Reports of the Academy of Sciences of the USSR

1. Volume 149, No. 4

MECHANICS

Academician L. I. SEDOV, V. V. LOKHIN

DESCRIPTION OF POINT SYMMETRY GROUPS BY MEANS OF TENSORS

In many problems of physics it is necessary to take into account the geometric properties of symmetry determined by point groups of orthogonal transformations. In particular, scalars, vectors, and tensors occurring in various equations that express physical laws must, in a number of cases, possess invariance properties with respect to prescribed groups of symmetry transformations.

The invariance properties specialize the form of scalar functions and of the components of the tensors under consideration. Many consequences of symmetry have been studied in detail in various applications. Existing data for various concrete examples are contained in the book by J. Nye (¹). A more detailed treatment of the question of tensor symmetry, the properties of scalar invariants, and the construction of examples of tensors with prescribed symmetry is given in works (², ⁹–¹²), especially in the works of A. V. Shubnikov (⁴, ⁵), Yu. I. Sirotin (⁶–⁸), and their collaborators.

The following proposition is valid: the geometric characteristics of the symmetry of textures and crystals (and, in other cases, those determined by the corresponding transformation groups) can be specified uniquely and completely by means of a small set of simple tensors.

The notation, the definition of the basic tensors, the basic tensors themselves, and the geometric diagrams explaining the symmetry of textures and the 32 classes of crystals are given in the appended table (see Fig. 1).

In view of the fact that only orthogonal symmetry transformations are considered, to each of the indicated bases one must adjoin the fundamental tensor $g$. In some cases the invariance of the fundamental tensor is a simple consequence of the invariance of the tensor basis indicated in the table (for example, for the groups $Oh$, $Th$, etc.).

It is easy to see that the choice of the corresponding sets of determining tensors can be made nonuniquely.

A proof of the formulated proposition is not difficult to give by directly checking that the requirement of invariance of the tensor basis is equivalent to specifying the system of transformation matrices that determine the given symmetry group.

The data of this table can be used to construct general formulas for the dependence of scalars and tensors on a number of other scalar and tensor quantities, taking into account their geometric symmetry properties. The corresponding formulas for tensor functions may be regarded as a generalization of the well-known Hamilton–Cayley formula, as applied to nonlinear tensor functions (³), to the case of several tensor arguments. A generalization of this formula to the case of dependence on several tensors of second rank is contained in works (¹³, ¹⁴) (see also (³)).

A more detailed development of the question of tensor functions of various tensor arguments will be given by us in another paper.

Received
7 II 1963

\(x^1, x^2, x^3\)—crystallophysical Cartesian coordinates

\(\xi^1, \xi^2, \xi^3\)—arbitrary coordinates

\[ a^i_j=\frac{\partial \xi^i}{\partial x^j}, \qquad \Delta=\left|a^i_j\right| \]

\[ e_i=\frac{\partial \bar r}{\partial x^i}, \qquad \mathfrak e_i=\frac{\partial \bar r}{\partial \xi^i} \]

\[ e_j=a^\alpha_j \mathfrak e_\alpha \]

\[ g=e_1^2+e_2^2+e_3^2=g^{\alpha\beta}\mathfrak e_\alpha \mathfrak e_\beta \]

\[ E=e_1e_2e_3-e_2e_1e_3+e_2e_3e_1-e_3e_2e_1+e_3e_1e_2-e_1e_3e_2= \]

\[ =\Delta\cdot(\mathfrak e_1\mathfrak e_2\mathfrak e_3-\mathfrak e_2\mathfrak e_1\mathfrak e_3+\mathfrak e_2\mathfrak e_3\mathfrak e_1-\mathfrak e_3\mathfrak e_2\mathfrak e_1+\mathfrak e_3\mathfrak e_1\mathfrak e_2-\mathfrak e_1\mathfrak e_3\mathfrak e_2) \]

