A note on the relation of anisotropic peak broadening with lattice symmetry in powder diffraction

Fabrykiewicz, P.

doi:10.1107/S2053273325003134

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Volume 81| Part 3| May 2025| Pages 245-247

https://doi.org/10.1107/S2053273325003134

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A note on the relation of anisotropic peak broadening with lattice symmetry in powder diffraction

Piotr Fabrykiewicz ^a,^b ^*

^aInstitute of Crystallography, RWTH Aachen University, Jägerstr. 17-19, 52066 Aachen, Germany, and ^bJülich Centre for Neutron Science at Heinz Maier-Leibnitz Zentrum, Forschungszentrum Jülich GmbH, Lichtenbergstr. 1, 85747 Garching, Germany
^*Correspondence e-mail: [email protected]

Edited by A. Altomare, Institute of Crystallography - CNR, Bari, Italy (Received 4 February 2025; accepted 7 April 2025; online 24 April 2025)

A bridge is established between the Gregorkiewitz & Boschetti [Acta Cryst. (2024), A80, 439–445] and Stephens [J. Appl. Cryst. (1999), 32, 281–289] formalisms of anisotropic peak broadening in powder diffraction. The paper by Gregorkiewitz & Boschetti presented formulas describing position shifts of low-symmetry peaks due to different lattice relaxation schemes. Anisotropic peak broadening caused by lattice relaxation can be parameterized by the variance of slightly dispersed peaks' positions. The calculated variances are compared with formulas from the widely used phenomenological model of anisotropic peak broadening by Stephens. Specific relations between anisotropic peak broadening parameters can be a hint of a possible unresolved peak splitting due to lattice symmetry lowering.

Keywords: Rietveld refinement; pseudosymmetry; microstructures.

Similar articles

The paper by Gregorkiewitz & Boschetti (2024 ) presented formulas for the 1/d²_hkl position of hkl contributions to the same (hkl) peak in a powder diffraction pattern considering six minimal lattice symmetry relaxation schemes: (1) cubic to tetragonal, (2) cubic to rhombohedral, (3) hexagonal to orthorhombic/monoclinic, (3a) hexagonal to orthorhombic, (4) tetragonal to orthorhombic, (5) orthorhombic to monoclinic, (6) monoclinic to anorthic (triclinic). Based on these formulas I calculated the variances $[\sigma^{2}(h,k,l)]$ of the 1/d²_hkl values of the hkl contributions. For all relaxation schemes the variances $[\sigma^{2}(h,k,l)]$ are expressed as fourth-order polynomials in h, k, l indices:

$[\sigma^{2}(h,k,l) = \sum_{HKL}S_{HKL}h^{H}k^{K}l^{L},\eqno(1)]$

with . Symmetry restrictions for each Laue class for S_HKL coefficients were given by Popa (1998 ). A phenomenological model of anisotropic peak broadening (Stephens, 1999 ; Popa, 1998) assumes that each crystallite in a powder sample is in general triclinic and that only average lattice constants over the whole sample fulfil restrictions of a given lattice symmetry. In this approach S_HKL coefficients are linear combinations of covariance matrix elements, which describe a lattice constant distribution in the polycrystalline sample (see Stephens, 1999). For each of the lattice relaxation schemes (1–6) listed above, the barycentre shift $[\Delta\langle 1/d^{2}_{hkl}\rangle]$ , the variance $[\sigma^{2}(h,k,l)]$ of 1/d²_hkl and the relations between S_HKL parameters are given in Table 1. The effective peak widths calculated in a Rietveld refinement depend on the instrumental resolution function, on $[\sigma^{2}(h,k,l)]$ and on possible other effects, e.g. crystallite size effects etc. Without specifying the peakshape function one can say that the total peak width is monotonic with $[\sigma^{2}(h,k,l)]$ .

Table 1
Number of split peaks for general (hkl), values of barycentre shift $[\Delta\langle 1/d^{2}_{hkl}\rangle]$ within linear approximation, variances $[\sigma^{2}(h,k,l)]$ of dispersed 1/d²_hkl positions within quadratic approximation and relations between anisotropic peak broadening parameters S_HKL as defined by Stephens (1999) for all relaxation schemes discussed by Gregorkiewitz & Boschetti (2024)

1. Cubic to tetragonal (3 peaks) $[c = a + \delta a]$

$[\Delta\langle 1/d^2_{hkl}\rangle]$ = $[-{{2}\over{3}}a^{-2}(h^2+k^2+l^2)(\delta a/a)]$

$[\sigma^2(h,k,l)]$ = $[{{8}\over{9}}a^{-4}(h^4 + k^4 + l^4 - h^2 k^2 - h^2l^2 - k^2l^2) (\delta a/a)^2]$

