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

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 15

Section Matrix method – application of Neumann's principle

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: Matrix method – application of Neumann's principle

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An operation of symmetry turns back the crystalline edifice on itself; it allows the physical properties of the crystal and the tensors representing them to be invariant. An operation of symmetry is equivalent to a change of coordinate system. In a change of system, a tensor becomes [t'^{\alpha \beta}_{\gamma \delta} = t^{ij}_{kl}A^{\alpha}_{i} A^{\beta}_{j}B^{k}_{\gamma}B^{l}_{\delta}.]If A represents a symmetry operation, it is a unitary matrix: [A = B^{T} = B^{-1}.]

Since the tensor is invariant under the action of the symmetry operator A, one has, according to Neumann's principle, [t'^{\alpha \beta}_{\gamma \delta } = t^{\alpha \beta}_{\gamma \delta}]and, therefore, [t^{\alpha \beta}_{\gamma \delta} = t^{ij}_{kl} A^{\alpha}_{i} A^{\beta}_{j}B^{k}_{\gamma}B^{l}_{\delta}. \eqno(]

There are therefore a certain number of linear relations between the components of the tensor and the number of independent components is reduced. If there are p components and q relations between the components, there are [p-q] independent components. This number is independent of the system of axes. When applied to each of the 32 point groups, this reduction enables one to find the form of the tensor in each case. It depends on the rank of the tensor. In the present chapter, the reduction will be derived for tensors up to the fourth rank and for all crystallographic groups as well as for the isotropic groups. An orthonormal frame will be assumed in all cases, so that co- and contravariance will not be apparent and the positions of indices as subscripts or superscripts will not be meaningful. The [Ox_{3}] axis will be chosen parallel to the threefold, fourfold or sixfold axis in the trigonal, tetragonal and hexagonal systems. The accompanying software to the present volume enables the reduction for tensors of any rank to be derived.

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