The Bell System Technical Journal 



Vol. XXII January, 1943 No. 1 



The Mathematics of the Physical Properties of Crystals 



By WALTER L. BOND 



SECTION 1 

 Introduction 



THE 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 coeflficient 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. 



Following these, we take up piezo electricity and the converse piezo 

 electric effects; these are important as they tell us the ways a crystal may be 

 driven. We have not seen anywhere a general proof that the modulii of 

 the converse effect are the same numbers as the modulii of the direct effect 

 — to the first order of small quantities, though Lippman predicted the 

 converse effect and demonstrated its magnitude to be about this; he ap- 



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