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

Authier, A.

doi:10.1107/97809553602060000628

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

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International Tables for Crystallography (2006). 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. Authier^a ^*

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 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 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 ): $[M_{i}= \Lambda _{i\alpha} T_{\alpha}.]$

1.1.4.10.7.1. Independent components according to the following point groups

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(i) Triclinic system

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

(ii) Monoclinic system

(a) Group 2:
(b) Group m:
(c) Group : all components are equal to zero.

(iii) Orthorhombic system

(a) Group 222:
(b) Group :
(c) Group : all components are equal to zero.

(iv) Trigonal, tetragonal, hexagonal and cylindrical systems

(a) Groups 3, 4, 6 and $[A_{\infty}]$ :
(b) Groups 32, 42, 62 and $[A_{\infty}\infty A_2]$ :
(c) Groups , , and $[A_{\infty}\infty M]$ :
(d) Group $[{\bar 4}]$ :
(e) Group $[\bar{4}2m]$ :
(f) Groups $[\bar{3}]$ , , $[\bar{6}2m]$ , $[\bar{3}m]$ , and : all components are equal to zero.

(v) Cubic and spherical systems

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

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 ) $[G = g_{ij}\alpha_{i}\alpha_{j},]$ 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:
(b) Group $[\bar{1}]$ : all components are equal to zero.

(ii) Monoclinic system

(a) Group 2:
(b) Group m:
(c) Group : all components are equal to zero.

(iii) Orthorhombic system

(a) Group 222:
(b) Group :
(c) Group : all components are equal to zero.

(iv) Trigonal, tetragonal and hexagonal systems

(a) Groups 3, 32, 4, 42, 6, 62:
(b) Group $[\bar{4}]$ :
(c) Group $[\bar{4}2m]$ :
(d) Groups $[\bar{3}]$ , , $[\bar{3}m]$ , , , , $[\bar{6}]$ , $[\bar{6}2m]$ and : all components are equal to zero.

(v) Cubic and spherical systems

(a) Groups 23, 432 and $[A_{\infty} \infty A_{2}]$ :
(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,b) 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.Google Scholar

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.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.1, pp. 29-31