PUBUCAT10NS'"'i

A11103 Tlllfl? I United States Department of Commerce

Technology Administration

National Institute of Standards and Technology

NIST Special Publication 846

Accuracy in Powder Diffraction II

Proceedings of the International Conference

May 26-29, 1992
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1 i^^z

NIST Special Publication 846

Accuracy in Powder Diffraction II

Proceedings of the International Conference

May 26-29, 1992

E. Prince and J. K. Stalick, Editors

Materials Science and Engineering Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899

Organized by:

Commission on Powder Diffraction of the

International Union of Crystallography

Sponsored by:

JCPDS — International Centre for Diffraction Data

International Union of Crystallography

National Institute of Standards and Technology

Issued October 1992

e. 5

U.S. Department of Commerce

Barbara Hackman Franklin, Secretary

Technology Administration

Robert M. White, Under Secretary for Technology

National Institute of Standards and Technology

John W. Lyons, Director
National Institute of Standards U.S. Government Printing Office

and Technology Washington: 1992

Special Publication 846

Natl. Inst. Stand. Technol.

Spec. Publ. 846

241 pages (Oct. 1992)

CODEN: NSPUE2

For sale by the Superintendent

of Documents

U.S. Government Printing Office

Washington, DC 20402
Foreword

Few areas of research endeavour in the physical sciences have experienced a renaissance on such a scale,

and few have had such an impact on related disciplines during the past 15 years, as has powder diffraction.

Sensing this imminent revolution back in 1979, the U. S. National Bureau of Standards (now NIST) hosted a

conference entitled Accuracy in Powder Diffraction. The proceedings of this symposium were subsequently

published as NBS Special Publication 567 and this document is still widely quoted in the literature, more

than 10 years later!

In the years since this benchmark meeting, new generations of high resolution and intense sources of

X-rays and neutrons have been designed and built, new and exciting methods for data collection and analysis

have been developed, and powerful computers have come within reach of, if not on, scientists' desks. Never

before have so many previously neglected materials come under such close scrutiny, and never before has

it been possible to study these materials under such a wide range of non-ambient conditions. Diffraction

patterns previously collected only for phase identification or unit cell determination are now used to study

such things as the subtleties of electron density distribution, the mechanisms of phase transformations, and

the distribution of cation ionization states. In the light of this renaissance, the lUCr Commission on Powder Diffraction (CPD), in conjunction with

the National Institute of Standards and Technology, JCPDS-International Centre for Diffraction Data, and

the International Union of Crystallography, organized a second conference, Accuracy in Powder Diffraction

II, as a continuation of the series started in 1979. Six sessions provided the framework for APD-II: phase

identification and quantification; accuracy and standards; new developments in software and data analysis;

profile fitting, decomposition, and microstructural effects; novel applications and structural science; new

developments in hardware, including detectors, and studies under non-ambient and time-resolved conditions.

The scientific sessions incorporated keynote, invited, oral and poster papers, a tribute to the late W.

Parrish, and two workshops organized by the JCPDS-ICDD, entitled ICDD round robin on powder diffractomeier sensitivity and Indexing methods. In addition, the program for the last session of the meeting was

specifically designed to interface smoothly with the Workshop on X-ray and Neutron Diffraction at High

Pressure, organized by the High pressure Group of the lUCr, which was held in Washington, DC, at the

Geophysical Laboratory of the Carnegie Institution of Washington. Many attendees obtained substantial

benefit from attending both of these meetings.

Generous support for the attendance of young scientists and invited speakers was obtained from the

lUCr and the JCPDS-ICDD, respectively. It need hardly be added that the continued financial support of

both of these organizations is extremely valuable to, and very much appreciated by the powder diffraction

community in general and the APD-II Organizing Committee in particular. The cooperation and generosity

of the NIST in providing support in the form of both personnel and facilities was essential for the organization

of the meeting, and for this we are also very grateful.

I take the opportunity here to express my thanks to the members of the Scientific Program Committee

for their hard work and cooperation; their efforts resulted in an enlightening and fruitful meeting.

R.J. Hill

Chairman, Scientific Program Committee

Acknowledgments

The timely publication of this volume of proceedings was made possible by the cooperation of the authors,

who provided machine-readable text, many of them in I^TgX or Wordperfect, so that the labor involved in

producing camera-ready copy was minimal. We are grateful to Carol O'Connor for invaluable assistance

in preparing the text for the printer. The success of the conference was assured by the efforts of the

international program committee, ably chaired by Rod Hill, and for the smooth running of the conference

we are indebted to Kathleen Kilmer and her staff in the NIST conference office, particularly Tammie Grice

and Lori Phillips. The conference would not have occurred at all without the impetus of Ray Young and

the lUCr Commission on Powder Diffraction and the generous sponsorship of the International Union of

Crystallography and the JCPDS-International Centre for Diffraction Data.

E. Prince

J. K. Stalick, Editors

111
Commission on Powder Diffraction

R. A. Young (USA), Chairman

Z. Bojarski (Poland)

D. E. Cox (USA)

J. Fiala (Czechoslovakia)

A. W. Hewat (France)

R. J. Hill (Australia)

J. I. Langford (UK)

D. Louer (France)

P.-E. Werner (Sweden)

T. Yamanaka (Japan)

D. K. Smith (USA), Consultant

L. Frevel (USA), JCPDS-ICDD representative

International Program Committee

R. J. Hill (Australia), Chairman

D. E. Cox (USA)

J. Fiala (Czechoslovakia)

H. Fuess (Germany)

D. Louer (France)

R. J. Nelmes (UK)

E. Prince (USA)

D. K. Smith (USA)

H. Toraya (Japan)

L. Zevin (Israel)

Local Committee

E. Prince, Co-chairman

J. K. Stalick, Co-chairman

T. Grice

K. Kilmer

C. O'Connor

L. Phillips

J. E. Post

C. T. Prewitt

Special thanks to:

Blake Industries, Inc. Charles Supper Company

Fein-Marquart Associates, Inc. Materials Data Inc. Molecular Simulations, Inc. Philips Electronic Instruments Company

Polycrystal Book Service

Scintag, Inc. Siemens Analytical X-ray Instruments

IV
Table of Contents

T. C. Huang. Contributions of William Parrish to Powder Diffraction 1

B. L. Davis. Quantitative Phase Analysis with Reference Intensity Ratios 7

L. Zevin. Standardless Method of Quantitative X-ray Diffractometry 17

R. L. Snyder. Reference Intensity Ratios, Whole Pattern Fitting and Standardless X-ray Quantitative Analysis 25

J. K. Stalick. Quantitative Phase Analysis with the Rietveld Method 34

H. Hermann and M. Ermrich. Microabsorption Corrections , 38

M. Nakhmanson. Computer-Aided Phase Identification 47

P.-E. Werner. On the Relative Merits of Cameras and Diffractometers for XRD 51

J.-F. Berar. Data Optimization and Propagation of Errors in Powder Diffraction 63

J. P. Cline. NIST XRD Standard Reference Materials: Their Characterization and Uses 68

L. B. McCusker. Integrated Software for Structure Solution from Powder Data 75

J. M. Newsam, M. W. Deem and C. M. Freeman. Direct Space Methods of Structure Solution

from Powder Diffraction Data 80

D. LOUER. Automatic Indexing Procedures and Applications 92

B. H. Toby and T. Egami. Accuracy in Pair Distribution Function Analysis for Crystalline and

Non-Crystalline Materials 105

J. I. Langford. The Use of the Voigt Function in Determining Microstructural Properties from

Diffraction Data by Means of Pattern Decomposition 110

V. Valvoda. Preferred Orientation Effects ." 127

T. M. HoLDEN and J. H. Root. Residual Stress Measurements with Neutrons 136

A. Le Bail. Modelling Anisotropic Crystallite Size/Microstrain in Rietveld Analysis 142

D. L. Bish. Structure Building with Rietveld Analysis 154

H. BoYSEN. Anharmonic Thermal Parameters, Disorder and Phase Transitions 165

J. P. Attfield. Recent Advances in the Use of Anomalous Dispersion Effects 175

L. W. Finger. Instrumentation for Studies at the Powder/Single-crystal Boundary 183

M. Sutton, S. Brauer, Y. S. Yang, H. E. Fischer, J. O. Strom-Olsen and G. B.

Stephenson. Time-resolved X-ray Diffraction Studies of Crystallization in Metallic Glcisses . . 189

V
R. J. Nelmes, J. S. LovEDAY AND J. M. Besson. New Developments in High-Pressure Neutron

Powder Diffraction 195

T. Yamanaka. Crystal Structure Analysis by Fitting of Powder Diffraction Data on Energy Dispersion 199

Abstracts. Session 1: Phase Identification and Quantification 206

Abstracts. Session 2: Accuracy and Standards 209

Abstracts. Session 3: New Developments in Software and Data Analysis 211

Abstracts. Session 4: Profile Fitting, Decomposition and Microstructural Effects 215

Abstracts. Session 5: Novel Applications and Structural Science 219

Abstracts. Session 6: New Developments in Hardware, Including Detectors, and Studies under

Non-ambient and Time-resolved Conditions 222

Author index 226

List of participants 228

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

Institute of Standards and Technology, nor does it imply that the products are necessarily the best available

for the purpose.

Abstract

The proceedings of the international conference Accuracy in Powder Diffraction II present the invited papers

and abstracts of the papers contributed to the conference, which was held at NIST, Gaithersburg, Maryland, during May 26-29, 1992. The conference was organized by the Commission on Powder Diffraction

of the International Union of Crystallography, and was jointly sponsored by NIST, JCPDS - International

Centre for Diffraction Data and the International Union of Crystallography. The proceedings contain 25

invited papers and 73 contributed abstracts. The program of the conference was divided into six topics:

Phase Identification and Quantification; Accuracy and Standards; New Developments in Software and Data

Analysis; Profile Fitting, Decomposition and Microstructural Effects; Novel Applications and Structrual

Science; and New Developments in Hardware, Including Detectors, and Studies under Non-ambient and

Time-resolved Conditions. A ceremonial session was devoted to a tribute to the late William Parrish and

his contributions to powder diffraction. In addition, there were two tutorial workshops organized by the

JCPDS - International Centre for Diffraction Data, one on diffractometer sensitivity and one on automatic

indexing methods.

Keywords: Accuracy; Instrumentation; Microstructure; Phase analysis; Powder diffraction; Software; Standards; Structural science.

vi
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Contributions of William Parrish to Powder Diffraction

T. C. Huang

IBM Research Division, Almaden Research Center

650 Harry Road, San Jose, CA 95120-6099, U.S.A.

Abstract

This paper summarizes some of the important contributions of the late Dr. William Parrish to powder diffraction. He was responsible for developing

many of the instruments and methods that became

commercial products widely used for powder diffraction analysis. His most important achievement was

the 1947 invention of the "Norelco" diffractometer

using a new X-ray optics geometry that gave high

intensity and resolution with good profile shape. It

is the basic instrument on which most powder work

has been done for almost half a century, and more

than 10,000 powder diffractometers are now in use throughout the world. After joining IBM in 1970, he

developed diffractometer automation methods that

eventually evolved into a series of IBM X-ray products. A large number of analytical programs were developed that added new capabilities to X-ray analysis. In the early 1980s he began a systematic study of

synchrotron powder diffraction, and this synchrotron

research provides the basis for a new and exciting era of powder diffraction. Many important advances will

be possible based on his pioneering efforts. Parrish

was not only a great powder diffractionist, but also a wonderful person, who will be greatly missed.

Introduction

Ever since its discovery by Debye and Scherrer more

than 75 years ago, the powder diffraction method has

played a significant role in the development of modern science and technology. The name of Bill Parrish

is linked closely to many of the most important steps

in the evolution of powder diffraction instrumentation, from relatively primitive powder cameras to the

highly sophisticated and automated diffractometers

we now use.

Invention of the powder diffractometer

Parrish 's most important achievement involved the

invention of the "Norelco" powder diffractometer in

Figure 1: The first "Norelco" powder diffractometer

invented by W. Parrish at Philips Laboratories, 1947.

1947 (figure 1). The instrument used a "new" Xray optics geometry to greatly increase the quality of

powder diffraction patterns over those of an earlier

diffractometer assembled by H. Friedman in 1945.

The latter device used the spot focus with an effec- tive size of 0.24 by 2.5 mm; the dimension of the line

focus in Parrish's device wais only 0.06 by 10 mm.

To improve asymmetric profile broadening due to the

large axial divergence in the Friedman diffractometer, Parrish used a set of long parallel equally-spaced

thin Mo foils (Soller slits) in the incident beam. The

long X-ray line source thus became effectively a number of short overlapping narrow sources parallel to

the specimen axis of rotation and had the additional

advantages of high intensity and high resolution. A

second set of parallel slits was placed in the diffracted

beam, and the large increase in resolution decreased

the profile widths of the peaks from about 0.35* to

0.10° (2^). Figure 2 shows the optics and line profiles

of the two instruments.

It is unfortunate that the Parrish diffractometer

1
2 Contributions of William Parish to Powder Diffraction

pulse amplitudes were proportional to the energies of

the X-ray quanta. This made possible the use of a

pulse height analyzer to accept only pulses of amplitude within preselected limits and thereby greatly re- duced the background with only a few percent loss of

characteristic line intensity. The combined action of

the Ni filter and electronic discrimination increased

peak-to-background ratio well beyond the range of

the Geiger counter used in early models. Figure 3

shows the effects of pulse height discrimination with

a scintillation counter on the spectrum of a Cu X-ray

tube.

'norm

Figure 2: X-ray optics of the early and new diffractometers (top) and their profiles: Si (422) and Cu

Ka. (bottom).

is often incorrectly referred as a Bragg-Brentano device. The counter tube diffractometer geometry was

uniquely Parrish's, and he was awarded a U.S. Patent

in 1951. More than 10,000 are now in use throughout the world; this alone could be recognized as a most important achievement because it became the

basic instrument on which most powder work has

been done for almost half of a century.

Within a few years, the new instrument had revo- lutionized powder diffraction techniques and made

possible major improvements in the measurement

of powder patterns. Following those developments,

Parrish devoted himself to many years of intensive

research aimed at improving both the instrumentation of the diffractometer and the understanding of

the basic principles of the new instrument. He was

a pioneer in the utilization of scintillation and proportional counters with pulse amplitude discrimination, making possible large improvements in X-ray

detection (Parrish and Kohler, 1956). Both detectors had dead times of the order of 0.25 //sec, and the

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Figure 3: Spectrum of Cu X-ray tube recorded with

a scintillation counter.

Parrish and his co-workers at Philips developed

principles and mechanical devices to aid in simplifying the alignment and calibration of the diffractometer (Parrish & Lowitzsch, 1959). A flat specimen

spinner for rotating the specimen in its own plane was introduced to reduce the effect of crystallite size

statistics and in-plane preferred orientation on absolute and relative intensities (de Wolff, Taylor k, Parrish, 1959). He was one of the first to investigate the

sources of systematic errors in counter diffractometry

(Parrish &; Wilson, 1959) and the use of the centroid

method for handling aberrations in precision lattice

parameter determination (Taylor, Mack & Parrish,

1964).

Parrish also extended the applications of the

diffractometer and made modifications of the stan-
T. C. Huang 3

Figure 4: X-ray optics of the transmission specimen

with a diffracted-beam focusing monochromator.

dard reflection diffractometer to permit transmission

and Seemann-BohUn measurements. The X-ray optical arrangement of the diffractometers, modified for

transmission specimen and curved focusing crystal

monochromator, is shown in figure 4. The transmission geometry is very useful for measuring large dspacings and as a supplementary technique to the

reflection method for measuring preferentially oriented samples such as clay minerals and polymers

(Parrish, 1958). The X-ray optics for the SeemannBohlin method using an incident-beam monochromator is shown in figure 5 (Parrish & Mack, 1967). This

method with a small fixed incident angle is most suitable for the measurement of thin films because of a

large path length and small penetration depth by the

incident beam.

Further advances in reflection specimen powder

diffractometry for recording complex powder patterns arising from low crystallographic symmetry and

large unit-cell dimensions, multiple-phase specimens

and structural imperfections were also made by Parrish (1968). A long fine focus X-ray tube was used;

the anode was set at a relatively large angle (> 10**)

to reduce target self- absorption for high incident intensity. The length and spacing of the incident-beam

parallel slits were increased so that the total intensity reaching the specimen was high, the distribution

was more uniform, and at the same time there was no

loss of resolution. The divergence slit and the antiscattering slit were set close to the specimen to effectively eliminate unwanted scattered radiation. A

simple vacuum chamber eliminated most of the airpath absorption losses. Cr Ka X-ray radiation was

used to increase dispersion and to reduce the difficulties caused by overlapping reflections.

Figure 5: X-ray optics for the Seemann-Bohlin

method with an incident-beam focusing monochromator.

Parrish somehow found time to write a large number of papers; many of these are contained in his

book X-ray Analysis Papers, which, for many years,

served as the Bible of powder diffractometry (Parrish, 1965).

Computer automation

Parrish was quick to recognize the potential power

of computers, and, in collaboration with his co- workers at IBM, he developed diffractometer au- tomation methods in the 1970s. A large number

of analytical programs were developed, these added

new capabilities to X-ray powder diffraction analysis.

The work eventually evolved into a series of IBM Xray products and was recognized by IBM Outstanding Contribution Awards.

The profile fitting method developed for the reduction of experimental patterns made possible the separation of the instrument function from the specimen

diffraction contribution, an essential step for obtaining accurate diffraction data. The precision of the

profile fitting method was found to be ±0.0004°2^

and 0.2% intensity (Parrish and Huang, 1980). Overlapping peaks could be resolved with better resolution than the original experimental data (Huang h

Parrish, 1975). Figure 6 shows the results of profile

fitting analysis of the quartz cluster at 67° — 69°

In collaboration with Professor G. Will of Germany, powder structures were refined using the twostage method in which the integrated intensities were determined by profile fitting and used in powder least

squares refinements (FOWLS) (Will, 1979). The

method was tested with Si, quartz, and corundum
4 Contributions of William Parish to Powder Diffraction

100

SC2,

Figure 6: Profile fitting analysis of the quartz cluster

of (12.2), (20.3), and (30.1) reflections.

gave R factors below 1% (Will, Parrish & Huang,

1983) . The method is different from the Rietveld

method, which refines the entire pattern and determines the structure in one step.

An accurate computer peak search program which

used first and second derivatives was also developed

for rapid determination of peak positions and intensities with accuracy up to 0.001° (Huang k. Parrish,

1984) . An advanced search/match algorithm was developed for automatic searching of the JCPDS Powder

Diffraction File (PDF) and accurate phase identification of an unknown mixture (Huang & Parrish,

1982). It was also demonstrated that only twelve dspacings in the PDF standard patterns were needed

for automatic phase identification (Huang, Parrish &

Post, 1983).

Synchrotron powder diffractometry

Parrish became interested in the remarkable new

storage ring radiation sources and began studies at the Stanford Synchrotron Radiation Laboratory in

1977. In collaboration with Professor M. Hart of

England, new parallel-beam optics with an incidentbeam channel monochromator and a set of horizontal

parallel slits in the diffracted beam were developed

(figure 7). High intensity patterns with narrow symmetrical profiles were obtained (Parrish, Hart, Erickson, Masciocchi h Huang, 1986; Parrish, Hart,

Figure 7: Synchrotron radiation parallel beam X-ray

optics used for powder diffractometry.

Figure 8: Fourier maps of a section of MgGe04 (left)

and MgSi04 (right).

Huang h Bellotto, 1987). The method could also

be modified for energy dispersive diffraction using a

selected, fixed 6 — 29 and step scanning the incidentbeam monochromator. Some of the finest X-ray powder diffraction patterns recorded to date were obtained with the synchrotron method (figure 9).

Synchrotron diffraction data were used successfully for powder-structure analysis. Figure 8 shows

electron density maps of orthorhombic MgGe04 cal- culated using Fourier coefficients taken directly from

the synchrotron intensities. These yielded excellent

agreement with electron density of MgSi04 calculated from a single-crystal data set containing 1349

reflections.

The advantages of using monochromatic and parallel synchrotron X-rays for the microstructural analysis of powder materials were also reported (Huang,

Hart, Parrish h Masciocchi, 1987). It was shown

that three dimensional crystallite-size and microstrain determinations of Pd powders were made successfully using 1 A synchrotron radiation. The symmetrical instrument profile shape had major advantages in the correction of instrumental broadening

and in the determination of a low level (lO"'* range)

of stacking-fault probabilities.
T. C. Huang

10

8

6

4

2

0

110

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321

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10

8

6

4

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2.0 1.75 1 50 1.25 1.0 0.75 A (A)

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10.00

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Figure 9: Synchrotron diffraction patterns of W powders: (a) 6-29 scan with A = 0.688A.; (b) energy

dispersive scan with 26 = 135°; and (c) expansion of

the energy dispersive pattern.

The precision of the derivation of d-values and

the accuracy of determination of lattice parameters

from symmetric and single-wavelength synchrotron

diffraction peaks were also reported (Huang, Parrish, Masciocchi Sz Wang, 1990). The precision in

determining d-values from synchrotron peaks in the

front reflection region Wcis high with Ad/d in the low

10~^ range. Using the precision lattice parameters

obtained from Cu Kai diffraction peaks in the back

reflection region as references, the accuracy in the

lattice parameter determination from alumina synchrotron peaks reached the 1 — 2 x 10~^ range.

This synchrotron research provides the basis for a new and exciting era of powder diffraction. Many

important advances will be possible based on his pioneering efforts. In recognition of his outstanding contributions

to powder diffraction, the JCPDS - International

Centre for Diffraction data awarded Parrish the

Hanawait Award in 1987, and the Annual Denver

X-Ray Conference presented him the first Barrett

Award in 1988.

Parrish was an indefatigable researcher. In a 1987

letter to the late Dr. J. D. Hanawalt, Parrish wrote

"I have always worked hard, but these past few years on the synchrotron research have been extremely difficult because of the very long hours and the multitude of papers to be written. It is so exciting I don't

realize how tired I am".

Parrish was not only a great and dedicated powder

diffractionist, but also a nice person, who will be

greatly missed.

References

Friedman, H. (1945a). Electronics 18, 132.

Friedman, H. (19456). U. S. Patent 2,286,785.

Huang, T. C, Hart, M., Parrish, W. & Masciocchi,

N. (1987). J. Appl. Phys. 61, 2813.

Huang, T. C. & Parrish, W. (1975). Appl. Phys. Lett. 27, 123.

Huang, T. C. & Parrish, W. (1982). Adv. X-Ray

Anal. 25, 213.

Huang, T. C. & Parrish, W. (1984). Adv. X-Ray

Anal. 27, 45. Huang, T. C, Parrish, W., Masciocchi, N., &

Wang, P. W. (1990). Adv. X-Ray Anal. 33, 295.

Huang, T. C, Parrish, W., & Post, B. (1983). Adv.

X-Ray Anal. 26, 93. Parrish, W. (1958). 7th National Conf. on Clays and

Clay Minerals, 230, New York: Pergamon.

Parrish, W. (1965). X-Ray Analysis Papers. Eindhoven: Centrex.
6 Contributions of William Parish to Powder Diffraction

Parrish, W. (1968). In X-ray and Electron Methods of

Analysis, H. van Alphen & W. Parrish, eds. 1, New

York: Plenum.

Parrish, W. & Hamaker, E. A. (1951). U. S. Patent

2,549,987.

Parrish, W., Hart, M., Erickson, C. R., MasciocCHI, N. & Huang, T. C. (1986). Adv. X-Ray Anal.

29, 243.

Parrish, W., Hart, M., Huang, T. C. & Bellotto,

M. (1987). Adv. X- Ray Anal. 30, 373.

Parrish, W. & Huang, T. C. (1980). Nat. Bur. Stand.

Spec. Pub. 457, 95. Parrish, W. & Kohler, T. R. (1956). J. Appl. Phys.

27, 1215.

Parrish, W. & Lowitzsch, K. (1959). Am. Mineral.

44, 765.

Parrish, W. & Mack, M. (1963). Acta Cryst. 23, 687.

Parrish, W. & Wilson, A. J. C. (1959). Inter. Table

of X-Ray Cryst. II, 216.

Taylor, J., Mack, M. & Parrish, W. (1964). Acta

Cryst. 17, 1229.

Will, G. (1979). J. Appl. Cryst. 12, 483.

Will, G., Parrish, W. & Huang, T. C. (1983). J. Appl. Cryst. 16, 611.

Will, G., Masciocchi, N., Hart, M. & Parrish, W.

(1988). J. Appl. Cryst. 21, 182.

DE Wolf, P. M. Taylor, J. & Parrish, W. (1959). J. Appl. Phys. 30, 63.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction 11, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Quantitative Phase Analysis with Reference Intensity Ratios

B. L. Davis

X-Ray Diffraction Laboratory, Engineering and Mining Experiment Station

South Dakota School of Mines and Technology

Rapid City, South Dakota 57701, U.S.A.

1. Introduction

The concept of introducing a ratio of analyte intensity to standard reference material intensity began with the

report of de Woolf and Visser (1964) which in effect

recommended establishment of a file of "reference

intensities" in the Joint Committee on Powder

Diffraction Standards manuals (Berry, 1970). This

early paper has recently been formally published (de

Woolf and Visser, 1988). This ratio has been given the

notation 1/1^, meaning analyte intensity I over corundum

standard intensity I^. More recently the notation RIR

(Reference Intensity Ratio) has become accepted with a somewhat broader implication that analyte peak

intensities may be referenced to intensities from other

than corundum standards (i.e., rutile, quartz).

Much earlier the internal standard method of

quantitative X-ray diffraction (XRD) analysis was placed on a solid theoretical and experimental foundation by

Alexander and Klug (1948). All subsequent work,

including the various developments of multicomponent

schemes for quantitative XRD analysis, have this

development as a foundation. However, the explicit

RIR form was not apparent in either the Alexander and

Klug or de Woolf and Visser presentations of

quantitative analysis. Indeed, many workers in the

period from 1948 to 1974 developed their methods

around a proportional constant, or combination of

constants, connecting mass absorption, density, and

scattering power (through fundamental calculable

structure and scattering parameters) with the weight

fraction of the analyte. These methods worked, and

within their mathematical framework the RIR may be

identified, but not always obviously. Table 1 presents a summary of the notation and equivalent parameters that

define an implicit RIR approach in methods for those

workers in quantitative multicomponent analysis whose

work is available in the literature during the past 40

years.

While these methods provided a means for arriving at

the desired result, namely the weight fraction of a given

analyte in a mixture of several components, they did not

provide the simple and powerful framework for direct

application of the RIR that came with the contribution of

F. Chung in his series of papers published in 1974-

1975. Chung's papers (where the notation kj is equivalent to RIR) demonstrated several of the powerful

features of RIR analysis that, in the opinion of this

author, had not been demonstrated prior to this time: (1)

that an internal standard need not be added to the

sample to be analysed in order to determine uniquely,

through a knowledge of predetermined RIR's, all component weight fractions in the absence of

unidentified or amorphous components, (2) that with

known RIR values and the addition of an internal

standard individual weight fractions could be determined

without regard to whatever else might be in the sample

and when all crystalline component weight fractions were determined, amorphous quantities (but not their

composition) could be determined as well; and (3) the

RIR, by nature of the base standard used for all

materials, "flushed" out absorption effects. This simply means that the use of the RIR in any sample will

correctly yield the weight fraction of that component

regardless of the nature of the sample matrix. Chung

(1975) also demonstrated that several RIR values could

be determined simultaneously from a single sample

mixture. Many of the features of the reference intensity

ratio analysis method are also summarized by Hubbard

and Snyder (1988).

Recent work now includes the use of RIR's in

Rietveld analysis (Bish and Howard 1988; Hill, 1991) as well as in whole-pattern analysis (Smith et al., 1988;

Smith, 1992). These techniques are very powerful for

phase analysis and structure refinement but may not be

the method of choice for routine analytical work simply

because of their greater dependence on sophisticated

computer algorithms and data input requirements; some

discussion of these methods will be presented below but

the emphasis of this paper is on application of basic

7
Quantitative Phase Analysis with Reference Intensity Ratios and Uses

Table 1. Equivalent Reference Intensity Ratio (RIR) notation

REFERENCE SYMBOL EQUIVALENT

Klug & Alexander, 1948 None (K,/pi)/(K,/pJ

Copeland & Bragg, 1958

Karlak & Burnett, 1966

None (Ky/pi)/(Ku/pj)

Bezjak, 1961 (Kj/pj)/(K,/pJ

Aa W/r^r^it Sir X/Joo^^ 1 OA/I IQfifi fle wooii ot visser, iyo'+, lyoo C'C" n nxi ^//'T /w \

Jahanbagloo & Zoltai, 1966 None ([INFI]i/pi)/([INFI],/p^

Chung, 1974; Davis, 1980 ki I?/^/I>c = (Ii/Xi)/(Ie/Xe)

Hubbard, et al., 1976 ki (Ii/Xi)/(Ie/X^ = (M*7/p)/(M*cTc/Pc)

Hubbard & Smith, 1977 None

Monshi & Messer, 1991 None

simple relationships and sample preparation methods to

quantitative multicomponent analysis using RIR values

based on corundum as the standard.

2. Brief summary of Rietveld and

whole-pattern methods

Both of these methods employ the matching of large

ranges of the component diffraction patterns in the

determination of quantities present. The Rietveld

method requires input of the structural parameters of

each phase and results in not only the component weight

fractions as output but refined structure and composition

as output. Disadvantages include the limited number of

variables that may be adjusted, and complications and

accuracy losses due to the presence of amorphous

phases or poorly crystalline phases. The working

equations given by Bish and Howard (1988) are analogous to the adiabatic and matrix flushing equations

of Chung (1974a,b) when the scaling factor is included.

The database for quantitation in the Rietveld method is a set of calculated patterns.

The whole-pattern method of Smith et al. (1988) can make use of either experimental pure-phase diffraction

patterns or calculated ones. The fitting (minimization)

equation is:

SA2e)=/^(A2e)-ER;ij/i^//A2e)

where the summation is over p profiles divided into

small (e.g. 0.1° 20) intervals A26. Wp is adjusted so

that this summation approaches as close as possible to

the observed intensity Iu„k of each A26 interval. The

reference intensity ratios must be generated

independently from the pure-phase traces and are

equivalent to peak-height ratios, rather than integrated

intensity ratios. As in the Rietveld method the

quantitation consists of matching and scaling the purephase patterns to the observed pattern with scale factors

proportional to the weight fractions.

Both methods require considerable operator set-up

time and the Rietveld method also involves substantial

computer time. Once the proper component phase

traces have been determined the accuracy of quantitation

should be on the order of 1-2% absolute; this is likely

to exceed the accuracy of routine single-peak RIR

analysis significantly.

3. Summary of Reference Intensity

Ratio methodology

The important simple relationships so useful in RIR

analysis and derived by Chung (1974a, 1974b, 1975)

will be stated and discussed briefly, with examples

given of their use. Moreover, the use of thin layer
B. L. Davis 9

(filter) technology, mass absorption, and solid solution

studies as used with RJR equations in our laboratory

will be integrated into this presentation. The more

compact Chung notation kj, for reference intensity ratio

will be used in the discussion.

3. 1 Analysis with added standard.

The working equation, given in the notation of Chung

(1974a) which provides the weight fraction Wj of

component jis,

W 'I. (1)

where kj is the reference intensity ratio for component

j of the mixture and Ij and are intensities measured

for the jth component peak and standard (corundum),

respectively, and is the weight fraction of the

corundum standard in the composite sample (original

sample + added corundum). This equation is straightforward to apply, but requires the addition of

standard material. The advantage of its use is that any

component weight fraction can be determined

independently of any other material, crystalline or amorphous, that may be in the original sample.

3. 2 Analysis without internal standard

Chung's (1974b) "adiabatic" equation permits analysis

of all identified components of a mixture without

addition of a standard providing all reference intensity

ratios, kj, are available. This equation is

-Ehi=\^i

(2)

where the notation is the same as for Eq. (1) above.

Here the sum is taken for all components so that the

equation is self-normalizing; i.e., all the Wj will sum to

unity. Consequently the absolute weight fractions will

be correctly determined through use of Eq. (2) only if

all components are identified and crystalline. However,

the ratios of the weight fractions remain correct in any

case.

sample measurements be devoid of significant primary

extinction, microabsorption, and preferred orientation.

It has been argued by some that maintaining a consistent

degree of preferred orientation in both k; and sample

intensity measurements is a satisfactory procedure.

Such an approach, however, will not result in the kj

being a universal constant and cannot be recommended

except perhaps for oriented clay particle analyses.

The obvious and most common procedure is to mix

analyte and standard together in a 1 : 1 weight proportion

and simply measure the intensities of the strongest

peaks, or a series of peaks from each and convert each

to the strongest reflection from calculated intensity

ratios, if available. Using the absorption-intensity

relationship of Leroux et al. (1953) it has been shown

(Davis, 1988, p. 75) that the kj can also be measured by

a pure phase, non-mixing method. The definition of the

kj and equivalency to this latter procedure is:

JO (3)

where the subscripts jo and co refer to intensities

collected from pure-phase preparations. Davis et

a/. (1990)* have demonstrated the equivalency of the two

methods in measuring the reference intensity ratios and

also provide experimental verification from actual

analysis by both procedures for different absorbing

materials. The value of Eq. (3) is that possible relative

losses of analyte or standard during aerosol suspension on filters, or of inhomogeneous mixing of bulk

preparations, is avoided; use of the second expression

does require, however, that there be no significant

differences between the conditions of the analyte and

standard intensity measurements (including instrument

alignment and performance).

3.4 Use of filter loading to suppress preferred

orientation

Two procedures for eliminating or greatly supressing

preferred orientation of crystallites of the sample have

been proposed. Calvert et al. (1983) describe a method

for encapsulating individual inequant particles into

3.3 Measuring the reference intensity ratio

For the highest accuracy of analysis it is vitally ' Two printer's errors should be noted in this paper. InEqs. (4)and

important that both reference intensity measurement and (5) the leftmost Ij should read kj.
10 Quantitative Phase Analysis with Reference Intensity Ratios and Uses

spherical chemical agents and collecting aggregates of

the spheres for analysis. We recommend (Davis, 1986)

suspension of fine particles in a vertical airstream for

collection onto glass microfiber filters. In either

method, any significant departure from "infinite"

thickness, as well as the presence of enclosing chemical

or filter element matrix, requires intensity correction.

For the filter technique the penetration, density, and mass absorption properties of several filter substrates

have been reported by Davis and Johnson (1982) and

Davis (1988). The corrections required in loading thin

aerosol layers onto filters are combined into the relation

(4)

where I**

J

is the intensity corrected from the raw

intensity I"; Wgg is the weight fraction of sample

material (B) in the "hybrid" layer, that layer containing

both substrate and sample components down to the depth

of penetration of the sample; and /xg are the mass

absorption coefficients of the pure sample material and

hybrid layer material, respectively; Mg is the mass per

unit area of the sample on and within the substrate

hybrid layer; and Q{6) is a function to convert intensities

from automatic divergence slit to fixed slit conditions.

The quantities /tg and Wgg are readily computed from

fundamental filter properties (Davis and Johnson, 1982)

using the relations

and.

1 +

(6)

where is the effective thickness (pore space removed)

of the substrate material within the hybrid layer, and Pq

and /iQ are the density and mass absorption of that

material, respectively.

The devices described by Johnson and Davis (1982)

and Davis (1986) may be constructed from easily

obtainable laboratory equipment. Another even simpler

device, from our experience in the past few months, has

shown to be as effective as the larger ones in producing

thin uniform sample layers. Figure 1 illustrates this

device; a 250-ml aspirator bottle is taped to a 25-mm

DELRIN filter cassette. The inlet is fitted with a short

rubber hose and plug (plastic 0.25" pipe cap works

well) drilled with a small hole to admit a 1-mm capillary

tube (slightly bent at one end). The sample is placed

within the inlet close to the plug. With the collecting

pump operating the capillary is moved outward and

rotated to dislodge the sample material.

Turbulance within the inlet tube disaggregates the

clumps and produces a fine aerosol. Very little practice

is required to produce uniform sample layers. Whatman

GF/C filters for Bragg-Brentano diffraction and Metricel

VM-1 for transmission diffraction are the filters of

choice for this technique.

4. Analysis of amorphous components

The determination of component weight fractions using

Eq. (1) is done independent of the presence of

amorphous (or imidentified crystalline) materials. In

this relationship, W^. + EWj = 1 , where the EWj come

from the original sample. The test for amorphous

Filter Cassette

250 ml Aspirator

Bottle

0 4 Scale - cm

Plug-, Capillary

Hose section

Sample goes here

Figure 1. Small suspension chamber for loading of

powders on filters.

material in the original sample is (Chung, 1974a)

Me

w

(7)

where the summation term is simply the sum of all expressions of Eq. (1) for all components for which
B. L. Davis 11

Table 2. Matrix flushing analysis of four-component sample

Component Intensity Known Wj Measured Wj

ZnO 4661 4.35 0.3443 0.3641

CaCOj 2298 2.98 0.2800 0.2620

Silica Gel 0.1613 0.1595

AI2O3 (c) 631 1.00 0.2144

weight fractions can be measured. The left and right

terms of Eq. (7) will be equal only if all components

have been identified, are crystalline, and are included in

the analysis. No information is generated about the

amorphous components. What is actually happening

when this "scattering power" balance is performed is

that through division of measured intensities by the kj

all intensity levels are converted to that value which

would exist if all material were corundum! This then is compared with an empirically equivalent parameter

computed from the corundum peak intensity and known

added weight fraction.

EXAMPLE

knowledge of the composition of the amorphous

component so that its mass absorption can be calculated.

Generally this is not too demanding a requirement, since

materials such as fly ash, carbonaceous matter, limonite,

and opal, to cite some very common ones, are generally

well enough characterized chemically to allow their mass absorption to be computed within reasonable

limits. Where the pure amorphous material is available

this parameter may be measured experimentally.

Experimental measurement is best done by direct beam

transmission (Davis and Johnson, 1987; Davis, 1988,

Ch. 9) or substrate diffraction (Frevel and Roth, 1982).

The procedure for such analysis makes use of the simple mass absorption balance

F. Chung (1974a) illustrates his "Matrix Flushing"

technique with added amorphous silica gel in a mixture

of ZnO, CaC03, and AI2O3 (standard). Table 2

summarizes the data for the analysis. The added

corundum is W^ of Eq. (7) and the value of T-Q^IV^ for

the two crystalline components, ZnO and CaC03, is observed to be 1843 cps; this is compared to the right

hand term of Eq. (7) which is 2312. The intensity sum

from the conversion of the sample intensities to equivalent corundum scattering power falls short,

indicating the presence of imaccounted for material, in

this case the silica gel. The weight fraction of silica gel

is thus 1 - W^ - EW- = 0. 1595, in good agreement with

the known input value of 0. 1613. Experience in our lab

using this procedure is not as good; while the method is

theoretically sound, there are many sources for

signiflcant error which render the results more

semiquantitative than often required.

The method frequently used at the EMES lab

combines the "adiabatic" equation (Eq. 2) with mass

absorption measurements to ascertain the amount of

amorphous material present. This procedure provides,

in our experience, more accurate results but requires a

;t,=WM.+W^<TM=W^^+(l -1V>^ (8)

where the total sample bulk mass absorption, /ig, is measured by one of the above two methods, /i^, the mass absorption coefficient of the crystalline portion of

the sample, is computed from the multicomponent

crystalline analysis, using the relation /i^ = Wj/tj, Wj

being the original sample component weight fractions;

is the known (measured or calculated) mass

absorption coefficient of the amorphous component. W^

and W^ are the weight fractions of crystalline and

amorphous portions of the whole sample, respectively,

and are to be determined from the mass absorption

balance.

EXAMPLES

Single Amorphous Component

We recently analyzed a series of graphite samples of

varying purity. Material from Sonora, Mexico

contained significant clay, carbonaceous matter,
12 Quantitative Phase Analysis with Reference Intensity Ratios and Uses

Table 3. Analysis of an impure graphite

Component

If 1?1 Wi

Chlorite 0.73 2455 11254 0.086

Kaolinite 0.84 312 1440 0.010

Muscovite 0.48 1620 9726 0.113

Quartz 0.72 3192 8143 0.063

Pyrite 1.62 979 2292 0.008

Orthoclase 0.73 1599 4673 0.036

Graphite 4.9 146691 380066 0.431

Carb (Amorph) 0.253

Mb = 0.00322 g cm'^ /Xg = 20.48 cm^ g"^; /i^ = 25.4 cm^ g'; /tcraphiu = 6.0 cm' g"*. All Data using CuKa

radiation. Data courtesy of Superior Graphite Co., Chicago.

chlorite, quartz, and other impurities. Table 3 presents

the results of the crystalline and amorphous component

analysis of the sample loaded on a 4.25-cm Whatman

GF/C filter by aerosol suspension, using Eq. 2. The

raw intensities (I") and the intensities corrected for

transparency and matrix effects (I"), for peak overlap,

and for automatic divergence slits are also given in the

table. The bulk sample mass absorption was measured by

direct-beam transmission. Because of the presence of

silicate impurities the value of /ig was substantially

higher than that for pure carbon (4.2 cm^*) but much

lower than that (ji^ for the crystalline portion of the

sample. The mass absorption for the "pure" graphite,

also by direct-beam transmission, was 6.0 cm^g'*,

substantially higher than that expected for pure carbon

and is presumably due to adsorbed volatile gaseous

materials such as water vapor, oxygen, nitrogen, and

other minor impurities within the graphite layering.

The results of the mass absorption balance reveals the

presence of non-crystalline carbonaceous matter

intimately mixed with the graphite.

Two Amorphous Components

Urban aerosols commonly contain both silicious fly ash

(SFA) and carbonaceous fly ash (CPA). When two

amorphous components exist in the sample Eq. 8 can be

expanded to /i,=W>^+W,,M<„,+W„^^ (9)

where + W^^j = / - ^ is the conservation

condition allowing solution of this equation if the

crystalline fraction W^ can be determined independently.

In many cases clearing of the collection filter with

immersion oils permits estimation of this parameter

from polarizing optical microscopy. For the industrial

region of Orem, Utah, several analyses of airborne

particulate demonstrated both types of amorphous

material. Table 4 illustrates one of the analyses (Davis

1981a) from this area. Note that the reference material

in this earlier study is CaF2, not AljOj. Variance errors

s(W;) are included in this example. They are calculated

through a formal variance error propagation procedure

(Davis, 1981b).
B. L. Davis 13

Table 4. Analysis of ambient industrial aerosol containing two amorphous components

Comnonent I'f (Counts) W-1

Muscovite 0.078 2986 0.083 0.015

Gypsum 0,225 6367 0.061 0.011

A.aoiuuie rt 179 U. 1 /Z yJ / u.uiz U.UU't

Quartz 0.829 48,812 0.128 0.020

Calcite 0.537 17,586 0.090 0.036

Hematite 0.342 4251 0.027 0.016

CFA 0.280 0.226

SFA 0.320 0.234

jLCn = 42.8 cm^ g \ /i^ = 59.8 cm^ g'^ /^cfa = 8 cm^ g'^; /Xgr^ = 52 cm^ g"' All data for CuKa radiation. Reference

material for k, is CaFj.

4. Reference Intensity Ratio determination

Undoubtedly the most critical parameter in quantitative

XRD analysis is the RIR value. The RIR values

published up until 1988 in the JCPDS volumes were

provided by analysts without use of guidelines for

uniform methodology. Hence, sample preparation,

particle size and orientation, and source of standard

AI2O3 were not controlled. Powder Diffraction

published a table of selected RIR values and other

parameters in 1988 with a second expanded table printed

in Dec. 1989 (Davis et al, 1989).

A sample "type" is provided with each entry and

measured mass absorption values are given, where

available. Most workers are still using "Well"

preparations, including "side drifting", which may only

partially supress preferred orientation, and mass

absorption measurements are as yet not commonly

provided.

5. Experimental and calculated RIR's

for solid solutions

Included in the following sections are brief descriptions

of very recently published, or as yet unpublished

experimental and calculated data from the SDSM&T

laboratory on kj and /ij measurements for some solid

solution minerals. These data illustrate with a few

examples the utility of such data as well as the need for

more experimental work on solid solution series

compounds.

5. 1 Biotite minerals

Figure 2 is taken from Kath et al. (1991). Eleven

biotites were studied by XRD, XRF, and AA/ICP to obtain species chemistry, observed and calculated mass

absorption coefficients, and kj values, here denoted Lk;

to distinguish the pure phase method of measurement

from the mixing method. The value of these data lie in

the ability to predict reasonably accurate kj values from

a knowledge of Fe/(Fe + Mg) ratios or mass absorption

coefficients. The regression

Fe/(Fe + Mg)] of Fig. 2 is:

for Lk; vs. Wp. [ =

Lkj = 6.359Wp^ + 1.513 (R = 0.91)

5.2 Asbestiform amphiboles

Figure 3 presents data obtained from Davis (1990a)

and Davis (1992). The monoclinic asbestiform

amphiboles demonstrate satisfactory correlation of both

/ij and Lkj with weight percent FeO. Except for

anthophyllite, with space group Prima (hkf = 610), the

amphibole symmetry is C2/m with the strong reflection

hkf = 1 10. Thus in the figure the anthophyllite points
14 Quantitative Phase Analysis with Reference Intensity Ratios and Uses

BIOTITE PURE PHASE k,-

1 10000

Fe/(Fe + Mg)

Figure 2. Reference constant and mass absorption data

for biotites.

ASBESTIFORM AMPHIBOLES

140 1

0 I 1 1 • 1 1 1

1 ' : 1 . 1 0 0 5 10 15 20 25 30 35 40 Wt % FeO

Figure 3. Reference constant and mass absorption data

for asbestiform amphiboles.

are plotted for comparision but are not included in the

regressions.

For the monoclinic amphiboles these regressions are:

Lk; = 0.045Wf^ + 0.982 (R = 0.82)

Hi = 1.712 Wp^ + 45.9 cm^ g^ (R = 0.994)

where Wp^Q is the weight percent FeO in the

amphibole. These data should be regarded as preliminary, as far as establishing kj and fi^ trends are concerned, since many more such values for amphibole

subspecies should be obtained.

5. 3 Olivines

Jahanbagloo and Zoltai (1966) completed a comprehensive study of the complete olivine solid

solution series, including XRD spacing tables and

intensity normalizing factors, from which kj values

could be calculated. Additional intensities and

absorption factors were calculated for olivines by Borg

and Smith (1969), Using the method of Hubbard et al.

(1976) and Hubbard and Smith (1977), we converted the

component intensities to kj values. Fig. 4 presents

calculated density (p), mass absorption {p), and kj

values, for the full Mg/Fe variation of this series.

The linear regression for all 10 olivine members is:

k; = 0.00754Fa + 0.762 (R = 0.98)

where Fa is the mol percent fayalite, Fe2Si04 in the

member.

OLIVINE SOLID SOLUTION SERIES

200

150

p gm/an'

Density

100 -3

. -j^.

50 '- 2

kj = RIR r

• Borg and Smith (1969)

"

. A Jahanbagloo and Zoltai (1966) -

0

, ! , 1,1,1. 1,1,1,1,1,- 0

" 10 20 30 40 50 60 70 80 90 100 Mole Percent FejSiO^

Figure 4. Calculated reference constant and mass

absorption data for olivines.

5.4 Plagioclase feldspars

Several strong plagioclase feldspar peaks cluster

together in the 27.5-28° 26 region (CuKa radiation).

We have measured a special type of RIR consisting of,

for example, the 202+220 + 040+002 intensities of

oligoclase peaks existing within this short spectral

range. Similar constants were measured for other

plagioclase members. The mass absorption coefficient

and kj are plotted in Fig. 5 versus weight percent CaO

O^Cao)- Note from the kj scale of the figure that little

variation of kj exists for this peak cluster over the NaCa solid solution limits. Using an average or median k- value for a plagioclase analysis will closely determine its weight fraction in the mixture; however, the

composition remains unknown, unless its mass

absorption or CaO composition can also be measured in

the pure phase.

The regression equations for /x; and kj are:

m = 0.877Wcao + 34.92 (R = 0.99)

kj = 0.0098Wcao + 1-545 (R = 0.95)
B. L. Davis 15

PLAGIOCLASE FELDSPARS

I

40

nS

o* 30

20

•

•

1 . . , . 1 . . . 1

1.7

00

1-6 J,

- 1.5

5 10 Wt % CaO

IS

Figure 5. Reference constant and mass absorption data

for plagioclase feldspars.

References and Bibliography

(* Indicates Direct RIR Application)

Alexander, L. & Klug, H. P. (1948). Basic Aspects

of X-ray Absorption in Quantitative Diffraction

Analysis of Powder Mixtures. Anal. Chem. 20, 886-

889.

*Bayliss, P. (1986). Quantitative Analysis of

Sedimentary Minerals by Powder X-ray Diffraction.

Powd. Diff. 1, 37-39.

*Berry, L. G, (Editor) (1970). Establishment of a Reference Intensity. In Inorganic Index to the

Powder Diffraction File; Joint Committee on Powder

Diffraction Standards, (PD15-21).

Bezjak, A. (1961). X-ray Quantitative Analysis of

Multiphase Systems. Croatica ChemicaActa33, 197-

200.

*Bish, D.L. & Howard, S.A. (1988). Quantitative

Analysis Using the Rietveld Method. J. Appl. Cryst.

21, 86-91.

Borg, I.Y. & Smith, D. K. (1969). Calculated X-ray

Powder Patterns for Silicate Minerals. Memoir 122,

Geological Society of America.

Calvert, L.D., Sirianni, A.F., & Gainsford, G.J.

(1983). A Comparison of Methods for Reducing

Preferred Orientation. y4<iv. X-ray Anal. 26, 105-110.

Copeland, L.E. & Bragg, R. H.(1958). Quantitative

X-ray Diffraction Analysis. Anal. Chem. 30, 196-

201.

*Chung, F. H. (1974a). Quantitative Interpretation of

X-ray Diffraction Patterns of Mixtures. I. Matrix

Flushing Method for Quantitative Multicomponent

Analysis. J. Appl. Cryst. 7, 519-525.

Chung, F. H. (1974b). Quantitative Interpretation of

X-ray Diffraction Patterns of Mixtures. II. Adiabatic

Principle of X-ray Diffraction Analysis of Mixtures.

J. Appl. Cryst. 7, 526-531.

*Chung, F. H. (1975). Quantitative Interpretation of

X-ray Diffraction Patterns of Mixtures. III. Simultaneous Determination of a Set of Reference

Intensities.,/. Appl. Cryst. 8, 17-19.

Davis, B.L. (1981a). Quantitative Analysis of Crystalline and Amorphous Airborne Particulates in the

Provo-Orem Vicinity, Utah. Atmos. Environ. 15,

613-618.

*Davis, B.L. (1981b) A Study of the Errors in X-ray

Quantitative Analysis Procedures for Aerosols Collected on Filter Media. Atmos. Environ. 15, 291-296.

Davis, B.L. (1986). A Tubular Aerosol Suspension

Chamber for the Preparation of Powder Samples, for

X-ray Diffraction Analysis. Powd. Diff. 1, 240-243.

*Davis, B. L., Johnson, L.R., & Mebrahtu, T. (1986).

X-ray Quantitative Analysis of Coal by the Reference

Intensity Method. Powd. Diff. 1, 244-255.

Davis, B.L. (1988). Reference Intensity Method of

Quantitative X-ray Diffraction Analysis. Second

Edition, Davis Consulting, 213 pp.

*Davis, B.L. (1990a). Quantitative Analysis of

Asbestos Minerals by the Reference Intensity X-ray

Diffraction Procedure. Am. Jnd. Hyg. Assoc. 7. 51,

297-303.

*Davis, B.L. (1990b). The Reference Intensity Ratio:

Its Measurement and Significance. Powd. Diff. 5, 76-

79.

*Davis, B.L. (1992). Powder Patterns and Reference

Intensity Ratios for Asbestos Amphiboles. Grants-InAid Final Summary Report, Jan. 1992. International

Centre for Diffraction Data, Swarthmore, PA

Davis, B.L., & Johnson, L.R. (1982). On the Use

of Various Filter Substrates for Quantitative

Particulate Analysis by X-ray Diffraction. Atmos.

Environ. 16, 273-282.

Davis, B.L., & Johnson, L.R. (1987). The Use of

Mass Absorption in Quantitative X-ray Diffraction

Analysis. Adv. X-ray Anal. 30, 333-342.

*Davis, B.L., Smith, D.K., & Holomany, M.A.

(1989). Tables of Experimental Reference Intensity

Ratios: Table No. 2, December, 1989. Powd. Diff.

4, 201-205.

Devlin, D.J., & Amin, K.E. (1990). A Method for

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National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Standardless Method of Quantitative X-ray Diffractometry

Lev Zevin

The Institutes for Applied Research, Ben-Gurion University,

P.O. Box 1025, Beer-Sheva, Israel

Abstract

Quantitative analysis of a system of n phases can be carried out without analytical standards provided

that at least n mixtures of these phases are available. The solution to the "one peak one phase" case

is given, as is that for the full-pattern approach. For

the latter, cluster decomposition by profile fitting is recommended. Precision of the method is affected to

the greatest extent by a lack of compositional diversity of the initial sample set and the probable pres- ence of an undetected (amorphous) phase. A simple

test is presented for estimating the suitability of the

sample set for analysis and for evaluating the analysis error. Current and potential fields of application

of the phase quantification technique are discussed.

1. Introduction

Some years ago Hermans & Weidinger (1948) proposed a method for evaluating the crystalline fraction

in cellulose that does not require a sample of a pure

crystalline or a pure amorphous material. Their idea,

which was rather simple, and is based on the following obvious relationships for the crystalline fractions,

X and y, in two samples of native and regenerated

cellulose, respectively:

X _ (In)c 1 - a: _ (In)a

y-(I,)c' l-J/"(Ir)a'

where (In)c and {ln)a are the intensities of the

diffraction lines for the crystalline and amorphous

phases in native cellulose, and (Ir)c and (Ir)a are the

same quantities for regenerated material. Equations

(1) are easily solved for x and y. This method was

then extended to polyethylene and other polymers

and has been used successfully for the past 40 years

(Zevin k Messalem, 1982).

The potential of this approach became better un- derstood in the late 60s and early 70s, when quantification was extended to complicated mineral mixtures, and scientists realized the importance of scaling experimental data against proper, but not al- ways available, analytical standards. The ideas of

Hermans Sz Weidinger (1948) were revived in the

works of Moore (1968), Zavyalova (1970), Driz k

Salin (1972) and Zevin k Zavyalova (1974). More

rigorous theory followed (Zevin, 1977; Fiala, 1980;

Brown 1986) and completed the mathematical formalism of the method. However, despite the very

attractive self-calibration aspect of this method, its application in practice has been rather sluggish as a

result of a number of inherent drawbacks:

1. The method is severely affected by phase- or sample-related effects, such as preferred orientation, microabsorption, etc.

2. When the composition of the samples in the basic sample set are not diverse enough, precision

is low; and

3. There may be technical problems in preparation

of the basic sample set.

In the present work we propose ways to surmount

the inherent difficulties of the method that hinder its wide implementation.

2. Basic principles

The intensity of the diffraction peak is related to the

phase abundance by the Klug-Alexander equation:

_ Csj

_ Csj , .

where J,j is the intensity diffracted by phase j in

the sample s, Kj is a calibration constant, dj is the abundance of the corresponding phase, /z* is the

mass absorption coefficient of the sample s, and is the mass absorption coefficient of phase j. Suppose

that l,j is the intensity corrected for sample absorption, and fi* is obtained by an independent method.

Then I,j = ^sjfJ'*,- If an internal standard is used,

then I,j = Jsi/(Ji)st, where (J,)8t is the intensity

17
18 Standardless Method of Quantitative X-ray DifFractometry

diffracted by the internal standard. A simple equation holds for the corrected intensity:

(3)

To obtain the calibration constant Kj , and to carry

out an analysis of an n-phase sample by simple scaling against a pure phase (analytical standard), we

need n — 1 pure phases. The remaining nth phase is obtained from mass balance equation (4):

Y^Csj = l, l<s<n (4)

If we assign the first n— 1 numbers to the pure phases

and the number s = n to the sample to be analyzed,

then equations (3) and (4) will give matrix equation

(5):

/ 111 0

0 I22

V Inl In2

0

0

/ ai \

0C2

Inn / \ OCn /

( 1 \

1

V 1 /

, (5)

where aj = 1/Kj. Let us substitute the pure phases

in the first (n — 1) rows of the data matrix with samples containing the same n phases in different proportions:

/ In

I21

I12

I22 Im \

l2n

\ Inl In2

/ "1 \

"2

Inn / \ "n /

/ 1 \

1

V 1 J

(6)

Each row s in the data matrix (6) represents a diffraction pattern of sample s, and each column represents

the intensities of the diffraction peaks j in the samples. Equations (6) can be solved for aj coefficients,

which emphasizes the principle of the standardless

approach: we obtain calibration constants without

the application of a pure phase.

If the number of samples m exceeds the number of

phases n, then the solution to equations (6) may be

found by least-squares minimization:

(7)

a =l ifc =l

where I is the matrix whose elements are l,j, and

is its transpose. As soon as ocj are calculated, the

weight fractions, , may be obtained from equation

(3). The results may be improved somewhat by appropriate weighting of the samples. Samples with one prominent phase, approaching the ideal pure phase

situation, can be given greater weight than samples

with even phase compositions or with a suspected

undetected (amorphous) phase. Weighting may be

achieved by simple repeating of rows of more reliable

samples in the I,j matrix.

Another form of analytical equation is obtained

from equation (3), using the obvious relationships of

the weight fractions of any phase j in two samples s and r, as follows:

(8)

where Lnj = Isj/lrj is an intensity ratio.

After summation of the left and right hand sides of

equation (8) over all phases j, we arrive at a system of

m linear equations with n unknown weight fractions

Crj in the sample r, as expressed in the form:

ELrsjCrj = 1 for 1 < s < m. (9)

Again, if the number of samples m > n, the solution

to equation (9) for phase abundances Crj in sample

r may be found by least squares minimization:

1=1 k=l

(10)

where Lr, is the matrix whose elements are Isj/Irj.

A situation that may arise in practice is that in

which a sample's absorption is unknown, but the absorption coefficients of the component phases can be

calculated. The solution to the standardless problem may also be found in this case. Starting from

the basic equation (2), we obtain for the ratio of the

uncorrected intensities corresponding to phase j in

samples s and r:

(11)

Multiplying both sides by and summing over all phases j we arrive at:

(12)

By variation of the sample number 1 < s < m, and

by including the mass balance equation, we obtain

a system of linear equations for n unknown weight

fractions Crj of phases j in sample r:

E; /^i[l - J.i/JrilCrj = 0, 1 < S < m,

E;=1 Cr; = 1.

(13)
L. Zevin 19

Table 1: Data matrix for the 13 four-phase test samples

Sample Quartz Kaolin Pyrite Calcite

101 100 001 002 311 200 104

1 0 0 56 44 0 0 100

2 12 4.2 51 41 8 7 95

3 20 7 49 38 12 11 99

4 27 9 49 39 19 14.5 83

5 30 12.5 44 35 21 19 69

6 40 14 38 29 26 22 56

7 46 16 34 26 34 28 43

8 55 19 30 23 40 32 32

9 64 23 18 14 48 42 23.5

10 74 26 10 9 54 46 7.5

11 34.5 12 55 45 12.5 11 17.5

12 26 9 52 40 30 24 25

13 17 6 47 37 33 29 46.5

Table 1!: Results of phase analysis of the 13 four-ph ase test samp] as Sample Quartz, % Pyrite, % Calcite, % Kaolinite, %

Actual Deter- Actual Deter- Actual Deter- Actual Determined mined mined mined

1 0 0.0 0 0.0 30 26.3 70 66.4

2 7 7.4 6 4.6 27 24.3 60 59.8

3 11 12.3 10 8.0 24 26.1 55 56.2

4 15 16.6 14 11.7 21 21.9 50 57.2

5 19 19.5 18 13.9 18 18.3 45 51.1

6 23 24.8 22 16.7 15 14.6 40 43.2

7 27 28.9 26 21.4 12 11.3 35 39.1

8 31 34.6 30 25.2 9 8.4 30 34.2

9 37 39.7 37 31.3 6 6.2 20 21.2

11 20 21.3 10 8.4 5 4.6 65 64.9

12 15 16.2 22.5 18.8 7.5 6.9 55 59.8

13 10 10.5 27.5 21.5 12.5 12.2 50 54.9

We have applied the standardless method to 13

four-phase mixtures of quartz, pyrite, calcite and

kaolinite. Integrated intensities (corrected for absorption) are presented in Table 1. For phases with

two measured diffraction peaks the sum of the two

intensities was used in the analytical equations. Table 2 presents the phase abundances resulting from

the implementation of equation (7). Most of the

observed weight fractions fall within 10% (relative)

limits around the actual values, although for pyrite

the discrepancies are greater. The results reflect

the main drawbacks of the technique, at least in its present state — only moderate precision.

3. Full-pattern approach in the standardless

technique

The modern tendency in quantitative phase analysis

is utilization of the full diffraction pattern of a sample

for phase quantification (Smith et al. 1988; Madsen

k. Hill, 1990). This approach aids in minimizing some

sample- and phase-related problems that severely af- fect the precision of the analysis, such as preferred

orientation and microabsorption. This approach also

holds true for standardless methods.

There are two different techniques for processing

digitalized patterns. In the first one, known as profile

fitting, overlapping peaks are separated into components (or individual peaks) (Parrish ei ai, 1976; Will

et ai, 1983) This technique requires some knowledge

of peak shape, but does not require any structural

data. As soon as cluster decomposition has been car-
20 Standardless Method of Quantitative X-ray Diffractometry

ried out, and more than one peak is available for any

phase j, then this phase is represented by combined

intensity

= XlUifc(i,j)jfc,

k

where Ujt is the weight given to the k-ih peak of

phase j. Combined intensities are handled in the

conventional way, as outlined in the previous section. In the second technique, known as the continuous

pattern approach, the sample pattern is presented

as a weighted sum of the patterns of the individual phases. For the standardless technique discussed

here, this approach is expressed in the equation:

n

i,i = X!a,K^f, (14)

where s (= 1, 2, . . .

, m) is the sample index, i (=

1,2,... , ^) is the angle index or step number, and

J (= 1, 2, . . . , n) is the phase index. Each row s in the I,,- matrix represents the diffraction pattern of a sample, and each row of the Kj, matrix represents

the diffraction pattern of a phase, j. The matrix I can be represented as the product of two matrixes C

and K in an unlimited number of ways, and in the

general case, with C,j and Kj,- unknown, there is no

unique solution to equation (15). However, equation

(15) can be solved in one particular case. Suppose

that:

i) the number of samples m is equal to the number

of phases n; and

n) there are at least p zero elements in each row of

the Kji matrix. In other words, each phase does

not contribute to at least p diffraction peaks

among t diffraction peaks measured (t elements

of a row in K).

The K matrix is expressed as K = WI, where W =

C~^. For each zero element in the K matrix:

nY^WjsU, 1 = 1,2, ...p. (15)

s =l

The total number of equations (16) is p x n (n =

the number of rows in K.) We add n mass balance

equations:

^Cj, = l, (16)

j

bringing the total number of equations to p x n + n.

If p > n — 1 , then the total number of equations is greater than or equal to n^, and the number of un- known elements of W is n x n. Inversion of W will

give the required weight fractions Cj,. The system

of equations (16) and (17) is not linear and must

be solved by iterations starting from some approximate solution. Data on zero-contribution angles can be acquired from the Powder Diffraction File or by

calculation of the powder pattern from known struc- tural data.

Let us now return to the general case expressed

in equation (15) and consider the difference between

the observed intensity at any scattering angle 20,-,

I,,-, and the sum of contributions of the individual

phases:

n £« =I,.-^C.,K,.,.. (17)

Our problem may be formulated as the minimization

of the sum of squares over all m samples and over all

t scattering angles:

m t I n \

"^i"EE h"-Ec^iK,' . (18)

5=1 .=1 \ j=l /

The weight fractions C,j and the coefficients Kj,- are considered to be parameters of the minimization,

subjected to the constraints:

0<C,,- <1; K,i>0; = 1;

i

1 < s < m; 1 < J < n; \<i<i. (19)

It is possible that the problems can be solved by

iterations, starting from some approximate concentration data Cjj^ (Fiala, 1980; Zevin ti. ai, 1981).

The procedure is repeated up to the point where the

difference between the successive iterations C,,- —

C^J^^^ falls below the expected error. The greatest uncertainty lies in the set of initial concentrations C^j^ In order to ensure convergence to correct concentrations, C^J^ must be sufficiently close

to the global minimum of the sum. There is al- ways some probability of falling into one of the lo- cal minima, which might not be close to the correct

phase composition. Approximate phase abundances

can sometimes be obtained from elemental chemical data, supposing some ideal chemical composition

of the phases. Another way is implementation of

conventional X-ray techniques, with analytical standards, which do not exactly correspond to the analyzed phases, but can nevertheless be used for scaling

and evaluation of the starting values C^J^.

We have applied this technique for the analysis of

Israeli phosphorites consisting of three major phases,

dolomite, calcite and francolite (carbofluor apatite
L. Zevin 21

Table 4: Analysis of the major phases in phosphate rock

Sample Francolite, % Calcite, % Dolomite, %

Starting X-ray Chem. Starting X-ray Chem. Starting X-ray Chem.

value anal. anal. value anal. anal. value anal. anal.

c(i) c(i) C(i)

1 40 42.0 43.5 30 35.2 36.9 30 21.8 18.5

2 20 18.0 15.2 60 72.7 76.8 20 9.2 7.6

3 70 68.4 69.7 30 29.2 27.5 0 1.8 1.9

4 50 51.4 50.9 30 36.4 37.8 20 10.3 9.0

5 80 81.8 82.5 0 11.7 12.4 20 6.3 7.0

6 50 37.1 50 47.5 0 12.1

Caio(P04)6-i;(C03)j;F2), and several minor phases,

quartz, gypsum and organic matter. Since the con- centration of organic matter was less than 1%, it was

ignored. The gypsum was removed by water treat- ment, and quartz was analyzed by conventional scaling against a pure quartz sample. The remaining

three phases were analyzed by applying minimization procedure (19), as follows. Twelve samples with

different proportions of phases were prepared from

several samples of phosphate rock by size separation.

The absorption correction was acquired by measure- ments of Compton scattering (Bar On ei ai, 1981).

The integrated intensity was collected in the 12 scatTable 3: Angular intervals and contributing phases

in analysis of phosphates. Phases are 1 - Francolite,

2 - Calcite, 3 - Dolomite, 4 - Quartz

Sample Angular range Phases

no. 29 , degrees

1 21.0- 23.6 1+2

2 25.0- 26.2 1+2

3 26.2- 27.0 4

4 28.5- 30.0 1+2

5 30.2- 31.2 3

6 31.2- 34.6 1 7 35.0- 36.4 1 + 2

8 39.6- 41.5 1-^2-1-3-1-4

9 42.6- 44.0 1

10 44.0- 49.10 1 + 2

11 49.1- 50.2 1 12 50.2- 53.5 1 + 3

tering angle intervals shown in table 3, together with an overlapping pattern. Initial concentrations were

obtained by scaling against pure phases, using separate diffraction peaks. Bulk chemical analysis was

carried out for five samples, and the phase composition was derived from chemical data, taking into ac- count PO4 —>C03 substitution in francolite (Zevin

ei al. 1988). Only a few iterations were required to

reach the final results given in table 4 for some of the

analyzed samples. We found that the minimization

converged to a composition very close to that derived

from bulk chemistry even, in the cases when the initial concentration was rather far from true values.

Two full-pattern approaches used in quantitative

phase analysis, i. e., pattern decomposition and the

continuous pattern approach, resemble two similar

techniques applied in structure refinement from powder diffraction data. In the first approach, conventional "single crystal" methods of structure analysis are implemented after pattern decomposition into

separate diffraction peaks. The second approach devised by Rietveld seems to be used overwhelmingly

in structural studies, but the first one is more attractive in the standardless quantification techniques discussed here. Indeed, weak peaks that are difficult to

separate have as much bearing in structural analysis

as the strong ones. In the quantification procedure, on the other hand, weak peaks are less suitable for

analysis, and greater weight is given to strong peaks,

which can be separated with better accuracy. The

continuous pattern approach, expressed in equations

(15), is certainly overburdened with a large number

{n X t) of Kji constants, thus increasing the chance

falling into a false minimum. In pattern decomposition the overlapping problem is treated independently of quantification, conferring an advantage on

this technique. Unfortunately, very little has been

done in experimental comparison of these two approaches, though full-pattern utilization must be the

main path to further development and hopes for standardless quantification.

4. Number of phases in analyzed samples

Although the number and identity of the phases

present in an analyzed sample are obtained by wellestablished manual or automatic identification procedures, it is sometimes useful to apply an independent
22 Standardless Method of Quantitative X-ray Diffractometry

check. Bezjak (1975) noted that if a set of m samples

each contains no more than n < m identical phases,

the intensity diffracted by any sample s at any scat- tering angle 29i is a linear combination of intensities

diffracted at the same angle by n other samples. The

number of phases may be established applying these

criteria. Relatively simple and easily programmable

mathematical tests (factor analysis) are available for

this purpose.

The number of factors (phases in our case) re- sponsible for the variability of the intensities in the

Isi matrix is equal to the minimum number of lin- early independent rows or columns, or to the matrix rank. Technically, it is simpler to determine the

rank of symmetrical square (m xm ot t xt) matrices

R' = II"^ or R" which is equal to the rank of

the original matrix I. The rank of R matrices is equal

to the number of non-zero rows in the reduced tri- angular form (Wallace, 1960; Wallace & Katz, 1969)

and to the number of non-zero eigenvalues (Ritter ei ai, 1976).

Due to inevitable errors in intensities, which contribute to the variability of the I,,- matrix, its rank

will be equal to m or n, whichever is smaller. This is why when we separate major factors (phases) that

affect the variability I„- from insignificant factors

caused by experimental errors, we have to take the

latter into account. An element of the R matrix is expressed as:

t Rjk = J2^ijlik, (20)

i=l

and the standard deviation a(Rjk) is obtained by the

general rules of error propagation. As a rough estimate, suppose that the mean element of the R matrix (R) m m(I)^, where (I) is the mean intensity in

the data matrix, and m is the dimensionality of the

R matrix. Then a{{R)) w 2m(I)cr((I)) = 2(R)£((I)),

where £((!)) is the relative error in the intensity measurements, which can often be supposed to be a constant. The R matrix is reconstructed from eigenvectors Xj and eigenvalues A,-. If there are r non-zero

eigenvectors, then

r

R'- = ^x.A.xf, (21)

i=l

where the superscript r in R*" indicates that reconstruction was carried out with the full number of non-zero eigenvectors. Now suppose that only the

first h eigenvectors of the full row of r eigenvectors,

arranged in order of decreasing eigenvalues, corre- spond to statistically significant factors. Then the

R'' matrix reconstructed by summation over only

the first h eigenvectors will be statistically equivalent to the original R matrix. If n is the number of

detected phases and h > n, then h — n phases were

undetected.

We have applied this test to the 13 x 7 I,,matrix

given in Table 1. Eigenvalues calculated for RtxT =

I I are shown graphically in figure 1. The steep

drop in eigenvalues from A4 to A5 emphasizes the

presence of four major factors, in our case the four

phases. The R"* matrix reconstructed from the

first four eigenvectors is statistically indistinguishable from the original matrix.

0 2 4 • I

nunii>«r

Figure 1: Eigenvalues of the product matrix R =

II . • - Nonvariant set of 13 4-phase samples, o - Amorphous phase with maximal abundance of 30%

is added to 8 of 13 samples.

This technique will not, however, reveal an amorphous phase that might otherwise go undetected.

The amorphous phase will dilute the sample and re- duce the intensities of all the crystalline phases in the

sample, but it does not change the rank of the intensity matrix. Various amounts of amorphous phase

were added artificially to 8 of 13 four-phase samples with the intensity matrix shown in table 1. The

abundance of the amorphous phase varied from 5% to

30%. The character of the eigenvalues' distribution

did not change (Fig. 1), emphasizing the presence of

the same four crystalline phases.
L. Zevin 23

5. Precision of the standardless technique

Three factors affect the precision of the standardless

technique:

i. Intensity errors,

ii. Uncertainty in the mass balance equation (4), and

Hi. Diverse phase compositions in the samples.

It turns out that all three factors can be brought

together in one compact formulation. Suppose the

total weight fraction of all the detected phases in

sample s is equal to bj < 1. Then a more general

form of equation (6) is:

^IsjCXj =hj, s = 1,2, ... , m. (22)

Estimation of the error in the elements of the aj

vector is given by Zevin k. Zevin (1989).

1-e (23)

where e = 61/1 is the relative error in intensity measurements, which is supposed to be constant through

all Ij,-, Ab, = y/Ylsi^^')'^ deviation of the ac- tual sum of the weight fractions from 1, m is the

number of samples and C(I) is the condition number

of matrix I, defined as:

C(I) = ^A^(I^I) X A^([I^I]-i), (24)

where A,„(I^I) is the maximal eigenvalue of product

matrix I I. Three sources of error noted in the beginning

are clearly separated in equation (24) with condition C(I) being responsible for the sample's diversity. Condition C(I) takes its minimal value of

\/{n — l)/n (« 1 for large n) in the case of the

most diverse samples, when we have one analyzed

sample and n — 1 pure phases (analytical standards)

[equation (5)]. Substitution for the pure phases by

unknown samples inevitably reduces the diversity of

the sample set, increases the condition number C(I),

and increases the analytical error. Equation (24) provides us with an excellent tool

for checking the "goodness" of the sample set. The

following example confirms this premise. The experimental data presented in table 1 were used to calculate the "true" calibration constants (Kjo) as slopes

of regression 1,^ against true phase abundances Cj.

Subsets of four or more samples were chosen at ran- dom from the general group of 13 samples, and for

each subset calibration constants Kj were calculated

from equation (7). The relative error in the calibration constants, 8K/K = J2j l^i ~ ^jo\/^jo was

plotted against the condition number of the intensity

matrix for the corresponding subset (Fig. 2). The

correlation of the analytical error with the condition

number is obvious. The value 6K/K remains within

eondltlon.numbar

Figure 2: Precision of analysis as a function of the

condition number of the intensity matrix.

the reasonable 10% range if the condition number

does not exceed 10. The presence of an undetected

(amorphous) phase was simulated by reducing the

intensities in randomly chosen rows of the intensity

matrix in table 1. This resulted in an dramatic in- crease of the analysis error. The value of 6K/K increased from ~ 0.09 to an unacceptably high value of

0.18 when the presence of an amorphous phase with

a maximum abundance of > 30% was simulated in 8

of 13 samples.

Intensity errors affect the results of standardless

analysis to a lesser degree than variability of the samples. We can expect, however, that at an approximately constant condition number, the precision of

the analysis will improve with increasing number of

samples due to increased precision in intensities. An

example is given in figure 3. The initial set of four

samples was well conditioned, with condition number ~ 10. Additional samples added subsequent to the initial set did not improve the condition number,

but contributed to intensity precision and improved

the results of analysis.

If the diversity of the sample set is unsatisfactory, it can be improved by using various materialhandling procedures known from mineral processing.

They include size separation by sifting, gravity sepa-
24 Standardless Method of Quantitative X-ray DifFractometry

2 •* « I 10 12

n um b«r_o f_sainp l*s

Figure 3: Improvement of precision of analysis by

subsequent addition of samples to the well conditioned initial set.

ration by sedimentation in various liquids, magnetic

separation, flotation, chemical treatment, etc. These

procedures are usually selective for certain phases

and produce samples with different proportions of

the same phases.

6. Conclusions

Despite their long history standardless techniques

have not yet found adequate application. Two-phase

systems, such as ordered and disordered phases in

polymers, are the sole object of its practical utilization. This can be partially explained by the lack

of theoretical development and full understanding of

the sources of main problems, which sometimes re- sult in confusing outputs. It seems that by now the

method has been more or less developed, including

requirements to the initial group of samples and intensity precision. We can now outline the fields and

modes of its application. It does, however require

thorough selection and very often special preparation

of the sample group used to evaluate calibration constants. The number of samples in this group must ex- ceed the number of phases, and compositions of samples must themselves be compositionally variable. As

soon as scaling has been achieved with a prepared

sample group, and calibration (scaling) coefficients

have been evaluated, analysis of other samples, composed of the same phases in different proportions,

may be carried out by conventional techniques of

phase quantification. Thus, the standardless methods discussed here can be considered as methods for

calibration or scaling of intensities, a necessary step

in phase quantification.

Taking into account the labor and skill required

for initial preparations, the standardless technique

is suitable for large-scale analysis when many samples must be analyzed, except for the initial group.

It is certainly unsuitable for sporadic analysis of a few samples. One cannot expect exclusive precision

for the standardless technique due to the multiple

intensity measurements needed for evaluation of single weight fraction. The virtues of this method lie in its potential accuracy, based on its self-calibrating

ability.

References

Bar-On, P., Zevin, L. & Lach, Sh. (1981). X-ray Spectrom., 10, 57-60.

Bezjak a. (1975). Anal Chem. 47, 790-793.

Braun G. E. (1986). J. Appl. Cryst. 19, 217-221.

Driz, V. A. & Salin, a. L. (1972). Int. Clay Confer.

Proceedings. 577-588, Madrid.

Fang J. H. & Zevin, L. (1985). J. Sedim. Petrol. 55,

611-613.

Fiala, J. (1980). Anal. Chem. 52, 1300-1304.

Hermans P. H. & Weidinger A. (1948). J. Appl. Phys.

19, 491-506.

Madsen, I. C. & Hill, R. J. (1990). Powd. Diffr. 5,

195-199.

Moore, C. A. (1968). Clays and Clay Minerals 16, 325-

336.

Parrish, W., Huang, T. C. & Ayers, G. L. (1976). Trans. Amer. Cryst. Assoc. 12, 55-73.

RiTTER, G. L., LOWRY, S. R., Isenhour, T. L. &; WiLKlNS, C. L. (1976). Anal. Chem. 48, 591-595.

Smith, D. K., Johnson, G. G. & Wims, A. M. (1988). Austr. J. Phys. 41, 311-321.

Wallace, R. M. (1960). J. Phys. Chem. 64, 899-901.

Wallace, R. M. & Katz, S. M. (1964). J. Phys. Chem.

68, 3890-3892.

Will, G., Parrish, W. & Huang, T. C. (1983). J. Appl. Cryst. 16, 611-622.

Zavyalova, L. L. (1970). Dissertation. Irkutsk Institute

of Rare Metals, Russia.

Zevin, L. S. (1977). J. Appl. Cryst. 10, 147-150.

Zevin, L. S. Zavyalova L. L. (1974). Quantitative Xray Phase Analysis, Nedra.

Zevin, L. Lach S., Levi Y. & Pregerson B. (1988).

Int. J. Min. Process. 24, 235-245.

Zevin, L. & Messalem, A. (1982). Polymer. 23, 601-

604.

Zevin, L. S., Shneck, R. & Zevin, L M. (1981).

Twelfth Intern. Congr. of Crystal. Collected Abstracts

C-280-281.

Zevin, L. S. & Zevin Sh. L. (1989). Powder Diffraction

4, 196-200.

j
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraciion II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Reference Intensity Ratios, Whole Pattern Fitting and Standardless X-ray

Quantitative Analysis

Robert L. Snyder

InstHuie for Ceramic Superconductivity, New York College of Ceramics at Alfred University

Alfred, NY 14802 USA

Abstract

A consistent set of notation is developed for each of

the RIR based methods for carrying out quantitative

analysis by X-ray powder diffraction. The so called

"standardless" analysis procedures are shown to be a

special case of the internal-standard method of analysis where the normalizing assumption is used. All

analytical methods, other than whole pattern matching procedures, require the use of explicitly measured

standards, typically in the form of RIR values; however, if only semi-quantitative results can be tolerated, the standards may be obtained by using published RIR and relative intensity values. The exciting new techniques of whole pattern fitting and

Rietveld constrained quantitative analysis are also

described in RIR notation and shown also to be the

internal- standard method with the normalization as- sumption. The quantitative results obtained from

Rietveld analysis are derived from computed standards in the from of computed, normalized, RIRn

values. The normalization assumption in Rietveld

analysis allows the exclusive use of computed standards and comes as close to a "standardless" analysis as one can achieve: completely relying on the

absence of amorphous material and on the validity

of the structural models. Relationships are given for

obtaining quantitative analysis from the RIR;^ val- ues obtainable from the least squares scale factors in

the Rietveld and pattern matching procedures.

Introduction

The internal-standard method is the most general

of any of the methods for quantitative phase analysis by X-ray powder diffraction. In recent years

this method has been generalized and recast into

Reference Intensity Ratio (RIR) notation (see for

example, Hubbard & Snyder, 1988, and Snyder

& Bish, 1989). Previous to these developments

Bezjak (1961), Karlak t Burnett (1966) and later

Chung (1974a,b) introduced a misnamed "standardless" analysis procedure which Chung called the "adiabatic principle" or "matrix flushing method" which

uses the RIR concept, but not the notation. This

method which we prefer to call the Normalized RIR

Method still relies on the RIR values as standards.

More recently (Snyder et al., 1981, 1982; Snyder k

Hubbard, 1984) the AUTO/NBS*QUANT84 system

implemented the internal-standard method with Ref- erence Intensity Ratio notation in addition to the

normalized RIR method. This system also implemented the use of overlapped lines and chemical constraints via an RIR restatement of the Copland Sz Bragg (1958) procedure. D. K. Smith and colleagues

(1986, 1987, 1988, 1989) have described the use of

the full pattern to even further enhance quantitative XRD analysis. The most recent developments

have been the introduction of structural constraints

while using the whole pattern via Rietveld analysis

of multi-phase mixtures (Hill & Howard, 1987, Bish

k Howard, 1988)

The purpose of this paper is to follow-up the Hubbard k Snyder (1988) paper and develop a consistent

set of notation to describe each of the RIR based

techniques in terms of this notation. It will also address the relatively recent misuse of the term "standardless analysis" : the normalized internal-standard,

whole pattern matching and Rietveld constrained

methods, will all be shown to be similar forms of the

internal-standard method, requiring the normalizing

assumption that the sum of the crystalline phases

contributing to the diffraction lines be unity. In addition, they must rely on standards in the form of

previously measured RIR values or crystal structure

parameters.

The internal standard method

The intensity of a diffraction line i from a pure phase

25
26 Reference Intensity Ratios

a in the form of a flat plate on a powder diffractometer using Bragg-Brentano geometry is given by:

Iia = (1)

The intensity of a diffraction line i from phase a in a mixture of phases is given by (Klug & Alexander,

1974):

(2)

where

• fi = the linear absorption coefficient,

• Xq = weight fraction of phase a,

• Pa = density of phase a.

The mass absorption (i. e., mass attenuation) coefficient of the mixture is.

#o/ phases

(3)

Kg is a constant for a particular experimental sys- tem,

a 9 /I Ke = -T-. cos' 2dm 1 , I , D47rr VmeC'

(4)

where

• 7o = incident-beam intensity,

• r = radius of the diffractometer,

• A = wavelength of the X-radiation,

•

{^ m^J^

is an electromagnetic radiation term

required for X-rays and omitted for neutrons,

• c = speed of light,

• e and me = charge and mass of an electron,

• IQm = the diffraction angle of the monochromator.

Kia is a constant for each diffraction reflection i from

the crystal structure of phase a,

2/ l-hcos^2g

V;2l^-I l^sin^^cos^;, (5)

• Mi = Multiplicity for reflection i,

• Va = volume of the unit cell of phase a,

• ( slili^^g cos g

)' ~ Lorentz and polarization cor- rections for the diffractometer (Lpi),

• Fia = the structure factor for reflection i, which

relates the intensity to the crystal structure parameters:

#o/ atoms in cell Fhki = ^

fj exp 2Tri{hxj + kyj + tzj),

(6)

where

• hkl = Miller indices of line i of phase a,

• Xj,yj, Zj = fractional coordinates for atom j,

• fj = the atomic scattering factor for atom j:

fj — fcorr exp

Bj sin^ 9

(7)

where

• fcorr = The atomic scattering factor corrected

for anomalous dispersion namely:

|/corrr = (/o+A/')' + (A/")f//\2

(8)

where

• Bj — Debye-Waller temperature factor for atom

i in phase a,

• ^ = Bragg diffraction angle.

The fundamental problem in quantitative analysis

arises from the (^)m term in equation 2. To solve

for the weight fraction of phase a we must be able to

compute (^)m and this in turn requires knowledge of

the weight fractions of each phase. This cyclic argument can be stated in mathematical terms as the sys- tem being underdetermined. The internal-standard

method is based on the elimination of the absorption

factor (^)m by dividing two equations of type 2 giving equation 9, which is linear in the weight fraction

of phase a:

Ij s (9)

where

Equation 9 is linear and forms the basis of the

internal-standard method of analysis. The addition
R. L. Snyder 27

of a known amount, (Xg), of an internal-standard to a mixture of phases, which may include amorphous

material, permits quantitative analysis of each of the

components of the mixture by first establishing the

values of k for each phase, (i.e. the slope of the

internal-standard plot of equation 9) from standards

of known concentration.

11Icorundum

The slope, k, of the plot of

Xs— VS.Xa

is a measure of the inherent diffracted intensities of

the two phases. Visser & DeWolff (1966) were the

first to propose that k values could be published

as materials constants if the concentration and the

diffraction lines of the phases a and s were specified

and s itself were agreed to by all researchers. Their

proposal was to use corundum as the universal internal standard and to use the 100% lines of phase

a and corundum mixed in a 50% by weight mixture.

This I/Icorundum Or I/Ic value has been widely ac- cepted and over 4,400 are published with patterns

for standards in the Powder Diffraction File of the

International Centre for Diffraction data.

The generalized Reference Intensity Ratio

The concept of the I/Icorundum valuc as a materials

constant leads naturally to a broader definition which

permits the use of reference phases other than corun- dum, more than just the 100% relative intensity lines

and arbitrary concentrations of the two phases a and

s. The idea of the RIR has developed in the literature over the last 15 years (Chung, 1974a,b; Hubbard

ei ai, 1976). The most general definition of it (Snyder & Hubbard, 1984; Hubbard k Snyder, 1988) is:

RIR<^.' = t'-^^' (10)

where

/J*'

is the relative intensity of line j scaled to

the 100% line. In equation 10 we see that the RIR

is a function of a and s only, not of i, j, or X,. It, of

course, remains the slope of the calibration curve for

phase a with internal-standard s but now has been

scaled so that it may be computed from any pair of

diffraction lines in a calibration mixture. We also see that I/Ic values are simply RIR values where s

is corundum. The RIR, when accurately measured

for two well crystallized and prepared specimens, is a

true constant and allows comparison of the absolute

diffraction line intensities of one material to another.

It also enables quantitative phase analysis in a number of convenient and useful ways.

The RIR value in equation 10 may be obtained by

careful calibration, in the traditional way that the

slope of the internal-standard curve would be determined, or it may be derived from other RIR values

via,

RIRct,s'

RIRa.s =

RIR,

(11)

where s' is any common reference phase. When s' is corundum, the RIRa,s' values are simply I/Ic values.

Hence, we can combine I/Ic values for phases a and

s to obtain the Reference Intensity Ratio for phase

a relative to phase s.

It is useful to express the generalized RIR in terms

of the fundamental constant Kia- This can be done

by combining equation 2 with 10 to give

T<r. n F^'

RIRa,, (12)

The RIR may also be obtained from pure standards run under the same conditions by combining equations 2 with 12

TO (H) jrTrel T3JT3 _ jMa 2_£_lf. ^i^a,s — JO /^x jrel > (13)

where 1° is the intensity of line i from a pure phase.

Measurement of RIRs

The simple 2-line procedure for measuring I/Ic is quick but suffers from several drawbacks. Preferred

orientation commonly affects the observed intensities

unless careful sample preparation methods, such as spray drying, (Cline & Snyder, 1983, 1985) are employed. Other problems include extinction and microabsorption (Cline k. Snyder, 1987), inhomogeneity of mixing, and variable crystallinity of the sample due to its synthesis, history and possible solid

solution (Gehringer et ai, 1983). Even the problem

of having a representative number of crystallites in

diffracting position is more serious than one would at

first presume: Smith (1992) has recently expanded

the classic particle statistics study of Alexander et

al. (1948) showing that, even for a relatively low absorbing quartz sample diffracting CuK^, crystallite

sizes of less than 1.0 /i are required to achieve a 1%
28 Reference Intensity Ratios

relative error at a 2.3<7 confidence. All of these ef- fects conspire to make the published values of ///<.

subject to substantial error. For greater accuracy Hubbard h Smith (1976)

recommended using multiple lines from both the

sample (a) and the reference phase (corundum, in

this case) and using multiple sample mountings.

Such an approach often reveals when preferred orientation is present and provides realistic assess- ment of the reproducibility in the measurement of

//7c. Traditional implementations of the internalstandard method usually use a single diffraction

line from each of the phases to be analyzed. The

AUTO/NBS*QUANT84 (Snyder ei al. 1981, 1982;

Snyder k, Hubbard, 1984) system implemented the

use of multiple lines from each phase to improve

statistics and help average out orientation errors. More importantly, this system rigorously propagated

counting and statistical errors to evaluate their influ- ence on the final analysis results.

Although comparison of the intensities in the observed pattern to those in a computed pattern will

allow a quantitative measure of the amount of preferred orientation normal to the sample surface (Snyder & Carr, 1974), it is never recommended to use

a computational procedure in place of proper experimental technique. In general spray drying the samples (Smith et al., 1979a, 6; Cline k. Snyder, 1985) is the best way to eliminate this serious source of error. In the spray-drying process the crystallites of the

powder sample are agglomerated into spheres. The

crystallites spread on the surface of these spheres ex- pose all crystallographic surfaces to the X-ray beam.

Effects of agglomerate inhomogeneity and microabsorption have been shown to be minimal (Cline &

Snyder, 1985). Spray dried powders have the disadvantage of acting like a somewhat viscous liquid but

this is offset by their ability to fill a sample holder

producing an X-ray infinite thick mount. Other techniques which minimize orientation, like powder on

tape, air dispersions captured in filter paper, or powder dusted into grease on a zero background holder,

introduce a 26 dependent sample transparency error, due to the thin layer of diffracting specimen, which

invalidates the general RIR equation 10.

Quantitative analysis with RIRs

The internal-standard method of quantitative analysis when stated in terms of an RIR comes from

rearranging Equation 10 (Hubbard k, Snyder 1988),

j\.o, —

re/

s

Ij, ir^' RIRo

(14)

Taking s to be an internal standard and substituting equation 11 into Equation 14 and rearranging we

have

_ lia RIR.

T. Trel pjp (15)

This equation is quite general and allows for the analysis of any crystalline phase with a known RIR in an unknown mixture by the addition of a known

amount of any internal standard. However, if all four of the required constants (I^^' , Ijf , RIR,^c and

RIRa^c) are taken from the literature the results

should be considered as only semi-quantitative, since

each of them may contain significant error when compared to the actual phases in a particular unknown.

Of course, the RIRs may also be calculated (Hubbard et al, 1976) when the crystal structures of the

two phases are known. However, the calculated val- ues show considerable variation from observed val- ues depending on assumptions in the computational

models. In addition the computational model cannot

allow for specific specimen conditions like poor crystallinity, microabsorption, extinction and preferred

orientation. In general, if the RIRs must be calculated the analysis should also be considered only

semi-quantitative. These comments also apply to the

Rietveld whole-pattern fitting method described below.

Equation 14 is valid even for complex mixtures which contain unidentified phases, amorphous

phases, or identified phases with unknown RIRs.

It, in fact, is simply a restatement of the internalstandard method in RIR notation.

The normalized RIR method

Hubbard k Snyder (1988) showed that the ratio of

the weight fractions of any two phases whose RIRs,

with respect to some reference s, are known may al- ways be computed from

Xa ha RIRp^a

Trel (16)

Ijp IV-J RIRa,s

If the RIR values for all n phases in a mixture are known, then n-1 equations of type 16 may be written.

It makes no difference if the reference phase, 5, is one

of the n phases being analyzed or if it is arbitrary like

corundum. Bezjak (1961), Karlak k Burnett (1966)
R. L. Snyder 29

and later Chung (1974a,b; 1975) pointed out that if no amorphous phases are present an additional equation holds:

(17)

k=l

This normalization equation permits analysis, without the addition of any standard to the unknown

specimen, by allowing us to write a system of n equations and solve for the n weight fractions via (Hubbard & Snyder 1988)

lia

rel RIRalia

if^of phases - -1

E (18)

Chung referred to the use of equation 17 as the matrix flushing or the adiabatic principle. The term

"Standardless Analysis" is coming into frequent mis- use or at least misleading use. Recent papers on the

RIR internal-standard method and its implementation in the whole pattern matching techniques have

used the phrase "standardless" in ways that have

caused some researchers to believe that the treat- ment of references, implicit in the RIR and nor- malized RIR approaches, are no longer required for

quantitative analysis. In fact, standards are still re- quired in all truly quantitative X-ray analysis procedures, and the quality of the assumptions which

go into the computation and measurement of RIRs

or their related pattern matching scale factors, along

with the assumption that no amorphous phases are

present, are often so poor as to make the method

semi-quantitative at best.

It is important to emphasize that the presence of

any amorphous or unidentified crystalline phase invalidates the use of equation 18. All crystallites have

some degree of amorphous character on their surfaces, due to the broken bonds in the surface. In

some materials this amorphous layer penetrates quite

deeply and produces a substantial concentration of

non-diffracting or diffusely diffracting material, invalidating the normalization assumption. This effect

is particularly important when calculated RIRs

are used, in that the computational model can- not allow for the effect of the bulk material's transformation into the amorphous layer. The loss of

crystallinity in the surface of the crystallites, where

the incident X-ray beam is the least absorbed, will

strongly distort the RIR from an ideal calculated

value. Of course, any sample containing unidentified

phases cannot be analyzed using the normalization

assumption in that the required RIRs will not be

known.

When the RIR values are known from another

source, for example published I/Ic, or computed val- ues, it may be tempting to use them along with

the F*' values on the PDF card to perform a completely (incorrectly called), "standardless" quantitative analysis using the normalization equation 17.

These Z""*' and I/Ic values are seldom accurate

enough to be used directly in quantitative analysis.

The analyst should accurately determine the relative

intensities and RIR values for an analysis by careful

calibration measurements from materials similar to

those observed in the specimens to be analyzed. It needs to be stated that this method is far from standardless. The essential standard information is simply being obtained from the literature or computations, adding substantial uncertainty to any derived

semi-quantitative results.

Constrained XRD phase analysis: the

generalized internal standard method

A quantitative analysis combining both the X-ray

diffraction and chemical or thermal results with a knowledge of the composition of the individual

phases can yield results of higher precision and ac- curacy than is generally possible with only one kind

of observation. The analysis becomes more complex

when several phases in a mixture have similar compositions and/or potential compositional variability,

but it is possible, with appropriate constraints during

analysis, to place limits on the actual compositions

of the constituent phases.

The most general formulation of these ideas

has been described by Copeland & Bragg (1958).

Their internal-standard equations for multicomponent quantitative analysis with possible line superposition and chemical constraints take the form of a system of simultaneous linear equations; each equation refers to the individual phase contributions to an integrated intensity over some 20 range. In terms

of RIR values each equation is of the form

^1 Jl'J

T. ( TV ^js

el RIRq,s)-^ +

rel X,.

+ ...

+(^i27/2.,,)^ + e. (19)
30 Reference Intensity Ratios

where

• is the integrated intensity of the i*'' 26 range

from the mixture of phases, which may contain

contributions from one or more lines from one

or more phases in the mixture,

• Ij, is the intensity of a resolved line j of the

internal-standard s,

• is the weight fraction of phase uj in the mixture of sample plus internal-standard,

• c is an error term.

Line overlap is allowed for by having each intensity ratio (7^) have contributions from multiple lines

from multiple phases. As many terms involving contributions to the intensity ratio as needed are included in each linear equation. The Copland-Bragg

analysis results in an overdetermined system of n

equations in m unknowns, which may be solved for

the weight fractions via least squares. Of course, the

normalizing assumption can be imposed here also.

Quantitative elemental data, obtained from X-ray

fluorescence analysis for example, can be added to

the system of equations, without increasing the number of unknowns, making it overdetermined. Coupling both the elemental and diffraction data will

result in more accurate quantitative analysis of multicomponent mixtures. More importantly, estimates

of the standard deviations of the results will be more

accurate. Both the line overlap and chemical analysis

features of the Copeland-Bragg formalism have been

included in the AUTO (Snyder ei al. 1981, 1982)

and NBS*QUANT84 systems (Snyder & Hubbard,

1984). An example matrix equation for such an analysis might look like:

where

Y = RX, (20)

Y =

/n.

%Si

%Al

%Fe

V %o )

x =

\x^ )

and R is the matrix

ircl '

rrel

V (%Oo)

(%Siy)

Full-pattern fitting

Quantitative analysis by X-ray powder diffraction

has evolved since the 1930's starting with simple

peak height measurements through the 1970's when

diffractometers could measure background corrected

integrated intensities. The arrival of automated

diffractometers in the 1980's permitted the evolution

of the AUTO/NBS*QUANT84 system which allowed

the measurement of multiple lines per phase and re- sulted in a substantially enhanced statistical precision. The next logical step in this evolution is to use the whole digitized pattern to further enhance

the statistical situation and average out some of the

preferred orientation effects. In recent years, D. K. Smith and his colleagues

(1986, 1987, 1988, 1989) have developed whole pattern matching algorithms which have produced some

of the best quantitative analyses ever reported. The

use of the whole pattern involves collection of standard data on pure materials using fixed instrumental

conditions. These standard patterns are processed to

remove background and any instrumental artifacts.

For standard materials that are unavailable in appropriate form, diffraction patterns can either be simulated from Powder Diffraction File data or calculated (Smith et al., 1987). In addition to collecting a

data base of standard patterns, RIR values for each

are also measured or computed. Patterns are then

measured for the unknown samples to be analyzed,

using conditions identical to those used in obtaining the standard data. Background and artifacts are

also removed from unknown-sample data. Lastly, the
R. L. Snyder 31

digital patterns are fit, point by point, to the standards with a least-squares procedure minimizing the

expression

at a given step is determined by summing the contributions from background and all neighboring Bragg

reflections as

6(29) = {lunkm -Y,XkRIRkhm)\ (21) j/.(e) ^sY,KiaG{Mi,j.)Pj + yii(c), (22)

where Iunfc(2^) and Ijt(2^) are the diffraction intensities at each 26 interval for the unknown and each

of the standard phases, k, respectively.

In practice each of the reference patterns in the

database is scaled point by point to its maximum

intensity. The same is done with the pattern of the

unknown mixture and the normalizing assumption is used to solve for the weight fractions.

This whole pattern fitting calibration procedure

becomes a method for measuring RIRs from whole

pattern standards. Once this is accomplished the

method becomes a conventional RIR quantitative

analysis procedure, which uses calibration standards.

Thus, the dilution of the unknown specimen with an

internal standard may be avoided, as usual, only by

applying the normalization assumption. The RIRs

determined before the rescaling of the reference patterns act in the final least squares adjustment as nor- malized RIRs described below.

Quantitative phase analysis using the

Rietveld method

At the same time that whole pattern matching procedures were developing. Hill & Howard (1987) and

Bish & Howard (1988) began applying crystal struc- ture constraints to the problem of quantitative analysis. The Rietveld method (Rietveld, 1969) was originally conceived as a method for refining crystal struc- tures using neutron powder diffraction data. Least

squares refinement is conducted by minimizing the

sum of the weighted, squared differences between

observed and calculated intensities at every step in a digital powder pattern. The variables are atomic

positions, site occupancies, lattice parameters, scale

factors, profiles, background and possibly a preferred

orientation function. The method consists of fitting

the complete experimental diffraction pattern with

calculated profiles and backgrounds, and obtaining

quantitative phase information from the scale factors

for each phase in a mixture.

The intensity of reflection i from a pure phase a

was given by equation 1 which contains all of the

structure sensitive parameters, which the Rietveld

procedure optimizes, in the Kia term. The intensity

where S is the Rietveld scale factor which puts the

computed intensities on the same scale as those observed, Pj is an optional preferred orientation function for the j*'* Bragg reflection, G(A^,ja) is the

profile shape function, and ytt(c) is the background

(Wiles k, Young, 1981). Comparison of equations

1 and 22 shows that the Rietveld scale factor, S, is given by

(23)

It is important to note that equation 23 will only

be true for a correctly converged Rietveld refinement. Many discrepancies between the structural

model and the actual structure will appear as errors

in the scale factor and make the quantitative analysis

incorrect. In the whole pattern-matching procedures

the same scale error will be caused by differences between the observed pattern and the reference pattern

due to such common effects as structural disorders or poorly modeled asymmetric profiles. It should

be emphasized that the common use of computed

symmetric profiles to model the asymmetric X-ray

profiles will produce intensity mismatches which are

directly incorporated into the scale factor and in turn

the determined concentrations. For an exact modeling of the asymmetric X-ray peaks a convolution of

the specimen profile shape function with a calibrated

instrumental function must be carried out (Howard

t Snyder, 1989).

For a multi-phase mixture equation 22 can be

rewritten summing over the k phases in a mixture

as,

Vic = Vib + 5Z 5* KjkGijk. (24)

k j

Comparison of equations 2 and 24 allows us to write

the scale factor for each phase as

= Ke (25)

Therefore, in a Rietveld analysis of a multicomponent mixture, the scale factors contain the desired
32 Reference Intensity Ratios

weight fraction information as,

(26)

However, the sample mass absorption coefficient is not known and we thus are forced to apply the usual

internal-standard analysis, measuring the RIRs of

the phases to be analyzed with an added internal

standard, or to use I/Ic values and apply the nor- malization assumption constraining the sum of the

weight fractions of the phases considered to unity.

Hill Howard (1987) were the first to show that the

Rietveld scale factor acts in the role of a Reference

Intensity Ratio permitting conventional quantitative

analysis. We should note that each phase in the Rietveld refinement need not be modeled in order to obtain correct scale factors for those that are refined.

In order to analyze the relationship between the

Rietveld scale factor and the RIR let's first look at a system where we apply the normalizing equation 17.

In this case.

{X^ + Xp + ...) (27)

Equation 27 can be solved for the weight fractions of

each of the diffracting phases by substituting equation 26 into it,

Xa —

SocPot

Zfc SkPk

(28)

We see that this procedure is exactly analogous

to the normalization assumption in the internalstandard method where the Reference Intensity Ratios are measured prior to analysis. Instead of measuring Reference Intensity Ratios to put all intensities on an absolute scale, the Rietveld method calculates what we should call Normalized values, RIR^,

in that each is referred to a pseudo-phase with an RIR of 1.0. Comparison of equations 28 and 18 shows

that the Rietveld scale factor is

lia

RIRN,aIl^'

(29)

The jj^ term permits the scaling of RIR^ a so as to

allow the computation of a conventional RIRap val- ues by taking the ratio of two RIRn q values. Equation 29 permits the computation of normalized RIRs

from published or computed RIR values.

