edmonds, J. W. and Henslee, W. W. , Application of Guinier camera, microcomputer

controlled film densitometry, and pattern search-match procedures to rapid routine

x-ray powder diffraction analysis, Advances in X-ray Analysis, 22, 143-150 (Plenum;

Press, New York, 1979).

[3] Klug, H. P. and Alexander, L. E. , X-ray Diffraction Procedures , 2nd Edition (John

Wiley & Sons, New York, 1974), p. 284.

[4] Jenkins, R. private communication.

[5] Klug, H. P. and Alexander, L. E. , ibid., p. 581.

[6] Parrish, W. , Results of the IUCr precision lattice-parameter project, Acta. Cryst.

13, 838-850 (1960).

[7] Calvert, L. E. , private communication.

[8] Swanson, H. E. , McMurdie, H. F. , Morris, M. C. , and Evans, E. H. , Standard x-ray

diffraction powder patterns, Nat. Bur. Stand. U.S. Monograph 25 Sec. 4, p. 3-4

(1966), National Bureau of Standard, Washington, D.C. 20234.

[9] Hubbard, C. R. , Swanson, H. E. , and Mauer, F. A. , A silicon powder diffraction

standard reference material, J. Appl. Cryst. 8, 45-48 (1975).

[10] Jenkins R. and Hubbard, C. R. , A preliminary report on the design and results of

the second round robin to evaluate search/match methods for qualitative powder diffractometry, Advances in X-ray Analysis, Vol. 22, p. 133-142 (Plenum Press,

New York, 1979)

500
[11] Powder Diffraction File Search Manual for Frequently Encountered Phases publication

FEP-28, International Centre for Diffraction Data, 1601 Park Lane, Swarthmore, PA 19081.

[12] Smith, D. K. , private communication.

[13] Jenkins, R. private communication.

[14] Sonneveld, E. J. and Visser, J. W. , Automatic collection of powder data from

photographs, J. Appl. Cryst. 8, 1-7 (1975).

[15] Brown, A., Optimal calibration curves for Guinier-type focusing cameras, Advances

in X-ray Analysis, 21, 289 (Plenum Press, New York, 1978).

J[16] Parrish, W. , private communication.

i [17] Suortti , P. IUCr commission on crystal lographic apparatus-accuracy of structure

factors from x-ray powder intensity measurements, Acta Cryst. A33, 1012 (1977).

[18] Klug, H. P. and Alexander, L. E. , ibid., 531-562.

j[19] Zevin, L. S. , A method of quantitative phase analysis without standards, J. Appl.

Cryst. 10, 147 (1977).

|[20] Clark, N. H. and Preston, R. J., Dilution methods in quantitative x-ray diffraction

analysis, X-ray Spectrometry, 3, 21 (1974).

[21] Copeland, L. E. and Bragg, R. H. , Quantitative x-ray diffraction analysis, Anal.

Chem. 30, 196 (1958).

[22] Bezjak, A. and Jelenic, I., The application of the doping method in quantitative

x-ray diffraction analysis, Croat. Chem. Acta, 43, 193 (1971).

[23] Popovic, S. and Grzeta-Plenkovic, G. , The doping method in quantitative x-ray

diffraction phase analysis, J. Appl. Cryst. 12, 205 (1979).

[24] Klug, H. P. and Alexander, L. E. , ibid. p. 549.

[25] Bezjak, A., Croat. Chem. Acta. 33, 197 (1961) as referenced in Karlak, R. F. and

Burnett, D. S. , Quantitative phase analysis by x-ray diffraction, Anal. Chem. 38

1741 (1979).

[26] Chung, F. H. , Quantitative interpretation of x-ray diffraction patterns of

mixtures-II, J. Appl. Cryst. 7, 526 (1974).

[27] de Wolff, P. M. and Visser, J. W. , Absolute intensities-outline of a recommended

practice, Report #641.109 Technisch Physische Dienst T.N.0. (1964).

[28] Swanson, H. , private communication.

[29] Scott, R. F. , The case for a universal x-ray diffraction intensity scale, Fifty

Years of Progress in Metal lographic Techniques , ASTM STP 430, Am. Soc. Testing

Matls. , p. 201-203 (1968).

[30] Powder Diffraction File, Alphabetical Index, Inorganic Compounds 1978 SMA-28,

JCPDS, 1601 Park Lane, Swarthmore, PA 19081.

[31] Standard X-ray Diffraction Powder Pattern, NBS Mongoraph 2b, Sec. 7-15, National

Bureau of Standards, Washington, D.C. 20234 (1969-1978).

[32] Hubbard, C. R. and Smith, D. K. , Experimental and calculated standards for quantitative analysis by powder diffraction, Advances in X-ray Analysis, 20, 27 (1977).

501
[33] Hubbard, C. R. , Evans, E. H. , and Smith, D. K. , The reference intensity ratio, I/Icj. for computer simulated powder patterns, J. Appl . Cryst. 9, 169 (1976).

[34] Calvert, L. D. , private communication.

[35] Brindley, G. W. , The effect of grain or particle size on x-ray reflections from

mixed powders and alloys..., Phil. Mag. 36, 347 (1945).

Discussion

Comment (Mortier): In your abstract, zeolite A is mentioned as a possible standard.

Generally, for all zeolite types, there are a number of problems such as the Si: Al ratio

the extent of the ion exchange and the proton content, and the crater content which

depends on the humidity of the environment. All of these factors do influence the unit

cell parameters in a rather sensitive way. If mixed with salts, ion exchange is probabl

Response (Hubbard): Comments received during and following the meeting indicate that ze lites, thallous salts and hs^O^ would be poor choices for an SRM for a variety of reason

1

I do not intend to consider these materials any further.

502
jkional Bureau of Standards Special Publication 567. Proceedings of Symposium

j| Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

,ssued February 1980)

STRUCTURE ANALYSIS FROM GUINIER FILM DATA

P.-E. Werner

Department of Structural Chemistry

Arrhenius Laboratory, University of Stockholm

Fack S-106 91 Stockholm, Sweden

Experiences from more than ten investigations at this Institute during the last four

irs have shown that structures containing up to about 50 structural parameters can be suc- sfully studied by use of Guinier-Hagg powder diffraction data. Photographs have been

en in a focusing camera of the Guinier-Hagg type, with strictly monochromatized CuKa-^ iation, single coated films and internal standard technique. A SAAB drum film scanner

l 1 and a software system for Guinier films [2] have been used to extract the intensity data

Sn'lable on the photographs. The basic principles for determination of peak positions from

t insmission data (-not from optical densities-), derivation of background curves, correction

p|cedures for non-linearity versus exposure and a method for parabolic least-squares fit of

ilernal standard lines are described in reference [2].

The drum film scanner system will now be replaced by a minicomputer controlled scanner

h semiconductor techniques and an improved software (to be published). An alphanumeric

play will be used not only to control the instrument by the minicomputer, but also for

ect transfer of data to a large-scale computer, where the magnetic tape version of the

der diffraction data base is available.

The Rietveld procedure [3] of profile refinement has made it possible to utilize all of

tj information present in a diffraction pattern. The original Rietveld program was written

II single phase analysis from neutron diffraction data. The program has been revised in steps the first of which comprised the application of profile analysis to x-ray diffracn Guinier data [4] and the second the introduction of a possibility to refine two simulteously occurring phases in a powder pattern [5]. Examples of structure determinations

f{ m Guinier-Hagg diffraction data are given in the references [6-9].

jj The Rietveld procedure is primarily a refinement method, but it can be very helpful in

later steps of a structure determination as it provides the user with the possibility

calculate the distribution of the observed intensities and subsequently to find light

m positions from Fourier maps. On the other hand, the structural information in the high

ijle region cannot be extracted unless the indexing problem can be solved and the Patterson

fjption can be calculated from a low angle region with well resolved diffraction peaks

( i. [6] and [7]). The most important advantage of using a focusing Guinier-Hagg camera

win dealing with complex structures is probably the good resolution of the diffraction lines.

^ures in brackets indicate the literature references at the end of this paper.

I 503
The accuracy in peak positions needed for solving an indexing problem can be discuss^

from the minimum condition that the De Wolff figure of merit, I^q, should be larger than

10 for a physically plausible indexing [10]. The following approximate relation can be

derived for the maximum verifiable unit cell volume for orthorhombic and lower symmetries

V =

20

A26

where m is the multiplicity factor for a general reflection, d

2

Q is the d-value for the 21 observed line and |A2G| is the average discrepancy between observed and calculated 26 val for the first twenty lines [11]. In table 1, |A20| -values needed for solving monoclinic j (m = 4) indexing problems are calculated for some unit cell volumes and d2Q- values.

Table 1.

V(A

J

)

MAXIMUM VERIFIABLE UNIT CELL VOLUME V.

d 2Q(A)

3.5 3.0 2. 5 2. 0

|A26|(°) (m = 4)

100 0.41 0.26 0. 15 0. 08

500 0.08 0.05 0. 03 0. 015

1000 0.041 0.026 0. 015 0. 008

2000 0.021 0.013 0. 008 0. 004

3000 0.014 0.009 0. 005 0. 0025

The F- index recently proposed by Smith and Synder [12] is probably superior to De Wo' M-index for ranking patterns, but it has a disadvantage when estimating the plausibility c< a found unit cell. The fact that the M-index increases systematically with symmetry is nc

a disadvantage. A cubic trial indexing of a powder pattern is more likely to be correct tyiv a triclinic, the fraction of observed lines and accuracy being equal.

504
The following statements are discussed and illustrated by the indexing of three aluminumjlphate minerals:

1. The volume of an unknown unit cell can usually not be estimated from the expression

= const • d

3^ [13] if V > 1000 A3

.

2. Although |A20| as a function of 26 is relatively constant, a small increase with 20

; usually observed, if the unit cell is large, and the diffraction data are accurate

|A2Q| < 0.01°; see fig. 1). This can be explained as overlap effects between observed

»nes in the high angle region. For low quality data overlap effects between observed

id calculated lines are often dominating and |A2G| decreases with 26.

0.010

0.005

Alum i n i te

20 30 40 50 SO 70 80 90 100 110

Figure 1. A20 as function of the number of observed lines for Aluminite.

3. For large unit cells the fraction of observed lines decreases strongly with 26. ready M^q (or F^q) cannot be used to prove the correctness of the indexing of the metauminite pattern (cf. table 2). It is therefore recommended that I^q is used in indexing

P|ograms. It is also concluded that the use of F^ (or M^) as data quality index can be

Ijsleading unless N is small (cf. table 2 and fig. 2).

| 4. The aluminite and meta-aluminite patterns have both been indexed by a trial-andjror indexing program, TREOR, written by the present author [14]. The program has also been

Ipecessary tool for the studies [6] and [7]. All indexing programs gain in efficiency with

e data quality. This is specially true for a trial-and-error program which is extremely

pendent on the parameters derived from the first lines in a pattern. A multitude of non505
systematic extinctions (i.e. large unit cells) will affect the relative efficiency of vari ous indexing programs differently. Provided that the data are accurate, relatively large

unit cells may sometimes be found by trial-and-error indexing.

As an extreme example the indexing of halotrychite is discussed. Although the unit ce °3 s , , volume is above 3000 A , a value of \A^q = 15 was obtained for the unit cell dimensions give

in table 2. If the space group is assumed to P2-./c the value of I^q is increased to 18. It a-axis 20.519 A given in the powder diffraction file, number 26-1425, cannot be verified fr the new data or from the data given in the powder file (I^q < 3).

5. As a general experience of the Guinier-Hagg film measuring system described in

reference [2] it can be stated that it is rare that | A20 1 i s above 0.01° for the first twent lines observed (cf. table 2). The resolution of the system is also illustrated by the fact that the number of resolved peaks in the aluminite pattern is 110, all indexed, whereas the number of d-values given in the powder diffraction file, number 8-55, is only 30. It should be pointed out that only peak positions have been used in the calculations

reported in table 2. For substances with known structures a least-squares fit of profile

functions, such as the Rietveld procedure, can be used to further reduce jA20| and determi^

more accurate unit cell parameters.

Two methods for quantitative analysis from Guinier data have been studied. The first one is based on the extension of the Rietveld program to treat two-component problems and

is described in reference [5]. The second method is based on integrated intensities of an} one line from two different amounts of samples of the same material. The ratio of the absc tion factors is determined from the masses and the integrated intensities. The absorption

conditions are analyzed by the theoretical expression derived by Sas and De Wolff [15] for

absorption in a Guinier camera. No internal standard need be added to the samples. The

method has been tested on total sample amounts in the range 0.1 - 1.5 mg [16].

506
Table 2.

INDEXED ALUMINUM SULPHATE MINERALS.

II.

III.

Meta-aluminite, A^SO^COH)^ 5H

2 0

a = 7.930(3) A, b = 16.879(4) A, c 7.353(5) A, p = 106.74(7)°

V °3942.4 A . Ref. [11].

Aluminite, A1

2 S04 (0H)4 8H

20

a = 11.444(2) A, b = 15.849(1) A, c = 7.4397(7) A, p = 106.85(l) c

V = 1291.5 A

3

. Ref. [11]

Halotrychite, FeAl

2 (S04

) 4

• 22H

20

a = 21.287(9) A, b = 24.291(5) A, c = 6.195(2) A, p = 100.27(2)°

V = 3151.8 A

3

.

Compound

II

III

N M

N

F NNposs

|A20|(°

20 17 28 75 0.0097

30 6 9 226 0.0149

36 5 7 349 0.0144

20 29 67 41 0.0073

30 27 72 68 0.0061

110 5 13 1023 0.0084

20 15 35 98 0.006

30 12 33 158 0.006

N = number of observed lines, N poss

= number of independent possible

N lines up to the Nth observed, M^ = De Wolff figure of merit [10], F F-index [12], | A20 | = average absolute discrepancy between observed and

calculated 29 values.

507
20 30 40 50 60 70 80 90 1~00 110

*>N

Figure 2. The figure of merits, F and M, as functions of the number of observed

lines for Aluminate.

