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$: Introduction - Shodhganga : a reservoir of Indian …shodhganga.inflibnet.ac.in/bitstream/10603/21359/12/012.pdfX-ray powder diffraction profile analysis Chapter 2 57 Introduction$
X-ray powder diffraction profile analysis Chapter 2

57

Introduction

One of the most important applications of X-rays powder diffraction, which has been

continued for many decades and is still continuing, is the use of diffraction line profiles to

study the nature of crystal imperfections introduced during growth and plastic deformation of

a material. The earlier X-ray diffraction studies on cold worked materials depict that plastic

deformation induces broadening of the X-rays diffraction profiles and such a peak broadening

increases as the sample is more and more deformed. With the use of modern techniques [1]

like focusing curved crystal monochromator, counter or solid state detector it is possible to

locate and measure the shape of the X-ray diffraction profiles with considerable accuracy. In

the past decades extensive theoretical and experimental studies have been made in this

direction by Wilson [2], Greenwough [1] and Warren [3]. A further development in this field

was made by Wagner [4], Warren [5] and Klug and Alexander [6], Enzo et al. [7], Langford

et al. [8], Mittemeijer et al. [9], Balzar et al. [10] and others. This chapter basically deals with

the various theoretical considerations in the X-rays diffraction line profile analysis for

microstructure characterization of polycrystalline materials in terms of various lattice

imperfections.

2.1 X-ray line profile analysis: Theoretical considerations

Diffraction is actually a scattering phenomenon. When X-rays interact with the atoms,

it gives rise to scattering in all directions. In some of these directions the scattered beams

will be completely in phase and so reinforce each other to form diffracted beams following

Bragg's law,

nλ sinθ2d (2.1)

where, is the wavelength of X-rays, directed towards the set of parallel planes in a crystal

at an angle and n is the order of reflection. The is called Bragg angle where the

maximum intensity occurs. At other angles there are little or no diffracted intensities because

of the destructive interference.


58

Each crystallographic phase in an ideal crystal has a characteristic set of d spacing,

which yields a particular diffraction pattern at respective Bragg angles , which may be

represented by discrete lines in the chart of the diffractometer. But in the experiments with

real crystals peaks are observed having considerable broadening instead of sharp lines. This

is due to the facts that it is hardly possible to have an incident beam composed of perfectly

parallel and monochromatic radiation and a real crystal always has some departures from an

ideal structure due to the presence of crystal imperfections.

As mentioned in the Chapter 1, small crystallite size, microstrain inside the

crystallites and stacking faults are mainly responsible for broadening of line profiles. Some

instrumental parameters like slit widths, sample size, penetration in the sample, imperfect

focussing, unresolved 1 and 2 peaks etc. also give some extraneous broadening to the line

profile. All these extraneous sources of broadening are grouped together under the name

"instrumental broadening".

The first step towards the determination of crystallite size from X-ray line profile was

made by Scherrer in 1918 [11]. He reported that the line breadth varies inversely with the

size of the crystallites according to the equation,

cosθ )i

βsam

(β

KλD (2.2)

known as Scherrer formula, where is the wavelength, sam and i are the measure of line

breadth of sample and instrumental „standard‟ respectively, is the Bragg angle, K is

Scherrer constant 89.0K0.1 and D is an apparent crystallite size along the direction

of hkl planes. The Scherrer formula actually gives the length of the crystal in the direction

of the diffraction planes and it is evident from equation (2.2) that size broadening is

independent of the order of reflection. It is important to note that the lattice strain effect,

which also contributes to the broadening is totally neglected in the Scherrer formula.

Therefore, this method is not suitable for the study of crystals where the strain broadening

plays an important role in peak broadening. Due to this disadvantage the use of Scherrer

formula is restricted in many sophisticated problems, but it is still being used in some simple


59

cases where the size of crystals is fairly less than hundred angstroms and the line broadening

is primarily due to small crystallite size.

2.2 Methods of separation of crystallite size and strain effects from X-ray diffraction profiles for deformed materials and alloys

For a polycrystalline specimen consisting of sufficiently large (`~10-4

cm) and strain

free crystallites, diffraction theory predicts that the lines of the powder pattern will be

exceedingly sharp. Hence an analysis of breadths of strain and/or size broadened diffraction

lines will give quantitative information about the crystallite size and strains of the deformed

crystallites.

The crystallite size and microstrain can be determined from the X-ray diffraction

profile with the help of the following methods:

1. Integral Breadth,

2. Variance,

3. Fourier analysis.

2.2.1 Integral Breadth method

This is the oldest method for determining crystallite size and microstrain. Scherrer [11]

defined the breadth of a diffraction line as its angular width in radians at a point where the

intensity has fallen to half of its maximum value. In 1926, Laue gave another definition of

the breadth of a diffraction line as the integrated intensity of a line profile above background

divided by peak height.

)()( 2d2II

1

p

(2.3)

In 1949, Hall assumed that both the size and strain broadening profiles can be

described by Cauchy functions and suggested that the total broadening can be expressed as:

DS (2.4)


60

where S is the line breadth measured by Scherrer (eqn 2.2) (size) and D is the

broadening arising from microstrain (distortion) which can be expressed as

tan4D (2.5)

Therefore, eqn (2.4) can be written in the form

tan4cosDKDS (2.6)

or, sincos 4D1 (2.7)

Thus from the Instrumental error corrected integral breadth of the X-ray line

profiles, a plot of cos against sin should be a straight line and the intercept on

the cos axis gives measure of D1 , the reciprocal of the crystallite size, while the

slope gives , the average microstrain.

2.2.2 Variance method

The variance (often called reduced second moment of the line profile) can be defined

as the squared standard deviation of the elements comprising the profile, mathematically the

most useful to measure the integral breadth [12]. It was directly associated with the centroid

2 as a line location, and is defined by,

2d2I

2d2I22222W

2

2 (2.8)

where centroid or centre of gravity, C.G., )(2d2I

2d2I22

The important and advantageous property of this method is that the variance are

additive [13-15] and therefore variance of line profile can be taken simply as the sum of

those due to crystallite size, microstrain, crystal defects and instrumental broadening.


61

Wilson [12, 16] derived expression for the variance of the line profile in terms of

crystallite size, microstrain and crystal defects and is given by,

costan

coscos

)()(

d2

24

p4

L

p2

2K2W

2

22

222

2

2 (2.9)

where p is the cube root of the true crystallite size, K and L are the constants for a

given particle shape, 2 is the range of integration, the fourth term involving gives the

contribution from defects other than stacking and twin faults.

From eqn. (2.9) it is apparent that the variances are extremely sensible to the angular

range over which the intensity distribution is measured. Thus, applying the method of

variance to separate the effect of crystallite size and microstrain, variances are to be

determined from the region of constant slope on using the property that the variances are the

linear functions of the range over which the variances are considered [8, 17-20]. Wilson in

1970, Edwards and Toman in 1970 and 1971 also examined the second order nonadditivity

properties of the line-profile variance to improve the method.

