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. 22-24

Section 1.1.4.9.9. Symmetric tensors of rank 4

A. Authiera*

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

1.1.4.9.9. Symmetric tensors of rank 4

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For symmetric tensors such as those representing principal properties, one finds the following, representing the nonzero components for the leading diagonal and for one half of the others.

1.1.4.9.9.1. Triclinic system

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[Scheme scheme39]

There are 45 independent coefficients.

1.1.4.9.9.2. Monoclinic system

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[Scheme scheme40]

There are 25 independent coefficients.

1.1.4.9.9.3. Orthorhombic system

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[Scheme scheme41]

There are 15 independent coefficients.

1.1.4.9.9.4. Trigonal system

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  • (i) Groups [3] and [{\bar 3}] [Scheme scheme109] with [t_{1111} - t_{1122} = t_{1212} + t_{1221}.]

    There are 15 independent components.

  • (ii) Groups [{\bar 3}m], [32], [3m] [Scheme scheme110] with [t_{1111} - t_{1122} = t_{1212} + t_{1221}.]

    There are 11 independent components.

1.1.4.9.9.5. Tetragonal system

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  • (i) Groups [4/m], [4], [{\bar 4}] [Scheme scheme111]

    There are 13 independent components.

  • (ii) Groups [4/mm], [422], [4mm], [{\bar 4}2m] [Scheme scheme112]

    There are 9 independent components.

1.1.4.9.9.6. Hexagonal and cylindrical systems

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  • (i) Groups [6/m], [{\bar 6}], [6]; [(A_{\infty }/M)C, A_{\infty}] [Scheme scheme113] with[t_{1111} - t_{1122} = t_{1212} + t_{1221}.]

    There are 12 independent components.

  • (ii) Groups [6/mm], [622], [6mm], [{\bar 6}2m]; [(A_{\infty }/M) \infty (A_{2}/M)C], [A_{\infty} \infty A_{2}] [Scheme scheme114] with[t_{1111} - t_{1122} = t_{1212} + t_{1221}.]

    There are 10 independent components.

1.1.4.9.9.7. Cubic system

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  • (i) Groups [23], [{\bar 3}m] [Scheme scheme42] with [t_{1111} - t_{1122} = t_{1212} + t_{1221}.]

    There are 5 independent components.

  • (ii) Groups [m{\bar 3}m], [432], [{\bar 4}3m], and spherical system: the reduced tensors are already symmetric (see Sections 1.1.4.9.7[link] and 1.1.4.9.8[link]).








































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