References

[I] Abrahamsson, S. , J. Sci. Instrum. 43, 931-933 (1966).

[2] Malmros, G. and Werner, P. -E. , Acta Chem. Scand. 27, 493-502 (1973).

[3] Rietveld, H. M. , J. Appl. Cryst. 2, 65-71 (1969).

[4] Malmros, G. and Thomas, J. 0., J. Appl. Cryst. 10, 7-11 (1977).

[5] Werner, P. -E. , Salome, S. , Malmros, G. and Thomas, J. 0., J. Appl. Cryst. 12,

107-109 (1979).

[6] Berg, J. -E. and Werner, P. -E. , Z. Krist. 145, 310-320 (1977).

[7] Waltersson, K. , Werner, P. -E. , andWilhelmi, K. -A., Cryst. Struct. Comm. 6,

225-230 (1977).

[8] Waltersson, K. , Werner, P. -E. and Wilhelmi, K. -A., Cryst. Struct. Comm. 6,

231-235 (1977).

[9] Marinder, B. -0., Chemica Scripta, 11, 97-101 (1977).

[10] De Wolff, P. M. , J. Appl. Cryst. 1, 108-113 (1968).

[II] Farkas, L. and Werner, P. -E. , Z. Krist. (to be published).

[12] Smith, G. S. and Snyder, R. L. , J. Appl. Cryst. 12, 60-65 (1979).

508
] Smith, G. S., J. Appl. Cryst. 10, 252-255 (1977).

] Werner, P. -E. , Fortran IV program. Available on request (unpublished) (1975).

] Sas, W. H. and De Wolff, P. M. , Acta Cryst. 21, 826-827 (1966).

] Werner, P. -E. and Ekstrb'm, T. (to be published).

509

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

OPTIMAL DEGREE OF AUTOMATION IN QUANTITATIVE

X-RAY DIFFRACTION PHASE ANALYSIS

A. Griger

ALUTERV-FKI

R and D Centre of the Hungarian Aluminium

Corporation, Budapest

During the development of an automatized hardware and software

system for the quantitative x-ray diffraction phase analysis of

multicomponent powder mixtures [1] the question emerged whether total

automation should be aimed in all cases or not. For hydrothermal

borehole samples P. Johnson [2] gave a definitely negative answer to this question. Using profile fitting evaluation techniques W. Parrish

and T. C. Huang [3] assert the feasibility of straightforward complete

computerization.

In our opinion two basic tasks should be differentiated:

(1) analysis of powder mixtures of 2-3 phases with well defined real

structures, producing no or little overlapping of the peaks,

(2) analysis of powder mixtures of more than 3-4 phases exhibiting

structural variability and causing serious overlappings.

The consequences of structural variability are variable line

breadths, shifts of the reflections, changes in the relative intensities, uncertainties of the intensity conversion factors. Some

pitfalls of unattended computer interpretation of powder patterns in

the (2) group will be demonstrated and thereby the statement supported

that interactive control is inevitable at some stages of dataprocessing corresponding to these materials.

The possibility of entirely computerized evaluation seems to be

restricted to group (1), while optimal proportions between automatic

and analyst-checked operations, respectively, must be established for

the interpretations of the patterns relevant to group (2).

References:

[1] Bardossy, Gy. , et al., to be published in Amer. Min. (1979).

[2] Johnson, P. D. , Adv. X-ray Anal. 21 (1977).

[3] Parrish, W. and Huang, T. C. , Conf. Appl. Cryst. , Kozubnik

(1978).

511

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

STANDARDS FOR THE PUBLICATION OF POWDER PATTERNS: THE AMERICAN CRYSTALLOGRAPHIC ASSOCIATION SUBCOMMITTEE'S FINAL REPORT

I

L. D. Calvert, J. L. Fl ippen-Anderson, C. R. Hubbard, Q. C. Johnson,

P. G. Lenhert, M. C. Nichols, W. Parrish, D. K. Smith, G. S. Smith,

R. L. Snyder, and R. A. Young

The American Crystal lographic Association

A subcommittee of the American Crystal lographic Association (ACA) was formed to recommend some guidelines the use of which could improve

the overall quality of published powder diffraction data. An analysis

of the Powder Diffraction File shows that many of the well-known techniques for obtaining good quality data are often not being applied, and

hence there is a need for the establishment of publication standards. A major recommendation from this subcommittee is the use of a standard dataform at the author/referee level of publication. The proposed data-form

is divided into five sections: sample characterization, technique, unit

cell data, references, and the powder data. The information requested

in these five sections represents what, in most cases, the referee and

editor can reasonably expect of experimenters seeking to publish their

powder data. An example of a filled-out form is shown, and instructions

for completing the form are given. Another recommendation is that the

ACA request that the International Union of Crystallography consider

implementing these recommendations in all IUCr journals.

1. Introduction

i Numerous analytical methods can supply elemental chemical information. What distinshes the diffraction method is its unique capability to characterize crystalline phases,

oubtedly, the most widely used technique in this area is x-ray powder diffraction. In

s technique, diffraction data from an unknown sample are compared with a reference set

powder patterns compiled from the literature. Thus, reference patterns prepared by other

stal lographers play an integral part in a typical analysis.

Regardless of its quality, it is possible for any published pattern to become a refnce pattern and be used world-wide for identification purposes. If the quality is good,

513
the pattern is useful; if the quality is poor, the pattern can become a hindrance. In tj future, this problem will become accentuated because the increasing availability and use|

small computers will produce excellent-quality data for the user; then the quality of th

1

reference patterns will become even more a limiting factor in the analysis. The situatt

is not satisfactory now and is expected to become worse unless corrective action is taktj

Guidelines for improving the overall quality of published powder patterns are therefore

deemed necessary.

In August 1977, the American Crystal lographic Association (ACA), formed a subcommi1><

under the sponsorship of the Apparatus and Standards Committee and the Crystal lographic i1 Committee, to recommend standards for the publication of powder patterns. This subj committee prepared a first report which was presented at the University of Oklahoma ACA meeting, March 1978. That report was modified to incorporate suggestions received durir and after that meeting. A second presentation was made at the International Union of

j Crystallography (IUCr) meeting in Warsaw, Poland, August 1978. At about the same time,: 1

the modified report was mailed to ^150 editors of journals known to publish powder

patterns. Suggestions received from those two sources were incorporated into this thin,

and final report.

2. Inaccuracy of Powder-Diffraction Data

The problem is basically one of minimizing both errors in recording powder diffra

data and errors and noise introduced during the publication and archiving of the data, jjl might better understand these errors if we compare a powder pattern to an electromagnet]

signal. In communication, a pure signal is contaminated by noise at its source and is il further degraded or distorted in transmission to the point where the receiver may not rjf

ognize it. Likewise, an experimental powder pattern is contaminated by such things as , f

I instrumental errors and is further distorted in the processes of publication and archive

[if to the point where an analyst may not recognize it as being the same as the pattern frO|

his unknown.

. .

i

1

' :J If one works with average-quality standards having considerable noise, it is generj^ more difficult to identify an unknown, especially if the unknown is a mixture of severajl

phases. This might be comparable to listening to a low-fidelity recording (e.g., an ea; Caruso record) on a poor AM radio during an electrical storm and trying to pick out spej words. While the communications industry has spent a great amount of research and deve

p

ment effort to minimize error/noise/distortion in broadcasting, transmission, and recei n

it might be fair to say the powder diffraction community has concentrated primarily on training of skillful receivers (analysts).

The ease of identifying an unknown material from its powder pattern can be relatecj

the quality of the standards (fig. 1). While it is difficult to quantify figure 1, tha to assess just how significant the problem is today in powder identification, almost ey analyst would agree that accurate data for both standard and unknown greatly simplify a"'

s

514
if

I fbtifi cation. In addition, computer search techniques using the existing data base have

Pched the point where further advances probably depend on higher-quality standards. Indeed,

sh investigators prepare their own reference-pattern libraries to enhance the ease of

fljntifi cation [l] 1. This approach is, in general, an inefficient way for the powder diffrjction community as a whole to operate.

3

Ease of

identification

Quality of pattern

II Figure 1. Schematic dependency of the ease of identification of an unknown material

from its powder pattern vs quality of the pattern.

The improvement of powder diffraction analysis as a routine analytical capability has

|n slow. On the other hand, there have been dramatic changes in the speed, accuracy, and

dnomy of other characterization methods during the last two decades. As examples from

- irfhods employing x-rays, x-ray fluorescence has been profoundly changed by energyR

Ipersive systems and structure determination ...by single-crystal diffraction has been

i ally affected by automation and direct methods. We are now at the threshold of major

I nge in powder identification by x-rays; this change is being brought about by new techI ij ues and instrumentation. As the signal from an unknown powder is improved by powerful

n hods of data processing [2-5] and as comprehensive search/matching [1,6,7] by low-cost

I puters becomes practical in the diffraction laboratory, the quality of the reference

Interns will become the limiting factor. If the quality of both the unknown and the refi nee diffraction data can then be improved, the overall process can become so much more

s icient that in normal cases it should be possible to characterize the phase of an unknown

ciistalline material in less than 10 minutes with powder diffraction methods,

jj How can we minimize the noise introduced into reference standards during the process of

Wlication and archiving? The problem is not technical--the methods for averting noise and

ijtortion are well established and are practiced by a number of laboratories. Reports by

3|liss [8] and Kennard, et al . [9] have already discussed the requirements of the publiChon and archiving processes. Basically, authors, editors, referees, and data compilers

d an increased awareness and consistent use of already established technique.

"LI

1 gures in brackets indicate the literature references at the end of this paper.

515
3. Problems with the Powder Diffraction File

A recent study [10] of the Powder Diffraction File (PDF), as distributed on magnet

tape by JCPDS-International Centre for Diffraction Data2

, has revealed several points c interest to our discussion. The summary, which follows below, applies to the 20428

active inorganic patterns in sets 1 through 26 of the PDF and to those 1638 cubic and

^ triclinic patterns in sets 1 through 24, which could be indexed using the reported lat^ parameters 3

. The problems encountered with data on the PDF tape fall into two categories: thos

to the archiving procedures and those inherent in the original published data. The 1 es serious topic of archiving procedures will be discussed first. These problems are unde standable in terms of historical development of the PDF and the uses for which it was

i ntended.

3.1. Problems in archiving procedures

3.1.1. Incompleteness of the data on the magnetic tape

1'1

Each pattern on the JCPDS magnetic tape is coded with the d-I values (i.e., values

each line of the d-spacing and intensity), name, formula, quality mark, and crystal -sys im indicator. To date, the crystal -system indicator has only been coded on 16 percent of ie patterns. The absence of space-group, lattice-parameter, and the other information tha i contained on the published PDF cards severely limits the use of the tape as a research j0

especially as a tool for information retrieval.

3.1.2. Incompleteness of individual patterns

Figure 2 shows the distribution of the number of d-I pairs given for each pattern}

sets 1-26 of the PDF. The spikes near 20, 40, 60, and 80 pairs reflect the 20-line-per

column format of a PDF card image. These spikes indicate that many patterns have been

truncated to fit the published card format. These truncations probably have little effjt on phase identification, however, if the pattern is otherwise complete and is sufficieny

accurate at the point of truncation.

HI

3.1.3. Rounded d values on the magnetic tape

All d values on the computer data base have been rounded to three places after the decimal. Figure 3 shows that for high-angle reflections this can cause an error of ove

2 1601 Park Lane, Swarthmore, PA, 19081, USA.

3 This lattice-parameter data base was originally prepared by G. G. Johnson, Jr., of

Pennsylvania State University and was updated by M. C. Nichols and B. Pimentel of Sandi

Laboratories, Livermore.

516
20. Thus, the derived d values are not an adequate substitute for the experimentally

ured 20' s, and precision is lost.

1200 | 1

1000 - C

100

Number of d - I" pairs

I F igUre 2. Distribution of d-I pairs for sets 1-26 of the Powder Diffraction File.

0.78 1.03 1.27 1.52 1.76 2.01 d (A)

Figure 3. The error resulting from rounding off d values to four, three, and two

digits after the decimal.

517
3.1.4. Editorial artifacts

Figure 4 shows the average error in 26 up to a particular line number versus line

number for the indexable cubic patterns in the PDF. The fall-off and steadying out of ].{$ values above line number 40 indicate that only better-than-average data continue onto

a second PDF card.

0.100

1? 0.075

| 0.050 oi 0.025

0

5 15 25 35 45 55

d number

Figure 4. The effect of the 40- line PDF card format on the quality of data reporti

(PDF cubic patterns). Authors who measure more accurate data are more likely to continue their listing on to a second card.

ij

3.1.5. Nonquantitati ve intensity notations j!

In the current version of the PDF on magnetic tape, the intensity notations, I < lnc

I « 1, are transcribed as unity; in previous tapes these values were set to zero. The«)|

approximations were made because the symbols, < 1 and « 1 are difficult to handle in a computer format. These symbols, furthermore, are not quantitative. The present scale jii of 1 - 100 should be expanded so that these nonquantitati ve notations are no longer nec<;£ A recommended procedure (see below) is to report the intensities on a scale of 0 - 100. Because the computer tape was originally compiled for the computer generation of m(x

search-manuals, which only require d's to three significant figures, it is not surprisir

that the current tape is not well suited for computer-assisted research. The JCPDS is currently investigating the creation of a complete data base, which will eliminate most |ir the above difficulties.

3.2. Problems with published data ] 3.2.1. Rounding or truncation of intensities

Figure 5 shows the distribution of the intensities for the cubic PDF patterns. Thi shows that the measured intensities of most patterns were rounded or truncated to ten p£ :s

in 100. Fewer patterns have been rounded to one part in 20. In general, the only attends at accuracy are made for intensities between 0 and 20.

518
4000

CO§ 3000

COa

"S 2000

-QE

3 1000

20 40 60

Intensity

80 100

Figure 5. Distribution of reported intensities in the PDF cubic data shows significant rounding of intensities.

