1

Stress and strain

by Nicolae C. Popa

National Institute of Materials Physics,

Atomistilor 105 bis, P.O. Box MG-7, Magurele, Ilfov, 077125, Romania

E-mail: nicpopa@infim.ro

Abstract

This chapter is a review of the basic concepts, models, methods and approaches in the

investigation by diffraction of the stress and strain in polycrystalline materials. The chapter

contains eight sections that can be grouped in three parts. In the first part the specific

quantities in single crystals and polycrystals are defined together with the mathematical

background. The state of art in the field and the classical models allowing determining the

macro strain and stress in the most of samples are described in the second part. The third part

is dedicated to the modern analysis by generalized spherical harmonics of the diffraction lines

shift and breadth caused by strain in textured polycrystalline sample.

Keywords

loading and residual strain/stress, elastic constants, type I, II, III, microscopic, macroscopic,

averaged and intergranular strains/stresses, strain/stress orientation distribution function,

mean and variance of the observed strain, strain pole distribution function, diffraction peak

shift and broadening in isotropic and textured polycrystal, the Voigt, Reuss, Kroner models,

the Ψ2sin method, diffraction elastic constants, hydrostatic strain/stress state, generalized

spherical harmonics, the Rietveld method.

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1. Introduction: the importance of stress determination and the diffraction method

The determination of elastic stress and strain state in polycrystalline samples is one of the

oldest applications of the powder diffraction technique. The stress that can be both, residual

or loading, can be beneficial or, by contrary, can provoke premature failures of

manufacturing materials and machine parts. Consequently the determination of stress and

strain state is of a major importance in engineering and technological applications and also in

geology, mining and earth science.

There are several methods of stress investigation: mechanical, acoustical, optical and

the diffraction of X – rays and neutrons, the last being the most appropriate for the crystalline

matter. The diffraction method is based on the variation of interplanar distance (d spacing)

caused by the strain induced by stress. There are two possible d spacing variation effects that

can be observed and measured by diffraction: the peak shift and the peak broadening. The

peak shift is observed if the strain averaged on the irradiated sample volume (the macroscopic

strain) is different from zero; the peak broadening is connected to the strain dispersion. There

is a rich literature on these subjects. The comprehensive monographs by Noyan and Cohen

(1987) and by Hauk (1997) are strongly recommended for the macro strain peak shift and the

books by Wilson (1962) and by Warren (1969) for the strain broadening.

2. Strain and stress in single crystals, elastic constants, transformations

The strain and stress are symmetrical tensor of rank two. If )(ru is a small displacement of

the point of position vector r in a single crystal, then the strain tensor is defined as follows:

( )3,1,),//)(2/1( =∂∂+∂∂= jixuxu ijjiijε , (1)

iu and ix being the components of u and r in an orthogonal coordinate system ).3,1,( =iix

The element ijσ of the stress tensor is defined as the i component of the force acting on the

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unit area normal to the vector jx . For ji ≠ we have 0=− ijji σσ , otherwise the body rotates

around the vector jik xxx ×= . Then, while the strain tensor is symmetrical by definition, the

symmetry of the stress tensor is imposed by the mechanical equilibrium.

The magnitudes of the strain and stress tensors elements are different in different

reference systems. Let us consider another reference system )3,1,( =iiy and denote by )(ga

the Euler matrix transforming this system into )3,1,( =iix :

∑=

=3

1

)(j

jiji ga yx (2)

( )

ΦΦ−Φ

ΦΦ+

−Φ−

−

ΦΦ+Φ−

=

00101

02021

21

021

21

02021

21

021

21

cossincossinsin

sincoscoscoscos

sinsin

coscossin

sincos

sinsincossincos

cossin

cossinsin

coscos

ϕϕ

ϕϕϕ

ϕϕϕϕϕϕ

ϕϕϕ

ϕϕϕϕ

ϕϕ

ga (3)

Here the triplet ),,( 201 ϕϕ Φ=g denotes the standard Euler angles, namely )2,0(1 πϕ ∈ is a

rotation of the reference system ),,( 321 yyy around 3y resulting ),,( 321 yyy ′′ , ),0(0 π∈Φ is

the rotation of this system around 1y′ resulting ),,( 321 xyy ′′′ and finally, )2,0(2 πϕ ∈ is the

rotation of the last system around 3x resulting ),,( 321 xxx . Let us denote the strain and stress

tensors in the system )3,1,( =iiy by the Latin characters ije and ijs , respectively. The

transformations of the strain and of the stress tensor elements at the coordinate system

transformation (2) are easily derived by starting from the tensors definitions. For strain tensor

these are the following, similar expressions being valid for stress:

∑∑= =

=3

1

3

1k mkmjmikij eaaε , ∑∑

= =

=3

1

3

1i jijjmikkm aae ε , (4a, b)

The quantity observable in a diffraction experiment is the spacing -d variation along

the reciprocal lattice vector ),,( lkhH of unit vector h :

4

hhhh eHHddhkldhkldhkld ≡=∆−=∆=− ε0000 //)(/)]()([ (5)

To calculate this quantity from the strain tensor elements ijε in )( ix we must define the

reference system ),,( hlk with the axis k in the plane ),( hx3 and normal to h and khl ×= .

Describing the unit vector h by the polar and azimuthal angles ),( βΦ the connection

between the systems ),,( hlk and )( ix is the following:

Φ=

3

2

1

),(

x

x

x

m

h

l

k

β ,

ΦΦΦ−

Φ−ΦΦ=Φ

cossinsincossin

0cossin

sinsincoscoscos

),(

ββββ

βββm (6), (7)

A comparison of (6) with (2) allows using a transformation similar to (4a) to calculate hhε :

∑∑∑∑= == =

=ΦΦ=3

1

3

1

3

1

3

133 ),(),(

k mkmmk

k mkmmkhh aamm εεββε (8)

Here )cos,sinsin,sin(cos),,( 321 ΦΦΦ= ββaaa are the direction cosines of h in )( ix . If

the polar and azimuthal angles of h in )( iy are ),( γΨ , then the direction cosines are

)cos,sinsin,sin(cos),,( 321 ΨΨΨ= γγbbb and in place of (8) we have:

∑∑∑∑= == =

=ΨΨ=3

1

3

1

3

1

3

133 ),(),(

k mkmmk

k mkmmkhh ebbemme γγ , (9)

Hereafter we denote hhε , hhe simply by hε , he . We have hh ε≡e as they define the same

quantity.

In the limits of elastically deformed body the stress and strain tensors are connected by

the Hooke equations. To derive these equations one starts from the elastic free energy per unit

volume. The differential of this quantity is ∑∑= =

=3

1

3

1i jijij ddF εσ , then the Hooke equations are:

)3,1,(,/ =∂∂= jiF ijij εσ (10)

The magnitudes of the strain tensor elements being small, the elastic free energy can be

expanded in powers of these quantities limiting the expansion to the second order terms. As

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the terms of order zero and of order one are zero because F and ijσ are zero when 0=ijε ,

we can write:

∑∑∑∑= = = =

=3

1

3

1

3

1

3

1

)2/1(i j k l

klijijklCF εε (11)

Here ijklC are the stiffness elastic constants forming a fourth rank tensor of 81 elements. The

strain tensor being symmetrical, the product klijεε is not changing if the indices ji, and lk, ,

as well as the pairs ij , kl are permuted. The tensor ijklC can be chosen to have the same

properties, as a consequence remain 21 independent constants for triclinic crystals, their

number decreasing with increasing the Laue symmetry. By applying (10) to (11) one obtains:

( )3,1,,3

1

3

1

==∑∑= =

jiCk l

klijklij εσ (12)

There are nine equations in (12), but only six are independent because for every ji ≠ there

are two equal strain and stress elements, respectively, and then two identical equations.

Inverting (12) we have:

( )3,1,,3

1

3

1

==∑∑= =

jiSk l

klijklij σε , (13)

where ijklS is the fourth rank tensor of compliance elastic constants having exactly the same

symmetry properties as the tensor of stiffness constants ijklC .

We can take advantage of the symmetry of the strain, stress and elastic constants

tensors to introduce the reduced indices defined as follows:

62112,53113,43223,333,222,111 →=→=→=→→→ (14)

By using the reduced indices the strain and stress tensors are represented as vectors of

dimension 6 and the elastic constants tensors as symmetrical matrices of dimensions 66× .

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We pass to the reduced indices representation according to the convention described in the

Wooster (1973) handbook1:

)6,1,(,

)6,1( ,

=→===

=→=

nmCCCCC

m

mnklijijlkjiklijkl

mjiij εεε (15)

and similarly for the stress tensor and compliance constants.

