United States Department of Commerce

I Technology Administration

National Institute of Standards and Technology

NISTIR 3998

MODELING OF X-RAY DIFFRACTION LINE

BROADENING WITH THE VOIGT FUNCTION:

APPLICATIONS TO HIGH-TC

SUPERCONDUCTORS

Davor Balzar

NISTIR 3998

MODELING OF X-RAY DIFFRACTION LINE

BROADENING WITH THE VOIGT FUNCTION:

APPLICATIONS TO HIGH-TC

SUPERCONDUCTORS

Davor Balzar*

Materials Reliability Division

Materials Science and Engineering Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

•Visiting Scientist, on leave from Department of Physics, Faculty of Metallurgy,

University of Zagreb, Sisak, Croatia

January 1993

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

Parts of this publication were published and presented at scientific meetings:

• D. Balzar, H. Ledbetter, and A. Roshko, Stacking Faults and Microstrain in

Lai 85M0.15CUO4 (M = Ca, Ba, Sr) by Analyzing X-ray Diffraction Line Broadening,

Physica C 185-189 (1991) 871-872. • D. Balzar, Profile Fitting of X-ray Diffraction Lines and Fourier Analysis of

Broadening, Journal ofApplied Crystallography 25 (1992) 559-570. • D. Balzar and H. Ledbetter, Microstrains and Domain Sizes in Bi-Cu-0

Superconductors: An X-ray Diffraction Peak-Broadening Study, Journal of

Materials Science Letters 11 (1992) 1419-1420. • D. Balzar and H. Ledbetter, Voigt-Function Modeling in Fourier Analysis of Sizeand Strain-Broadened X-ray Diffraction Peaks, Journal ofApplied Crystallography

26 (1993), in press. • D. Balzar, H. Ledbetter, and A. Roshko, X-ray Diffraction Peak-Broadening

Analysis of (La-M)2Cu04 High-7^ Superconductors, Powder Diffraction 8 (1993),

in press. • D. Balzar, H. Ledbetter, and A. Roshko, Stacking Faults and Microstrain in

LaL85Mo. 15 Cu04 (M = Ca, Ba, Sr) by Analyzing X-ray Diffraction Line Broadening,

M2S-HTSC 111, Kanazawa, Japan, May 1991. • D. Balzar and H. Ledbetter, 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 Functions, Accuracy in Powder

Diffraction 11, Gaithersburg, Maryland, U.S.A., May 1992.

111
'
Contents

List of Tables viii

List of Figures ix

List of Principal Symbols xii

Abstract 1

1 Introduction 3

1 . 1 Powder X-ray Diffraction 3

1.2 Diffraction-Line Broadening 4

1.3 Superconductivity and Defects 5

1.4 Purpose of the Study 6

2 Literature Review 8

2.1

Size and Strain Broadening 8

2.1.1

Determination of the Pure Specimen-Broadened Profile 8

2 . 1. 1 . 1 Deconvolution Method of Stokes 9

2. 1 . 1 .2 Integral-Breadth Methods 10

v
2.1.2

Separation of Size and Strain Broadening 11

2. 1.2.1 Warren-Averbach Method 13

2. 1.2.2 Multiple-Line Integral-Breadth Methods 16

2 . 1 . 2 . 3 Single-Line Methods 1

2.2 Diffraction-Line-Broadening Analysis of Superconductors 18

3 Experiment 20

3.1 Materials 20

3.1.1 Preparation of Specimens for X-ray Diffraction 21

3.2 Measurements 21

3.3 Data Analysis 22

4 Methodology 24

4.1 Separation of Size and Strain Broadenings 24

4.2 Size Coefficient 26

4.3 Distortion Coefficient 31

4.4 Discussion 31

4.5 Random Errors of Derived Parameters 36

5 Application 39

5.1 Correction for Instrumental Broadening 39

5.2 Applicability of the Method 42

5.2.1 Silver and Tungsten Powders 43

5.2.2 La2.xSrJCCu04 Powders 51

5.3 Comparison with the Integral-Breadth Methods 52

5.4 Reliability of Profile Fitting 57

5.5 Remarks 60

6 Analysis of Superconductors 62

6.1 (La-M)2 Cu04 Superconductors 62

vi
6.2 Bi-Cu-0 Superconductors 69

6.3 Remarks 73

7 Conclusions 75

Acknowledgments 78

References 79

vn
List of Tables

Table 1.1 Use of diffraction line-profile parameters 4

Table 5.1 Parameters of the pure-specimen Voigt function, as obtained from profilefitting procedure for tungsten and silver powders 47

Table 5.2 Microstructural parameters for tungsten and silver powders 47

Table 5.3 Parameters of the pure-specimen Voigt function, as obtained from profilefitting procedure for La2.;c SrJt Cu04 powders 51

Table 5.4 Microstructural parameters for 85 Sr0 15Cu04 and La2 Cu04 powders. . 52

Table 5.5 Comparison of results obtained with the integral-breadth methods: CauchyCauchy (C-C), Cauchy-Gauss (C-G), Gauss-Gauss (G-G), and single-line (S-L)

analysis 56

Table 5.6 Comparison between two specimens run separately and mixed together. 59

Table 6.1 Lattice parameters and Tc at zero resistivity (AC method at 10 mA

current). 64

Table 6.2 Results of line-broadening analysis for (La-M)2Cu04 specimens 67

Table 6.3 Results of line-broadening analysis for Bi-Cu-0 superconductors 72

Vlll
List of Figures

Figure 2.1 Observed profile h is a convolution of the instrumental profile g with the

specimen profile/. Adapted from Warren [59] 9

Figure 2.2 Voigt functions for different values of Cauchy and Gauss integral breadths.

Adapted from Howard and Preston [4] 12

Figure 2.3 Representation of the crystal in terms of columns of cells along the a,

direction [59] 14

Figure 2.4 Surface-weighted domain size is determined: (a) by the intercept of the

initial slope on the L-axis; (b) as a mean value of the distribution function. ... 16

Figure 3.1 Optical arrangement of an x-ray diffractometer. Adapted from Klug and

Alexander [24] 22

Figure 4.1 (upper) The ’hook’ effect of the size coefficients A

s (full line) at small L;

(lower) it causes negative values (set to zero) of the column-length distribution

functions 28

Figure 4.2 The ratio of volume-weighted and surface-weighted domain sizes as a

function of the characteristic ratio of Cauchy and Gauss integral breadths k. . . 30

Figure 4.3 Mean-square strains <e2 (L)> for two approximations of the distortion

coefficient: (upper) Voigt strain broadening; (lower) pure-Gauss strain broadening. 34

ix
Figure 5.1 A split-Pearson VII profile. The two half profiles have same peak position

and intensity. Adapted from Howard and Preston [4]. . 41

Figure 5.2 Refined FWHMs and shape factors (exponents) m for low-angle and highangle sides of LaB6 line profiles. Second-order polynomials were fitted through

points 42

Figure 5.3 Observed points (pluses), refined pattern (full line), and difference pattern

(below): (110) W untreated (upper); (200) Ag ground (lower) 44

Figure 5.4 Fourier coefficients for the first- (pluses) and second-order (crosses)

reflection, and size coefficients (circles): [111] Ag untreated (upper); [100] Ag

ground (lower) 45

Figure 5.5 Mean-square strains <e2 > as a function of 1/L 46

Figure 5.6 Surface-weighted and volume-weighted column-length distribution functions,

normalized on unit area: [111] Ag untreated (upper); [100] Ag ground (lower). 48

Figure 5.7 Surface-weighted and volume-weighted column-length distribution functions

for [010] La2Cu04 , normalized on unit area 49

Figure 5.8 Observed points (pluses), refined pattern (full line), convoluted profiles

(dashed line), and difference plot (below) for part of La2 Cu04 pattern 50

Figure 5.9 Fourier coefficients for the first- (pluses) and second-order (crosses)

reflection, and size coefficients (circles) for [110] Lat 85 Sr0 15Cu04 53

Figure 5. 10 Fourier coefficients for first- (pluses) and second-order (crosses) reflection,

and size coefficients (circles) for [010] La2Cu04 54

Figure 5.11 (upper) Laj 76Sr0 24Cu04 (110) peak; (middle) ^Sro.^CuC^ (020) and

(200) peaks; (lower) (110), (020), and (200) peaks 58

Figure 6. 1 Crystal structure of (La-M)2 Cu04 showing one unit cell 63

Figure 6.2 Diffraction patterns of Lat 85M0 15Cu04 specimens: (a) M = Sr; (b) M =

Ba; (c) M = Ca; (d) M = La 65

Figure 6.3 Arithmetic average of [110] and [001] root-mean-square strains and linear

function of stacking-fault and twin-fault probabilities as a function of Tc 68

x
Figure 6.4 Average crystal structure of Bi2Sr2Cam. 1 Cum04 +2/n for: (a) m = l; (b) m=2;

(c) m=3 [164] 70

Figure 6.5 Part of Bi,Pb,Mg,Ba-2223 refined pattern. There are 48 fundamental

reflections in this region 71
List of Principal Symbols

a , b, c, m,

u, V, w,

U\ V, W

/, X, Z

A

a*

D

d

e

FWHM

f, F

g,

G

h, H

hkl

I

Jc

K

k

L

lMSS

N

n

P

R

RMSS

sTc w

m\

General constants

General variables

Fourier coefficient

Edge of orthorhombic cell, orthogonal to diffracting planes

Domain size orthogonal to diffracting planes

Interplanar spacing

Upper limit of strain

Full width at half maximum of profile

Pure-specimen (physically) broadened profile and its Fourier transform

Instrumentally broadened profile and its Fourier transform

Observed broadened profile and its Fourier transform

Miller indices

Intensity

Critical superconducting current density

Scherrer constant

r 1/2j8 c), characteristic integral-breadth ratio of a Voigt function

na3 , column length (distance in real space) orthogonal to diffracting planes

Order of reflection

Mean-square strains

Average number of cells per column

Harmonic number

Column-length distribution function

Relative error

Root-mean-square strains

2sin0/X = 1 Id, variable in reciprocal space

Critical superconducting transition temperature

Observation weight

xu
z

a

a'

P

7

8

<e\L)>

Displacement of two cells in a column

Stacking-fault probability

Twin-fault probability

/3(20)cos0o/\, integral breadth in units of s (A 1 )

Geometrical-aberration profile

Fraction of oxygen atoms missing per formula unit

Mean-square strain, orthogonal to diffracting planes, averaged over the

distance L

V

6

Oo

X

cr

03

’Apparent strain’

Bragg angle

Bragg angle of Kcxj reflection maximum

X-ray wavelength

Span of profile in real space

Wavelength-distribution profile

Superscripts

D Denotes the distortion-related parameter

S Denotes the size-related parameter

Subscripts

C

D

f

G

g

h

S

sV

wp

Denotes Cauchy component of Voigt function

Denotes distortion-related parameter

Denotes pure-specimen (physically) broadened profile

Denotes Gauss component of Voigt function

Denotes instrumentally broadened profile

Denotes observed broadened profile

Denotes size-related parameter

Denotes surface-weighted parameter

Denotes volume-weighted parameter

Denotes weighted-residual error

Operators

Convolution: g{x)*f{x) = \ g(z)f[x-z)dz

Xlll
u

.

'
Abstract

A method to analyze powder-diffraction line broadening is proposed and applied to some

novel high-7^ superconductors. Assuming that both size-broadened and strain-broadened

profiles of the pure-specimen profile are described with a Voigt function, it is shown that

the analysis of Fourier coefficients leads to the Warren-Averbach method of separation of

size and strain contributions. The analysis of size coefficients shows that the ’hook’ effect

occurs when the Cauchy content of the size-broadened profile is underestimated. The ratio

of volume-weighted and surface-weighted domain sizes can change from — 1.31 for the

minimum allowed Cauchy content to 2 when the size-broadened profile is given solely by

a Cauchy function. If the distortion coefficient is approximated by a harmonic term, meansquare strains decrease linearly with the increase of the averaging distance. The local strain

is finite only in the case of pure-Gauss strain broadening because strains are then

independent of averaging distance. Errors of root-mean-square strains as well as domain

sizes were evaluated. The method was applied to two cubic structures with average volumeweighted domain sizes up to 3600 A, as well as to tetragonal and orthorhombic

(La-Sr)2Cu04 , which exhibit weak line broadenings and highly overlapping reflections.

Comparison with the integral-breadth methods is given. Reliability of the method is

discussed in the case of a cluster of the overlapping peaks. The analysis of La2 Cu04 and

Lai. 85M0.i 5 CuO4 (M = Ca, Ba, Sr) high-Tc superconductors showed that microstrains and

incoherently diffracting domain sizes are highly anisotropic. In the superconductors,

1
stacking-fault probability increases with increasing Tc \ microstrain decreases. In La2 Cu04 ,

different broadening of (hOO) and (0&0) reflections is not caused by stacking faults; it might

arise from lower crystallographic symmetry. The analysis of Bi-Cu-O superconductors

showed much higher strains in the [001] direction than in the basal a-b plane. This may be

caused by stacking disorder along the c-axis, because of the two-dimensional weakly

bonded BiO double layers. Results for the specimen containing two related high-Tc phases

indicate a possible mechanism for the phase transformation by the growth of faulted regions

of the major phase.

Key words: diffraction line broadening; lattice defects; profile fitting; superconductors;

Voigt function; Warren-Averbach analysis; x-ray diffraction

2
1 Introduction

X-ray diffraction is one of the oldest tools to study structure of matter. In 1912,

Laue [1] demonstrated in a single experiment that crystals consist of regularly repeating

elementary building blocks, and that x-rays show wave nature. Since then, x-ray diffraction

became one of the basic and the most widely used methods for characterization of a broad

range of materials.