3.2.2. Inaccuracies of the intensities

Preferred orientation in the sample and conventional estimation of intensity by visual

niques are two main contributors to inaccuracies in published intensities. Diffraction

i'niques have developed to a level such that visual estimates of film intensities should

>;be published in most cases. Except in rare cases, the nonnumerical descriptors (very

J ng, medium, very weak) have also outgrown their usefulness. (Paradoxically, some patterns

>lg these descriptors contain very accurately measured line positions!) Preferred orition, perhaps a more serious problem, can be minimized by special techniques [11,12].

t, if not all, of these techniques require careful comminution of the powder.) Com5on of the reported PDF intensities with calculated values shows [11] orientation and/or

liicle statistics effects which, in the best cases, are on the order of 4 to 8 percent,

tijhe worst cases, these effects cause intensity discrepancies of several hundred percent.

i 3.2.3. Incompleteness of d's

| An analysis of the pattern-completeness factor (N

Q ^ s /Np0ss = number of lines observed/

wjer that possibly could be observed for a particular space group) shows clearly that many

irf|)rs only report the d spacings of the most intense reflections for a pattern. This fact

eues more noticeable as the total number of lines present increases with decreasing

||try.

3.2.4. Inaccuracies of d's

I' We computed the average A26 (26 . -26 , ) for each of the 1638 cubic patterns in

I I ODS C9 I

etjjl to 24, using the reported lattice parameters. We found the average value of |A26

519
for all these cubic patterns is 0.091°. For just the cubic patterns published by the i National Bureau of Standards, the average value of | A20 | is 0.015°. Thus, although th technology for producing accurate d values has existed for over 20 years, it is not iri routine use. A quantitative figure of merit, F^, for evaluating powder patterns has tjj been proposed [13] and has the form:

F N =

A2ei

'obs

1 poss

This figure of merit combines both the accuracy of the line measurements and the patter

completeness factor mentioned above. It allows a quantitative estimate of the metric jsi of a powder diffraction pattern. Figure 6 shows the distribution of F^ values for th&N

cubic patterns and the NBS cubic patterns. This function clearly allows a quantitativ

comparison of the quality of powder diffraction patterns. The use of a quantitative

assessment of powder patterns is not new [13,14,15]. We only stress the need for routu

use of such a measure. i

° 50

| 40

30

20

10

(b) 1

I III I tllll.VlJll-llll.llll.liVl.llllll.llli-l-l r ill

20 40 60 80

Average 20 merit value (F N

)

100

Figure 6. Quality of powder diffraction standards, (a) PDF cubic data and (b) Nl cubic data. Note that (a) includes (b), and that all patterns having F^ > 100 r< put in the F^ = 100 entry.

520
The problems shown above indicate a need for the establishment of uniform publication

Standards for powder diffraction data. In the sister field of single-crystal structure

;termi nations, there has been very marked progress over the past 20 years: experimental

id computational techniques (e.g., automated diffractometers, computerized refinements, and,

vmany cases, computerized determinations) which were readily adopted for widespread use,

le mandatory reporting of an acceptable figure of merit (the R factor), and documentation

I the experimental and calculational procedures so that other investigators could, if

;cessary, repeat the experiment and verify the results. Such documentation is required

I practically all leading journals publishing structure determinations and, as a result,

j; by now almost completely standardized. Today, the great bulk of published single7stal structure determinations have an R value less than 10 percent. We assert that the

jime sort of "field-wide" standards can and should be achieved in the area of powder

I f fraction.

4. Recommendations

j

J 1. We recommend the establishment of publication standards for powder diffraction data

irough the use of a standard data-form. We have designed such a form to be used at the

ithor-referee level to ensure that published data are of good quality.

An example of a filled-out form is shown in Appendix A. Suggestions for filling out

e data- form are given in Appendix B. The data-form itself is divided into five sections: Sample Characterization, TechIque, Unit Cell Data, References, and the Powder Data. Items on the form that are con-

, dered essential documentation are indicated by bold-face print. The other items are

ghly desirable information; however, it is recognized that some experimental procedures

11 not yield data on some of these items. Hence, omissions of data for these items should

II of themselves preclude publication of the powder data.

This list of items is not all-inclusive: an all-inclusive list might be longer than

le paper accompanying it. But this list does include the data most important to today's

Istrumentation. It represents, we believe, what the referee and editor can reasonably

Kect of experimenters seeking to publish their data. However, we do not intend that the

st be used to stifle either new techniques in powder diffraction or the publication of

tfder data obtained under difficult conditions, e.g., low and high temperatures, high

jessure, ultra-small quantities of materials.

i We have several specific recommendations pertaining to the reporting, dissemination,

|1 archiving of powder diffraction data:

| (a) The published powder pattern should be as complete as possible; for example, it

should include weak as well as strong diffraction lines. An element of doubt

often arises in phase identification when the observed strong diffraction lines

match those of a particular reference pattern, but some weak lines are observed

that are not listed in the reference pattern. Complete reference patterns would

be a benefit in these cases and generally would make identification easier.

Ill Furthermore, the ease of computer- indexing of powder patterns depends, in most
cases, on the accuracy and completeness of the data. Ordinarily, 40 lines

pj ready characterization of the material. In the absence of 40 lines, the dat(

should extend to (at least) 100° 29 (Cu Ka radiation).

(b) Researchers should report the experimentally observed 20 values, in degrees

corrected for systematic instrumental errors. The d values are usually not adequate substitute for the primary data, the 20 values. Reporting the 20's not distort information, but reporting the derived quantities, d's and Q's ( 2 H 1/d ), usually does. For example, even the d values reported by NBS (for th

example in Appendix A) do not match exactly their observed 20 values. We ha

add additional significant figures to the reported d values (in Appendix A)

accurately accomplish an otherwise simple transformation: 20's to d's. Hen we believe that the magnetic-tape version of the PDF should contain 20 value!5 which can be readily converted to d's, if necessary, by a user program. (An- using the magnetic-tape version, would have computer capability.) The PDF tj cards, on the other hand, should probably continue to list d values.

(c) Intensities should be reported numerically, and on a scale from 0 to 100. TP will avoid the nonquantitati ve notations I < 1 and I << 1 that sometimes occl with the present scale of 1-100. For example, 0.7 might be used instead of

This new scale will require no additional digits. The recommended scale doc not preclude the use of a scale from 1-10 or 1-100, when appropriate, nor stfjj it be understood to imply an increased accuracy for I < 1. (d) The reproducibility of the measured 20's and I's should be indicated. Thesetll should be obtained by multiple mountings of the sample material. PI (e) Indexing of the powder diffraction data should be required for all but the ij<ra and best-defended cases. Without indexing, there is little proof that the iW

pattern is for a single phase. Patterns can often be automatically indexed fyl appropriate computer codes [4,5,16]. Authors should report a figure of mer^l based on accuracy of the 20 measurements and completeness of their data. Afj i mentioned above, a figure of merit has been proposed for this purpose.

(f) Information concerning line breadth of the sample should be supplied. This

could be full-width at half-maximum (FWHM) of a resolved Ka^ line in the rej<

30-60° 20. (g) Additional information of value to future users should be supplied. Such di could be standard deviations, Chemical Abstracts Service Registry number, Cf|

Data index number, etc. We believe the use of a standard data-form has a number of benefits includii

• This data- form can be used as a checklist. Authors can use the form to ensi the necessary characterization of their data. Also, familiarity with the

standard form will encourage authors to record complete information on thei. measurements.

522
• The completeness of the experimental data will be easy to evaluate. Because

information pertaining to the experimentation will be collected in one place

in the manuscript, referees or journal editors will be able to readily judge

the future usefulness of the contribution. Moreover, an editor could send

the powder data to an expert in powder diffraction for refereeing and send the

rest of the manuscript to another reviewer.

• The data form could be recast into a camera-ready format for direct printing

in the journals. In this way, data-base managers could receive the powder data

exactly as sent by the authors 4,

j 2. This subcommittee has also given some thought as to what constitutes publishable

wder diffraction data. We believe that to justify being published, powder diffraction

ta must be potentially useful to someone else and must constitute in some way an original

ntribution to the literature. To constitute an original contribution, the data must be

|e first published for a well-characterized phase, must be a significant correction to or

I improvement on published data, or must relate to the phase in a previously uncharacterized

jindition, e.g., at elevated temperatures or pressure. A powder pattern calculated from

ngle-crystal structure data does not in itself meet the criterion of originality. Such

ita, we hope, will continue to be contributed directly to the data-base managers under

lidelines being established by a JCPDS committee.

3. To help gain acceptance for publication standards within the world-wide powder

ffraction community, we recommend that the ACA request the International Union of Crysallography (IUCr), through the Commission on Journals, to now consider implementation of

lese recommendations in all IUCr journals. As a first step in that process, the Commission

Journals may elect to refer these recommendations to the Commission on Crystal lographic

ita for review and approval. It is envisioned that a mutually acceptable version could be

icluded in Notes for Authors and in the Handbook for Co-Editors . If the IUCr journals

;iact publication standards in the near future, other journals will in time follow that lead

id the quality of published powder diffraction data will improve.

We wish to thank our many colleagues who have taken the time to provide helpful

jmments during the course of this subcommittee's activities.

References

[1] Frevel , L. K. , Adams, C. E. , and Ruhberg, L. R. , A fast search-match program for

powder diffraction analysis, J. Appl. Cryst. 9, 199-204 (1976).

JCPDS has recently offered to referee powder data and to receive powder data directly

f they are not to be published by the journal.

523
[2] Parrish, W. , Huang, T. C. , and Ayers, G. L. , Profile fitting: a powerful method of

computer x-ray instrumentation and analysis, Transactions of the American Crystallographic Association, 12, 55-73 (1976).

[3] Goehner, R. P. and Hatfield, W. T. , Automation of the Siemens D500 Powder Diffracto{

meter, Paper 1-1, Winter Meeting, American Crystal lographic Association, University

of Oklahoma, Norman, Oklahoma (1978).

[4] Visser, J. W. , A fully automatic program for finding the unit cell from powder data|

J. Appl. Cryst. 2, 89-95 (1969).

[5] Smith, G. S. and Kahara, E. , Automated computer indexing of powder patterns: the

monoclinic case, J. Appl. Cryst. 8, 681-683 (1975).

[6] Nichols, M. C. , Performance of the Search Program (1966-1978- ) Past-PresentFuture, Paper D6, Winter Meeting, American Crystal lographic Association, University!

Hawaii, Honolulu, Hawaii (1979).

[7] Johnson, G. G. , Jr., Resolution of powder patterns, in Laboratory Systems and Spectt

i scopy, J. S. Matson, H. B. Mark, Jr., and H. C. MacDonald, Jr., eds. , pp. 45-87, ch^

3 (Marcel Dekker, New York, Basel, 1977).

[8] Bayliss, P., X-ray diffraction powder data, Amer. Mineralogist, 61, 334-336 (1976).,

[9] Kennard, 0., Hanawalt, J. D. , Wilson, A. J. C. , deWolff, P. M. , Frank-Kamenetsky , V. Internation union of crystallography. Commission on crystal lographic data. Powder

data, J. Appl. Cryst. 4, 81-86 (1971).

[10] Snyder, R. L. , Johnson, Q. C. , Kahara, E. , Smith, G. S. , and Nichols, M. C. , An Anal ; of the Powder Diffraction File , Lawrence Livermore Laboratory, Rept. UCRL-52505 (197)

[11] Smith, S. T. , Snyder, R. L. , and Brownell, W. E. , The minimization of preferred

orientation in powder diffraction patterns by spray drying, Advances j_n X-ray Analyj!;

G. J. McCarthy, C. S. Barrett, J. F. Newkirk, and C. 0. Ruud, eds., Vol. 22 (Plenum,

New York, 1979).

[12] Smith, D. K. and Barrett, C. S. , Sample handling problems in x-ray diff ractometry,

Advances in X-ray Analysis , G. J. McCarthy, C. S. Barrett, J. F. Newkirk, and C. 0. 1

Ruud, eds., Vol. 22 (Plenum, New York, 1979).

[13] Smith, G. S. and Snyder, R. L. , F^: a criterion for rating powder diffraction patte>

and evaluating the reliability of powder pattern indexing, J. Appl. Cryst. 12, 60-65

(1979).

[14] Donnay, G. and Donnay, J. D. H. , The symmetry in the high- temperature al kal i-feldspa

series, Amer. J. Sci., Bowen Vol., 115-132 (1952).

[15] deWolff, P. M. , A simplified criterion for the reliability of a powder pattern

indexing, J. Appl. Cryst. 1, 108 (1968).

[16] Shirley, R. , Indexing powder diagrams, Crystal! ographic Computing , Proceedings

of 1978 Summer School (Delft University Press and Oostkoeks, 1978).

