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. 22-24

## Section 1.1.4.9.9. Symmetric tensors of rank 4

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France

#### 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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There are 45 independent coefficients.

#### 1.1.4.9.9.2. Monoclinic system

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There are 25 independent coefficients.

#### 1.1.4.9.9.3. Orthorhombic system

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There are 15 independent coefficients.

#### 1.1.4.9.9.4. Trigonal system

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 (i) Groups and with There are 15 independent components. (ii) Groups , , with There are 11 independent components.

#### 1.1.4.9.9.5. Tetragonal system

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 (i) Groups , , There are 13 independent components. (ii) Groups , , , There are 9 independent components.

#### 1.1.4.9.9.6. Hexagonal and cylindrical systems

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 (i) Groups , , ; with There are 12 independent components. (ii) Groups , , , ; , with There are 10 independent components.

#### 1.1.4.9.9.7. Cubic system

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 (i) Groups , with There are 5 independent components. (ii) Groups , , , and spherical system: the reduced tensors are already symmetric (see Sections 1.1.4.9.7 and 1.1.4.9.8).