Accounting for (14) and (15) the transformations (4) for strain and for stress become:

∑=

=6

1jjiji eQε , ∑

=

=6

1jjiji Pe ε , (16a, b)

∑=

=6

1jjiji sQσ , ∑

=

=6

1jjiji Ps σ , (16c, d)

The matrices Q and 1−= QP of dimensions (6, 6) in (16) can be expressed by other four

matrices of dimensions (3, 3) that are calculated from the elements of the Euler matrix (3):

=

ON

MLQ

2,

=

tt

tt

OM

NLP

2, (17a, b)

≠≠=

+=+=

===kji

lkji

aaaaOaaaaO

aaNaaMaL

jiijjjiikkjkkijikkij

jlilklljlilkklkl 3,1,,,,

,

,,2

. (17c)

Further the following notations will be used:

)2,2,2,1,1,1(),.....,( 61 =ρρ , (18)

),,,,,(),....,( 21313223

22

2161 aaaaaaaaaEE = , (19)

),,,,,(),....,( 21313223

22

2161 bbbbbbbbbFF = . (20)

With these notations the strain along the reciprocal lattice vector (8) and (9) become:

∑=

=6

1jjjjE ερεh , ∑

=

=6

1jjjj eFe ρh . (22a, b)

The free energy expression (11) and the direct Hooke equations (10), (12) become:

1 Part of literature uses a different convention which is described in Nye (1957) handbook. Wooster convention is preferred as it keeps the values of the tensors elements and for symmetry higher than triclinic gives identical structures for the matrices C and S .

7

∑ ∑∑= +==

+=5

1

6

1

6

1

22

2

1

m mnnmnmmn

mmmmm CCF εερρερ , (23)

)3,1(/ =∂∂= mF mm εσ ; )6,4(/2 =∂∂= mF mm εσ 2 (24a, b)

=+++++==+++++=

)6,4(,4442222

)3,1(,222

665544332211

665544332211

mCCCCCC

mCCCCCC

mmmmmmm

mmmmmmm

εεεεεεσεεεεεεσ

(25)

The inverse Hooke equations obtained by solving (25) have exactly the same form, only the

vectors mε and mσ change places and the matrix S replaces C . From (25) can be observed

that the inversion relations between the matrices C and S are 1)( −′= CS and 1)( −′= SC ,

where C′ and S′ are obtained by partitioning C and S in four 33× blocks multiplied as

follows: the upper left block by 1, the upper right and down left by 2 and the down right

block by 4. The relation between C and S once established, the factor 2 in the second

equation (25) can be dropped and the Hooke equations written unitary as follows:

∑=

=6

1jjjiji C ερσ , ∑

=

=6

1jjjiji S σρε , )6,1( =i (26a, b)

The matrices C for all Laue classes are given in the Table 1. The matrices S have exactly

the same structure. For a given Laue group the matrix C is found from the invariance

conditions of the free energy (23) to the operations of this group3. It results a system of 21

homogenous equations determining a number of linear constraints between the matrix

elements and consequently, the number of independent elastic constants specific to the Laue

group. The simplest structure of C with a lot of elements equal to zero is obtained if the

crystal reference system )( ix is taken with 3x along the foldn − axis and 1x along the

fold−2 axis normal to the foldn − axis for the groups where this axis exists or in an

2 The factor 2 in the left side of (24b) is due to the fact that for ji ≠ there are two contributions to the

differential of the free energy, ijijjijiijij ddddF εσεσεσ 2=+= , and then

ijijF σε 2/ =∂∂ 3 Invariance to inversion is assured by the quadratic form.

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arbitrary direction4 for the groups where this axis doesn’t exist. Passing to the system )( iy

the Hooke equations and the single crystal elastic constants change as follows:

∑=

=6

1

)()()(j

jjiji gegCgs ρ , ∑=

=6

1

)()()(j

jjiji gsgSge ρ , (27a, b)

∑∑= =

−=6

1

6

1

1 )()()(j k

klijkjklil gQgPCgC ρρ , ∑∑= =

−=6

1

6

1

1 )()()(j k

klijkjklil gQgPSgS ρρ (28a, b)

3. Strain and stress in polycrystalline samples.

3.1. Types of strains and stresses. Strain/stress orientation distribution function. A textured

polycrystalline sample is formed from a large number of small single crystal blocks with

different orientations following a certain distribution called the orientation distribution

function (ODF). The elastic strain and stress state of an individual crystallite is determined by

the Hooke equations together with the boundary conditions. In a polycrystalline sample the

boundary conditions are the result of the interaction of the crystallite with its neighbors and

this interaction depends on the crystallite shape and orientation. To describe both, the texture

and the strain/stress state of a polycrystalline sample two reference systems should be

defined, one linked to the crystallite, the second one to the sample. The crystallite reference

system is )3,1,( =iix defined for every Laue group as described in the previous section. The

sample system )3,1,( =iiy can be defined by using some remarkable surfaces and directions

resulted in the manufacturing process or given by the sample geometrical characteristics. In

the most general case the strain and stress in a crystallite are not homogenous and are

described by functions depending on the crystallite orientation g and on the position vector

in crystallite. Let us denote by kR the position vector with respect to the sample system of

the crystallite k having the orientation in the range ),( dggg + . If r is the position vector of

4 a/1 ax = is appropiate in all cases if the fold−2 axis, when it exists, is taken along the unit cell vector a .

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a point in this crystallite in the crystal reference system, then the elastic strain at this point in

this system is ),( gki rR +ε . Denoting by kV the crystallite volume, the averaged strain in this

crystallite is:

rrRR dgVg kikki ∫ += − ),(),( 1 εε (29)

If gN is the number of crystallites of orientation g the second average can be defined::

∑=

−=gN

kkigi gNg

1

1 ),()( Rεε (30)

Further, the third average can be defined by integrating (30) over the crystallites orientations.

Denoting by )(gf the orientation distribution function (ODF) this average is:

∫∫∫= )()( ggfdg ii εε , (31)

The functions )(giε are the type I strains and the following differences define the type III,

type II and the intergranular strains:

type III: ),(),(),( ggg kikiki RrRrR εεε −+=+∆ (32)

type II: )(),(),( ggg ikiki εεε −=∆ RR (33)

intergranular: iii gg εεε −=∆ )()( (34)

The type I strain as well as its average and the intergranular strain are macroscopic quantities.

For simplicity both the type II and the type III strains are called together microscopic strains

although the type II is mesoscopic. Obviously the following averages are zero:

0),(1 =+∆∫− rrR dgV kik ε , 0),(

1

1 =∆∑=

−gN

kkig gN Rε , 0)()( =∆∫∫∫ ggfdg iε . (35a,b,c)

The strain in the point r in the crystallite k of orientation g can be written as a sum of type

I, type II and type III strains:

),(),()(),( gggg kikiiki rRRrR +∆+∆+=+ εεεε , )6,1( =i (36)

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Expressions similar to (29) – (36) can be written for any elastic strain or stress component in

any reference system. In these expressions iε is, in fact, a placeholder for iε , ie , iσ , is .

Similarly to the texture we can call )(giε the strain/stress orientation distribution

functions (SODF). In contrast to ODF, the average of SODF over all variables is not unity but

the averaged macroscopic strain/stress iε . In general both, the averaged value and

intergranular strain/stress are necessary for a complete description of the macroscopic

strain/stress state in a material. The intergranular strains/stresses have various origins like

elastic and plastic deformations, phase transformations, thermal treatments, mismatch of d -

spacing in composite materials and differences in the coefficients of thermal expansion

(Behnken, 2000). It is advantageous to group all these elastic intergranular strains/stresses of

different origins in only two terms, elasticiε∆ and plastic

iε∆ then the SODF can be written as

follows:

)()()( ggg plastici

elasticiii εεεε ∆+∆+= (37)

The first two terms in the right side of (37) represent the elastically induced part of SODF,

the last term being the plastically induced part.

3.2. The mean and variance of the observable strain: the peak shift and broadening.

According to the equation (22a) the strain along h and its square in terms of the strain tensor

elements in the crystal reference system are the following:

∑=

+=+6

1

),(),(i

kiiik gEg rRrRh ερε (38)

∑∑= =

++=+6

1

6

1

),(),(),(i j

kjkijijik ggEEg rRrRrR2h εερρε , (39)

The peak shift and the integral breadth caused by strain are obtained from (38) and (39),

respectively, after substituting (36) and performing three averages. The first two averages are

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over r and then over k for a given orientation g . Taking account of (35a,b) the following

macroscopic quantities depending only on the crystallite orientation are obtained:

∑=

=6

1

)()(i

iii gEg ερεh , (40)

[ ]∑∑= =

∆+=6

1

6

1

)()()()(i j

ijjijiji gggEEg εερρε 2h , (41)

∑ ∫∑==

+∆+∆+∆∆=∆gg N

kkjki

kg

N

kkjki

gij ggd

VNgg

Ng

11

),(),(11

),(),(1

)( rRrRrRR εεεε (42)

The third average is performed only over those crystallite orientations +g for which h is

parallel to y , the unit vector of the scattering vector. This means that only the crystallites in

Bragg reflection are considered. Under this constraint the Euler matrix (3) can be written as

follows:

),()(),(),,()( 201 γωβϕϕ ΨΦ=Φ= ++++ mnmaa tg (43)

where m is given by (7), ),( βΦ and ),( γΨ are the polar and azimuthal angles of h in )( ix

and of y in )( iy , respectively, and where )(ωn is a simple rotation matrix of angle

)2,0( πω ∈ . Now, taking account that the sample could be textured and denoting by

( ) ∫ +=yh

h y||

)(2/1)( gfdp ωπ the (texture) pole distribution function, the average over +g of

the observable strain (n=1) and of its square (n=2) are the following:

( ) )(/)()(2/1)(||

yy hyh

hh pggfd nn∫

++= εωπε (44)

This equation contains as normalizing factor the texture pole distribution )(yhp that is not

accessible to the diffraction measurements. This can be replaced by the reduced pole

distribution [ ])()()2/1()( yyy hhh −+= ppP because the peak positions for h− and h are not

distinguishable. Therefore in place of (44) we have:

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∫

∫∫

±

±±

−

−−++

=

+=

yhh

yhh

yhhhh yy

||

||||

)()(2

1

)()(2

1)()(

2

1

2

1)()(

gfgd

gfgdgfgdP

n

nnn

ωεπ

ωεπ

ωεπ

ε. (44’)

Here the significance of the symbol )(± is the average of the two terms, the Euler matrix for

yh ||− being calculated by using (43) with ),( βΦ replaced by ),( πβπ +Φ− . Now,

substituting (40) and (41) in (44’) one obtains the following expressions for the mean

(equation (45)) and for the variance (equations (46)) of the observable strain:

∑ ∫= ±

±±=6

1 ||

)()(2

1)()(

iiii gfgdEP

yh

hh yy ωεπ

ρε . (45)

)()()()()(2

yyyyy hhh2hh

mM VVV +=−= εε , (46a)

∆∆−

∆∆

=∫∫

∫∑∑

±

±±

±

±±−−

±

±±±−−

= =

yhyhh

yhh

hy

y

y

||||

22

||

11

6

1

6

1 )()()()()2)((

)()()()2)((

)(gfgdgfgdP

gfggdP

EEVji

ji

i jjiji

M

εωεωπ

εεωπρρ (46b)

∆= ∫∑∑

±

±±−−

= = yhhh yy

||

116

1

6

1

)()()2)(()( gfgdPEEV iji j

jijim ωπρρ (46c)

The mean determines the peak shift, the variance determines the peak breadth caused by

strain. The variance has two components: the microstrain variance )(yhmV and the

macrostrain variance )(yhMV . The contributions to the peak shift of the second and of the

third terms from (36) are zero because the averages of these terms over r and k ,

respectively, are zero. The peak shift is exclusively caused by the type I strain. The strains of

type II and III contribute only to the peak broadening. The broadening effect of microstrains

is explained by their correlations different from zero appearing in the equation (42); can be

observed that only correlations of strains of the same level are involved. Finally, the

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macrostrain broadening is produced only by the intergranular strain and, as will be seen later,

can be considered a small correction of the microstrain broadening.

There is an alternative derivation of the peak shift and broadening, namely starting from

(22b), the strain along h in terms of the strain tensor elements in the sample reference

system. The resulted mean and variance of the observable strain keep exactly the same

structure as the equations (45) and (46) with ’ε ’ in the right side5 replaced by ’e’ and ’ iE ’

replaced by ’ iF ’. For referring in the text below, the labels (45’) and (46’) are assigned to this

alternative set of equations without copying it. For a given problem the appropriate choice of

one or other alternative can save important volume of calculations.

The peak shift (45) can be also called the strain pole distribution function. The strain

pole distribution is for SODF what is the pole distribution for texture, with an important

difference: in place of one distribution, six independent SODF’s in a determined linear

combination (equations (22)) are projected on the space ),( γΨ .

4. Status of the strain/stress determination by diffraction.

4.1. The macro strain/stress. Although the diffraction is able, in principle, to determine both,

the averaged strain/stress tensors ii se / as well as the intergranular strain/stress

)(/)( gsge ii ∆∆ , the principal aim for many decades was to determine only the averaged

tensors. The determination was based on the supposition that the elastic strain and stress

tensors )(gei , )(gsi in a crystallite are connected to the average tensors is , ie as follows

(see e.g. Behnken, 2000):

∑=

=6

1

# )()(j

jjiji sgSge ρ , ∑=

=6

1

# )()(j

jjiji egCgs ρ . (47a,b)

5 Remembering that hh ε≡e , in the left side of (45’) and (46’) we have:

nne )()( yy hh ε≡ .

14

Here )(# gSij and )(# gCij are fourth-rank tensors describing the elastic behavior of the

crystallites in the polycrystalline materials, not necessarily the single crystal compliance and

stiffness tensors like in (27). Equations of type (47) are obtained if the plastically induced

term in (37) is zero and the elastically induced term of the strain/stress is different from zero

only if the averaged stresses/strains are different from zero. If the equations (47) are valid

then the mean strain measured by diffraction has the following expression:

∑=

ΦΨ=6

1

),;,()(j

jj sR βγε yh . (48)

The coefficients jR are called diffraction stress factors and can be calculated analytically or

numerically by using (45) or (45’). Classical models like Voigt (1928), Reuss (1929) and

Kroner (1958) fit perfectly to this category. The average stresses can be obtained by fitting

(48) to the measured data for a number of peaks and directions in sample. For isotropic (not-

textured) samples (48) becomes linear in Ψ2sin and Ψ2sin and is the basic equation of the

traditional "sin" 2 Ψ method (Hauk, 1952, Christenson & Rowland, 1953). Most of the

experimental data can be processed by this method, even if the sample has a weak texture.

For textured samples the relation between the peak shifts and Ψ2sin becomes non-

linear. Sometimes analytical expressions can be found by approximating the texture pole

distribution by δ functions on some prominent sample directions (Dolle, 1979). In general

this could be a rough approximation, numerical calculations of the diffraction stress factors

being preferable. A lot of applications on thin films can be found in the review paper by

Welzel et al. (2005).

The determination of the stress in textured sample requires a prior, accurate

determination of the texture. To eliminate this time consuming step and to increase the

accuracy of the stress determination Ferrari and Lutterotti (1994) proposed to include the

stress analysis into the Rietveld method, the stress parameters being refined together with the

15

texture spherical harmonics coefficients and the structural parameters. Balzar et al. (1998)

also used the Rietveld method with a Voigt type formula implemented in GSAS by Von

Dreele (2004) to determine the average strain tensor from multiple time-of-flight neutron

diffraction patterns on Al/SiC composites.

Sometimes the dependence of the measured strain on Ψ2sin becomes strongly non-

linear, especially in metals after plastic deformation and cannot be explained by the texture or

by the stress gradient effect. In general the equations (47) are too restrictive because they

don’t take into account the plastically induced part of the strain and stress. They must be

replaced by the exact equations (27), in another words the averaged strain and stress tensors

and the whole intergranular strains and stresses must be considered. There are two

possibilities to account for the whole intergranular stress. The first one is to calculate the

plastically induced part of the stress starting from the models of the plastic flow of crystallites

within a polycrystalline sample (Van Akeret al., 1996). The second possibility is to construct

SODF )(giε by inverting the strain pole distributions )(yhε measured for several peaks

and in a large number of sample directions. No model is necessary to assume for the elastic or

plastic interactions of crystallites. By contrary, the determination of SODF on elastically

loaded or plastically deformed samples gives essential information on the mechanisms of

crystallite interactions.

To determine SODF Wang et al. (1999, 2001, and 2003) and Behnken (2000)

proposed an approach based on the representation of these functions by generalized spherical

harmonics. These approaches presume a prior determination of the ODF (texture) and

individual peak fitting to find the peak position. Only isolated peaks can be used for accurate

position determination, and then a large part of information contained in the diffraction

pattern is lost. To eliminate these drawbacks Popa and Balzar (2001) proposed a variant of

16

the strain representation by generalized spherical harmonics appropriate for implementation

in the Rietveld method.

In the following, three subjects concerning the macro strain/stress determination are

developed: the classical approximations for the macro strain/stress in isotropic polycrystals,

isotropic polycrystals under hydrostatic pressure and the spherical harmonics analysis in the

variant implementable in the Rietveld programs .

4.2. Status of the micro strain determinations. The microstrain is part of the microstructure,

the departure caused by imperfections of the real polycrystal from the ideal one. The

microstructure affects the profile of the diffraction peaks, and then its study is performed by

the line profile analysis (LPA) techniques. A nice historical overview of LPA techniques can

be found in a paper by Langford (2004).

The LPA is quite as old as the powder diffraction itself due to the Scherrer (1918)

equation. In terms of the physical quantities used today, the Scherrer equation for constant

wavelength and for time of flight diffraction method is the following:

]cos)(/[ HVDH D θλβ h= , ]sin)(2/[2 θλβ hVH

DH D=

Here HH λθ , are the Bragg angle and wavelength, )(hVD is the volume averaged column

length along the reciprocal lattice vector and DHβ is the corresponding integral breadth of the

diffraction line. During two decades after Scherrer paper, diffraction phenomena not

interpretable with his formula have been observed but only after 1940 these have been

understood and treated theoretically. Stockes and Wilson (1944) realized that a peak

broadening caused by strain may coexist with the broadening due to the crystallite size and

that in the reciprocal space the former increases linearly with the reflection order while the

later is independent on this order. Consequently, considering a Gaussian for the distribution

of the observable strain, the dependence of the strain breadth SHβ on the strain variance

[equation (46)] is the following:

17

[ ] 2/1)(2tan2 yhS VHH πθβ = , [ ] 2/1)(2 yh

S VHH πλβ = (49a, b)

The sample peak profile is a convolution of the size and strain profiles and the central

problem of LPA is to separate the two contributions. The separation is possible due to the

different dependence of the breadth on the scanning variable. There are two kinds of methods

for separation: the integral breadth methods requiring models for the peak profiles and the

Fourier methods, not requiring such models.