It is clear that the weight fractions obtained from

equation 28 will be incorrect when an amorphous or unmodeled phase is present, in that the sum in the

denominator of equation 28 should contain terms for

the unmodeled phases. From equation 28 we further

see the analog to equation 18,

Xa —

RIRn,, Trel

#o/ phases

Efc =l RiRN,krj%

(30)

The evaluation of weight fractions using equation 30 produces results similar to the normalized

internal-standard method described above but without the need to measure RIR values in advance. If conventional RIRs are not to be used or mixed in

to the set, then the somewhat artificial

-jfj^ terms

tor can be ignored and the RIRf^ k values will shift to

another scale. The weight fractions resulting from

the application of equation 28 or its RIR equivalent

equation 30 or a version of 30 without the scaling

ji^ terms, will all be correct as long as the normalizing assumptions hold. Thus, this procedure is much

closer to the "standardless" analysis all researchers

desire. It should be noted that the whole pattern

matching procedures of Smith et al. can be cast in

this same notation framework. Both procedures are subject to the systematic error inherent in the nor- malizing assumption as well as the additional errors, due to poor fitting between the reference and un- known due to poor profile modeling {e.g., due to

anisotropic peak broadening and unmodeled peak

asymmetries) and errors in the structural model, all of which will be swept in to the scale factors and, in

turn, into the derived weight fractions.

The internal standard method in

pattern-fitting techniques

To avoid the normalizing assumption, the conventional internal-standard method of analysis may be

applied to Rietveld or pattern-matching quantitative

analysis. This requires that a known weight fraction of a crystalline internal standard be present in, or added to, the unknown mixture. Thus, if X, is known, then the Normalized RIR^, with or without

the scaling jjf^ term, can be used directly, to determine the weight fractions for other phases in the

sample. For example, the weight fraction for the a

phase is determined by

^ _ RIRn,s

RIRn,a

(31)
R. L. Snyder 33

The normalized RIR^ values are effectively refined

parameters in the least squares, so the weight fraction of phase a can be directly determined. This

internal-standard method does not constrain the sum

of the weight fractions, as does the normalized RIR

or whole pattern fitting methods. The total weight

fraction of any amorphous components can also be

determined with the internal-standard Rietveld or pattern matching methods by explicitly fitting the

pattern of the glass and using its RIRi^ value. Alternatively, if the amorphous pattern can be fit and

subtracted using the background polynomial, then

the difference between the sum of the weight fractions of the crystalline components and 1.0 is the

total weight fraction of the amorphous components.

O'Connor &; Raven (1969) used this method to con- clude that their quartz contained an 18% amorphous

component.

The only thing differentiating whole-pattern fit- ting RIR method from the normal RIR-internalstandard method is its use of the whole pattern to

establish the RIR^ values greatly improving the statistical precision and averaging over some of the preferred orientation. When measured, conventional,

RIRs are incorporated into either whole-pattern fit- ting technique then the procedure becomes a con- ventional internal-standard RIR procedure with the

advantage of using all diffraction lines rather than a

selected few. Despite the many potential problems

with these whole pattern fitting methods the results

reported to-date are among the best quantitative re- sults in the literature.

References

Alexander, L. E. Klug, H. P &; Kummer, E. (1948).

J. Appl. Phys. 19, 742.

BisH, D. L. & Howard, S. A. (1988). J. Appl. Cryst.

21, 86-91.

Bezjak, a. (1961). Croatica Chemica Acta 33, 197-199.

Chung, F. H. (1974a). J. Appl. Cryst. 7, 519-525.

Chung, F. H. (1974b). J. Appl. Cryst. 7, 526-31.

Chung, F. H. (1975). J. Appl. Cryst. 8, 17-19.

Cline, J. P. &; Snyder, R. L. (1983). Adv. in X-ray

Anal. 26, 111-118.

Cline, J. P. & Snyder, R. L. (1985). Advances in Material Characterization II, 131-144, R. L. Snyder, R. A.

Condrate & P. F. Johnson, eds., New York: Plenum

Press. Cline, J. P. & Snyder, R. L. (1987). Adv. in X-ray

Anal. 30, 447-456.

Copeland, L. E. & Bragg, R. H. (1958). Anal. Chem.

30, 196-206.

Gehringer, R. C, McCarthy, G. J., Garvey, R. G.,

& Smith D. K. (1983). Adv. X-ray Anal. 26, 119-128.

Howard, S. A. & Snyder, R. L., (1989). /. Appl. Cryst.

22, 238-243.

Hill, R. J. & Howard, C. J. (1987). J. Appl. Cryst.

20, 467-474.

Hubbard, C. R., Evans, E. H. & Smith, D. K. (1976).

J. Appl. Cryst. 9, 169-174.

Hubbard C. R. & Snyder, R. L., (1983). Powder

Diffraction 3, 74. Hubbard C. R., Robbins, C. R. & Snyder, R. L. (1983). Adv. X-ray Anal. 26, 149-157.

Hubbard C. R. &; Snyder, R. L. (1988). Powder

Diffraction 3, 74-78.

Karlak, F. &; Burnett, D. S. (1966). Anal. Chem.

38[12], 1741-1745.

Klug, H. P. & Alexander, L. E. (1974). X-ray

Diffraction Procedures, 2nd ed. New York: John Wiley

and Sons

O'Connor, B. H. & Raven, M. D. (1988). Powder

Diffraction 3, 2-6. Rietveld, H. M. (1969). /. Appl. Cryst. 2, 65-71.

Smith, D. K., Nichols, M. C. &; Zolensky, M. E.

(1982). POWDIO. A FORTRAN IV program for calculating X-ray powder diffraction patterns - version 10. The Pennsylvania State University, University Park,

Pa. Smith, D. K., Johnson, Jr., G. G. & Ruud, C. O.,

(1986). Adv. X-ray Anal. 29, 217-224.

Smith, D. K., Johnson, Jr., G. G., Scheible, A.,

WiMs, A. M., Johnson, J. L. k. Ullmann, G. (1987).

Powder Diffraction 2, 73-77.

Smith, D. K., Johnson, Jr., G. G. & Wims, A. M.

(1988). Aust. I. Phys. 41, 311-321.

Smith, D. K., Johnson, Jr., G. G. Kelton, M. J.

k. Anderson, C. A. (1989). Adv. X-ray Anal. 32 in press. Smith, D. K., (1992). Adv. X-ray Anal. 35, in press. Smith, S. T., Snyder, R. L. & Brownell, W. E.,

(1979a). Adv. X-ray Anal. 22, 77-88.

Smith, S. T., Snyder, R. L. &; Brownell, W. E.

(1979b). Adv. X-ray Anal. 22, 181-191.

Snyder, R. L. & Bish, D. L. (1989). Modern Powder

Diffraction, D. L. Bish k, J. E. Post ed. plOl-144. Washington: MineraJogical Society of America.

Snyder, R. L. k Carr, W. L. (1974). Surface. Surfaces and Interfaces of Glass and Ceramics, ed., V. D.

Frechette p. 85-99, New York: Plenum Press

Snyder, R. L., Hubbard, C. R. k PanagiotopouLOS, N. C. (1981). AUTO: A Real Time Diffractometer Control System. NBSIR 81-2229, Gaithersburg: National Bureau of Standards.

Snyder, R. L., Hubbard, C. R. k Panagiotopoulos,

N. C. (1982). Adv. X-ray Anal. 25, 245-160.

Snyder, R. L. k Hubbard, C. R. (1984). NBS*-

QUANT84: A System for Quantitative Analysis by

Automated X-ray Powder Diffraction, Internal Publication, Gaithersburg: National Bureau of Standards.

ViSSER, J. W. k DEWolff, P. M. (1964). Absolute Intensities Report 64 1.1 09 Delft: Technisch Physische Dienst. Wiles, D. B. k Young, R. A. (1981). J. Appl. Cryst.

14, 149-151.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraciion II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Quantitative Phase Analysis with the Rietveld Method

J. K. Stalick

Materials Science and Engineering Laboratory, National Institute of Standards and Technology,

Gaithersburg, MD 20899, U.S.A.

Abstract

Quantitative phase analysis of multiphase mixtures

using Rietveld refinement has recently been shown

to be a viable alternative to conventional discretepeak methods, particularly for the analysis of mixtures with complex patterns or phases with broadened peaks. The analysis of the entire pattern re- duces problems due to extinction, preferred orientation, and other systematic errors. The use of neutron rather than X-ray diffraction further enhances

this technique, owing in part to the elimination of microabsorption and preferred orientation effects. Neutron Rietveld refinement has been used to determine

low-concentration impurity phases in the high-Tc su- perconductor YBa2Cu307_i by including the scat- tering contributions of the impurity phases as additions to background. In this system, CuO was detected at 0.36(6) wt. %, even though no impurity

peaks were visually observed.

Introduction

Rietveld analysis of powder diffraction data involves

the fitting of the entire experimental diffraction pattern to a calculated profile (Rietveld, 1969) based on

a known structural model. When used for quantitative phase analysis, the scale factors of two or more

phases are refined along with background, unit cell parameters, peak widths and shapes, and perhaps

crystal structure parameters. The recent review by

Hill (1991) gives an excellent summary of the advantages of this technique, including (1) utilization

of the entire diffraction profile, thus reducing sys- tematic errors; (2) the ability to fit the background

over the entire pattern; (3) the ability to correct for

extinction and preferred orientation in each phase;

(4) more efficient treatment of overlapping peaks; (5)

the ability to refine crystal structure and peak profile parameters; and (6) the correct propagation of

error into the results, using the standard deviations

of the scale factors estimated from the least squares

refinement.

Hill & Howard (1987) have shown that for a mixture of crystalline materials, the mass of phase p, rrip,

is given by

mp a Sp{ZpMpVp)

where Sp is the Rietveld scale factor, Zp the number

of molecules per unit cell, Mp the molecular weight,

and Vp the unit cell volume. Bish & Howard (1988)

derived a similar relationship, nip oc Sppp, where pp

is the calculated density with pp = ZpMp/NVp, thus

giving rup oc Sp(ZpMp/Vp). However, since most

Rietveld software does not include the normalizing

term 1/Vp in the calculated intensity, but rather incorporates it as part of the scale factor, the formulation of Hill k. Howard is more generally applicable.

Thus the weight fraction, Wp, of a component phase

p in a mixture of n phases is given by

_ Sp{ZMV)p

'~

1:7=1 SiizMvy

Hill (1990) refers to the quantity ZMV (or pV^) as the pattern intensity constant for a particular phase;

this is the Rietveld equivalent of the more conventional Reference Intensity Ratio (RIR).

The model

The application of this technique is chiefly limited

by the accuracy of the model used for the calculated diffraction patterns. The crystal structures of

all phases must be known; while it is possible to refine the structures of one or more of the phases as part of the analysis, it is important that this refinement be both necessary (phases with variable composition) and warranted by the data (major constituents with reasonably well-differentiated reflections, and neutron diffraction data if possible). Using neutron data. Hill &; Howard (1987) were able

to refine all crystal structure parameters along with

34
J. K. Stalick 35

Table 1: Theoretical and Experimental Phase Compositions (wt. %)

Measured

Mixture Weighed X-ray Neutron

TiOs 70 69.0(20) 70.5(3)

AI2O3 30 31.0(12) 29.5(3)

TiOs 50 48.8(15) 50.5(3)

AI2O3 50 51.2(19) 49.5(3)

TiOa 30 26.5(7) 30.4(3)

AI2O3 70 73.5(19) 69.6(3)

TiOa 10 8.1(3) 9.8(2)

AI2O3 90 91.9(21) 90.2(2)

Ti02 50 48.9(18) 50.7(3)

Si02 50 51.1(21) 49.3(3)

the phase abundances in Ti02/Al203, Ti02/Si02,

and Si/Si02 mixtures with good results. However,

with X-ray data the structure parameters are not

generally refined (Bish &; Howard, 1988; O'Connor &

Raven, 1988; Madsen & Hill, 1990). Table 1, taken

from Madsen & Hill (1990), illustrates the results of

neutron and X-ray refinements. The agreement between the weighed and measured wt. % is in general

quite good, with the absolute errors and standard

deviations being lower in the neutron refinements.

Other important considerations are primary extinction, preferred orientation, and microabsorption

corrections, as well as the presence of amorphous or partly amorphous constituents (Hill, 1990). The use

of the entire diffraction pattern minimizes the effects of extinction, and normalized preferred orientation corrections are included in most Rietveld soft- ware. Alternatively, sample preparation techniques

(spray-drying etc.) may be used to eliminate or re- duce preferred orientation. The Rietveld method

also permits the determination of amorphous content through the addition of a standard of known

crystallinity; for example, O'Connor Sz Raven (1988)

and Jordan, O'Connor & Deyu (1990) have successfully determined the amorphous content of quartz

specimens. The most serious remaining problem appears to be that of microabsorption corrections for

phases with differing particle size distributions and

mass absorption coefficients. Madsen &; Hill (1990)

found reasonably good agreement between the theoretical and experimental X-ray Rietveld phase compositions for mixtures of Ti02, AI2O3, and Si02 with

mass absorption coefficents for Cu Ka radiation of

129.3, 31.1, and 34.5 cm^g-i, respectively (Table 1).

However, mixtures of LiF and Pb(N03)2 {fJ-/p = 20

and 231 cm^g~^ for Co Ka radiation) resulted in

large quantification errors without the application of a Brindley absorption correction (Taylor Matulis,

1991).

Determination of impurity phases

The problems arising from extinction, preferred orientation, and microabsorption are all effectively eliminated with neutron diffraction data. Furthermore,

the large irradiated sample volume, along with the

Gaussian line shape and greater angular range of

data, permits the quantitative phase analysis of minor impurity phases.

This technique was applied in two studies of the

high-Tc superconductor YBa2Cu307_i. The first study was undertaken to optimize processing conditions for the production of pure-phase YBa2Cu307

superconducting material. The second study involved the correlation of superconducting properties with structural changes upon partial substitution (up to 10 %) of other elements for Cu.

There are significant structural changes in this sys- tem with varying oxygen content and site substitution; for example, the crystallographic space group

changes from tetragonal (P4/mmm) to orthorhombic (Pmmm) with increasing oxygen content in the

unsubstituted material. Crystallographic symmetry

also varies with site substitution; furthermore, the

determination of the amount of secondary phases is necessary as a check on the amount of substituent not

incorporated in the superconductor. The Rietveld

method provided the ability to refine the structure

of the primary phase while simultaneously determining the phase abundance of impurity phases in these

two studies.

Experimenial- Potential impurity pheises were

identified from phase diagram studies (Roth, Rawn,

Beech, Whitler & Anderson, 1988), and experimental neutron diffraction patterns were obtained

for CuO, BaCu02, BaC03, and Y2BaCu05 on

the five-detector high resolution powder diffractometer at NIST. The structures of CuO, BaCOs, and

Y2BaCu05 were refined using the Rietveld method

in order to obtain the best values for the structural
36 Quantitative Phase Analysis with the Rietveld Method

parameters (in particular the neutron thermal parameters); the structure of BaCu02 could not be

satisfactorily refined based upon published results

(Kipka &i Miiller-Buschbaum, 1977; Weller Lines,

1989), and the observed pattern was used in the impurity phase analyses. For all other potential impurity phases calculated patterns based upon known

structures were used.

High-resolution neutron powder diffraction patterns were obtained for several preparations of

YBajCuaOT-i and YBa2(Cu,M)307±i , M = Al, Fe,

Ga, Co, Ni, Mn, Zn, Au. The samples were contained

in vanadium cans, and data were collected at room

temperature with a 29 angular range of 5-120°, with

a step size of 0.05° and a monochromatic wavelength

of 1.5525(1)A. Structures were refined using a modified version of the Rietveld program (Prince, 1980).

The intensity contributions of impurity phases were added as a contribution to background and scale

factors Sp were refined for each potential impurity

phase. The total calculated intensity for each point

is given by

n

il/c)Yi^calc) + Bi+Y^SpYip

p=l

where 1/c is the primary phase scale factor and Bi

is the contribution at point i of background as represented by a series of Chebychev polynomials. Whenever possible the impurity phase contributions Yip were calculated from the known structure parameters; for BaCu02 a background-subtracted observed

pattern was used. The weight fractions Wp were then

calculated as previously described.

Results- The impurity phase found in nearly all samples studied was CuO. Note that neutron diffraction is approximately twice as sensitive to the CuO

phase as is X-ray diffraction, owing to the greater rel- ative scattering of neutrons by atoms of low atomic

number. Table 2 gives the results for a series of

YBa2Cu307 samples prepared under different processing conditions. For sample A excess CuO was known to be present; however, samples B - E appeared phase-pure based on X-ray powder patterns.

Below about one wt. % the CuO peaks were not discernable in the neutron patterns, even though the

fraction of CuO was statistically significant. Figure 1 shows the experimental, calculated, and difference

patterns in the region of the strongest CuO peak for

sample B, with 1.27(6) wt. % CuO. The CuO (111)

reflection is clearly evident in the difference pattern,

although it is smaller than the noise owing to approximations in the model for the YBa2Cu307 phase. TaTable 2: Determination of CuO in YBa2Cu307 by

Rietveld Refinement

Scale Factor Scale Factor Weight

Sample YBa2Cu307 CuO % CuO

A 6.90(3) 2.93(3) 8.7(2)

B 7.29(3) 0.42(2) 1.27(6)

C 7.81(2) 0.19(2) 0.54(5)

D 3.73(2) 0.06(1) 0.36(6)

E 3.24(2) -0.03(2) 0

Figure 1: Experimental (circles), calculated (solid

line), and difference (observed - calculated) patterns

for YBa2Cu307 with 1.27(6) wt. % CuO in the region

40-56° 29. The CuO (111) reflection is indicated on

the observed and difference patterns.
J. K. Stalick 37

Table 3: Refinement of Impurity Phase Scale Factors

Material Scale Factor Calc. Wt. %

YBa2Cu307 7.81(2) 99.35(13)

CuO 0.19(2) 0.54(5)t

YzBaCuOs -0.001(1) 0

BaCOa 0.004(3) 0.11(8)

finclusion of CuO as an impurity phase decreased

the agreement factors R^p and Rb from 6.50 and

3.19 to 6.37 and 3.13, respectively.

ble 3 presents the results for sample C; the X-ray and

neutron diffraction patterns exhibited no impurity

peaks. Only CuO was found to be present; the inclusion of CuO impurity scattering in the calculated

model led to statistically significant improvements in

the agreement factors for the Rietveld refinement.

The results for the site-substitution studies parallel those presented here for the processing of the

pure superconductor material, and will not be discussed in detail. In general, the inclusion in the

model of scattering from impurity phases lowered the

standard deviations for the site occupancies, and in

the case of the Au-substituted material confirmed

the displacement of Cu atoms by Au atoms in the

structure (Cieplak et ai, 1990).

Discussion

Quantitative phase analysis with the Rietveld

method has been shown to be a viable alternative

to conventional discrete-peak methods. In the determination of impurity phases in the system discussed

here, the Rietveld technique is far superior to other

methods owing to the variability of the structure of

the major superconductor phase and the very low

concentration of impurity phases. Another system

in which the Rietveld method appears to offer significant advantages is in the determination of mineral

content of rock specimens. Two computer programs

incorporating databases of known mineral structures

have recently been reported. The program QPDA

by Madsen & Hill (1990) was designed for use by

powder diffractionists with little or no experience in

structure refinement. QPDA can be run in either

single-phase or multi-phase refinement mode, and

calculates phase abundance and size/strain information from the refined parameters. The database in

this case also includes structures of common inorganic materials. The second system of computer

programs, SIROQUANT, features corrections for

amorphous content and for microabsorption using

Brindley particle absorption contrast factors (Taylor, 1991). The database contains structure information on 90 common minerals, and new minerals can be added. Both programs include correction for

preferred orientation based upon the March model.

The Rietveld method for quantitative phase analysis

has been shown to give good results in known mixtures of mineral specimens; thus the application of

this technique to unknown mixtures appears to be

forthcoming.

Acknowledgments Many thanks to those who assisted with this project:

E. Prince, J. J. Rhyne, and W. Wong-Ng of NIST,

C. L. Chien and G. Xiao of The Johns Hopkins University, S. Johnson of SRI, C. Parker of Allied-Signal

EMRC, and J. Hodge of CPS Superconductor Inc.

References

BiSH, D. L. & Howard, S. A. (1988). J. Appl. Crystallogr. 21, 86-91.

Cieplak, M. Z., Xiao, G., Chien, C. L., Bakhshai,

A., Artymowicz, D., Bryden, W., Stalick, J. K.

Rhyne, J. J. (1990). Phys. Rev. B 42, 6200-6208.

Hill, R. J. (1991). Powd. Diffr. 6, 74-77.

Hill, R. J. &; Howard, C. J. (1987). /. Appl. Crystallogr. 20, 467-474.

Jordan, B., O'Connor, B. H. & Deyu, L. (1990).

Powd. Diffr. 5, 64-69.

KiPKA, R. &; Muller-Buschbaum, Hk. (1977). Z. Naturforsch. B 32, 121-123.

Madsen, I. C. & Hill, R. J. (1990). Powd. Diffr. 5,

195-199.

O'Connor, B. H. & Raven, M. D. (1988). Powd.

Diffr. 3, 2-6. Prince, E. (1980). U. S. Tech. Note 1117, 8-9. Rietveld, H. M. (1969). J. Appl. Crystallogr. 2, 65-71.

Roth, R. S., Rawn, C. J., Beech, P., Whitler, J. D. & Anderson, J. O. (1988). In Ceramic Superconductors II (M. F. Yan, Ed.), Amer. Chem. Soc,

13-26.

Taylor, J. C. (1991). Powd. Diffr. 6, 2-9. Taylor, J. C. & Matulis, C. E. (1991). J. Appl. Crystallogr. 24, 14-17.

Weller, M. T. &; Lines, D. R. (1989). /. Chem. Soc. Chem. Commun. 484-485.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Microabsorption Corrections

H. Hermann*and M. Ermrich

Institute of Solid State and Materials Research,

D-0 8027 Dresden, Germany,

and

STOE & CIE GmbH,

D-W-6100 Darmstadt, Germany

Introduction

The intensity of X-rays diffracted by multiphase powder specimens is governed not only by the parameters

of the X-ray device used for the experiment and the

crystal structure and physical state of the scatter- ing phase, but also by absorption processes that the

X-ray is subjected to as it passes through the sample. The magnitude of the absorption effect depends on the geometrical arrangement of X-ray source and

detector, the macroscopic shape and size of the scat- tering sample, and the microstructural distribution

of the absorbing constituents in the sample (Klug k,

Alexander, 1974; Koch k MacGillavry, 1985; Koopmans k Rieck, 1985).

Here we consider the absorption of X-rays reflected

from a fiat specimen. For homogeneous materials,

the processes are assumed to be described by the

linear absorption coefficient, n, of the absorbing material and an exponential decay of the intensity with

the path length of the ray in the material.

For heterogeneous materials the situation is much

more complicated. The beam penetrates not only

the scattering phase, i, in an n-phase material but

also regions filled with particles of the other phases.

If all particles are very small compared with the penetration depth of the X-rays, the beam, on average,

will pass through many regions of all phases. Then

the path length through particles of the scattering

phase is given by the geometrical path length of the

beam in the whole sample multiplied by the volume

fraction of the scattering particles. Considering a

series of samples of the same composition but with

increasing particle size, the length of the path of

the beam through the scattering phase exceeds the

above-mentioned value more and more, and in the

limit of very large particles the whole path of the

'Institut fur Festkorper- und Werkstofforschung Dresden,

Helmholtzstr. 20, D-O-8027 Dresden, Germany

beam through the sample is within a single scatter- ing particle at the surface of the sample. Supposing

the linear absorption coefficients //fc, k = l,...,n

to be different, it is obvious that the strength of the

absorption effect varies with the size of scattering

particles. In the following we analyze the absorption

effect for flat samples and its dependence on bulk microstructure and surface morphology of the sample.

Experimental evidence for microabsorption

One of the first to investigate the microabsorption ef- fect was Brindley (1945). He pointed out that in multiphase materials both the size of the scattering particles and the absorption contrast (//,— ^l) (where /lis

the mean linear absorption coefficient of the sample)

play a decisive role. The granular structure of powder specimens causes an angle-dependent contribution to the absorption of the incident and diffracted

beam. This effect was studied by de Wolff (1956)

and by Suortti (1972) who measured the reduction

of specimen fluorescence radiation due to granularity. Trucano k Battermann (1970) investigated the

effect of porosity of amorphous powders, analyzing

diffuse scattering.

Although the experimental results differed in some

detail, the intensity reduction due to granularity

could be described by the sum of a constant term

proportional to the volume fraction of pores, the lin- ear absorption coefficients and mean chord lengths

of particles, and a smooth function of scattering angle. The former is determined by the bulk properties

of the granular material, whereas the latter is essentially caused by the surface roughness of the sample.

Hermann k Ermrich (1989) and Ermrich k Hermann (1990) carried out quantitative phase analyses of silicon-molybdenum powder mixtures. They

showed that the neglect of the microabsorption ef- fect may cause a considerable deviation of the estimated volume fraction from the true value, which

38
H. Hermann & M. Ermrich 39

can reach the order of magnitude of the true value

itself, whereas the error is well below 5% when the

scattered intensities are corrected for microabsorption. Due to the region of scattering angle, 29, considered {29 not too small) and low degree of surface

roughness it was sufficient to consider only the bulk

contribution of the microabsorption effect as a cor- rection to the quantitative phase analysis.

The surface roughness of scattering specimens has

been shown to reduce the intensity of scattered Xrays especially in the low-angle region (Suortti, 1972;

Sparks et al, 1991). As a consequence of this ef- fect, structural refinements based on Rietveld analyses may result in unphysically low and even negative

thermal parameters (Sparks et al., 1991; Masciocchi et al., 1991; Pitschke et al., 1991; Kumar et al,

1991).

The physical processes governing the attenuation

of X-rays in matter are quite clear. Considering

diffraction by flat specimens, the problem of estimating the microabsorption effect quantitatively by

means of

Ii/Io,i= / exp[-fiiXi- H(^i)X(^i)]dV/Qo (1)

Jv

consists mainly in the mathematical description of

the geometrical distribution of scattering and absorbing particles, in the calculation of the path length of a given ray through the regions covered by the scat- tering phase, and in averaging of all rays reflected by

the sample. Here, the scattering phase is denoted by

i, Xi is the path length of a ray through phase i,

is the same for the region outside phase i, and //(,) is the mean linear absorption coefficient of all regions

of the sample that do not belong to phase i. lQ^i/2ni

is the intensity from an i-type homogeneous smooth

reference specimen obtained under the same experimental conditions as the scattered intensity /,• from

the heterogeneous sample. Qo is the cross section of

the incident beam and V is the sample volume. (See

also figure 1.) It is obvious that a mathematically well-defined

structure model is needed to solve the integral (1)

for representative classes of powder structures. The

model should be variable with respect to volume fraction, and size and shape of scattering particles. The

model should include both bulk properties and sur- face morphology, and it should be possible to calculate the geometrical characteristics required for the

evaluation of (1). Furthermore, the model should

be easy to handle. In the following we describe an approach to tackling the problem outlined above.

It is based on methods that have been developed

within the framework of the mathematical branch of

Stochastic Geometry (see, e.g., Stoyan et al, 1987)

0

Figure 1: Scattering of a X-ray by a plane sample.

The broken line represents the path length x,- within

phase i.

and have already proved to be suitable for practical

applications (Hermann, 1991).

Structure model for random powders

Bulk structure

We simulate the powder sample by a random twophase model given by a random set A and the complement B = A'^ oi A. The one set represents the

scattering particles and the other one the pores or the

distribution of both pores and particles that do not

belong to the scattering phase for respectively single- or multi-phase powders. The model is generated in

two steps. In the first step a random (Poisson) field of

points Pi, P2, . .

. , with density A is generated. Then

a sequence of isotropic, identically distributed, con- vex geometrical objects (for example spheres, ran- dom polyhedra etc.), Ai,A2, . ., is produced. The

P^s are the 'germs' and the A'^s are the 'primary

grains'. The A'f.s are shifted to the points Pk whereby

the grains may overlap. The union of all A'f.s forms

a random set A. The model is given by all points

in the space that are covered by one or more primary grains. Figure 2 illustrates the construction

algorithm.

Generally the grains are characterized by the mean

volume, V{Ak), the mean surface area, S{Ak), and

the mean breadth h{Ak). (The breadth, hk, of a

grain, Ak, in a given direction is the distance between

the two supporting parallel planes that are j)erpendicular to this direction. The mean value b{Ak) is the average of 6jfc over all directions and all grains

Ak,k=\,2,..).

Furthermore, the distance probability function of
40 Microabsorption Corrections

Figure 2: Generation of a planar Boolean model with

identical circular grains. R - radius of the grains,

Pk - germs of the model. The bold line marks the

boundary of the random structure A generated by

the union of the grains.

a grain Ak is

tk{r) = ^

j^j^Sk{u)sk{u + r)dVdQ, (2)

eAk

^Ak

where the integration over Q corresponds to the av- erage over all spatial directions. The function

(3)

is the mean distance distribution function averaged

over all grains, Ak . The volume fraction of the ran- dom set A is

CA = l-exp[-A7(Aik)]. (4)

The covariance C>i(r) of A, which is defined by the

probability of finding two arbitrary points with distance r both in region A, is

Ca^v) = 2c^ - 1 -h (1 - CAf exp[A7°(r)], (5)

CA'{r) = CA{r) + l-2cA. (6)

The intersection of a Boolean model with straight

lines generates a sequence of chords. The mean value

of the chord lengths x outside the random set A is (Stoyan et al, 1987)

7b = 4/A5(^fc) (7)

for three-dimensional models. For models with con- vex grains the chord lengths xb are random variables

with an exponential distribution

f{xB) = Al exp(-AiXs), Al = 1/7^. (8)

Here we use random Poisson polyhedra with mean

breadth h as grains. The following relations hold

(Stoyan ei a/., 1987)

T°(r) = F(^;fc)exp(-3r/26),

S{Ak) = m^/Zn,

V{Ak) = 166^/97r.

(9)

(10)

(11)

The results for covariance and mean chord length are

CA{r) = 2c^-l-h(l-c^)exp{[-ln(l-CA)]

xexp(-3r/26)}, (12)

Is = 26/[-31n(l-c^)], (13)

U = ca1b/{1-ca). (14)

Surface morphology

The Boolean model is infinitely extended in all directions. A model with a planar surface is generated by

a planar cut through the infinitely extended model.

If the planar cut is carried out smoothly, the sur- face of the model is called smooth too. We consider

the volume fraction a{t) of particles in the model

bounded by a planar surface and its dependence on

the distance t from the surface. Then a sample with

a smooth surface (Fig. 3a) is characterized by

do,

0,

t > 0

t < 0

(15)

where ca = I — ocq denotes the volume fraction of

the pores if the random set A is used to simulate the

pore distribution. A rough surface is generated by a

planar section of the infinitely extended model that is smooth at large dimensions but fuzzy at dimensions

comparable to the size of the pores and the grains of

the model (Fig. 36). The volume fraction of a sample

with a rough surface deviates from the bulk value oq

in the vicinity of the surface and may be described

by the expression (Hermann &: Ermrich, 1987)

ait)

ao[l - exp(- t/to)],

0,

t > 0

t < 0. (16)
H. Hermann & M. Ermrich 41

(a)

(b)

Figure 3: Simulation of surfaces of porous materials

by intersecting a Boolean model, (a) Smooth surface

produced by a smooth planar section through the

infinitely extended model, (b) Rough surface generated by a fuzzy planar section of the Boolean model.

The difference between surfaces of the types shown

in figures 3a and 36 will be discussed in some detail, since a precise definition of the term "surface

roughness" is required for the correct interpretation

of experimental data.

For this end, we calculate the mean profile depth,

t = {PQ)j and the mean fluctuation of the surface

profile. At = {\PQ — t\), of a smooth (^o = 0), porous

sample ({...) denotes the average over all points P of

the surface). We consider a point P ^ A (B = A'^) at the surface (t = 0) of the sample and the line

PQ that is parallel to the direction of the coordinate

t (Fig. 4). The point Q is the intersection point of

that line with the boundary of B. The length y of

the line PQ is randomly distributed. The density

hi{y) of the probability distribution of y = PQ is approximated by

hiiy) = Aiexp(-Aiy),

1/Ai = (l-ao)7/ao (17)

where I = Ib. (The exponential form of hi{y) applies

exactly for P £ B. See Stoyan ei al, 1987). Averaging over all points P ^ A and P E B and taking into

account (15, 17) one obtains:

rOO i = (1 - ao) / yhi{y)dy

Jo

2 At I unT^^MTma wTTj^^jMii^m^/m^^

(b)

Figure 4: Surface (a) of a porous sample without

surface roughness {to = 0) and its depth profile (b).

The surface is situated at i = 0, ? is the mean profile

depth, At is the mean depth fluctuation. B represents the powder particles, and the complement of B

for t > 0 describes the intersected pores.

= (l-ao)27/ao. (18)

The mean fluctuation At of the distance PQ is defined by

Ai=aot + {l- ao) / \y - t\hi{y)dy.

Jo

One obtains

At = 2(1 - ao)^7/aoexp(l - ao).

(19)

(20)

The quantities t and At given by (19, 21) characterize

the surface of a porous sample produced by a smooth

planar section according to figure 3a. The surface

roughness parameter to defined in (16) vanishes for

samples prepared in this way, so that, if the surface

profile of a porous sample is characterized by t, At

and (18, 20), the degree of the surface roughness is zero. This definition is reasonable, since t and At

are determined completely by the bulk parameters ao and 7 of the sample.

A sample with the property to > 0 can be simulated by_the model shown in figure 5. The quantity

z = PP' is assumed to be random with the probability density

/(z) = (l/to) expi-z/to), z > 0 (21)

according to (16). The probability density for the

profile depth x = y + z on the condition that P' ^ B

is /+0O

•oo hi{v)f(x — v)dv

= Ai[Ai<o-l]-'

X [exp(—x/^o) — exp(—Al x)] . (22)
42 MiCROABSORPTION CORRECTIONS

0

t

Figure 5: Model for a rough surface of a porous material. The surface is generated by a fuzzy section

(bold line) through the Boolean model. Considering

dimensions which are large compared to the size of

pores and particles the section is planar and smooth.

The mean profile depth t of the rough surface of a porous material is

t = ao xf{x)dx + (l — ao) I xg{x)dx

Jo Jo = to + {l-aofl/ao. (23)

The mean fluctuation is defined by

At = ao /\x -t\f{x)dx

Jo

+(l-ao) / \x-t\g{x)dx. (24)

Jo

The integrals can be evaluated easily, but the corre- sponding expressions are somewhat complicated. An

approximation is given by

At = 2(1 - ao)[(l - ao)! + aoh] (25)

for

to<l, 0.5<ao<l.

Now we estimate the order of magnitude of the quantities to and 7 that can be expected for real samples.

We use a hard-sphere structure (Fig. 6), which is, of course, not a Boolean model, but yields limiting

values for the ratio to/ 1 for random structures. The

model with vanishing surface roughness consists of all regions with t > 0 including the spherical segments

(shaded areas in figure 6). However, in a real sample prepared from equally sized spherical particles

there would not be any intersected particle. Assuming tetrahedral packing, the height of the spherical

segments is R/2 (R is the sphere radius). If the intersecting particles are removed, the density of the

sample is locally reduced in the vicinity of the surface. This reduction can be approximated by (16)

Figure 6: System of spherical particles with vanishing (shaded areas included) and non-vanishing (without shaded areas) surface roughness.

with parameter to = 0.3R. In a polydisperse system

of spheres with mean radius R the value to = 0.3i2

is an upper limit, since holes generated by removing

large intersecting particles situated at the surface can be filled by small particles. The mean chord length

of the pores in a hard-sphere system is related to the

mean chord length 7, = 4R/5 of a single sphere by

7 = (1 — c^)7,/c^. For a packing fraction of about 0.6 one obtains 7 = 0.9R. The reduced surface roughness

parameter to /I is therefore expected to be limited by

to/1 < 0.3 (26)

for real powder samples with particles having approximately spherical shape. Consequently, (25) can be

used to estimate At through the quantities 7, ao and

to.

Absorption correction

Single-phase powders

We consider X-rays diffracted symmetrically by a flat single-phase powder (Fig. 1 and /? = 7 = 0 , where

20 is the scattering angle). Let L be the geometrical

length of a beam within the sample, and La{xi = La

in (1)) the corresponding path through the powder

particles. La is, even for fixed L and scattering angle

20, a stochastic variable. It depends on the position

of the volume element dV within the plane parallel

to the surface of the powder specimen. The mean

value La is given by the integral over all increments

dr on L lying in powder particles, on condition that
H. Hermann & M. Ermrich 43

the point of reflection, Q, (Fig. 1) is also in a powder

particle. Simulating the distribution of pores by the

random set A explained in the previous section, one

can write

La = CA-{r)dr/a+ / CA-{r)dr/a

Jpi Jq

= 2 / CA^ir)dr/a. (27)

The width of the distribution of La for fixed L is of

the order of the value b describing the mean linear

dimension of the Poisson polyhedra from which the

model is constructed.

Equation (1) can be written in terms of the variable L,

1 /"^

I/Io =

-J^

a{exp{-fiLa))dL, (28)

where {A) denotes the average of the quantity A over the distribution of La for fixed L. If

La>fe (29)

the width of the distribution of the random variable

L is small compared to L, and (exp(—/xLa)) can be

replaced by exp{—/iLo) • If, on the other hand, the

condition (29) does not hold, one must require

/i6<l. (30)

Then, expression (exp(—//La)) can be approximated

by (1 — fiLa) which is tantamount to the approximation (exp(—//La)) = exp(—//Lo). Therefore

1 f°° -

I/Io = ^

aexip{-nLa)dL (31) ^ Jo

is a good approximation for (28) in cases of 'fine' and

'medium' powders (cf. conditions (29, 30) according

to the classification of powders introduced by Brindley, 1945). Expanding (12) for a low concentration

of pores, ca •< 1, and inserting (12) and (16) into

(27) one obtains (Hermann & Ermrich, 1987)

fiLa/c = rz/sin©

-|-(1 — ao)[l — exp{—Tz/ sin6)]/ao

-(r/sine)[l-exp(-z)]

1 — exp[— (r -f- sin Q)z/ sin 6] r -|- sin 0

'

e = 2///?ao, r = to/l3, (3 = 26/3. (32)

Of course, the special cases La = L (for oro = 1)

and La = aoL (for f3/L 0) follow from (32) for

vanishing surface roughness (to = 0).

According to Harrison & Paskin (1964) and Suortti

(1972), we write

/ = 7o(l - P)/2fi = loJ, (33)

where P is the term resulting from both bulk granularity and surface roughness (7o = normalized intensity). Now the problem is reduced to the evaluation

of the integral

J = f exp{-fiLa)dV/Qo

Jv

a{z)exY)[—^iLa{z)]PTdz/smQ. (34)

Assuming weak surface roughness and expanding

(34) up to terms of order e and r^/ sin^ 0 one obtains

P = Po + P„ (35)

Po = 2/i7(l - ao)2, (36)

and

P, = 2;.7aoln-l^(l - ^). (37) ao sm 0 sm 0

Figure 7 illustrates the dependence of the bulk con0-3 0 2 01 0 0

1 I 1 00

001

002 / / / /

/

003 /

/ 0 04 /

/

0 OS

Figure 7: Dependence of bulk contribution, Pq, to absorption correction on volume fraction (1 — ao)

of pores for different values /i/ (full curve = 0.1;

broken curve /// = 0.2).

tribution, Pq, on the volume fraction of pores for

different values of ///, and figure 8 shows the surface

contribution, P, , and its variation with scattering

angle for different degree of surface roughness.
44 Microabsorption Corrections

on

0 10 20 30 40 50 60 70 80 90 0 00/— 1 1 1 1 1 1 1 1 1

Figure 8: Dependence of surface contribution, Pa(0), on the degree, r, of surface roughness (/i7 =

0. 15, ao = 0.70). 1 : r = 0.1; 2 : r = 0.2; 3:

r = 0.4; 4:r = 0.6.

Multiphase powders

The sample is assumed to consist of n — 1 scattering

phases, and the pores of the powder are denoted as phase n. We consider a beam diffracted by phase

1. The distribution of t-type particles is described

by a Boolean model with volume fraction c,- and co- variance C,j(r) given by C>i(r) in (12). Taking into

account that

and

n T'^l^CjTZj, (39)

i=i

Equation (1) can be evaluated in the same way as for

single-phase powders but using

Li = 2 Cii{r)dr/ci (40)

^0

instead of (27) and replacing (30) by

|/i.-7i|7<l. (41)

Considering the bulk contribution one obtains for a low volume fraction, c,- , of phase i

1 f°° - -

Ii/Io,i = -^Ci J

exp[-HiLi-fi(^i){L-Li)]dL, (42)

W/o, = §, (43)

= (44)

l-2(l-c.)(/^.-7l)/.' ^

^

and

Ec, = l. (45)

Equations (43 to 45) can be used to correct for experimental scattering intensities obtained from samples

with negligible surface roughness. However, generally both Ci and /,• , i = 1 , . .

.

, n — 1 , are unknown

quantities. That means that one set of scattered intensities, /,-, i = 1, . .

.

, n — 1, will not be sufficient to

carry out the complete microabsorption correction.

Hermann &: Ermrich (1989) suggested recording a second set of intensities using radiation of a different

wavelength. Then the absorption contrast, (/ij — Ji), will be modified, a second, independent set of equations can be established to eliminate the parameters

or to estimate both c,- and /,• , i = 1 , . .

.

, n — 1 , and

the microabsorption correction can be carried out.

Ermrich Hermann (1990) applied the method of

different wavelengths to study the microabsorption

effect in quantitative phase analysis. Figure 9 illusI 1 I I 1 0 0.0S 0.10 015 020 025

J^Mo 'Mo

Figure 9: Volume fraction cmo versus hImo es- timated by quantitative phase analysis of siliconmolybdenum powder including microabsorption cor- rection.

trates the effect. A mixture of molybdenum and sil- icon powder was analyzed using five different wavelengths. The scattering intensities were corrected for

by means of (43 to 45) assuming various values for / . (The value for molybdenum was estimated from electron micrographs: Imo = 2.5 ± 0.5/xm, particle size

distribution from 0.1 to 30/im). If the correct /-value
H. Hermann &: M. Ermrich 45

is tried, the estimated volume fraction should not depend on the wavelength used, but should approach

the true value {cmo = 19.4 vol% in figure 9). In- deed, this is the case for assumed Imo = 3 to 3.5/im,

and the result for the volume fraction of molybdenum particles obtained from the corrected intensities is 19.8 vol%. (The results of the quantitative

phase analysis without microabsorption correction

but with assumed fx'^ = JI in (43) varied from 7.5%

for A = 0.2291 Inm to 15.2% for A = 0.07107nm.)

Influence of particle shape

Since expression (44) has been derived for a special

structure model, the question arises whether it is also

typical for other random structures. Therefore , we

consider a Boolean model with equally sized spheres

as grains. This model represents a good contrast to

the structure based on the polydisperse set of ran- domly shaped Poisson polyhedra used for the derivation of (35) and (44) The covariance for a Boolean

model composed of spheres with radius r,- is {2ci-l + ... ...(l-c.)exp{-[^-^]ln(l-c.)},

r < 2r,- c?,r > 2r,-.

(46)

Inserting (46) into (40) and (42) one obtains for Cj -C

1 and condition (41)

li/Io.i = ^[1 - k{l - c.)(/i.- - mi (47)

with

^ ^ 1

9/8, spheres, .^g.

1^ 2, polyhedra. ^

^

The mean chord length, of particles is, in the limit

of small volume fraction of phase i (c,- <C 1), U =

6,/(l — c,) for Poisson polyhedra and /,• = 4r,/(l — c,)

for spheres.

Expressions (47) and (48) show that the intensity

of an X-ray beam reflected symmetrically by a planar

sample does not only depend on the absorption contrast (/I,- — 71) and the mean chord length 7,- but also on the shape of the scattering particles (Hermann et ai, 1991). Because of the restriction c,- C 1, within

the random set produced by the Boolean model,

only a few overlaps of grains appear. Therefore, the

shape of the scattering regions of the present struc- ture model is similar to that of the grains used for

the construction of the random set, and we may con- clude from (47) and (48) that the absorption caused

by a random arrangement is remarkably enhanced

with increasing polydispersity and irregularity of the

particles if systems with equal mean chord lengths

of scattering particles are considered. The typical

dependence of the absorption effect on the linear absorption coefficients, the volume fraction and the size

of i-type particles is, however, retained.

Grazing incidence

For non-symmetric geometry of incident and re- flected X-rays (Fig.l: ^ ^ j) and vanishing surface

roughness (r = 0) Eq.(l) yields the well-known formula

J//o = l/[//(l + sin/?/sin7)]. (49)

Considering a single-phase powder of packing fraction ao and r = 0 one obtains

I/Io = (1 - P)/[/x(l -hsin/?/sin7)], (50)

and

P = Po = nl{\ - ao)^ sin /? < /i7. (51)

That means the bulk contribution, Pq , of the microabsorption effect takes half the value of Pq for

symmetric (Bragg-Brentano) geometry. The following arguments show that, compared to the symmetric beam geometry, the strength of the surface contribution is reduced as well when using the grazing

incidence method: If the angle of incidence is very

small,

sin/?</i7, (52)

the sample is irradiated down to the depth, of the

order of magnitude of t, = sin/?/ao/i. Taking into

account (52) and the estimate (26) for the maximum

value of surface roughness one obtains

ts<3to. (53)

That means, at least for strong surface roughness,

that the diffracted X-ray does not feel the total

width. At = 2(1 — ao)aotQ , of the surface roughness

profile fluctuation (see (25)) on its path through the

sample. Consequently, at grazing incidence the systematic error of the scattering intensity caused by

the 0 -dependent surface roughness contribution of

the microabsorption must also be expected to be well

below the error appearing for Bragg-Brentano geometry. The grazing incidence geometry is therefore

proposed as a tentative method of minimizing the

bulk and surface roughness microabsorption effect.

Conclusion

Microabsorption affects the intensity of X-rays

diffracted by highly absorbing granular materials

considerably. The deviation of the measured intensity from the intensity expected for vanishing microabsorption may amount to the order of magnitude
46 MiCROABSORPTION CORRECTIONS

of the latter itself. This has been proved experimentally both for the bulk contribution and the surface

roughness effect. The systematic error of scattering

intensities due to microabsorption can be estimated

by means of analytical expressions derived for ran- dom powders and variable microstructural parameters.One should try to reduce the effect by suitable

sample preparation {e.g. powder particles as fine as possible) and minimization of the absorption contrast (suitable choice of the X-ray wavelength). The

grazing incidence method is also expected to bring

about a weakening of the microabsorption effect. For high-accuracy measurements and for those

cases were the microabsorption effect cannot be

avoided the analytical expressions presented above

should be used to correct for the measured X-ray

intensities. Then, the microstructural parameters

'mean chord length of particles', 'volume fraction'

and 'surface rouhgness' must be determined by other

experimental techniques, e. g. electron microscopy

and quantitative metallography, or by carrying out

scattering experiments at different wavelengths. It should also be possible to incorporate the expression

describing the surface roughness contribution into

Rietveld programs and to fit the unknown parameters. Of course, one can also make use of the microabsorption eff'ect to estimate microstructural parameters of granular bulk materials and rough surfaces.

References

Brindley, G. W. (1945). Philos. Mag. 36, 347-369.

Ermrich, M. & Hermann, H. (1990). Phys. Status Solidi A118, Kl-4.

Harrison, R. J. & Paskin, A. (1964). Acta Cryst. 17,

325-331.

Hermann, H. (1991). Stochastic Models of Heteroge- neous Solids. Zurich: Trans Tech Publ. Ltd.

Hermann, H. &; Ermrich, M. (1987). Acta Cryst. A43,

401-405.

Hermann, H. &; Ermrich, M. (1989). Powder Diffraction 4, 189-195.

Hermann, H., Pitschke, W. & Mattern, N. (1992).

Phys. Status Solidi A132, 103-114.

Hermann, H., Roske, K. & Hentschel, M. (1991).

Acta Cryst. A47, 446-447

Klug. H. P & Alexander, L. E. (1974). X-Ray Diffraction Procedures 2nd Ed. New York: Wiley.

Koch, B. & MacGillavry, C. H. (1985). International

Tables for X-Ray Crystallography, C. H. MacGillavry

& G. D. Rieck, eds. 3, 157-192.

KOOPMANS, K. & RiECK, G. D. (1985). International

Tables for X-Ray Crystallography, C. H. MacGillavry

& G. D. Rieck, eds. 3, 194-199.

Kumar, R., Sparks, C. J., Shiraishi, T., Specht, E.

D., ZscHACK, P., Ice, G. E. &; Hisatsume, K. (1991).

Mat. Res. Soc. Symp. Proc. 213, 369-374.

Masciocchi, N., Toraya, H. & Parrish, W. (1991).

Mater. Sci. Forum79-82, 245-250.

Pitschke, W., Hermann, H. &; Mattern, N. (1991).

Powder Diffraction, to appear.

Sparks, C. J., Kumar, R., Specht, E. D., Zschack,

P. & Ice, G. E. (1991). Advances in X-Ray Analysis

- PIXAM Conf., Hawaii, to appear.

Stoyan, D., Kendall, W. S. & Mecke, J. (1987).

Stochastic Geometry and Its Applications. Chichester:

Wiley.

SUORTTI, P. (1972). J. Appl. Cryst. 5, 325-331.

Trucano, p. & Batterman, B. W. (1970). J. Appl.

Cryst. 41, 3949-3953.

Wolff, P. M. de (1956). Acta Cryst. 9, 682-683.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Computer-Aided Phase Identification

M. Nakhmanson

Comphys Lab Enterprise, NPO "Bourevestnik"

St. Petersburg, 195272 Russia

Considered in this paper are two problems connected

with the use of powder diffraction standards base:

identification and classification. Identification of Xray patterns or qualitative phase analysis for a long

time was considered a single problem deserving con- templation. As a matter of fact the PDF file has

been created just for solution of this problem. The

available database which is a result of many-year ef- forts of JCPDS usually allows solution of some other

non-traditional problems as well. One of these problems, which is directly connected with identification,

is considered in the second part of the paper.

1. Identification

Let us consider mathematical formulation of identification problem. Assume that there is a set A of some

objects Qi, 02, • • •

, a,-, •

, a„, where n is the number

of objects:

A = (a,) = (ai, 02, • •

, a,-, • • • ,a„) (1)

In our case objects (a,) are various substances (compounds). Each object can be described by a collection of attributes or parameters (pik)'-

P.- = {Pik) = {Pii,Pi2, ,Pik, • ,Pimi), (2)

where m,- is the number of attributes for the z-th

object and P,- is an image of object a,-. In our case, for example, values of interplanar spaces d and/or

intensities I in a stick diagram serve as parameters P;

atomic numbers of elements constituting compound

a,- can also be added to them.

A set of all the n images P = {p, } (empirical standards) forms a database that contains information

about objects of the set A. The totality of all the

possible values pik forms a space of search parameters R in which point or vector P,- corresponds to

each image P,-. The space R can be defined by means

of several well- known methods that I shall not dwell

upon. They are considered in works of Fiala (1972;

1976) and in the book (Nakhmanson k Feklichev,

1990).

Let us consider the identification problem as a first approximation limiting it by the following idealized

conditions:

A. All parameters p,jt are known absolutely exactly,

i.e. there exists one-to-one correspondence between sets A and P: one and only one image Pf

corresponds to each object a,-, and vice versa. This means that all the points in R are absolutely distinguisable.

B. All parameters p,fc have different numerical val- ues or, at least, they are distributed in R uniformly.

C. If an unknown material is a mixture of some objects from the set A, its image Q = {q,} is an

exact linear combination of corresponding images from P:

Summed up in (3) are images of objects forming

the unknown mixture. Condition C, in particular,

means that when image Q is obtained in experiment

there are no losses of information even if some coefficients Cj in equation (3) are very small.

If conditions A, B, C are satisfied, the identification procedure comes to comparison of parameters

of each sample P,- with sequence {gg} and selection

from the database of those standards whose parameters exactly coincide with some numbers of

In this case there are no such situations as ambiguity of selection (because of conditions A and B) and

loss of low-concentration phases (because of condition C). There is only one problem left, connected

with the speed of the identification process. It can be increased sufficiently if search and comparison are carried out in several steps by means of a number of

auxiliary files separated from the main database.

Let us turn to a second approximation, which better corresponds to reality. Let us estimate conse- quences resulting from violation of conditions A,B,C.

47
48 Computer-Aided Phase Identification

Violation of condition A. Parameters of our standards are not known with absolute accuracy. If we take values d (database images that contain the

greater amount of information) as pik, the com- pounds with similar crystallochemical characteristics

have almost the same sets of d. Sometimes, because

of error in experiment these sets in practice are indistinguishable (e.g. for magnetite Fe304 and magnesioferrite MgFe204). Therefore one-to-one corre- spondence between A and P is broken, and some

points in the space R are indistinguishable, which is the first cause of identification ambiguity.

Violation of condition B. Many interplanar spaces d

belonging to various samples P,- have the same nu- merical value, and their frequency distribution U(d)

is not uniform. The same relates to parameters I and

N and their frequency distributions, V(I) and W(N).

Distribution U(d) is a unimodal curve with a maximum within 1 - 2.5 A that sharply decreases towards

small d and smoothly decreases towards large d; distributions V(I) and W (N) are of hyperbolic form.

So for d belonging to P that are within the range 1

-

2.5 A there exists a high probability of random coincidences (within a certain range of d) with values d

belonging to Q. If in the database P there are images

- X-ray patterns - for which the set of d totally or mostly is within 1 - 2.5 A, such "short" X-ray patterns will very often coincide with X-ray patterns of

objects under study. Similar X-ray patterns are con- tained in the database, and their amount is at least

3 - 5 % of the total number of objects n (e.g. 4-783,

Ag; 4-784, Au; 4-787, Al; 4- 806, W; 4-836,Cu; 6-642,

TiN; 6-675, C; etc.) If by chance two or three of the

strongest lines of "short" X-ray patterns from P co- incide with some lines of an unknown pattern they in

practice cannot be distinguished from low-concentration phases. This is the second cause of identification

ambiguity.

Violation of condition C, which always takes place

in real experiments (e.g. the loss of weak lines of

low-concentration phases), brings about additional

uncertainty in identification results. Thus, if two

strongest lines of diamond 6-675 with d,I equal to

2.06 A, 100 and 1.26 A, 30 coincide with two lines of

the unknown that can be lines of some other phases, one can consider that:

1. This coincidence is accidental, and there is no diamond in the mixture;

2. This coincidence is not accidental, and there is diamond in the mixture as a low-concentration

phase, but its two strongest lines overlap lines of

basic phases and the other four weak lines were

simply not recorded in the experiment.

Violations of conditions A, B or C lead to the re- sult that the identification problem becomes mathematically incorrect (instable by Adamar), and its solution has a probabilistic but not accurate character. In the initial stage of development of automatic identification systems in 1965 - 1985 there were many attempts to create complex algorithms for so- lution of the problem and to construct complex cri- teria (FOM) permitting determination of correspondence between the standard and identified X-ray patterns. The credit for development of these methods

is shared by J. Hanawalt, L. Frevel, G. G. Johnson,

R. Jenkins, R. L. Snyder, J. Fiala and other scientists. However, I would like to emphasize that ambiguity of identification is connected not so much with

imperfection of their methods and programs as with

objective characteristics of the database: availability

in it of similar X-ray patterns as well as of "short"

X-ray patterns of low information value.

Let us try to lay down ways to decrease identification ambiguity, taking into account the above analysis. Before this we should improve formulation of

the goals of qualitative analysis.

Practically all modern systems of automatic identification can determine the basic phases reliably ex- cept for the cases when compounds with very similar X-ray patterns (such as magnetite and magnesioferrite) are amongst them. But this is the simplest

problem. Much more difficult is the problem of correct differentiation of minor phases without including objects having anomalous short X-ray patterns.

I think that this is the main goal of automatic qualitative analysis.

Until now we have usually taken an X-ray pattern,

d and I, to be the parameters {pik}- The traditional

method of making the identification results more exact uses an expansion of the search space R, adding

information about the chemical compositions of the

database standards and the unknown material. However, this method does not always permit getting rid

of identification ambiguity. For example, it is impossible to make a choice between Fe304 and MgFe204

if some other phases of the mixture include magnesium (a mixture of magnetite, Fe304, and sellaite,

MgF2 is a typical example).

The possibility of expansion of the space R is limited because only those parameters of substances

that obey the additive relationship (3), e.g. infrared

spectra, can be included in it. In particular, for multiphase objects, which are our main interest, optical

and thermodynamical data are omitted from consideration.

An important step in increasing identification reli- ability is the use of local methods. By means of an Xray microanalyzer one can obtain data about the el-
M. Nakhmanson 49

emental composition of each phase; this information

is much more valuable then information about chemical composition of the unknown mixture as a whole.

In this case there is a partial localization in R of a number of subspaces corresponding to phases of the

mixture, and, as a result, it is possible to solve such

problems as choice between Fe304 and MgFe204 or

reliable classification of diamond as a false or true

phase of low concentration. Actually local methods

can eliminate ambiguity of both the first and sec- ond type with the same reliability with which they

provide information about elemental composition of

separate phases of the mixture.

Special preparation of analyzed material, i.e. its division into several samples, each being enriched

with one of the constituent phases, is a highly

promising method of increasing analysis reliability.

In this case several X-ray patterns are entered into

the automatic identification system and comparison

of their intensities permits determination of minor

phases and elimination of ambiguity of the second

type. Using the data of quantitative elemental analysis, one can also try to eliminate ambiguity of the

first type. This method has been suggested by J. Fiala.

Last in our store of methods we have the increase

in accuracy of diffraction measurements and replacement of database standards by more reliable ones. This method corresponds to an approximation to the

idealized condition A.

Let us sum up this theoretical discussion of the

identification problem to form a clear view of the

situation. The article of Johnson (1979) contains a partly humorous but very exact description of a number of requirements that should be met by both

the X-ray pattern of material under study (image Q)

and the skill of the operator who uses the automatic

identification program. In particular, Johnson has

written the following: "The system is not a cure-all.

When the computer prints out the results, it is at that point that the intelligent analyst starts to analyze the results - not just say this is the answer".

I would like to add a pair of additional points to

Johnson's words, proceeding from my own experience:

1. There always exist X-ray patterns that can be

identified by none of qualitative analysis programs.

2. There always exists a user who thinks that a computer should do all the work for him.

2. Clusterization

Let us try to carry out a more profound analysis of

the first cause of identification ambiguity. It would

be very interesting to clear up "the arrangement" of

the database from a crystallochemical point of view:

how many groups of similar compounds are in it?

what are sizes of these groups? and, last, whether

they exist in reality or only in our imagination. The

problem formulated in such a way actually comes

down to classification of the database, i.e., to separation of similar images P,- in groups (clusters). I, myself, always would like to see these clusters; this,

to a great degree, influenced the graphical design of our cluster programs.

First of all, I want to note that classification in

space R is inexpedient. Sets of d spacings for any

object of the set A are totally determined by a crystal system and the values of the cell parameters, so

it is considerably simpler to realize clusteriazation in a six-dimensional space of linear and angular parameters of lattice cells. The first program created for

this purpose permits one to carry out the following:

1. Selection of crystal system, two parameters (in

the general case out of six) and measurement

limits for these parameters, e.g. amin, Cmax)

2. Selection from the database of all the compounds in the given crystal system whose parameters are within the given range, e.g. all compounds in the hexagonal system for which

2 A< a < 20 A, 2 A< c < 20 A.

3. Display on a monitor screen each compound by

a point in Cartesian axes x and y corresponding to the first and second of the given crystal

parameters (in our example for the hexagonal

system x = a, y = c).

4. An opportunity for the user to outline with a closed line any part of the screen and to

obtain the following information about pointscompounds in it:

(a) chemical formulas of compounds;

(b) their names and numbers in the PDF;

(c) stick diagrams (which were displayed on

the screen).

A better program that is under development now

will provide two more important functions to this

list:
50 Computer-Aided Phase Identification

Table 1: Clusters for minerals in the tetragonal system, classified by the a and c lattice parameters.

No Group Ranges of lattice parameters

a, A

9 c, A

1 Rutile type 4.2 - 5.0 2.6-3.4

2 MgSb206 type 4.6 - 4.75 9.15 - 9.25

3 Ag2FeSnS4 type 5.3 - 5.75 10.4- 11.0

4 CaW04 & PbMo04 types 5.2 - 5.45 11.4- 12.1

5 CaCr04 & YVO4 types 7.0 - 7.25 6.25 - 6.35

6 Ca(U02)2As04 6.95 - 7.2 8.6-9.1

type 16.8-21.0

7 Melilites 7.5 - 7.85 5.0-5.1

10.5 - 10.6 9.9

8 KMnsOie type 9.85- 10.1 2.85 - 2.95

9 Scapolites 12.0 - 12.35 7.35 - 7.6

5. The possibility of selecting objects not only ac- cording to crystal systems but also according to

space groups, chemical composition and a number of other attributes as well.

6. The possibility of selecting as axes x and y not

only crystal parameters but any of their functions, such as cell volume, a/c ratio, etc.

These programs permit one to obtain on the computer screen a database section for any parameters;

this plane image is very convenient for subsequent

interpretation by the user. Studies have revealed clusters of various shapes

and sizes, ranging from strongly localized (within 0.5

A for linear crystal parameters) to spread out or ex- tended along one axis (over 1 - 3 A). The main clus- ters for minerals in the tetragonal crystal system are presented as an example in table 1. Let us note that in the book by Nakhmanson Sz. Feklichev (1990) the results of studies that have been

carried out are presented in detail. This book is published in Russian but, tables and graphs are intelli- gible even to English speaking readers. These re- sults lead to the following two conclusions the first of which concerns the topic of this paper, i.e. the

identification problem, and the second one is of more

general character.

1. Compounds found to be in strongly localized

clusters with similar values of crystal parameters

have, naturally, very similar X-ray patterns. Although the number of such clusters is not so large,

some common or industrially important compounds:

spinels, tourmalines, micas, chlorites and a number

of others are among them. These compounds are common in the practice of X-ray phase analysis, but

operation with them requires special accuracy be- cause of possible identification ambiguity.

2. I can not imagine yet in full measure the capabilities of cluster programs, but it is clear now that

they will permit extension of the field of use of the

PDF and other similar databases far beyond the limits of the traditional identification problem. In particular, using these programs it is possible to carry

out classification of compounds, to find interchangeable materials, and to construct various statistical

distributions. However, it seems to me that the most

curious question is why some bond lengths between

atoms exist in natural and synthetic materials while

others do not. We can suppose that this is connected

with allowed ionic radii.

References

FiALA, J. (1972). J. Phys. D: Appl. Phys., 5, 1874-1876.

FlALA J. (1976). J. Appl. Cryst., 9, 429-432.

Johnson, J. J. (1979). Norelco Reporter, 26, No.l.

Nakhmanson, M. & Feklichev V. (1990). Diagnostika

Sostava Materialov Rentgenodifrakcionnimi i spectralnymi metodami, Mashinostroenie
On the Relative Merits of Cameras and DifFractometers for XRD

P.-E. Werner

Department of Structural Chemistry, Arrhenius Laboratory,

S-106 91 Stockholm, Sweden.

Abstract

Obviously there are a large number of studies when

Debye- Scherrer and Guinier film techniques are so

clearly inferior that no comparison with diffractometers are meaningful. This is not the case for determination of diffraction angles, however. There are only

marginal differences in the quality of d-spacings obtainable by modern diffractometers and Guinier systems, but preferences are extremely divergent. However, a film is a detector and a long-time mass stor- age, whose information might be evaluated any time

by a densitometer system or visually inspected. So,

the exchange of a film against another recording device requires some good arguments. It will be shown

that modern automatic Guinier film scanner systems

need not be liable to more subjective influences than

automatic diffractometers. The speed and accuracy

of a computerized evaluation system for Guinier photographs will be demonstrated. All X-ray diffraction instruments need careful adjustment, but a well

adjusted and carefully handled Guinier camera is usually stable for years. Only diffractometers with

focussing incident beam monochromators may compete with the resolution of a Guinier photograph, but

for profile refinements Guinier and Debye-Scherrer

photographs suffer from poorly defined peak shapes.

For preliminary studies, including structure determinations and Rietveld refinements, the Guinier technique is a cheap and efficient alternative. It will be

argued that a modern laboratory working in the field

of X-ray powder diffraction should use diffractometer

and camera techniques in parallel.

Introduction

Although not new, the technique of X-ray diffraction is still developing and improving. New X-ray

sources and detectors, and new ways of using them

in computer-controlled diffractometers, make more

advanced experiments possible. During the last few

years we have seen time-resolved studies made by

electronic position-sensitive detectors, PSDs. Investigations of crystallite size, microstructural imperfections, stacking faults and compositional inhomogenities are, with few exceptions, made with powder

diffractometers. Why then should anybody use oldfashioned film methods in powder diffraction work?

Obviously there are a large number of problems

where Debye-Scherrer and Guinier film techniques

are so clearly inferior that no comparison with

diffractometer methods are meaningful. If it were

only of interest to judge in general terms how to

collect the most accurate powder diffraction data,

it should be enough to quote from the preface, by

Delhez and Mittemeijer, to the Proceedings of the

first European Powder Diffraction Conference held in

Munich, Germany, March 1991: "Although this has

been the subject of rather recent debate, it is now

possible to state that the accuracy achievable with a modern diffractometer in principle surpasses that

of the photographic techniques." At first sight this

statement seems to be final and sufficient. There are some points in this statement, however, that need to

be discussed. What is meant by "in principle," and

what is a "modern diffractometer"?

Is a modern diffractometer really the conventional

automatic powder diffractometer using a monochromator on the reflected beam side, or is it a diffractometer with an incident-beam focusing monochromator yielding pure CuKai radiation? Is it a diffractometer with Bragg-Brentano geometry and a scintillation detector, or is it not regarded as more modern

to use a PSD? In principle one cannot use a PSD

in the Bragg-Brentano geometry, as the position of

the focal circle depends on the angle between the

sample and the primary beam. The diffracted beam

is only focused at 29^ , i. e. at one specific detector

position. All other beams reflected from the sample are increasingly defocused with increasing differ- ence from the exact 203 position. Thus, the use of a PSD in Bragg-Brentano mountings is a compro51
52 Relative Merits of Cameras and Diffractometers

mise between resolution and measuring speed. So

far only two questions about the most frequently oc- curring diffractometer geometry, the Bragg-Brentano

mode, have been raised. As will be shown later,

however, even the oldest sample mounting, the Debye setting, may be successfully used in a modern powder dilfractometer equipped with a focusing

incident-beam monochromator. Using a fine-focus

X-ray tube, a large focal distance, an internal standard, and a small linear PSD, even this kind of

diffractometer may surpass the accuracy in diffraction angles measured by the most accurate Guinier

technique. Furthermore, intensity measurements are always more precise with an electronic counter and

also more accurate unless the instrument set-up and

the corresponding instrument functions are in error. As we shall see, however, this does not mean that the

Guinier camera is an historical object that should be

relegated to the museum collections.

d-spacings and Bragg intensities.

X-ray powder diffraction is a well established technique for the phase analysis of solids. Structural

studies of polycrystalline samples, not available as

single crystals; multiphase analysis, qualitative as

well as quantitative, have all benefited from the introduction of powder-pattern characterization. A

related application, the measurement of the dimensions and angles of the crystallographic unit cell, has

afforded a powerful means for studying and understanding the phenomenon of solid solubility. At a fundamental level these cell parameter measurements

serve to convert the atomic parameters, obtained in

crystal structure studies, into spatial realities of bond

distances and angles for subsequent correlation with

physical and chemical properties.

As stated by Mighell & Stalick (1979), identification of materials by cell parameters is extremely

efficient. The NBS Crystal Data File contains unitcell data for many more substances than the total

number of patterns, indexed and non-indexed, in the

PDF-2 data base. It may also be true that the main

reason for a large number of structural investigations

not being made is the simple fact that the powder

diffraction patterns of the substances have not been

indexed.

In general, only the diffraction angles, the Bragg

angles, are needed to solve the indexing problem and

determine the unit cell parameters. This is fortunate because of the fact that, regardless the method

used to collect powder diffraction data, the intensity measurements are often much more dependent

on instrument function and sample preparation and

thus much less reliable than the observed diffraction

angles, especially if an internal standard is used.

The intensity round-robin directed by the JCPDS

- International Centre for Diffraction Data, and published by Jenkins & Schreiner (1989), illustrates

this very clearly. From the tests made by twentytwo individual experimentalists using a variety of

diffractometer types and a range of configurations

of any specific instrument model, a number of interesting conclusions were drawn. It was concluded

that although the raw intensity precision for the single person/single laboratory was within a factor of

two of the statistical counting limit, systematic intensity variations of 30-60 % with 26 angle were found between measured and calculated data. It was

stated that systematic errors in powder diffraction

data remain abundant, but that, being systematic,

these can frequently be addressed by applying ex- ternal calibration procedures. Naturally all intensity

measurements in the round-robin were made with

diffractometers, in one case using synchrotron ra- diation. Usually scintillation detectors, but in two

laboratories PSDs, were used. Except for the synchrotron user only one of the two PSD users had

an incident-beam monochromator. Although not explicitly stated in the report, it seems clear that all instruments were set up in Bragg-Brentano, or at

least in reflection, mode. This was of course done in

order to avoid absorption corrections.

There is no reason to believe that any photographic detector is capable of competing successfully

with powder diffractometers when it comes to ac- curacy in intensity measurements. The primary fo- cus of the intensity round-robin was to study measured intensities obtained from modern computercontrolled powder diffractometers and therefore no

internal standards were used. The tests, however, were designed to yield information also on the performance of data treatment software packages. Therefore angular accuracy studies were also made. The

general conclusion from this part of the study was

rather disappointing. Quoting from the report: "Automation of powder diffractometers has not significantly improved d-spacing accuracies of raw data."

In many cases very significant (> 0.05° 20) systematic errors were found. For details see the complete

report given by Jenkins k, Schreiner (1989).

In the report a special study is made of the M0O3.

sample, because the penetration depth in this sample is negligible and the powder packs extremely well,

making a smooth surface easy to obtain and a dis-
P.-E. Werner 53

placement error less likely. In spite of this, large systematic trends were found in many data sets. Jenkins

& Schreiners made the comment: "One wonders how,

with automated systems, such gross errors can go uncorrected!" There may be different ways to explain

this. It should be noted, however, that the linear representation of the systematic angular errors (Fig. 12

in the report) is a simplified picture of the situation.

From the data (Table 27 in the report) it is obvious

that the angular error versus 16 is often a strongly

curved function.

An incident-beam monochromatized focusing

powder diffractometer

The powder diffractometer used in the following tests

is a Stoe Stadi/P instrument. It is a computerized high-resolution focusing monochromatic-beam

diffractometer. The diffractometer enables the user

to choose between transmission measurement using

flat samples or capillaries, and reflection measure- ment (by rotating the sample 90deg). In order to

provide high resolution, the radius of the recording

circle, i. e. the distance between sample and detector, is as large as 220 mm. Thus, a curved germanium

monochromator with a focal distance of 440 mm is used to produce pure CuKai radiation.

In a study, by Louer & Langford (1988), of the use of incident-beam monochromators in conventional Xray powder diffractometry a number of advantages

have been summarized:

(a) The removal of CuKq;2. reduces the number of

peaks, and consequently the resolution of the re- maining lines is improved. The peak positions

are not biased by any Kq;2. component. Although the bias can be analytically removed, it may introduce systematic errors unless a precise model for the Kai - Kag- doublet and its

satellite group is used (Parrish, Huang & Ayers, 1976; Will, Parrish k Huang, 1983). The

Kai and Ka2 lines have different widths and

are skewed by different amounts, which is rarely

taken into account.

(b) The background has no contribution from filter absorption edges and residual K/? lines.

(c) Instrumental line profiles have a simple, well

characterized shape whose parameters vary with

angle in a straightforward manner.

Louer & Langford discuss also the advantages of

incident-beam monochromators for indexing and determination of cell dimensions, line-profile analysis.

structure determination and structure refinement.

Although the article deals mainly with a BraggBrentano diffractometer, much of the content is also

valid for the Stoe diffractometer discussed above as

well as for a conventional Guinier camera.

d-spacings

Some general comments on intensity and d-spacing

measurements have been given above. In the present

section we will concentrate on d-spacings alone. The

Bragg-Brentano powder diffractometer is generally

regarded as the principal source of precision lattice

parameters. For several reasons it was also preferred

by the National Bureau of Standards (NBS) in the

preparation of diffraction patterns for the now defunct NBS monographs Standard X-ray Diffraction

Patterns. One reason for the choice of instrument

may have been the good possibilities of registering accurate absorption-independent intensities, al- though this was not the primary aim. Experimental

techniques and calculations are well described in the

first part of the monographs and will not be repeated

here. The description should also be valid for most

of the patterns now published in the journal Powder Diffraction. The use of internal standards is recommended, and most of the low-quality patterns

in the PDF-2 data base would probably have been

much better if this had been a more generally applied procedure.

Several studies of precision measurements of lattice parameters have been published since the article on this subject by Parrish & Wilson (1959) appeared in the International Tables for Crystallography, Vol.11. It is notable, however, that the main

parts of the article deal with Debye-Scherrer cameras and counter-diffractometer methods. Some comments are given about the back-reflection flat-plate camera. The conventional Guinier camera (1937,

1939) is not discussed. This is a pity, since there

may be reason to agree with the statement by G.

Hagg (1986): "In 1937 Professor Guinier, in a different arrangement, ingeniously combined the Johansson element with a focusing powder camera and thus

attained what I would call the most perfect instrument existing for powder diffraction." Studies, con- cerning precision and reproducibility of lattice parameters from Guinier powder patterns have been

published by Brown, Edmonds k, Foris (1981) and

by Brown k Foris (1983).

As we are indebted to A. J. C. Wilson for the most

extensive mathematical theory of powder diffractometry, I would like to show a quotation from his
54 Relative Merits of Cameras and DifFractometers

textbook Elements of X-ray Crystallography (1969,

p. 46): "The general principles of the calculation of

lattice parameters from the line positions are the

same for counter diffractometers as for powder cameras. However, the greater accuracy resulting from

the use of diffractometers makes the results both

more accurate and less liable to subjective influences."

It is interesting to compare this statement with

the following comment by R.Shirley (1979): "The

paramount importance of resolution for indexing

work explains the high success rate for focusing camera data, especially from Guinier-Hagg instruments,

whose resolution can only be described as superb.

It is rather less common (and considerably more ex- pensive) to obtain as good resolution with diffractometer data.", and the conclusion by A.Brown

(1986): "Comparison tests, made by members of the

Joint Committee for Powder Diffraction Standards,

demonstrate that Guinier data are at least equal, and

often superior to diffractometer measurements."

As we shall see, these statements by Wilson,

Shirley and Brown are not necessarily in contradiction. In order to show this, and also to illustrate the

relative merits of diffractometers and focusing cameras, two data sets of the same compound, the solid

solution (Zno.75Mgo.25)3(P04)2, were collected. The

sample is not a high-symmetry standard compound.

It is a normal sample from an ordinary crystallographic study using Rietveld refinements to determine cation distribution in solid solutions (Benard,

Nord, Werner Westdahl, 1992). A reasonably complicated powder pattern has been chosen in order

to illustrate the importance of good resolution and

make it possible to use statistical arguments.

It is the intention to show that a normal routine

Guinier film evaluation, one of approximately 2500

per year in our laboratory, gives average deviations

between observed and calculated diffraction angles

less than 0.01° in 29. Lattice parameters for the

compound will be given in order to illustrate the ac- curacy in d-values. It should be kept in mind that

the majority of the patterns obtained in the laboratory are used for identification, indexing, or phase

analysis. A few patterns are also collected for struc- ture determinations and refinements. Because of the

large number of photographs evaluated per year, it has been found essential to minimize the amount of

human interactions and to this end a computerized

film scanner system has been designed.

One powder pattern was collected with the Stoe

Stadi/P diffractometer described above, using a sample mounted in a 0.5 mm rotating capillary. The

diffraction data were collected with a small linear

PSD covering 7.0° in 29. The PSD was moved in

steps of 0.2°, thus giving a sum intensity from 35

measurements at each 29 position (19°< 29 < 135°).

Partly in order to make the following discussion less dependent on the type of diffractometer used, sili- con (SRM 640, a = 5.43088 A) was added to the

sample as an internal standard. The raw data were

analysed by a peak-picking program based on the

Morrey fourth derivative algorithm (1968) followed

by a program to decompose overlapping peaks and

adjust the entire profiles of the lines. The strategy

of the program is based on the maximum neighborhood method of Marquardt (1963), which overcomes

the problem of highly correlated parameters in nonlinear least-squares refinement. Both programs are

available in the Stoe software package. The silicon

line positions were then used to derive a second-order

polynomial correction function for the 2^-scale. Finally, 84 uniquely indexed lines of the 128 observed were used in a least-squares refinement of the cell parameters. The cell parameters obtained and some

relevant figures of merit are given in table 1 . Table 2

shows a small part of the list of observed and calculated diffraction angles. For comparison, a Guinier

photograph of the same compound was produced,

also with the addition of silicon as internal standard.

The camera used was a Guinier-Hagg camera, XDC

700 (R= 50 mm) .The photograph was measured and

analysed by the automatic Guinier film scanner sys- tem. In order to minimize subjective influence (cf. the statement by Wilson above) no manual changes were made in the default input parameters to our

Guinier film analysing program, SCANPI. The film

scanner, LS18 (Johansson, Palm h Werner, 1980), is controlled by a PC/AT computer. The SCANPI program evaluates background, performs noise analysis

by a procedure similar to the one described by Sonneveld & Visser (1975), determines peak positions and

integrated intensities, identifies the internal-standard

lines and calculates sample displacement, zero point

error and film-shrinkage factor (see appendix).

Usually some time should have been spent to check

the result by visual inspection with a plot module in

the program. To rule out any "subjectivity", this

check was omitted; and 105 lines, silicon lines au- tomatically excluded, were printed within 6 minutes

from the time the Guinier photograph was placed

in the film scanner. The cell parameters were indexed and least-squares refined with the same program, PIRUM (Werner, 1969), that was used for the
P.-E. Werner 55

Table 1. Unit cell data for (Zno.75Mgo.25)3(P04)2 obtained from different data sets and by different methods.

N = number of lines. E.s.d.s given in parentheses,

a) Unit cell dimensions from indexed powder diffraction patterns.

Data N a(A) 6(A) c(A)

Stoe 128 7.55854 8.36322 5.05213 95.0164

(19) (26) (13) (24)

Guinier 104 7.55831 8.36311 5.05172 95.0128

(39) (46) (23) (38)

» 50 7.55880 8.36227 5.05156 95.0147

(48) (50) (43) (65)

» 30 7.55885 8.3628 5.0514 95.014

(96) (11) (10) (15)

b) Unit cell dimensions from Rietveld refinements.

Data a(A) 6(A) c(A) /?° Stoe 7.55825(7) 8.36223(7) 5.05195(5) 95.0148(6)

Guinier 7.5564(2) 8.3597(3) 5.0502(2) 95.014(2)

c) Figures of merit.

M = de Wolff figure of merit. (de Wolff, 1968)

Index for the physical plausibility of a unit cell derived solely from powder diffraction data.

F = Quality index for an indexed powder diffraction pattern. (Smith & Snyder, 1979)

Data N M20 F20 Fjv

Stoe 128 126 205(0.0036, 27) 33(0.0070, 565)

Guinier 104 69 96(0.0078, 27) 53(0.0078, 251)

» 50 70(0.0073, 98)

30 73(0.0084, 49)

Stoe data set. One of the three weakest lines was

not indexable, but visual inspection of the photograph revealed that it was a noise contribution. The

unit cell parameters obtained are shown in table 1 and observed 29 values in table 2, adjacent to those

obtained from the Stoe data.

If the plot program module had been used, it would

also have been easy to see that the strongest reflec- tion (210) was overexposed, and its position should

normally have been corrected. It has a negligible influence on the cell parameters and STDs shown in

table 1, however, and it has therefore been left unadjusted in order to illustrate this kind of problem

in photographic methods. It may also be instructive

to see how clearly differences between observed and

calculated 26 values reveal the existence of an error. The absence of systematic errors is demonstrated

in table 1 in two ways. Refinements of unit cell parameters based on different numbers of lines do

not show significant differences in the cell dimensions, only STDs increasing with decreasing number

of lines. Furthermore, no significant deviations are seen between cell parameters obtained from the Stoe

and Guinier data sets. As can be seen from table 1, Wilson's statement is confirmed, although he had probably a quite different diffractometer geometry in mind. The standard

deviations of the unit cell parameters and the aver- age differences between observed and calculated 26,

especially for the first 20 lines, are smaller for the

diffractometer data. This should be no surprise, as the Guinier data have not only a more limited 26

range but also rather irregular and asymmetric peak

shapes; that makes it more realistic to rely on the
56 Relative Merits of Cameras and DifTractometers

Table 2. Observed and calculated 20{'') for (Zno.75Mgo.25)3(P04)2.

I-obs Guinier data integrated by SCANPI

I-obs Stoe data from peak program

Systematic absences: (hOl) h + l = 2n+l and (OifcO) ib = 2n + 1.

Guinier data

I-obs 9fl 9fl , ^fcalc 9fl u 9fl 1

Toi DO 9n 9n 90 ^t1 90 ^^9

Oil 90 on c;7c; ZU.O 1 0 90 t^ai 9n (^80

020 aD 91 997 9n 9^1 inx\j 91 998 91 9^n

101 99 n^iQ 9(1 nfio 28 22 Ofifi 99 nfi8

Til 99 99 Qf^Q 90 ORG 27 99 QfiS

120 9 94 "^1 94 "^90 4 24 ^"^4 210 1 nn zo.yuo 9"% 0^4 100 25

021 9n 97 700 9^^ 71 24 27^1 .1712x£t 97 714

T21 7 29.568 29.563 11 2Q 561 20 562

121 2 30.812 30.811 4 30.811 30.810

ZZy) 11 31.963 31.962 14 31.961 31.961

211 6 32.703 32.702 8 32.699 32.701

130 11 34.277 34.273 6 34.270 34.273

002 68 35.649 34.653 59 35.653 35.650

221 34.660 35.658

031 44 36.824 36.828 37 36.825 36.827

310 18 37.366 37.379 16 37.359 37.378

221 5 37.758 37.758 8 37.757 37.758

112 3 38.238 38.223 38.220

131 38.282 4 38.261 38.280

131 7 39.281 39.283 8 39.282 38.282

311 11 40.161 40.164 13 40.161 40.162

452 2 89.441 89.446 2 89.448 89.444

315 4 91.198 91.187

044 91.210

823 2 134.255 134.246

653 134.264

peak positions alone. It should be noted, however,

that only one of the Stoe data cell parameters deviates marginally more than one STD from the Guinier

data cell parameters. One may well ask when such a

small improvement is worth the effort, however.

Internal standard and absorption.

Systematic errors depending on the use of a weakly

absorbing material to correct the 26 scale for a

strongly absorbing material, as found for the BraggBrentano geometry (Brown, 1986), should not oc- cur with a Guinier camera or a dilfractometer with

transmission geometry and a focusing monochromator. For demonstration, a flat sample of lead nitrate with silicon as internal standard was prepared

for data collection in transmission geometry. Since

Brown k, Foris (1983) found that the measurement

of unit cell dimensions of lead nitrate is sensitive to

the specimen preparation, the same flat transmis-
P.-E. Werner

A. J

Obs.

'J

57

5000-

2500-

I

ii

5/

i' ULjLJUL_JLJJli_^ —I

60.0

1

eo.o

-I r- 20.01 40.0 100.0

2Theta

Figure 1: Stadi/P data set of Pb(N03)2 with Si added as internal standard.

sion sample was used for data collection by the Stoe

diffractometer and the automatic Guinier film scan- ner system. The small linear PSD was used in the

same way as described above. For comparison with

the Guinier data set, the upper limit in 29 was set to

100°. The data sets were treated as described in the

previous section. The diffraction pattern obtained

with the Stoe diffractometer is shown in figure 1. It should be noted that all silicon reflections are weak.

For a Bragg-Brentano measurement the strongest re- flections from the sample and the internal standard

should preferably be of equal intensity. Such samples

may take some time to prepare.

For the Stoe data the maximum difference between observed and calculated 26 was only 0.003°,

F20 = 321(0.0012,54). and the cubic cell parameter

a = 7.85722(3) A. The corresponding figures for the

Guinier data were 0.019°, F20 = 62(0.006,54). and

7.8574(2) A. Thus, no significant systematic error is indicated. It is not possible, however, to know if the

accuracy in the Stoe data set equals the precision. It

is possible that the accuracy in the internal standard

sets the limit.

d-spacing quality needed.

A recent article by Jenkins & Nichols (1990) about

problems in the derivation of d-values from experimental digital XRD patterns, deals mainly with con- ventional reflected-beam monochromatized BraggBrentano data. A large number of problems and a number of useful recommendations are described in

the article. These will not be repeated here. Only a few points related to the choice of technique will be

discussed.

Jenkins & Nichols (1990) state that, in spite of

numerous sources of error in the measurement of 29

values and subsequent conversion to d-spacings, the

derived d's are generally of sufficient quality to allow

qualitative phase identification. This was concluded

from the result of the round-robin test (Jenkins k

Schreiner, 1989), where it was found that the multiperson/multi-laboratory precision for a binary mixture of CaCOa/ZnO. was about 0.048°. This may

not be good enough for powder indexing, however.

It is a revealing fact that most of the laboratories

that have made contributions in the field of powder

indexing have also been deeply involved in the prob-
58 Relative Merits of Cameras and DifFractometers

lem of accurate determination of diffraction angles,

whether camera or dilfractometer methods have been

used. This is not strange; on the contrary, it is a logical part of the indexing work. Although it is difficult

to give any quantitative measure of the data quality

needed, it is certainly true that there are few fields

of endeavour in which careful data preparation is as dramatically rewarded as in computer-based powder

indexing (Shirley, 1979).

It should be pointed out, however, that finding the

unit cell, i. e. solving the indexing problem, and verifying the physical plausibility of the indexing are

different problems. For the latter purpose we may

use the criterion that the De Wolff figure of merit

M2Q (De Wolff, 1968) should at least reach a value

of 10. Approximate calculations show that for the

data sets in the NBS monographs. Standard X-ray

Diffraction Patterns, the average A{29) needed to

obtain M20 = 10 is strongly symmetry dependent. - Supposing the observed patterns to be equally complete (or incomplete), the following required precision ranges for 29 are obtained:

• 0.035° - 0.06° for triclinic patterns,

• 0.045° - 0.07° for monoclinic patterns, and

• 0.070° - 0.15° for orthorhombic patterns.

Solution of the indexing problem usually demands

much smaller deviations, however.

The statement by Visser (1969) that the errors in

diffraction angles should not exeed 0.03° (2^) and

that the first three lines especially should be as accurate as possible may be regarded as 'rules of thumb'.

Programs using different strategies may also be sensitive in different ways to accuracy and completeness.

The DICVOL program by Louer and Louer (1972,

new program version by Boultif and Louer 1991) is probably less dependent on accuracy in the first loworder lines than for example the TREOR program

(Werner, Eriksson k, Westdahl, 1985, new program

version 1990), which is more sensitive to accuracy in

this range but less dependent on completeness than

many other indexing programs.

It is well known that it is possible to obtain accurate d-values also by an external standard technique.

From the frequently occurring low data quality in the

PDF-2 data base, however, it may be concluded that

unless an internal standard technique is used, the ex- perimentalists often have very limited knowledge of

the data quality obtained.

Film sc£inner system.

It may be concluded that modern powder diffractometers using an incident-beam monochromator can combine the high resolution of a Guinier pattern

with mathematically well defined peak shapes. Furthermore, the step intensities are more accurately

collected with a diffractometer than by densitometer measurements, which need corrections for nonproportionality between exposure and optical density. However, as long as powder diffractometers not

only give much broader peaks, but also peak doublets

from CuKai and a2, Guinier data may be preferable

for structure determinations although diffractometer

data may be preferred for final Rietveld refinements.

This is because high resolution is the most important factor for indexing and for deriving integrated

500 1000 1500 2000

Exposure Cs)

Figure 2: Corrected and uncorrected relative optical

density from a density wedge. The relative densities,

D^i^g are obtained by subtraction of the local background.

intensities for the Patterson function and for preliminary Fourier calculations using incomplete structure

models.

a) Intensities from photographs.

As stated above, film scanner data need correction

for non-proportionality between exposure and optical density. It is well known that it is possible to use different exposure times and to scale the intensities from two or more photographs using only the

linear parts of the optical density versus exposure
P.-E. Werner 59

Table 3. Data from Rietveld refinements of (Zno.75Mgo.25)3(P04)2.

Stoe data set Guinier data set Number of parameters: 46 39

Profile function: pseudo-Voigt modified-Lorentzian

Background: refined not included

Step size (2^): 0.02* 0.0229*

Max. half-width: 0.137* 0.135*

Min. half-width: 0.099* 0.078*

E.s.d. in coordinates: 0.0001-0.0.0017 0.0004-0.0027

Rf 0.040 0.124

Rb 0.042 0.081

curve. This is usually not needed, however, if a simple second-order correction procedure is applied to

the relative optical densities D', i. e. the optical density above the local background.

A A;-value between 0.3 and 0.5 (cf. Fig. 2) may be rec- ommended (Werner, 1971). The local background

should be determined with the scanner system. A

disadvantage in using intensities derived from densitometer data, however, is that one has to use a weight

limit in the Rietveld program, as absolute intensity

data are not available.

b) Peak-finding on film data.

As stated by Jenkins & Nichols (1990), one major source of confusion is often the recognition of

peaks by a given automated system. In the intensity round-robin (Jenkins & Schreiner, 1989) diiferent users reported from 25 to 53 peaks for a-A^Oa.

over the same 20 range, whereas the calculated number is 42. In order to avoid the extreme sensitivity of the conventional second-order derivative peakhunting method, which often gives too many lines, a

different procedure has been used in our film evaluation program.

A peak-finding strategy that works well for photographs with low background and well-defined peak

shapes may not be useful for photographs with ir- regular line broadenings, many overlaps and/or high

background levels. Therefore the peak-finding algorithms are not based on any assumption of a general

profile function or a smooth variation of the halfwidths with 29. In order to increase the range of

peak types that can be analysed, three different peak

finding algorithms are available.

The first one is, in principle, a normal first-order

derivative method with relatively strong constraints.

These serve to avoid regions where end-point maxima

occur, or where several closely adjacent very small

peaks are created by the addition of noise to weak

but significant signals. Such regions are analysed in a second pass, which uses maximum deviations from

the background for peak definition. These two procedures are normally used.

The third, less frequently used algorithm is a stepwise movement of an integration window of predefined width across all regions where peaks of the first two types are not already defined. The method is similar to that used in some single-crystal diffractometer programs for calculation of integrated intensities. Two different noise levels are calculated and used

by the program. One dynamic noise level (i. e. a

function of the local optical density) is defined for the

individual step intensities and a second noise level is used for integrated intensities.

Rietveld refinements.

As stated above, Rietveld refinements using photographic data may be useful in the preliminary stage

of a structure determination, although the results are

less accurate than refinements involving diffractometer data. There is no need to discuss the well-known

Rietveld method in any detail here, but it may be

instructive to show why Guinier data should not be

underestimated. As an illustration, the two data sets from the solid solution (Zno.75 Mgo.25)3(P04)2 discussed above were refined with a local version of

the Rietveld program DBW3.2S (Wiles, Sakthivel

& Young, 1988). A pseudo-Voigt profile function

was used for the Stoe data. The profile function for
60 Relative Merits of Camercis and DifFractometers

J

9000 .

GOOO

4000

2000 .

I

I

I M I I I M I

I

I

I

I I I i M I

I

I

I II I I I I

I

I

I

I I I I I I I I

I M > I I

20 . 50 . 'tO . 50 . 60

^

1 1 1 1 1 1 1 1

1

1

1 1 1 1 1 1 1 1

10 . so

20

Figure 3: Final Rietveld plot of (Zno.75Mgo.25)3(P04)2 from a Stadi/P data set. The upper curve shows

observed data, while the lower curve is the difference between observed and calculated step intensities.

the Guinier refinement was a modified Lorentzian.

Some non-structural parameters were naturally dif- ferent. The amounts of silicon were somewhat diff'er- ent, and normally no internal standard should have

been added in a sample used for profile refinement.

The number of positional parameters was 18; further

information about procedure and results is available

in the reference (Benard, Nord, Werner Sz Westdahl,

1992).

The observed intensity curve and the differences

between observed and calculated intensities for the

Stoe data set are shown in figure 3. Some information about the refinements is given in table 3. Of course the diff"ractometer data set gives a much

more accurate final result. Otherwise it would have

been difficult to justify an additional, vastly more

expensive and time-consuming data collection sys- tem. It should be noted, however, that in spite of

the fact that a powder diffractometer with exeptionally high resolution has been used, the Guinier lines

have smaller half-widths, especially in the low angle range. Although this is of no importance for the

final result, it is extremely important for ab initio

structure determinations.

The formally refined unit cell dimensions are shown in table lb. It should be realised, however,

that the cell dimensions obtained by Rietveld refinements are parameters that partly correct the data for

non-linear 2^-scales. The small standard deviations

usually obtained for unit cell parameters by Rietveld

refinements are therefore often strongly misleading.

Figure 4 shows a small part of the observed and

calculated intensity sets from the two Rietveld re- finements.
P.-E. Werner 61

a)

'^B Vs . so. si'. 52 ! SS . 5^ ! ss

!

SLoo eJciLo I -coLc .

'OOO .

5000 .

?ooo

.

Gu ddLa t —ob« . ^ ^ ^

Figure 4: Observed and calculated intensities within

the range 48° < 26 < 55.5° from Rietveld refinements of (Zno.75Mgo.25)3(P04)2-

a) Stadi/P data set. Pseudo-Voigt profile function.

b) Guinier data set. Modified Lorentzian profile function.

Final remarks.

Different types of equipment for X-ray powder

diffraction work have different advantages and disadvantages. A conventional automatic Bragg-Brentano

diffractometer with diffracted-beam monochromator usually yields a high peak-to-background ratio,

absorption independent intensities, short exposure

times and, as most diffractometers, well defined peak

shapes. As it cannot compete with the high resolution of a Guinier photograph, however. Another kind

of diffractometer has been used in the present work.

This diffractometer has a very good resolution at the

expense of long measuring times, in spite of the use of a small linear PSD. Operated in the symmetric transmission mode it also needs much less sample than a Bragg-Brentano diffractometer. Thus, it is hardly

meaningful to discuss powder diffractometers as a homogeneous group of instruments. Diffractometers

are also strongly dependent on software systems.

Although, as shown above, it is true that a modern diffractometer may surpass the accuracy of a Guinier camera, it is also true that it is rather difficult, and comparatively expensive to obtain Guinierquality resolution in diffractometer data. All Xray diffraction instruments need careful adjustment,

but a well adjusted and carefully handled Guinier

camera is usually stable for years. The film is an

efficient position-sensitive detector and a long-time

mass storage of information that may be visually inspected and evaluated at convenience with a densitometer system. Moreover, it is certainly not true

that a computerized Guinier film evaluation system

is more liable to subjective influence than an automatic diffractometer.

As illustrated by the present work, both precision and accuracy in 29 angles are normally better

than 0.01° in routine measurements of Guinier photographs, if an internal standard and an automated

film scanner system are used. This is usually suffi- cient for phase analysis, indexing and unit cell refinement. Thus, although modern powder diffractometers are indispensable for studies of line broadening, structure refinements, quantitative analyses and

time-resolved experiments, it may be strongly recom- mended to use advanced camera techniques for most

of the routine measurements in a laboratory working

in the field of powder diffraction.

References

Benard, p., Nord, A., Werner, P.-E. & Westdahl,

M. (1992). J. Solid. State Chem. 99, 290-296.

BOULTIF, A. & LOUER, D. (1991). J. Appl. Cryst. 24,

987-993

Brown, A. (1986). Chemica Scripta 26A, 77-83.

Brown, A., Edmonds, J. W. & Foris, C. M. (1981).

Adv. X-ray Anal. 24, 111-120.

Brown, A. & Foris, C. M. (1983). Adv. X-ray Anal.

26, 53-62.

Delhez, R. & MiTTEMElJER, E. J. (1991). Materials

Science Forum, 79-82, Pt.l

Guinier, A. (1937). Compt. Rend. 204, 1115-1116.

Guinier, A. (1939). Ann. Phys. 12, 161-237.

Hagg, G. (1986). Chemica Scripta26A, 3-6. Jenkins, R. & Nichols, M. (1990). Adv. X-ray Anal.

33, 285-293.

Jenkins, R. & Schreiner, W. N. (1989). Powd. Diff. 4, 74-100.
62 Relative Merits of Cameras and Diffractometers

Johansson, K. E., Palm, T. & Werner, P.-E. (1980).

/. Phys. E: Sci. lustrum. 13, 1289-1291.

LOUER, D. & Langford, J. I. (1988). /. Appl. Cryst.

21, 430-437.

LoUER, D. & LoUER, M. (1972). J. Appl. Cryst. 5, 271-

275.

LouER, D. & Vargas, R. (1982). /. Appl. Cryst. 15,

542-545.

Marquardt, D. M. (1963). J. Soc. Indust. Appl. Math.

11, 431-441.

MiGHELL, A. D. &: Stalick, J. K. (1979). Natl. Bur.

Stand. (US) Spec. Publ. No. 567, 393-403.

MORREY, J. R. (1968). Anal. Chem. 40, 905-914.

Parrish, W., Huang, T. C. & Ayers, G. L. (1976). Trans. Am. Crystallogr. Assoc. 12, 55-73.

Parrish, W. & Wr.son, A. J. C. (1959). Int. Tables

for X-Ray Crystallogr. Vol.11, 216-234

Shirley, R. (1979). Natl. Bur. Stand. (US) Spec. Publ.

No. 567, 361-382.

Smith, G. S. & Snyder, R. L. (1979). /. Appl. Cryst.

12, 60-65.

Sonneveld, E. J. & Visser, J. W. (1975). J. Appl.

Cryst. 8, 1-7. Visser, J. W. (1969). J. Appl Cryst. 2, 89-95.

WER.NER, P.-E. (1969). Arkiv for kemi 31, 513-516.

Werner, P.-E. (1971). /. Phys. E: Sci. lustrum. 4, 351-

353.

Werner, P.-E., Eriksson, L. & Westdahl, M.

(1985). /. Appl. Cryst. 18, 367-370.

Will, G., Parrish, W. & Huang, T. C. (1983). /. Appl. Cryst. 16, 611-622.

Wilson, A. J. C. (1970). Elements of X-ray Crystallography, Addison-Wesley Publ. Comp. Inc. Wolff, P. M. DE (1968). /. Appl. Cryst. 1, 108-113.

Appendix

Influence of systematic errors in 29 on a Guinier film

(Brown, 1986).

S = Measured distance on Guinier film between primary beam line and reflection

R = Camera radius ^ = Angle between primary beam and normal to

the specimen

e = Sample displacement

F = Film shrinkage factor

ASq. — Zero point error in S

2^corrected = Aq + AiS + A24>

where

(f) - sin(2^)/ cos(^) cos(2^ -

Ai = 90/{TrRF),

Ao = -eAi,

and

A2 = -ASoAi.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Data Optimization and Propagation of Errors in Powder Diffraction

J.-F. Berar

Laboratoire de Chimie-Physique du Solide (UA 453 CNRS), Ecole Centrale, 92295 Chdtenay, France

and LURE, Universite Paris-Sud, 92405 Orsay, France.

Errors in powder diffraction were widely discussed

in the first "Accuracy in Powder Diffraction" meeting

(Young, 1979; Albinati, Cooper, Rouse, Thomas k,

Willis, 1979; Cooper, Sakata &; Rouse, 1979). In the

following years, related papers have been published.

The significance of the value of estimated standard

deviations (e.s.d.'s) obtained by the Rietveld (1969)

method was discussed (Sakata &; Cooper, 1979; Pawley, 1980; Scott, 1983): they are not in accordance

with those obtained by intensity refinements, their

values being smaller. Still it has been proved that

e.s.d.'s are an unbiased estimate of the parameter

precision when the assumptions on which the method

of least squares is based are satisfied (Prince, 1981;

1985), that is when the model is perfectly suited to

the data set. Data collection strategies have been

developed to overcome this problem (Hill & Madsen,

1984; 1986; 1987). Methods have also been proposed

to test the validity of the statistical assumptions in

Rietveld refinements (Hill k Flack, 1987). These

strategies can be seen as an answer to the question:

If we take into account the quality of the results obtained by powder diffraction, is it reasonable to improve the quality of our measurements? Such experiments are time consuming, and wasting time is very

expensive in big science.

Data collection strategies

Since that meeting, in which most of the results

come from neutron diffraction, the Rietveld method

is now commonly used on data obtained from X- ray

sources, either classical tubes or synchrotron beams.

In such cases, or with the new neutron diffractometers, higher resolution can be attained. The number

of data collected increases following this resolution

improvement. The consequences of this progress in

the quality of the measurements have to be considered.

This trend from neutrons to X-ray raises other

questions. Can we use the same strategy to optimize data collection with both sources? What do

we really want to measure through this diffraction

experiment? Are we are really interested in all parameters? Is the same experimental method the right

way to get all the parameters that we can physically

reach?

If we consider a high resolution synchrotron experiment, the full width at half maximum (FWHM) of

lines has a value of some hundredths of a degree, typically 0.03° 29. The full pattern has to be recorded

with a step width of some thousandths of a degree,

and the data set will contain more than 30000 val- ues. If we want to perform a full analysis of these

data we need a program able to refine not only the

structural model (cell parameters, atomic positions,

thermal factors) but also other variables related to

size, shape, defects and instrumental functions. If we do not know valid and accurate models for all these factors, the refinement will suffer from bias,

and it will be difficult to know with accuracy the

structural parameters, even if we get very precise

values for them. On the other hand, if we wish to

know only the cell parameters to study thermal ex- pansion, we need to measure fewer lines, but those

measurements should be made with higher angular

accuracy. The knowledge of structure depends only on the integrated intensities and can often be obtained using lower resolution and a poor statistical

level. For typical structures with fewer than 50 parameters good precision can be achieved with 2500

measurement points.

These relations among resolution, recording step

and counting time have been extensively studied by

Hill k Madsen (1984; 1986; 1987). Their conclusions can be roughly summarized as follows: the step

width should be between one half and one fifth of

the FWHM of the sharpest well resolved line, while

the time per step is chosen to obtain some thousand

counts at the maximum of the data set. If some evidence of correlation still exists in the data set, a subset should be extracted. This will increase e.s.d.'s,

but will reach a point where the statistical assumptions are valid.

The quality of results that can obtained in very

short recording time can be surprising (0.05s is given

63
64 Data Optimization and Propagation of Errrors

as an example by Hill &; Madsen, 1984). Our tendency is to use a visual reference to assess the quality

of the data. In figure 1 , the highest count is no higher

1 1 1 -

1

1

1 1 1 ] n » I

J I n I J 11 1

1

1 1 1 1

1

; 1 1 1 1 » 1 1 1 n 1 1 r r 1

1

1 11 1

1

1 1 1 [1 1 1 1 ] nn n n I ri 1 1 1 1 1 1 i>

22. D 24.0 26.0 £6.0 30.0 32.0 3-1.0 36.0 2Chat3

Figure 1: Expected curve (a) in a low counts experiment (x) compared with the result obtained with a

digital filter(6).

than 6. Nevertheless the information contained in

the data can be extracted even with a simple digital filter adapted to the instrumental function. The

resulting curve looks like the one obtained with a

recording time a hundred times longer.

As the strategy of collecting data depends on the

sample, some remarks can be added to the previ- ous rules. In many problems, we are not only con- cerned with the structure, but also with its evolution.

In such cases it is possible to obtain a very precise

knowledge of the parameters if we take into account

the dependence of the structure on external factors

(time, temperature, . .

. ). This can be done either by

refining all the collected data simultaneously using a modified program that includes such external dependence (Berar, 1990), or by fitting the parameter

dependence.

