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. 17-20

Section 1.1.4.8. Reduction of the components of a tensor of rank 3

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

1.1.4.8. Reduction of the components of a tensor of rank 3

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1.1.4.8.1. Triclinic system

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1.1.4.8.1.1. Group [1]

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All the components are independent. Their number is equal to 27. They are usually represented as a [3\times 9] matrix which can be subdivided into three [3\times 3] submatrices: [\left(\matrix{111 &122 &133\cr \noalign{\vskip3pt} 211 &222 &233\cr \noalign{\vskip3pt} 311 &322 &333\cr} \left|\matrix{123 &131 &112\cr \noalign{\vskip3pt} 223 &231 &212\cr \noalign{\vskip3pt} 323 &331 &312\cr}\right| \matrix{132 &113 &121\cr \noalign{\vskip3pt} 232 &213 &221\cr \noalign{\vskip3pt} 332 &313 &321\cr}\right).]

1.1.4.8.1.2. Group [{\bar 1}]

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All the components are equal to zero.

1.1.4.8.2. Monoclinic system

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1.1.4.8.2.1. Group [2]

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Choosing the twofold axis parallel to [Ox_{3}] and applying the direct inspection method, one finds [Scheme scheme13]

There are 13 independent components. If the twofold axis is parallel to [Ox_{2}], one finds [Scheme scheme14]

1.1.4.8.2.2. Group m

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One obtains the matrix representing the operator m by multiplying by −1 the coefficients of the matrix representing a twofold axis. The result of the reduction will then be exactly complementary: the components of the tensor which include an odd number of 3's are now equal to zero. One writes the result as follows: [Scheme scheme15]

There are 14 independent components. If the mirror axis is normal to [Ox_{2}], one finds [Scheme scheme16]

1.1.4.8.2.3. Group [2/m]

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All the components are equal to zero.

1.1.4.8.3. Orthorhombic system

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1.1.4.8.3.1. Group [222]

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There are three orthonormal twofold axes. The reduction is obtained by combining the results associated with two twofold axes, parallel to [Ox_{3}] and [Ox_{2}], respectively. [Scheme scheme17]

There are 6 independent components.

1.1.4.8.3.2. Group [mm2]

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The reduction is obtained by combining the results associated with a twofold axis parallel to [Ox_{3}] and with a mirror normal to [Ox_{2}]: [Scheme scheme18]

There are 7 independent components.

1.1.4.8.3.3. Group [mmm]

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All the components are equal to zero.

1.1.4.8.4. Trigonal system

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1.1.4.8.4.1. Group [3]

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The threefold axis is parallel to [Ox_{3}]. The matrix method should be used here. One finds [Scheme scheme19]

There are 9 independent components.

1.1.4.8.4.2. Group [32] with a twofold axis parallel to [Ox_{1}]

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[Scheme scheme20]

There are 4 independent components.

1.1.4.8.4.3. Group [3m] with a mirror normal to [Ox_{1}]

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[Scheme scheme21]

There are 4 independent components.

1.1.4.8.4.4. Groups [\bar{3}] and [\bar{3}m]

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All the components are equal to zero.

1.1.4.8.5. Tetragonal system

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1.1.4.8.5.1. Group [4]

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The method of direct inspection can be applied for a fourfold axis. One finds [Scheme scheme22]

There are 7 independent components.

1.1.4.8.5.2. Group [422]

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One combines the reductions for groups 4 and 222: [Scheme scheme23]

There are 3 independent components.

1.1.4.8.5.3. Group [4mm]

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One combines the reductions for groups 4 and 2m: [Scheme scheme24]

There are 4 independent components.

1.1.4.8.5.4. Group [4/m]

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All the components are equal to zero.

1.1.4.8.5.5. Group [\bar{4}]

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The matrix corresponding to axis [\bar{4}] is [\pmatrix{0 &\bar{1} &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}] and the form of the [3 \times 9] matrix is [Scheme scheme25]

There are 6 independent components.

1.1.4.8.5.6. Group [\bar{4}2m]

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One combines either the reductions for groups [\bar{4}] and 222, or the reductions for groups [\bar{4}] and 2mm.

  • (i) Twofold axis parallel to [Ox_{1}]: [Scheme scheme26]

    There are 6 independent components.

  • (ii) Mirror perpendicular to [Ox_{1}] (the twofold axis is at [45 ^{\circ}]) [Scheme scheme27]

The number of independent components is of course the same, 6.

1.1.4.8.5.7. Group [4/mm]

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All the components are equal to zero.

1.1.4.8.6. Hexagonal and cylindrical systems

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1.1.4.8.6.1. Groups [6], [A_{\infty}], [622], [A_{\infty} \infty A_{2}], [6mm] and [A_{\infty} \infty M]

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It was shown in Section 1.1.4.6.2.3[link] that, in the case of tensors of rank 3, the reduction is the same for axes of order 4, 6 or higher. The reduction will then be the same as for the tetragonal system.

1.1.4.8.6.2. Group [\bar{6} = 3/m]

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One combines the reductions for the groups corresponding to a threefold axis parallel to [Ox_{3}] and to a mirror perpendicular to [Ox_{3}]: [Scheme scheme28]

There are 2 independent components.

1.1.4.8.6.3. Group [\bar{6}2m]

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One combines the reductions for groups 6 and 2mm: [Scheme scheme29]

There is 1 independent component.

1.1.4.8.6.4. Groups [6/m], [(A_{\infty}/M)C], [6/mm] and [(A_{\infty}/M) \infty (A_{2}/M)C]

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All the components are equal to zero.

1.1.4.8.7. Cubic and spherical systems

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1.1.4.8.7.1. Group [23]

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One combines the reductions corresponding to a twofold axis parallel to [Ox_{3}] and to a threefold axis parallel to [111]: [Scheme scheme30]

There are 2 independent components.

1.1.4.8.7.2. Groups [432] and [\infty A_{\infty}/M]

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One combines the reductions corresponding to groups 422 and 23: [Scheme scheme31]

There is 1 independent component.

1.1.4.8.7.3. Group [{\bar 4}3m]

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One combines the reductions corresponding to groups [{\bar 4}2m] and 23: [Scheme scheme32]

There is 1 independent component.

1.1.4.8.7.4. Groups [m{\bar 3}], [m{\bar 3}m] and [\infty(A_{\infty}/M)C]

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All the components are equal to zero.








































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