Reduction of the components of a tensor of rank 3

Authier, A.

doi:10.1107/97809553602060000628

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

pdf | chapter contents | chapter index | related articles

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. 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

1.1.4.8. 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

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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

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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

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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

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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

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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

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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

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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 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 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

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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

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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

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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

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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 $[Ox_{1}]$ :

There are 6 independent components.
(ii) Mirror perpendicular to $[Ox_{1}]$ (the twofold axis is at $[45 ^{\circ}]$ )

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

1.1.4.8.5.7. Group

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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 , $[A_{\infty}]$ , , $[A_{\infty} \infty A_{2}]$ , and $[A_{\infty} \infty M]$

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It was shown in Section 1.1.4.6.2.3 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]