2.2.3 Fourier analysis

2.2.3.I Stokes' method of deconvolution

The observed line-profile is a convolution of the instrumental broadening and

broadening due to defects in the sample itself. A deconvolution of instrumental broadening

from diffraction broadening is necessary to obtain the pure diffraction profile which is to be

used for the determination of particle size, strain, and stacking fault effect in the samples. In

Fig. 2.1 if h(x) represents the observed profile, xf , the true profile, i.e., profile broadened

due to defects in specimen, and xg represents the instrumental contributions to the

broadening, the observed line profiles, xh can be written as:

xgxfxh (2.10)


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Fig. 2.1 h(x), the observed profile, f(x), the true profile (profile broadened due to defects in

specimen), and g(x), the instrumental contributions to the broadening.

For ease of deconvolution of the various components, earlier work on line broadening

analysis was based on the assumption that the constituent line profiles are either Cauchy

(Lorentzian) or Gaussian. It was evident that neither of these functions accurately models the

experimental profiles and a more rigorous approach is thus necessary to deconvolute the

instrumental and sample contributions. Stokes in 1948, suggested a very general but rigorous

method to correct the powder diffraction pattern for instrumental correction based on Fourier

transform method. He considered F , G and H as the Fourier transforms of xf , xg and

xh respectively and due to multiplicative property of the Fourier transforms of the

convoluted functions one can write

yFyH * yG (2.11)

2.2.3.II Determination of coherent domain sizes and rms strains

Fourier analysis can be used for interpreting the precise shape of diffraction peaks in

terms of microstrains, particle size and stacking faults, and can be applied when more than

one of these causes is active.

The first step is to subtract the background intensity from measured intensity value to

obtain a corrected line profile, This may involve some uncertainty for powder diffraction

peak of a cold-worked metal often has tails that extend many degrees from the center of the

peak. If peaks are very broad, it may be desirable to first correct the diffracted intensity

values by the Lorentz-polarization factor, the absorption factor and the atomic scattering


63

factor. It may then be preferable to obtain the profile of the 1K peak uninfluenced by

overlap with the 2K peak -unless a precision monochromator has already removed the

2K component.

The next step is to correct the instrumental broadening. This may be done by the

method of Stokes‟ [21]. Removing instrumental broadening is an unfolding operation that

involves the Fourier coefficients representing the profile for a standard sample that lacks

broadening from distortion, particle size, or faults. For the standard, a sample (often a part of

the material under investigation) is annealed enough to sharpen lines but not enough to

produce grain growth that could reduce intensities by extinction. When proper background

corrections are made, Warren [22] finds that cold working does not lessen the integrated

intensity of the reflections from a material.

After having corrected profile each reflection is expressed by its separate Fourier

series, a brief presentation of which is given below:

Let us consider a general distortion in which the position of a unit cell m1m2m3 is given

by the vector R, in terms of unit cell vectors a1, a2, a3 and R can be written as

32mmm1

R = m1a1 + m2a2 + m3a3 + m (2.12)

where the displacement vector, m = Xma1 + Yma 2+ Zma3 is in general different for

each cell m= m1m2m3.

Let the direction of the primary and diffracted beams be represented by the unit

vectors S0 and S, so that their difference gives

S0 – S = ( h1b1 + h2b2 + h3b3) (2.13)

where b1, b2, and b3 are the vectors of the reciprocal lattice and h1, h2, h3 are

continuous variables. The intensity from the crystal is then related to the displacement at the

pairs of cells, i.e., cells at 32mmm1

R and 32mmm1

R by the relation:


64

I (h1h2h3) = 1 2 3 1 2 3m m m m m m

2 eF (2 i/ )(S- 0s )(R m -R m ) (2.14)

The distribution of intensity in reciprocal space is then related to throughout

reciprocal space so as to give the power in the corrected plot of intensity distribution vs. 2

in a powder diffraction line. Bertaut [23] suggested to visualize a crystal to be made up of

unit cells along the a3 direction i.e.; in the direction perpendicular to the reflecting planes 00l

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

m3

a1

Fig. 2.2 Representation of the crystal in terms of cells along the a3 direction.

L

= n

a3

a3

m1

The columnar distribution is illustrated schematically in Fig. 2.2. Hence for convenience,

crystal axes a1, a2 and a3 are chosen so that each reflection is of the type 00l. In this way, the

contributions to intensity are thought of as arising from columns of cells in the crystal, each

column being parallel to a3 (normal to the reflecting plane).

The summation with respect to m3 and m3 is carried out for all pairs in a given column,

and the other summations are to add the contributions from the different columns. Following

Warren‟s notation [5], let N3 be the average number of cells in a column of a coherently

diffracting domain, averaged over all diffracting domains in the sample, and let n = m3-m3 .

Let us consider as arbitrary displacement, varying with position m and having a Z component

equal to Z(m3) and Z(m3 ) at m3 and m3 respectively. Let Zn= Z(m3)- Z(m3 ) and N1N2N3=N


65

be the average number of cells per domain, and let Nn (m1m2) be the number of cells with an

nth neighbor in the same column. Then the distribution of diffracted power per unit length of

the diffraction line, P(2 ), is

P(2 ) = K ( )N n

An cos2 nh3 + Bn sin2 nh3 (2.15)

where An = 3N

Nn cos2 lZn

Bn = 3N

Nn sin2 lZn

K( ) = G (u+b) b3 sin2

h3 = 3b λ

2sin=

λ

sin 2 3a

n = harmonic number corresponding to the separation, L=na3, between a pair of cells in a

column perpendicular to the 00l plane as shown in Fig 2.2.

In the expression of K( ), G contains the dependent function that are due to the

variation of the atomic scattering factor f and to the Lorentz-polarization factor. It also

contains a numerical to take account the number of components, b, of a given peak that are

affected by stacking faults, and the number u, that are unaffected. Components are defined as

a reflection from one set of parallel planes of a crystal, for example (111), as distinct from,

say (11-

1) [5].

Both the Fourier coefficients An and Bn contain the information on coherent domain

size and lattice microstrain and characterize the deformed state of the material under

consideration. If for a given n there is equal probability for finding positive and negative

values of Zn, the sine coefficient Bn=0, otherwise Bn is small enough to be neglected. The

cosine coefficient An is analyzed further to extract information on coherent domain size and

lattice microstrain.