1.1 Powder X-ray Diffraction

Many materials are not available in a monocrystal form. Moreover, powders and

bulk materials are more easily obtainable, practical, and less expensive. A powderdiffraction experiment requires an order-of-magnitude shorter time than a monocrystal

experiment. Thus, powder diffraction is used very often. However, because data are of

lower quality and peaks are generally highly overlapped at higher diffracting angles, until

25 years ago powder diffraction was mostly used for qualitative phase analysis. Through

advances by Rietveld [2,3], powder-diffraction patterns become used in structure analysis,

so-called structure (Rietveld) refinement. Development of fast on-line computer-controlled

data acquisition has allowed a quick analysis of the whole diffraction pattern. Table 1.1

summarizes uses of different diffraction line-profile parameters in various types of analyses

(after Howard and Preston [4]). We shall focus on line-profile analysis, to obtain

information about microstructural properties of materials: microstrains in the lattice and

size of incoherently diffracting domains in crystals.

3
Table 1.1 Use of diffraction line-profile parameters.

Position Intensity Shape Shift Method Results

/ Indexing Cell parameters

/ / Phase

analysis

Identification and

quantity

/ Peak-shift

analysis

Internal strain

(residual stress)

/ / Profile

analysis

Microstrain,

crystallite size,

lattice defects

/ / / Structure

refinement

Atomic positions,

Debye-Waller

factors, others

1.2 Diffraction-Line Broadening

Diffraction from crystal planes occurs at well-defined angles that satisfy the Bragg

equation

X = 2dhu sinew . (1.1)

Theoretically, intensity diffracted from an infinite crystal should consist of diffraction lines

without width (Dirac delta functions) at some discrete diffraction angles. However, both

instrument and specimen broaden the diffraction lines, and the observed line profile is a

convolution of three functions [5, 6]

4
h(26) = [(w*7)V](20) + background . ( 1 . 2)

•

Wavelength distribution and geometrical aberrations are usually treated as characteristic for

the particular instrument (instrumental profile):

g(28) = («*7)(28) . (L3>

To obtain a specimen’s microstructural parameters, the specimen (physically) broadened

profile / must be extracted from the observed profile h.

Origins of specimen broadening are numerous. Generally, any lattice imperfection

will cause additional diffraction-line broadening. Therefore, dislocations, vacancies,

interstitials, substitutions, and similar defects manifest themselves through the lattice strain.

If a crystal is broken into smaller incoherently diffracting domains by dislocation arrays,

stacking faults, twins, or any other extended imperfections, then domain-size broadening

occurs.

1.3 Superconductivity and Defects

Since discovery of the ~ 90 K superconductor YBa2Cu3 07^ [7], it became clear that

the novel superconductivity relates closely to defects in structure. Both point and extended

defects relate closely to the physical properties of superconductors [8, 9]. Defects play an

important role both in the critical superconducting transition temperature Tc [10] and in

the critical current density Jc [11]. Some theories also connect Tc with lattice distortion

[12], with strains around dislocations [13], and with interaction of current carriers and the

elastic-strain field [14].

The Tc of YBa2 Cu30M, for instance, depends strongly on the oxygen stoichiometry,

that is, number of oxygen vacancies in the charge-reservoir layers and their arrangement

(see Jorgensen [15] and references therein). Superconductivity in La2 Cu04 appears either

by the partial substitution of La with Sr, Ba, Ca [16, 17], or by the introduction of

5
interstitial oxygen defects in the La2 02 layer [18]. Some substitutions, especially on Cu

sites, destroy the superconductivity.

For classical superconductors, Jc can be drastically increased by introducing defects

to pin magnetic flux vortices. The layered structure of high-Tf cuprates causes the vortices

to be pinned in the form of pancakes, rather than long cylinders [19]. Because of relatively

small coherence length of vortices, pinning can not be increased in the classical way by

introducing second-phase precipitates. Instead, submicroscopic lattice defects caused by

local stoichiometry fluctuations, vacancies, substitutions, Guinier-Preston zones, and the

strain field of small coherent precipitates are much more effective. Especially in highly

anisotropic Tl-based and Bi-based cuprates, substitutions are very successful. Even a 5 %

Mg for Ba substitution in Tl2Ba2CaCu208 increases Jc by 25 % [20].

1.4 Purpose of the Study

We know that defects have very important role in novel high-7^ superconductivity.

Defects can be characterized and quantified by analyzing the x-ray diffraction broadening.

Basically, there are two approaches:

(i) The Stokes deconvolution method [21] combined with the Warren and Averbach

analysis [22] give the most rigorous and unbiased approach because no assumption about

the analytical form of diffraction-peak shape is required. However, when peaks overlap and

specimen broadening is comparable with the instrumental broadening, the Stokes method

gives unstable solutions and large errors or can not be performed at all. To obtain reliable

results, proper corrections have to account for truncation, background, sampling, and the

standard’s errors [23].

(ii) The simplified integral-breadth methods (summarized by Klug and Alexander

[24]) are more convenient and easier to use, but they require that size and strain

broadening are modeled by either Cauchy or Gauss functions. Experience has shown,

however, that in most cases both size and strain profiles can not be satisfactorily

represented with either function. However, there is some theoretical and experimental

6
evidence that the effect of small-domain-size broadening produces long profile tails of the

Cauchy function, and that the lattice-strain distribution is more Gauss-like. Langford [25,

26] used the convolution of Cauchy and Gauss functions (Voigt function) to model

specimen broadening. However, the results obtained by the integral-breadth and WarrenAverbach analyses are usually not comparable; the first methods give volume-weighted

domain sizes and upper limit of strain; the second gives surface-weighted domain sizes and

mean-square strain averaged over some distance perpendicular to diffracting planes.

Unfortunately, most high- Tc superconductors show weak peak broadening (because

of high annealing temperatures) and strong peak overlapping (because of relatively

complicated crystal structures), which makes it very difficult to apply the Stokes

deconvolution method to extract pure specimen broadening. The aim in this study is

twofold:

(i) To develop a reliable method for analysis of a pattern with highly overlapping

reflections and weak structural broadening, and to compare it with the previously described

approaches. It will be shown that the Voigt-function modeling of the specimen broadening

concurs with the Warren-Averbach approach.

(ii) To apply the method to the some high-Tc superconductors and conclude how

much information about defects can be extracted from analysis of the x-ray diffraction

broadening.

7
2 Literature Review

2.1 Size and Strain Broadening

Some important methods to extract specimen size and strain broadening and

information about domain sizes and strains will be reviewed briefly. An excellent review

about Fourier methods and integral-breadth methods is given by Klug and Alexander [24].

A survey of single-line methods was authored by Delhez, de Keijser, and Mittemeijer [27].

The use of variance (reduced second moment of the line profile) in the analysis of

broadening will not be treated here. Wilson described the contributions to variance by

crystallite size [28] and strain [29].

2.1.1 Determination of the Pure Specimen-Broadened Profile

As mentioned in Paragraph 1.2, before the specimen’s size and strain broadening

can be obtained, the observed profile must be corrected for instrumental broadening.

Mostly used methods are the Fourier-transform deconvolution method [30, 21] and

simplified integral-breadth methods that rely on some assumed analytical forms of the peak

profiles. The iterative method of successive foldings [31, 32] is not used extensively, and

will not be considered here.

8
2. 1.1.1 Deconvolution Method of Stokes

From Equations (1.2) and (1.3), it follows that deconvolution can be performed

easily in terms of Fourier transforms of respective functions:

m = H(n)

G(n) * (2 . 1 )

Figure 2.1 Observed profile h is a convolution of the instrumental profile g with the

specimen profile/. Adapted from Warren [59].

Hence, the physically broadened profile /is retrieved from the observed profile h without

any assumption on the peak-profile shape (see Figure 2.1). This method is the most

desirable approach because it is totally unbiased. However, because of the deconvolution

process, there are many problems. Firstly, Equation (2.1) may not give a solution if the

Fourier coefficients of the / profile do not vanish before those of the g profile.

Furthermore, if physical broadening is small compared with instrumental broadening,

deconvolution becomes too unstable and inaccurate [33, 34]. If the h profile is 20 %

broader than the g profile, this gives an upper limit of about 1000 A for the determination

of effective domain size [34]. Regardless of the degree of broadening, deconvolution

produces unavoidable profile-tail ripples because of truncation effects. To obtain reliable

results, these errors have to be corrected, along with errors of incorrect background,

sampling, and the standard specimen [23, 35, 36]. The largest problem, however, is peak

9
overlapping. If the complete peak is not separated, the only possible solution is to try to

reconstruct the missing parts. That would require some assumption on the peak-profile

shape, that is introduction of bias into the method. The application of the Stokes method

is therefore limited to materials having the highest crystallographic symmetry.

2. 1.1. 2 Integral-Breadth Methods

The basic assumption of these methods is that diffraction profiles can be

approximated with some analytical function. In the beginning, two commonly used

functions were Gauss

I(x) = 7(0) exp (2 . 2)

and Cauchy

I(x) = 7(0)

1

From the convolution integral, it follows easily that

(2.3)

$hc = tic

+

0/c (2- 4>

for Cauchy profiles, and

tic - tic tic (2 - 5)

for Gauss profiles.

However, the observed x-ray diffraction line profiles can not be well represented with a

simple Cauchy or Gauss function [24, 37]. But they are almost pure Cauchy at highest

angles because the dominant cause of broadening becomes the spectral distribution in

radiation [24]. Different geometrical aberrations of the instrument are difficult to describe

with simple analytical functions. In the case of closely Gaussian broadening of 7, following

10
Equation (1.3), the instrumental line profile can be best described by a convolution of

Cauchy and Gauss functions, which is the Voigt function. Experience shows that the Voigt

function [38] (or its approximations, pseudo-Voigt [39, 40] and Pearson-VII [41, 42]) fits

very well the observed peak profiles [25, 43, 37]. The Voigt function is usually represented

following Langford [25]

l(x) = 7(0)

0 1 Re erfi

r

+ £

KJ Pc

Here, the complex error function is defined as

erfi(z) = exp(-z 2

) [1 - erf(-iz)] .

(2 . 6)

(2.7)

Its evaluation can be accomplished using Sundius [44] or Armstrong [45] algorithms with

eight-digit accuracy. Useful information about the Voigt function can be found in papers

of Kielkopf [46], Asthana and Kiefer [47], and de Vreede et al. [48]. Figure 2.2 presents

a Voigt function for different values of Cauchy and Gauss integral breadths.

Integral breadth of the Voigt function is expressed through its constituent integral

breadths [49]

0

O exp(-fc

2

)

G

tTfc(k)

(2. 8)

Here, erfc denotes the complementary error function.

Because convolution of two Voigt functions is also a Voigt function, integral

breadths are easily separable conforming to Equations (2.4) and (2.5).

2.1.2 Separation of Size and Strain Broadening

After removing the instrumental broadening from the observed line profile, it is

possible to analyze the pure-specimen (physically) broadened line profile, to consider the

origins and amount of broadening.

11
Figure 2.2 Voigt functions for different values of Cauchy and Gauss integral breadths.

Adapted from Howard and Preston [4].

In 1918 Scherrer [50] recognized that breaking the crystal into domains smaller than — 1000 A causes diffraction-line broadening

<D> = — . (2.9)

v 0(20) cos0

The constant K depends on the crystallite shape [51, 52, 53, 54, 55, 56], but generally is

close to unity. The main characteristic of size broadening is that it is independent of the

reflection order, that is, independent of diffraction angle.

Most of the work on x-ray diffraction line broadening was done on metals and

alloys. It is widely accepted that plastic deformation in metals produces dislocation arrays,

which divide crystallites into much smaller incoherently scattering domains. These

dislocations produce strains within the domains, causing strain broadening. It was

elaborated in Paragraph 1.2 that any lattice imperfection (vacancies, interstitials, and

12
substitutions) would broaden the diffraction peaks. These effects would be interpreted in

the frame of this theory as a strain broadening, too. Stokes and Wilson [57] defined

’apparent strain’ as

n = p(26) cote . (2 - 10)

Strain broadening is angle dependent. Therefore, the angle dependence of the line

broadening gives a possibility to distinguish between contributions of size and strain.

However, when we speak of size and strain broadening, it is regarded that they may

include other contributions. For instance, stacking faults and twins will contribute to

broadening similar to size effects.

2. 1.2.1 Warren-Averbach Method

This method was developed originally for plastically deformed metals, but since its

introduction [58, 22] it found successful application to many other materials. The method

is extensively described in Warren’s publications [59, 60]. Each domain is represented by

columns of cells along the a3 direction [61] (see Figure 2.3). The crystal has orthorhombic

axes with the direction a3 normal to the diffracting planes (00/). The experimentally

observable diffraction power may be expressed as a Fourier series

oo H26) = J2 A„ cxp(2-nna}s) . (2 -U)

sin 6 n =-oo

Here, experimentally measurable coefficients An are

A =

n <exp(2'7ri/ZJ> . (2.12)

n <D>S

The An coefficients are the product of two terms. The first term depends only on the

column length (size coefficient); the second depends only on distortion in domains

(distortion coefficient):

13
Figure 2.3 Representation of the crystal in terms of columns of cells along the direction

[59].

An AnSAnD (2.13)

Ans = ; An°

= <exp(27ri/Zn)> . (2.14)

It is more convenient to express the distortion coefficient in terms of the strain component.