524
APPENDIX A: EXAMPLE OF COMPLETED DATA FORM

POWDER DIFFRACTION DATA FOR PHASE CHARACTERIZATION

(Data from Swanson, H. E., et al. (1971). NBS Monograph No. 25, Section 9, p. 25)

BOLDFACE ITEMS ARE CONSIDERED ESSENTIAL SAMPLE CHARACTERIZATION

NAME (CHEMICAL, MINERAL, Tri vial) Magnesium Aluminum Oxide (Spinel) EMPIRICAL FORMULA MgAj2Q4 CHEMICAL ANALYSIS NO _x YES, SOURCE/PREPARATION Synthetic; Fusion of binary oxides

CHEMICAL ABSTRACT REGISTRY NO. 1 2068-5 1-8 PEARSON PHASE DESIGNATION cF56 OTHER Index of Refraction =1.718 (Isotropic)

TECHNIQUE

RADIATION TYPE, SOURCE x-rays, Cu X VALUE USED 1.54056 A k a , X DISCRIM. (Filters Mono, Etc.) Diffracted beam, curved LiF mono. X DETECTOR (Film, Scint, Position Sensitive etc.) Geiger INSTRUMENT DESCRIPTION (Type, Slits, etc.) 17 cm vertical Diffractometer DIV _T REC 0.003" SOLLER Yes No. _] Position inc. Aperture g=1.2 INSTRUMENTAL PROFILE BREADTH 0.10 "20 TEMP (°C) 25 ± 1 SPECIMEN FORM/PARTICLE SIZE Edge loaded powder/< 10 (j.m particle size for I's, packed for 20's RANGE OF 20 FROM j

"20 to 165.0 °26 SPECIMEN MOTION None INTERNAL/EXTERNAL 20 STD (if any) Ag (internal) LATTICE PARAMETER OF 20 STD 4.08641 A

20 ERROR CORRECTION PROCEDURE Linear interpolation from nearest 2fl's of std. INTENSITY MEAS. TECHNIQUE strip chart record (peak heights) ERROR (~) _5%_ PEAK _x INTEGRATED

MINIMUM INTENSITY THRESHOLD (IN RELATIVE INTENSITY UNITS) _0J

INTENSITY STD USED a- ai 7 o, hkJl's OF INTENSITY STD _H3

INTENSITY RATIO l/l c 1.70 (5) CONVERSION FACTOR IF CORUNDUM NOT USED RESOLUTION (FWHM) FOR THIS MATERIAL: 0.10 °20 AT 59.37 "20 26 REPRODUCIBILITY FOR THIS MATERIAL: ±0.02 °20 AT All "20

UNIT CELL DATA

METHOD OF CELL DETN. Cell and structure known from Ref. 1 CELL REFINEMENT METHOD Least-squares. See Ref. 2

a = 8.0831 (1) & ; b = ( ) A ; c = ( ) A

<*= _°( °) ; p =

0

( °) , Y = __"( °)

Z

= JL ' D m=

' )gcm"3

, D x = 3.578 g cm 3

, V = 528.1 A

3

: Formula Wt = 142.25 CRYSTAL SYS. Cubic SPACE GROUP Fd3m [227] CRYSTAL DATA INDEX NO. 8.0831 FIGURE OF MERIT TYPE F N

. See Ref. 3 VALUE F

?fl = 58(0.015, 33)

REFERENCES

1. Bragg, W. H. (1915). Nature, 95, 561

2. Appleman, D. E., Evans, H. T. (1973). NTIS Document No. PB-21 61 88

3. Smith, G. S., and Snyder, R. L. (1979). J. Appl. Crystallography, 12, 60.

( ) INDICATES STANDARD DEVIATION IN LEAST SIGNIFICANT DIGIT(S)

525
POWDER DATA

ESSENTIAL DESIRED

26 EXP d EXP A20*

\u l u n l to/ l/l 0D hk!L

19.02 35 4.66 ill + .019 31.27 40 2.858 220 - .003 36.84 100 2.437 311 - .009 38.53 3 2.335 222 - .021 44.83 65 2.020 400 + .016

55.64 9 1.650 422 - .020 59.37 45 1.5554 511 + .008 65.24 55 1.4289 440 - .001 68.64 3 1.3662 531 + .006 74.13 3 1.2780 620 + .003

77.32 8 1.2330 533 - .029 78.40 1 1.2187 622 - .013 82.64 5 1.1666 444 + .006 85.76 2 1.1320 711 - .012 90.97 5 1.0802 642 - .009

94.10 12 1.0524 731 - .005 99.34 7 1.0104 800 - .006 107.90 2 0.9527 822 - .020

1 1 1 .22 8 0.93343 751 - .014

1 12.32 1 0.92738 662 - .035

1 16.91 6 0.90384 840 - .025 120.50 1 0.88722 911 + .004 121.69 0.9 0.88203 842 - .021 126.76 0.8 0.86161 664 + .013 130.74 8 0.84737 931 - .011

138.07 17 0.82488 844 + .033 142.97 0.4 0.81232 933 + .024 152.70 2 0.79266 10.2.0 - .033 160.65 1 1 0.78139 951 + .025

*20

EXp -20CALC

526
Appendix B. Information to Aid in the Completion of

the Powder Diffraction Data Form

. The underlined items below correspond to each of the items on the Powder Diffraction

lata-Form. Authors may not be able to complete every item, but are urged to be as complete

is possible.

AMPLE CHARACTERIZATION

Chemical Name . Names should be consistent with the conventions of the journal in which

he pattern is to be published. Such nomenclature is often in accord with IUPAC conventions

5.-3]. This name should include polymorphic phase identification (e.g., 0-manganese dioxide).

Mineral Name . The mineral name should be included for all minerals or synthetic comounds known to have mineral counterparts (e.g., lead sulfide, galena).

Trivial . Common names like Rochelle salt or methylene blue, which may be of value,

hould be included.

Empirical Formula . The empirical or stoichiometric formula should, if possible, connote

Structural information such as functional groups (e.g., (PO^^). Such stoichiometric

formulas can be readily converted to strictly empirical formulas by computers. The

toichiometric formula should, however, be consistent with the conventions of the journal in

hi ch the pattern is to be published.

Chemical Analysis . The results of a partial or full chemical analysis should be

ommunicated either on this form or in the journal article.

Source . List the source of the material. If it is a natural mineral, the location must

e given. If the material is commercially obtained, state the supplier.

Preparation . State method and pertinent conditions of preparation.

Other Data . Give any other data that will help to assure that the specimen is well

haracterized apart from the diffraction pattern. (A high-quality diffraction pattern of an mproperly identified material is of less than no use.) Such data might include: melting

oint or transformation temperatures, the fact that single-crystal studies were done; tructure type (e.g., NaCl type), color, indices of refraction, etc.

Chemical Abstracts Service Registry Number . This number is uniquely assigned to each

jompound by Chemical Abstracts. It is very useful in cross correlating between various

527
computer data bases. This number should be included it if is known to the author. For further information see reference [4].

Pearson Phase Designation . A method for classifying structures of metals and alloys

described by W. B. Pearson [5]. The "Pearson Symbol" is quite useful and, if known, shoulc

be included (e.g., orSe has the symbol mP32, Al^Zn has the symbol tI16). The three parts ^ the Pearson symbol are: the first, lower-case letter (a, m, o, t, h, c) designating crysU

system, the second, capital letter (P, C, F, I, R) designating Bravais lattice type, and a^ number indicating the number of atoms in the conventional unit cell: NOTE: This is not tf number of formula units.

TECHNIQUE

Radiation Type . X-rays, neutrons, electrons, etc.

Radiation Source . X-ray target material, neutron source, electron accelerating volta)

etc.

K Value Used . The numerical value of the wavelength used in calculating d values. L A(Ka^) if the a-^ component was fully resolved throughout the pattern or if analysis leads

20 's for the component. List A(Ka) if the Ka doublet was not resolved. If both A(Ka)

and A.(Ka,) were used, indicate the dividing line and list both wavelengths.

JA Discriminator ( Filters , Mono , etc. ). Give the method used to monochromatize the bei State whether the monochromator or filter was used in the incident or the diffracted beam.

\ Detector ( Film , Scint . , Position Sensitive , etc.). The type of radiation detector

used and any unusual electronic processing should be given, (e.g., pulse-height discrimination for a scintillation or proportional detector is conventional and need not be

stated.

)

Instrument Description ( Type , SI its , etc.). State the type and size of instrument us'

1

(e.g., 114.6-mm Debye-Scherrer camera, 17-cm diffractometer, 200-mm Guinier camera, etc.)

the conditions for the experiment. For diffractometers, state the divergence angle (Div) < the incident beam, receiving-slit width ( Rec ) , and whether or not different or continuous V

1

varying divergence slits were used for various 26 ranges.

Sol ler ( Number , Position , Apertures ). For diffractometers, give the number of sets o[ Soller slits, their positions (e.g., in incident or diffracted beam or in both) and aper- ture^), because axial divergence can be a source of large profile displacement or broaden

ing. Aperture is best characterized by q = RA/h, where R is the radius of the diffractometer; h is 1/2 the axial extension of the sample, and

528
spacing between Soller foils

length of Soller foils

Instrumental Profile Breadth . This parameter is the full width at half maximum (FWHM)

a reference sample that has minimal intrinsic broadening. It allows the instrument used

be compared with others and indicates the resolving ability. A recommended procedure is

o measure the 311 reflection (d,,, = 1.638 A) of a well-annealed Si specimen whose particle

lize is between 1 and 20 pm. The FWHM of only the Ka-, peak should be reported (for Cu Ka

Temperature . State the temperature of the sample when the pattern was obtained.

Specimen Form . Indicate if the sample is a loose powder, sintered compact, metalloJraphic mount, etc. State how the specimen was prepared for diffraction analysis (e.g., side

baded or vertically packed into a dif fractometer holder, dusted onto a substrate, packed

nto a capillary). If different preparations were used in the measurement of d values and

ntensities, indicate both methods.

Particle Size . Give the average or maximum particle size in the specimen (e.g., 10 pm,

^25 mesh, <20 pm).

Range of 26 . Indicate the range of 26 that was examined. All peaks above the stated

itensity threshold (given below) in the reported 26 range should be included under Powder

ata. Include a minimum of 40 peaks or all data up to 26 = 100°.

Specimen Motion . State type of sample motion during the diffraction experiment (e.g.,

ample spinner, rotated cylinder, Gandolfi motion, none).

Internal/External 26 Standard (if any). If the instrument was calibrated with a known

tandard before or after the pattern was obtained, state "external--(material of standard)."

f an internal standard was mixed with the sample to check for a specimen displacement and/or

9 zero error, state "internal --(material of standard)." A standard material recommended for

bth procedures is National Bureau of Standards SRM silicon [6].

Lattice Parameter of 26 Standard . State the numerical value of the lattice parameter

,|sed in calculating the expected 26 positions of the lines of the standard.

If another material or peak is chosen, it should be

resolved, report that fact along with the FWHM for

529
26 Error Correction Procedure . State the method used to correct the observed 28 val for systematic instrumental error. A recommended procedure is to use one or more external

standards such as Si (ref. [6]) and determine the A28 (26 - 26 , ) vs 26 curve. Thes 6Xp C3 I

data may be fitted : using a least squares polynominal regression (e.g., a third-degree

2 3 equation: A26 = a^ + a-^26 + a£28 + a^26 ). If this procedure is used, state the coef

ficient (a^'s) of the polynominal. A cubic standard having a large cell dimension (>10 A recommended for patterns having many lines at low angles.

Intensity Measuri ng Technique . Indicate the method used to determine diffracted intensities: peak heights or integrated areas from a strip chart, or densitometer or stepdata. Indicate if monochromator polarization effects have been removed from the I/Iq valjjji reported under Powder Data.

Minimum Intensity Threshold ( in Relative Intensity Units ). Indicate the relative inp

tensity (see I/I n below) not considered to be a real peak. Typically this threshold is t) u as the background reading plus 2.5 times VDac kground. NOTE: Because this value will dif

in regions of high and low background, limiting values should be reported.

Intensity Standard Used . Indicate the material used in determining the reference in tensity ratio. Intensity standards allow for the direct comparison of intensities in dif

ferent materials. Methods involving the use of corundum as an intensity standard for dir

quantitative phase analysis have been described [7-9].

hkl ' s of Intensity Standard . Give the hkl(s) of the line(s) used for determining th I/I standard value,

c

Intensity Ratio I/I ^. Give the value of the ratio of the strongest line in the patt; to the strongest line of corundum (d-^ = 2.085 A) in a 50-50 wt percent mixture of the t\ [7-9], and give the reproducibility, a(I/I ). c

Conversion Factor if Corundum not used . If a-A^O^ is not used as the intensity stat dard, then the conversion factor between the standard used and corundum should be listed.:

This conversion factor can be used to calculate the I/I

c for the specimen.

Resolution ( FWHM ) for this Material . The full width at half maximum, in degrees 26, oi

a well-resolved line (if possible) should be given along with the 26 of this line. Th peak should be chosen between 30° and 60° 26 to avoid wavelength broadening effects. Thi: value, of course, includes the instrumental profile breadth given above. If and an

not resolved, this fact should be stated.

530
29 Reproducibility for this Material . State any measured reproducibility of the

ijositions of the diffraction maxima; if measured for one line, state the 26.

NIT CELL DATA

Method of Cel 1 Determination . If the unit cell was obtained from the literature, give

he reference. If the unit cell was derived from the powder data, state the method or

irogram used.

Cel 1 Refinement Method . If the cell was refined, state the method and computer program

ised (if any).

Cel 1 Parameters a, b, c, a, and These values should be reported to the proper

iiumber of significant digits; put the standard deviations, in terms of the least significant

jligit, in parentheses (parameters in angstroms, angles in degrees).

Dm

: experimentally determined density in g/cm^

D : calculated density; from D = Z FW/N V, where Z is the number of formula units in

24 each unit cell, FW is formula weight, N = Avogadro's number (0.60225 x 10 ), 0

3 and V is the volume of the unit cell in cm

Space Group . If the space group has been determined by single-crystal studies, give the

lermann-Mauguin symbol followed in brackets by the space-group number as listed in reference

10]. For space group determinations that are not your own, a reference should be given

below. For unit cells determined solely from powder diffraction data, give the diffraction

aspect as defined in reference [11] (pp. S-8 to S-18).

Crystal Data Index Number(s) . List the axial ratios, derived from the cell parameters,

that would be used to locate this material in Crystal Data Determinative Tables [11].

Warning: these axial ratios are based on a unit cell defined by Crystal Data , which may not

be the cell for which the parameters are given above. Reference [10] describes how the

Crystal Data cell is obtained. These rules have been incorporated in the powder pattern

evaluation program AIDS [12].

Figure of Merit Type and Val ue . List the type of figure of merit used (e.g. , M

2

q or

F^|— see references [13] and [14]) and its value. The figure M

2Q

is defined as:

M

20 /i^Y i \

^2 |AQ|/\N poss /

531
2 where Q 2Q

is the Q value (= 1/d ) of the 20th observed line, |AQ| is the average absolute

discrepancy between Q Qbs

and Q ca-| c

> and Nposs

15 tne number of independent diffraction li possible up to the 20th observed line. The figure is defined as:

FN=

(|A2e|)(Np0Ss)

3

where |A20| is the average absolute discrepancy between observed and calculated 26 value

and NpQssis the number of independent diffraction lines possible up to the Nth observed

1 ine.