66

2.2.3.III Warren-Averbach method for separating the coherent domain size and rms strain

A method for separating the size and distortion components was proposed by Warren

and Averbach [22, 24]. In the Warren-Averbach method, the cosine coefficient An is taken as

the product of two quantities, 3N

Nn and cos2 lZn . Since 3N

Nn depends only on column

length, it is the size coefficient and represented by AnS

= 3N

Nn . The other quantity cos2 lZn

depends on the distortion in the domain, and is represented as the distortion coefficient AnD =

cos2 lZn . The cosine coefficient An, which we determine experimentally can now be

expressed as

An = 3N

Nn cos2 lZn = AnS An

D (2.16)

The size coefficient AnS is independent of order of reflection l, whereas the distortion

coefficient AnD(l) is a function of l and it approaches the value of unity as l goes to zero.

Hereafter we will write An as An(l) and AnD as An

D(l). From this difference in the dependence

on l, it is possible to separate the two coefficients. Suppose that we have measured several

orders of (00l), such as (001), (002), (003), etc., and that we have corresponding cosine

coefficients An(l) = AnS An

D(l). For the small values of l and n, so that the product lZn is small,

the cosine part can be expanded as

cos2 lZn 1-22l2

Zn2

.

and the logarithm can be written as

ln cos2 lZn = ln (1-22l2

Zn2

) = -22l2

Zn2

.

For small values of l and n the logarithm of the measured Fourier coefficient is given by

ln An(l) = ln AnS-2

2l2

Zn2

(2.17)


67

For the fixed values of n, we plot the values of ln An(l) against l2 as indicated by Fig.

2.3. The intercepts at l = 0 give the values of size coefficients AnS and the slope gives

22

Zn2

.

The distance L=na3 represented in Fig. 2.2, is the undistorted column length. Due to

distortion this distance is changed by L= a3Zn. The ratio L L = L, is a component of strain

along the a3 direction, averaged over the length L. Hence Zn n = L, and the value of the slope

can be written as 22

Zn2

. From the initial slopes of the lines of Fig. 2.3, we obtain mean

square values of an average component of strain.

0

0 1 4 9

l2

Fig. 2.3 The logarithm plot that is used to separate particle size and distortion effects when multiple

orders (00l) are available.

Using multiple orders, the extrapolation is made by plotting ln AL(1 d) against (1 d)2.

The values of ALS are then plotted against L as indicated by Fig. 2.4. The intercept of the

initial slope on the L axis gives directly the average column length D perpendicular to

the set of planes under consideration

ln A

n(l

)

n = 0

n = 1

n = 2

n = 3

n = 4


68

1.0

AS

L

<D> L

Fig. 2.4 Pot of the particle size coefficient ASL against L

2.2.3.IV Errors in the Fourier method

The principal sources of error in the Fourier (Warren-Averbach) analysis are:

(a) counting statistics;

(b) standard used to obtain g (instrumental) profiles;

(c) background determination;

(d) truncation of profiles at finite range;

(e) sampling interval (step length);

(f) choice of origin;

(g) limitations of approximations used in analysis.

These affect the analysis in different ways and by different amounts. Young et al. [25]

simulated the effect of (c), (d) and (e) and many other authors have since discussed the

treatment of errors in the Fourier method. Delhez et al. [26, 27] have given correction for (a),

(b) and (f). Procedures for improving the reliability of the line profile analysis by the Fourier

method have been represented by Zorn [28] and Delhez et al. [29].

2.2.3.V Demerits of Warren-Averbach method

The main drawback of Warren-Averbach method is the appearance of the "hook effect".

Two experimental errors giving rise to "hook" effect have been pointed out are: (i) truncation

of line profile and (ii) estimation of high background. Crystallographers from all over the


69

world tried their best to correct the "hook" effect, but not a single established and valid

procedure is found till date for the correction of the "hook" effect.

Another drawback is that the method fails if the peaks are seriously overlapping.

Many materials having interesting technological applications display diffraction patterns

with overlapping peaks. Warren-Averbach method of line profile analysis fails to

characterize the microstructure of such materials.

2.3 Consideration of stacking faults

Barrett [30] first suggested that stacking faults could be produced on the close-packed

(111) planes in f.c.c. metals as a result of cold working, which then lead to certain changes in

the observed X-ray diffraction pattern. Paterson [31] showed it that for f.c.c. metals,

deformation faults produce peak shift and symmetric broadening while growth faults

produce asymmetry in the line broadening. Among several methods for handling stacking

fault problems, the adopted difference equations method which lead to the expression for the

probability of mth

neighbor layer is the same or different from the original layer, was

originally used by Wilson [32] and then by Paterson [31]. Difference equation for combined

effect of deformation and growth faults was given by Gevers [33]. Warren has used the

Gevers [33, 34] technique in a simplified form with an assumption that the probabilities for

both faults are sufficiently small for f.c.c. structure. Probability of individual stacking faults

can be estimated from the following consideration:

2.3.1 F.c.c. structure

(i) Peak shift analysis:

The shift in peak position of a cold worked material from its annealed specimen is

mainly due to combined effect of stacking faults ( , ), change in lattice parameter

( 0aa ) and long range residue stress ( ) and can be written as

'''''''''''' lkh

hkl

lkh

hkl0

lkh

hkl

lkh

hkl HKaaA2 (2.23)


70

where

annlkhhklcwlkhhkllkhhkl

lkh

hkl 2222222 '''''''''

''' ,

the relative peak shifts of the neighboring pairs of reflections hkl and lkh from cold

worked and annealed profiles.

'''

''' tantan lkhhkl0

lkh

hkl 360aaA

''''''

''' tantan lkhlkh1hklhkl1

lkh

hkl SS360K

''''''

''' tantan lkhlkhhklhkl

lkh

hkl GG360H

where, 1S is the coefficient arising out of elastic compliances, Ghkl is the faulting

coefficient, and are intrinsic and extrinsic stacking fault probabilities respectively.

(ii) Peak asymmetry analysis:

The asymmetry in the cold worked profiles is primarily due to the presence of

extrinsic stacking fault ( ) and twin or growth fault ( ). Considering the pair of

reflections (111-200), the relevant expression may be written as

200111

200111

tan14.6tan11

224.5 CGCG θ

(2.24)

where PMCGCG 222 111 ; CG denotes the centre of gravity of the peak and

PM, the peak-maxima.

(iii) Peak broadening analysis:

The peak broadening of a cold worked profile is due to combined effects of small

coherently diffracting domains D, microstrain and stacking faults .,, For (111-

200) pair of reflections the relevant expression will be


71

100111

10011176.15.1

DD

DDa (2.25)

The eqns. (2.23), (2.24) and (2.25) are solved to determine the individual values of

stacking fault probabilities, , and .