Based on the Stockes and Wilson (1944) dependence of the size and strain breadths on

the reflection order, Hall (1949) proposed an integral breadth method of separation known

today as Williamson – Hall plot. It consists in plotting λθββ /cos*HHH = versus hH dmd /* = ,

where m is the reflection order, hd is the interplanar distance of the lattice planes )(hkl and

Hβ is the integral breadth of the experimental line profile corrected for the instrumental

resolution. Presuming SH

DHH βββ += which means that both, size and strain profiles are

Lorentzians we have: ( ) *2/1* 2)(/1 HVH dVD hh πβ += , and then the crystallite size and the strain

variance are obtained from the intercept and the slope of this line. Similar linear plots can be

drawn for Gauss sample profile and for Voigt sample profile (Lorentz for size, Gauss for

strain) if the coordinates ),( 2*2*HHd β are used in place of ),( **

HHd β .

Based also on Stokes and Wilson (1944) paper, Averbach and Warren (1949) firstly

proposed a Fourier method for separating the strain and size. According to these authors, the

logarithm of the cosines Fourier coefficients of the sample line profile which are the product

of the size and strain Fourier coefficients can be written as follows:

hVmnAmAAmA Dn

Sn

Dnn

2222)ln()](ln[)](ln[ π−== .

where n is the harmonic index and m the order of reflection. Rarely more than two orders

are available. In the plot )ln( nA versus 2m for different values of n the slope and intercept

give the strain variance and respectively, the parameters connected with the crystallite size

18

namely the area and volume averaged column lengths and the faulting probabilities if it is

the case. Details can be found in Warren (1969).

Over the years, the Warren-Averbach method has been modified and improved

becoming a standard method in LPA; for details see Langford (2004). Unfortunately, like the

Williamson – Hall plot, this method has a serious limitation: it requires isolated diffraction

peaks, without overlapping, condition rarely fulfilled.

The overlapping is not an impediment for the integral breadth method called the whole

powder pattern fitting (WPPF) or pattern decomposition. The WPPF means to fit the whole

diffraction pattern by using analytical functions for the peak profiles with at least one

obligatory constraint: the breadth of a given peak is not free, refinable parameter, but is

calculated with a parameterized function depending on the scanning variable. For the strain

breadth this function is (49a) or (49b) and one expects to have a parameterized expression

describing the dependence of the strain variance on the directions in crystal and sample. Such

expressions for all Laue groups were recently derived by Popa and Lungu (2013) by using a

spherical harmonics approach described below in section 8. They have as classical limits the

expressions of strain variance dependent only on crystal direction derived by Popa (1998) and

independently by Stephens (1999), implemented today in all popular Rietveld programs.

5. Macro strain/stress in isotropic samples, classical approximations

As was stated in the previous section, if the aim of the diffraction experiment is the

determination of the averaged strain and stress tensors, the classical models Voigt, Reuss,

Kroner are enough good to process most of the experimental data. Moreover, the

determination becomes a very simple routine if the sample is isotropic. The term isotropic for

a polycrystalline sample denotes the absence of the preferred orientation which means

1)( =gf . Concerning the elastic properties, such sample is isotropic only in average, but the

19

behavior of crystallites, with rare exceptions, is anisotropic. From this reason it is much

appropriate to call this sample quasi-isotropic. As a consequence of isotropy, two averaged

elastic constants specific to the model can be derived from the single crystal elastic constants.

5.1. The Voigt model. According to Voigt (1928) the intergranular strain in the sample

reference system is zero and then the macroscopic strain tensor in crystallite is identical with

the averaged strain:

ii ee =Φ ),,( 201 ϕϕ (50)

To find the averaged stress tensor in the same system we substitute (50) into (27a) and

integrate over the Euler space. The integral acts only on the single crystal stiffness tensor

elements (28a) and can be calculated analytically. The averaged stress is:

∑=

=6

1jjj

Viji eCs ρ . (51)

In (51) VijC are the Voigt averaged elastic stiffness constants for the isotropic polycrystal.

They are calculated from the single crystal stiffness constants as follows:

15/)222(25/)( 66554423131233221111 CCCCCCCCCCV ++++++++= (52a)

15/)(415/)222( 23131266554433221112 CCCCCCCCCCV +++−−−++= (52b)

To find the peak shift for the Voigt model the equation (50) must be substituted in

(45’). The integral in (45’) is trivial and one obtains:

∑=

=6

1

)(j

jjj eF ρε yh (53)

The equation (53) can be arranged to be similar to (48). By inverting (51) and substituting

resulted je in (53) this equation becomes:

))((2])([

])([])([)(

665544121131221113

2123111211232111

sFsFsFSSsSFFSF

sSFFSFsSFFSFVVVV

VVVV

++−++++

+++++=yhε (54)

In (54) VijS is the compliance tensor obtained by inverting the stiffness tensor V

ijC :

20

])(2)(/[)( 212121111121111VVVVVVV CCCCCCS −++= (55a)

])(2)(/[ 2121211111212VVVVVV CCCCCS −+−= . (55b)

We can see from (53) or (54) that in the frame of the Voigt model of the crystallite

interactions the relative peak shift does not depend on the Miller indices, fact frequently

contradicted by experiment. The Reuss model gives such dependence.

5.2. The Reuss model. In the Reuss (1929) hypothesis the intergranular stress in the sample

system is zero and then:

ii ss =Φ ),,( 201 ϕϕ (56)

To find the averaged strain, equation (56) is substituted into (27b) which is integrated on the

Euler space and one obtains:

∑=

=6

1jjj

Riji sSe ρ . (57)

Here RijS are the Reuss averaged compliance constants for the isotropic polycrystals. They

are calculated from the single crystal compliances with formulae similar to (52):

15/)222(25/)( 66554423131233221111 SSSSSSSSSSR ++++++++= (58a)

15/)(415/)222( 23131266554433221112 SSSSSSSSSSR +++−−−++= (58b)

Note that RijS and V

ijS are different. Also the stiffness constants RijC obtained by inverting R

ijS

are different from VijC defined before.

To calculate the peak shift there are two possibilities that should give identical results.

According to Behnken and Hauk (1986), the equations (56) and (28b) are substituted into

(27b), (27b) is substituted into (22b), then the peak shift is calculated with (45’) which

becomes:

∑ ∑ ∑∑= = = =

=6

1

6

1

6

1

6

1

)(i l j k

klijkjklii QPSsF ρρε yh

21

Alternatively, according to Popa (2000), the equation (56) is substituted into (16c), (16c) into

(26b), (26b) into (22a), then the peak shift is calculated with (45) which becomes:

∑ ∑ ∑= = =

=6

1

6

1

6

1

)(i l j

jljijlii QSsE ρρε yh . (59)

Obviously the last possibility which is adopted here is much cheaper as there are to calculate

only 36 averages jlQ in place of 1296 klij QP . Taking account of (17), (43) and (7) the

integrals

∫±±± Φ=

π

ϕϕωπ2

0

201 ),,()2/1( ijij QdQ . (60)

can be calculated analytically and one obtains:

=−

=−+−=

6,5,4for 3

3,2,1for 2/)1(2/)13(

jFEF

jFEFQ

jiij

jiij

ij δδ

, (61)

)0,0,0,1,1,1(),.....,( 61 =δδ

Inserting (61) in (59) and rearranging the terms the peak shift becomes:

2/)3(2/)()( 42 rtsrst ss −+−= yyh yε . (62)

Here st and ys are the trace of s and, respectively, the averaged stress along y :

321 sssts ++= , ∑=

=6

1iiii sFs ρy . (63a,b)

The dependence of the peak shifts on the Miller indices is given by the factors 2r and 4r that

are quadratic and respectively quartic form of the direction cosines ia . For the triclinic

symmetry these factors are the following:

213626163135251532342414

23332313

22232212

211312112

)(2)(2)(2

)()()(

aaSSSaaSSSaaSSS

aSSSaSSSaSSSr

+++++++++++++++++=

, (64)

22

321262

3116

331353

3115

332343

3224

23214536

3221462532

215614

22

216612

23

215513

23

224423

4333

4222

41114

44

4444)2(4

)2(4)2(4)2(2

)2(2)2(2

aaSaaS

aaSaaSaaSaaSaaaSS

aaaSSaaaSSaaSS

aaSSaaSSaSaSaSr

++

++++++

++++++

++++++=

. (65)

For higher symmetries 2r and 4r can be found by reader taking account of the constraints for

ijS from Table 1; they are also given in the Tables 12.6 and 12.7 from Popa (2008). The

expressions in these tables are identical to those derived by Behnken and Hauk (1986) except

for two Laue classes, erroneous at these authors.

5.3. The Hill average. The Voigt and the Reuss models are two extreme cases of crystallite

interactions that roughly describe the strain/stress state of isotropic polycrystalline samples.