$[[S_{400} = S_{040} = S_{004}]=[-S_{220} = -S_{202} = -S_{022}]]$ = $[{{8}\over{9}}a^{-4} (\delta a/a)^2]$

(HHH), locally narrowest; (H00), locally broadest

2. Cubic to rhombohedral (4 peaks) $[\alpha = ({\pi}/{2}) + \delta \alpha]$

$[\Delta\langle 1/d^2_{hkl}\rangle = 0]$

$[\sigma^2(h,k,l)]$ = $[4a^{-4}(h^2 k^2 + h^2l^2 + k^2l^2)(\delta \alpha)^2]$

$[[S_{220} = S_{202} = S_{022}]]$ = $[4a^{-4}(\delta \alpha)^2]$ and $[[S_{400} = S_{040} = S_{004}] = 0]$

(H00), locally narrowest; (HHH), locally broadest

3. Hexagonal to orthorhombic/monoclinic (6 peaks)

$[b = a+\delta a]$ and $[\gamma = ({2\pi}/{3}) + \delta \gamma]$

$[\Delta\langle 1/d^2_{hkl}\rangle]$ = $[ - {{4}\over{3}}a^{-2}(h^2+hk+k^2)[(\delta a/a) - {{\sqrt{3}}\over{3}}\delta\gamma]]$

$[\sigma^2(h,k,l)]$ = $[{{32}\over{27}} a^{-4} (h^2+hk+k^2)^2 [(\delta a/a)^2 + (\delta \gamma)^2]]$

$[[S_{400} = S_{040} = S_{310}/2 = S_{130}/2 = S_{220}/3]]$ = $[{{32}\over{27}}a^{-4} [(\delta a/a)^2 + (\delta \gamma)^2]]$

$[[S_{202} = S_{022} = S_{112}]=[S_{004}]]$ = $[ [S_{301}/2 = -S_{031}/2 = S_{211}/3 = -S_{121}/3] = 0]$

(00L), locally narrowest; (HKL), locally broadest

3a. Hexagonal to orthorhombic (3 peaks)

$[\cos\gamma = -({a}/{2b})]$ and $[b = a+\delta a]$

$[\Delta\langle 1/d^2_{hkl}\rangle]$ = $[-{{16}\over{9}} a^{-2} (h^2 + h k + k^2)(\delta a/a)]$

$[\sigma^2(h,k,l)]$ = $[{{128}\over{81}}a^{-4}\left(h^{2} + h k + k^{2}\right)^{2} (\delta a/a)^2]$

$[[S_{400} = S_{040} = S_{310}/2 = S_{130}/2 = S_{220}/3]]$ = $[{{128}\over{81}} a^{-4} (\delta a/a)^2]$

$[[S_{202} = S_{022} = S_{112}]=[S_{004}]]$ = $[ [S_{301}/2 = -S_{031}/2 = S_{211}/3 = -S_{121}/3] = 0]$

(00L), locally narrowest; (HKL), locally broadest

4. Tetragonal to orthorhombic (2 peaks) $[b = a+\delta a]$

$[\Delta\langle 1/d^2_{hkl}\rangle]$ = $[-a^{-2}(h^2+k^2)(\delta a/a)]$

$[\sigma^2(h,k,l)]$ = $[a^{-4}(h^2 - k^2)^2 (\delta a/a)^2]$

$[[S_{400} = S_{040}] = [-2 S_{220}]]$ = $[a^{-4} (\delta a/a)^2]$

$[[S_{004}] = [S_{202} = S_{022}] = 0]$

(HHL), locally narrowest; (H0L), locally broadest

5. Orthorhombic to monoclinic (2 peaks) $[ \gamma = ({\pi}/{2}) + \delta \gamma]$

$[\Delta\langle 1/d^2_{hkl}\rangle = 0]$

$[\sigma^2(h,k,l)]$ = $[4a^{-2}b^{-2}h^2k^2 (\delta \gamma)^2]$

$[[S_{220}] = 4a^{-2}b^{-2} (\delta \gamma)^2]$

$[[S_{400}] = [S_{040}]=[S_{004}]]$ = $[[S_{202}] = [S_{022}] = 0]$

(H0L), (0KL), locally narrowest; (HKL), locally broadest

6. Monoclinic to anorthic (triclinic) (2 peaks)

$[\alpha = ({\pi}/{2}) + \delta \alpha]$ and $[\beta = ({\pi}/{2}) + \delta \beta]$

$[\Delta\langle 1/d^2_{hkl}\rangle = 0]$

$[\sigma^2(h,k,l)]$ = $[4l^2c^{-2}\cos^{-4}\gamma [a^{-1}(\delta\beta-\delta\alpha\cos\gamma)h + b^{-1}(\delta\alpha-\delta\beta\cos\gamma)k]^2]$