2.4 Whole Powder Pattern Decomposition method

Pattern decomposition or whole powder pattern decomposition (WPPD) can be defined

as a method for decomposing a powder diffraction pattern into individual components of

Bragg reflection in one step [35]. A fundamental difference between the pattern

decomposition method and Rietveld method is that the integrated intensity parameters, jI , in

the fitting function are independent variables in the former while they are a function of

structural parameters in the latter. Thus one can refine crystal structures and microstructures

in Rietveld method, while the pattern decomposition can only be conducted in the WPPD

procedures, although integrated intensities obtained as output from the latter can be used for

refining crystal structures. The WPPD method was first proposed by G.S. Pawley applying it

for analysis of neutron data [36]. The method was further developed by Toraya [37], Le Bail

[38] and Hall Jnr. et al. [39]. Developments on WPPD techniques are still in progress.

2.4.1 Iterative method

The WPPD method based on the iterative method was first proposed by Le Bail et al.

[38] and is known as Le Bail method. The integrated intensity of the thj reflection in Le Bail

method is calculated as

icalci

jicalcj

i

iobsiobsj

BI

PIBII

22

222'' (2.31)

where ji

calc

j PI 2. is the contribution to the ''obs

jI from the

thj reflection.


72

In this method refinement is started assigning uniform integrated intensities (for

example, j

I = 100) to all reflections and approximate values to all profile parameters.

Calculated profile intensity will therefore have equal peak intensities at the beginning. If the

two reflections 1 and 2 with an intensity ratio of 2:1 are partially overlapping, then after the

first cycle, the numerical integration on observed profile intensity will derive ''obs

1I and

''obs

2I

with an intensity relation ''obs

1I >

''obs

2I . In the second cycle

''obs

1I and

''obs

2I thus obtained will

be used in place of calc

jI and the process is iterated until the intensity ratio of

''obs

1I and

''obs

2I

approaches 2:1.

2.4.2 Merits and Demerits of WPPD methods

The WPPD methods have several merits in extracting the information about, for

example, the phase abundance, the unit-cell parameters and the crystallite size and

microstrain etc. Since the whole pattern is used for analysis, the refined parameters are more

precise and accurate, particularly in the analysis of complex powder patterns, compared to

those obtained from the single profile fitting methods. Because, no structural model is

required in calculating the intensity, the computation requires less information and runs in 10

or 20 seconds in the present-day work station. If the computer software is made more robust

against interruption by ill-conditioning of the normal matrix, a series of intensity datasets will

be analyzed successively without operator interaction just as if the intensities were measured

by using an auto-sample changer. In these respects, the WPPD method is well suited for

laboratory automation and high volume analysis.

There are some demerits also. Overlapping reflections at the same Bragg angle

(intrinsic overlapping) can not be perfectly decomposed by the WPPD methods. Thus the

industrially important materials, which usually exhibit severely overlapping lines are

difficult to characterize using this method. It has been found that the effective separation

limit in pattern decomposition (measured as the shortest 2 distance between the two

adjacent peaks) ranges from 0.1 to 0.5 times the FWHM (generally~0.25) and it is influenced

by several factors, namely, counting statistics, the 2 -range of data included in the analysis,

and the resolution of the data.


73

2.5 The Rietveld method

The method of total pattern fitting introduced by Rietveld [40] requires two models at

the start of the refinement, a structural model based on approximate atomic positions and a

non-structural model which describes the Bragg reflections in terms of analytical or other

differentiable functions. Both must be considered in order to obtain an optimum

representation of the observed pattern. In the Rietveld method, the least-square refinements

are carried out until the best fit is obtained between the entire observed powder diffraction

pattern taken as a whole and the entire calculated pattern based on the simultaneously refined

models for the crystal structure(s), diffraction optics effects, instrumental factors, other

specimen characteristics. A key feature is the feedback, during refinement, between

improving knowledge of the structure and improving allocation of observed intensity to

partially overlapping individual Bragg reflections. Since the Rietveld method is a structure

refinement method and not a structure solution method, a reasonably good starting structure

model is needed.

The calculated intensities Ici are determined from the structure factor [Fk]2

values

calculated from the structural model by summing of the calculated contributions from

neighbouring (i.e. within a specified range) Bragg reflections plus the background:

K

biKKi

2

KKci IAP2θ2θΦFLsI (2.36)

where s is the scale factor, K represents Miller indices, hkl for a Bragg reflection, KL

contains Lorentz polarization and multiplicity factor, K is a reflection profile function

which approximate the effect of both instrumental features and specimen features such as

aberration due to absorption, specimen displacement, crystallite size and microstrain effects

etc., KP is preferred orientation function, KF is structure factor for thK Bragg reflection,

biI is background intensity at thi

point. The simulated pattern obtained by adding up the

calculated intensities ciy , is fitted to the observed pattern. The quality of fit is established by

the residue, Sy,


74

i

2

cioiiy IIwS , (2.37)

where oii I1w and Ioi is the observed intensity at ith

point which is preferred to be

minimum.

The model parameters that may be refined include not only atom positional, thermal

and site-occupancy parameters but also parameters for the background, lattice parameters,

instrumental geometrical optical features, specimen aberrations (e.g. specimen displacement

and transparency), an amorphous component, extinction parameters and specimen reflection-

profile-broadening agents such as crystallite size and microstrain. Phase contents of

individual phases in a multiphase material may be refined simultaneously and comparative

analysis of the separate scale factors to the overall scale factor for the phases offers the phase

contents of these phases. It is found to be the most reliable method for doing quantitative

phase analysis. In order to improve fitting the model parameters of the simulated pattern

namely, atomic positions, thermal and site occupancy parameters, parameters for

background, lattice parameters, parameters representing instrumental geometrical-optical

features and specimen aberration (e.g. specimen displacement and transparency), parameter

for amorphous component, specimen reflection profile broadening parameters (crystallite

size and microstrain) and often parameter for extinction effect are refined in steps with the

calculation of residual at every steps. The best fit is said to obtain when yS reduces to

minimum.

Preferred orientation

The grains or crystallites in many powder samples tend to have a shape which does

not approximate to a sphere, due to cleavage or growth mechanism. When compacted into a

sample holder, such crystallites tend to orient preferentially in a particular crystallographic

direction. Preferred orientation produces systematic distortions of the reflection intensities,

after-the-fact corrections can be made for it, i.e. the distortions can be mathematically

modeled with “preferred orientation functions”, Pk in equation (2.36), of the form

Pk=G2+(1-G2)exp(-G12

k )


75

where G1 and G2 are refinable parameters and k is the angle between d*k and the preferred

orientation direction.

Recently, Dollase [41] showed the superior performance of the March function [42]

and March-Dollase function incorporated in the Rietveld refinement codes is

Pk = 2

32

1

22

1 sin)/1(cos GG (2.38)

Quantitative phase analysis by the Rietveld method

It has been reported [43] that there is simple relationship between the individual scale

factor determined in a Rietveld structure refinement of a multiphase system and the phase

concentration in a mixture. The weight fraction of the phases present in a mixture is thus

obtained directly from the relation:

Wi= si(ZMV)i/j

jj(ZMV)s (2.39)

where si, Zi, Mi and Vi are the scale factor, the number of molecules per unit cell, the

molecular weight and unit cell volume of phase i, respectively and the summation is carried

out over all phases present. The density of each phase i can also be evaluated using this

relation:

i=ZiMi/NAVi (2.40)

where NA ia Avogadro‟s constant.

