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

International Tables for Crystallography (2013). Vol. D, ch. 1.1, pp. 16-17

Section Reduction of the components of a tensor of rank 2

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: Reduction of the components of a tensor of rank 2

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The reduction is given for each of the 11 Laue classes. Triclinic system

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Groups [{\bar 1}], 1: no reduction, the tensor has 9 independent components. The result is represented in the following symbolic way (Nye, 1957[link], 1985[link]): [Scheme scheme1] where the sign • represents a nonzero component. Monoclinic system

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Groups 2m, 2, m: it is sufficient to consider the twofold axis or the mirror. As the representative matrix is diagonal, the calculation is immediate. Taking the twofold axis to be parallel to [Ox_{3}], one has [t^{1}_{3} = t^{3}_{1} = t^{2}_{3} = t^{3}_{2} = 0.]

The other components are not affected. The result is represented as [Scheme scheme2]

There are 5 independent components. If the twofold axis is taken along axis [Ox_{2}], which is the usual case in crystallography, the table of independent components becomes [Scheme scheme3] Orthorhombic system

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Groups mmm, 2mm, 222: the reduction is obtained by considering two perpendicular twofold axes, parallel to [Ox_{3}] and to [Ox_{2}], respectively. One obtains [Scheme scheme4]

There are 3 independent components. Trigonal, tetragonal, hexagonal and cylindrical systems

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We remarked in Section[link] that, in the case of tensors of rank 2, the reduction is the same for threefold, fourfold or sixfold axes. It suffices therefore to perform the reduction for the tetragonal groups. That for the other systems follows automatically. Groups [{\bar 3}], [3]; [4/m], [{\bar 4}], [4]; [6/m], [{\bar 6}], [6]; [(A_{\infty}/M)C], [A_{\infty}]

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If we consider a fourfold axis parallel to [Ox_{3}] represented by the matrix given in ([link], by applying the direct inspection method one finds [Scheme scheme5] where the symbol ⊖ means that the corresponding component is numerically equal to that to which it is linked, but of opposite sign. There are 3 independent components. Groups [{\bar 3}m], [32], [3m]; [4/mm], [422], [4mm], [{\bar {4}}{2}m]; [{6}/mm], [622], [6mm], [{\bar{6}}{2}m]; [(A_{\infty}/M) \infty (A_{2}/M)C], [A_{\infty}\infty A_{2}]

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The result is obtained by combining the preceding result and that corresponding to a twofold axis normal to the fourfold axis. One finds [Scheme scheme6]

There are 2 independent components. Cubic and spherical systems

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The cubic system is characterized by the presence of threefold axes along the [\langle 111 \rangle] directions. The action of a threefold axis along [111] on the components [x_{1}, x_{2}, x_{3}] of a vector results in a permutation of these components, which become, respectively, [x_{2}, x_{3}, x_{1}] and then [x_{3}, x_{1}, x_{2}]. One deduces that the components of a tensor of rank 2 satisfy the relations [t^{1}_{1} = t^{2}_{2} = t^{3}_{3}.]

The cubic groups all include as a subgroup the group 23 of which the generating elements are a twofold axis along [Ox_{3}] and a threefold axis along [111]. If one combines the corresponding results, one deduces that [t^{2}_{1} = t^{3}_{2} = t^{1}_{3} = t^{3}_{1} = t^{1}_{2} = t^{2}_{3} = 0,]which can be summarized by [Scheme scheme7]

There is a single independent component and the medium behaves like a property represented by a tensor of rank 2, like an isotropic medium. Symmetric tensors of rank 2

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If the tensor is symmetric, the number of independent components is still reduced. One obtains the following, representing the nonzero components for the leading diagonal and for one half of the others. Triclinic system

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[Scheme scheme8]

There are 6 independent components. It is possible to interpret the number of independent components of a tensor of rank 2 by considering the associated quadric, for instance the optical indicatrix. In the triclinic system, the quadric is any quadric. It is characterized by six parameters: the lengths of the three axes and the orientation of these axes relative to the crystallographic axes. Monoclinic system (twofold axis parallel to [Ox_2])

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[Scheme scheme9]

There are 4 independent components. The quadric is still any quadric, but one of its axes coincides with the twofold axis of the monoclinic lattice. Four parameters are required: the lengths of the axes and one angle. Orthorhombic system

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[Scheme scheme10]

There are 3 independent components. The quadric is any quadric, the axes of which coincide with the crystallographic axes. Only three parameters are required. Trigonal, tetragonal and hexagonal systems, isotropic groups

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[Scheme scheme11]

There are 2 independent components. The quadric is of revolution. It is characterized by two parameters: the lengths of its two axes. Cubic system

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[Scheme scheme12]

There is 1 independent component. The associated quadric is a sphere.


Nye, J. F. (1957). Physical Properties of Crystals, 1st ed. Oxford: Clarendon Press.
Nye, J. F. (1985). Physical Properties of Crystals, revised ed. Oxford University Press.

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