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

© International Union of Crystallography 2006

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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. 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

1.1.4.10.4. 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 $[Ox_{2}]$ :

There are 8 independent components.
(ii) Group m:

There are 10 independent components.
(iii) Group : all the components are equal to zero.

1.1.4.10.4.3. Orthorhombic system

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(i) Group 222:

There are 3 independent components.
(ii) Group :

There are 5 independent components.
(iii) Group : all the components are equal to zero.

1.1.4.10.4.4. Trigonal system

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(i) Group 3: 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 $[Ox_{1}]$ : with the same conventions. There are 4 independent components.
(iii) Group , mirror perpendicular to $[Ox_{1}]$ : 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 $[A_{\infty}]$ :

There are 4 independent components.
(ii) Groups 422, 622 and $[A_{\infty}\infty A_{2}]$ :

There is 1 independent component.
(iii) Groups , and $[A_{\infty }\infty M]$ :

There are 3 independent components.
(iv) Groups , and $[(A_{\infty }/M)C]$ : all the components are equal to zero.
(v) Group $[\bar{4}]$ :

There are 4 independent components.
(vi) Group $[\bar{6} = 3/m]$ : with the same conventions as for group 3. There are 2 independent components.
(vii) Group $[\bar{4}2m]$ – twofold axis parallel to $[Ox_{1}]$ :

There are 2 independent components.
(viii) Group $[\bar{4}2m]$ – mirror perpendicular to $[Ox_{1}]$ (twofold axis at $[45^{\circ}]$ ):

The number of independent components is of course the same.
(ix) Group $[\bar{6}2/m]$ : with the same conventions as for group 3. There is 1 independent component.
(x) Groups , 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 $[{\bar 4}3m]$ :

There is 1 independent component.
(ii) Groups 432 and $[\infty A_{\infty }]$ : it was seen in Section 1.1.4.8.6 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.

References

© International Union of Crystallography 2006

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

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