Hill (1952) observed that the real elastic constants are in general close enough to the

arithmetic average of the constants calculated within the two models. Consequently a very

good description of the peak shift is obtained in practice by using the arithmetic average of

the Voigt and Reuss peak shifts (54) and (62). Even better is to use a weighted average with

the weight w ( 10 << w ) as a refinable parameter in a least square analysis (Popa, 2000):

RV

ww )()1()()( yyy hhh εεε −+= . (66)

5.4. The Kroner model. A model for crystallite interaction better than the Voigt or the Reuss

models was proposed by Kroner (1958). According to Kroner every crystallite is an inclusion

in a continuous and homogenous matrix that has the elastic properties of the polycrystal. For

the isotropic polycrystal the strain in inclusion is the following:

[ ]∑=

+=6

1

)()(j

jjijR

iji sgtSge ρ . (67)

In this expression similar to (47a) the first term is the strain of the isotropic matrix given by

(57). The second term is the strain induced in crystallite by the matrix and is given by the

Eshelby (1957) theory of the ellipsoidal inclusion. The tensor )(gtij accounts for the

23

differences between the compliances of the inclusion and of the matrix and has the property

0=ijt . To calculate the peak shift, (67) is substituted in (45’). Analytical calculations can be

performed only for a spherical crystalline inclusion that has a cubic symmetry. For the peak

shift an expression similar to (54) is obtained but with different compliances. According to

Bollenrath et al. (1967), the Voigt compliance constants in (54) must be replaced as follows:

Γ−+→ 0111111 2TTSS RV , Γ++→ 0121212 TTSS RV , (68a,b)

4412110 2TTTT −−= , 22

21

23

21

23

22 aaaaaa ++=Γ . (68c,d)

The matrix compliances RS11 and RS12 are given by (58) for the case of cubic symmetry. The

induced compliances 11T , 12T and 44T are also calculated from cubic single crystal

compliances by bulky algebraic expressions that are reproduced in a lot of papers: Bollenrath

et al. (1967), Dolle (1979), Gnaupel – Herold et al. (1998) and Welzel et al. (2005).

5.5. The method “ Ψ2sin ” . The peak shift equations (54) and (62) can be arranged in the

following form, convenient for experimental data processing:

Ψ++Ψ+−++

+++=

2sin)cossin()2/1(

sin)2sinsincos()2/1(

)2/1()()(

542

263

22

212

323211

γγγγγ

ε

ssS

ssssS

sSsssSyh

. (69)

The factors 1S and 2S are called diffraction elastic constants. For the models examined

above they are the following:

(cubic)Kroner - )3(2,

Reuss- 3,2/)(

Voigt- )(2,

0121112112012121

242421

12112121

Γ−−+−=Γ++=

−=−=−==

TTTSSSTTSS

rrSrrS

SSSSS

RRR

VVV

(70)

Except for the Voigt model, the diffraction elastic constants are dependent on the Miller

indices.

If consider the peak shifts for γ and πγ + at the same value of Ψ equation (69) is

splited in two linear equations, one in Ψ2sin and the other one in Ψ2sin :

24

Ψ+−++

+++=+Ψ+Ψ=+

263

22

212

323211

sin)2sinsincos(

)(2),(),(

γγγ

πγεγεε

ssssS

sSsssShhh , (71a)

Ψ+=+Ψ−Ψ=−2sin)cossin(),(),( 542 γγπγεγεε ssShhh (71b)

Consequently, if the peak shifts for one or more peaks are measured as function of Ψ in the

range )2/,0( π at γ and πγ + for three fixed values of γ (for example 0 , 4/π and 2/π )

the stress tensor elements is can be determined from the intercepts and the slopes of these

lines. It is presumed that the single crystal elastic constants are known and the diffraction

elastic constants in equations (71) can be calculated following one of the models presented

before. This is the conventional “ Ψ2sin ” method. Alternatively equation (69) can be used in

a least square analysis or implemented in the Rietveld codes. If diffraction patterns measured

in a number of points ),( γΨ are available the stress tensor elements is can be refined

together with the structural and other parameters.

5.6. Determination of the single-crystal elastic constants. The dependence of the diffraction

elastic constants on the Miller indices can be exploited to find the single crystal elastic

constants from powder diffraction data. Indeed, let us presume that an axial, known stress 3s

is applied on a polycrystalline sample. All other components of the stress tensor are zero and

then the equation (69) becomes:

32

21 ]cos)2/1([)( sSS Ψ+=yhε . (72)

By measuring the peak shift for 0=Ψ and 2/π=Ψ both 1S and 2S can be determined. If

the measurement is repeated for many peaks the single-crystal elastic constants can be

calculated by minimizing a 2χ calculated with the differences between the measured

diffraction elastic constants and those calculated with one of the models presented above

(except Voigt). For a given Laue group the number of measured diffraction peaks must be

greater than the number of independent single-crystal elastic constants. A comparison of the

25

single-crystal elastic constant determined in this way on aluminum, copper and steel

(Gnaupen – Herold et al. 1998) with those measured on single crystal by ultrasonic pulse

proved the reliability of the diffraction method.

6. Hydrostatic state in isotropic polycrystals

A hypothesis only recently examined in literature (Popa, 2008) is that the intergranular strain

in the crystallite reference system is zero and then:

ii εϕϕε =Φ ),,( 201 . (73)

To obtain the strain tensor in the sample reference system, (73) is substituted into (16b); to

obtain the stress tensor, (73) is substituted into (26a) and (26a) into (16d). We have:

∑=

=6

1

)()(j

jiji gPge ε , ∑ ∑= =

=6

1

6

1

)()(l j

jlijlli CgPgs ερ . (74a,b)

To calculate the macroscopic strains and stresses, (74) are averaged on the Euler space.

Presuming isotropic polycrystal, the average acts only on the matrix P and one obtains:

>=

=3or if 0

3,1, if 3/1

ji

jiPij . (75)

The macroscopic strains and stresses are then the following:

3/)( 321321 εεε ++==== eeee , 0654 === eee . (76a)

∑=

++====6

1321321 )()3/1(

llllll CCCssss ερ , 0654 === sss . (76b)

The structure of equations (76) is specific for the strain/stress state in a sample under a

hydrostatic pressure.

To calculate the peak shift we insert (73) in (45) and one obtains:

216315324233

222

211 222 aaaaaaaaa εεεεεεε +++++=h . (77)

26

As expected, the peak shift is independent on the direction in sample. Crystal symmetries

higher than triclinic impose constraints on the strains iε that can be found by setting

invariance conditions of the peak shift (77) to these symmetry operations. Hence for

monoclinic m/2 , 11 aa −→ and 22 aa −→ by fold−2 axis along c and then one founds

054 == εε . The peak shifts for all Laue classes higher than triclinic are given at the row

2=l of the Tables 3. These formulae can be easily implemented in the Rietveld codes with

iε refinable parameters. In fact they were already implemented in GSAS (profile #5) but the

derivation presented in the GSAS manual (Von Dreele, 2004) is different, the concrete

connection of the refined parameters with the macroscopic hydrostatic strain and stress being

not revealed.

The hydrostatic state can be also modeled presuming zero the intergranular stress in the

crystal reference system:

ii σϕϕσ =Φ ),,( 201 (78)

Substituting (78) into (16d) and also into (26b) which is further substituted into (16b) one

founds:

∑=

=6

1

)()(j

jiji gPgs σ , ∑ ∑= =

=6

1

6

1

)()(l j

jlijlli SgPge σρ . (79a, b)

and averaging on the Euler space one obtains:

3/)( 321321 σσσ ++==== ssss , 0654 === sss (80a)

∑=

++====6

1321321 )()3/1(

llllll SSSeeee σρ , 0654 === eee . (80b)

The peak shift is given by the same equation (77) but with iε calculated from iσ :

∑=

=6

1jjjiji S σρε (81)

27

The hypotheses (73) and (78) fully describe the hydrostatic strain/stress state in

isotropic samples. Indeed, from the refined parameters iε or iσ the averaged strain and stress

ie , is can be calculated and also the intergranular strains and stresses )(gei∆ , )(gsi∆ , both

different from zero. Note that anything was presumed concerning the nature of the crystallites

interaction, which can be elastic or plastic. From the equations (73) and (78) cannot be

obtained relations of the type (47) but only of the type (27). From this reason a linear

homogenous equation of Hooke type between the averaged stress and strain cannot be

established.

7. The macroscopic strain/stress by spherical harmonics.

As discussed in section 4, in many cases the classical models of crystallite interactions cannot

explain the strongly non-linear dependence of the diffraction peak shift on Ψ2sin , even if

the texture is accounted for. A possible solution to this problem is to renounce to any physical

model and to find the strain/stress orientation distribution functions SODF by inverting the

measured strain pole distributions )(yhε .

Similar to the ODF (texture), the SODF can be subjected to a Fourier analysis by

using generalized spherical harmonics. However there are three important differences. The

first is that in place of one distribution (ODF), six SODF’s are analyzed simultaneously. The

components of the strain or of the stress tensor in the sample or in the crystal reference

system can be used for analysis. The second difference concerns the invariance to the crystal

and the sample symmetry operations. The ODF is invariant to both, crystal and sample

symmetry operations. By contrast, the six SODF in the sample reference system are invariant

to the crystal symmetry operations but they transform similarly to equations (16) if the

sample reference system is replaced by an equivalent one. Inversely, SODF in the crystal

reference system transform like (16) if an equivalent one replaces this system and remain

28

invariant to any rotation of the sample reference system. Consequently, for the spherical

harmonics coefficients of SODF one expects selection rules different from those of ODF.

Finally, in the average over the crystallites in reflection (45) the products of SODF with ODF

are integrated, which in comparison with the average for texture entails a supplementary

difficulty.

