NISTIR 88-3850

XRAYL: A Powder

Diffraction Profiie

Refinement Program

C. R. Hubbard, National Institute of Standards and Technology;

J. M. Stewart and Y. Zhang, University of Maryland:

B. Morosin and E. L. Venturini, Sandia National Laboratory

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

Electronic Materials Group

Ceramics Division

Gaithersburg, MD 20899

October 1988

National Bureau of Standards became the

National Institute of Standards and Technology

on August 23, 1988, when the Omnibus Trade and

Competitiveness Act was signed. NIST retains

all NBS functions. Its new programs will encourage

improved use of technology by U.S. industry.

U.S. DEPARTMENT OF COMMERCE

C. William Verity, Secretary

NATIONAL INSTITUTE OF STANDARDS

AND TECHNOLOGY

Ernest Ambler, Director

NI STIR 88-3850

XRAYL: A Powder

Diffraction Profile

Refinement Program

C. R. Hubbard, National Insitute of Standards and Technology:

J. M. Stewart and Y. Zhang, University of Maryland;

B. Morosin and E. L. Venturini, Sandia National Laboratory

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

Electronic Materials Group

Ceramics Division

Gaithersburg, MD 20899

October 1988

7S Y««r» SdmulitlrB AmerKb s Progr»s>

XRAYL: A Powder Diffraction Profile Refinement Program

J. M. Stewart and Y. Zhang

University of Maryland

Department of Chemistry

College Park, MD 20742

C. R. Hubbard

Ceramics Division

Institute for Materials Science and Engineering

National Bureau of Standards

Gaithersburg, MD 20899

B. Morosin and E. L. Venturini

Sandia National Laboratory

Albuquerque, NM 87185

ABSTRACT

In many applications of X-ray powder diffraction in material studies, when

overlap occurs it is necessary to separate the total (or composite) profile

into its components. A computer program, XRAYL, has been developed to fit

analytical functions to powder diffraction lines. After fitting, the

powder diffraction data, given as counts at steps in 16 , is represented,

diffraction line by diffraction line, by parameters of a profile function.

The resulting profile parameters may then be used to generate "idealized"

powder diffraction lines, effectively free of statistical noise and

contributions from overlapping lines, extending to background intensity on

each side. Because of this capability XRAYL may be employed in X-ray

powder diffraction profile analysis as a preprocessor program, that is,

separating peaks and feeding the "resolved" data to subsequent analysis

programs. This NBS-IR contains: (1) profile functions and the non-linear

least-squares algorithm utilized in XRAYL; (2) user options and execution

of the program; and (3) several examples.
' ^v

, : 3

"'

^^....v^v

t» :-n'><

l-V.t -

0 11 J: J r|

',f, ^'

f i X t I

- .(- . >i'

•r«

*• n

. ,1 ' I

•

>Y- f .' J ...

i. -

, ;.•& 'K'O ..j i -Y

*• ' V'i * ^ *'.. ^“i

•'w ,..u

'/ , *. .-ij w'V * i

.’ <

, ,. ’v jsttl 4'k^M

i,;,„.: \-, ir:

•

'

'. ««•.»: if*’''.- . i . /yo’i ^y:;.:

..,>•:

• c.- s . V.lfc v'i ‘^ui

r -\f . ' - ;{ ^.i.; Xr 1'4

.*

",

, .,.. .

. r -.%.v , ,.. •’>•• 4

'’., .1 .,0 .. ji,'' i( .. ' .S^SiX *5.,

.

.' :’

i y *[=^- ’ ’

“*• ls>

'

I ., 7

'3

-

iW

y^

'%yj:

, vijf

'; "slii

.....

"

'

; :;v ’-M;'i

.

, , . 7" '.i;i5SS

^

y.: .. .:;
TABLE OF CONTENTS

I. Introduction 1

II. BackEround 3

Parametric Representation of XRD Profiles 3

The General Description of Program XRAYL 4

Non-linear Least-squares Refinement 5

III. Program Files and Execution 7

User Instruction File 7

The Input Raw Intensity Data Files 14

The Output Generated Data Files 15

The Output File of Refined Profile Parameters 16

IV. Execution Examples 17

Gaussian Generated Profiles 17

Resolved XRD Profiles 19

Overlapped XRD Profiles 20

V. Acknowledgements 21

VI Appendix 22

VII. References 23
j M » .•

I *> I ' f> / ! 0 i 7 Ci’i 'f a 5 ' ii U i 49 1 e<t if> ta (Vt\ '£^

.;> i

' ’* fc^'57. «.> Pi.ii *4 «'W

' "• .

'

'

:

^ //.-.r v;-:'B-i I''w^

<t\

i*. '.V

•'

,- ,.t ;; ' . -rX '

• M &»yri

y.

.!

'., , ,h ft V.

''' ‘Wl’, ,7 >t;-.

:

:> r?4.,v , ,.>'
I. Introduction

A powder diffraction profile analysis preprocessor program, XRAYL, has been

written in FORTRAN 77. The purpose of this program is to separate

overlapping x-ray powder diffraction lines with fidelity to their shape.

X-ray powder diffraction is a widely used method to identify and

characterize crystalline phases. The diffraction pattern consists of a series of intensity readings measured as a function of 26 , the scattering

angle. The pattern from a crystalline material consists of a slowly

varying background and a number of narrow lines whose positions are defined

by the unit cell parameters and crystal symmetry. With a small cell and

cubic symmetry, for example, the diffraction lines are well separated and

can individually be measured. However, as the syTiuiietry of a material is

reduced, more lines are observed, and many lines overlap. Further, for the

majority of diffraction systems the Ka^ ,Kq2 doublet remains, causing

additional overlaps in complex patterns. The shape of a diffraction line

is a convolution of the instrumental effects, including monochromatization,

instrumental alignment, and microstructure of the specimen in terms of

crystallite size and residual lattice strain. One of the major goals of

program XRAYL is to separate the individual diffraction lines from the

overlapping total pattern and to present these separated line profiles for

further analysis.

In many applications of X-ray powder diffraction in material studies, when

overlap occurs it is necessary to separate the total (or composite) profile

into its components. Examples of applications requiring separated lines

include measurement of accurate line positions for lattice parameter

determination, of line intensity for quantitative phase and texture

analysis, and of line profile shape for crystallite size and residual

microstrain analysis. The profile fitting technique^ ’ ^ is effective for

decomposing overlapping reflections into individual component profiles

without prior knowledge about the crystal structure of the material.

However, profile fitting can not be applied unless certain specific shapes

are assumed for the line shapes. A few analytical functions have been

reported^ . ^ . 5 represent adequately the shape of a real powder

diffraction line. Functions frequently implemented include the Gaussian,

Lorentzian, Voigt, Pearson VII and "Rational" functions.

Another approach to obtaining microstructure as well as atomic structural

information is the Rietveld^ , or whole pattern, approach to profile fitting

method. This method overcomes most of the limitations in obtaining

accurate structure information from raw powder diffraction data It has

become widely used when single crystals are unobtainable. Parametric

representation of diffraction profiles has been applied with great success

in the Rietveld method. Recently, Young and Wiles^ reviewed the profile

shape functions as applied to neutron studies. Their analysis has direct

bearing in the work in the area of powder diffraction profile fitting for

X-ray data. In the Pvietveld method, a complete pattern is usually collected

and analysed. The typical analysis involves fitting a functional

representation of the instrumental breadth® , a slowly varying background.

