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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: [email protected]
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.
2
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
3
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
5
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× .
6
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.
8
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 .
9
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)
10
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
11
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:
12
∫
∫∫
±
±±
−
−−++
=
+=
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
13
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
References
Averbach, A. L. and Warren, B. E. (1949) Interpretation of X-ray patterns of cold-worked
metals, J. Appl. Phys. 20, 885-886.
Balzar, D. , Von Dreele, R. B., Bennett, K. and Ledbetter, H. (1998) Elastic strain tensor by
Rietveld refinement of diffraction measurements, J. Appl. Phys., 84, 4822-4833.
Balzar, D., Popa, N. C., and Vogel, S. (2010). Strain and stress tensors of rolled uranium
plate by Rietveld refinement of TOF neutron-diffraction data,Mat. Sci. Eng. A-
Struc. Mat. Prop. Microstr. Proc. 528, 122-126.
Behnken, H. and Hauk, V. (1986) Berechnung der rontgenographischen
Elastizitetskonstanten (REK) des Vielkristalls aus den Einkristalldaten fur beliebige
Kristallsymmetrie, Z. Metalkde, 77, 620-626.
Behnken, H. (2000) Strain-Function Method for the Direct Evaluation of Intergranular
Strains and Stresses, Phys. Stat. Sol., A177, 401-418.
Chateigner, D. (2010) Combined Analysis, ISTE, U.K. & Wiley, USA.
Christenson, A. L. and Rowland, E. S. (1953), X-ray measurement of residual stress in
hardened high carbon steel, Trans. A.S.M., 45, 638 – 676
Dolle, H. (1979) The Influence of Multiaxial Stress States, Stress Gradients and Elastic
Anisotropy on the Evaluation of (Residual) Stresses by X-rays, J. Appl.
Cryst. 12, 489-501.
Eshelby, J. D. (1957) The Determination of the Elastic Field of an Ellipsoidal Inclusion, and
Related Problems, Proc. Roy. Soc. Lond., A241, 376-396.
Ferrari, M. and Lutterotti, L.(1994) Method for the simultaneous determination of anisotropic
residual stresses and texture by X-ray diffraction, J. Appl. Phys., 76, 7246-7255.
Gnaupel – Herold, T., Brand, P. C. and Prask, H. J. (1998). Calculation of Single-Crystal
Elastic Constants for Cubic Crystal Symmetry from Powder Diffraction Data,
45
J. Appl. Cryst. 31, 929-935.
Hauk, V. (1952) Röntgenographische und mechanische Verformungsmessungen an
Grauguss, Arch. Eisenhüttenwesen, 23, 353–361.
Hauk, V. (1997) Structural and Residual Stress Analysis by Nondestructive Methods
Elsevier Science, Amsterdam.
Hill, R. (1952) The elastic behavior of a crystalline aggregate, Proc. of the
Physical Society A65, 349-35.
Kroner, E. (1958), Berechnung der elastischen konstanten des vielkristalls aus den
konstanten des einkristalls, Z Phys., 151, 504-518.
Langford, J. I. (2004) Line Profile Analysis: A Historical Overview, p. 3 – 13 in
Diffraction Analysis of the Microstructure of Materials, eds. E. J. Mitemeijer and
P. Scardi, Springer, Berlin.
Lutterotti, L., Matthies, S. and Wenk, H.-R. (1999) MAUD (Material Analysis Using
Diffraction) a user friendly Java program for Rietveld texture analysis and more,
National Council of Canada, Ottawa, http://www.ing.unitn.it/~luttero/maud/ .
Noyan, I.C. and Cohen, J.B. (1987) Residual Stress: Measurement by Diffraction and
Interpretation, Springer- Verlag, New York.
Nye, J. F. (1957) Physical Properties of Crystals, University Press, Oxford.
Popa, N. C. (1992) Texture in Rietveld refinement, J. Appl. Cryst., 25, 611-616.
Popa, N. C. (1998) The (hkl) Dependence of Diffraction-Line Broadening Caused by Strain
and Size for all Laue Groups in Rietveld Refinement, J. Appl. Cryst., 31, 176-180.
Popa, N. C. (2000) Diffraction-line shift caused by residual stress in polycrystal for all Laue
groups in classical approximations, J. Appl. Cryst., 33, 103-107.
Popa, N. C. and Balzar, D. (2001) Elastic strain and stress determination by Rietveld
refinement:generalized treatment for textured polycrystals for all Laue classes,
46
J. Appl. Cryst., 34, 187-195.
Popa, N. C. (2008) Microstructural Properties: Texture and Macrostress Efects, chap. 12 in
Powder Diffraction, Theory and Practice, eds. Dinnebier, R. E. and Billinge, S. J. L.,
R.S. C. Publishing, Cambridge.
Popa, N. C. and Balzar, D. (2012) Elastic strain and stress determination by Rietveld
refinement: generalized treatment for textured polycrystals for all Laue classes –
Corrigenda, J. Appl. Cryst., 45, 838 – 839.
Popa, N. C. and Lungu, G. A. (2013) Dependence of the strain diffraction line broadening on
(hkl) and sample direction in textured polycrystals, J. Appl. Cryst., 46, 391 – 395.
Reuss, A. (1929), Berechnung der Fließgrenze von Mischkristallen auf Grund der
Plastizitätsbedingung für Einkristalle, Z. Angew. Math. Mech., 9, 49–58.
Stephens, P. W. (1999) Phenomenological model of anisotropic peak broadening in powder
diffraction, J. Appl. Cryst., 32, 281-289.
Stokes, A. R. and Wilson, A. J. C. (1944) The diffraction of X-ray by distorted crystal, Proc.
Phys. Soc. (London) 56, 174-181.
Van Acker, K., Root, J., Van Houtte, P. and Aernoudt, E. (1996) Neutron diffraction
measurement of the residual stress in the cementite and ferrite phases of cold
drawn steel wires, Acta Mater., 44, 4039-4049.
Voigt, W. (1928) Lehrbuch der Kristallphysik, Teubner Verlag, Berlin-Leipzig.
Von Dreele,R. B. (2004) GSAS Manual
http://www.ccp14.ac.uk/ccp/ccp14/ftp-mirror/gsas/public/gsas/manual/ .
Wang, Y. D., Lin Peng, R. and McGreevy, R., (1999) High anisotropy of
orientation dependent residual stress in austenite of cold rolled stainless steel,
Scripta Materialia 9, 995-1000.
Wang, Y. D., Lin Peng, R. and McGreevy, R., (2001) A novel method for constructing
47
the mean field of grain-orientation-dependent residual stress,
Phylos. Mag. Lett., 81, 153-163.
Wang, Y. D. Wang, X. L. Stoica, A. D., Richardson J. W. and Lin Peng, R. (2003)
Determination of the stress orientation distribution function using pulsed neutron source,
J. Appl. Cryst., 36, 14-22.
Warren, B. E. (1969) X – Ray Diffraction, Addison – Wesley, Reading, MS.
Welzel, U., Ligot, J., Lamparter, P., Vrmeulen, A. C. and Mittemeijer, E. J. (2005)
Stress analysis of polycrystalline thin films and surface regions by X-ray diffraction,
J. Appl. Cryst. 38, 1-29.
Wilson, A. J. C. (1962) X-ray Optics, 2nd ed., London, Methuen.
Wooster, W. A. (1973) Tensors and Group Theory for Physical Properties of Crystals,
Clarendon Press, Oxford.