In literature three different approaches were reported based on the spherical

harmonics representation of SODF: by Wang et al. (1999, 2001 and 2003), by Behnken

(2000) and by Popa and Balzar (2001). Wang et al. represented by spherical harmonics the

stress tensor in the sample reference system )(gsi . Consequently the harmonic coefficients of

0=l are the averaged stresses is , but to calculate the averaged strains ie the coefficients

with 4,2,0=l are necessary. Behnken proposed to expand in spherical harmonics both )(gei

and )(gsi , independently. In this case ie and is are the coefficients with 0=l of the two

series but the volume of calculations by least square to find the harmonic coefficients is

higher. Both, Wang and Behnken used (45’) to calculate the strain pole distribution )(yhε .

When the calculation starts from the harmonic series of )(gsi , the single crystal compliances

in the sample reference system appear in (45’) as supplementary factors to SODF and ODF.

Behnken performs the integrals (45’) numerically. Wang et al. used the spherical harmonics

representation of ODF and the Clebsch-Gordon coefficients to express the product of the

SODF, ODF and the single crystal compliances in a series that is further integrated like the

ODF for texture. Both Wang and Behnken considered only the case of the cubic crystal

symmetry and orthorhombic sample symmetry and constructed the corresponding

symmetrized spherical harmonics according to the invariance and non-invariance properties

in the Euler space. The third approach reported by Popa and Balzar (2001) is similar to those

of Wang and Behnken, but with an important distinction that makes the problem of

29

determination of the strain tensor equivalent to the texture problem and significantly

simplifies the mathematical formalism. This approach, described below, is extended to any

sample and crystal symmetry and is appropriate for implementation in the Rietveld method.

Lutterotti implemented it in his Rietveld program MAUD (Lutterotti, 1999) and Chanteigner

integrated in a system of “Combined Analysis” described in the book with the same title

(Chateigner, 2010).

7.1. The strain expansion in generalized spherical harmonics. In the approach by Popa and

Balzar (2001) the representation by spherical harmonics is performed on the product of

SODF and ODF that is the SODF weighted by texture (WSODF):

),,(),,(),,( 201201201 ϕϕϕϕεϕϕτ ΦΦ=Φ fii (82)

In this product the strain tensor components in the crystallite reference system are used for

SODF. With this choice the calculation of the averaged strains and stresses ie and is requires

only the harmonic coefficients of 0=l and 2=l (see 7.3. below). The WSODF’s are

expanded in series of generalized spherical harmonics:

∑ ∑ ∑∞

= −= −=

Φ=Φ0

102201 )exp()()exp(),,(l

l

lm

l

ln

mnl

mnili inPimc ϕϕϕϕτ . (83)

where, if denote 0cosΦ=x , the functions )( 0ΦmnlP are defined as follows:

[ ]mlmlnl

nl

mnmnl

mnmlmn

l

xxdx

d

xxnlml

nlml

ml

ixP

+−−

−

+−−−−−

+−×

+−

−++−

−−=

)1()1(

)1()1()!()!(

)!()!(

)!(2

)1()( 2/)(2/)(

2/1

(84)

The functions mnlP are real for nm+ even and imaginary for nm+ odd and have the

following properties:

)()1()( Φ−=Φ +∗ mnl

nmmnl PP , )()()( , Φ=Φ=Φ −− nm

lmn

lnm

l PPP , (85a,b)

)()1()( Φ−=Φ− −++ mnl

nmlmnl PP π , ∫ ′

∗′ +

=ΦΦΦΦπ

δ0 12

2sin)()( ll

mnl

mnl l

dPP (85c, d)

30

The last equation says that the functions mnlP of different harmonics indices l are orthogonal.

By using the equation (85a) and taking account that iτ are real functions one obtains the

following condition for the complex coefficients mnlc :

∗+−− −= mnil

nmnmil cc )1(, (86)

The integral on the Euler space of (83) gives iiic ετ ==000 , and then the term 0=l (the

ground state) represents the hydrostatic strain/stress state of the isotropic polycrystal

discussed in section 6. The rest of terms represent the deviation of the real strain/stress from

the hydrostatic state of isotropic polycrystal (the perturbation of the ground state). To

calculate the peak shift is used (45) with iτ replaced by (83). Further the integral over the

crystallites in Bragg reflection in (45) is performed taking account of the following equation

from Popa and Balzar (2012):

nn

lm

l

mnl

PinPiml

inPimd

)1()()exp()()exp()]12/(2[

)exp()()exp()2/1(||

102

−⋅ΨΦ−+=

Φ∫

γβ

ϕϕωπyh (87)

In this equation mlP is the adjunct Legendre function defined as follows ( Φ= cosx ):

lml

mlm

l

mlm

l xdx

dx

ml

mll

lxP )1()1(

)!(

)!(

2

12

!2

)1()( 22/2

2/12/1

−−

−+

+−= −

−−

−

.

There is an obvious relation between the functions 0mlP and m

lP :

[ ] )()12/(2)()( 2/100 Φ+=Φ=Φ − ml

mml

ml PliPP . (88)

By substituting (88) into equations (85) one obtains the following properties for mlP :

)()1()( Φ−=Φ −ml

mml PP , )()1()( Φ−=Φ− + m

lmlm

l PP π , (89a,b)

∫ ′′ =ΦΦΦΦπ

δ0

sin)()( llm

lm

l dPP (89c)

31

Accounting for the properties (86) and (89a,b), the terms in the expression of the peak shift

resulted after integration over ω are rearranged to have only positive indices and one obtains:

[ ] even,),()12/(2)()(0

=+=∑∞

=

lIlPl

Ml yhyy hhε (90)

∑=

=6

1

),(),(i

iliiMl tEI yhyh ρ (91)

[ ]∑=

Φ++Φ=l

m

ml

mil

millilil PmBmAPAt

1

00 )(sin)(cos)()()(),( ββ yyyyh , (92)

( ) ),0(,)(sincos)()(1

00 lmPnnPAl

n

nl

mnil

mnill

mil

mil =Ψ++Ψ= ∑

=

γβγααy , (93)

( ) ),1(,)(sincos)()(1

00 lmPnnPBl

n

nl

mnil

mnill

mil

mil =Ψ++Ψ= ∑

=

γδγγγy . (94)

The coefficients mnilα , mn

ilβ , mnilγ , mn

ilδ are obtained from the coefficients mnilc according to the

Table 1 in Popa and Balzar (2012). The equations (90) to (94) are valid for triclinic sample

and crystal symmetries and for a given value of l the total number of coefficients is

2)12(6 +l . If the crystal and sample symmetries are higher than triclinic, the number of

coefficients is reduced, some coefficients being zero and some being correlated.

7.2. The selection rules for all Laue classes. To find the selection rules for all Laue classes

the invariance conditions to rotations are applied to the peak shift weighted by texture

)()( yy hh Pε . As the terms of different l in (90) are independent, the invariance conditions

must be applied to every MlI .

We begin with the selection rules imposed by the crystal symmetry. An fold−r axis

along 3x transforms Φ , β , 1a , 2a as follows: Φ→Φ , r/2πββ +→ ,

)/2sin()/2cos( 211 raraa ππ −→ and )/2cos()/2sin( 212 raraa ππ +→ . By applying the

invariance conditions to (91) one obtains a system of six linear equations:

32

∑=

Φ=+Φ6

1

),,()(),/2,(k

klikil trfrt yy βπβ . (95)

These equations are just the transformations (16) for a particular value of r . Further, if (92)

is substituted into (95) one obtains a system of homogenous equations in milA and milB that has

a non-trivial solution only for certain values of m. If, besides the fold−r axis in 3x , there is

an fold2− axis along 1x , then milA and m

ilB must fulfill supplementary conditions resulted

from the invariance of MlI to the transformations Φ−→Φ π , ββ −→ and

),(),( 3232 aaaa −−→ . The selection rules imposed by the crystal symmetry for the non-cubic

Laue groups are given in the Tables 4 to 7 and 9 to 12 in Popa and Balzar (2001). Note that

coefficients belonging to different strain tensor components are correlated, but in all

correlations are involved only two coefficients. The fold3− axis added on the big diagonal

of mmm and mmm/4 prism to obtain the cubic groups 3m and mm3 introduces

supplementary correlations between the coefficients milA and m

ilB of the orthorhombic and

tetragonal group, respectively. These correlations involve more than two coefficients and are

found by evaluating MlI in terms of the direction cosines ia and setting the condition of

invariance to the transformation ),,( 321 aaa → ),,( 132 aaa . The supplementary correlations

added to mmm and mmm/4 by the cubic fold3− axis are given in the Tables 13 and 14 in

Popa and Balzar (2001).

Concerning the selection rules imposed by the sample symmetry one expects to be

identical to those for the texture of the same sample symmetry. Indeed, the invariance

conditions act directly on (93) and (94) that are identical to the spherical harmonics

coefficients of texture pole distribution (see e. g. Popa, 2008). Hence the selection rules in the

index n for the coefficients mnilα , mn

ilβ , mnilγ and mn

ilδ are those from the Table 2 in Popa and

Balzar (2001).