1
atomic positions, atomic thermal motion parameters, and in defect

structures a site occupancy®. In some cases, Rietveld analysis is used for

quantitative analysis^® of multiphase samples (usually two phase samples).

Occasionally, microstructure parameters, such as mean coherent domain size

and mean residual microstresses, are also introduced^ ^

.

Typical examples of the difficulties which both the Rietveld and individual

profile fitting methods resolve are; statistical noise in the intensity

data, determination of background intensity, separation of complex patterns

due to large cell parameters or a low symmetry crystal system, and

resolution of overlapping lines due to multiple phases. However, in many

materials the microstructure is quite anisotropic and the determination of

mean values of microstrain with the Rietveld method provides only limited

information. For these cases, the fitting of an individual profile

function for selected peaks in the pattern is required. This profile

fitting approach, compared to use of raw data, provides several important

additional benefits, namely elimination of truncation errors and freedom to

choose the data collection range at will^^

.

XRAYL is a computer program for fitting analytical functions to powder

diffraction lines. After fitting, the powder diffraction data, given as

counts at steps in 26 , are represented, diffraction- line-by-diffractionline, by parameters of a profile function. The resulting profile

parameters may then be used to generate "idealized" powder diffraction

lines effectively free of statistical noise and contributions from

overlapping lines and extending to background intensity on each side.

Because of this capability XRAYL may be employed in X-ray powder

diffraction profile analysis as a preprocessor program, that is, separating

peaks and feeding the "resolved" data to subsequent analysis programs. To

date, we have used XRAYL to analyze selected diffraction regions and to

write the "resolved" data for use in our crystallite size - residual

microstrain package CRYSIZ^^. When broadened and reference profile data

sets are to be processed by XRAYL, they should be analyzed in the same run,

not two separate runs. Analyzing the two data files within a single run

provides the program the ability to prepare the generated data files with

the identical 26 range and with the profile centered at the same 26

(requirements for meaningful Fourier deconvolution) . With this capability

the reference and broadened profile data files may be collected with

different step size and range. XRAYL could be easily modified to write

data files for analysis by some other data processing software, for

example, by the quantitative analysis package NBS*QUANT^^.

This report contains the following information to guide the user in the use

of the program XRAYL: (1) introduction to profile functions, (2) the nonlinear least-squares algorithm utilized by the program, (3) user options

and execution of the program; and (4) several examples.

2
II. Background

A. Parametric Representation of XRD Profiles

In X-ray powder diffraction, the most useful profile functions are those

that can be applied over the entire angular range of data collection. The

profile parameters of these functions must include angle-dependent breadth

and shape parameters. Ideally, the profile parameters should be easily

interpreted by the user and should change slowly as a function of the

scattering angle. The breadth and shape parameters should accurately

account for XRD line asymmetry. The profile breadth is most simply

expressed as the full width at half maximum (FWHM) . It is theoretically

desirable to select profile functions based on the convolution of each

component in line broadening^ ^ . Unfortunately, no parametric

representation for the convolution is available. However, representation

of an X-ray diffraction peak with a simple mathematical function is

convenient even if there is no physical justification for the selection. So

far, the widely used mathematical functions for XRD line profiles are

Gaussian (G) , Lorentzian (L) , modified Lorentzian (ML) , intermediate

Lorentzian (IL) , Voigt (V), pseudo-Voigt (P-V) , and Pearson VII (P-VII)

.

The Rational polynomial function is another model that has been proposed.

Table 1 lists the functions implemented in XRAYL and the refining

parameters in the least- squares refinement. In these functions, x = {26-

26

y

the distance between any 26 position and the maximum position of the

peak, 26^ . 7q and 7l are the FWHM of the peak for the Gaussian or

Lorentzian forms*. N's are the corresponding normalization factors for the

functions. For example, the normalization factor for a Pearson VII function

is: Np_y j j=I«r(m)/[y(7rma) •F(m-l/2) ] ; where I=Jy(x)dx and F(m) is the gamma

function and m and a are the refining parameters.

For the implementation of these functions for profile refinement by XRAYL,

several things should be pointed out:

(1) The forms of Lorentzian, Gaussian, and Pearson VII functions

implemented in XRAYL are expressed in terms of the FWHM of the peak. A

clear benefit of expressing these profile functions in terms of FWHM

is that the starting values for the profile parameters may be obtained

easily and the refined parameters are readily interpreted by the user.

(2) In XRAYL, the asymmetric nature of powder diffraction profiles is

modeled by using "split" functions, that is, by using different

profile parameters of the same functional form for each side of the

peak. The "split" function, left side and right side, shares the same

peak ma::imum and peak positions. The FWHM used in the scaling for the

An e:.ample of the derivation for writing the parametric functions in

terms of FWF21 is given in the Appendix.

3
Table

1.

Profile

Functions

Implemented

in

XRAYL

t>o

w

C 5-1 •r-l (1) <U i- -3 o C <L ?- E

4-1 ci:

u

0) VO o

0^ c: is D- E

CNI 1

CM o X

X

CN • h4 OJc \ -1 CM cf X 1 T3O a + X 0) o p CD E

•H 0) •—1 CO •U E Ui P (1) CO U -4 "w' • O P • CO 2: E • E 1—1 O hJ CM t_i CO c • • CM r—1 CM N

*r^o o o

4-1 ?- \ \ 0) X -3 £

/'“S 1 CM • 'w' r—. N CM • CM 1 CM CS3 'w'

w

a e

/~s CM /'~N

X CM 1—1 c CM X N CD C CM

CO

O 1 '— II •H ^ C CNJ

O «H 3 + •—1

•r-l 4-> + a

N e CM 'w' +

D- cO

X (V II II II >-, cu Dd 3 N e CCl H O \ • • • z 'z Z z

1—1 c M

CO > c •H c o N CO P

•f—1

i-' •H o

4J

CW 4-1

w

O

O

w bO

u c }-i p •rW p p O cO p CD

p-l

O D-i

Rational

N/(

1+Ax^+Bx^ )
split function is taken as the arithmetic mean of the left and right FWHM

parameters

.

(3) In addition to modeling the diffraction profiles, several other

contributions in XRD profiles are also considered. They are a linear

background with constant intensity (b) and slope (m)

,

the scaling factor

(S)

,

and the intensity ratio of the and Q2 lines (R)

.

As the result, the

calculated intensity at every 26 position within a limited 26 range may be

written as:

Icai(2«i) = XS-R-Y(2^,-2«„) + b + m-(i-l)-A(2^),

where, Y. denotes the sura over the contributions to intensity from all

the overlapped peaks and both Qj and Q2 lines; i is the step number

from the beginning of the region; Y is the calculated intensity; and

L{26) is the step size in degrees 26.

(4) The raw data input is normalized such that the strongest intensity

within the data region has the value 1.0. On this scale the background

intensity typically has a normalized value of 0.05 to 0.2. The profile

models, however, are normalized functions that have an integrated area of

1.0. The scaling parameter (S) is the product of the height at maximum

above background of the normalized data times the reciprocal of the maximum

of the model function. A separate scaling parameter is needed for each

peak within the region. XRAYL will estimate S using a symmetric Gaussian

model if the starting value of S in the instruction file is 0.0.