Rietveld refinement process by iterative method will adjust the refinable parameters

until the residual (eqn. 2.34) is minimized in some sense i.e., a best fit of the entire calculated

pattern to the entire observed pattern is obtained. Some often-used numerical criteria of fit are

represented by residual factor (R) of different kinds:


76

‘R-structure factor’; RF = 21

2121

/

i

oi

i

/

ci

/

oi

)(I

)(I)(I

(2.41)

‘R-Bragg factor’; RB =

i

oi

i

cioi

I

II

(2.42)

‘R-weighted pattern’, Rwp =

21

2

2/

i

oii

i

cioii

)(Iw

)I(Iw

(2.43)

‘R-expected pattern’, Rexp =

1/2

i

2

oii )(Iw

P)(N (2.44)

where N is the number of experimental observation and P is the number of refinable

parameters. Here Ii is the intensity to the ith

Bragg reflection at the end of the refinement

cycles. The „Goodness of Fit‟(GoF) indicator, S, is

S = Rwp/Rexp (2.45)

From a purely mathematical point of view, Rwp is the meaningful of these R‟s because the

numerator is the residual being minimized. For the same reason, it is also the one that best

reflects the progress of the refinement.

For angle dispersive data, the dependence of the breadth H of the reflection profiles

(measured as Full-Width-at-Half-Maximum, FWHM ) was modeled as [44]

H=[U tan2θ +V tanθ +W]

1/2 (2.46)

where U, V, W are the refinable parameters.

This formula (termed as Caglioti formula) worked satisfactorily for the initial developed

medium (or less) resolution powder diffractometers where the instrumental function was


77

purely Gaussian. Even with the X-ray diffractometer operating on sealed-off x-ray tube and

rotating anode, whose instrumental profiles are neither Gaussian nor symmetric, Caglioti

function was widely used for the lack of anything better. But with the reflection profiles from

modern x-ray diffractometer with Guinier camera and high-resolution neutron powder

diffractometer various complications arise. Their instrumental profiles are sufficiently

narrow so that broadening of the intrinsic diffraction profile from specimen defects is a

significant part of the total and do not have Caglioti dependence of the angle.

An essential step in Rietveld method applying on data from modern X-ray

diffractometer is to examine the variation of FWHM (or integral breadth) with 2 or d

and to compare this with the resolution curve of the instrument used. If the two curves are

identical, indicating that sample effects are negligible, then Caglioti formula can be used to

model breadth variation. If the curves differ, but the scatter for the sample curve is not

greater than that would be expected from counting statistics or there is no marked

'anisotropy', on the average, then also one can model the breadth of the profile with Caglioti

formula, but this time U , V and W should be treated as refinable parameters. But if the

sample curve exhibits a scatter which is 2 or d dependent, then the nature of 'anisotropic'

breadth variation must be ascertained and the dependence of breadth on hkl has to be

modeled with special care.

In general, sample induced line broadening includes contributions which are

independent of 2 or d , known as 'size effect' and which depends on 2 or d , known as

'strain effect'. There have been various attempts to make allowance for smoothly varying

(isotropic) microstructural effects in Rietveld programs. David and Matthemam [45]

modeled experimental line profile by means of a Voigt function and assigned the 'Lorentzian'

and 'Gaussian' components to the 'size' effects and the instrumental broadening respectively.

A different approach was adopted by Howard and Snyder in the program SHADOW [46],

who convoluted a Lorentzian simple line profiles, assumed to be due to 'crystallite size'

and/or 'microstrains', with experimentally determined instrumental profiles, to match the

observed data. The simultaneous presence of isotropic 'size' and 'strain' effects was

considered by Thompson, Cox and Hastings [47]. They used a pseudo-Voigt function to

model the overall line broadening and assigned the Lorentzian components of the pseudo-


78

Voigt functions to 'size' effects and Gaussian components to the combined 'strain' and

instrumental contributions.

An early attempt to model anisotropic line broadening in the Rietveld method was

made by Greaves [48], who assumed that the crystallites had the form of platelets with

thickness H and infinitely large lateral dimensions. In this case the contribution to the

integral breadth of reflection from plates parallel to the surface, in the reciprocal unit, is

simply H1 . In order to allow for the direction dependence of microstrain, some assumptions

are made regarding the stress distribution. If microstrain is assumed to be statistically

isotropic, then the anisotropy of the elastic constants leads to an hkl dependence of strain.

Thompson, Reilly and Hestings [49] expressed microstrain as a function of hkl and refined

appropriate strain parameters based on elastic compliances. Simultaneous anisotropic „size‟

and „strain‟ broadening was incorporated in the Rietveld method by Le Bail [50] and

Lartigue, Le Bail and Percheron-Guegan [51]. The hkl dependent nature of these quantities

was modeled by means of ellipsoids and Fourier series were employed to represent the line

profiles. The number of microstructural parameters to be refined was restricted by adopting

Lorentzian function for „size‟ contributions and an intermediate Lorentz-Gauss function for

„strain broadening. In a similar approach Lutterotti and Scardi [52] included crystallite size

and microstrain as refinable parameters, in the place of usual angular variation line-profile

width [eqn. (2.57)]. Microstructural analysis based on approximate single line Fourier

method was introduced by Nandi et al. [53].

2.6 MAUD: a user-friendly Java program for Materials Analysis Using

Diffraction

In the field of materials analysis, there is a constant demand for sophisticated tools,

which can process enormous number of data in a short time, deal with cases showing sample-

induced anisotropy and extract more information about the samples. With the Rietveld

program Maud we have already proved it is an efficient way to analyze diffraction spectra to

obtain a complete characterization of the sample under investigation. The program has been

customized for the pharmaceutical field including some specific methods and routines.


79

The Maud project [54] started in 1996 as a mere exercise using the one year old Java

language [55]. Java at that time was just in its infancy and was not ready for a scientific

application in need of speed and highly demanding mathematical performances. At the

beginning not, still in 1996, with version 1.0 of the jdk (Java Development Kit), there were

few bugs preventing a full computation of a powder pattern when the memory requirement

was more than a 1 or 2 Mb. But one thing was already clear, writing a moderately complex

interface was more easier than with other languages (C/C++

) which provides a little bit less

speed for the user but much less painful debugging for the developer. The first public version

of Maud was released on the web in the early fall of 1998 with the jdk 1.1 and that was the

first really usable Java version for applications. This program can run virtually in any

computer environment supporting the Java Virtual Machine with installers available for the

Macintosh and Windows platform. In 1999, Lutterotti et al. developed a new version of

MAUD program, which is easy to use, can be applied to wide variety of materials and helps

the user in obtaining more information from data collected by traditional and new

instrumentation [56]. Recently (2011) a new version of MAUD has been released which is

more user friendly and free from errors.

