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. 3.2, pp. 259-260

Section 3.2.3.4. Anisotropic strain broadening

P. W. Stephensa*

aDepartment of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794–3800, USA
Correspondence e-mail: peter.stephens@stonybrook.edu

3.2.3.4. Anisotropic strain broadening

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Strain broadening is often observed not to be monotonically dependent on the magnitude of the d-spacing, but to have a more complicated dependence on the direction of the diffraction vector G. An illustration of the effect in cubic Rb3C60 is shown in Fig. 3.2.5[link] (Stephens, 1999[link]). The lower part of the figure shows an apparently irregular trend of diffraction peak widths on diffraction angle 2θ.

[Figure 3.2.5]

Figure 3.2.5 | top | pdf |

Diffraction peak width versus diffraction angle from a powder pattern of face-centred cubic Rb3C60 (Stephens, 1999[link]). The inset at the top shows a limited range of the data, fitted by a model with two different peak widths (upper trace) and with the widths constrained to be equal (lower trace).

Whatever the nature of random internal stresses and strains, it can be argued on general grounds that strain broadening depends on a combination of fourth-rank tensors. On the assumption that the distribution of random strains is Gaussian, each diffraction peak has Gaussian shape, and the variance of the inverse d-spacing squared is a quartic form in the reflection indices (hkl). This can be expressed as[\sigma^2=\langle d_{hkl}^{-4} \rangle -\langle d_{hkl}^{-2}\rangle^{-2} = \textstyle\sum\limits_{HKL} S_{HKL} h^H k^K l^L,\eqno(3.2.17)]where the sum is over terms with [H+K+L=4]. This leads to a contribution to the width of a diffraction peak proportional to [\Gamma_{2 \theta} = d^2\sqrt{\sigma^2}  \tan \theta] for angle-dispersive measurements, and [\Gamma_t = t d^2 \sqrt{\sigma^2}] for time-of-flight neutron measurements. Strain broadening in real samples frequently leads to peak shapes that are closer to Lorentzian than Gaussian, and the justification of this expression breaks down because the second moment of the distribution of d−2 is then not defined. Nevertheless, this method is often successful at modelling the phenomenology of anisotropic peak broadening.

The parameters [S_{HKL}] arise from the nature of the strain and the values of elastic constants for any particular sample. Typically, they must therefore be regarded as phenomenological, and can be adjusted in fitting a model to observed data. They are constrained by the symmetry of the Laue group of the sample, which leads to restrictions on the [S_{HKL}] terms for the various Laue classes, listed in Table 3.2.1[link] (Popa, 1998[link]).

Table 3.2.1| top | pdf |
Restrictions and reflections of anisotropic strain parameters for the various Laue classes

[\bar 3, \bar 3m1, \bar31m]: hexagonal indices.

Class[\langle1/d^2\rangle^2]
[\bar{1}] [S_{400} h^4 + S_{040}k^4 + S_{004}l^4 + S_{220}h^2k^2 + S_{202}h^2l^2] + [S_{022}k^2l^2 + S_{310}h^3k + S_{130}hl^3 + S_{301}h^3l + S_{103}hl^3] + [S_{031}k^3l + S_{013}kl^3 + S_{211}h^2kl + S_{121}hk^2l + S_{112}hkl^2]
2/m (b-axis unique) [S_{400} h^4 + S_{040}k^4 + S_{004}l^4 + S_{220}h^2k^2 + S_{202}h^2l^2] + [S_{022}k^2l^2 + S_{301}h^3l + S_{103}hl^3 + S_{121}hk^2l]
2/mmm [S_{400} h^4 + S_{040}k^4 + S_{004}l^4 + S_{220}h^2k^2 + S_{202}h^2l^2 + S_{022}k^2l^2]
4/m [S_{400} (h^4 + k^4) + S_{004} l^4 + S_{220}h^2k^2 + S_{202} (h^2 + k^2) l^2] + [S_{310} (h^3k - h k^3)]
4/mmm [S_{400} (h^4 + k^4) + S_{004} l^4 + S_{220}h^2k^2 + S_{202} (h^2 + k^2) l^2 ]
[\bar{3}] [S_{400} (h^2 + k^2 + hk)^2 + S_{202} (h^2 + k^2 + hk) l^2 + S_{004} l^4] + [S_{211}(h^3 - k^3 + 3 h^2k)l + S_{121}(-h^3 + k^3 + 3 hk^2)l]
[\bar{3}m1] [S_{400} (h^2 + k^2 + hk)^2 + S_{202} (h^2 + k^2 + hk) l^2 + S_{004} l^4] + [S_{301}(2 h^3 +3 h^2 k -3 h k^2 -2 k^3)l]
[\bar{3}1m] [S_{400} (h^2 + k^2 + hk)^2 + S_{202} (h^2 + k^2 + hk) l^2 + S_{004} l^4] + [S_{211}(h^2 k + h k^2)l]
Hexagonal [S_{400} (h^2 + k^2 + hk)^2 + S_{202} (h^2 + k^2 + hk) l^2 + S_{004} l^4]
Cubic [S_{400} (h^4 + k^4 + l^4) + S_{220} (h^2k^2 + h^2 l^2 + k^2 l^2)]

References

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








































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