If L = na3 is the undistorted distance between a pair of cells along direction a3 , and

distortion changes distance by AL = a

the component of strain in the % direction

14
(orthogonal to reflecting planes) averaged over distance L can be defined as e(L) = A(L)/L.

Because a3/l = d, interplanar spacing, the distortion coefficient can be rewritten

A

d

(L) = <exp(2iriLe(L)ld)> . (2 - 15)

To obtain the strain component, it is necessary to approximate the exponential term. For

not too large L

< exp(2xiL€(L)/d) > * exp(-2x2L

2 <e2(L)> /

d

2). (2-16)

This relationship is exact if the distributions of e(L) for all L values follow the Gauss

function and is generally true as far as terms in e

3 (L) because these distributions are usually

sufficiently symmetrical [57].

Now, Equation (2.13) can be approximated as

\r\A(L) = InA S(L) - 2

i

2 <e2(L)>L2 /d2

. (2 - 17)

Warren and Averbach [22] derived this relationship in a similar way. It separates size and

strain contributions to the broadening, and allows for their simultaneous evaluation.

If the size coefficients are obtained by applications of Equation (2. 17), it is possible

to evaluate the average surface-weighted domain size and the surface-weighted columnlength distribution function [59]:

' d A

S(L)

d L <D>

(2.18)

P/L) «

d

2 AS(L)

dL2 (2.19)

Figure 2 . 4 shows how <D> S can be obtained from both the size coefficients A

S

(L) and the

column-length distribution function.

15
Figure 2.4 Surface-weighted domain size is determined: (a) by the intercept of the initial

slope on the L-axis; (b) as a mean value of the distribution function.

2. 1.2. 2 Multiple-Line Integral-Breadth Methods

To separate size and strain broadening by using integral breadths, it is necessary to

define the functional form for each effect. In the beginning, size and strain contributions

were described by Cauchy or Gauss functions. Using Equations (2.9) and (2.10) on the s

scale, and additive relations for the integral breadths following Equations (2.4) and (2.5)

16
0 (Cauchy -Cauchy) , (2 . 20) =

1 <D>

+ 2es

1 <D> v

+ 4e 2j

2 T~ (Cauchy -Gauss) , (2 . 21 )

0

2 = + 4e 2j

2 <D>

(Gauss -Gauss) (2 . 22)

Here, e = 77/4

* Mid is the upper limit for a strain. Equation (2.21) uses the Haider and

Wagner [62] parabolic approximation for the integral breadth of the Voigt function

expressed by Equation (2.8):

Pc., {PcV

p p

' (2.23)

Experience shows, however, that neither Cauchy nor Gauss functions can model

satisfactorily size or strain broadening in a general case. Langford [26] introduced the socalled multiple-line Voigt-function analysis. Both size-broadened and strain-broadened

profiles are assumed to be Voigt functions. Using Equations (2.9) and (2.10), it follows

symbolically for Cauchy and Gauss parts that

$C @SC

+ * (2.24)

&G = 0SG + $DgS 2

• (2.25)

This approach disagrees with the Warren-Averbach analysis, that is, the two methods give

different results (see Paragraph 4.4) [63, 64].

2. 1.2. 3 Single-Line Methods

There are cases where only the first order of reflection is available or higher-order

reflections are severely suppressed (extremely deformed materials, multiphase composites,

17
catalysts, and oriented thin films). Many methods exist to separate size and strain

broadening from only one diffraction peak. However, it was said in Paragraph 2.1.2 that

the different size and strain broadening angle dependence is a basis for their separation;

hence, using only one diffraction line introduces a contradiction. Consequently, single-line

methods should be used only when no other option exists.

The single-line methods can be divided in two main parts: Fourier-space and realspace methods. Fourier-space methods are based on the Warren-Averbach separation of

size and strain broadening following Equation (2.17). The functional form of <e2 (L)> is

assumed either to be constant [65, 66, 67, 68], or assumed to depend on L as <e2 (L)> = dL [69, 70, 71, 72]. Then, Equation (2.17) can be fitted to few points of A(L) for the

small averaging distance L, to obtain size and strain parameters. All Fourier-space methods

have the serious problem that the Fourier coefficients A(L) are usually uncertain for small

L, because of the so-called ’hook’ effect [60] (see Paragraph 4.2). Zocchi [73] suggested

that fitting the straight line through the first derivatives of the Fourier coefficients, instead

through the coefficients themselves, would solve the ’hook’ -effect problem.

All real-space methods [74, 75, 76] are based on the assumption that the Cauchy

function determines size and that the Gauss function gives strain. The most widely used

method of de Keijser et al. [76] gives size and strain parameters from Cauchy and Gauss

2.2 Diffraction-Line-Broadening Analysis of Superconductors

In this field, very few studies exist. Williams et al. [77] reported isotropic strains

in YBa2Cu3 07^ powder by the simultaneous Rietveld refinement of pulsed-neutron and xparts of the Voigt function, respectively:

(2.26)

(2.27)

\

18
ray diffraction data. Using a GSAS Rietveld refinement program [78], both size and strain

broadening were modeled with the Gauss functions for the neutron-diffraction data [79,

80], and with the Cauchy functions for the x-ray diffraction data (modified method of

Thompson, Cox, and Hastings [81]). Interestingly, both the neutron and x-ray data gave

identical values for the isotropic strain (0.23 %) and no size broadening. Singh et al. [82]

studied internal strains in YBa2Cu307^ extruded wires by pulsed-neutron diffraction. They

separated size and strain parameters by means of Equation (2.22) (Gauss-Gauss

approximation). Size broadening was found to be negligible, but (isotropic) microstrains

range from 0.05 % for the coarse-grained material to 0.3 % for the fine-grained samples.

Eatough, Ginley, and Morosin [83] studied Tl2Ba2 Ca2 Cu3 O10 (Tl-2223) and Tl2Ba2 CaCu208

(Tl-2212) superconducting thin films by x-ray diffraction. Using the Gauss-Gauss

approximation, they found strains of 0.14-0.18 % in both phases, and domain sizes of

1200-1400 A for Tl-2212, but 500 A for Tl-2223.

We are aware of only two more unpublished studies [84, 85] involving size-strain

analysis in high-Tf superconductors. The probable reason is that any analysis is very

difficult because of weak line broadening and overlapping reflections. This precludes

application of reliable analysis, such as the Stokes deconvolution method with the WarrenAverbach analysis of the broadening. Instead, simple integral-breadth methods are used,

which gives generally different results for each approach. Moreover, for x-ray diffraction

broadening, application of the Gauss-Gauss approximation does not have any theoretical

merit, although reasonable values, especially of domain sizes, may be obtained [86]. We

showed [87, 88, 89] that reliable diffraction-line-broadening analysis of superconductors

can be accomplished and valuable information about anisotropic strains and incoherently

diffracting domain sizes obtained.

19
3 Experiment

3.1 Materials

The materials used for this study were tungsten and silver commercially available

powders with nominal grain sizes 4-12 /tm, La^r^CuC^ (x = 0, 0.06, 0.15, 0.24)

powders, Ia 85Mo. 15 Cu04 (M = Ca, Ba) powders, Bi2Sr2 CaCu2 08 (Bi-2212) sinter,

(BiPb)2Sr2 Ca2Cu3 O10 (Bi,Pb-2223) sinter, and (BiPb)2 (SrMg)2 (BaCa)2Cu3 O10 (Bi,Pb,Mg,Ba2223) sinter.

Powders with nominal compositions La^Sr^CuC^ (x — 0, 0.06, 0.15, 0.24) and

La185M015CuO4 (M = Ca, Ba) were prepared at the National Institute of Standards and

Technology, Boulder, Colorado, by A. Roshko, using a freeze-drying acetate process [90].

Acetates of the various cations were assayed by mass by calcining to the corresponding

oxide or carbonate. The appropriate masses of the acetates for the desired compositions

were dissolved in deionized water. The acetate solutions were then sprayed through a fine

nozzle into liquid nitrogen to preserve the homogeneous cation distributions. Frozen

particles were transferred to crystallization dishes and dried in a commercial freeze dryer,

to a final temperature of 100 °C. After drying, the powders, except the Laj ^Bao 15 Cu04 , were calcined in alumina (99.8 %) crucibles at 675 °C for 1 h in a box furnace with the

door slightly open to increase ventilation. Because BaC03 is difficult to decompose, the

Lai 85 Bao 15 Cu04 was calcined under a vacuum of 2 Pa at 800 °C for 4 h, then cooled slowly

in flowing oxygen (2 °C / min). The calcined powders were oxidized in platinum-lined

20
alumina boats in a tube furnace with flowing oxygen at 700 °C. After 3 h at 700 °C the

powders were pushed to a cold end of the furnace tube where they cooled quickly (20 °C

/ s) while still in flowing oxygen.

The cylindrical specimens (23 mm in diameter and 9 mm thick) of Bi-2212, Bi,Pb2223, and Bi,Pb,Mg,Ba-2223 were prepared at the National Research Institute for Metals,

Tsukuba, Japan, by K. Togano [91]. Starting oxides and carbonates were: Pb3 04 , Bi2 03 , CuO, SrC03 , MgC03 , and BaC03

. They were calcinated in air at 800 °C for 12 h. Powders

were then pressed and sintered in 8 % oxygen-92 % argon mixture at 835 °C for 83 h.

Specimens were furnace cooled to 750 °C, held for 3 h in flowing oxygen, and then furnace

cooled in oxygen to room temperature.

3.1.1 Preparation of Specimens for X-ray Diffraction

The bulk specimens were surface polished, if necessary, and mounted in specimen

holders. Coarse-grained powders of La2-xSrJt Cu04 (x = 0, 0.06, 0.15, 0.24) and

La185M0. 15 CuO4 (M = Ca, Ba) were ground with a mortar and pestle in toluene and passed

through a 635-mesh sieve (20-^m nominal opening size). Silver and tungsten powders were

dry ground with a mortar and pestle. All powders were mixed with about 30% silicone

grease and loaded into rectangular cavities or slurried with amyl acetate on a zerobackground quartz substrate.

3.2 Measurements

X-ray-diffraction data were collected using a standard two-circle powder goniometer

in Bragg-Brentano parafocusing geometry [92, 93] (see Figure 3.1). A flat sample is

irradiated at some incident angle to its surface, and diffraction occurs only from

crystallographic planes parallel to the specimen surface. The goniometer had a vertical 6-26

axis and 22-cm radius. CuKa radiation, excited at 45 kV and 40 mA, was collimated with

Soller slits [94] and a 2-mm divergence slit. Soller slits in the diffracted beam, 0.2-mm

21
receiving slit and Ge solid-state detector were used in a step-scanning mode (0.01° / 10 s

for standard specimen, 0.02°-0.05° / 30-80 s for other specimens, depending on the amount

of broadening).

Figure 3.1 Optical arrangement of an x-ray diffractometer. Adapted from Klug and

Alexander [24].

3.3 Data Analysis

The diffractometer was controlled by a computer, and all measurements were stored

on hard disk. Data were transferred to a personal computer for processing.

22
We used computer programs for most calculations. X-ray diffraction patterns were

fitted with the program SHADOW [95]. This program allows a choice of the fitting

function and gives refined positions of the peak maximums, intensities, and functiondependent parameters. It also has a possibility to convolute the predefined instrumental

profile with the specimen function to match the observed pattern. Choice of the specimen

function includes Gauss and Cauchy functions. We added the possibility to model the

specimen broadening with an exact Voigt function and implemented SHADOW on a

personal computer. In the fitting procedure, for every peak in the pattern, the program

firstly generates the instrumental profile at the required diffraction angle. The instrumental

profile is determined from prior measurements on a well-annealed standard specimen (see

Paragraph 5.1). Then it assumes parameters of the specimen profile. For an exact Voigt

function, parameters are peak position, peak intensity, and Cauchy and Gauss integral

breadths of the Voigt function. By convoluting the instrumental profile with the specimen

profile, and adding a background, the calculated pattern is obtained, Equation (1.2).

Parameters of the specimen profile are varied until the weighted least-squares error of

calculated and observed patterns, Equation (4.29), reaches a minimum. This process avoids

the unstable Stokes deconvolution method. It is possible that the refinement algorithm is

being trapped in a false minimum [96], but it can be corrected by constraining some

parameters. Refined parameters of the pure-specimen profile are input for the size-strain

analysis of the broadening. A program for this analysis was written in Fortran.

Lattice parameters of powder specimens were calculated by the program

NBS*LSQ85, based on the method of Appleman and Evans [97]. A program in Fortran

was written to apply corrections to observed peak maximums by using an NIST standard

reference material 660 LaB6 as an external standard. Lattice parameters of bulk specimens

were determined by the Fortran program, which uses a modified Cohen’s method [98, 99,

100] to correct for systematic diffractometer errors. Lattice parameters were also calculated

by the Rietveld refinement programs GSAS [78] and DBW3.2S [101, 102].

23
4 Methodology

When instrumental and specimen contributions to the observed line profile must be

modeled separately, adopting a specimen function is a critical step. Yau and Howard [103,

104] used Cauchy, and Enzo et al. [105] pseudo-Voigt function. Benedetti, Fagherazzi,

Enzo, and Battagliarin [106] showed that modeling the specimen function with the pseudoVoigt function gives results comparable to those of the Stokes deconvolution method when

combined with the Warren-Averbach analysis of Fourier coefficients. De Keijser,

Mittemeijer, and Rozendaal [107] analytically derived domain sizes and root-mean-square

strains for small averaging distance L in the case of the Voigt and related functions.