The principal features of MAUD software are:

(i) Written in Java and can run on Windows, MacOSX, Linux, Unix (need Java VM 1.4

or later)

(ii) Developed for Rietveld analysis, simultaneous multi spectra and different

instruments/techniques supported

(iii) Different optimization algorithms available (LS, Evolutionary, Simulated Annealing,

Metadynamics)

(iv) Works and input images from 2D detectors (image plates, CCD)

(v) Simultaneous crystal structure refinement, line-broadening, texture (stress under

implementation) and quantitative phase analyses can be performed.

(vi) Multiple samples from different instruments can be analyzed at one time.

(vii) The diffraction patterns of a sample from different instruments, e.g. X-ray tube,

synchrotron, neutron constant wavelength and time of flight can be analyzed

simultaneously.


80

(viii) There is option for wizard or manual mode of refinement; the wizard mode allows the

user to select what kind of analysis the user needs to perform in quantitative phase

analysis, crystal structure analysis or texture analysis.

(ix) There is option for adding different methodologies to the program by the user without

the need to recompile it or to know the internal structure of the program. The plug-in-

structures are included in instrument geometries and correction/calibrations, data

formats, line-broadening methods, texture algorithms, peak intensity extraction, etc.

(x) A CIF (Crystal Information File) user-friendly program is included. The program

uses, imports and supports CIF formats.

(xi) Various data format, Philips, Rigaku, Siemens, GSAS, D1B etc. can be used for input

files (only ASCII).

(xii) Unlimited number of data files can be analyzed at a time. Till now some analyses

have been done loading simultaneously more than 1000 datafiles (from the SKAT

diffractometer at Dubna).

(xiii) Space group and symmetries relationships computed using SgInfo [57] linked as a

native library to the package for ease of analysis.

(xiv) Popa model has been included to analyze the cases having anisotropic crystallite size

and microstrain [58].

(xv) In order to give better accuracy in the results, the square root of actual intensity is

plotted against 2 in the fitted pattern, which gives magnified image of the weaker

peaks.

Maud runs everywhere if there is a 1.4+ (1.5 preferred) Java Virtual Machine. Complete

data display and plotting routines are included in this program. On running the program, a

menu bar is displayed from which pull-down menus can be activated by clicking the mouse

onto corresponding levels. All mouse functions can also be obtained by pressing

simultaneously the 'Alt' key and the initial letter of the options on the computer keyboard.

Maud to run optimally needs to be tune up to the user instrumentation and needs. After that

the instrument broadening and characteristics remain fixed for every analysis. A standard

sample spectrum (for profile; well known size-strains or no broadening) was refined to

determine instrumental broadening. Before starting refinement, the instrumental function was

determined following the method described by Enzo et al. [7] by measuring the intensity


81

across 8 peaks of a Si standard specimen, which has large crystallite size and free of defect

broadening. The profiles were fitted with a pseudo-Voigt, pV function created for

asymmetry by convolution with an exponential function

00s 2tan22aexp2A (2.47)

and the dependence of the shape parameters in 2 was described by the formula of Caglioti

et al. [eqn.(2.42)] for HWHM and by a linear interpolation of , the Gaussian content. The

asymmetry parameter as showed a parabolic trend in 2 .

To start the refinement, one has to write a parameter (*.par) file using the 'New

Parameter' option. The 'New Parameter' routine needs some information about instrument

used and the sample to be studied. These include,

(i) Coefficients of polynomial fitting (degree 4/5) to represent the background intensity,

(ii) Intensity, positions and width of amorphous halos (if present),

(iii) Three (U , V , W ) parameters for peak width (eqn. 2.42) decreases linearly with

increasing 2θ), two for , the Gaussian content of pseudo-Voigt function and two for

asymmetry (increases linearly with increasing 2θ). (All these parameters were

predetermined from Si-standard sample),

(iv) Sample positioning (zero-shift error), absorption, beam divergence parameters, which

are often necessary to model some errors influencing peak positions,

(v) Scale factor, lattice parameters, number of atoms per unit cell, atomic occupation

numbers, atomic coordinates, number of phases, temperature factors (isotropic and

anisotropic), preferred orientation parameters [41,42] and crystallite size and

microstrain along different crystallographic directions for each concerned phases are

needed to start refinement.

Once the parameter file has been created, 'Simulate' command synthesizes an XRD spectrum

corresponding to the structure described by the input parameters. The experimental data is

loaded from the 'data file', which consists of step scan data of diffracted intensity and general

measurement parameters: number of points, starting angles and step of measurement. There


82

is provision for inclusion of comment line in the first line of the data file. The quality of

fitting is judged through the calculation of weighted residual error, wpR , (eqn. 2.43) which is

then minimized through a Marquardt least-squares program [59]. The e.s.d. is calculated

following the method proposed by Scott [60] and finally, the goodness of fit ( GOF ) is

established by comparing wpR with the expected error, expR (eqn. 2.44).

The simulated pattern is described by a convolution equation according to Enzo et al.

[7]

bkg2θAIB2θI ci (2.48)

where is convolution symbol and bkg is a polynomial function of degree four reproducing

the background .

The true line broadening function B and the symmetric part of the instrumental

function I is represented by a pV function with both 1

K and 2

K peaks,

21 α,α

212

nt Sln2ηexpS1η1I2θpV (2.49)

where, HWHM22S 0 , HWHM is the half width at half maximum of the X-ray

peaks.

To simplify the procedure, first the convolution IB between two pV functions is

carried out using the procedure reported by Keiser et al. [61]. Then the convolution with the

exponential function 2A [eqn.(2.43)] is performed numerically for the whole pattern.

The shape parameters HWHM and for the B profile function, varying with the

scattering angle, have been generated from the crystallite size ( D ) and r.m.s. microstrain

(<2>

1/2) values of the sample as the fitting parameters following the procedure as adopted

by Lutterotti et al. [62].


83

To refine the microstructure, e.g. the crystallite size and microstrain of an

experimental sample, the pattern of an instrumental standard be refined first. This is

necessary to define the instrumental peak width, shape and asymmetry as a function of 2 .