B. The General Description of Program XRAYL

Program XRAYL is designed to operate on "windows" in 26, that is, on

regions in 26 space. The program dimensions permit each window to contain

up to five powder lines; this number may be changed by increasing the array

dimension and recompiling the program.

There are two types of profile parameters. The first are "regional" ones as

they are the same for all peaks within the window. The "regional"

parameters include the background intensity, slope of the background

intensity, and the ratio of 02 to peak intensity. The second type of

parameter depends upon the functional form selected and whether or not an

asymmetric refinement is selected. The maximum nuiTiber of parameters is six

per line: 26 of peak maxima, intensity scale, low and high FvaUM, and low

26 and high 26 shape parameters. The selection of profile model and

symmetry/asymmetry is under user control. The initial values of the

parameters are either estimated by computation within the program or they

may be supplied by the user. Every parameter may be either refined or held

constant

.

The program reads either one or two intensity data files. The first ftle is

called the "reference" file and is used to establish values for an

unbroadened diffraction pattern. The second is a file containing data for

a sample v.^hich shows broadening and is called the "broadened" file. The

4
reference and broadened file may have been collected with different step

size and 26 range. XRAYL will write the generated profile reference and

broadened data file with identical step size (as specified by the user) and

26 range (a3 , as specified by the user). Neither of the designations is

required and one broadened file may be supplied alone if parameter

refinement is carried out appropriately. Another input file is used to

select the profile model and control of the refinement process. This file

is referred to as the "instruction" file.

In the course of the calculations the data input files are read, the

intensity data are stored in immediate access memory, and a survey is made

of the starting parameters. When starting parameters are not supplied in

the instruction file, estimates are made from the input intensity data.

Then, based upon the chosen function to be used in the fitting process, a

non- linear-Marquardt least-squares fitting is carried out. The fitting

subroutines used by XRAYL are those from the International Mathematical and

Statistic Library (IMSL).^^ This fitting process usually produces

analytical curves which match the input raw data quite well.

However, a number of caveats must be given. First, if the initial values of

the parameters are too far from the best values, the fitting may fail. The

most sensitive parameters are those of the 26 maxima. The initial value for

the 26 maximum must be very close to the actual maximum position of the

peak. This problem is greatest when several peaks overlap strongly within

the window. In general, sharp lines of a reference pattern should be used

to establish 26 values for the broadened lines, and the 26 maxima of the

broadened lines should not be refined. If problems arise during

refinement, close attention should be given to the values of all starting

parameters. A symmetric refinement prior to an as3nnmetric refinement can

be useful to estimate good starting parameters in difficult problems. The

estimated FWHM must also be given if the peaks are not cleanly separated

above the half maximum intensity level. A general statement may be made

that any starting parameters that is too far from the refined value may

cause non-linear least-squares to fail.

C. Non-linear Least-squares Refinement

In the least-squares refinement, the function minimized is i ~^c i > where 1^^ and I^^ are the observed and calculated intensities at 26 ^,

respectively, and w^ is a weighting factor. Each intensity may be weighted

by the square root of the number of counts. This means that the fitting

will be closest on the peaks and less dependent on the background or small

peaks within the window. If desired, the weighting can be set to 1.0, and

is recommended when the purpose of refinement is to separate a weak peak

from a much stronger peak.

The IMSL program ZXSSQ^^ continues to refine the param.eters until any one

of three convergence criteria is met. The first criterion is that all

parameters in two successive cycles agree to n (a user-given integer

number) significant digits. The second one is that the relative difference

of the residual sum of squares estimates on two successive iterations is

less than or equal to a specified value, SV. The third criterion is that

5
the norm of the approximate gradient is less than or equal to another

specified value, delta. All these criteria are under user control. The

maximum number of cycles allowed is 1500.

Once the least- squares fitting has been completed, the profile parameters

of the chosen equation are used to generate output intensity data. For

each profile written, the output may have any step size t\16 and 26 range.

Lines, such as 02 or those due to spurious phases may be omitted from the

output file. Components of overlapped regions are individually written to

the output files as completely resolved lines. Besides the separation of

overlapping lines and the smoothing of the raw data, this operation serves

another purpose. First, carefully screened data with constant a.^ values

(regions of 1/d space) are generated for use by the subsequent program

CRYSIZ, which estimates crystallite size and residual strain. The output

parameter data file serves two purposes. First, it may be edited by the

user and then provided as an instruction file for a subsequent refinement.

Second, the parameter file can be used to fit a 2^ -dependent model to each

profile parameter. With the 26 dependence modeled, one can generate

reference lines at any 26 for use with substances where a sample free of

strain and size broadening is not available^®.

The estimates of the errors of the refined profile parameters are obtained

from the Jacobian matrix produced by the IMSL program ZXSSQ^^ . As overall

evaluation for the fitting, two quantities are calculated. One is called

the correlation of fit (COF) , which is also based upon the Jacobian matrix

Its definition can be seen from the formula:

COF = la -I ) -Z(i obs ave obs cal'

la K a ) obs ave

where lots' ^ave > ^cai observed, average, and calculated

intensities, respectively. For a perfect fit, COF should equal 1, and for a

random fit, it should equal 0. The other measure of fit is the conventional

R factor, which has been widely used in both single crystal and powder

diffraction. The definition of the R factor in XRAYL is:

R =

2 ^

obs

Data for Size and Strain Error Estimates

For the purpose of estimating errors in the subsequent data analysis, such

as in size and strain analysis, two additional files may be written. One

is for profiles generated using the refined parameter values plus X'^sigma

and the other one is for X^sigma below, where X is any number and sigma is

the standard devi-ation. The values of X and the number of profile

paramieters which will be modified may be specified by the user. These

files permit calculation of a numerical estimate of the limit of errors in

predicting size and strain parameters by the program CRYSIZ.

6
III. Program Files and Execution

The user's instructions and one or two raw data files are the input

required by XRAYL. The program XRAYL produces up to four "idealized"

diffraction data files and one refined profile parameter file. The program

writes each of these files to a "logical unit". The corresponding unit

numbers are:

User instruction file unit 8;

Input raw data files unit 20 - obs . reference profiles

unit 21 - obs . broadened profiles

Output parameter file unit 9;

Output generated unit 10 - gen. reference profiles;

data files unit 12 - gen. broadened profiles;

Output generated unit 11 - gen. broadened profiles

data files for error • with minus X's^sigma;

analysis unit 13 - gen. broadened profiles

with plus X^sigma

Following is the detailed description about the contents and the formats of

these files.

A. User Instruction File.

The user instruction file, unit 8, contains several lines that provide the

problem title, wavelength of the radiation used, and convergence criteria

for the non-linear refinement. In order to distinguish the records easily,

the first field of each record is reserved for the record name. The last

line of unit 8 must contain the three characters "END" which indicates the

end of the instruction file. All of the input items are in free format, and

therefore none of these items can be omitted.

Each of the following records must be included in unit 8. Records 1 and 2

provide the general information for the problem and will be used for all

diffraction angle regions.

