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Remarks on Texture Coefficients of Polycrystals with Improper Crystallite Symmetry

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Abstract

The orientation distribution function (ODF) in classical texture analysis is defined on the rotation group SO(3). For polycrystalline aggregates with crystallite symmetry defined by a crystallographic point group \(G_{\mathrm{cr}}\) which is not a subgroup of SO(3), the improper group \(G_{\mathrm{cr}}\) is routinely replaced by its proper peer (i.e., a subgroup of SO(3)) in the same Laue class. In this note we examine how the texture coefficients obtained from such a practice are related to their counterparts that pertain to the corresponding ODF defined on the orthogonal group O(3) as it should.

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Notes

  1. A survey [7, Sect. 3.5] of circa 127,000 inorganic and 156,000 organic crystals show that a vast majority of them have their symmetries described by improper point groups.

  2. The discussion is similar for the case where the orientation space is \(\text{O(3)}/G_{\mathrm{cr}}\).

  3. See Sect. 2.1. Note that \(T_{f}: \mathscr{M}(X) \rightarrow \mathbb{R}\), as defined by (6), is vaguely continuous for each \(f \in \mathscr{C}(X)\).

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  1. Department of Mathematics, University of Kentucky, Lexington, KY, 40506-0027, USA

    Chi-Sing Man & Ding Zhao

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  1. Chi-Sing Man
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  2. Ding Zhao
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Correspondence to Chi-Sing Man.

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Man, CS., Zhao, D. Remarks on Texture Coefficients of Polycrystals with Improper Crystallite Symmetry. J Elast 138, 111–124 (2020). https://doi.org/10.1007/s10659-019-09732-0

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