$[a^2(\delta\beta-\delta\alpha\cos\gamma)^{-2}[S_{202}]]$ = $[b^2(\delta\alpha-\delta\beta\cos\gamma)^{-2}[S_{022}]]$ = $[{{1}\over{2}}ab(\delta\alpha-\delta\beta\cos\gamma)^{-1}(\delta\beta-\delta\alpha\cos\gamma)^{-1}[S_{112}]]$ = $[4c^{-2}\cos^{-4}\gamma]$

$[[S_{400}] = [S_{040}] = [S_{004}]]$ = $[[S_{220}] = [S_{310}] = [S_{130}] = 0]$

(HK0) and (00L), locally narrowest

The anisotropic peak broadening method is often used to improve the Rietveld refinement, but the values of the Stephens parameters S_HKL are not always discussed. The equalities between S_HKL parameters in Table 1 included in square brackets, e.g. $[[S_{400} = S_{040} = S_{004}]]$ , are identical to those given by Stephens (1999). The relations between S_HKL parameters included in two neighbouring brackets, e.g. $[S_{004} = -S_{220}]$ (see cubic to tetragonal in Table 1) are additional restrictions due to the specific lattice relaxation scheme and they were not provided by Gregorkiewitz & Boschetti (2024). The equations in Table 1 form a bridge between the Gregorkiewitz & Boschetti (2024) and Stephens (1999) formalisms, giving the values of the S_HKL parameters in terms of lattice parameter increments. One has to keep in mind that definitions of anisotropic peak broadening parameters used in Rietveld refinement software do not always agree with the definitions from the Stephens (1999) paper. For a quantitative analysis of the S_HKL parameters one should refer to the documentation of the specific Rietveld refinement program in use.

Leineweber (2017 ) showed a relationship between anisotropic peak broadening parameters and the strain tensor within symmetry-related low-symmetry domain states. Applying his general formula [see equation 10 of Leineweber (2017)] to the cubic to tetragonal lattice relaxation scheme leads to the same relationship between S_HKL parameters as given for this scheme in Table 1 [see equation 17 of Leineweber (2017)].

The study of Leineweber (2017) points out that the high-symmetry model with anisotropic peak broadening parameters and the low-symmetry model with isotropic peak broadening frequently yield in Rietveld refinements similar agreement factors. Therefore the choice of the proper model should be made in combination with possibly available additional information. Statistically relevant non-zero values of S_HKL parameters fulfilling the relations given in Table 1 obtained in Rietveld refinement of powder diffraction data can be a signature of a possible unresolved peak splitting due to lattice symmetry lowering. It should be possible to implement an automatic check of this relationship in Rietveld analysis software, which will alert the user to when there are grounds to consider a low-symmetry lattice model. Users could also manually constrain S_HKL parameters to follow equalities given in Table 1.

Additional remark: in the Gregorkiewitz & Boschetti (2024) supporting information I found an error in the equations of barycentre shift for relaxation from cubic to rhombohedral and from orthorhombic to monoclinic. Instead of $[B_{r} = (h^{2}+k^{2}+l^{2})\sin\delta\alpha]$ and $[B_{m} = (h^{2}/a^{2}+k^{2}/b^{2})\sin\delta\gamma]$ , it should be $[B_{r} = 2a^{-2}(h^{2}+k^{2}+l^{2})\sin^{2}\delta\alpha+O(\sin^{3}\delta \alpha)]$ and $[B_{m} = (h^{2}/a^{2}+k^{2}/b^{2})\tan^{2}\delta\gamma]$ , respectively.

Acknowledgements

Thanks are due to Martin Meven (RWTH Aachen University and Forschungszentrum Jülich GmbH), Radosław Przeniosło and Izabela Sosnowska (University of Warsaw) for inspiring discussions. Open access funding enabled and organized by Projekt DEAL.

References

Gregorkiewitz, M. & Boschetti, A. (2024). Acta Cryst. A80, 439–445. CrossRef IUCr Journals Google Scholar
Leineweber, A. (2017). Powder Diffr. 32, S35–S39. Web of Science CrossRef CAS Google Scholar
Popa, N. C. (1998). J. Appl. Cryst. 31, 176–180. Web of Science CrossRef CAS IUCr Journals Google Scholar
Stephens, P. W. (1999). J. Appl. Cryst. 32, 281–289. Web of Science CrossRef CAS IUCr Journals Google Scholar

This is an open-access article distributed under the terms of the Creative Commons Attribution (CC-BY) Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.

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A note on the relation of anisotropic peak broadening with lattice symmetry in powder diffraction

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