7
Record 1: TITLE, an alphanumeric title for the problem

Field #1

TITLE

Field #2: Alphanumeric title information

Record 2: GENREF, provides the general information for the least-squares

refinement for all the diffraction data. It contains the

following fields:

Field #1

GENREF

Field #2: wavelength of Kq^ in units of A; The default value is

that for Cu target ( A (K^ ^ )=1 . 5405981A)

;

Field #3

Relative intensity weight of . The default is 1.0;

Field #4: Wavelength of Kq2

• Tbe default is that for Cu target

(A(K,2)=1.5444342A):

Field #5: Relative intensity weight of Kq2

• Th® default is 0.5;

Field #6

Signal for whether lines alone or and Q2 components to the profile are to be generated and

placed in the output files. If Y (for yes) the Q2 be included with weight as specified by Field #5. If N

(for no) the 02 component will be omitted from the

output files.

Field #7: The first convergence criterion of least-squares

fitting. It is an integer used in the significant

digit test based on the difference between parameter

values of two successive fitting cycles. That is, if

any two successive cycles agree, parameter by

parameter, to within this number of significant digits,

the fitting is considered to have converged. The

default value is 5.

Field #8

The second convergence criterion of least-squares

fitting. If the difference of the residual sum of

squares estimates of any two successive cycles is less

than or equal to this number, the fitting is considered

converged. The default value is 1.0x10"^.

Field #9

The third convergence criterion of least-squares

fitting. If the (Euclidean) NORN of the approximate

gradient is less than or equal to this number, the

fitting is considered converged. The default value is

0.0.

8
Field #10: A signal to create additional output files for error

analysis. 1/0 for generated/not generated. When the

input signal is 0, only one output file will be

generated for the broadened data; the refined profile

parameters are used in the data generation. This best

estimate output data file of generated profiles will be

written to unit 12. If the input signal is 1, two

additional data files will be generated for the

broadened data. One of them is generated using the

parameters obtained by subtracting n^^sigma from the

refined parameter j , where n is a number specified by

the user, sigma is the least-squares estimated standard

deviation, and j is the j-th parameter for the region.

This file is written to output unit 11. The other file

is generated using the parameters obtained by adding

nj*sigma to the j th refined parameter. This file is

written to output unit 13. The purpose of this feature

is to provide the user the means to estimate the errors

in the diffraction data analysis with a subsequent

program.

Following the GENREF record, the instruction file provides the information

needed for the least-squares refinement, window by window. For each window

in 2^, there must be a record that specifies the number of reflections in

the window and some refinement instructions applicable to all the peaks in

the window (WINREF) . This record must then be followed by four records per

reflection within the window which give specific information and other

refinement instructions for each reflection. Note that in these records,

all diffraction angle related quantities must be in the units of degrees

26 .

The record WINREF provides the universal instructions for the refinement

for the window. It contains the following items:

Field #1: WINREF

Field #2: The number of reflections in the window. Maximum

number is 5

.

Field #3

Field #4:

Field ?=5

The refinement function to be employed.

1 Lorentzian

2 Gaussian

3

Voigt

4 Pearson VII

5 Rational

A signal to indicate if a split function is to be used,

1/0 for split function/symmetric function refinement.

A signal for whether one or two input raw data files

are to be treated. 0/1 for one/two raw data files.

9
Field # 6 :

Field #7

Field # 8 :

Field #9

Field #10:

Field #11:

Field #12:

The number of background points on each side of the 16

region. These background points will be used to

calculate the starting values for the background

intensity and its slope.

A signal for whether background intensity will be

excluded or included in the output profile data. 0/1

for exclude/include the background intensity in the

output

.

The 33 value, the period or spread in Angstroms, for

the output profile data. The value of a.^ sets the 1/d

scan range in the output data. If 33 is outside the

range 15 to 100.0 A, the minimum value of 33 in the

experimental input file will be used.

The step size in degrees 26 for the generated output

profile data.

A signal for printer plots.

0

no plots

1 plot fitted curves only

2 plot generated curves only

3 plot input raw data only

4 plot everything

A signal for whether background intensity will be

refined or held constant. 1/0 indicate refined/held

constant. Default is 1.

A signal for whether background intensity slope will be

refined or held constant. 1/0 for refined/held

constant. The default value is 1.

Field #13: A signal for whether the intensity ratio of the and

Q2 lines will be refined or held constant. 1/0 for

refined/held constant. The default is 0. It is

recommended that this parameter be held constant for

peaks at low 26 containing and 02 .

Field #15: A signal for whether the line position of the

diffraction peaks in the "broadened" data file will be

held constant in the multi- line refinement. When it is

requested to hold them constant, the corresponding

refined 26 positions of the "reference" peaks will be

assigned to them. This feature is useful when rather

severe overlap occurs to the "broadened" peaks due to

line broadening. Y/N for hold them constant/not hold

them constant.

10
Field #14: A signal for whether the intensity difference in the

least squares refinement will be weighted by a counting

statistics factor or not. Y/N for weighted/not

weighted.

Following the general information for the window, there must be four

additional instruction records for each of the reflections in the window.

The first line of these four, denoted as LINERF, contains reference line

information:

Field #1

Field #2

Field #3:

Field #4:

Field #5:

Field #6

Field #7:

Field #8

Field #9:

LINERF

hki , the indices of the peak, 312.

26 position of the peak. Note that the 26 value

given must be very close to the actual values in the

input raw data file. If this condition is not met the

least-squares may fail to converge.

User provided starting value for the intensity

compression scaling factor. When input is 0.0, this

factor will be estimated in the program.

Estimated FWHM in degrees 26 for the reference peak,

symmetric case. If Fields 4, 5 and 6 are all 0.0, then

the necessary FWHM values will be estimated within the

program. Input of 0.0 is recommended if there is only

one peak in the window. If multiple peaks are present,

reasonable estimates by the user will aid convergence.

For the rational polynomial function. Field #4 is the

refining parameter A for the symmetric case.

Estimated FWHM for the left side, reference peak.

(or Aj^** for Rational polynomial model)

Estimated FWHM for the right side, reference peak.

(or Ag for Rational polynomial model)

Estimated starting shape parameters of the reference

peak, symmetric case. Default value is 1.0. (Pearson

VII, Voigt and Rational functions. Equals B, another

refining parameter for Rational polynomial)

.

Estimated shape parameter, left side of reference peak,

(or Bg for Rational model)

Ag , Ag , Bl

,

Bg ate the low and high angle side refining parameters

for a split Rational function.

11
Field #10:

Field #11:

Estimated shape parameter, right side of reference

peak, (or B|^ for Rational model)

Signal for whether the current peak is to be included

in the output data set or not.

0 current peak will be included in the output.

1 current peak is treated as impurity peak and

will be excluded from the output.

The second line of these four, denoted as LINEBR, contains broadened line

information. Only the parameter portions should be different than the

values for the reference line. If only a reference data set is to be

analyzed, the input of this line is still required. However, in such

cases, this line may be simply duplicated from LINERF. The fields of

LINEBR are:

Field #1: LINEBR

Field #2: hki , the indices of the peak (same as for reference

line)

Field #3: 2d position of of the peak. The value will be reset

to the refined value for the reference line and not

refined

.