With regard to the figures of merit, some guidelines for the counting of possible

independent diffraction lines are in order:

Systematic absences caused by symmetry elements and lattice type are excluded

the tallying J a of N poss

Only one plane from the complete set of planes related by crystal symmetry is counted in Np osg

. For example, in the cubic system, the 100 line is counted ai independent line although it is composed of diffracted intensities from all si> planes of that crystal! ographic form.

Some forms, though not related by symmetry, have exactly the same spacing and v* give rise to the same line in the powder pattern (e.g. , 333 and 511 in the cubi system). Forms of this kind are also counted as one independent line. Note th this rule means that the higher- symmetry Laue group is always assumed. When a lower- symmetry Laue group is definitely known from single-crystal studies, thi si rule is not strictly correct for some of the diffraction lines. However, for nil of these cases, the effect on the value of the figure of merit for the overall

pattern should be insignificant.

For the case of accidental degeneracy (i.e., nonequivalent forms which have spa' ings so nearly identical that the individual lines would not be experimentally

resolved), all lines in such a cluster are counted as possible independent line^ the line having the smallest A28 is used in the calculation of |A29| for the

pattern, and the other line or lines are listed as not observed, thereby increa^

N poss-

\ The function is reported in the form _ FKI = Value (IA26I , N ). N I I poss

II For example,

532
F,n = 73.5(0.012, 34).

We recommend that N in F^ be the 30th observed line or the last line if there are fewer than

30 lines in the pattern.

Note that if the space group or the diffraction aspect is not determined, ^QSS

can or, ly

be reported as a maximum value; i.e., no allowance would have been made for systematic

absence, a fact which could appreciably reduce M^q or F^. [REFERENCES

Cite pertinent literature references for previous x-ray or preparative studies.

POWDER DATA LISTING —exp'

RePort tne experimentally observed 26 values, in degrees and corrected for

(Systematic instrumental errors. If multiple determinations of each peak were made, then the

njstandard deviation of each 26 value should follow in parentheses.

I/I q. List the relative intensities I/Ig of the diffraction lines on a numeric scale;

I/In = I/I x scale, where I = the numeric value chosen for the most intense refection.

(J max max

The scale value should be chosen such that the maximum intensity value is not greater than

100. Intensities less than 1 are reported as decimal fractions. If the standard deviation

Iiof each intensity was computed, place the value in parenthesis after each I/Iq value.

-exp* "" 1st ^e c" va ^ ues in angstroms, derived from the observed 26 values using Bragg 1 s

law: d = A/[2 sin (26/2)], where A is the value of the wavelength stated under Technique.

In reporting d's, the number of significant figures given should be sufficient to allow

recomputation of the experimental 26's to their measured accuracy, i.e., the number of

significant figures should be in accordance with a(d) = (d/2) cot 6a(26).

hkl . List the Miller indices of the diffraction lines determined from the known unit

cell. If several peaks are overlapped such that separate peak position measurements cannot

be made, the hkl 1 s should be grouped together and given for the single intensity value. The

hkl 1 s of all peaks can be listed, or a "+" can be post-scripted to the last one given to 'indicate others not listed.

II

A26 . List the difference, with sign, between the experimental and calculated 26 values:

A28 = 26 - 26 -, . A useful rule of thumb is that, if systematic errors have been removed

exp calc

completely, the absolute value of each A26 should be less than 3a(26). Alternatively, a small bar-graph can be presented showing A26 (with sign) versus line number. The reader, if

interested, could reconstruct more accurate values of A26 from the experimental 26's and the

I reported lattice parameters.

533
References, Appendix B

[1] International Union of Pure and Applied Chemistry, Inorganic Chemistry Section,

Definitive Rules for Nomenclature of Inorganic Chemistry 1957 (Butterworths , Londct, 1959).

[2] Pure and Applied Chem. 28, 1 (1971).

[3] Fletcher, J. H. , Dermer, 0. C. , and Fox, R. B. , Nomenclature of Organic Compounds ,? Principles , and Practice , Advances in Chemistry Series No. 126 (American Chemical

Society, Washington, DC, 1974).

[4] Chemical Abstracts Service Chemical Registry Structure Conventions (Chemical Abstn

Service Publishers, Columbus, Ohio, 1968).

[5] Pearson, W. B. , Handbook of Lattice Spacings and Structures of Metals and Alloys , Vtlji (Pergamon Press, Oxford, 1967).

[6] Hubbard, C. R. , Swanson, H. E. , and Mauer, F. A. , A silicon powder diffraction star^rji reference material, J. Appl . Cryst. 8, 45 (1975). '

[7] Visser, J. W. and de Wolff, P. M. , Absolute intensities, Report No. 641.109, Technkhi

Physische Dienst. , Delft, Netherlands (1964).

[8] Chung, F. H. , Quantitative Interpretation of x-ray diffraction patterns. III. Sim -

taneous determination of a set of reference intensities, J. Appl. Cryst. 8, 17 (197 .

[9] Hubbard, C. R. , Evans, E. H. , and Smith, D. K. , The reference intensity ratio, I/I

c

fi computer simulated powder patterns, J. Appl. Cryst. 9, 169-174 (1976).

[10] International Tables for X-ray Crystal lography , N. F. M. Henry and K. Lonsdale, eds. Vol. 1, 2nd edition (Kynoch Press, Birmingham, England, 1965).

[11] Donnay, J. D. H. and Ondik, H. M. (eds.), Crystal Data Determinative Tables , Third E Vols. 1-4 (National Bureau of Standards and Joint Committee on Powder Diffraction

Standards, Publishers, 1972, 1973, 1978).

[12] Mighell, A., et al., Single crystal and powder data evaluation and standardization 1

program, U.S. National Bureau of Standards, to be published.

[13] de Wolff, P. M. , A simplified criterion for the reliability of a powder pattern

indexing, J. Appl. Cryst. 1, 108 (1968).

[14] Smith, G. S. and Snyder, R. L. , F^: a criterion for rating powder diffraction pattei

and evaluating the reliability of powder pattern indexing, J. Appl. Cryst. 12, 60

(1979).

Discussion

Question (Zwell): Many analysts have asked for publication of all diffraction patterns.

Wouldn't your proposal decrease the number of patterns being published and thereby reduce

534
nformation available? (Would the improvement in quality of pattern compensate for the loss

if data?)

lomments (Calvert, et al.): It would be hard to predict the effect with any degree of conffidence. A similar set of publication recommendations made earlier by the single-crystal

:ommunity resulted in a marked improvement in the average quality of papers published in Acta

( ind many other journals. Some journals continue to accept lower quality data but the

Irestige of publication in the lead crystal lographic journals is such that authors generally

[trive to meet these standards. That result can't be bad.

I

i juestion (Zwell): It appears that it is implied that 26 would be for Cu radiation. Wouldn't

I the use of such angles reduce the value of data for those persons using other radiations

ilhromium, cobalt, molybdenum? (d's are constant and independent of radiation.)

omments (Calvert, et al.): Two theta would be recorded as measured. These data would not

e transformed to a different wavelength since that process could potentially introduce

oise.

535

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1079

(Issued February 1980)

SUGGESTIONS FOR A QUANTITATIVE EVALUATION OF POWDER PATTERNS

Gabrielle Donnay

Department of Geological Sciences

McGill University

Montreal, P.Q. Canada H3A 2A7

The use of the residual D Nr = | A26 | , the mean absolute deviation (D) between 26

obs

land 26 1 for all N resolved peaks (Nr), was proposed long ago [l] 1 for much the same La I C

I reasons that it is now proposed again [2]. The observed 26 values are the primary

I experimental data; they all have the same reading error, a constant for peaks of equal

quality. Conversion to any wavelength-independent quantity, be it Q

= L*. .-, 2 or d = 1/

9 o I nKI I k*hkl ' recl u

'' res a function of 6 that is far from linear, 4 sin 6 A or \/(2 sine),

respectively. The objection of journal editors to the inclusion of a A26 column must be

J force of habit, since the width of the table of powder data (table 1) would not have to

j exceed the half-page width currently used in most journals. Adding a ad column cor- responding to a given constant reading error in 26, for each observed, resolved peak

would take as much space and is not nearly as informative.

A RESIDUAL (ideally, close to zero) was chosen in 1952 rather than a FIGURE OF MERIT

[3,4,2] for two main reasons: (1) the residual represents a type of evaluation crystal

lographers are familiar with (cp. R in structure determinations); (2) the residual

J A26

|

is directly obtained from the measurements, whereas a figure of merit requires using its

reciprocal. We could not conceive of a number expressed in reciprocal units of angle as being more readily appreciated than one expressed in degrees.

The reason that only "resolved" reflections are included in the quality residual

(D^

r

), contrary to the proposals made in 1968 and 1969, is the following: the evaluation

under discussion concerns itself only with the cell geometry, not with the calculation of

1 peak intensities and peak shapes. Thus, peaks due to more than one possible reflection

1 with unknown relative intensities cannot be reported with any meaningful standard deviation and thus should be excluded from the least-squares refinement of cell dimensions and

from the evaluation procedure. (Peaks due to several reflections such as, for example,

the cubic ones 710, 550, 543, with distinct I's but identical 26 values are, of course, to be included in Nr.)

1 Figures in brackets indicate the literature references at the end of this paper.

537

MB
The resolved peaks with observed A20 values exceeding the value by an agreed-upd

factor, say 3, should be marked with a dagger in the data representation (table 1). If

such reflections have reasonable high intensities, so they can be read as well as the

properly interpreted ones, the pattern cannot be considered indexed in a reliable fashion An impurity phase would have to be called on to account for the uninterpretable peak or

peaks. Such patterns should certainly be eliminated from the JCPDS Powder Diffraction

File if they are now included and should not be incorporated in the future.

DNr and A20's > 3DNr permit us to judge the quality of experimental work that produced a given pattern. A third quantity, N o^/N^g. give qualitative information

on the type of crystal structure responsible for the observed pattern. ^p0SS represents

the number of symmetry- independent lattice points permitted by the space group, whereas

N obs 9 1ves tne sum °f re so l ved and unresolved observed peaks. Note that two overlapping

or nearly coincident peaks count for 2 in N but for 1 in N . . When N /N . is poss obs poss obs

close to unity, no substructure of heavy atoms and no pseudo-repeats are present. The

pattern can then be considered a reliable fingerprint of its phase. If, on the other

hand, the ratio is large (values up to 60 have been observed for mineral patterns), it is clear that the powder pattern by itself does not contain sufficient data to be considerec

a trustworthy means of identification. It is still worth recording, but it may not be

unique to the phase under study. Whenever the ratio is near unity, it is highly recom- mended that all possible r hkl and d , values be included in the tabulation. It means

calc

very little additional length of tabulation (paper, after all, is cheaper than highlytrained brain power! ) and saves a great deal of calculation time for later users: they j will almost certainly notice, on their pattern additional faint lines which may or may

not be accounted for by the interpretation in the literature.

Smith and Snyder [2], like De Wolff [4] before them, advocate the use of a single

evaluation number, F^^, but actually give this "Figure of Merit" always with two separate factors following it in parenthesis. The example "F„

n = 101 (0.009, 22),"

i quoted from Smith and Snyder [2], means that the first 20 observed reflections on the lov 26 side give a mean absolute A26 of 0.009° and that a total of 22 reflections are possibl

in this 26 range. The evaluation number 101 equals 20/(0.009x22). What good is it? Yoi' are not really given a single evaluation number when three independent values must be

specified for it to be meaningful. The only relation between the factors involved

requires ^pQS5

to be equal or greater than N Qbs (here 22 > 20).

The sample tabulation given here (table 1) shows how little space is needed to list

all three evaluation parameters: D Nr , daggers to mark the seriously mismatched or unmatched resolved peaks, Np

OSS

/N

o ^ s

. More importantly, every qualified user of powder 4 patterns will have no problem appreciating their significance. After careful consideration it will become "crystal clear" that it is actually advantageous to work with three !

1 independent evaluation parameters rather than only one uninterpretable number. Three

examples, the first two hypothetical and the third an actual case, will show how the use to which a powder pattern is put must determine whether one or another of the three

parameters is to be weighted most heavily.

538
Table 1. Powder Pattern of Barite, BaS0

4 (26<26. 6°)

a = 7.1565, b = 8.8811, c = 5.4541 A, Pbnm

X = CuKa(l. 54178 A); D 97 = 0.009°; no A29>3D„;

N /N . = 45/31 = 1.45

poss obs

k26(°)

1 I/I.

1 Q . obs

dcal ch k 1

0.02 2 5.58 5.572 1 1 0

0.00 16 4.440 4.441 0 2 0 0.00 30 4.339 4.338 1 0 1

0.01 50 3.899 3.898 1 1 1

0 12 3.773 3. 773 1 2 0

0. 01 30 3. 577 3. 578 2 0 0

0.01 100 3.445 3.444 0 2 1

0 70 3.319 3.319 2 1 0

0 95 3.103 3.103 1 2 1 0.01 50 2.836 2.835 2 1 1

- - - 2.786 2 2 0 0.01 15 2.735 2.736 1 3 0 0.02 45 2.729 2.727 0 0 2

0. 02 13 2.482 2.481 2 2 1

2.447 j 2.450 1 1 2

2

j 2.445 1 3 1

0.02 14 2.325 2.324 0 2 2 0.02 6 2.305 2.304 3 1 0

U. uc

oo

9 9Q9 9 9P1 L , Zol 9

3 nu 2.220 0 4 0

0.02 25 2.211 2.210 1 2 2

- - - 2.186 3 0 1

0 3 2.169 2.169 2 0 2

80 2.121

(9 199 0j1J. 1

j. 19 1 91

j C. . L<LL 1

/I 1

nu

/o

(2.107 2 1 2 )2.104 2 3 1

- 2.102 3 2 0 0.02 19 2.057 2.056 0 4 1

1.9764 1 4 1

1.9609 3 2 1

(J. Ul 1 1. r^OO -\1. QAQQ 3483 <L 0

L. 0. 01 -71

T QQ1 "7 1. yoio 1

0

0 9

X. OODD 9 u

U lo 1. OJ/D 1. OD/D 0 0 u 0.01 4 1. 7889 1.7891 4 0 0

1 . 7829 2 4 1 0.01 8 1.7617 1.7621 1 0 3

10 1.7584 U.7599 3 1 2

j 1.7583 3 3 1

0.00 8 1.7540 i:7539 4 1 0 1.7496 2 3 2

0 4 1.7284 1.7284 1 1 3

0 5 1.7239 1.7239 1 5 0

1.722 0 4 2

539
(1) A monoclinic powder pattern of (K.Na^lSi^Og in the literature has a relatively

high D^

r value, say 0.04° for Nr = 20; none of the resolved peaks, however, exceeds 0.12c

in A20, Np0SS

/N0DSeo. ua l s 1-23. Thus, the indexed pattern looks trustworthy on the whol

The user has obtained a similar pattern from his unknown sample; he wishes to determine

precise K to Na ratio. Uncorrected experimental errors in the literature pattern are thi most likely cause of a high and the user will therefore want to include a cal ibratiorj

standard such as quartz or calcite with his sample. The calibration-corrected measuremer indexed on the basis of the literature pattern will give the user the preliminary cell

dimensions needed for the least-squares refinement of his own data. His evaluation paran eters turn out to be: D ori = 0.005°, no peak with A26 > 0.015° and N N . = 35/29 = 1. 20 ' K - poss obs I The user can trust his cell dimensions sufficiently to make use of the plots of cell

dimensions vs K/Na ratio for alkali feldspars in the literature.