In X-ray experiments, data collection is commonly

performed by scanning the 29 circle with the counter.

In such cases, it seems possible to use algorithms

that are able to modify the step width and use larger

steps in the background. This will allow some time

to be spared during measurements, but information

contained in the weak lines may be lost. For instrumental reason, we commonly record each

step with a fixed time or fixed monitor counts. Other

choices can be made. In X-ray experiments the

atomic scattering factors decrease very quickly. If we want to extract valuable information out of data

recorded at high angles, we have to increase the quality of these data. It can be useful to record the data

using diff"erent counting times at low and at high

angles. This implies that we use either the same

-

-T^ A. .1 , 111

n rm rm-rrVl?! 1 iT-rri-t n-rl m i ii r

. . I;

Ti 1 1 1 r 1

1

1

1

—

L

1 M 1 tt-irrt

Figure 2: Two possibilities for recording the same

X-ray data.

scale factor and adapted weights or observed weights

and two related scale factors. Otherwise we will not

only waste time, but the data recorded at low angles would be known with unnecessay precision. This

compels us to take into account more sophisticated

corrections (extinction, ...). Both diagrams in fig- ure 2 need the same recording time. The refinement

qualities are similar, but in the case of the diagram

recorded with two difi"erent step times having a one

to eight ratio, the e.s.d.'s on most of the parameters

are three times smaller. Commonly with isotropic

thermal factors reliability factors Rj or Rp reach

values of a few percent. In order to get a balance between lines, most of the integrated intensities should

be chosen in order that the ratios of their e.s.d.'s to

their values are in the same range, that is no more

than thousand counts, with the peak maxima having fewer counts. Studying real samples, similar but

more elaborate strategies should be applied.

E.s.d.'s in the Rietveld method

In the Rietveld method for structural refinements,

the calculated model is directly compared to the observed pattern by a least squares minimisation

S = J2MYoi-Y.i)' = J2''iD^ = J2^i (1)

in which Yd and Yd stand for the observed and the

calculated counts at angle 20i, Wi is the associated

weight, Di their difference and a,- the normalised or weighted one. The summation is carried over the N

points in the data set. In this kind of method, there

is no requirement for the data to be recorded with

fixed step width or counting time. This method does

not try to get information out of extracted values of
J.-F. Berar 65

integrated intensities. It just samples the diffraction

pattern, each data point being an independent measurement. Therefore it is important for the weighting

scheme to have a realistic meaning. With the use of

multidetectors, the fact that all the data set is often

not really covered with the same number of detectors should be taken into account. The use of the

Poissonian variance l/Yoj was unappropriate in old

programs in which the background was substracted.

However it seems possible to use weights defined us- ing the calculated counts, This can be seen as an approximation to the maximum likehood method,

which has been applied with success in fitting ex- perimental data by curves (Antoniadis, Berruyer &; Filhol, 1990).

The calculated intensities Yd can be expressed by:

Yd = Yti + ^

X! KjPj'^i^i - 0k)IjkCjk (2)

j k

P(Ad) being a normalised profile function whose

shape depends on the phase j and the line k. h

is the intensity, which is simply the square of the

structure factor for non-magnetic structures. Cjk

is a correction factor that includes factors like the

Lorentz-polarization and, if necessary, orientational

factors. Kj is the scale factor of the phase. The value

Yii of the background may be represented by polynomials, splines or interpolation functions, which can be refined.

The S sum is minimized using a linear approximation, shifts Xp of the P independent parameters are calculated solving the linear system:

MX = V,

where

i

Vj, = 'lY,WiDidYd/dp

(3)

(4)

(5)

When the minimum of S is reached, the e.s.d. of

parameter q is given by

s] = {M-'\,SKN-P) (6)

It follows that for N large enough, e.s.d. 's behave

like \/y/N.

Such a decrease, which is the normal behaviour for

parameters obtained by least square analysis, leads

to very small values for e.s.d. 's. Sometimes they can be even smaller than values obtained with single crystals. Here was the source of the debate. E.s.d. 's are statistical values, so they cannot tell anything

that is not in the model. They are estimates of

the precisions of parameters and not of their accuracies. We can obtain very precise but inaccurate

(wrong) results (Schwarzenbach, Abrahams, Flack,

Gonschorek, Hahn, Huml, Marsh, Prince, Robertson,

RoUet k Wilson, 1989). In the case of the Rietveld

method. Prince (1985) has demonstrated that e.s.d. 's are unbiased estimates, but it is assumed that the

model is right. Then the remaining differences between the observed and the calculated values are due

only to the Poissonian noise. In such case the Goodness of Fit {GoF) has its square distributed like a with (N — P) degrees of freedom.

GoF= \/S/{N - P) (7)

Because the expected value of GoF is 1, its value in a fit should be smaller than 1.4 if (A^ — P) is large

enough.

In the case of Rietveld refinements, other kinds

of discrepancy frequently appear. They are due to

serial or local correlations and can be characterized

by the Durbin-Watson d distance (Hill, Flack, 1987):

N N

(8)

t=2 »=1

For random data, d has an expected value of 2, and

if (A^ — P) is large enough, d will not deviate from

2 by more than Z/y/N . Values of the distribution

are given in Hill k Flack (1986) and Theil k Nagar

(1961).

As a consequence of (1), it is normal that the Rietveld e.s.d. 's Sq are lower than those calculated by integration or individual fit. The sampling realized by

the fit gives a better estimate of the integrated intensity (Table 1). If we add a lot of background data.

Table 1: Precision of intensity measurement for var- ious method.

Summation Individual fit points

parameters

intensity

background

50

1 992(33)

19.2(2)

50

5

975.2(8)

19.200(6)

200

5

975.2(8)

19.200(3)

the e.s.d. 's on the intensity are close to those previously obtained because we do not modify the Mpq

matrix elements that are directly related to this es- timate. Only the significant points really contribute

to the e.s.d. 's. In a Rietveld or a whole pattern fit, most of the parameters are common to all lines. Then

the equivalent number of parameter for each line is close to 1 and e.s.d. 's are reduced more than twice in

comparison to single line fits.
66 Data Optimization and Propagation of Errrors

Estimated probable error

With X-ray diffraction, we frequently encounter data

that do not fit well. These data exhibit serial or local

correlation. Their d values significantly differ from 2. This was the objection given by Cooper (1982) about

the e.s.d.'s, but it was mainly thought that the local

correlations where associated with the intensities of

lines. This kind of local correlations vanishes when

the structural model is right. However, in many data

sets there are local correlations that are due to insufficient knowledge of profile functions. It is quite easy

COUNTS

1 0OOOJ

aoooj

nooo;

?.00'

Figure 3: Local correlation associated with profile or

intensity misfit.

to characterize the two cases. If the local correlations

are due to the profile, they appear on the difference

(Fig. 3) as small peaks with opposite sign, whereas if

it is an intensity error the correlation length is similar

to the line width.

In a pragmatic approach, when this occurs, we

can extract subsets out of the recorded data up to a

limit where the correlations vanish. Data are then in

agreement with the model, and the e.s.d.'s values are meaningful. Following this procedure, mean values

of parameters and e.s.d.'s can be directly calculated

from equivalent subsets (Table 2).

Table 2: Mean value and e.s.d. obtained with 5

aPbO simulated data sets (zpj = .2344) including

the same profile misfit.

Step width ("26) .005 .020 .080

s

(4)

.2348

5

3

6

.2339

7

6

8

.2339

13

11

12

We can obtain similar results assuming that all correlated differences are due to the same cause

(Berar & Lelann, 1991). Their contribution are added linearly. The m independent errors are then

added quadratically. It is possible to define a level of

significance above which differences have to be considered to be correlated. It seem more efficient to

split the differences in two terms according to the

probability that such differences will be correlated.

A modified S' can be calculated.

5' = j;(l-z2)a2 + j;^(J3,,a,)^ (9)

ml

In a first try, we have considered that z is proportional to the probability of being inferior to

X = of + a]_i in the distribution with two degrees of freedom.

0

if a,a,_i > 0

otherwise (10)

But with such a choice, if you look at random data,

S' will be 20% higher than 5" (Ross, 1991). This discrepancy can be reduced to 10% if we use the ratio of

probability of x to taking a value in the distribution with one or two degrees of freedom. The remaining which could be corrected is due to the probability

that X can take a value in random data.

-{

xy/2irx/{2 + 2x + -\/2Trx) ifa,a,_i>0

0 otherwise

The probable error Sj. on the parameter k is in

the ratio p = y/S'/S according to the previously

calculated e.s.d.'s

Sfc = psk (12)

that is, the value obtained in the limit where the

correlations vanish. In practical cases, the range of

p is 1.5 to 3. As we know in many Rietveld refinements there are some discrepancies. It may seem better to minimise

the S' sum. Such methods have already been used

in other techniques (Mijlhoff, 1991, Alcock, 1991). It corresponds to the use of a non-diagonal weighting

scheme (Murata &; Morino, 1966; Beagley, Robiette

k. Sheldrick 1968a; 19686). Our first attempts we are not at present sufficient to draw any conclusions.

Precision and accuracy

We have to remember that e.s.d.'s and the associated probable errors are only precision indicators.

They are not concerned with the accuracy of results.

The essential problem of accuracy is not satisfactorily resolved, but accuracy is needed when we have
J.-F. Berar 67

to compare Rietveld results to others. Accounting

for systematic errors as if they are random noise can produce some bias in results, even if statistical as- sumptions seem to be valid (GoF « 1, rf « 2). If we want to make real progress in accuracy, the only

way is to improve the Rietveld model in software or

to obtain simpler line shapes, as in the synchrotron

case. At present many of the discrepancies can be

imputed to profile functions.

Better profile functions have to take into account

the real emission spectra, which can include the line

commonly referred to as 03 (Howard & Preston,

1989), some WLa lines, and part of the continuous

spectrum. A better angular dependence of asymmetry and width can be used. In the low angles range

instrumental broadening (axial divergency) behaves

like l/sin2^. The specific problems related to size

and shape can be avoided using an (hkl) or an angular dependence of line width parameters; it is easy to

expand them using spherical harmonics. A similar

expansion allows easier handling of the orientational

correction. The software must really take into ac- count the geometrical arrangement in cases of non- standard geometry (position sensitive detector, . .

. ). Some other causes of systematic error have to be

checked: anomalous dispersion effects; extinction,

which has been noticed in powders (Sabine, 1988);

and the poor knowledge of background. Thermal diffuse scattering is really nonuniform in space and is a problem in absolute intensity measurements (Suorti,

1977). It will introduce systematic errors into Rietveld refinements. It is mainly this background to

signal ratio that promotes single crystal experiments,

not the fact that there are overlap problems to solve.

When we want more accurate results, we also need

good, well characterized samples.

To conclude, a lot of progress has to be made to obtain more accurate values, but our insufficient knowledge of the profile is not so dramatic. It can be seen with simulated data that the induced bias on struc- tural parameters is very small. Furthermore it seems

possible to profit from the remaining local correlations to improve the profile function and then the

accuracy of structural parameters. At the present

the Rietveld model is not wrong, but it needs to be

completed.

References

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

M. W. & Willis, B. T. M. (1979). Natl. Bur. Stand.

(US) Spec. Pub. No. 567, pp.203-210.

Alcock, N. W. (1991). private communication.

Antoniadis, p., Berruyer, J. & FiLHOL, A. (1990).

Acta Cryst. A46, 692-711.

Beagley, B., Robiette, A. G. &; Sheldrick, G. M.

(1968a). J. Chem. Soc. A, 3002-3005.

Beagley, B., Robiette, A. G. k. Sheldrick, G. M.

(19686). J. Chem. Soc. A, 3006-3008.

Berar, J.-F. (1990). /. U. Cr. Satel. Meeting "Powder

Diffraction", Toulouse.

Berar, J.-F. & Lelann, P. (1991). J. Appl. Cryst. 24,

1-5. Cooper, M. J. (1982). Acta Cryst. A38, 264-269.

Cooper, M. J., Sakata, M. & Rouse, K. D. (1979).

Natl. Bur. Stand. (US) Spec. Pub. No. 567, pp.167-187.

Flack, H. D. Vincent, M. G. (1980). Acta Cryst.

A36, 495-496.

Hell, R. J. & Flack, H. D. (1987). J. Appl. Cryst. 20,

356-361.

Hill, R. J. & Madsen, I. C. (1984). J. Appl. Cryst. 17,

297-306.

Hn.L, R. J. & Madsen, I. C. (1986). J. Appl. Cryst. 19,

10-18.

Hill, R. J. Madsen, I. C. (1987). Powd. Diff. 2,

146-162.

Howard, S. A. &; Preston, K. D. (1989). Review in

Mineralogy (US) 20, 217-275.

MlJHOLF, F. C. (1991). private communication.

MURATA, Y. & MORINO, Y. (1966). Acta Cryst. 20, 605-

609.

Pawley, G. S. (1980). J. Appl. Cryst. 13, 630-633.

Prince, E. (1981). J. Appl. Cryst. 14, 157-159.

Prince, E. (1985). Structure & Statistics in Crystallography, A. J. C. Wilson, ed. pp 95-103, Guilderland, NY:

Adenine Press. Ross, C. R. (1991). private communication. '

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

Sabine, T. M. (1988). Aust. J. Phys. 41, 413-422.

Sakata, M. &; Cooper, M. J. (1979). J. Appl. Cryst.

12, 554-563.

Schwarzenbach, D., Abrahams, S. C, Flack, H. D.,

Gonschorek, W., Hahn, Th., Huml, K., Marsh, R.

E., Prince, E., Robertson, B. E., Rollet, J. S. &

Wilson, A.J.C. (1989). Acta Cryst. A45, 63-75.

Scott, H. G. (1983). J. Appl. Cryst. 16, 159-163.

Suorti, P. (1977). Acta Cryst. A33, 1012-1027.

Theil, H. L., & Nagar, a. L. (1961). J. Am. Stat. Assoc. 56, 793-800.

Young, R. A. (1979). Natl. Bur. Stand. (US) Spec. Pub.

No. 567, pp.143-165.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

NIST XRD Standard Reference Materials: Their Characterization and Uses

J. P. Cline

Materials Science and Engineering Laboratory, National Institute of Standards and Technology,

Gaithersburg, MD 20899 U. S. A.

SRM production and certification

Standard Reference Materials (SRMs) from the National

Institute of Standards and Technology (NIST) are designed to increase the accuracy and precision of

measurements which are pertinent to science and

industry. Powder diffraction SRMs consist of stable

materials which have one or more diffraction properties

measured and certified. The success of an SRM is dependent on the material and certification measure- ments being of such quality that the accuracy and

precision of the certified values allows for the

calibration of subsequent measurements made by the user community. The standardization of such

measurements is achieved by the performance of a

material rather than the standardization of equipment

design and/or measurement procedures.

The certification of a NIST SRM begins with the

isolation of a measurement error which may be

corrected with the proper use of a standard. A

preliminary study is imdertaken to determine the

feasibility of producing an SRM which will address the

problem. In addition, the potential demand for, and

impact of, the SRM is evaluated. A proposal for the

production of the SRM is then submitted to the Standard

Reference Materials Program (SRMP) for review and

subsequent funding. The SRMP of NIST operates as a

non-profit business with the costs of developing and

producing an SRM being fiiUy recovered by the sale of

that SRM. Thus the demonstration of the demand for

the SRM is critical to a favorable response.

Once the decision to produce the SRM has been

reached, an "idealized character" of the SRM material

is determined from a review of the literature. The

effects of deviations from this "ideal" character

exhibited by available materials are the subject of

subsequent experiments. The characteristics investigated are those judged as capable of having an effect on the

critical property which is at least as large as the smallest

conceivable measurement error of that property. The

study may include the development of improved

measurement techniques which increase the accuracy

and precision of the property determinations.

Throughout this work the procurement or production of

several kilograms of material for the SRM itself is under consideration. The material selected for the SRM

must be optimal with respect to the desired property,

homogeneous, stable, and of such character as to minimize errors in the measurements made by the end

user. When the required quantity of material has been

obtained and verified to be acceptable, it is riffled and

bottled by the SRMP. During bottling, a representative

sampling of bottles is removed from the stock for

certification measurements. The certification process is designed to determine the homogeneity of the material

and also determine, via methods traceable to fundamental physical constants, the value of the certified

property. However, in the case of certain SRMs, such

as the line broadening SRM 660, the desired property

does not lend itself to absolute measurement. In such

cases the desired property is determined to be

appropriate for the demands placed on the SRM by

supplemental research or a round robin study, in which

case the certification process is used to determine

homogeneity only. Modem statistical procedures are used to design and analyze experiments to insure that

the measurements are as accurate as possible and the

uncertainties fairly reflect the precision of those

measurements.

Line position SRMs

The primary line position SRM, silicon SRM 640b, is the most popular of the NIST powder diffraction SRMs.

SRM 640b consists of a silicon powder, jet milled to a median particle size of about 5 fim from electronic

grade single crystal boules, certified with respect to

lattice parameter. A second line position SRM 675,

fluorophlogopite, is used for low angle measurements.

The popularity of SRM 640b reflects the widespread

adoption of the internal and external standard methods

for use in lattice parameter measurements. These

procedures involve the determination and application of

a delta d curve which represents the end result of the

various optical aberrations of the particular diffraction

equipment. A polynomial is then fitted to the curve and
J. p. Cline 69

applied either to subsequent measurements in the case of

the external standard, or to the line positions of the

unknown with which the SRM has been mixed in the case of the internal standard. The external standard

method is less reliable because it cannot account for a

possible sample displacement error of the unknown.

Adoption of the internal standard method has resulted in

consistent and routine measurements of lattice parameter

to 1 part in 10" (Edmonds et al, 1989).

SRM 660, LaBg, is certified with respect to lattice

parameter. However, its primary virtue is a lack of

size- and strain-induced peak broadening. The

suitability of this material for the determination of

instrumental broadening functions prerequisite for size

and strain analysis was determined with a round robin

study (Fawcett et al., 1988). The SRM displays evenly

spaced, non-overlapping, high intensity diffraction lines

whose full-width-half-maximum (FWHM) values

measured on fixed slit diffraction equipment closely

follow the expected quadratic dependence on 26 angle.

These properties also render it suitable for use in

determining calibration parameters for a Rietveld

refinement.

The Rietveld method uses models of the origins of the

specific sources of optical aberration to refine the degree

to which they contribute to a delta d curve. This

technique is applied to data from the unknown only; no

standards are used. The delta d curve derived from an unknown may differ significantly from that of an admixed standard due to variations in character between

the unknown and the standard. Therefore, an increase

in accuracy could be realized with the use of the

Rietveld method. The chief factor preventing the

realization of this potential is the lack of fully developed

models appropriate for conventional divergent beam x- ray equipment. These models will have to address

aberrations which are inherently more complex than

those associated with parallel beam equipment. In the case of the software Generalized Structure Analysis

System (GSAS) (Larson & Von Dreele, 1987), the

refined delta d curve is obtained with a specimen

transparency and sample shift correction. However,

examination of actual delta d curves from equipment of

various manufacturers indicates that these corrections

alone cannot account for the observed aberration curves. With the application of the pseudo-Voight profile

function of GSAS to conventional x-ray diffraction

(XRD) data, it is reasonable to assume that the

instrumental contribution is primarily Gaussian, while

the sample contribution is Lorentzian. Thus refinements

of SRM 660 can be used to obtain a set of initial profile

parameters; only the Lorentzian terms are released in

subsequent refinements. The value of the Lorentzian

terms obtained with SRM 660 are then subtracted from

those refined for unknowns in size and strain

estimations. In this application the models describing

optical aberrations affecting peak position can be tested

using a material of known lattice parameter. However,

in order to perform this fixnction the lattice parameter of

the SRM and the wavelength of the x-rays must be

known to an accuracy and precision greater than the

error introduced by improper or incomplete model(s)

describing the optical parameters.

The wavelength of Cu Ka, radiation has been

determined to be 0. 15405945 nm by combining single

crystal lattice spacing measurements (x-ray/optical

interferometry) and diffraction angle measurements

(Deslattes & Henins, 1973; Kessler, Deslattes &

Henins, 1979). The quoted value is taken from "The

1986 adjustment of the fimdamental constants" and is considered accurate to 0.7 part in 10* (Cohen & Taylor,

1987). The lattice parameters of the silver and tungsten

used as internal standards for the certification of SRMs

660, 640b, and 675 were measured in 1966 with a high

precision, temperature controlled, back reflection camera (Swanson, Morris & Evans, 1966). Linkage to the fundamental length standard was through the

wavelength of the radiation used.

The procedure by which the lattice parameters of

existing SRMs were measured and certified is neither

precise nor accurate enough to yield SRMs suitable for

development of the Rietveld method as applied to divergent beam x-ray diffraction equipment. A project

has begun to develop a new high accuracy/precision

lattice parameter SRM(s) linked to the fundamental

length standard via procedures which are independent of

the x-ray wavelength determination. This project

involves the lattice parameter determination of a silicon

single crystal relative to the iodine stabilized HeNe laser

length standard with an X-Ray Optical Interferometer

(XROI). A second experiment will transfer the lattice

parameter measurement from the single crystal of

silicon to a crystalline powder via x-ray diffraction

equipment of parallel beam optics. Supplemental studies

will include an investigation of the effects of crystallite

size on lattice parameters. It is anticipated that this

project will result in an SRM in about 2 years.

Instrument sensitivity SRM

Line position SRMs are used to calibrate equipment

with respect to the angular variable of the goniometer.

SRM 1976 is a new SRM which is designed to calibrate

equipment for the intensity variable as a function of 20

angle, i.e., the instrument sensitivity. The need for this

SRM was defined by the Jenkins, Schreiner & Dismore

(1992) round robin which was, in turn, prompted by the

consideration of a new generation of database by the

ICDD. The new database will consist of complete
70 NIST Xrd Standard Reference Materials: Their Characterization and Uses

digitized diffraction patterns as opposed to reduced d

and I values. The removal of any instrumental

aberrations from the reference data reported to the

ICDD for use in this database will require as yet

undeveloped algorithms. The round robin was used to assess the level of variation in instrument performance,

which indicates the degree of difficulty to be faced in

the building of the new database. Highly uniform,

sintered a alumina, corundum structure, plates were

circulated among selected respondents with instructions on the collection of a range of intensity data. Among

other things, variations in instrument sensitivity of up to 30% were discovered.

SRM 1976 consists of a sintered a alumina plate

originally manufactured as a micro-chip carrier. The

plates exhibit considerable, though highly uniform and

cylindrically symmetrical, levels of preferred orientation. This cylindrical symmetry allows the use of a sample spinner to reduce the errors of particle counting

statistics. The certified values include the lattice parameters, 12 relative intensity values from 25 to 145

degrees 29, and absolute variation in intensity. The

certification of the last of these parameters allows for

comparison of absolute diffraction intensity from

equipment at differing locations.

The round robin study indicated only that a problem

existed with regards to variation in instrument

sensitivity. With the exception of groupings in

performance by manufacturer, it indicated little as to its

origins. The certified relative intensity data of SRM

1976 were collected on a Philips' diffractometer of

conventional optical layout which included a graphite

diffracted beam monochromator (Cline, 1992). This

"reference" machine was determined to be in "correct" alignment with respect to the results of the Jenkins,

Schreiner & Dismore (1992) round robin. However,

due to a lack of understanding of the mechanisms

responsible for the effects observed, this machine could

not be determined to be any more suited as a reference

than any other properly aligned, conventional x-ray

diffractometer. Early experiments found no dependence

of instrument sensitivity on the size of the incident and

receiving slits, or the presence or absence of soUer slits,

either incident and receiving.

The use of SRM 1976 for evaluation of instrument

sensitivity entails the collection of data on the test instrument in a manner analogous to that used in the

certification. The uniformity of the SRM material with

'Certain commercial equipment, instruments, or materials are

identified in this paper in order to specify the experimental

procedure adequately. Such identification is not intended to imply

recommendation by NIST, nor is it intended to imply that the materials or equipment identified are necessarily the best available

for the purpose.

respect to size and strain broadening (minimal) permits

the use of peak heights. They were included in the

certification. Evaluations performed with the use of

peak heights will reflect the lower precision of this type

of intensity measurement. Certified integrated

intensities were measured using the Siemens version of

the NBS*Quant (Hubbard, Robbins & Snyder, 1983)

algorithm. The use of this algorithm is not required,

but a credible method of determining a backgroimd

subtracted integrated intensity measurement should be

employed. Graphical evaluation of the performance of

the test instrument is accomplished by plotting the ratio

of the 12 relative intensities measured from the test instrument to the certified values as a function of 29

angle. Such a plot is shown in figure 1.

zm

1.08

1.02

o.es

S 0.90

IE 0.84

0.78

t uncertainty bounds

i lor SRM 1976

20 30 40 60 60 70 80 90 100 110 120 130 140 160

Two-Thata AngIa

Figure 1. Graphical presentation of test machine

results from four samples of SRM 1976.

The upper and lower bounds displayed in figure 1 represent the coverage of 90% confidence, 95%

coverage tolerance interval for the certified relative

intensity data. Tolerance intervals were used due to the

expected heterogeneity of the levels of preferred

orientation and thus the intensity values measured; no

true mean for the population as a whole exists. The

preferred orientation is expected to uniformly affect all measurements, thus data analysis assumed one true

relative variance (Cline, Schiller & Jenkins, 1991).

Interpretation of the plot may require some scientific

judgement on the part of the operator. Pattemless

scatter within the error bounds indicate the test machine

performs in a manner analogous to the reference

machine and needs no correction. If the scatter of the

data is such that more than one peak falls outside the

error bounds, a lack of precision on the part of the user measurements is indicated. Patterns observed in the

data, even within the error bounds, indicate that a

correction may be necessary. Procedures to test this

hypothesis can be found in the appendix of the SRM

1976 certificate (Cline, 1992). The data of figure 1

clearly indicate that a correction curve should be applied

to intensity data collected from this machine.
J. p. Cline 71

During a realignment of the test machine of figure 1,

it was determined that slight changes in the angle of the

graphite monochromator crystal would cause changes in

instrument sensitivity of up to 20%. The machine was

a second Philips diffractometer similar to the reference,

with the exception of being equipped with an automatic

sample changer as opposed to a sample spinner. The

effect occurred through a rocking angle of the graphite

crystal within which the intensity of the (104) reflection

(35.1° 26) from an SRM 1976 sample did not vary

significantly. The tuning of the monochromator was

performed to minimize contributions from the Cu K/3

and W L lines. Alignment of the crystal to minimize

this spectral contamination resulted in the calibration curves of figure 1. The test machine could be made to perform identically to the reference machine with an

appropriate setting of the monochromator crystal. This

observation indicates the slight variations in the graphite

monochromator angle are responsible for at least part of

the effects observed in the Jenkins & Schreiner round

robin. A more exhaustive study of the causes of

variation in instrument sensitivity is underway utilizing a survey of selected equipment of different manufacturing origin (Cline, Schiller & Jenkins, 1992).

Quantitative analysis SRMs

Quantitative analysis SRMs are powders selected to offer a diffraction intensity as close to the "ideal" as possible. They are of high phase purity, small

crystallite size, and of isometric particle morphology.

They may be specific to given systems, as in the case of

SRMs 1878 and 1879, which are designed for quantification of quartz and cristobalite, respectively, in

airborne dusts. Alternatively, SRM 674a consists of a

set of five powders of varying absorptivity: a AI2O3,

/i= 126cm'; ZnO, /i=279cm-'; TiOj, /i=536cm';

CrjOj, /i=912''; and CeOj, /*= 2203cm"' (absorptivities are for Cu Ka radiation). This selection of materials

allows for the matching of the absorptivity of the

standard to that of the unknown, minimizing the effects

of microabsorption.

Recently SRM 676 was certified. It consists of an a

alumina powder designed for measurement of I/I^ values. The concept of the I/I^ as a constant relating a

material's diffraction intensity to that of a alumina,

corundum, was proposed by Visser & deWolf (1964).

It is a specific case of the more general Reference

Intensity Ratio (RIR) later proposed by Chung (1974

a,b). The I/I^ concept has been subsequently adopted by

the ICDD and is one of the material constants included

in the PDF2 database. The RIR of any two phases can be determined from the ratio of their respective

values. Thus the listing of these values in the database

allows for quantitative information to be gained in a more routine manner. The selection of the alumina powder for SRM 676

involved a study of the dependence of diffraction

intensity on specific powder characteristics (Cline &

Von Dreele, 1992). The first of these, obviously, is phase purity; a basic problem surfaces immediately for

quantitative XRD SRMs based on commercial powders.

The ceramics industry requires active powders for use

in the manufacture of sintered ware. These powders are generally in the tenths of a micron size range and are of

a high surface area. The resultant line broadening is undesirable, but cannot be considered a major factor in

selection of quantitative analysis SRMs; a credible

intensity measurement will consist of an integrated

intensity. The high surface area is detrimental due to the potential increase in the mass fraction of an "amorphous" non-diffracting, surface phase. This

surface phase, while possessing some degree of order,

does not contribute to Bragg diffraction from the bulk

crystal structure. The exact amount of the impurity

phase is dependent on the lattice energy of the bulk

crystal structure, the specific surface of the material,

and its processing history.

The consideration of surface phase contamination is particularly problematic for aluminum oxide powders.

Commercial alumina powders are products of the Bayer

process wherein, initially, an aluminum hydroxide

precursor is chemically generated. This precursor

material is subsequently calcined through a series of

well known transition aluminas until the high

temperature a phase is reached (Wefers & Bell, 1972).

Highly active, sinterable, alumina powders are often

incompletely calcined and are phase contaminated by

several of the transition aluminas. Complete conversion

to the a alumina requires a long duration calcination to

at least 1200C. Powders produced by this process can

not be considered ideal for XRD purposes either due to

their large grains, state of aggregation, and platy crystal

habit. Early experiments indicated that a prerequisite

for precise l/l^ measurements are fiilly deaggregated

powders. Extensive experiments evaluated four highly

calcined, deaggregated, phase pure powders of relatively

small crystallite (particle) size, for suitability as this

SRM material.

The experimental design considered the effects of

extinction, the "amorphous" non-diffracting surface

layer, and preferred orientation. The first two of these

effects will manifest themselves in terms of a reduction

in diffraction intensity. The isolation of the material

offering the maximum diffraction intensity was

accomplished by first spray drying mixtures of the test materials and silicon, SRM 640b, to eliminate the

effects of preferred orientation (Cline & Snyder, 1983).

An I/I^ determination of SRM 640b relative to the test
72 NIST Xrd Standard Reference Materials: Their Characterization and Uses

materials was performed via Rietveld refinements of

XRD data collected from these and conventionally

prepared specimens. Equipment consisted of a Siemens

D500 diffractometer equipped with a focusing Ge

incident beam monochromator, sample spinner, and a

position sensitive detector. The conventionally prepared

specimens provided data on the levels of preferred

orientation via refinements of the March-DoUase

parameter (DoUase, 1986). Extinction effects were

measured using the Sabine model, (Sabine, 1985;

Sabine, Von Dreele & Jorgensen, 1988) in conjunction

with Rietveld analysis of time-of-flight (TOP) neutron

measurements. Additional measurements included

particle size via laser scattering, specific surface

measurements, and examination with a scanning electron

microscope.

Results from these four powders as well as the final

SRM 676 material are shown in table 1. A low I/I^ value was obtained from Linde C which exhibited no

extinction but had the largest surface area of the four

test materials. This may indicate that the effect of its presumably higher amorphous content was offset by the

extinction effects of the other candidates. However, this

conclusion is not supported by the observation that the

Sumitomo material exhibited considerable extinction and

also displayed a favorable I/I^ value. This may indicate an as yet undetermined variable is operative or that the

magnitude of the effects is below the measurement

precision. The reiined March-DoUase parameters,

which equals one for a randomly oriented powder,

indicated the Linde C material did not display preferred

orientation. The SRM itself was prepared from a second lot of the Linde C product. SRM 676 was

certified with respect to lattice parameters and seven

relative intensity values from 24 to 78 degrees 29. The

certification of the relative intensity values allows the user to measure any one of the eight lines for an l/l^ determination (use of more than one line is reconmiended). The machine used for collection of the

certified relative intensity data was the "test" machine

disciissed in the section concerning SRM 1976. In

addition to an adjustment of the reported values, the

reported uncertainty also had to reflect the measurement

uncertainty of SRM 1976 leading to the calibration curve (figure 1). This was done using the statistical technique of bootstrapping. Figure 2 indicates the

dramatic increase in the imcertainty from the 30

certification measurements of the (024) reflection of

SRM 676 with the incorporation of the uncertainty of

test machine measurements of SRM 1976. The three

vertical lines represent the size that a 95% confidence

interval for the mean would be when different sources of error are incorporated. The smallest (leftmost)

interval includes only the variability in the 30

measurements of SRM 676. In the middle interval, the

uncertainty in the calibration curve has been included as

well. Finally, the third interval incorporates the

uncertainty in the certified values of SRM 1976, as well

as the uncertainty in the calibration curve and the

variability of the measurements of SRM 676.

Incoiporaled Unceimlnltes

1* Uncoirecied Observllloni of 676

2* Unconected Obiervalions of 676 Ten Device Calibnllon

3* Uncorrected Observllloni of 676 Ten Device Callbrallon 1976 Ceitiricallon

See Legend for Iiicoipocated Unoeriatiily

Figure 2. Comparison of 95 % confidence intervals of

SRM 676 certification data when different sources of

error are included.

These data indicate that the imcertainty in relative

intensity values, once converted to an interlaboratory

scale via SRM 1976, is due as much to the uncertainty

in the conversion as to the variability in the imcorrected

Table 1 . Results from measurements of SRM 676 candidate materials

Material or

Supplier

Mean Particle

Size, fim

Laser

Refined

Domain Size,

lita TOF data

BET Surface

Area, m^/gr

I/I^ Values for

Si, SRM 640b

March -

Dollase coef.

Linde C 1.5 3.34 4.105 .998

IBM 3.3 1,4 1.27 4.173 .886

Sumotomo 4.3 2.5 .93 4.097 .818

Alcan 4.0 2.7 1.07 4.176 .767

SRM 676 1.4 4.98 4.104 .998
J. p. Cline 73

measurements themselves. This suggests that users of

SRM 1976 need to incorporate the uncertainty of this

correction into the total of their measurements. The

bootstrapping approach used here is not practical for user at other laboratories, but a simpler approximation

is being developed at NIST and will be described in an upcoming publication (Cline, Schiller & Jenkins, 1992).

Another new quantitative analysis SRM, which is presently being certified, is specific to analysis of silicon

nitride (Cline & Von Dreele, 1992). The ceramics

industry is interested in silicon nitride due to its desirable high temperature properties. Ubiquitous to processing this material is the transformation of the

starting low temperature, a phase to the high

temperature, jS phase. Approximately 90% of typical

commercial powder designed for the production of

sintered ware will consist of the a phase, with the

remainder being amorphous material and the /3 phase.

Upon sintering, the a phase transforms into the jS, resulting in an interlocking microstructure of acicular jS grains. This interlocking microstructure, along with the

intrinsic properties of the 0 phase of Si3N4, are responsible for the desired high temperature

performance and fracture toughness of this material.

The a to /? ratio of both starting material and the

subsequent sintered ware is known to affect properties

and is the primary issue addressed by this SRM.

However, after discussions with industrial

representatives, it became apparent that the ability to analyze for amorphous content was also desired. The

SRM itself will consist of two powders, one high in a

content, the other high in /?. Each is to be certified

with respect to a/0 ratio and the amorphous content and

thus will be suitable for use as a spiking material for user measurements.

The analysis of amorphous content can be performed

with the assumption that a spiking phase is 100%

crystalline. Samples of the commercial "a" material

selected as the SRM powder were spiked with various

concentrations of a phase pure jS whisker material.

XRD samples were spray dried. Rietveld refinements

of XRD and TOF data allowed the refinement of a histogram scale factor and the unit cell fraction of the

two phases. The use of the Rietveld method

circumvents any difficulty caused by the complete

interference of the /? phase's diffraction profiles with

those of the a. The failure of measured unit cell fractions to follow the expected dependence on the

concentration of the spiking phase can be entirely

attributed to material in the analyte which is not

contributing to Bragg diffraction from either phase,

defined in this case as the "amorphous phase."

Equations are derived wherein a measured unit cell fraction and the mole fraction of the spikant are related

to the mole fractions of the a and /? phases in the

analyte:

a similar equation may be written for S^. Also:

X„ + X^ + Xa = 1

where:

f„ = mole fraction of "pure" a phase = mole fraction of jS phase

contents of "pure" a phase; unknowns

x„ = mole fraction a phase

Xp = mole fraction (3 phase

Xa = mole fraction amorphous phase

Z„ = 4 S„ = unit cell fraction of a phase = 2 = unit cell fraction of /? phase

Rearranging the first equation and substituting 1-S„

for S^ gives:

(l-/„) (l-5„)

f ' 2S ^

Graphical presentation of this linear relation for the

XRD and TOF data is shown in figure 3. The slope of

this plot indicates the a content while the y intercept

indicates the jff content and the difference between the

two indicates the amount of the amorphous phase. The

XRD data indicate a higher amorphous content than the

TOF data which is believed to be caused by a humidity

induced atmospheric reaction with one or both phases.

This reaction can be assumed to affect the specimens of

Gaithersburg, MD more than those of Los Alamos NM.

These data indicate this silicon nitride powder consists

of 93.8% a phase, 3.4% j(3 phase, and 2.8% amorphous

phase. This technique can be extended to the measure- ment of the amorphous or impurity level of any material

relative to a second material which can be considered

phase pure. The "impurity level" can be considered as the amount of any material which is not contributing to Bragg diffraction from either phase. The extension of

this technique to other quantitative SRMs is a topic of

current investigation.
74 NIST Xrd Standard Reference Materials: Their Characterization and Uses

4.5

3.9

. 2.5

- 1.5

o.s

-0.5

— 1' 1 —-B Neutron TOF

e X-ray

T ' r T

•-

-O.S 0.0 o.s i.O i.S 2.0 2.9 3.0 3.9 4.0 4.9

(l-S.) / 2S,

Figure 3. Graphical analysis of TOF and XRD data for the silicon nitride SRM

powder of high a phase content.

Acknowledgments

SRMs 640b, 675, 660, 1879, and 1878 were certified

under the direction of Camden R. Hubbard.

Collaborators, in order of participation:

Robert B. Von Dreele

Ron Jenkins

Susannah B. Schiller

Hung-Kung Liu

Steve Danforth

Daniel Louer

Bob Newman

Richard D. Deslattes

Ernest G, Kessler, Jr.

SRMP project monitor: Nancy Trahey

SRMP Sales Office: (301) 975 6776

FAX (301) 948 3730

References

Chung, F. H. (1974a). J. Appl. Cryst. 7, 525-531.

Chung, F. H. (1974b). J. Appl. Cryst. 7, 519-525.

Cline, J. P. (1992). SRM 1976 Certificate, NIST.

Cline, J. P. & Snyder, R. L. (1983). Adv. X-ray

Anal. 26, 111-116.

Cline, J. P., Schiller, S. B. & Jenkins, R. (1992). in

progress,

Cline, J. P., Schiller, S. B.& Jenkins, R. (1991). Adv.

X-ray Anal. 35, 341-352.

Cline, J. P. & Von Dreele, R. B. (1992). in progress.

Cohen, E. R. & Taylor, B. N. (1987). Rev. Mod. Phys.

59, 1121.

Deslattes, R. D. & Henins, A. (1973). Phys. Rev. Lett. 31, 972.

Dollase, W. A. (1986). J. Appl. Cryst. 19, 267-272.

Edmonds, J. et al. (1989). Methods and Practices in Xray Powder Diffraction, Section 13.1 ICDD.

Fawcett, T, G. et al. (1988). Powder Diffraction, 3,

209-218.

General Structure Analysis System, (Rietveld software

package) A. C. Larson & R. B. Von Dreele, (1987),

LANSCE, MS-H805, Los Alamos National

Laboratory, Los Alamos, NM 87545.

Hubbard, C, R., Robins, C, R. & Snyder, R, L.

(1983). Adv. X-ray Anal. 26, 149-157.

Jenkins, R., Schreiner, W, & Dismore, P. (1992) to be

submitted to Powder Diffraction.

Kessler, Jr., E. G., Deslattes, R. D. & Henins, A.

(1979), Phys. Rev. A19, 215,

Sabine, T, M, (1985). Aust. J. Phys. 38, 507-18.

Sabine, T, M., Von Dreele, R, B. «& Jergensen, J. E,

(1988). Acta Cryst. A44, 374-79.

Swanson, H. E., Morris, M. C. &. Evans, E. H.

(1966). NBS Monograph #25, sec 4, June 28.

Visser, J. W. & Wolff, P. M. (1964). de Absolute

Intensities. Report 641.109, Technisch Physische,

Delft, Netherlands.

Wefers, K. & Bell, G. M. (1972). Technical Paper #19,

Alcoa Research Laboratories, East St. Louis, Illinois,
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Integrated Software for Structure Solution from Powder Data

Lynne B. McCusker

Institute of Crystallography, ETH, Zurich, Switzerland

Introduction

The possibility of applying the extremely welldeveloped and exhaustively-tested single-crystal

structure determination techniques to powder data

has aroused much interest recently in both crystallographic and materials science circles. Many materials of industrial importance can only be prepared

in polycrystalline form, yet knowledge of their struc- tures is essential to the understanding of their properties and to the further development of related materials. In principle, the steps necessary to generate a

pseudo-single-crystal data set from powder data are straightforward. In practice, there are many pitfalls

awaiting the unwary.

The steps involved in producing a list of hkVs and

their respective integrated intensities are: 1) the collection of the powder diffraction data, 2) the correc- tion of the data (scale, 29 zero correction, estimation

of the background, etc.), 3) the determination of the

peak positions, 4) the indexing of the pattern, 5) the

determination of possible and likely space group(s),

6) the extraction of integrated intensities from the

powder pattern, and 7) the treatment of severely

overlapping reflections. Difficulties can be encountered in any or all of these steps, but experience has

shown that the critical points tend to be the data collection itself and steps 5-7. Some practical guidelines

for avoiding some of the more common mistakes, and

for improving the chances of success are discussed.

Although a number of computer programs are

available to perform the necessary calculations for

each of these steps (Smith & Gorter, 1991), no single

software package covers the whole procedure. Consequently, data transfer from one system to the next is necessary, and must be done correctly. A concerted

effort is presently underway to alleviate this problem by including intensity extraction, at least one

treatment for overlapping reflections, and Rietveld

refinement in the (until now) single-crystal package

XTAL. A summary of the plans for and progress with

this aspect of XTAL3.2 (Hall k Stewart, 1992) are presented.

Data collection

Since data collection is handled in detail elsewhere

{e.g. Bish & Post, 1989 or Hill k Madsen, 1987), only

those aspects critical to structure solution will be

mentioned here. If ab initio structure determination

is to be attempted, it is essential that the 29 values be

well calibrated, the measured intensities be accurate,

and the resolution of the pattern be optimized.

The calibration of the 29 values using an internal standard facilitates the indexing process and

allows accurate lattice parameters to be determined. This information can then be used with

confidence in the extraction of integrated intensities from data collected without an internal standard using a whole-profile refinement program such

as ALLHKL (Pawley, 1981), that of Le Bail, Duroy

and Fourquet (1988), EXTRACT {BsjexXoditt, 1990),

or LSQPROF (Jansen, Peschar k Schenk, 1992a).

Crystallographic methods of structure solution re- quire that the relative intensities of individual reflec- tions be correct. If the crystallites of a powder sample are not randomly oriented during data collection,

this requirement is not fulfilled. Such preferred orientation problems should be solved experimentally

if at all possible. Putting the sample in a capillary

which is then rotated during data collection seems

to be the simplest solution, if such a facility is available. If a flat plate sample holder is used, loading

the sample either from the back or from the side and

avoiding pressing is often sufficient to obtain the nec- essary random distribution. A further alternative is to prepare the sample using spray drying techniques.

One must also consider the X-ray absorption coefficient of the material under investigation. In general, reflection mode is preferred for strong absorbers,

and transmission mode for weak ones. If a flat plate

sample is used, care must be also be taken that the

sample is deep enough to approximate infinite thickness, and that the X-ray beam remains on the sample

(i.e. does not touch the sample holder). This means

that a smaller divergence slit is needed at lower angles than at higher angles, and that the data must

be corrected accordingly.

To maximize the information content of a powder

pattern, the overlap of reflections in 29 needs to be

minimized. This can be done by taking care that the

75
76 Integrated Software for Structure Solution

sample itself is highly crystalline, and by optimizing

the resolution of the diffractometer used. If the sample broadening is low (i.e. the full width at half maximum (FWHM) of the peaks is determined primarily

by the instrumental function), a measurement at a synchrotron radiation facility may be appropriate.

The high intensity and parallel nature of synchrotron radiation allow very high resolution powder

diffraction data to be collected. At present, diffractometers with Debye-Scherrer geometry and a simple slit system (Cernik & Bushnell-Wye, 1991), with

Parrish-Hart multiple foils (Parrish, Hart, Huang &; Bellotto, 1987), and with a post-sample analyzer

crystal (Cox, Hastings, Cardosa & Finger, 1986 and

Arnold ei ai, 1989) are available. The first yields

simple instrumental peak shapes and good resolution, the second is restricted to flat plate samples

but provides high intensity and moderate resolution,

and the third results in more complex peak shapes

but excellent resolution. Since the last two diffraction geometries are also independent of sample displacement, they provide accurate 26 values which are

ideal for indexing purposes. The optimal choice of geometry depends upon the quality of the sample and

the purpose of the investigation.

Space group determination

A powder diffraction pattern rarely allows a space

group to be determined uniquely. The number of single reflections is limited, so systematic absences are

difficult to establish with certainty. Electron diffraction can be very useful in reducing this ambiguity,

but, even so, several possible space groups usually

emerge. These must then be probed systematically.

Certainly for the application of direct methods of

structure solution, it is sensible to explore the cen- trosymmetric possibilities first, but the others should

not be forgotten.

Pseudosymmetry can also be exploited in the

structure determination process. For example, the

calcined form of the aluminophosphate molecular

sieve AIPO4-I8 can be indexed on an orthorhombic

unit cell (a = 13.711, b = 12.731, and c = 18.570 A)

with the possible space groups Cmcm or Cmc2i , al- though the structure itself has only monoclinic symmetry (Simmen ei ai, 1991). The framework topology was established in the space group Cmcm as- suming AI = P. Only then was the strict alternation

of AI and P in the tetrahedral sites of the framework taken into account, and the symmetry reduced

accordingly to C2/c. By using this approach, the

structure determination problem was reduced from

an 18-atom to a 9-atom one. Two similar cases (the

aluminophosphates MCS-1 and SAPO-40) are shown

in Table 1.

Extraction of reflection intensities

Individual reflection intensities are extracted from a powder pattern by adjusting lattice parameters, peak

shape parameters, 20 zero corrections, and hkl intensities to give the best whole-profile fit to the observed diffraction pattern. Many Rietveld refinement

(Rietveld, 1969) programs have now been modified to

allow whole-profile refinement without a structural

model. Originally all parameters were refined us- ing a least-squares algorithm with slack constraints

for the intensities of nearly overlapping reflections

(Pawley, 1981), but more recently, the simpler and

more stable modification first described by Le Bail,

Duroy and Fourquet (1988) has been adopted as the

favored approach. In this case, the reflection intensities are integrated numerically in an iterative process

rather than refined. Other variables are refined as in

the Pawley program.

The maximum sensible 29 value to use in an ex- traction procedure depends upon the amount of over- lap in the pattern. This is a function of the wavelength used, the FWHM of the peaks, the symmetry, and the lattice constants. At some point, so many reflections overlap that the background is no

longer well defined, and the extracted intensities are no longer useful. The determination of this point is not unequivocal, and depends to some extent upon

the complexity of the structure under investigation.

Several 20 limits may have to be tested before an

appropriate one is found.

If direct methods are to be used for structure determination, all reflections in the range chosen, whether

weak, intermediate, strong or overlapping, should be

included in the data set. Weak reflections also contain structural information, and are particularly important for negative quartets. Since many programs

set the intensities of missing reflections to zero, the

exclusion of overlapping reflections from the data set can result in false information being introduced.

Not only do extraction programs generate a

pseudo-single-crystal set of hkl's and intensities, they can also be used effectively to evaluate different space

group possibilities, to refine lattice parameters, to

adjust profile parameters prior to a Rietveld refinement, and to obtain a better estimate of the background.

Treatment of overlapping reflections

Reflections are considered to overlap in a powder

diffraction pattern when their 29 values are too close

to allow the reflections to be distinguished from one

another. This is normally expressed as some fraction

of the FWHM, with the fractions varying from 0.1

to 0.5. When several reflections contribute to what
L. B. McCusker 77

Table 1: Some Examples of Structures Solved from Powder Data

Sample LaMosOg" Sigma-2* Cimetidine*^ MCS-l'' SAPO-40''

Space Group P2i/a IAll ami P2i/n Phcn Pmmn

a (A) 9.91 10.24 10.21 9.65 22.04

6(A) 9.09 10.24 17.97 18.03 13.70

c(A) 7.57 34.38 6.52 13.91 7.12

pn 109.05 111.28

v(A3) 645 3605 1114 2420 2150

Reflections (CuKq;i)

Total (up to 75° 2^) 330 281 568 633 621

Overlapping-^ 124 66 217 257 248

(within 0.5-FWHM) (38%) (23%) (38%) (41%) (40%)

Method Direct Direct Fourier FIPS/Direct FIPS/Direct

Methods Methods Cycling Methods Methods

Atoms/Asym. Unit

Found Initially lLa+5Mo 4Si-f40 1S+3C^' 6(Al,P)+90 4(Al,P)+50

Subsequent Fouriers 80 30+5C+1N 6N+7C 30 50+6C+1N

Used in Refinement 14 17 33'' 36' 62-'"

"Hibble, Cheetham, Bogle, Wakerly & Cox, 1988

''McCusker, 1988

'=Cemik, Cheetham, Prout, Watkin, Wilkinson WiUis, 1991

''Simmen, 1992

*Estennann, McCusker &l Baerlocher, 1992

^assuming FWHM = ^0.0092 tan 5 + 0.0023 (i.e. FWHM ranges from 0.05 to 0.10°25)

^not recognizable as correct positions initially ''16 H-atoms included

'Space Group Pca2\

'Space Group P2/n, c doubled

appears to be a single peak in the diffraction pattern,

the total intensity is usually equipartitioned. That

is, the intensity ratio of the reflections involved is set to one. If the percentage of overlapping reflections

is low, such a data set will usually allow the struc- ture to be solved (see, for example, Sigma-2 in Table

1), but as the percentage increases, the chances of

success diminish.

Some exciting developments in the treatment of

overlapping reflections have been reported in the last

year. In their program DOREES, Jansen, Peschar

and Schenk (1992b) use a combination of triplet and

quartet relationships and a Patterson function calculated using only accurately determined Fhki's to

obtain a more correct estimate of the intensity distribution for overlapping reflections. On the basis of

this information, they can judge whether a particular Fhki is likely to be strong, intermediate or weak.

The method has been tested on two known struc- tures (one organic and one inorganic), and the data

sets were significantly improved.

Cascarano, Favia and Giacovazzo (1992) take advantage of the power of modern computers in their

program SIRPOW .91, which is written specifically

for structure solution from powder data. They permute the possible intensity ratios of overlapping re- flections and apply a multisolution procedure to each

data set. The solutions are then analyzed with the

help of appropriate figures of merit to identify the

correct one. This method has been applied successfully to a number of known structures.

Estermann and Gramlich (1992) have taken a different approach. They generate a Patterson map us- ing equipartitioned data, square the map to emphasize the main features, back-transform this squared

map to obtain new Fourier coefficients, and ex- trapolate these new coefficients to redistribute the

relative intensities of overlapping reflections (nonoverlapping ones remain unchanged). This cycle

is repeated until the intensity distribution of the

overlapping reflections approaches that of the nonoverlapping ones. This Fast Iterative Patterson

Squaring (FIPS) method is remarkably fast, and

has been applied successfully to many test struc- tures (Estermann, 1991) and to two unknown struc- tures which could not be solved from equipartitioned

data (Simmen, 1992 and Estermann, McCusker &; Baerlocher, 1992).
78 Integrated Software for Structure Solution

By producing a data set whose relative intensities are more realistic (i.e. closer to single-crystal values), these methods significantly enhance the success

rate for structure solution from powder data. They

represent the most important recent advance in the

field.

Examples

Some data for a few of the more complex structures

solved ab initio from powder data are given in Table 1. The bottom row, which shows the number of

atoms in the asymmetric unit of the final structure,

illustrates the structure complexity now within the

scope of the powder diffractionist.

The two earlier structures, LaMosOg and Sigma-2,

are relatively simple ones, and were solved by direct

methods using equipartitioned powder data. However, this approach did not yield recognizable struc- tures for cimetidine, MCS-1 or SAPO-40.

In the case of cimetidine, only 4 of the 17 highest peaks on the E-ma.p were correct, and it was not

clear initially which these were. Nonetheless, the positions of all 17 non-hydrogen atoms were determined

by an iterative least-squares refinement/ Fourier synthesis technique. The procedure was fully automated

in the CRYSTALS package (Watkin, Carruthers k

Betteridge, 1985). The inclusion of 16 H-atoms in

the model then significantly improved subsequent Rietveld refinement.

For MCS-1 and SAPO-40, the FIPS method was

used to redistribute the intensities of the overlapping

reflections more realistically, and then the structures were determined by reapplying direct methods . The

redistribution of intensities by the FIPS method appears to allow a larger number of atoms to be found on the initial ^'-map. In both cases, the pseudosymmetry was exploited in the structure solution step.

By assuming that Al and P are indistinguishable, the

higher space groups could be used to determine the

aluminophosphate framework topologies. The strict

alternation of Al and P in the tetrahedral sites was

introduced only at the structure refinement stage.

The number of overlapping refiections given in Table 1 were calculated and included for comparison

purposes. In fact, the data collection conditions

for the five structures vary significantly. For ex- ample, the data for cimetidine were collected at a synchrotron facility (FWHM varying from 0.05 to

0. 07*261), whereas those for SAPO-40 were collected on a laboratory instrument equipped with a position sensitive detector (FWHM varying from 0.10 to 0.14° 2^). The eff'ect of this difference is to reduce

the percentage of overlapping reflections for cimetidine to approximately 29%, and to increase that for

SAPO-40 to 65%. Consequently, a direct comparison of methods is not possible. Nonetheless, there

appears to be a progressive increase in the complexity of structures that have been solved from powder

data, and this trend can be expected to continue over the next decade as new methods are developed, improved and combined.

Software

In figure 1 some of the more commonly used programs for each of the steps required for structure

solution using powder data are shown. With the

exception of the structure determination program

SIMPEL (Peschar, 1990), references for all of these

programs have already been cited, or are given in the

program list compiled by Smith and Gorter (1991).

The purpose of the diagram is to illustrate the variety of programs available without providing an ex- haustive list, and to indicate some of the program

combinations that have already been used successfully (those connected with arrows). Several laboratories have developed their own inhouse set of soft- ware with appropriate bridging programs to transfer

information correctly from one system to another,

but a completely integrated system is still lacking.

Structure solution from powder data usually involves the use of at least four different program systems. The diff"ractometer software normally includes

data collection, some data correction (a2-stripping,

background correction, 29 zero correction), and a

peakfinding routine. The latter is sometimes cou- pled to an autoindexing program, but if it is not,

the transfer of 20 values to stand-alone programs is relatively straightforward. An attractive alternative

to using the peakfinding routine in a diffractometer

package is the program recently developed by David,

Ibberson and Sivia (1992), which is a Bayesian approach to Bragg peak location. In difficult cases, this

algorithm may yield peak positions which improve

the chances of success in the indexing process.

Most intensity extraction programs are modifications of the Rietveld refinement program used in the

last step, but the intermediate steps for the treat- ment of overlapping reflections and the structure determination itself are separate and require a criti- cal transfer of data. The user must know exactly

what form of intensities or structure factors are generated by the extraction program in order to transfer the information correctly. That is, have the intensities been corrected for the Lorentz-polarization

factor; have the multiplicities been factored out; is the peak width listed the FWHM, the FWHM/2 or the FWHM^; are the values output hkh F\ku ^hki

or something else? The hkl indices must also cor- respond to the cell input to the structure determination program. The extraction program may have

performed a cell transformation (e.g. changed the
L. B. McCusKER 79

1) Data Collection

2) Data Correction

3) Peak Positions

4) Indexing

5) Space Group Determination

Diffractometer

Diffractometer

Diffractometer

DICVOL ITO TREOR

6) Intensity Extraction EXTRACT LSQPROF Le Bail ALLHKL

7) Overlap FIPS DOREES SIRPOW

8) Structure Determination XTAL SIMPEL SIRPOW CRYSTALS SHELX

9) Rietveld Refinement XRS GSAS DBWS

Figure 1: Some Programs Commonly Used in Structure Solution from Powder Data

hexagonal axes of an R-centered cell to the rhombohedral ones) before generating the hkl values. Although these may appear to be trivial and obvious

points, they are easily overlooked, and that can lead

to long periods of unnecessary frustration.

It is obvious that an integrated system that would

perform all the necessary calculations and require no data transfer would be welcome in the powder

diffraction community. The editors of the single crystal package XTAL are presently in the process of

adding a Rietveld refinement program to the system,

and plan to follow that with an extraction program

and at least the FIPS method of treating overlapping reflections (the program series shown in the left column of Figure 1). It can be hoped that this trend

will continue and that eventually peakfinding and

autoindexing routines will also be included. In the

meantime, there are plenty of programs available to

the crystallographer who is willing to write miniprograms to bridge between the various program systems, and certainly there is no shortage of polycrystalline materials whose structures need to be solved.

References

Arnold, H., Bartl, H., Fuess, H., Ihringer, J., Kosten,

K., LocHNER, U., Pennartz, P.U., Prandl, W. &

Wroblewski, T. (1989). Rev. Sci. Instrum., 60, 2380-2381.

Baerlocher, Ch. (1990). extract. A Fortran Program for

the Extraction of Integrated Intensities from a Powder Pattern. Institut fuer Kristeillographie, ETH, Zurich, Switzerland.

BiSH, D.L. & Post, J.E. (1989). Editors. Modern Powder Diffraction. Washington: The Mineredogiceil Society of

America.

Cascarano, G., Favia, L. &i Giacovazzo, C. (1992). J. Appl.

Cryst. 25, 310-317.

Cernik, R.J. &c Bushnbll-Wye, G. (1991). Mater. Sci. Fo- rum, 79-82, 455-462.

Cernik, R.J., Cheetham, A.K., Prout, O.K., Watkin, D.J.,

Wilkinson, A.P. & Willis, B.T.M. (1991). J. Appl. Cryst.

24, 222-226.

Cox, D.E., Hastings, J.B., Cardosa, L.P. &c Finger. L.W.

(1986) . Mater. Sci. Forum, 9, 1-20. David, W.I.F., Ibberson, R.M. & Sivia, D.S. (1992). , in preparation.

Estermann, M.A. (1991). Ph.D. Thesis, ETH, Zurich,

Switzerland.

Estermann, M.A. & Gramlich, V. (1992). , in preparation.

Estermann, M.A., McCusker, L.B. &c Baerlocher, Ch.

(1992). J. Appl. Cryst. 25, 539-543.

Hall, S.R. Si Stewart, J.M. (1992). Editors. XTAL3.2, in preparation.

Hill, R.J. & Madsen, I.C. (1987). Powder Diffraction, 2, 146-162.

HiBBLE, S.J., Cheetham, A.K., Bogle, A.R.L., Wakerly,

H.R. & Cox, D.E. (1988). J. Am. Chem. Soc. 110, 3295-

3296.

Jansen, J., Peschar, R. & Schbnk, H. (1992a). J. Appl.

Cryst. 25, 231-236.

Jansen, J., Peschar, R. &i Schenk, H. (19924). J. Appl.

Cryst. 25, 237-243.

Le Bail, A., Duroy, H. & Fourquet, J.L. (1988). Mater.

Res. Bull. 23,447-452.

McCuSKER, L.B. (1988). J. Appl. Cryst. 21, 305-310.

Parrish, W., Hart, M., Huang, T.C. & Bellotto, M.

(1987) . Adv. X-ray Anal., 30, 373-381.

Pawlby, G.A. (1981). J. Appl. Cryst. 14, 357-361.

Peschar, R. (1990). In MolEN, Molecular Structure Solution

Procedures, Vol. 3, pp. 59-82. Delft: Enreif-Nonius.

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

Simmen, a. (1992). Ph.D. Thesis, ETH, Zurich, Switzerland.

SiMMEN, A., McCusKER, L.B., Baerlocher, Ch. <Si Meier,

W.M. (1991). Zeolites, 11, 654-661.

Smith, D.K. & Gorter, S. (1991). J. Appl. Cryst. 24, 569-

602.

Watkin, D.J., Carruthers, J.R. & Betteridge, P.W.

(1985). CRYSTALS User Guide. Chemical Crystallography

Laboratory, Univ. of Oxford, England.
National Institute of Standards and Teciinology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Direct Space Methods of Structure Solution From Powder Diffraction Data

J. M. Newsam, M. W. Deem*and C. M. Freeman

Biosym Technologies Inc. 9685 Scranton Road, San Diego, CA 92121-2777, USA

Abstract

Complementing improvements in instrumentation

and in more traditional crystallographic approaches,

direct space methods are now facilitating the solution of crystal structures ba^ed on powder diffraction

data. Two such methods are discussed here. In an

extension of physical model building methods, unit

cell size and symmetry constraints are used to control the combination of particular structural motifs

through a graphical interface; model validation or re- finement can be achieved by interactively updating

simulated analytical data (such as diffraction patterns or NMR spectra) as the developing model is adjusted. Model development can be automated when

the reasonableness of a given atomic arrangement is quantified by cost functions or 'energy' terms. These

may include first-neighbor distance, bond angle and

coordination number constraints etc. Viable models

are derived from an initially random distribution of

the required number of atoms of each type within

the observed unit cell by simulated annealing. The

effectiveness of this approach is demonstrated by test applications to the known structures of zeolites and

other more condensed oxide phases. It is straightforward to include additional 'energy' terms in this

process, such as micropore dimensions in the case of

zeolites or, more generally, the degree of match between model and target powder diffraction patterns.

Particularly when diffraction pattern matching constraints are included, simulated annealing appears

to be an efficient and potentially general route to ah

initio structure solutions.

1. Introduction

How valuable is a knowledge of structure at an

atomic level? The answer to this question helps

quantify, at one level, the level of resources one is willing to devote to the solution and refinement of a

particular crystal structure. Although perhaps diffi- cult to define precisely, particularly in advance of the

'Present address: Department of Chemical Engineering,

University of California, Berkeley, CA 94720.

availability of the data, the benefits of detailed atomistic structural knowledge are almost universally sufficient to warrant a substantial characterization ef- fort. A large proportion of the interesting new materials being synthesized, however, are only available

in polycrystalline form. Even where single crystals can be made, their production at a quality suitable

for accurate single crystal diffraction analysis may be

time-consuming. As powder diffraction data is usually available immediately, there are often opportunities for ground breaking studies using powder diffraction methods. For example, many of the early studies

of YBa2Cu307_x used powder diffraction methods

(See Johnston, Jacobson, Newsam, Lewandowski,

Goshorn et ai, (1987) and references cited therein).

Improvements in experimentation have helped to improve dramatically the quality of powder diffraction

data that are now obtainable. In parallel, computer

hardware improvements permit us to exploit computational approaches that even a few years ago would

have been perceived as restrictively expensive. In

this report we discuss primarily some ways in which

modeling or simulation can aid in initial structure

solution. Although not addressed here, there have

been equally impressive advances in structure refinement methods. Computational tools are also a wellestablished and important means of extracting the

maximal value from the atomic structural data that

diffraction measurements ultimately provide.

2. Graphical and geometrical manipulations

The building of physical models has traditionally

been an important means of developing new struc- tural models, particularly of framework structures

such cis those of zeolites. Physical models capitalize

directly on the defined geometrical constraints that

the structure class obeys. The selection of those

models that might be appropriate for a new material in question is based on the matching of the unit

cell sizes, symmetries and computed diffraction patterns with the observed data available. The physical

model building approach does not allow the known

symmetry or unit cell size to be used directly as a

80
J. M. Newsam, M. W. Deem and CM. Freeman 81

Figure 1: Symmetry elements for space group Pna2i , shown superposed on the framework structure of zeolite Li-A(BW).

constraint. It can be slow and almost inevitably proceeds according to the particular preconceptions of

the builder.

Mathematical or computational model building

methods can offer substantial advantages. The combination of geometrical constraints and a known unit

cell size and space group can, for high symmetry

cases, limit greatly the possible atom positions or

configurations. Such symmetry arguments (Bragg &

Warren, 1928) have been used often in solving simple

inorganic crystal structures. Symmetry constraints can be programmed in a general way so that as an atom or fragment is moved in the structure via a

graphical interface, the symmetry-related atoms or fragments are moved in consort. Recognizing the

character and location of the symmetry elements that

are present in the defined space group (Fig. 1) allows

the motion to be constrained so that atoms lying on special positions such as 2-fold rotation axes or mirror planes remain on those symmetry elements.

After molecular fragments or structural motifs such

as polyhedra are combined, subject to geometrical

rules, unit cell dimensions and space group symmetries can be determined automatically (Fig. 2).

Further, model validation or refinement can then be

achieved directly by interactively updating simulated

analytical data (such as diffraction patterns, EXAFS

or NMR spectra) as the developing model is adjusted

(Fig. 1) (Newsam, Deem, Freeman, Gorman, Ho et ai, 1992).

3. Automatic model development - Background

A structure determination from diffraction data involves the development of an initial model, that is the

structure solution step, and the completion and re- finement of the model. Why are these steps distinct?

Today's structure refinement technology can operate

effectively only when the starting model is relatively

close to the actual model. Classical least squares

techniques are unable to locate global minima in parameter spaces that are complex, multi-dimensional

and which have many local minima, unless the cur- rent model is close to this global minimum. Structure

completion tasks, likewise, rely most extensively on

interpretations of scattering density maps produced

by Fourier inversion of the diffraction data. Such

maps must provide at least a recognizable level of

definition, placing requirements on both the resolution and the quality of the phasing.

With powder data, as with single crystal data from

materials with very large unit cells, the methods that

work well for small molecule single crystal work usually cannot be applied directly. Powder diff'raction

data, even when obtained at high instrumental reso- lution, necessarily suffer from systematic or accidental peak overlaps, yielding only intensity sums for a significant proportion of the reflections. Although

ways are being developed to aid in the proper apportioning of the measured intensity to each of the

contributing reflections (David, 1987; Estermann &

Gramlich, 1991), the precision of the decomposed

intensities cannot match that obtainable by separate measurements, as is possible with single crystals.

The set of reflection intensities obtained by decomposition of the powder diffraction data is, additionally,

almost always a 'weak' data set. The precision with

which the structure factor amplitudes are defined is modest and the proportion of 'observed' reflections is usually small. This situation, even for single crystal

data, often gives rise to problems in the application

of Direct Methods. Thus, although significant headway is being made in applying Direct Methods to structure solutions from powder diffraction data (see,

e.g., Gilmore, Henderson Bricogne, 1991; Morris,

Harrison, Nicol, Wilkinson &: Cheetham, 1992), the

number of new structures solved this way remains

small (McCusker, 1992).

In principle, were our practical description of the

interatomic interactions to be sufficiently accurate

and were sufficiently powerful numerical optimization tools available, we could truly predict crystal

structures from little more than composition data.

For some ideal molecular systems such structure prediction, or, more accurately post-diction, has proven

possible (Williams, 1967; Gdanitz, 1992). In gen-
82 Direct Space Methods of Structure Solution

Figure 2: Edge-shared pairs of octahedra clipped from the corundum structure (a), are optimized (6), and

then combined graphically to develop a piece of a hypothetical structural model (c), from which a primitive

unit cell (d) and, subsequently, its space group symmetry (e) are automatically determined.

eral, however, we have at hand substantially more

information than the chemical composition alone.

A successful indexing of the powder diffraction pattern, which can often be done automatically, yields

the unit cell dimensions and information on possible space groups. The chemical analysis, combined

where necessary with other analytical probes of the

unit cell composition, provides the number of atoms

or molecules of each type that are contained within

this unit cell. The global optimization problem is then that of positioning these species within the unit

cell such that the various diffraction data are reproduced. We can also require that the model make

reasonable chemical and physical sense and perhaps

also match other experimental data. This type of optimization, the derivation of that set of parameters

which will optimally match a set of constraints or observations is truly generic; many scientific problems

can be expressed in terms of a function whose minimum value is the solution sought. Any improvements

made in algorithms that better address this problem

can have immediate impacts in crystallography.

4. Simulated annealing as an optimization

method

Stimulated in part by the evolution of readily available computer resources, the methodology and application of numerical optimization have progressed
J. M. Newsam, M. W. Deem and CM. Freeman 83

dramatically. Gradient optimization methods that

seek first derivative stationary points were originally

developed in the 1950s and 1960s (Powell, 1982;

Fletcher t Powell, 1963; Gill, Murray k Wright,

1981; Scales, 1985; Press, Flannery, Teukolsky &

Vetterling, 1989; Sedgewick, 1988), and are now

quite robust and sophisticated. However, all such

methods are, in general, confounded by functions

which possess many local minima.

Simulated annealing (Press, Flannery, Teukolsky

& Vetterling, 1989) a comparatively recent development, places little emphasis on the local curvature of

the function to be minimized. The resulting procedure is, therefore, less susceptible to simply rolling

downhill to the nearest function minimum and provides a higher probability of finding the function's

global minimum. The price paid for improved avoid- ance of local minima is the higher number of function

evaluations which must be made to achieve successful

minimization. However, if the single point function

evaluations do not present a prohibitive computational burden (and the size of burden that is tolerable

grows steadily, in consort with hardware performance

improvements), the method can show great success (Kirkpatrick, Gelatt & Vecchi, 1983). Although the

present work has uniformly used simulated annealing as the optimization method, combined on occasion with the use of conventional gradient optimizers

for final adjustments when viable models have been

produced, other interesting approaches to the global

optimization problem are also being pursued (Piela,

Kostrowicki & Scheraga, 1989; Somorjai, 1991).

All minimization procedures start from a given

proposed solution and iteratively attempt to improve

on this starting point. Simulated annealing achieves a smaller degree of local bias than gradient optimizers in its search from the starting point, by accepting

with a finite probability configurations which possess

higher function values. This property actually permits the optimization to commence from a random

starting point. An analogy can be drawn with an en- semble of physical particles at finite temperature. In

such a physical system the equilibrium distribution

of states accessible at a given temperature characterizes a well defined phase for that system: solid,

liquid or gas, for example. Each phase is made up of a collection of 'states' and their associated probabilities. For a physical system the relative probabilities

of two states are simply related to the temperature

via Boltzman statistics. At high temperature there

will be little diff"erence in the probability of any two

states, while at low temperature the state with the

lowest potential energy will dominate. The 'temperature' of a system may, then, be used as a convenient

control of the extent to which states higher in energy

than the global minimum are populable.

In the simulated annealing approach to optimization, a 'state' is a possible system configuration; its 'energy' is a measure of the degree to which the configuration matches the target constraints or observations. Thus in fitting a measured diffraction pattern, one configuration of the atoms in the unit cell would

be one state, and the conventional weighted residual,

Rw, might be taken as its 'energy'. The 'temperature', the amount of thermal 'energy' in the system,

determines the relative probabilities of the system

adopting two states of different energies.

To locate the sought minimum requires defining a path over the parameter space surface. This path

might be a molecular dynamics (MD) trajectory, in

which the positions and velocities of the particles in

the system are allowed to evolve by solving Newton's equations of motion. The simulation, initially

commenced at high temperature, is subjected to an

annealing schedule which allows the system to settle into an energy-minimum configuration. Because

of the need to ascribe masses to the components of

the system molecular dynamics has found greatest use in its application to molecular systems, where

the choice of mass is unambiguous, for example in

the refinement of protein structures (Brunger, Leahy,

Hynes Fox, 1991) and the determination of ter- tiary molecular structure from NMR derived distance

constraints (Crippen Sz Havel, 1988). Molecular Dynamics, with suitable dummy variables, has also been

exploited in combined structural and electronic calculations (Car k Parinello, 1985; Field, 1991).

In this work, the path in parameter space

is driven by the Metropolis importance sampling

method (Metropolis, Rosenbluth, Rosenbluth, Teller

& Teller, 1953). A particle is selected at random and

displaced; the direction is chosen randomly and the

displacement extent is also chosen at random from

within defined bounds. The 'energy' of the new trial

state, Enew, is evaluated and the state is accepted if

Enew ^ Ejnitial j (1)

or if

exp < -

Er -Einitial KT }

<rand(0, 1), (2)

where Einitial is the total 'energy' of the system before

the particle displacement, T is the temperature, K

a constant term and rand(0,l) represents a random

number between 0 and 1. At low values of T, the

probability of accepting higher energy configurations

is small; at high temperature, however, the probability becomes high, allowing substantial tracts of the

parameter space to be sampled. When the energy
84 Direct Space Methods of Structure Solution

function relates to a true thermodynamic energy, the

system equilibrium properties at a defined temperature, T, can be evaluated by numerically averaging

over the states sampled by the Metropolis scheme, as the method samples the equilibrium states according

to the Boltzmann probability distribution.

In simulated annealing, the temperature is raised

from an arbitrary point in increments or powers until

the system become 'molten'. The 'melting' transition can be recognized by a maximum in the heat capacity of the system as a function of temperature. The

heat capacity, Ct , is a measure of the degree of en- ergy fluctuation possible in the system at a particular

temperature:

=

(E?)-(E.)-

KT^ V ;

Alternatively the 'molten' state can be defined as that in which more than, say, 50% of the attempted

moves are accepted (equation 2). From the 'molten'

state, the temperature is reduced slowly according to a defined schedule, the particulars of which might depend on the details of the system under investigation

(Press, Flannery, Teukolsky k Vetterling, 1989). Often, for example, the cooling rate is reduced in the

vicinity of the change in system heat capacity, Cj,

that indicates the 'freezing' transition. The annealing procedure corresponds to the physical process of

solidification. In the limit of low temperature, only

energy reducing moves are accepted (equation 2). Although the terminology of simulated annealing is derived from that of physical simulations, the method

can be applied quite generally to functions to be minimized (Kirkpatrick, Gelatt k, Vecchi, 1983). Simulated annealing has, for example, been applied to the

traveling salesman problem (Press, Flannery, Teukolsky & Vetterling, 1989), to integrated circuit design

(Kirkpatrick, Gelatt k Vecchi, 1983), to studying

atomic ordering in alloys (Binder, 1987) and aluminosilicates (Soukoulis, 1984; Newsam, 1992), and to

curve-fitting.

4. Cost or 'energy' functions

4-1 Geometric and poieniial functions

The degree to which a model structure is chemically

or physically sensible can be quantified at various

levels of sophistication. One of the simplest is a measure of the degree to which unrealistic atomic

overlaps are avoided; this criterion can be used in

the first stage of model assessment once atoms are introduced at random into the unit cell. At a some- what more sophisticated level, the measure might

capture the reeisonableness of the interatomic distances. Bonding criteria require that the distances

from a given element in a defined oxidation state to

its first and second neighbors will lie within reason- ably narrow bounds. Distance targets are therefore a

viable and well-used basis for crystal structure optimization. This distance least-squares (DLS) method

(Meier k Villiger, 1969) takes as input a trial or ran- dom set of atomic coordinates, the crystal symmetry

and possibly atomic site symmetry constraints, and

the atomic connectivity or the set of distances to be

optimized. It has become a standard method in zeolite structure analysis and has also been successfully

employed in studies of various other inorganic crystal

structures (Khan, 1976; Tillmanns, Gebert k Baur,

1973). The function to be minimized is a sum of

squares; generally, either the coordinates of the initial model are within the convergence capture radius, or the prescribed distance constraints are sufficiently

restricting that the global minimum and the path(s)

to it are sufficiently unambiguous that least-squares can be successful. The DLS approach, nevertheless,

often has problems with convergence and does not

recognize unacceptable atom conflicts unless the cor- responding separations are explicitly defined in the

initial set of distance targets.

We have been able to use 'energy' functions constructed from simple geometrical terms successfully

for structure solutions, particularly for zeolite struc- tures (Fig. 3). These 4-connected framework struc- tures are built from TO4 tetrahedra (T= Tetrahedral

species, Si, Al, Ga etc.) sharing apices. The first phase of framework structure solution requires definition only of the approximate T- atom positions, for

the centers of the T-T vectors are sufficiently close to

the actual oxygen atom positions that DLS or refinements against diffraction data can proceed from this

starting point. The geometrical constraints on the Tatom configurations obeyed by zeolites and related

4-connected framework structures can be quantified

by constructing an 'energy' function that is a sum

of terms based on the T-T distances, and the T-T-T

angles and average angles observed in known zeolite

structures (Fig. 3) (Deem k Newsam, 1989), and a

coordination number term that reflects the requirement of framework 4-connectedness. The degree to which these combined constraints are satisfied by a

given arrangement of T-atoms within the unit cell then provides a measure of the reasonableness of that

arrangement. In the simulated annealing procedure,

the required number of unique T-atoms is introduced

at random into the unit cell and the T-atom coordinates are then adjusted so as to optimize the matching with the defined geometrical constraints. Given

the openness and flexibility of zeolite structures, we

have found it advantageous to limit the number of independent variables to as large a degree as possible.
J. M. Newsam, M. W. Deem and CM. Freeman 85

(c) T-T-T angle (°) average T-T-T angle (°)

Figure 3: Contributions to the total system 'energy' for a zeolite framework model from each T-T distance

(a), individual (c) and average (d) T-T-T angles, and coalescence or merging of symmetry related T-atoms

(6) (Deem & Newsam, 1992).

We therefore impose rigid space group constraints

and adjust the positions only of the unique T-atoms;

the positions of the other T-atoms that in the coor- dination shells of the unique T-atoms are generated

by the space group symmetry operations.

In many cases, high space group symmetries are involved and there is a tendency for T-atoms to reside on special positions. This possibility is supported

by treating all of the T-atoms as being on general

positions, but allowing pairs of symmetry related Tatoms to 'merge' when they become separated by

less than a defined capture distance during the simulated annealing. In cases where the location of Tatoms on special positions has been inferred from

compositional or, perhaps, NMR data, the merging can be promoted by subtracting a contribution

to the total 'energy' sum when it occurs (Fig. 3).

Other geometrical terms can be included. Projection and pore terms are two that we have found ef- fective. The projection term provides for a bias in

favor of structures that, when viewed in projection

down a defined crystallographic direction, appear as 3-connected nets (a property common to many of the

known zeolite structures). The pore term is a simple

mechanism for favoring structures that have cylindrical or spherical pores; sorption experiments on zeolites often yield data on the character of the pore

architecture, particularly the free pore volume and

the minimum ring sizes. As noted further below, the

full zeolite 'energy' function comprised of these var- ious components, when coupled with the simulated

annealing procedure is a very effective means of developing new structural models (Deem & Newsam,

1989 and 1992).

Beyond the interatomic distances themselves,

functions of these distances, that might more realistically approximate the structure energetics can be

used as the basis for the 'energy' function. Pannetier

ei al. (Pannetier, Bassas- Alina, Rodriguez-Carvajal

& Caignaert, 1990) used an 'energy' function com-
86 Direct Space Methods of Structure Solution

prised of a combination of bond valence and electrostatic terms in determining the structures of NbF4,

K2NiF4, BaSiFe and LiCoF4; they have also used

simulated annealing as a route to solving magnetic

structures (Lacorre & Pannetier, 1987).

Molecular or crystal mechanics methods, that

use effective interatomic potential functions, have

achieved wide success in studies of molecular con- formation (Burkert & Allinger, 1982) and molecular

crystals (Williams, 1967). Several 'predictions' or refinements of inorganic crystal structures have been

reported (Catlow k Mackrodt, 1982; Maddox, 1988;

Hawthorne, 1990; Catlow k Price, 1990), including

several studies of zeolite structures (Bell, Jackson &; Catlow, 1990; Tomlinson, Jackson & Catlow, 1990).

Both DLS and lattice energy minimization (using

the METAPOCS code (Parker, Cormack & Catlow,

1984)) were used in a complementary fashion in determining of the complex structure of zeolite NU-87

(Shannon, Casci, Cox k Andrews, 1991).

The convergence characteristics of potential en- ergy surfaces based on rigid ion or shell model potentials are poor when the starting model is far from

the global minimum; the form of the short range re- pulsive terms makes it impossible for atoms to interchange positions directly. This problem can be cir- cumvented by staging the application of the potential

functions (Freeman k Catlow, 1991). The starting

point is again the generation of random ionic positions within the experimental unit cell. A straightforward distance criterion is used to avoid excessive

steric overlap, potentially overlapping positions being rejected and repeated random insertions made

until the cell is filled with the required number of

atoms or ions of each type. No assumptions of spacegroup symmetry are made throughout the procedure,

though translational periodicity is implicit. Having obtained a starting point, simulated annealing,

using Metropolis Monte Carlo, is applied to relieve

any unreasonably close inter-ionic contacts. The 'en- ergy' function employed in this stage is the computationally convenient r~^^ form, with repulsion between ions of similar charge being twice that between

pairs of formally dissimilar charges. The effect of

this procedure is to evenly disperse the ions within

the cell, with locally preferred cation- anion ordering. At the end of the annealing schedule a switch is made to the quadratically convergent conjugate gradient minimization method to optimize the model

with respect to the crude 'energy' function. The fi- nal stage of the procedure takes the final annealing

result and employs full lattice energy minimization

using the Born model of the solid, as implemented

in the METAPOCS code (Parker, Cormack & Catlow, 1984; Catalysis 2.0, 1992). The cell dimensions

may be maintained throughout at the experimental

values. In practice, as in the other simulated an- nealing procedures, several distinct crystal structure

predictions are usually obtained for a given composition and unit cell. The scheme is, therefore, performed repeatedly in order to sample an adequate

set of structures possible for the material (Freeman

k Catlow, 1991).

1 1 1

(a)

J 1

,. .. .1

(b)

, , 1 1.1 1

1

(c)

. 1 / il . 1 . ,

1

(d)

. \ i il .

00

c

c oc

OJTO

0 0 0

6.0 36.0 Two Theta (")

Figure 4: Simulated annealing 'solution' of the crystal structure of benzene using only the fit between

target and simulated PXD patterns in the 'energy'

function. The rigid benzene molecules are moved

and rotated from a random starting configuration

(top) using the Metropolis algorithm. The struc- ture obtained by simulated annealing (second from

top) was further adjusted using conjugate gradient

optimization (third from top). The measured crystal

structure is shown at bottom.

4-2 Matching experimental diffraction or scattering

data

Experiences to date indicate that with chemical composition and unit cell dimensions and, optionally,

symmetry as input, simulated annealing using geometrical or potential function terms in the 'energy'

function will generate potentially viable structural

models. When a number of structures have been

produced, the most likely model might be selectable

based on the corresponding value of the 'energy'
J. M. Newsam, M. W. Deem and CM. Freeman 87

Figure 5: Stereo pair comparing the measured benzene crystal structure (full lines) with the results of

simulated annealing using PXD profile matching as the 'energy' function (dashed lines).

function. More typically, this selection must be made

based on comparison between model characteristics

and experimental data; usually the observed and

simulated powder diffraction patterns are compared

(Deem & Newsam, 1989). The degree of matching

with experimental data can also be incorporated as an additional term in the 'energy' function expression

(Deem &; Newsam, 1992). When the fitting of a tar- get powder X-ray diff"raction (PXD) pattern is added

to the zeolite cost function, the number of hypothetical but inappropriate structural models is dramatically reduced (Deem & Newsam, 1992). To test the

generality of the methodology, 'solutions' of a large

number of known zeolite structures were attempted

in which the 'energy' function comprised only the

degree of match to the target PXD pattern. Successful results were obtained in a substantial number

of cases, even with simulated PXD patterns based

only on the T-atom coordinates (Deem & Newsam,

1992).

This approach is now being extended to attempted

solutions of molecular crystal structures, both or- ganic and organometallic. As an illustration of the

potential of the method we show in figure 4 results

for the benzene crystal structure. The input data were the observed unit cell dimensions, a = 7.44,

b = 9.55, c = 6.92A and the density (which indicates

that the unit cell contains four benzene molecules).

Triclinic symmetry, PI, was assumed. The PXD profile (6°< 29 < X = 1.5418A) simulated based on the known crystal structure was used as a tar- get. Simulated annealing was used to adjust the positions and orientations of each of the four independent benzene molecules, treated as rigid molecules,

with the profile residual (the normalized sum of the

squares of the differences between target and computed PXD profiles point by point across the pattern) being taken as the 'energy' function. The simulation was performed using the Catalysis software

(Catalysis 2.0, 1992), augmented with a macro utility. As figure 5 illustrates, the measured crystal

structure is reproduced well by this automated struc- ture solution procedure. Applications to other sys- tems are in progress. Simulated annealing has also

been applied to the interpretation of experimental

scattering data from molten and amorphous materials (McGreevy & Pusztai, 1990; Keen k. McGreevy,

1990; McGreevy & Pusztai, 1988).

5. Some examples of geometric or potential

function constraints

5.1. Framework structures

There are currently some 85 known zeolite struc- ture types (Meier & Olson, 1992; Newsam &; Treacy,

1992). The number of hypothetical 4-connected 3-

dimensional nets is also growing rapidly (see, e.g., Deem & Newsam, 1992; Smith, 1988). The simulated annealing method, using only T-T distance,

T-T-T angle and average angle, coordination, and

merging terms in the zeolite 'energy' function (as described above, figure 3), was applied to 84 known zeolite structures, using the crystallographic data given

for the representative structure in ZeoFile (Newsam

& Treacy, 1992). The unit cell dimensions and space

group, and the numbers of crystallographically distinct T-atoms, nunique i and the total number of Tatoms in the unit cell, n^, were taken from those of

the representative structure. A structure type was

considered solved if the coordination sequence for one

of the final set of structures generated by the simulated annealing matches that of the representative

structure of the known material. (The coordination

sequences out to a defined shell, k, from the central atoms (Meier Sz Moeck, 1979) is defined as the

graph of the framework connectivity; for the unit cell
88 Direct Space Methods of Structure Solution

(a) 34

-13098

(b) 36

-11615

(C) 36

-10238

(d) 40

-4450

(e) 36

-10164

Figure 6: Some of the results of the simulated annealing procedure applied to the known crystallographic

data for ZSM-18 (P63/m, a = 13.2, c = 15.8A, tar- get number of T-atomsper unit cell, = 34) (Deem

&: Newsam, 1992); the values of for each model,

and the calculated 'energies' are shown. The actual

MEI-framework is (a).

volumes typical of most zeolites, if the coordination

sequences out to ^ = 10 for two structures are identical, the topologies are identical with a probability

that approaches one).

The single first simulated annealing run gives the

correct structure in 25 (30%) of the total of 84 cases. Some of the structures that were obtained based on

the input data appropriate for the MEI framework

of zeolite ZSM-18 are shown in figure 6; an interesting feature of this structure type is the presence of

3-rings (Newsam & Treacy, 1992). Of those struc- tures not solved, those having six or fewer unique Tatoms were selected for further runs, with only the

initial random number seed being changed between

runs. The method was found to reproduce 57 out

of these 65 distinct structures. The full set of cal- culations consumed about a week of CPU time on a

cluster of half-a-dozen 20 MHz Silicon Graphics Personal Irises; a single run consumes typically 10 mins.

Figure 7: The framework structure of a new sodium

zinc arsenate phase solved by simulated annealing

using solely geometrical terms (Fig. 3) in the 'energy'

function (Nenoff, Harrison, Nicol, Stucky & Newsam,

1992).

An analogous set of calculations was also performed using, in addition, a PXD diffraction matching term in the zeolite 'energy' function. As experimental PXD data for most of the materials for which

structural data are known (Newsam k, Treacy, 1992)

are not available, synthetic data were used. The

PXD patterns were calculated for 26 <30'* assuming

the CuKa wavelength of 1.54 18A, from the known

representative structures assuming, where possible, a chemical composition of Si02. The maximum intensity in the calculated PXD was scaled to 1000,

and standard deviations of 1, 2, 3, and 4 were as- signed to intensities greater than 0, 40, 80, and 120

in an effort to mimic approximately the uncertainties

expected in the experimental data. The PXD pattern measured from a real material will, of course,

reflect the scattering from the full unit cell contents,

both framework species, non-framework cations, and

sorbates or occluded template molecules. However,

unless a substantial proportion of heavy elements are present in well-localized positions, the general distribution of intensities will be a reasonable discriminator for possible framework models. Using data carefully measured from real materials we would expect a degree of success similar to that obtained with the

synthetic data. The method was again successful in
J. M. Newsam, M. W. Deem and CM. Freeman

Figure 8: Comparison of the crystal structure of

brookite viewed along [001] (full lines) and that generated by simulated annealing using rigid ion model

potentials (Freeman, Newsam, Levine & Catlow,

1992).

the majority of cases, with the PXD pattern matching term acting as a powerful discriminator against

inappropriate hypothetical structures.

The simulated annealing method, with the general form of the zeolite 'energy' function described

here has been used to solve the structures of some

new materials (Deem Sz Newsam, 1989), notably the

product that results from a framework reconstruction on dehydration of Na6(ZnAs04)6 -81120 with the so- dalite structure at 190 °C (Fig. 7) (Nenoff, Harrison,

Nicol, Stucky Sz Newsam, 1992). Although initially

conceived and presented here as an aid to structure

solution, the method is also now being used as a true

structure design tool.

5.2. TUania polymorphs

The use of staged potential functions, culminating in a full lattice energy minimization worked well in 'predicting' the Ti02 rutile structure (Freeman &; Catlow, 1991). As a further test of the procedure we

have applied the method to the anatase and brookite

polymorphs of Ti02 (Freeman, Newsam, Levine

Catlow, 1992). The potential model employed in the

lattice energy relaxation phase of the calculation was

that derived by fitting to the experimental structure

89

Figure 9: Comparison of the crystal structure of

Ti02(II) (full lines) and that generated by simulated annealing using shell model potentials and the

brookite unit cell dimensions (Freeman, Newsam,

Levine Sz Catlow, 1992) (viewed along [010] in the

brookite unit cell).

and properties of rutile. As in the rutile study, 50 tri- als were performed and the resulting structures were ranked in terms of predicted lattice energy.

For rutile the correct rutile structure is generated

in 43 from 50 trials (Freeman Sz Catlow, 1991). In the case of anatase the correct structure was generated in 4 of the 50 trial runs, and is the lowest energy structure obtained, whilst for brookite

the number of instances of correct structure generation was 3. For brookite it is interesting to note

that the correct brookite polymorph was found to be ~ 0.1 eV less stable than an alternative structure,

found in 20 trials. This corresponds to the Ti02(II)

polymorph (Hyde Sz Andersson, 1989), but with a doubling of the a lattice vector to match that of

brookite's unit cell (Brookite has space group Pbca,

with a = 9.116A, b = 5.436A, c = 5.135A; Ti02(II)

Pbcn, with a = 4.563A, 6 = 5.469A, c - 4.91lA).

Figures 8 and 9 provide a comparison, for representative fragments, of the calculated (dotted line)

and the experimental structures (solid) for anatase

and brookite respectively. For the higher Ti02 polymorphs the frequency of correct structural prediction was significantly less that in the case of rutile. This

is unsurprising given rutile's smaller unit cell and

therefore fewer variable parameters to be optimized.

Nevertheless the results clearly indicate that, when

combined with cell parameters derived from an indexed powder pattern, the method has the potential

to assist in the process of structure solution for con- densed inorganic materials.
90 Direct Space Methods of Structure Solution

6. Conclusion

When diffraction data from a high quality single crystal are available, conventional structure solution and

refinement methods work very effectively. With less precise data, or with data only available from a powder sample, new approaches to structure solutions

may have substantial value. Graphical model building methods, coupled with the interactive simulations of diffraction or other spectral data and their

comparison against experiment, can be used to develop crystal structure models from discrete molecular components or structural motifs. Unlike physical models, unit cell size and symmetry constraints can be used directly as constraints in the model construction process. By quantifying the reasonableness

of a given atomic arrangement by cost or potential

functions, viable models can be derived from an ini- tially random distribution of the required number of

atoms of each type within the observed unit cell by

simulated annealing. The effectiveness of this approach has been demonstrated by test applications to

the known structures of zeolites and other more con- densed oxide phases. Additional cost functions can be included in this process, such as micropore dimensions in the case of zeolites or, more generally, the

degree of match between model and target powder

diffraction patterns. Particularly when diffraction

pattern matching constraints are included, simulated

annealing appears to be an efficient and potentially

general route to ab initio structure solutions.

Acknowledgments

The Biosym Catalysis and Sorption Project is supported by a consortium of industrial and academic

institutions; we thank the membership for their guidance, input and for many stimulating discussions.

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October 1992)

Automatic Indexing: Procedures and Applications

D. LOUER

Lahoraioire de Cristallochimie (URA CNRS 1495), UniversHe de Rennes I, Avenue du General Leclerc, 35042 Rennes cedex, France.

Abstract

Indexing a powder-diffraction pattern by computerbased methods has advanced to the state that it is now possible to determine unit-cell dimensions, re- gardless of the crystal symmetry, for an unknown material solely from the one-dimensional d-spacing information. It is an essential requirement in a number

of applications of the powder diffraction technique;

the recent progress in ab initio structure determination from powder data have furthermore demonstrated the fundamental role of the indexing of a powder diffraction pattern. A survey of the different

approaches used in computer-based methods is given,

with emphasis on data accuracy required. The influ- ence of perturbing factors is discussed, e. g. random

and systematic errors, and some means of reducing

the latter are described. The respective merits of

modern instruments, operating with monochromatic

radiation from conventional or synchrotron X-ray

sources, are discussed. Applications are presented,

including exceptional examples, such as cases with a dominant zone and unit cells with a large volume.

1. Introduction

The indexing of a powder pattern is an essential re- quirement in a number of applications of the powder

diffraction technique. Pattern indexing and the determination of symmetry and unit cell parameters al- low a unique characterization of the powder material,

providing evidence that the sample is a single phase.

The recent developments in ab initio structure determination from powder diffraction data have highlighted the fundamental importance of indexing a powder pattern. It is an easy task with accurate

data, but the problem is usually complicated by ran- dom and systematic errors. Efficient computer-based

methods, regardless of symmetry, are currently used;

their rate of success is very high, provided that data

are accurate. However, their use in routine work requires a minimization of systematic errors by knowing perfectly the accuracy attainable by careful alignment of instruments.

The interest in ab initio indexing of a powder

diffraction pattern started a few years after the discovery of X-ray diffraction in 1912. Runge (1917)

derived some useful specific relations in reciprocal

space; later, from similar principles, Ito (1949) and

de Wolff (1957) reported a more general method for

the geometrical reconstruction of the reciprocal lattice of a crystal from the observed d-spacings. The

key role of data accuracy for solving the crystallographic problem was clearly expressed by de Wolff

(1957), as is shown by this quotation: "It would be

quite an easy puzzle if errors of measurement did

not exist." In 1968, a simple criterion for selecting

correct indexing from false solutions was proposed

(de Wolff, 1968); the introduction of this factor was

a milestone in the slow progression of powder pattern indexing. At the same time, a computer program, based on the Runge (1917), Ito (1949) and de

Wolff (1957, 1968) principles, was written by Visser

(1969). With the development of computer technology, new approaches using trial- and-error methods

were devised: the method based on the permutation

of Miller indices by Werner (1964) and the method

based on the variation of the lengths of cell edges and

interaxial angles by means of a successive dichotomy

procedure by Louer Sz Louer (1972). The efficiency of

these modern computer-based methods was reviewed

by Shirley (1978).

The present paper summarizes the basic principles of the indexing of a powder pattern, describes

the modern computer-based methods, emphasizes

the role of data accuracy and presents the features

of modern instruments that reduce the influence of

perturbing effects. Also, examples, including cases with dominant zones and large unit-cell volumes, are presented.

92
D. Louer 93

2. Basic principles

The objective of indexing a powder diffraction pattern is to reconstruct the three-dimensional reciprocal lattice of a solid from the one-dimensional dspacing information available in the pattern. Fig- ure 1 shows how the reciprocal lattices of the ran- domly oriented crystallites constituting the powder

generates, by a rotational projection, a radial distribution of lengths d*(= 1/d) of diffraction vectors. The geometrical restoration of the node positions of

Figure 1: Two-dimensional representation of the distribution of diffraction lines from the projection of

randomly directed reciprocal vectors, d*.

the reciprocal lattice from this radial distribution is a crystallographic problem based on algebraic relations; it is also dependent on computing facilities,

chemical knowledge of the material, in order to ex- plain (possible) spurious lines due to impurities, and

the instrument used for collecting data, since accu- racy is required for a successful indexing. The outcome of the indexing will depend on the measure- ment of d-spacings, derived from 26 Bragg angles in

angle-dispersive experiments. Consequently, indexing a powder diffraction pattern cannot be dissociated either from the quality of instrument adjustment (or calibration), or from the degree of diffraction line overlap. The latter is related to geometrical

coincidences, to the instrument resolution characteristics, and to the microstructure of the material. In- deed, every observed diffraction line {h profiles) is the

convolution of instrumental aberrations [g profiles),

including the wavelength distribution and sampleinduced geometric and physical effects, with contributions that are specific to the microstructure of the

sample (/ profiles). These effects reduce the reso- lution of the pattern, so that some lines cannot be

observed.

Each vector d^^., is defined in terms of the basis

vectors of the reciprocal lattice (a*, b*, c*) and the

Miller indices: d^^./ — +^b* -|-/c*. The basic re- lation used for indexing a powder diffraction pattern

is obtained by squaring the d* vector:

Q{hkl) = |d*|2

+2klQiy + 2hlQE + 2hkQp, (1)

where = a* • a*, Qb = b* • b', Qc = c* c*,

Qd = h* -c^Qe = c* a* and = a* • b*. This

quadratic form corresponds to the general triclinic

case; it simplifies considerably for the other systems.

The one-dimensional distribution of lengths of

reciprocal-lattice vectors is completely defined by the

geometry of the unit cell, the lengths of the basis

vectors (a*, b* , c*) and the inter-axial angles of the

reciprocal lattice (a*, /3* In practice, some lines are missing, due to symmetry conditions, accidental

or exact overlap, and diffraction lines with low intensity. Although the powder diffraction pattern suffers

from the collapse of the three-dimensional crystallographic data to the one-dimensional diffraction pattern, some useful mathematical relationships remain:

(i) Algebraic relationships in reciprocal space can be

derived from equation (1). Among them are the

following general properties:

• Colinear vectors d^j^i are easily detected by

the integer value n of the ratio of their

length to the length of the shortest vector i^h.kiij in the [hkl] direction: d^j^, =

''^^hikiii- This property can also be used

for the detection (and correction) of systematic errors in the pattern.

• Relations for a zone of lattice planes can be derived from the additive properties of

Qhki, particularly in the case of triclinic

symmetry, e. g. the Q(OA;/) and Q{Okl) val- ues can be used for the determination of

the interaxial angle a*. For the detection

o{Q{Okl) and Q(Ofc/), it can be shown that

they are symmetrical with respect to a nonexistent value (Qoit/) calculated for an or- thogonal lattice.

Some other general relations have also been derived by de Wolff (1957);

• Additive properties of equation (1) can also

be used for finding the OkO row in monoclinic cases; an example of a relation that
94 Automatic Indexing: Procedures and Applications

serves to locate the 020 reflection is (Smith

& Kahara, 1975):

2Qo20 + Q/»10 = Q/»30-

(iz) The theoretical number N of diffraction lines

in the pattern can be evaluated in the general

triclinic case. It is easily demonstrated that the

number of radial distances less than d*^ is given

by the approximate relation:

N « ATrV{d%f/3m, (2)

where V is the cell volume of the direct lattice.

In fact, the actual number of observed lines is much less than N. The reduction is caused by

the multiplicity factor m of Bragg reflections

[m is given as 2 for triclinic, 4 for monoclinic

symmetry, etc. by de Wolff (1961)], the space

group extinctions and the non-observed reflec- tions. From this equation. Smith (1977) has derived an empirical relation useful for estimating

the unit-cell volume in triclinic cases, based on

the d value of the 20th observed line:

V = 13.39d^n.

3. Assessing the reliability of indexing

The solution to equation 1 is not unique and is also

strongly dependent on the accuracy of data. In general, a satisfactory agreement between calculated and

observed Q values is not sufficient to be certain of

the unit cell. To assess the reliability of an indexed

powder diffraction pattern, de Wolff (1968) recom- mended the use of a criterion based on the ratio of

two average quantities: the ratio Q20 /^co/c and the

actual discrepancy between observed and calculated

Qi values. This index of reliability is defined as:

M20 = Q2o/2(AQ)Ar,,,„ (3)

where Q20 is the value of Q for the 20th observed

line, (AQ) is the average absolute discrepancy between Qiobs and the nearest Qcaic values and Ncaic

is the number of distinct Q values smaller than Q20)

calculated according to equation (1), excluding any

systematic absences, if they are known. M20 will be

larger if (AQ) is small, and if NcaU is as close as possible to 20. De Wolff stated that incorrect indexing

may have a value of M20 up to 6, and that correct

indexing usually has M20 greater than 10. Examples

obtained from data collected with different instruments are discussed below in section 7. They show

that values greater than 20 are obtained in routine

work, and that they can be considerably greater with

high resolution instruments.

The figure of merit M20 can be expressed in terms

of the cell volume V of the direct lattice, by combining equations 2 and 3 (Werner, 1976):

V = 3md2o/87r(AQ)M2o. (4)

For a given average discrepancy (AQ), M20 and the

cell volume are inversely proportional. This result

is interesting because it gives a useful argument for

searching the smaller cells first in automatic indexing; it is applied in the strategy based on the successive dichotomy method (Louer & Louer, 1972).

Another figure of merit that is simple to use, Fn,

has been proposed by Smith and Snyder (1979).

Originally intended for rating powder diffraction patterns, it is defined by:

FN = N/{2e)Nposs,

where Npo,s is the number of possible diffraction lines

up to the A'^th observed line, and (2^) is the average

absolute angular discrepancy between observed and

calculated 29 values. This index is reported in the

form Fjv = value((2^), Nposs). Again, the higher the

accuracy of data and the more complete the pattern,

the larger is Ftv- Of special interest is the reported

average angular discrepancy (2^), which is another

evaluation of the quality of data. The two figures

of merit, M20 and Fjv, are usually calculated in indexing programs. The M20 factor depends strongly on the crystal symmetry, large M20 values being obtained for higher symmetries. Some authors (Shirley,

1980; Werner, 1980) have discussed the relative merits of these factors and argued that M20 is to be

preferred for testing the reliability of a proposed indexing model; Werner concluded that if (26) and

N/Nposs are identical for a cubic and a triclinic indexing, the cubic indexing is obviously to be preferred.

Recently, Taupin (1988) has proposed another fig- ure of merit based on the information conveyed by

the hkl indices assigned to a given diffraction line,

namely individual discrepancies between computed

and observed line positions, unit-cell volume and systematic extinction rules. This factor has been incorporated in the author's program.

Finally, from the experience of the present author,

based on the indexing a great number of powder

diffraction patterns, the M20 figure of merit has al- ways been found satisfactory in assessing the reliability of a proposed solution. The F20 index and the
D. Louer 95

reported average angular discrepancy between observed and calculated 26 positions constitute other

useful indicators of precision. Also, the information

contained in equation 4 (a similar equation would be

obtained from the index) should be taken into

consideration in indexing problems. With accurate

data, the solution with the smallest volume is usually the correct one. In some exceptional cases, diffi- culties can arise from {i) geometrical ambiguities or

{ii) when a zone dominates in the twenty first lines.