Field #4: User-provided starting value for the intensity

compression scaling factor. When input is 0.0, this

factor will be estimated in the program.

Field #5

Estimated FWHM in degree 26 (or A of Rational

polynomial) for the corresponding broadened peak,

symmetric case. If Fields 4, 5 and 6 are all 0.0, the

needed values will be estimated within the program.

Field #6

Estimated FWHM (or A) for the left side, broadened

peak.

Field #7

Estimated FWHM (or A) for the right side, broadened

peak.

Field #8: Estimated starting shape parameter (or B of Rational

polynomial), broadened peak. Default value is 1.0.

(Pearson VII, Voigt and Rational functions)

Field #9

Estimated starting shape parameter (or B) for the left

side of the broadened peak.

Field #10: Estimated starting shape parameter (or B) for the right

side of the broadened peak.

Field #11: Same as the Field #10 in LINERF.

12
The next record, PARCON, is the profile parameter refinement control line

for the current reflection. It provides signals indicating whether each of

the profile parameters will be refined or held constant. When it is a 1

,

then the corresponding parameter will be refined. When it is a 0 the

parameter will be held constant. The reference and broadened data

refinements will be refined identically, except that the 26 values for the

broadened profiles will be held constant. The sequence of these signals

and their default values are as follows:

Field #1: PARCON

Field #2: Refine 26 position of the current peak. Default is 1.

Field #3Refine intensity scale. Default is 1.

Field #4: Refine FWHM of symmetric mode. Default is 0.

Field #5Refine FWHM of the left side of the peak when split

function mode. Default is 0.

Field #6: Refine FWHM of the right side of the peak when split

function mode. Default is 0.

Field #7: Refine shape parameter of symmetric function mode.

(Pearson VII, Voigt and Rational functions). Default

is 0

.

Field #8

Refine shape parameter of the left side of the peak

when split function mode. Default is 0.

Field #9: Refine shape parameter of the right side of the peak

when split function mode. Default is 0.

The last record for a diffraction lines is SETSIG which is paired with the

PARCON record,

multipliers of

specified when

value between -

It contains eight numbers, which are the user specified

sigma for each of the parameter errors. These must be

output data files for the error analysis are requested. A

3.0 to 3.0 is recommended. These numbers are for the eight

profile parameters defined above for record PARCON. The sign of the

multiplier affects the profile shape differently for different parameters

For example, using +n for the FWHM parameter will broaden the peak. With

the Pearson VII model, the shape parameter should be changed by -n to

broaden the peak. Thus the greatest change of a Pearson VII profile is

achieved by using +n for the FWHM and -n for the shape parameter.

Field #1SETSIG

Field #2 - #9: "n" multiplier value for each parameter defined under

13
PARCON. Default values are 0.0, Leave blank if field

#10 of GENREF is 0.

The records LINERF, LINEBR, PARCON and SETSIG repeat diffraction peak by

diffraction peak for lines within the 26 window. Each new region or window

begins with a WINREF record. The last line in the instruction file will be

a record which contains only three characters, END, in its first three

columns

.

Table 2 is an example of a short instruction file. This instruction file

is for least-squares fitting of the raw data collected for FegO^. This raw

data set contains two diffraction regions or windows. The first region has

only one peak, (111), and the second region has two peaks, (311) and (222).

Table 2, Example of Instruction File (unit 8)

TITLE Fe3 04 reference powder. bottle #3456.

GENREF 1.5405981 1.0 1.54433 0. 5 Y 4 1. OE-5 l.OE-8 1 WINREF 1 4 1 1 10 0 30.0 0.005 4 1 0 0 N Y

LINERF 1 1 1 18.269 0.0 0 .0 0 .0 0 .0 0-.0 0 .0 0.0 0

LINEBR 1 1 1 18.269 0.0 0 .0 0 .0 0 .0 0'.0 0 .0 0.0

PARCON 1 1 0 1 1 0 1 1 SETSIG 0.0 0.0 0.0 3.0 3.0 0.0 -3.0 -3.0

WINREF 2 4 1 1 10 0 30.0 0.005 4 1 0 0 N N

LINERF 3 1 1 35.422 0.0 0.0 0.0 0.0 0.0 0.0 0. 0 0

LINEBR 3 1 1 35.422 0.0 0.0 0.0 0.0 0.0 0.0 0. 0

PARCON 1 1 0 1 1 0 1 1 SETSIG 0.0 0.0 0.0 3.0 3.0 0.0 -3.0 -3.0

LINERF 2 2 2 37.052 0.0 0.0 0.0 0.0 0.0 0.0 0. 0 0

LINEBR 2 2 2 37.052 0.0 0.0 0.0 0.0 0.0 0.0 0. 0

PARCON 1 1 0 1 1 0 1 1 SETSIG

END

0.0 0.0 0.0 3.0 3.0 0.0 -3.0 -3.0

B. The Input Raw Intensity Data Files

The raw data files, containing the observed intensity data for the

reference and broadened materials, are read from logical units 20 and 21,

respectively. In the case of one input raw data file it should be assigned

to unit 20.

The format for the raw data files is given below. The file structure

begins with a title record for the first region followed by a region

definirion record and the intensity data for each step across the region.

The region information records contain the indices, sample ID, beginning

and ending angles, counting time at each data point, and the step size.

Every intensity datum occupies an individual line. Record types 1, 2, 3

are repeated for each region in the data file.

14
RECORD 1 TITLE

Field #1 TITLE (A5)

Field #2 Sample ID (A12)

Field #3 hkl indices in parentheses, 312.

Field #4 hhl indices in parentheses, 312.

RECORD 2 REGION DEFINITION

Field #1 Number of data points

Field #2 Count time in seconds

Field #3 Starting 26 for region

Field #4 Ending 26 for region

Field #5 Step size in 26

(15)

(F10.2)

(F10.4)

(F10.4)

(F10.4)

RECORD 3 INTENSITY DATA (repeats for the number of data points)

Field #1 Intensity (F11.2)

TITLE Ba2SM600C (030) (030)

161 7.90 22.0000 23.6000

3438.97

3444.11

3420.28

3451.28

3439.04

3420.00

3428.00

3513.00

3453.00

3579.00

3508.00

3427.00

3397.00

3414.00

.0100

C. The Output Generated Data Files.

The generated data are written to units 10, 11, 12, and 13, depending on

data input and user options. Units 10 and 12 contain the data generated by

using the refined profile parameters directly from the least-squares

results. Unit 10 contains the generated profiles corresponding to the

reference data read from unit 20. Unit 12 contains the generated profiles

for the broadened data read from unit 21, if input. In the case of a

single input data file, the output will be written to unit 10. The data

15
written to units 11 and 13 are the modified profile data for use in

subsequent error analysis algorithms.

The input raw intensity data file and the generated output file have

essentially the same structure given in Section B above. The only

difference between the input and the output files is that in the generated

output file each region contains only one reflection while in the input raw

data each region may contain one or several peaks. Thus the output may

have several regions, each with a different "isolated" peak, covering the

same 16 region.

