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. 22-24

## Section 1.1.4.9.9. Symmetric tensors of rank 4

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

#### 1.1.4.9.9. 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

| top | pdf | There are 45 independent coefficients.

#### 1.1.4.9.9.2. Monoclinic system

| top | pdf | There are 25 independent coefficients.

#### 1.1.4.9.9.3. Orthorhombic system

| top | pdf | 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 ).