International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 14-16
Section 1.1.4.6. Symmetry of tensors imposed by the crystalline medium^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France |
Many papers have been devoted to the derivation of the invariant components of physical property tensors under the influence of the symmetry elements of the crystallographic point groups: see, for instance, Fumi (1951, 1952a,b,c, 1987), Fumi & Ripamonti (1980a,b), Nowick (1995), Nye (1957, 1985), Sands (1995), Sirotin & Shaskol'skaya (1982), and Wooster (1973). There are three main methods for this derivation: the matrix method (described in Section 1.1.4.6.1), the direct inspection method (described in Section 1.1.4.6.3) and the group-theoretical method (described in Section 1.2.4 and used in the accompanying software, see Section 1.2.7.4 ).
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 If A represents a symmetry operation, it is a unitary matrix:
Since the tensor is invariant under the action of the symmetry operator A, one has, according to Neumann's principle, and, therefore,
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 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 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.
If one takes as the system of axes the eigenvectors of the operator A, the matrix is written in the form where θ is the rotation angle, is taken parallel to the rotation axis and coefficient is equal to +1 or −1 depending on whether the rotation axis is direct or inverse (proper or improper operator).
The equations (1.1.4.1) can then be simplified and reduce to (without any summation).
If the product (without summation) is equal to unity, equation (1.1.4.2) is trivial and there is significance in the component . On the contrary, if it is different from 1, the only solution for (1.1.4.2) is that . One then finds immediately that certain components of the tensor are zero and that others are unchanged.
All the diagonal components are in this case equal to −1. One thus has:
By replacing the matrix coefficients by their expression, (1.1.4.2) becomes, for a proper rotation, where r is the number of indices equal to 1, s is the number of indices equal to 2, t is the number of indices equal to 3 and is the rank of the tensor. The component is not affected by the symmetry operation if where K is an integer, and is equal to zero if
The angle of rotation θ can be put into the form , where q is the order of the axis. The condition for the component not to be zero is then
The condition is fulfilled differently depending on the rank of the tensor, p, and the order of the axis, q. Indeed, we have and
It follows that:
The inconvenience of the diagonalization method is that the vectors and eigenvalues are, in general, complex, so in practice one uses another method. For instance, we may note that equation (1.1.4.1) can be written in the case of by associating with the tensor a matrix T: where B is the symmetry operation. Through identification of homologous coefficients in matrices T and , one obtains relations between components that enable the determination of the independent components.
The method of `direct inspection', due to Fumi (1952a,b, 1987), is very simple. It is based on the fundamental properties of tensors; the components transform under a change of basis like a product of vector components (Section 1.1.3.2).
Examples
It is not possible to apply the method of direct inspection for point group 3. One must in this case use the matrix method described in Section 1.1.4.6.2; once this result is assumed, the method can be applied to all other point groups.
References
Fumi, F. G. (1951). Third-order elastic coefficients of crystals. Phys. Rev. 83, 1274–1275.Fumi, F. G. (1952a). Physical properties of crystals: the direct inspection method. Acta Cryst. 5, 44–48.
Fumi, F. G. (1952b). The direct-inspection method in systems with a principal axis of symmetry. Acta Cryst. 5, 691–694.
Fumi, F. G. (1952c). Third-order elastic coefficients in trigonal and hexagonal crystals. Phys. Rev. 86, 561.
Fumi, F. G. (1987). Tables for the third-order elastic tensors in crystals. Acta Cryst. A43, 587–588.
Fumi, F. G. & Ripamonti, C. (1980a). Tensor properties and rotational symmetry of crystals. I. A new method for group 3(3_{z}) and its application to general tensors up to rank 8. Acta Cryst. A36, 535–551.
Fumi, F. G. & Ripamonti, C. (1980b). Tensor properties and rotational symmetry of crystals. II. Groups with 1-, 2- and 4-fold principal symmetry and trigonal and hexagonal groups different from group 3. Acta Cryst. A36, 551–558.
Nowick, A. S. (1995). Crystal properties via group theory. Cambridge University Press.
Nye, J. F. (1957). Physical properties of crystals, 1st ed. Oxford: Clarendon Press.
Nye, J. F. (1985). Physical properties of cystals, revised ed. Oxford University Press.
Sands, D. E. (1995). Vectors and tensors in crystallography. New York: Dover.
Sirotin, Y. I. & Shaskol'skaya, M. P. (1982). Fundamentals of crystal physics. Moscow: Mir.
Wooster, W. A. (1973). Tensors and group theory for the physical properties of crystals. Oxford: Clarendon Press.