D. The Output File of Refined Profile Parameters

The refined profile parameters are written, peak by peak, into an output

file on unit 9. Unit 9 also duplicates the control information for the

least-squares process. The format of this file is identical with that of

the user instruction file (logical unit 8) defined above. Thus, the output

parameter file can be used to establish the starting values of a second

refinement. The ability to restart a refinement is particularly useful

when the refinement is constrained on the first few cycles (for example,

refining only background and 16 values). To prepare an instruction file

for subsequent refinements unit 9 can be edited with a text editor and used

as the instruction file, unit 8, for the next refinement.

16
IV. Execution Examples

In this section, three example analyses are presented as a guide to use of

program XRAYL. The first one uses a generated diffraction data set of

Gaussian form. This generated profile, reference and broadened, is refined

using all five implemented profile functions. The other two examples are

the Pearson VII refinement results for experimental data collected for LaBg

powder and a gadolinium barium cuprate powder. The former material gives a very good distribution of narrow diffraction peaks without any overlap,

evenly spaced in 26 . Many of the reflections of the latter material are

heavily overlapped. Therefore, the last example will show the peak

separation feature of XRAYL.

1. Gaussian Generated Profiles:

In this example, two sets of input "raw" data were generated utilizing a

symmetric Gaussian function. One set was the reference data which is

assumed to be free of strain and size broadening in its diffraction peaks.

The other set was the broadened data which was assumed to have a size of

lOOA and strain of 1x10"^***. These two files were then fed into the

program XRAYL to be refined by both symmetric and split profile models of

Gaussian, Pearson VII, Voigt, and Rational forms. The purpose of these

refinements was to test the program and to assure that the least-squares

refinement works appropriately. In these test runs, two parameters,

background intensity slope and the q2 and q1 intensity ratio, were held

constant. Table 3 lists the refinement results for the various symmetric

profile models on the simulated Gaussian "broadened" data. Table 4 lists

the results for the asymmetric models.

The refinement results show that, in addition to a Gaussian function, a

Voigt function, either symmetric or split, also fits the input "Gaussian"

profile perfectly. This result is not surprising because the Voigt

function is a convolution of Gaussian and Lorentzian functions. The

results in Tables 3 and 4 also show that the Pearson VII and the Rational

function profiles exactly match the input profile as well. Note that the

resultant shape parameters, m's in P-VII case, are reasonable in the sense

that they reflect the Gaussian nature of the input data (according to

theory, for m=l the P-VII is Lorentzian, and for m=large number, the P-VII

is Gaussian), All of the refinements have a perfect COF (=1) and a very

small R factor except when the Lorentzian function was used.

If the crystallite size and strain distributions are assumed to be in

Gaussian form, the following relationship among the FWHM of the overall

obser\^ed peak, size, and strain values exists: (^cos^/A)^ = (1/D)^ +

16e^ (s'n^/A)^

;

where /3, D, € are the FWHM, size, and strain values,

respectively

.

17
FUNC B

Table 3. Syminetric Refinement Results for Simulated

Symmetric Gaussian Profile

26 (^) N 2 S COF R

Sy-G .0001

+.00001

28.538

T.OOO

.4881 .9171

T.OOOO T.OOOO

1.0 .000

Sy-L -.3554

T.0051

28.538 1.2761 1.1915

T.0004 T.0096 T.0049

.9997 .014

Sy-V .0000

T.OOOO

28.538

T.OOO

.4064 .9172

T.OOOO T.OOOO

.0000

T.OOOO

1.0 .000

Sy-P -.0244

T.0012

28.538

T.OOO

.5180 .932

T.0014 T. 01971

12.11

T.51

1.0 .002

A B

Sy-R - .0751

T.0012

28.538

T.OOO

1.0733 11.620

T.0012 T.024

95.62

T.89

1.0 .003

Table 4. Asymmetric Refinement Results for Simulated

Symmetric Gaussian Profile

FUNC B 26 (^) N -Yt Tr COF R

Sp-G .0001

+.00001

28.538

T.OOO

.4881 .9172 .9171

T.OOOO T. 00001 T. 00001

1.0 .000

Sp-L -.3553

T. 00521

28.538

T.OOl

1.2764 1.1915 1.1914

T.0096 T.0057 T.0057

.9997 .0141

Sp-V .0000 28.538 .4064 .9172 .9172 .0000 .0000 1.0 .000

+.0000 T.OOO T.OOOO T.OOOO T.OOOO T.OOOO T.OOOO

Sp-P -.0225

+.0010

28.538

T.OOO

.5155 .931 .931

T.0012 T.022 T.022

13.13 13.13

TO. 61 TO. 61

1.0 .002

Aj_ Ag Bj_ Bg

Sp-R -.0751

+.0012

28.538

T.0006

1.9732 11.619 11.621

T.0012 T.073 T.073

95.63 95.62

T.97 T.97

1.0 . 0032

Sy = Symmetric, Sp = ^-lit; G = Gaussian;

P = Pearson VII; R = P>.ational function; S

symmetric function; Sl and Sg = Low

L = Lorentzian; V = Voigt;

= Shape parameter for a

and High angle side shape

parameters

.

18
In the test where the input data were generated in the form of a symmetric

Lorentzian profile, the refinement using a Lorentzian function gave a perfect fit (results not shown) while a relatively poor fit occured when a

Gaussian function was used. Again the Voigt, Pearson VII and Rational

functions were excellent model profiles.

2. Resolved XRD Profiles;

In example 2, experimental X-ray powder diffraction data collected for a

LaBg powder^® (single input) was refined by using a split Pearson VII

function. In the LaBg pattern, each peak is well resolved and no overlap

occurs. Therefore, the starting values for the FVHM, 7, of all the peaks

were estimated by the program. The shape parameters, however, were set to

2.5 for both sides of the peak. Table 5 lists the refined profile

parameters along with the 26 positions (uncorrected for instrumental

aberration). The intensity R factor for every peak is less than 2%, and

most of them are around 1.5%, indicating an excellent fit. Figures 1 a-c

show the quality of the fit for a low, intermediate and high 26 peak.

From this example, it is clear that with the least- squares algorithm used

in XRAYL, the P-VII function can fit. real experimental diffraction data

very well. Although the correlation of the refinement among different

peaks is not yet clear at this stage, a good match for each individual peak

may be obtained by an appropriate refinement. Moreover, smooth variations

of the FWHM parameters versus 26 are observed. This suggests that such

fitting provides the possibility to establish a model of the profile

parameters as a function of 26 . Such dependencies of the profile

parameters on 26 would be very useful in generating a reference peak at any

arbitrary 26 position. To meet this need, a LaBg powder for use as a

instrumental profile standard reference material in X-ray powder

diffraction analysis is being certified by NBS

.

Table 5. Pearson VII Fitted Profile Parameters for LaBg

Peak o| CM 'Yt 'Yh

110 30.374 .1461 .0800

111 37.426 .1314 .0821

200 43.489 .1171 .0823

210 48.940 .1246 .0842

211 53.972 .1270 .0837

220 63.195 .1305 .0912

311 75.819 .1328 .0988

222 79.843 .1342 .1029

400 95.636 .1330 .1100

420 111.888 .1418 .1395

332 120.676 .1559 .1709

422 130.364 .1821 .1979

510 141.718 .2177 .2493

Sl Sh COF R

0.902 2.249 .9995 .0190

1.101 2.275 .9993 .0209

1.148 2.899 .9995 .0174

1.218 1.925 .9996 .0160

1.490 1.768 .9997 .0147

2.491 2.070 .9996 .0135

2.060 1.474 .9996 .0149

1.450 1.206 .9995 .0143

2.060 1.303 .9997 .0103

1.117 1.035 .9997 .0119

0.975 0.890 .9997 .0il5

1.008 0.899 .9996 .0112

0.942 0.892 .9995 .0164

19
3. Overlapped XRD Profiles:

The material for this example is GdBa2 Cu307 , a rare earth high

superconductor powder (SpGr=Pmmm, a= 3,8466, b= 3.8958, c= 11,6794 A).^°

In the pattern of this material, there are several overlapped peaks.