33

7.3. Determination of the averaged strains and stresses. For the calculation of both ie and

is , only the coefficients mnilα , mn

ilβ , mnilγ and mn

ilδ with 0=l and 2=l are needed. This is easy

to see by combining (83) and (16b) into (31) written for ie and (83), (16d) and (26a) into (31)

written for is . The integrals of the terms with 1=l and 2>l are zero because the elements

of the matrix P can be written as linear combination of spherical harmonics with 0=l and

2=l and these functions are orthogonal. So, keeping from (83) only the terms with 0=l and

2=l , and rearranging to have only positive indices nm, , the following truncated WSODF is

substituted into (31) in place of (83):

∑=

Φ=Φ′25

0201201 ),,(),,(

kkiki Rg ϕϕϕϕτ (96)

Here the functions ),,( 201 ϕϕ ΦkR are linear combinations of terms like

)cos()cos( 012 Φ±± mnlQnm ϕϕ or )cos()sin( 012 Φ±± mn

lQnm ϕϕ , where mnl

mnl PQ = for nm+

even and mnl

mnl iPQ = for nm+ odd. They are listed in Popa and Balzar (2012). The elements

of the matrix g are the harmonic coefficients with 2,0=l . The row i of this matrix is the

following:

=

222

222

212

212

202

222

222

212

212

202

122

122

112

112

102

122

122

112

112

102

022

022

012

012

002

000

,,,,,,,,,,,,

,,,,,,,,,,,,,

iiiiiiiiiiiii

iiiiiiiiiiiiii δγδγγβαβααδγδ

γγβαβααβαβαααg (97)

From the combination of (96) with (16b) or with (16d) and (26a) into (31), it results a number

of 936 integrals on the Euler space of products between the functions ijP and kR . Although

an analytical calculation is possible they were calculated numerically. Only 73 are different

from zero and the macroscopic strain tensor becomes:

)222)(20/1(

)222)(30/2/3(

)2)(30/2/3(

)2)(30/1())(3/1(,

24,619,219,114,49,5

21,616,216,111,46,5

4,34,24,1

1,31,21,10,30,20,121

ggggg

ggggg

ggg

ggggggee

+−++±+−++−

−+

−++++=

m (98a)

34

)222)(15/2/3(

)2)(15/1())(3/1(

21,616,216,111,46,5

1,31,21,10,30,20,13

ggggg

gggggge

+−+++

−+−++= (98b)

)222)(20/1(

)2)(30/2/3(

23,618,218,113,48,5

3,33,23,14

ggggg

ggge

+−++−−+=

(98c)

)222)(20/1(

)2)(30/2/3(

22,617,217,112,47,5

2,32,22,15

ggggg

ggge

+−++−−+=

(98d)

)222)(20/1(

)2)(30/2/3(

25,620,220,115,410,5

5,35,25,16

ggggg

ggge

+−+++−+−=

(98e)

The elements of the averaged stress tensor is have exactly the same expressions (98) only the

matrix g must be replaced by g′ defined as follows:

∑=

=′6

1llkljljk gCg ρ . (99)

7.4. Simplified harmonics representation of the peak shift. When it is not necessary to find

WSODF’s and the average strain and stress tensors is not of interest, one can choose a

different approach that corrects only for the line shifts caused by stress. In this case an

alternative representation for MlI with fewer parameters is possible. To arrive at this

representation, the quantities iE in equation (91) and the angles ),( βΦ in equation (92) are

replaced by the direction cosines ia and one obtains:

∑=

=6

13212 ),,(),(),(

ννν aaajtI l

Ml yhyh (100)

( )∑=

=l

aaajCt lilil

µ

µµ

µ

1321 ,,)(),( yyh (101)

In equation (100) ),(),( yhyh liiil tt ρδνν = , iνδ being Kroneker symbol, while )(yµilC in (101)

are linear combinations of )(ymilA and )(ym

ilB . In both, (100) and (101) the functions

( )321 ,, aaaj lµ are the following monomials of degree l in the variables ),,( 321 aaa :

35

( ) )0,(,,, 321321321321 ≥=++= illl

l lllllaaaaaajµ , (102)

For a given value of the harmonic number l there are lµ monomials. The complete sets of

monomials for evenll == ),6,2( , is given in Table 2.

Substituting (101) into (100) and taking account that llll jjj ′+′ = ,λµν where

],1[ ll ′+∈ µλ one obtains:

∑+

=+=

2

13212, ),,()(),(

l

aaajEI llMl

µ

µµµ yyh (103)

The functions )(ylEµ are linear combinations of )(yµilC and then are linear combinations of

)(ymilA and )(ym

ilB . Consequently, according to (93) and (94) we have:

( )∑=

Ψ++Ψ=l

n

nl

nl

nllll PnnPE

1

00 )(sincos)(2

1)( γζγηη µµµµ y (104)

The equations (103) and (104) stand for the reduced representation of ),(yhMlI ,

alternative to (91) up to (94), with a smaller number of parameters. These equations are valid

for triclinic crystal symmetry and respectively for triclinic sample symmetry. If the crystal

symmetry is higher than triclinic the equation (103) should be replaced by the following:

∑+

=+=

2

13212, ),,()(),(

lM

llMl aaaJEI

µµµ yyh (105)

Here lJ ,µ are homogenous polynomials of degree l in the variables 1a , 2a , 3a , invariant to

the symmetry operations of the crystal Laue group and lM their number. They are derived by

setting these invariance conditions on the function ),( yhMlI given by equation (103). Details

of derivation can be found in the paper by Popa and Balzar (2001). The polynomials for

)6,2(=l are listed in Tables 3 for all Laue classes. For sample symmetry higher than triclinic

the coefficients nlµη and n

lµζ in equation (104) follow the selection rules of the texture with

36

the same sample symmetry. For not-cubic Laue groups these selection rules can be found, for

example, in Table 3 of Popa (1992).

The maximum number 2+lM in equation (105) must be equal or smaller than the total

number of functions milA , m

ilB in (91) and (92), but for crystal symmetry higher than triclinic,

it is frequently much smaller. For example, for the Laue class 3 and 4=l the total number of

milA , m

ilB is 18 but 106 =M . This is important in Rietveld refinement, as the number of

refinable parameters is kept to minimum. On the other hand it is not possible to obtain

WSODF and the average strain and stress tensors from the coefficients nlµη and n

lµζ .

7.5. The implementation in the Rietveld codes. In the practical applications reported up to

now by Behnken (2000) by Wang et al. (2001, 2003) and recently by Balzar et al. (2010) the

least square method was used to fit the calculated peak shifts with the measured peak shifts

determined by individual peak fitting. This procedure presumes a prior determination of the

pole distribution )(yhP . The procedure is time consuming and only a limited number of

peaks can be used because the extraction of position becomes inaccurate for overlapped

peaks. The variant of spherical harmonics analysis of WSODF presented in this chapter is

similar to those of ODF and consequently is suitable for implementation in the Rietveld codes

that would allow using the whole information contained in the diffraction patterns. There are

three possible levels of implementation. The easiest is to implement (90)+(105) for any value

of l . This allows fitting the peak positions shifted by stress, but the average strain and stress

tensors as well as the WSODF’s are not accessible. A mixed implementation with the term

2=l according to (90) + (91), the rest by using (90) + (105), allows to fit the peak positions

and to determine ie and is , but without reconstruction of WSODF’s. Implementation of

(90)+(91) for any value of l allows a full determination of the average and of the

37

intergranular strain and stress tensors. By using the coefficients obtained from the Rietveld

refinement, WSODF’s can be calculated directly from (83) and then )(gei , )(gsi .

7.6. Limitations of the spherical harmonics approach, further possible developments. The

coefficients ikg from the series (90) + (91) refined in a least square program (Rietveld

included) could be unreliable if the number of terms in this truncated series is too large. One

expects to have a large number of terms in three cases: for low sample and crystal symmetry;

if the peak shift is a strongly varying function in its arguments; if the strain/stress state is far

from the ground state. For the approach presented here the ground state is the hydrostatic

strain/stress state of isotropic polycrystal. It seems feasible to derive at least two other

spherical harmonic representations of WSODF having as ground state the Voigt and

respectively the Reuss strain/stress state of isotropic polycrystal. In comparison to Wang et

al. (1999) and Behnken (2000) approaches these new representations will be extended to any

sample and crystal symmetries and implementable in the Rietveld programs. The best

between the hydrostatic, Voigt and Reuss ground state representation can be determined by

trials. Nevertheless, for low crystal and sample symmetries and strongly varying WSODF the

number of parameters may still remain large. A simplified representation similar to

(90)+(105) has a smaller number of parameters but, unfortunately, is not possible to

reconstruct WSDOF from these parameters. Will be then useful to examine the possibility to

find a direct method allowing calculating WSDOF by inverting the strain pole distributions

determined with such simplified representation. This direct method would be similar to the

direct method WIMW (Williams, Imhof, Matthies and Vienel) used in the texture analysis

(see texture chapter) to determine ODF from the texture pole distribution.

38

8. The spherical harmonics approach for strain broadening

8.1. Ignoring the macrostrain variance. The strain diffraction line integral breadth in WPPF

calculated with one of equations (49) is proportional with the square root of the variance of

strains. According to equations (46) this quantity has two components, the microstrain

variance )(yhmV and the macrostrain variance )(yh

MV . Like in the section 7 the strain tensors

in the right side of these functions are defined in the crystallite reference system. Being

determined only by the intergranular strain the macrostrain variance is small if the macro

strain/stress state is not too far from the hydrostatic state. Starting from the fact that the

spherical harmonics series of intergranular strain begins with the harmonic index 2=l it is

easy to show that the macrostrain variance can be put in a series of polynomials in 321 ,, aaa

beginning with the degree 8. Anticipating, the microstrain variance can be represented by a

series of polynomials beginning with the degree 4. The macrostrain variance has a

contribution to the high order terms )8(≥ which in practice are ignored; then we can set:

)()( yy hhmVV ≈ .