(2) An orthorhombic pattern in the Powder Data File has D^q = 0.003° and shows one

j

resolved peak v with dagger Ma for which A20 = 0.13° (I = 30); N /N , = 1.30. The user's

' poss obs

pattern shows all the peaks with I > 20 of the literature pattern, except the marked peak

There is thus little doubt that an impurity phase is to be blamed for the problematic pea in the literature pattern. The user can trust the identity of his unknown with the phase

described in the literature.

(3) A new rhombohedral mineral, MngSigO-^COhOgC^, Z = 23, gives a powder pattern

extremely close to that of friedelite, a mineral with similar composition but less chlori

per formula unit. The appearance of the two minerals is different, however, and single

o crystal work shows the c cell dimension of the new mineral to be 85.9 A, four times that

o S

friedelite. The a axes, 13.46 A, are the same within 0.3 percent; the space group is R3m or R3m for both minerals. The evaluation parameters for the Debye-Scherrer pattern taken

with FeKa radiation are: D on = 0.032°, A26t = 0.21 at 20 = 73.4°, N /N . = 1535/26 => 20 ' ' poss obs

59.04. Except for three close-in very weak reflections, all observed, resolved peaks

have indices of the type 4h 4k • 41. In the crystal structure, the Mn atoms are arranged

in basal layers of the brucite type; they are situated near the nodes of a sublattice wit

cell dimensions a/4, c/12. To a first approximation the Mn atoms control the powder pat

tern. Since the Mn sites are the same in the new mineral as in friedelite, powder patter

cannot tell the two minerals apart. r The high3 value of N /N . tells us that the powde

poss obs r pattern cannot be used as a determinative criterion.

The above examples cannot illustrate all the various problems that the user of powde

patterns may encounter. They should, however, suffice to show that the revived proposals

deserve serious and objective consideration. These evaluation criteria have been used

routinely in our work with mineral, inorganic and organic powder patterns for over thirty

years. They have proved highly satisfactory.

References

[1] Donnay, Gabriell and Donnay, J. D. H. , The symmetry change in the high- temperature

alkali-feldspar series, Amer. J. Sci., Bowen vol., 115-132 (1952).

540
[2] Smith, Gordon S. and Snyder, Robert L. , F^: a criterion for rating powder diffraction patterns and evaluating the reliability of powder-pattern indexing, J. Appl

Cryst. 12, 60-65 (1979).

[3] Wolff, P. M. de, Reliability of unit cells derived from powder diffraction patterns,

Acta Cryst. 14, 579-582 (1961).

[4] Wolff, P. M. de, A simplified criterion for the reliability of a powder pattern indexing, J. Appl. Cryst. 1, 108-113 (1968).

541

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

X-RAY POWDER DIFFRACTION

L. D. Calvert

National Research Council of Canada

Ottawa, Ontario, Canada, K1A 0R9

Recent years have seen rapid developments in the field of powder diffraction arising on

one hand from the widespread availability of small computers and on the other hand from

l| devices such as solid state detectors (SSD's), position sensitive detectors (PSD's),

it^nse sources of x-radiation (synchrotrons, storage rings, flash-tubes, possible laser

ijpces). These also, of course, make great use of on-line computers. An era of development

parable to that following the introduction of the counter-dif fractometer [l] 1 seems

si )able.

The small powerful computer, cheap enough to be dedicated to a given experiment has

Ken rise to the automation of existing standard techniques both commercially, as summarized

lently at the Oklahoma meeting of the American Crystal lographic Association [2] and

Hl/ately [see, for example 3, 4, 5, 6 and references therein]. A key feature is the

)i nation in a single system of the routine steps in data acquisition, data reduction and

e identification. This latter process, first programmed for big computers, is now

uded in routines operating on small computers [7,8,9]. The combination of x-ray

rescence and x-ray diffraction in a single automated unit may offer advantages in some umstances [10,11] and may be developed further. The general process can be described as ncrease in the portability of sophisticated techniques which can be exported from

itialist centres as a hardware plus software combination. Along these lines we may

I irjcipate developments of combined techniques previously precluded by the excessive amount

iljabour required. Thus, the combination of reflection and transmission experiments on the

Hi sample [12,13] may become routine for monitoring the often-present and vexing problem

n referred orientation in powder specimens or to obtain directly, parameters now inferred.

Itjitively little has been implemented so far in the area of feed-back between the experiment

the observed data. With the advent of fully- integrated software systems it will be

ible to monitor the data as it is acquired and vary or repeat the data-collection as

ired. Similarily automatic search-and-match routines [14] tied to automatic peak-finding

profile fitting routines [15] allow the possibility of using powder diffraction methods

l F)ures in brackets indicate the literature references at the end of this paper.

543
for on-line process control in say, fibre-making or sheet-rolling. For relatively large 1 ; specimens we have the possibility of using micro-dif fractometers [16] for grain-by-grain

phase analysis or the possibility of automated grain-by-grain orientation analysis [17, 1? with grain and particle size estimation [19] and stress analysis [20] to provide various L types of process control or monitoring.

With respect to SSD's [21,22] PSD's [23,24] and intense radiation sources [25,26] t^ are wide-spread opportunities for applications especially in time-limited experiments, f'l of these have been dealt with in this symposium so that it is unnecessary to repeat detaj

here. The use of flash x-ray tubes to investigate shock-wave phenomena [27] is one area

laser systems are another area [28] in which future development can be expected.

Further developments in the Rietveld method [29,30,31] can be expected as can be !1 judged from the emphasis on this topic both in the present symposium and elsewhere [32];

again it is unnecessary to repeat details already given.

Attempts to predict the future are, of course, subject to great uncertainly and the

present attempt is limited to selected topics and references and also to a conservative

extrapolation from present experience.

References

[I] Friedman, H. , Geiger counter spectrometer for industrial research, Electronics, J_8

,

132-137 (1945). |

[2] Session I. Hubbard, C. R. , Chairman, Winter meeting, American Crystal lographic Ass March 19-24, 1978. Program and Abstracts, Vol. 6, No. 1, p. 21. [3] Richesson, M. , Morrison, L. , Cohen, J. B. , and Paavola, K. , An inexpensive computer

control for an x-ray diffraction laboratory, J. Appl. Cryst. 4, 524-527 (1971).

[4] Slaughter, M. , A. Modular automatic x-ray analysis system, Advances in X-Ray Analys

135-147 (1972).

[5] Segmuller, A., Automated x-ray diffraction laboratory system, Advances in X-ray

Analysis, 15, 114-122 (1972).

[6] King, P. J. and Smith, W. L. , A computer-controlled x-ray powder diffractometer,

J. Appl. Cryst. 7, 603-608 (1974).

[7] Jenkins, R. , Hahm, Y. , Pearlman, S. and Schreinar, W. N. , The APD, a new dimension

( qualitative and quantitative x-ray powder diff ractometry, Norelco Reporter 26, 1-15 (1979) (This version includes the entire PDF on one disc.)

[8] Johnson, G. G. , A search-match pogram for the PDP-11 (Private communication).

[9] Edmonds, J. W. and Henslee, W. W. , A rapid search-match technique on a small compute

J. Appl. Cryst. (1979) to be published. j|J

[10] Jenkins, R. , The role of the energy dispersive detector in x-ray di ffractometry, Tr i:

Amer. Cryst. Ass. 12, 43-54 (1976).

[II] Mantler, M. and Parrish, W. , Advances in X-ray Analysis, 20, 171-186 (1976).

544
|2] Ladell, J. and Nicolosi, J., Asymmetric texture sensitive powder dif fractometer, Amer.

I Cryst. Assn., Winter Meeting, March 20-30, 1979, Program and abstracts, Vol. 6,

I No. 2, p. 54.

;3] Huang, T. C. and Parrish, W. , The transmission specimen powder diffractometry and the

profile-fitting method, Amer. Cryst. Assn. , Winter Meeting, March 20-30, 1979, Program

and abstracts, Vol. 6, No. 2, p. 55.

I] Frevel , L. K. , Adams, C. E. , and Ruhberg, L. R. , A fast search-match program for

, powder diffraction analysis, J. Appl. Cryst. 9, 199-204 (1976).

,p] Ayers, G. L. , Huang, T. C. , and Parrish, W. , High-speed x-ray analysis, J. Appl.

Cryst. TJ, 229-233 (1978).

j>] Goldsmith, C. C. and Walker, G. A., X-ray microdi f f ractometry: new applications in

phase and stress analysis, Winter meeting Amer. Cryst. Assn., March 19-24, 1978,

I Program and abstracts, Vol. 6, No. 1, p. 31.

if] Riquet, J. P. and Bonnet, R. , Depoui 1 lement par Ordinateur des Cliche s de diffraction

I obtenus par le Methode de Laue, J. Appl. Cryst. ]2, 39-42 (1979).

'!] Bilderback, D. H. , A real-time back reflection Laue camera, J. Appl. Cryst. 1_2, 95-99 (1979).

j] Hilliard, J., Statistical analysis of the measurement of grain and particle size by

x-rays, Symposium on Accuracy in Powder Diffraction, National Bureau of Standards,

I Washington, D.C., June 11-15, 1979.

i] James, M. R. and Cohen, J. B. , Study of the precision of x-ray stress analysis,

a Advances in X-Ray Analysis 20, 291-307 (1976).

] Heath, R. L. , The application of high-resolution solid state detectors to x-ray

j spectrometry: a review, Advances in X-Ray Analysis, T5, 1-35 (1972).

[' ] Gobel , H. , Fast and time-resolved XRPD using linear position sensitive detectors,

Winter meeting, Amer. Cryst. Assn., March 20-30, 1979, Program and abstracts, Vol. 6,

I No. 2, p. 54.

[', ] Hendricks, R. W. , One-and-two-dimensional position-sensitive x-ray and neutron-detectors

I Trans. Amer. Cryst. Assn. 12, 103-146 (1976).

[] Arndt, U. W. and Gilmore, D. J., X-ray television area detectors for macromolecular

structural studies with synchrotron radiation sources, J. Appl. Cryst. ]1, 1-9 (1979).

[jp Hodgson, K. 0., Phillips, J. C. , and Wlodawer, A., Applications of synchrotron radiation

II to x-ray diffraction studies of proteins, Trans. Amer. Cryst. Assn. 1_2, 1-10 (1976).

Hi] Hastings, J. B. , Synchrotron radiation facilities: opportunities for x-ray research

in the United States, Winter meeting: Amer. Cryst. Assn., March 20-30, 1979, Program

J and abstracts Vol. 6, No. 2, p. 56.

[}p Johnson, Q. C. and Mitchell, A. C. , Flash x-ray diffraction study of Li F to 11 mega

|| bars, Winter meeting, Amer. Cryst. Assn., March 26-30, 1979, Program and abstracts

|! Vol. 6, No. 2, p. 75.

!

|

ll

i

I 545
[28] Forsyth, J. M. , High intensity soft x-ray pulses from laser produced plasmas, Tranj Amer. Cryst. Assn. 12, 11-26 (1976).

[29] Rietveld, H. M. , Line profiles of neutron powder peaks for structure refinement,

Acta. Cryst. 22, 151-152 (1967).

[30] Young, R. A., Mackie, P. E. , and von Dreele, R. B. , Application of the patternfitting structure-refinement method to x-ray powder diffractometer patterns,

J. Appl. Cryst. 10, 262-269 (1977).

[31] Werner, P. E. , Salome, S. , Malmros, G. , and Thomas, J. 0., Quantitative analysis o1 multi component powders by the full-profile refinement of Guinier-Hagg film data,

J. Appl. Cryst. 12, 107-109 (1979).

[32] Conference on Diffraction Profile Analysis, Cracow, Poland, August 14-15, 1978. (/ meeting it was agreed that the method be called the Rietveld method.)

t

1

f

i :ll
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

THE FUTURE OF POWDER NEUTRON DIFFRACTION

M. H. Mueller

Materials Science Division

Argonne National Laboratory

Argonne, IL 60439

The objective of this Symposium was to assess the current position of powder difraction--both x-ray and neutron. A good international scientific representation was

i resent with sixteen countries represented with a total attendance of 125. The follow-

(ijng comments are primarily confined to neutron diffraction. During this week a number

if aspects of neutron powder patterns were addressed such as: the various scattering

i

j jontri buti ons to the total pattern, the problem of profile fitting, use of special or

lultiple detectors and distortion of peak shape.