The aim here is to study more thoroughly the consequences of assumed Voigt

specimen function on the size-strain analysis of the Fourier coefficients of the broadened

peaks. It is shown that some experimentally observed interrelations between derived

parameters (particularly volume-weighted and surface-weighted domain sizes) and their

behavior (the ’hook’ effect and dependence of mean-square strains on the averaging

distance) can be explained by this simple assumption. Moreover, the discrepancy between

the integral-breadth methods and Warren-Averbach analysis results from different

approximations for the strain broadening and the background experimental errors.

4.1 Separation of Size and Strain Broadenings

The normalized Fourier transform of a Voigt function is easily computable [46]:

24
An = exp

(4.1) 2

o(26)

2 &c(26)

-

m

1 26)

It is convenient to express Fourier coefficients in terms of distance L, by immediately

making the approximation A(26) = X As / cos#0

:

A(L) = exp[ -2£/Sc - tL 2

fi2

a]

. (4- 2)

Equation (4.2) is a good approximation even for large specimen broadening. Even for a

profile span of A(20)=8O°, the error made by replacing this interval by an adequate A(sin0)

range is 2%. However, strictly speaking, the profile will be asymmetrical in reciprocal

space, and Fourier-interval limits will not correspond to the 20! and 2

0

2 peak-cutoff values

in real space. It is important to keep Fourier interval limits identical for all multiple-order

reflections; otherwise serious errors in the subsequent analysis will occur [108]. If higher

accuracy for a considerable broadening is desired, profile fitting can be accomplished in

terms of the reciprocal-space variable s, instead of in a real 20 space.

Assuming that only the Cauchy function determines domain size [AS(L) = exp

{-LI <£>>,)] and only the Gauss function gives root-mean-square strain (RMSS) [AD

{L) =

exp (-27i2L

2 <e2 >/d1)], Equation (4.2) leads to the Warren-Averbach Equation (2.17) for

the separation of size and strain contribution [62]. Experience shows that Cauchy and

Gauss functions can not satisfactory model specimen broadening. Balzar and Ledbetter [64]

postulate that the specimen function includes contributions of size and strain effects, both

approximated with the Voigt functions. Because the convolution of two Voigt functions is

also a Voigt function, Cauchy and Gauss integral breadths of the specimen profile are

easily separable:

25
@C @SC

+ @DC » (4 . 3)

•

P

2C = l?SG + tic . <4 * 4>

Langford [26] separated the contributions from size and strain broadening in a similar way.

(See Equations (2.24) and (2.25).) Note, however, that Equations (4.3) and (4.4) do not

define size and strain angular order-dependence.

Because Fourier coefficients are a product of a size and a distortion coefficient, from

Equations (4.2), (4.3), and (4.4), we can obtain the separation of size and strain

contributions to the pure specimen broadening:

A\L) = exp( ~2L(3sc - irL 2&

2sa) ; (4 -5)

A

D

(L) = exp( -2L/3dc -tL 2Q

2dg)

. (4.6)

Wang, Lee, and Lee [109] modeled the distortion coefficient, and Selivanov and Smislov

[110] modeled the size coefficient in the same way.

To obtain size and distortion coefficients, at least two reflections from the same

crystallographic-plane family must be available.

4.2 Size Coefficient

Surface-weighted domain size is calculated from the size coefficients following

Equation (2.18). From Equation (4.5) we obtain

<D> = _i 5 2/3sc

(4.7)

Therefore, surface-weighted domain size depends only on the Cauchy part of the sizeintegral breadth.

26
The second derivative of the size coefficients is proportional to the surface-weighted

column-length distribution function, Equation (2. 19). The volume-weighted column-length

distribution function follows similarly [111]:

PV(L) « L

&

2 A

s(L)

dL

2

By differentiating Equation (4.5) twice, we obtain

(4.8)

d — = [(2irL;8|G +2/3so)2 - 2x^c] A

S(L) . (4.9)

dL

2

Because the column-length distribution function should always be positive [59], the Cauchy

part must dominate. Inspection of Equation (4.9) shows that for small L we must require

>

(4.10)

Otherwise, the ’hook’ effect will occur in the plot of size coefficients A

S(L) versus L, that

is, the plot will be concave downward for small L (Figure 4. 1). The ’hook’ effect is usually

attributed to experimental errors connected with the truncation of the line profiles, and

consequently overestimation of background [59]. This is a widely encountered problem in

the Fourier analysis of line broadening. It results in overestimation of effective domain

sizes and underestimation of the RMSS [36]. Some authors [106] claim that the preset

specimen-broadening function eliminates the ’hook’ effect. However, Equation (4.9) shows

that, effectively, too high background causes underestimation of the Cauchy content of the

Voigt function, because the long tails are truncated prematurely. It has to be mentioned that

Wilkens [112] proposed tilted small-angle boundaries to be the source of the ’hook’ effect.

Liu and Wang [113] defined a minimum particle size present in a specimen, depending on

the size of the ’hook’ effect. Figure 4.1 shows that negative values of the column-length

distribution functions (set to zero), do not affect the shape, but shift the entire distribution

toward larger L values.

Note also that the surface-weighted column-length distribution function ps(L) will

usually have a maximum at L—0; but for the particular ratio of integral breadths,

27
pM

0 100 200

L (A)

Figure 4.1 (upper) The ’hook’ effect of the size coefficients A

s (full line) at small L\

(lower) it causes negative values (set to zero) of the column-length distribution functions.

28

Pv(L)
determined by Equation (4.10), it can be zero. The volume-weighted column-length

distribution function

p

v(L) will always have a maximum for L^O.

If the column-length distribution functions are known, it is possible to evaluate

mean values of respective distributions:

00

f LpJL)dL

<D> = i (4.11)

J, v oo

f PsJL)dL

Integrals of this type can be evaluated analytically [114]:

i

x

m exp(-bx 2 -cx)dx = (- 1 )' X d

m

exp

c

2 erfc

- " c

b dc m 4b

b A 2Jb b j

(4.12)

Surface-weighted domain size <D>S must be equal to the value obtained from Equation

(4.7). The volume-weighted domain size follows:

<D> -

exP<* 2)

erfc(k) = 1 . Pso f

(4.13)

Using Equations (4.7) and (4.13), we can evaluate the ratio of domain sizes:

<D> y <D>S

= 2\frkQxp(k2

) erfc (k) . (4.14)

Theoretically, k can change from zero to infinity. However, the minimum value of k is

determined by Equation (4.10):

1/^2 < k < * . (4- 15>

Hence, the ratio of domain sizes can change in a limited range (see also Figure 4.2):

29
1.31 *

'Jlirt erfc

V < (4.16)

<D>

<D>

< 2 .

k

Figure 4.2 The ratio of volume-weighted and surface-weighted domain sizes as a function

of the characteristic ratio of Cauchy and Gauss integral breadths k.

It may be noted that most experiments give the ratio <D> v/<Z)> J in this range (see for

instance review by Klug and Alexander [24]). When k goes to infinity the size broadening

is given only by the Cauchy component and <D>V = 2 <D>S

. This is a case of pure

Cauchy size broadening, described by Haider and Wagner [62] and de Keijser, Mittemeijer

and Rozendaal [107]. It is possible to imagine a more complicated column-length

distribution function [27] than Equation (4.9), which would allow even larger differences

between surface-weighted and volume-weighted domain sizes. However, we are not aware

of any study reporting a difference larger than 100 %.

30
4.3 Distortion Coefficient

In Paragraph 2. 1.2.1 it was shown that the distortion coefficient can be

approximated by the exponential

A

d

(L) = exp(-2x2s

2L2 <e2(L)>) . (4 - 17)

Comparing with Equation (4.6), we can write

< e

2(L) >

1 &DG + PDC 1 7 [TZ ~Vl(4.18)

Therefore, mean-square strains (MSS) decrease linearly with averaging distance L. This

behavior is usually observed in the Warren-Averbach analysis. Rothman and Cohen [115]

showed that such behavior would be expected of strains around dislocations. Adler and

Houska [116], Houska and Smith [117], and Rao and Houska [118] demonstrated for a

number of materials that MSS can be represented by a sum of two terms, given by Cauchy

and Gauss strain-broadened profiles.

However, for /Sqc = 0, the MSS are independent of L:

<e2 >'n = PTM(26)cote = J_e , (4.19)

2\]2t v/27

where the upper limit of strain e is defined as 77/4 (see Paragraph 2. 1.2.2). This is a

limiting case of pure-Gauss strain broadening, described by de Keijser, Mittemeijer and

Rozendaal [107].

4.4 Discussion

To calculate domain sizes and strain, it is necessary to define size and distortion

integral-breadths angular order-dependence. From Equation (4.13) it follows that the

domain size <D> V is always independent of the order of reflection:

31
/35C = const. ; (3SC = const. (4.20) •

However, from Equation (4.18) we find

= const. ; = const. (4-21)

j

2 J

An important consequence is that ’apparent strain’ rj will be independent of angle of

reflection only in the case of pure-Gauss strain broadening because (3DC and &DG depend

differently on the diffraction angle. If we compare Equation (4.21) with the multiple-line

Voigt-function analysis [26], given by Equations (2.24) and (2.25), it is evident that they

disagree. Therefore, recombination of constituent integral breadths fiDC and $DG to Voigt

strain-broadened integral breadth (3

D and the subsequent application of Equation (4.19) to

calculate strain [26, 119] will concur with the Fourier methods if strain broadening is

entirely Gaussian and for the asymptotic value of MSS (<e2(oo)>). However, neither

volume-weighted domain sizes <D> V will agree, although they are defined identically in

both approaches, because size-broadened and strain-broadened integral breadths are

dependent variables.

If at least two orders of reflection (/ and /+ 1) of the same plane (hkl) are available,

using Equations (4.20) and (4.21) we can solve Equations (4.3) and (4.4):

32
« = (4.22)

'SC

s

2(hD^D - sHMJl+l)

s

2(l+ 1 ) - s

2(l)

tic =

s

2 (hl)fat) - s\W2<Al+ 1 )

s

2(l+ 1 ) - s

2(l)

Pnc = s\D

DC

S

2(l+ 1 ) - S

2(l)

(4.23)

(4.24)

fio -

" 2(0

T-- [/&M) - ^c(0] • (4- 25)

J

2 (/ + l) - s

2 (l)

If we substitute these expressions into Equations (4.5) and (4.6), we see that this approach

leads exactly to the Warren-Averbach Equation (2.17) for two orders of reflections. This

is expected because the distortion coefficient is approximated with the exponential

(Equation (4.17)). Delhez, de Keijser, and Mittemeijer [23] argued that, instead of

Equation (2.17), the following relation would be more accurate:

A(L) = A

S(L) (1 - 2ir2 <e2(L)>L 2ld2

) . (4- 26)

These two approximations differ with fourth-order terms in the power-series expansion. In

terms of this approach, Equation (4.18) has to be rewritten:

2 _

1 - exp(-2L/?DC -7rL 2/3pG) (4.27)

2tVL2

This means that even if the strain-broadened profile is given entirely by the Gauss function,

the MSS depend on distance L (see Figure 4.3). In this approximation no simple relation

for the distortion integral-breadths angular order-dependence exists. For not so large L,

however, Equation (4.21) holds, and approximations (2.17) and (4.26) do not differ much

(see Figure 4.3).

Generally, it was shown that in the size-broadened profile the Cauchy part must

dominate. No similar requirement for the strain-broadened profile exists. However,

33
Figure 4.3 Mean-square strains < e

2 (L) > for two approximations of the distortion

coefficient: (upper) Voigt strain broadening; (lower) pure-Gauss strain broadening.

34
experience favors the assumption that it has to be more of Gauss-type. The WarrenAverbach approach is exact if strain broadening is purely Gaussian, so Equations (2.17)

and (4.18) are better approximations as strain profile is closer to the Gauss function. In any

case, both approaches, given by Equations (2.17) and (4.26), are good up to the third

power in strain, and the Warren-Averbach relation (2.17) does not assume that MSS are

independent of distance L [27]; it also represents a harmonic approximation.

If at least two orders of reflection from the same plane (hkl) are available, we can

use Equations (4.22), (4.23), (4.24), and (4.25) to calculate size-related and strain-related

integral breadths. Subsequent application of Equations (4.7), (4.13), and (4.18) gives

directly domain sizes <D>S and <D> V , and mean-square strains <e2 (L)>. This

approach is more straightforward and much simpler than the original Warren-Averbach

analysis. Great care should be given to the possible systematic errors. The easiest way to

observe the ’hook’ effect is to plot column-length distribution functions as a function of

averaging distance L. If they show negative values for small L (see Figure 4.1), all derived

parameters will be in error. This is because (i) if only the positive values of the columnlength distribution functions are numerically integrated or (ii) if the intercept on the L-axis

of the linear portion of the /l

s-versus-L curve is taken (see Figure 2.4), always larger

values of domain sizes will be obtained than by the application of Equations (4.7) and

(4.13). Therefore, we conclude that the discrepancy between integral-breadth and Fourier

methods is always present by the appearance of the ’hook’ effect in the A

s-versus-L curve.

An analogous discrepancy exists between integral-breadth and variance methods [119]. In

such cases, correction methods for truncation can be applied [35, 119, 120], but the best

procedure is to repeat the pattern fitting with the correct background.

