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. 17-20

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

A. Authiera*

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

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1.1.4.8.1.1. Group | top | pdf |

All the components are independent. Their number is equal to 27. They are usually represented as a matrix which can be subdivided into three submatrices: 1.1.4.8.1.2. Group | top | pdf |

All the components are equal to zero.

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1.1.4.8.2.1. Group | top | pdf |

Choosing the twofold axis parallel to and applying the direct inspection method, one finds There are 13 independent components. If the twofold axis is parallel to , one finds 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: There are 14 independent components. If the mirror axis is normal to , one finds 1.1.4.8.2.3. Group | top | pdf |

All the components are equal to zero.

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1.1.4.8.3.1. Group | top | pdf |

There are three orthonormal twofold axes. The reduction is obtained by combining the results associated with two twofold axes, parallel to and , respectively. There are 6 independent components.

1.1.4.8.3.2. Group | top | pdf |

The reduction is obtained by combining the results associated with a twofold axis parallel to and with a mirror normal to : There are 7 independent components.

1.1.4.8.3.3. Group | top | pdf |

All the components are equal to zero.

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1.1.4.8.4.1. Group | top | pdf |

The threefold axis is parallel to . The matrix method should be used here. One finds There are 9 independent components.

1.1.4.8.4.2. Group with a twofold axis parallel to | top | pdf | There are 4 independent components.

1.1.4.8.4.3. Group with a mirror normal to | top | pdf | There are 4 independent components.

1.1.4.8.4.4. Groups and | top | pdf |

All the components are equal to zero.

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1.1.4.8.5.1. Group | top | pdf |

The method of direct inspection can be applied for a fourfold axis. One finds There are 7 independent components.

1.1.4.8.5.2. Group | top | pdf |

One combines the reductions for groups 4 and 222: There are 3 independent components.

1.1.4.8.5.3. Group | top | pdf |

One combines the reductions for groups 4 and 2m: There are 4 independent components.

1.1.4.8.5.4. Group | top | pdf |

All the components are equal to zero.

1.1.4.8.5.5. Group | top | pdf |

The matrix corresponding to axis is and the form of the matrix is There are 6 independent components.

1.1.4.8.5.6. Group | top | pdf |

One combines either the reductions for groups and 222, or the reductions for groups and 2mm.

 (i) Twofold axis parallel to : There are 6 independent components. (ii) Mirror perpendicular to (the twofold axis is at ) The number of independent components is of course the same, 6.

1.1.4.8.5.7. Group | top | pdf |

All the components are equal to zero.

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1.1.4.8.6.1. Groups , , , , and | top | pdf |

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 | top | pdf |

One combines the reductions for the groups corresponding to a threefold axis parallel to and to a mirror perpendicular to : There are 2 independent components.

1.1.4.8.6.3. Group | top | pdf |

One combines the reductions for groups 6 and 2mm: There is 1 independent component.

1.1.4.8.6.4. Groups , , and | top | pdf |

All the components are equal to zero.

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1.1.4.8.7.1. Group | top | pdf |

One combines the reductions corresponding to a twofold axis parallel to and to a threefold axis parallel to : There are 2 independent components.

1.1.4.8.7.2. Groups and | top | pdf |

One combines the reductions corresponding to groups 422 and 23: There is 1 independent component.

1.1.4.8.7.3. Group | top | pdf |

One combines the reductions corresponding to groups and 23: There is 1 independent component.

1.1.4.8.7.4. Groups , and | top | pdf |

All the components are equal to zero.