Although Rietveld [1.2] 1

, proposed a profile method for refining powder

eutron patterns in two articles, it took several years before the method was extensively used. However, by 1977, Cheetham and Taylor [3], pointed out in a review

|iirticle that more than 175 compounds (listing included in the article) had been

reated by the profile refinement technique.

As a result of our discussions this week, indeed a number of areas of developing

esearch which offer improved or new vistas to the field have been pointed out in neuron scattering. Some of these areas are: higher resolution, especially with pulsed

eutron instruments; use of multiple detectors; refined methods of fitting profiles; a etter assessment of errors; availability of generalized computer programs; use of

Standards such as A1

203 for interlaboratory comparisons; development of sample environments; a consideration and understanding of the total pattern, use of deconvol ution

ethods and application of Fourier maps to confirm the structure and refine some

features, including missing atom locations; a better experimental treatment of backgrounds and an understanding of its components and the combination of x-ray and neutron

lowder patterns for some applications.

The consensus of opinion expressed during the Conference is that the higher resolution: will be an advantage for Fourier applications; should make possible solutions of

Structures involving 200 variables; will make possible an examination of scattering in

he vicinity of the Bragg peaks. The high resolution powder di f f ractometer (HRPD)

I ~ -4

Resigned for the SNS project will provide resolutions of Ad/d = 3 x 10 . Multiple

letectors are indeed an advantage and a must for both steady state and pulsed neutron

tources. The problem of profile fitting to the diffraction peaks has been given

Figures in brackets indicate the literature references at the end of this paper.

547
considerable attention the past several years so that generalized computer programs are becoming available for both the steady state and pulsed neutron techniques. Although

there is some disagreement on assessment of errors associated with profile refinements,

the use of the program SCRAP by M. Cooper may assist in the evaluation.

It is quite an accepted fact that environmental changes can be made with the

required neutron scattering instrumentation so that temperatures from 0.1 to 2500 K are possible and pressure up to 40 K bars are within range. For example, a recent

study was carried out at ILL by Aldebert and Traverse [4], on high temperature struc- tures of La20^ and Nd,,0 3 at temperatures up to 2225 °C. It is apparent that the use of standards for comparison between various installations and instruments is very important. You may recall that the IUCr had a project

in progress during the early 70 's to collect powder pattern results from a standard

A^Og sample which was furnished to the several laboratories. It was not intended

that the investigation be a competition between the different laboratories but rather

a means for everyone... "to decide whether in their particular case optimum conditions

have been achieved; or whether they should spend more time trying to improve the situation," Andresen and Sabine [5]. The results of least squares treatment of the Al^O^

data was presented at the Cracow, Poland IUCr satellite meeting last summer and will

be published in J. Appl. Cryst. in the near future. It may also be desirable to use other materials as standards of comparison which may have a more complex structure or j may be a material such as MgO where it is possible to calculate and account for the

total scattering.

Development and understanding of high resolution neutron powder patterns not only

offer possibilities in the solution of certain atom site locations in the structure,

but also may be used to better assign physical characteristics to many of the materials.1

For example, it should be possible to detect and characterize: stacking faults; partij cle size broadening; lattice strains, including anisotropic characteristics; short rang^

order; partial site occupation, displacement disorder; thermal diffuse scattering; extirj

tion effects; incoherent scattering; diffuse scattering; and paramagnetic scattering.

The general conclusion of the Conference was that: (1) considerable progress has

been made in the past five years in the application of neutron diffraction powder pat- terns (2) future plans look promising and exciting; (3) contributions can be made to structure analysis to a characterization of many materials where single crystals are not available; and that future meetings should be held covering the topical area of

neutron diffraction powder patterns. It has been tentatively proposed and accepted

to have a satellite topical meeting at Argonne National Laboratory on neutron scattering with emphasis on powder techniques in conjunction with the next IUCr Congress and

Assembly to be held in Ottawa, Canada, August 1981. It was evident in the meeting as recently stated by Dr. Pehka Suortti , "powder diffraction as a method of structure

analysis has been rediscovered in the last few years... ." This has certainly been pos-' sible for the neutron case because of increased flux, improved resolution and introduction and refinement of Rietveld profile analysis methods and techniques as well as the use of x-ray and neutron results in combination.

548

1
We are indeed indebted to the organizers of this Symposium for the stimulating

Irogram and the fine arrangements which included a tour of the NBS Reactor. We look

orward to an early appearance of the Symposium Proceedings.

References

1] Rietveld, H. M. , Line profiles of neutron-powder diffraction peaks for structure

refinement, Acta Cryst. 22, 151-152 (1967).

Rietveld, H. M. , A profile refinement method for nuclear and magnetic structures,

J. Appl. Cryst. 2, 65-71 (1969).

Cheetham, A. K. and Taylor, J. C. , Profile analysis of powder neutron diffraction

data: its scope, limitations, and applications, in solid state chemistry, J. Solid

State Chem. 21, 253-275 (1977).

4] Aldebert, P. and Traverse, J. P. , Etude par diffraction neutronique des structures

de haue temperature de La^O^ et ^£0^ (Study by neutron diffraction of the structures of La

203 and Nd203 at high temperature), Mat. Res. Bull. 14, 303-323 (1979).

5] Andresen, A. F. and Sabine, T. M. , An intercomparison of neutron powder diffraction

instruments, J. Appl. Cryst. 10, 497-501 (1977).

549

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

JCPDS - INTERNATIONAL CENTRE FOR DIFFRACTION DATA: PRESENT AND FUTURE ACTIVITIES

G. J. McCarthy 1 and D. K. Smith2 Materials Research Laboratory

The Pennsylvania State University

University Park, PA 16802

Highlights of recent and current activities of the JCPDS-ICDD

subcommittees, grants-in-aid and NBS Associateship are pesented.

Future areas of emphasis will include new subfiles and computer databases, interaction in the Chemical Information System, topical reviews

and generation of new powder data and assessment of the potential

merging of the single crystal and powder databases.

1. Introduction

While some diff ractionists may not be familiar with the name "JCPDS -International

itre for Diffraction Data" (JCPDS - ICDD), they will recognize its main publication, the

ider Diffraction Fi le (PDF) . For almost four decades, the JCPDS - ICDD has collected (from

p literature, from contributions, or from grant research), edited and published powder data

card, magnetic tape and book form, along with search manuals and computer search/match proims for accessing this data. We present here an overview of recent past, present, and

;ure activities of the JCPDS - ICDD.

I 2. Subcommittee Activities

I Like many scientific organizations, the JCPDS - ICDD operates both as a professional

.'h'ety having volunteer members drawn from industry, universities, and government labora-

[Hes and as a publishing house. Both functions have expanded several-fold in the last

(fade. Technical activities are carried out primarily by subcommittees of the Technical

tjimittee, with support from the headquarters staff as required. Some of the highlights of

Use activities can be cited:

'iiairman, Technical Committee, JCPDS - ICDD; address after September 1, 1979: Departments

(|Chemistry and Geology, North Dakota State University, Fargo, ND 58102.

^airman, JCPDS - ICDD; also a member of the Department of Geosciences.

551
2.1. Mineral subcommittee

Founded in 1971. Members of this group reviewed and upgraded all existing powder d for minerals and prepared the Mi neral Subf i le consisting of a data book and search manua

The group has just completed work on an update to be published in 1980. A very high per age of all known minerals will be represented in this update.

2.2. Computer subcommittee

Since 1974, this group has been largely responsible for improvements in the format,

documentation and quality control of the magnetic tape database. The complexity and cos establishing the database and search/match programs at users' sites has been greatly red

by tailoring these for three of the major computer types. The computer subcommittee has instrumental in keeping the JCPDS - ICDD current in computer hardware and software advan

2.3. Education subcommittee

The obvious need for a JCPDS - ICDD role in primary and continuing education in

diffraction analysis and identification led to the formation of this group in 1972. Sin

then approximately a dozen symposia on these topics have been conducted at conferences.

Workshops on the use of the PDF have been developed and total attendance at these is approaching 1000. Some 20 members of the JCPDS - ICDD have volunteered their time as i structors in these workshops. The workbook designed for the PDF Workshop is available

also for other groups teaching powder diffraction methods.

A joint undertaking of the education and computer subcommittees was the recent set "round robins" aimed at assessing the effectiveness and relative merits of the manual

and computer search/routines and the importance of input data quality on successful ideti fication. Current joint efforts include development of a computer search/match workshop

an American Chemical Society Audio Course.

2.4. Search procedures subcommittee

One of the oldest of the subcommittees, this group has concentrated on the concepts'

S pattern searching, especially by manual methods, in a continuing evaluation of present a

j alternative versions of JCPDS - ICDD Search Manuals. The subcommittee has had a long ti concern with the effect of a growing PDF (^35,000 patterns; 2000 added per year) on the!

and effectiveness of these manuals. Realizing that certain crystalline phases occur as answers in a high percentage of searches, the group has sought to define a set of "Freqij

Encountered Phases" (FEP). A FEP Search Manual incorporating Hanawalt, Fink, and Alphat

cal sections has been available for several years. Approximately 2300 phases are incluci! the FEP list. Continuing definition of the list and upgrading of the powder data of phc*

the list are performed by the JCPDS - ICDD Associateship at the National Bureau of StanC

552

in

it »

'50
2.5. Metals and alloys subcommittee

Recently, members of this group performed an extensive review and update of all

nil able data on metals and alloys. This effort resulted in publication of the Metals and

loys Subfile in 1978. Included in the subfile were a number of calculated patterns of

ifises for which powder data were not available. Work is underway on a supplement to be

If |)1 i shed in the early 1980' s. As the PDF continues to grow, the importance of high quality data for well-characterized

jises and of smaller sets of data--subfiles--will increase. Other subcommittees (organic,

"ensic, ceramic, etc.) are at work defining subfiles, reviewing the quality of data that

1 within these subfiles and taking steps to calculate, solicit or otherwise obtain missing

;a. These groups will be generating the new products of the 1980' s.

3. Grants-in-Aid and the JCPDS - ICDD Associateship at the NBS

! While the major historical source of data for the PDF has been the scientific 1 iterate, the JCPDS - ICDD has long recognized that these data appear in rather random fashion

I are of quite variable quality. Thus, for three decades, it has allocated a portion of

t budget to the generation of high quality data for well -characterized materials.

The first effort of this type was an Associateship at the NBS that still operates today

jthe Crystallography Section. Since 1951, nearly 1200 experimental patterns of high purity

;erials, along with hundreds of calculated patterns, have been generated by the Associatep. A collection of these data through 1976 has been published in book form and a book of

lawalt, Fink, and Alphabetical search manuals has been prepared. Most of these data are of

[Cimon laboratory chemicals, minerals and other frequently encountered phases. These books

istitute the JCPDS - ICDD educational package for universities and industrial continuing

ication programs.

In a typical year, the JCPDS - ICDD awards 15 to 18 grants-in-aid to universities and

fi-profit institutes for determination of powder data or for other powder diffraction

ivities. Current grantees are located in Japan, the United Kingdom, France, The Netherids, Germany, and Israel as well as the U.S. Grants are often focussed on groups of

ierials on which the grantees are specialists. For example, current grantees are deterging powder data for common crystalline organic substances, hydrous and anhydrous cement

jises, rare earth-containing intermetal 1 ics, homogeneous catalysts, agricultural chemicals

| phosphate minerals. Other grants calculate powder data for phases not represented in the

1. One such grant provides calculated patterns for all metals and alloys being entered

0 Structure Reports . The fact that 30-35 percent of all data issued in recent sets comes from grants-in-aid

1 the Associateship is evidence of their importance to the PDF .

553
4. Future Activities

In addition to the continuing subfile and data review activities described above, tti JCPDS - ICDD is increasing its efforts in several related areas. The interaction of

computers in all aspects of preparation and utilization of powder data is the center of t of these efforts. The editorial activities have been streamlined by computer storage an^

1

analysis of the data which are employed by the editor when the pattern evaluation procedi are initiated. The computerized data will then be transferred via phone lines to the of; and other editors and will ultimately be used to compose the master database and the PDF ! products in either hard or computer readable format. This approach will help minimize d; handling and consequent errors as well as allow an expansion of the present database to include considerably more information than is presently incorporated.

The expansion of the database is part of the effort to maintain a state-of-the-art , status with the PDF. Recent incorporation of the PDF in the Chemical Information System

CIS , is one of several projected uses of the database. The particular advantage of the

network is its ability to cross index with the other databases on the system through the

compound codes, the chemical abstracts registry number. Thus, one database can augment

information contained in another database. A more direct link is being established betw

the PDF and the Crystal Data file that will allow cross indexing through several data

channels including the crystal cell. This close correlation is possible because of the

similarity of information in the two databases, and coordination is especially important

because of possible partial or total merging of the two databases in the future. The \ JCPDS - ICDD is already the publisher for Crystal Data in the four volume book form, and plans to continue in this role with\ future volumes.

With the projected increase in the usage of automatic powder diffractometer, APD, | systems, the PDF will become even more important in computer readable form. Currently,! one manufacturer has incorporated the complete PDF as part of its system. Other manufac

turers have included or are contemplating provisions for using "mini" PDF files tailorec

to the special needs of the user. The everchanging hardware requirements and desires o1 the users have resulted in the addition of a computer specialist to the JCPDS - ICDD pei

manent staff who will be responsible for implementing these activities. Major efforts < towards simplifying the preparation of mini files on hard media that are compatible wit! the many instruments in use. The acquisition by the JCPDS - ICDD of an in-house compute

will facilitate these efforts.

Although the powder diffraction method is one of the oldest instrumental analytica

techniques available to the chemist, geologist, and materials scientist, it is still se< increasing usage in research and industry. Improvements in accuracy and sophistication

instrumentation, the ever expanding scope of compounds being studied and types of infor

desired, such as precise cell constants, have placed very high demands on the database ; in completeness of coverage and in accuracy of the data itself. The demands necessitat

554
|jstant review and change of the PDF , and no diminishing of the activities is foreseen in

ft

1

: near future. In fact, the bibliographic searches are still uncovering about the same Hber of published powder data each year which, along with Grant- in- Aid generated data,

Itinues to keep the editorial flow of patterns very active.

