Tables for
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 Dislocations

M. Leonia*

aDepartment of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy
Correspondence e-mail: Dislocations

| top | pdf |

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.


Armstrong, N., Leoni, M. & Scardi, P. (2006). Considerations concerning Wilkens' theory of dislocation line-broadening. Z. Kristallogr. Suppl. 23, 81–86.Google Scholar
Berkum, J. G. M. van (1994). Strain Fields in Crystalline Materials. PhD thesis, Technische Universiteit Delft, Delft, The Netherlands.Google Scholar
Dragomir, I. C. & Ungár, T. (2002). Contrast factors of dislocations in the hexagonal crystal system. J. Appl. Cryst. 35, 556–564.Google Scholar
Groma, I., Ungár, T. & Wilkens, M. (1988). Asymmetric X-ray line broadening of plastically deformed crystals. I. Theory. J. Appl. Cryst. 21, 47–54.Google Scholar
Kaganer, V. M. & Sabelfeld, K. K. (2010). X-ray diffraction peaks from correlated dislocations: Monte Carlo study of dislocation screening. Acta Cryst. A66, 703–716.Google Scholar
Kamminga, J.-D. & Delhez, R. (2000). Calculation of diffraction line profiles from specimens with dislocations. A comparison of analytical models with computer simulations. J. Appl. Cryst. 33, 1122–1127.Google Scholar
Klimanek, P. & Kuzel, R. (1988). X-ray diffraction line broadening due to dislocations in non-cubic materials. I. General considerations and the case of elastic isotropy applied to hexagonal crystals. J. Appl. Cryst. 21, 59–66.Google Scholar
Krivoglaz, M. A. (1969). Theory of X-ray and Thermal Neutron Scattering by Real Crystals. New York: Plenum Press.Google Scholar
Krivoglaz, M. A., Martynenko, O. V. & Ryaboshapka, K. P. (1983). Influence of correlation in position of dislocations on X-ray diffraction by deformed crystals. Phys. Met. Metall. 55, 1–12.Google Scholar
Krivoglaz, M. A. & Ryaboshapka, K. P. (1963). Theory of X-ray scattering by crystals containing dislocations. Screw and edge dislocations randomly distributed throughout the crystal. Phys. Met. Metall. 15, 14–26.Google Scholar
Kuzel, R. & Klimanek, P. (1989). X-ray diffraction line broadening due to dislocations in non-cubic crystalline materials. III. Experimental results for plastically deformed zirconium. J. Appl. Cryst. 22, 299–307.Google Scholar
Leoni, M., Martinez-Garcia, J. & Scardi, P. (2007). Dislocation effects in powder diffraction. J. Appl. Cryst. 40, 719–724.Google Scholar
Martinez-Garcia, J., Leoni, M. & Scardi, P. (2007). Analytical expression for the dislocation contrast factor of the <001>{100} cubic slip-system: Application to Cu2O. Phys. Rev. B, 76, 174117.Google Scholar
Martinez-Garcia, J., Leoni, M. & Scardi, P. (2008). Analytical contrast factor of dislocations along orthogonal diad axes. Philos. Mag. Lett. 88, 443–451.Google Scholar
Martinez-Garcia, J., Leoni, M. & Scardi, P. (2009). A general approach for determining the diffraction contrast factor of straight-line dislocations. Acta Cryst. A65, 109–119.Google Scholar
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.Google Scholar
Scardi, P. & Leoni, M. (1999). Fourier modelling of the anisotropic line broadening of X-ray diffraction profiles due to line and plane lattice defects. J. Appl. Cryst. 32, 671–682.Google Scholar
Scardi, P. & Leoni, M. (2004). Whole powder pattern modelling: theory and application. In Diffraction Analysis of the Microstructure of Materials, edited by E. J. Mittemeijer & P. Scardi, pp. 51–91. Berlin: Springer-Verlag.Google Scholar
Stephens, P. W. (1999). Phenomenological model of anisotropic peak broadening in powder diffraction. J. Appl. Cryst. 32, 281–289.Google Scholar
Stokes, A. R. & Wilson, A. J. C. (1944). The diffraction of X-rays by distorted crystal aggregates – I. Proc. Phys. Soc. 56, 174–181.Google Scholar
Ungár, T. (2001). Dislocation densities, arrangements and character from X-ray diffraction experiments. Mater. Sci. Eng. A Struct. Mater. 309–310, 14–22.Google Scholar
Ungár, T., Dragomir, I., Révész, Á. & Borbély, A. (1999). The contrast factors of dislocations in cubic crystals: the dislocation model of strain anisotropy in practice. J. Appl. Cryst. 32, 992–1002.Google Scholar
Ungár, T. & Tichy, G. (1999). The effect of dislocation contrast on X-ray line profiles in untextured polycrystals. Phys. Stat. Solidi A Appl. Res. 171, 425–434.Google Scholar
Wilkens, M. (1970a). The determination of density and distribution of dislocations in deformed single crystals from broadened X-ray diffraction profiles. Phys. Stat. Solidi A Appl. Res. 2, 359–370.Google Scholar
Wilkens, M. (1970b). Fundamental Aspects of Dislocation Theory, edited by J. A. Simmons, R. de Wit & R. Bullough, Vol. II, pp. 1195–1221. Washington DC: National Institute of Standards and Technology.Google Scholar
Wilkens, M. (1987). X-ray line broadening and mean square strains of straight dislocations in elastically anisotropic crystals of cubic symmetry. Phys. Stat. Solidi A Appl. Res. 104, K1–K6.Google Scholar

to end of page
to top of page