Case (i) has been discussed by Mighell k, Santoro

(1975); it can occur if the lattices have high symmetry, e. g. an hexagonal pattern can be indexed by an orthorhombic unit cell having the cell dimensions

a/2, a\/Z/2, c and a volume half the hexagonal cell volume. This situation can be detected by analysing

separately each crystal system, from high to low symmetries and by the metric relations between parameters of the two cells. In case {ii) solutions with

similar unit cell volumes and M20 factors can be observed. This case is illustrated by an example in

section 9. Nevertheless, whatever the definition of

the figure of merit used, the problem of a dominant

zone remains unsolved.

4. Modern methods for the indexing of

powder diffraction patterns

With the availability of computers, automatic programs were devised for indexing powder patterns of

solids with an unknown crystal structure. These were

reviewed by Shirley several years ago (1978, 1980).

He also reported an analysis of the success rate of

these programs. Since this study, some of the tested

programs have been considerably improved. Nowadays, three main important procedures for indexing a powder pattern, regardless the symmetry, are cur- rently used:

(z) The Runge-Ito-de Wolff" method is based on al- gebraic relations in the reciprocal lattice, of

which some are given in section 2. A computer

program using these principles was written by

Visser (1969); it was the first general program,

regardless of the symmetry, written for powder

pattern indexing. It is well known for its ef- ficiency for low symmetry cases; the most fre- quent reason for failure is the incompleteness

of the input data. The program has frequently

been used with success since 1970. It helped

to demonstrate that powder pattern indexing can be used with confidence for a unique characterization of powder materials. For example.

the Visser program was used to identify four

cadmium hydroxide sulphate phases obtained

by slow hydrolysis of a neutral cadmium sulphate solution, by indexing the powder diff"rac- tion patterns with tetragonal and monoclinic

unit cells, with high M20 figures of merit (37,

54, 20 and 29) (Walter-Levy, Groult k Visser,

1974). Later, the correctness of the reported

results was demonstrated by single-crystal studies (Labarre, Louer, Louer k. Grandjean, 1976a,

1976b; Louer, Louer k, Grandjean, 1982; Louer,

Labarre, Auff"redic k Louer, 1982).

(n') The trial- and-error method, based on a permutation of Miller indices for selected base lines,

was introduced by Werner (1964), and the principles have since been applied by several authors

(Taupin, 1973; Kolhbeck k Horl, 1976; Wu,

1989). Among the most popular recent indexing programs, regardless of the symmetry, are TREOR (Werner, Eriksson k Westdahl, 1985)

and the programs by Taupin (1989) and Kolhbech k Horl (1976, 1978). From its principles,

the method has to be sensitive to a dominant

zone in the powder data, and provision for such

cases can be implemented in programs, as has

been done in the program TREOR; this part

of the program has been further developed in

the program version TREOR90. An objective

analysis of the success rate of TREOR has been

reported by Werner et al. (1985) on the basis of

all powder diffraction data reported in Section

17 of Monograph 25 {Natl. Bur. Stand. (U. S.),

1980). The overall success of the program has

been found to be better than 90%, considerably

more for orthorhombic and higher symmetries;

moreover, it was noted that for a minority of

patterns, data were of low quality, mainly due

to unstable phases.

(Hi) The successive dichotomy method is based on

the variation of the lengths of cell edges and in- teraxial angles by finite ranges, followed by a

progressive reduction of intervals by means of a dichotomy procedure. This method, introduced by Louer k Louer (1972), was first applied to high symmetries and then extended to monoclinic (Louer k Vargas, 1982). Recently, a considerably improved version, DICV0L91, ex- tended to all crystal systems, has been published (Boultif k Louer, 1991). The method

was described by Shirley (1980) as probably the

optimal exhaustive strategy in parameter-space
96 Automatic Indexing: Procedures and Applications

within input parameter limits. The property

of equation 4 is used for finding the solutions

with smaller volumes first. In contrast to other

approaches, the calculated patterns for possible solutions are here based on ranges [Q^(hkl),

Q+(hkl)] for every line, these ranges being progressively reduced by a dichotomy strategy. In

the course of the calculation the experimental

error of every line, AQj , is rigorously taken into

account. Due to the principles of the method, it

is not too sensitive to the presence of a dominant

zone, unless considerable ambiguity is present

in the data used for solving the problem. The

strategy for indexing starts from the cubic end

of the symmetry sequence and finishes with triclinic symmetry. For an unbiased test of effi- ciency of the program, data reported in section

17 of Monograph 25 (Natl. Bur. Stand. (U. S.),

1980), comprising powder data for all symmetries, were analysed by DICV0L91, as well as

all triclinic patterns reported in the complete

Monograph 25. All orthorhombic or higher symmetry solutions were found; all solutions with

monoclinic symmetry were also found, but for

four of them the input absolute error, AQj, of

observed peak positions needed to be increased

slightly owing to inaccuracy of the reported

data; the six powder patterns of triclinic materials in section 17 were also correctly indexed.

Powder patterns for triclinic materials, reported

in the complete monograph, were indexed, ex- cept for three cases; one of them corresponds to a situation (the first possible lines are missing)

not implemented in the program, and for the

two others the explanation for failure was found

to be the low precision of data.

5. Some considerations on methods and data

accuracy

All authors of indexing programs have emphasized

the dramatic effect of data inaccuracy in the search

for the correct unit cell and have pointed out that it

is the central parameter for a successful outcome of

pattern indexing. From the mathematical point of

view, equation 1 would be easy to solve if errors did

not exist. In fact, taking into account the absolute

error, AQi, in every observed Qi value, the indexing

of a powder pattern requires that, for a set of parameters Q^, Qb, Qc, Qd, Qe and Qf, there exist

three hkl integers satisfying the inequality:

Qi - AQi < Qihkl) <Q, + AQi , (5)

where Q(hkl) is given by equation (1). A rigorous

analysis of the influence of errors in observed Qi val- ues is difficult. Its disastrous effect is well known

and is one of the main reasons, along with incompleteness of data, for failure in indexing powder patterns. By rule of thumb the error in the observed

data should be less than 0.03° (20) for a wavelength

such as copper Kai. Before discussing experimental

procedures for reducing data inaccuracy, the effect

of errors in modern approaches for indexing will be

discussed briefly.

In methods of type (ii), solutions are obtained

from a set of base lines, against which the remaining

lines in the pattern are checked. These solutions are column vectors P (with elements Q^, Qb, Qc, Qd,

Qe and Qp) derived from a set of linear equations

formed by the base lines; in practice, each of them

includes implicitly an unknown individual error ^Q;

thus, the problem should be formulated by the matrix form:

P-f ^P = M-^(Q + <5Q),

where M is a square matrix whose elements are based on the Miller indices. In this equation, for a reasonable error 6P, there is no isomorphic relation between

the vectors P -|- (5P and Q + SQ, which implies that a small error (^Q) in Q does not necessarily give a

small error (^P) in P. On the other hand, from a

given error vector ^P in P, it is possible to calculate

exactly the corresponding error (^Q) in Q. (This

feature of non-bijectivity is at the origin of the successive dichotomy method.) In approach (ii), when

the vectors P have been obtained, they are used to

check the second data set formed by the equations

describing the remaining diffraction lines. As a con- sequence of the parameter error (^P), expanded error ranges AQ* must be used, instead of the original ex- pected error AQi. In practice, this problem is solved

in current programs and has been discussed by different authors (Werner, 1964; Taupin, 1973; Kohlbeck

& Horl, 1978). It should also be noted that further

confldence in a solution is provided if identical solutions are found from different base line sets. In any

case, accurate data for the base lines are of special

importance for reducing the effect of error propagation. A similar comment is also true for approach (i). The lines used to obtain linear and angular reciprocal

lattice parameters must be as accurate as possible in

order to have an agreement, within the experimental

errors, between observed and calculated Qi values.

Also, it is important that there should not be too

many lines missing from the pattern.
D. Louer 97

The successive dichotomy method

As reported above, the error vector (5Q can be calculated from a given vector SV, and this calculation can be extended to all possible hkl indices of the

input data. This property is the basis of the successive dichotomy method. The parameter axes Pj

(Pj = Q^,...,Qj^ or related parameters in direct

space) are scanned by linear ranges APj {= Pj_,

Pj+), within limits selected by the user. The volume space is thus divided into discrete j-dimensional

domains (j=l, . .

. , 6) formed from the ranges APj.

Each domain is analysed separately. A calculated

pattern is generated, each line being represented in

Q-space by an interval [Q_(/iA;/), Q^{hkl)\, the domain is then retained if the observed lines Qi lie within the limits

Q_ {hkl) - AQi < Qi < Q+(hkl) + AQi , (6)

which include the majcimum expected experimental

error, AQi. Otherwise it is discarded. At this stage

the correct solution can be included in the domain,

which will subsequently be analysed in detail. It can be noted that the hkl indices of the exact solution

for every observed line constitute a subset of the

set of hkl values that have been found possible at

this stage for each line. This is an interesting point,

since the introduction of a recursive procedure for

the storage of hkl reduces computing times. If all observed Qi values are located within the first calculated limits, each domain is divided into 2j subdomains by halving the intervals [Pj_ , Pj+]. Again, calculated lines [Q^(hkl), Q+(hkl)] are generated and

compared with the observed data, within the limits of

experimental error (relation 6). If a solution still exists, the dichotomy approach is applied successively,

up to a maximum of seven times. It is an arborescent

procedure that progressively encloses the solution (s).

It has been found preferable to apply this technique

to direct parameters; the lengths of cell edges and

inter-axial angles are scanned by finite ranges (0.40

A and 5**, respectively), which are progressively re- duced. With this stategy, solutions are searched ex- haustively in a j-dimensional space. (However, for

the triclinic case, j = 6, the procedure is applied in Qspace with some constraints.) The method is simple;

it is only necessary to derive the mathematical relations giving the limiting values Q-{hkl) and Q^(hkl)

for each symmetry. For orthogonal and hexagonal

systems this calculation from the intervals [Pj_ , Pj+]

is trivial (Louer Sz Louer, 1972); for low symmetry

cases, the calculations are more difficult and have

been reported recently (Boultif &; Louer, 1991). Finally, it should also be noted that the method ac- cepts individual experimental error bounds for every

observed diffraction line. This feature can be useful

in cases where, for some reason, the measurement of a few lines in the pattern is less accurate.

6. Precision and accuracy of data

In angle-dispersive powder diffraction, the precision

of the position of Bragg reflections is defined by the

error A2^. The maximum precision in d-spacing at- tainable by assuming a constant error A2^ is obtained by differentiating the Bragg equation, which

shows that the error in d, proportional to cot^A^,

increases as 0 decreases. Thus, the d values derived

from low-angle lines, which play a fundamental role

in indexing, are most affected by errors. These ef- fects can be illustrated by simulating errors in calculated patterns, as shown in this section. It is beyond the scope of this paper to analyse the origin

of random and systematic errors that may occur in

powder diffraction; this has been discussed by vari- ous authors, e.g. Wilson (1963). In collecting data

for indexing a powder pattern, it is strongly recom- mended that these factors be considered. The effects

of all instrumental factors on the quality of data and

their influence on the derived interplanar d-spacings

must be properly understood to obtain highest accuracy. For example, reproducibility of measurements

is an indication of the precision, but does not account

for possible systematic errors that can affect the ac- curacy of data. In practice, it is essential to adjust

the instruments correctly and to check the quality

of alignment by means of reference materials, which

will give an indication of the absolute angular error of data. In what follows the influence of random and

systematic errors in input data is described by means

of two calculated patterns corresponding to unit cells with a small and a large volume. They correspond

to the calculated patterns for a cadmium hydroxide

nitrate (a = 3.4206 A, 6 = 10.0269 A, c = 11.0282 A,

V = 378 A3; Auffredic, Plevert k Louer, 1990) and

barium titanyl oxalate hydrate (BTO) (a = 14.044

A, 6 = 13.812 k,c= 13.382 A, /? = 91.48", V = 2595

A^; Louer, Boultif, Gotor & Criado, 1990).

The effects of simulated random and systematic errors on solutions can be easily evaluated by

means of the successive dichotomy method (program

DICV0L91), since the treatment of error is rigorous

in this approach. The results are presented until the

correct solution is obtained. The default absolute er- ror value for every line (i. e. 0.03°2^) was used for
98 Automatic Indexing: Procedures and Applications

the case of maximum random or systematic errors lower than 0.03° (2^), and was increased for greater

simulated errors.

(i) Random errors Figure 2 shows the effect on the figures of merit

(FoM), of a degradation of cadmium-hydroxidenitrate data by a random error. It is remark150

100

50

0.04

A

0.03

0.02

0.01

0.02 004 0O6

Mnxiiiiiiin random error 008 0.10

Figure 2: The effect of random error ["(2^)] on the

figures of merit (FoM) M20 (o) and F20 (+), and

on the average angular discrepancy (2^) (°) (x), for

cadmium hydroxide nitrate.

able that the correct solution is obtained with

large random errors, up to 0.10°(2^) for the cadmium hydroxide nitrate; it is 0.07°(26') for BTO.

A similar conclusion was reported by Shirley

(1980), who estimated the error limits to be

±0.08°. This result would indicate that the indexing problem is not too sensitive to random

errors. The increasing Ad error as 6 decreases,

for a constant A26, explains that, for patterns

with low angle lines, like for BTO, the maximum random error acceptable for obtaining the

correct solution is less than for the cadmium

compound, but it is still large compared with

the quality of data that it is possible to obtain

by modern powder diffractometry. However, it must be added that this interesting result should

not be overemphasized because, with increasing

error bounds, the risk of obtaining false solutions is considerably higher. It should be noted

that the de Wolff figure of merit is greater than

20 for maximum random errors less than 0.06°

and 0.023°(2^) for cadmium hydroxide nitrate

and barium titanyl oxalate respectively.

(ii) Systematic errors An example of a systematic error can be simulated by a shift of the zero 26 position. Data

degraded by a zero-point error were used for the

pattern of the above materials and again input

to DICV0L91. The effect on the figures of merit can be seen in figure 3 for BTO. As for the ran-

- 0.03

- 0.02

0.05 0.10 Zero point error °(29)

0.15

Figure 3: The effect of zero-point error [°(20)] on

the figures of merit (FoM) M20 (o) and F20 (+) and

on the average angular discrepancy (2^) (°) (x), for

barium titanyl oxalate hydrate (BTO).

dom error case, the 'correct' solution is obtained

for quite large zero-point errors, with the obvi- ous condition than the error bounds in every

line tolerated by the program are similar to the

angular shift of the zero point. Indeed, the so- lution is found for shifts as large as 0.15° and

0.30°(2^) for BTO and the cadmium compound,

respectively. Again, the successive dichotomy

method would not be too sensitive to this sys- tematic error. Figures of merit M20 greater than

20 are obtained for zero point errors of 0.12° and

0.07°(2^) for the cadmium and BTO compounds

respectively. It should be noted that the pres- ence of a zero-point error is easily detected from

the refined solution by the angular discrepancies

A2^i (= 29iobs — 2^ica?c) displaying a systematic

bias as Bragg angles increase.

From these studies, it appears that, for indexing a pattern, the error limit for every diffraction line

could be greater than 0.03° (2^) when the successive
D. Louer 99

dichotomy method is employed. However, again this

point should not be overemphasized, because it also

increases the risk of finding pseudo-solutions and of

missing the correct solution. A result with a high

figure of merit always inspires confidence, and that

is the consequence of a correctly adjusted instrument.

7. Data collection strategy for indexing

The most frequent reasons for failure to obtain a cor- rect solution are inaccuracy and incompleteness of

data and also the presence of spurious lines due to

impurities in the sample. The presence of a second

phase in fact implies a mixture of two patterns, with one usually having weaker reflections. The problem

of indexing ab initio the two patterns simultaneously

seems, in the author's opinion, unsolvable by automatic indexing methods. Some programs will tol- erate a few spurious lines; nevertheless, unindexed

diffraction lines should always be explained by a

chemical reason or by imprecision. In order to en- sure a successful outcome of a powder pattern indexing, it is recommended that accurate data from an

instrument carefully adjusted or calibrated be used.

The user of automatic indexing programs is urged to

consider this fact when obtaining the 26 positions.

As a consequence, a number of critical checks of the

instrument adjustment must be made prior to collecting data. High instrumental resolution and accu- racy are of paramount importance. Higher resolution

allows a less ambiguous decomposition of the powder diffraction pattern, and, although overlapping re- flections are not eliminated, the overlap is reduced.

High resolution was a common feature of the instruments used by the first workers to practice powder

pattern indexing. In particular, monochromatic ra- diation coupled to a careful calibration of a Guinier

camera was used by the Dutch (de Wolff and Visser)

and the Swedish (Werner) teams, while the French

group (Louer) collected accurate data by means of

conventional powder diffractometer with monochromatic X-rays. Because diffractometers, with BraggBrentano geometry, are the instruments commonly

employed in powder diffraction, it is of interest to

know the precision of peak position that can routinely be achieved. Also, high resolution powder

diffractometers with synchrotron radiation have at- tractive possibilities. Some features of these two

currently employed instruments are now presented

briefly.

(i) Resolution

The Instrumental Resolution Functions (IRF),

i. e. variation of instrumental line profile

FWHM with diffraction angle, have been described by Louer and Langford (1988) for a con- ventional diffractometer with an incident-beam

monochromator and by Langford, Cernik and

Louer (1991) for the 9.1 High-Resolution Powder Diffractometer at the SERC Daresbury Laboratory. The IRF curves for flat samples have

been discussed for slightly different wavelengths,

1.5405981 A and 1.4188 A. For the conventional diffractometer, the IRF has a minimum

of 0.065°(26') at about 30* and has twice this

value at 130''(26i); a minimum of 0.028°(26') is obtained at 30° and twice this value at 100° (2^)

for the 9.1 HRPD. In both cases, instrumental

line profiles can be modelled, to a close approximation, by a single analytical function. The

benefit of high resolution can be seen in figure

4, giving a limited range of patterns registered

with the two instruments, for a triclinic zirconium hydroxide nitrate. Partly resolved peaks,

frequently vital for successful indexing, are more

easily detected with higher resolution, and the

separation of close lines, by fitting techniques,

becomes more precise. Part of this precision is

lost if the Kaidg doublet is employed.

(ii) Accuracy

Random errors are minimized by using fitting

methods for locating peak positions, but systematic errors need to be reduced by a careful alignment (or calibration) of the instrument.

Because data for indexing collected by means of a powder diffractometer are often regarded with

suspicion due to zero-point errors, it is of interest to present some results that can be obtained

with a Bragg-Brentano geometry and monochromatic radiation, whose features are given else- where (Louer, 1991). Before collecting data, two

kinds of control are recommended:

• The zero-2^ position, which must be close

to the true zero, can be established by

means of parallel glass-plates located at the

centre of the goniometer rotation axis. In general, the value of the zero error can be

less than 0.005°(2^).

The accuracy of the angular position of

the entire goniometer circle must be evaluated. Indeed, the biggest sources of ex- perimental error are due to specimen displacement and, to a considerably less ex- tent with modern diffractometers, misset-
100 Automatic Indexing: Procedures and Applications

{a)

u ; u

<18.000 TWO-THETA 29.000>

(b)

\3

<16.500 TWO-THETA P6.500>

Figure 4: Partial angular ranges of the powder

diffraction pattern of zirconium hydroxide nitrate obtained by means of high-resolution powder diffractometers using (a) a conventional X-ray source with

monochromatic radiation (A = 1.5405981 A) and (6)

synchrotron radiation (A = 1.4188 A).

ting of the 9/20 adjustment. This can be easily evaluated by using standard ref- erence materials; for this purpose highly

oriented crystallites of a layer-type material, deposited in a thin layer on a silicon

plate to reduce transparency effects, can be used for checking the adjustment of the

diffractometer. For example, the synthetic

fluorophlogopite mica (SRM 675: Natl.

Bur. Stand., 1982) is suitable for diffractometry by reflection; the basal spacing is d(OOl) = 9.98104 ± 0.00007 A. With a high

degree of preferred orientation of crystallites, ten 00/ reflections are observed in the

angular range 8-135° 20 for A = 1.54059

A. Each diffraction line must produce an

identical value of the interlayer spacing

d(OOl) = /xd(00/). An example of calibration is shown in figure 5, where the angular

discrepancies between observed and calculated 20 angles are given for ten diffraction

lines. It demonstrates that the angular er-

•0.01

A

^ o

o . o o

- 50

o o

100 o

Figure 5: A2^(°) discrepancies between 20 angles

calculated and observed peak positions for 00/ reflec- tions of SRM 675.

ror on peak positions is less than 0.007" (2^) over the entire goniometer circle. At low

angles, axial divergence can produce asymmetry and a shift of lines; this effect is re- duced by using Soller slits located in the

diffracted beam. The use of high orders of a reflection (if available) can be used for a correction of the positions of low-angle

peaks. This practice can be useful when

methods are very sensitive to the precision

of low-angle lines.

With a correct adjustment of the diffractometer, there is no need to use an internal standard

material. Of course, the front surface of the sample has to be very smooth and attention has to

be paid to the thickness of the sample in case of materials with only light atoms. In current

practice, the absolute error on peak positions can be less than O.O1°(20) for well crystallized

materials.

{in) Preparation of data for indexing programs

In modern powder diffraction it is a current

practice to use fitting techniques to decompose

the pattern into Bragg components. Some efficient programs are commercially available and

can be used with PCs. In particular, a zoom option is useful to examine low intensity lines. The

preparation of data should include the following

information for every Bragg component: peak

position, relative intensity, FWHM and an 'as- pect' indicator (e.g. asymmetry, shoulder, etc.).
D. Louer 101

Table 1: M20 and F20 figures of merit for indexed powder diffraction patterns, by DICV0L91, from data with

different origins : photographic recording (a), conventional diffractometer with monochromatic radiation

(A = 1.54059 A) (6) and 9.1 HRPD (A = 1.40394 A) (c).

Compound Reference Symmetry Measure type Mon

a-Cs2Cdl4 TLL monoclinic a 23 39(0.011,46)

/?-Cs9Cdl4 TLL orthorhombic u 20 37f0 009 601

NdfOH^oNOqHoO LDHG monoclinic a 34 48f0 008 511

KCaPO/iH^O LDO monorlimr X X X V-'XXV^ ^X X X 1. X v.- Q 19 28(0 012 581

CL b 73 107(0.0038,49)

CL c 156 260(0.0016,47)

La(OH)2N03H20 LLLG monoclinic b 44 67(0.0061,49)

CL c 86 129(0.0032,49)

Cd5(OH)8(N03)2-2H20 BL monoclinic b 82 151(0.0050,25)

Ba(TiO)(C204)2-4H20 LBGC monoclinic b 51 173(0.0036,32)

CL c 76 248(0.0025,32)

Zr(OH)2(N03)2-4H20 BLL triclinic b 54 112(0.0059,30)

CL c 295 635(0.0012,26)

PbFa CL orthorhombic b 105 113(0.0052,34)

CL c 699 821(0.0006,34)

TLL: Touchard, Louer & Louer, (1986); LDHG: Louer, Deneuve, Herviou & Gourlaouen, (1986); LDO:

Louer, Deneuve & Ouillon, (1987); CL: Cernik & Louer, (1993); LLLG: Louer, Louer, Lopez Delgado &i Garcia Martinez, (1989); BL: Benard & Louer, (1990); LBGC: Louer, Boultif, Gotor k Criado, (1990); BLL:

Benard, Louer & Louer, (1991).

In a first indexing attempt, the first twenty lines,

not including (less precise) lines with very low

intensities, should be used as input data. In a second run the lines with low intensity must be

incorporated in input data. If the same solution is obtained, then cissurance that the indexing is correct is increased. This strategy can be

slightly different if programs tolerate spurious

lines.

{iv) Precision of indexing solutions

The fundamental stage for indexing a pattern

is the measurement of peak positions. As ex- plained previously, the accuracy is related to the

resolution of the instrument, its adjustment, the

degree of diff"r action line overlap, and the technique used for measuring the peak positions.

A few years ago, graphic recording of powder

patterns was frequently used, and angular positions were taken as the maximum intensities observed. These measures (type a) are obviously

biased in case of line overlap. In modern powder crystallography, step scanning data collection combined with fitting methods gives better

estimates of the peak positions. However, the

accuracy still depends on the instrument resolution function. In table 1 some indexing results

obtained from experimental measures of type a are compared with those obtained by means of a

fitting technique applied to data collected with a conventional diff'ractometer with monochromatic radiation (type 6) and with data collected

by means of 9.1 HRPD (type c). Figures of

merit are significantly greater for data obtained

by fitting procedures. The mean of A26 is 0.01",

0.005° and 0.0018°(2^) for measures of type a,

b and c, respectively. These results show the

advantage of using synchrotron radiation for ab

initio pattern indexing. However, data collected

by means of diffractometers with conventional

sources and monochromatic radiation lead to excellent results; compared with synchrotron data,

the precision of the indexing is lowered by a factor varying from 2 to 7.

8. Reliability of indexing solutions

Although the success rate of computer-based indexing programs is very high, to be accepted with a

higher degree of assurance the proposed (mathematical) solutions must satisfy several conditions, which

can be summarized as follows:
102 Automatic Indexing: Procedures and Applications

• High values of the figures of merit Mat and Fjv

(values of M20 greater than 20 are a good indication).

• Since the calculations are usually based on the

first twenty lines, all remaining diffraction lines

in the pattern must also be indexed, within the

error bounds. For this purpose, it is convenient to review the complete observed dataset by

means of the computer program NBS*AIDS83

(Mighell, Hubbard k Stalick, 1981). A benefit

of this careful review is an easier derivation of

the possible space groups.

• If the density of the material has been measured

accurately, an integral number of chemical formula units per unit cell should be obtained.

• If necessary, the use of programs, e. g. CDFSRCH, can be employed to find the normalized

unit cell parameters. A subsequent advantage

of this analysis is the possibility of interrogating the NIST Crystal Data Base (or another

database) to search for apparently isostructural

chemically-related materials.

• Finally, due to recent advances in ah initio struc- ture determination, a successful determination

of the crystal structure of the material would

be an elegant check on the validity of the cell obtained by indexing.

9. Discussion of particular cases

The presence of a dominant zone in a pattern introduces a complementary degree of freedom, which

can cause problems, with the consequence of possible pseudo-solutions with high figures of merit. Also,

the indexing of unit cells with large volumes has al- ways been regarded with suspicion, although there

is no mathematical limit for an indexing provided

that the information contained in the pattern is sufficiently complete and accurate. A few examples are now discussed.

Cases with a short dimension

The dominant zone is due to one of the cell edges

being significantly shorter than the other two, as can

occur for materials with a layer-type structure. The

pattern is then characterized by a zero value for one common Miller index for all the first lines of the pattern. This feature can be a problem for methods us- ing base lines for generating solutions and requires

Table 2: Examples of patterns with a dominant zone,

indexed by DICV0L91 ; N : number of successive

low-angle diff'raction lines having a common zero index.

Compound Reference N

La(OH)2N03-H20 LLLG 6 lines with k — 0

Nd(OH)2N03-H20 LDHG 7 lines with k = 0

Cd3(OH)5N03 APL 8 lines with h = 0

Zn(OH)N03-H20 ELW 13 lines with k = 0

LLLG : Louer, Louer, Lopez Delgado & Garcia Martinez, 1989 ; LDHG : Louer, Deneuve, Herviou k,

Gourlaouen, 1986 ; APL : Auffredic, Plevert & Louer,

1990 ; ELW : Eriksson, Louer k Werner, 1989.

further consideration; in order to solve this problem eff'ectively, special algorithms for finding possible dominant zones must be incorporated in these

approaches (Werner et al., 1985). The successive dichotomy method is, by definition, not sensitive to

this situation, unless the number of lines with a common zero-index is exceptionally high. Examples of

patterns indexed by DICV0L91 are shown in table

2. The last example, Zn(OH)(N03) H20 contains 13

first lines with ^ = 0, which caused some instability, as is shown by the three similar solutions [694 A^,

M20 = 17; 728 A3, M20 = 16; 757 A^, M20 = 18]

obtained from the 20 first lines and an absolute angular error of O.O3°(20). To select the correct solution, more lines are necessary, or it is better to re- view the complete observed dataset by the program

NBS*AIDS83, which reveals that only the last is correct. Moreover, since these data, collected by means

of a Guinier-Hagg camera, are accurate, the reduction of the error bounds from 0.03 to 0.01°(2^) leads

only to the correct solution. This again demontrates

again the importance of accuracy as a stabilising parameter in indexing problems.

Cases with a large unit-cell volume

If enough crystallographic information is present in

the powder diff'raction pattern, high instrument res- olution, possibly coupled with a judiciously selected

wavelength, is required to solve the problem of high

line density at low angles. For very high volumes

{e.g. greater than 4000 A^), synchrotron facilities

(resolution, accuracy and tunability of the radiation) are a decided advantage. However, examples

using data collected with conventional instruments,

have been reported. Among them are the indexing of the patterns of monoclinic halotrycite [V =

3152A3, M20 = 15, F20 = 35(0.006,98); Werner,
D. Louer 103

1980] and barium titanyl oxalate hydrate (BTO)

[V = 2595A3, M20 = 46, F20 = 107(0.0056,50);

Louer et al., 1990]. The benefit of using synchrotron

radiation for the indexing of patterns of materials

with large unit cell volumes can be appreciated with

the powder pattern indexing of the zeolitic material Sigma-2 (A = 1.5468A) [V = 3604A3, McCusker, (1988)]. For testing the successive dichotomy

method, data for two powder patterns, collected on

the 9.1 HRPD, were entered in DICV0L91. The first sample was the zeolite ZSM-5 (A = 1.4963A), and

the solution found is orthorhombic [a = 20.096(6)A,

b = 19.955(5)A, c = 13.433(4)A, V = 5384A3,

M20 = 24, F20 = 69(0.0060,48)], in accordance

with a single-crystal study (Olson, Kokotailo, Lawton &: Meier, 1981). The second sample was a synthetic gallophosphate with very large unit cell volume studied by Estermann, McCusker, Baerlocher,

Merrouche & Kessler (1991); data were collected

with the wavelength A = 1.70469A to have better res- olution and accuracy at low angles. From the first 28

reflections input to DICV0L91, a cubic solution was

found [a = 51.715(2)A, V = 138309A3, Mas = 32,

F28 = 105(0.0013,212)], in agreement with the re- ported results.

10. Conclusion

Computer-based methods are nowadays powerful for

indexing powder diff'r action patterns, whatever the

crystal symmetry, provided that data are accurate.

Recent developments in high-resolution diffractometers using conventional or synchrotron sources offer the possibility of obtaining the required precision

routinely. This represents important progress, since

the indexing of a powder pattern is required in most

applications of powder diffraction. It has become a method for the unique characterization of a single

phase, and reported data for a pure material with an unknown structure should always be as precise as possible and analysed by indexing methods, unless a good reason is given; from such a practice the quality

of databases would be increased considerably. Recent applications in powder diffraction demonstrate

the fundamental role of indexing a powder pattern.

For example, its power in materials characterization can be seen by the indexing of the pattern of baryum

titanyl oxalate hydrate, used as a precursor in the

synthesis of high-purity barium titanate. The pattern was long described as complicated and used ex- tensively only for qualitative identification. From

accurate data, the pattern was easily indexed with

the cell given in Table 1, showing that this material

was in fact a pure phase. Another recent spectacular consequence of powder pattern indexing is the

rapid development of ab iniiio structure determination from powder data. The reconstruction of the

reciprocal lattice by means of indexing methods is a

crucial and inevitable stage for a subsequent struc- ture analysis, from which the structural chemistry of

materials will benefit greatly in the near future.

The author wishes to thank Prof. P.-E. Werner

and Dr. J. I. Langford for their many suggestions

and critical reading of the manuscript, Dr. C. Baerlocher, who provided the synchrotron powder diff"rac- tion data for the two examples with large unit-cell

volume, and Dr. R. J. Cernik, who collected synchrotron data for a few samples.

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National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Accuracy in Pair Distribution Function Analysis for Crystalline and

Non-Crystalline Materials

B. H. TOBY^ AND T. Egami^

^Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, PA 18195, U. S. A

and

"^Department of Materials Science and Engineering and

Laboratory for Research on the Structure of Matter,

University of Pennsylvania, Philadelphia, PA 19104-6272, U. S. A.

Abstract

Pair distribution function (PDF) analysis of neutron

and x-ray powder diffraction data is useful for analysis of short-range structure in both amorphous and

crystalline materials. The sources for errors in PDF

determinations are explored. The significant error

sources: termination of the Fourier transform, inability of the instrument to completely resolve the

diffraction pattern, inaccurate corrections for experimental artifacts and counting statistics are discussed

in detail. Methods for estimating these errors are presented. With care, PDF's can be computed very

accurately, as demonstrated by the excellent agreement shown for a standard sample.

Introduction

The atomic Pair Distribution Function (PDF) has

been traditionally used for structural studies of noncrystalline materials, where crystallographic analysis cannot be applied (for example see Cargill, 1975,

or Wagner, 1972). More recently, analysis of PDF's

obtained for crystalline materials has demonstrated

features not reflected in the crystal structure (for ex- ample see Dmowski et ai, 1988, or Toby et ai, 1990).

While much attention has been applied to understanding the accuracy and precision of the crystallographic results, relatively little attention has been

given to understanding and evaluating the precision

and accuracy of PDF analysis.

The PDF is determined by direct Fourier transformation of neutron or x-ray powder diffraction data,

after application of corrections and shows the probability of finding a pair of atoms at a given interatomic

distance (Warren, 1968; Klug and Alexander, 1968).

The PDF is similar to a spherically averaged Patter- son map. However, for Patterson analysis only Bragg

scattering is used, while the PDF also includes dif- fuse scattering. Thus, the PDF is sensitive to local

order while the Patterson map, and indeed all techniques which only utilize Bragg scattering, can only

reflect the long-range symmetry in a material.

It is commonly and erroneously thought that the

PDF is an approximate technique and the PDF is incapable of observing structure beyond the first few

coordination spheres. The PDF of polycrystalline

aluminum shown in figure 1 refutes that belief. In

this figure, the experimental result is superimposed on the expected result computed from the known

crystal structure. It should be noted that the observed and computed PDF's are in excellent agreement and the PDF clearly shows structural features

out to at least 40 A. This paper will quantify the factors that limit the accuracy of the PDF and will also

show how to obtain the minimum statistical error for a given data collection time by a strategic choice of

the scan rate.

Background

The intensity for scattering of a^probe particle usually written as I{Q,uj) where Q is the momentum

transfer vector, Q = kf — ki, and u! is the energy

transfer, u = Wi — uj, (subscripts i and / denote

the incident and final momentum or energy, respectively). If diffracted intensity is detected without

energy discrimination but at fixed Q, I(Q,w) will be

integrated over all u. This yields a PDF indicative

of the instantaneous correlations of atomic positions.

^(^) = ^ E ((«^p I'Q - Mt)]] ))

where double angle brackets (((...))) indicate a time

average, single angle brackets indicate a composi105
106 Data Pair Distribution Function Analysis

a)

0.6

0.4-

0.2

r. (A)

Figure 1: An observed (solid line) and model (dotted line) pair distribution function for polycrystalline

Al powder measured at 50 K over the range 0-20 A

(fl) and 20-40 A (6). The lower box shows the deviation between the observed and calculated results

(solid line) and plus and minus one estimated standard deviation computed from the counting statistics

(dashed line). The lower box contents are magnified

4:1 for (a) and 2:1 for (6).

tional average, bj is the scattering length for the

jth atom, and the summation is performed for all pairs of atoms in the sample (Lovesey, 1984; Warren, 1969; also see Toby and Egami, 1992 for a more

complete discussion). It should be noted for detection at fixed 20, Q will vary with u>, since Bragg's

law, Q = \ Q \ = AnsinO/X, is true only for elastic

scattering. Thus in practice, data collected without energy discrimination will not directly measure

7(Q) since Q is not constant as u changes for constant 9. The Placzek approximation is customarily

applied to correct for this shift in Q with oj (Placzek,

1952). In addition, intensity contributions from multiple scattering and attenuation due to absorption,

which depend on the experiment geometry, have been

neglected here. Corrections for these factors as well as polarization, background and scaling for the flux

incident on the sample must be applied to the observed intensities.

For powder diffraction, a sample composed of a

large number of randomly oriented crystallites effec- tively averages the measurement over all orientations

of Q. The powder-averaged scattering intensity is usually defined as the powder-averaged structure factor, S{Q),

<6>2 <6>2

Note that the definition of the structure factor used

here differs from the definition for the structure factor, Fhki, used in crystallography. For an ideal crystal, S{Q) can be expressed in terms of the crystallographic structure factor, Fhki,

SiQ) =

1 Q+SQ

2hkh

where the sum is over all reflections with momentum

transfer between Q and Q + 6Q [Q = 27r(/ia* -|- kb* +

/c*)], Vc is the unit cell volume, ric is the number of

atoms in the unit cell. The PDF, p(r), for neutron diffraction is defined

for the instantaneous structure as

bjbk

N ^<b>^ {{Hr-m)-rkm )).

The PDF may be obtained from the powder-averaged

S{Q) using

P(0 -Po = 7rr [q [S{Q) - 1] sin Qr dQ.

The equations given above are strictly accurate for

neutron diffraction, but are approximate for x-ray

diffraction where the scattering factor, fj{6), is substituted for the neutron scattering length, bj (Warren

et al, 1936).

The PDF for a collection of atoms may be simulated by counting the interatomic distances between

every possible pair of atoms, weighted by bj for each

atom. This delta-function PDF is then broadened to

simulate thermal motion. For comparison between

observed and simulated PDF's, an agreement factor,

A, is defined in a manner analogous to Rietveld analysis,

1 "

A"^ = 2 [Pobs(ri) - Pmodej{ri)f .
B. H. Toby and T. Egami 107

Agreement factors in the range of 0.05 to 0.1 represent very good fits with only minor discrepancies

visible to the eye. The agreement between the observed and simulated PDF's given in Fig. 1 are 0.156

and 0.045 over the ranges 2-20 A and 20-40 A, re- spectively.

Error sources in the PDF measurement

All diffraction measurements are subject to random

statistical errors. The following section will demonstrate how to estimate these errors. In addition there are three obvious sources for systematic errors in the

PDF: (1) termination of the Fourier integration at a finite value of Q, (2) instrumental broadening of

the diffractogram, (3) inaccuracies in the experimental corrections applied to the diffraction data. These

factors will also be discussed below.

Counting Statistics

All diffraction measurements either directly or indirectly count quanta and are subject to statistical

fluctuations where the estimated standard deviation

(e.s.d.) is proportional to the square- root of the

number of quanta counted. To determine the e.s.d. for the observed PDF, one can propagate errors from

the intensity measurements to the computed PDF

values.

The estimated errors for a set of observations may

be propagated to estimate the error on a function

determined from these observables. In general form,

the e.s.d. for a function, f, evaluated using a set of independent observations, 0i,02,...0n, can be

estimated from the e.s.d. of each of the observations

using,

computed from

N = E <TOj dOj

(Hamilton, 1964). If the S{Q) values are computed

independently, e.g., no observation is used to compute more than one S(Q) value, then the S{Q) val- ues may be treated as an observable. The e.s.d. for

the S{Q) values, (Ts^q), is then.

M

( dS{Qk)

V dOj

where M independent intensity measurements for

sample, background, container scattering, etc., are averaged for each S{Q) value. Note that smoothing removes that independence of S{Q) values and

should be avoided. The e.s.d. for p(r) may be then

(27r2r)'

YlWs{Q^)QkAQk sin Qkrf

The dashed lines in the lower box of Fig. 1 show

the e.s.d. for the observed PDF. These small e.s.d. 's were obtained for only two hours of data collection

time for a relatively weak scatterer of neutrons using

the Argonne National Laboratory SEPD instrument

when the IPNS enriched uranium target was in use.

Systemaiic Errors

To quantify the effects of termination and instrumental broadening on the PDF, the ideal total structure

factor, S{Q), was computed for a FCC metal and an

ideal PDF was then computed from this S{Q). Experimental effects were then simulated by appropriately modifying S{Q) and repeating the PDF computation.

Termination of the Fourier integral at finite Q is equivalent to multiplying S{Q) by a step function.

This is equivalent to convoluting the PDF with the

Fourier transform of the step function. Indeed, a damping function is commonly applied so that the

convolution function appears more Gaussian. Thus,

the degree that termination will introduce error into

the PDF depends on two factors, the amount of thermal motion (including zero-point vibration) and the

intrinsic sharpness of the PDF peaks due to the range

of interatomic distances in the material. The effect

of termination on the simulated PDF was studied as a function of termination and thermal motion conditions, and the results are shown in figure 2. The

smallest values shown for the thermal amplitude,

0.05 and 0.06 A, which are equivalent to isotropic

temperature factors, B, of 0.1 and 0.14 A^, respectively, correspond to the estimated zero-point vibration for Ni and Cu, respectively {International Tables, 1968). The remaining three a values, 0.084,

0.100 and 0.153 A (B = 0.28, 0.39 and 0.92 A^),

correspond to the estimated vibration amplitudes for

crystalline aluminum at T = 0, 100 and 295 K {Ibid.).

Except at very low temperatures, most materials exhibit thermal vibration greater than that for Al at 0

K, so termination at 30 A~^ will be sufficient for ac- curate determination of the PDF. It should be noted

that these values represent upper limits for termination errors. Most crystalline materials and particularly amorphous and quasicrystalline materials tend

to have much broader features in their PDF's than

FCC metals and their truncation errors are less se- vere. The broadening of peaks due to lack of instrumental resolution can seriously limit the usefulness
108

Termination Effect on PDF for FCC

Data Pair Distribution Function Analysis

Peak Broadening with AQ/Q Constant

00d

o

o

c

0)E

<

CDd

1

1 1

' Thermal Motion.

_ A CT=0.05 A

act=O.Ud a

CT=0.084 A

. A

xcj=0.10 A

- X i)(CT=0.153 A . X

A

X

X

A

A

1> 9

20 40

Qmox. A

-1

Figure 2: The amount of error introduced into the

PDF due to termination of S{Q) with different values

of the Debye-Waller temperature factor. Agreement

is computed over the range r = 2 to 20A by compari- son to a PDF modeled with appropriate broadening.

of diffraction data for conventional crystallographic

analysis. In contrast, instrumental resolution has

much less impact on PDF computation. In the simplest case, consider an instrument with a Gaussian

response function that broadens all peaks by a constant amount in Q, AQ. This convolution of 5(Q) is equivalent to multiplication of the PDF by a damping

function decreasing with r. For moderate broadening

values, AQ, of 0.05, the damping of the PDF will be

negligible for r below 20 A. For larger values of AQ,

an exact correction may be applied to the PDF. The

only loss due to the instrument response function is a decrease in the signal to noise ratio in the PDF at large r. For instruments where the broadening in- creases with Q, for example pulsed sources, broadening is not a simple convolution of S{Q). The results

for a simulation with Gaussian broadening, increasing with Q, are shown in figure 3. For reasonable

broadening values, AQ/Q of 5%, again the effect on

the PDF is slight. Correction, either by broadening the PDF at large r, or by multiplication of the

model PDF by a damping function reduces the error

further. Consequently, while lack of instrument res- olution may limit the range and utility of diffraction

data for crystallographic computations, lack of instrumental resolution seldom impedes PDF analysis

and can often be compensated by improving statistics and application of corrections to the PDF.

Inaccurate corrections for absorption, multiple

oqd

o CD .P o

ce

0)0)

l_ <

d

- 1

' ' 1

1

' 1 1 '

1

' 1 1 1 1 1 1 . 1 1 .

Uncorrected -

Corrected * * * * *

*0

*

9, o o

. 1 . o

o o"

. 1 .... I

AQ/Q. %

Figure 3: The agreement factor over the range r =

2 to 20 A between a model PDF and one computed

from a model S{Q) broadened with a instrument re- sponse function increasing with Q. Data indicated by

squares are uncorrected and those shown with asterisks and circles are after correction by an r-dependent

broadening term and multiplication by a Gaussian

envelope function, respectively.

scattering, inelastic or Placzek scattering, sample

container and background scattering can also introduce systematic error into the PDF computation.

These effects typically vary slowly with Q. Thus, any

deviations between the computed corrections and

the accurate values should also be broad, smoothlyvarying functions of Q. Adding such a slowly varying error function to S(Q) will add to p{t) an error function with rapid oscillations that will decrease in

magnitude as r increases.

The errors in the PDF for aluminum given in Fig. 1 are approximately three times larger than the ex- pected statistical error, even after compensation for

instrumental broadening. Termination errors do not

account for more than a third of this error either.

This leads to a surprising conclusion that the errors

in the PDF due to inaccuracies in the experimental corrections must be of comparable magnitude to

termination and statistical errors. Performing PDF

computations on well standardized samples is important for characterization of the accuracy of the analysis software.

Optimizing data collection to reduce errors

In practice, access to synchrotron and neutron
B. H. Toby and T. Egami 109

sources is very limited, and it is very important to obtain the most accurate result for a given total amount

of time for a diffraction measurement. Defining a

scan rate function as R{Q), where R{Q) is determined by the source intensity, instrument configuration and is proportional to the count time for each

point. Thus for a given instrument and a fixed period of time for the complete measurement, the sum

of R{Q) must remain constant, but the time for an

individual point may be increased or decreased. The

number of counts for an individual measurement will

be given by

NobsiQk) = RiQk)^QkliQk).

If the contributions to the e.s.d.'s from experimental corrections such as background subtraction are considered invariant with Q, and S(Q) and R{Q) are treated as continuous functions, one obtains

Further simplification is possible using the approximation that sin^Qr averages to 1/2 over a wide range

in Q yielding

SiQ)Q'

RiQ) < b >2 dQ.

The optimum scan rate, R{Q), to yield the smallest

e.s.d.'s will be obtained when

RiQ) DC <b> <6>2 '

because

dQ

In contrast, any other scan rate, such as scanning

at a constant rate or waiting for a constant number

of counts at each point will yield less accuracy for a

given data collection period.

Conclusions

PDF computation is considered by most crystallographers to be inaccurate, a method of last resort. This conclusion is in error. In fact the PDF may

be determined quite accurately with modern instrumentation. The PDF is less sensitive to experimental

broadening than crystallographic techniques that fail when diff'r action peaks may no longer be resolved.

This work does show that there is room for improvement in the methods used for computation of experimental corrections in S{Q) computations, now that

the experimental state-of-the-art has caught up to the software. It should be routine practice to verify the accuracy of software by detemination of the

PDF for an appropriate standard. Finally, inclusion

of e.s.d.'s with all published PDF's should become

routine practice, just as inclusion of e.s.d.'s has become routine for crystallographic computations.

Acknowledgments

Work at the University of Pennsylvania was supported by the National Science Foundation through

Grants DMR90-01704 and DMR88-19885. The In- tense Pulsed Neutron Source is operated as a user facility by the U.S. Department of Energy, under Contract W-31-109-Eng-38. We also thank Dr. Diane P. Toby for proofreading this paper.

References

Cargill, G. S. Ill (1975). Solid State Physics, 30, 227-

320.

Dmowski, W., Toby, B. H., Egami, T., SubramaNiAN, M. A., Gopalakrishnan, J. & Sleight, A.

W. (1988). Phys. Rev. Lett. 61, 2608-2611.

Hamilton, W. C. (1964). Statistics in Physical Science,

pp. 32-34, New York: Ronald Press. International Tables for X-Ray Crystallography (1968).

Vol. III. Birmingham: Kynoch Press. Klug, H. p. & Alexander, L. E. (1968). X-Ray

Diffraction Procedures for Polycrystalline and Amorphous Materials, 2nd ed. New York: Wiley.

LovESEY, S. W. (1984). Theory of Neutron Scattering

from Condensed Matter, Volume 1, Oxford: Clarendon

Press. Placzek, G. (1952). Phys. Rev. 86, 377-388.

Toby, B. H., Egami, T., Jorgensen, J. D. & SubraMANIAN, M. A., (1990). Phys. Rev. Lett. 64, 2414-2417.

Toby, B. H. & Egami, T., (1992). Acta Cryst. A48, In

press. Wagner, C. N. J. (1972). Liquid Metals, Chemistry and

Physics, edited by S. Z. Beer, pp. 257-329. New York:

Marcel Dekker.

Warren, B. E., Krutter, H., & Morningstar, O.

(1936). J. Am. Ceram. Soc. 19, 202-206.

Warren, B. E. (1969). X-Ray Diffraction. Reading:

Addison-Wesley; also, New York: Dover.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

The Use of the Voigt Function in Determining Microstructural Properties from

Diffraction Data by means of Pattern Decomposition

J. I. Langford

School of Physics & Space Research, University of Birmingham, Birmingham B15 2TT, UK

Abstract

Techniques for studying the microstructural properties of polycrystalline materials by applying pattern

decomposition to complete diffraction patterns have

advanced substantially since the first conference on

Accuracy in Powder Diffraction was held in 1979.

In particular, there have been a number of improvements to the method of line-profile analysis based on the Voigt function. The basic concepts of this approach, procedures for applying it and the limitations

of the method are reviewed, and developments during

the intervening years are illustrated by considering

specific applications. The review is self-contained,

in that all formulae required to study microstructural properties by means of the Voigt function are included.

1. Introduction

When the procedure based on the Voigt function for

analysing diffraction line broadening in terms of microstructural properties Wcis introduced (Langford,

1978), shortly before the first conference on Accuracy

in Powder Diffraction, powder crystallography was

enjoying something of a renaissance. This had largely

been brought about by the advent, a decade or so

earlier, of the Rietveld method for the refinement of

crystal structures by means of powder data, but a

contributing factor was the introduction of patterndecomposition programs in the mid 1970s, whereby

a non-structural model is fitted to a complete diffraction pattern to obtain parameters describing the individual Bragg reflections. However, the method

was then only applicable to relatively straightforward

cases; nowadays, with more intense sources, instruments with higher resolution, greatly improved soft- ware and a better understanding of peak shapes, a wealth of information about the microstructure of

polycrystalline materials can be obtained. Even for

fairly complex patterns with peak overlap, the mean

size and shape of crystallites (coherently diffracting

domains) can be determined, various structural 'mistakes' can be studied, and the directional dependence

of microstrains and other 'lattice distortions' can be

obtained. It is thus timely to review the role of the

Voigt function in characterising these quantities and

in other applications of powder diffraction.

A brief resume of the properties of the Voigt function will be given, followed by its use in patterndecomposition methods. A procedure for dealing

with instrumental effects will then be considered,

and techniques for carrying out line profile analysis

based on the Voigt function will be discussed. The

last will be illustrated by considering specific cases of sample broadening. It will be demonstrated that,

if sufficient reflections are recorded, data are of adequate quality, and a systematic analytical procedure

is followed, then meaningful and precise information

about sample imperfections can be obtained by the

Voigt approach. Examples will be based on data

obtained from fixed-wavelength radiation (CuKai,

A = 1.540598 A), angle-dispersive experiments, but

the principles involved apply equally well to energydispersive data.

2. The Voigt function

The Voigt function, the convolution of Lorentzian

and Gaussian functions, can be represented by

(Langford, 1978, equation (18))

I{x) = {P/Pg)Io^W{z)}, (1)

with

Z = yfUxj^G + (2)

and

y = pL/{^l3G\ (3)

where is the peak intensity, /? is the integral

breadth (= area//o), Pl and Pa are the integral

breadths of the Lorentzian and Gaussian components, and Re{w(z)} is the real part of the complex

error function.

A Voigt function can be characterised by a shape

parameter <j>, defined as the ratio of the full width at

110
J. I. Langford 111

half the peak intensity (FWHM or 2w) to the integral

breadth, or

<f> = FWHM/yS. (4)

From the definition of the Voigtian, it follows that

the convolution of two or more such functions is also a Voigt function. The main reason for its introduction as a means of investigating the nature of diffraction line profiles is that Pl and /3a can readily be

determined from /? and (f) {vide infra, (11) Sz (12)),

and that the breadths of the constituent line profiles can then be separated in the usual way; i. e., from

the additive property of the breadths of Lorentzian

functions [e. g., (14)] and the squares of the breadths

of Gaussian functions [e.g., (15)]. The Lorentzian

and Gaussian functions are clearly limiting cases of

the Voigtian; from Langford (1978), for the former

<f> = 2/ir = 0.6366, (5)

and for the latter

<t> = 2V(ln2)/7r = 0.9394. (6)

2.1 Applicabiliiy of the Voigt approximation

There is no more theoretical justification for modelling diffraction line profiles by means of the

Voigtian than there is for other analytical functions

commonly used in line-profile analysis, except perhaps in the case of the instrumental contribution

(Louer & Langford, 1988; Langford, Cernik & Louer,

1991). However, in practice this function frequently

fits symmetric observed line profiles well within the

experimental error, or to an adequate degree of apTable 1: Goodness of Fit (GoF) for Voigt and

pseudo-Voigt functions fitted to line profiles from

tungsten filings.

hkl Voigt Pseudo-Voigt

110 2.3 2.5

200 1.4 1.4

211 1.7 1.6

220 1.2 1.2

310 1.3 1.2

222 1.0 1.0

312 1.5 1.4

400 1.2 1.2

proximation {t. g., figure 1 and table 1; see also Langford & Wilson, 1978, figures 2 and 3 for fits of the

Voigt function to theoretical line profiles for crystallites having various shapes). The quality of the fit, indicated by the Goodness-of-Fit parameter {e.g..

Hill & Madsen, 1987) in table 1, depends to some

extent on the magnitude of the sample broadening

relative to that introduced by the instrument. If the

latter is dominant, then the fit tends to be inferior

for line profiles at lower angles.

Clearly a single Voigt function cannot be applied

if the experimental data are markedly asymmetric.

Also, (j) must lie within the Lorentzian and Gaussian

limits or, from (5) and (6),

0.6366 <<!>< 0.9394. (7)

Cases have been reported, for data from high resolution instruments, where the intensity in the tails of

line profiles decreases more slowly than a Lorentzian

{(f) < 0.6366; e.g., Plevert k Louer, 1990), the so- called 'super-Lorentzian' (Wertheim et al., 1974).

Such cases have hitherto been rare and, if line-profile

analysis is to be applied, then probably the best that can be achieved by using existing techniques is to

assume that the reflections are Lorentzian. If (f) is significantly greater than 0.9394, then it is possible

that the fitting procedure is at fault or that the data

are suspect.

3. Experimental considerations and pattern

decomposition

3.1 Data acquisiion

Line-profile analysis is one of the most exacting applications of powder diffraction, and, for the results

to be meaningful, high quality data are essential.

Any sample treatment which may affect the properties of interest is clearly to be avoided, but otherwise standard procedures for sample preparation

should be followed. These are described in 'Methods & Practices in X-ray Powder Diffraction', published by the JCPDS-ICDD. (See also Will, Parrish

k Huang, 1983.)

The data should not be unduly 'noisy', and ideally the standard deviation in the peak intensity of

the weakest recorded reflection should not be greater

than a few percent. The effects of counting statistics on parameters derived from an analysis based on the Voigt function have been considered by Langford (1980), and these can be appreciable. Since additional systematic errors due to pattern decomposition and assumptions made in the analysis of the

data are inevitable, it is desirable to minimise ran- dom errors, in so far as is practicable, at the datacollection stage.

The angular increment during data collection is not critical provided that, in addition to the usual

criterion based on the aperture of the receiving slit,

there are 5 to 10 data points within the FWHM of
112 The Use of the Voigt Function in Determining Microstructural Properties

(a) 110 (b) 211

oo • o

g

C

o - d

in • • Coin 01 "

1 1 1 3B.O0 39.00

1

1 1 1 1 1 1 40.00 41.00 42.00 43.00 Two Theta

I I I I I I—I—I—I—I—I—

I

70.50 71.50 72.50 73.50 74.50 75.50 Two Theta

(c) 310 (d) 321

o

O I o

in • c °

IS c

in - ColO C

"-I S

I—I—I—I—I—I—I—I—I—I—I—

I

95.00 97.00 S9.00 101.00 103.00 105.00

I I I i I I I I I 1 1

1 126.00 12B.00 130.00 132.00 134.00 136.00 Two Theta Two Theta

Figure 1: Line profiles for tungsten filings, CuKori radiation, with fitted Voigt functions and difference plot.

the narrowest line. As many lines as are feasible

must be recorded, up to an angle at which the peak

intensity has fallen to a few percent of that of the

strongest reflection; if possible, data for at least 20-

30 reflections should be obtained if a detailed analysis

of sample imperfections is required. It is often convenient to scan the desired angular range in two or more sections, with an increase in the counting time,

and perhaps the step length, at higher angles.

It is desirable to use monochromatic radiation
r

J. I. Langford 113

for line-profile analysis. This can of course be

achieved automatically if synchrotron or reactor neutron sources are used, and, in the case of Ka radiation from conventional sealed-tube X-ray sources, the Ka2 component can be removed completely by

means of an incident-beam focusing monochromator (e.g., Louer &; Langford, 1988). The resulting

monochromatic Kai spectrum can then readily be

modelled with a simple analytical function, the number of peaks in a diffraction pattern is halved, and

the lower background can be assumed to vary lin- early over a wide range of angle. Standard analytical

procedures for removing the Ka2 component, usually

based on modifications of the method introduced by

Rachinger (1948) (e. g., Delhez & Mittemeijer, 1975), are adequate for many applications of powder diffraction, but introduce systematic errors in line-profile

analysis. These arise because the different widths,

shapes and asymmetries of the Kai and Ka2 components and the Ka satellite lines are not normally

taken into account. Such errors are undesirable, but

whether or not they are significant depends on the

relative magnitudes of the sample and instrumental

broadening and the resolution of the instrument.

3.2 Instrumental effects

A correction for the contribution from the instrument

to the observed data must be made in all studies of

sample imperfections by means of line-profile analysis, and inadequate instrumental data can result

in severe systematic errors. Instrumental line profiles are normally obtained by means of data from a

standard reference material (SRM), and, for samples

which are essentially opaque to the radiation used,

LaBe is the SRM marketed by NIST (Fawcett et al.,

1988). Alternatively, annealed BaF2 has been found

to be eminently suitable for this purpose (Louer

& Langford, 1988). The broadening introduced by

sample transparency can be estimated {e. g., Wilson

(1963), chapter 3 for flat samples), and, if this is not negligible, an annealed sample of the material of

interest should be used. However, in such cases it

is important to confirm that line broadening due to

residual sample imperfections is inappreciable.

If the breadths of the instrumental line profiles

g{x) are obtained by means of an SRM, in general the

'standard' reflections will be at different angles from

those of the sample and it is then necessary to interpolate 2wg and Pg by some function to describe their

variation with angle. The nature of the function is unimportant, provided that the fit is adequate, and

that frequently used is a quadratic in tan^:

( Breadth)^ = U tan^ 9 + V iaxiO -^-W. (8)

U, V and W are then constant for a given instrumental configuration and can be checked periodically

to ensure that alignment has been maintained. A

typical instrument resolution function (IRF: FWHM

versus 26), obtained by a least-squares fit of (8) to data from annealed BaFg, is shown in figure 2. An

iX

10.00 T 1 30.00 90.00 70.00 90.00 Two Theta

tto.oo —I—I—

tso.oo tso.oo

Figure 2: Instrument Resolution Function (IRF) and

FWHM for ex-oxalate ZnO, CuKai radiation.

alternative function, with more theoretical justification for X-ray diffractometers, and which may model

breadths at low angles more satisfactorily if, for ex- ample, axial divergence has a measurable effect, is

(Breadth)^ = A tan^ e + B + C cot^ 9. (9)

If necessary, a fourth term to account for sample

transparency can be added to (9). For a detailed discussion of this function, and a comparison with (8), see Langford (1987), section 2.2. It should be noted

that U, V, W or A, B, C are required for both 2wg

and l3g in the Voigt approach to line-profile analysis,

in order to obtain <f)g from (4).

3.3 Pattern decomposition

In order to investigate the nature of structural imperfections, reliable parameters defining the position,

breadth and shape of all accessible Bragg reflections are required. Various programs available for obtaining these quantities by means of pattern decomposition have been reviewed by Smith & Gorter (1991).

The programs employed to obtain data used in the

examples given below were PROFIT (Langford et ai,

1986) and the Socabim program FIT, available in the

PC software package DIFFRAC-AT. The interactive

nature of the latter makes it particularly convenient

to use, since, with the 'zoom' facility for displaying

line-profile tails, erroneous 'fits' are usually apparent. Alternatively, a logarithmic intensity scale can be used.

Ideally, experimental line profiles should be modelled with Voigt functions, as given by (1), and a
114 The Use of the Voigt Function in Determining Microstructural Properties

Fortran program for evaluating the complex error function u}{z) is listed in Appendix 1. /?/,

fSjG and <l>f are then given directly. However, most

pattern-decomposition programs employ some other

function, notably the pseudo-Voigt and the Pearson

VII, when only and <f)f are obtained. The vari- ous functions most commonly used in powder diffraction were defined and reviewed by Wiles Young,

1982. The pseudo-Voigt is a close approximation

to a Voigtian (Wertheim ei al., 1974), and fits obtained by using these functions are frequently identical, within the experimental error (Table 1). In

general the Pearson VII is not such a good approximation to a Voigtian. The effects of using parameters obtained from a pseudo-Voigt and Pearson VII,

rather than a true Voigtian, and the errors introduced thereby, have been considered by Keijser, Mittemeijer & Rozendaal (1983). An advantage of these

functions is that computational time is usually less than that for the Voigtian.

Ahtee et al., (1984) have shown that, for a Voigt

function, <^ is related to y (3) by

<i> = E{\+Ay+By'')/{\ + Cy^Dy''), (10)

where A = 0.9039645, B = 0. 7699548, C = 1.364216,

D = 1.136195 and E = 2y/(\n2)/n = 0.9394372.

(10) is an approximation, but the maximum differ- ence from the exact value is only 0.16% (at y = 0.15).

Thus, if 2w and (3, and hence <f>, have been obtained

by fitting a pseudo-Voigt or some other function to

the observed reflections, then

y = {{C<j>/E-A) + [C(f>/E-Af-A{D<t>/E

-B){<f>/E - 1)Y^'}/[2{B - DHE)]. (11)

^G can then be obtained from (Langford, 1978, equation (25))

0G=fSexp{y^)[l- erf(y)], (12)

and /?i is given by (3).

4. Preliminary analysis of line breadths and

shapes

4-1 Preliminary assessment

Before one embarks on a detailed analysis of line

breadths, the nature of any microstructural properties in the sample must first be ascertained. The

first stage in this procedure is to compare the FWHM

{2wh) of the observed line profiles h(x) with the IRF.

2wh for a sample of ZnO obtained by thermal decomposition of the oxalate (Langford ei ah, 1992) is plotted in figure 2, and it is clear that 2wh ^ 2wg

for all reflections, and that there is thus appreciable sample broadening. Also, the scatter for the 2wh

data is considerably greater than would be expected

from random effects (cf the data for the IRF), and

the broadening due to the sample thus exhibits a marked 'anisotropy' or lattice-direction dependence.

Plots such as figure 2 serve to indicate whether or not

the data merit further analysis in terms of structural

imperfections. As part of the general assessment of

the data, the diffraction pattern can be reviewed by

using the observed peak positions with the program

NBS*AIDS83 (Mighell, Hubbard k Stalick, 1983), or some other program which obtains refined unit-cell

dimensions. This will indicate if there are any significant sample-induced peak displacements, such as can

arise from some structural imperfections. The magnitude and direction of systematic peak shifts can be

of importance in analysing and interpreting sample

line broadening, particularly if there is evidence of a

contribution from structural 'mistakes'.

4-2 Instrumental correction

The integral breadths /?/ of the line profiles f(a;) due

to sample imperfections are required in order to characterise microstructural properties. If all line profiles are assumed to be Voigtian, since h(a;) is the convolution of f(x) and g(a;), i. e.,

h(x) = f(x)*g(a;), (13)

then

PfL= PhL-/3gL, (14)

and

Ph=/^lG-l3h- (15)

If Voigt functions have been used in the pattern decomposition, then (14) and (15) can be applied directly to obtain and P/g- If some other function

is used to model reflections, then yh and yg are given

by (11) with (j) = (f)h and <f) = <j)g, respectively. ^hG

and f3gG are then obtained by inserting yh, Ph or yg, Pg in (12), and the corresponding Lorentzian components are given by (3). This procedure is shown

schematically in Langford et al, (1988) figure 3. Depending on the nature of any imperfections

present, Pjl and /?/g may be needed in a subsequent

analysis of line broadening, but the Voigtian breadth

/?/ is required in the preliminary evaluation of the

data. This is obtained from (12), with (Bg =

and y =z yj = /?/i,/(v^/?/g). Additionally, 4>f is given by (10), again with y = yf. Hitherto, breadths

have been expressed in angular units (2^), but the

subsequent analysis is simplifed ii /3j is given in terms

of reciprocal units, where

13} = cos 9/ X. (16)
J. I. Langford 115

In what follows, reciprocal units are implied throughout, hut for convenience /3j will be denoted by

4.3 Williamson-Hall plot

The next stage in the analysis is to ascertain if the

breadths depend on d* (= 2sin^/A) and to investigate the nature of any [hkl] (i. e., lattice-direction)

dependence. For this purpose the Williamson-Hall

plot (Williamson Sz Hall, 1953) is used, in which

(in reciprocal units) is plotted as a function of d*

,

with the addition of the Miller indices hkl for each

point. It should be noted that the Williamson-Hall

plot should not be used quantitatively, other than as a first approximation, if (3j has both (i*-independent

and rf*-dependent contributions, since the implicit

assumption is that the corresponding line profiles are Lorentzian, which is unlikely to occur in practice. Nevertheless, such a plot gives a valuable insight into the nature of any structural imperfections

present and the procedure to be adopted in subsequent analyses. The use of the Williamson-Hall plot can best be demonstrated by considering various specific cases of sample broadening.

(a) Ex-hydroxide nitrate ZnO, sample A [Fig. 3(a)]

From the Williamson-Hall plot for a sample of ZnO

obtained by the thermal decomposition of the hydroxide nitrate (Louer et al., 1983), /?/ is the same,

to within the experimental error, for difi'erent orders

of reflections (/lOO, hhO, 00/), indicating that there is not a measurable contribution from microstrains or other 'lattice distortions', which introduce an order

dependence. Furthermore, the intercept for the 00/

reflections is much less than for the hkO, suggesting

that on average the crystallites or domains have a

prismatic form, with the axis in the c direction and

an axial dimension greater than the base diameter.

Since there is not a significant difference in /?/ for the

hOO and hkO reflections, the prisms may be regarded

as having a circular section. A subsequent analysis

of the breadths for this sample would thus be based on strain-free crystallites whose shape, on average,

is a cylinder. The data for the Williamson-Hall plot are taken from Langford et al., (1986).

(b) Tungsten filings [Fig. 3(b)]

The breadths for a sample of tungsten filings (Table

2) increase linearly with d* , and there is a non-zero

intercept, indicating that microstrains are appreciable, and that there is a measurable 'size' effect. The

points do not display a significant scatter, so the

strain is isotropic, as would be expected for coldworked tungsten. There is also a grain or sub-grain

structure within the filings, and this may represent

regions bounded by dislocations.

(c) Ex-oxalate ZnO [Fig. 3(c)]

For the ex-oxalate ZnO data (Langford et al, 1992,

Table 3), it is immediately apparent from the

Williamson-Hall plot that the values of /?/ are distributed among three groups, according to the values

of hkl:

Group 1 (/?/i) hkO and h — k = 3n.

Group 2 (/?/2) h - k = 3n±l, I odd.

Group 3 (/?/3) h — k = 3n±l, I even.

Furthermore, (3fi < /3f2 < Pf3 for almost all the

data. This behaviour is a classic case of broadening

due to stacking faults in materials belonging to the

hexagonal system, which had not been been reported

previously for ZnO. From the theory of stacking-fault

broadening {e.g., Wilson, 1962, chapter 6), group 1 reflections are unaffected by 'mistakes'. For these

lines, j3f is independent of d* , again indicating that

microstrains are negligible. Since the plot for group 1 reflections has a non-zero intercept, the crystallites or domains are of relatively small size. Also, there is lit- tle scatter for these data and the lack of 'anisotropy'

suggests that on average the crystallites may be re- garded as spherical, or as prismatic with equal apparent dimensions in the axial and basal directions. (See

section 5.3.) The ex-oxalate ZnO thus represents a

case of combined 'size' and 'mistake' broadening, but

with no measurable strain contribution.

(d) Ex-hydroxide nitrate ZnO, sample B [Fig. 3(d)]

This sample has been considered by Langford et al. (1986) and by Sonneveld et al. (1986). From the

Williamson-Hall plot, lines for hOO and 00/ re- flections have non-zero and slightly different slopes,

indicative of the presence of microstrains which display a degree of anisotropy. There is also a 'size' effect (non-zero intercepts) and the crystallites are prismatic (00/ intercept < hOO intercept). /3j for the

110 reflection is close to the hOO line, again suggesting that on average the crystallite shape is cylindrical. A curious feature for this sample is that /?y for

all other hkl lies on, or close to, the hkO line. This

may be due to an inverse correlation between domain size and strain. There is, however, no evidence

of measurable broadening due to stacking faults. For

ZnO sample B, a subsequent analysis would thus be

based on combined 'size' and strain broadening, with

anisotropic strain and possibly a cylindrical model

for the crystallite shape.
116 The Use of the Voigt Function in Determining Microstructural Properties

Figure 3: Williamson-Hall plots, (a) ex-hydroxide nitrate ZnO, sample ^. (6) tungsten filings, (c) ex-oxalate

ZnO; + hkO and h — k = 3n, A h — k = ± 1, I odd, Q h — k = Sn±l, I even, (d) Ex-hydroxide nitrate

ZnO, sample B.

4-4 Shape of sample line profiles

To conclude the preliminary assessment of the data,

the nature of the shape parameter is considered, i. e., whether the profiles are predominantly Lorentzian or Gaussian, and if there is any dependence of on d* or lattice direction. The line profile shape for the

ZnO sample A data is closely Lorentzian (Langford

ei ai, 1986), as is that for the tungsten filings [from

table 2, (<^/) = 0.673(11)], and there is no evidence

of any d* or lattice-direction dependence for either of

these samples. For the ex-oxalate ZnO, on the other

hand, {<f)f) = 0.778(19) for group 1 reflections and

0.714(25) for groups 2 and 3. There is thus a significant Gaussian component in this instance which is greater for the group 1 ('size only') reflections than

for groups 2 and 3. </)/ increases steadily for ZnO
J. I. Langford 117

Table 2: Line breadths and shape parameter for tungsten filings; apparent size and strain from single-line

method.

h £ e xio^A-i xio^A-i xio^A-^ A xlO^ xlO^

0.447 4.153 3.407 1.503 0.697 294 6.70 1.34

0.632 5.664 5.046 1.561 0.673 198 4.95 0.99

0.774 6.774 6.109 1.763 0.670 164 4.55 0.91

0.894 7.407 6.448 2.239 0.680 155 5.00 1.00

0.999 8.203 7.455 2.054 0.667 134 4.10 0.82

1.094 9.010 8.088 2.398 0.671 124 4.40 0.88

1.182 9.679 9.020 2.074 0.659 111 3.50 0.70

1.264 10.107 9.273 2.399 0.664 108 3.80 0.76

sample B (Langford ei al., 1986, Fig. 4), which is consistent with a markedly Gaussian character of the

strain component of f(x).

5. Microstructural properties from line

breadths

5.1 Ex-hydroxide nitrate ZnO, sample A: crystallite

size only

The analysis of the data for ZnO sample A is representative of an important application of line-profile

analysis, since strain-free materials occur frequently

in practice, particularly ceramics and other compounds which have been produced at a high temperature. In such cases, if there is no evidence of 'mistake' broadening (section 5.3), /?/ can be attributed

solely to the effects of crystallite or domain size and

shape, and

/?/ = l/£, (17)

where e is an apparent size, equivalent to the volume average of the thickness, measured perpendicular to the reflecting planes (Appendix 2 and Wilson,

1962). It should be noted that an analysis of line

breadths based on Fourier coefficients {e. g., Warren,

1969) gives a different measure of the apparent size,

namely an area average of the crystallites, projected on to the reflecting planes (Wilson, 1963, chapter 9).

In general this will be less than the apparent size obtained from the integral breadth. An indication of

the mean dimensions of the crystallites or domains

can be obtained by fitting a suitable model to the

values of e (Appendix 2).

In the case of ZnO sample A, the Williamson-Hall

plot suggests that the morphology of the crystallites

approximates to a cylinder, with its axis parallel to

c, which is not unreasonable for a hexagonal material. (A case for which a hexagonal prism is a better

model, again for a sample of ZnO, has been considered by Vargas, Louer & Langford, 1983.) e is re- lated to the mean diameter D and the height H of a cylinder, and the angle <f)z defining the direction

of the diffraction vector, by (A4) and (A5) of Appendix 2. These equations are fitted to the observed

values of e by the method of non-linear least squares,

and the result of this procedure is shown in figure 4,

which depicts a cross-section through the 'average'

cylinder. From this analysis, (D) = 180(10) A, and

{H) = 270(60) A. From figure 4, it can be seen that a cylinder is a close approximation to the 'average'

morphology of the crystallites. The results of TEM

measurements are in broad agreement with this re- sult and demonstrate that the particles of the powder

are in fact single crystals (Louer et ai, 1983).

5.2 Tungsten filings: isotropic size and strain broadening

If order-independent and order-dependent broadening occur simultaneously, then the analysis is less straightforward than in the 'size only' case. The two

contributions to are normally regarded as 'size' and 'strain' broadening, though the former can include a contribution from structural 'mistakes' and

the mean separation of dislocations, and the latter can arise from a compositional gradient within the

sample. f(i) is then the convolution of S(x), the line

profile due to all 'size' effects, and D(x), that due to

strain, actual or apparent, or

i{x) = S{x)*r>(x). (18)

There are various approaches to separating Ps and

0D, the integral breadths of S(x) and D(x), all based on assumptions regarding the nature of the line profiles due to 'size' and 'strain', and these are considered below.
118 The Use of the Voigt Function in Determining Microstructural Properties

0<

50 A

Figure 4: Least-squares fit of (A4) and (A5) (cylindrical model) to line breadths from ex-hydroxide nitrate, sample A (Langford ei ai, 1992); —Cobs,

Seal ) Tobs •

(a) 'Average' Size-Sirain Plot

If the Williamson-Hall plot does not indicate a marked anisotropy of size or strain broadening, as is the case for the tungsten sample, the 'average' sizestrain plot maybe employed to obtain {13s) and (/?£>). Haider & Wagner (1966) have shown that an approximation to the integral breadth of a Voigt function is given by

P' = (3lP + /31, (19)

which is accurate to within about 5%, /3 being less than the true value for a given y (Langford, 1978). If

it is assumed that /3f is Voigtian, that the Lorentzian

component is due solely to size effects, or

Pl=I3s = 1/e,

and that j3a arises from strain, or I3g=Pd= vd*/2, (21)

where 77 is a measure of the strain, then (19) may be

written as

(22)

The implications of assuming that the 'size' and

'strain' profiles are Lorentzian and Gaussian respectively have been discussed by Delhez ei al., (1993,

Appendix B). From (22), the intercept of the plot

of (P/d*y versus /3/(d*y gives the mean value of 77, and the slope gives the mean apparent size (e). Despite the approximation and assumptions made, this

plot has the advantage that data for reflections at low

and intermediate angles are given more weight than

those at higher angles, which are often less reliable.

From the 'average' size-strain plot for the tungsten filings (Fig. 5), (e) = 640(70) A, and {rj) =

so® S

H

S -

—I

1 1 1 1 1 1 1 1 1 1 1 2.00 6.00 10.00 14.00 IS. 00 22.00

Figure 5: 'Average' size-strain plot for tungsten filings.

14.9(3) X 10"^. Since the line breadths show no

[hkl] dependence, the domains may be regarded as having spherical form, on average, with a diameter

(D) = 860(90) A [Appendix 2, (A2)]. The value re- ported for the strain depends on the stress model

invoked (Stokes &; Wilson, 1944; see also Langford et

al., 1988). If the rms strain e (= (e^)^/^) is considered to be appropriate, then

(20) e = »7/(2v^) ^ 7?/5, (23)

and, for the tungsten filings, (e) = 2.98(5) x 10"^.
J. I. Langford 119

(6) Multiple Line Analysis

If the values of /?y show little or no [hkl] dependence, as is the case for the tungsten sample, 0s

and 0D may be separated by assuming that both

are the breadths of Voigtian functions, with components (SsL,(^SG and Pdl,Pdg respectively. The

average values of these quantities can be obtained

from plots of l3fL versus d* [Fig. 6(a)] and ver- sus (d*y [Fig. 6(b)]. If such plots are linear, as are those of figure 6, then the intercept of figure

6(a) gives /?l = 0.36(3) x IQ-^k'^, and, from fig- ure 6(b), /?5G = 1.38(21) X psL and 0sG

can then be combined by means of (12) and (3), with

J/ = /?5L/(v^/35G),to|ive/?5 = 1.43(25) xlO-^A-i,

with {e) = 620(100)A. Implicit in the above as- sumption is that 7/ also has Lorentzian and Gaussian components r)L and j]a- The slope of figure

6(a) is then rjL/2, and that of figure 6(b) is {qa/^)^-

By again using (12) and (3), with y = riL/is/T^rja),

{t}) = 15.2(6) X 10-3, and e = 3.04(12) x 10"^.

It should be noted that, as with approach (a), the

multiple-line analysis gives average values of crystallite size and microstrain and does not furnish information on any [hkl] dependence of these quantities.

It is also of interest to examine the Voigt shape parameter <f) for the size and strain broadening. From

(10), ifs) = 0.884(38), and {^ic) = 0.660(5). Thus,

if f(x) is in fact a Voigt function, then its size component is strongly Gaussian and the strain component

is almost pure Lorentzian. However, from table 2

((^y)(= 0.673(12)) is close to the Lorentzian limit, as would be expected, since the line broadening is dominated by the microstrain component.

(c) Multiple- Order Analysis

If data for two or more orders of a reflection are available, then a Voigt approach which yields information on crystallite shape and anisotropy of microstrains can be employed. However, this depends on the as- sumption that, for a given series of reflections, j3ji

and 0fG vary linearly with d* and (d*)"^ respectively.

While this is clearly the case for all reflections for the

tungsten sample (Fig. 6), in general at least three or- ders of a reflection are required to confirm that this

is so. By using

0fL = PsL + (»7L/2)d* (24)

and

l3h=PlG + iriGm\d*)\ (25)

and the procedure outlined in (6) for combining

PsL^PsG and Pdl,Pdg, e = 720(110)A, and e =

2.90(9) X 10-3 for the 200 and 400. These values

confirm the lack of an [/iAr/]-dependent broadening

for the tungsten sample, as before.

(d) Single-Line Analysis

In the single-line method of Keijser et al. (1982), it is normally assumed that 0jl is due only to size eff'ects,

and 0fG is attributed to strain. Since this approach

is applied to data for each line separately, information on crystallite shape and anisotropy of strain can

in principle be extracted, but the results will clearly

be influenced markedly by this assumption. In the

single-line method

l3fL = l/£, (26)
120 The Use of the Voigt Function in Determining Microstructural Properties

and

0JG = rid*l1. (27)

Values of e, 7/ and e obtained from (25) and (26) are given in table 2, from which (e) = 160(60)Aand e = 0.93(2) X 10~^, which are significantly less than

the values obtained previously. Also, from table

2, £ and e decrease steadily, whereas both quantities should be constant, aside from variations due

to crystallite shape or elastic anisotropy, for all val- ues of d* . However, is close to the Lorentzian

limit for all lines (section 4.4), and the WilliamsonHall plot [Fig. 3(b)] indicates that is dominated

by the effects of microstrains. This suggests that the

line profile due to the latter may be predominantly

Lorentzian in character. If and in (25) and

(26) are interchanged, then (e) = 515(100)A, and

(e) = 3.00(11) X 10-3.

The average values of e and e obtained by applying the four approaches to the tungsten data are summarised in table 3. There is good agreement between

the results of (a) to (c); strain is determined with

fairly high precision, though the standard deviation

in the size estimates is somewhat larger. From the

customary procedure for (d), £ and e are considerably less than for other methods, but this is to be

expected, since it has been shown that Lorentziansize and Gaussian-strain line profiles are not close

approximations for the tungsten data. However, this

represents a special case which is not necessarily applicable generally. It should be noted that, if data

can only be obtained for a few lines and there are no multiple orders, then a single-line analysis is the

only method available.

5.S Ex-oxalate ZnO: crysiallUe-size and 'mistake'

broadening

The microstructure of this sample has been investigated by Langford et al. (1992). The WilliamsonHall plot revealed that there are three catagories of

reflections, and those of group 1 {hkQ and h — k =

3n), which are unaffected by stacking faults, are used

to obtain the mean crystallite size and shape. By

assuming a spherical morphology, (£)) = 453(17)A,

and, if the form is assumed to be cylindrical, (D) =

404(7)A, and {H) = 351(9)A. Although there is little to choose between the two models from statistical

considerations, the latter is probably more realistic on physical grounds. As noted previously (section

4-4)! f(2^) for group 1 reflections has approximately

the same shape for all hkl and has a significant Gaussian component. The ex-oxalate ZnO is strain-free,

but if measurable microstrains were present, then the

analysis of 5.2 would be applied to the group 1 reflections.

There are no measurable peak displacements for

this sample, which suggests that the lattice 'mistakes' are stacking faults, since other forms of fault can introduce a peak shift. The next stage in the

analysis is to separate the breadth contribution Pm

due to stacking faults from that arising from size ef- fects (/?s for groups 2 and 3 reflections). For this

purpose, /?5 is calculated from (A2) or (A4) and

(A5), according to whether the spherical or cylindrical model is used. There are then two approaches

for obtaining /3m- The simpler method is to assume

that the 'size' and 'fault' profiles are both Lorentzian,

when

0m = Pf-l3s. (28)

The theoretical line profile for stacking-fault broadening is approximately Lorentzian (Wilson, 1962,

chapter 6), but it has been demonstrated that the

'size' profiles are not. Thus, an approach based on

the Voigt function can again be applied, where

Pm = l3fL - PSL, (29)

and

f3sG=l3fG. (30)

Since /3fG and (f>s are known from the group 1 re- flections, PsL can be obtained by solving (11) for y

(Langford et al., 1992, Appendix I). is then given

by (28).

The final stage is to obtain the stacking-fault probability a from , and for hexagonal materials (from

Wilson, 1962, chapter 6) = acos(^^/2c, /i - iSr = 3n± 1, / odd, (31)

and

j3m = cos<f)z/2c, h — k = 3n ± 1, I even. (32)

From the Lorentzian approximation, cylindrical

model, (a) = 0.0069(11) for group 2 reflections and

(a) = 0.0068(3) for group 3. These values corre- spond to a mean separation between stacking faults

(= c/2a) of 380(30) A, or about one fault per crystallite, on average; i. e., according to this approximation, the crystallites are in fact domains bounded by

fault planes.

By assuming that the 'size' profiles are Voigtian,

(a) = 0.014(3) for the group 2 reflections and (a) =

0.009(1) for group 3, with about two faults per crystallite, on average. The difference between the two

estimates of a suggests that the line profiles due to

stacking faults are not strictly Lorentzian. An analysis based on Lorentzian functions gives highly consistent results, though there are clearly systematic

errors. That based on the Voigt function is probably more accurate, though less precise. For a more
J. I. Langford 121

Table 3: Summary of size-strain analysis for tungsten filings

Method

A

iv)

xlO^

{<f>n) e xlO^

'Average' s-s plot 640(70) 14.9(3) 2.98(5)

Multiple line 620(100) 0.884(38) 15.2(6) 0.660(5) 3.04(12)

Multiple order (a) hhO

(b) hOO

720(110)

580(90)

0.872

0.808

15.4(6)

14.5(5)

0.675

0.663

3.07(11)

2.90(9)

Single line (average)

(a) L size G strain

(b) G size L strain

160(60)

520(100)

4.70(10)

15.1(5)

0.93(2)

3.00(11)

detailed discussion of the analysis of the ex-oxalate

ZnO data, see Langford et ai, 1992.

Other cases of 'mistake' broadening are treated

similarly. For example, twinning in the orthorhombic

phase of the high Tc ceramic YBa2Cu307_i does not

affect the hhl. planes. In this case a parallelepiped,

given by (A8), can be used to model the domain

shape (Rand, Langford & Abell, 1992). A theoretical treatment of the line broadening and peak displacement due to various types of mistake is given in

Wilson, 1962. (See also Warren, 1969, chapter 13).

5.4 Ex-hydroxide nitrate ZnO, sample B: size and

strain broadening

The Williamson-Hall plot indicates that, at least to a first approximation, there are two main groups

of reflections for ZnO sample B, namely hkl with

h,k 0 and 00/. The 'average' size-strain plot and

multiple-line analysis can be applied to the former

group, since the [hkl] dependence of the breadths

of these reflections is not large. From the intercept

of the former [(22) and Fig. 7] (77) = 9.8(5) x 10"^,

and (e) = 1.96(11) x 10"^. From the slope, (e) =

240(12)A, and, if spherical domains are assumed,

{D) = 320(16)A. However, from a multiple-order

analysis (Sonneveld et ai, 1986), there appears to

be a small [hkl] dependence of both size and strain

broadening, and, from the former, the domain shape

can be modelled by a cylinder of diameter about

375A and height 420A. The strain estimated by this

method was found to be 2.9 x 10~^ in the basal plane

of the cylinder, with 2.0 x 10~^ in the direction of its axis. The results of a single-line analysis are also

given by Sonneveld et al. and, excluding 00/ data,

(e) = 270(50) A and (e) = 2.4(5) x lO'^ from this approach, and, for a spherical model, (D) = 360(70)A.

The agreement between the values estimated from

00

0

mw«00

0

0rl e

M 0

00 * 94 '9

CO

0

0

M

+

1—1—I—I—I

I—I—I—i—I—I—I—

4.00 12.00 20.00 2B.00 36.00 44.00

Figure 7: 'Average' size-strain plot for ex-hydroxide

nitrate ZnO, sample B.

the single-line approach and those from other methods is thus better than for the tungsten filings. This

is to be expected, since it has been shown that the

line profiles due to microstrains are more Gaussian

in character for ZnO sample B (section 4-4)-

6. Summary

After obtaining data of adequate quality for the sample of interest and for an instrumental standard, fol- lowed by pattern decomposition, the procedure for

carrying out line profile analysis by means of the

Voigt function is as follows:

1. Compare the angular variation of the FWHM

with the IRF to ascertain if there is measur-
122 The Use of the Voigt Function in Determining Microstructural Properties

able sample broadening, and if there is any

[hkl] {i. e., lattice-direction) dependence (section 4-i)- If an annealed version of the sample

is used as an instrumental standard, then the

FWHM for this should also be compared with

the IRE. Use NBS*AIDS83 or an equivalent program to ascertain if there are any measurable

peak shifts.

2. Check that the Voigt function is appropriate for

analysing line breadths (criterion (7) and inappreciable asymmetry: section 2.1)

3. Correct data for instrumental effects to obtain

breadths /?/ and 2wf for the sample line profiles

[(14) Sz (15)] and hence the shape factors <pf (4).

Obtain fSfL and P/q from (11), (12) and (3) and

convert all integral breadths to reciprocal units

(section 4-^)-

4. Make Williamson-Hall plot (Pf versus d*) and

add Miller indices. Ascertain the nature of the

sample broadening and note any [hkl] dependence of line breadths (section 4- 3)- Examine

<f}j to determine the Lorentzian/Gaussian character of f(a;), and whether the profile shape has

any d* or [hkl] dependence (section 4-4)-

5. Carry out a detailed analysis of line breadths in

accordance with the information obtained from

step 4. This should be based on one or more of

the procedures given in section 5:

5.1: Size-only broadening

5.2: 'Isotropic' size and strain broadening

5.3: Size and 'mistake' broadening

5.4: 'Anisotropic' size and strain broadening

7. Conclusions

The Voigt function can be used in the majority of

cases to study microcrystalline properties by means

of diffraction data. It is only inapplicable when

line profiles are markedly asymmetric, and when the

shape factor 4>h is outside of the Voigtian range. The

success of the method depends critically on the quality of the data, the number of lines recorded (except for high symmetry materials) and the calibre

of the pattern-decomposition program used. The

last can be a limitation at present, and pattern decomposition is an area where further development is needed. Erom the examples given in section 5, it is evident that a considerable amount of information on

structural imperfections can be obtained from an approach based on the Voigt function, and that, particularly for the ex-oxalate ZnO and tungsten samples, a relatively high degree of precision can be achieved.

However, the assumptions made in the analysis introduce systematic errors, particularly when 'size' and

'strain' broadening occurs simultaneously, and the

results are not necessarily accurate. Nevertheless, if meaningful data are available for a large number of

lines, the degree of self-consistency usually gives an

indication of the validity of the results. The foregoing

has been concerned with studies of microstructural

properties by using data from observable reflections.

However, after obtaining parameters defining struc- tural imperfections, it is a relatively simple matter to reverse the procedure given above to model all line

profiles of a diffraction pattern, as is required in the

Rietveld method for the refinement of crystal structures. This technique has been discussed further by

Delhez et al. (1993).

Much of this work originated from collaboration over a period of many years with Dr. D. Louer, of

the University of Rennes, France. I am indebted

to him for providing the ZnO data and also to Dr.

J. P. Auffredic, who prepared the samples. I am

also grateful to Professor E. J. Mittemeijer and his

colleagues of TU Delft, the Netherlands, for many

helpful discussions and assistance in developing the

method of line profile analysis based on the Voigt

function.
J. I. Langford 123

Appendix 1. Evaluation of complex error function.

FUNCTION CDWERF(Z) 01

C Evaluation of complex error function uj{x + iy) = exp(—2^)(1 — erf (—iz)) 02

C J.I. Langford, School of Physics & Space Research, University of 03

C Birmingham, Birmingham B15 2TT, UK. 04

C 15.iii.82; based on a CERN Computer Centre algorithm 05

C 06

IMPLICIT DOUBLE PRECISION (R-Y),C0MPLEX*16 (D,Z) 07

DOUBLE PRECISION VCONST 08

C0MPLEX*16 CDWERF 09

REAL LAMDA 10

LOGICAL B 11

VCONST = 1.0/DSQRT(DATAN(0.1D01)) 12

XX = DREAL(Z) 13

YY = DIMAG(Z) 14

X = DABS(XX) 15

Y = DABS(YY) 16

IF (Y.LT.4.29.AND.X.LT.5.33) GO TO 1 17

W = 0. 18

NC = 0 19

NU = 8 20

LAMDA = 0. 21

B = .TRUE. 22

GO TO 2 23

1 S = (1.0 - Y/4.29)*SQRT(1.0-(X/5.33)**2) 24

W = 1.6*S 25

W2 = 2.0*W 26

NC = 6 + INT(23.0*S) 27

NU = 9 + INT(21.0*S) 28

LAMDA = W2**NC 29

B = LAMDA.EQ.O. 30

2 Rl = 0. 31

R2 = 0. 32

51 = 0. 33

52 = 0. 34

N = NU + 1 35

3 N = N - 1 36

VN = N + 1 37

Tl = Y + W + VN*R1 38

T2 = X - VN*R2 39

U = 0.5/(Tl*Tl + T2*T2) 40

Rl = U*T1 41

R2 = U*T2 42

IF (W.LE.O.O.OR.N.GT.NC) GO TO 4 43

Tl = LAMDA + SI 44

51 = R1*T1 - R2*S2 45

52 = R2*T1 + R1*S2 46

LAMDA = LAMDA/W2 47

4 IF (N.GT.O) GO TO 3 48

:fc % :|e % % :|c 3(c :|c % :f: :|c % % % r^c :|c :fc 3|c :^ 3|e :|c 3(c :|c :f: :|c %
124 The Use of the Voigt Function in Determining Microstructural Properties

Q:(c :(c :|c :|c % :|c :|c :(c % 3|c 3|c 4: :|c :|c :|c :|c :|c :|c :|c :tc :|c :|c :t: *********** IF (B) GO TO 5 49

RSI = SI 50

RS2 = S2 51

GO TO 6 52

5 RSI = Rl 53

RS2 = R2 54

6 RSI = VC0NST*RS1 55

IF (Y.EQ.0.0) RSI = DEXP(-X*X) 56

CDWERF = DCMPLX(RS1,VC0NST*RS2) 57

IF (YY.LT.0.0) GO TO 7 58

IF (XX.LT.0.0) CDWERF = DCONJG(CDWERF) 59

RETURN 60

7 CDWERF = 2.0*CDEXP(-DCMPLX(X,Y)**2) - CDWERF 61

IF (XX.GT.0.0) CDWERF = DCONJG(CDWERF) 62

RETURN 63

END 64

Q:t:***** *******************************************************************

Appendix 2. Apparent and 'true' size of

crystallites or domains

The reciprocal of the integral breadth of a diffraction line profile due to the finite size of crystallites or domains is the volume-weighted average thickness

£, measured in the direction of the diffraction vector.

e is thus an apparent size and, if < is a distance measured in this direction, then (Wilson, 1962, chapter

4)

where V(0) is the volume of the crystallite. By taking into account the hkl dependence of £ and assuming that on average the crystallites have a particular form, the mean dimensions can be obtained by

a non-linear least-squares procedure, in which £obs

is compared with Seal, the value calculated from parameters describing the selected shape and its orientation with respect to the crystallographic axes. A

useful construction for providing a visual indication

of the validity of the model is given by a comparison of the observed and calculated thickness, Tgbi

and Teal, measured from a suitable origin in the direction of the diff"raction vector. Equations for e and

r are given below for crystallites having the form of

spheres, cylinders and parallelepipeds.

Spheres:

For spherical crystallites of diameter D [Wilson

(1962), chapter 4, (12)]

and

£ = 3Z)/4,

T = D

(A2)

(A3)

for all hkl.

Cylinders:

£ for cylindrical crystallites of diameter D and height

H for a reflection whose normal makes an angle <j)z with the cylinder axis may be obtained from Langford and Louer (1982), (10), (17) and (18):

D

£ =

IT CSC (^^

1 .

8

-f- 2q cos ^ q

, 0 < < (A4)

or

£ = D CSC I _8_

37r 4q

TT (A5)

where q = Htan<j)z/D and $ = ta.n~^{D/H). In

this instance it is convenient to take as origin the

intersection of a diameter of the base and a generator

of the cylinder, when

and

T = H sec<j)z, 0 < <f)z

TT T = Dcsc<j)z, $ < < 2

•

(A6)

(A7)

While crystallites are unlikely to be cylindrical in

reality, a hexagonal prism may be approximated by

this form.

Parallelepipeds: The procedure introduced by Wilson (1962, chapter 4) can be used to obtain £ for
J. I. Langford 125

a parallelepiped. Consider a parallelepiped defined

by the vectors A, B, and C, with angles a =B*C,

P =C"A, and 7 = A~B, and with a vertex as the

origin. If A is taken along the x axis, B is in the

x-y plane, and <^x, <j>y, <l>z are the angles between

the direction of the diffraction vector and x, y and z

respectively, then

2r

, (A8)

where

P

q

r

cos <l)x/-A.,

cos <f>y CSC 7/5,

cos<i)z sin '^/CD,

with

Z) = (1—cos a—cos /3—cos 7+2 cos a cos /? cos 7) '

.

The value of r to be used in (A8) depends on which

face is intersected by a vector through the origin and

parallel to the diffraction vector. If face A is defined

as the face parallel to the plane containing B and C

,

at a distance A from it, with similar definitions for

faces B and C, then.

if T terminates in face A,

Asm J

T =

sin 7 cos (l)x — cos 7 cos <j)y — F cos ^zl

if T terminates in face B,

5 sin 7

cos <j)y — E COS <i>z/

and if r terminates in face C,

CD CSC 7

COS <l>z

'

; (A9)

(AlO)

(All)

where E = cosa — cos j3 cos 7 and F = cos (3 —

cos 7 COS a. The above equations simplify considerably for a

rectangular parallelepiped, which can be used to

model a 'tetragonal' form and a cube. The orientation of the parallelepiped relative to the orthogonal

axes X, y, z was chosen arbitrarily in the above. In any particular application it may be necessary to ro- tate A, B, C by means of the Euler transformation.

References

Ahtee, M., Ungnius, L., Nurmela, M., & Suortti,

P. (1984). J. Appl. Cryst. 17, 352-357.

Delhez, R., Keijser, Th. H. de, Langford, J. I., LOUER, D., MiTTEMEIJER, E. J. & SONNEVELD, E.

J. (1993). In The Rietveld Method. R. A. Young, ed. Oxford: Oxford University Press, pp. 132-166, in press. Delhez, R. & Mittemeijer, E. J. (1975). J. Appl.

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National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Preferred Orientation Effects

V. Valvoda

Charles Universiiy, Faculty of Mathematics and Physics,

121 16 Prague 2, Czechoslovakia

Abstract

Methods of texture correction in samples with axial

symmetry introduced by spinning the sample during measurements are reviewed and experimentally

tested. The most accurate methods are based on a

representation of the orientation distribution of crystallites by a set of symmetrized harmonics with coefficients that can be determined from the orientation

distribution of a small number of crystallographic

planes (as measured on a texture goniometer) or from

the relative intensities of all reflections (as measured

on a powder diffractometer) when the structure of

the sample is known, at least approximately. The

latter case is met in quantitative phase analysis and

in structure refinement procedures. In special cases of fiber textures simpler analytical functions (like ex- ponential functions) are sufficient for texture description and handling. The Harris texture index can also

be used for texture characterization and correction

when the texture is not too strong.

1. Introduction

Non-random orientation distribution of crystallites

in powder specimens or compact polycrystalline samples can influence almost all accurate diffraction measurements. This regards mainly quantitative phase

analysis and structure determination because both

these methods are based on intensities of reflec- tions from a randomly oriented assembly of particles. However, a strong preferred orientation (texture) also causes problems in phase identification,

stress measurements and profile analysis, especially

in thin films. On the other hand, as has been pointed

out by Bunge, Dahms & Brokmeier (1989), texture can also help to separate completely overlapped re- flections, like 330 and 411 reflections in the cubic

7-brass structure, if the sample has an approriate

preferred orientation.

The effect of preferred orientation in powder specimens can be reduced by special preparation techniques (see, e.g., Smith & Barrett, 1979). However, such techniques cannot be used in the case of

compact polycrystalline samples and, as has been

shown by Suortti &; Jennings (1977), the inhomogeneity of the packing of the crystallites in a loosely

pressed specimen will cause an uncontrollable variation in the intensity distribution, making accurate

intensity measurements impossible. Another experimental method of texture suppression is a measure- ment that integrates diffracted intensities over all (or

almost all) sample orientations (Will, 1991). This

approach is time consuming, and accuracy is usually

limited by incomplete pole figure integration together

with systematic errors appearing at large inclinations

of flat specimens.

The present paper is mainly focused on methods

of correction of integrated intensities for the effect of

preferred orientation in samples with axial symmetry introduced by spinning the sample during measurements. These methods have found application in

accurate powder diffractometry directed to structure

refinement and phase analysis or to studies of charge

distribution and thermal vibrations of atoms.

2. Theoretical background and corrections

based on texture goniometer measurements

The sample statistics in powder diffraction are improved by spinning the sample. The rotation of the

sample introduces in every crystallite a special direction P called the polar axis (in the case of the

Bragg-Brentano diffractometer this polar axis coincides with the normal to the flat sample surface).

The polar axis density, W{0, (p), is defined as proportional to the volume fraction, dV/V, of the diffracting crystallites whose polar axis falls within the solid

angle dCl about the direction {0, (p) in the crystallographic coordinate system (Fig. 1),

In a randomly oriented sample W{6, (p) = Wran = 1-

127
128 Preferred Orientation Effects

Figure 1: Left: Polar axis P in a solid angle df2 in the direction of in a crystallographic coordinate

system. Right: During integration of reflection h = hkl the polar axis P is circulating along the circle s whose radius corresponds to the angle o; of sample inclination.

When measuring the intensity of reflection hkl

with the sample inclined by angle a (which is the

angle between the reciprocal lattice vector h = [hkl]

and the polar axis P) only those crystallites whose

polar axes P are pointing in a direction that forms

angle a with the direction h can contribute to the

measured intensity (Fig. 1, right). When the sample

is spinning, the direction P turns uniformly along

the circle s on the unit sphere, and the integrated

intensity of reflection h is

lH^) = 7^^^<fw(^,'P)ds, (2) zTTsm a J

where 7^(j„ is the intensity corresponding to a ran- domly oriented sample. Because the integral on the

right side of equation 2 depends solely on the value

of angle a we can rewrite this equation in the form

j''(a) = /,\„Py''(«), (3)

where ^^''(q;) represents an average polar axis density along the circle of integration. The intensity

7^g„ of the random sample can be thus obtained from

the intensity 1^(0) measured in ordinary diffraction

geometry {i. e. with a = 0) if we know the value

W''iO) = WieH,>ph):

I\0) = 7,^„Py''(0). (4)

In principle, W(6h,(ph), as well as the pole density at any other location, could be determined from complete texture analysis by measuring pole density distributions of a number of diffracting planes followed

by inversion of the data to yield pole densities for

all other, unmeasured, planes (see, for example. Roe

h Kriegbaum, 1964 or Bunge, 1985). Such measure- ments require a multiaxis pole-figure goniometer and

correction for such effects as absorption, defocusing

and surface roughness.

Method A. A variant of this approach, specific for a spinning sample, has been applied by

Jarvinen, Merisalo, Pesonen & Inkinen (1970), Pesonen, Jarvinen &: Kurki-Suonio (1973) and Pesonen

(1979). This method involves an analytical description of the polar axis density W by means of the

expansion in terms of a set of properly symmetrized

harmonics Yij :

Wid,<p) = J2CijYij{^,^). (5)

Performing the integration (2) results in the equation

W\a) = ^CijYij{^h,<Ph)Piicosa), (6)

where Pj(cosa) is a Legendre polynomial and

i^h , 'Ph ) are the spherical coordinates of the reciprocal lattice vector h with respect to crystallographic
V. Valvoda 129

Table 1: Picking rules for indices of symmetrized harmonics

Choice of Indices of

T 1 Laue class CO ordinate symmetrized

axes

T any (2A,/x,±)

2/m z 1 2,z 1 m (2A,2,x,±)

mmm z 1 m, y ± m, a:

II m (2A, 2/^,-1-) 4/m z 1 4,z ± m (2A,4/x,±)

z 1 4,3/ ± m (2A,4/x,+)

3 z (2A,3/*,±)

3m z 1 3,1/ ± m (2A,3//,+)

a; ± m (2A,3/x,+

for even ^,

—for odd /i) 6/m 1 6, z ± m (2A,6,x,±)

S/mmm 1 6, 0 J. m, y J. m (2A, 6/^,-1-)

In the cubic classes m3 and m3m the symmetrized

functions up to the 8th order are as follows:

Laue class Cubic Harmonics

m3 Ko K\ Kei Ks2 K%

mSm Ko Ki K& — K%

axes a, b, c. The needed value of W^{Q) is then

given by the expression

because Pi>i = 1 for a 0.

Practical application of method A consists of three

steps:

(i) Determination of indices of non-vanishing symmetrized harmonics for given Laue class according

to table 1 and their calculation according to tables 7

and 8 (see A-ppendix).

(ii) Determination of unknown coefficients C,j by

simultaneously fitting functions W^{a)/W'*(0) to

the ratios (a)/1^(0) measured in steps (say 10"

) of angle a for a few reflections (usually 2 to 4). The

standard linear least-squares method is suflBcient in

these calculations.

