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. 3.3, pp. 267-268

Section 3.3.5.2. Microstrain broadening

R. B. Von Dreelea*

aAdvanced Photon Source, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439–4814, USA
Correspondence e-mail: vondreele@anl.gov

3.3.5.2. Microstrain broadening

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The existence of imperfections (e.g. deformation faults) within the crystal lattice produces local distortions of the lattice and thus a broadening of the points in reciprocal space. To a first approximation these points are broadened proportionally to their distance from the origin, e.g. Δd*/d* = Δd/d ≃ constant.

As for crystallite size, there is normally dispersion in the density of defects and thus the peak shape will be intermediate between a Gaussian and a Lorentzian form, and it is well described by the Voigt, pseudo-Voigt or Pearson VII functions. Usually, the Lorentzian form dominates this type of broadening and it is the most common form of sample broadening in powder diffraction. It usually arises because of defects introduced during sample preparation (especially during grinding). The Lorentzian width contribution from microstrain broadening is[{\Gamma }_{sL}={{180}\over{\pi }}s\tan\theta, \eqno(3.3.28)]where s is the dimensionless microstrain; it is frequently multiplied by 106. A similar expression for neutron TOF is[{\Gamma }_{sL}=Csd,\eqno(3.3.29)]where C is defined by equation (3.3.21)[link].

In many cases, the microstrain broadening is not isotropic; presumably this is a consequence of the interaction between the defects and the elastic properties of the crystals. A phenomenological description of these effects by Popa (1998[link]) and Stephens (1999[link]) is obtained by considering the variance of[{{1}\over{{d}^{2}}}={M}_{hkl}= {\alpha }_{1}{h}^{2}+{\alpha }_{2}{k}^{2}+{\alpha }_{3}{l}^{2}+{\alpha }_{4}kl+{\alpha }_{5}hl+{\alpha }_{6}hk \eqno(3.3.30)]with respect to each of the coefficients αi.[{\Gamma }_{sL}^{2}=\sum\limits_{i,j}{S}_{ij}{{\partial M}\over{{\partial \alpha }_{i}}}{{\partial M}\over{{\partial \alpha }_{j}}}\eqno(3.3.31)]where[{{\partial M}\over{{\partial \alpha }_{i}}}{{\partial M}\over{{\partial \alpha }_{j}}}=\left[\matrix{{h}^{4}& {h}^{2}{k}^{2}& {h}^{2}{l}^{2}& {h}^{2}kl& {h}^{3}l& {h}^{3}{k}\cr {h}^{2}{k}^{2}& {k}^{4}& {k}^{2}l^2 &{ k}^{3}l& {hk}^{2}l& {hk}^{3}\cr {h}^{2}l^2& {k}^{2}l^2& l^{4} &{kl}^{3}& h{l}^{3}& hk{l}^{2}\cr {h}^{2}kl& {k}^{3}l& k{l}^{3}& k^{2}l^2& h{k}l^{2}& hk^2{l}\cr {h}^{3}l& h{k}^{2}l& h{l}^{3} &{hkl}^{2}& h^{2}l^2& h^2{kl}\cr h^{3}k& {hk}^{3}& {hkl}^{2}& h{k}^{2}l& {h}^{2}kl& {h}^{2}{k}^{2}}\right].\eqno(3.3.32)]

Examination of this sum for the triclinic case collects terms to give[\eqalignno{{\Gamma }_{sL}^{2}&={S}_{400}{h}^{4}+{S}_{040}{k}^{4}+{S}_{004}{l}^{4}+3({S}_{220}{h}^{2}{k}^{2}+{S}_{202}{h}^{2}{l}^{2}+{S}_{022}{k}^{2}{l}^{2})&\cr &\quad +2({S}_{310}{h}^{3}k+{S}_{103}h{l}^{3}+{S}_{031}{k}^{3}l+{S}_{130}h{k}^{3}+{S}_{301}{h}^{3}l+{S}_{013}k{l}^{3})&\cr &\quad +4({S}_{211}{h}^{2}kl+{S}_{121}h{k}^{2}l+{S}_{112}hk{l}^{2})&(3.3.33)\cr}]with 15 coefficients [{S}_{hkl}]. The subscript hkl in Shkl refers to the powers used for [h,k,l] in equations (3.3.33)[link]–(3.3.44)[link].