If the peak-broadening and asymmetry effects of the sample are negligible in comparison to

instrumental standard, the crystallite sizse and microstrain parameters of the sample

parameter file may be set initially zero or the values obtained for the instrumental standard

having large crystallite size (>1 μm) and negligible strain. In this case, the software can be

used solely for structure refinement and U, V, W parameters of the sample parameter file

may be refined to fit the observed profile as other Rietveld refinement software [GSAS,

FULLPROF, DBWS etc.] do. But, if the peak-broadening and asymmetry effects are

significant (plastically deformed materials) as in the case of nanocrystalline materials, the

crystallite size and microstrain parameters of the sample parameter file have to be refined by

setting the instrumental peak-broadening (U, V, W) and asymmetry parameters as

unrefinable parameters. The crystallite size and microstrain parameters would be refined

along with other structural parameters to fit the excess broadening and asymmetry of an

experimental profile arise due to various kinds lattice imperfections in the experimental

sample. The size-strain analysis is done following the single-peak method of Fourier analysis

because in lower symmetric crystals, multi order reflections may not be available in the

range of 2θ, developed by Nandi et al. [53] The basic formulae used in size-strain analysis

are:

D1dLdT 0LpV (2.50)

21exp)d2(D1T 2222

2DLpV (2.51)

where d is interplanar spacing. In the case of a Gaussian microstrain distribution, eqn. (2.51)

can be written as

21)d2(DexpT 2222

2DLpV (2.52)

Taking into account that the normalized Fourier transform of the pV function, used for fitting

the peak profile, is [53]


84

L2Z112LZ1ZLT 222

pVexplnexp (2.53)

where 21

21Z ln

and 00 HWHM2 sinsin

From eqns. (2.51), (2.52) and (2.53) one can write

21Z1Z1216Z1Z 2 exp)ln(exp

21d2D 2222exp (2.54)

0D41ZHWHM sinarcsin (2.55)

21

2ZZ ln (2.56)

This approach to the evaluation of size-strain data does not introduce any hypothesis

about the trend of 2

vs. L , which depends on the type of microstructural disorder. This

makes the method more suitable for reproducing the broadening profile parameters.

The most important feature of Maud is that it is applicable to the cases of anisotropy

of size and strain as a consequence of which profiles with different Miller indices are

broadened in a different manner. To solve the anisotropy problem two tensors for the

crystalline (D) and microstrain (<ε2>

1/2) in the different crystal directions are used in the

analysis mentioned above:

D(h1h2h3) =

21

2

/

i,j

jiij

i,j

jiij hhδhhD (2.57)

<ε2>

1/2 (h1h2h3) =

21

2

/

i,j

jiij

i,j

jiij

hhδhhε (2.58)

where δij = 0 if Dij = 0 or i j

2= 0 and in other cases δij = 1


85

Another important feature of this program is the quantitative determination of the

weight fraction of phases concerned. It has been demonstrated [63] that there is a simple

relationship between the individual scale factors determined considering preferred

orientations of crystallite in a Rietveld structure refinement of a multicomponent sample and

the phase concentration in the mixture. The quantitative information on the weight fractions

( iW ) for each phase is obtained from the scale factor of each phase obtained from the

refinement, using the eqn (2.39)

Finally the values of the crystallite sizes and microstrains obtained from the

program are used to evaluate the values of stacking fault probability, compound fault

probability and dislocation densities, stacking fault energy.

Later some modifications are made in crystallite size-microstrain separation, texture

analysis and defect parameters study. The basic methodology for the crystallite size-

microstrain separation are performed using the theory developed by Delhez et al. [64] and

the calculation of crystallite size and microstrain is performed using the recently published

method of Popa [58]. The equations used for crystallite size hR and microstrain 2

are:

For Laue group 1 :

.....sincossincos 2xPR2xPRxPRxPRRR 2

25

2

24

1

23

1

210h (2.59)

lhEkhElhElkEkhElEkEhEEHhh

3

8

3

7

22

6

22

5

22

4

4

3

4

2

4

1

42 44222

hklE4hlkE4klhE3klE4hlE4lkE4hKE4 2

15

2

14

2

13

3

12

3

11

3

10

3

9

(2.60)

For Laue group m2 :

............sincos 2xPR2xPRxPRRR 2

23

2

22

0

210h (2.61)

2

9

3

8

3

7

22

6

22

5

22

4

4

3

4

2

4

1

4

H

2

hhhklE4lhE4khE4lhE2lkE2khE2lEkEhEE

(2.62)


86

For Laue group mmm2 :

............cos2xPRxPRRR 2

22

0

210h (2.63)

22

6

22

5

22

4

4

3

4

2

4

1

4

H

2

hhlhE2lkE2khE2lEkEhEE (2.64)

For Laue group mmm4 (tetragonal):

............cos4xPRxPRxPRRR 4

43

0

42

0

210h (2.65)

)()( 222

4

22

3

4

2

44

1

4

H

2

hhhklE2khE2lEkhEE (2.66)

For Laue group mmm6 (hexagonal):

............cos4xPRxPRxPRRR 4

43

0

42

0

210h (2.67)

)()( hkhklE2lEhkkhEE 222

2

4

3

222

1

4

H

2

hh (2.68)

For Laue group m3m (cubic):

............,,, xKRxKRxKRRR 1

63

1

62

1

410h (2.69)

)()( 222222

2

444

1

4

H

2

hhhllkkhE2lkhEE (2.70)

To analyze the texture of multiphase samples, in addition to the classical March-

Dollase formula, harmonic texture [65] and the WIMV method [66] are included in order to

obtain the entire orientation distribution function, provided a sufficient number of spectra at

different tilting angles are available for the refinement. The formulae used for harmonic

texture are

,, hkl

random

hklhkl PI (2.71)


87

,,, n

hklhkl

mmn

0 n m

hkl kkC12

4P (2.72)

where mnC are the harmonic coefficient used as fitting parameters. All the parameters

correspond exactly to those reported in the text of Bunge [67].

Finally, three kinds of planar defects, namely two deformation (stacking) faults,

expressed through the intrinsic, and the extrinsic, probabilities and twin faults,

expressed in terms of probability , which contribute to the peak shift, anisotropic

broadening and to the asymmetry respectively, can be obtained as the direct output of the

program. The calculations of the stacking and twin fault probabilities for crystals having

cubic and hexagonal structures are done following Warren's method [48]:

b

02

0

2L

buh

3902

tan (2.73)

b

o

0efff

Lbuah

51

D

1

D

1 . (2.74)

22o

12xc

1

L

L

bu3

54Ab2yy

. (2.75)

2

020eff

2xD4

1csinsin

(2.76)

The , and probabilities can be used as refinable parameters in this software, in

order to allow for slight variation in them. However, for the present dissertation, stacking

fault probabilities are estimated using the equation (2.24) as stated in section (2.4.2.1).

Thus it can be concluded that MAUD is an elegant method for material

characterization. The program has been successfully applied by Lutterotti to analyze the

Y2O3 CPD Round Robin sample, to analyze the texture of various multiphase samples, to


88

refine spectra with anisotropic peak broadening, quantitative analysis of polymers and

samples containing silica glass etc. [56].