In the Fourier analysis it is usually observed that the mean-square strains diverge

as the averaging distance L approaches zero. This also follows from Equations (4.18) and

(4.27). However, because the MSS dependence on distance L is not defined in WarrenAverbach analysis, it was suggested [27, 107, 121] that local strain can be obtained by

taking the second derivative of the distortion coefficient, or by a Taylor-series expansion

of local strain. Therefore, we obtain from Equation (4.6):

35
(4.28) < e

2(0) >

1 &G

s

2 2tt

It is evident that this relation is wrong. It holds only for a special case of pure-Gauss

strain-broadened profile, when the MSS are equal for any L. Otherwise, if the Cauchy

function contributes to strain broadening, all derivatives of strain in L - 0 are infinite, and

local strain can not be defined. If the main origin of strains is dislocations [115], strains

are defined after some distance from the dislocation (cutoff radius) to be finite. Averaging

strains over a region smaller than the Burgers vector is probably not justified. For instance,

Equation (4.18) gives, even for a small averaging distance L = 1 A, and considerable

strain broadening (/3D(£20) = 0DC(2d) = 10°), root-mean-square strain <e2 (L = 1 A) >

1/2 * 0.2, that is, about the elastic limit.

4.5 Random Errors of Derived Parameters

Errors in size and strain analysis of broadened peaks are relatively difficult to

evaluate. Following Langford [26], sources of the systematic errors include choice of

standard specimen, background, and type of analytical function used to describe the line

profiles. The first two errors should be minimized in the experimental procedure. Errors

caused by inadequate choice of specimen function would systematically affect all derived

results, but they can not be evaluated. Random errors caused by counting statistics have

been computed by Wilson [122, 123, 124] and applied to the Stokes deconvolution method

by Delhez, de Keijser, and Mittemeijer [23], as well as by Langford [26] and de Keijser

et al. [76] using single-line Voigt-function analysis. Nevertheless, the approximate error

magnitude can be calculated from estimated standard deviations (e.s.d.) of the parameters

refined in the fitting procedure. In the program SHADOW, the weighted least-squares error

is minimized:

36
RES Rt

~ ~£w,7, (obs)

(4.29)

i K 1

Here

RES = ^wi [/l (obs)-/,(cal)]: (4.30)

i-1

and weights are the reciprocal variances of the observations:

w. = l//.(obs) . (4.31)

Each line profile has four parameters varied independently: position, intensity, and Cauchy

and Gauss integral breadths of the Voigt profile. In least-squares refinement, e.s.d.’s are

computed as

K

RES

m -m 1 (4.32)

Here bti are diagonal elements of the inverse matrix of the equation coefficients, m is the

number of observations, and m! is the number of refined parameters.

The main source of errors is integral breadths. Errors in peak position, peak intensity, and

background are much smaller and can be neglected in this simple approach. For two

independent variables, &G and /3C, covariance vanishes, and from Equations (4.7) and

(4.18), for the two orders of reflection, / and /+1, it follows that

37
R (4.33) 2(<D>S) s\hl)fyl)R2(fi^D) +s\t)fahl)R2(l3Jhl))

[^(M)/?^/) -s 2mj[hl)]2

R2(<e2(L)> m

)

m

+&M)R2 (W+\))

{2 USetf+l) -PM *TL[J3

2G(h\) -fyl)]}2

+

‘x2L

2 [fio(r)R2 (Pa

(l)) +&c(l + l)R2 (PG

(l+l))]

{2[^c(/ + l)-/3c(/)] + ^^2G(/ + l)-/3kO]}2

' (4.34)

Here /?(*) are ’relative standard deviations’.

Error of <D> V would be complicated to evaluate, but because 1.31 <D>S < <D> V <

2 <D>S , Equation (4.33) gives a good estimation for error of <D>V as well.

Alternatively, to see how errors depend on the Fourier coefficients, errors can be estimated

from the Warren-Averbach Equation (2.17) [86]. From Equation (4.2) it follows that

R\A(L)) = WL*fas)R\PG

(2e)) + 4L 2fys)R 2{$(i26)) . (4-35)

Errors in root-mean-square strains and domain sizes are

R\<e2(L)> m

)

R

2(A(L,D) +R2(A(L,h\)) . 4 in2 A(L,l)

A(L,l+ 1)

(4.36)

R

2 (<D>) A

S(L) 1

2 s

4(l+l)R 2(A(L,Q) +si (l)R 2(A(L,l+\)) (4 37)

1 -A S(L)

J [y

2(/+l)

j

!

(I)]

!

Here, <D>S is approximately defined with A

S

(L) = 1 - L/<D> S

.

Errors in Fourier coefficients increase with L, while factors in Equations (4.36) and (4.37)

lower the errors for large L. In general, errors of domain sizes and strains are of the same

order of magnitude as errors of integral breadths [86].

38
5 Application

5.1 Correction for Instrumental Broadening

Before specimen broadening is analyzed, instrumental broadening must be

determined. This is accomplished by carefully measuring diffraction peaks of some wellannealed ’defect-free’ specimen. It is then assumed that its broadening may be attributed

only to the instrument. The usual procedure is to anneal the specimen. However, in some

instances that is not possible, because either the material undergoes an irreversible phase

transition on annealing, or the number of defects can not be successfully decreased by

annealing. Another possibility is to measure the whole diffraction pattern of the material

showing the minimal line broadening, and then to synthesize the instrumental profile at the

needed diffraction angle. This approach requires the modeling of the angle dependence of

the instrumental (standard) parameters. Cagliotti, Paoletti, and Ricci [125] proposed the

following function to describe the variation of the full width at the half maximum of profile

with the diffraction angle:

FWHM2 (20) = £/tan20 + VtanS + W . (5- 1 )

Although this function was derived for neutron diffraction, it was confirmed to work well

also in x-ray diffraction case [126, 127]. A more appropriate function for the x-ray angledispersive powder diffractometer, based on theoretically predicted errors of some

instrumental parameters [128] may be the following [129]:

39
FWHM2 (20) = W+ Vsin220 + t/tan20 + U'cot26 . (52) .

This function may better model the increased axial divergency at low angles and correct

for the specimen transparency [129]. However, contrary to the requirement on the

specimen function, most important for the instrumental function is to correctly describe the

angular variation of parameters, regardless of its theoretical foundation.

When specimen broadening is modeled with a Voigt function, the simplest way to

correct for the instrumental broadening is by fitting the line profiles with the Voigt

function, too. Cauchy and Gauss integral breadths of the specimen-broadened profile are

then easily computable by Equations (2.4) and (2.5). However, because the instrumental

broadening is asymmetric [24], modeling with the symmetric Voigt function can cause a

fictitious error distribution, resulting in errors of strain up to 35% [76]. Another approach

is to model the instrumental-broadening angle dependence by fitting the profile shapes of

a standard specimen with some asymmetrical function; split-Pearson-VII [95] or pseudoVoigt convoluted with the exponential function [105]. The instrumental function can then

be synthesized at any desired angle of diffraction and convoluted with the assumed

specimen function to match the observed profile by means of Equation (1.2).

In the program SHADOW, instrumental parameters are determined by fitting the

split-Pearson VII function (see Figure 5.1) to line profiles of a standard specimen:

with

I(x) = 7(0).

1 + me

(5.3)

c « . (5.4)

\Jmir Y(m -1/2)

Here, m = 1 or m = oo yields a Cauchy or Gauss function, respectively.

Refined full widths at half maximum (FWHM) and shape factors m for both low-angle and

high-angle sides of the profiles are fitted with second-order polynomials. To fit the

FWHM, we used Equation (5.1), and for the shape factors

40
1200

Figure 5.1 A split-Pearson VII profile. The two half profiles have same peak position and

intensity. Adapted from Howard and Preston [4].

m(26) = U' (26)

2 + V' (26) + W' . (5 - 5)

The resulting coefficients U, V, W, £/', V, and W permit synthesis of the asymmetrical

instrumental line profile at any desired angle.

The basic requirement on the standard specimen is, however, to show as small a

line broadening as possible. To minimize physical contributions to the line broadening of

the standard specimen, a few moments were emphasized as follows. Because diffractionline width depends strongly on degree of annealing, it is preferable to use some reference

powder-diffraction standard. Furthermore, asymmetry in the peak profiles is introduced by

axial divergence of the beam, flat specimen surface, and specimen transparency [24].

Choosing a standard specimen with low absorption coefficient would cause transparency

effects to dominate. If the studied specimen has a large absorption coefficient (compared

to the standard), this might produce a fictitious size contribution and errors in microstrains.

All of the specimens studied have absorption coefficients exceeding 1000 cm'1

, so an NIST

41
0 50

2 9 (deg)

100 150

1

E-Z

EdZ

o

cuX

Ed

tan 9

Figure 5.2 Refined FWHMs and shape factors (exponents) m for low-angle and high-angle

sides of LaB6 line profiles. Second-order polynomials were fitted through points.

standard reference material 660 LaB6 was chosen to model the instrumental broadening (ji = 1098 cm'1 ). According to Fawcett et al. [130], LaB6 showed the narrowest lines of all

studied compounds. Furthermore, LaB6 has a primitive cubic structure (space group Pm3m)

i resulting in relatively large number of peaks equally distributed over 26. This allows for

better characterization and lower errors of FWHMs and shape factors of split-Pearson VII

functions (Figure 5.2).

5.2 Applicability of the Method

To study the applicability of method described in Paragraph 4, we first studied

simple cubic-structure materials such as silver and tungsten. Tungsten has very narrow line

42
profiles, allowing us to obtain the upper limit of domain sizes that can be studied. Silver

is easily deformed, which provides a possibility to apply the method to broad line profiles.

To test the case of relatively complicated patterns and weak line broadening, the method

was also applied to Laj 85 Sro. 15 Cu04 and La2 Cu04 powders. In this paragraph, only the

mechanical aspects of the line broadening are discussed. Discussion about the origins of

broadening of superconductors can be found in Paragraph 6.

5.2.1 Silver and Tungsten Powders

Figure 5.3 shows observed and refined peaks of tungsten untreated and silver

ground powders. In Table 5.1 are listed results of fitted pure-specimen Voigt profiles for

silver and tungsten specimens; Table 5.2 and Figure 5.4 give results of the line-broadening

analysis. Untreated tungsten powder shows relatively weak broadening. Instrumental profile

FWHMs at angle positions of (110) and (220) tungsten lines are 0.059° and 0.081°,

respectively, close to values measured for tungsten: 0.065° and 0. 100°. Results in Table 5.2

reveal that small broadening is likely caused by domain sizes, because microstrains have

negligible value. This pushes the limit for measurable domain sizes probably up to 4500-

5000 A. However, one must be aware that weak specimen broadening implies higher

uncertainty of all derived parameters. Moreover, the choice of the instrumental standard

becomes more crucial.

Both silver and tungsten line profiles become more Cauchy-like after grinding,

which probably increases dislocation density in the crystallites. This is consistent with the

presumption that small crystallites and incoherently diffracting domains separated by

dislocations within grains affect the tails of the diffraction-line profiles [60, 115].

Figure 5.5 illustrates the dependence of MSS on the reciprocal of the averaging distance

L.

Errors in integral breadths allow estimation of errors in strain and size parameters

(Paragraph 4.5), but in some cases the refinement algorithm gives unreliable errors of

integral breadths. When a particular parameter is close to the limiting value (for instance,

10‘5 ° has been put as the minimum value for integral breadths), errors become large.

43
30.5 , 40.5

20 (deg)

20 (deg)

Figure 5.3 Observed points (pluses), refined pattern (full line), and difference pattern

(below): (110) W untreated (upper); (200) Ag ground (lower).

44
FOURIER

COEFFICIENTS

FOURIER

COEFFICIENTS

0 2000 o 4000

L(A)

Figure 5.4 Fourier coefficients for the first- (pluses) and second-order (crosses) reflection,

and size coefficients (circles): [111] Ag untreated (upper); [100] Ag ground (lower).

45
100

CDO

rH

X

A 50

CM

CO

V

0

0 50 O 100 150

l/L(A-i)X103

Figure 5.5 Mean-square strains <e2 > as a function of 1/L.

— • [1 001 Ag ground --- — [1 1 1 ]Ag ground

[1 1 1 ] Ag untreated

% *

% ^

**

However, in Equations (4.33), (4.34), (4.35), (4.36), and (4.37), only the product of

integral breadth and accompanying error is significant,, which is roughly equal for Cauchy

and Gauss parts. Errors of domain sizes and strains are of the same order of magnitude as

errors of integral breadths.

The possible source of systematic error is potential inadequacy of Voigt function

to accurately describe specimen broadening. This effect can not be evaluated analytically;

but it would affect all derived parameters, especially the column-length distribution

functions. The logical relationship between values of domain sizes (see Table 5.2) for

different degrees of broadening indicates that possible systematic errors can not be large.

Equation (4.11) allows computation of volume-weighted and surface-weighted

average domain sizes if respective column-length distribution functions can be obtained.

Figure 5.6 gives surface-weighted and volume-weighted average column-length distribution

46
Table 5.1 Parameters of the pure-specimen Voigt function, as obtained from profile-fitting

procedure for tungsten and silver powders.

Specimen hkl 260O 0cO 0oO FWHM(°) K(%)

W untreated 110 40.28 0.0065(7) 0.020(2) 0.021 9.0

220 87.05 0.0301(6) <ia5 0.019 4.4

W ground 110 40.29 0.100(2) 0.038(4) 0.080 9.0

220 87.05 0.222(3) 0.069(6) 0.166 4.2

Ag untreated 111 38.06 0.056(1) cio-5 0.036 10.5

222 81.51 0.105(4) 0.0008(21) 0.067 9.7

200 44.25 0.118(2) cio-5 0.075 10.5

400 97.86 0.274(21) <ia5 0.175 12.2

Ag ground 111 38.07 0.175(3) 0.038(12) 0.121 9.9

222 81.51 0.410(26) <10-5 0.261 14.5

200 44.24 0.365(10) 0.076(29) 0.250 9.9

400 97.83 1.079(82) <10-5 0.687 14.5

Table 5.2 Microstructural parameters for tungsten and silver powders.

