International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 3.6, pp. 293-295

Section 3.6.2.6.6. Dislocations

M. Leonia*

aDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
Correspondence e-mail: Matteo.Leoni@unitn.it

3.6.2.6.6. Dislocations

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Dislocations are often the main source of strain broadening. The magnitude of this broadening depends not only on the elastic anisotropy of the material, but also on the relative orientation of the Burgers and diffraction vectors with respect to the dislocation line (Wilkens, 1970a[link],b[link]). This problem was analysed in the 1960s by Krivoglaz and Ryaboshapka (Krivoglaz & Ryaboshapka, 1963[link]; Krivoglaz, 1969[link]) and then subsequently reprised and completed by Wilkens (1970a[link],b[link]). Further elements have been added to put it into the present form (see, for example, Krivoglaz et al., 1983[link]; Groma et al., 1988[link]; Klimanek & Kuzel, 1988[link]; van Berkum, 1994[link]; Kamminga & Delhez, 2000[link]). For the purpose of WPPM, the distortion Fourier coefficients caused by dislocations can be written as[A_{\{ hkl \} }^D(L) = \exp [-\textstyle{1 \over 2} \pi b^2{\overline C}_{ \{ hkl \}} \rho d_{\{ hkl \}}^{*^2} L^2 f(L/R_{e}')], \eqno (3.6.34)]where b is the modulus of the Burgers vector, [{\overline C}_{ \{ hkl \} }] is the so-called average contrast factor of the dislocations, ρ is the density of the dislocations and [R_{e}'] is an effective outer cutoff radius. Only the low-L trend of equation (3.6.34)[link] is well reproduced by Wilkens' theory: a decaying function [f(L/R_{e}')] has thus been introduced to guarantee a proper convergence to zero of the Fourier coefficients for increasing L. Actually, the function [f^*(\eta)] is mostly quoted in place of [(L/R'_e)], where [\eta = (e^{-1/4}/2)L/R_{e}']: the multiplicative term can however be dropped, considering that the cutoff radius is an effective value [some discussion of the meaning of the f and [f^*] functions and of the effective cutoff radius can be found in Scardi & Leoni (2004[link]), Armstrong et al. (2006[link]) and Kaganer & Sabelfeld, 2010[link])].

The most complete definition of [f^*(\eta)] is from Wilkens (1970a[link],b[link]):[\eqalignno {f^*(\eta) &= - \ln \eta + {7 \over 4} - \ln 2 + {{256} \over {45\pi \eta }} + {2 \over \pi }\left(1 - {1 \over {4\eta ^2}} \right) {\int \limits_0^\eta} {{\arcsin y} \over y}\,{\rm d}y \cr &\ \quad -\ {1 \over \pi} \left({{769} \over {180\eta}} + {{41} \over {90}} \eta + {{\eta ^3} \over {45}} \right) (1 - \eta ^2)^{1/2} \cr &\ \quad -\ {1 \over \pi } \left({{11} \over {12\eta^2}} + {7 \over 2} + {{\eta ^2} \over 3} \right) \arcsin \eta + {{\eta ^2} \over 6}, & \eta \le 1, \cr && (3.6.35)}][\eqalignno{f^*(\eta) &= {{256} \over {45\pi \eta }} - \left({{11} \over {24}} + {1 \over 4}\ln 2\eta \right) {1 \over {\eta^2}}, & \eta \ge 1. \cr &&(3.6.36)}]For η < 1, the integral in (3.6.35)[link] can be calculated in terms of special functions as[\eqalignno {&{\int \limits_0^\eta} {{\arcsin y} \over y}\, {\rm d}y &\cr&\quad= {i \over {12}} \bigg\{ \pi^2 - 6 \arcsin^2 \eta - 12i\ln \left [2\eta \left(\eta - i\sqrt {1 - {\eta ^2}} \right)\right] \arcsin \eta &\cr&\quad\quad- 6{\rm Li}_2 \left (1 - 2\eta^2 + 2i\eta \sqrt {1 - {\eta ^2}} \right) \bigg\} \cr &\quad= \ln (2\eta)\arcsin \eta + {1 \over 2}{\rm Im}\left[{\rm Li}_2 \left(1 - 2\eta^2 + 2i\eta \sqrt {1 - {\eta ^2}} \right) \right] \cr &\quad= \ln (2\eta)\arcsin \eta + {1 \over 2}{\rm Cl}_2 (2\arcsin \eta), & (3.6.37)}]where Li2(z) and Cl2(z) are the dilogarithm function (Spence's function) and the Clausen integral, respectively:[{\rm Li}_2(z) = \textstyle \sum\limits_{k = 1}^\infty z^k/k^2, \eqno (3.6.38)][{\rm Cl}_2(z) = \textstyle\sum\limits_{k = 1}^\infty \sin (kz)/z^2 = - \textstyle \int \limits_0^x \ln [2\sin (t/2)]\,{\rm d}t. \eqno (3.6.39)]The approximation proposed by van Berkum (1994[link]) for (3.6.35)[link] and (3.6.36)[link],[f^*(\eta) = \openup2pt\cases {\displaystyle - \ln \eta + {7 \over 4} - \ln 2 + {{\eta ^2} \over 6} - {{32\eta ^3} \over {225\pi}}, & $\eta \le 1$ \cr \displaystyle{{256} \over {45\pi \eta}} - \left({{11} \over {24}} + {1 \over 4}\ln 2\eta \right){1 \over {\eta ^2}}, & $\eta \ge $1,} \eqno (3.6.40)]should not be employed, as the derivative is discontinuous at η = 1. A simpler approximation, valid over the whole η range, was provided by Kaganer & Sabelfeld (2010[link]):[f^*(\eta) = - \ln \left({\eta \over {\eta _0 + \eta }} \right). \eqno (3.6.41)]With η0 = 2.2, the results of equation (3.6.41)[link] are similar to those of (3.6.35) and (3.6.36).

