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. 20-24

## Section 1.1.4.9. Reduction of the components of a tensor of rank 4

A. Authiera*

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

| top | pdf |

#### 1.1.4.9.1. Triclinic system (groups , )

| top | pdf |

There is no reduction; all the components are independent. Their number is equal to 81. They are usually represented as a matrix, where components are replaced by ijkl, for brevity: This matrix can be represented symbolically by where the matrix has been subdivided for clarity in to nine submatrices.

#### 1.1.4.9.2. Monoclinic system (groups , , m)

| top | pdf |

The reduction is obtained by the method of direct inspection. For a twofold axis parallel to , one finds

There are 41 independent components.

#### 1.1.4.9.3. Orthorhombic system (groups , , )

| top | pdf |

There are 21 independent components.

| top | pdf |

#### 1.1.4.9.4.1. Groups and

| top | pdf |

The reduction is first applied in the system of axes tied to the eigenvectors of the operator representing a threefold axis. The system of axes is then changed to a system of orthonormal axes with parallel to the threefold axis: with

There are 27 independent components.

#### 1.1.4.9.4.2. Groups , , , with the twofold axis parallel to

| top | pdf |

with

There are 14 independent components.

| top | pdf |

#### 1.1.4.9.5.1. Groups , ,

| top | pdf |

There are 21 independent components.

#### 1.1.4.9.5.2. Groups , , ,

| top | pdf |

There are 11 independent components.

| top | pdf |

#### 1.1.4.9.6.1. Groups , , ; ,

| top | pdf |

with

There are 19 independent components.

#### 1.1.4.9.6.2. Groups , , , ; ; ,

| top | pdf |

with

There are 11 independent components.

| top | pdf |

#### 1.1.4.9.7.1. Groups ,

| top | pdf |

There are 7 independent components.

#### 1.1.4.9.7.2. Groups , ,

| top | pdf |

There are 4 independent components. The tensor is symmetric.

| top | pdf |

#### 1.1.4.9.8.1. Groups and

| top | pdf |

with

There are 3 independent components. The tensor is symmetric.

#### 1.1.4.9.9. Symmetric tensors of rank 4

| top | pdf |

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

| top | pdf |

 (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

| top | pdf |

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

#### 1.1.4.9.9.6. Hexagonal and cylindrical systems

| top | pdf |

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

#### 1.1.4.9.9.7. Cubic system

| top | pdf |

 (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).