(Hi) Calculation of the values W^{0) according

to equation 7 for all reflections measured in normal

Bragg-Brentano geometry. Dividing the observed

/''(O) values by W^{0) gives the required intensities

/^an of the sample with random orientation of crystallites (See equation 4).

3. Corrections based on analysis of

intensities of all measured reflections

It has been shown (Valvoda, 1987) that the coefficients dj determining the polar density distribution can also be obtained from the whole set of integrated intensities 1^(0) as measured in ordinary

Bragg-Brentano arrangement. The following three

methods, B, C and D, assume that we know the

structure of the sample under study exactly, or we

are at the stage of structure refinement.

Methods B and C. If we know the relative intensities Rh of the sample with random orientation of

crystallites then

Iran = kR\ (8)

where k is an unknown scale factor. The combination

of (4), (7) and (8) for two reflections hi and I12 leads

to the relation

'''{0)R''' /I'''{0)R''']YI'j^ - Y/j'}Cij = 0. (9)

•J

The system of linear equations (9) for different pairs

hi and /12 can be solved with respect to dj either

directly (when the number of selected independent

equations is equal to the number of coefficients re- quired) or by linear least-squares (when the number

of independent equations exceeds the number of coefficients demanded). These two computational procedures will henceforth be designated as methods B

and C, respectively.

Method D. In structure refinement with an overall

Debye parameter B and relative intensities R^ of the

static structure,

/''(O) = k.R^„ exp(-25A-2 sin^ 0)

^

CijYi). (10)

The system of equations (10) for all measured re- flections contains the following unknown parameters:

a scale factor fc, structural parameters appearing in

Rg, a Debye parameter B and texture coefficients

Cij. All these parameters can be refined by a nonlinear least-squares method starting with estimates

of the texture coefficients derived by method B.

In the example shown below the program POWLS

(Will, Frazer & Cox, 1965) was used. On the other

hand, equation (10) is also the basic equation for

structure refinement by means of a Rietveld-type

program with simultaneous texture refinement. In
130 Preferred Orientation Effects

such a case the list of unknown parameters is ex- tended by lattice parameters, line profile parameters

and instrumental parameters (zero shift, angular dependence of line breadths, etc.). A Rietveld refinement program with texture correction according to

equation (10) was introduced by Ahtee, Nurmela,

Suortti & Jarvinen (1989).

4. Experimental tests of texture corrections

based on the description of polar density in

terms of symmetrized harmonics

Method A was tested by Jarvinen, Merisalo, Pesonen

k. Inkinen (1970), by Pesonen, Jarvinen & KurkiSuonio (1973), and by Pesonen (1979), on samples of

crystals with cubic, hexagonal and tetragonal symmetry, using a texture goniometer in reflection geometry. The results have shown that measurements

of 7(a)/7(0) for three reflections are sufficient for the

determination of accurate texture corrections using

only four terms in the expansion for the polar axis

distribution W. The corrected intensity values were

in agreement, within 1% or less, with those of a specimen in which the orientation distribution was ran- dom within the limits of experimental error. Paakari, Blomberg, Serimaa &; Jarvinen (1988) applied this method in the case of monoclinic crystals of

cellulose using only two measurements of five reflec- tions, one in symmetrical reflection (a = 0°) and one

in symmetrical transmission (a = 90°). It has been

found that the effect of texture can be reduced by an

order of magnitude, which is sufficient accuracy for

applying the method to quantitative determination

of the crystallinity of cellulose.

A comparative test of methods >1 - D on a powder

sample of titanium carbide with the NaCl type of

structure (Valvoda, 1987) verified the equivalence of

both basic approaches in this case. The results of this

test are summarized in tables 2-4 and in figures 2 and

3. The integrated intensities, measured with a specimen spinner and Cu Kq radiation on a Philips X-ray

powder diffractometer, are presented in table 2. The

texture coefficients derived from these intensity val- ues by methods B, Cand D are given in the last three

columns of table 4. The relative intensities R'^ of a random sample used in methods B and C were cal- culated from the known structure of the sample with

the use of a Debye-parameter value B = 0.34 A . This value was estimated from the pairs of reflections

111/222 and 200/400. Only the first three texture

coefficients were determined by method B with the

reflection 111 as hi and the reflections 200, 220 and

311 as h2 in (9). The parameters refined in method D

Table 2: Comparison of intensities (lobs/^) cor- rected for the effect of texture from the data of Table 4 {Iran are the intensities calculated for a random

specimen with k = 7.49 and B = 0.33A^)

hkl lobs Iran A B C D

111 133-8 139-9 140-6 140-2 139-9 140 8

200 190-8 146-1 146-5 146-1 145-8 146 9

220 79-7 85-9 86-5 85-9 86-7 86 3

311 47-1 46-8 46-5 46-7 46-6 46 5 222 24-7 25-9 25-9 25-8 25-8 25 9

400 15-9 12-1 12-2 12-1 121 12 2 331 18-9 20-6 20-2 20-3 20-2 20 1 420 38-8 39-8 39-2 38-9 39-4 39 6

422 39-9 41-7 41-9 42-2 41-8 41 6

333 361 33-9 34-5 34-4 34-4 34 7

Table 3: Correlation matrix

C4 k B

1-00 - 0-27 - 0-27 0-78 0-07 0-09

Ce - 0-27 100 0-15 -0-41 -0-36 - 0-28

-0-27 0-15 1-00 - 0-02 -0-77 - 0-65 Cio 0-78 - 0-41 - 0-02 1-00 -0-10 -0-06

k 0 07 - 0-36 - 0-77 - 0-10 1-00 0-8 8

B 009 - 0-28 - 0-65 - 0-06 0-88 1-0 0

'I I II

1 1 1— 0 10 20 30 40 50 aC)

Figure 2: Plane-normal orientation distribution of

the first three reflections of the specimen TiCo.ge as measured on the Schulz texture goniometer. Solid curves were obtained by fitting cubic harmonics ex- pansions to all experimental values simultaneously

(method A). • Reflections 200, x 111, o 220.
V. Valvoda 131

Table 4: Texture coefficients determined from the powder diffraction data of TiCo.ge (methods B-D) and

their comparison with the values obtained from texture measurements (method A)

A: is a scale factor, B the overall Debye parameter, r the correlation coefficient and R^j is the residual index.

A B C J-/ n

Linear least squares 1IVnn- 1 KJXi. liTipar L±1L\^CLL goniometer {Bestim. = 0 - 34A ) least squares

Reflections 111,200,220 111,200,220,311 All All

C4 0-192 0199 0-190 0-181(8)

Ce 0-072 0 067 0-077 0-073(5)

Cs 0-063 0 069 0-067 0-065(8)

Cio -0-011 -0-014 - 0-020(7)

k 7-49(5) 7-46(6) 7-47(5) 7-50(6)

0-33(2) 0-32(2) 0-33(1) 0-33(2)

r 0-988 0-985 0-989 0-984

R^i X 100 0-83 0-95 0-83 0-88

Estimated standard deviations corresponding to the last digits are given in parentheses.

by the program FOWLS (Will, Frazer k Cox, 1965)

using (10) are given in the last column of table 4; the correlation matrix of these parameters is given

in table 3. Table 2 compares intensities corrected for the tex- ture effect with the observed values lobs and values

Iran Corresponding to a random specimen (calculated

according to equation (8) with scale factor and Debye parameter values corresponding to method A).

These last two parameters were determined by linear

regression of the data presented in Wilson plots, i. e. as ln[7''(0)//2^H^''(0)] versus sm"^ 6, and their values

are given, together with the values of the correlation

coefficients, in the lower part of table 4. An example of such a Wilson plot is given in figure 3. The

goodness of the fit of the corrected and calculated intensities is also expressed in table 4 by means of the

residual index R^i based on weighted intensities.

Inspection of table 2 shows that methods B, C

and D, which are based on the powder diffractometer

data and on the known relative intensities of a ran- dom specimen, lead to corrected intensities deviating

from those obtained by method A by less than 1%.

Method D has also been tested in a Rietveld re- finement program by Ahtee, Nurmela, Suortti b,

Jarvinen (1989) on samples of Ni, Mg and NaNOa.

The results demonstrated clearly that the wholepattern fit was substantially improved when the tex- ture correction was included in the refinement, and

that the model is able to handle correction factors

that range between 0.5 and 2.3.

Figure 3: Wilson plots for the specimen TiCo.geA Before texture correction [/ = ^(0)]; o after tex- ture correction [7 = /(0)/W^(0)], based on the coefficients determined by method C.
132 Preferred Orientation Effects

5. Less exact methods and special cases

Method E: Texture Index. A texture index concept was originally introduced by Harris (1952). It is applied in quantitative phase analysis for correcting the

effect of texture, or in thin film studies as a measure

of degree of preferred orientation. The texture index

T of the reflection hi = is defined as

nhi) = -

Iihi)/R{hi)

(11)

Using equations (4) and (8), we can reformulate (11)

in terms of polar densities W as follows:

T(hi) = W(hi)/^j2^{hj). (12)

i=i

In the case of random orientation, W{hi) = 1 for all

hi , and the value of texture index is one. Harris as- sumes that, for a large set of reflections, the value of

the denominator in (12) is near one. In such a case T{hi) = W{hi), and the texture index can be used

as a texture correction instead of W^{0) in equation

(4). It has been shown by Valvoda & Jarvinen (1990)

that this is true in cases of a weak texture, whereas

in cases of stronger textures this assumption is invalid. This is because of incorrect averaging of pole

densities in the form of an arithmetic mean in the denominator of equation (12) instead of an appropriate

integral.

Method F: Single Pole-Density Profile. This

method can be used for texture correction of intensities diffracted from cylindrically symmetric flat-plate or capillary specimens, composed of effectively rodor disc-shaped crystallites. Several analytical functions used to approximate a single pole-density profile W'^{a) are presented in table 5. Dollase (1986) has tested these functions on two

kinds of trigonal samples. In the case of CaCOa

the texture was too weak to demonstrate a prefer- ence for any of the tested functions. In the case of stronger preferred orientation of (1011) planes in

huntite, CaMg3(C03)4, an R factor of 29% and unreasonably high temperature factors were obtained

without a texture correction. Correction based on

functions 1 and 2 resulted in R factors of 10% while

the remaining functions gave R factors of 6 to 7%

in the structure refinement procedure. In these re- finements functions from table 5 replace the sum on

the right side of equation (10), and the adjustable

coefficients G, 6 or r of these functions, which reflect the strength of the preferred orientation, are

refined together with structural parameters. None

of the functions (other than 6) is normalized to unit

Table 5: Mathematical forms of proposed pole-figure

profiles

(a), (b)

(c), (d)

(e)

(e)

(f). (g)

No. Function Reference

1 exp[-Ga2]

2 exp[G(7r/2-a)2]

3 exp[—Gsin^a]

4 exp[-G(l - cos^ a)]

5 b+{l-b)exp[-Ga^]

6 (r^cos^a-l-r-^sin^a)-^/^

(a)- Uda (1967), (b)- Rietveld (1969), (c)- Parrish & Huang

(198 3), (d)- WiU, Parrish & Huang (1983), (e)- Capkova &

Valvoda (1974), (f)- Sasa &l Uda (1976), (g)- Toraya &

Marumo (1981), (h)- March (1932)

integral, which means that a change in an adjustable

coefficient must be counterbalanced by a change in

the overall scale factor k. The angle a represents angular separation of a reciprocal lattice vector h of a measured reflection and the reciprocal lattice vector ho corresponding to the direction of preferred

orientation.

Functions 3 and 4 were also tested by Capkova &

Valvoda (1974) on samples of Mg and MgsCd prepared in different ways by measurements on a texture

goniometer. The single pole-density profiles can also

be constructed from the complete set of /''(O) val- ues measured on an ordinary powder diffractometer

because

W'''{a) 1^(0) R^'

V7fto(0) J/'o(0) R^

(13)

as follows from combination of equations (3), (4) and

(8) (See Valvoda, 1986). The results of both approaches are compared in figure 4. Regardless of the analytical form of ^(a), the pole

densities W^°{0) for correction of the intensity of

reflection ho can be obtained from the normalization

condition

1 r ri"^ — / W^''{a)dQ, = W^''{0) / g{a) sin ada,

47r Jo = 1. (14)

and the pole densities W'^{0) for other reflections can be derived from equation (13) because W'^{0) =

W'^°{a). Figure 5 shows the integrands of equation

(14) for the case of magnesium and magnesium alloys

with [002] preferred orientation and table 6 compares

the pole densities W°°^(0) obtained by a graphical

integration of profiles in figure 5 with those obtained

by method A.
V. Valvoda 133

Ol 1 I I L. 0 20 AO 60 80 a (deg)

Figure 4: Approximation of the normalized pole distribution fia) = 7°°2(a)/7002(o) (given by full lines)

by the set of discrete g{a) values (given by full and

open circles) calculated according to (13) from intensities measured at zero inclination angle of the

specimen, (a) Mg-specimen B, (b) Mg-specimen C,

(c) MggCd-specimen D and (d) MgaCd-specimen E.

0.6

Figure 5: The functions g{a) sin a obtained from Fig- ure 4 for specimens B, C, D and E of magnesium and

MgaCd alloys.

Table 6: Comparison of pole densities W°°'^{0)

of magnesium and MgaCd alloys as obtained from

Equations (13) and (14) with the results of method

A (after Valvoda, 1986)

Sample B C D E

Equation (13) and (14) 1-51 5-26 2-70 3 03

Method A 1-62 4-97 2-64 3 02

6. Thin films

Medium and strong textures appear in thin films eis a

rule. Strong textures complicate phase identification

and profile analysis of weak reflections. In some cases an asymmetric reflection geometry can reveal more

well defined peaks. Texture effects can also be partially eliminated by using DOSOPHATEX (doseur

oscillant de phase texture) diflTraction system introduced by Terrat (1987) which uses sample rotation

of 360" around its polar axis and a-inclinations from

-60° to -f-60° in each step of a ^ — 2^ scan. Strong

textures also complicate stress analysis by the appearance of so-called 'snake tails' in sin^ i/i plots of

lattice spacings (Marion & Cohen, 1987).

Apart from a complete texture analysis by means

of a texture goniometer, there also exist less accurate

ways of texture characterization:

(i) oj-scan of the reflection ho from the preferentially oriented planes gives a profile that can serve as a first approximation to the true single pole-density

profile (see example shown in figure 6). Its full width

at half maximum is inversely proportional to the degree of texture.

t r

= 13.6* V

1

1 15 10 5 0 -5 -10 " = ©111

Figure 6: Characterization of normal texture by the

breadth of an w-scan profile. Refiection 111 of thin

filmof TiN.
134 Preferred Orientation Effects

{ii)Texture index (Equation (11)) is another quantity that can also be used for an approximate characterization of preferred orientation in thin films. The

values of texture indices of individual reflections divided by their sum should be in some cases regarded

as a measure of the volume fraction of crystallites

grown with given planes parallel to the substrate

surface rather than a measure of deviation of orientation distribution of crystallites from the random

orientation. This point of view seems to be realistic, especially in films composed of columnar grains.

However, great care has to be given to the effect of

absorption, which is no longer independent of angle

in thin films, even in the case of a symmetrical re- flection geometry. The accuracy of the method can be estimated by checking whether the sum of texture

indices of all reflections divided by their number is actually equal to one, as it should be according to

equation (12). In cases of strong textures this condition can be strongly violated due to the reasons

discussed in section 5.

7. Conclusions

The flexibility of the expansion in symmetrized harmonics as a description of the orientation distributions of crystallites and the axial symmetry introduced by specimen spinning, which has reduced the

problem to only one variable (angle a), are two basic tools of the currently most accurate methods of

texture correction.

These corrections can be calculated from the complete set of measured intensities of reflections if the

structure of the sample is known, at least approximately. In the opposite case, when the structure is unknown, the orientation distributions of three crystallographic planes, as determined by a texture goniometer measurement, is usually sufficient for calculation of corrections for all reflections. The accuracy

of these methods is comparable with the accuracy

of intensity measurements. The most promising application is probably in structure determination and

refinement from powder data, using programs based

either on fitting of integrated intensities or on fitting

of the whole diffraction profile (as in the Rietveld

method).

Even the less accurate methods based on a particular choice of function type in describing single

pole-density profiles or on the whole set of measured

reflections, are applicable in cases when preferentially oriented systems of planes have no other symmetry equivalent members. This is usually the case

for samples composed of disc- or rod-shaped crystallites. When properly understood, the Harris texture

index can also serve as a first approximation in tex- ture characterization and correction.

References

Ahtee, M., Nurmela, M., Suortti, P., & Jarvinen,

M. (1989). J. Appl Cryst. 22, 261-268.

BuNGE, H. J. (1985). Preferred Orientation in Deformed

Metals and Rocks. Orlando: Academic Press. BuNGE, H. J., Dahms, M., & Brokmeier, H.

G. (1989). Collected Abstracts of the XII. Hungarian Diffraction Conference. Budapest: Roland Eotvos

Physical Society.

Capkova, p., & Valvoda, V. (1974). Czech. J. Phys.

B24, 891-900.

Dollase, W. a. (1986). J. Appl. Cryst. 19, 267-272.

Harris, G. B. (1952). Phil. Mag. 43, 113-123.

Jarvinen, M., Merisalo, M., Pesonen, A.,& InkiNEN, O. (1970). J. Appl. Cryst. 3, 313-318.

March, A. (1932). Z. Kristallogr. 81, 285-297.

Paakari, T., Blomberg, M., Serimaa, R., &; Jarvinen, M. (1988). J. Appl. Cryst. 21, 393-397.

Parrish, W., & Huang, T. C. (1983). Adv. X-ray Anal.

26, 35-44.

Pesonen, A. (1972). J. Appl. Cryst. 12, 460-463.

Pesonen, A., Jarvinen, M., & Kurki-Suonio, K.

(1973). Phys. Fenn. 8, 81-91.

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

Roe, R. J., & Kriegbaum, W. R. (1964). /. Chem.

Phys. 40, 2608-2615.

Sasa, Y., & Uda, M. (1976). J. Solid State Chem. 18,

63-68.

Smith, D. K., & Barrett, C. S. (1979). Adv. X-ray

Anal. 22, 1-12. Suortti, P., & Jennings, L. D. (1977). Acta Cryst.

A33, 1012-1027.

Terrat, J. p. (1987). 6th International Conference on

Ion and Plasma Assisted Techniques, Brighton. Edinburg: CEP Consultants Ltg., p. 411.

Toraya, H., & Marumo, F. (1981). Mineral. J. 10,

211-221.

Uda, M. (1967). Z. Anorg. Allg. Chem. 350, 105-109.

Valvoda, V. (1986). Powder Diffractionl (No 2), 28-32.

Valvoda, V. (1987). J. Appl. Cryst. 20, 453-456.

Valvoda, V., & Jarvinen, M. (1990). Powder Diffraction 5 (No 4), 200-203.

Will, G., Frazer, B. C, & Cox, D. E. (1965). Acta

Cryst 19, 854-857.

Will, G., Parrish, W., & Huang, T. C. (1983). /. Appl. Cryst. 16, 611-622.

Will, G. (1991). Materials Science Forum 79-82, 207-

220.
V. Valvoda 135

Appendix

Table 7: Real spherical harmonic functions Yij±('0,(p). The norm is chosen so that max \ Y\\ \ = 1

00 1X

20 0.5(3 cos^ &-!)

21+ 2 cos 1? sin cos <p 21— 2 cos ^ sin & sin (p

22+ sin^ cos 2(p

22— sin^ 1? sin 2v?

40 0.12500(35 cos* ^ - 3cos^i? + 3)

41+ 0.94695(7 cos^ -- 3) cos J? sin 1? cos (p

A t 41 — 0.94695(7 cos^ ^ -- 3) cos & sin 1? sin v? 42+ 0.77778(7 cos^ 1? -- l)sin^ '&cos(p

42— 0.77778(7 cos^ I? -- 1) sin^ d sin y

43+ 3.07920 cos??sin^ I? cos 3^

43— 3.07920 cosi?sin^ sin 3v

44+ sin* I? cos 4y)

44— sin* & sin 4^)

60 0.06250(231 cos^ i? - 315 cos* i? + 105 cos^ d - 5)

61+ 0.69140(33 cos* ^ — 30 cos^ J? — 5) cos d sin i? cos <^ 61- 0.69140(33 cos* i? — 30 cos^ — 5) cos d sin i? sin

63+ 1.41685(11 cos^ I? — 3) cos 1? sin"* 1? cos 3^

63- 1.41685(11 cos^ I? — 3) cos d sin^ i? sin Ztp

64+ 0.81675(11 cos^i? — l)sin* i?cos4^

64- 0.81675(11 cos^i? — 1) sin* d sin

65+ 3.86393 cosi?sin^ d cos b<fi 65- 3.86393 cosi?sin^ d sin 5y)

66+ sin^ & cos Gtp

66- sin^ & sin 6(p

Table 8: Cubic harmonics Kij{'d, (p)

ij Ki,{d,ip)

0 1

41 2.5(1* + 3/* + ^*) - 1.5

61 21.65625(1^ + / + z^) - 29.53125(a;* + j/* + 0*) + 8.4375

62 6V3[(x*J/2 _ ^ (j^4^2 _ ^4j,2) ^ (^4^2 _ ^4^2)J

81 ki^T-K^" + / + - 182(1" + / + 2«) + 105(1* + J,* + z*) - 17.5]

I = sin J? cos <p y = sin I? sin z = cos d
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Residual Stress Measurements with Neutrons

T. M. HOLDEN AND J. H. RoOT

AECL Research, Chalk River Laboratories,

Chalk River, Ontario, Canada, KOJ IJO

Abstract

Mechanical methods of residual stress measurement

currently claim an accuracy of ±(10 — 20) MPa for a

given stress component. This corresponds to an ac- curacy of ±0.3 X 10"'* in each of three principal strain

components. For example, the changes in the (110)

plane spacing of iron, 2.02 A, have to be obtained

with an accuracy of ±0.00006 A, which is a worthy

goal in any high resolution neutron diffraction measurement and not too far from being achievable.

The determination of residual strain fields involves a difference measurement, under identical diffractometer conditions, between a Bragg peak position

in the sample displaying the strain field and a ref- erence sample which is determined to be strain free. For confidence in the strain results we measure interplanar spacings, not just shifts in spacings, and this

requires calibration of the neutron wavelength with

a standard powder sample under identical diffractometer conditions. This permits intercomparisons

to be made among different spectrometers at dif- ferent times and between conventional and time-offlight diffractometer measurements. Presently the

typical accuracy achieved is ±(0.5 — 1.0) x 10~^%

in calibrated plane spacing measurements.

The need to measure strain tensors associated

with intergranular or type II stresses, for example

in Zr2.5%Nb alloys, poses more severe requirements.

In this case d-spacings are measured as a function

of orientation on a sample completely bathed in the

neutron beam. Strains are derived by comparison

with reference d-spacings calculated from the liter- ature, or from high temperature measurements, or from single crystals. For this situation, a high res- olution configuration with Soller collimators is re- quired. The crucial requirement is accurate center- ing of the sample on an Eulerian cradle and careful

checks on the centering by measuring first and sec- ond orders of several reflections at different sample

orientations. The accuracy in repeat measurements

of separately mounted specimens each properly calibrated against a standard powder is again in the

range ±(0.5 - 1.0) x 10-2%.

1. Introduction

The accuracy of residual stress measurements by neutron diffraction methods must eventually be able to

match the current accuracy, ±(10 — 20) MPa, claimed

for each stress component by mechanical methods.

This corresponds to an accuracy of ±0.3 x lO""* in

each of the strain components, and this is close to

being achieved.

The accuracy of elastic residual strain measure- ments depends on the factors that govern the accu- racy of any neutron diffraction experiment. These

include: optimization of the angular resolution by

appropriate choices of coUimation, monochromator

mosaic spread and take-off angle; adequate knowledge of the peak shape; and a good signal-to-noise

ratio in the diffraction peaks. However, in measure- ments of lattice strain, additional factors must be

considered.

Studies of both macroscopic residual strain fields

in engineering components, as well as intergranular

strains, often referred to as microstrains, are made

at Chalk River. Macroscopic strains occur when the

manufacture of an object entails an inhomogeneous

distribution of plastic deformation. An example is provided by a weldment, where plastic deformation

occurs on heating and cooling in and near the fusion

zone, but elastic deformation occurs further away.

The strain field has dimensions of order tens of mm

and may extend over the whole sample. Microstrains occur because individual grains in a polycrystal may

be plastically deformed in manufacture while others are not because of the anisotropy of deformation

properties in the crystal lattice. Microstrains vary on the length scale of the grain size.

136
T. M. Holden and J. H. Root 137

2. Macroscopic strain fields

The measurements of macroscopic strain fields at depth below the surface of engineering components

are possible because neutrons readily penetrate most

materials of interest. Narrow beams of neutrons, defined by slits in absorbing cadmium in both the incident and scattered beams, are arranged to inter- sect over the centre of rotation of the sample table.

The region of overlap is termed the gauge volume by

analogy with strain gauges. The sample, mounted

on an XYZ translator table, can then be moved so that any desired location lies in the gauge volume.

The diffraction peak measured at this location then

gives the average interplanar spacing, dhu, over the

gauge volume for grains whose plane normals (h,k,[)

lie along the scattering vector. High accuracy is achievable because a set of diff^erence measurements

are made under identical diff'ractometer conditions

between a Bragg peak position in the sample displaying the strain field and a reference sample which is determined to be strain free. The following sections

deal with the principal sources of error, systematic

and random.

2.1 Beam centering

The gauge volume should be centered over the ro- tational axis of the sample table so the same location

in the sample is examined for any choice of scattering

vector direction. A fine polyethylene pin is centered on the table by checking its precession on rotation of

the table through ^60" with a theodolite (Smith ei

al., 1988), which also provides a reference line of site. A plumb-bob lies over the pin as a useful secondary

reference. Since neutrons scatter incoherently from

hydrogenous materials, the detector may be set at any angle and the incident slit is traversed across the neutron beam to find the slit position at which

the count rate is maximized. The process is repeated

with the slit in the scattered beam.

2.2 Wavelength calibration r

The average wavelength passing through the slit system is calibrated with a standard Si powder sample obtained from the National Institute of Standards

and Technology (NIST). A calibration with three or more peaks yields the eff'ective zero-angle offset, 0o,

of the mechanical scale as well as the mean wavelength from Bragg's law,

A = 2dsin(^-^o). (1)

Without careful centering of the gauge volume, the

calibration measurements made over a wide angular

range will yield an incorrect wavelength and zero- angle offset. Intercomparison between the results

for the same sample at different spectrometers is then possible. For example, measurements at Chalk

River by conventional diffraction on a diametrally

compressed ring in 1986 agreed within the quoted

errors of ±0.0003A and ±0.0002A for the (111)

and (002) reflections, with measurements by time-offlight diffraction at Los Alamos in 1989 (Holden ei

al., 1990) with errors of ±0.00015A and iO.OOOieA

for the same reflections. A similar slit system had

been set up at the time-of-flight diffractometer at Los Alamos, using the 26 = 90° banks, and the slit system had been calibrated with a standard powder.

Assurance of the interplanar spacings generates confidence in the correctness of the strains.

RESIDUAL STRAIN IN A BENT PLATE

0 1 2 3 if 5 6 7 8 9 10 11 12 13 K

DISTANCE FROM CONCAVE SURFACE (mm)

Figure 1: Residual elastic strain components in a bent plate, determined with the aid of the (110),

(112) and (220) reflections of the body-centered cu- bic steel alloy.

Figure 1 shows the results of experiments on a 14.5

mm thick plate of HY80 steel that had been bent so that the regions furthest from the neutral axis had

yielded plastically. On removing the load, the plate

had retained a permanent set. Measurements are shown of strains measured in the longitudinal, trans- verse and normal directions of the plate with the
138 Residual Stress Measurements with Neutrons

aid of the 110, 112 and 220 reflections of the bodycentered structure. Two separate experiments were

carried out at two neutron wavelengths to collect this

data. Typical errors in individual d-spacing measure- ments were 1.01373 ± O.OOOOSA, 1.17100 ± O.OOOOSA

and 2.02990 ± 0.00012A for the 220, 112 and 110

reflections. The independent sets of measurements,

when converted to strain, agree within the quoted

experimental errors.

2.3 Peak Fitiing, reproducibilHy and setting accuracy

In conventional difFractometry, the angular instrumental resolution is close to Gaussian. To obtain

the peak position, width and intensity, a Gaussian

peak shape with a sloping background is fitted to

the experimental data. The precision of peak fit- ting depends on the peak width as well as the signal

strength and is ±0.003° at best. This figure is also

typical of the reproducibility when measurements are repeatedly made at the same spatial locations with

the same set-up. The precision of setting the detector angle is also in this range for our diffractometer.

2.4 Reference lattice spacings

The identification of a strain-free reference, with

respect to which the strains are calculated from interplanar spacings, is potentially a source of systematic error. The basis of mechanical methods of measuring stress fields is that cutting a small section of

material from the workpiece removes the field in that

section. In many cases, a small sample is cut from

the workpiece. Measurements are made on this ref- erence sample under exactly the same conditions as the sample of interest, in different orientations and

in several slightly different locations. This provides an average, d^j(.j, and a standard error with which to

compute the strains and their errors from dhu measured at locations within the object, through the formula,

^hki = {dhki - dlkd/dlki- (2)

If the reference sample is annealed to relieve the

stress thermally, great care must be taken not to

change the microstructure or the state of solution

of minor alloying additions.

Sometimes a localized region of the sample will

have been subjected to thermal or plastic distortion,

and then it is possible to take as reference the spacings in the sample well removed from the localized

region.

The reference lattice spacing has also been identified by computing the equilibrium offerees in the sec- tion and finding the value which most closely satisfies

the condition of no net force (Prask & Choi, 1987).

Since these conditions are for the behaviour of the

ensemble of all orientations of grains, and since different crystal orientations do display different stresses

in the same orientation, this criterion must be used

judiciously and would ideally be computed from the

average behaviour of all the different planes.

Brand (1991) has pointed out that the curvature

of the diffraction cones can be significant when the

slit defining the scattered beam is close to the gauge

volume for scattering angles not equal to 90°, and

that this can introduce effective peak shifts. This

source of systematic error can be minimized by en- suring that the slit arrangement is not changed between measurements on the object and the reference

specimen.

2.5 Grain size effects

The accuracy of lattice spacing measurements is strongly affected by grain size. The reproducibility

at precisely the same spatial location is as stated in

section 2.3, but the scatter among the results from lo- cation to location in the sample may be two or three

times worse than this for grain sizes of order 100-150

/im, which are common for austenitic stainless steel

(Holden, Root, Fidleris, Holt k Roy, 1988). To en- sure that the same grains contribute at every angle

under a Bragg peak in these conditions, it is very important to do ^ — 26 scans, otherwise the peaks will

be ragged and shifted randomly due to the statistics

of grain selection. However, even if the sample is half-angled, there can be a marked scatter in strain.

The exact strain in each grain is determined by the

boundary conditions with its juxtaposed neighbours.

For small grain sizes, the ensemble of all grain en- vironments is sampled adequately. For large grains,

only 10-20 grains may contribute to the average in a 10 mm^ gauge volume. Thus, a poor sampling of

the environment and subsequent scatter occur. The

grain size may actually be deduced from the r.m.s. variation of intensity in the Bragg peaks as we tra- verse through the sample. It is often possible to use the symmetry of the sample to make mesisurements

at many equivalent points and thereby increase the

number of grains contributing to the average result

(Holden, Root, Fidleris, Holt k Roy, 1988).

2.6 Near surface effects

When the gauge volume is only partly within a component surface, such as in attempts to measure

near-surface strains, there are possibilities for large

systematic errors. These originate from two causes:
T. M. Holden and J. H. Root 139

the offset of the effective " centre of gravity" of the

scattering volume from the centre of the spectrometer table and the selection of only part of the wavelength distribution across the neutron beam. The

latter error can be several times larger than the former if special care is not taken to minimize it. To

control the wavelength distribution across the beam,

there must be good (~ 0.3°) collimation between the source and the monochromator, as can be seen from

considerations of the incident resolution (Cooper k,

Nathans, 1967). In the work of Ezeilo et al. (1990),

the effect of the long guide before the monochromator

at the DIA station at the ILL reactor may have had

the same effect as a collimator before the monochromator.

S. 7 MuUideieciors

In order to increase the rate of data acquisition, a

position- sensitive detector [see for example, Krawitz,

Rudnick, Butler & Cohen (1986)] or a multidetector

such as the one used at Chalk River with, for example, 15- 25 closely spaced wires or counters are employed. A complete diffraction peak can then be collected with a small number of settings of the counter

position. Collimation is achieved by the distance of

the counting element to a slit placed within 20-30

mm of the sample. Parallax occurs for wires offset

from the central wire, which not only spreads the

gauge volume along the incident beam, but also cor- responds to a different counter angle than the nominal setting. Parallax is minimized by placing the

exit slit as close as possible to the gauge volume, al- though this is sometimes prohibited by the desired

movement of the sample. The precise angle of the

counter element can be computed from the offset of

the exit slit from the centre of the gauge volume.

It is important to rock the sample angle, 6, during

the measurement so that a common subset of grains

contributes to the scattering in each element of the

counter.

Near-surface measurements should not be made

with a multidetector because the parallax will cer- tainly introduce errors.

3. Grain interaction strains

Grain interaction strains arise because of the differ- ence in plasticity of grains of different crystal orientations juxtaposed in the material. For Zr and Ti,

the [0002] direction is elastically and plastically stiff while directions in the basal plane such as [1010] and

equivalent directions are relatively soft (MacEwen,

Table 1: Comparison of lattice spacing measure- ments derived from two experiments on the same

Zircaloy-2 sample

Direction Experiment #1 Experiment #2

A A

[0002] c = 5.1500 ±0.0004 5.1489 ±0.0002

[1010] a = 3.2306 ±0.0002 3.2306 ± 0.0002

[T210] a = 3.2322 ±0.0001 3.2320 ±0.0001

[1011] d = 2.4584 ±0.0001 2.4583 ±0.0001

[1012] d = 1.8948 ±0.0001 1.8947 ±0.0001

Faber k Turner, 1983). On imposing and removing

a load that exceeds the yield point for some grains,

the (0002) orientations will tend to be left in a state

of tensile residual strain, and the (1010) in residual

elastic compression. Strong crystallographic texture,

which usually occurs for Zr, can generate macroscopic differences in strain relaxation, even though

the strains are on the scale of the grains. This, in

turn, affects engineering properties such as in-reactor

growth (Fidleris, Tucker & Adamson, 1987).

The very presence of strong crystallographic texture, however, opens up the possibility of measuring

the complete strain tensor for an assembly of grains

that have nearly the same orientations, termed a family of grains in this article, because there is a

one-to-one correspondence between direction in the

sample and crystallographic direction. Grain interactions are intensive in the sense that a sample may

be cut in two halves and still contain the same information. A composite sample may thus be assembled from small pieces, free of macroscopic stress. The composite sample may have a volume of order

lO^mm^, completely bathed by the neutron beam at

all orientations.

A prerequisite is the determination of the texture

of the sample and calculation of the orientation distribution function. This permits identification of the

various "families" of grain orientation of technological interest and subsequent calculation of the sample

orientation to pick up particular reflections, say, for

example, the [1210] reflection, from each family of

grains.

3.1 Sample centering

The goal is to measure the strain tensor, Cij , for a

given family of grains from measurements of strains

in perhaps 20 orientations. The goal in accuracy is the measurement of dhku to ±0.000lA, which cor-
140 Residual Stress Measurements with Neutrons

Table 2: Comparison of strain tensors derived by fitt two experiments on a Zircaloy-2 sample.

#1 (10-^)

/ -0.3 ±0.4 -0.5 ±0.4 0.4±0.5 \

-0.5 ±0.4 3.9 ±0.4 -0.9 ±0.5

V 0.4 ±0.5 -0.9 ±0.5 -4.6 ±0.9 y

responds to an accuracy in strain of better than

±1 X 10""*. A typical high-resolution diffractometer set-up is required with the sample mounted on

an Eulerian cradle. The measurements at Chalk

River (Holden, Root, Fidleris, Holt k Roy, 1988) were made with a Ge(331) monochromator, of mosaic 0.2°, at a take-off angle of 100° and collimation

of 0.3°, before and after the sample. The wavelength was chosen to allow measurements of the {1010},

{lOll}, {0002}, {10T2}, {2020}, {2022} and {0004}

reflections. Crucial to the accuracy is the careful centering of the sample at the centre of rotation of the

spectrometer table and the centre of the Eulerian cradle. This is done optically and checked by computing

the effective offset angle, O'q, in the Bragg equation,

(1), from the first and second orders of at least a dozen reflections at different orientations of the cylindrical sample in the beam. A typical average value of

6'q for an individual sample is —0.003 ±0.005°, where

the error is the r.m.s. spread for all the different

reflections. Systematic variations in 6q must be absent. Since very high intensities can be achieved with

large volume samples and marked texture, errors of

±0.003° in the fitted Gaussians are usual.

3.2 Wavelength calibration

The wavelength and scale offset are measured with

the standard Si powder sample. The offsets mentioned in the previous paragraph will be deviations

from the Si powder, for each sample. The calibration is vital since absolute d-spacing measure- ments are being sought here. The reference values

for lattice spacing are obtained in a separate experiment, or from the literature. The repeatability of the

measurements, taking the sample off the spectrometer and measuring it again subsequently on the same

or a different spectrometer, but calibrated with the

same standard, was within the goal set in section 3.1.

Results for a Zircaloy-2 sample are shown in table 1.

ing equation 2 to 25 data points measured in each of

#2 eij (10-^)

/ -1.1 ±0.3 0.3 ±0.2 0.6 ±0.4 \

0.3 ±0.2 4.2 ±0.3 -2.2 ±0.7

^ 0.6 ±0.4 -2.2 ±0.7 -3.3 ±0.6 j

3.3 Reference d-spacings

The major source of error in strain measurements

of this kind is the uncertainty in the strain-free ref- erence, d°j(.,j. Because of the coefficient of thermal

expansion in the c direction of the hep lattice, ac, is twice that in the a direction, cxa, thermal strains are always generated on cooling from some elevated an- nealing temperature, where strains might have been

eliminated. To circumvent this problem, the temperature of an alloy was raised in steps, and the plane

spacing meeisured in different directions at each step,

until all directions gave the same spacing to within

the experimental errors. This is a strain-free spacing at the annealing temperature. Then the single crystal thermal expansion between the strain-free

temperature and room temperature was used to find

the unconstrained alloy parameters at room temperature. The reference spacings for alloys were also

found by taking the concentration derivative of the

interplanar spacing change, d/dc{dhkii), from sets of polycrystal measurements and then applying this

change to single crystal, i. e., unconstrained, plane

spacings. The two methods agree within an uncertainty of about ±1 X 10~^ for the a lattice parameter

for both Zr-2.5Nb and Zircaloy-2 but less well for the

c-parameter ±(2 — 3) x 10"'*.

3.4 Strain tensors

The strain tensor, €,j, determined with respect

to Cartesian crystallographic coordinates, (X,Y,Z),

[mo], {X,Y,Z), IT2TO] and {X,Y,Z), [0002], is obtained by fitting the measured strains to the expression,

^ = J2eijlilj, (3)

where /j and Ij are direction cosines for the direction of measurement. Six values with zero uncertainty are required, in principle, to determine a strain

tensor. For the uncertainties characteristic of neutron strain measurements, ±(10 — 30)%, we require
T. M. Holden and J. H. Root 141

many measurements to be made, and, in our case, at least twenty strain measurements are fitted to obtain the tensor elements with a precision approaching

±1 X lO"'*. Table 2 shows values of the strain tensor

determined on two separate occasions on the same

sample which agree within the errors.

4. Acknowledgments

We would like to acknowledge discussions of the ac- curacy of strain measurements with R. A. Holt, P. J. Webster and J. Schroder.

References

Brand, P. C. (1991). Thesis, University of Twente,

ECN-R-91-006.

Cooper, M. J. & Nathans, R. (1967). Acta Cryst. 23,

357- 67. EzEiLO, A. N., Webster, P. S., Webster, G. A. &

Webster, P. J. (1991). Proceedings of the NATO Advanced Research Workshop on Measurement of Residual and Applied Stress using Neutron Diffraction.

Edited by M. T. Hutchings and A. D. Krawitz. Amsterdam: Kluwer Academic Publishers, B. V.

FiDLERis, v.. Tucker, R. P. & Adamson, R. B.

(1987) . ASTM STP 939, 49- 85

HoLDEN, T. M., Root, J. H., Fidleris, V., Holt,

R. A. &; Roy, G. (1988). Materials Science Forum

27/28, 359-370.

HoLDEN, T. M., HosBONS, R. R., MacEwen, S. R.,

Flower, E. C, Boubke, M. A. Goldstone, J. A. (1991). Proceedings of the NATO Advanced Research Workshop on Measurement of Residual and Applied Stress using Neutron Diffraction. Edited by M.

T. Hutchings and A. D. Krawitz. Amsterdam: Kluwer

Academic Publishers, B. V.

Krawitz, A. D., Rudnick, P. J., Butler, B. D. &

Cohen, J. B. (1986). Advances in X-ray Analysis.

Edited by C. S. Barrett, J. B. Cohen, J. Faber, Jr., R.

Jenkins, D. E. Leyden, J. C. Russ and P. K. Predecki,

29, 163-71.

MacEwen, S. R., Faber, J., Jr. & Turner, A. P. L. (1983). Acta. Metall. 31, 657-76.

Prask, H. J. &; Choi, C. S. (1987). In Residual Stress

in Design, Process and Materials Selection. Edited by

W. B. Young, p. 21-26, ASM International.

Smith, D. J., Leggatt, R. H., Webster, G. A.,

MacGillivray, H. J., Webster, P. J. & Mills, G.

(1988) . J. Strain Anal. 23, 201-11.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Modelling Anisotropic Crystallite Size/Microstrain in Rietveld Analysis

A. Le Bail

Lahoratoire des Fluorures, Universite du Maine, 12017 Le Mans Cedex, France

Abstract

Studies aiming at modelling crystallite size/microstrain (particularly anisotropic) by a direct approach

in Rietveld analysis are reviewed. Possible evolutions

are suggested.

Introduction

The Rietveld method (Rietveld, 1967, 1969) offers a powerful approach for characterizing crystallized

powdered materials (for a review concerning mainly

neutron diffraction: Hewat, 1986). Application of

this method has dramatically increased in recent

years, partly due to the efforts to find new high Tc

superconductors. Problems in solid state chemistry

frequently involve powders, which may be easily verified in any recent issue of a specialized journal like

the Journal of Solid State Chemistry, where it is not

rare that more than half the papers make use of

modern powder methodology. Organometallic and

organic chemistry may be behind in this domain,

but this could change in the future. An increasing

number of dedicated ultra-high resolution powder instruments is available using synchroton sources (Cox,

1991); their development will probably have some impact on possible amelioration of in-laboratory diffractometers. Repeated demonstration of the feasability of ah initio structure determination from powder

data is certainly another cause of increasing numbers of applications (Cheetham & Wilkinson, 1991).

There is a new golden age for the powder method,

so more problems delaying the routine use of pattern decomposition and profile analysis in structure

determination are now coming to light. Anisotropic

line broadening intrinsic to the sample is one such

problem, particularly noticeable when using ultrahigh resolution.

Real materials may be affected by a variety of defects (concentration variation, macrostrain, microstrain, size, stacking fault, antiphase domains ...),

all able to influence the shape and width of the powder diffraction pattern reflections. Moreover, the

three-dimensional order may be highly perturbed, as

in polymers and liquid crystals, with the most ex- treme situation corresponding to amorphous materials. The subject of this paper is mainly restricted to

the attempts to interpret line broadening in terms of

anisotropic crystallite size/microstrain (S/M) effects

in Rietveld refinement (RR). Reviews on this subject (at least partly) may be found in Louer (1988),

Young k, Desai (1989), Delhez, de Keijser, Langford,

Louer, Mittemeijer k, Sonneveld (1989). Without the

structure constraint, an alternative is the determination of microstructural properties by means of pattern decomposition (for a review, see Langford, 1987

and in this volume; Toraya, 1989). Some definitions

about S/M will be given. Then earlier works taking

account of isotropic effects in RR will be briefly discussed before showing how the general anisotropic

case has been treated by various authors. Special

attention will be given to experimental applications.

A test case will be considered, and the results from

different methods will be compared. Finally, a projection for the future will be attempted.

Statistical approach to size/microstrain

There is an enormous specialized bibliography about

effects of imperfections of all kinds on diffraction phenomena, and some controversy. Because theories are generally not directly applicable, quite a large number of numerical methods were derived to approach

experimental cases. They cannot be detailed here;

the reader will be generally directed towards recent

review articles. Several models have been proposed

to quantitatively estimate lattice distortions. For instance, different formulae have been given for the ex- perimentally observable integral breadth. According

to the work of Warren & Averbach (1950) (WA), the

integral breadth, for "microstrain" broadening is a function of the mean-square lattice distortion (e^);

/?<! is related to an index of the "paracrystalline" distortion in the work of Hosemann k, Bagchi (1962); /?d

is related to the "maximum distortion" e defined by

142
A. Le Bail 143

Wilson (1949); a direct relationship exists between

e and (e^) (Buchanan, McCullough k Miller, 1966).

In the more recent controversy about the S/M view

of WA for cold-worked metals, one can find a theory developed for symmetrical X-ray diffraction lines

broadened by dislocations (for a review, see Wilkens,

1979). Ergun (1970) stated that highly defective

lattices must be characterized by a "mean defectfree distance" rather than a domain size. Nevertheless, the method most applied (when possible, i. e. if harmonics are isolated, see below) remains the WA

method. Its formulation was based on the work of

Bertaut (1949) for the definition of the size distribution. It is important to note that the WA theory is general and must not be confused with the so-called

WA multiple line method for S/M separation, which

uses some assumptions.

According to the formalism of Warren (1959,

1969), the experimentally observable power per unit

length P'(26) of the intersection of the diffraction cone with a receiving surface at a distance R

from the sample (equation 13.24 in Warren, 1969)

may be expressed as a Fourier series, whose coefficients are A(n) = (cos 27r/Z„)A^„/A^3 and B(n) =

{sm2'KlZn)Nn/Nz for a family of 00/ reflections.

Nn = Yl'iLinli'^ ~ inl)P{i) is the average number

of n pairs of cells per column; P(i) is the fraction

of the columns of length i cells (i. e. the surfaceweighted size distribution) and A^3 is its mean (a

related "volume-weighted" size distribution function

may be defined); Zn is the component (along the

normal to the diffracting plane) of the displacement

vectors for a pair of n apart cells. In fact, this

Fourier series represents the "true profile" /, which,

by convolution with the "instrumental profile" g, gives the experimental profile h = f*g. At this

stage, no assumption about the distribution Pn{Z)

of local strain is made. Thus asymmetry of a re- flection is possible in this rather general formulation (asymmetry cannot be due to size effect but

to strain according to the sine term). This formulation clearly shows that strain effect on broadening is dependent on /, whereas size effect is not;

the possibility of separating the two contributions

originates from this different behaviour in 29 for the

same hkl family of reflections. However, some simplifications are necessary. Concerning microstrain,

the assumption that (cos27r/Z„) = exp(—27r^/^(Z^))

is exact if local strain has a Gaussian distribution

Pn{Z) = (a/-y/^) exp(—a^Z^), whatever the value of

n. This assumption (or expanding (cos27r/Zn) as a

series approximated by a few of the first terms) provides a way to separate size and microstrain effects

if at least two orders of a reflection family are available. As this is rare, methods were developed to separate S/M effects from one reflection, called "single

line" methods (SL). Assumptions used in SL methods come from the fact that the profile shape is governed by some particular variations of {Z^) versus n.

If the local strain distribution Pn{Z) is Gaussian, the

associated reflection profile shape is Cauchy-like if (Z^) = n{Zf); it is Gaussian if {Z^) = n^Z^). Concerning the size effect, Cauchy-like profile has been

widely adopted, though this corresponds to a very

particular size distribution function. To be clear, the

same profile shape associated with size and microstrain effects renders the separation impossible with

SL methods but possible with whole pattern fitting

like RR because of the different 29 dependency. Recent reviews on SL methods may be found in Delhez,

de Keijser Sz, Mittemeijer (1982); Nandi, Kuo, Schlosberg, Wissler, Cohen k Crist (1984).

Because of these approximations, the "size" and

"microstrain" parameters as obtained from the WA

or SL methods need to be used cautiously. "Sense

and nonsense" of these parameters were discussed

recently (Delhez, de Keijser, Mittemeijer & Langford, 1988); the question of accuracy of S/M values

from X-ray diffraction line profiles using Fourier se- ries was treated by Delhez, de Keijser & Mittemeijer

(1980). It is a paradox that, as will be apparent below, current methods aiming at extracting S/M by

direct approach in RR use mainly the basic principles of SL methods, while the Rietveld method was

developed to minimize overlapping effects.

Profile shapes in Rietveld analysis

Many analytical profile shapes are routinely used in

RR. Some are fortunately compatible with oversimplifications associated with the SL methods. They

were mainly introduced for RR applications to Xray powder data (for a review: Young k Wiles, 1982).

For constant wavelength powder data, the list of such

profile shapes includes Gaussian (G), Lorentzian (L)

(or Cauchyan (C)), intermediate Lorentzian (IL),

Voigtian (V) (but see below) and its approximations

the pseudo-Voigtian (pV) and Pearson VII (PVII).

Other descriptions than analytical have been used in

RR: learned profile shape by numerical fitting (Hepp

k Baerlocher, 1988) or by Fourier coefficient representation (Le Bail, 1984); analytical Fourier coefficients were also used (Le Bail, Duroy k Fourquet,

1988).
144 Modelling Anisotropic Crystallite Size/Microstrain

Modelling isotropic crystallite

size/microstrain in Rietveld analysis

Formulae enabling extraction of S/M parameters af- ter a RR using V, pV and PVII were listed by de

Keijser, Mittemeijer & Rozendaal (1983). The possibility of anisotropic effect was not considered, and

no application was given. The cases of PVII and pV

were treated by giving equivalence of breadth with

V (the Voigt function was introduced in RR later).

Extraction of isotropic particle-size and microstrain broadening have been taken into account in a modified Rietveld program by Cox (1984), introducing the constraint of the different 16 dependency

for size and microstrain using pV.

Considered as the best approach to profile-shape

for S/M interpretation in a simplified view, the

Voigt function was used much earlier in SL methods (Langford, 1978; de Keijser, Langford, Mittemeijer k. Vogels, 1982) than in the Rietveld method

(Ahtee, Unonius, Nurmela & Suortti, 1984; David

k, Matthewman, 1985). The latter took advantage

of the possible physical meaning of V by extraction of a mean crystallite size for a de-intercalated

Lii_j;Mn204 spinel. The L part of V, with a 1/ cos^

dependency, was assumed to come from the size effect.A neutron powder diffraction study of materials

presenting various peak shapes and crystallite sizes was realized by Hill &: Howard (1985), showing that

at least a part of isotropic S/M effects is handled empirically by standard Rietveld procedures with pV

or PVII. Convolute profiles were obtained numerically by Howard & Snyder (1985,1989). The algorithm was implemented in a modified Rietveld program able to extract isotropic S/M parameters; applications were made on La-substituted SrTiOa and

Al specimens (Yau k, Howard, 1989; Howard, Yau k,

Anderson, 1989).

Young k Desai (1989) reviewed the literature (including anisotropic) and developed an approach al- lowing both the size and the microstrain broadening

to have both L and G components, using the pseudoVoigt function of Thompson, Cox k Hastings (1987).

Preliminary results with MgO (no anisotropy modelled) suggested that the assignment of the L component to size and the G component to microstrain

broadenings cannot be a general procedure.

The most amazing applications of the Rietveld

method including isotropic microstrain effects were

for modelling the structures of some amorphous materials. The simultaneous fit of three neutron patterns for Pb2M2F9 fluoride glasses (M = Zn,Fe;

Mn,Fe; Mn,V) was made by using different crystalline models, supposing isomorphous substitution

(Le Bail k Jacoboni, 1984; Le Bail, 1985a; Le Bail,

Jacoboni k De Pape, 1985). The simultaneous fit of

the neutron and X-ray patterns for a BaZr2Fio glass was also done (Le Bail, Boulard k Jacoboni, 1987).

Moreover, the structure of liquid alloys Ali-^Mi; (M = Mn or FeCr) was simulated in this way (Maret,

Chieux, Dubois k Pasturel, 1991). Instrumental ef- fects were neglected in these studies. Atomic coor- dinates and cell parameters of the crystalline models were refined (thermal motion effects were considered as included in a refined disorder parameter

having the (Zf) meaning) using G profile shapes

(their widths were so large that more than 1000 re- flections could overlap, depending on the structuremodel complexity).

Rietveld application to amorphous materials may

be used to eliminate wrong structural models. However, it is insufficient to establish the validity of a model, even if the fit is quite "good." Very good fits (Rp < 4%) were generally obtained with unrealistic

individual distances between pairs of atoms in the

final refined model. However, the mean distances for

particular types of pairs were generally credible. After all, this holds also for other methods of simulating

amorphous materials structures, such as molecular

dynamics or model-building with relaxation.

Modelling anisotropic S/M in Rietveld

analysis: chronology of attempts

Attempts to model anisotropic S/M effects in RR began in almost the same years that attempts to model

isotropic effects began. The exhaustivity of the list of studies presented below, although they are not so numerous, cannot be guaranteed; not all were in the

Journal of Applied Crystallography. Many of them

were found in congress proceedings.

1985- Greaves (1985) encountered a problem of

anisotropy in the neutron study of Ni(0D)2. The

RR was unsatisfactory with Rt^p = 14.1% (using

G). Individual fits of some reflections indicated that

the crystallites were plate-like with mean size 400A

and I8OOA along [001] and [110], respectively. The

variation law of the FWHM was modified by adding

a term depending on the angle between [hkl] and

[001]. This way, R^p decreased to 8.5%. This may

be considered to be the first empirical approach to

anisotropic broadening (G functions correspond to

unphysical P(n) < 0 for low n values).
A. Le Bail 145

The idea of approaching anisotropy for both size

and microstrain by rank 2 symmetrical tensors was

soon tested (Le Bail, 1985a, 6) using a modified

Rietveld-type program specially conceived to treat such effects. A three-dimensional mean apparent co- herency domain and a three-dimensional mean microstrain were defined, in which the mean size M;,jt/

and {Zf)hki were both approximated by using an

equation giving the distance from the origin of an el- lipsoid to its surface. This way, some self-consistency

in the anisotropy is forced that would not be obtained by simple independent fits of widths. Applications were to ZnO, NiF2, and W samples (the

latter being isotropic) that were highly deformed by

grinding. The instrumental g contribution to the

h profile was obtained from well crystallized samples (with same formula). The process uses a description of profiles by Fourier coefficients. This

choice was made in order to simplify the convolution of size and microstrain associated profiles between themselves and with the g profile (multiplication in Fourier space). This method will be denoted

the "Rietveld Fourier" method (RF). Some oversimplifications were used: A^{hkl) was modelled to cor- respond to L profiles and Af^{hkl) to profiles continuously varying between L and G by using the hypothesis {Z^)hki = {ZDhki and refining K (Adler &

Houska, 1979). A quasi size-only sample was also

studied: Ni(0H)2 prepared by precipitation. Profile

shapes associated with the K value for microstrain were found intermediary between G (K = 2) and L (K = 1); K took the value 1.4 for both cold-worked ZnO

and W. However, the shape was a "super-Lorentzian"

for NiFz (K = 0.8).

1986- An application of the RF method to the neutron powder diffraction patterns (50K and 4.2K) of

MnNbFe (Bizot, Chassaing, Pannetier, Leblanc, Le

Bail & Ferey, 1986) greatly improved the fit, decreasing Rp values by more than 4%. The anisotropy effect

was complex (reflections with / indices larger than h

or k are sharp, whereas the others are significantly

broadened) and no attempt was really made to further understand its physical origin.

Using the same RF method, Gibaud, Le Bail &; Bulou (1986) presented a study of antiphase domain size

effects for KAIF4 (easily produced by simple handpulverizing, with no strain effect). The broadened re- flections were due only to some F atoms (K, Al and a

part of the F atoms did not contribute to these reflec- tions because of their particular positions, leading to

special extinction conditions). For such a problem,

there are at least two sets of reflections to be treated

differently in line shape and width; the g contribution was obtained from the sample itself in a refinement

excluding the broadened reflections.

Uniaxial anisotropy of strain together with the

differently constrained S/M 29 dependence were

used with neutron time-of-flight (TOF) data for

Pbo.76Cao.24Ti03 (Rotella &; Richardson, 1986,

1987).

Thompson, Reilly, Corliss, Hastings k, Hempelmann (1986) studied a deuterided LaNis sample ac- cording to a two stage process. First the profile

parameters for S/M effects were deduced from a Williamson-Hall (1953) plot; second they were imposed in the Rietveld refinement made with pV, us- ing G (with imposed 29 dependence) for the g function (from a ZrNiAl reference sample), L for the size

effect (isotropic) and G for the distortion (anisotropic

with no effect along [001]).

1987- Thompson, Reilly k Hastings (1987) re- ported a study of de-deuterided LaNis, assuming an

isotropic size effect (L shape); the microstrain was

allowed to vary anisotropically (G shape) through

the elasticity equations, reducing to 4 unknowns in

the hexagonal system (21 components in the most

general case). The g profile was fixed in the refinement and the f*g convolution was realized through

the possibilities of pV.

Lartigue, Le Bail & Percheron-Guegan (1987) published another application of the RF method to

LaNisDe.?. This material is strongly affected by re- peated operations of deuterium (or hydrogen) absorption and desorption. The distortion along [hkO]

directions of this trigonal phase was found to be

much larger than along the [00/] direction, and this

was consistent with the size effect (the mean apparent domain sizes for the volume distribution were

estimated to be 160 and > 5000A, respectively, in

these directions). Both size and strain anisotropy were modelled by ellipsoids of revolution along the c axis. The major interest of this study weis to propose a repartition of deuterium atoms from the simulta- neous structure refinement.

TOF neutron diffraction study on Pd2MnSn

was reported by Kamiyama, Shinohara, Tomiyoshi,

Asano, Yamamoto k, Watanabe (1987). The simultaneous effects of crystallite size, microstrain and antiphase domain boundaries were taken into the standard Rietveld method; full details were given later

(1990; see below).

1989- Delhez, de Keijser, Langford, Louer, Mittemeijer k Sonneveld (1989) reviewed the litera-
146 Modelling Anisotropic Crystallite Size/Microstrain

ture about crystal imperfection broadening and peak

shape in the Rietveld method, and proposed a two

stage approach. In stage I, pattern decomposition is used to obtain line-shape parameters for a number

of more or less resolved reflections in the pattern;

with the help of these data, the hkl dependence of

the line-shape parameters is modelled. In stage II, crystal-structure refinement by the Rietveld method

is performed using the line-shape model obtained in

stage I. This is not exactly a modelling of S/M in

RR since the approach is not direct. Application was made to ZnO-B data, (Langford, Louer, Sonneveld & Visser, 1986) for which stage I allowed the

conclusion that two different laws could fit the angular width variation of 00/ reflections apart from all the others (with PVII) well. These laws, once es- tablished, were kept fixed in stage II. However, the

authors tried ultimately to refine the set of line shape

parameters corresponding to the non-00/ reflections.

1990- The RF method was also applied to improve

the fit during the ab initio structure determination of

Cu3V207(OH)2 •2H2O (Lafontaine, Le Bail & Ferey,

1990). The anisotropic broadening was found to be

mainly due to size effect. This was not unexpected,

because of the method of synthesis (precipitation)

and the layer structure adopted by this material.

The full details of the TOF neutron diffraction

study on Pd2MnSn were reported by Kamiyama,

Shinohara, Tomiyoshi, Minonishi, Yamamoto, Asano

&: Watanabe (1990). All the contributions of defects (size: Lorentzian profile, anisotropic; microstrain: Voigtian profile, anisotropic; antiphase domain boundary with particular hkl dependency) and

the gprofile were expressed in terms of Fourier coefficients.

Simultaneous structure and S/M refinement by the

Rietveld method was also proposed by Lutterotti

& Scardi (1990). They used pV functions; the

anisotropy of both the crystallite size (corresponding to L profile shape) and microstrain were modelled

by using two tensors in a procedure similar to that

used for the temperature factors. Applications were made to tetragonal Zr02, tetragonal Zro.82Ceo.i8O2

and Q;-Al203.

1991- A study of La2Ni04 presenting an anisotropic

microstrain effect (no size effect) was made by

Rodriguez-Carjaval, Fernandez-Diaz &i Martinez

(1991), using the FULLPROF program (RodriguezCarjaval, 1990). The particular hkl dependence of

the contribution to the FWHM of reflections was described with only one adjustable parameter. FULLPROF is one of the most versatile Rietveld programs

for fixed wavelength, based on the code of the Wiles

k Young (DBW) program (1981). In its 2.1 version, FULLPROF proposes no less than nine possible pre-established models for domain size and eight

models for microstrain eflfects. The general formulation of the anisotropic size effect assumes that the

average diameter can be expressed as an ellipsoid

with refinable parameters (six maximum), with L

profile shape. Up to ten parameters (corresponding to the fluctuations and correlations of direct cell parameters) may be refined to describe the general

anisotropic strain broadening case, with G profile

shape.

Scardi, Lutterotti, Di Maggio k Maistrelli (1991)

improved the method proposed by Lutterotti ei al. (1990) by considering a more general expression for

the crystallite size distribution leading to both G

and L components of profiles. Applications were to

Zro.85Ceo.15O2, Ni(0H)2 prepared by precipitation

and a two-phase sample Ni(OH)2-NiO obtained by

thermal treatment of Ni(0H)2.

Millini, Perego &; Briickner (1991) implemented in a RR program a theoretical approach relating the

FWHM (L profile) of each hkl reflection to the av- erage crystal lengths parallel to the crystallographic

axes. The instrumental part was treated with G profiles (X-ray powder data), and the convolution f*g was taken into account by pV profiles. Applications were made to layered zirconium phosphites and zir- conium phosphate-phosphite derivatives with large

improvements in agreement factors when compared

to RR without crystallite size optimization.

Modelling anisotropic S/M in Rietveld

analysis: comments

The evolution of methods in the above list of applications to real cases, since 1985, is to increase complications from the most simple approaches for which

both S/M effects were each assigned to have a quite

precise profile shape (L and/or G) to the most complex for which both effects would be described as having both L and G components to profile shapes.

Two very distinct types of profile-shape representations were used: analytical or by Fourier coefficients.

However, up to now, these two approaches use similar approximations to model the size and microstrain

effects. This could change in the future: all that is done with analytical profiles may be done easily with

Fourier representation, the reverse being far from

true (see below).
A. Le Bail 147

No example of application of modelling anisotropic

S/M in RR for ultra-high resolution synchrotron

powder data was found in the list above. Of course, there is a degree of line broadening associated with

S/M effects rendering such ultra-high resolution useless. However, below such a limit very fine effects are expected to be observed from synchrotron data, and

they will probably be very difficult to model.

Extraction of S/M effects from whole-profile fitting

procedures has advantages relative to the WA multiple line method. The series expansion of the cosine

term is not truncated; as it is a fit, there is no profile

truncation effect in the usual sense of the WA analysis (there is no Fourier inversion). However the location of the background has no exact solution. Also,

there is no need to deconvolute to obtain /; instead,

profiles are convoluted. Nevertheless, convolution of

analytical profiles may prove to be difficult when the

g (and also /) profiles are asymmetric. The limitations are those of the oversimplifications retained: if the sample shows effects not exactly described by the

formulae used (this would generally be the case), the

kind of error is not predictable; large discrepancies

in results are expected when applying different approaches to the same data set. Actually, there is no example where an asymmetry effect intrinsic to the

sample has been taken into account, with extraction

of a physical meaning, in RR.

The simplest and most efficient way to model

anisotropic effects is certainly to describe the variation in real space (in directions normal to the re- flecting planes) of mean size and microstrain by ellipsoids. The nature of powder diffraction, which

is to superpose the equivalent hkl reflections with a

multiplicity p, leads to limitations of such a procedure. Lutterotti & Scardi (1990) noticed that it was

necessary to apply some restrictions similar to those

used for the anisotropic temperature factors in order

to respect the symmetry properties of the crystal, so they used Mn = M22 ^ M33 and M13 = M23 ^ M12

(four unknowns) for a tetragonal case belonging to

the 4/mmm Laue class. Restrictions could in fact

be more drastic, and only two unknowns should be

refined in this case unless certain conditions are fulfilled, or unless one is not trying to obtain an ellipsoid. As the anisotropic thermal parameters cannot,

in general, be used with the factored form of the

structure factor expression (the contribution of each

atom to the structure factor must be calculated separately and added), the six parameters corresponding to the general ellipsoid description cannot be re- fined unless the hkl contributions are considered separately and added (with the exception of Friedel's

pairs): i. e., the classical multiplicity factor cannot

be used if the symmetry is not triclinic. Anisotropic

temperature factors have the same effect on equivalent reflections, anisotropic S/M effects do not. To

illustrate this, imagine an orthorhombic phase with a

needle-like shape elongated in the a direction showing a topotactic phaise transition to a tetragonal

phase with correspondence of cell parameters. In

such a case, the inevitable superposition of hOO and

0^0 reflections on the powder pattern would probably lead to the conclusion that the size distribution

function is bimodal (Young & Sakthivel, 1988), with

profiles quite difficult to reproduce. Can we expect

to treat such a problem with an ellipsoid that would

not be of revolution along the c axis? To try that,

it would at least be necessary to calculate separately

the shape and width of the hOO and 0^0 reflections.

Instability of such a calculation is probable; it would

be necessary to start refinement by giving different

values to Mn and M22- Now if the refinement is performed without separating the equivalent reflections

(Laue class 4/mmm) it is a priori nonsense to refine

more than two parameters (Mn = M22 ^ M33), i. e. an ellipsoid of revolution along the c axis. In this sense it is not possible to detect any anisotropy for a

cubic compound (special effects on some hkl classes,

corresponding to stacking faults, antiphase domain

boundaries and so on, are of course not concerned in

this discussion of a generalized anisotropic effect).

So, what does one do with the 21 parameters of the

elasticity equations in the procedure of Thompson

ei al. (1987) or the ten parameters of FULLPROF

in the case of microstrain effects? Restrictions were

not given so the user must be careful. I recognize

not having respected these necessary restrictions in one case, when applying the RF method to NiF2 (Le

Bail, 1985 a, 6). This material has the rutile structure

belonging to the 4/mmm Laue class; I tried to refine

independently three unknowns for both S/M effects,

the 11 = 22, the 33 and the 12 terms of the symmetrical rank two tensors. Since I did not separate the

hkl from the hkl during refinements, I must now con- clude that, for being self-consistent, the broadening

effect should be the same for the hkl and hkl reflec- tions, and thus the cross term in my formula ought

to be |/ifc|Si2 and not hkSi2- Thus, considering the

refined values I had obtained, it was not an ellipsoid

that defined the mean domain size; it was a mathematical object of which the section in the 00/ plane was not an ellipse but a sort of four-branched star, a

quite curious shape for the "mean crystallite."
148 Modelling Anisotropic Crystallite Size/Microstrain

Quite excellent results have been obtained in most

of the experimental cases cited above (i. e. low Rp,

indicating some quality of the fit, comparable to

those obtained when no S/M effect is present). The

excellence of these results is in fact questionable,

and it must be emphasized that it concerns generally "school cases": materials with simple struc- tures (having few reflection overlapping problems)

sometimes specially prepared to show the feasibility of such studies. The cold-worked materials, like

ZnO for instance, probably really present nearly

centered and symmetrical Gaussian local distributions of microstrain (corresponding to the approximation made), or materials with size-effect only may

have size distributions well approaching that corre- sponding to Lorentzian profile shape (three independent workers studied anisotropic size effect by RR

of Ni(0H)2 or Ni(0D)2; several teams studied ZnO

as a test case). In spite of these "excellent" results, one must realize that a common point to all proposed methods is an extreme oversimplification unfavorable to any accuracy in the deduced crystallite

size and microstrain parameters. However, the accu- racy in structure parameters certainly benefited from

the account of anisotropic effects. In most of these

studies, the comparison of results with and without

S/M modelling evidenced a considerable decrease in

Rp and Rb (up to 10% in some cases).

The first version of the RF method (Le Bail,

1985a, 6) had the inconvenience of modelling g from

a reference reflection, which must be chosen isolated on the pattern of a well-crystallized sample. The

whole-pattern fit was realized (Le Bail, 1984) by con- voluting the reference profile with an analytical one

having a pseudo-Caglioti 29 dependence (Caglioti,

Paoletti k, Ricci, 1958). This was a problem when

no reference reflection could be isolated. This inconvenience was suppressed in a further version in

which the g function could be modelled more easily

from analytical Fourier coefficients (Le Bail, Duroy

&c Fourquet, 1988). However, the first version may

sometimes work better for modelling g (particularly

when asymmetry is present at low diffracting angle).

The two stage process of Thompson et al. (1986)

is very similar to that proposed later by Delhez et al. (1989). The first stage would be hardly practicable

in case of important overlapping. On the other hand,

it is evidence that interpretation of S/M in RR is not

realizable in one stage for any sample. There cannot

be any routine in such a kind of study; each case is a special case. In fact, the first stage may be: one

observes that standard RR gives poor results; stage

two: one tries to identify what kind of problem it

is; stage three . .

.

, but why not refine the profile

parameters in the last stage? Remember also that

doing a RR supposes that the structure is known, at

least for a large part.

Some selected RR cases without S/M

modelling

Problems of anisotropic line broadening were encountered in many published papers using standard RR in

which some efforts merit to be emphasized. A very

pronounced anisotropic broadening effect considerably limited the quality of the Rietveld refinement in

the case of PbC204 (Christensen, Cox k, Lehmann,

1989), although it did not make impossible the ah

initio structure determination. The structure being

triclinic, PbC204 would be a very interesting example for testing the general anisotropic case with ellipsoids. An approach to S/M was presented from individual peak fits of the synchrotron data that revealed

all of the reflections to be broadened appreciably by

comparison with the instrumental resolution. Those

with scattering vectors close to the [001] axis showed

an additional systematic broadening consistent with

stacking faults in this direction. Rb and Ru,p (%)

were respectively 10.0 and 21.3 for synchrotron and

10.1 and 17.9 for neutron data. This shows how the

problem was not anecdotal.

Some authors have sometimes detected problems

of stacking fault, antiphase domain and so on (effects on some classes of reflections), and have tried

to fit two independent sets of linewidths (or more)

in order to improve the structure parameters. This

has been done for two ah initio structure determinations from powder data with qualitative interpretation of faulting. For instance, reflections hkl, I odd

were fitted with different profile parameters than

those with / even in the case of Rb(V02)(HP04)

and Cs(V02)(HP04) (Amoros, Beltran-Porter, Le

Bail, Ferey k, Villeneuve, 1988). Reflections hkl,

h odd were treated separately from the others in

the case of Cd3(OH)5(N03) (Plevert, Louer k, Louer,

1989). Some other works where a significant amount

of anisotropic broadening was noticed are: the study

of La2Cu04+i at 297K (ZoUiker, Cox, Parise, McCarron III k Farneth, 1990); the study of the

VOP04 -2D20 structure (Tachez, Theobald, Bernard

k Hewat, 1982).
A. Le Bail 149

Table 1: Size and microstrain parameters for ZnO-B

(D)v (A) e X 10^

lines A B C D E A B C D E

(100) 392 3.8

(hOO) (200) 352 318 287 260 288 2.7 2.9 2.3 2.0 7.1;2.5

(300) 186 1.9

(hOh) (101) 327 305 313 337 316 3.3 3.2 2.7 2.9 7.3;2.5

(202) 238 2.5

(00/) (002) 484 417 408 490 598 2.3 2.0 1.6 2.0 7.8;2.2

(004) 391 1.7

A : PVII, single line (Sonneveld ei ai, 1986)

B : PVII, multiple line (Sonneveld et ai, 1986)

C : PVII, multiple line (Langford ei ai, 1986)

D : pV, multiple line (Langford et ai, 1986)

E : RF method (Two approximate "e" values are given at distances 7.36A

and at (D)v.)

Application of the RF method to ZnO-B and

comparison with previous results

A rather well-documented "size-strain" test case is ZnO-B. The conventional X-ray powder pattern has

been studied by means of Total Pattern Analysis

(Langford et ai, 1986; Sonneveld, Delhez, de Keijser, Langford, Mittemeijer, Visser & Louer, 1986).

Results have been extensively discussed by Langford

(1987) , Langford, Delhez, de Keijser &; Mittemeijer

(1988) and Delhez et ai (1989). The latter made a RR on the ZnO-B data (see above), with final residuals Rb = 0.028 and R^ = 0.039 (background (probably) not subtracted (BNS); to understand the in- terest of giving "background subtracted" (BS) Rp,

see Hill & Fisher, 1990). The RF method was thus

applied in its original form (Le Bail, 1985a, 6) to the

same data set for comparison purposes. The g profile

component of h was modelled from the ZnO standard

(Langford et ai, 1986) in a preliminary step by a RR

(Le Bail, 1984) leading to the residuals Rb = 0.022

and Rp = 0.089 (BS). Results on g would probably

have been better without the presence of a small 0^2 component residue (< 3%). Good results are really

essential on the g part in order to model correctly

h = f*g in the next step. It is noteworthy that the

best RR results on this standard ZnO data set obtained by using the FULLPROF program were with

PVII, leading to residuals Rb = 0.031 Rp = 0.124

(BS) or Rp = 0.108 (BNS). Then, the ZnO-B pattern WcLS fitted keeping the g part fixed, using el- lipsoids of revolution along the c axis for both the

S/M effects, with L shape for the size and intermediate G/L shape for the microstrain. Final reliabilities were Rb = 0.017 and Rp = 0.063 (BS), with quite

comparable structure parameters as those quoted by

Delhez et ai (1989).

The S/M values are compared in Table 1 with

those from previous studies. As the RF method gives

the mean of the area-weighted size distribution function (D)a, it was multiplied by two to be compared

with {D)v (volume weighted), the relation being ex- act for L profiles. On the other hand, the RF method

gives {Z^) = lnl^{Zf), and K was refined to 1.43(6);

thus the comparison with e given in previous studies is not straigthforward, since e = (l/2)\/27r(e^)^/^

(de Keijser et ai, 1983) only if (e^) = (e^) (note that

(^n) — {•^n}/'^^); ^- if ^he microstrain profiles are
150 Modelling Anisotropic Crystallite Size/Microstrain

G (corresponding to K= 2). Thus, two values calculated as (l/2)v^((-^i}/n/^-2)i/2 were given as an

approximation for e (at n = 1, corresponding to the

choice of a 7r359A reference length, associated with

the interval of definition of the reflections, and at the n value corresponding to (-D)v). The discrepancies in

Table 1 give an idea of the poor accuracy now attainable on S/M parameters. All methods use large as- sumptions, and it is not possible to find out whether

one result is better than another. One cannot expect

to obtain more than an order of magnitude for the

extracted S/M parameters (in this sense, the results

of Table 1 are quite similar).

In the future

It is not yet possible to recommend a particular

choice of a procedure for analysis of anisotropic line

broadening effects by RR from the jungle of possibilities listed above. What will be useful in the future

is somewhat unpredictable. However, any progress

in line broadening understanding will certainly come

from the fact that whole pattern fitting, with or without a structure model, will allow the suppression of

some oversimplifications, since overlapping problems

are partly overcome in these methods. Some directions for doing that may be suggested that could be

taken from the recent literature or by a complication

of formulae used currently, going crescendo towards

the general 13.24 equation in Warren (1969).

There still exist activities in the S/M domain

which have been said to be in infancy. Some selected

works are listed in the following. An "Information

Theory" approach was devised in order to obtain

crystallite-size distributions from X-ray line broadening by Guerin, Alvarez, ReboUo Neira, Plastino

& Bonetto (1986). In very recent works, some approaches to flexible size distribution functions were made. It has been shown that the Voigt functions

may be considered to be a good approximation to

some particular size distribution functions P(n) (Seli- vanov & Smyslov, 1991). Moreover, exactly the same

formulation was proposed for the strain effect (Yuming, Shansan & Yenchin, 1982), so the description

of both S/M effects by V functions may have physical justifications. Other recent work that could serve

as a basis for modelling size effects quite different

from "Cauchy-like" is that of Rao Sz Houska (1986).

They showed (for a normal distribution of diameters

of statistically spherical subgrains) that the diffraction profile is not very sensitive to the fine details

of the size distribution, but is rather sensitive to the

mean column height N3 and the column-height variation coefficient, V^c, the latter giving a measure of

the spread of the distribution. Two parameters may

thus describe A^ifikl). The question is how to insure

some self consistency if subgrains are statistically

ellipsoidal instead of spherical? Asymmetry problems may have a solution: random-layer line profiles

may be synthesized by Fourier transform methods,

as shown by Ruland (1967). A theory has been developed for asymmetric line broadening of plastically

deformed crystals (Groma, Ungar &: Wilkens, 1988).

A new method of synthesizing the reflection profile,

based on the calculation of Fourier coefficients of instrumental functions, was recently presented (Kogan

& Kupriyanov, 1992). Ideas may be found in the

domain of modelling powder patterns; for instance, a general recursion method for calculating diffracted

intensities from crystals containing planar faults has

been proposed recently (Treacy, Newsam & Deem,

1991).

Improvements could be obtained by the suppression of some practiced approximations having catastrophic effects when reflections are strongly broadened. Actually, the Lorentz-polarization factor variation over one peak, which may be quite large at low

angle, is not taken into account. Instead, the value

of LP at the 26o position is applied to the entire reflection.

Now, the possibilities of going more closely to

the 13.24 equation of Warren (1969) will be considered. Profile shape description by analytical functions, such as L or G or convolution of these two

(Voigtian), seems to be in a "golden age," but may

have no future in the sense that evolution is hardly

possible because these shapes correspond to very special cases of S/M. There are not many other analytical profile shapes corresponding to physically realistic S/M effects, and the inevitable convolution

h =f*g, which may have a simple analytical solution for L, G and V, could cause some problems with

other ones, although convolute profiles can be obtained numerically (Howard & Snyder, 1985, 1989).

Quite good fits, according to actual criterion of what

is a good fit may be Hp < 12% (BS), are obtained

with V, PV and/or PVII even for cases with strong

anisotropy. A possible limit to the future evolution

with analytical profiles appears to be consideration of

S and M as having both L and G components, using

ellipsoids or something else to describe anisotropy.

However, this affirmation may be wrong.

The alternative to analytical profile shapes, i. e. the description of profile shapes in terms of Fourier
A. Le Bail 151

coefficients, has large possibilities of evolution for two

main reasons: the mathematical convolution is obtained from a simple multiplication, and a great variety of S/M effects may be modelled through flexible expressions for A^{hkl) and An{hkl). Towards

the 13.24 equation (Warren, 1969), one can imagine

quite a lot, starting from the current state of the

RF method. The variation law of (Z^) with n could

be, in a first step, made dependent on hkl, providing up to 6 more parameters in the general case with

K(hkl). More flexible size distributions allowing definition of Af^{hkl) with a few parameters could be

introduced (see Rao et ai, 1986; Scardi ei ai, 1991).

The (sin27r/Zn) term could be expanded as a series

to try to handle asymmetry and could have hkl dependence. Currently, two ellipsoids are used to simulate the mean N3(hkl) of the size distribution and

{Zi){hkl). We could go further and imagine that it would be possible to approximate the size and microstrain distributions (P{n)hki and Pn(Z)/ii;/) themselves by ellipsoids for each n and Z value. With

Fourier series, the number of steps that may be ex- pected in order to describe the Z variable of the microstrain distribution depends on the number of harmonics in each hkl family, whereas the number of

steps to describe n depends on the number of points

defining the reflection profiles: typically one could

have 50 to 100 n values (defining each reflection on

100 to 200 points) and maybe 5 to 10 Z values (if the

pattern is recorded with low wavelength up to large

diffracting angle) corresponding to a total of 300 to

1100 unknowns to be multiplied by 1 to 6 depending on the restrictions on the ellipsoids. Before claiming that this is impossible, remember that Fourier

transforms have a role to play here (moreover with

physical meaning), and that one profile defined by

201 steps (for instance) may be transformed into a

Fourier series having 101 harmonic numbers (if symmetrical), all being significant if the profile is a "sizeonly" / profile. Since the problem depends on the

profile shape, all data points contribute (15000 points

for a pattern with O.OV20 between 10 and 160°2^),

so, is it really impossible? One can imagine considerably reducing the number of unknowns if each

microstrain distribution is assumed to be Gaussian,

so only the width and centroid position would have

to be found for each n and hkl value. The number of

unknowns reduces to 150 to 300 (or 100 to 200 if the

Gaussian distributions of microstrain are centered on

the mean cell parameters), to be multiplied by 1 to 6. The problem has to be correctly formulated with constraints to force P(n) to be positive or zero. Nevertheless, the three-dimensional self consistency would

be perhaps difficult to ensure. Being a chemist, I am

afraid that I have not enough mathematical qualifications to try to do it! On the other hand, as stated by Langford (1987), for many applications it

is often not necessary to obtain microstructural parameters with a high absolute accuracy. One may

be mainly interested in improving the structure parameters and not wish to give a meaningful physical sense to functions used for modelling profile shapes

(to model empirically, but perfectly, powder patterns

for structure refinement purposes appears not so useless, particularly for ultra-high resolution data from

synchrotron X-rays). To do that, one can imagine

using direction-related ellipsoids for both L and G

parts, both having different 29 dependence, and not

taking into account the instrumental part g. This

would probably provide a "good" fit favorable to the

structure accuracy.

Conclusion

A question is: does simultaneous structure-S/M re- finement increase the credibility of all parameters?

Moreover, why do it when more or less perfect fits may be obtained without structure constraints, without space group constraint, and even without cell constraints? It is clear that the fewer the constraints,

the more the fit may be perfect. It may even be ar- gued that profile decomposition techniques are free

from the structural model errors and the preferred

orientation correction, unlike the Rietveld method;

I believe in turn that the structure constraint is a

"garde-fou" in case of serious overlapping of peaks.

Some Rietveld programs now offer the possiblity to

switch off the structure constraint when desired, and

to perform iterations of the Rietveld decomposition

formula leading to a cell and space-group constrained

full-pattern profile fitting (Le Bail, 1992). Nevertheless, the high quality of the fit obtained for CuO by

pattern fitting, without structure, using PVII (Langford & Louer, 1991) is not attainable now by any

S/M modelling in Rietveld analysis. In this case, residual effects after annealing at 1273K were an hkl

dependent skewing and line broadening due to a distribution of d spacing.

When I have sample effects, unless there is a special interest in studying these effects, I do my best

to obtain a well-crystallized sample by some chemical or physical means, but this is not always possible.

The state of the art has to progress, and real accuracy

concerning S/M will certainly not be obtained unless
152 Modelling Anisotropic Crystallite Size/Microstrain

Rp < 4 or maybe 2% (BS). We are far from routinely

obtaining such low values. It would be necessary

to improve instrumentation, sample alignment, theories of sample effects, eic. Modelling g in the Rietveld method must be at the same level of accuracy

as modelling h (Langford, Cernik & Louer, 1991).

However, if such low Rp are obtained by adding say

200 profile parameters to be refined, I am not sure that this would be highly credible. On the other

hand, a rigorous treatment of the problem theoretically involves much more (by Fourier transform) , and

the results would be credible without the overlapping

problem. The study of defective single microcrystals

using synchrotron radiation is also a possibility of the

present decade. A full account (three-dimensional) of

the / profiles of all reflections would be of the highest

interest. I am afraid for the next generation of users

(solid state chemists or physicists, for instance) that

the level of sophistication will seriously increase in

the future.

Acknowledgments

The author is grateful to J. I. Langford, D. Louer,

E. J. Sonneveld and J. W. Visser for kindly supplying the patterns of ZnO-B and well-crystallized ZnO

studied in their 1986 paper.

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National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Structure Building with Rietveld Analysis

D. L. BiSH

MS D469, Los Alamos National Laboratory

Los Alamos, NM 87545, U. S. A.

1. Introduction

The Rietveld method was originally conceived (Rietveld,

1967, 1969) to refine crystal structures using neutron

powder diffraction data. By its very nature, structure refinement relies on the existence of a reasonably

complete structure model. Diffraction data are then

used to improve, or to refine, the model so that

optimum agreement between observed and calculated

observations is obtained. Such refinement methods have

been critically important over the past few decades in

expanding our knowledge of the details of thousands of

crystal structures. The Rietveld method has been

adapted over the years to be used with X-ray powder

diffraction data, in addition to neutron diffraction data,

thereby greatly expanding the application of the method.

However, the user of the method is still faced with the

necessity of having a general knowledge of the crystal structure of the phase (or phases) in question before

beginning refinement.

The Rietveld method has been used in recent years

with other ancillary techniques to expand the number of

crystalline phases that can be analyzed and the amount

of information obtainable. High-resolution transmission

electron microscopy (HRTEM) images (e.g.. Post and

Bish, 1988; Richardson and Vogt, 1992) have been used

in some cases to provide starting models for structure refinement, and distance least-squares (DLS) methods

(e.g., McCusker and Baerlocher, 1984) are very

commonly used. The Rietveld method was used in these cases to evaluate and to refine proposed structure models. In several other applications, observed and

calculated structure factors extracted from Rietveld

refinements have been used as input to Fourier

syntheses (e.g., Adams and Hewat, 1981; Lager et al.,

1987). Difference-Fourier (AF) syntheses have been

used very effectively to locate the positions of extra- framework atoms or to investigate the possibility of

multiple or split atom positions in a variety of materials,

including zeolites and clay minerals. Thus, although the

Rietveld method was originally designed for structure refinement, it is very well suited to structure solution.

This paper will review the results of partial and

complete structure solution using the Rietveld method.

emphasizing both the strengths and weaknesses of the

combined methods. In addition, it will review the

benefits and difficulties of using several types of

observations, e.g. , neutron and X-ray powder diffraction

data, in Rietveld refinements. It will not review ab

initio structure determinations that use a combination of

profile fitting to obtain individual reflection intensities,

direct or Patterson methods to obtain an approximate

structure, and final Rietveld refinement (e.g., Clearfield

et al., 1984) as this combination of methods warrants a

separate paper.

2. Starting models

Because any crystal structure refinement relies on a

reasonable beginning crystal structure, determination of

the starting model is a crucial part of any Rietveld

refinement. Geometrical constraints and crystal

chemical intuition have played a part in solving

structures since the first crystal structure determinations

in the early part of this century. However, as the

complexity of structures increases, additional information is usually required. In recent years, fiirther data

have come in the form of HRTEM images; these images

have been used to provide structure models. In several

cases, the structure models derived from HRTEM

images were then fiirther optimized using DLS modeling

to provide a useful starting model for Rietveld

refinement.

2.1. High-resolution TEM data

Within the past 20 years, transmission electron

microscopy has advanced to the point where it can

provide images of the atomic arrangement in a crystal,

often easily yielding insights into general structural

schemes and, sometimes, solving apparently intractable

problems (e.g., Veblen, 1985; Buseck and Veblen,

1988). It is important to note, however, that HRTEM

images must be used with caution. Minor changes in

focus in the microscope can cause large changes in the

high-resolution image, greatly changing the images

(e.g., Guthrie and Veblen, 1990). To interpret the

images unambiguously, it is necessary to couple

154
D. L. Bish 155

HRTEM observations with modeling of the images

using a multi-slice calculation; such calculations require

at least an approximate knowledge of the structure (e.g.

O'Keefe, 1984).

In an example of the use of HRTEM images to elucidate the structure of an unknown material, Turner and

Buseck (1981) and Post and Bish (1988) recognized

similarities between the chain structures of hollandite

(BaMugOie) and romanechite (BaMnjOio'HjO) and

HRTEM images of todorokite. The hollandite structure consists of double chains of Mn-0 octahedra forming

2X2 tunnels, and romanechite has both double and triple

chains, forming 2X3 tunnels (Figure 1). HRTEM

images (Turner and Buseck, 1981) suggested that

todorokite has a 3X3 tunnel structure, although other

authors had suggested a layer structure. Post and Bish

derived a starting model for todorokite using DLS

modeling, assuming a 3X3 tunnel structure inferred

from HRTEM images and observed Mn-0 and O-O

bond distances from hollandite and romanechite. A

reasonable refinement was obtained, in spite of the very

poor quality of the X-ray powder diffraction pattern due

to structural disorder; the refinement confirmed the 3X3

tunnel structure. In a similar situation, Richardson and

Vogt (1992) used HRTEM images, symmetry information, DLS modeling, and comparisons with the structure of the zeolite VPI-5 to obtain a reasonable starting

model for AIPO4-8. They successfiiUy refined their

model structure using neutron powder diffraction data.

It is important to note, however, that just as with the

first structure solutions in the early part of this century,

crystal chemical intuition was important with both structure determinations, as was informed interpretation of

the HRTEM images.

The structure determination of Ti02(B) by Banfield et

al. (1991) using HRTEM and DLS modeling illustrates

the utility of this combination of methods and is an example of the importance of HRTEM image simulations. A model structure was obtained from HRTEM

images of semicoherently intergrown lamellae of the B

phase in anatase (also TiO^. The authors confirmed the

validity of the model structure by comparing computercalculated images with their observed images. They

then refined their model structure using DLS methods,

assuming bond lengths taken from the well-known

polymorphs of TiOj, anatase and rutile. Image

simulations for their model structure agreed closely with

the observed images, and it is noteworthy that their final

model structure reproduced the polyhedral distortions

predicted for the ideal Ti02(B) structure using

electrostatic minimization procedures. Although

Banfield et al. did not proceed with a Rietveld structure refinement due to a lack of material, their procedure

could be applied in detail to virtually any unknown

structure.

• - y=0

O - y=^2

• + O - A cations

• + O - Oxygen atoms

• + o — B cations

Figure 1. Projection of the hollandite structure down

the monoclinic b axis, illustrating the 2X2 Mn-0

octahedral structure (from Post, 1981).

2.2. Distance least-squares modeling

DLS modeling (Meier and Villiger, 1969) is a powerful

method for determining whether a particular model is geometrically possible. The method is similar in many

respects to conventional crystal structure refinement,

particularly in that it requires a reasonable starting

model. These models can be obtained through analogy

with other structures or by crystal chemical reasoning

(or from HRTEM images). Fortunately, the DLS

method is very robust and will usually converge, even

given very poor starting models. An optimum distance

model is obtained by varying atomic coordinates and

lattice parameters to minimize discrepancies between

interatomic distances calculated for the model structure and prescribed (input) distances that comprise the

observations. Input distances are usually obtained from

published, well-refined analogous structures and can

often be estimated to within several hundredths of an Angstrom. Weights applied to distances are usually

either related to Pauling bond strengths or to some

empirical measurement, such as stretching force

constants; weights based on the latter weight shorter

distances and those involving highly charged ions
156 Structure Building with Rietveld Analysis

much more heavily than the former method. Known

lattice parameters can be used often to constrain the

refinements further, resulting in more reasonable

structures. When beginning with unknown or poorly

known structures, lattice parameters can usually be

obtained by indexing the powder or electron diffraction

pattern using well-established techniques.

DLS modeling of a completely unknown structure such as todorokite requires some imagination and

adjunct information (such as HRTEM images), but

modeling a variant of a known material can be quite

easy. For example, if an intercalation reaction in a layered material such as kaolinite, Al2Si205(OH)4, gives

rise to a larger unit repeat in the direction perpendicular

to the layers, DLS modeling can use the parent structure with the new lattice parameters to represent the new

structure. Of course, one must either provide lattice parameters or use distance observations sufficient to constrain the structure in three dimensions. After any

DLS modeling, calculated diffraction patterns usiially

provide a quick idea of how reasonable the structure model is, easily revealing whether the symmetry and

cell are correct and how closely the model structure approximates the real one. This procedure is similar to

that used by Banfield et al. (1991) with HRTEM

images, namely obtaining a model structure and

determining how well observations agree with those

calculated based on the model. In some cases, a DLS

model structure can minimize the problems of false- minima in a Rietveld refinement, as appears to be the case for kaolinite (Bish and Von Dreele, 1989).

Rietveld refinement of kaolinite using a previously

published structure as a starting model consistently

resulted in a similar structure with imusual bond

distances. Refinement of the kaolinite structure using a DLS structure as the starting model resulted in

convergence to a reasonable structure with normal bond

distances. It is noteworthy in this case that application

of heavily-weighted soft constraints during refinement

led to the same false-minimum structure when starting

with the published structure.

3. Partial structure solution using AF maps

Difference-Fourier synthesis is a method used for

refining crystal structures and is particularly well suited

for locating scattering density missing from the structure model. A AF synthesis is imaffected by series

termination errors and uses structure factors (F^)

obtained from observed intensities and those calculated

using the model structure (F^J. It involves calculating

the ftinction

D(x,y,z) = 1/V E,E^,(FAV -

FA''i'c)cos27r(hx +ky + Iz).

There are many factors to consider when calculating AF

maps, whether from powder or single-crystal diffraction

data. First, in this calculation, the sign (or phase) of

F^ is assumed to be the same as F^^. Therefore, if a

significant amoimt of scattering density is missing or

misplaced, the calculated signs (or phases) may be

incorrect. For example, Rietveld refinements and AF

calculations of synthetic Cs A and Y zeolites (Bish,

unpublished) were fraught with difficulties because the

omitted Cs atoms resulted in many incorrect structure

factors (magnitudes and phases). Further, during the

initial stages of refinement when some atoms are

omitted, it is likely that temperature factors and scale

factors will be incorrect, often by a large amoimt. An

additional important consideration when calculating AF

maps during Rietveld refinement is that decomposition

of overlapping peaks is performed assuming the ratios

of the calculated intensities of the contributing

reflections. Thus, results from a set of Rietveld

refinements and AF-map calculations may be very

different from those that would be obtained from profile

refinement, where observed intensities are generally

obtained independent of any model. In general,

electron-density (or neutron-density) maxima are lower

for Rietveld-derived AF maps than for those obtained

from single-crystal data.

Because of these complications, location of missing

atoms is always an iterative process, where atoms are found, added to the model if reasonable (in scattering

power and position), and another AF map is calculated

after extraction of observed F's. As more of the correct

scattering material is added to the model, the calculated

F's (magnitudes and phases) and scale factors become

more correct, and the decomposition of overlapping

peaks is improved. These complications also dictate

how a Rietveld refinement must be performed to obtain

reasonable calculated F's. Typically, a minimum of

parameters is varied, and structural parameters are usually not refined. If structural parameters are varied,

they may adjust in an attempt to compensate for the

missing scattering power, yielding an unreasonable

structure and complicating the interpretation of the AF

map. In a Rietveld refinement/AF map calculation, the

scale factor(s) and background parameters are usually

varied first, followed by the lattice parameters. In some

cases it is preferable to obtain the scale factor from

high-angle data, fix it, and then calculate a AF map.

Some profile parameters may be refined, although

heavily overlapped patterns may require either fixing or manual adjustment of these parameters if observed and

calculated patterns are significantly different. In

practice, it appears that AF maps are relatively

insensitive to a preferred orientation correction,

although it is preferable to use data unaffected by

orientation.
D. L. Bish 157

A test case using the known structure of the zeolite

analcime, NaAlSijOg'HjO, illustrates some common

features of partial structure solution during Rietveld

refinement. Rietveld refinement of analcime was

conducted using the tetragonal structure of Mazzi and

Galli (1978) and X-ray powder diffraction data. For the

test case, both the Na and HjO sites were purposely left vacant, and initial cycles varied only the scale,

background, and lattice parameters. Five large positive

peaks appeared on the first AF map (Figure 2)

calculated; the two tetrahedral Al,Si sites (which were

fiiUy occupied in the model), the two Na sites, and the

HjO site. The refined scale factor was 239.2.

Interestingly, the Na positions in Figure 2 reveal the

same anisotropy observed by Mazzi and Galli (1978)

using single-crystal X-ray diffraction data. After

insertion of the two Na atoms, refinement was continued

and another AF map was calculated. Three large

positive peaks appeared on the resultant map; the two

tetrahedral Al,Si sites and the H2O site, but little evidence of the Na sites. The refined scale factor was

291.3. At this stage, the HjO molecule (represented by

an O atom) was inserted and occupancies of the Na and

H2O sites and temperature factors of all atoms were

refined. A AF map calculated after this stage revealed no large peaks, and the scale factor was 456.4.

Removing the HjO from this refined structure yielded a AF map that had only a single peak due to HjO (Figure

3). The appearance of density at the tetrahedral atom

positions during the initial stages of refinement and AF

map calculation was probably due to a combination of

an incorrect scale factor and incorrect temperature

factors. Determination of an initial scale factor using

only high-angle data may have partially compensated for

these discrepancies. Results of this simple test refinement should be considered when performing

Rietveld refinement coupled with calculation of AF

maps on a poorly known structure. Several further

examples will illustrate the potential of these methods

when applied to incompletely known structures.

3. 1 Todorokite

The structure of todorokite, (Na,Ca,K)oj5

(Mn''+,Mn^+)s6Mgo.40,2*4H20, was first solved and

refined by Post and Bish (1988) as outlined above.

After determination of the Mn-0 framework structure using a DLS starting model and Rietveld refinement, a remaining question was the positions of the included Na,

Ca, K, and HjO. Post and Bish used Rietveld-derived

observed intensities and the DLS model structure to calculate AF maps for two different todorokites. All of

the electron density on the maps was located in the

tunnel region (Figure 4) and was similar for both

Figure 2. Difference-Fourier map of analcime at z=0,

calculated without Na or HjO in the model. Contours

are drawn at 2.0, 4.0, 6.0, and 8.0 electrons.

)

+ T(1) 0 (

(

+ 1

Figure 3. Difference-Fourier map of analcime at z=0,

calculated after full refinement and subsequent removal

of HjO. Contours are drawn at 1.0, 3.0, and 5.0

electrons.

samples, suggesting that the information was not an

artifact. Since most of the non-framework material in

todorokite is HjO, the three largest peaks in the AF

maps were assigned to HjO, and Rietveld refinement was continued. The positions and occupancies of the

three tunnel sites were refined successfully, yielding a

significant reduction in the R^ values. Although it is often difficult to distinguish between contributions from

tunnel cations and H2O molecules. Post and Bish

conducted electrostatic minimizations strongly

suggesting that the two largest regions of electron

density correspond to HjO whereas the density near
158 Structure Building with Rietveld Analysis

(1/2,1/2,1/2) is due to tunnel Ca and Na. Given the

disordered nature of todorokite and the resultant poor

quality of the diffraction data, it is remarkable that such

a cohesive picture of the todorokite structure emerged,

attesting to the jjower of the DLS-Rietveld-energy

minimization combination. The final refined structure

is shown in Figure 5.

a b

Figure 4. Difference-Fourier maps at y=0

superimposed onto projections of the 3X3 tunnel

structure of todorokite, looking down b, for samples

from (a) South Africa and (b) Cuba. Filled circles are atoms at y=0 and open circles are those at y=l/2.

Large circles are O and small circles indicate Mn.

Contour interval is 0.4 e'/A^. The todorokite unit cell

is indicated by the dotted lines (from Post and Bish,

1988).

c

Figure 5. Projection of the final refined structure of

todorokite down b. Filled circles represent atoms at y=0 and open circles are atoms at y=l/2 (from Post

and Bish, 1988).

3. 2 Location ofhydrogen atoms in kaolinite and dickite

One of the more interesting aspects of structural studies

of layer silicates and clay minerals is the location of the

H atoms that so significantly affect their properties.

The structure of kaolinite, Al2Si205(OH)4, was outlined

as early as 1930, but it was not until 1983 that

determinations of the locations of the H positions were

first made (Adams, 1983; Suitch and Young, 1983).

Before that date, Giese and Datta (1973) and Giese

(1982) had determined H positions using electrostatic

modeling methods. Both Adams (1983) and Suitch and

Young (1983) used published structures as starting

models. Adams determined the H positions using AF

maps, but his results appear to have been affected by

disorder and two-dimensional diffraction effects in the

sample. Suitch and Young assumed H positions for

their starting model and they assumed a lower space

group symmetry (PI) than accepted for kaolinite (CI),

resulting in an apparent false-minimum structure. More

recently, Young and Hewat (1988) re-refmed the

kaolinite structure, again assuming a complete structural

starting model and the lower space group symmetry and

obtaining a structure similar to that obtained by Suitch

and Young. Bish and Von Dreele (1989) refined the

structure of kaolinite using X-ray powder diffraction

data, but they used a DLS model in CI as the starting

structure. Their final refined structure appeared

accurate, with reasonable bond lengths. In 1990, Bish and Von Dreele re-examined the

kaolinite structure using constant-wavelength neutron

powder diffraction data and the non-hydrogen structure of Bish and Von Dreele (1989) as a starting model to

rectify the differences between the earlier refinements.

They calculated AF maps, and the four largest negative

regions of neutron density appeared reasonable for the

four H positions (Figure 6). They then refined the

complete structure, using AF maps and anisotropic

refinements to examine the nature of the H sites. There was no significant anisotropy in any of the four H

positions, although the inner-hydroxyl H was slightly

smeared along the z direction and the three inner-surface

hydroxyl H atoms had larger displacement parameters

within the plane of the layers. These results do not

agree with those of Adams (1983), Suitch and Yoimg

(1983) , or Young and Hewat (1988), probably because

of the difficulties with those refinements mentioned

above. It is noteworthy that Guthrie and Bish (1991)

reproduced the refined H positions using an electrostatic

minimization procedure. Calculated energy contours

also showed that the inner-surface hydroxyl H atoms are

situated in disk-shaped minima approximately parallel to

(001) and the iimer-hydroxyl H minimum is approximately perpendicular to (001), in agreement with the

results of the Rietveld refinement.
D. L. Bish 159

a)

b)

{ ("""-I )

Figure 6. Difference-Fourier maps for kaolinite

illustrating the positions of a) HI within the kaolinite

layer at y = 0.329, contours drawn at -0.20, -0. 15, and

-0.10, and -0.05; and b) H2, H3, and H4 in the

kaolinite interlayer; contours drawn at -0.3, -0.2, and

-0.1. The a and b axes are approximately horizontal

and vertical, respectively, in both figures.

Until recently, there was a similar lack of information on H positions in the related mineral, dickite, a polymorph of Al2Si205(OH)4. In probably the first application of the Rietveld method to a layer silicate,

Adams and Hewat (1981) examined the dickite structure using Fourier methods and neutron powder diffraction

data. Their H positions, obtained on a sample

exhibiting some two-dimensional diffraction effects, do

not agree well with subsequent single-crystal X-ray

determinations. Therefore, Bish and Johnston (1992)

re-investigated the structure of dickite using lowtemperature time-of-flight neutron powder diffraction

data, the Rietveld method, and Fourier syntheses, in

addition to variable-temperature Fourier-transform

infrared (FTIR) spectroscopy. Using a recent X-ray

structure as a starting model, with no H atoms included.

the four H atoms were unambiguously located as the

four largest negative regions of density on AF maps

(Figure 7). The minor anisotropy noted for the H atoms

in kaolinite was not observed for dickite. The observed

H positions were most similar to the recent singlecrystal X-ray refinement of Joswig and Drits (1986),

although the O-H bond distances found by Bish and

Johnston appear to be more accurate. OH geometries,

and the changes occurring from room temperature to 12

K, are consistent with the FTIR spectra as a function of

temperature and lend credence to the Rietveld-refined

results. The relative ease with which the H positions were obtained in a powdered clay mineral such as

dickite is remarkable when one considers that barely 25

years ago such information would have been out of

reach.

a)

-O.Si(b,;^««t 0-lU 0.197

b)

Figure 7. Difference-Fourier maps for dickite illustrating the positions of a) HI within the dickite layer;

contours drawn at -0.5, -0.4, -0.3, and -0.2; and b) H2,

H3, and H4, in the dickite interlayer; contours drawn at

-0.6, -0.5, -0.4, -0.3, and -0.2. The a and ft axes are approximately horizontal and vertical, respectively, in

both figures.
160 Structure Building with Rietveld Analysis

It is important to make one final comment regarding

the use of AF methods to locate H (or D) using neutron

powder diffi'action data. It is common practice in such

experiments to attempt to replace H with D to reduce

incoherent scattering. If the replacement can be made

to go to completion, it should facilitate improved

analysis due to a lowered background and improved

signal-to-noise ratio. However, one can easily envision

a situation in which only partial substitution of D for H

is accomplished, on a statistical basis. Such a result

could produce AF maps revealing little or no information about the positions of H or D due to averaging of

the negative neutron scattering length of H and the

positive neutron scattering length of D. One could

create a material with an average neutron scattering

length of zero in the H sites.