8.2. The double dependent anisotropic strain breadth (DDASB). The variance of microstrains

)(yhmV is susceptible to an analysis by generalized spherical harmonics similar to those

developed before for the peak shift, equation (45). The functions to be expanded in

generalized spherical harmonics are the elements of the microstrains correlation matrix

)(gij∆ defined by equation (42), weighted by ODF:

)()()( gfgg ijij ∆=τ (106)

Then, in place of )(giτ defined by (82), one starts from )(gijτ defined by (106) and follow

the derivation from the section 7.1. by replacing the index i with the pair ij . The series

expansion of )(gijτ similar to (83) is substituted into equation (46c) and, ignoring the

macrostrain variance, one obtains:

39

)(),()]12/(2[)()(0

evenlIlPVl

ml −+=∑

∞

=

yhyy hh (107)

∑∑= =

=6

1

6

1

),(),(i j

ijljijiml tEEI yhyh ρρ (108)

[ ]∑=

Φ++Φ=l

m

ml

mijl

mijllijlijl PmBmAPAt

1

00 )(sin)(cos)()()(),( ββ yyyyh , (109)

Expressions similar to (93) and (94) can be written for the coefficients )(ymijlA and )(ym

ijlB .

Although similar, there are important differences between the spherical harmonics

representations of )(giε and of )(gij∆ . Firstly, the symmetrical tensor ij∆ has 21 distinct

elements and iε only 6, and then the number of harmonic coefficients is much greater for the

first one. Secondly, the selection rules of )(ymijlA and )(ym

ijlB due to the crystal symmetry

operations should be different from those of )(ymilA and )(ym

ilB for the problem of the peak

shift. Finally, if the determination of the strain orientation distribution functions )(giε is

many times required, there is no practical interest to determine the functions )(gij∆ . For the

problem of the strain broadening only the dependence on h and y is required and for this

aim an alternative representation of ),(yhmlI with a smaller number of parameters can be

used. In this case the selection rules for )(ymijlA and )(ym

ijlB become useless.

To arrive to the alternative representation one replaces the quantities iE in (108) and

the angles ),( βΦ in (109) by the direction cosines ),,( 321 aaa . After replacement these

equations become:

∑=

=15

13214 ),,(),(),(

ννν aaajtI l

ml yhyh (110)

( )∑=

=l

aaajCt lijlijl

µ

µµ

µ

1321 ,,)(),( yyh (111)

40

The functions ),( yhltν in (110) are some linear combinations of ),(yhijlt while )(yµijlC in

(111) are linear combinations of )(ymijlA and )(ym

ijlB . Substituting (111) into (110) one

obtains:

∑+

=+=

4

13214, ),,()(),(

l

aaajEI llml

µ

µµµ yyh (112)

For crystal symmetries higher than triclinic the monomials in (112) should be replaced by

symmetrized polynomials and we have:

∑+

=+=

4

13214, ),,()(),(

lM

llml aaaJEI

µµµ yyh (113)

The equations (49) + (107) + (113) + (104) describe the dependence of the strain

diffraction line breadth on both crystal and sample directions. These equations are

appropriate for implementation in WPPF programs, Rietveld included, if (107) is truncated to

keep the dependence on both, h and y and to have a reasonable number of parameters;

truncating at 2=l could be a fair choice.

8.3. The ‘classical’ limit of DDASB. If ijij g ∆=∆ )( , where ∫∫∫ ∆=∆ )()( ggfdg ijij , then in

the right side of equation (107) there is only the term of harmonic index 0=l . This condition

is similar to (73) defining the hydrostatic macro strain/stress state. If 0=l the coefficients

(104) are constants and the strain variance depends on sample direction only through the

texture pole distribution. If, moreover, the sample is isotropic, the strain variance becomes

independent on sample direction and we have:

∑=

=4

13214,0 ),,(

M

aaaJVµ

µµηh (114)

This is the phenomenological model of strain broadening anisotropy reported by Popa (1998)

and independently by Stephens (1999). This model is implemented today in all popular

Rietveld programs. In GSAS (Larson and Von Dreele, 1994), to process simultaneously

41

diffraction patterns recorded in multiple sample directions, an independent set of ‘Stephens

model’ parameters is available for every pattern (R. B. Von Dreele, personal

communication). Implementing the DDASB model, a single set of strain breadth parameters

will be necessary to process such multiple patterns.

This work was funded by the Romanian National Authority for Scientific Research through

the CNCSIS Contract PCE 102/2011

42

Table 1. The matrix C for all Laue groups represented by the specific constraints. The

constraints for the Laue group m3 are those of )3( to which is added 015 =C . The number

of independent elastic constants is given in the last column

1 21

)(/2 cm 05646353425241514 ======== CCCCCCCC 13

mmm ( m/2 ), 045362616 ==== CCCC 9

m/4 ( m/2 ), 04536 == CC , 1122 CC = , 1323 CC = , 1626 CC −= , 4455 CC = 7

mmm/4 ( m/4 ), 016 =C 6

3 0453635342616 ====== CCCCCC , 1122 CC = , 1323 CC = , 1424 CC −= ,

1525 CC −= , 1546 CC −= , 4455 CC = , 1456 CC = , 2/)( 121166 CCC −=

7

m3 (3 ), 015 =C 6

Hexag. ( m3 ), 014 =C 5

Cubic ( mmm/4 ), 1213 CC = , 1133 CC = , 4466 CC = 3

Isotropic ( Cubic ), 2/)( 121144 CCC −= 2

Table 2. The monomials ( )321 ,, aaaj lµ for 6,4,2=l

43213

42132

41

23

22

213

32

213

22

31

332

21

232

31

33

221

23

321

32

31

33

31

33

32

42

21

22

41

43

21

23

41

43

22

23

42

5212

51

5313

51

5323

52

63

62

616

23213

22132

21

22

21

23

21

23

22

3212

31

3313

31

3323

32

43

42

414

21313223

22

212

,,,,,,

,,,,,,,,,

,,,,,,,,,,,,:28,6

,,

,,,,,,,,,,,,:15,4

,,,,,:6,2

aaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaaaal

aaaaaaaaa

aaaaaaaaaaaaaaaaaaaaal

aaaaaaaaal

==

==

==

µ

µµ

43

Table 3. The polynomials lJµ for 6,2=l for all Laue groups. In Tables 3a to 3d the list in

brackets should be added to the list for dihedral group to obtain the list for the cyclic group.

Table 3a: Laue groups mmm and m/2

],,,,,[

,,,,,,,,,,:]16[10,6

],,[,,,,,,:]9[6,4

][,,,:]4[3,2

4321

23

321

232

31

32

31

5212

51

23

22

21

43

22

23

42

43

21

23

41

42

21

22

41

63

62

616

2321

3212

31

22

21

23

21

23

22

43

42

414

2123

22

212

aaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaMl

aaaaaaaaaaaaaaaaMl

aaaaaMl

==

==

==

Table 3b: Laue groups mmm/4 and m/4

])(,)[(,)(

,,)(,)(,,:]8[6,6

])[(,,)(,,:]5[4,4

,:2,2

2142

41

2321

22

21

22

21

22

21

23

22

21

43

22

21

23

42

41

63

62

616

2122

21

22

21

23

22

21

43

42

414

23

22

212

aaaaaaaaaaaaa

aaaaaaaaaaaaMl

aaaaaaaaaaaaMl

aaaMl

−−+

+++==

−++==

+==

Table 3c: Laue groups m3 and 3

])3)(3(,)3(,)3)([(

,1515,)3(

)3)((,)(,)(,,)(:]10[7,6

])3[(,)3(,)(,,)(:]5[4,4

,:2,2

2122

21

22

21

331

22

2131

22

21

22

21

62

42

21

22

41

61

332

22

21

3222

21

22

21

43

22

21

23

222

21

63

322

216

3122

2132

22

21

23

22

21

43

222

214

23

22

212

aaaaaaaaaaaaaaaa

aaaaaaaaaa

aaaaaaaaaaaaaaaMl

aaaaaaaaaaaaaaMl

aaaMl

−−−−+

−+−−

−++++==

−−++==

+==

Table 3d: Laue groups mmm/6 and m/6

])3)(3[(,1515

,)(,)(,,)(:]6[5,6

)(,,)(:3,4

,:2,2

2122

21

22

21

62

42

21

22

41

61

43

22

21

23

222

21

63

322

216

23

22

21

43

222

214

23

22

212

aaaaaaaaaaaa

aaaaaaaaaMl

aaaaaaMl

aaaMl

−−−+−

+++==

++==

+==

Table 3e: Laue groups mm3 and 3m

22

43

21

42

23

41

21

43

23

42

22

41

22

43

21

42

23

41

21

43

23

42

22

41

23

22

21

63

62

616

22

21

23

21

23

22

43

42

414

2

,:3

:3

,,:]4[3,6

,:2,4

1:1,2

aaaaaaaaaaaam

aaaaaaaaaaaamm

aaaaaaMl

aaaaaaaaaMl

Ml

++++

+++++

++==

++++==

==

44

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