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

International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 25-26

Section 1.1.4.10.4. Independent components of the matrix associated with a third-rank polar tensor according to the following point groups

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.4.10.4. Independent components of the matrix associated with a third-rank polar tensor according to the following point groups

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1.1.4.10.4.1. Triclinic system

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  • (i) Group 1: all the components are independent. There are 18 components.

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

1.1.4.10.4.2. Monoclinic system

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  • (i) Group 2: twofold axis parallel to [Ox_{2}]: [Scheme scheme47]

    There are 8 independent components.

  • (ii) Group m: [Scheme scheme48]

    There are 10 independent components.

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

1.1.4.10.4.3. Orthorhombic system

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  • (i) Group 222: [Scheme scheme49]

    There are 3 independent components.

  • (ii) Group [mm2]: [Scheme scheme50]

    There are 5 independent components.

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

1.1.4.10.4.4. Trigonal system

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  • (i) Group 3: [Scheme scheme51] where the symbol ⊖ means that the corresponding component is equal to the opposite of that to which it is linked, ⊙ means that the component is equal to twice minus the value of the component to which it is linked for [d_{ijk}] and to minus the value of the component to which it is linked for [e_{ijk}]. There are 6 independent components.

  • (ii) Group 32, twofold axis parallel to [Ox_{1}]: [Scheme scheme52] with the same conventions. There are 4 independent components.

  • (iii) Group [3m], mirror perpendicular to [Ox_{1}]: [Scheme scheme53] with the same conventions. There are 4 independent components.

  • (iv) Groups [\bar{3}] and [\bar{3}m]: all the components are equal to zero.

1.1.4.10.4.5. Tetragonal, hexagonal and cylindrical systems

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  • (i) Groups 4, 6 and [A_{\infty}]: [Scheme scheme54]

    There are 4 independent components.

  • (ii) Groups 422, 622 and [A_{\infty}\infty A_{2}]: [Scheme scheme55]

    There is 1 independent component.

  • (iii) Groups [4mm], [6mm] and [A_{\infty }\infty M]: [Scheme scheme56]

    There are 3 independent components.

  • (iv) Groups [4/m], [6/m] and [(A_{\infty }/M)C]: all the components are equal to zero.

  • (v) Group [\bar{4}]: [Scheme scheme57]

    There are 4 independent components.

  • (vi) Group [\bar{6} = 3/m]: [Scheme scheme58] with the same conventions as for group 3. There are 2 independent components.

  • (vii) Group [\bar{4}2m] – twofold axis parallel to [Ox_{1}]: [Scheme scheme59]

    There are 2 independent components.

  • (viii) Group [\bar{4}2m] – mirror perpendicular to [Ox_{1}] (twofold axis at [45^{\circ}]): [Scheme scheme60]

    The number of independent components is of course the same.

  • (ix) Group [\bar{6}2/m]: [Scheme scheme61] with the same conventions as for group 3. There is 1 independent component.

  • (x) Groups [4/mm], [6/mm] and [(A_{\infty }/M) \infty (A_{2}/M)C]: all the components are equal to zero.

1.1.4.10.4.6. Cubic and spherical systems

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  • (i) Groups 23 and [{\bar 4}3m]: [Scheme scheme62]

    There is 1 independent component.

  • (ii) Groups 432 and [\infty A_{\infty }]: it was seen in Section 1.1.4.8.6[link] that we have in this case [d_{123}= - d_{132}.] It follows that [d_{14} = 0], all the components are equal to zero.

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








































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