For Laue symmetries other than triclinic, there are restrictions on the allowed [{S}_{hkl}] terms and, as a practical matter, additional equivalences from symmetry-forced reflection overlaps for trigonal and tetragonal Laue symmetries.

Monoclinic (2/m, b axis unique; others similar, nine coefficients):[\eqalignno{{\Gamma }_{sL}^{2}&={S}_{400}{h}^{4}+{S}_{040}{k}^{4}+{S}_{004}{l}^{4}+3{S}_{202}{h}^{2}{l}^{2}&\cr &\quad +3({S}_{220}{h}^{2}{k}^{2}+{S}_{022}{k}^{2}{l}^{2})+2({S}_{301}{h}^{3}l+{S}_{103}h{l}^{3})&\cr &\quad +4{S}_{121}h{k}^{2}l.&(3.3.34)}]

Orthorhombic (mmm, six coefficients):[{\Gamma }_{sL}^{2}={S}_{400}{h}^{4}+{S}_{040}{k}^{4}+{S}_{004}{l}^{4}+3({S}_{220}{h}^{2}{k}^{2}+{S}_{202}{h}^{2}{l}^{2}+{S}_{022}{k}^{2}{l}^{2}).\eqno(3.3.35)]

Tetragonal (4/m, five coefficients):[\eqalignno{{\Gamma }_{sL}^{2}&=S_{400}({h}^{4}+{k}^{4})+{S}_{004}{l}^{4}+3{S}_{220}{h}^{2}{k}^{2}&\cr &\quad +3{S}_{202}({h}^{2}{l}^{2}+{k}^{2}{l}^{2})+2{S}_{310}({h}^{3}k-h{k}^{3}).&(3.3.36)\cr}]

The last coefficient (S310) cannot normally be determined owing to exact reflection overlaps. Thus, equation (3.3.37)[link] is normally used for both 4/m and 4/mmm Laue symmetries:

Tetragonal (4/mmm, four coefficients):[{{\Gamma }_{sL}^{2}=S}_{400}({h}^{4}+{k}^{4})+{S}_{004}{l}^{4}+3{S}_{220}{h}^{2}{k}^{2}+3{S}_{202}({h}^{2}{l}^{2}+{k}^{2}{l}^{2}).\eqno(3.3.37)]

Trigonal ([ \overline{3}], rhombohedral setting, five coefficients):[\eqalignno{{\Gamma }_{sL}^{2}&={S}_{400}({h}^{4}+{k}^{4}+{l}^{4})+3{S}_{220}({h}^{2}{k}^{2}+{h}^{2}{l}^{2}+{k}^{2}{l}^{2})&\cr &\quad +2{S}_{310}({h}^{3}k+{k}^{3}l+h{l}^{3}) +2{S}_{130}({h}^{3}l+k{l}^{3}+h{l}^{3})&\cr &\quad +4{S}_{211}({h}^{2}kl+h{k}^{2}l+hk{l}^{2}).&(3.3.38)\cr}]

The pair of coefficients S310 and S130 cannot normally be independently determined owing to exact reflection overlaps. Thus, equation (3.3.39)[link] is normally used for both rhombohedral symmetries:

Trigonal ([ \overline{3}m], rhombohedral setting, four coefficients):[\eqalignno{{\Gamma }_{sL}^{2}&={S}_{400}({h}^{4}+{k}^{4}+{l}^{4})+3{S}_{220}({h}^{2}{k}^{2}+{h}^{2}{l}^{2}+{k}^{2}{l}^{2})&\cr &\quad +2{S}_{310}({h}^{3}k+{k}^{3}l+h{l}^{3}+{h}^{3}l+k{l}^{3}+h{l}^{3})&\cr &\quad +4{S}_{211}({h}^{2}kl+h{k}^{2}l+hk{l}^{2}).&(3.3.39)\cr}]