3.3 The interlayer structure ofkaolinite intercalates

The determination of the interlayer structure of the

kaolinite-hydrazine intercalate is a good example of the

utility of combined DLS modeling, AF methods, and

Rietveld refinement. Although kaolinite, a layer silicate consisting of sheets extending in two dimensions but

with weak bonding between the sheets, does not

normally expand, it interacts strongly with several

liquids to create expanded intercalates. When kaolinite

is exposed to anhydrous hydrazine, N2H4, it expands to a material with either a 9.5 or a 10.4 A basal (001)

spacing, depending on the vapor pressure of water.

Although the starting kaolinite may not possess good

three-dimensional order, the 9.5 A complex usually

exhibits three-dimensional order sufficient to apply the

Rietveld method. As a first step, the structure of the

9.5 A intercalate was modeled using DLS, assuming the

structure and a, b, a, and y lattice parameters of a 7 A,

non-expanded kaolinite, and a value of (3 of 114",

yielding a value for c of 10.5 A obtained from a measured dooi of 9.52 A. It was assumed in the DLS

model that the stacking of layers in the intercalate was

the same as in unexpanded kaolinite. The DLS struc- ture was used as the starting model for Rietveld

refinement, with the N2H4 molecule omitted. The two

largest peaks in a AF map were in the interlayer region

and were 1.40 A apart, consistent with an N2H4

molecule. The molecule is canted within the interlayer,

and there is evidence from the AF maps for positional

disorder of the N atom closest to the hydroxyl interlayer

surface (Figure 8). Inclusion of these two N atoms in

the structure model and continued isotropic refinement

yielded an R^ of 25 % , compared with 36 % without the

N atoms, although the refinement is still preliminary.

This structure determination/refinement resolved the

question of the exact orientation of the N2H4 molecule

within the kaolinite interlayer, and the refinement result

Figure 8. Difference-Fourier map for kaolinitehydrazine showing the N2H4 molecule represented by the

two largest peaks at (0.38, 0.39, 0.723) and (0.467,

0.483, 0.655) in the interlayer space. Contours are drawn at 0.15, 0.30, 0.45, and 0.65 electrons. The c

axis is approximately vertical.

suggests that the intercalate has the same layer stacking

as kaolinite. Given the low atomic number of the N and

the comparatively poor quality of the material, it is remarkable that such a result was obtained.

Comparable results were obtained using this

methodology with a similar kaolinite intercalate.

Costanzo et al. (1980) created a stable 8.4 A hydrate of

kaolinite by expanding the material with dimethylsulfoxide and then reacting the expanded material in a

solution containing NH4F, thereby partially or completely fluorinating the interlayer hydroxyl groups.

The resultant material has an enlarged c lattice

dimension and three-dimensional order sufficient to apply the Rietveld method. A starting structure model was obtained using DLS methods, and the location of

interlayer water molecules was determined from

Rietveld refinement and AF maps (Bish et al., 1992).

The largest peak on the AF maps was midway between

the layers, with only two other minor peaks above zero

density. Rietveld refinement, incorporating an O atom

at the position of the largest peak, yielded an occupancy

of HjO of 0.75, positioned almost exactly midway

between the silicate layers. Subsequent AF maps

revealed no significant areas of electron density.

3. 4 Exchangeable cations and water in sepiolite

The study of sepiolite by Bish and Post (1987) is an example of the use of Rietveld/AF methods for

elucidating the nature of the occupants of cavities in

open-structure materials for which large single crystals

cannot be obtained. A schematic structure of sepiolite,

Mg4Si60,j(OH)2*6H20, was proposed by Nagy and
D. L. Bish 161

Bradley (1955) and BrauDer and Preisinger (1956) using

so-called fiber diagrams (two-dimensional data).

However, the details of the atom distribution in the

structural tunnels between the silicate chains remained

poorly understood. Bish and Post used X-ray data, with

the Brauner and Preisinger framework as a starting

model, to locate the tunnel occupants. Several iterations

of Rietveld refinement and AF map calculation revealed

the presence of several HjO sites and several Mg sites

(as judged by interatomic distances). The greatest

difficulty faced in interpreting the AF maps was

deciding which peaks were real; all of the tunnel

occupants are low-atomic number atoms, and most or all of the sites are probably partially occupied. In cases such as this, one must rely on crystal chemical

reasoning, using information such as bond distances and

observed stoichiometry. Refinements of Bish and Post

confirmed the presence of all water molecules postulated

by Brauner and Preisinger (1956) but did not support

several split-atom sites determined by Rautureau and

Tchoubar (1974) using two-dimensional electron

diffraction data. The Rietveld refinement of sepiolite

constituted essentially the first three-dimensional

structural study of sepiolite.

4. Other structure reflnonent methods

In addition to the use of the above model-building and

structure-solution methods in connection with Rietveld

refinement, there are several other techniques that can be used profitably during or after a refinement to improve and/or to supplement the information gained.

These techniques include the use of so-called soft and

hard distance constraints during refinement (see

Baerlocher, 1986), use of multiple observations such as both X-ray and neutron diffraction data, analysis of the

sample by other methods to obtain supporting information, and modeling of the refined crystal structure using

energy minimization methods. These different methods

have the effects of stabilizing difficult refinements,

nninimizing the problems of false-minimum structures,

and supporting and rationalizing the final refined

structure.

4.1 Use of distance constraints

Soft distance constraints, in the form of prescribed

interatomic distances and angles obtained from wellrefined similar structures, can be used to supplement

diffraction observations, but they do not reduce the

number of variables. Their weight is typically chosen

manually, often higher during beginning cycles and

lower (or even zero) during the later stages of

refinement. Such geometric restrictions have been used

to great advantage in refinements of high-symmetry

materials with numerous atoms, particularly those with

poor ratios of observations/variables (e.g., Baerlocher,

1984). This procedure is exactly analogous to performing a combined DLS/Rietveld refinement. The

use of soft constraints often improves the behavior of a refinement in the initial stages, but the constraints

cannot always be completely removed (Baerlocher,

1986). Although the use of incorrect soft constraints can lead to false-minima structures, Baerlocher (1986)

argued that such structures can be detected. Arguments

have also been advanced that the use of soft constraints

will unreasonably bias the outcome of a refinement, but

the additional observations are usually known with high

precision and one is simply exploiting the wealth of

accurate crystal chemical and structural information

available. In spite of this, however, it is important to

state that the use of soft distances constraints in the final

stages of a Rietveld refinement will affect the derived

bond lengths to varying degrees, depending upon the

quality of the refinement and the weighting of the soft

constraints. A poor refinement with poor data can still yield bond distances that appear acceptable if numerous

soft distance constraints are used with high weights in

the final stages of a refinement.

Hard constraints, known as rigid-body constraints in

single-crystal refinements (e.g.. Finger and Prince,

1975), differ from soft constraints in that they reduce

the number of variables. These constraints are usually

used with molecular groups, such as CO^, in which the

molecular conformation (distances and angles) is fixed

and only the position and orientation of the group are

refined. This method is very effective when studying

the structures of complex materials containing

essentially isolated molecular groups, such as an

adsorbate in a zeolite cavity. As with soft constraints,

the intra-molecular distances and angles can often be

predicted with high precision and accuracy, and these

constraints should not unduly bias a refinement.

However, also as with soft constraints, highly weighted

hard constraints can unduly influence the results of a refinement.

4. 2 Joint X-ray and neutron refinements

A very effective method that has seen little application

in Rietveld analysis is the simultaneous use of both Xray and neutron powder data. The use of combined data

generally provides a more complete structure determination, in part because of the very different

variation of scattering length (scattering factor) with

atomic number for X-rays compared with neutrons. For

example, there is no sin^/X fall off of scattering factor

for neutrons as is seen for all atoms with X-rays,

yielding lower intensities at high angles. In addition,

whereas adjacent atoms in the periodic chart have
162 Structure Building with Rietveld Analysis

virtually identical X-ray scattering factors, they may

have very different neutron scattering lengths. Of

course, the large variation of neutron scattering length even among isotopes of the same element is often

exploited in structural studies that would be impossible

with X-ray diffraction data. The strengths of neutron

diffraction are illustrated by attempts to differentiate

between Fe and Mn on different sites. Although their

X-ray scattering factors are almost identical, the neutron

scattering length for Mn is -0.373xlQ'^ cm compared

with that for Fe of 0.954x10'^, providing for very easy

discrimination between the two species. Sometimes, the reverse is true, where atoms widely separated in the

periodic chart have very similar neutron scattering

lengths, but very different X-ray scattering factors. The

ultimate result of using combinations of X-ray and

neutron observations is that refinement of individual

anisotropic thermal parameters and site occupancies can be accomplished for complex structures. This method

also minimizes problems with false-minima refined

structures since a model must agree with very different

types of data.

Williams et al. (1988) illustrated the utility of this

combination in their study of superconducting

YBa2Cu307.,. Although the structure of this material

had been studied several times and was well known

before their study, most of the structural studies utilized

neutron diffraction data. Because the neutron scattering

lengths of Y (0.775xia'2) and Cu (0.7718x10 '^ are

virtually identical, the earlier neutron studies were

unable to identify uniquely the distribution and ordering

of these two cations. The structure refined using both

X-ray and neutron powder diffraction data was much

more precise than previous determinations, and the

refinement showed definitively that there was no

disorder between the Y and Cu sites. In addition, the

refinement provided a complete picture of the

anisotropic thermal motion of all the atoms (Figure 9).

Clearly, the combined use of multiple types of

observations makes the Rietveld method much more

powerful than originally conceived, potentially allowing

for determination of intricate details of site occupancy

and thermal motion of powdered materials. The only

caveat that must be recognized is that all sets of data

must be obtained under identical conditions, so that the

lattice parameters and occupancies of any environmentally sensitive sites are the same.

4.3 Supporting observations

A few final comments should be made concerning

observations that can (and often should) be made in

support of a structural determination by Rietveld

methods. For example, it has become routiae to determine cation distributions in Fe-bearing materials

using Mossbauer spectroscopy, and Nord and Ericsson

(1982, 1985) and Nord (1986) used this combination

very effectively. These authors gave several examples

illustrating the benefits of using the Rietveld/Mossbauer

combination, in particular elucidating Fe-Mn ordering.

This combination allows an imderstanding of complex

structures that could not be unraveled with only one

technique. Another useful combination that has not

been exploited is the use of magic-angle spinning

nuclear magnetic resonance (MAS-NMR) spectroscopy

with Rietveld refinement. ^Si MAS-NMR spectroscopy

has proven particularly valuable in elucidating Al-Si

ordering in silicates and should be used more often in

conjunction with structure determinations and Rietveld

refinements.

Cu(2)

0(4)

Figure 9. Structure of orthorhombic YBa2Cu307.,, with

thermal ellipsoids drawn at 99% probability surfaces

(from Williams et al., 1988).

A final method that is proving particularly useful in

supporting determinations of H-atom orientations and

extra-framework species positions is electrostatic energy

minimization. This method combines calculation of the

Coulomb portion of the lattice energy with some

formulation of the short-range repulsive energies to determine the minimum-energy configuration of a

particular arrangement of atoms. Guthrie and Bish
D. L. Bish 163

(1991) applied this method to a study of the H

orientations in kaolinite, for which H positions had just

recently been determined using Rietveld refinement/AF

methods. It is extremely interesting that the results

ofthe electrostatic minimizations reproduced both the H

positions and the general shape of the potential wells

within which the H atoms resided. Thus, this method

explained and supported the anisotropic thermal motion

of the H atoms observed in the Rietveld refinement.

Giese (1982) and Guthrie and Bish (1991) also used

these methods to predict the H positions and orientations

in layer silicates for which complete structure refinements have not been performed. Catlow et al. (1986)

described numerous uses of electrostatic methods,

including studying cation distributions in zeolites, and

they commented on the possibilities of coupling energy

minimizations with structure refinements using

diffraction data.

5. Summary

With careful and imaginative application of several

ancillary techniques, the Rietveld method can provide

structural information for powdered materials that was,

until recently, out of reach. Structure building with

HRTEM images and DLS methods can provide useftil

starting models for refinement and can allow solution of

previously unknown structures. In addition, the use of

DLS models can minimize the problems of false- minimum refined structures. One of the most powerful

combinations for structure determination is the use of

Rietveld refinement with difference-Fourier syntheses.

A variety of important structural problems can be solved

with this combination, including location of missing

atoms, including H (D) atoms and extraframework

atoms in open framework or layer-structure materials.

Supplementing diffraction observations with soft

distance constraints can stabilize the initial stages of a refinement and can sometimes allow refinement of structures that otherwise could not be refined. However,

their indiscriminate use can significantly affect derived

bond distances. A potentially very beneficial enhancement of the Rietveld method is the use of both neutron

and X-ray powder diffraction data with a single

refinement. Such a combination can provide superior

structural details, including useful anisotropic

temperature factors and high-precision site occupancies.

All structural results can benefit from support from

other observations, including spectroscopic data and

electrostatic energy minimizations that should allow a

large amount of structural information to be extracted

from powdered materials.

Acknowledgment

It is a pleasure to acknowledge the never-failing

assistance of Dr. R. B. Von Dreele throughout my

studies, and Dr. G. D. Guthrie for helpful comments on

the manuscript. In addition, I have also benefitted from

ongoing discussions with Dr. J. E. Post concerning

Rietveld refinements of mineral structures.

References

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356.

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Baeriocher, C. (1986). Zeolites 6, 325-333.

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Bish, D. L. and Johnston, C. T. (1992). Proc. 29th

Annual Clay Minerals Soc. Meeting.

Bish, D. L. and Post, J. E. (1987). Proc. 24th Annual

Clay Minerals Soc. Meeting, 33.

Bish, D. L. and Von Dreele, R. B. (1989). Clays &

Clay Minerals 37, 289-296.

Bish, D. L. and Von Dreele, R. B. (1990). Progr. &

Abstracts, 27th Annual Clay Minerals Soc. Meeting,

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Bish, D. L., Giese, R. F., Jr., and Costanzo, P. M.

(1992). Proc. 29th Annual Clay Minerals Soc.

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Mitt. 6, 120-140 (1956).

Buseck, P. R. and Veblen, D. R. (1988). Mineralogy,

chapter 9, Oxford Univ. Press, New York, 308-377.

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M. J., and Parker, S. C. (1986). Materials Sci.

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(1984). Inorg. Chem. 23, 4679-4682.

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Jr. (1980). Clays & Clay Minerals 28, 155-156.

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(U.S.) Tech. Note 854, 133 pp.

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471-479.

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63.

Guthrie, G. D. and Veblen, D. R. (1990). Amer.

Mineral. 75, 276-288.
164 Structure Building with Rietveld Analysis

Joswig, W. and Drits, V. A. (1986). N. Jb. Miner. Mh.

19-22.

Lager, G. A., Armbruster, Th., and Faber, J. (1987).

Amer. Mineral. 72, 756-765.

Mazzi, F. and Galli, E. (1978). Amer. Mineral. 63,

448-460.

McCusker, L. B. and Baerlocher, C. (1984). Proc. 6th

Int. Zeolite Conf., Reno, Nevada, 812-822.

Meier, W. M. and ViUiger, H. (1969). Zeit. Krist. 129,

411-423.

Nagy, B. and Bradley, W. F. (1955). Amer. Mineral.

40, 885-892 (1955).

Nord, A. G. and Ericsson, T. (1982). Amer. Mineral.

67, 826-832.

Nord, A. G. and Ericsson, T. (1985). Amer. Mineral.

70, 624-629.

Nord, A. G. (1986). Chemica Scripta 26A, 115-118.

O'Keefe, M. A. (1984). Electron Optical Systems, J. J. Hren, F. A. Lenz, E. Munro, and P. B. Sewell,

eds., SEM Inc., AMF O'Hare, Chicago, IL, 209-

220.

Post, J. E. (1981). Ph.D. thesis, Arizona State University, 321 pp.

Post, J. E. and Bish, D. L. (1988). Amer. Mineral. 73,

861-869.

Rautureau, M. and Tchoubar, C. (1974). C. R. Acad.

Sc. Paris 278B, 25-28.

Richardson, J. W., Jr. and Vogt, E. T. C. (1992).

Zeolites 12, 13-19.

Rietveld, H. M. (1967). Acta Cryst 22, 151-152.

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

Suitch, P. R. and Young, R. A. (1983). Clays and Clay

Minerals 31, 357-366.

Turner, S. and Buseck, P. R. (1981). Science 212,

1024-1027.

Veblen, D. R. (1985). Mineralogical Assoc. Canada,

63-90.

Williams, A., Kwei, G. H., Von Dreele, R. B.,

Larson, A. C, Raistrick, I. D., and Bish, D. L.

(1988). Phys. Rev. B 37, 7960-7962.

Young, R. A. and Hewat, A. W. (1988). Clays & Clay

Minerals 36, 225-232.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraciion II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Anharmonic Thermal Parameters, Disorder and Phase Transitions

H. BOYSEN

Institui fiir Kristallographie, Universitdt Miinchen,

8000 Miinchen 2, Theresienstr. 41> Germany

Abstract

Anharmonic atomic displacement parameters may

become quite important in structure refinements

from powder data in many practical situations, es- pecially at high temperatures. Valuable information

is obtained for materials like fast ionic conductors,

polymorphic substances close to the phase transition and statically disordered crystals. A standard

Rietveld program has accordingly been modified us- ing a Gram-Charlier series expansion of the DebyeWaller factor up to the fourth order. The basic theory, the refinement procedures, several examples and

possible future developments are discussed.

1. Introduction

In conventional crystal structure analysis from Bragg

diffraction data the effect of thermal motion of the

atoms is taken care of by the harmonic Debye-Waller

factor (HDWF), which corresponds to a Gaussian

probability distribution of the atoms in real space.

Anharmonic thermal vibrations, although evidenced

by many physical properties like thermal expansion,

heat conductivity, specific heat anomalies, interaction of phonons etc., are usually considered to have

a negligible influence on the Bragg intensities. If the

amplitudes of the thermal motion of the atoms become large, however, e. g. at high temperatures, this

is no longer true, and anharmonic Debye-Waller factors (ADWF) have to be considered. Clearly, for an atom in a non-centrosymmetric site symmetry a

description by a Gaussian distribution is not adequate. More generally, the local environment, e. g. the directions of weak and hard bonds, will determine the form of the distribution. Examples of materials exhibiting large amplitude motions are ionic

conductors and substances undergoing phase transitions close to the transition point. On the other

hand, static displacements of the atoms in disordered

crystals can also often be successfully described by

ADWF's. Therefore, to emphasize the applicability of DWF's to both static and dynamic disorder

the term "atomic" (ADP) instead of "thermal" displacement parameters should be used (Dunitz cited

in Kuhs, 1992).

In the past most of the work concerning ADWF's

has been done with single crystals, although single

line powder intensities have also occasionally been

analyzed (e. g. Matsubara, 1975). With the high

intensity and resolution of powder diff"ractometers

available nowadays, however, such anharmonicities

may become important and should be considered in

the refinement. In particular, the increasing interest

in high temperature crystallography and the related

physical properties, together with the availability of

good diffraction furnaces, clearly demands the inclusion of ADWF's in the refinement programs.

In section 2 of this paper a brief outline of the

theory of ADWF's will be given, following an entirely phenomenological approach. For a more detailed discussion, and also the physical background,

see the recent review by Kuhs (1992) and the refer- ences therein. Refinement procedures, evaluation of

the results, and a comparison with the alternative

split atom model, are discussed in section 3. The

application of ADWF's is illustrated in section 4 by

several examples including fast ionic conductors, (order/disorder) phase transitions and static disorder.

Finally, the conclusions in section 5 also give an outlook on what can be expected from future developments.

2. Theory

In a scattering experiment (X-ray or neutron) the

amplitude of the scattered wave A(Q) (where Q is the scattering vector, Q = 47rsin(0)/A, 20 the scat- tering angle and A the wavelength) is given by the

Fourier transform of the electron density or scat- tering length weighted nuclear density, respectively,

p{r). As long as an averaged structure can be defined, p{t) can be divided into

p(r) = (p(r))+A/;(r), (1)

where the definition means (Ap(r)) = 0. (p) is the

projection of all unit cells into a single one (divided

165
166 Anharmonic Thermal Parameters, Disorder and Phase Transitions

by the number of unit cells) repeated strictly periodically in space. Then the intensity of the scattered

beam I = AA* is given by the Fourier transform of

p{r) * p{-t) = (p) * (p) +Ap* Ap, (2)

where * denotes convolution. Thus it can be divided

into a Bragg (Is) and a diffuse (Id) part

I = Ib + Id. (3)

The average structure may be written

(Pi^)) = '^nk6{r-rk)* Pk{r)*pdfk{vL) * /(r),

. k (4)

where the contents of the unit cell {k runs over all atoms in the unit cell) are convoluted with the lat- tice function /(r) = 6{r — r/), where / runs over

all unit cells, njt takes care of the occupation of a

specific site by different atom species (including vacancies), pk is the atomic electron distribution for

X-rays or 6jfc6(r') for neutrons (bk is the coherent

nuclear scattering length) and pdfk the probability

density function that describes the space and time

averaged distribution of the atoms around the equilibrium position rjfe (u = r — rjt). From the definition

/ pdf{v)dv = 1. Fourier transformation gives

where {v?) is the mean square displacement, and B =

87r^(u^) is the usual isotropic atomic displacement

parameter.

In lattice dynamics (applicable only for dynamic

displacements) this result is derived in the "harmonic" approximation by truncating the crystal potential

^ = ^0 + I 5Z S ^ckla'k'l'Ua{kl)Ua>(k'l')

akl a'k'l' +V3 + + (8)

after the quadratic term. Inclusion of higher order

terms to account for the anharmonicity is extremely

complicated and has only been carried out for a few

simple cases. Instead, at high temperatures, one can make a drastic simplification and neglect all interactions between the atoms, i. e. we regard each atom

as an Einstein oscillator performing independent vibrations in a one particle potential (OPP)

V = Vo + \cc\n\^ + /?/3(u) + 7/4(u) -f (9)

where /a and /i denote homogeneous functions of

order 3 and 4, respectively. This approximation can

be further justified as we are only interested in the

average distribution of the atoms. Moreover, the pd/

can be calculated by applying Boltzmann statistics

A{Q) = Y^nke'^"^" fk{Cl)Tk{Cl) m). (5) prf/(u) = e-'^W/*^/^ = pd/(0)e-^W/*^, (10)

/*;(Q)> the X-ray scattering amplitude has to be re- placed by hk for neutrons. /^(Q) is the lattice function in reciprocal space and Tlt(Q) the temperature

factor. The term in square brackets represents the

usual structure factor, i^(Q). The main result is that

the temperature factor is the Fourier transform of the

pdf in a very general sense, i. e. for any distribution,

static or dynamic

r(Q) - J

pd/(u)e'Q"d^ u. (6)

For a trivariate Gaussian pdf the temperature factor is also Gaussian and we obtain the usual HDWF.

In the isotropic case

pdfiu) = (27r(n2))-3/2exp ("2^) (7a)

and

r(Q) = exp ( -Iq'{u')\ = exp (-5^

where Z is the partition function and the second

equality is obtained by setting V^(0) = 0. In summary, equations (6) and (10) define the re- lations among the potential, the pdf and the temperature factor. Hence, each of these quantities can be used, and is used in the literature, as the start- ing point for approximation of ADWF's. After Dawson's generalized structure factor formalism (Dawson, 1967) introducing imaginary parts to account

for anti-symmetry, the OPP expansion (Willis, 1969;

Tanaka Sz Marumo, 1983), which starts from equation (9), the expansion of the pdf adapted to a specific problem [e. g. spherical harmonics for orientationally disordered crystals (Press & Hiiller, 1973;

Prandl, 1981)], and the moment or cumulant expansions (Johnson, 1969; Johnson Sz Levy, 1974), which

are based on statistics, have been employed. Here,

for purely mathematical reasons (analytical calculation of pdf and OPP, applicability to all symmetries,

easy inclusion into refinement programs), the quasimoment expansion has been chosen. This is given by

(76)
H. Boysen 167

T(Q) = T'''"-"'(Q)[1 + j^C^'^QiQkQi

+'^,iy'''"'QjQkQiQm + ..] (11a)

(summation convention), which is known as the

Gram-Charlier series. After Fourier transformation

pdfivi) = p#''--(u)[l + ^a"Hjuiu)

+^iy''"'Hjum{u) + ...], (116)

where Hjk.. are the so-called Hermite polynomials.

Up to the fourth order there are 10 coefficients C^^'

and 15 coefficients These coefficients are (covariant) components of tensors and are subject to

restrictions imposed by the local site symmetry of

the atom under consideration, i. e., their number is reduced in many practical cases. The Hermite polynomials and the symmetry constraints for tensors up

to the fourth order may be found in Johnson &; Levy

(1974). It should be emphasized that the refinable

parameters C-'*' and D^^^"^ have no direct physical

meaning, and, moreover, there are often correlations

with the second order (harmonic) coefficients and

even the positional parameters. The refined positional parameter does not necessarily coincide with

the center of gravity of the distribution. Therefore,

to interprete the results, one always has to calculate

the pdf and/or the potential.

Since (/>) is a space and time average one can in

general not distinguish between static and dynamic

disorder. Some hints may, however, be obtained by

analyzing the temperature dependence of the DWF.

In the harmonic case with purely dynamic disorder

the mean square displacements {v?)dyn should be

proportional to the temperature at high enough temperatures (classical regime), as may be derived easily

from equations (9) and (10). Therefore, a plot of {v?) versus T should extrapolate to zero at 0 K. Assuming

that any static disorder is independent of temperature, there will be a constant additional {v?)stai,

which can be read off the ordinate at 0 K by the

same extrapolation procedure. Care has, however,

to be taken with this widely used method, since (1)

the static disorder may also in fact be temperature

dependent; (2) there may be couplings between the

static and dynamic components; (3) at too low temperatures (T < 0£), where 0d is the Debye temperature) quantum mechanical effects cause deviations

from linearity (zero point motion!); and (4) at too

high temperatures anharmonicities again cause deviations from linearity. The latter may be simply seen from the quasi-harmonic approximation, which ac- counts for thermal expansion by replacing the normal

mode frequencies u> through = u) '^{l-\-2fGxT),

where fa is the Griineisen contstant, and x is the

volume expansion coefficient. Then = B^'''"'{\ + 2'iGXT), (12)

i. e., proportional to T-f- const, x T^.

If ADWF's have been refined, similar extrapolations hold for C-'*' and D^^^"^ , which should be

proportional to and , respectively, but should

again be used only with care as discussed above.

A better method seems to be the consideration of

the temperature dependence of the potentials (Bachmann k, Schulz, 1984). Clearly, in the purely dynamic case this should be constant, whereas in the

presence of static contributions it makes no sense at

all and consequently will change with temperature

(becoming virtually steeper with increasing temperature, since the relative contribution of thermal motion increases).

3. Data evaluation

The Gram-Charlier series expansion (equation (11a))

up to the fourth order has been included in the multiphase Rietveld program of Thomas & Bendall (1978).

The restriction to four orders seems to be sufficient

in the wide majority of cases (also in single crystal work). The program consists of two parts, the

preparation program PREP and the refinement program PROF. At present only Gaussian line profiles are available for neutrons. This has been found to

be adequate in most cases. In fact, the deviations at the tails of the reflexions (calculated too low) which

are observed in the presence of strong thermal diffuse

scattering (TDS) are considered to be even advantageous, since a (partial) correction for this effect is made by this procedure. This has been confirmed by

test refinements on lead. On the other hand, simple,

modified and intermediate Lorentzians are supplied

for X-rays to account for the intrinsic different line

shape (intermediate Lorentzians gave the best fits for the local diffractometer). Problems may arise,

however, in both cases, if strong particle size and/or

strain broadening is present in the sample.

Further analysis of the results is done with the

single crystal system PROMETHEUS (Zucker ei a/., 1983), which allows the calculation oi pdf maps, potentials and Fourier maps. Higher orders (up to six)

and other expansions (Edgeworth) may be tried also.

For this purpose an output file is provided which contains the "observed" integrated (Bragg) intensities

r /u\ yicaUyh) /io ^ Iobs{h) = 2_^yiob, (13a)

Vicalc

{h stands for the reflexion indices hkl, yiobs/cau are the observed and calculated profile intensities, and
168 Anharmonic Thermal Parameters, Disorder and Phase Transitions

the summation runs over all y,- that can contribute

to a specific reflexion h) and the corresponding vari- ances

CT{Iobs{h)) = ,E[<T(y.c.6,)^^^^]2. (136)

Y

^

Vicalc

Further input files for PROMETHEUS are prepared

from the refined parameters of PROF by a special

program RIPR. Note that pdf maps and potentials

only need the final refined parameters of PROF,

while Fourier maps and other expansions require the

list of (pseudo) Bragg intensities. The latter are of course biased by the weighting scheme in equation

(13) (ambiguous for many overlapping reflexions), so that care has to be taken in interpreting these results. Difference Fourier maps have, however, been

used successfully, e. g., to locate interstitial atoms, as long as the reflection overlap is not too large. Misleading results have sometimes been obtained with

strongly overlapping diagrams.

To estimate the significance of the results one

should distinguish two situations:

(1) For low symmetry space groups and/or low

atomic site symmetries the number of structural parameters p, (including the scale factor) is usually

much larger than the number of "instrumental" parameters pi (half width parameters, lattice constants,

etc.). During the refinement the number of anharmonic coefficients should be increased stepwise fix- ing non-significant ones to zero at each step. The

conventional agreement factors

(14a)

and the corresponding goodness of fit (= Rwp/Rexp)

gofp = ^iiViob' - yicalcYlirriy - p) (146)

are adequate in this case. Here ruy is the number of

profile intensities, p (= p, -\- Pi) the total number of

refinable parameters and Wi = l/(7^(j/s) the weight.

(2) For high symmetries (both space group and site

point group) p, is small and about the same as p,-. Then the consideration of anharmonicity requires

only a few additional parameters. Consequently, the

eff'ect on the conventional gofp is small, since the denominator in equation (14b) is only slightly changed.

Instead, a much larger effect may be obtained, if one

uses a weighted Bragg R-factor

Y

/i h

(15a)

and

where wj = l/a'^{Iobs{h)) and mj is the number of

reflections. Clearly, this procedure is statistically not

fully correct, since it neglects the correlations with

the instrumental parameters and those between the

"observed" intensities. In general, however, the cor- relations between structural and instrumental paramaters are small (although this has to be checked in

each application) and, for high symmetries, the re- flection overlap is not too severe. The latter limits,

however, the applicability of equations (15) for low

symmetries, and therefore equations (14) are recom- mended in this case. It may be added that a weighted

Bragg R-factor is generally preferred to the usually

quoted unweighted Rb, since it also avoids unrealistically large values encountered sometimes, when near

the end of the dataset or near excluded regions some

reflections contribute to the fit only with their tails. In any case, to avoid spurious results, one should

keep only those terms in the refinement that are very

clearly significant (more than, say, four or five estimated standard deviations (e.s.d.)).

Of course one always starts a refinement with

HDWF's. Apart from the natural expectation that

allowance should be made for anharmonicity for diagrams taken at very high temperatures or for weakly

bonded crystals, other criteria might also indicate its necessity, e. g. very large B values and a temperature

dependence following equation (12). A very strong

indication has been found to be occupation numbers

n coming out considerably less than one (unless this

is due to nonstoichiometry etc.). Another strong hint

is a large and modulated diffuse background, which

always indicates strong disorder (compare equations

(2) and (3)).

As has been shown by Bachmann k Schulz (1984), a mathematically equivalent description of non- Gaussian pdf's is the use of split atoms. Strictly this

is true only if an infinite number of split positions

is used, and the anharmonic expansion is carried to

infinity. In practice both models must be truncated,

so similar but slightly different distributions are obtained. Depending on the actual problem one or the

other can provide a better description. For example,

for a disorder type where two atoms are distributed

randomly over two closely spaced equilibrium positions, the split model is superior. Very often, however, one encounters severe convergence problems,

due to very strong correlations in the split model,

unless the starting values are already close to the fi- nal ones. Such correlation problems are usually less severe in the anharmonic model. The latter finds the
H. Boysen 169

correct distribution simply by refining the higher or- der coefficients, whereas in the split model a number

of possible displaced positions (magnitude and direction) have to be tried as long as nothing is known

from the beginning. Hence, even if the split model

turns out to be superior at the end, first hints may of- ten be obtainend more easily by refining anharmonic

terms first. Moreover, quite often the number of free

parameters is less in the anharmonic model.

A final remark concerns the use of Fourier maps

calculated from the "observed" structure factors,

which in principle can give the same information.

Apart from the ambiguity in the "observed" intensities (see above), they always suffer from truncation

effects, which are avoided in the pdf approach.

4. Examples

In order to demonstrate the necessity to employ

ADWF's in the refinements from powder diffraction

patterns even for "normal" substances at elevated

temperatures, an experiment has been carried out

on zincblende (ZnS) up to 1223 K (Boysen, Steger,

Hewat & Buevoz, 1981). Both atoms lie in tetra- hedral site symmetry (43m). The appropiate single

third order anharmonic coefficient becomes significant (larger than one e.s.d.) above f« 900 K. It is in good agreement (within one e.s.d.) with a thorough single crystal investigation at room temperature (Cooper, Rouse k. Fuess, 1973) and is proportional to T^, as expected (although the e.s.d. 's were

too large to really confirm this without doubt). The

corresponding pdf has lobes pointing towards the

corners of a tetrahedron opposite to the Zn-S bonds.

This example clearly shows the importance of third

order anharmonic terms for atoms lying in non- centrosymmetric site symmetries. In this investigation the OPP expansion (which is equivalent to the

Gram-Charlier formalism in this special case), as well

as single line fits, were employed. The anharmonicity

could be traced back to intensity changes (as compared to the harmonic case) of individual reflections

that amounted to as much as about 6%. This provides an estimate of the required statistical precision

of the measurement.

Much larger anharmonic effects were found for the

special materials in the following examples. Most

of the experiments discussed below were carried out

with neutrons, since they usually allow more reliable access to temperature factors. With X-rays one also

has to keep in mind that asphericities of the atomic

electron density can cause similar effects and can in

general not be easily distinguished from thermal motion.

(a) Fast ionic conductors

Particularly interesting materials exhibiting large

atomic motions are superionic conductors. The pdfs

of the mobile ions will indicate the diffusion pathways. The maximum potential along these paths

may be compared with activation energies obtained

e. g. from conductivity measurements: they should

be equal, if no additional energy is required for the

formation of defects. Note that the potential barrier

derived from the scattering experiment only contains

the activation energy for migration.

[110]

Figure 1: pdf map of Cu in Cul at 698 K for an- harmonic (top) and split (bottom) models. Contour

lines refer to 3, 6, 12, 24, 48 and 96% of maximum.

Negative values are dashed. Tetrahedral and octahedral positions are marked by a cross and a star, re- spectively. Straight dashed lines indicate the traces

of the faces of the surrounding I-tetrahedron.

Cul (Yu, Boysen k Schulz, 1990) has the

zincblende structure at room temperature (7-phase,
170 Anharmonic Thermal Parameters, Disorder and Phase Transitions

space group F43m) and transforms to the a-phase

(disregarding an intermediate hexagonal /?-phase) at high temperatures. Here the anions keep their ar- rangement as in the 7-phase, while the cations are randomly distributed over all tetrahedral sites. The

space group is Fm3m. In figure 1 the pdf maps of

the Cu cations at 698 K are compared for a split and

an anharmonic model. Although they look quite dif- ferent at first sight, they both indicate a similar dif- fusion path, first along [111] through the face of the

surrounding I-tetrahedron into the octahedral cavity,

where it turns to [001] and reenters the neighbouring

tetrahedral site along the equivalent [111] direction.

At this temperature both models yielded practically

the same R-values. Hence the real pdf is probably in

between. On the other hand, a distinction could be

made in favour of the split model at 633 K and the

anharmonic model at 763 K. Altogether these findings may be understood by a transition from a jump

to a diff"usion like behaviour (shorter and longer res- idence times on interstitial positions). At the highest temperature the difi"usion path has moved a little more inwards towards the direct connection between

the tetrahedral sites (through the edges). The potential barrier along these diff'usion paths turns out to be

independent of temperature and is in excellent agreement with determinations of the activation energy

from conductivity and NMR measurements. Very

similar results have recently been obtained for the 7-

phase of CuCl (Graneli ei ai, 1992) and CuBr (Altorfer & Biihrer, 1992). Measurements at a number

of temperatures nicely revealed the expected proportionality between the single third order anharmonic

coefficient CI23 and T^.

Table 1: Agreement factors of diff"erent models for

Cul at 763K (h=harmonic, s=:split, 3=3'"'' order,

4=4*'' order anharmonic)

h s 3 4

9.40 8.76 8.63 8.62

gofp 1.30 1.21 1.19 1.19

Rt.B(%) 7.04 5.26 4.50 4.60

gpfs 2.33 1.87 1.54 1.69

Ps 3 5 4 6

Pi 6 6 6 6

Table 1 summarizes the refinement statistics for

diff"erent models at 763 K. As anticipated before, the

preference for the (third order) anharmonic model

is hardly seen in the profile R-factors, whereas it is quite obvious in the weighted Bragg R-factors: the

lower RwB is obtained with fewer structural parameters! Fourth order terms are clearly not significant:

while R^p is slightly reduced, RwB becomes even

larger. This is, of course, a consequence of the fact

that the fit is still done on the profile intensities. It thus demonstrates the statistical shortcomings of this

procedure. Finally, table 1 shows that the harmonic

model (isotropic by symmetry) is definitely not adequate even with powder data of moderate quality:

this investigation was carried out on a low flux re- actor (instrument MAN I at the FRM in Garching).

The errors in the pdf maps are about 3% of the maximum (the lowest contour line in figure 1; i. e., the

negative regions are not significant) as estimated by

calculating pdf maps with the parameters changed by

one e.s.d. This procedure for estimating the errors can, however, no longer be applied, if the number

of parameters becomes larger. Statistically relevant

error maps may be calculated with the help of the

Monte-Carlo method and have recently been incorporated in the program package PROMETHEUS by

Kuhs (1992).

Doped zirconia (Zr02) and ceria (Ce02) are examples of high temperature anionic conductors. They

may serve here to illustrate the temperature dependence of the harmonic and anharmonic coefficients

and the discrimination of static and dynamic contributions (Martin et ai, 1992). Figure 2(a) shows

the harmonic B values of oxygen in CSZ (15% CaO)

derived from powder and single crystal data. Extrapolations to 0 K indicate the presence of static

contributions, which become larger at lower temperatures. This is consistent with an analysis of the

diff'use scattering in single crystals (Neder, Frey

Schulz, 1990), which revealed correlated clusters the

number of which decreases strongly around 900 K

(Proff'en et al., 1992). As anticipated before, the

presence of static disorder may also be inferred from

the temperature dependence of the potentials, [See

Fig. 2(6)]. They are not, of course, real potentials

in this case. Care has to be taken in interpreting

potentials unless the absence of static disorder has

been proven. The deviation from a linear temperature dependence of the third order anharmonic coefficient C123 when plotted versus T^, shown in figure

2(c), again reflects the presence of static disorder.

On the other hand, the expected behaviour for (almost) purely dynamic disorder was found for Y-TZP

(tetragonal zirconia polycrystal, 3% Y2O3) and ceria

(3% Y2O3), as is also shown in Fig. 2(c). It may be

worth mentioning that the pdf maps of TZP (which can only be prepared in polycrystalline form!) did

not indicate any enhanced ionic conductivity at low

temperatures as compared to CSZ, although this was

found in conductivity measurements. Hence, one has

to conclude either that the conduction takes place

predominantly along grain boundaries or intergrown

glassy regions of the sintered material rather than in

the bulk, or, more probably, that the larger static
H. Boysen 171

(c)

500 1000 1500 2000

T/K

>a)

o

o

\

1

\

(

\

\

/

/

\ ;

\ /

V ;

\

/

V

/

V /

\

/ \ /

\ / V \ / / \ \ / / \ \ / / \ \ \ \

/ / / / \ \ / /

/ /

\ ^— \

0.0

[xxx]

short range order found in CSZ is responsible for the

reduced conductivity in that material.

A comparative study of tetragonal zirconia with

X-rays and neutrons (Martin, 1992) yielded rather

good agreement between the two methods provided

a careful absorption correction was made for the

X-ray data (by using two diffraction geometries,

Debye-Scherrer and Bragg-Brentano, and adjusting

the absorption coefficient). Non-perfect corrections

resulted in wrong isotropic B values. Nevertheless,

the anharmonicity and therefore the /orm of the 'pdj

was described correctly.

3000

Figure 2: Temperature dependence of various parameters in zirconia. (a) 5(0) of CSZ from different

powder (circles and squares) and single crystal (triangles) measurements. The dashed line indicates the

extrapolation of the high temperature values to 0 K.

(6) Section through the potential at room temperature (solid line) and 770 K (dashed line) for CSZ.

(c) Anharmonic third order coefficient plotted versus for CSZ (circles), Y-TZP (squares) and ceria

(triangles).

(h) Phase transitions

Anharmonicity is a prerequisite for any structural

phase transition. Large amplitude atomic motions

may be appreciated from the fluctuation-dissipation

theorem

X = {AA)/kT, (16)

where x is the susceptibility (which should diverge

at a second order phase transition) and A is the amplitude of the critical normal mode (or spin normal

mode in the case of order-disorder transitions). Note,

however, that the ADP's represent an average over

all normal modes, i. e. the additional contribution

of the critical mode may be relatively small. For a

general discussion of DWF theories near structural

phase transitions see Bruce & Cowley (1981). A

cusp-like increase of the HDWF near Tc is predicted

and has been observed e. g. by Ivanov ei al. (1979)

in ferroelectric perovskite-type structures and by Kihara (1990) at the a- /? transition in quartz. Again,

more information about the critical mode could be

obtained from "anharmonic" refinements.
172 Anharmonic Thermal Parameters, Disorder and Phase Transitions

1 t

o ^

1^'

o

>

1

\

^

[010]

Figure 3: pd/-map of NiTiOa at 1520 K. Ni and Ti

are on the threefold axis, while oxygens are not all exactly on the section shown. The planes of oxygens

are indicated by the dashed lines. Contour lines as

in figure 1

.

At room temperature NiTiOa possesses the or- dered ilmenite structure (/23) which may be envisaged as a nearly hexagonal close packing of the oxygens with the cations occupying 2/3 of the octahedral voids in an ordered sequence along the threefold

axis. At Tc 1560 K) it undergoes a phase transition to a disordered corundum structure (i23c) with

a random occupation of the same cation sites (Lerch

ei al, 1992). Besides the order parameter describing the cation exchange there are further displacive

degrees of freedom belonging to the same symmetry

species. A neutron powder investigation of the struc- tural changes with temperature (Lerch ei al., 1992)

revealed not only a critical increase of the HDWF as T —y Tc, but also unexpected coupling relations between the different order parameters. The reason for

this could be found from "anharmonic" refinements.

As shown in the pdf map at about 40 K below Tc

(Fig. 3) the cation densities are deformed and, most

important, there is some additional density even in

the normally {i. e., far below and above Tc) empty

octahedron. Hence, a description in terms of a pure

site exchange process is by far too simple. In more

accurate phase transition theories this (probably dynamic) delocalization of the atoms has to be taken

into account. A rather similar situation has also been

found in the ferroelectric-paraelectric transition in

LiNbOa (Boysen & Altorfer, 1992), i. e., it seems to

be a more general phenomenon. In both cases trials

using split models failed due to very strong correlations.

[110]

Figure 4: pdf map of K in leucite at 1073K. Arrows

point to the positions of the potential barriers along

the diffusion path. Two equivalent K positions re- sulting from the tetragonal refinement are marked

by squares and triangles, respectively. Contour lines as in figure 1, but starting at 1.5%.

(c) Static disorder

Static displacement distributions in disordered crystals, (e. g. solid solutions) are almost never truly

Gaussian (Krivoglaz, 1969) and require the inclusion of higher order terms. One example (CSZ) has

already been mentioned. Another example is the

framework silicate leucite, K(Al,Si2)06, although it also exhibits aspects of ionic conductivity and phase

transitions. The structure consists of corner-linked

(Al,Si)04-tetrahedra forming 4-, 6- and 8-fold rings

perpendicular to (100), (HI) and (110), respectively.

K atoms are located in non-intersecting tunnels along
H. Boysen 173

(111) created by the 6-fold rings. The structure is cubic (laSd) above Tc « 940 K and tetragonal below

(I4i/acd and 74i/a). In the cubic phase the space

group symmetry requires statistical occupation of Al

and Si on the tetrahedral sites, while partial ordering

is allowed (and is realised) in the tetragonal phases

(Boysen, 1990). A conventional harmonic refinement

yielded extremely large 5(K) f« 29 A^, which could

be reduced to B{K) « 16 A^ if the occupancy of

K was also refined: n(K) f» 0.84. As mentioned,

both findings are strong indications of anharmonicity. After introducing the appropiate anharmonic

terms n(K) returned to full occupancy. The cor- responding pdf of K at 1073 K is shown in figure

4. Two features may be discerned: (1) there is a continuous probability density all along the tunnel,

and, (2) there are lobes along (110). (1) may be re- lated to ionic conductivity, which has also been found

from dielectric measurements (Riischer ei ai, 1987),

again with a remarkable agreement of the activation

energy with the potential barriers at the positions

marked by the arrows in figure 4. (2) may be interpreted as static short range order: if the Al/Si

distribution is not completely random (local tetragonal symmetry) the K ions are displaced towards the

potential Al-sites. In fact, an equally good fit, again

with n(K) « 1, could be obtained in space group

lAi/acd with a strictly cubic metric. The resulting positions of the K atoms are marked in figure 4

for two equivalent distortions. It may be concluded

that both models are approximations to the real one: tetragonal distortions on a very fine scale within a

strictly cubic framework. It should be added that (of course unphysical) negative pdfs like those in figure

4 (and figure 3) are frequently encountered near very

steep features in the true pdf, a consequence of the

truncation of the series.

5. Conclusions

The above examples should have demonstrated that

valuable information can be obtained from an analysis of anharmonic temperature factors also in the case of powder diffraction. For a proper understanding of

the physical and chemical properties of matter a detailed knowledge of the structure is essential, and this

includes not only average positions of the atoms, but

also their static and dynamic displacements and disorder. All of the examples represent relatively simple structures, with the consequence that the reflec- tion overlap is not too severe. In more complicated

structures, however, the strong overlap and the large

number of refinable parameters cause problems that

eventually lead to unphysical results. Such a situation was, e. g., encountered in the tetragonal leucite

phases and in (Pbo.92Bao.o8)3(P04)2 (Hensler ei al,

1992), where an anharmonic refinement was possible

in the high temperature trigonal phase but not in the

low temperature monoclinic one. It may therefore

be hoped that with even higher quality data (resolution and intensity) such problems may be tackled in

the future. Furthermore, the extension of the scat- tering diagram to large Q-values (small wavelenghts

and large scattering angles) is also generally helpful,

since the influence of the anharmonic terms increases

with Q (see equation (11)).

Although the Gram-Charlier expansion is suffi- cient in a wide range of applications, there are limitations in some special cases. Note that the truncation of the series restricts its application to relatively

small anharmonicities. In an anharmonic refinement

of non-stoichiometric Li^jAl ( 0.8 < x < \) (Tadin

et al., 1991) lobes in the pdf of Li indicated a difi'u- sion along certain zig-zag-chains. The fit could then

be considerably improved by placing interstitial Li

atoms all along this path, leading to a nearly uniform density distribution. However, this could only

be achieved by fixing the positions, occupancies and

thermal parameters. Extreme correlations between

these parameters prevented a refinement. Therefore, a problem adapted expansion of the pdf that uses only a small number of parameters {e.g. cylinder

functions) would be helpful in this case. Another

example, where problem adapted expansions might

be superior is rotational motions of rigid groups in

molecular crystals. Such a treatment based on spherical harmonics (Press & Hiiller, 1973; Prandl, 1981)

has been used, e.g., by Hohlwein et al. (1987) for

the orientational disorder in CSNO2.

A particularly interesting field of future applications of ADWF's seems to be the critical behaviour

near phase transitions (in a wide sense, i. e., including melting, decomposition and chemical reactions).

Current theories are based on anharmonicity (mainly

in the quasiharmonic approximation) and disorder.

Very little has been done so far even with single

crystals. More experimental evidence could challenge

further theoretical studies. Powder difi"raction can be

advantageous because it is less susceptible to systematic errors {e.g. the change of mosaicity/extinction)

and the ease of data collection, allowing structure determinations at many temperatures in a reasonable

time. High quality furnaces with very good temperature homogeneity and stability (a few hundredths

of a degree) are necessary for such investigations.

More work is needed for the correction of thermal

diff"use scattering (TDS) and multiple scattering in

strongly anharmonic systems. Another point con- cerns the relation of anharmonicities to line broadening effects like microstrains and particle sizes.

In all of the examples given in section 4 a strong
174 Anharmonic Thermal Parameters, Disorder and Phase Transitions

and modulated diffuse background immediately signaled the presence of disorder. This background by

itself contains valuable information about the cor- relations of the deviations from the average struc- ture (compare equations (2) and (3)). Attempts

to calculate this diffuse background include the use

of the Warren-Cowley short range order formalism

(Warren, 1969; for an example (NaTl) see Schneider, 1988), Fourier-filtering (Richardson &; Faber,

1986) and low dimensional disorder phenomena (corresponding to diffuse streaks and planes; Boysen,

1985). The final goal will be to refine this background simultaneously with the sharp Bragg peaks.

In any case, an accurate determination of the background is important to define a reliable baseline un- derneath the Bragg peaks, especially in the presence

of sharper features (TDS peaks, low dimensional disorder). It has been found, however, that varying the

background mainly affected the harmonic part of the

DWF and to a lesser extent the anharmonic one. Although, as shown before, a distinction between

static and dynamic disorder can often be obtained

from an analysis of the temperature dependence of

the harmonic and anharmonic ADP's, more definite

conclusions could be drawn by using both elastic

(with an analyzer in the diffracted beam) and integral (i. e. energy integrated) neutron diffraction patterns (Boysen k. Schneider, 1986). Not only can the

static or dynamic nature of the diffuse background

and therefore the origin of the disorder be determined, but also an estimate of the TDS contribution

to the Bragg intensities can be made.

This work was supported by the BMFT under

project No. 03-SC2LMU3.

References

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LNS-161 , ETH Ziirich, 114-115.

Bachmann, R. & SCHULZ, H. (1984). Acta Cryst. A40,

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Boysen, H. (1985). J. Appl. Cryst. 18, 320-325.

BoYSEN, H. (1990). in Phase transitions in ferroelastic

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174, 26-27.

Boysen, H., Steger, G., Hewat, A. W. & Buevoz,

J. L. (1982). Neutron Scattering- 1981. AIP Conference

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Boysen, H. & Altorfer, F. (1992). in preparation.

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Cooper, M. J., Rouse, K. D. & Fuess, H. (1973). Acta

Cryst. A29, 49-56.

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Graneli, B., Fischer, P. & Dahlborg, U. (1992).

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Hensler, J., Boysen, H., Bismayer, U., Vogt, T.,

Francke, W. & Heitjans, P. (1992). Z. Kristallogr.,

submitted.

HOHLWEIN, D., HosER, A. & Prandl, W. (1987). Z. Kristallogr. 177, 93-102.

Ivanov, S. a., Tchornei, S. A., Michaichenko, B.

P. & Benevtsey, J. A. (1979). Ukrainian Physical

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Johnson, C. K. & Levy, H. A. (1974). International

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Krivoglaz, M. a. (1969). The Theory of X-ray and

Thermal Neutron Scattering by Real Crystals. London:

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Lerch, M., Boysen, H., Neder, R., Frey, F. &

LaquA, W. (1992). J. Phys. Chem. Solids 53, 1153-

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Lerch, M., Boysen, H., Frey, F. & Vogt, T. (1992).

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Martin, U., Frey, F. & Boysen, H. (1992). Acta

Cryst., submitted.

Martin, U. (1992). Thesis. Universitat Miinchen.

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1079.

Neder, R. B., Frey, F. & Schulz, H. (1990). Acta

Cryst. A46, 792-809.

Prandl, W. (1981). Acta Cryst. A37, 811-818.

Press, W. & Hijller, A. (1973). Acta Cryst. A29, 252-

256.

Proffen, T., Neder, R., Frey, F., Keen, D. A. &

Zeyen, C. (1992). Acta Cryst., submitted.

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RuscHER, C, Papendick, M., Boysen, H., Putnis, A.

& Salje, E. (1987). Z. Kristallogr. 178, 195-a96.

Schneider, J. (1988). Materials Science Forum 27-28,

63-68.

Tadin, M., Schneider, J., Boysen, H. & Frey, F.

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Warren, B. E. (1969). X-Ray Diffraction. AddisonWesley.

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16, 358.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Recent Advances in the Use of Anomalous Dispersion Effects

J. P. Attfield

Department of Chemistry, University of Cambridge,

Lensfield Road, Cambridge CB2 lEW,

and Interdisciplinary Research Centre in Superconductivity,

Madingley Road, Cambridge, CBS OHE, U.K.

Abstract

Recent advances have centred upon the use of synchrotron X-ray wavelengths that are very close to

(usually within 50 eV of) an elemental absorption

edge, giving rise to very large anomalous dispersion

effects. Rietveld refinements using such resonant Xray diffraction data enable the distribution of the

absorbing element over crystallographically distinct

sites within a structure to be determined precisely, even when other elements with very similar atomic

numbers are present e. g. Fe in FeNi2(B03)02 us- ing data collected at the Fe K edge. Distributions

of more than two cations can be found by simultaneously fitting a structural model to resonant profiles collected at several absorption edges. Resonant

diffraction data may also be used to study different

electronic states of the absorbing element if these are ordered within a crystal structure, e. g. Eu^"*" and

Eu^"*" in EU3O4, due to differences in the absorption

edges of the two states. This leads to the possibility of obtaining site-resolved XANES spectra. The

accuracy of the results depends upon the accuracy

with which the anomalous dispersion coefficients f

and f are known; they can be calculated theoretically or determined from X-ray absorption spectra or

crystallographic refinements. Strategies for collecting resonant diffraction data will also be discussed.

Introduction

The intense white X-ray beams produced by electron

synchrotrons have recently been used in anomalous

scattering powder diffraction experiments. These

studies are typically performed close to elemental absorption edges and so are termed 'resonant' X-ray

diffraction experiments. Resonant diffraction experiments tend to fall into three classes:

(i) Fundamental studies of simple materials in which

reference values of the anomalous dispersion coefficients are determined and strategies for powder methods are explored.

(ii) Elemental contrast experiments to determine

the distribution of the resonant element within a crystal structure in the presence of other ele- ments with similar atomic numbers.

(iii) Electronic contrast experiments in which

the differences between inequivalent states of

the resonant element, usually different valence

states, are studied through differences in their

anomalous dispersion coefficients.

In this review, the origin of anomalous scattering, experimental considerations, and examples of

the above experiment types will be discussed, especially in the latter two areas where anomalous dispersion gives information about elemental or electronic

distributions that is often not available from other

techniques.

Anomalous scattering and X-ray absorption

The atomic scattering factor for X-rays, f, contains

both a normal (Thomson) term, f", and an anomalous dispersion contribution consisting of real, f, and

imaginary, f, terms.

f = r f (E) -h if"(E) (1)

The normal scattering depends upon the electron distribution and varies only with sm9/\, whereas the

anomalous contribution results from spectroscopic

processes, and so is highly energy dependent. The

curves for f(E) and f'(E) are directly related to

the linear photoelectric absorption coefficient /i(E)

through the optical theorem:

f"(E) = {mc/2he'^N)En{Y), (2)

where m = electron mass, c= velocity of light, h =

Planck's constant, e = electron charge, N = number

density of atoms; and the Kramers-Kronig dispersion

relationships:

f(E) = (2/7r) / [E*f"(E*)/(E2 - E*^)]dE* (3)

^0

175
176 Recent Advances in the Use of Anomalous Dispersion Effects

^ > >

Figure 1: Schematic energy variations of the photoelectric absorption coefficient, /z, and the anomalous

dispersion terms f and f (continuous lines). The same quantities are also shown for a higher oxidation

state of the same element (broken lines).

f"(E) = (2E/7r) /[f (E*)/(E2 - E*2)]dE (4)

by taking the Cauchy principal values of the integrals

in (3) and (4).

X-ray absorption spectra of solids contain discontinuities ('edges') at the threshold energies for electronic transitions from core levels to excited states,

e. g. Is -> np for the K edge. Accessible X-ray

wavelengths are available for elements with Z > 20

at K-edges from Ca (3.07 A) to Ag (0.49 A) and

L and M edges beyond Cd (Lj = 3.09 A). XANES

(X-ray Absorption near-Edge Structure) and EXAFS

(Extended X-ray Absorption Fine Structure) effects are also apparent in such spectra. XANES features

around the edge arise from electronic transitions to

localised excited states, which may result in a very intense 'white line' absorption peak (Brown, Peierls &; Stern, 1977). The EXAFS modulations result from

the backscattering of the emitted photoelectron by

surrounding atoms and persist for several hundred

eV above the edge.

Through the above equations, the features in /i(E) are present in the f'(E) curve and, less straightforwardly, in f(E), as shown in figure 1. This diagram

also illustrates the use of resonant X-ray energies for

elemental and electronic contrast studies. The former studies are made possible by the large negative

values of f around the edge, which significantly re- duce the overall atomic scattering factor of the reso- nant element. The edge position tends to vary with

the electronic state of the resonant element, and in- creases by several eV for each unit increase in ox- idation number, and so discernible differences between the f or f' curves for different electronic states may be found from resonant X-ray diffraction experiments. The anomalous terms account for an increasing proportion of the total scattering as 29 increases, as f and f show virtually no fall-off with sin^/A

(Templeton, Templeton, Phizackerley &; Hodgson,

1982).

Anomalous dispersion curves can be determined

experimentally from X-ray absorption spectra using

equations (2) and (3) (Bonse & Hartmann-Lotsch,

1984; Hoyt, De Fontaine & Warburton, 1984; Dreier,

Rabe, Malzfeldt, Sz Neimann, 1984). Good agreement has been found between such spectroscopic

results and values refined from single crystal data

(Templeton & Templeton, 1982, 1985, 1988, 1990;

Templeton, Templeton, Phillips Sz Hodgson, 1980;

Templeton, Templeton, Phizackerley k. Hodgson,

1982). Anomalous coefficients can also be determined from powder diffraction data, as described below, or by X-ray interferometry Begum, Hart, Lea, Sz Siddons, 1986). Theoretical methods (Creagh, 1988)

such as the relativistic dipole calculations of Cromer

&; Liberman (1970, 1981) are commonly used to compute f and f (Cromer, 1983), but these do not take

account of the valence shift and local edge features

described above, and so are of limited accuracy at resonant wavelengths. Only crystallographic methods allow anomalous coefficients to be measured independently at different sites within the same structure, giving rise to the new developments described

here.

ExperimentjJ considerations

The use of existing synchrotron diffractometers for

angle-dispersive resonant powder X-ray experiments

has been discussed by Wilkinson, Cox Sc Cheetham
J. p. Attfield 177

(1991). Specific considerations for resonant studies are the wavelength and sample absorption. The en- ergy spread and mean wavelength must be known

accurately, and it is desirable to be able to select the

wavelength accurately and to vary it in relation to

the absorption edge position. If data are collected

above the edge, then large sample absorption and

background fluorescence make the choice of sample

geometry and detection system important.

The energy spread within the X-ray beam is determined by the monochromator band-pass, given by:

AE/E = ^(A<;icot ^B)2-h(Ad/d)2, (5)

where A(f> is the angular divergence of the beam in

the scattering plane, 9b is the monochromator Bragg

angle, and Ad/d is the divergence of the monochromator, which for a perfect crystal is related to the

intrinsic Darwin width of the reflection plane. Typical values for the Ad/d contribution at 1.5418 A

(8.04 keV) are AE = 1.1, 0.5, 2.6, and 1.2 eV for

Si(lll), Si(220), Ge(lll), and Ge(220) perfect crystals, respectively (Freund, 1987). The wavelength

thus needs to be determined with an accuracy of

AA/A ~ 10"^ at such energies, usually by fitting several peaks from a Si standard. Recalibration is typically performed each time the wavelength is changed.

This can be done conveniently from the positions of

the sample peaks.

For many experiments the wavelength is chosen

relative to the positions of features in the absorption

edge spectrum. The widths of absorption edge features tend to increase with energy, from Cu =1.5

to Sn ~ 8 eV for K edges (Hamalainen, Manninen,

Suortti, Collins, Cooper & Laundy, 1989; Bambynek,

Craseman, Fink, Freund, Mark, Swift, Price & Rao,

1972), placing a limit upon the necessary experimental energy resolution. As the experimentally observable absorption curve is a convolution of the true curve with the monochromator band-pass, it is highly

desirable to be able to record the spectrum in situ. rather than relying on spectra recorded on other instruments. This can be done by a direct transmission

experiment, but a more convenient alternative is to

monitor the sample fluorescence at an angle away

from the Bragg peaks. With a detector of poor en- ergy resolution, the variation of background intensity with energy gives a measure of the fluorescence,

and an approximate fi(E) spectrum can be derived

(Attfield, 1991). Alternatively, solid state detectors

offering good energy resolution can be used to measure the fluorescence spectrum directly (Wilkinson,

Cox k Cheetham, 1991).

When resonant diffraction is performed above the

threshold of the absorption edge then sample absorption becomes severe. Geometry-dependent absorption corrections must be applied to the diffraction

data, except for flat plate samples in reflection mode,

which are therefore preferable for most experiments.

However, even in this geometry, surface roughness can provide a small angle dependent correction, and

at very absorbing wavelengths the penetration depth

may be so small that incomplete powder averaging

or surface texture may result in systematic errors in

the diffraction intensities. The high fluorescent background at such wavelengths can be reduced by using

SoUer slits and a solid state detector with a high

energy resolution (200-400 eV) or a crystal analyser

with a gas proportional or scintillation detector. The

latter combination gives a very high peak to background ratio, but the low acceptance angle at the

detector may exacerbate the systematic errors at absorbing wavelengths.

Results of resonant powder diffraction

experiments

i) Fundamental Studies

The sensitivity of powder refinements to f and f in

comparison to single crystal studies can be demonstrated by considering the structure factors F± for a

Friedel pair of reflections ±hkl with only one anomalously scattering element in the unit cell: F± = A + a{f + if")

±i[B + b{f + if')] (6)

where A/B and a/6 are the summed cos /sin terms

for the normal and anomalous contributions, respectively. Hence,

F^ = A^ + B^ + ia^ + P^f'^ + f"^)

+2{Aa + Bb)f ± 2{Ba + Ab)i" (7)

In the centrosymmetric case this reduces to: Fl= A^ + a^^^ +i"^) + 2Aa^ (8)

In a powder pattern F^ and Fi are overlapped,

whereas they can be measured independently in single crystal studies. For a centrosymmetric structure

F^ = F^ so there is no loss of information in the

powder pattern compared to the single crystal case, beyond the usual compression of three-dimensional

data into a radial pattern. For small anomalous coefficients, the refinements are sensitive to f through

the cross term AaP but are relatively insensitive to f.

Single crystal studies of non-centrosymmetric materials are able to determine f from the difference

between F^ and F^ (equation (7)), whereas in powder diffraction only their sum is measured:

F^ -)- Fi = 2A^ + 2B^

+2{a^ + b^){f^ + f"^)

+4{Aa + Bb)r. (9)
178 Recent Advances in the Use of Anomalous Dispersion Effects

Hence the Bijvoet effect information that is used

to solve complex non-centrosymmetric structures

(Karle, 1989) and determine absolute configurations

(Flack, 1983) is lost in a powder pattern. Again,

for small anomalous coefficients the powder data are much more sensitive to variations in f than in f.

Strategies for determining anomalous dispersion

coefficients from powder patterns have been discussed for Ni (Suortti, Hastings &c Cox, 1985) and

YbaOa (Will, Masciocchi, Hart k Parrish, 1987). A

variety of reference materials have also been used

to obtain f values for a given geometry or valence

state as part of the elemental and electronic contrast

experiments described below, e. g. Fe203 (Perkins

& Attfield, 1991) and Fe3(P04)2 (Warner, Wilkinson, Cheetham Sz Cox, 1991) for studies of iron

oxo-salts and CU2O, CuO and KCUO2 (Kwei, Von

Dreele, Williams, Goldstone, Lawson &; Warburton, 1990) for the investigation of copper valences

in YBa2Cu306+r- In all cases, data were collected

below the absorption edge and used to refine f by

conventional integrated intensity or Rietveld (1969)

methods with f fixed at a calculated value. For ex- ample, in the study of Yb203 by Will, Masciocchi,

Hart & Parrish (1987), f values up to -21.2 e/atom

at 8 eV below the Lm edge were obtained, due to a

white line at the absorption edge.

Although high precision in f may be obtained, ac- curacy may be limited by correlations with temperature and scale factors. The former problem may be

minimised by collecting data over as wide a sinO/X

range as possible, or by determining temperature

factors from another off-edge or neutron diffraction

profile. When the resonant element is the dominant X-ray scatterer, then f tends to correlate with

the scale factor, requiring an intensity standard or measurements of absolute diffraction peak intensities

(Suortti, Hastings h Cox, 1985).

Resonant powder diffraction data can also be used

to study the anisotropy of the anomalous scattering

coefficients that results from anisotropy in //(E). This

can lead to the breakdown of extinction rules such as

for the n-glide planes in CU2O with PnSm symmetry

(Kirfel, Eichhorn k Wroblewski, 1988).

(ii) Elemental contrast

These experiments provide the most straightforward

and useful application of resonant X-ray diffraction.

When two or more elements of similar atomic number

are disordered within a structure, then resonant data

collected at one of the elemental absorption edges can be used to refine the individual distributions. Examples are given in table 1. In all cases contrast is achieved by using a wavelength just below the absorption edge, where sample absorption is low and

so f is small and calculable, but large negative val- ues of f can be achieved, giving marked changes in

Table 1: Materials studied by elemental contrast res- onant powder X-ray diffraction, with the resonant

element(s) underlined. 'N' indicates a simultaneous

refinement using resonant X-ray and neutron profiles.

EuSm204

Feo.5oCoo.48Yo.02, N

FeCo2(P04)3

FeNi2(B03)02

Lap . 9 Gdo. 9 Stq. 2 CUO4

Yo.i9Zro.8lO1.90 YBa2(Cu,Ga)307-r, N

YBa2(Cu,Zn)307-^

YBa2(Cu,Zn)307_x, N

Attfield (1990).

Williams et al. (1990).

Warner et al. (1991).

Perkins k Attfield (1991).

Kwei, Von Dreele,

Cheong et al. (1990).

Moroney et al. (1988).

Kwei, Von Dreele,

Williams et al. (1990).

Rowland et al. (1989).

Kwei, Von Dreele,

Williams et al. (1990).

the peak intensities.

An illustrative example is the study of

FeNi2(B03)02 (Perkins k Attfield, 1991) in which

a single resonant pattern collected 16 eV below the

Fe K edge was used to refine the Fe/Ni occupancies

of four crystallographically distinct octahedral sites

(Fig. 2). The profile fit is shown in figure 3, and the

results are summarised in table 2. The refinement

gives typical precision for the fractional coordinates

and other structural parameters, and also Fe/Ni oc- cupancies precise to 0.6% for the extra effort of se- lecting a resonant wavelength. The refined value off

from an Fe203 standard was -6.94(3) (calculated f =

-6.61 e/atom), which greatly reduced the Fe scatter- ing factor (f"(0) = 26 e/atom). A variation on this

type of study was carried out by Moroney, Thompson

k Cox (1988) on yttrium-stabilised cubic zirconia,

Yo.19Zro.8iO1. 90- In this case the Y/Zr occupancy of

the single cation site was fixed, and resonant data

were used to find how each species was disordered

around the ideal position by determining its temperature factor independently. The isotropic B-factors were found to be Y = 0.64(15) A^, Zr = 1.18(4) A^.

Contrasting elements with similar atomic numbers by resonant diffraction compares favourably

with the traditional neutron diffraction method.

Resonant diffraction is general for elements with

Z > 20 and can be used for highly neutron absorbing materials, e. g. EuSm204 (Attfield, 1990)

or for elements with similar neutron scattering

lengths, e. g. Fe and Ni in FeNi2(B03)02 above.

Warner, Wilkinson, Cheetham k Cox (1991) have

shown that independent Rietveld refinements of the

Fe/Co distribution in FeCo2(P04)3 with resonant

X-ray and neutron diffraction data give comparable results. The values of x for the distribution
J. p. Attfield 179

Table 2: Results of the resonant X-ray refinement of orthorhombic FeNi2(B03)02 (Perkins & Attfield, 1991).

The cation occupancies were constrained to give the correct overall stoichiometry.

Number of points = 9062 Number of reflections = 189

R-Factors (%): Rrvp = 12.6 9 5 R/ = 4.8 Rfjx — o.o Cell Parameters (A): a = 9.21048(7) b = 12.23794(9) c =

Atomic Parameters (space group Pbam):

Site Multiplicity X y ii Bi,.(A2) %Fe /OiM

M(l) 2 0 0 0 0.13(3) 8.8(6) Q1

M(2) 2 1/2 0 1/2 0.13 38.5(6) 61.5(6)

M(3) 4 -0.0001(2) 0.2814(4) 0 0.13 14.4(5) 85.6(5)

M(4) 4 0.7378(2) 0.3848(1) 1/2 0.13 61.9(4) 38.1(4)

B(l) 4 0.274(1) 0.3602(9) 1/2 0.3(2)

0(1) 4 0.8516(5) 0.0438(4) 1/2 0.13(4)

0(2) 4 0.3910(5) 0.0763(4) 0 0.13

0(3) 4 0.6276(5) 0.1394(5) 1/2 0.13

0(4) 4 0.8497(6) 0.1444(5) 0 0.13

0(5) 4 0.8497(6) 0.2395(4) 1/2 0.13

^^(Coa;Fei_i;)^(Co2-irFea;)(P04)3 over the five and

six-coordinate cation sites were found to be 0.502(8)

and 0.486(10) for time-of-flight neutron and resonant

X-ray diffraction, respectively. The contrast gained

by inducing f(Fe) = -7.8 e/atom, 9 eV below the

Fe K edge, was approximately equivalent to that between the neutron scattering lengths of b(Fe) = 0.954

and b(Co) = 0.253 x lO"^'' m.

A further advantage of resonant X-ray diffraction

for elemental contrast studies is the ability to collect data at the edges of several elements to solve

complex disorder problems involving more than two

elements. Simultaneous refinements of a combination of resonant synchrotron X-ray diffraction profiles, laboratory X-ray data, and neutron patterns

have been used by Kwei, Von Dreele, Williams, Goldstone, Lawson & Warburton (1990) to obtain precise

structural models for several compounds listed in ta- ble 2. For example, the distribution of cations over the eight and nine-coordinate sites in the T* type

superconductor (Lao gGdo gSro 2)Cu04 was found to

be ^^(Lao.8Sro.2)^'"(Lao.iGdo.9)Cu04 (Kwei, Von

Dreele, Cheong, Fisk & Thompson, 1990).

(iii) Electronic contrast

Studies of differences in anomalous scattering from a

resonant element at two or more crystallographically

distinct sites are shown in table 3, and have recently

been reviewed (Attfield, 1991). The ability of res- onant X-ray diffraction to contrast ordered valence

states was illustrated by a study of EU3O4 using a

single powder diffraction pattern collected at the Eu

Liii edge with A = 1.7771(1) A (6976.9 eV) (Attfield, 1990). Eu2+(Eu3+)204 is a localised valence

compound in which Eu^"*" occupies one of three crysTable 3: Compounds studied by electronic contrast

resonant powder X-ray diffraction, with the different

states of the resonant element(s) underlined.

a-Ce, valence fluctuations

Eu2+Euf+04

Ga+Ga^+CU

YBa2Cu(l)Cu(2)206+x,

X = 0.09,0.38,0.72,0.95

YBa2Cu(l)Cu(2)206+x,

X = 0.27

Y3(i^Ga3+)3(ViGa3+)20i2

Kwei, Von Dreele,

Williams et al. (1990).

Attfield, (1990).

Wilkinson, Cheetham k Cox, (1991).

Kwei, Von Dreele,

Williams et al. (1990) . Attfield (1991).

Wilkinson, Cox

& Cheetham

(1991) .

tallographically distinct Eu sites and Eu^"*" fills the

other two. In order to determine the valence distribution from the anomalous scattering alone, Rietveld

fits were performed for the three possible ways of or- dering the valences over the three sites by refining

both f and f" for Eu2+ and Eu^+ independently. The

value of the integrated peak intensity residual, Rj, was significantly lower for the correct model (11.8%)

than for the other two (18.5 and 17.8%). The re- fined anomalous coefficients were compared to published curves for f(E) and f"(E) around lanthanide

Liii edges (Templeton, Templeton, Phizackerley &; Hodgson, 1982), enabling the energy shifts of the Lm
180 Recent Advances in the Use of Anomalous Dispersion Effects

Figure 2: (001) polyhedral view of FeNi2(B03)02

with the unit cell marked (a horizontal, b vertical)

and the boron atoms shown as circles. The four crystallographically distinct octahedra are labelled, and

those centred in the z = 0 / 1/2 planes are lightly /

heavily hatched.

edges of Eu^"*" and Eu^"*" relative to the beam energy

to be estimated as -2.3(10) and 4.2(8) eV respectively

(Fig. 4). The valence shift of 6.5(13) eV is in excellent agreement with the difference of 6.4(3) eV from

the X-ray absorption spectra of EuO and EU2O3.

In the study of EU3O4, large differences between

the anomalous scattering coefficients of different oxidation states were observed as the chosen wavelength

lay between the edge energies of the two oxidation

states. However, the high sample absorption re- sulted in low counting statistics and poor precision

in the refined anomalous coefficients. To avoid these

problems, other electronic contrast experiments have

been performed at pre-edge energies to obtain good

counting statistics, from which significant differences

in refined f values have been obtained. For example, a study of Ga+Ga^+CU (Wilkinson, Cheetham

k Cox, 1991) revealed differences up to 3.2 e/atom

in f between the Ga+ and Ga^"*" sites. The studies in

table 3 show that resonant powder X-ray diffraction

QjUiijjL,

\ iiiiiiiiiiiiiiiiiiBiiiiiifiiiiiiiiiiiigiiiiiiiiin

H

40 60 80

29 (Degrees)

100

Figure 3: Observed (dashes), calculated (full line),

and difference resonant X-ray diffraction profiles for

FeNi2 (603)02 with the reflection positions marked

(Perkins & Attfield, 1991).

can be used to demonstrate that different valence

states, and perhaps differently coordinated sites, of

the same element present in the same structure may

have significantly different anomalous scattering coefficients. However, the interpretation of such results

is not always straightforward. In the case of EU3O4,

a single pattern was used to refine four anomalous

terms, which could be related to the published f(E)

and f'(E) curves only by assuming that the edge

shifts but does not change shape in going from Eu^"*" to Eu^"*". A better approach is to use f (E) or f'(E)

curves derived from appropriate standards, for ex- ample Cu -t-1, +2, and -f-3 oxides to interpret results

from YBa2Cu306+r (Kwei, Von Dreele, Williams,

Goldstone, Lawson Sz Warburton, 1990), but even

this involves a certain level of approximation, as the XANES features are sensitive to local electronic

structure as well as oxidation state (Alp, Goodman,

Soderholm, Mini, Ramanathan, Shenoy, k. Bommannavar, 1989). The influence of site geometry has

been investigated in the garnet Y3Ga50i2 (Wilkinson, Cox, k. Cheetham, 1991). A small difference in f

between octahedrally and tetrahedrally coordinated

Ga^"*" was found, but the range of observations was

too limited to determine the origin of the difference.

In view of the problems in interpreting electronic

contrast results, the ultimate aim of such experiments might be to produce resolved X-ray absorption edge spectra for each crystallographically independent site. In principle, this could be achieved by

collecting data at energies across the edge, refining

f and f for each site, and scaling the f"(E) curves

to /i(E) via equation (2). This is not yet practicable,

as sample absorption necessitates very long count-
J. p. Attfield 181

a)

35

30

E

0

4J 25

(0 \

0) 20

15

10

5

_ -5

i"o -10

to

4-1

-10

-20

-25

-30

Figure 4: Curves for the variation of f and f' with

AE (= Efieam— EEdge) around the Eu LIII absorption

edge. The points show the refined values for Eu^"*" (AE = -2.3(10) eV) and Eu^+ (AE = 4.2(8) eV) in

EU3O4 (Attfield, 1990).

ing times at highly absorbing energies, and correlations between the refined f and f values may be

severe. However, a simple experiment using limited

crystallographic data to resolve an X-ray absorption

spectrum has been demonstrated for YBa2Cu306.27

(Attfield, 1991). The variations of |Fooi| and |Fiii|

in the near-edge region were measured (Fig. 5) and

used to divide the bulk Cu K edge spectrum into curves for the two Cu sites (Fig. 6). Although the re- sults are of very limited precision, they do reveal that

the Cu(l) edge is broader, with unresolved pre-edge

features, and at a lower energy than that of Cu(2).

This is consistent with the accepted view that both

Cu"*" and Cu^"*" are present on the Cu(l) (chain) sites,

but only Cu^"*" is on the Cu(2) (plane) sites for this

oxygen content.

Resonant X-ray diffraction may also be able to investigate systems in which valence fluctuations are too rapid to give rise to a structural distortion. Kwei,

Von Dreele, Williams, Goldstone, Lawson & Warburton (1990) report a study of the low temperature a

form of cerium metal. If the previously determined

average valence of +3.6 results from fluctuations between -|-3 and +A states that are ordered for a sufficiently long time to give coherent Bragg diffraction

14

13

o

o

Ua 12

11

8960 8980 9000 9020 9040 9060 9080

E (eV)

21

20

19

18

17

16

8960 8980 9000 9020 9040 9060 9080

E (eV)

Figure 5: The energy variation of |F(001)| and

|F(111)| for YBa2Cu306.27 around the Cu K edge

from Attfield (1991). The smooth curves are a fit to

the data by eye.

(~ 10"^^ — 10~^^ s) then superlattice peaks should

be observable at resonant wavelengths. A long scan

7 eV below the Ce Lm edge, revealed a weak peak

at the position expected for the otherwise forbidden

100 reflection, but this result awaits further confirmation.

Conclusions

Recent resonant X-ray diffraction studies have shown

that large anomalous scattering effects may be observed close to absorption edges using high resolution powder synchrotron instruments. Accurate f

values may be determined from pre-edge data, but

powder refinements are less well suited for studying

f and f at absorbing energies within the edge. The

large f values obtained in the pre-edge region en- able the resonant element to be distinguished from

neighbouring elements, and such elemental contrast

refinements give site occupancies of comparable precision to those typically obtained by powder neutron

diffraction.
182 Recent Advances in the Use of Anomalous Dispersion Effects

400

300

200

100

8960 898() 9000 9020 9040 9060 9080

E (eV)

Figure 6: Site-resolved Cu K edge X-ray absorption

spectra for YBa2Cu306.27; Cu(l) = solid line; Cu(2) = dashed line (Attfield, 1991).

Electronic contrast experiments show that significant differences in the anomalous scattering from

different oxidation states in the same crystal struc- ture can be measured experimentally. In many cases, these differences can be related to valence effects in

the X-ray absorption edge and suggest the possibility of deriving site-resolved X-ray absorption edge

spectra.

These results establish resonant powder X-ray

diffraction as a tool for studying elemental distributions and the valence states of atoms at different sites

within crystalline materials. Further improvements,

particularly in electronic contrast experiments, will

result from the future use of very intense synchrotron

sources.

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KwEi, G. H., Von Dreele, R. B., Williams, A.,

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National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Instrumentation for Studies at the Powder/Single-crystal Boundary

L. W. Finger

Geophysical Laboratory and Center for High Pressure Research,

Carnegie Institution of Washington, Washington DC 20015-1305, U. S. A.

Abstract

Crystal structures have been refined for single crystals that have volumes on the order of 1 /xm^ and un- known structures have been solved for crystals with

volumes of the order of 10^ /im^. Study of microcrystals gives rise to a number of special problems, such

as maximization of signal to noise by reduction of

background, development of an incident beam that

is homogeneous on the scale of fim, and reduction

of data collection time by use of parallel recording

techniques. Solutions to these problems have been

attempted at a number of beam lines at various synchrotrons, using both monochromatic and polychromatic sources. On the latter type of beam, the flux

tends to be more homogeneous; however, the analysis

is more complicated. For monochromatic radiation,

the experiment is similar to that performed in the

laboratory, but requires considerable beam time for

the data collection. With current sources, the scat- tering power required for structural refinement has

been reduced by approximately six orders of magnitude relative to the laboratory. The next generation

of machines should be bright enough to reduce the

scattering power by another three orders of magnitude, if means are found to handle the crystals.

1. Introduction

It is now possible to obtain accurate intensities from

crystals that are the size of an individual grain

in a powdered sample. This method is the only

single-crystal technique available for materials whose

single crystals are smaller than the 50 - 100 fim

size needed for conventional diffractometry. Microcrystal diffraction studies that have used monochromatic synchrotron beams are typified by Eisenberger

et al. (1984), Bachmann et al. (1985), Rieck ei

al. (1988), and King et al. (1991a,b). Despite significant advances in deconvolution of a complex powder

pattern, and solution of the crystal structure with

standard structure solution techniques such as di- rect methods (McCusker, 1988), or by the Patterson

method (Attfield et al., 1988), there are still struc- tures for which single-crystal intensities are needed.

Small crystals are also needed to obtain accurate intensity data that are as free as possible from the ef- fects of extinction and absorption. Such data are needed if accurate electron densities are to be obtained.

Ability to study small volumes has also been important in the field of high-pressure investigations,

where the maximum pressure attainable is, in part,

determined by the area ratio between the external

load device and the sample chamber. Because the

thickness of the sample is also constrained, sample

chamber volumes on the order of 1000 //m^ are common. Although the bulk of high-pressure, X-ray

studies have been performed with powdered samples, a number of recent investigations have involved single crystals (Mao et al, 1988; Hemley et al, 1990;

LeToullec et al, 1992a,b).

As the crystal being studied in a single-crystal ex- periment approaches a size appropriate for an individual grain in a powdered sample, i.e. on the or- der of 1 /ym, a number of experimental requirements

become quite severe. For example, the process of

selecting and mounting a microcrystal is not trivial - a micromanipulator is essential. In addition, the

amount of extraneous material such as fiber and glue

in the beam must be carefully controlled; otherwise

the amorphous scattering will swamp the Bragg scat- tering from the crystal. To control background from

air scattering, the beam size must be greatly reduced.

If possible, beam paths should be filled with He or evacuated. The crystal must be centered extremely

carefully; otherwise it may precess out of the beam as the diffractometer axes are moved. Harder to control,

the so-called "sphere of confusion" , which is the volume of space swept out by a perfectly-centered, point

sample must also be very small. One is also con- fronted with the "normal" problems of synchrotron

183
184 Instrumentation for Studies at the Powder/Single-crystal Boundary

experimentation, i.e. an unsteady source intensity

and too little beam time available for the experiment.

This contribution reviews the progress made in

the study of microcrystals and presents some of the

details of apparatus under development at present sources and planned for the next generation of synchrotrons. Particular attention will be directed to- ward the effectiveness of the design in solving the

problems outlined above.

2. Effective crystal size

In any discussion of what constitutes a "small" crystal, the fact that materials differ in their ability

to scatter X-rays, primarily due to effects such as atomic number and density, must be taken into ac- count. For a single crystal, monochromatic experiment. Warren (1969, p. 46) shows that the total

diffracted energy for a reflection hkl is proportional

to {Fhki/Vu)^Vc^^, where Fhki is the structure factor for hkl, Vu is the unit-cell volume, Vc is the

volume of the crystal, and A is the wavelength of

the radiation. Lorentz-polarization factors and absorption are ignored. The majcimum for this power

would be obtained at a scattering angle of zero, where

all atoms would scatter in phase and thermal effects

would be negated. We can, therefore, use i^ooo ill" stead to yield the scattering power, S, defined as

5=(^)V.A3.

which is a dimensionless measure of how well a crystal scatters radiation. For experiments that use polychromatic radiation, the equation needs to be modified; however, it still serves as an effective metric for

the scattering expected for a given crystal. Note that

all factors, except Vc, are fixed by the crystallography of the material.

With a conventional source, S must be no smaller

than 10^^ - 10^^ for a successful diffraction experiment to be performed. In a study of solid hydrogen at 5.4 GPa pressure and room temperature by Hazen et

al. (1987) (5 = 7x10^^ , Table 1), the intensity of the

strongest diffraction maximum was only 68 counts

per second for an Mo, fine-focus. X-ray tube operated

at a power of 1.4 kW. Signal to noise was less than 7. Although the intensity data were sufficient to demonstrate that H2 has a hexagonal, not cubic, structure

under these conditions, it is clear that the experiment

would be impossible if the scattering power were re- duced by another order of magnitude. At 6 GPa,

the crystal either fragmented or became broadened

Table 1: Crystal Volume and Scattering Power.

Year Volume, S Ref

1987 9 X 10^ 7 X 10^5 (1)

1984 800 3 X 10^"^ (2)

1985 200 1.3 X lO^'' (3)

1988 2.2 7.1 X 10^2 (4)

1991 0.02 7

(5)

1991 0.03 1.8 X 10^° (6)

(1) Hazen et al. (1987) - H2 - Sealed Source.

(2) Eisenberger et al. (1984) - Na6Al6Si6024 - Monochromatic Synchrotron

(3) Bachmann et al. (1985) - CaF2

- Monochromatic Synchrotron

(4) Rieck et al. (1988) - CaF2

- Monochromatic Synchrotron

(5) Ohsumi et al. (1991) - Mo

- Polychromatic Synchrotron

(6) Skelton et al. (1991) - Bi - Polychromatic Synchrotron

by strain, and the diffraction could no longer be detected with the laboratory source. Note that the integrated intensity is proportional to S; however, the

detectibility of a given hkl is more dependent upon

the peak intensity, which is inversely proportional to

the peak width. The optimum peak to background is obtained if the beam divergence matches the mosaic

spread of the crystal (Rieck k Schulz, 1991); however, most real crystals have a much broader spread

than synchrotron beams.

With synchrotron sources, the increases in both

total flux and, more importantly, brightness, have

resulted in a decrease in the minimum size of crystals

and scattering power for crystals that can be studied.

Table 1 lists some of the notable experiments and the

date published.

3. Monochromatic synchrotron sources

As noted in Table 1 (references 2-5), there has

been a considerable decrease in crystal volume and

S for studies using monochromatic synchrotron radiation. In the first of these investigations, Eisenberger
L. W. Finger 185

01 02\

MH

Figure 1: A possible experimental station for microcrystal examinations: Sp: storage ring; MG: monochromator housing; M: curved monochromator; S: focusing mirror; MH: experimental hutch; Bl, B2, B3: pinholes

or diaphragms; V: beam shutter; Hel, He2: helium or vacuum paths; MP: beam monitor; He: helium or vacuum source; St: stage; LL: collimator or X-ray guide tube; Gl, G2: collimator positioning goniometers;

O: optical bench; D: diffractometer; P: beamstop; F: area detector; Z: telescope. Adapted from Rieck &; Schulz (1991).

et al. (1984) did not measure a complete set of intensities for structural analysis, but used two hkl's at two wavelengths each to estimate the minimum size

of crystals that could be studied. Their prediction

that "complete data sets could be measured from

crystals having volumes less that l^m^" has already

been realized. In a review of the literature for X-ray

studies of microcrystals using monochromatic synchrotron sources, Rieck & Schulz (1991) present a diagram of a diffractometer suitable for microcrystal

studies, reproduced here as Figure 1. Such an instrument would provide the high flux and brightness

required for examination of small crystals, particularly if mounted on a wiggler or undulator beam line.

In addition, the use of small pinholes and helium or vacuum beam paths will reduce background due to

general scatter; thereby increasing signal to noise.

The use of an area detector will also facilitate rapid

data collection, which is necessary given the diffi- culty of obtaining long periods of time on such beam

lines. In addition, this detector will also help in the

determination of crystal orientation. If the unit-cell

constants are unknown, this step can be rather time

consuming. Note that for the smallest crystals, it is not possible, in general, to perform alignment in an

off-line experiment.

Variations in the flux density of the beam on a fim

scale is a matter of concern. Although the beam incident upon the monochromator is usually very homogeneous, the scale of imperfections in the bent

crystal and in the mirror make it difficult to ensure

that the flux density incident upon the sample is not

changing as it moves within the sphere of confusion of

the diffractometer. King et al. (1991a,b) have used

slits several meters apart to define an X-ray beam

with a narrow angle of acceptance, which reduces

fluctuations arising from small changes in the electron orbit. The resulting beam is relatively uniform

over a 250 //m diameter. Variations in the total flux,

after beam monitor corrections, are on the order of

5% throughout the ring current decay and refill cycle (King, personal communication 1992). This variation is definitely not random, but has a minimum

just after the fill, followed by a steady increase as the beam decays. A maximum is reached in midcycle, followed by a slight decay near the end of the

fill. Such behavior is tentatively ascribed to thermal

variations in the mirror between the ring and the

monochromator. As shown by King et al. (1991a,b),

such a system may be used to measure accurate intensities for crystals with dimensions on the order of

10 fim.

The data of Rieck et al. (1988), as presented by

Rieck & Schulz (1991) are quite different. For a wide

variety of refinements, the temperature factors for

the Ca and F ions in the CaF2 structure are nearly

identical with the results of lattice dynamics calculations (Elcombe & Pryor, 1970). For the 2.2 fxm^ crys-
186 Instrumentation for Studies at the Powder/Single-crystal Boundary

SOiBO^in 80>80fim

1 2350-

16600 mm 18950

Loue comtfo linyocuum)

SOO^m*?18.2mm»

19330 19550

e f

Figure 2: The optical system employed in the Laue

difFractometer at Photon Factory. Letters a, b, c indicate the first and second slit, and a pinhole, re- spectively. A beam tube is indicated by d, and letters

e and / indicate the sample and image plate detector. The numbers below the horizontal line are the

distances from the source. Adapted from Ohsumi

(1992).

tal, however, the factors are suddenly much larger,

indicating some uncorrected systematic error, which

may be caused by crystal imperfections noted below.

It is also possible that microvariations in the flux

density of the beam are the cause of the systematic

error. King (personal communication, 1992) suggests

that the present generation of high-brightness synchrotrons may be more susceptible to this problem

than the first generation of machines, because their

electron bunch size is smaller making it easier to detect the effects of variations in the electron orbit.

4. Polychromatic sources

Because it is impossible to ensure that any beam

arising from a monochromator and/or focusing mirror is homogeneous on a very small scale, other investigators have chosen an alternative approach that

eliminates all active optical elements. This is accomplished by using the "white" radiation from either a bending magnet or a wiggler to illuminate the sample

directly. The only elements between the storage ring

and the sample are filters, windows, and pinholes; none of which should introduce appreciable beam inhomogeneity. Ohsumi et al. (1990) describe a Laue

diffractometer that uses an image plate as a detector (Fig. 2). This instrument has exceedingly low

background as the slit system has been designed to

eliminate the scattered radiation arising from the interaction of the polychromatic beam with the beam

defining aperatures. This difFractometer has been

used to study spheres of Mo, 0.8 ^m in diameter.

One such sphere was found to be twinned and the

Diamond

anvils

Figure 3: Schematic diagram of Laue two-circle, Xray difFractometer used for single- crystal diffraction

studies of crystals with small values of scattering

power. The motions on x and u) are used to position a crystal into diffracting condition. Approximately

30% of the allowable diffraction data is accessible.

For samples that are not contained within a diamond

anvil cell, a modified <^-cixis, which allows complete

access to reciprocal space is added. Figure adapted

from Mao et al. (1988)

volume fraction of the twin individuals were varied

by least-squares analysis. The volume of the lesser

part refined to a value of 0.02 fim^, too small to be

seen in an optical microscope. Despite this small volume, they were able to refine the temperature factor

of Mo to a value of 0.08(6) A^, a value compatible

with that obtained with powder diffraction on the

same material.

For a Laue difFractometer that does not have any

means of energy resolution, it can be difficult to index the pattern. In addition, only axial ratios, not

absolute axial lengths, may be determined. For this

reason, Ohsumi et al. (1992) have added a solid-state

detector. An additional benefit is that the X-ray flu- orescence effect may be used to measure the heavy

element composition of the sample.

A different type of single-crystal difFractometer

(Figure 3) for a polychromatic incident beam has

been developed at NSLS, initially on beamline X7a,

and more recently, on beamline X17C, which has a superconducting wiggler source. This instrument,

initially used to study the crystallized form of nor- mally gaseous materials such as hydrogen, deuterium, etc. (Mao et al., 1988; Hemley et al., 1990;

LeToullec et al., 1992a, 1992b; Vos et al., 1992), has

been extended to study the crystallography of a series of Taylor wires (Taylor, 1924), which consist of

small metallic filaments with submicrometer diameters contained within glass capillaries (Skelton et al., 1991). Such samples are particularly difficult because
L. W. Finger 187

the glass container has a diameter of approximately

10 /im, i.e. a volume that is 2 to 3 orders of magnitude larger than the crystal. The background level

from this amorphous material is very large, however,

this instrument has been able to detect diffraction

from a Bi wire only 670 A in diameter, and to

determine the lattice constants of a previously un- known form of Bi (Finger et ai, in preparation). A

severe difficulty of this instrument is that very large

amounts of time are required to locate the diffraction

maxima needed to determine the orientation and lattice of the crystal, particularly, when the scattering

power of the sample is very small. To overcome this

problem, an imaging plate is being added, which will

be used to locate the position of diffraction maxima,

and to calculate changes in angles uj and x necessary

to bring the diffraction into the detector. Additional

efforts are still needed to reduce background and improve signal to noise.

D. Hausermann (personal communication, 1992)

has begun work on a Laue diffractometer of a new

design for ESRF. This instrument uses a solid-state

detector with a coaxial design in which the active

crystal has a hollow center. This design permits the

use of an active region large enough for a scattering

angle of roughly 7° while still maintaining acceptible energy resolution. The collimator assembly consists of inner and outer tungsten carbide cones separacted by a few /im. For powdered samples, the entire

Debye-Scherrer ring is collected at once, greatly increasing the data collection rate and reducing some of

the effects of preferred orientation. With single crystals, the search process is reduced to a single scan on u>. If an additional thin slit is available to mask

all but a narrow region of the cone, single crystal

reflections can be located and indexed.

5. Crystal perfection

Although it might seem likely that small crystals are more likely to be free of many effects such as twinning

and low-angle grain boundaries, the limited experi- ence suggests that microcrystals are no better than

larger ones. For example, Ohsumi et ai (1990) found

that one of their two Mo spheres was twinned. Similarly, the 2.2 fjim^ crystal of CaF2 studied by Rieck

et al. (1988) showed the presence of extra crystallites

is some scans (Fig. 4).

6. Future sources

Although the developments in microcrystal diffraction have been spectacular on first-generation synI . I . . . . I . , . . I . . . . I . . J

23.6 23.8 24.0 24.2

O (deg)

Figure 4: Omega scan (rocking curve) of the 202 re- flection of a CaF2 crystal with volume of 2.2 [irn^.

Note the left-side shoulder indicating a second crystallite. Figure from Rieck k, Schulz (1991).

chrotrons, and significant results are now obtained on the second-generation machines, one can expect even more advances with the increased brightness ex- pected for the third-generation machines now being

built at Grenoble (ESRF - 6 GeV), Argonne National

Laboratory (APS - 7 GeV), and Japan (SpringS -

8 GeV). These machines are optimized for insertion

devices such as wigglers, which produce a polychromatic beam, and undulators, which have peaks in

the energy spectrum and are used with monochromators. Once the problems of increased heating of

optical components have been solved so that a uniform fiux is achieved at the sample position, it should

be possible to reduce the scattering power by another

three orders of magnitude. At this point, linear dimensions of the crystals will be approaching the limit

at which Scherrer broadening (Warren, 1969, p. 253)

will make it no longer possible to detect the diffracted

intensity. At this size, it will be impossible to resolve

the crystals with an optical microscope; therefore, it

will be necessary to mount the crystals in a scanning

electron microscope.

7. Conclusions

As shown by Bachmann et al. (1985) and King et

al. (1991a, b), it is possible to measure accurate intensity information for crystals that have scattering

powers as large as lO^'* on monochromatic beamlines

at both first- and second-generation synchrotrons.

As the scattering power of the crystal is reduced into
188 Instrumentation for Studies at the Powder/Single-crystal Boundary

the 10^^ range, diffracted intensity can be observed;

however crystal structure refinements on such small

crystals have not yielded satisfactory results to date.

One possibility is that the incident beam may not be

sufficiently homogeneous at the 1 /zm scale.

Although correction of measured intensities for instrumental effects are not trivial for Laue geometry,

it has been shown that such calculations are suffi- ciently accurate to yield meaningful structural parameters, even for crystals with scattering powers in

the 10^^ range. In addition, the simple X-ray optics should lead to a more homogeneous beam, even

at the submicron scale. Although either a solidstate detector, or an image plate would be sufficient

to measure complete intensities at a polychromatic

beamline, there are considerable advantages in speed

and flexibility if both are available.

References

Attfield, J. P., Cheetham, a. K., Cox, D. E. &

Sleight, A. W. (1988). J. Applied Crystallography,

21, 452-457.

Bachmann, R., Kohler, H., Schulz, H. & Weber,

H-P. (1985). Acta Crystallograpbica, A41, 35-40.

ElSENBERGER, P., NeWSAM, J. M., LEONOWICZ, M. E. & Vaughan, D. E. W. (1984). Nature, 309, 45-47.

Hazen, R. M., Mao, H. K., Finger, L. W. & Hemley,

R. J. (1987). Physical Review B, 36, 3944-3947.

Hemley, R. J., Mao, H. K., Fbstger, L. W., Jephcoat, a. p., Hazen, R. M. & Zha, C. S. (1990).

Physical Review B, 6458-6470.

King, H. E. Jr., Mlhsidi, L. A., Strohmaier, K. G.

k. Haushalter, R. C. (1991a). J. Solid State Chem.,

92, 1-7. Kestg, H. E. Jr., Mundi, L. A., Strohmaier, K. G.

& Haushalter, R. C. (1991b). J. Solid State Chem.,

92, 154-158.

LeToullec. R., Loubeyre, P., Pinceaux, J. P., Mao,

H. K. & Hu, J. (1992a). High Pressure Research, 6,

379-388.

LeToullec. R., Loubeyre, P., Pinceaux, J. P., Mao,

H. K. & Hu, J. (1992b). High Pressure Research, 8,

691-696.

Mao, H. K., Jephcoat, A. P., Hemley, R. J., Finger,

L. W., Zha, C. S., Hazen, R. M. & Cox, D. E.

(1988). Science, 239, 1131-1134.

McCusker, L. B. (1988). J. Applied Crystallography,

21, 305-310.

Ohsumi, K., Hagiya, K. & Ohmasa, M. (1990). Acta

Crystallographica, A46, C-26.

RiECK, W., EuLER, H., Schulz, H. & Schildcamp, W.

(1988). Acta Crystallographica, AAA, 1099-1101.

RiECK, W. & Schulz, H. (1991). Handbook on Synchrotron Radiation, Vol. 3, G. Brown, D.E. Moncton,

Eds., 267-290.

Skelton, E. F., Ayers, J. D., Qadri, S. B., Moulton,

N. E., Cooper, K. P., Finger, L. W., Mao, H. K.

& Hu, Z. (1991). Science, 253, 1123-1125.

Taylor, G. F. (1924). Physical Review, 23, 655-660.

Vos, W., Finger, L. W., Hemley, R. J., Hu, J. Z.,

Mao, H. K. & Schouten, J. A. (1992). Nature, in press. Warren, B. E. (1969). X-ray Diffraction. Reading,

Mass: Addison-Wesley.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction JI, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Time-resolved X-ray Diffraction Studies of Crystallization in Metallic Glasses

M. Sutton, S. Brauer, Y. S. Yang, H. E. Fischer and J. O. Strom-Olsen

Centre for the Physics of Materials and the Department of Physics, McGill University,

3600 University Street, Montreal, Quebec, Canada H3A 2T8

G. B. Stephenson

IBM Research Division, Thomas J. Watson Center,

Yorktown Heights, NY, U. S. A. 10598

Abstract

The IBM/MIT beamline X-20C at the National

Synchrotron Light Source has been optimized for

time-resolved scattering measurements. Using a multilayer monochromator and a fast linear positionsensitive detector system, scattering patterns from

a transforming material can be acquired with 3 ms

time-resolution. In conjunction with a fast pyrometric temperature controller which has been developed to change and regulate the sample temperature

with microsecond response, we have been studying

the kinetics of crystallization of metallic glasses on

time-scales varying from minutes to milliseconds. In a typical measurement, two synchronized positionsensitive detectors provide complementary in situ information about the transformation. The first is used

for large-angle scattering and the second detector measures small-angle scattering. At relatively slow

transformation rates, quantitative measurements of

crystal volume fractions as small as lO"'* are possible. In many systems, as the transformation rate

is increased, the crystallization mechanism changes

such that new metastable crystal phases are formed

en route to the equilibrium structure.

Introduction

X-ray diffraction has played a key role in our understanding of the structure of materials. However, one would often like to have structural information for

phenomena in systems whose structures evolve with

time. To acquire this information, the time required

to measure a diffraction pattern must be of suitable

duration.

One way to study these systems is to compare standard X-ray measurements made before and after any

changes. Also, it is often possible to arrest the development in mid-process and thus obtain information on intermediate structures. Classic examples of such

ex situ measurements are diffraction measurements

made after heat treatments in which the samples

have been annealed at high temperatures for varying amounts of time. For fast processes, such studies are often not possible or are fraught with systematic

errors in which the structure has evolved in unexpected ways during the processing used to arrest the

time evolution. In spite of these limitations this technique has been and is still very useful. Provided that

the processing is fast enough, this method allows for

measurements on time scales shorter than the time

to measure a diffraction pattern.

Another possible technique relies on the ability to cycle a system repeatedly through its time evolution

sequence. This allows measurements to average over many cycles and build up a diffraction pattern slowly over time. This powerful technique is not always

applicable, since not all processing can be cycled or cycled reproducibly enough. When applicable, this

technique will also give time resolutions which are not limited by the time to accumulate a complete

diffraction pattern.

The method we will emphasize in this article is to make in situ measurements by recording a se- quence of diffraction patterns in real time. This often

sets the most severe experimental requirements, but

it has the advantage that the analysis of the data

is more straight forward and systematic errors are greatly reduced. The time resolution obtainable in

this way directly depends on the time to accumulate

each diffraction pattern. With conventional X-ray

sources, as found in most laboratories, the time to obtain a diffraction pattern ranges from minutes to days. With the advent of the new synchrotron X-ray

sources, this time can be reduced to milliseconds.

