The Mathematics of Physical Properties of Crystals

T HE use of crystals as oscillating elements and as light valves in electric circuits has given the mathematics of crystalline media an engineering importance. Soon after the first simple quartz oscillators were made it was noticed that some ways of cutting the block from the natural crystal gave lower temperature coefficients of frequency than did other ways. This led to studies of the change of elastic modulii with direction and temperature and finally to the discovery that there are directions in quartz for which the shear modulus does not change with temperature. Such computations are rather involved, and there is, in the English language, no general reference book on these new problems. The existing works were evidently not written with the idea in mind that anyone would ever actually do much numerical work with directional properties of crystals, since the methods used are not the best suited to this. The matrix algebra has the advantages of a symbolic algebra and is also, through the concept of matrix multiplication, a scheme for computing numerical results. As the problem of temperature coefficients of frequency involves the temperature coefficient of expansion, the temperature coefficient of density and the temperature coefficient of elastic modulii, these problems must be put into the language of matrix algebra so that they will fit into the general structure being built for more difficult problems. For this reason, after an introduction to the idea of linear vector functions, through consideration of the relation between the electric field and the induction in a crystal, and a hasty sketch of symmetry types found in crystals, we proceed to the consideration of stress and strain and their relations to each other.