Trigonal ([ \overline{3}], five coefficients):[\eqalignno{{\Gamma }_{sL}^{2}&={S}_{400}({h}^{4}+{k}^{4}+2{h}^{3}k+2h{k}^{3}+3{h}^{2}{k}^{2})+{S}_{004}{l}^{4}&\cr &\quad +3{S}_{202}({h}^{2}{l}^{2}+{k}^{2}{l}^{2}+hk{l}^{2}) +{S}_{301}(2{h}^{3}l-2{k}^{3}l-6h{k}^{2}l)&\cr &\quad +4{S}_{211}({h}^{2}kl+h{k}^{2}l).&(3.3.40)\cr}]

The coefficient S301 cannot normally be independently determined owing to exact reflection overlaps. Thus, equation (3.3.42)[link] is normally used for [ \overline{3}] Laue symmetry.

Trigonal ([ \overline{3}m1], four coefficients):[\eqalignno{{\Gamma }_{sL}^{2}&={S}_{400}({h}^{4}+{k}^{4}+2{h}^{3}k+2h{k}^{3}+3{h}^{2}{k}^{2})+{S}_{004}{l}^{4}&\cr &\quad +3{S}_{202}({h}^{2}{l}^{2}+{k}^{2}{l}^{2}+hk{l}^{2})&\cr &\quad +{S}_{301}(3{h}^{2}kl-3h{k}^{2}l+2{h}^{3}l-2{k}^{3}l).& (3.3.41)\cr}]

The coefficient S301 cannot normally be independently determined due to exact reflection overlaps. Thus, equation (3.3.43)[link] is normally used for [ \overline{3}m1] Laue symmetry.

Trigonal ([ \overline{3}1m], four coefficients):[\eqalignno{{\Gamma }_{sL}^{2}&={S}_{400}({h}^{4}+{k}^{4}+2{h}^{3}k+2h{k}^{3}+3{h}^{2}{k}^{2})+{S}_{004}{l}^{4}&\cr &\quad +3{S}_{202}({h}^{2}{l}^{2}+{k}^{2}{l}^{2}+hk{l}^{2}) +4{S}_{211}({h}^{2}kl+h{k}^{2}l).&\cr &&(3.3.42)\cr}]

Hexagonal (6/m and 6/mmm, three coefficients):[\eqalignno{{\Gamma }_{sL}^{2}&={S}_{400}({h}^{4}+{k}^{4}+2{h}^{3}k+2h{k}^{3}+3{h}^{2}{k}^{2})+{S}_{004}{l}^{4}&\cr &\quad +3{S}_{202}({h}^{2}{l}^{2}+{k}^{2}{l}^{2}+hk{l}^{2}).&(3.3.43)\cr}]

Cubic ([m\bar 3] and [m\bar3m], two coefficients):[{\Gamma }_{sL}^{2}={S}_{400}({h}^{4}+{k}^{4}+{l}^{4})+3{S}_{220}({h}^{2}{k}^{2}+{h}^{2}{l}^{2}+{k}^{2}{l}^{2}).\eqno(3.3.44)]

These equations can be used with the refined values of the coefficients to produce a surface representing the extent of the microstrain in reciprocal space. The surface resulting from Stephens' (1999[link]) analysis of powder diffraction data from sodium parahydroxybenzoate is shown Fig. 3.3.4[link]. At the present time, the connection between the elastic properties and defects with these microstrain surface models is unclear. Some aspects of this for cubic and hexagonal systems are discussed in Chapter 5.1[link] .

[Figure 3.3.4]

Figure 3.3.4 | top | pdf |

Microstrain surface for sodium parahydroxybenzoate multiplied by 106.

References

Finger, L. W., Cox, D. E. & Jephcoat, A. P. (1994). A correction for powder diffraction peak asymmetry due to axial divergence. J. Appl. Cryst. 27, 892–900.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
Stephens, P. W. (1999). Phenomenological model of anisotropic peak broadening in powder diffraction. J. Appl. Cryst. 32, 281–289.Google Scholar
Von Dreele, R. B., Jorgensen, J. D. & Windsor, C. G. (1982). Rietveld refinement with spallation neutron powder diffraction data. J. Appl. Cryst. 15, 581–589.Google Scholar








































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