In the present study, MAUD has been extensively used to analyze the

microstructure of various industrially important materials. The detailed results of each

analysis will be described in the respective chapters of the dissertation.

Some unique features of Maud:

Automatic refinements

Live parameters adjustment

Multithreading computation

Thin films and multilayers

As a Rietveld program:

No defined peak profile function (for distributions)

Residual stress models

Most of the texture models (WIMV, Standard function)

Works with diffraction images

Planar defects, microabsorption correction

Amorphous modelling

Evolutionary, Simplex and Metadynamic algorithms

Maximum Entropy Electron Map fitting

Per iteration structure solution

Peak fitting and evolutionary indexing

Full profile indexing with evolutionary or least squares (or each one of the

available refining algorithm) fitting

Extensible (by user) models through plugins

CIF input/output based

Evolutionary computation can be spread over computers

Network (XGrid on Mac/Linux/Unix/Windows)

Sample position errors (and CCD/IP tilting and centering)


89

refinement

Absorption models (layers and flexible 3D sample shape)

Modelling for instruments like Hippo or GEM or D20)

2.7. Particle size and strain distributions

There is no priori reason to believe that a simple Voigt model can successfully

describe all size and strain broadening related effects because these profiles might correspond

only to very special cases of size and strain broadening. Recently, Langford, Louer and

Scardi [68.] have applied a better approach to study the influence of normal and lognormal

size distributions of spherical crystallites on the diffraction line profile. However, they have

not considered the case of a large dispersion where the void function or its approximations

fail to accurately describe the size broaden profile. Balzar et al [69] derive an analytical

approximation that is a sum of up to three Gauss and/Lorentz functions that can be

convoluted analytically with the strain and instrumental profile, thus facilitating an easy

introduction in the whole pattern fitting programs, such as Rietveld refinement [70]. In the

present study, the MAUD software has been extensively used to analyze the particle size and

strain distributions of nanocrystalline powders synthesized by ball milling.

2.7.1 Lognormal size distribution: isotropic case:

The lognormal distribution for spherical crystallites is characterized by two

parameters, the average radius R of the particles and the dispersion ζR2.

. It is convenient to

define the dimensionless ratio 2

2

Rc R to characterize the distribution. Then lognormal

distribution can be written as follows [70]:

f (R) = R-1

[2π ln(1+c)]-1/2

exp })]1ln(2[

])1([ln{

2/112

c

cRR (2.77)

One can calculate the volume and area- averaged dimensions as

Dv

2

)1(3 3cR (2.78)


90

DA

3

)1(4 2cR (2.79)

Unfortunately, the size-broaden profile cannot be calculated analytically either in direct or

Fourier space. For the numerical computation, by using simple transformations of the

integration variable [V

hsVPhsP

),(),( ] can be reduced to a standard quadrature formula:

)Rπs(φc))(R

((s)P 212

3 3 (2.80)

]}c)([t(c)(Φ{)Xt(dtπ(χχΦ //

α

α

/ 2127221 1ln2exp1exp (2.81)

Here, we assume an isotropic case with spherical crystallites with

4

22 2sinsin

χ

χ)χχ(χΦ(χ) (2.82)

an interference function for a sphere. The integral in eqn. (2.81) can be computed by a

standard Gauss-Hermite quadrature. One can determine the distribution parameters from

either Fourier analysis of line profile or the parameters of the least squares refined Fourier

transform of size-broadened profile.

2.7.2 Importance of particle size distribution

Mineral particle size distributions may yield geological information about a mineral's

provenance, degree of metamorphism, degree of weathering, etc. We currently are using this

program for research applications in the earth sciences. However, this program also would be

useful to many types of manufacturers who use or synthesize clay (i.e. very fine grained) or

other kinds of crystalline materials, because a material's particle size and structural strain may

strongly influence its physical and chemical properties (e.g. its rheology, surface area, cation

exchange capacity, solubility, reflectivity, etc.).


91

Nanosized crystals are too fine to be measured by light microscopy (~2 to 100 nm in

thickness). Laser scattering methods give only average particle sizes, and particle size cannot

be measured in a particular crystallographic direction. Also, the particles measured by laser

techniques may be composed of several different phases, and some particles may be

agglomerations of individual crystals. Individual particle dimensions may be measured by

electron and scanning force microscopy, but it often takes several days of intensive effort to

measure a few hundred particles per sample, which may yield an accurate mean size for a

sample, but is often too few measurements to determine an accurate size distribution.

Furthermore, such instrumentation is usually not available outside a research setting.

Measurement of size distributions by X-ray diffraction (XRD) solves these

shortcomings. An X-ray scan of a sample is automated, taking a few minutes to a few hours.

The resulting XRD peaks average diffraction effects from billions of individual nanosized

particles. The size that is measured by XRD may be related to the "fundamental" particle size

of a mineral, i.e. to the size of the individual crystalline domains, rather than to the size of

particles formed by the agglomeration of crystals. Furthermore, one can determine the size of

an individual phase within a mixture, and the dimension of particles in a particular

crystallographic direction. Crystallite shape can be determined by measuring crystallite size

in several different crystallographic directions.

The XRD method is based on the regular broadening of XRD peaks as a function of

decreasing crystallite size. This broadening is a fundamental property of XRD, described by

well-established theory. In this work the distributions of size and strain broadening of XRD

pattern of different types of nanocrystalline material have been measured by the Rietveld

analysis using MAUD software.

2.8 References

1. G. B. Greenough, Progr. Metal. Phys., 3 (1952) 176.

2. A. J. C. Wilson, Proc. Roy. Soc., London, A180 (1942) 277.

3. B. E. Warren, Prog. Met. Phys., 8 (1959) 147.

4. C. N. J. Wagner, 'Local Atomic Arrangements Studied by X-ray Diffraction', ed. J. B. Cohen

and J. E. Hilliard, N.Y., Gordon Breach (1966).


92

5. B. E. Warren, 'X-Ray diffraction', Addison-Wesley, (1969).

6. H. P. Klug and L. F. Alexander, 'X-Ray diffraction Procedures', John-Wiley and Sons., N.Y.

(1974).

7. S. Enzo, G. Fagherazzi, A. Benedetti and S. Polizzi, J. Appl. Cryst., 21 (1988) 536.

8. J. I. Langford and A. J. C.Wilson, Proc. Symp. On Crystallography and Crystal Fection, ed. G.

N. Ramashandran, N. Y. Academic press, (1963) 207.

9. E. J. Mittemeijer and R. Delhez, J. Appl. Phys., 49 (1978) 3875.

10. D. Balzar, J. Res. Natl. Inst. Stand. Tech., 98 (1993) 321.

11. P. Scherrer, Gittinger Nachrichten, 2 (1918) 98.

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