Can a Crystal Be Both Isotropic and Anisotropic?

A. V. Shubnikov

Persons becoming acquainted for the first time with physical crystallography, and sometimes even qualified crystallographers, often cannot understand how one and the same crystal can be simultaneously anisotropic and discontinuous with respect to some properties, and isotropic and continuous with respect to other properties; can have the symmetry of a cube with respect to some properties and the symmetry of a sphere with respect to other properties. The fact that, for example, crystals of rock salt have the same refractive indices in all directions, the same electrical conductivity, the same dielectric constant, etc., such persons are inclined to explain either by the imperfection of the measurement technique or by the fact that up to now we have dealt only with such crystals in which the quantities named change only insignificantly with a change of direction. At the same time they point out that, for example, the surface of the elastic moduli of crystals of the cubic system is not a sphere and has the symmetry of a cube; that earlier the surface of the magnitude of magnetization of crystals of the cubic system was considered a sphere, whereas now it has turned out that for ferromagnetic crystals it is not a sphere, but a surface having the symmetry of a cube, etc.

In this note we wish to show that, despite a certain soundness of the doubts indicated and similar to them, the coexistence in one and the same crystal of such seemingly contradictory and mutually exclusive properties as discontinuity and continuity, isotropy and anisotropy, etc., is logically necessary.

Let, for example, we have some piece of an ideally constructed crystal of rock salt (Fig. 1). Like any other crystal, our crystal has a lattice structure, possesses the properties of anisotropy, and belongs to one of

32 crystallographic symmetry classes, in the present case—to the class \(\bar{6}/4\).

Let us mentally select from our crystal, along its cleavage planes, a region in the form of a cube \(ABCD\), and see what will happen to it under thermal expansion, i.e. under uniform heating of the crystal that does not cause phase transformations. It is obvious that the cube \(ABCD\) must thereby be transformed into a cube \(A'B'C'D'\) of somewhat larger size. In an ideal crystal, no curvature of the faces of the cube can occur in this process, since otherwise it would lose its lattice structure, i.e., in exact accordance with the generally accepted definition of the concept “crystal,” it would cease to be a crystal. It cannot be admitted that a crystal of rock salt has a strictly lattice structure only at some one temperature and changes it upon heating and cooling.

Fig. 1.

Having accepted this, let us determine what the coefficient of expansion of the crystal must be along its different directions. From the drawing it is seen that in the direction \(OE\) the linear coefficient of thermal expansion \(\alpha\) is equal to \(\dfrac{EE'}{OE \cdot \Delta t}\), where \(\Delta t\) is the difference between the final and the initial temperatures. This follows from the fact that, upon heating, the point \(E\) passes into the point \(E'\) and the vector \(OE\) into the vector \(OE'\). For the same reason the coefficient of expansion in the direction \(OF\) must be equal to \(\dfrac{FF'}{OF \cdot \Delta t}\), and in the direction \(OB\) it must be equal to \(\dfrac{BB'}{OB \cdot \Delta t}\). From the similarity of the triangles \(OEF\) and \(OE'F'\), and of the triangles \(OEB\) and \(OE'B'\), it follows that

\[ \frac{EE'}{OE}=\frac{FF'}{OF}=\frac{BB'}{OB}. \]

This means that the coefficients of expansion of our ideal crystal in all directions are exactly equal to one another. This also means that the surface of the coefficients of expansion, i.e. the geometrical locus of the endpoints of radius vectors, equal in magnitude and direction to the coefficients of expansion, is an ideally constructed sphere.

In our drawing, the origin of coordinates was taken to be a point coinciding with a lattice node, and the directions along which the coefficient of expansion was determined were taken to be rational crystallographic directions. It is not difficult to see that these conditions are not necessary. To show this, let us repeat our argument for a cube \(ABCD\), oriented in the crystal lattice quite arbitrarily (Fig. 2). As before, as a result of thermal expansion such a cube must be transformed into a cube \(A'B'C'D'\) of somewhat larger size. At the same time all rows of nodes of the first cube must pass, without curvature, into rows of nodes of the second cube, and the faces of the first cube, in the general case irrational, must pass without curvature into the corresponding faces of the second cube.

Fig. 2.

The last remark should be understood as follows. If the irrational face \(BC\) passes through the points \(E\), \(F\), which are not lattice nodes, then the face \(B'C'\) must pass through the analogously situated points \(E'\), \(F'\) in it. Under thermal expansion the point \(E\) passes into \(E'\), the point \(F\) passes into \(F'\), and, in general, every point on the surface of the original cube passes into the corresponding point on the surface of the second cube. From the similarity of the triangles \(OEF\) and \(OE'F'\) it follows, however, that

\[ \frac{EE'}{OE}=\frac{FF'}{OF}, \]

and this, as before, means that the coefficients of expansion in all directions again turn out to be equal to one another. This also means that in this case too the surface of the coefficients of expansion is a sphere, and indeed the very same sphere, although its center is now a point that does not coincide with a lattice node.

It follows from what has been set forth that an ideal crystal of rock salt, and in general all crystals of the cubic system, while manifestly anisotropic with respect to a number of properties (elasticity, strength, etc.), are isotropic with respect to thermal expansion.

Since, in describing the phenomenon of thermal expansion, any point of the crystal lattice (nodal or non-nodal) may be considered the center of the spherical surface of the coefficients

expansion, i.e., all points of the lattice are equivalent and indistinguishable from one another, then all crystals can and must be considered, with respect to the phenomenon indicated, as media of a homogeneous continuous structure.

All crystals of the cubic system, with respect to thermal expansion, have the symmetry of a sphere, \(\infty/\infty \cdot m\).

Similarly, one can prove that the surface of the coefficients of thermal expansion of crystals of the hexagonal and tetragonal systems has a circular section in the plane normal to the principal axis of the crystals.

What has been set forth shows once again that symmetry and dissymmetry, isotropy and anisotropy, continuity and discontinuity of crystals are relative properties. There is no symmetry of crystals “in general”; rather, there is symmetry of crystals with respect to one or another of their properties. There is no anisotropy of crystals “in general”; rather, there is anisotropy of crystals with respect to certain of their properties. There is no discontinuous (lattice) structure of crystals “in general”; rather, there is a discontinuous structure with respect to certain of their properties.