Specimen Direction <D>AA) <D>XA) < e

2 (a3) >1/2 <e2(<D>J2)> m

W untreated [110] 3200(200) 3500(200) 0.00038 0.00008(2)

W ground [110] 620(20) 1030(30) 0.0023 0.00054(2)

Ag untreated [111]

[100]

1000(20)

510(20)

2000(20)

1030(20)

0.0010

0.0022

0.00024(1)

0.00047(5)

Ag ground [111]

[100]

380(20)

210(10)

650(20)

350(20)

0.0047

0.0100

0.00095(7)

0.00180(14)

functions following Equations (4.9) and (4.8). Width of the distribution function determines

the relative difference between <D>S and <D> V

. The broader the distribution, the larger

47
Pb(L)

p

s

(L)

I n : n Ji 0 2000 4000 6000

L(A)

0 500 0 1000

L(A)

Figure 5.6 Surface-weighted and volume-weighted column-length distribution functions,

normalized on unit area: [111] Ag untreated (upper); [100] Ag ground (lower).

48

Pv(L)

pv(L)
Figure 5.7 Surface-weighted and volume-weighted column-length distribution functions

for [010] La2 Cu04 , normalized on unit area.

the differences, because small crystallites contribute more to the surface-weighted average.

That is much more evident comparing the surface-weighted column-length distribution with

the volume-weighted. If they have similar shape and maximum position, as in Figure 5.7,

differences are small. Conversely, if the surface-weighted distribution function has a sharp

maximum toward smaller sizes, differences are larger (see Figure 5.6). The difference

between <D>s and <D> V , and the actual mean dimension of the crystallites in a

particular direction, also depends strongly on the average shape of the crystallites [52, 131,

132].

If experimental profiles are deconvolved by Stokes method, even for considerable

specimen broadening, size coefficients A

s usually oscillate at larger L values [59, 106],

49
42

26 (deg)

Figure 5.8 Observed points (pluses), refined pattern (full line), convoluted profiles

(dashed line), and difference plot (below) for part of La2Cu04 pattern.

preventing computation of the column-length distribution function. Few techniques were

used to deal with this problem: successive convolution unfolding method [32, 133],

smoothing, and iterative methods [134, 135, 136, 137]. Figure 5.6 and Figure 5.7 show

very smooth column-length distribution functions. However, they follow from size

coefficients A

s that depend on the accuracy of the approximation for the distortion

coefficient, given by Equation (4.17). Equation (4.17) is exact if the strain distribution is

Gaussian, but in general holds only for small harmonic numbers n, if strain broadening is

not negligible.

50
5.2.2 La2.jeSrxCu04 Powders

To test the applicability of the discussed method to more complicated patterns, two

compounds with lower crystallographic symmetry were studied. Lax 85 Sr0 15 Cu04 has a

tetragonal K2NiF4-type structure, space group IMmmm. La2 Cu04 is orthorhombic at room

temperature. Both compounds show slight line broadening and relatively highly overlapping

peaks (see Figure 5.8), which makes it very difficult, if not impossible, to perform a

Stokes analysis.

Table 5.3 and Table 5.4 list results from the fitting procedure and the analysis of

specimen-broadened integral breadths. There are no qualitative differences in results for

tungsten and silver powders. However, average errors are higher, as expected because of

overlapping peaks, while weighted errors R^

are surprisingly smaller. This fact illustrates

the unreliability of R

when only a segment of the pattern is being refined, because it

depends on the number of counts accumulated in points, as well as on the 26 range of the

refinement.

Table 5.3 Parameters of the pure-specimen Voigt function, as obtained from profile-fitting

procedure for La2.xSrtCu04 powders.

Specimen hkl 20„(°) «°) 0oO FWHM(°)

Lai.MSro.wCuO, 004 26.95 0.046(4) 0.024(8) 0.042 3.4

006 40.92 0.083(5) <10-5 0.053 6.2

110 33.52 0.110(5) <10-5 0.070 12.5

220 70.46 0.171(9) <10 5 0.109 4.3

La2 Cu04 004 27.13 0.061(8) 0.045(9) 0.067 3.9

006 41.19 0.124(7) <10'5 0.079 7.7

020 33.16 0.074(10) 0.078(10) 0.102 6.9

040 69.57 0.210(30) 0.026(81) 0.135 2.9

200 33.45 0.074(8) 0.061(10) 0.087 6.9

400 70.28 0.130(10) 0.023(95) 0.089 2.9

51
Table 5.4 Microstructural parameters for 85 Sr0 15 Cu04 and La2 Cu04 powders.

Specimen Direction <D>,(A) <£»„(A) <e2(a3)> 1/2 <e\<D>J2)> m

Lai.85Sr0.i5CuO4 [001] 1700(300) 2000(200) 0.0024 0.00049(9)

[110] 470(20) 940(20) 0.0017 0.00044(6)

La2 Cu04 [001] 1100(100) 1200(100) 0.0038 0.0008(1)

[010] 680(50) 760(50) 0.0033 0.0007(2)

[100] 680(50) 810(50) 0.0016 0.0003(3)

Figure 5.9 and Figure 5.10 represent Fourier coefficients of Laj 85 Sr0 i 5 Cu04 [110]

and La2 Cu04 [010] directions. In the second plot, the /l

s(L)-versus-L plot shows a concavedownward part near L = 0, the so-called ’hook’ effect. It was shown in Paragraph 4.2 that

the ’hook’ effect originates because of underestimation of background, connected with the

truncation of profiles. In the profile refinement, all peaks separated less than 4° 26 have

been included in the refinement region to avoid possible overlapping of peak tails, and

specimen profiles have been truncated below 0.1 % of the maximum intensity. However,

because the polynomial background was determined prior to profile refinement, it may be

overestimated for complicated patterns containing many overlapping peaks. If background

is refined with other profile parameters, undesirable correlation with integral breadths

occurs.

5.3 Comparison with the Integral-Breadth Methods

Knowing the specimen integral breadths, simplified methods can be applied. In

Paragraphs 2. 1.2.2 and 2. 1.2.3 we reviewed multiple-line and single-line integral-breadth

methods, respectively. We shall compare the simplified multiple-line methods (Equations

52
0 1000 O 2000

L(A)

Figure 5.9 Fourier coefficients for the first- (pluses) and second-order (crosses) reflection,

and size coefficients (circles) for [110] Laj 85 Sr0 15Cu04

.

(2.20), (2.21), and (2.22)) and the single-line method [76] (Equations (2.26) and (2.27))

with results obtained for silver, tungsten, and La2-xSr.*Cu04 powders (Table 5.2 and

Table 5.4). We did not compare with the multiple-line Voigt-function method [26] because

it was shown in Paragraph 4.4 that it is incompatible with the Warren-Averbach analysis.

In Table 5.5, integral breadth was computed according to Equation (2.8). This value was

compared with @A , integral breadth computed from Fourier coefficients [58]:

a, E ^C*)

Z, = -oo

(5.6)

Results obtained by the Warren-Averbach analysis are significantly smaller for both size

and strain. Integral-breadth methods give the upper limit for strain, so comparison is

53
Figure 5.10 Fourier coefficients for first- (pluses) and second-order (crosses) reflection,

and size coefficients (circles) for [010] La2Cu04.

limited. Considering the approximation e = 1.25 <e2 (L)> 1/2 [138] with L = <D>J2 or

L = <D>J2, strains resemble more closely than crystallite sizes. It must be noted,

however, that this approximation makes sense only in the case of pure-Gauss strain

broadening. Benedetti et al. [106] reported excellent agreement for both size and strain

using Warren-Averbach and Cauchy-Gauss (Equation (2.21)) methods. Results from

Table 5.5, on the contrary, indicate that the Gauss-Gauss approximation gives values

closest to the Warren-Averbach method for crystallite sizes, whereas the Cauchy-Gauss and

especially the Cauchy-Cauchy approximation tend to give too large values. For strain, the

trend is opposite. The Cauchy-Cauchy approximation resembles most closely a WarrenAverbach method. These results concur with the Klug and Alexander [24] comparison of

the published size and strain values obtained by Warren-Averbach (Stokes) and integral54
breadth methods. However, for x-ray diffraction, the Cauchy-Gauss assumption for the

size-strain broadening is theoretically and experimentally more favored than the other two

models. The fact that the Gauss-Gauss method for crystallite size and the Cauchy-Cauchy

method for strain give more realistic results may mean that the presumption that size and

strain broadening are exclusively of one type is an oversimplification.

The single-line method seems much less reliable. If Gaussian breadth is very small,

no information about strain is obtained. Moreover, second-order reflections give much

lower values of both size and strain than do basic reflections. However, if multiple

reflections are not available, the single-line method can give satisfactory estimations.

55
Table

5.5

Comparison

of

results

obtained

with

the

integral-breadth

methods:

Cauchy-Cauchy

(C-C),

Cauchy-Gauss

(C-G),

Gauss-Gauss

(G-G),

and

single-line

(S-L)

analysis.

Specimen

hkl

0,x

10 3

0

Ax

10 3

<D>

V

<D> (V A)

<e\<D>J2)>

m

ex

10 3

(A

1)

(A

1)

(A)

C-C

C-G

G-G

S-L

xlO 3

C-C

C-G

G-G

S-L

W

untreated

110

0.259

0.259

3500

3700

3810

3810

14500

0.08

-0.01

-0.03

-0.05

0.24 »n

Tf

ts

* * o ©

•©

o d

m

o

mON mON © ©

mOn

ONtn in

00 d d ©

oOn m CN

00 d © ©

00 © <N

00 ow |o * * o © s #o Tf NO On00 Tf © © © © — o © ©

00 00 o f" o in

ON © 00 G© ts

Tf © © ©

tS in 00 cs © , , OO © r~- NO ON

w~> Tf © © d ©

ON in NO cs r- On tS

r~~ © t- m © ©

—

Tf © © © ©

©

Tf

tSo

rTfdmOndo00 ON Tf

Tf Tf d d

oooddr- cn

© © © © © © © © © © © o © © © © © © © © Tf m r- ON fTj 00 NO ts f— Tf m On vO Tj- <3=*~S m m

On m NO 00 Tf tN CN On 00 vo Tf r- ts ts 00

1 © © © o © © © O © © CS ON m V—

ts VO p" cn VO

CS ts On m VO On vo 00 00 fS ts © © © © o © © o © o

in On r- Tf 00 ON Tf 5*=> Tf m Tf tN t" ts © On On 00 CN Tf tn *“* © © © © © © © © g o Tf Tf r- 00 00 tn Tf 00 A) w—

tS m ts © cn ts ts On CN 00 ts ci CN CN On in ts

om

o

rTf

tsd

mts in© 8VO 8ON Tfts

Tf© 8

ts

in 8

cn©

©mVO ©00

00 Os 00in

00

oo

On tn

ts

'f 8

Tf

ts ©ts ts © © ts ts rn Tf od © © ©

» ts

t—

Tfts s 8VO ©On d ts © ©

© o © ts

ts ts ts

ts ts ' ts

Tfts

8

fS

3

(N

8

OO tS

Os in

pr> m m

© ©

Tf od

r- —

in oo

VO 00 o o

ON OO OO

ONOno

CSm

mTf

ooONvo m ts ts

— cs — ts — ts 88ts tj- ^ ^

o ©

s

8cs 8

T3a

o

ot-00 £

•8

>«—

$<*=» c3

00 <

•oc

3

£

00

00 <

O

3

uv» i? o C/5 3

s2 uU

5 5

56

*

Not

possible

to

evaluate

because

of

too

small

Gauss

integral

breadth.
5.4 Reliability of Profile Fitting

The profile fitting of a cluster of even severely overlapping peaks can be

accomplished with a low error and excellent fit of total intensity. But how reliable is

information obtained about separate peaks in the cluster? In the fitting procedure, it is

possible to put constraints on the particular profile parameter to limit intensity, position,

and width of the peak. If anisotropic broadening or different phases are present, constraints

may not be realistic. Morever, acceptable results can be obtained even using a different

number of profiles in refinement [4]. Hence, the first condition for the successful

application of this method is knowledge of actual phases present in the sample and their

crystalline structure.

In this regard, two powders were measured: orthorhombic 1-^ ^Sro ^CuC^ has a

slightly distorted K2NiF4-type structure (space group Bmab, a = 5.3510(2) A, b =

5.3692(2) A, c = 13.1931(7) A) and tetragonal Lai^Sro.^CuC^ (space group 74/mmm, a = 3.7711(3) A, c = 13.2580(8) A). Data were collected for each specimen separately and

for a specimen obtained by mixing the same powders. Because diffracted intensity of each

phase was lower for the mixture, counting time was proportionally increased to obtain

roughly the same counting statistics. Figure 5.11 shows two partially separated

orthorhombic (020), (200) and (040), (400) peaks, overlapped with tetragonal (110) and

(220) reflections. During the profile fitting of the mixed powders, (200) orthorhombic peak

tended to ’disappear’ on account of neighboring (020) and (110) reflections. Its intensity

was constrained to vary in range ±30 % of (020) peak intensity, which can be justified if

the crystallographic structure is known.

57
600

Figure 5.11 (upper) La, 76Sr0 24CUO4 (110) peak; (middle) La, ^Sro ^CuC^ (020) and (200)

peaks; (lower) (110), (020), and (200) peaks.

58
Table

5.6

Comparison

between

two

specimens

run

separately

and

mixed

together.