\[ \Omega=e_1e_2-e_2e_1=(a^\alpha_1a^\beta_2-a^\alpha_2a^\beta_1)\mathfrak e_\alpha\mathfrak e_\beta =a^\alpha_1a^\beta_2(\mathfrak e_\alpha\mathfrak e_\beta-\mathfrak e_\beta\mathfrak e_\alpha) \]

\[ e_3^2=a^\alpha_3a^\beta_3\mathfrak e_\alpha\mathfrak e_\beta \]

\[ O_h=e_1^4+e_2^4+e_3^4= \]

\[ =(a^\alpha_1a^\beta_1a^\gamma_1a^\delta_1+a^\alpha_2a^\beta_2a^\gamma_2a^\delta_2+a^\alpha_3a^\beta_3a^\gamma_3a^\delta_3)\mathfrak e_\alpha\mathfrak e_\beta\mathfrak e_\gamma\mathfrak e_\delta \]

\[ T_h=e_1^2e_2^2+e_2^2e_3^2+e_3^2e_1^2 \]

\[ T_d=e_1e_2e_3+e_2e_1e_3+e_2e_3e_1+e_3e_2e_1+e_3e_1e_2+e_1e_3e_2 \]

\[ D_{2h}=\lambda^{11}e_1^2+\lambda^{22}e_2^2+\lambda^{33}e_3^2 =\lambda^{ij}a_i^\alpha a_j^\beta \mathfrak e_\alpha\mathfrak e_\beta =d^{\alpha\beta}\mathfrak e_\alpha\mathfrak e_\beta;\quad \lambda^{11}\ne\lambda^{22}\ne\lambda^{33}\ne\lambda^{11};\quad \lambda^{ii}\ne0,\ d^{\alpha\beta}=d^{\beta\alpha} \]

\[ C_i=D_{2h}+\omega^{ij}e_i e_j=C^{\alpha\beta}\mathfrak e_\alpha\mathfrak e_\beta,\quad \omega^{ij}=-\omega^{ji}\ne0 \]

\[ D_{6h}=(e_1^3-e_1e_2^2-e_2e_1e_2-e_2^2e_1)^2 \]

\[ D_{3h}=e_1^3-e_1e_2^2-e_2e_1e_2-e_2^2e_1 \]

\[ D_{3d}=e_3(e_1^3-e_1e_2^2-e_2e_1e_2-e_2^2e_1) \]

References Cited

¹ J. Nye, Physical Properties of Crystals, IL, 1960.
² S. Bhagavantam, T. Venkatarayudu, Theory of Groups and Its Application to Physical Problems, Moscow, 1959.
³ L. I. Sedov, Introduction to the Mechanics of a Continuous Medium. Moscow, 1962.
⁴ A. V. Shubnikov, Izv. Acad. Sci. USSR, Phys. Ser., 13, 3, 347 (1949).
⁵ A. V. Shubnikov, Symmetry and Antisymmetry of Finite Figures, Moscow, 1951.
⁶ Yu. I. Sirotin, Dokl. Acad. Sci., 133, No. 2, 321 (1960).
⁷ Yu. I. Sirotin, Crystallography, 5, 2, 171 (1960).
⁸ Yu. I. Sirotin, Crystallography, 6, 3, 331 (1961).
⁹ F. G. Smith, R. S. Rivlin, Quart. Appl. Math., 15, 308 (1957).
¹⁰ F. G. Smith, R. S. Rivlin, Trans. Am. Math. Soc., 88, No. 1, 175 (1958).
¹¹ W. Döring, Ann. Phys., (7), 1, 1–3, 104 (1958).
¹² A. C. Pipkin, R. S. Rivlin, Arch. Rat’l Mech. Anal., 4, No. 2, 129 (1959).
¹³ A. J. M. Spencer, R. S. Rivlin, Arch. Rat’l Mech. Anal., 2, No. 4, 309 (1959); 2, No. 5, 435 (1959); 4, No. 3, 214 (1960); 9, No. 1, 45 (1962).
¹⁴ A. J. M. Spencer, Arch. Rat’l Mech. Anal., 7, No. 1, 64 (1961).