Together with dislocation density and outer cutoff radius, a parameter traditionally quoted for the dislocations ensemble is Wilkens' dislocation arrangement parameter [M = R_e\sqrt \rho ] (Wilkens, 1970a[link]). By combining the information on dislocation screening and dislocation distance, it gives an idea of the interaction of dislocations (strength of the dipole character; Ungár, 2001[link]). A value close to or below unity indicates highly interacting dislocations (for example, dipole configurations or dislocation walls), whereas a large value is typical of a system with randomly dispersed dislocations (weak dipole character).

The anisotropic broadening caused by the presence of dislocations is mainly taken into account by the contrast (or orientation) factor Chkl. The contrast factor depends on the strain field of the dislocation and therefore on the elastic anisotropy and orientation of the scattering vector with respect to the slip system considered. The average of the contrast factor over all equivalent slip systems, [{\overline C}_{hkl}], is often used in the analysis of powders. The averaging is usually performed under the assumption that all equivalent slip systems are equally populated. The calculation of the contrast factor can be lengthy: full details can be found in the literature (Wilkens, 1970a[link],b[link], 1987[link]; Krivoglaz et al., 1983[link]; Kamminga & Delhez, 2000[link]; Groma et al., 1988[link]; Klimanek & Kuzel, 1988[link]; Kuzel & Klimanek, 1989[link]) for the cubic and hexagonal cases. For a generalization, the reader is referred to the recent work of Martinez-Garcia et al. (2007, 2008[link], 2009[link]). It is possible to show that the contrast factor of a given material has the same functional form as the fourth-order invariant of the Laue class (Popa, 1998[link]; Leoni et al., 2007[link]):[\eqalignno {&d_{\{ hkl \}}^4 C_{ \{ hkl \}} &\cr&\quad = E_1 h^4 + E_2 k^4 + E_3 l^4 +\ 2(E_4 h^2 k^2 + E_5 k^2 l^2 + E_6 h^2 l^2) \cr & \quad\quad + 4(E_7 h^3 k + E_8 h^3 l + E_9 k^3 h+ E_{10} k^3 l +\ E_{11} l^3 h + E_{12} l^3 k) \cr & \quad\quad +\ 4(E_{13} h^2 kl + E_{14} k^2 hl + E_{15} l^2 hk). & (3.6.42)}]In the general case, 15 coefficients are thus needed to describe the strain anisotropy effects. Symmetry reduces the number of independent coefficients: for instance, two coefficients survive in the cubic case, and the average contrast factor is (Stokes & Wilson, 1944[link]; Popa, 1998[link]; Scardi & Leoni, 1999[link])[{\overline C}_{ \{ hkl \} } = (A + B H) = A + B {{h^2 k^2 + h^2 l^2 + k^2 l^2} \over {(h^2 + k^2 + l^2)^2}}. \eqno (3.6.43)]The values of A and B can be calculated from the elastic constants and slip system according to the literature (Klimanek & Kuzel, 1988[link]; Kuzel & Klimanek, 1989[link]; Martinez-Garcia et al., 2007[link], 2008[link], 2009[link]). Excluding the case of [{\overline C}_{ \{ h00 \}} = 0], the parameterization [{\overline C}_{ \{ hkl \}} = {\overline C}_{ \{ h00 \}} (1 + qH)] proposed by Ungár & Tichy (1999[link]) can also be used. Some calculated values for cubic and hexagonal materials can be found in Ungár et al. (1999[link]) and Dragomir & Ungár (2002[link]), respectively.

As the calculation of the contrast factor for a dislocation of general character is not trivial, it is customary to evaluate it for the screw and edge case and to refine an effective dislocation character ϕ (Ungár et al., 1999[link]),[\eqalignno{{\overline C}_{ \{ hkl \} } &= [\varphi {\overline C}_{E,\{ hkl \}} + (1 - \varphi){\overline C}_{S,\{ hkl \}}] &\cr&= [\varphi A_E + (1 - \varphi)A_S] + [\varphi B_E + (1 - \varphi)B_S] H, & (3.6.44)}]where the geometric term H is the same as in equation (3.6.43)[link]. Although not completely correct, the approach proposed in equation (3.6.44)[link] allows the case where a mixture of dislocations of varying character are acting on equivalent slip systems to be dealt with. For a proper refinement, however, the active slip systems as well as the contrast factors of the edge and screw dislocations should be known.

It is worth mentioning that the invariant form proposed by Popa (1998[link]) has been reprised by Stephens (1999[link]) to describe the strain-broadening anisotropy, for example, within the Rietveld method: the formula correctly accounts for the relative broadening (i.e. for the anisotropy), but it does not give any information on the actual shape of the profiles. This is the major reason why the Stephens model can be considered as only phenomenological (it captures the trend but not the details): when the source of microstrain broadening is known, we can obtain the functional form of the profile (as proposed, for example, here for dislocations) and the model can become exact.

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