Specimen

hkl

20

o

(°)

/3

C

<°)

(3J°)

<D>

s (k)

<£>>

V (A)

<t\<D>J

2)>

,,2

xl0

3

©

CO

o

CO

OO

o o

CN

'w' o

S’S

©wo

00

8 8

CN —

'

CN

CO

CN

CO

© © g 8 w©oo

Os

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59
Table 5.6 contains results of fitted specimen-Voigt functions, as well as size and

strain values obtained by the analysis of broadening. Comparing integral breadths of the

starting and mixed powders, some profiles change from more Cauchy-like to Gauss-like

and vice versa. This is especially evident for the (400) reflection, meaning that accurate

determination of the profile tails is indeed affected in the cluster of overlapping peaks.

However, integral breadth does not change significantly, so results of the multiple-line

methods are not much influenced. On the contrary, the single-line method gives quite

different results. This may support the conclusion of Suortti, Ahtee, and Unonius [43] that

the Voigt function fails to account properly for size and strain broadening simultaneously

if only Cauchy integral breadth determines crystallite size.

Following Table 5.6, results of Warren-Averbach analysis of starting and mixed

powders agree in the range of standard deviations. However, errors are much larger,

especially for the hidden (200)-(400) reflections. From Equation (4.32) it follows that more

counts (longer counting times) and more observables (smaller step-size) would lower

standard deviations. Therefore, in the case of highly overlapping patterns, to obtain

desirable accuracy, much longer measurements are needed, although that will not change

the fact that higher overlapping implies intrinsically higher errors of all refinable

parameters. To further minimize possible artifacts of the fitting procedure, it is desirable

to include in the analysis as many reflections in the same crystallographic direction as

possible.

5.5 Remarks

By modeling the specimen size and strain broadening with the simple Voigt

function, it is possible to obtain domain sizes and strains that agree with experiment and

show a logical interrelationship. Furthermore, the Voigt function shows the correct

1/A(20)

2 asymptotic behavior of peak tails, as expected from kinematical theory [139].

However, if the specimen profile is significantly asymmetric (because of twins and extrinsic

stacking faults [140, 59]) or the ratio FWHM//3 is not intermediate to Cauchy (2hr) and

60
Gauss (2(ln2/?r)

1/2) functions, then the Voigt function can not be applied [25]. Suortti,

Ahtee, and Unonius [43] found good overall agreement by fitting the Voigt function to the

pure-specimen profiles of a Ni powder, deconvolved by the instrumental function.

However, more accurate comparison is limited because of unavoidable deconvolution

ripples of specimen profiles. Furthermore, the Voigt function might not be flexible enough

to model a wide range of specimen broadening, as well as the different causes of

broadening. Therefore, the question whether the Voigt function can satisfactorily describe

complex specimen broadening over the whole 26 range in a general case needs further

evaluation.

This method of Fourier analysis, with the assumed profile-shape function, is most

useful in cases when the classical Stokes analysis fails; therefore when peak overlapping

occurs and specimen broadening is comparable to instrumental broadening. It was shown

that for a high degree of peak overlapping, the fitting procedure can give unreliable results

of Gauss and Cauchy integral breadths; that is, the peak can change easily from

predominantly Cauchy-like to Gauss-like, and contrary. This can lead to illogical values

of size-integral and strain-integral breadths, following Equations (4.22), (4.23), (4.24), and

(4.25), and to irregularities in the behavior of Fourier coefficients for large L.

Simultaneously, if the size-broadened profile has a large Gauss-function contribution, the

’hook’ effect will occur. However, volume-weighted domain sizes <D>V and MSS

< e

2(L) > for small L are much less affected, because they rely on both Cauchy and Gauss

parts of the broadened profile. Any integral-breadth method that assigns the size and strain

broadening to be described exclusively by a Cauchy or Gauss function, in this case, fails

completely. For specimens with lower crystallographic symmetry, when many multipleorder reflections are available, but the peaks are overlapped, it is therefore desirable to

include in the analysis as many reflections as possible, to minimize the potential artifacts

of the fitting process.

61
6 Analysis of Superconductors

6.1 (La-M)2Cu04 Superconductors

Bednorz and Muller [16] found the first high-temperature oxide superconductor in

the La-Ba-Cu-0 system. Soon after, substituting Ba with Sr and Ca, Kishio et al. [17]

found superconductivity in two similar compounds. Later, many compounds with much

higher transition temperatures ( Tc ) were found. The (La-M)2 Cu04 system remains favorable

for study because the cation content, responsible for the superconductivity, is easier to

control and measure than the oxygen stoichiometry. Also, this system provides a wide

range of substitutional cation solubility in which the tetragonal K2NiF4-type structure is

maintained. Compared with much-studied Y-Ba-Cu-O, this structure is simpler because it

contains only one copper-ion site and two oxygen-ion sites.

The undoped compound La2 Cu04 is orthorhombic at room temperature. It becomes

superconducting either with cation substitution on La sites or increased oxygen content to

more than four atoms per formula unit. Doping with divalent cations increases the itinerant

hole carriers in the oxygen-derived electron band [141], which probably causes the

superconductivity. However, different dopants influence the transition temperature Tc

.

Although the unit-cell c-parameter simply reflects ionic dopant size [142], the in-plane unitcell parameter (and accordingly the Cu-0 bond length) correlates with Tc for different

dopants and for different doping amounts [142, 143].

62
0( 1

)

Figure 6.1 Crystal structure of (La-M)2Cu04 showing one unit cell.

Lai.85Mo.i5Cu04 (M = Ba, Ca, Sr) have a tetragonal K2NiF4-type structure, space

group IMmmm with four atoms in the asymmetric unit [17]. One site is occupied with La,

partially substituted with dopant ions (Figure 6.1). La2Cu04 is orthorhombic at room

temperature. The best way to correlate these two structures is to describe La2Cu04 in space

63
group Bmab; the c axis remains the same, tetragonal [110] and [110] directions become

[010] and [100] orthorhombic axes, and size of the unit cell is doubled [144].

Table 6.1 Lattice parameters and Tc at zero resistivity (AC method at 10 mA current).

Specimen a (A) b (A) c (A) TAK)

Laj 85 Sr0 15 Cu04 3.77814(7) — 13.231(2) 36

La185Bao.i5Cu04 3.7846(2) — 13.288(4) 28

La! 85 Cao j 5 Cu04 3.7863(7) —13.17(2) 19

La2 Cu04 5.3558(2) 5.4053(2) 13.1451(8) —

„

Figure 6.2 shows parts of the x-ray diffraction patterns, observed and refined, of

Lai.85M0.

5 Cu04 (M = Sr, Ba, Ca) and La2 Cu04

. Refined lattice parameters are given in

Table 6.1. Table 6.2 shows results of the line-broadening analysis. Relatively weak peak

broadening was observed, which can not be attributed to the proximity of the phase

transformation; 1^ 85 Sr0 15Cu04 is tetragonal down to

* 170 K [145], La! 85Bao 15 Cu04 to

* 140 K [146, 147], and Lai ssCao 15Cu04 to

* 130 K [148]. Another possible reason for

peak broadening might be continuous compositional variations in dopant content. The

scanning Auger microprobe analysis of La2.xSrcCu04 compounds [149] showed that grainboundary segregation of Sr is weak and limited to a narrow region near the grain surface.

In the case of the separation of two phases with different Sr content and similar lattice

parameters, asymmetry of some reflections would be observed [150]. Moreover, in the case

of compositional inhomogeneity, microstrains < e

2 (L) > would be roughly independent of

the averaging distance L in the grains [151]; therefore distribution of strains would be

Gaussian. For all specimens, microstrain behavior is identical; it decreases linearly with

distance L. Rothman and Cohen [115] showed that such behavior of microstrains is caused

by dislocations that might arise during grinding. Assuming the same applied stress,

64
40 41 42 43 44 45 40 41 42 43 44 45

28 (deg) 20 (deg)

20 (deg?

-30

-V"

26 (deg?

Figure 6.2 Diffraction patterns of Lat 85Mo 15 Cu04 specimens: (a) M = Sr; (b) M = Ba;

(c) M = Ca; (d) M = La.

65
microstrain would be approximately inversely proportional to the elastic moduli of these

materials. If mean microstrain is defined as the arithmetic average of strains for the [001]

and [110] directions, results agree with values of the Young’s moduli for polycrystalline

La2 Cu04 , Lsl1 85M0 15 Cu04 (M = Sr, Ba, Ca) [152]. However, other factors may contribute

to the microstrain. Coordination-6 ionic radii of La3+

, Sr2+, Ba2+

, Ca2+

are 1.06, 1.16,

1.36, 1.00 A, respectively [153]. Substitution of La3+ with ions of different sizes affects

the lattice parameter [142] and may cause much larger strains in the [001] direction when

some of the ions are replaced with Ba2+ than with Sr2+

, in accord with results in Table 6.2.

Substitution of Ca2+ ions produces an entirely different effect; microstrains in the [110]

direction are larger. This may be explained by comparing effects of substitutional ions on

the unit-cell £-parameter. The Ba-doped specimen has a larger ^-parameter than the Srdoped, but the Ca-doped ^-parameter is largest, although Ca2+ has the smallest ionic size.

This should increase the cell distortion in the a-b plane.

Table 6.2 reveals high anisotropy in these materials, consistent with other physical

properties such as thermal expansion [154] and upper critical magnetic field [155].

Differences of domain sizes might arise from a layered structure in the [001] direction and

consequently easy incorporation of stacking faults between regularly stacked layers. The

stacking-fault and twin-fault probabilities can be estimated from the average surfaceweighted domain sizes [59]:

1 <D>r —— + c

lhm fun(a,c/) <D> (6. 1 )

Here, c

lhU]

is a constant for the particular [hkl\ direction, and fun(a,a') a linear

combination of stacking-fault probability a and twin-fault probability a'

The exact form of Equation (6.1) was derived only for the cubic and hexagonal crystal

structures. On condition that the ’true’ domain dimension D is isotropic or so large that the

second term of Equation (6.1) dominates, a linear function of stacking-fault and twin-fault

probability is

66
Table 6.2 Results of line-broadening analysis for (La-M)2Cu04 specimens.

Specimen hkl W) 0G

(°) <o>,(A) cz» v(A) <t\<D> J2)>

m

f-*tl.85^0.15^-'UO4 004 0.046(4)

006 0.083(5)

0.024(8)

CIO"5 1700(300) 2000(200) 0.00049(9)

110 0.110(5)

220 0.171(9)

CIO-5 CIO-5 470(20) 940(20) 0.00044(6)

Laj 85Bao.i5 Cu04 004 0.049(6)

006 0.110(7)

0.076(6)

0.089(9)

1100(200) 1100(100) 0.0011(2)

110 0.087(4) 0.00042(36)

220 0.116(3) CIO-5 560(20) 1110(20) 0.00024(5)

Lai.85Cao.i5 Cu04 004 0.120(13)

006 0.146(12)

0.032(22)

0.033(27)

440(70) 780(90) 0.0007(4)

110 0.045(25)

220 0.343(33)

0.200(19)

CIO"5 340(30) 360(20) 0.0013(2)

La2 Cu04 004 0.061(8)

006 0.124(7)

0.045(9)

CIO-5 1100(100) 1200(100) 0.0008(1)

020 0.074(10)

040 0.210(30)

0.078(10)

0.026(81)

680(50) 760(50) 0.0007(2)

200 0.074(8)

400 0.130(10)

0.061(10)

0.023(95)

680(50) 810(50) 0.0003(3)

fun(a,(/) = c

1 1 <D>\[110] <d> i;

[001 ] (6. 2)

Here, c denotes a constant that depends only on geometrical factors and is the same for all

three doped compounds.

Figure 6.3 shows a possible correlation between Tc and fault-defect probabilities computed

from Table 6.2. TEM observations of La2-xSr*Cu04 revealed that stacking-fault density

increased with Sr content up to x = 0.15 [156]. That study suggests that a shear

mechanism forming the fault-boundary dislocations implies oxygen vacancies and/or cation

deficiency. We did not measure oxygen content, but neutron-diffraction studies showed that

67
Figure 6.3 Arithmetic average of [110] and [001] root-mean-square strains and linear

function of stacking-fault and twin-fault probabilities as a function of Tc

.

oxygen deficiency starts to appear at much higher Sr content [157]. However, the increase

of fault probability with the simultaneous decrease of microstrain in Figure 6.3 supports

the assumption that defects are introduced to accommodate lattice strains [156]. A similar

increase in Tc was found in Y2Ba4Cug02(W thin films [158]. Compounds with asymmetric

broadened and shifted peaks showed higher Tc, which was attributed to stacking faults.

Although extrinsic stacking faults may cause asymmetric peak broadening [140], it is

probably caused by twinning. Twinning in La-M-Cu-0 materials is less abundant than in

Y-Ba-Cu-0 compounds, but it was observed in (La-Sr)2Cu04 monocrystals on (110) planes

[159]. In the profile-fitting procedure, peak asymmetry is incorporated only in the

instrumental function; therefore we could not model asymmetric peak broadening.

However, any significant asymmetry of the specimen profile would cause an uneven fit of

68
low-angle and high-angle sides of peaks, which was not observed. This gives much more,

importance to stacking faults in Equation (6.2).

In Figure 6.3, average microstrains, defined as the arithmetic average of [001] and

[110] directions, correlate inversely with Tc

. However, we did not find a correlation of Tc and microstrains in a particular direction. [001] microstrains simply reflect difference in

size of dopant and host La3+ ions (see Table 6.2).

