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. 20-24

Section 1.1.4.9. Reduction of the components of a tensor 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. Reduction of the components of a tensor of rank 4

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1.1.4.9.1. Triclinic system (groups [{\bar 1}], [1])

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There is no reduction; all the components are independent. Their number is equal to 81. They are usually represented as a [9\times 9] matrix, where components [t_{ijkl}] are replaced by ijkl, for brevity: [Scheme scheme102] This matrix can be represented symbolically by [Scheme scheme33] where the [9\times 9] matrix has been subdivided for clarity in to nine [3\times 3] submatrices.

1.1.4.9.2. Monoclinic system (groups [2/m], [2], m)

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The reduction is obtained by the method of direct inspection. For a twofold axis parallel to [Ox_{2}], one finds [Scheme scheme34]

There are 41 independent components.

1.1.4.9.3. Orthorhombic system (groups [mmm], [2mm], [222])

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[Scheme scheme35]

There are 21 independent components.

1.1.4.9.4. Trigonal system

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1.1.4.9.4.1. Groups [3] and [{\bar 3}]

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The reduction is first applied in the system of axes tied to the eigenvectors of the operator representing a threefold axis. The system of axes is then changed to a system of orthonormal axes with [Ox_{3}] parallel to the threefold axis: [Scheme scheme103] with [\left. \matrix{t_{1111} - t_{1122} = &t_{1212} + t_{1221}\cr t_{1112} + t_{1121} = &- (t_{1211} + t_{2111}).\cr}\right\}]

There are 27 independent components.

1.1.4.9.4.2. Groups [{\bar 3}m], [32], [3m], with the twofold axis parallel to [Ox_{1}]

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[Scheme scheme104] with [t_{1111} - t_{1122} = t_{1212} + t_{1221}.]

There are 14 independent components.

1.1.4.9.5. Tetragonal system

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1.1.4.9.5.1. Groups [4/m], [4], [{\bar 4}]

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[Scheme scheme105]

There are 21 independent components.

1.1.4.9.5.2. Groups [4/m m], [422], [4mm], [{\bar 4}2m]

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[Scheme scheme106]

There are 11 independent components.

1.1.4.9.6. Hexagonal and cylindrical systems

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1.1.4.9.6.1. Groups [6/m], [{\bar 6}], [6]; [(A_{\infty}/M)C], [A_{\infty}]

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[Scheme scheme107] with[\left. \matrix{t_{1111} - t_{1122} = t_{1212} + t_{1221}\hfill\cr t_{1112} + t_{1121} = - (t_{1211}+ t_{2111}).\cr}\right\}]

There are 19 independent components.

1.1.4.9.6.2. Groups [6/mm], [622], [6mm], [{\bar 6}2m]; [(A_{\infty}/M) \infty]; [(A_{2}/M)C], [A_{\infty}\infty A_{2}]

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[Scheme scheme108] with[t_{1111} - t_{1122} = t_{1212} + t_{1221}.]

There are 11 independent components.

1.1.4.9.7. Cubic system

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1.1.4.9.7.1. Groups [23], [{\bar 3}m]

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[Scheme scheme36]

There are 7 independent components.

1.1.4.9.7.2. Groups [m{\bar 3}m], [432], [{\bar 4}3m]

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[Scheme scheme37]

There are 4 independent components. The tensor is symmetric.

1.1.4.9.8. Spherical system

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1.1.4.9.8.1. Groups [\infty (A_{\infty}/M)C] and [\infty A_{\infty}]

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[Scheme scheme38] with [t_{1111} - t_{1122} = t_{1212} + t_{1221}.]

There are 3 independent components. The tensor is symmetric.

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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