Without appropriate peak separations, no diffraction profile analysis would

be possible. Therefore, this material serves as a good example of using

XRAYL to separate heavily overlapped peaks. Figure 2 shows the raw pattern

of this material, the composite Pearson VII curve and the difference in the

region of 32.5 degrees 26 where peaks (013) and (103) are severely

overlapped. The least- squares refinement was carried out by using split

Pearson VII function. The starting values for this refinement are given in

Table 6. In the refinement, two parameters, the slope of background

intensity and the ratio of the Q2 to intensities, were held constant.

It was noticed that the starting values of the 26 positions of the two

peaks are particularly important because altering the 26 by 0.05° would

cause the intensity R factor to increase dramatically. Table 6 also shows

the refined parameters and their errors as well as the correlation of fit

and the intensity R factor. Figure 3 shows the separated (013) peak and

(103) peak along with their composite raw data curve. These two profiles

were generated by using the fitted profile parameters with the background

intensity removed.

From the results shown in Table 6 and the Figures 2 and 3, it can be

concluded that a very severe overlapped pattern may still be separated into

its component profiles. This separation is essential to the subsequent

data analysis, such as crystallite size and residual microstrain. In our

example, the refinement using a split P-VII function gave very good values

for COF and the R factor (1.1%). These values suggest that the generated

curves reliably represent the raw data curves very closely. It should be

pointed out, however, the fitting errors of the high 26 or "right" side

shape parameter of the first peak, (013), and the low 26 or "left" side

shape parameter of the second peak, (103), are relatively large. This is

due to the overlap because the shape param.eter reflects mainly the tailing

part of the profile and in this case, the right side tail of (013) and the

left side tail of (103) are superimposed. Therefore, the refinement is

less reliable. Such difficulties of refinement revealed in this example may

suggest a further enhancement to XRAYL, namely, introducing constraints

into the program. For example, if the shapes were assumed to be identical

within the window, then the number of variables would be reduced and the

ability to reasonably separate closely overlapped peaks enhanced.

20
Tab 1 e 6 . Profile Fitting the (013) and (103) Peaks of GdBa2 Cu3 07

Starting Final Fitting Fitting

Parameters Line Values Values Errors Control

B .0523 .0498 .0015 Fitted

ra .0031 .0031 — Held constant

Ratio .50 .50 — Held constant

26 {°) (013) 32.540 32.548 .002 Fitted

N (013) .1042 .039 .011 Fitted

7l (013) .5225* .179 .023 Fitted

7h (013) .5225* .159 .056 Fitted

Sl (013) 1.000 1.36 .32 Fitted

Sh (013) 1.000 1.96 1.43 Fitted

2^(°) (103) 32.720 32.769 .001 Fitted

N (103) .149 .073 .007 Fitted

7l (103) .5225* .203 .033 Fitted

7h (103) .5225* .143 .007 Fitted

Sl (103) 1.000 1.51 .52 Fitted

Sh (103) 1.000 1.85 .16 Fitted

Correlation of Fit: .9998 Intensity R factor: 0113

* estimated in program

V. Acknowledgements

This work is from a dissertation to be submitted to the graduate school at

the University of Maryland by Y. Zhang in partial fulfillment of the

requirements for the PhD degree in Chemistry. Computing support for Y.

Zhang was furnished by the University of Maryland Computer Science Center.

This work is partially supported by the U.S. Department of Energy, Contract

#DE-AC04-76DP00789 , U.S. Department of Commerce, NBS Grant 70N-ANB-7H0710

,

and the NBS Office of Standard Reference Materials.

21
VI . Appendix

The following is an example of the derivation to write the parametric

functions in terms of their full width at half maximas (FWHM's). The

function used in this example is a Gaussian:

I =

1 exp( )

From equation (1): when x=0 , I

when x=x^, I^

( 1 )

1

Hence

aj'l'n ’

1

and

X

oJT.'k

exp(

2af-)

where x^ and are the position and intensity at FWHM

2

If 1 X - ^ = — = exp(- -^)

o 2a

( 2 )

and by definition, FWHM = 2x^, FWHM^ = 8a^' ln2

Then it follows that:

> FWHM

" 2jln2

2 " = ~"81n2

Substituting (3) and (4) into (1) gives:

(3)

(4)

'/(41n2)

yir-FWHM exp [ -41n2 • (‘ X

FWHM

)^]

22
VII. References

1. Taupin, D., (1973), J. Appl. Cryst., 6, p266

2. Parrish, W. , Huang, T. C., and Ayers, G. L. , (1976), Trans. Am.

Crystallogr, Assoc., 12, p65

3. Mignot, J. and Rondot, D., (1976), J. Appl. Cryst., 9, p460

4. Hall, M. M. Jr., Veeraraghavan , V. G., Rubin, H., and Wincheil, P. G.,

(1977), J. Appl. Cryst., 10, p66

5. De Keijser, Th. H., Langford, J. I., Mittermijer, E. J., Vogels, A. B.

P. , (1982) , 15, p308

6. Rietveld, H. M. , (1969), J. Appl. Cryst., 2, p65

7. Young, R. A. and Wiles, D. B., (1982), Profile Shape Functions in

Rietveld Refinements, J. Appl. Cryst., 15, p430

8. Cagliotti, G., Paoletti, A. and Ricci, F. P., (1958), Nucl. Instrum., 3,

p223.

9. Beech, F. , Miraglia, S., Santoro, A. and Roth, R.S. (1987), Phys . Rev.

B, 35, p 8778.

10. Hill, R. Presentation at the International Union of Crystaliographers

,

Perth, Australia, 1987.

11. Howard, S.A. (1984), Ph.D Thesis "Incorporating (W*G) Deconvolution in

X-ray Rietveld Structure Refinement, Volume 1", Alfred University.

12. Zhang, Y. , Stewart, J. M. , Hubbard, C. R. , Morosin, B., (1987),

Advances in X-ray Line Pprofile Analysis to Determine Microstructure

Information about Ceramics, in The Proceedings of the First International

Conference on Ceramic Powder Processing Science , in press.

13. Zhang, Y. , Stewart, J., Hubbard, C.R., Morrison, B., (1988), CRYSIZ: A

Computer Code for X-ray Powder Diffraction Profile Analysis, paper in

preparation.

14. Hubbard, C.R. and Snyder, R. L. , (1988), Powder Diffraction, 3, p74.

15. Klug, H. P. and Alexander, L. E., (1974), X-Ray Diffraction Procedures,

John Wiley and Sons, New York, N.Y., 2nd Edition.

16. National Bureau of Standards, (1984), Guide to Available Mathematical

Software, NBSIR84-2824

.