U In conclusion, the JCPDS - ICDD looks forward to continued employment of diffraction

liiods of identification for the coming decade and views its chief role as providing quality

Ja of wide coverage among all crystalline substances and in the various forms necessary for

Jicient manual and computer utilization. It also looks forward to increased cooperation

J compatibility with other organizations and databases.

555

National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979

(Issued February 1980)

DESIGN OF A HIGH-ACCURACY GONIOMETER FOR X-RAY

POWDER DIFFRACTOMETRY AT CONTROLLED TEMPERATURE

J. F. Berar, G. Calvarin, J. Chevreul , M. Gramond and D. Weigel

Laboratoire de Chimie-Physique du Solide

Ecole Central e des Arts et Manufactures

Grande Voie des Vignes

92290 Chatenay-Malabry, France

Precise measurements of lattice parameters, at numerous temperatures, by means of x-rays powder diff ractometry give valuable information about thermal expansion coefficients and phases transitions of a material. In this purpose, we have designed in our laboratory a goniometer, easy to use, and efficient to measure diffraction angles with an accuracy of 10 3 (° 6), even if the sample is attached into a large and

heavy device (cryostat, furnace...).

Owing to an original mechanical design, rotation angle (26) of the

detector is exactly measured by means of an incremental photoelectric

angle encoder connected to the goniometer axis. 6 and 26 rotations are independently executed by means of step by step motors electronically

operated. So, by means of the variable angular positions "zero" for 6 and 20, the different adjustments are executed quickly and easily with

a very good reproductibi 1 ity.

Another important particularity of this goniometer is its variable

radius. So, in using a large radius (500 mm) and a set of high efficiency undivergent slits, angular resolution is drastically improved,

principally for 0 < 20°. Nevertheless, use of a 12 kw rotating anode

x-rays generator permits to hold an appreciable intensity for diffraction peaks.

Lots of diagrams have been recorded in continuous scanning (0.06°

0/min) with Pb (N03 ) 2 and Pb3 0 4 samples at room temperature. In the

usable angular range (- 130 < 20 < + 160), reproductibi 1 ity of mea- urements is of ± 10 3 (° 0); moreover for the same diffraction peak, the

difference between the negative and positive 26 values does not exceed

± 2 x 10 3 (° 6). Lattice parameters refinements, made in taking account

of a simple correction function, show the differences 6., - 6 are

less than ± 10 3 (° 6).

557
Such a high-accuracy goniometer, owing to its easy and quick

working up and its competitive price, is a new and very efficient unit

for all x-rays powder diffraction studies.

558
National Bureau of Standards Special Publication 567. Proceedings of Symposium

on Accuracy in Powder Diffraction held at NBS, Gaithersburg, MD, June 11-15, 1979.

(Issued February 1980)

THE EFFECT OF TWINNING ON PARTICLE SIZE BROADENING IN SOME

OXIDES DERIVED BY DEHYDRATION REACTIONS

Fumio Watari

Department of Physics

Arizona State University

Tempe, AZ 852811

1. Results

| Dehydration products generally consist of small crystallites, which causes peak

j-oadening in x-ray diffraction. In most cases the reflection peaks of the decomposition

poducts have more or less the same width, except for the gradual increase with

iffraction angle. This is the case for the reactions:

Co (OH)

2 16Q* CoOOH

!

I Co (0H)

2 Co

30

4

Y

- FeOOH y - Fe

2

0

3 "^*T a - Fe^ .

the other hand, the reflection peaks are classified into two groups, sharp and broad

aks for the products of the following reactions:

ct - FeOOH

25QC

a - Fe

2 03 goethite hematite

a - A100H

55Q^ a - Al^

diaspore corundum

le investigation of the goethite-hematite system [l] 2 was performed at the University of

:werp, RUCA, B2020 Antwerp, Belgium.

gures in brackets indicate the literature references at the end of this paper.

559
For these cases it was revealed by high resolution electron microscopy that the product

oxides are composed of highly oriented twinned crystallites as a result of the equal

possibilities of two iron atom configurations in the hexagonal close packed oxygen

lattice. This results in the appearance in electron diffraction patterns among

crystallites [2]. The same hexagonal unit cell can be used to describe the structure of

the two twinned variants of the rhombohedral oxides [3]. Sharp powder line peaks appear

for the reflection with 1, h-k = 3n, which are common to both components of twin. Broad

peaks occur for the reflections with 1, h-k i 3n, for which, in the powder diffraction

pattern, there is an overlapping of the particular reflections contributed from only one component and forbidden for the other.

Figure 1 shows the half peak widths of synthetic goethite and its decomposition

product hematite for various dehydration temperatures.

The peak widths of goethite appears uniform, although the shape of an individual

particle is a needle in synthetic goethite and usually a cleavage plate in mineral

goethite [2].

Diffraction peaks common to both twin components of hematite, represented by the

(110) peak, have almost the same widths as for goethite and they do not vary with the

dehydration temperature. Non-common reflections, represented by the (104) peak, are much

broader and they become sharper, approaching the width of the (110) reflections, as the

dehydration temperature increases. The corundum formed by decomposition of mineral

diaspore [4,5], involving isomorphs of phases in the geothite-hematite system, shows the

same behavior except that the decomposition temperatures are higher.

2. Explanation

Careful examination of electron diffraction patterns had revealed the different

intensity behavior of the common and non-common spots of hematite. This was interpreted

using the concept of a mosaic spread in electron diffraction [2]. The same figure used to explain the above would apply for the present case. The configuration of the scatterers

giving rise to the different types of reflections in figure 1 is schematically represented

in figure 2. Before decomposition, each individual particle constituting the powder sample is a single crystal, typically with the size of 200 x 1000 x 10^ in the case of synthetic

goethite (fig. 2a).

After transformation, twinned oxide crystallites and voids are formed in a topotactic

way [6] by loss of compositional water. As revealed by high resolution electron

microscopy [2], the lattices are coherent between twins and the two different components

cannot be distinguished when imaged using the common diffraction peaks (fig. 2b). Non- common reflections are produced by only one component of crystallites and so come from

distinct regions separated in the specimen by the existence of other components (fig.

2c). This situation is clearly illustrated in the dark field image of hematite in [151]

560
561
EOo

i-<D

4-> fOB

-t-> sto

S- O) O T3 M— t-X

V) o

S-<D

s-

<u

+-> +-> 10o

to

<u o3

TDO

s-o.co

+J +->

«4- S- o -o

>» c x: 0 <U

•i- T3

+J

<0 "4- S- o 3

CT! -

•r- o

«t-

— coo o

01 -r-

»-> o

cu

«4- i— o <t-QJ

sCO coI

4-> <0 4Jc

10

CD

s_Q.

<U

S-

(0 -~2 E -a

ai ^ x:o 10 oo co

CMO

ai Q) fS- 4- 3 CO

562
o3

to -o

cu

-l-> tooo

•o

CD JC

to

•i- cn

u> •<-

•r- SZ

t-> o

tO i

•r- a> -O -Q

<D XICO

to

CDCO

u

XI

cu

+->

(0 o

io oo

Q-

•rO

i— CL i— to

id

+-> c

tO

O

>> -r- S- +-> u o

re cd s- x: <4- I- M-

+-> •r- te o

to

•r

re CD

+-> ro +-> c <o

+-> o c

fOE +j cd CD x: x: OS

+-> H— Q. to O EO

MO CD o .c

CD to

l+- E o 4->

to se o too

CD a> >> XI Th

S- -a

rO cu 1—

TD rm

GS oni i C o cr> o c to

•r— CD

+-> to J- »-> ~o M— I—

"o o +J

to > 3 rtj cd O +J

S- o x: to +J 4-> >> x: s. en TJ o

3C o cu cu D. to to

to o s_

CD x; cu CO i~

i4-> ev

a>

O E Ss_ a o 3 s- o

CD CM 4- -t->

Ll_

563
orientation (fig. 3). Because non-common spots in this orientation are allowed only from

one component but not from the other, the lattice fringes appear in the one component of

crystallites which is responsible for these spots [2].

The decrease of the widths for the non-common diffraction peaks with dehydration

temperature, shown in figure 1, results from a grain growth process among the twin

crystallites during heat treatment. The size of the crystallites of hematite estimated

o from non-common reflections is about 50 A just after transformation near 250 °C. For

o higher temperatures it approaches about 300 A, the size estimated for synthetic goethite

single crystals and for aggregates of twinned crystallites of hematite. These values are consistent with the observation by electron microscopy [2] and the results by other

methods [7,8], which supports the model of figure 2. Thus, these product oxides show different natures depending on the reflections used

to study them. They behave like single crystals with voids in them for the common reflections and like well-oriented polycrystals for the non-common reflections.

References

[1] Watari , F., Van Landuyt, J., and Amelinckx, S., to be published.

[2] Watari, F., Van Landuyt, J., Delavignette, P., and Amelinckx, S., Electron

microscopic study of dehydration transformations, part 1. Twin formation and mosaic

structure in hematite derived from goethite, J. Sol. Stat. Chem. (1979), in press.

[3] Henry, N. F. M. and Lonsdale, K., International tables for x-ray crystallography,

Vol. 1, p. 20 (Kynoch Press, Birmingham, 1969).

[4] Watari, F., Delavignette, P., and Amelinchx, S., Electron microscopic study of

dehydration transformations, part II. The formation of "super-structures" on the

dehydration of goethite and diaspore, J. Sol. Stat. Chem. (1979), in press.

[5] Watari, F., Periodic regular texture of oxides derived from oxyhydroxide minerals by

dehydration reaction, Modulated Structures 1979 (Kailua Kona, Hawaii), J. M. Cowley,

ed. (American Institute of Physics, New York, 1979), in press.

[6] Gunte, J. R. and Oswald, H.-R., Attempt to a systematic classification of topotactic

reactions, Bull. Inst. Chem. Res. Kyoto Univ., 53, 249 (1975).

[7] Shinjo, T., Mbssbauer effect in ant i ferromagnetic fine particles, J. Phys. Soc. Japan, 21, 917 (1966).

[8] Szytula, A., Burewicz, A., Dimitrijevic, Z., Drasnicki, S., Rzany, H., Todorovic, J.,

Wanic, A., and Wolski, W., Neutron diffraction studies of a-Fe00H, Phys. Stat. Sol.,

26, 429 (1968).

564
List of Participants

Alberto Albinati

Institute of Chemistry - Polytechnic

32,P.za L. Da Vinci

20133 Milano

ITALY

Joseph Angilello

IBM Research Center

Yorktown Heights, NY 10598

Christian Baerlocher

Institut fiir Kristallographie und

Petrographie, ETH Sonneggshasse 5 CH-8092 Zurich

SWITZERLAND

James H. Baird

W. R. Grace & Co. 7379 Route 32

Columbia, MD 21044

D. W. Beard

Siemens Corp.

2 Pin Oak Lane Cherry Hill, NJ 08034

Stanley Block

Center for Materials Science

National Bureau of Standards

Washington, DC 20234

H. Burdette

National Bureau of Standards

Washington, DC 20234

L. D. Calvert

National Research Council of Canada

Ottawa, Ont. K1A0R9

CANADA

Donald A. Carpenter

Union Carbide Corp. -Nuclear Div.

Oak Ridge Y-12 Plant

P.O. Box Y, Bldg. 9203 MS 1 Oak Ridge, TN 37830

John A. Carrabine

John Carroll University

University Heights

Cleveland, OH 44118

Odis B. Cavin

Oak Ridge National Lab

P. 0. Box X Oak Ridge, TN 37830

J. B. Cohen

Materials Science & Engineering Dept.

The Technological Institute

Northwestern University

Evanston, IL 60201

William Boettinger

National Bureau of Standards

Washington, DC 20234

Robert W. Broach

UOP Inc.

10 UOP Plaza

Des Plaines, IL 60016

M. J. Cooper

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Materials Physics Division

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Brookhaven National Lab

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Kenn Gardner

DuPont

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National Bureau of Standards

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British Museum (Natural History)

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State University of New York

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IBM - Route 52

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Texas A & M University

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Corning Glassworks

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Center for Matls. Science

National Bureau of Standards

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SS-114A (REV. 9-78)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

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1. PUBLICATION OR REPORT NO. SP 567

2. Gov't Accession No.

TITLE AND SUBTITLE

ACCURACY IN POWDER DIFFRACTION

(Proc. of Symp. on Accuracy in Powder Diffraction, NBS,

Gaithersburg, MD, June 11-15, 1979)

5. Publication Date February 1980

$. Performing Organization Code

AUTHOR(S) S. Block and C.R0 Hubbard, Editors8. Performing Organ. Report No.

PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

10. Project/Task/Wort Unit No.

11. Contract/Grant No.

. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS fSfree*. City, state. ZIP) nternational Union of Crystallography, 5 Abbey Square,

Ihester, CHI 2HU England; Chemistry Division of the National

Research Council of Canada, Ottawa, Canada K1AOR6. and the

fational Bureau of Standards, Washington, DC 20234

13. Type of Report & Period Covered Final

14. Sponsoring Agency Code

, SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 80-600010

Document describes a computer program; SF-185, FIPS Software Summary, is attached. .ABSTRACT (A 200-word or less tactual summary of most significant information. It document includes a significant bibliography or

literature survey, mention it here.) he proceeding of the Symposium on Accuracy in Powder Diffraction presents the

lapers, abstracts and discussions of the symposium held at the NBS, Gaithersburg,

jaryland, on June 11-15, 1979. The symposium was jointly sponsored by the NBS, the

ational Research Council of Canada, and the International Union of Crystallography,

hese proceedings contain a total of 24 invited and contributed abstracts. Many

apers are followed by an edited discussion. The proceedings are divided into the

allowing topics: Total Pattern, Instrumentation and Automation, Profile Fitting,

nalysis of Peak Shape, Lattice Parameters and Indexing, Applications, and Future

pportunities in Powder Diffraction.

KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons) Applications; lattice parameters; peak shape; powder diffraction; profile fitting;

x-ray neutron.

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