189
190 Time-resolved X-ray Diffraction Studies of Crystallization in Metallic Glasses

Metallic glasses

In this article, we will use crystallization studies

of metallic glasses as examples in which the time

evolution of structure is important. Metallic glasses are formed by cooling molten metal alloys rapidly

enough to inhibit the formation of any crystalline

order. The absence of lattice periodicity leads to

several characteristics which make glasses interesting both from the point of view of our fundamental understanding of materials and for materials applications (Luborsky, 1983). In the simplist picture, metallic glasses are in a thermally arrested

metastable state. When heated, a rich assortment

of non-equilibrium phenomena occur as they transform into their equilibrium phases by nucleation and

growth.

A single transformation of an amorphous material into a crystalline phase can be classified into

three types depending upon the phases involved in

the transformation. The least complicated transformation is called a polymorphic crystallization. This

involves the transformation of a metallic glass into a crystal phase with the same composition and does

not require the diffusion of material over large distances. A more complicated type, called primary

crystallization, occurs when the composition of the

crystal phase is different from the starting composition. As primary crystallization proceeds, the

composition of the remaining amorphous material

changes with time and it eventually crystallizes into

another phase or phases with different compositions.

This type of crystallization is thus inherently a multistep process. The third type is called eutectic crystallization and involves the simultaneous formation

of two crystal phases with the correct compositions

so as to leave the remaining amorphous material at

its original composition. Such a transformation involves diffusion on length scales comparable to the

separation of the two crystal phases and often results

in a lamellar microstructure.

For a particular alloy and composition, crystallization may not proceed directly to the equilibrium

structure on the phase diagram and may involve

many competing transformations, metastable phases

and the formation of phases which must relax to

reach equilibrium. In situ time-resolved diffraction

provides an ideal tool for investigating these irre- versible processes. It allows one to monitor the appearance and disappearance of phases to give a qualitative understanding of the transformation processes.

Furthermore, it is possible to measure the amounts

and morphologies of the various phases. This allows

quantitative comparison to theoretical models of the

kinetics. In this paper we will give several examples to demonstrate the different types of information

that can be obtained.

ExperimentEil setup

The IBM/MIT beamline X-20C at the National Synchrotron Light Source (NSLS) has been optimized

for time-resolved scattering measurements. The details of the experimental configuration is discussed

elsewhere(Stephenson, 1988, Stephenson et ai, 1989

and Brauer et ai, 1992). This beam line is equipped

with a Pt coated silicon 1:1 focusing mirror, which

collects up to 4 mrad of bending-magnet synchrotron

radiation and focuses it into a 1 mm x 1 mm spot

22 m from the source. The X-ray wavelength is se- lected by a double crystal monochromator, typically

in the range 1.5 to 2.0 A. The beam line vacuum is separated from ring vacuum by a Be window at one end and the X-rays exit the beam pipe from another

Be window near the sample position. Three aspects

of this setup are emphasized here.

The first aspect is the use of a 1% bandpass

monochromator based on synthetic multilayers of

tungsten and silicon. This greatly increases the incident X-ray intensity although at some loss of diffraction resolution. The resulting increase in intensity

for diffuse scattering is much greater than the cor- responding increase for sharp Bragg peaks, and so

this loss of diffraction resolution actually dramatically improves measurements of diffuse scattering.

The second aspect is the use of fast linear detector

arrays to collect many diffraction angles in parallel

at count rates approaching a million counts per sec- ond per angle. In our recent measurements we are using two such detectors to simultaneously collect

small-angle and large-angle X-ray scattering. The

two measurements give complementary information on crystal structure, the phases present and on the

microstructure. With this setup, powder diffraction

profiles covering ten to twenty degrees of 29 can be

collected with a time resolution of a few milliseconds.

The third aspect is the development of in sUu techniques to rapidly measure and control the sample en- vironment on a time scale fast enough to take advantage of the millisecond time resolution(Stephenson et ai, 1989). Since one is comparing diffraction profiles taken under almost identical conditions, one of

the primary advantages of in situ measurements is that quite small structural changes can be detected.

Furthermore, one can observe and often understand

the nature of the temporal evolution in the structure even though a complete structural determination is not possible.

Our metallic glasses are made by melt spinning.

This produces a ribbon of material which is 10 to 30

fim thick and 1 to 2 mm wide. For our experiments a
M. Sutton, S. Brauer, Y. S. Yang, H. E. Fischer, J. O. Strom-Olsen and G. B. Stephenson 191

5

-

o

oo

a

o

o

Q> cnh

0)e.B

a

o

o

4

-

3

-

CM o O CM CM o evi

I

time (s)

I 2.6 2.8 3.0 3.2

q (A ^)

3.4

Figure 1: Time-resolved diffraction of C02B at Ta =675 K. Each successive pattern is offset by 5000

counts/second for clarity. The time per diffraction

pattern is 6.6 s. The crystal structure of C02B is M/mcm{D\l) with a=5.015 A and c=4.22 A. The

indexing of the peaks is shown in the figure. The

solid lines are fits to the data aS discussed in the

text.

30 mm length is suspended with one end held under

slight tension to compensate for thermal expansion.

The ribbons are heated directly by passing current

through them and their temperature is controlled us- ing an infrared pyrometer. This thermal geometry

with its small thermal mass allows us to increase the

temperature of our samples by 1000 *'C in under a millisecond (Brauer ei. al, 1990) and then hold the

temperature constant at the anneal temperature TaBy surrounding the sample with He, cooling rates of over 1000 *C/s can be obtained. A typical measurement involves acquiring 100 to 500 consecutive

diffraction patterns. The temperature step is timed

to occur after several low temperature patterns are taken and to end so that the last few patterns can be

taken at room temperature. This enables us to monitor what happens during the heating and cooling of our samples. We define t = 0 as the time at which Ta

is reached. In our current setup, up to four voltage

signals are digitized at a rate 10 times faster than the

time per diffraction pattern. Two of these are used

to measure incident beam intensity and the temperature. Measurements for large-angle scattering can be

made in reflection or in transmission. Using reflec- tion tends to give counting rates 2 to 5 times higher

than transmission.

C02B

Figure 1 shows a sequence of diffraction patterns

for different times during the polymorphic crystallization of C02B for Ta =675 K. The measurements

were taken in a transmission geometry. The time

t = 0 s profile shows the characteristic scattering

of a metallic glass. As the material crystallizes this

scattering reduces in intensity and is replaced by the

characteristic diffraction pattern of a polycrystalline

material shown by the t = 1604 s profile. Since the

intensity of X-rays from a synchrotron changes with

time, all intensities have all been normalized to a beam current of 100mA. The X-ray ring at NSLS

typically operates between 150 and 250 mA. Figure 1 shows that we routinely obtain count rates of 40,000

to 60,000 cts/sec/angle for transmission. For reflec- tion this becomes 100,000 to 250,000 cts/sec/angle.

The minimum time to read out a scan from the linear

detector is under 3 ms. From these numbers one can

see that diffraction patterns with maxima of several

hundred counts can be taken with a time resolution

of 3 ms. To determine the crystalline volume fraction for

this data, each diffraction profile is fit by least

squares to a simple linear combination of the initial

amorphous scattering Ia{q), and the final crystalline

scattering Ic{q)- Explicitly, the scattering intensity
192 Time-resolved X-ray Diffraction Studies of Crystallization in Metallic Glasses

time (s) (10^)

Figure 2: Crystallized volume fraction of C02B ver- sus time. The solid line is a fit to x{t) = 1 — e""' , with a = 1.5 X 10~^^ s~'^. Inset is same data with

linear scales.

for a given wave vector q and time t, is

Iiq,t) = [l-x{t)]Ia{q) + xii)hiq), (1)

where x{t) is a measure of the volume fraction.

Providing that the scattering profile of each phase

is independent of time, the most sensitive way to

extract x is to use the measured initial and final

scattering patterns, i.e. Ia{q) = Iiq,tinitial) and

Ic{q) = I{q,tjinai)- This procedure implicitly tests whether the increase of crystalline scattering is compensated for by a corresponding loss of amorphous

scattering. Furthermore, any systematic deviations

of the model from the data directly indicate when the

model breaks down. The results of such fits are plotted in figure 2. Note that volume fractions smaller

than 1 part per 10^ are easily seen for data which

have a large number of counts per diffraction angle.

A fit to a simple model of nucleation and growth

(Luborsky, 1983) is also plotted in the figure to show

how kinetic parameters can be obtained from the

data.

As with all powder X-ray diffraction measurements

care must taken to avoid insufficient powder averaging. In our scattering geometry, large crystallites re- sult in peaks which are sharper than resolution the

intensities of which are unusually sensitive to sample

and beam motion. The use of melt spun metallic ribbons seems to be particularly advantageous in giving

small grain sizes and avoiding these problems.

FeZr2 and CoZr2

Figure 3 shows the polymorphic crystallization of

FeZr2. Again, the initial amorphous pattern disappears with time to be replaced by a polycrystalline

powder pattern. This time, however, the crystal scat- tering continues to evolve in time; the peak heights

grow and the peak widths decrease as the sample

continues to anneal. The crystal structure of this

U ! I ! \ I !l 2.4 2.5 2.6 2.7 2.8 2.9 3.0

Figure 3: Time-resolved diffraction of FeZr2 at ra=690 K. The time per diffraction pattern is 0.2

s up until 40 s and 2.0 s thereafter. Solid lines are

fits to a model discussed in the text and the crystal

Bragg peaks are labeled.

phase of FeZr2 is known to be cubic (Fd3m(0^))

with a lattice constant of 12.2 A. Because the crystalline scattering evolves with time, the simple linear

decomposition of line shapes used in equation 1 must

be modified. The solid lines in the figure are based on

fitting the crystal profiles as the sum of three Gaussian peaks. The positions and the integrated intensities of these peaks are obtained by fitting the final

scattering pattern. Intermediate patterns are then

fit to equation 1 with the Ic{q) replaced by this functional form, /c(?, where w{i) is a common peak

width. A least squares fitting procedure is used to

determine x{t) and w{i) and none of the other parameters are varied. In general, one could imagine

the need for models of the scattering which cover the

spectrum from simple ones, as those used here, to ones as complicated as those required for a complete

Rietveld analysis.

From the fits, it is found that for this run, x{t) —
M. Sutton, S. Brauer, Y. S. Yang, H. E. Fischer, J. O. Strom-Olsen and G. B. Stephenson 193

0.10

0.00

10'

time(s)

lO"-

Figure 4: Peak width (half width half maximum) as a function of time for FeZr2. Gap in data results

from the transfer of the data of the first part of the scan and starting the second part.

1.0 is reached within the first 20 seconds, and the

peak width continues to evolve for the duration of

the measurement. The measured width is plotted

in figure 4. This phase of FeZr2 is known to have

Figure 5: A three dimensional representation of the

scattering as a function of time for CoZr2 annealed

at Ta=720 K. The time per diffraction pattern is 0.1

s. At this temperature crystallization occurs in well

under 0.1 s and the first pattern is due to the cubic

structure.

small crystallite sizes when prepared from melt spun

ribbons (Altounian, Volkert and Strom-Olsen 1985).

Assuming the width is only due to finite sized crystallites and using the Scherrer equation, particles are «50A at i = Is and grow to over 125A at 500s.

The crystallization of CoZr2 is shown in figure

5. The first hot diffraction pattern indicates that

the amorphous phase has already transformed. This

crystal phase has the same cubic structure as deFigure 6: The time dependence of the volume fractions for the data in figure 5. The dotted, dashed

and solid lines represent the volume fractions for the

cubic, the unknown and the tetragonal phases re- spectively.

scribed above for FeZr2. From figure 5 it is immediately obvious that two other phases are involved

in the crystallization. One can see that the cubic

phase lasts for less than 10 seconds. A second, previously undiscovered, metastable phase appears for w

40 seconds. This phase does not appear in crystallization at low temperatures. The final phase which

appears is the known equilibrium tetragonal phase

of CoZr2, I4/mcm{D\f^). Extending equation 1 to include more phases is straight forward and produces volume fractions versus time for each of the

phases, as shown in figure 6. With our ability to rapidly quench samples, we have been able to obtain

specimens with large volume fractions of this new

unknown phase. It is hoped that with more complete powder diffraction profiles, TEM and selected area diffraction more information can be obtained to

uniquely determine this crystal structure.

Eutectic CosgZrii

Figure 7 shows the crystallization of CoggZrn.

These data look very similar to those in figures

1, 3 and 5. In this crystallization both /?— Co

{FmZm{Ol)) and CogZr {F4Zm{T^)) appear. The

occurrence of the (200) of /?-Co and the (220) of

CosZr in isolated regions of the profile enable us to

monitor the appearance of each of the phases independently. The extra structure in the final scattering

pattern is due to the formation of a small amount of

a-Co.

Figure 8 shows the time evolution of the smallangle scattering for the same composition ribbon at a slightly lower temperature but for a much longer

annealing time. A well defined small-angle peak near
194 Time-resolved X-ray Diffraction Studies of Crystallization in Metallic Glasses

1.4

Figure 7: Crystallization of CoggZrn at T « 8S0K,

large- angle scattering. The peak at 3.05 is the

superposition of the (311) of CosZr and the (111) of

P-Co.

q — .07 is clearly seen. This peak corresponds to

the eutectic formation of Co and CosZr with an interlamellar spacing of just under 100 A. From the high

angle data we can measure the order of appearance of

the phases, and the relative volume fractions. From

the small-angle data we can study the evolution of

the eutectic structures. By making these measure- ments at several compositions around the eutectic

composition and at several temperatures, we will not

only be able to test theories for the time evolution

of eutectic crystallization, but we will also be able to measure the cross-over from eutectic crystallization

to primary crystallization. The ability to measure

both small and large-angle scattering is important

for these studies.

Conclusions

Typically, further measurements are needed to fully

understand the structures seen in our in situ work.

We often perform ex situ measurements using Xray diffraction and electron microscopy to help identify phases. From our time-resolved data we can determine the appropriate times for quenching our

samples. Furthermore, in situ quenching gives us

confidence that the quenched-in structure has not

< 1.0

o

o 0.8

« 0.6 o< 0.4 m

u

0.2 K

"4 04 0.00 0.04 . 0.08

q (A~^)

0.12

Figure 8: Small-angle scattering from CoggZrn crystallizing at r « 820K.

changed so that further ex situ measurements do indeed represent the transient structures of the transformation under study.

In this paper we have tried to emphasize the wide

range of structural information that can be measured

by time-resolved X-ray diffraction. Metallic glasses

provide a particularly rich set of structural transformations and clearly demonstrate both the power of

time-resolved diffraction in unraveling these transformations and the necessity for using it. We have

also emphasized in situ measurements since these al- low one to use the continuity in the time evolution

of the scattering to help ensure that some aspect of

the transformations is not missed.

References

Altounian, Z., Volkert, C. A. & Str6m-Ol.sen,

J. O. (1985). J. Appl. Phys. 17, 1777-1782.

Brauer, S., Ryan, D. H., Strom-Olsen, J. O., Sutton, M. & Stephenson, G. B. (1990). Rev. Set. In- strum. 61, 2214-2219.

Brauer, S., Strom-Olsen, J. O., Sutton, M., Yang,

Y. S., Zaluska, a., Stephenson, G. B. & Koster,

U. (1992). Phys. Rev B, 45, 7704-7715.

LuBORSKY, F. E. (1983). Amorphous Metallic Alloys,

London: Butterworths & Co. Ltd.

Stephenson, G. B. (1988). Nucl. Inst, and Meth.

A266, 447-451.

Stephenson, G. B., Ludwig Jr., K.F., JordanSweet, J. L., Brauer, S., Mainville, J., Yang,

Y. S. & Sutton, M. (1989). Rev. Set. Inst., 60, 1537-

1540.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction II, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

New Developments in High-Pressure Neutron Powder Diffraction

R. J. Nelmes*and J. S. Loveday

Department of Physics, The University of Edinburgh, Edinburgh, EII9 3JZ, U.K.

AND J. M. Besson

Physique des Milieux Condenses, Universite P. et M. Curie, Jussieu, Paris cedex 05, France

For high-pressure neutron powder diffraction studies, time-of-flight methods have considerable advantages. The fixed scattering geometry greatly simplifies the construction of the cell in reducing the

the size of the windows required for incident and

diffracted beams. Diffraction at a fixed scattering

angle also allows the construction of simple coUimation to screen out background scattering from the

material around the sample.

Amongst the earliest work to exploit this technique

is that of Brugger and co-workers (Brugger, Bennion

& Hall, 1965), who developed an alumina-insert cell capable of reaching 3 GPa and an opposed-anvil cell that achieved 7 GPa (Brugger, Bennion, Worlton &; Myers, 1969). A number of successful studies were

performed, including an investigation of Bi II (Brugger, Bennion &: Worlton, 1967).

This project, and particularly the work on the

opposed-anvil cell was curtailed by the closure of the

Idaho Falls reactor, but a programme based around

the alumina-insert cell was continued at Argonne

National Laboratory. Many notable high-pressure

studies have been carried out there including work

on ReOa and deuterated ice (Jorgensen k, Worlton, 1984; J0rgensen, Jorgensen, Batlogg, Remeika

& Axe, 1986). Over the past few years the Argonne group has also been responsible for much of

the successful high-pressure diffraction work on highTc superconductors (Pei et ai, 1989; Jorgensen et ai, 1990; Yamada et ai, 1990). This recent work,

however, has generally made use of a large sample

volume in a gas cell (Yamada et ai, 1990), which

gives higher precision on the initial (small) rates of

change of bondlengths with pressure.

Work using angle-dispersive diffraction on reac- tor sources has all been largely confined to pressures

10 It 1.2 1.3 1.4 1.5 1.6 1.7 I.e 1.9 2.0 d-spacing (A)

c= 1500

« 1000- c

dumb-bell cell

—r1.3 —I

1.9 d-spacIng (A)

*To whom correspondence should be addressed.

Figure 1. Powder patterns collected from samples of

La2-2;Srj;Cu04 at ILL, Grenoble. The upper pattern was collected in an aluminium gas-cell with a sample

volume of 3000 mm^ on the DIA instrument. The

lower pattern was collected using a steel "dumb-bell"

cell on DIA with a sample volume of approximately

500 mm^. The asterisks mark non-sample peaks.

below 1 GPa, where larger sample volumes allow

the background scattering from the cell materials to be tolerated. The problems of background scatter- ing are illustrated by the patterns shown in figure 1, which are dominated by this scattering even with the

large sample volume available with the gas cell.

195
196 New Developments in High-Pressure Neutron Powder Diffraction

One of the successful studies using a reactor source

at pressures above 1 GPa is the series of experiments

described in Kuhs, Finney, Vettier &; Bliss (1984) on deuterated ice. Long count times were employed

to obtain high quality data, and values of the 0-D

bondlength were obtained with a precision ofO.OOTA.

It was also possible to observe indications of multisite disorder of the oxygen site in ice VII.

Thus we see that until recently, with the exception of the Brugger opposed-anvil cell, which was not

pursued, neutron diffraction has been limited to the 3

GPa achievable with clamp cells and often much less. However there are now two independent attempts in

progress to extend this pressure range significantly.

One development is based around a reactor source

in Russia and employs cells giving a small sample

volume of only 1 mm^ or less (Glazkov, Naumov,

Somenkov h Shirstein,1988; Goncharenko, Glazkov,

Irodova & Somenkov, 1991). With such small samples, the pressure cell has to be almost completely

surrounded by detectors in the (horizontal) scat- tering plane. Each individual detector is carefully

collimated to view just the region of the sample

and is shielded to minimise contaminant and background scattering. Successful structure refinements

of AID3 up to 7 GPa have recently been reported

(Goncharenko, Glazkov, Irodova & Somenkov, 1991).

However the very small sample volume used in this

technique means that it is not expected to yield

full structure refinements other than for the most

strongly scattering materials.

The other new development is our own, using timeof-flight techniques on the UK pulsed neutron source,

ISIS, at the Rutherford Appleton Laboratory. This

project is based around a novel opposed-anvil cell specifically designed to achieve pressures of 10 GPa

and above with a sample volume of the order of 100

mm^.

The cell is shown in figure 2 and consists of a sample mounted between two tungsten carbide or sintered-diamond anvils and surrounded by a gasket. Pressure is applied by a compact 200 tonne

hydraulic-ram (HR), which allows in situ variation

of the pressure. The complex anvil profile is important to the operation of the cell. The quasi-spherical

sample volume (S) has the role of increasing the sample volume (compared with flat anvils), and its shape

also ensures a more homogeneous pressure distribution. The key feature is the outer toroidal groove

(T), which inhibits side extrusion of the gasket and

hence allows higher pressures to be reached. Details

of the cell and ram design may be found in papers

200 mm

Figure 2: A schematic cross section through the

Paris-Edinburgh cell showing the hydraulic ram

(HR). The enlargement shows the toroidal gasket

groove (T) and the sample space (S).

by Besson ei al. (1992a, b). The performance of

the equipment has been tested and proved in making an accurate study of the equation of state of LiD

(Besson et al, 1992b).

The cell is designed for use on the POLARIS powder diffractometer at ISIS. It is mounted with its axis horizontal so that the the incident beam enters through the hole in the front of the cell. To en- sure that the beam illuminates the minimum amount

of non-sample material, the diameter of the beam

is reduced to slightly less than that of the sample

by a collimator fixed to the front of the cell. The

diffraction pattern is collected by detectors placed

at 2^=90° , and background scattering from the illu- minated parts of the anvils is screened out by coating the inner (opposing) faces of the anvils with a neutron absorbing material (either Gd203 or Cd).

Further shielding is placed around the cell and the

scattered-beam flight paths to reduce the background

due to other sources. After careful optimisation

of the shielding it is now possible to achieve signahbackground ratios of up to 100:1 with no detectable diffraction peaks due to the anvil materials,

as shown in figure 3. Once good quality data are obtained, any systematic effects on the relative intensities must be corrected. In the case of the Paris-Edinburgh cell the

principal systematic error is that due to attenuation

by the cell materials. Finite-element calculations and
R. J. Nelmes, J. S. Loveday and J. M. Besson 197

Iron test sample

T T

1.2 14 le d-spacing (A)

Figure 3. A time-of-flight spectrum collected from

a sample of iron in the Paris- Edinburgh cell. The

marks show the expected positions of the diffraction

peaks due to the anvils.

70- _

^ 60- ^ 50- g

'cn

.52 40 H

E

w

30-

0)

cn 20- o

>

10-

Tungsten carbids dosed anvils Tungsten caiMe cajculation Diamond dosed anvNs Diamond caleuJalioa

I i 1 I I I I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

neutron wavelength (A)

Figure 4. The inverse transmission factor (defined as

lo /I) as a function of neutron wavelength for tungsten carbide anvils (circles,solid line), and sintereddiamond anvils (squares, dotted line) .The symbols

show measurements made using an iron pellet sample, with the anvils fully closed together, while the

lines are the results of the corresponding finiteelement calculations.

measurements show that this can vary by as much

as a factor of 20 (Fig. 4) over the wavelength range

of interest (0.7-3 A). If left uncorrected, such se- vere attenuation significantly biases the refined val- ues of the positional parameters. The form of at- tenuation curve is a function of the anvil separation,

and hence pressure, so attenuation may introduce

spurious pressure dependences to the refined values

of structural parameters. Accurate corrections using

the finite-element calculations are thus essential.

The experience gained on the neutron scattering

aspects of the cell has been used in a high-pressure

study of the structure of deuterated ice VIII. This was chosen as it is a problem of suitable difficulty

and interest that is appropriate to study by neutron

diffraction. In addition, previous neutron diffraction

studies in the 2-3 GPa range (Kuhs, Finney, Vettier

k Bliss, 1984; Jorgensen k Worlton, 1984) provide

a check on the accuracy of data in the low-pressure

region. Ice VIII is stable only below 0 °C, and so the sample and cell were cooled by a stream of cold

nitrogen gas.

a

c

Ice VIII 2.7 GPa

iiMiiiiiiig nil mil HI ii II I

•.•-•.v^|.',U.4,-'.^-.>>.jS!|i'.K'li' —I 1 1 1 1 1 1 1 1 1

0.4 0.6 oe to 1.2 1.4 1.8 t.S 2.0 22 2.4 2.6 d-spacIng (A)

Figure 5: The measured spectrum of D2O ice VIII

at 2.7 GPa and -5 °C (dots) and the results of a Rietveld profile refinement (solid line).

Ice VII1 11.7 GPa

iiiHiiiiiiii II n 11 n II II II It I II M

1 1 1 1 1 1 1 1 1 1 0.4 06 O.B 10 1.2 1.4 1.6 1.B 2.0 2.2 2.4 2.6 d-spacIng (A)

Figure 6: The measured spectrum of D2O ice VIII

at 11.7 GPa and -5 °C (dots) and the results of a Rietveld profile refinement (solid line).

Figures 5 and 6 show the measured and calculated spectra obtained from a profile refinement of

corrected data collected at pressures of 2.7 and 11.7
198 New Developments in High-Pressure Neutron Powder Diffraction

GPa. To illustrate the precision achieved, the val- ues obtained for the 0-D bondlength in preliminary

profile refinements are shown in figure 7, along with

tector bank, currently being constructed, will make

such studies feasible.

References

» 0.94- n

om

Q

6

o'^-

f

1 i

5 5'

,.J,J.

• Paris/Edinburgh cell

Jotgensen and Worlton t964

X Kuhs. Finney, VaWer and BHss 1 964

—

r10 tt

pressure (GPa)

Figure 7: The pressure dependence of the intramolecular 0-D bondlength in ice VIII. The val- ues shown as circles were measured in the ParisEdinburgh cell while those shown as a cross and an

open-square were taken from previous work (Kuhs,

Finney, Vettier & Bliss, 1984; Jorgensen h Worlton,

1984). The dotted line is a fit to our values.

the results of previous studies (Jorgensen &; Worlton, 1984; Kuhs, Finney, Vettier k Bliss, 1984). The

pressures were derived from the measured unit-cell

volume and the equation-of-state of ice VII (Hemley

ei ah, 1987), which differs very little in density from

ice VIII. The level of random variation of the val- ues suggests that the true e.s.d.'s are approximately

0.005 A, as shown. The overall internal consistency

of the values, which were obtained from three separate loadings, along with the good agreement with

previous measurements, suggests that the contributions from non-random errors are rather small. The

rate of change of 0-D with pressure appears to be

around 0.001 A/GPa, which compares with a predicted variation of 0.0018 A/GPa (Klug & Whalley,

1987). The very small size of these changes illus- trates the need to obtain precise data over a wide

range of pressures in order to draw meaningful conclusions about such structural variations.

The results achieved to date give considerable cause for optimism concerning the feasibility of performing structural studies routinely to 20 GPa. Further work is needed to reduce the background to the

levels necessary for work with smaller sample volumes (to attain higher pressures) or to study weaker

scatterers. The factor of 10 improvement in count

rate that will become available with a new, larger deBesson, J. M., Nelmes, R. J., Hamel, G., Loveday,

J. S., Weill, G. & Hull, S. (1992a). Physica B, 180,

in press. Besson, J. M., Hamel, G., Weill, G. , Nelmes, R.

J., Loveday, J. S. & Hull, S. (1992b). Phys. Rev. B,

45, 2613.

Brugger, R. M., Bennion, R. B. Hall, H. T.

(1965). Bull. Am. Phys. Soc. Series II, 10, 172.

Brugger, R. M., Bennion, R. B. & Worlton, T. G.

(1967). Phys. Lett. A, 24, 714.

Brugger, R. M., Bennion, R. B., Worlton, T. G.

& Myers, W. R. (1969). Proceedings of the Symposium on Crystal Structures at High Pressure 141. Seattle: American CrystaUographic Assoc.

Glazkov, V. P., Naumov, I. V., Somenkov, V. A. &: Shil'stein, S. Sh. (1988). Nucl.Inst and Meth., A264,

367.

GoNCHARENKO, I. N., Glazkov, V. P, Irodova, A. V.

& Somenkov, V. A. (1991). Physica B, A264, 117.

Hemley, R. J., Jephcoat, A. P., Mao, H. K., Zhu,

C. S., Finger, L. W. & Cox, D. E. (1987). Nature,

330, 737.

J0RGENSEN, J. E., JORGENSEN, J. D., BaTLOGG, B.,

Remeika, J. P. & Axe, J. D. (1983). Phys. Rev. B,

33, 4793.

Jorgenson, J. D. &: Worlton, T. G. (1984). J. Chem.

Phys. 81, 3211.

Jorgensen, J. D., Pei, S., Lightfoot, P., Hinks, D.

G., Veal, B. W., Dabrowski, B., Paulikas, A. P.,

Kleb, K. & Brown, I. D. (1990). Physica C, 171, 93. Klug, D. D. & Whalley, E. (1984). J. Chem.Phys., 81,

1220.

Kuhs, W. F., Finney, J. L., Vettier, C. & Bliss, D.

V. (1984). J. Chem. Phys., 81, 3612.

Pei, S., Jorgensen, J. D., Hinks, D. G., Dabrowski,

B., Lightfoot, P. & Richards, D. R. (1990). Physica C, 171, 175.

Yamada, Y., Jorgensen, J. D., Pei, S., Lightfoot,

P., Kodama, Y., Masumoto, T. & Izumi, T. (1991).

Physica C, 173, 101.
National Institute of Standards and Technology Special Publication 846. Proceedings of the international

conference Accuracy in Powder Diffraction 11, held at NIST, Gaithersburg, MD, May 26-29, 1992. (Issued

October 1992)

Crystal Structure Analysis by Fitting of

Powder Diffraction Data on Energy Dispersion

T. YamANAKA

College of General Education, Osaka University,

Machikaneyama Toyonaka Osaka 560, Japan

Abstract

A profile fitting method based on energy dispersive

(ED) spectra has been established. The ED measurement has a large advantage for diffraction studies under various physical conditions, such as at high

temperature and pressure, because of the fixed Xray diffraction system. However, ED spectra present

numerous difficulties in respect to diffraction intensities. The diffraction intensities must be corrected

for source intensity distribution, absorption, escape

efficiency and background as a function of energy.

Energy resolution and accuracy of the structure re- finement have also been tested. The structure re- finements were performed on ED spectra taken with

synchrotron radiation at the Photon Factory using a diamond anvil pressure cell and multi-anvil pressure

apparatus.

Introduction

Crystal structure analysis bcised on the powder dif- fraction method has been advanced by the progress

in profile fitting methods (Rietveld, 1969; Langford,

1981). Numerous discussions have been presented on the analysis of not only neutron but also X-ray

diffraction profiles (Parrish, Huang & Ayers, 1976;

Young, Mackie & Von Dreele, 1977; Malmros &

Thomas, 1977; McNelles k Cambell, 1975). In re- cent years, synchrotron radiation (SR) sources have

induced extensive powder diffraction studies because

of their excellent beam characteristics. Thus the field

of application of powder diffraction has been greatly

broadened.

Energy dispersive (ED) diffraction experiments us- ing a white X-ray source, as well as angle dispersive

diffraction, have become more popular. ED experiments can be made under fixed X-ray optical conditions, where X-ray source, sample and detector are

set in place. The method has, therefore, a great advantage for in situ diffraction studies under variable

physical conditions such as high pressure and high or low temperature.

Discussions on the ED diffraction intensity were

presented by Buras, Olsen, Gerward, Selsmark k,

Andersen (1975). Wilson (1973), Gerward, Morup

& Topsoe (1976) and Fukamachi, Hosoya k Terasaki

(1973) have discussed the profile shape. The application of profile analysis of the ED spectrum to struc- ture refinement was discussed by Glazer, Hidaka k

Bordas (1978).

Various difficulties encountered in the profile fit- ting of ED spectra have been overcome by empirical

corrections to the diffraction intensities. Yamanaka

k Ogata (1991) detailed correction of the intensity

for source intensity distribution, absorption, escape

efficiency and background as a function of energy.

The profile fitting is performed by the profile decomposition method, using the whole ED pattern (Pawley, 1981).

1. Intensity measurement of ED spectra

Diffraction intensity measurements based on ED

spectra require a white X-ray source having uniform

and large intensity. Synchrotron radiation and tungsten or silver rotated anode targets in laboratory

sources are extremely well-suited for ED diffraction

studies. Radiation emitted from the positron stor- age ring at the Photon Factory (PF) with 2.5 GeV

and 300 mA and from the Accumulator Ring (AR)

of TRISTAN with 6.5 GeV and 50 mA provides Xray sources having maximum energies of 40 keV and

120 keV, respectively. In the latter case, it is possible to detect diffraction peaks falling in the range

0.7113 k <dH < 14.227 A with the detector set at 29 = 10°. By changing the detector angular position,

we can shift the detectable energy region. Because

of the nature of the synchrotron radiation beam, a parallel-beam mode can be used instead of a focusedbeam mode. Therefore a very small divergence of the

incident source gives a sharp diffraction profile.

199
200 Crystal Structure Analysis by Fitting of Powder DifTraction Data on Energy Dispersion

In the present experiment intensities were measured using a Ge (intrinsic) solid state detector

(SSD). In cases where a sample contains elements

with larger atomic number than Ge (Z=32), X-ray

fluorescence interferes with the diff'raction pattern.

The fluorescence increases the photon count error due to pulse pileup and a large dead time. For the

elimination of the fluorescence, a Zn foil absorption

filter was used.

The energy-resolution of the SSD was tested in advance. The full width half maximum (FWHM) of the

fluorescent peak of MnKa indicated a resolution of

about 150 eV. Figure 1 presents the energy resolution of the Ge SSD. This resolution is related to the

accuracy of the interplanar distance dhkl (-^)- That

the precision of dhkl varies with the angular position

(6) of the detector can be inferred from the formula

dhkl =

12.398

(1) 2{E + AE) sine

A higher angular position 9 of the detector ensures a more precise determination of d. Figure 2 shows the

systematic error of Ad depending on the energy of

diffraction peaks for a SSD with an energy resolution

of 200 eV. It ensures a precision higher than ±0.005

A, but indicates ±0.001 A at best. The value of

is less than 0.001. It is necessary to take into

account the energy resolution for the determination

of the lattice constants.

An intensity measurement was made using a multichannel analyzer (MCA) and computer aided measurement and control (CAMAC) module. A linear

relationship between the channel and energy was con- firmed by the known fluorescent X-ray energies of

several elements. The zero position and the channel/energy relationship were tested by the leastsquares method, using as standards the fluorescence

of the elements Ti, Ni, Y, Mo, Mn, Zn, Se and Zr. The linearity of the quantum efficiency of the SSD

at each channel was also confirmed using an Am ra- dioactive source. The photon counting efficiency of

the SSD was tested in the energy region of the measured spectrum. In this test several parameters of

MCA, AD converter and amplifier have to be optimized. Since the frequency of AD converter clock

pulses was 100 MHz, the shaping time was 10 fis. The X-ray source intensity was adjusted in order to prevent the number of photons from exceeding 1.5

kcps per channel throughout all 2048 or 4096 channels. The dead time was kept within 5%.

The photon counting eflRciency is discontinuous at the K absorption edge (11.103 keV) of Ge. Therefore,

great care should be taken as to the angular position

of the SSD in order to keep the diff'raction peaks from

occupying the above-mentioned position.

2. Correction of the difTraction intensity

(a) Intensity distribution of the white X-ray source For ED experiments the energy distribution of the

SR X-ray source ideally should be continuous, without any characteristic X-ray lines such as are found

in Bremsstrahlung X-ray sources. The source, however, does not have a uniform intensity throughout

the whole energy region. An experimental investigation of the intensity distribution of the source was made by varying the energy positions of the reflec- tions, using a fixed-time intensity measurement on a
T. Yamanaka 201

2100 2800 5500 ctanel

10

I' l l I I

I

15 20 25 30 K ICev

10.0 5.6 4.0 1 1—

3.0 2.0 1.5 A

Figure 3: Intensity correction of the photon count

distribution of the source and the absorption of

exp(-^r). The SR source is emitted from the stor- age ring of electrons with the energy of 2.4 GeV and

300 mA. The absorption corrections are calculated

for the diamond anvil cell and Ge02 sample.

Si crystal plate cut parallel to (111). An approximation to the SR source intensity distribution was also

made using a Bessel function. An example of the SR

source from a bending magnet with 2.5 GeV and 300

mA is shown in figure 3. The maximum brightness

of the source is located at 2.9 keV.

(b) Correction of the escape peak intensity

When photons have larger energies than that of

the K absorption edge of Ge, energy escape arises.

Then the accompanying escape peaks appear beside

Bragg peaks at positions £^Bragg - 9-87 keV. The es- cape yield £{E) depends on energy. The observed

intensity, yob){E), is related to the true intensity,

yo(E), of the diffraction peak by:

Vob.iE) = yoiE)[l - siE)] (2)

The intensity of the escape peak is given as

yobs{E - Eo) = eiE - Eo)yoiE - Eo) (3)

where Eo is the energy of the characteristic X-ray of

Ge-K.

It is necessary to determine the correction curve

for the energy dependence of €(E). The integrated

intensity of reflection hhh from a Si single crystal

plate was measured using a Ge (intrinsic) SSD. The

correction curve for £{E) was previously shown in

Yamanaka & Ogata (1991). Measurement was car- ried out by varying the angular position of the SSD.

An empirical equation of the correction curve for the

escape efficiency was determined by least-squares calculation, as expressed below:

e{Eo) = exp(-0.455 x E{keV) - 2.2236) (4)

(c) Background intensity

Profile analysis demands an accurate measurement

of the true Bragg reflection intensity, excluding the

background intensity which is caused by X-ray scat- tering other than Bragg reflection. An attempt has

been made in the present program to express the

background intensities in the whole energy range by

several different functions in order to ensure highly

precise structure determination. Among many trials,

the best function for fitting the background shape was found to be a sum of a linear function and a modified pseudo-Voigt function:

y(Ei )= aEi + b + P\T] exp (^^)

+ il-rj){l + 4^y (5)

where a, 6, P, E, F and tj are variable parameters.

A part of photon energy is lost by the release of

recoil electrons, and it produces Compton scattering.

The incoherent scattering due to the Compton effect

generates an aperiodic intensity. Thus the ED spectra include the Compton scattered spectra.

(d) X-ray absorption correction

An incident X-ray having an energy greater than

the ionization energy induces photon absorption by

liberation of an atomic electron. The observed intensity / is derived from / = lo exp{—nr), where is the incident intensity, fi (cm~^) is the linear absorption coefficient, and r (cm) is the X-ray path length

in the material. The linear absorption coefficient as a function of energy can be calculated from the mass

absorption coefficient ^(cm^g~^), because the Xray absorption varies with the photon energy of the

wavelength. The mass absorption coefficient of an

element is calculated by

A A

(6)

where A is the atomic mass and Cp and tr, are defined

by the photoelectric cross-sections for individual electron orbits derived from quantum mechanics. The

mass absorption coefficient of each atom has been

given as a function of energy of the primary X-ray

wave (Sasaki, 1990).
202 Crystal Structure Analysis by Fitting of Powder Diffraction Data on Energy Dispersion

In a previous paper (Yamanaka & Ogata, 1991),

the mass absorption coefficient derived from the empirical equation of Victoreen (1948, 1949) was used:

(7)

The coefficients Ci , Di, and a were taken from the

International Tables for X-ray Crystallography, Vol.

III. In equation (7), N denotes Avogadro's Number, Z indicates an atomic number, and A represents atomic mass. When a sample contains an ele- ment with its absorption edge in the measured energy

range, Cz and Dz should also be taken into account.

3. ED diffraction profile

(a) Integrated intensity of diffraction peaks

Several functions for describing the powder diffraction profile have been proposed for angular dispersion

spectra. The profile decomposition method was applied to the ED spectra. Then each integrated intensity in the superposed diffraction peaks was assigned

separately.

For the least-squares profile fitting of the ED spectra, the initial parameters of the crystal structure were required. The fitting procedure optimized the

structure parameters by the least-squares method so

as to ensure a good agreement with the ED diffraction intensity.

The energy positions of the ED diffraction peaks

(Ei) can be determined from Eh = 12.398/2c/^f sin B, where 6 denotes the angular position of the detector.

The integrated intensity of reflection H can be derived from the following equation:

Ih= Io{Eh)-p\ HoT"^\ ]\^^\

I sm 29 • COS0 J

X exp

25 • E]j_sm^ 0

2„2 h'c

A {Eh) (8)

Io{Eh) ' source intensity at' the energy Eh

p : multiplicity parameter

Fh ' structure factor of reflection H

B : overall temperature factor

A{Eh) ' X-ray absorption term at the energy Eh

SR X-rays have an approximately horizontal polarization and thus the polarization factor, (1 -f cos 29),

is neglected.

(h) Profile function

The full width at half maximum (FWHM) of ED

diffraction peaks is generally larger than is found in

angular dispersion data. This is particularly noticeable at high energies. The diffraction profile varies

not only with the crystallized state but also such optical and instrumental conditions as X-ray source,

slit system and SSD resolution. Therefore, an optimum function had to be selected for the description

of profile shapes.

In the case of X-ray angular dispersion diffraction,

functions such as Gaussian, Lorentzian, Pearson VII

and Voigt have been attempted to ensure accord with

the observed profile. Glazer et al. (1978) applied the

Gaussian function in profile fitting to the ED pattern.

However, the present program uses the pseudo-Voigt

function, which satisfactorily models the profiles.

yi2AEiH) = C jr/exp {-X') + (1 - '7)y^}

(9)

AEiH

X = 2

Y = 2

= Ei — Eh

F/f (Gaussian)

(Lorentzian)

Eh ' energy of the reflection H

Ei : variable energy

r^(Gauss), F/f (Lorentz) : variable energy

The FWHM of the diffraction profile is not constant,

but increases with energy Eh- The energy dependency of F was expressed by the empirical formula

F = UEjj + VEh + W (10)

using the observed data presented in figure 4. This

procedure is in conformity with the procedure of

Glazer et al. (1978). U, V and W are variable parameters in the least-squares fitting.

Unlike the angular dispersion diffraction profiles,

the profile of the ED spectrum remains asymmetric,

and consequently the following correction term has

been introduced:

a (AEiH) = 1-A i^AE], - (^)'} ' ^EiH

(11)

The profile decomposition divides the superposed

diffraction patterns into the respective diffraction

profiles in accordance with the corresponding profile functions, thereby determining the integrated

intensity, peak position, profile pattern parameters

(FWHM, peak height, Gaussian fraction) and background intensity.
T. Yamanaka 203

10 20 30

ENERGV (keU)

40

FUIHM = 0.21311 ' 0.00508E R = 0.970

FUJHM " 0.17194 0.00888E - 0.00007^ ° 0.986

Figure 4: Variation of FWHM with energy. Linear

and quadratic functions have been fitted to the observed variation in the Si ED spectrum.

(c) Program

The integrated intensity Ih derived from equation (8) is converted into the diffraction intensity

ydEi) of the respective energies in accordance with

the following formula:

y,{Ei) = B{Ei) + Y:y(^AEiH)lH (12)

where

B{Ei) : background

y(2AEiH) '• profile function defined by eq. (9).

The structure factors Fg are derived from the observed diffraction intensities yo{Ei) of the respective

energies in the 4096 individual channels of the MCA.

Structural parameters, including lattice constants,

atomic coordinates and temperature factors, are optimized by the least squares method so as to minimize the standard residual:

(13)

The scale factor, profile parameters and background

are also optimized by the refinement. Figure 5 indicates the flow chart of the computer program. The

Gauss-Newton method is applied as the nonlinear

minimum square method for the parameter refinement.

Three R factors are calculated for the determination of the results of refinement.

RwD —

Rd =

Rs —

Y:wi{yo{Ei-E,)y

Y.Wi{yo{Ei)f

Y.\yo{Ei) - y,{Ei)\

EVoiEi)

T.\Io{H) - h{H)\

(14)

(15)

(16)

4. Structure analysis and kinetic study

of the transition at high pressures

Profile fitting of ED spectra of two Ge02 polymorphs, hexagonal (P3221, Z=Z) and tetragonal

(P4^/mmm, Z=2), were performed under various

high pressures and temperatures using a diamond

anvil pressure cell with a platinum ring heater (Yamanaka &: Ogata, 1991). Synchrotron radiation with

2.5 GeV and maximum current of 300 mA at the

Photon Factory in Tsukuba was used as the X-ray

source. The diamond anvil cell was set on the spinner, which was rotated at the rate of 1/3 rpm around

the <p axis and simultaneously oscillated in the range

of ±5" around the uj axis for a randomization of the

sample orientation and for reducing the effect of grain

growth. A white X-ray beam of 50fim<f> was applied.

Figure 6 shows the profile fitting of the ED spectrum

of hexagonal Ge02 at 2.9 ~ 3.0 GPa and 493 K.

;5oo -

5000 -

2500

3600 Channel

20 25 50 J5 keV

5.0 <i.O -I f- 5.0 2.5 2.0 —»—

1.5A

Figure 6: Profile fitting of the ED spectrum of hexagonal Ge02 at 2.9 GPa and 493 K.

Structure refinements by profile fitting were made

after the many intensity corrections mentioned
Crystal Structure Analysis by Fitting of Powder Diffraction Data on Energy Dispersion

(l

READ : INTENSITY DATA

PLOT SPECTRUM

DATA REDUCTION

Absorption correction

Normalization on the source intensity

Escape peaK

Efficiency of the SSD

PLOT SPECTRUM

BACKGROUND FITTING

READ : REGION

INITIAL PARAMETERS OF

BACKGROUND SHAPE FUNCTION

CALCULATE : BACKGROUND SHAPE

Least squares

WRITE REFINED PARAMETERS

^RETRY

PLOT BACKGROUND

READ : INITIAL PARAMETERS (Pi) Lattice parameters

Atomic coordinates

Overall temperature parameters

FWHM. Gaussian fracton

return)

READ : SYMMETRY OPERATION

PARAMETERS

ATOMIC SCATTERING FACTORS

DISPERSION FACTORS

SET OF REFLECTIONS

HKL. Fc, Multiplicity

LEAST SQUARES FITTING

OF THE WHOLE PATTERN

CALCULATE : CHANNEL NUMBER

d - VALUES

FWHM OF EACH REFLECTION

WRITE REFINED PARAMETERS

Structure parameters

Fo - Fc table R • factor. Standard deviation

CALCULATE : INTEGRATED INTENSITY

AT EACH REFLECTION

INTENSITY (Yi) AT EACH CHANNEL

LEAST SQUARES ROUTINE

derivatives dYi/dPi

M - matrix,

inverse matrix of M

damped Gauss-Nevton Method

Simplex Method

Marquared Method

CALCULATE : Pj

NEW Pj

Pj - Pj * Pj

PLOT THE CALCULATED PATTERN

RETRY

RETURN }

Figure 5: Flow chart of the program of the ED spectrum profile fitting.
T. Yamanaka 205

above. The final residual factor of the least-squares

calculation was Rd = 0.088. From the lattice parameters as a function of pressure determined by the

profile fitting, volumetric compressibility, equation

of state and isothermal bulk modulus were calculated (Yamanaka & Ogata, 1991). Profile fitting was

applied to the kinetic study of the structure transition from hexagonal to tetragonal structure (Yamanaka, Sugiyama k Ogata, 1992). The transition

rate was calculated from the mass fractions that were

obtained from the precise measurement of the diffraction intensities of both phases in the time resolved

ED spectra at several fixed pressures and temperatures. The above structure study had a drawback due to

the small eff'ective energy region, within 40 keV. The

small number of photon counts at higher energy emitted from the bending magnet at the Photon Factory

storage ring could not provide large enough intensities to use in the refinement. However, a multianvil pressure apparatus MAX80 (Shimomura ei ai,

1989) set on the beam line of the Accumulator Ring

of TRISTAN with 6.5 GeV and 50 mA overcomes

the difficulty for the structure refinements. The ED

spectrum with an energy range of 10 ~ 150 keV is available for profile fitting at high pressures. The

pressure apparatus, with anvils composed of sintered

diamond instead of tungsten carbide, can give a maximum pressure of 13 GPa and temperature of 1700

°C; the sample volume in the anvil is 2 mm in diameter and 3 mm high.

Structure refinements of Mo (Im3m, Z=2) were examined by profile fitting of the ED spectrum (fig- ure 7 ) falling in the range of 5.0 ~ 0.6 A observed

at 5.9 GPa and 800 °C using the high pressure apparatus (Kikegawa et ai, personal communication).

The measurement duration was 1171 sec with the

ring current 8 mA. The observed peaks in the spectrum are much more intense than those from the nor- mal bending magnet with 2.5 GeV and 300 mA, and

higher accuracy of the profile fitting can be expected

than that obtained from the ordinary ED measure- ment.

References

BuRAS, B., Olsen, J. S., Gerward, L., Selsmark, B.

& Andersen, A. L. (1975). Acta Cryst. A 31, 327-

333C.

FUKAMACHI, T., HOSOYA, S. & Terasaki, O. (1973).

J. Appl. Cryst. 6, 117-122.

Gerward, L., Morup, S. & Topsoe, H. (1976). J. Appl. Phys. 47, 822-825.

Glazer, a. H., Hidaka, M. &; Bordas, J. (1978). J. Appl. Cryst. 11, 165-172.

Langford, J. I. (1981). Acta Cryst. A 37 Supplement.

stmpJe : Mo powder

X-iijf sounx: SR ( 6. 3CeV 30mA u AR ) P md Tcooditkm: P - 5.9GPl T-300K by MAX80

Ocieaor: Ge-SSD, 29 -8*. lime • 1 171 jec Energy cilibniian E (keV)- 0.0736 x NCdi)-!- 1.0746 muchu>2048

10000

^ 8000 s

6000

4000

3

^ 2000

1

'

1

' 1

s

~ at~ 8

" o

1

. Ci 1 ;i i

' _ 1 ' 1 ' I '-—i- -

-.3

o 5 ^

~ O — r»»» <«<A<«<o o o o o o o

.

' i a

"

. i i 8 a ssa 2

.. . 1 . 1 . 1 . 1 . I . 1 . 1 , 20 40 60 80 100 120 140 160 Energy I keV

250

200

'53

3

O

150

too

50

d

8

d

or^m

en — o s- o

O

110 115 120 125 130 135 140 145 150 Energy I keV

Figure 7: ED spectrum of Mo.

Malmros, G. & Thomas, J. O. (1977). J. Appl. Cryst.

10, 7-11.

McNelles, L. a. & Cambell, J. L. (1975). Nucl. Instr. Meth. 127, 73-81.

Parrish, W., Huang, T. C. & Ayers, G. L. (1976). Trans. Am. Cryst. Assoc. 12, 55-73.

Pawley, G. S. (1981). J. Appl. Cryst. 14, 357.

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

Sasaki, S. (1990). X-ray Absorption coefficients of the

elements, KEK Rep. 90-16.

Shimomura, O., Taniguchi, T., Kanda, H., Akaishi,

M., Yamaoka, S., Nagashima, T., Kikegawa, T.,

Kato, T. & Yaoita, K. (1990). PF Activity, Rep. 290.

Victoreen, J. A. (1948). /. Appl. Phys. 19, 855-860.

Victoreen, J. A. (1949). J. Appl. Phys. 20, 1141-1147.

Wilson, A. J. C. (1973). J. Appl. Cryst. 6, 230-237.

Yamanaka, T. & Ogata, K. (1991). J. Appl. Cryst. 24,

111-118.

Yamanaka, T., Sugiyama, K. &; Ogata, K. (1992). /. Appl. Cryst. 25, 11-15.

Young, R. A., Mackie, P. E. & Von Dreele, R. B.

(1977). J. Appl. Cryst. 10, 55-73.
206 Accuracy in Powder Diffraction II

01.1 THE USER-ASSISTED AUTO CALIBRATION PROCESS

IN THE ^PDSM MICRO POWER DIFFRACTION SEARCH/MATCH.

R. Gary Marquart, Fein-Marquart Associates, Inc., 7215 York Road,

Baltimore, Maryland 21212, USA

jiPDSM (micro Power Diffraction Search/Match) continues to be improved

and augmented. The latest addition is the ability to use multiple identified phases as internal standards. Even with well-aligned instruments and

carefully prepared and mounted samples, some amount of systematic error

is usually noted. The JCPDS database size is large and ever increasing;

reducing systematic error permits the use of tighter match parameters,

thereby enhancing the search/match specificity.

However, it is possible that large systematic errors will preclude automatic assignment of lines from the proposed internal standards to the correct corresponding problem pattern lines. For those cases, p.PDSM provides for user interaction, both numerically and graphically, in the correction/

calibration process, with the user identifying the fiduciary lines and

assigning their correct values, both directly and by reference to the lines of

selected PDF patterns.

The final internal standard calibration is iterative and uses all matched lines from the user-selected standards, weighing the contribution of each by its relative significance as defined by the relationship of its intensity to the user-specified standard error at that intensity.

01.3 X-RAY PHASE ANALYSIS IN TEXTURED MATERIALS.

H. Oettel and P. Klimanek, Institut fur Metallkunde, Bergakademie Freiberg,

D-O-9200 Freiberg, BRD

The integral intensities of X-ray reflections strongly depend on the texture,

respectively the pole density in the direction of investigation. Neglecting this

effect, the error in X-ray phase analysis can reach more than 100 %! To

suppress texture effects, it is possible to apply special preparation procedures

(powders) or to average results from different directions of investigation. For

general treatment of the problem it is necessary to combine the formalism of quantitative texture analysis (description of quantitative pole figures [H. J. Bunge et al.: Cryst. Rev. 2 (1989) 65]) and the optimization algorithm of

the multicomponent quantitative phase analysis [J. Fiala: J. Phys. D: Appl.

Phys. 5 (1972) 1874]. This can be realized in practice [H. Oettel, P. Klimanek:

1st Europ. Powder Diffraction Conference, Munich, 14 - 16th March 1991], if the number of the coefficients needed for describing the pole figures of different

reflections is not too high. To keep the number of coefficients small enough, it

is useful to rotate the specimen around an axis perpendicular to the surface while measuring with different inclinations, so enforcing a fibre texture [M.

Jarvinen et al.: J. Appl. Cryst. 3 (1970) 313]. The uniform optimization

model proposed here yields both the volume part of the phases and the texture

coefficients of their enforced fibre textures. The efficiency of the procedure will be demonstrated by experiments.

01.2 THE LIMITING EFFECT OF CRYSTALLITE SIZE ON .AC- CURACY IN QUANTITATIVE POWDER DIFFRACTION ANALYSIS.

Deane K. Smith, Department of Geosciences. University Park, PA 16802, USA.

The effective number of diffracting crystallites in a sample is usually the limiting sample parameter on the potential accuracy obtainable in quantitative

powder diffraction analysis. A detailed analysis of the interactions between

the number and orientations of the crystallites and the optics of the diffrac- tometer indicate that the typical quantitative limit in present practices is of the order of 1 weight percent. Reduction of crystallite sizes to an average

of 1 micrometer is critical to achieving this accuracy. This limit applies to whole-pattern matching and to the Rietveld method.

Detailed analysis of the ray paths of the diffracting crystallites reveals the primary angular limit of crystallite diffraction. The rocking range of the indi- vidual crystallites and the image size of the focal spot of the X-ray source and

the receiving slit are the main controlling properties. The angular opening of

the divergence slits have little effect except on the size of sample irradiated.

Simple modifications to the diffractometer and to its operation during data col- lection can improve crystallite statistics significantly, Broad focus optics and

sample oscillation have more effect on increasing crystallite statistics than in- creased beam diverence or sample spinning. Optimum diffractometer optics are not the same as for high resolution, thus the modem diffractometer oper- ating in most laboratories is not optimized for quantative or Rietveld analysis.

QUANTITATIVE EVALUATION OF CRYSTALLINE

AND AMORPHOUS COMPONENTS IN MIXTURES BY RIETVELD

ANALYSIS OF X-RAY POWDER DIFFRACTION PROFILES.

M. Bellotto, CISE, P.O. Box 12081, 20134 Milan, Italy; and G. Artioli,

j Istituto di Mineralogia e Petrologia, University di Modena, 41100 Modena,

'

Italy.

Accurate quantitative assessment of amorphous components in polycrystalline mixtures is a very complex and difficult task. Semiquantitative evaluations are generally carried out by combining results from

chemical analysis and X-ray diffraction techniques, but reliable estimates of

the amorphous component are virtually impossible when crystalline and noncrystalline fractions in the mixture have similar chemical compositions.

Such a case frequently occurs in materials having important industrial applications, as in natural zeolite-rich rocks, where the original volcanic glass has nearly the same chemical composition as the secondary zeolite minerals.

Both the Rietveld refmement and the Reference Intensity Ratio (RIR)

techniques have been recently employed to successfully evaluate crystalline

phase fractions in polycrystalline mixtures. We here propose a method for the quantitative determination of the amorphous component, based on a

combination of the Rietveld analysis of powder diffraction profiles and of

the internal standard matrix flushing technique. By successive dilution of

the original sample with known amounts of a glassy component and a fixed amount of a standard material, it is possible to check for internal consistency of the extracted weight proportions. Linear regression on the concentration values obtained for the series of diluted samples leads to an independent

estimate of the fraction of non-crystalline component in the original sample.

The method has been tested on two artificially prepared materials, and then

applied to two clinoptilolite-rich tuffaceous samples from the Zlatokop

deposit (Yugoslavia). The results obtained with both methods are in good

agreement and allow a reliable estimate of the volcanic glass fraction in the rock.
Abstracts, Session 1: Phase Identification and Quantification 207

PI. 2 MODAL ANALYSIS OF EIGHT MAJOR ROCK TYPES

BY RIETVELD ANALYSIS OF X-RAY POWDER DIFFRACTION DATA.

R.J. Hill, G. Tsambourakis and I.C. Madsen, CSIRO Division of Mineral Products, PO Box 124, Port Melbourne, Victoria 3207, Australia.

Mineral modes have been determined for specimens of eight natural rock

types by multiphase Rietveld analysis of CuKq X-ray powder dlflFractometer

data. The samples include two granites, a granodiorite, adamellite, gabbro,

basalt, trachyte amphibolite and a granulite. The normally severe problems

encountered with preferred orientation (especially of the feldspar components)

when using a Bragg- Brentano, flat-specimen configuration were eliminated by

packing the samples in glass capillaries and using Debye-Scherrer geometry

(Hill & Madsen, 1991). Up to eight individual mineral components were measured in each sample, with a detection limit of 1 to 2 weight percent for the

accessory minerals, depending on the minerjJ assemblage. The X-ray analyses compare favourably with (i) optical modes determined for the medium-coarse

grained samples by point counting, (ii) normative calculations undertaken with

locally enhanced catanorm and mesonorm software, and (iii) corresponding

Rietveld modes determined, for two samples, from neutron powder data. The

good agreement of the X-ray and neutron modes shows that microabsorption

is of little concern in these rocks. The results confirm one of the major advantages provided by Rietveld modal analysis over the more traditional "reference

intensity" methods, namely, that the crystal chemistry (and thus the calibration constants) of the individual phases can be adjusted dynamically during

each individual analysis (HiU, 1991). This not only provides more accurate phase analyses, but also gives important supplementary information about the

unit cell dimensions, chemical composition and structural state of the major

components.

Hill, R.J. (1991). Powder Diffraction, 6, 74-78. Hill, R.J. k Madsen, I.C. (1991). Zeit. Kristallogr. 196, 73-92.

PI.4 POWDER DIFFRACTION IN PHARMACEUTICALS. L.Zevin,

J. Bernstein, and S. Lach, Ben-Gurion University of the Negev, P.O. Box 1025, Beer-Sheva, 84110, Israel

During the past 6 years we have prepared more than 150 standard diffraction

patterns for various phaimaceuticais in high demand. Though most of the studied

materials are low-symmetry, large unit cell, low-absorbing, grinding-sensitive,

orientation-prone and low-mass-available compounds. X-ray diffraction proved to be an excellent tool for their identification and quantification.

The experimental pattern was compared with a simulated one computed from the

positional parameters obtained by single crystal structure analysis, when available. The X-ray technique is unique for the identification of the often numerous

polymorphs, which have diverse physical and chemical properties crucial for pharmaceutical application, such as melting point, rate of dissolution, thermal

stability, etc. Curiously, we have found that the polymorphs serving as standard pharmaceuticals sometimes vary from country to country. Polymorphs of anhydrous material are often accompanied by hydrated and solvated forms

(pseudopolymorphs) which are also readily identifi^ by powder diffraction.

Due to rapid decrease of the atomic form factor, appreciable thermal vibration and

the sharp drop of the Lorentz factor, the characteristic powder diffraction patterns of the studied materials are confmed to within sin 6/X=0.lA, i.e. in the range of maximal instrumental peak distortion. Nonetheless a high degree of precision is generally achieved in the lattice constants (relative standard deviation is less than

5-10"* and A26s0.02°). Comparison of observed and calculated peak intensities usually gives an R-factor of -25%. This figure is quite satisfactory given the peak intensity measured and simplifications made in pattern simulation. In many

cases we detected significant deviation between the lattice parameters reported by

different authors and those observed in this study. This phenomenon, which is formally similar to solid solution effects, is probably caused by the presence of

various amounts of retained solvent. Numerous examples are given of powder

diffraction studies of pharmaceuticals with emphasis on polymorphism.

^^•5 SIGNIFICANCE OF PHASE ANALYSIS IN

HETEROGENEOUS CATALYSIS. S. Mehdi and M. Reddy, X-Ray

Section, nCT Hyderabad, 500 007, AP, India.

Phase analysis plays an important role in the study of catalysis where the nature and state of the catalyst, its carrier and promoter greatiy influence its

activity. Results of phase analysis study of a series of TiO, supported

M0O3 and WO3 and Zr03 supported M0O3 catalysts, prepared by wet

impregnation technique are presented in this paper. Variation of the M0O3-

TiOj- M0O3- ZrOj and WO3- TiOj XRD intensity ratio as a function of

weight percent of M0O3 and WO3 respectively lead to estimates of

crystalline contents in the catalyst. In the case of the WO3- Ti02 catalyst a good correlation was found between the intensity of crystalline WO3

loaded. For M0O3- TIO2 no such correlation is noted because M0O3 was

found to be in an amorphous state and the intensity of Ti02(anatase) remains

unaffected by the increase in M0O3 loading. Variation of the XRD intensity

ratio in Mo03-Ti02 is different from the other two catalysts. It passes through a minimum at 6 wt% M0O3 and increases again. Formation of

interstitial or substitutional solid solution could be inferred from the changes

observed in the cell parameters of the active species or the support.

PI.5 NEUTRON POWDER DIFFRACTION STUDY OF

LEAD ZIRCONATE TITANATE CERAMICS. S. L. Town, C. J. Howard, E. H. Kisi' k P. Bryantt, ANSTO, PMB 1, Menai, NSW

2234, Australia, *Division of Science & Technology, Griffith University, Nathan. Queensland 4111, Australia, fGEC Marconi, Faraday Park,

Meadowbank, NSW 2114, Australia.

Lead zirconate titanate ceramics are widely used in piezoelectric trans- ducer applications''^. These materials have their best piezoelectric properties, and the samples pole more easily, near the morphotropic phase

boundary. Although there have been some neutron diffraction studies''''' on these materials, there has been little work done on and around the phase boundary. Here we report room temperature neutron powder

diffraction data collected on samples with the following compositions: Zr 42.5%, 47.5%, 49%, 50%, 52.5%, 55%, 57.5%, 60% and 62.5%. Details

of QPA results are given.

1. P. Bryant, Ceramic Developments, Materials Science Forum Vol- umes 34-36 1988, p285.

2. P. Bryant, Ceramic Developments, Materials Science Forum Vol- umes 34-36 1988 p 249.

3. H. Fujishita k S. Hoshino, J. Phys. Soc. Jap., Vol 53, No 1, 1984,

p226.

4. A. M. Glazer, S. A. Mabud k R. Clarke, Acta Cryst, B34, 1978,

pl060.
208 Accuracy in Powder Diffraction II

SEARCH/MATCH PROCEDURES WITH WHOLE-PAT- TERN MATCHING. Deane K. Smith, Susan Q. Hoyle and Gerald G. Johnson. Jr., Materials Research Laboratory, The Pennsylvania State University,

University Park, PA 16802, USA.

Computer-based identification procedures in powder diffraction have employed

numerical matching schemes relying on the d-I data sets since 1962. With the

availability of digitized diffraction traces, it is now feasible to use pattern- matching techniques based on the characteristic regions of the full diffraction traces of the unknowns and the reference patterns.

Databases containing digitized diffraction patterns may be composed of ex- perimental data or data calculated from crystal structures or simulated from

d-I data sets. Search procedures use matching algorithms which emphasize

the probablility of the reference pattern being contained in the pattern of the unknown regardless of the other patterns which comprise the unknown. Rapid

matching algorithms and sensitive figures-of-merit are critical to success.

Successful searching requires techniques compensating for sample and instru- ment aberrations and variations in materials such as solid solution. Successful searching yields candidates for matching and stripping from the initial data

trace for further search analysis. This matching involves profile fitting then

subtraction, which effectively removes the phase contribution to the unknown

pattern. Sequential search/match/stripping allows more phases to be detected than are usually obtained with d-I matching, and the levels of detection are

usually lower for the minor phases.
Abstracts, Session 2: Accuracy and Standards 209

02.1 ACCURATE DETERMINATION OF UNIT-CELL PARAMETERS USING AN INTERNAL STANDARD REFERENCE MATERIAL

AND WHOLE-PATTERN FITTING. Hideo Toraya, Ceramics Research Laboratory, Nagoya Institute of Technology, Asahigaoka, Tajimi 507 Japan

A new technique for peak-shift correction achieved the accuracy of a few p. p.m.

in routinely determining the unit-cell parameters of a sample with internal standard reference material (Toraya & Kitamura, J. Appl. Cryst., 23, 282- 285, 1990; Toraya & Parrish, PICXAM, 1991). This algorithm is based on 1) the simultaneous least-squares determination of both unit-cell parameters and unknown parameters in the calibration function and 2) the use of observed profiles with symmetric shape. The first idea in the above two has already been

realized by the whole-powder-pattern fitting without reference to a structure model (Toraya, Advances in Ceramics, Vol 21, 811-819, 1987). However, the

profile asymmetry usually observed in conventional powder diffractometry in- duced a strong parameter correlation between the peak positions (unit-ceU pa- rameters) and parameters in the function for modeling profile asymmetry, and

the required accuracy could not be realized. Recent use of a high-resolution

powder diffractometer and a synchrotron radiation source make it possible to obtain nearly symmetric profile and greatly improve the accuracy of determining the peak positions. The use of whole-powder-pattern fitting ha5 a great advantage in that the method can easily be extended to the analyses of samples with low crystallographic symmetry. In the present study, the unit-ceU parameters of samples of cubic to orthorhombic symmetry were measured by

using whole-powder-pattern fitting on the above two bases. The procedure

will have potential uses in the routine detrmination of unit-cell parameters

with both conventional tube- generated X-rays and synchrotron radiation.

P2.2 INSTRUMENT ALIGNMENT METHODS FOR RESIDUAL

STRESS MEASUREMENTS BY MEANS OF NEUTRON DIFFRACTION.

P.C. Brand and H.J. Prask, National Institute of Standards and Technology,

Gaithersburg, MD 20899, U.S.A.

Neutron powder diffraction in general, and residual stress measurements by means of neutron diffraction in particular, require very well aligned instru- mentation.

Unlike regular powder diffraction where the specimen is fully immersed in the neutron beam, stress measurements by means of neutron diffraction require the presence of small beam apertures that define the size of the gauge volume

in the specimen. It is the gauge volume for which one determines the stress

state during a stress measurement.

The correct positioning of the beam apertures is crucial for successfully applying the stress measurement technique. Therefore we have developed a sequence

of alignment procedures to assure correct positioning. These employ diffraction in a small test specimen situated at a known location on the neutron

diffractometer, and a new analysis method of the diffraction data taken from

the test specimen. These will be presented. The presented alignment methods

have proved extremely helpful in obtaining very well aligned beam apertures

in a reproducible manner.

P2 . 1 A UNIVERSAL FILE FORMAT FOR POWDER DIFFRACTION

DATA ARCHIVE AND INTERCHANGE: THE lUCr CRYSTALLOGRAPHIC

INFORMATION FILE (CIF) FORMAT. B. H. Toby, Air Products k Chemicals Inc, 7201 Hamilton Boulevard, Allentown, PA 18195, USA; J. I. Langford, University of Birmingham, School of Physics & Space Research, Birmingham B15 2TT, UK; and

S. R. Hall, Crystallography Centre, University of Western Australia, Nedlands 6009,

Australia.

Powder diffraction data items have been defined for use with the Crystallographic Information File (CIF) approach to data storage and exchange. These will permit

the ready exchange of processed and raw powder data generated by diffractometers of

every type: commercial sealed-tube instruments; CW synchrotron; neutron; energydispersive; and time-of-flight. International collaboration is vital to ensure that the

definitions meet the needs of the powder diffraction community.

The Crystallographic Information File (CIF) has been developed by the lUCr for storage of crystallographic data ranging from measured intensities through structural diagrams and manuscript text [S. R. Hall, F. H. Allen ii I. D. Brown, Acta Cryst. A47, 655-685.] A CIF is composed of ASCII characters which may edited with a normal text editor, and transmitted via electronic networks (e.g. email or ftp). A

CIF is completely self-descriptive in that data items are identified by unique tags

(referred to as a data name). These data names may be defined locally or globally. Global data items (i.e. those used outside the local environment) are carefully defined

in a CIF data name dictionary that is also electronically stored in a CIF-like format. Powder diffraction data are usually measured by computer-controlled diffractometer or densitometer and are stored in a data file format specific to the measurement

instrument. Typically these data are then processed using computer programs modified to use data in the local format. If the data will be published or archived, the diffractogram will usually be reduced to a peeik table of position and intensity val- ues. Apart from the need to standardise the data processing software, the need for the

global exchange of unprocessed powder diffraction data is increasing. This need arises because of the use of whole pattern methods for (i) crystalline phase identification and quantification, (ii) ab initio structure determination, (iii) structural refinements, and (iv) characterization of materials with limited ordering (e.g. polymers and clays), where the diffractogram cannot be represented as a set of peaks.

Definitions for CIF data items used in powder diffraction are proposed. These are

in addition to those data items in the proposed 1991 Core CIF Dictionary. Discussion

of these definitions is to be encouraged prior to their formal adoption at the 1993 lUCr Congress.

P2.3 A REFERENCE SPECIMEN FOR

HIGH RESOLUTION LINE PROFILE ANALYSIS.

PREPARATION, CHARACTERIZATION AND APPLICATION.

J.G.M. van Berkum, G.J.M. Sprong, Th.H. de Keijser, E.J. Sonneveld,

A.C. Vermeulen & R. Delhez. Delft University of Technolosy, Laboratory

of Materials Science, Rotterdam seweg 137, 2628 AL Delft, The Netherlands.

High resolution line profile analysis requires a high qualitj' standard reference specimen to characterize the instrumental line profile. A universal reference has to meet very stringent criteria, of which negligible strucmral broadening is

tlie most important. Si SRM 640a/b powder has teen selected as an appropriate starting material. Reference specimens have teen prepared according to a standardized procedure, which includes (i) removal of the smallest particles

(<5 \xm), (ii) sedimentation in 2-propanol of a mono-layer of particles onto a

Si (510) single crystal substrate (for low background) and (iii) annealing for 20 h at 1273 K. The sedimented powder sticks to the substrate and is very homogeneously distributed. Loose fractionated powder, annealed in a rotating

ampulla, can be applied as an internal standard. Using a conventional powder diffractometer the reference specimens

yield narrower line profiles than those reported in the literature (down to 0.050° 26 full width at half maximum) and the breadth vs. 26 plots show

ver%' smooth curves. As a comparison LaBg SRM 660, also sedimented onto

a Si substrate (but no other treatment), yields systematically broader lines over the complete 26 range and the breadth vs. 29 plot shows deviations from a smootfi curve due to {hklj -dependent structural broadening. Crystal

statistics is the limiting factor for the line profile reproducibility. For a

spinning specimen with 2-4x10^ particles per mm^ the integral intensity and

the breadth can te reproduced within 1%. An improvement is possible by

preparing thicker specimens, having several layers of powder particles, but only at the cost of extra 'transparency' broadening. Essentially mono-layered

non-spinning specimens are not suited as a reference. Using the new reference specimens line profiles with a very small amount of size and strain broadening can te interpreted reliably. As an

example, line profiles from ball-milled panicles (1-10 ^tm) of various

materials have been studied. Although the integral breadths of the measured

lines may te smaller than 0.1° 26 at low angles, they can te analysed reliably

using Stokes deconvolution. Besides high resolution line profile analysis, the new class of standard specimens also provides a means for assessing the state of any powder diffractometer. The smallest deviations from the perfect

aligiiment, from the ideal shape of a focus or from the average room or

cooling-water temperattu-e result in detectable breadth changes.
210 Accuracy in Powder Diffraction II

P2.4 SELFCORRECTING METHOD FOR PRECISE

DETERMINATION OF LATTICE PARAMETERS. M. Ya Gamamik,

Institute of Geochemistry and Physics of Minerals, Academy of Sciences, Palladin Pr. 34, Kiev 252680, Ukraine

The standardless method is proposed for lattice parameters determination by

X-ray diffractometer. Goniometer aberrations, distorting Bragg's reflections, can be accounted for by the method. The displacement of the sample

reflecting plane from the goniometer axis, the zero detector position, the

longitudinal and transversa] components of goniometer eccentricity are determined from reflections locations for both large (analytical reflections) and small (correcting reflections) Bragg angles. These aberrations are

calculated from the equation systems including also the lattice parameters as unknown quantities. The method accuracy is determined by that of the angle measurement and by the positions of analytical and correcting

reflections. The method can be used for polycrystal and monocrystal

samples of any crystallographic system.

P2.6 ACCURACY IN POWDER DIFFRACTION: OPTIMISATION OF DATA COLLECTION STRATEGIES. W.I.F.David, ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon., 0X11 OQX, U.K.

X-ray and neutron constant wavelength powder diffraction are conventionally

collected in step scan mode with constant time increments or with a constant monitor count. From an instrumental viewpoint these methods are clearly the simplest - additionally, of course, the resulting pattern has a Bragg peak

intensity variation that is similar to that obtained by photographic methods.

For neutron diffraction data this generally is an adequate procedure since the

high angle Bragg peaks do not suffer from a form-factor fall-off. However, in X-ray powder diffraction the atomic scattering factor variation as a function of

sinS/A may lead to intensity differences that exceed two orders of magnitude

between low and high angle data. Standard least-squares for X-ray profile re- finement is often dominated by fitting the first few Bragg peaks (which contain

the least structural information!). The principle of optimised powder diffrac- tion data collection strategy that will be discussed is similar to well-known

leverage techniques that are used in single crystal data collection. Simple

theoretical models will be presented and initial results discussed.

P2.5 USE OF AN AUTOMATED POWDER DIFFRACTOMETER TO PERFORM LATTICE PARAMETER REFINEMENT.

R. A. Newman, Analytical Sciences Laboratory, 1897 G Building,

Dow Chemical, Midland, Michigan 48667, U. S. A.

It is well known that careftjlly jjerformed experiments using specialized

powder diffraction instrumentation to analyze pure, crystalline materials can

yield lattice parameters with a precision of 1x10"' or better. However, this

is seldom the case when using standard laboratory diffraction equipment,

especially Bragg-Brentano geometry powder diffractometers. The errors associated with the reflection geometry, sample size/preparation, etc. for

typical diffractometers limit lattice refinement performed on these units to precision on the order of 1x10"^ in the best cases.

At Dow, we have improved our ability to perform diffractometer-based

lattice refinements by using the combination of [1] a Siemens D-5(X) diffractometer equipped with an incident beam monochromator, [2] the use of zero-scatter sample holders, and [3] our in-house developed profile-fitting software. With this system, we are able to obtain data from a Ka, source, which greatly simplifies the diffraction pattern. The use of a shallow depth

zero-scatter sample holder reduces the effect of sample transparency and flat sample errors. Finally, the ability to perform interactive profile

decomposition base on a split Pearson VII algorithm improves our accuracy

in peak centroid determination. Utilizing this approach, we are able to routinely obtain lattice refinement results on the order of 1x10"^. The

various steps in the data collection and analysis procedure will be illustrated, and examples of lattice refinements will be shown.

P2.7 ON THE PRECISION AND ACCURACY OF POWDER

DIFFRACTION STUDIES OF KAOLINITE AND DICKITE. L. Smrcok, In- stitute of Inorganic Chemistry, Slovak Academy of Sciences, 842 36 Bratislava, Czechoslovakia.

Kaolinites are known to exist in four polytypic forms which differ only in the

stacking mode of simple layers formed by SiO^ tetrahedral and A1(0,0H)6

octahedral sheets. Since kaolinite, in contrast to dickite, h<is not, with one

exception, yielded single crystals suitable for structure analysis, (the results

are, however, of lower qualit\') all structure refinements carried out so far have been based on powder data. Such refinements are hampered mciinly by

heavy overlap of the majority of individual reflections (kaolinite is triclinic) and by kaolinite's extreme tendency to orient preferenticilly. In order to eval- uate the precision and/or accuracy of the refined structure parameters, we

compared the interatomic distances (i(Si-0) and (f(Al-0) calculated from the data reported in ten powder refinements of both well ordered and partially disordered kaolinite specimens. The main tool used in our considerations were median-based box-and-whisker plots. We have found, that the d(Si-0) are within the interval < 1.43,1.87 >A while d(Al-0) vary from < 1.64,2.12 >A;

the expected values being 1.617 and 1.91 A, respectively. A comparison of

these individual refinements with two reliable single crystal analyses of dickite has shown that the spread of distances is in only one case comparable with

the single crystal results. Severed authors have reported rather precise but

inaccurate distances, or vice versa. As far as the medians are concerned, all refinements have provided very close results, but the sizes of the boxes vary

significantly from one refinement to another. Potential sources of difficulties are (not ordered according to their relative importance): preferred orientation wrongly accounted for; weights emphasizing the role of small intensities, which dominate in the region of heavy overlap and are measured frequently with very limited precision; small ratio of the number of Bragg reflection to the number of refined structure parameters; and limited ability of the method

to decompose unambiguously profiles that are heavily overlapped. It is rather

surprising that the accuracy of the interatomic distances is almost independent

of the presence or absence of disorder in the structure.
Abstracts, Session 3: New Developments in Software and Data Analysis 211

03.1 THE FAST ITERATIVE PATTERSON SQUARING (FIPS)

METHOD - AN IMPROVED TREATMENT OF SEVERELY OVERLAPPING RE- FLECTIONS IN A POWDER PATTERN FOR THE APPLICATION OF DIRECT

METHODS. M. Estermann and V. Gramlich, Institute of Crystallography, ETH, CH8092 Zurich, Switzerland.

Single crystal techniques have been used to solve a number of structures ab initio from powder data (see, for example, refs. (1) and (2), and references therein). Obviously, these techniques require a single-crystal-like data set, which has to be extracted from the powder pattern. Extraction programs cope well with partial overlap of diffraction intensity in the powder pattern, but not with severe or exact

overlap. Usually the intensity ratio of these severely overlapping reflections is set to one (equipartitioning). Unfortunately, direct methods, which are so successful for routine single crystal structure determination, often fail when used with an extracted powder data set. The

limiting factors are the complexity of the crystal structure and the number of severely

or exactly overlapping reflections in the powder pattern.

In our laboratory, we have recently developed a new method, the Fast Iterative Patterson Squaring (FIPS) method (3,4), for unravelling the intensities of severely or

exactly overlapping reflections. Starting from an equipartitioned data set, a Patterson map is generated, each point in the map is squared, this new map is backtransformed

to obtain new Fourier coefficients, and these coefficients are then extrapolated to give a new set of |F (hkl)|^ and a new intensity distribution for the overlapping reflections (non-overlapping ones remain unchanged). The cycle is repeated until the intensity

statistics of the overlapping reflections approximate those of the non-overlapping ones. Tests with powder data obtained from materials with known crystal structures proved

to be successful. The molecular sieve SAPO-40 of unknown structure, which could only be synthesized in the form of a polycrystalline powder, provided a real test case for the FIPS

method (5). Approximately 65% of all the reflections were aff'ected by severe overlap, and the structure could not be solved from the equipartitioned data. However, af- ter the intensities had been redistributed using the FIPS method, the structure was

solved by direct methods.

(1) McCusker, L. (1988). / Appl. Cryst. 21, 305-310.

(2) Hiraguchi ei al. (1991). J. Appl. Cryst. 24, 286-292.

(3) Estermann, M. (1991). Thesis.

(4) Estermann, M. and Gramlich, V. (1992). in preparation.

(5) Estermann, M. and McCusker, L. (1992). in preparation.

P3.1 USE OF THE CSD PROGRAM PACKAGE FOR STRUCTURE DETERMINATION FROM POWDER DATA. L. G. Akselrud, P. Yu.

Zavalii, Yu. N. Grin, V. K. Pecharski. Institute of Inorganic Chemistry, Lviv

State University, 290005 Lviv, Ukraine & B. Bauragartner, E. Wolfel, STOE

& Cie, 6100 Darmstadt, Germany

The CSD program package has been developed to combine the possibilities

of single crystal and powder diffractometry for crystal structure research and

offers to the user tools for solution, refinement and interpretation of the crystal structure independent of the data collection method. It runs on a PC

XT/AT/PS2 or fully compatible computer systems under DOS3.3 and higher.

The power of this program system has been tested using the powder data of

various samples with already known structures. These crystal structures have

been investigated using STOE powder diffractometers for data collection and

CSD as the computational system, starting from reflection indexing and fol- lowed by structure solution and integrated-intensity or full-profile structure

refinement. Different kinds of data collection (transmission and Bragg-Brentano geometries) have been compared in respect to their advantages for struc- ture solution. The samples comprise both inorganic and organic compounds

with crystal systems ranging from triclinic to cubic, thus presenting usual experimental situations. These include fully overlapped peaks as a result of symmetry of the unit cell, partially overlapped peaks dependent on unit cell parameter relations, and different variation of peak-half- widths with theta for

different reflection zones.

The precision and accuracy of these results are discussed.

03.2 COMBINED RIETVELD AND SHORT RANGE ORDER

PARAMETER REFINEMENT. J. Schneider, Institut fur Kristallographie

und Mineralogie, Universitat MQnchen, Theresienstrasse 41, D-8000

Munchen 2 Germany.

Although the Rietveld method' has become a widely accepted technique in powder diffraction to refine structural parameters from Bragg peak profiles,

less attention has been given to refinement of concurrent diffuse disorder

gcattering. Substitutional disorder in binary alloys produces a modulation

of monotonic Laue scattering intensity if short range order is present. For

cubic polycrystalline alloys this may be written as a series expansion*:

IsRO = N x^xg (fA-ffl)' ZiCiOiSin(kri)/(kri) (1)

N is the number of lattice sites, x^ and Xg are the fractions of the A and B

components, respectively, f/^ and fg their formfactors (scattering lengths),

C; is the number of neighbors in shell i, r^ it's radius, k = 47tsin9/A. and O;

are the well known Warren-Cowley short rango order parameters. Eq. fl)

(i = 1..6) was incorporated into the local PC version-' of a standard Rietveld program*.

Application of the program to neutron data' of Na deficient Na^Tli.^, x =

0.47 showed that the observed diffuse intensity can well be described by eq.

(1). The major contribution arises from a term which yields r, = 3.227(9) A. This agrees well with the next neighbor distance d = v'S a^ =

3.2250(4) A derived from the Rietveld refinement. Assuming that the heavy

Tl atoms do not take part in substitutional disorder, this result suggests short range order between Na atoms (and vacancies) on the Na sublattice.

'Rietveld, H. M., J. Appl. Cryst. 2 (1969) 65. ^Warren, B. E., X-ray diffraction, Addison-Wesley 1969. ^Schneider J., Dinnebier R. E., Material Science Forum 79/82 (1991) 277.

^Sakthivel A., Young R. A., Int. Workshop on the Rietveld method, Petten 1989. ^Schneider J., Material Science Forum 27/28 (1988) 63.

P3.2 SOFTWARE PACKAGE PATCOM FOR SIMULATION AND

.MANIPULATION OF POWDER DIFFRACTION PATTERNS AND

STANDARDS. Ya. E. Cherner, Instinite of Physics, 194 Stachki Ave., Rostov-on-Don, 344104, Russia.

The PATCOM software package is intended for simulating, comparing, displaying, and printing powder diffraction patterns of single- or multiphase samples and solid solutions using crystal structure data for the substance or its isomorph. It consists of five programs.

SCREEN is a program for preparation of the input files. The SITES and HKL

programs calculate, by means of synmietry operations, the atomic position

multiplicities and coordinates of all atomic sites, as well as the Miller indices, dspacings, angular positions, and multiplicity factors of all lines within the desired angle range.

INTENS computes both the integrated and relative intensities of Bragg reflections.

It also calculates the unit cell weight, volume, density, and the mass absorption

factor. INTENS allows the creation of lattice vacancies, and can place up to 5

different atoms (ions) into a single atomic position.

PICTURE is an interactive color graphics program for:

- displaying or printing up to 5 theoretical or measured patterns, or powder diffraction standards simultaneously as a 2-, or 3-dimensional

picture Oine- and continuous diagrams are available);

- simulating and storing the patterns of multiphase substances;

- comparing patterns of different origin by means of superposition and/or subtraction.

Five profile functions are available for peak shape simulation and the profile parameters may be changed. STRART enables crystal structure to be displayed or

plotted.

Options include rotation of a structure model around any direction, shift of any

atoms and recalculation of die theoretical pattern for a modified atomic

arrangement, drawing of the unit cell projection on a desired plane, etc.
212 Accuracy in Powder Diffraction II

P3.3 SOFTWARE PACKAGE FOR POWDER DIFFRACTION

DATA PROCESSING. Ya.E. Chemer, Institute of Physics, 194 Stachki Ave., Rostov-on-Don, 344104, Russia.

The software package PD-ASSISTANT is intended for handling measured

powder diffraction patterns. It consists of four programs.

The PATTR program offers many facilities for displaying and treating

patterns. It also creates the input parameters files for the BCGR and PEFIT

programs.

The BCGR program for background intensity elimination is based on our own algorithm. The background is interpolated with a polynomial of degree up to 6. The effect of spread in the raw background data is eliminated by means of the use of an intensity-deviation band. PATTR has a routine for choosing a band width directly on a display. BCGR can create output files containing a list of Bragg peak regions, refining polynomial coefficients, running protocol, and other useful information. The program runs in both

autopilot and interactive mode. A user has the opportunity to mark on the

pattern some areas belonging to background or peaks and to exclude areas using PATTR. This improves the result when the measured pattern is not

sufficiently good.

The PEFIT program allows the separation of overlapped Bragg peaks and

the refinement of position, integral intensity, and FWHM of both single and

overlapped j)eaks. This routine employs the Powel method. The list of

profile functions available for description of the peak shape includes Gaussian, two Lorentzians, pseudo-Voigt and Pearson VTI. Each Bragg

reflection is approximated by the K^l-Kj2 doublet; the ratio of the

intensities of its components can be set by the user (the default is 2:1).

The COMP utility program allows the comparison of two measured

patterns, to shift them with respect to each other, and to display the difference diagram on a separate panel.

P3.5 A VERSATILE PROGRAM FOR RIETVELD REFINEMENT. J.F. Berar(-|-*) and P. Ganiier(-I-), (+) Chimie-Physique du SoUde

(UA 453 CNRS), Ecole Centrale, 92295 Chatenay, France and (*)LURE, Universite Paris-Sud, 91405 Orsay, France.

The XND program for crystal structure refinement of powder data was writ- ten in order to take into account the correlations between data obtained in real-time experiments for which the statistical quality of each diagram is very

low.

Line profiles are represented by Voigt functions, new terms have been added

to describe the evolution of line width and asymmetry. Preferred orientation

effects are implemented using spherical harmonic functions. These functions

are also used to allow an anisotropic broadening due to size or strain effect for the sample dependent part of the line width.

This allows us to reach very good profile agreements even if data have been

collected with a conventional X-ray diifractometer ( = 0.035, GoF = 1.2 have been obtained with various MVjOs samples).

Some other possibilities (anomalous scattering, bond length penalties and rigid bodies) allow its use for complex structural problem : for example zeolites or stearate salts.

P3.4 PDN-PROGRAM FOR RIETVELD ANALYSIS, A. B. Tovbis and Ya. E. Chemer*, Institute of Crystallography, 59 Leninsky

Ave., Moscow, 117333,* Institute of Physics, 194Stachki Ave., Rostov-on- Don, 344104, Russia.

PDR IS a program for Rietveld analysis of X-ray and neutron powder

diffraction patterns. It runs on IBM PCs and compatibles (540 kB memory

required). The raw pattern file can contain up to 10,000 data Points, the allowed number of Bragg reflections is less than 4, (XX), and up to 10 phases

can be refined simultaneously. The list of profile-shape functions includes Gaussian, three Lorentzians, pseudo-Voigt, Pearson vn, and Thomson-CoxHastings pseudo-Voigt. PDR takes automatic account of symmetry

dependent atomic coordinates, anisotropic temperature factors and unit-cell parameters. The program has an extremely easy and flexible way of

creating and editing input parameter flies in an interactive manner and a

routine for set-up of the input files using a data entry screen (for beginners).

A color graphics utility enables the display and plotting of the raw and

refined patterns, the difference diagram and a structure model.

The PDR's runtime for a test data set is half the corresponding runtime of

the well known program DBW 4.1 (890101).

The use of PDR together with PD-ASSISTANT and PATCOM software packages provides the best starting model and an optimal strategy for the Rietveld refinement.

P3.6 A RIETVELD PC-PROGRAM SYSTEM ADAPTED FOR

THE GUINIER METHOD. N. O. Ersson, Solid State Chemistry, Uppsala

University, Box 531, S-751 21 Uppsala, Sweden

The incomparable sharpness of the diffraction lines obtained with a Guinier

film camera may give problems when the observed data, as obtained with

a film scanner,' are used in a Rietveld program. The problems in finding a profile ftinction describing the complex angle dependence are well known.^ Another complication is the deviation from linear relationship

between film distance and Bragg angle that is always present on a film. Even small distortions may cause an incomplete overlap when observed and

calculated data are compared.

The use of an internal calibration substance can eliminate the latter complication. We use high-purity Si, the NIST standard SRM 640b. In the presented system an angular correction function is first calculated. This

function is then used to "rubber-band stretch" the theta-scale, so that the distance between two observed points is no longer constant. In practice this :

is obtained by interpolating the observed intensities and maintaining a

| constant step-length for the input data to the Rietveld refinement.

The present Rietveld program is a modified version of DBWS-9(X)6PC ! written by Young and Satthivel.'' Besides some changes in data input (to ! provide for different formats) functions for Lorentz factors for the Guinier

geometry have been incorporated. A display program for easy graphics

control of the output has also been written using the QuickBasic language,

as for the correction programs. The Rietveld program is still written

entirely in FORTRAN-77 language.

|

'Johansson, K.-E., Palm, T., and Werner, P. E., Rev. Sci. Instrum. 13, 1289-1291 (1980).

[ -Ersson, N.O., J. Appl.Crystallogr. 12, 295-302 (1979). I

^Sakthivel, A., Young, R. A., User's Guide to program DBWS-9006PC (1991). ,
Abstracts, Session 3: New Developments in Software and Data Analysis 213

P3.7 MACINTOSH POWDER DIFFRACTION SOFTWARE.

A. W. Hewat, Institut Laue-Langevin, 156X Grenoble, 38042 FRANCE.

Personal computers now rival workstations for processing speed, and have the advantage of a common, user-friendly windowing interface. For example, the new Macintosh Quadra machine, using the 20.MIPS 68040 processor, runs powder profile refinement programmes at typically 10-20 seconds per refinement

cycle, rivalling the ILL-Vax central computer !

Standard mainframe Fortran, Pascal and C programmes can be compiled and

run as easily as on any other computer, since there are no limitations to the

size of the code, and a simple windowing environment is added automatically.

With a little work, more sophisticated windowing environments can be in- cluded, greatly simplifying the input of data and the examination of results. An example of this second class of programme is Mac-Lazy for the calculation

of powder patterns. Much of the data can be entered simply by pointing and

clicking, and the results plotted in independent graphic windows. The struc- ture can be drawn in another window, rotated, scaled and 'animated' in read time to emphasise its 3D nature.

Personal computers can be justified if only for 'word-processing', but can in practice be used for much more. Macintosh versions of some of the 'classical' powder diffraction software will be shown, together with applications using more sophisticated user interfaces, such as Mac-Lazy.

P3.8 COMPUTER PROGRAM "CUVFIT" FOR DECOMPOSITION OF MULTIPHASE POWDER DIFFRACTION PATTERNS. S. L. Wang, Department of Chemistry, National Tsing Hua University, Hsinchu,

Taiwan 30043, R. 0. C.

The program CUVFIT has been developed for analyzing powder diffraction patterns which may contain several phases. Determination of the intensities of completely overlapping reflections is feasible by using this program. Up to ten peaks can be fitted and refined simultaneously, including phases that can be

stipulated to have only certain reflections. The program handles various peak

shapes such as Gaussian, Lorentzian, pseudo-Voigt, and Pearson VII. Angular

dependence of the peak's FWHM for each phase can be best-determined by

the program and used in the next fitting cycle. Either shifted Tchebychev

polynomials (degree 2) or the usual tangent function can be chosen for de- scribing the behavior of the width. Other parameters include peak position,

asymmetry factor, and polynomial background. The changes, such as the zero

offset of the 2-B value, and constraints between various parameters, can be made easily. The program is essentially an individual profile-fitting one which

aims at apportioning observed intensities to the individual components analytically. Details of the program and its applications to several complex systems

will be presented.

P3.9 EXTRACTING STOUCTURE FACTORS FROM POWDER

DIFFRACTION DATA BY ITERATING FULL PATTERN PROFILE FITTING. A, Le Bail, Laboratoire des Fluorures, Unlverslte du Maine. 72017 Le Mans Cedex,

France.

New strategies using old formulae sometimes lead to efficient algoritlims. Ever/ user of the Rietveld method has worked with the so-called 'lobs' (or

'I Fobs I') either by the simple presentation of the Bragg R factor or by

Fourier difference calculation or so on. Some special uses of the Rietveld's formula for ' I Fobs I ' extraction were made (to be classified among the twostages alternatives to the Rietveld method), however the starting 1 F i s were

always the iFcalcIs corresponding to a (complete or partial) structure model CToroyo. Marumo and Yamase. Acta Cryst.. 1984, B40. 1^150 ; Taylor, Miller and Bibby, Z. Kristallogr, 1986. 176, 183-192; Taylor, Z. Kristallogr., 1987, 181, 151-160).

In the original Rietveld's work, the "I Fobs I' are estimated by partitioning

among reflections contributing to each profile point of the pattern

according to the I Fcalc I , thus they are biased. However, it will be shown

that iterating the Rietveld's formula provides a nice and simple v/ay to

extract | Fobs I as accurately as possible, starting from initial |f|s aitjitrarily set to have the same value. The first study applying such an algorithm for structure determination purpose (using the unpublished program ARITB) was

from Le Bail, Duroy and Fourquet, Mater. Res. Bull., 1988, 23,447-452. Near 20 ab initio structure determinations have been now performed in this way,

either from conventional or synchrotron X-ray powder data. The algorithm has been implemented as an option in two other programs (FULLPROF.

GSAS); any Rietveld program can be easily modified to accommodate this possibility to extract structure factors. The strategy necessary for success will be exposed. The best results are from cell and space group constrained

conditions: only all equal starting | F | s can ensure that the structure factors

of strictly overlapping reflections are undifferentiated: they keep the same

value after each iteration; tt is strongly recommended to start from very

precise cell parameters (not approximated ones) In order to avoid false minima in case of complicated patterns with strong overlapping: some

iterations are necessary without modifying cell and profile parameters in order to start their least-squares refinement at further cycles in good

conditions. A definite advantage of the procedure is the very low number

of parameters to be refined (15 maximum in ARITB) allowing It to handle problems of any size. Extracting in one run the several thousand iF|s (for' instance from synchrotron powder data with 0.02'2e-FWHM) implicit for large and/or low symmetricol cells is not a Utopia.

P3.10 MAXIMUM ENTROPY METHOD ANALYSIS OF X-RAY

AND NEUTRON POWDER DIFFRACTION DATA. M. Sakata, T. Uno, M.

Takata, Department of Applied Physics, Nagoya University, Nagoya, Japan; and C. J. Howard, Australian Nuclear Science & Technology Organisation, Lucas Heights, NSW 2234, Australia

It is often said that X-ray and neutron diffraction are complementary. In Xray powder diffraction. X-ray photons are scattered by electrons, while neutron beams are diffracted by nuclei in neutron powder diffraction, assuming that

there are no magnetic interactions. What can be observed from X-ray and

neutron powder diffraction data is, therefore, the electron and nuclear density

distribution of the crystalline materials, respectively. It is, however, not known

how to restore such electron and/or nuclear density distributions directly from

the observed structure factors by powder diff'raction without using a structural model.

Recently Sakata & Sato [M. Sakata & M. Sato (1990) Acta Cryst. A46, 263] have successfully applied the Maximum Entropy Method (MEM) to restore

the precise electron density distribution from structure factors accurately de- termined by the PendeDosung method. It is possible to measure accurate Bragg intensities free from extinction effects by a powder diffraction experi- ment. In this study, the MEM analysis of both X-ray and neutron powder

diffraction data wiU be given. In the neutron diffraction case, it was necessary

to overcome the difficulty of negative scattering length of some atoms, such as H, Ti, Mn. Both the electron and nuclear density distributions of rutile were

obtained from X-ray and neutron diffraction data. In the electron density

distribution, apical and equatorial bonds of rutile are clearly shown, while it

is shown in the nuclear density distribution that the nuclei of Ti and 0 are

located within very confined regions around atomic sites.
214 Accuracy in Powder Diffraction II

P3.ll A BAYESIAN APPROACH TO BRAGG PEAK LOCATION

IN POWDER DIFFRACTION PATTERNS. W.I.F.David, R.M.Ibberson and

D.S.Sivia, ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot,

Oxon., 0X11 OQX, U.K.

This paper presents an ab-initio Bayesian approach to Bragg peak location in powder diffraction patterns. It addresses the question: "Given that a region

of a powder diffraction pattern is presumed to contain a few Bragg peaks and

that the instrumental resolution function is known, what is the most probable

number of peaks and what are their positions and intensities?". One of the major advantages of this Bayesian approach is that the relative probabilities of

the number of possible peaks is quantified. Two applications are illustrated:

(i) the location of Bragg peak positions in an unknown powder diffraction

pattern prior to auto-indexing and (ii) the unit cell determination of pseudosymmetrical structures.

P3 . 1 3 THE EFFECT OF THE THIRD ANGULAR PARAMETER IN TWOCIRCLE DIFFRACTOMETRY. Cong Qiuzi, Laboratory of Solid Lubrication, Lanzhou Institute of Chemical Physics, Chinese Academy of Sciences, Lanzhou

730000, People's Rep. China

On the basis of the e-and 2e-angies in a two-circle difftactometer, a third a-angle between the sample surface and the incident X-ray beam is introduced, so that three X-ray scan modes of 9/ 29, 26 and 9 are normalized into a common form of 9(«)/

26. Subsequently, based on the asymmetrical Bragg reflection geometry, equations of the diffraction intensities and the observed crystal-plane azimuthal angles are derived as the following common forms;

I = I(«,26,t) -P G) and, = 6^1,1 -a (2) where t is the film thickness or effective X-ray penetration depth, P is the

reflection profile funaion, and e^^j is the Bragg angle, for the given reflection.

Changing one or two parameters in eq. 1 we can obtain three groups of equations

corresponding to three scan modes respectively.

a. For 6(9) / 26 (called CBD) scan mode,

Ic = Ic(9,29,t) -Pc (3) and, (9^^1)0 = 0° (4)

b. For 9(Oo) /29 (called STD) scan mode.

Is = Is(a„,29,t) .Ps (5) and, (*hki)s = ^m-'o (6) where is an angular parameter in the range from 0.6° to 9^^, for Cu K,

radiation.

c. For 9(9)/ 29^1^1 (used for ADA) scan mode, Ta = Ta(9,29hi,,t) .Pa (7) and, (*hki)a = 9y,i-9 (8) where 9 is the rotating angle of die specimen, and 29^^! is the diffraction angle for the given (hkl) reflection.

The effect of each scan mode in X-ray diffractometry is well known, for example,

the "a" mode is most used to determine phases, as is the "b" mode. Furthermore,

the effective X-ray penetration dep± can be controlled with "b" mode by adjusting

the sample-tilting a^-angle; at the same time, the film-layer or film thickness can be fitted with a combination of "c" and "b" modes.

ro.i^ TRE0R90: AN IMPROVED VERSION OF THE TRIALAND-ERROR INDEXING PROGRAM TREOR. L. Eriksson, P.-E. Werner, and M. Westdahl, Department of Structural Chemistry, Arrhenius

Laboratory, University of Stockholm, S-106 91 Stockholm, Sweden.

Although high quality data is the most important prerequisite for powder

indexing, the rate of success of a semi-exhaustive program for powder

indexing is most strongly dependent on the built-in crystallographic

experience. Accumulated experience from the use of the trial-and-error indexing program TREOR'-" has been incorporated in the new version,

TRE0R90. Dominant zone tests, originally used only for the monoclinic

symmetry have been added for the orthorhombic and triclinic symmetries.

Dominant zone tests are made prior to the conventional trials, thus preventing the most frequently occurring situation when figure of merit tests

in powder indexing may faU.

Other new features included in the program are: successive reduction of

trial cell volumes by statistical criteria, automatic unit cell reduction and

conversion to conventional cells according to the metric symmetry, a condensed output file and an increased over-all efficiency of the indexing

algorithms.

The original key-word system is retained, but at the expense of a relatively small increase in computing time the user may normally accept a default

setting of parameters to run all symmetries from cubic to triclinic.

Computing times by the vectorized TRE0R90 for triclinic and monoclinic

patterns have never exceeded 5 and 2 minutes, respectively, on a CONVEX

210, which is about 5% of the time required on a Micro-VAX U.

'P.-E. Werner, Z. Kristallogr., 120, 375-387, 1964. 2p.-E. Werner, L. Eriksson and M. Westdahl.J. Appl. Crystallogr. 18, 367-370, 1985.

Pc5.14 rifraN "92: THE IBM PC SYSTEM FOR AUTOMATIC

X-RAY POWDER DIFFRACTION PHASE ANALYSIS. T. Havlik, M.

Skrobian, and F. Petricko, Technical University of Kosice, Czechoslovakia.

An efficient identification system RIFRAN '92 was developed for rapid

qualitative and quantitative X-ray powder diffraction phase analysis of

polycrystalline materials. The system is represented by a set of interactive computer programs written in the Boriand Turbo C++ and Borland Turbo

Pascal 6.0 computer languages and implemented on the IBM PC computer

working under MS DOS.

The identification procedure for the analysis is based on comparison of the

diffraction patterns of standards with those of the unknown sample,

utilizing statistical criteria and chemical analysis for qualitative analysis,

and the method of mathematical optimization for quantitative analysis.

The JCPDS database is used for qualitative analysis, as well as the

specialized database for quantitative analysis in which the standards were

calculated previously'.

The system enables the specialized database preparation according to the

requirements of the user, using chemical composition, mineralogical

name, subfiles, PDF numbers, etc. Thus, the analysis is more effective

and faster. Specification of the chemical criteria with the aid of Boolean

operands is used to further enhance the procedure.

The analysis takes only a few seconds in the average case. The results are

sorted according to statistical criteria, or chemical analysis. Output of results

is printed on the screen, as well as to the printer or a file. Both numerical

and graphical output of the results are possible.