23
17. Zhang, Y. , Stewart, J. M. , Morosin, B., Graham, R. A., and Hubbard, C.

R, , (1987), X-ray line broadening study on shock-modified hematite.

Advances in X-ray Analysis, in press.

18. Stewart, J. M. , Zhang, Y. , Hubbard, C. R. , Morosin, B,, (1988), A

General Reference Material for X-Ray Powder Diffraction Profile Analysis,

Paper in preparation.

19. Fawcett, T. G., Crowder, C. E., Brownell, S. J., Zhang, Y. , Hubbard,

C. R, , Schreiner, W. , Hamili, G. P., Huang, T. C., Sabino, E., and

Langford, J. I., (1988), Establishing an Instrumental Calibration Standard

for Powder Diffraction Analysis: International Round Robin Conducted by the

JCPDS-ICDD and the U.S. National Bureau of Standards, Powder Diffraction,

in press.

20. Paretzkin, B. amd Wong-Ng W. , private communication. Data to be

submitted to Powder Diffraction.

24
RAW

DATA

CURVE

OF

INPUT

REFERENCE

PEAKS

CO

CN

A1ISN31NI 3AllV13d

THETA

IN

DEGREES
o

o

)

/

cJ

GO

o

AilSNjINI 3AllV13y

THCTA

IN

DEGREES
RAW

DATA

CURVE

OF

INPUT

REFERENCE

PEAKS

A1ISN31NI 3AliV13d

138

140

142

144

146

THETA

IN

DEGREES
ro

A11SN31NI 3AllV13d
RAW

AND

GENERATED

CURVES

FOR

REFERENCE

DATA

32.

33.

34.

THETA

IN

DEGREES
:BS«114A <REV- 2-8C1

1. PUBLICATIOI^ OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

NISTIR 88-3850 OCTOBER 1988

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

' SHEET (See instruction s) L TITLE AND SUBTITLE

XRAYL: A Powder Diffraction Profile Refinement Program

5. AUTHOR(S) Stewart and Y. Zhang, University of Maryland; C. R. Hubbard, National

Bureau of Standards; B. Morosin and E. L. Venturini, Sandia National Laboratory

>. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

t. Type of Report & Period Covered

I. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, State, ZIP)

0. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FlPS Software Summary, is attached.

1. ABSTRACT (A 200-^ord or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

In many applications of x-ray powder diffraction in material studies,

when overlap occurs it is necessary to separate the total (or composite)

profile into its components. A computer program, XRAYL, has been developed

to fit analytical functions to powder diffraction lines. After fitting,

the powder diffraction data, given as counts at steps in 20 , is

represented, diffraction line by diffraction line, by parameters of a

profile function. The resulting profile parameters may then be used to

generate "idealized" powder diffraction lines, effectively free of

statistical noise and contributions from overlapping lines, extending to

background intensity on each side. Because of this capability XRAYL may be

employed in x-ray powder diffraction profile analysis as a preprocessor

program, that is, separating peaks and feeding the "resolved" data to

subsequent analysis programs. This NBS-IR contains: (1) profile functions

and the nonlinear least-squares algorithm utilized in XRAYL; (2) user

options and execution of the program; and (3) several examples.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Least-squares; powder diffraction; preprocessor; profile fitting; profile functions;

x-ray

13. AVAILABILITY

! I X I Unlimited

I I For Official Distribution. Do Not Release to NTIS

I I Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

36

15. Price

$11.95

USCOMM'DC e043.PS0
'f t. m 7V< I -riitfliri' 1

'' 'jlfSr

^c^ .' ' ^f®t:'H'

';

"'^xi ?Tf ’, tr-^' r

’ ^

vr'

— t . ./-,>4Hrt>* MV-* » '3j.TJTaui-<i»MMM

< '*'

t rtoiJ, :v.„s >

I

-

, ,

. ,.,.'"H»lj|«i»». .. 0. 'm.^ii.w ...f

' .. ^ ^ ;\‘>iif

>, -t

*

'

^ ^ ‘ ' I * i . -••-' 4 V': hr‘. Ki'H *3 . :> 'tc y .i ;.' * - 'wi#? M ^

,•* -v.. i ^.:„* M--- -i -\ *»

: •'^ • .u ««..'. f..?! 4 <f

;

*^'

'?•.!'.» K ?4Ki.flU' l^rr.ij .fjslj' 4

r{(f / !•*. 1 ^ ' (S'vw, .« [

;>c. vV v*S: \ W

I

' 4^' s.«*u,t5i>!ju,a TO «A|tw|“^vi<?:mi^

icum^-un:- m

«!3et aM.featfe2sa}

%' ^ '\' •• '-'.’rf.* •,f-^'*'«» —«»rM, '<.».v*^ - <^-fr fi, rw.,f> 1 *^

^ ^ .' .; -’•.‘J" '?H,^ •3^fe^.V}

."V

--".'/I S'

ft ^

• , '• *i

* "-^Ci -^•^^0 = r '. 'CX

hS f 'i- - .'> .

••' ::» a r"- •r.2'.'?l .

'

. r;;c.vt.:2 o& :/'• 1 ?.^ .|i»'i

-.V r>t:' / f-:;^ 1 «•' - • , •a.

3;.4 ^

/ -' ^ , : , - ^ a -»• <• '! k : v' '^'.1 a ^'4 T1 « 1, '

. t ' ^

' V •• o

t’

•-

, /• i

' ,_ 'ti- ».' '. 4p=

' • - ti,, ^,C . ' j:

--

r

' t

'-

- -

•Ji > ^ * •

' , ,.

• -’t:* - . ' a ^ i4 =(,.• •'!=

•,a 4' /*.

i. ,•.*/>

'»

(' a- - . 1 „l. ^i\ ^

1'

i",: S t ' ’

Ifti

4 :- V ; fri •!.. _ 5 J a u

a»‘: •.} *;'./ j ,.iif5ay :a Sa.?.: f . i?'

'j' a :* vi!

•Ujh: r* j<'

yt r.*'“r’.K

r

• ^ ,

... ^ • .;,\v>-H ' xifTiS ^ tit a-'^ ’'^V' '' ;;/S.rtYS?

;'f '--s^ f

: . v; ' ;t 'i ^

-

‘fu . vgysi;^a^i5 :^fcf^y •ti'f ;

.;a K_n

\

I

,

' ‘'i>'jV“^i' . i \ j

. " - r- ,,: V

'

'•- • ;v / ..

^

tT.

.V

, ^

4 <£k4I^HMi4.( • K-vi<»>^ *wju«nU9t'A

: ; '. L<i >-. ' ?


1/ c3

U)

too CO _ oo as dj

OO "— ^ Oaft> tan

00

00

003“ C

cr

O' z o ^ G H o

73 o

ft)

c/5 ft) ^ 03 0\ o C iX

X

s.

r 3O

C. 00

C3" OO 3 OJ ^ CO < o>o

vO

OOOO

n ^a

?a o<

c/5

r-^ fD

033.

D.

ft)

03o

o’3 N a 3-

03 o 3 3]

go fD

fD

N» > 3 o

fD3 o

S£

fD3 5 '

•* T3TO s> . OO3 3 3 C5 ’ m