20 BELL SYSTEM TECHNICAL JOURNAL 



But because of the symmetry k is unchanged by this transformation, 

 hence 



k = k' 



Two matrices can be equal only if corresponding terms are equal, hence 



kn = ^22 , kn = — kn = 0, ^23 = — ^31 and ku = ^23 



whence 



^23 = hi = 0. 



We are left then, with the dielectric constant matrix for the tetragonal 

 bisphenoidal class: 



/kn \ 



^ = I y^ii I 



\0 W 



Applying other transformations possible for tetragonal crystals gives no 

 further simplification. 



If we go through all the symmetry transformations possible for the 32 

 classes we find that cubic crystals require but one dielectric constant, hex- 

 agonal, trigonal and tetragonal crystals require two constants, orthorhombic 

 monoclinic and triclinic crystals require 3. 



As the triclinic class has no fixed axes or planes of symmetry the reduction 

 of its 6 constants to 3 is not so obvious. It may be seen by expanding into 

 ordinary xyz coordinates, that pckp = 1 is the equation of an ellipsoid, (p 

 is the radius vector) where the six ^'s are the coefficients oi x , y , z , yz, 

 zx and xy respectively. If we choose the coordinate axes along the axes of 

 the ellipsoid the yz, zx and xy terms drop out and only three ^'s are needed. 

 With triclinic crystals then, if we determine the axes of the ellipsoid, then 

 choose the coordinate axes along them, only three dielectric constants are 

 needed to completely specify the polarization in terms of the electric field. 

 The determination of the ellipsoid axes must be made experimentally as 

 there are no symmetry elements to guide us. It is possible to compute the 

 positions of the axes from the 6 ^'s by solving a cubic equation. 



The values of the ^'s depend on the frequency of the applied field. In 

 crystals of low symmetry the ellipsoid axes for different frequencies do not 

 necessarily coincide. 



Another vector quantity of interest is the polarization, P = D — —- E. 



47r 