In La2 Cu04 , (OK)) reflections are significantly broader than (MX)). This has also

been observed in low-temperature orthorhombic La2 85 Sr0 15Cu04 and Lat ^Bao 15 Cu04 [160],

and explained by either stacking faults or possible lower symmetry. Our results reveal that

domain sizes <D>S have identical values for [010] and [100] direction (Table 6.2). From

Equation (6.2), it follows that stacking-fault probabilities in these directions have the same

values. However, microstrains in the [010] direction are roughly twice as large as in the

[100] direction, which is responsible for the different broadening. In the case of doped

La2 Cu04 compounds, different microstrains might be caused by oxygen vacancies ordered

along one direction in the a-b plane, similar to effects found in YBa2Cu3 07_*. However,

undoped La2Cu04 is not expected to have significant oxygen deficiency. Other possibilities

are lower crystallographic symmetry and/or La vacancies, although they were not reported

for this system, but La deficiency was found in an La2 Co04 compound [161].

We can conclude that lattice strains and incoherently diffracting domain sizes

confirm high anisotropy in these materials. This simple approach, however, does not allow

unambiguous identification of the strain’s origin. Probably, the strains originate from

dislocations and from different sizes of host and dopant ions. Domain sizes of all

compounds are larger in the c-direction than in the a-b plane. This indicates existence of

stacking faults and twins. However, because estimated errors are relatively large and only

three materials were studied, the possible connection between stacking faults and Tc needs

further study.

6.2 Bi-Cu-O Superconductors

Among the high-Tc superconductors, the Bi-Cu-0 compounds appear intriguing,

especially because their incommensurate structure modulation remains incompletely

69
Figure 6.4 Average crystal structure of Bi2Sr2CaTB.iCuITI 04+2m for: (a) m=l; (b) m=2; (c)

m=3 [164].

understood [162, 163]. The crystal structure consists of perovskite-like SrO-(Cu02 Ca)m. 1 - Cu02-SrO layers separated by NaCl-like BiO double layers (see Figure 6.4 [164]). The

number, m, of CuC^ layers alters the superconducting transition temperature Tc (for m =

1, 2, and 3, Tc * 10, 90, and 110 K, respectively). This two-dimensional layered structure

makes these materials highly anisotropic and favours the creation of defects.

Bulk sintered specimens showed relatively weak peak broadening. Moreover,

diffraction patterns contained many overlapping peaks because the compound crystallizes

70
300 -

rocu

o

200

CO

5S

wH 100

35

-35

42 44 46 48 50 52 54 56 58 60 62

2© (deg)

-^r

Figure 6.5 Part of Bi,Pb,Mg,Ba-2223 refined pattern. There are 48 fundamental

reflections in this region.

in an orthorhombic low-symmetry space group A2aa [165], and incommensurate

modulation gives superlattice reflections. All patterns revealed a strong [00/] texture

(Figure 6.5), making reflections other than (00/) difficult to analyze. The lattice parameters

vary depending on the exact compound stoichiometry and processing history, but

characteristically they have almost identical a and b parameters, and large c parameter: a = 5.4095 A, b = 5.4202 A, c = 30.9297 A for Bi2Sr2CaCu208 (Bi-2212) [166] and a =

5392 A, b = 5.395 A, c = 36.985 A for (BiPb)2Sr2Ca2Cu3O10 (Bi,Pb-2223) [165], Thus,

we treated (MX)) and (OkO) reflections as single peaks. Table 6.3 gives the results for three

specimens: Bi-2212, (BiPb^fSrMg^fBaCa^C^Cho (Bi,Pb,Mg,Ba-2223), which were onephase specimens, and Bi,Pb-2223, containing approximately equal amounts of 2212 and

2223 phases. The in-plane lattice parameters of 2212 and 2223 phases differ little, hence

only the [00/] directions of the mixed-phase specimen were evaluated.

71
Table 6.3 Results of line-broadening analysis for Bi-Cu-0 superconductors.

Specimen Phase hkl (} c O

Pc

o

<D>

,

(A)

<t\<D>J2)> m

Bi-2212 2212 008 0.163(8)

00.12 0.22(3)

0.01(6)

0.05(6)

420(80) 0.0021(6)

020 +200 0.104(4)

040+400 0.18(2)

<10-5 0.0004(6300)

580(30) 0.0007(1)

Bi,Pb,Mg,Ba2223

2223 00.10 0.097(1) 0.0002(1700)

00.20 0.21(2) 0.007(150)

770(60) 0.0012(1)

020+200 0.12(27)

040+400 0.15(95)

0.13(10)

0.21(33)

380(430) 0.0009(71)

Bi,Pb-2223 2212 008 0.23(2)

00.12 0.25(5)

0.002(350)

0.001(1000)

220(30) 0.0013(26)

2223 00.10 0.22(8)

00.20 0.23(99)

0.092(67)

0.18(47)

230(190) 0.0012(197)

Interpretation of the results in the case of these complicated compounds must be

considered carefully. Microstrains are probably caused by dislocation arrays. However,

substitution of different-sized ions and vacancies would also contribute to the strains, as would other kinds of disorder, especially stacking faults and twins. Table 6.3 shows larger

strains in the [001] direction that in the basal plane. This can be explained by stacking

disorder along the c-axis, which was frequently observed in electron-diffraction patterns

as streaking [167, 168]. Easy incorporation of stacking faults is possible because of the

two-dimensional weak-bonded layers (BiO double layers are spaced about 3 A apart). The

difference in microstrain between in-plane and the c-direction is much smaller for the

Bi,Pb,Mg,Ba-2223 phase, probably because Pb lowers the faulting by stabilizing the

higher-Tc 2223 phase [167]. For the Bi,Pb-2223 specimen, with two phases, there are two

possibilities: Either the specimen consists mainly of a mix of two kinds of one-phase

grains, or both phases occur within the same grains. The second possibility was confirmed

experimentally [169, 170], and our results concur because microstrain in the 2212 phase

72
is much smaller than in the one-phase Bi-2212 specimen. Processing conditions and cation

content probably favour development of the macroscopic intergrowth of both phases within

the grains; that is, microscopic defects grow to sufficiently large regions to give a

diffraction pattern of another phase with the different number of Cu02 layers. As a result,

strain is relieved, and both strains and domain sizes, being quite different in one-phase

specimens, tend to equalize.

For the Bi-Cu-0 specimens, the mean surface-weighted domain sizes are generally

smaller than for (La-M)2Cu04 compounds. Therefore, the computation of the stacking- and

twin-fault probabilities by using Equation (6.1) may be wrong. It would be very instructive

to compare the different broadening of satellite and fundamental reflections, which indicates

a more pronounced stacking disorder in the modulated structure [171].

In summary, we found that for both Bi-2212 and Bi,Pb,Mg,Ba-2223

superconductors, the c-axis strain exceeds the in-plane strain. The Bi,Pb-2223 specimen,

containing both 2212 and 2223 phases within the single grains, shows smaller incoherently

diffracting domains and lower c-axis strain than the one-phase specimens. This fact may

direct us toward the mechanism of the second-phase formation in these materials.

6.3 Remarks

It was shown that this particular method of the line-broadening analysis can give

information about the size of incoherently diffracting domains and lattice strains in (LaM)2Cu04 and Bi-Cu-0 superconductors. Moreover, results show high anisotropy of both

size and strain parameters, as one may expect, because of the layered structure of all

superconducting cuprates. Generally, structural line broadening is not large; that is a result

of relatively large domain sizes (200-2000 A) and small strains (0. 02-0.2 %). It is

interesting to compare these values with the published results for YBa2Cu307^ (isotropic

strains of 0.23 % [77] and 0.05-0.3 % [82]). Although the different materials were studied,

they seem too large, especially because they are the average of strains in different

directions.

The main drawback of the line-broadening theory is the uncertainty about the

origins of the line broadening; that is, it is difficult to distinguish the main effects

73
responsible for the broadening. For the novel superconductors, particularly, the numerous

defects (substitutions, vacancies, stacking faults, dislocations, and similar) can occur

simultaneously, which makes it very difficult to determine their individual contributions.

Furthermore, the crystal structures of most novel superconductors are relatively

complicated. The average structure has mostly a slightly distorted tetragonal symmetry;

Very similar a and b lattice parameters, and large c parameter, but c « 3b for

YBa2 Cu307^, for instance. This results in a large number of overlapping peaks, and

consequently in large errors of all results. This showed especially in the case of Bi-Cu-0

superconductors (see Table 6.3). Following the discussion of reliability of profile fitting

in Paragraph 5.4, it would be desirable to include more than two reflections in the analysis

of broadening, but higher-order peaks are usually too weak and overlapped. One way to

partially solve this problem would be a whole pattern fitting [172, 173] and Rietveld

structure refinement [2, 3] in terms of size and strain parameters. This requires an accurate

preset function to model specimen broadening and precise definition of angular dependence

of size and strain parameters. The Rietveld programs that model the line broadening in

terms of size and strain parameters are based either on the simplified integral-breadth

methods [78, 95] or on the single-line method [174].

74
7 Conclusions

A method for analyzing the pure-specimen (structural) broadening of x-ray

diffraction line profiles is proposed. By modeling the specimen size and strain broadening

s

with the simple Voigt function, it is possible to obtain domain sizes and strains that agree

with experiment and show a logical interrelationship. Furthermore, some common

consequences and problems in the Fourier-method analysis follow easily. The specimen

function is convoluted with the instrumental function to match the observed x-ray

diffraction-line profile. This avoids the Stokes deconvolution method, thus allowing

analysis of patterns with highly overlapping peaks and weak structural broadening.

Therefore, the method was applied to some novel high-7^. superconductors. We reached the

following conclusions:

(i) Surface-weighted domain size depends only on the Cauchy integral breadth of

the size-broadened profile.

(ii) The ’hook’ effect occurs when the Cauchy-content of the size-broadened profile

is underestimated, that is, for @sc < (tt/2)

1/2/3sc. Usually this happens when the background

is estimated too high, so the long tails of the Cauchy function are prematurely cut off.

(iii) It also follows from conclusion (ii) that the volume-weighted domain size can

not be more than twice the surface-weighted domain size (limiting value <D> V =

2 <D>S is obtained for the pure-Cauchy size-broadened profile). Minimum value for

<D> J <D>s should not be less than —1.31.

(iv) If the distortion coefficient is approximated with a harmonic term, this leads

exactly to the Warren-Averbach method of separation of size and strain broadenings.

(v) In that case, mean-square strains decrease linearly with the distance L.

75
(vi) It is possible to evaluate local strain ( < e

2 (0) >

1/2) only in the case of pureGauss strain broadening. Then, root-mean-square strain is independent of the distance L

and related to the upper limit of strain obtained from the integral breadth of the strainbroadened profile.

(vii) If strain broadening is not described exclusively with the Gauss function, the

multiple-line Voigt integral breadth and Warren-Averbach analyses can not give the same

results for volume-weighted domain size, although <D> V is defined identically in both

approaches, because the ’apparent strain’ ij = /?D

(20)cot0 is not angular-independent.

(viii) Smooth size coefficients give column-length distribution functions without

oscillations and make possible a computation of volume-weighted average domain sizes up

to -5000 A.

(ix) Comparison with the simplified integral-breadth methods, in most cases, has

shown not large, but systematic discrepancy of results, indicating that size and strain

broadening can not be accurately modeled by a single Cauchy or Gauss function.

Moreover, the application of the single-line method is much less reliable, and it should not

be used where multiple reflections are available.

(x) The knowledge of all phases present in the sample and their crystallographic

structure is required to successfully fit a cluster of overlapping peaks. Although it is very

difficult to accurately determine the Cauchy-Gauss ratio of an overlapped peak, final results

of domain size and strain are much less affected. Therefore, connecting the crystallite size

exclusively with the Cauchy content and strain with the Gauss content of a specimen profile

might be doubtful.

(xi) Evaluated errors of domain sizes and strains increase substantially with a degree

of peak overlapping. Hence, for complicated patterns, very accurate measurements are

necessary and one needs to analyze as many reflections in the same crystallographic

direction as possible.

(xii) Line-broadening analysis of (La-M^CuC^ and Bi-Cu-0 compounds confirms

high anisotropy in these materials.

(xiii) Strains in (La-M)2 Cu04 probably originate from dislocations and from

different sizes of host and dopant ions.

76
(xiv) Stacking-fault probability in (La-M)2 Cu04 (M = Ba, Ca, Sr) increases with

increasing Tc , whereas the average strain decreases.

(xv) In La2 Cu04 , different broadening of (hOO) and (0&0) reflections is not caused

by stacking faults.

(xvi) All Bi-Cu-0 superconductors show larger strain in the c-direction than in the

a-b plane, which can be explained by stacking disorder along the [001] direction.

(xvii) In the specimen with approximately equal amounts of Tl-2212 and Tl,Pb-2223

phases, it is likely that a secondary phase develops by the growth of the microscopic

faulted regions of the primary phase.

77
Acknowledgments

This publication represents the thesis submitted by Davor Balzar to the University of

Zagreb, Croatia, to satisfy part of the requirements for the degree of Ph. D. of Natural

Sciences in Physics. Dr. Ledbetter directed the thesis research.

I am thankful to Dr. Harry McHenry for his support and especially to Dr. Hassel Ledbetter

for his wise guidance and most useful help during my stay at NIST. I am also indebted to

my home institution, Faculty of Metallurgy of the University of Zagreb, for allowing me

the leave of absence at NIST, and to Professor Antun Bonefadic (Physics Department,

Faculty of Science, University of Zagreb, Croatia) for his help.

78
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