International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 29-31

Section 1.1.4.10.7. Reduction of the number of independent components of axial tensors of rank 2

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.4.10.7. Reduction of the number of independent components of axial tensors of rank 2

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It was shown in Section 1.1.4.5.3.2[link] that axial tensors of rank 2 are actually tensors of rank 3 antisymmetric with respect to two indices. The matrix of independent components of a tensor such that [g_{ijk}= - g_{jik}]is given by [\left(\matrix{&122 &133\cr -121 & &223\cr -131 &-232 &\cr}\left|\matrix{123 &131 &\cr &231 &-122\cr -233 & &-132\cr}\right|\matrix{132 & &121\cr 232 &-123 &\cr &-133 &-231\cr}\right).]The second-rank axial tensor [g_{kl}] associated with this tensor is defined by [g_{kl} = {\textstyle{1\over 2}} \varepsilon_{ijk} g_{ijl}.]

For instance, the piezomagnetic coefficients that give the magnetic moment [M_{i}] due to an applied stress [T_{\alpha }] are the components of a second-rank axial tensor, [\Lambda_{i\alpha }] (see Section 1.5.7.1[link] ): [M_{i}= \Lambda _{i\alpha} T_{\alpha}.]

1.1.4.10.7.1. Independent components according to the following point groups

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  • (i) Triclinic system

    • (a) Group 1: [Scheme scheme82]

    • (b) Group [{\bar 1}]: all components are equal to zero.

  • (ii) Monoclinic system

    • (a) Group 2: [Scheme scheme83]

    • (b) Group m: [Scheme scheme84]

    • (c) Group [2/m]: all components are equal to zero.

  • (iii) Orthorhombic system

    • (a) Group 222: [Scheme scheme85]

    • (b) Group [mm2]: [Scheme scheme86]

    • (c) Group [mmm]: all components are equal to zero.

  • (iv) Trigonal, tetragonal, hexagonal and cylindrical systems

    • (a) Groups 3, 4, 6 and [A_{\infty}]: [Scheme scheme87]

    • (b) Groups 32, 42, 62 and [A_{\infty}\infty A_2]: [Scheme scheme88]

    • (c) Groups [3m], [4m], [6m] and [A_{\infty}\infty M]: [Scheme scheme89]

    • (d) Group [{\bar 4}]: [Scheme scheme90]

    • (e) Group [\bar{4}2m]: [Scheme scheme91]

    • (f) Groups [\bar{3}], [4/m], [\bar{6}2m], [\bar{3}m], [4/mm] and [6/mm]: all components are equal to zero.

  • (v) Cubic and spherical systems

    • (a) Groups 23, 432 and [\infty A_{\infty}]: [Scheme scheme92]

      The axial tensor is reduced to a pseudoscalar.

    • (b) Groups [m\bar{3}], [\bar{4}3m], [m\bar{3}m] and [\infty (A_{\infty}/M)C]: all components are equal to zero.

1.1.4.10.7.2. Independent components of symmetric axial tensors according to the following point groups

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Some axial tensors are also symmetric. For instance, the optical rotatory power of a gyrotropic crystal in a given direction of direction cosines [\alpha_{1,} \alpha_{2}, \alpha_{3}] is proportional to a quantity G defined by (see Section 1.6.5.4[link] ) [G = g_{ij}\alpha_{i}\alpha_{j},]where the gyration tensor [g_{ij}] is an axial tensor. This expression shows that only the symmetric part of [g_{ij}] is relevant. This leads to a further reduction of the number of independent components:

  • (i) Triclinic system

    • (a) Group 1: [Scheme scheme93]

    • (b) Group [\bar{1}]: all components are equal to zero.

  • (ii) Monoclinic system

    • (a) Group 2: [Scheme scheme94]

    • (b) Group m: [Scheme scheme95]

    • (c) Group [2/m]: all components are equal to zero.

  • (iii) Orthorhombic system

    • (a) Group 222: [Scheme scheme96]

    • (b) Group [mm2]: [Scheme scheme97]

    • (c) Group [mmm]: all components are equal to zero.

  • (iv) Trigonal, tetragonal and hexagonal systems

    • (a) Groups 3, 32, 4, 42, 6, 62: [Scheme scheme98]

    • (b) Group [\bar{4}]: [Scheme scheme99]

    • (c) Group [\bar{4}2m]: [Scheme scheme100]

    • (d) Groups [\bar{3}], [3m], [\bar{3}m], [4/m], [4mm], [4/mm], [\bar{6}], [\bar{6}2m] and [6/mm]: all components are equal to zero.

  • (v) Cubic and spherical systems

    • (a) Groups 23, 432 and [A_{\infty} \infty A_{2}]: [Scheme scheme101]

    • (b) Groups [m\bar{3}], [\bar{4}3m], [m\bar{3}m] and [\infty (A_{\infty}/M)C]: all components are equal to zero.

In practice, gyrotropic crystals are only found among the enantiomorphic groups: 1, 2, 222, 3, 32, 4, 422, 6, 622, 23, 432. Pasteur (1848a[link],b[link]) was the first to establish the distinction between `molecular dissymmetry' and `crystalline dissymetry'.

References

Pasteur, L. (1848a). Recherches sur les relations qui peuvent exister entre la forme cristalline, la composition chimique et le sens de la polarisation rotatoire. Ann. Chim. (Paris), 24, 442–459.
Pasteur, L. (1848b). Mémoire sur la relation entre la forme cristalline et la composition chimique, et sur la cause de la polarisation rotatoire. C. R. Acad Sci. 26, 535–538.








































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