'Smrcok L., Weiss Z.: J. Appl. Cryst.-in press.
Abstracts, Session 4: Profile Fitting, Decomposition and Microstructural Effects 215

P4.1 NONDESTRUCTIVE RESIDUAL STRESS

MAPPING BY NEUTRON AND X-RAY DIFFRACTION

METHODS. C.R. Hubbard*, T.A. Dodson, S.A. David, Metals and Ceramics Division, and S.Spooner, Solid State Division, Oak

Ridge National Laboratory, Oak Ridge IN 37831-6064, Phone

(615) 574-4472.

Advanced X-ray and neutron diffraction facilities to map macro

and micro residual stresses have been established at Oak Ridge

National Laboratory. For mapping stresses within the specimen

by neutron diffraction, attachments for a triple axis spectrometer

at HFIR have been built and automated which include beam

collimators, an XYZ-chi specimen stage, and a linear position

sensitive detector. Sampling gauge volumes of 20 to as little as

2 mm3 have been successfully used in mapping stresses in

several specimens including a ferritic steel plate with a multipass weld, weld simulation test samples, and a brazed zirconia to iron

joint. Results of mapping macro residual stresses in these specimens show significant stress gradients. Micro residual stresses in ceramic and metal matrix composites have been

studied with both high resolution neutron powder diffraction and

the triple axis spectrometer attachment. Comparison of the

sensitivity and time required for measurement of microstresses

will be presented. For mapping stresses at the surface a new Xray mapping facility was developed which consists of an 18 KW

rotating anode generator and a Scintag PIS goniometer.

Capabilities of this system will be presented, and data collected by both neutron and x-ray method will be compared.

Research soonsored by the Lajoralory Directed Research and Develoaiient Program of Oak

Ridge Natior^ Laboratory anc the Offtce of Transportation Technoiog es. Conservation and

Renewaole Energy, Department of Energy, under contract DE-A05-8^OR21400 wth Martin

Manena Energy Systems. Inc..

P4.2 ABOUT ANALYTICAL MODELS FOR TEXTURE CORRECTION. M. Jarvinen, Department of Information Tecnology, Lappeenranta

University of Technology, SF-53851 Lappeenranta, Finland.

Powder diffraction is commonly used for obtaining information on the properties of materials. Accurate intensity measurements are needed, for instance,

for the determination of the volume fractions of phases in a sample or for refinement of the structures of different materials.

The accuracy of results depends significantly on the treatment of sample effects. Loose samples can suffer from inhomogeneity and dense samples can be bothered by preferred orientation or texture. In practice, a sample with random orientation is very difficult to prepare, so that usually dense samples

are used in measurements, and the texture effects are corrected mathematically. For this procedure analytical models are needed for presentation of the orientation distribution in the sample. j

Very elegant theoretical and experimental methods have been created for de- termination of the texture factors of a sample, but often these methods are too complicated and superfluous for practical use. That is why simplified methods

have been developed for some frequently occuring cases, for instance, a Gaussian function or March function is used for the description of fiber texture.

In the present paper the use of Gaussian and March functions are compared

with the use of symmetrized harmonics by making model calculations and

analyzing Ni and Mg powder data. The accuracy of these analysis methods

are evaluated, and the advantages and the limitations of different models are

discussed.

P4.3 SAMPLE-INDUCED ERRORS IN QUANTITATIVE TEXTURE

ANALYSIS FROM DIFFRACTION POLE FIGURES. A. Miicklich and P. Klimanek,

Institute of Ion Beam Physii:s and Materials Research, Research Centre Rossendorf, DO-8051 Dresden, Germany, and Institute of Metal Science, Mining Academy Freiberg, DO-9201 Freiberg/Sa., Germany

During the last decade many efforts have been mtide to improve the reproduction of

orientation distribution functions (Odf) of polycrystalline materials from X-ray and

neutron diffraction pole figures. The advances achieved in this field are in contrast with a relatively unsatisfactory level of data evaluation due to the neglect of a number

of systematic errors, which can significantly influence the reliability of quantitative

texture analysis. In this connection two types of uncertainties have to be taken into

account:

• experimental errors which, however, can be kept sufficiently small by suitable

choice of the conditions of investigation, and

• sample-induced errors caused by the microscopical architecture (e.g. lattice perfection of the crystalUtes, grain size and orientation distribution) of the material taken under investigation.

Errors of the second type are characterized by the fact that they modify the actual absorption and/or scattering behaviour of the sample material. This is particularly important in investigations of structurally inhomogeneous polycrystalUne materials

[P. Klimanek in: X-Ray and Neutron Structure Analysis in Materials Science (Ed. J. Hasek).Plenum Press, New York 1989, 125; EPDIC 1 - European Powder Diffraction (Ed. R. Delhez, E.J. Mittemeijer), Mater. Sci. Forum, Vol. 79-82, Pt.l, 73. Trans

Tech Publ., Aedermannsdorf/Switzerland 1991]. After a brief discussion of general

aspects of pole figure evaluation, the present paper considers three effects that give

rise to errors of the second type in the case of neutron diffraction:

1. absorption weighting of diffraction pole figures from polycrystalUne layer structures with texture gradients or different texture components at various distances from the sample surface,

2. structure-induced extinction in materials whose texture components clearly differ in grain size and/or lattice perfection of the crystallites [A. Miicklich, P. Klimanek: Mater. Sci. Forum Vols. 79-82 (1991) 185] and

3. texture-induced secondary extinction in polycrystals with strong preferred ori- entation [A. Miicklich, P. Klimanek in: Cryst. Res. Technol. 23 (1988) K105;

Mater. Sci. Forum Vols. 70-82 (1991) 185].

The practical importance of the effects is illustrated by computer simulation of neutron diffraction pole figures as well as by experimental results obtained with copper and iron.

P4.4 CHARACTERIZATION OF PREFERRED ORIENTATION IN CRYSTALLINE MATERIALS BY X-RAY POWDER DIFFRACTION

WITH THE MARCH FORMULA APPRMSAL OF THE LINE RATIO

METHOD. H. Sitepu, Centre of Basic Sciences, Indonesian Institute of Technology, Serpong - Tangerang, Indonesia. 15320; B. H. O'Connor and Li Deyu, Department of Applied Physics/Centre for Materials Technology Curtin

University of Technology, Perth, Australia.

Texture, i.e. preferred orientation, can cause large systematic errors in quantitative

analysis of crystalline materials using x-ray powder diffraction (XRPD) data. Various mathematical forms have been proposed for the application of preferred

orientation corrections. The most promising of these appears to be the single- parameter March (1932) model proposed by DoUase (1986).

Li et al. (1990) and O'Connor et al. (1991) applied the March model to determine

the level of preferred orientation in various gibbsites using two procedures. The

first involved the Rietveld (1969) least squares pattern fitting method. Each pattern was Rietveld-analyzed in two ways, initially assuming random orientation of the

crystallites and subsequently with the March model. The second procedure for preferred orientation analysis, described here as the line ratio method, determines

preferred orientation factors according to the intensity ratios of carefully selected

line pairs.

In this study the procedures proposed by Li et al. and O'Connor et al. for texture

analysis have been evaluated with XRPD data sets for molybdite, calcite and

kaolinite. The results indicate that, while the March formula improves agreement between the calculated and measured patterns in Rietveld analysis, other forms of

systematic error in the intensity data appear to limit the effectiveness of the March

formula in general. It has been found also that the line ratio method improves

agreement between the data sets, but less effectively than the Rietveld method. It

is proposed that extinction is likely to be the most influential source of systematic

error competing with texture.

Dollase, W. A., 1986, J. Appl. Crvst. . 19:267.

Li, D. Y., O'Connor, B. H., Roach, G.I.D and Cornell, J. B., 1990, Powd. Diff. . 5:79. March, A., 1932, Zeith. fur Kristallo^r. . 81:285. O'Connor, B. H., Li, D. Y and Sitepu, H., 1991, Adv. in X-rav Analv. . 34:409.

Rietveld, H. M., 1969, J. Appl. Crvst. . 2:151.
216

P4.5 MICROSTRUCTURAL EFFECT ON THE CORROSION

OF TUNGSTEN IN SF^. W. Wong-Ng, L. P. Cook, and C. S. Choi,

Materials Science and Engineering Laboratory, National Institute of

Standards and Technology, Gaithersburg, Maryland 20899, U.S.A.

Stored chemical energy release systems based on liquid metal combustion

in general require the injection of a gaseous oxidant through a nozzle of

chemically resistant material. Tungsten (W) has been the traditional metal of choice. Understanding of the kinetics of reaction of W with the oxidant SFg as a function of temperature is of primary importance for predicting service lifetime. Scanning electron microscopy study of tungsten rods manufactured by a swaging process and exposed to SF^ indicated a signifi- cant orientational effect on the corrosion process, possibly originating from

the considerable granular elongation parallel to the rod axis produced during

the fabrication process. X-ray analysis of these rods showed preferred

orientation. Texture measurement was subsequently conducted using neutron diffraction. This paper discusses the result of the pole figure study

and the correlation of the relative rate of reaction with the grain alignment.

P4.6 HTTING X-RAY PROFILES IN TERMS OF PHYSICALLY

RECOGNIZABLE PARAMETERS. R.W. Chean' and A. Coelho. Depanmenl of Applied Physics, Universily of Technology Sydney. PO Box 123, Broadway, NSW.

Australia 2007.

X-ray powder line profiles are generated by convoluiing the insirumenlal aberration functions of a Bragg-Brentano diffractomeier with the Cu Ka emission spectrum and the

diffraction function arising from the presence of either small crystallites or strain in the specimen. These profiles are filled to experimental profiles using a combination of

multilinear regression and Newton-Raphson non-linear least squares fitting based on

numerical differentials. In addition to the 26 values and integrated intensities of the lines, the fitted instrumental parameters include the receiving slit width, the length of the receiving slit, the x-ray target width, the divergence angle of the incident beam and a "top

hat" mis-alignment function which can be used to represent defocussing effects or specimen tilt. The wavelengths and lifetime widihs of the five components of the emission spectrum are fixed by Uie spectrometric measurements of Berger(l). The relative

intensities of these components are calibrated with an Si crystal or a standard 2nm

polycrysialline MgO disc to allow for differential attenuation by either a diffracted beam or incident beam monochromalor. Crystallite size effects are assumed to produce a Lorentzian profile whilst strain produces a Gaussian shape; the refined parameters are the apparent crystallite size and the RMS strain. In practice each instrumental parameter is constrained to be the same for all profiles from a panicular specimen. As some of the parameters are either highly correlated with one another or do not contribute significantly

to the profile shape, it is only possible to refine on two or three instrumental parameters even when diffraction data over a wide range of 29 values are being fitted. Refining on all

the instrumental parameters simultaneously normally produces results with little physical

validity. The successful application of this convolution fitting procedure to a standard powder diffractomeier has already been discussed(2). In this work we will demonstrate how

the fitting procedure can be used for slandaidless line profile analysis and the measurement

of crystallite size and stacking fault defects. We will also examine the consistency of the

fitted instrumental parameters when applied to two additional powder diffractomeier

configurations. In the first case the effects of incorporating a high resolution Ge

monochromalor (CuKojonly) in the incident beam are considered. In the second system,

the effects of replacing the receiving slit/detector with a position sensitive detector are considered and how the broadening of the line profile under non-focussing conditions is incorporated into the filling procedure.

(1) Berger, H. (1986) X-ray Spectrom. 15,241.

(2) Cheary. R.W. & Coelho, A, (1992) J. Appl. Cr>st. 25, In Press

Accuracy in Powder Diffraction 11

P4.7 INSTRUMENTAL BROADENING DETERMINATION FOR

XRD PROFILE ANALYSIS. P. Scaidi & L. Lutterotti, Dipartimento di In- gegneria dei Materiali, Universita di Trento, 38050 Mesiano (TN), Italy; Paul

Maistrelli, Dipartimento di Fisica, Universita di Trento, 38050 Povo (TN),

Italy.

A standard sample for XRD instrumental profile determination was prepared

by pressing -32+45fim KCl powder to obtciin a tablet. After a suitable annealing treatment, the tablet gave very sharp profiles (FWHM=0.07° and 0.04°

at 20 = 28.4° for the normal and high resolution set-up, respectively). The

experimental patterns were modelled by non-linear least squares fitting using

pseudo-Voigt (pV) or Pearson VII (PVII) functions, corrected for asymmetry,

and the trend of fitting parameters describing the curves (a> and t] for pV, lj and m for PVII) as a function o{26 was measured. These trends, which can be

described by fitting data with a low degree polynomial, can be used for a fast and reliable 26 dependent instrumental correction in line broadening analysis, both using raw experimental data and modelled profiles. The features of the experimentally determined instrumental function were

studied using several diffractometers, always in Bragg-Brentano geometrj', with different set-ups. In particuleir, the effect of using a monochromator

in the diffracted beam and changing the slits width was studied. It was found

that the goniometer radius, beside having a remarkable influence on the resolu- tion (peak width) of the diffractometer, also affects low angle peak asymmetry.

High resolution geometry (narrow slits and large radius) is highly advisable

if sharp and fully symmetric profiles are needed, especially at low angles. In

spite of the loss in intensity, involving an increzise in the data collection time

necessary to obtain a given precision, this kind of set up can greatly enhance

the limit for crystallite size and microstrain determination through line broad- ening analysis. Also, structure refinements from X-ray powder diffraction data

give better results when the instrumental peak is sharp and symmetric.

I

P4.8 A PREDICTABLE DIFFRACTION PATTERN - METHOD

AND SUBSTANCE. V. A. Kogan, Physical Dept. of Rostov State University,

Zorge St. 5, Rostov-on-Don, 344104, Russia

The purpose of this work is to study the possibility that X-ray powder diffraction profiles can be computed independently. The problem may be divided

into two parts:

i) development of the method of computation

ii) synthesis of a stable substance in which the factors of physical broadening

are small and may be described on an unempirical scale.

The Fourier-synthesis method [Kogan & Kupriyanov, 1992] implemented in

the computer program for fvill profile refinement, KET, was used to compute

the reflection profiles theoretically. This programme makes it possible to synthesize powder patterns for Bragg-Brentano geometry by taking into account

the dimensions of the geometrical system, the spectral distribution, porosity

effects and a model for imperfections in the sample. The bum fraction of stron- tium titanate was used as a sample. The crystallites obtained were of cubic

j habit and were not subjected to any external influences after synthesizing. It was assumed that the coherent scattering regions coincide with the geometrical

crystallite sizes, and that micro-deformations of the crystal lattice are absent. During the experiments the diffraction profiles were recorded over a wide angular range for two different geometrical conditions. Full-profile refinement without introducing varying 'profile' parameters (Rprof=.05-.07) provided ad- equate values of the structure parameters in different experiments.

References

V. A. Kogan k M. F. Kupriyanov, (1992), J. Appl. Cryst. 25, 16-25.
Abstracts, Session 4: Profile Fitting, Decomposition and Microstructural Effects 217

P4.9 X-RAY TRACING, A TOOL FOR IMPROVED ACCURACY IN POWDER DIFFRACTOMETRY J. Timmers (Philips Research Laboratories, P.O. Box 80.000, NL-O600 JA Eindhoven, The Netherlands), R. Delhez, F. Tuinstra (Delft University of Technology, The Netherlands), F. Peerdeman (Philips IE Analytical, Almelo, The Netherlands)

The dimensions of X-ray optical components as source, specimen and slits have a

significant influence on the positions and shapes of powder diffraction lines and es- tablish a well-known source of error (the so-called geometrical aberrations) in powder

diffraction analysis. With the advancing mechanical goniometer accuracy in mind,

corrections for geometrical aberrations, in order to arrive at improved accuracy of peak positions and improved interpretation of line shapes, become increasingly rele- vant. The majority of geometrical aberration sources can be quantified analytically (see

e.g. A.J.C. Wilson, Mathematical Theory of X-ray Powder Diffractometry, Eind- hoven Centrex 1963) in terms of centroid shift and variance of the geometriczil part of

the instrumental line profile. However, for the full geometrical line profile neither the

shift of the peak maximum nor the shape can be obtained analytically. Therefore, a versatile computer program has been developed in order to simulate X-ray powder

line profiles from the known di- mensions of the diffrac- tometer and its optical components. Diffraction geometries to be simulated include the BraggBrentano, Seemann-Bohlin and

Schulz (texture) geometries as

well as all non-focusing geometries. As an example, peak shifts have been calculated for two Philips diffractometer systems: the In- tegrated Powder Diffractometer (PW1800) and the new X'Pert

diffractometer. Simulated and measured peak shifts for the NBS silicon powder standard Figure 1. Numerically calculated peak (-0-) and (SRM640b) are compared in fig- centroid (- A-) shifts as well as measured peak

ure 1. A clear discrepancy exists shifts (-!-) as a function of Bragg angle for the between peak and centroid shifts NBS sihcon standard SRM640b. All data refer to

in the low-angle region. the Philips Integrated Powder Diffractometer.

P4.ll X-RAY PROFILE ANALYSIS IN STRUCTURALLY INHOMOGENEOUS POLYCRYSTALS. P. KUmanek, Department of Materials Sci- ence, Academy of Mining, P.O.B. 47, DO - 9200 Freiberg/Sa., Germany

X-ray diffraction peaks of real polycrystals as, for instance, bulk materials or

real powders interesting in materials science and technology have, in general, to be considered as diffraction multiplets (i.e. as weighted sums of partial

reflections) due to the volume fractions of lattice disorder-statistics that are formed by crystallite groups with significantly different degrees of structural

perfection. Accordingly the macroscopicaUy observable line broadening of such

reflections must be interpreted in terms of a generalized sample statistics rep- resenting the microscopical architecture of the scattering crystallite ensemble,

which encloses

1. the type of the lattice-disorder statistics, and

2. the mean grain size, the orientation distribution (texture), and the spatial arrangement of the structurally different crystallite groups.

After a brief representation of the formalism needed in this case (Klimaiiek,P.

in: Freiberger Forschungsheft B 265 (1988) 76; X-Ray and Neutron Structure

Analysis in Materials Science (Ed. J.Hasek). Plenum Press, New York 1989,

125; EPDIC 1 - European Powder Diffraction (Ed. R.Delhez, E.J.Mittemeijer), Mater. Sci. Forum, Vol. 79-82, Pt.l, 73. Trans Tech Publ., Aedermannsdorf/Switzerland 1991) some important methodological aspects of the analysis of X-ray multiplet profiles are discussed and illustrated by experimental results obtained with model specimens (powders) and real polycrystalline materials.

^^^^ ACCURACY IN CRYSTALLINITY DETERMINATION BY

MEANS OF X-RAY POWER DIFFRACTOMETRY. G. Berti, Dipartimento Di Scienze Leila Terra - Univ di Pisa -Via S. Maria 53 - 56100 Pisa - Italy

According to the symmetry rules of the crystalline system used to describe

the crystallographic features of the sample, the crystallinity may be

considered as a natural performance of the ideal crystalline state. The

crystallinity is a feature pertaining to the sample, not to the crystalline

system. The achievement of a high level performance is ruled by the

crystallization conditions which act on the crystal evolution and produce

mosaicity and also other observable results. Unfortunately, the effects

related to the compositional variation, phase transformation, and planar

defectivity are not directly visible on the diffraction pattern. They have to be deduced by reconstructing both the reciprocal average lattice and the shape of the nodes. The starting point of the reconstruction is the profile

fitting of the experimental pattern by means of suitable analytical functions. The accurate determinations of positions and shapes of the peaks in the powder pattern are useful to find the proper lattices and shapes of nodes. So the crystallinity determination requires a phenomenological description as accurate as possible where, besides the many instrumental aberrations,

crystallinity and structure factors are taken into account. To this aim the mathematical theory of X-ray powder diffraction suggests simple and

effective relationships.

Some samples of quartz show that the crystallinity and the crystallite volume

are closely related when an equivalent average lattice is used to describe the assemblage of defective and/or disordered cells.

x-^.x^ P4 12 A STUDY OF CRYSTALLITE SIZE AND STACKING FAULTS IN EX-OXALATE ZnO BY PATTERN DECOMPOSITION. D. Louer*. A. Boultif*, J. P. Auffr^dic* and J. I. Langfordt Laboratoire de Cristallochimie, University de Rennes I. Avenue du Gdneral Leclerc, 35042 Rennes cedex, France, and tSchool of Physics ans Space Research, University of Birmingham, Birmingham B15 2TT, U.K.

Previous X-ray diffraction studies of ex-hydroxide mtrate powder ZnO demonstrated that the form of the crystallites was prismatic, on average approximating to cylinders (Louer

et al., J. Appl. Cryst. 16, 183-191, 1983). Recently, data were obtained for the first 27

lines of the diffraction pattern for ZnO obtained from thermal decomposition of the oxalate, by using the same sample as had been previously investigated by means of electron micro- scopy and adsorption calorimetry (Bolis et al., J. Chem. Soc. Faraday Trans. 85, 855-867,

1989). Integral breadths obtained by total pattem fitting again have a marked hkl dependence; an indexed Williamson-Hall plot indicated that microstrains are negligible and

that there are three groups of reflections, according to the values of hkl. This is a classic example of the effects of stacking faults combined with broadening due to crystallite size and it is the first time that the former has been reported for ZnO.

Twelve reflections with 1=0 or h-k=3n are unaffected by slmcniral 'mistakes' and these were used to obtain mean crystallite dimensions, based on spherical and cylindrical morphologies. From the dimensions obtained [<D> =453(17) A for spheres and <D> =404(7) A, <H> =351(9) A for cylinders)], the 'size' contribution to the remaining reflections (h-k=3n± 1) can be calculated. This can then be separated from the

total breadth for these lines to obtain the 'mistake' component and hence the stacking-fault

probability. For this purpose it is customary to assume that the constinient line profiles are Urentzian (eg. Warren, X-ray diffraction, 1969, p. 303), but here all sample line profiles, including those broadened solely by 'size' effects, have an appreciable Gaussian component. The Voigt hinction may therefore be a better approximation to the 'size- profiles and an alternative analysis based on this hmction (eg. Langford, J. Appl. Cryst.

11, 10-14, 1978) was also carried out. From the Lorentzian model and cylindrical form,

the mean probability of the occurrence of a stacking fault was found to 0.007(3), corresponding to a separation between fault planes approximately equal to the mean height of the crystallites. By assuming Voiglian 'size' profiles, the probability was about

0.014(9), equivalent to ~ 1 fault per crystallite, on average. This study has also demonstrated that with high quality data, detailed self-consistent and

fairly precise information about structural imperfections can be obtained from a combination of line-profile analysis and total pattern fitting. The ex -oxalate ZnO analysis represents a fairly demanding case, since the fault probability is small and the crystallites are relatively large, and a significant factor in obtainmg such detailed information was the use of a high resolution diffractometer. On the other hand, it revealed some of the

limitations of the procedure used to model the diffraction line profiles.
218 Accuracy in Powder Diffraction II

P4.13 DETERMINATION OF CRYSTALLITE-SIZE AND

LATTICE-STRAIN PARAMETERS FROM X-RAY DIFFRACTION LINE- PROFILE ANALYSIS BY APPROXIMATING BOTH SIZE AND STRAIN

BROADENED PROFILES WITH THE VOIGT FUNCTION. D. Balzar' and H. Ledbetter, Materials Science and Engineering Laboratory,

National Institute of Standards and Technology, Boulder, Colorado

80303, U.S.A.

When the pure-specimen (physical) broadened profile can be

satisfactorily described with a Voigt function, by applying the

Langford (1980) method, it is possible to separate exactly the size A^{L) and strain A°{L) Fourier coefficients. The column-length

distribution functions are analytically derived. It is shown that <D>s< <£»> v,<2<D>s, where <D>s and <D>^, refer to surface-weighted and volume-weighted average domain sizes. Distribution of average strain is evaluated. For not too large a harmonic number, mean-square strain decreases linearly with

distance L, but generally a more complex relationship holds with a

finite value for L=0. Errors are estimated, and the method is compared with the Warren-Averbach (1952) analysis and its modification (Delhez et al., 1980) for separation of size and strain parameters. Some problems with the application are described.

References

Delhez, R., de Keijser, Th.H. & Mittemeijer, E.J. (1980). Accuracy

in Powder Diffraction. NBS Special Publication No. 567, edited by

Block, S. & Hubbard, C.R., pp. 213-253. Washington, D.C.: National Bureau of Standards.

Langford, J.I. (1980). ibid., pp. 255-269.

Warren, B.E. & Averbach, B.L. (1952). J. Appl. Ptiys. 23, 497.

'On leave from Department of Physics, Faculty of Metallurgy,

44000 Sisak, Croatia.

P4.15 X-RAY DETERMINATION OF THE CRYSTALLITESIZE DISTRIBUTION FUNCTION FOR ALKOXY-DERIVED BARIUM

TITANATE POWDER MATERIALS. V. V. Zhurov, S. A. Ivanov, M. I. Yanovskaya, V. K. Lymar', Karpov' Institute of Physical Chemistry,

103064 Moscow K-64, Russia.

A detailed knowledge of the crystallite-size distribution function (CSDF) is required in many fields of materials science. A careful analysis of X-ray

diffraction line broadening provides a convenient method for obtaining this information. The purpose of our work was to determine the CSDF in barium titanate powders prepared by hydrolysis is of alkoxides with or without further heat treatments.

X-ray powder diffraction experiments (URD-63 diffractometer, Cu-radiati- on, T = 300 K) were combined with electron microscopy (JEM-IOOOX) and

analysis with Fritsoh laser partical sizer. Some peaks in the diffraction patterns of our samples were carefully recorded by the step-scanning

technique. The instrumental and spectral broadening was obtained from a

single crystal standard. The regularization method of Fourier X-ray line profile analysis was applied for the CSDF determinations. All calculations were performed by special programs adapted to an IBM-PC computer.

Some of the CSDF curves obtained are shown in the figure. There is a

quite reasonable agreement between the sizes obtained from X-ray profile

analysis and other physical methods up to 1 (im. The particles above 1 jim

detected by electron microscopy are

not monocrystalline. Values of

crystallite size were computed for

several crystallograhic directions. The

shape of crystallites is isotropic for all tested samples. i

i

1) initial pwwders

| 2) after heating at 800°C | 3) after heating at 1000°C ]

P4.14 A COMPARATIVE STUDY OF WARREN-AVERBACH,

MITRA-MISRA AND de KEUSER METHODS FOR CRYSTALLITE

SIZE AND STRAIN ANALYSIS IN NiO POWDER. L.D. Pradhan and

B.S. Achaiya, R.R.L. Bhubaneswar-751013, Oriss, INDIA.

Out of all the methods available for analysis of crystallite size and strain in polycrystalline materials, the three most widely used methods are Warren- Averbach (WA), Mitra-Misra, and the recentiy proposed metiiod of de

Keijser et al. In many cases it is difficult to use the WA method because

of the nonavailability of multiple orders of reflections and the fact that the shape of the profile from Cauchy or Gaussian. The method proposed by de

Keijser et al. takes into account the shape of the profile function, whether

it is Cauchy, Gaussian, Voigt, Pseudo Voigt, or Pearson VII. Nickel oxide

powder, the material used in the present investigation, gives an interesting case study for comparison of the above three methods. This is due to the

fact that multiple order reflections are available, and the shape of the profile

is Voigtian. Line profile analysis of the powder has been performed

repeatedly at different temperatures, including 300, 773, 873, and 973 K,

and the crystallite size and strain values have been calculated by using the WA, Mitra-Misra, and de Keijser methods. In calculating strain from the de Keijser method, the Voigt, Pseudo Voigt, and Pearson VII functions have been used. It has been observed that the de Keijser method does not yield

meaningful results, whereas the WA and Mitra-Misra methods do give good

outcomes. The cause of this variation is not known. However, it is tentatively recommended that even if die profiles fit the equations for Voigt,

Pseudo Voigt, or Pearson VII, one should perform all the above analytical

methods before arriving at any conclusion.

P4.16 X-RAY DIFFRACTION STUDY OF THE EFFECT OF

SiC ON THE DEFECT CHARACTERISTICS OF Al-Mg-Zn ALLOY.

R. Sen, S. K. Chattopadhyay, and S.K. Chatterjee, Regional Engineering

College, Durgapur-7 13209, INDIA

Silicon carbide was introduced into the matrix of an Al-Mg-Zn alloy (5.98

wt% Zn, 2.91 wt% Mg, and traces of Cu, Mn, and Fe) and X-ray

diffraction line profile analyses were performed to evaluate the role of SiC

in the composite matrix. Peak shift peak asymmetry and Fourier line shape

analysis carried out in detail in the cold worked systems have revealed that the concentration of twin faults introduced into the matrix increases with the increase in SiC concentration (from 0 to 20 wt%). The other fault parameters, such as intrinsic and extrinsic faults, have been found to be

negligible, as expected in aluminium based systems having high stacking

fault energy. The observations obtained from X-ray analysis have been

supported by scanning electron and optical metallographic studies.

!
Abstracts, Session 5: Novel Applications and Structural Science 219

2000

05.1 A HIGH RESOLUTION X-RAY POWDER DIFFRACTION AND

MOLECULAR MODELLING STUDY OF THE STRUCTURAL CHANGES

ACCOMPANYING ABSORPTION OF TRANS-STILBENE INTO H-ZSM-5

ZEOLITE. J. B. Parisc, SUNY Stony Brook, NY 11974-2100, J. Hriljac and

D. E. Cox, BNL, Upton NY11972, Dr. R. Corbin and V. Ramamunhy,

DuPont Company, Wilmington De 19880-0262, USA

The restricted pore geometry available in molecular sieves makes them excellent candidate materials for a number of selective chemical reaction. These have recently included the first stabilizations of the cation radicals of the rrani-polyenes^ We have

studied stilbene in H- ZSM-5' and have suggested the molecule

absorbs into the straight channels on the basis of NMR data'. High

resolution X-ray data

collected at room tem- • perature have been

modelled, using a commercial software

package, and are consistent with the inter- pretation of the NMR

data. Sorption of

fra/zi-stilbene causes

the monoclinic H-ZSM5 to transform to a pseudo tetragonal

phase at room temperature with no detectable distortion of

the unit cell. Upon

cooling to 20K a pronounced orthorhombic splitting is evident. The

figure above shows scans through the (400/040) reflections from

tra/i5-stilbene/H-ZSM-5 at selected temperatures up to 300 K collected

at a wavelength of 1.3064 A. Acknowledgements; Work at X-7A supported through DOE, DMR

under contract DE-AC02-76CH00016; partial support for JBP DuPoni

and NSF DMR-9024249

1. Ramamurthy et al.J. Amer. Chem Soc. 1991, 113, 594)

14.6 14.8

28 15.0 15.2

05.3 A PHYSICAL MODEL FOR X-RAY ANALYSIS OF

LAMELLAR SYSTEMS: APPLICATIONS TO MYELINATED NERVE

SHEATHS. R. Vargas and L. Mateu, Centro de Bioflsica yBioqulmica,

IVIC, Apdo. 21827, Caracas 1020-A, Venezuela, and V. Luzzati, Centre de G6ndtique Moleculaire, CNRS, 91198 Gif-sur-Yvette, France.

In order to undertake the study of order-disorder phenomena in myelinated nerve sheaths, a physical model has been proposed and an algorithm

developed for the fast and accurate analysis of the whole diffraction pattern of lamellar systems [1].

The model consists of planar lamellae, all stacked into crystals with a

variable repeat distance. Within each crystal the lamellae are parallel to each

other. The following parameters are considered in the model: the average

lattice repeat and its variance, the average number of unit cells per

crystallite, the continuous intensity curve of a single motif the intensities of

the diffuse scattering, and the fraction of lamellae that does not belong to the coherent diffraction domains.

The algorithm has been applied to a variety of X-ray scattering spectra

from rat sciatic and optic nerve myelin [2,3]. Such data have been recorded and analyzed in a few minutes using a position sensitive detector and a

personal computer. A precise determination of the structural parameters of

myelin and as evolution as a function of the age of the rats has also been

analyzed.

References

[1] Luzzati, V. and Mateu, L., J. Mol. Biol. (1990) 215, 373-384.

[2] Mateu, L., Luzzati, V., Vargas, R., Vonasek, E. and Borgo, M., J. Mol.

Biol. (1990) 215, 385^2.

[3] Mateu, L., Luzzati, V., Borgo, M., Vonasek, E. and Vargas, R., J. Mol.

Biol. (1991) 220, 351-357.

X-RAY LINE BROADENING FROM INTERGROWTHS AND

ANTIPHASE DOMAINS IN BARIUM HOLLANDITE COMPOUNDS. R.W. Chcary

and R. Squadrito, Depanmenl of Applied Physics. University of Technology Sydney. PO

Box 123. Broadway. NSW, Australia 2007.

Barium hoUandiles Bax (AyTig.y) Oig generally form over the composition range x=8/7

to x=4/3 where A is normally a 2* or ion such Al. Mg or a metal from the first transition series. As the tunnel sites accommodating the Ba are only partially occupied the Ba and vacant sites usually adopt an ordered configuration along the tunnels and

superlatlice lines appear in the x-ray powder pattern. These compounds undergo an order- disorder transition and above the transition temperature T^ the superlaltice lines disappear. For the hoUandites Bax (MgxTig-x) Oje , Tc goes from 710*C at x=8;7 to 900"C at x=4/3. When these materials are first cooled below the transition temperature Tq- the superlatlice lines are broadened through the formalion of iniergrowihs and antiphase domains. On further annealing the superlaltice lines become sharp and also undergo a 2B

shift. No significant changes are observed in the hollandile lattice lines on ordering. At x=4/3, Ba ions occupy 67% of the sites and the ordering sequence is Ba-Ba-V with a 3choU supercell where V represents a vacancy. V> hen the occupancy drops lo 60%

(x=1.20) the ordering sequence is Ba-Ba-V-Ba-V with a 5c),oU supercell. At intermediate occupancies, the suucture consists of intergrowths of regions with Ba-Ba-V order and Ba-Ba-V-Ba-V order. For x<1.20, the structure switches between regions of Ba-Ba-V-Ba-V

order and Ba-V order. In all compositions lateral ordering occurs between tunnels. This is manifested as unbroken corrugated layers of vacancies perpendicular to the tunnels. To

fully describe partially ordered hoUandites it is also necessary to include antiphase domains

into the model with boundaries parallel to and perpendicular to the tunnels.

In this work a genera) diffraction theory is developed for interpreting the breadths and shifts of the superiatlice hnes from partially ordered polycrs stalline hoUandites. This will allow

the kinetics of ordering to be studied through the lime and temperature dependence of the superlatlice line breadths. Analytical expressions for x=4/3 hoUandites that only contain antiphase domains have recently been published(l). This work has now been generahsed

to incorporate inlergrowths in the model as well as antiphase domains. In all cases the diffraction profiles are Lorentzian. Results will be presented for a range of hoUandiles al different Ba concenuaiions. These include the 20 shifts and apparent crystallite sizes corrected for instrumental effects for differeni superlaiuce lines from both fully ordered and quenched specimens. We have been able lo measure the mean dimensions of the ordered

regions and identify both the boundary layer structure and the hkl planes defining the boundaries. In monoclinic hoUandiles the boundaries form on the 101 and 101 planes

parallel lo the tunnels and on the 010 perpendicular lo the tunnels, but after extended anneahng only 010 boundaries remain.

(I) Cheary, R.W. i: Squadrito. R. (1992) Ada Cr>si. A48, In Press

P5 . 1 CRYSTAL STRUCTURE SOLUTION OF MG-LEUCITES

FROM SYNCHROTRON X-RAY POWDER DIFFRACTION DATA.

R. J. Cemik', A.M.T. Bell' -, P. E. Champness^ A. N. Fitch^-', C.M.B.

Henderson^, S. C. Kohn"*, and S.A.T. Redfem'.

A previously unknown low-symmetry leucite structure has been determined

from synchrotron x-ray powder diffraction data collected on a hydrothermally synthesized leucite analogue of stoichiometry K2MgSi50i2.

Laboratory based x-ray powder diffraction and ^'Si nmr work showed that

this structure is a distorted version of the known cubic (Ia3d) leucite

structure and that all 12 tetrahedral (T) Mg and Si sites are ordered. Electron diffraction showed that the low-symmetry structure was monoclinic

(P2,/c). The cubic structure in the monoclinic cell was used as a starting

model for Rietveld refinement of this structure. The refinement was able

to determine on which two of the T sites Mg were situated from differences

in the T-0 bond lengths. The synchrotron x-ray diffraction data were able

to distinguish quite clearly between the Mg and Si sites.

Rietveld refinement has also been carried out on a dry synthesized, cubic

(Ia3d) KjMgSijOij leucite analogue, in this structure the T sites are

disordered.

'SERC Daresbury Laboratory, Warrington, WA4 4AD, UK.

^University of Keele UK.

^University of Manchester UK.

''University of Warwick UK.

'Now at ESRF, Grenoble, France.
220 Accuracy in Powder Diffraction II

P5.2 CRYSTAL STRUCTURE REFINEMENTS OF A/sAljOs

( A/ = Ca, Sr, Ba) USING NEUTRON POWDER DIFFRACTION DATA

- AN EXAMPLE STUDY FROM THE NEW HB-4 DIFFRACTOMETER

AT THE HFIR, OAK RIDGE*. B. C. Chakoumakos, J. A. Fernandez-Baca,

and G. A Lager', Solid State Division, Oak Ridge National Laboratory, Oak

Ridge, TN 37831, U. S. A., 'Department of Geology, University of Louisville, LousiviUe, KY 40292, U. S. A.

CasAljOe is a major component of Portland cements, and the conversion of the aJuminate to the hydrogarnet structure is an important hydration reaction

in the setting of these cements. Detailed structural studies of isostructural

alkaline earth aluminates are underway to elucidate their reactivity and to examine the systematics of Al-O bond length and Al-O-Al angle variations among tetrahedral aluminates. Room temperature powder diffraction patterns were collected for the title compounds in the form of dense ceramic rods using a step size of 0.05° over the range of 11° to 135° 20 with a wavelength of 1.400 A. These compounds are cubic, Pa3, Z = 24, and are described by 48 structural parameters using isotropic thermal vibration parameters. Their

crystal structure consists of isolated 6-membered tetrahedral aluminate rings

cross- linked by the alkaline earth ions. Rietveld structure refinements were

completed, employing a Pseudo-Voigt peak shape function, and a total of 12 profile and background parameters. For M = Ca, a = 15.2588 (4) A,

Vctii = 3552.7 (2) A^, R„p = 0.0496, R^p = 0.0618, for 1444 reflections. Comparison of the refined structural parameters with a previous single crystal x-ray structure determination of the Ca analog (Mondal & Jeifery, 1975, Acta

Crysi. B 31, 689) show excellent agreement. For M = Sr, a = 15.8544 (4) A, = 3985.2 (2) A^, = 0.0294, R^^ = 0.0446, for 1614 reflections. The present refinement of the Sr analog is essentially the same as an earlier refinement employing the Voigt peak shape function (Chakoumakos et td., 1992, Acta Cryst. C48, in press). For M = Ba, a = 16.512 (2) A, V„,, = 4502

(1) A^, R^p = 0.0497, R^ = 0.0955, for 1829 reflections. This refinement is not as good as those for the Ca and Sr analogs. The assumption that the Ba

analog is isostructural (based on the similarity of the powder pattern) may

be incorrect, or the sample purity may not be as good. Both possibilities are being explored.

'Supported by the Division of Materials Sciences, U. S. D. 0. E. under contract DE-AC05-84OR21400 with Martin Marietta Energ,v Systems, Inc.

P5.4 STRUCTURAL CHARACTERIZATION OF WEBERTTERELATED ANTIMONATES AgSbA (A = Pb,Sr,Ca,Cd). S. A. Ivanov,

Karpov'Institute of Physical Chemistry, 103064 Moscow K-64, Russia.

The discovery of ferroelectricity in weberite-related PbgSbsO; crystals

initiated a search for similar compounds with unusual dielectric properties.

The title compounds appear to be good candidates for application as

ferroelectrics. Investigation of the crystal chemistry of these antimonates may be beneficial in developing a better understanding of the origin of

ferroelectricity. With this in mind, the features of the crystal structure of

A2Sb207 complex oxides have been thoroughly studied by X-ray powder

diffraction. Diffraction patterns were collected with a URD-63 diffracto- meter at room temperature between 12 and 132°26 using CuK,-radiation and analyzed with the Rietveld profile refinement technique (the DBW-3.2s

computer program). The structures of the A2Sb207 samples are basically

similar: the Sb ions are octahedrally coordinated whereas the A2-t- cations are B-coordinated and lie in square prisms and hexagonal bipyramids. The

existence of ferroelectric properties at room temperature only in PbaSbjOv

can be related to the features of crystal structure. The important role played

by the effective size of the A2-I- cations in determining the structural parameters, and the electrostatic effect of the lone pair of electrons on

Pb2-I- are also discussed.

A^* a, A b, A c, A S.G. <a> R,/R

Pb 7.484(1) 7.857(1) 10.426(2) I2cm 5.351 1.93 Sr 7.455(1) 10.369(2) 7,684(1) Imma 5.295 1.85 Ca 7.298(1) 10.213(2) 7.459(1) Imma 5.179 1.68 Cd 7.302(1) 10.207(2) 7.249(1) Imma 5.131 1.61

P5.3 STRUCTURE DETERMINATION OF THE LOW- AND

HIGH-PRESSURE FORMS OF MgaBNj FROM X-RAY POWDER DATA.

H. Hiraguchi, H. Hashizume, S. Nakano' and 0. Fukunaga', Research Laboratory of Engineering Materials, Tokyo Institute of Technology, Nagatsuta,

Midori, Yokohama 227, Japan. "Fac. of Engineering, TIT.

The crystal structures of magnesium boron nitride MgsBNs in the low- and

high-pressure forms have been determined ab initio from X-ray powder data. The strategy used is to decompose the powder pattern to estimate individual reflection intensities, caJculate Patterson functions and build trial-anderror structure models, which are refined by the Rietveld method. 54 and

21 reflections observed on a 0-9 diffractometer were used to solve the structures of Mg3BN3(L) and Mg3BN3(.ff) respectively. Low-pressure Mg3BN3(i)

has a hexagonal cell (space group P^zlmmc, Z=2) with 0=3.54453(4) and

c=16.0353(3) A while the high-pressure Mg3BN3(.ff) cell is orthorhombic

{Pmmm, Z=\) with o=3.0933(2), 6=3.1336(2) and c=7.7005(5) A. The two

structures commonly include linear N-B-N molecular anions, considered to play a role in the catalytic properties of the material in the reaction converting hexagonal boron nitride into a cubic form. A very similar structure has been obtained by applying standard direct methods to the same Mg3BN3(i) intensity data. High-quality electron density

maps have been calculated for this form using the maximum entropy method

and the phased structure factors. A separate application of the maximum

entropy program MICE [C. Gihnore et ai: Acta Cryst. A47, 830 (1991)]

to the unphased structure factors yielded a similar map, in support of the determined structure of Mg3BN3(L).

P5.5 CHARGE TRANSFER AND COMPRESSION EFFECTS IN DOPED

Y-123. S. Eriksson'^'*, C. Strom^ P. Berastegui'', L.-G. Johansson'^, A. Simon*. Hj. Mattausch*, ^Dept. of Inorganic Chem., S-412 96 Goteborg, SWEDEN, *MPI hir Festkorperforschung, 7000 Stuttgart 80, GERMANY.

In most high-T(, superconductors the number of positive holes in the CUO2. sheets to a

great extent determines the superconducting properties. By introducing dopant atoms or by

preparation of oxygen deficient samples drastic structural changes occur. The changes can

in many cases be correlated with a charge transfer in the structure and thus to a change in

The POLARIS instrument at ISIS, RAL, England has been used for collection of high

quality TOF neutron powder diffraction data. In all cases solid state sintered single phase

samples of doped Y-123 have been analyzed by the Rietveld method. The structural changes caused by the various dopant atoms have been correlated with the change in properties (See table 1). Compression effects (Sr^*) can be separated from charge transfer

effects (Co , Al'"*", Fe^"*") and the two effects seem to be additive. Raman spectroscopic

investigations, susceptibility and Hall effect measurements confirm the picture obtained from the analysis of the neutron data.

Table 1. Some characteristic distances in Y]Ba2.j,SryCu3.^COjj07_j. Cul is the chain

copper, Cu2 the plane copper, 02 and 03 the plane oxygens and o4 the apex oxygen.

x/y (Cu2-02,3) (Ba-04)^ Cul-04 Cu2-04 Tc

0.00/0.00 0.269 0.299 1.853(2) 2.291(5) 92

0.00/0.20 0.270 0.297 1.847(3) 2.282(3) 90

0.00/0.60 0.269 0.295 1.846(2) 2.250(3) 86

0.00/1.00 0.267 0.299 1.843(3) 2.224-{3) 81 0.05/0.00 0.259 0.304 1.854(5) 2.306(6) 89

0.15/0.00 0.228 0.343 1.836(5) 2.345(6) 76

0.30/0.00 0.216 0.349 1.838(6) 2.352(7) 30

Data will also be presented for the materials YBa^.^SryCu^^ej^O-y.^ ^e= Co^ "'' , Al^ '*')

and Y[.j,CayBa2Cu2 7Cog 30^^. The introduction of Ca" at the Y site reintroduces

holes in the Cu02.planes in Y-123 doped with cobalt. From the large number of samples

investigated it has been possible to obtain a firm understanding of the structure-property

relations in the Y-123 system.
Abstracts, Session 5: Novel Applications and Structural Science 221

P5.6 APPLICATION OF HIGH-RESOLUTION POWDER

NEUTRON DIFFRACTION TO THE DETERMINATION OF THE

NUCLEAR AND MAGNETIC STRUCTURES OF (Yi.xCax)Ba2Fe308^. Q.

j Huang, P. Karen, V. Karen, J.W. Lynn, A.D. Mighell, I. Natali-Sora, N. Rosov & A. Santoro. Materials Science and Engineering Laboratory, National

Institute of Standards and Technology, Gaithersburg, Maryland 20899,

U.S.A.

The nuclear and magnetic structures of (Yi.^Ca,()Ba2Fe30g^ have been

determined as function of temperature, calcium doping, and oxygen

stoichiometry, using data collected with the high-resolution five-counter neutron diffractometer at the Reactor of the National Institute of Standards and Technology. The general atomic configuration in the unit cell of this compound is similar is that of the superconductor YBa2Cu307. For 5 = 0 the symmetry of (Yi.xCax)Ba2Fe30g is tetragonal and the iron ions corresponding

to the Cu chains have octahedral coordination since all the oxygen sites on the basal plane are fully occupied. For 5 > 0, the oxygen vacancies occur on the basal plane and in such a way that some of the iron ions remain octahedrally coordinated and some assume a square-planar coordination similar to that of

the chain copper ions in the 1,2,3 compound. This means that oxygen is eliminated preferentially only in one direction of the basal plane and, as a consequence, the symmetry of the compound becomes orthorhombic. The

magnetic structure is based on a unit cell related to that of the nuclear

structure by an axis transformation of matrix (110/110/002). The iron moments are coupled antiferromagnetically within each Fe02 layer. The

coupling along the c-axis, however, depends on the values of x and 5.

P5.7 A NEUTRON POWDER DIFFRACTION STUDY OF

THE STRUCTURAL CHANGES IN Tlj.^^BajCuOg^ AND TI2.,, BajCaCuOg^ RELATED TO THE OXYGEN AND THALLIUM

CONTENT. C. Strom*, S. Eriksson*t, and L.-G. Johansson*, *Dept. of

Inorganic Chem., S-412 96 Goteborg, SWEDEN. fMPI fur Festkorperforschung, 7000 Stuttgart 80, GERMANY.

The thallium-based high T^ superconductors with two Tl-O layers have

tetragonal symmetry except for TljBa^CuO^ (Tl-2201), which exists in both tetragonal and orthorhombic form. Tl-2201 shows a remarkably wide

range in T^ (0-90K) depending on preparative conditions. It is known that changes in T^ and cell parameters can be correlated with changes in oxygen

content. Tl^.^BajCaCuOg^ (Tl-2212) shows no phase transition but it does

show a large variation in the c axis length dependent on the method of

preparation.

In this investigation we have studied the structural changes by neutron

diffraction of samples with different thallium and oxygen content in order

to simulate different synthetic conditions.

High quality neutron diffraction data were collected at the spallation source

ISIS, RAL, UK, using the high flux, medium resolution powder diffracto- meter POLARIS. Data were collected from twelve solid state sintered

single phase samples of Tl2.j(Ba2CuO^ and Tl2.xBa2CaCuOg^ with different X and d, and analyzed by the Rietveld method.

Results from the analysis show that the tetragonal-orthorhombic phase

transition in Tl-2201 is a function of both the oxygen and thallium content. The samples with high thallium deficiency are tetragonal. Orthorhombic

samples with small thallium deficiency become tetragonal after oxygen

treatment. The structural changes are largest in the Tl-O layers. Tl-2212 shows a shortening of the c-axis when the thallium and oxygen content

increase. However, the changes in bondlengths are not additive. Structural and T^ data will be presented for Tl-2201 and Tl-2212.
222 Accuracy in Powder Diffraction II

K6.1 DEVELOPMENT OF mCH PRESSURE POWDER

DIFFRACTION ON SYNCHROTRON SOURCES. Y. Fujii, Institute of

Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305 and now

Institute for Solid State Physics, The University of Tokyo, 7-22-1 Roppongi,

Minato-ku, Tokyo 106, Japan.

By using pressure to directly control inter-atomic interactions, we have

studied crystal structures newly stabilized at high pressures and related phase transition phenomena. At high pressures, hydrostaticity is no longer

maintained because of the freezing of the pressure-transmitting fluid. TTiis fact necessarily leads us to the use of powder specimens for diffraction experiments to obtain reliable intensity data. The relatively small aperture of a high-pressure apparatus seriously limits the accessible reciprocal space;

therefore, powder diffraction has another advantage of allowing the

collection of as much diffraction data as possible in such a restricted geometry. In order to enable us to obtain reliable intensity data from the powder specimen, the following systems dedicated to synchrotron radiation x-ray diffraction experiments at the Photon Factory have been developed

cooperatively by the High-Pressure Group in Japan:

(1) SR (BM, MPW on the 2.5 GeV Storage Ring, 10-20 keV)

- Diamond-anvil cell (at low/high temperatures)

- Imaging-plate, SSD (energy-dispersive mode)

- Data acquisition software,

(2) SR (BM on the 6 GeV Accumulation Ring, 10-100 keV)

- Multi-anvil apparatus (high temperatures, MAX-80 & -90)

- SSD (energy- or angle-dispersive mode)

The precise structure determination of molecular crystals and metals has enabled us to study precursory behavior, structural instability, and

structural-similarity/scaling-rule uniquely driven by pressure.

^^^ HIGH TEMPERATURE X-RAY DIFFRACTION

DETERMINATION OF PHASE TRANSITIONS IN LARGE

GRAIN ALLOYS. O. B. Cavin*. C.R. Hubbard, and P.J. Maziasz. Metals and Ceramics Division, Oak Ridge National

Laboratory, Oak Ridge TN 37831-6064, Phone (615) 574-5121.

HTXRD studies of the phase equilibria ol alloys as a function of

temperature depend, upon the ability to accurately follow the

intensity of particular lines. Large grain size introduces the

additional complication of spotty patterns due to poor particle

statistics yielding ambiguous or uncertain intensities. To

overcome this limitation, a rocking curve method has been used

which involves sample oscillation about the goniometer axis. We

have studied inherently large grained intermetallic aluminides, some of which have two ordered lattice types (DO3 and B2) and have determined the transition temperatures of each type. These temperatures are required for modeling the alloys and for

predicting ideal forming temperatures. By carefully choosing the

alloying additions, the properties of these alloys can be

controlled. Changes in the transition temperatures have been observed which makes them more attractive for specific

applications.

Research sponsofed by ihe U.S. Depanmeni ol Energy, Auistant Seaeta/y for Coruervatiofi

and Renewable Energy. Office ol Industrial Tecfinologies, Advanced Industrial Concepts

Materials Program, under contract DE-AC05-84OR21400 with Martin f»(anetta Energy

Systems, Inc

06.2 NEUTRON POWDER DIFFRACTION AT THE HIGH

FLUX ISOTOPE REACTOR -AT OAK RIDGE *. J. A. Fernandez-Bacai, H. A. Mock', H. S. White', B. S. Hoffheins^, and B. C. Chakoumakos\ 'Solid State Division, 'Computing and Telecommunications Division, ^Instrument '

and Control Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee

37831, U. S. A.

A new multidetector neutron powder diffractometer has been installed at the HB4 beam tube at the High Flux Isotope Reactor at Oak Ridge National

Laboratory. The diffractometer design currently includes a 12' collimator in the primary reactor beam, a flat Ge (1 1 5) monochromator crystal and 32

equally spaced (2.7°) ^He counters, with 6' Soller-slit collimators, that can be scanned to obtain a diffraction pattern between 11° and 135° 28. A new,

vertically focusing, Ge monochromator will be installed in the near future. The monochromator angle is usually fixed at 29\f = 80° to select neutrons 1 of a wavelength of 1.4 A. The instrumental resolution for this configuration I has a minimum of Ad/d= 2 x 10"^. The instrument control is performed

from a microVax computer, which is also used for data visualization and analysis. In order to allow for different users' preferences various Rietveld analysis codes are available. These include RIETAN (by F. Izumi), LHPMl (by R. J.

Hill and C. J. Howard) and GSAS (by A. C. Larson and R. B. Von Dreele). Various tests of the performance of the diffractometer have been completed i successfully and this instrument is now available for routine use. These tests included the crystal structure refinements for a number of powder diffraction standards (Ni, Si, AI2O3, ZrOj, etc), and a variety of oxides: M3AI2O6, M = | Ca, Sr. Ba; Sr3Al2(D404)3; MCuSi40io, M = Ca, Sr, Ba; and PbAljSijOg.

The best fits were obtained using the pseudo-Voigt peak shape function.

'Supported by the Division of Materials Sciences, U. S. D. 0. E. under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc.

P6.1 A NEW FOURIER DIFFRACTOMETER AT THE lBR-2 1 PULSED REACTOR - HIGH RESOLUTION, HIGH INTENSITY, HIGH

| ACCURACY. A.M.Balagurov, Laboratory of Neutron Physics, JINR, 141980 1 Dubna, Moscow Reg., Russia. i

1 Two neutron Fourier diffractometers are in practical operation at present: the mini-SFINKS facility/''' in Gatchina (Russia) and FSS/'^ in Geesthacht (Germany). They use the Reverse Time Of Flight (RTOF) method/^/ of data

acquisition and have good prospects in structural and residual stress studies.|

It is anticipated that a new high resolution Fourier diffractometer (HRFD) I

will commence operation in Dubna (Russia) at the IBR-2 pulsed reactor in

| 1992^''''. At this diffractometer the RTOF mode will be also used, and precise structural studies of powders with very high resolution and high intensity

are proposed as the main area of interest. The Ad/d resolution of 0.0005 is ,

provided by the high frequency of neutron beam modulation (~150 kHz); a

| high intensity is due to the high transmission of the Fourier chopper (~0.25) •

and the large solid angle (0.1 sr) of the time focused detector system. The '

additional modulation of the beam by neutron source pulses leads to a considerable (by 10 or more times) decrease in the correlation background level . compared to a Fourier diffractometer at a steady state reactor. The high total

neutron flux at the sample position enables studies of very small samples and

time resolved experiments. The measurement time for a complete diffraction ; pattern is determined by the Ad/d resolution and in the low-resolution limit ,

(1%) is much less than one minute. The opportunities for structural studies

with the new Fourier diffractometer are expected to be similar to those of the HRPD at the RAL and, in addition, the new instrument offers the possibility of optimizing the experimental conditions with respect to resolution and

intensity.

j

1. V.A.Trunov et al, Sol.St.Comm., 59 (1986) 95 !

2. H.G.Priesmeyer, J.Schroder, Mat.Res.Soc.Symp.Proc, 166 (1990) 299

3. P.Hiismaki et al, J.Appl.Cryst., 21 (1988) 349 I

4. A.M.Balagurov, Physica B, 174 (1991) 542
Abstracts, Session 6: New Developments in Hardware and Studies under Special Conditions 223

P6.2 THE HIGH RESOLUTION NEUTRON POWDER DIF- FRACTOMETER AT JRR-3M. Y. Morii, K. Fuchizaki(*), S. Funahashi, N. Minakawa, Y. Shimojyo and A. Ishi(ia(**), Department of Physics, Japan

Atomic Energy Research Institute, Tokai, Ibaraki 319-11, Japan

The characteristics of the High Resolution Neutron Powder DiflFractometer (HRPD) installed and studied recently at the JRR-3M are reported. Some

experiments performed with the HRPD are also presented. The HRPD is an ILL-D2B type diffractometer which has the following features. (1) a fine

collimation system: 6712' - 20740' - 6', (2) pseudobent germanium 331 and

copper 311 monochromators with take-off angles of 89° or 130°, (3) Sixty four

3rd collimator and detector systems placed at every 2.5° of diffraction angle.

Typical diffraction patterns can be taken over scattering angles from 5° to

160° in a few hours, depending on the sample material, and analyzed by the

Rietveld profile refinement method or others. For the case of a monochromator take-off angle of 130°, the angular resolution, FWHM, of the diffraction

peaks is less than 0.35° over the whole scattering angle range. The best resolution is 0.2° at the focusing position, which corresponds to 0.2% in Ad/d.

Various kinds of materials such as high Tc superconductors, ionic conductors,

metals, carbon clusters etc. have been studied with the HRPD. Among them,

La2Ni04.oo was investigated between 610K and 6K. The crystal structure be- low 60K was determined to be orthorhombic, and the magnetic moment of Ni'+ was evaluated to be about 1.5 fia at 80 K and 1.7 hb at 6K through a

Rietveld analysis.

(*) present address: Department of Physics, Faculty of Science, Kyusyu University; Fukuoka 812 Japan

(*) present address: Nuclear Plant Department, NKK corporation; Yoko- hama 230 Japan

P6.4 NEUTRON POWDER DIFFRACTION TO 20 GPa.

J. S. Loveday, R. J. Nelmes, Department of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK, J. M. Besson, G. Hamel, Universite

Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris, Cedex 05, France,

and S. Hull, the ISIS Facility, Rutherford Appleton Laboratory, Chilton,

Oxfordshire, UK

The maximum pressure accessible to neutron diffraction techniques has

until recently remained in practice at the « 3 GPa available with

McWhan type alumina clamp ceUs. This has limited the study of a

range of classic neutron problems (magnetism, structures containing light

atoms, disorder, etc). To address this problem we have been developing an opposed-anvil cell, able to take samples of sufficient size for crystallographic studies to pressures in excess of 20 GPa. This cell is now in use on the POLARIS powder diffractometer at the UK spallation neutron

source ISIS.

The cell consists of a compact 200t hydraulic ram applying pressure to a sample contained between two opposed anvils. These are made either of tungsten carbide or, for the highest pressures, sintered-diamond. A

toroidal anvil profile is used and allows a sample of « 100 mm^ volume

to be taken to 10 GPa or 30 mm^ to be taken to 20 GPa.

This geometry has required the development of new shielding procedures

to obtain the optimum signal to background and a means to correct for the complicated attenuation by the anvils. This work is still in progress, but the results of a number of preliminary structural studies will be

presented, including the pressure dependence of the structure of D2O ice VIII and the high-pressure phases of bismuth .

ro.o THE HIGH RESOLUTION POWDER DIFFRACTOMETER

AT THE NIST REACTOR. E. Prince, A. Santoro, I. G. Schroder & J. K.

Stalick. Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, U. S. A.

A new, state-of-the-art, high-resolution, neutron powder diffractometer has

recently been installed at the NIST research reactor. Because optimum powder patterns require differently shaped curves of resolution as a function of d spacing for crystals with different unit cell sizes, this instrument has provision

for producing monochromatic beams with ~ l.sAwavelength at three different monochromator takeoff angles. A 75° takeoff angle is most useful for large unit

cell crystals such as zeolites, where there is comparatively little information

at short d spacings. On the other hand, a 120° takeoff angle is necessary for crystals with small, but low symmetry, unit cells. The instrument has 32 detectors spanning an angle of 155°. Focusing monochromators allow optimum

use of reactor neutrons without significant degradation of peak shape. The

instrument will be used for the determination and refinement of the structures of a wide range of technologically important materials.

1^0-0 HIGH-PRESSURE STRUCTURAL STUDIES USING

ANGLE-DISPERSIVE TECHNIQUES. M. I. McMahon, R. J. Nelmes,

P. D. Hatton, R. 0. Piltz, J. Grain and D. R. Allan, Department of Physics, The University of Edinburgh, Edinburgh, EH9 3JZ, UK.

During the past year, an image-plate (IP) system has been developed at SRS Daresbury with the objective of performing full Rietveld refinement

of high-pressure crystal structures using angle-dispersive powder-diffraction techniques. Considerable care has been taken to reduce background

scatter and eliminate contaminant diffraction lines, and this hcis enabled

us to collect data of unrivalled quality. In our initial studies of the III-V semiconductor InSb, we have discovered two new high-pressure phases,

and have shown that the long-accepted structures of two more high- pressure phases are incorrect. Furthermore, the very high sensitivity of the IP system, coupled with anomalous dispersion techniques, has enabled us to determine whether the high-pressure phases of InSb are ordered or not - a question that has remained unanswered for 30 years. The latest results on InSb will be described, along with new results from

another III-V material, GaSb. Very recently, we have managed to obtain

high-quality diffraction patterns from the low-Z semiconductor B4C. The

effect of pressure on this interesting material will also be presented.
224 Accuracy in Powder Diffraction II

P6.6 AUTORIET: SOFTWARE FOR "AUTOMATING" RIET- VELD ANALYSIS OF TIME-OF-FLIGHT NEUTRON POWDER DIF- FRACTION DATA. F. J. Rotella, Intense Pulsed Neutron Source (IPNS)

Division, and R. L. Hitterman, Materials Science Division, Argonne National

Laboratory, Argonne, Illinois 60439, U. S. A.

In certain types of experiments (e.g., time-resolved diffraction measurements),

the rate at which the data are collected far outstrips the speed with which

the said data can be analyzed by standard (user-intervention) Rietveld techniques. In an effort to deal with this situation, computer software has been

written to automate Rietveld analysis at IPNS by optimizing data analysis

time, much in the same way that computer control of instruments has optimized data collection times and sample throughput. AUTORIET is similar

to software which has been used for the automated analysis of time-resolved neutron powder data from the Institut Laue-Langevin for the past eight or so

years. It is a VAX/VMS DCL command file that, with auxiliary FORTRAN

programs, creates the appropriate files that will be used for input to the IPNS

Rietveld analysis software, controls the execution of the software, and creates summaries of the analysis for all sets of data considered. The user analyzes

the initial data set via standard Rietveld techniques; AUTORIET will analyze

the remaining sets in a user-specified order, based on the assumption that the

starting parameters for any given data set are the results from the analysis of

the preceding set. The summaries that are produced may be collated into files from which the variation of any parameters with respect to one or more other parameters in the model may be obtained in tabular form and/or graphically

using visualization software on the IPNS VAX Cluster or on a workstation

or personal computer. Although AUTORIET was designed to be used specifically with software in the IPNS Rietveld analysis package, adapting it for

use with other Rietveld analysis software in a VAX/VMS environment should be fairly easy. The use of AUTORIET in the analysis of in situ diffraction data from a working electrochemical cell and in the determination of thermal

expansion coefficients of bismuth-based superconductors will be presented.

This work was supported by the U. S. Department of Energy under Contract W-31-109-ENG-38.

^"•O HIGH RESOLUTION SYNCHROTRON X-RAY POWDER

DIFFRACTION WITH A LINEAR POSITION-SENSITIVE DETECTOR. D. E. Cox and G. C. Smith, Brookhaven National Laboratory, Upton, N.Y. 11973,

U.S.A., L. W. Finger, Carnegie Institution, Washington, D. C. 20015, U.S.A., and

A. P. Jephcoat, Oxford University, Oxford OX13PR, England.

Some synchrotron x-ray powder diffraction experiments are described in which a

multiwire proportional detector was operated in the "escape-peak" mode with a 90% Kr- 10% CO2 gas mixture at 4 bar. Under these conditions, it is possible to achieve intrinsic position resolution of about 30 nm for 17.4 keV X-rays, compared

to about 150 \im for the "photo-peak" mode (1). A gating signal can be applied to the jwsition-sensing electronics to provide energy discrimination. The detector, which has an active length of 10 cm, was mounted vertically on the scattering arm of the diffractometer at beam-line X7A at the NSLS about 45 cm from the sample axis. With well<rystalli2ed reference materials such as Ce02 and AI2O3 and an incident 0.7A x-ray beam with horizontal and vertical dimensions

of 4.0 and 0.2 mm respectively, the angular resolution obtained in the center of the detector was about 0.03° FWHM, although there was considerable degradation at the ends because of parallax effects. High quality data suitable

for precision lattice parameter determination, auto-indexing or Rietveld

analysis can he obtained in as little as a few minutes by stepping the detector at 2° intervals and combining the central segments into a composite data set. Some

applications will be described, including diffraction exjjeriments in diamond - anvil cells.

[1] J. Fischer, V. Radeka, and G. C. Smith, IEEE Trans. Nucl. Sci. NS-33, 136

(1986); Nucl. Instrum. and Meth. A252, 239 (1986).

Work supported by the Division of Materials Sciences, U.S. Dept. of Energy,

Contract DE-AC02-76CH00016 (DEC and GCS), National Science Foundation

grant EAR-8817263 (LWF), and Nato CRG 890944 and Natural Environment

Research Council grant GR3/7984 (APJ).

t^D- ' HIGH-PRESSURE X-RAY DIFFRACTION

STUDY OF INDIUM UP TO 93 GPA K. Takemura.

National Institute for Research in Inorganic Materials, Tsukuba,

Ibaraki 305 JAPAN

Structural change of indium under pressure has been

precisely studied up to 93 GPa at room temperature using a diamond-anvil cell, synchrotron radiation and an imaging plate. The

c/a axial ratio of the face-centered tetragonal (let) structure increases with pressure, reaching a maximum around 24 GPa (

(c/a)rr,ax = '^^^'^ ) and decreases with further compression [1]. At around 45 GPa, indium undergoes a structural phase transition. The powder diffraction pattern of the high-pressure phase can be

indexed to a face-centered orthorhombic cell with the lattice constants of a=3.771, b=3.841, c=4.142 A at 93 GPa. By assuming

plastic deformation of indium under uniaxial stress, the observed

diffraction intensity can be successfully explained, including the asymmetric peak profiles. The model assumes a successive slip of the (111) plane in the [01 T] direction, which often occurs in the fee metals subjected to uniaxial stress. The estimated volume change

at the fct-fco transition is close to zero, offering a possibility that the

transition is of the second order. The transition was reversible with respect to pressure. The lowering of the lattice symmetry of indium under pressure can be understood on the basis of the universal phase diagram for the group lllb elements calculated by Hafner and Heine with the pseudopotential method [2].

[1] K. Takemura, Phys. Rev. B44, 545 (1991).

[2] J. Hafner and V. Heine, J. Phys. F13, 2479 (1983).

P6.9 TIME RESOLVED POWDER DIFFRACTION. H. Fuess and P. U. Pennartz*, TH Darmstadt FB21/Strukturforschung.

*now: ENRAF/NONIUS-Solingen Germany.

Complete powder diffraction patterns were recorded at the synchrotron source DORIS II (HASYLAB, Hamburg) in time intervals of mUliseconds. Recording of the diffracted intensity was achieved by a commercial diodearray detector (EG & G-PARC) in Debye-Scherrer geometry. A time

resolution of 200 msec for a snapshot of 40° (26) was obtained by a toroidal mirror and a conventional (n,-n) silicon (111) double crystal

monochromator.

The detector consists of 1024 single silicon (p/n) diodes each 25 \im wide

and 2 mm high. The detecting window is therefore only 25 mm long. In order to record a large region of the diffraction pattern the detector is mounted very close to the sample. The angular resolution of this setup is comparable with that of a standard x-ray powder pattern from a conventional source, but the time resolution is about 10 times higher than with

conventional systems. Enlarging the detector to sample distance will increase the angular resolution and decrease the integral time resolution. A multilayer monochromator (OVONICS, 200AV/Si double layers of 2.05 nm thickness each) decreased the measurement time for a whole pattern to 2.5 msec. The multilayer system, however, increased the energy-bandpass

by two orders of magnitude. Thus the angular resolution is about 10 times worse than the angular resolution of conventional powder diffractometers (Pennartz et al., J. Appl. Cryst, 1992, in press).

In contrast to previous work (Wong et al. 1990, Science, 249, 1404) we

aimed at the recording of a complete powder pattern and not just a small region or a single Bragg-reflection. Therefore we tested two other detectors (Pennartz et al. Z. Kristallogr. 1989, 186, 227-230), including the blade- anode ENRAF-NONIUS/INEL 120° curved CPS120. The performance of

these detectors has been tested using conventional sealed x-ray tube sources and rotating anodes, and also at the synchrotron.
Abstracts, Session 6: New Developments in Hardware and Studies under Special Conditions 225

P6. 10 A NEW METHOD FOR POWDER DIFFRACTION

STUDIES. Mark A. Beno and Gordon S. Knapp,

Material Science Division, Argonne National Laboratory,

Argonne II 60439, U.S.A.

Powder difftactometers are among the most widely used of all analytical instruments. Seemann-Bohlin diffractometers, the most widely used type, measure data serially, one angle at a time with many angles scanned to observe the entire diffraction pattern. Linear and area detectors lack energy

resolution and so can suffer from high backgrounds particularly if materials containing elements which fluoresce are to be studio! In addition, the one

dimensional detectors developed to date do not possess sufficient resolution

for highly accurate structural studies particularly in the case of synchrotron based x-ray powder diffraction instruments. The method we employ uses a focusing diffracted beam monochromator in combination with a multichannel detector which allows us to reduce the background and to take data in parallel, providing a large improvement over

conventional difftactometers. m

our technique the incident x-rays multichannel detector (from a laboratory source or

synchrotron) fall on a flat plate or

capillary sample and are intercepted

by a bent focusing monochromator

which has the focus of the bend

at the sample surface. The beam

emerging from the bent crystal monochromator falls on a linear or 2-dimensional detector. Initial tests show that this method can be used

to acquire the data much faster and

with much higher sensitivity. Existing instruments can be easily

modified to employ this method.

Results from laboratory and synchrotron sources will be presented for

crystalline and glassy materials along witii schemes for rapid data collection employing other monochromator and detector configurations.

Work at Argonne National laboratory is sponsored by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences, under contract W-3 1 - 1 09-ENG-38.

diffraction

line one

Focusing^

crystal

P6.ll HIGH TEMPERATURE NEUTRON POWDER DIF- FRACTION FURNACES, S. L. Town, R. L. Davis' k E. H. Kisit,

ANSTO, PMB 1, Menai, NSW 2234, Australia, *AINSE, PMB 1, Menai,

NSW 2234, Australia, fDivision of Science &: Technology, Griffith University, Nathan, Queensland 4111, Australia.

A neutron powder diffraction furnace, capable of temperatures to llOCC

has been tested and commissioned for use on the medium and high reso- lution powder diffractometers at ANSTO. The design of this is discussed,

and data from YjBaCuOs and CaFj are presented; problems in refining

thermal parameters are highlighted. A new furnace, capable of temperatures to 1750°C is currently being developed, and the design is detailed

here.
226

Acharya, B. S. P4.14

Akselrud L G P3.1

Allan, D. R P6.5

Artioli G Pl.l

Attfield .1 P 175

Auffredic .7 P P4.12

Balaffurov A IVI P6.1

Ra 1 7 a T P4.13

BaumEfartner B P3.1

Bell, A. M. T. P5.1

Bellotto M Pl.l

Beno M A P6.10

Berasteffui P P5.5

Berkum, J. G. M. van P2.3

Berar J F 63Wj P3X • 5vV Bernstein J P1.4

Berti, G. P4.10

Besson, J. M. 195,P6.4

Bish D L 154

Bniiltif A P4 12

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Brand, P. C. P2.2

Brauer, S. 189

Bryant, P. PI.

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Ghatterjee, S. K. P4.16

GhattoDadhvav S K P4.16

Ghearv R W P4 6 05 2

Gherner, Ya. E. P3.2, P3.3, P3.4

Ghoi, G. S. P4.5

Gline, J. P. 68

Goelho A P4.6

Cone: 0 P3.13

Gook, L. P. P4.5

Gorbin, R. 05.1

Gox, D. E. 05 1 P6 8

Grain \^X UfXXX ^ J * P6.5

David, S. A. P4.1

David xv ' X vx • WfT*x*I Fx» P2.6, P3.ll

Davis, B. L. 7

Davis XV C-v Y XkJ

J RX \/ • LXJ * P6.ll

Deem M W 80

Delhez XVV^X XX

J RX \/ « P2X ^ t 3W « P4X X * 9V Dodson, T. A. P4.1

Egami, T. 105

Eriksson, L. P3.12

Eriksson, S. P5.5, P5.7

Ermrich, M. 38

Ersson, N. 0. P3.6

Estermann, M. 03.1

Accuracy in Powder Diffraction II

P5 2 Ofi 2

Fineer L W 183 P6 8

Fischer, H. E. 189

Fitch, A. N. P5.1

TVppTTi X X^^lllCLXlj a,n C] * IVT ITX 80

Pfi 2

Fuess X \X V^OkJ

J HXX * P6 9

Fujii, Y. K6.1

Fukunaga, 0. P5.3

Fiinahashi .S Pfi 2

Gamarnik VJi U-XXXUrX XXXXV j IVT X*X « YaX U>< P2.4

Gamier VJi OvX XXXV^X J PX * P3 5

rrramlirVi V 03 1 Grin Yn N P'i 1

Hall XX CI>1 i. J S • RX \y * P2X ^ . 1X Hamel G Pfi 4

Hashizumi, H. P5.3

Hattnn XXCL V J PX * DXv> Pfi 5

Havlik XXcl V XIIV J TJ. . P^ 14

XXC7X1U.\^1 OVJII J> IVX • XJ > P5X tv < 1X TTprmann XXViX XXX<X'XXXX TT

J XX* 38

Hewat, A. W. P3.7

Hill, R. J. PI.

2

TTira XX XX \Ju^^criirlii U ^XXX TT

J XX * P5X KM *tj3

Hitterman, R. L. Pfi.

6

Hoffeins, B. S. 06.2

Holden, T. M. 136

Howard, G. J P1.5, P3.10

Hovle S 0 PIX X • 6

TTriliar .T 05 1 Hubbard, G. R. P4.1, 06.1

Huang, Q. P5.6

Huane T G 1 Hull, S. P6.4

Ibberson, R. M. P3.ll

Ishida, A. P6.2

Ivanov, S. A. P4.15, P5.4

Jarvinen, M. P4.2

.Tenhcoat V V^IVXX WVUiU J JX* A PX • P6.8

.TnhanssoTi V vXXCvXXOOWXX \t Cr

J XV t • P5 5 P5 7

Johnson, G. G., Jr. PI.

6

Karen, P. P5.6

Karen, V. P5.6

XVdIOCX Keiiser J ThX IX* HXX devx^ P2.3

Kisi, E. H. P1.5, P6.ll

Klimanek, P. 01.3, P4.3, P4.ll

Knapp, G. S. P6.10

Kogan, V. A. P4.8

Kohn, S. G. P5.1

Lach, S. P1.4

Lager, G. A. P5.2
Author Index 227

T.ancfnrnTT 110 P9 1 P4 1 

XXU,X £i 0 X

y

X ^.X^

Le Bail, A. 142, P3.9

Ledbetter, H. 'P4.13

Li, D. P4.4

09ij £i

y P4X ^ . 12X ^ TiOVPn^iv 1 SI 1 Qfi Pfi 4

Lutterotti, L. P4.7

T.vmarV TC P4X ^ . 1X ^^ Lvnn 7 W P5 fi

A/TaHQPTiT C^. PIXT X . 9^

IVTaist.rplHP P4X 71 Marquart, R. G. 01.1

O.^ 'i Mattausch Hi P5.5

IVTaziasz

PJ 06.1

McCusker, L. B. 75

McMahon, M. I. P6.5

Mehdi

S PI. 

MiehellAD P5 6

MinakawaN P6.2

MookHA 06.2

MnriiY P6.2

Miicklich, A. P4.3

P5 '\ Nakhmanson, M. S. 47

Natali-Sora, I. P5.6

1Q5 P64 Pfi 5

Npwman X1V<>**111 U>J.X

J

RX ^AX • P2.5

X 1 VV OCLXXX

J

V • XtX • 80

O'Connor, B. H. P4.4

OettelH 01.3

PariseJB 05.1

Pecharski, V. K. P3.1

Peerdeman, F. P4.9

Pennartz, P. U. P6.9

Petricko, F. P3.14

Piltz X XXUUjRM.\>tO\^

t P6.5

Pradhan X X u^vxxxvwxx * Lxy > Dx^ * P4.14

Prask X X UOA

y

HXX • JV • P2.2

Prince, E. ' P6.3

PaTnarniirt.lrv X VUfXXX UiXXXUX VHJf

y

VT • 05.1

Reddv X I/^UIU V

y

MXTX • PI. 

Redfern X\/C^VXXC^X XX

J SAT • xV > X. > P5.1

Root, J. H. 136

Rosov, N. P5.6

Rotella, F. J. P6.6

Sakata, M. P3.10

Santoro, A. P5.6, P6.3

Scardi, P. P4.7

Schneider, J. 03.2

^rnTTiMPT OV.'lil^-'UClT It

y X. VJi • Pfi X \J .O Sen R P4.16

Shimojyo, Y. P6.2

Simon, A. P5.5

Sitepu, H. P4.4

Sivia, D. S. P3.ll

Skrobian, M. P3.14

Smith, D. K. 01.2, P1.6

Smith, G. C. P6.8

Smrpr^K T. kJXXXl ^^k^JV J XJ > P2X ^ . 7i ^"nvH^iT Oil R T,

J* UCi , XL* Xj . 25

Sonneveld, E. J. P2.3

Sprong, G. J. M. P2.3

LidUX i \*\J

y XL* 05 2

Stalirk OUCIXXI./A.,

u7 . KXV. 34 Pfi 3 O Uv^ L^XXV^XXoV^XX J VJI • XJ r 189

Strom ik^t^xv^xxx* Cj\^ • P5.5 P5 7

Strom-Olsen, J. 0. 189

Sutton k-/ U. u VV^XX

y

MXTX « 189

Takata X Ct> ACL u Cl>

J

MXtX • P3X %J . 10X\j

Takemura, K. P6.7

Timmers, J. P4.9

Toby, B. H. 105, P2.1

Toraya, H. 02.1

Tovhis X\JVUiOy XL* ARXJ * P3

4

'Pown X \J TT XXS T»

J • XJ • PIX X •%!5 y P6X vliXX 11

Tsambourakis, G. P1.2

Tuinstra, F. P4.9

UnoT P3X o inxu

ValvnnaV 127 X^ 1 Varffas T %A/M. Syt-t/U y RX^* 05 3

Vermeulen, A. C. P2.3

Wane T * UXX

y

Sk_/tL-XJ • P3X V • 8(J Werner, P.-E. 51, P3.12

Westdahl, M. ' P3.12

White, H. S. 06.2

Wolfel TV \^XX^^X,EXJ • P3X O • 1X Wone-Ne W P4 5

Yamanaka, T. 199

Yang, Y. S. 189

Yanovskaya, M. I. P4.15

Zavalii, P. Yu. P3.1

Zevin, L. 17, P1.4

Zhurov, V. V. P4.15
228 Accuracy in Powder Diffraction II

Tariq Ahmedali Jean Francois B^rar Xianhui Bu

McGill University Ecole Centrale Paris Chemistry Department

3450 University St. voie des Vines Univ. of California (SB)

Geol. Sciences 92295 Chatenay Santa Barbara, CA 93106

Montreal, H3A 2A7 France U.S.A.

Canada

J.G.M. van Berkum O. B. Cavin

Charles Albert Laboratory of Materials Sci. Oak Ridge National Lab.

SCM Chemicals Rotterdamseweg 137 PO Box 2008

3901 Fort Armistead 2628 AL Delft Bldg 4515, MS 6064

Baltimore, MD 21226 The Netherlands Oak Ridge, TN 37831

U.S.A. U.S.A.

Giovanni Berti

David Allan Univ. Pisa Bryan Chakoumakos

Univ. of Edinburgh Via S. Maria 53 Oak Ridge National Lab.

Mayfield Rd. 1-56100 Pisa PO Box 2008

Edinburgh EH9 3JZ Italy Solid State Division

UK Oak Ridge, TN 37831

David L. Bish U.S.A.

J. Paul Attfield Los Alamos National Lab.

Univ. of Cambridge Mail Stop D469 Lawrence Chatfield

Dept. of Chemistry Los Alamos, NM 87545 Alfred Univeristy

Lensfield Rd. U.S.A. Inst, for Ceramic SuperCambridge CB2 lEW conductivty BMH-1-55

UK Julie A. Borchers Alfred, NY 14802

NIST U.S.A.

Neil A. Ball Bldg. 235, Rm E151

University of Manitoba Gaithersburg, MD 20899 Robert Cheary

Dept. of Geological Sci. U.S.A. University of Technology

Rm 24, Wallace Bldg. Broadway

Winnipeg, Manitoba R3T 2N2 Alun Bowen Sydney, 2007

Canada Defence Research Agency Australia

Materials & Structures Dept.

Davor Balzar RAE Famborough Yakov E. Chemer

NIST Hants, GU14 6TD Institute of Physics

325 Broadway 853 UK 194 Stachki Ave.

Boulder, CO 80303 344104 Rostov-on-Don

U.S.A. Hans Boysen Russia

Inst, fur Kristallographie

Maurizio Bellotto Universitat Munchen Feng Chen

CISE-Tecnologie Innovative Theresienstr. 41 Rider College

Via R. Emilia, 39 - Segrate 8000 Munchen 2 2083 Lawrenceville Rd.

(MI) PO Box 12081 Germany Chemistry Dept.

Milan, 20134 Lawrenceville, NJ 08648

Italy Paul Brand U.S.A.

University of Maryland/NIST

Mark Beno Bldg. 235, Rm. E151 Rosaria Chiappetta

Argonne National Lab. Gaithersburg, MD 20899 University of Calabria

9700 S. Cass Ave. U.S.A. Arcavacata Di Rende

Bldg. 223, A225 87030 Cosenza

Argonne, IL 60439 Bob Broach Italy

U.S.A. UOP

50 East Algonquin Rd.

Des Plaines, IL 60017

U.S.A.
List of Participants 229

Steve Chipera Briant L. Davis Jaime Femandez-Baca

Los Alamos National Lab. X-Ray Diffraction Laboratory Oak Ridge National Lab.

Mail Stop D469 South Dakota School of PO Box 2008 MS 6393

Los Alamos, NM 87545 Mines and Technology Oak Ridge, TN 37831-6393

U.S.A. Rapid City, SD 57701 U.S.A.

U.S.A.

Chang Sun Choi Paul Fewster

NIST W. Depmeier Philips Research

Bldg. 235, Rm E151 University of Kiel Cross Oak Lane

Gaithersburg, MD 20899 Mineralogisch-Petrogr. Inst. Redhill, RHl 5HA

U.S.A. Olshausenstr 40 UK

D-W-2300 Kiel

Jim Cline Germany Larry Finger

NIST Geophysical Lab.

Materials Bldg., Rm A256 Janet Douglas 5251 Broad Branch Rd., NW

Gaithersburg, MD 20899 Freer Gallery of Art Washington, DC 20015

U.S.A. 12th & Independence Ave. U.S.A.

Washington, DC 20560

Alan A. Coelho U.S.A. C. M. Foris

University of Technology Dupont Central Res. & Develp.

51 Stanley St. East Lars Eriksson Experimental Station

Sydney 2010 NSW Stockholm University Bldg. 228

Australia Arrhenius Laboratory PO Box 80228

S- 106 91 Stockholm Wilmington, DE 19880

Palola Comodi Sweden U.S.A.

Piazza Universita

Dipartimento di Scienze della Sten Eriksson Joseph Formica

Terra University of Goteborg Siemens Analytical X-Ray

1-06100 Perugia Dept. of Inorganic Chem.

6300 Enterprise Lane

Italy Sven Hultius gata Madison, WI 53719-1173

S-41296 Goteborg U.S.A.

David E. Cox Sweden

Brookhaven National Lab. Mario Fomoff

Physics 51 OB Nils O. Ersson International Centre for

Upton, NY 11973 Uppsala University, Inorg. Diffraction Data

U.S.A. Chemistry, Box 531 1601 Park Lane

S-75121 Uppsala Swarthmore, PA 19081

M. K. Crawford Sweden U.S.A.

DuPont Co.

Experimental Station r Michael Estermann Yasuhiko Fujii

E356/209 ETH Zurich, Inst, of Cryst. University of Tokyo

Wilmington, DE 19889-0356 ETH-Zentrum NOP Institute for Solid State

U.S.A. CH-8092 Zurich Physics

Switzerland 7-22-1 Roppongi Minato-Ku

Yongshan Dai Tokyo 106

American Museum of Natural Howard T. Evans Japan

History (Mineral Sciences) U.S. Geological Survey

Central Park West at 79th St. National Center 95Q Klaus Futterer

New York, NY 10024-5192 Reston, VA 22092 Technische Univ. of Berlin

U.S.A. U.S.A. Inst, fur Mineral. & Krist.

Emst-Reuter-Platz 1

Bill David Melanie E. Feather D-1000 Beriin 12

Rutherford Appleton Lab. Smithsonian Institution Germany

Chilton, Didcot, Oxon CAL-MSC

OX11 OQX Washington, DC 20560

UK U.S.A.
230 Accuracy in Powder Diffraction II

Bill Giessen

Northeastern University

Boston, MA 02115

U.S.A.

Ganesh Golikeri

Unilever Research

45 River Rd.

Edgewater, NJ 07020

U.S.A.

Donald Goodwin

Charles Supper Co.

25 Tech Circle

Natick, MA 01760

U.S.A.

Berton Greenberg

Philips Laboratories

345 Scarborough Rd.

Briarcliff Manor, NY 10510

U.S.A.

Boualem Hammouda

NIST

Bldg 235,RmE151

Gaithersburg, MD 20899

U.S.A.

George Harlow

American Museum of Natural

History

Dept. of Mineral Sciences

Central Park West at 79th St. New York, NY 10024-5192

U.S.A.

Richard Harlow

DuPont Co.

CRP E228/316D

Wilmington, DE 19880

U.S.A.

Hiroo Hashizume

Tokyo Institute of Technology

Research Lab. of Eng. Matls.

Nagatsuta, Midori

Yokohama 227

Japan

Daniel Hausermann

European Synchrotron

Radiation Facility

B.P. 220

F-38043 Grenoble Cedex

France

Tomas Havlik

Technical University

Letna 9/A

Dept. Non Ferrous

CS-04385 Ko§ice

Czechoslovakia

Peter Heaney

Princeton University

Guyot Hall

Princeton, NJ 08544

U.S.A.

Helmut Hermann

Inst, of Solid State Res.

Helmholtzstr. 20

Postfach 6

D-0-8027 Dresden

Germany

Alan Hewat

I.L.L.

156 X Grenoble 38042,

France

Rafael Hidalgo

Northeastern University

Boston, MA 02115

U.S.A.

Roderick Hill

CSIRO Mineral Products

PO Box 124

Port Melbourne

Victoria 3207

Australia

T. M. Holden

Chalk River Nuclear Lab.

Atomic Energy of Canada

Chalk River, Ontario KOJ IJO

Canada

Mark Holomany

International Centre for

Diffraction Data

1601 Park Lane

Swarthmore, PA 19081

U.S.A.

Joe Hriljac

Brookhaven National Lab.

Dept. of Physics

Bldg. 510B

Upton, NY 11973

U.S.A.

Jingzhu Hu

Geophysical Laboratory

5251 Broad Branch Rd., NW

Washington, DC 20015-1305

U.S.A.

Qingzmen Huang

University of Maryland/NIST

Bldg. 235, Rm E151

Gaithersburg, MD 20899

U.S.A.

T. C. Huang

Alamaden Research K34/802

IBM Research Division

650 Harry Rd.

San Jose, CA 95120

U.S.A.

Camden Hubbard

Oak Ridge National Lab.

PO Box 2008

Bldg. 4515, MS 6064

Oak Ridge, TN 37831

U.S.A.

Diane Hues

NIST

Bldg. 222, Rm 1113

Gaithersburg, MD 20899

U.S.A.

Matti Jarvinen

Lappeenranta Univ. of

Technology

PO Box 20

SF-53851 Lappeenranta

Finland

William Jackson

Shell Development Co.

3737 Bellaire Blvd.

Room 210

Houston, TX 77025

U.S.A.

Ron Jenkins

International Centre for

Diffraction Data

1601 Park Ave.

Swarthmore, PA 19081

U.S.A.
List of Participants 231

Quintin Johnson

Materials Data Inc. POBox 791

Livennore, CA 94551

U.S.A.

Arthur Jurgensen

Westinghouse

Savannah River Lab.

Aiken, SC 29808

U.S.A.

James Kaduk

Amoco Corporation

PO Box 3011 F-9

Naperville, IL 60566

U.S.A.

Walter Kalceff

University of Technology

PO Box 123 Broadway

Sydney NSW 2007

Australia

Vicky L. Karen

NIST

Bldg. 223, Rm A215

Gaithersburg, MD 20899

U.S.A.

Richard M. Kirchner

Manhattan College

Chemistry Dept.

Bronx, NY 10471

U.S.A.

Peter Klimanek

Mining Academy Freiberg/Sa.

PO Box 47

DO-9200 Freiberg/Sa,

Germany

Vladimir Kogan

Rostov-on-Don State Univ.

Zorge St. 5,

344044 Rostov-on-Don

Russia

Susan Krueger

NIST

Bldg. 235, Rm E151

Gaithersburg, MD 20899

U.S.A.

George A. Lager

University of Louisville

Department of Geology

Louisville, KY 40292

U.S.A.

Vratislav Langer

Chalmers Univ. of Technology

Dept. of Inorg. Chemistry

S-41296 Goteborg

Sweden

J. Ian Langford

University of Birmingham

School of Physics & Space

Research

Birmingham B15 2TT

UK

Armel Le Bail

Laboratoire des Fluorures

University du Maine

72017 Le Mans Cedex

France

Kurt Leinenweber

SUNY at Stony Brook

Earth & Space Science

Stony Brook, NY 11794-2100

U.S.A.

Kenneth Liao

National Steel Corp.

1745 Fritz Dr.

Trenton, MI 48183

U.S.A.

Ulrich Loechner

Forchenweg 1 7100 Heilbronn

Germany

Daniel Louer

University de Rennes

Laboratorie de Cristallochimie

Avenue de Gyn^ral Leclerc

35042 Rennes Cedex

France

John Loveday

Rutherford Appleton

Bldg. R3

Chilton, Oxon 0X11 OQX

UK

Charlotte Lowe-Ma

Naval Air Warfare Ctr-WPNS

Code 3851

China Lake, CA 93555

U.S.A.

Luca Lutterotti

Univerisiti di Trento

Dept. of Materials Engineering

Via Mesiano, 77

Trento, 38050

Italy

Jeff Lynn

NIST

Bldg. 235, Rm E151

Gaithersburg, MD 20899

U.S.A.

Paul Maistrelli

Dipartimento Di Fisica

Povo

Trento, 38050

Italy

Massimo Marezio

AT«&T Bell Labs/CNRS

IC-211

Murray Hill, NJ 07974

U.S.A.

Curtis Martin

NSWC

10901 New Hampshire Ave.

Code R31

Silver Spring, MD 20874

U.S.A.

Lynne McCusker

Inst, of Crystallography

ETH-Zentrum

CH-8092 Zurich

Switzerland

Malcolm McMahon

The Univ. of Edinburgh

Dept. of Physics

Edinburgh EH9 3JZ,

UK

Jan van Mechelen

Koninklijke/Shell Lab

Badhuisweg 3 AG-Dept.

1031 CM Amsterdam

The Netherlands
232 Accuracy in Powder Diffraction II

Ron \ledrud Fang-Ling Needham ben rost

v^nevron iv&scdrcn oc iniemaiionai i^nire lor iniemauonai i^nire lor

iccnnoiogy kjj. DiTiraciion uata Diffraction Data

luu i^nevron way loul i^arK L.ane ioul rarK i^ane

owannmore, rJ\ lyuoi dwartnmore, rA lyuoi

jxicnmona, i^A y4ouz TT C A U.o.A. TT C A U.O.A.

U.S.A.

Diane Nelson Jeitrey rost

Apurva Mehta Air Products and Chemical Smithsonian Institution

Princeton University /ZUl rlamilton olva lOtn & Constitution Ave., INW

11 J Uuyot Hall Allentown, rA lolyj-1301 NHd lly

r^ruiceion, rNJ uoj*!*! TTU .o.C A.A wasmngion, uc zujOU

U.S.A.

Robert Newman

U.S.A.

/vian iviigneii i^ow v^nenucai nenry rrasK

XNlO 1 loy / yj Dlug. iNlo 1

iiiag. zzj, Km AZiD iviiQianu, JVLi 4000/ blag. Zjj, Km rSljl

TT S A vJaimcFoDurg, ivii^ zxfoyy

U.S.A.

John M. Newsam

U.S.A.

dOdynx 1 cciinoiugica inc. i^Darico ricwiii

yooj ocFdnion jvu. ucupnysicdi LJAO. Prkv 111 oan uiego, c-a yzizi jzd 1 broau brancn ko. , jn w

LyncnDurg, va /,**j\)o TTU C A .o. A. wasnmgton, uc zuuij

U.S.A.

Jackie Nicol

U.S.A.

Yukio Mom Umversity or Maryland/INlol Edward Prmce

Japan Atomic Energy bldg Z5j, blDl

ivesearcn insi. vjaimersDurg, wiu zuoyy DlUg. Z^JJf Kill 121.71 Tnlrai Tfiaralri "^10-11

XXCXXXXldl V/CiiCl

USA

/\mui xviucKiK^n lecn umv. r^reioerg Angciine ivcnieiui-v^dKicraKi

JVCoCnl^XX \^CXXI-1C IXUooClXUUXl OlCilldld

!> O TJ 10 r reiDerg/ oa, /j(A) is.arisrune zi

lllHliy VJCXXXuiiXjr

vjci ixuiny

Rosemary Pacalo James Rice

1 C XT 1 u Michael S. Naknmanson oUiN 1 ai olony urooK oun Clin r"/^ P\yf/P-feT^ i\jyi/i\ociy

Comphys Lab Enterpise Earth & Space Sciences pr* 1 1

K

rU box 1 1 Jj

NPO Bourevestnik olony jjrooK, ini ii/y^-ziuu iviarcus tiooK, rA lyuoi

68 Mapookhtinsky Ave. TT C A U.S.A. TT Q A U.o.A.

i7^z/z oi. x^eiersDurg

JvUSSla jonn Jr arise LJX' ixUudXdun

ouxN X ai olony xjfouk

isaDeiia rNamii-oora Earth & Space Sciences Diug. h-juu, ivio 0x10

l^Xi3 X OtUlXy iJlLNJKy X X X / 7^ Half Ridae TN ^78'? 1

iiiUg. ZJJ, jVIIl ClJi TTU .o.<; A.A TT C A

Gaithersburg, MD 20899

U.S.A. Damodara Poojary Stephen B. Robie

Texas A&M University Scintag, Inc. College Station 707 Kifer Rd.

Dept. of Chemistry Sunnyvale, CA 94086

College Station, TX 77840 U.S.A.

U.S.A.
List of Participants 233

Nick Rosov Lyle Schwartz Eric Steel

NIST Materials Bldg./B309 NIST

Bldg. 235, Rm E151 NIST Bldg. 222, Rm 1113

Gaithersburg, MD 20899 Gaithersburg, MD 20899 Gaithersburg, MD 20899

U.S.A. U.S.A. U.S.A.

Daphne Ross Julia Sigalovsky Carin Strom

Smithsonian Institution Massachusetts Institute of University of Goteborg

10th & Constitution Ave.,NW Technology Dept. of Inorganic Chemistry

E408/NHB 119 77 Massachusetts Ave. S-41296 Goteborg

Washington, DC 20560 Rm 12-022 Sweden

U.S.A. Cambridge, MA 02139

U.S.A. Mark Sutton

Frank J. Rotella McGill University

Argonne National Laboratory Husinsyah Sitepu 3600 University St. 9700 S. Cass Ave. Indonesian Institute of Tech. Montreal H3A 2T8

IPNS Division - Bldg. 360 JL Raya Puspiptek Canada

Argonne, XL 60439 Serpong Tangerang

U.S.A. Indonesia J. Timmers

Philips Research

J. Michael Rowe Ewa Skrzypczak Bedrokvem B/V Bldg. WY

NIST Univeristy of Toledo 2.22

Bldg. 235/A106 2801 W. Bancroft St. PO Box 80.000

Gaithersburg, MD 20899 Chemistry Dept 5600 JA Eindhoven

U.S.A. Toledo, OH 43606-3390 The Netherlands

U.S.A.

John J. Rush Brian Toby

NIST Deane Smith Air Products

Bldg. 235, Rm E151 Perm State University 7201 Hamilton Blvd.

Gaithersburg, MD 20899 239 Deike Bldg. RSDl R1102

U.S.A. University Park, PA 16802 Allentown, PA 18195

U.S.A. U.S.A.

Makoto Sakata

Nagoya University Lubomir SmrCok Hideo H. Toraya

Chikusa-Ku, Furo-cho 1 Institute of Inorganic Nagoya Institute of

Nagoya, T464-01 Chemistry Technology

Japan Slovak Academy of Sciences Asahigaoka

Bratislava Tajimi 507

Antonio Santoro Czechoslovakia Japan

NIST

Bldg. 235, Rm E151 Robert Snyder Sue Town

Gaithersburg, MD 20899 NYS College of Ceramics ANSTO PMB 1 U.S.A. Alfred University Menai

Alfred, NY 14802 NSW 2234

David Schiferl U.S.A. Australia

Los Alamos National Lab.

MS C345 Judith Stalick Judy Triphahn

Los Alamos, NM 87545 NIST UOP

U.S.A. Bldg. 235, Rm E151 50 East Algonquin Rd.

Gaithersburg, MD 20899 Des Plaines, IL 60017

Julius Schneider U.S.A. U.S.A.

Inst, fur Kristallographie

Universitat Munchen

Theresienstr. 41

D-8000 Munchen 2

Germany
234 Accuracy in Powder Diffraction II

Shirley Turner P.-E. Werner Marek Zakrzewski

NIST Arrhenius Lab. Procter & Gamble

BIdg. 222, Rm 1113 Stockholm University Pharmaceuticals 13815-0191

Gaithersburg, MD 20899 S- 10691 Stockholm Norwich, NY 13815

U.S.A. Sweden U.S.A.

Vaclav Valvoda Marianne Westdahl Pier Francesco Zanazzi

Charles Univ., Prague Stockholm University Piazza Universita

Ke Karlovu 5 Arrhenius Lab. Dipartimento di Scienze

121 16 Prague 2 S-10691 Stockholm della Terra

Czechoslovakia Sweden 1-06100 Perugia

Italy

Rodolfo Vargas Angus Wilkinson

I.V.I.C. University of California Lev Zevin

Apartado 21827 Materials Dept. Ben-Gurion University

Caracas 1020 A Santa Barbara, CA 93106 Institute for Applied Research

Venezuela U.S.A. PO Box 1025

Beer-Sheva

Mary Vickers George Will Israel

BP University Bonn

Chertsey Rd. Poppelsdorfer Schloss Xinyu Zhang

Sunbury-on-Thames 5300 Bonn Northeastern University

TW16 7LN Germany Boston, MA 02115

UK U.S.A.

Eric Windsor

Jan W. Visser NIST

Henry Dunantlaan 81 Bldg. 222, Rm 1113

2614 GL Delft Gaithersburg, MD 20899

The Netherlands U.S.A.

Robert Von Dreele Winnie Wong-Ng

Los Alamos National Lab. NIST

Lansce, MS H805 Materials Bldg., Rm A215

Los Alamos, NM 87545 Gaithersburg, MD 20899

U.S.A. U.S.A.

Wilem L. Vos XT* 1_ 1 1

* Nicholas Wnght

Geophysical Laboratory Edinburgh University

5251 Broad Branch Rd., NW Edinburgh

Washington, DC 20015-1305 Scotland

U.S.A.

Takamitsu Yamanaka

Sue-Lein Wang Osaka University

National Tsing Hua Univ. College of Gen Educ.

Dept. of Chemistry 1-1 Machikaneyama

30043 Hsinchu, Taiwan Toyonaka Osaka, 560

Republic of China Japan

Donald Weidner R. A. Young

SUNY at Stony Brook Georgia Inst, of Tech.

Earth & Space Sciences School of Physics

Stony Brook, NY 11794-2100 Atlanta, GA 30332

U.S.A. U.S.A.
NIST-114A

(REV. 3-90)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

1. PUBUCATION OR REPORT NUMBER

NIST/SP-846

2. PERFORMING ORGANIZATION REPORT NUMBER

BIBLIOGRAPHIC DATA SHEET 3. PUBUCATION DATE

October 1992

4. TITLE AND SUBTITLE

Accuracy in Powder Diffraction II

Proceedings of the International Conference

May 26-29, 1992

5. AUTHOR(S)

E. Prince and J. K. Stalick, Editors

6. PERFORMING ORGANIZATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

GAITHERSBURG, MD 20899

7. CONTRACT/GRANT NUMBER

8. TYPE OF REPORT AND PERIOD COVERED

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (STREET, CITY, STATE, ZIP) JCPDS-International Centre for Diffraction Data, 1601 Park Lane, Swarthmore, PA

International Union of Crystallography, 5 Abbey Square, Chester CHI 2HU, UK

National Institute of Standards and Technology, Gaithersburg, MD 20899, USA

19081, USA

10. SUPPLEMENTARY NOTES

1 1 . ABSTRACT (A 200-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. IF DOCUMENT INCLUDES A SIGNIFICANT BIBUOGRAPHY OR

UTERATURE SURVEY, MENTION IT HERE.)

The proceedings of the international conference Accuracy in Powder Diffraction II present the

invited papers and abstracts of the papers contributed to the conference, which was held at NIST,

Gaithersburg, Maryland, during May 26-29, 1992. The conference was organized by the

Commission on Powder Diffraction of the International Union of Crystallography, and was jointly

sponsored by NIST, JCPDS—International Centre for Diffraction Data and the International Union

of Crystallography. The proceedings contain 25 invited papers and 73 contributed abstracts. The

program of the conference was divided into six topics: Phase Identification and Quantification;

Accuracy and Standards; New Developments in Software and Data Analysis; Profile Fitting,

Decomposition and Microstructural Effects; Novel Applications and Structural Science; and New

Developments in Hardware, Including Detectors, and Studies under Non-ambient and

Time-resolved Conditions. A ceremonial session was devoted to a tribute to the late William

Parrish and his contributions to powder diffraction. In addition, there were two tutorial workshops

organized by the JCPDS—International Centre for Diffraction Data, one on diffractometer

sensitivity and one on automatic indexing methods.

12. KEY WORDS (6 TO 12 ENTRIES; ALPHABETICAL ORDER; CAPITAUZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)

accuracy; instrumentation; microstructure; phase analysis; powder diffraction; software;

standards; structural science.

13. AVAILABIUTY

UNUMITED

FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NTIS).

X

ORDER FROM SUPERINTENDENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE,

WASHINGTON, DC 20402. ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NTIS), SPRINGFIELD, VA 22161.

14. NUMBER OF PRINTED PAGES

241

15. PRICE

ELECTRONIC FORM * U.S. G.P . 0. : 1992-341 -832 : 60544

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