MATHEMATICS OF PHYSICAL PROPERTIES OF CRYSTALS 19 



'an an a^^ 



Ar 



a = I 021 ail a^z I ^^ Oa 



Vsi 032 033/ *^i,^5»* 



It is shown in the sec. 4 that any vector V = I ^2 I can be written on the 



\Vzl 

 new system of axes as V where V — aV, conversely V = a W'; a ^ is the 

 matrix reciprocal to a. Since the induction D and the electric field E are 

 simple vector functions they transform as the vector V, that is: 



D' = aD (6.3) 



E' = aE (6.4) 



But by (6.1) 



whence: 



or 



D = ^kE 



Aw 



aD = — - akoc aE 

 At 



D' = ^ k'E' (6.5) 



4 



IT 



k' = akac (6.6) 



We see that the form of (6.5) is the same as that of (6.1) for any set of 

 axes if (6.6) is used to define the new dielectric matrix k. 



To apply this relation (6.6) to a particular crystal let us consider a tetrag- 

 onal crystal (which has its properties unchanged by a rotation of 90° 

 about a four fold axis) . Let us choose the four fold axis as xz and then rotate 

 the axis 90° about Xz . In this case 



and the reciprocal matrix a 



whence equation (6.6) becomes: 



^n^i2-^3i\ /O -10\ / h2 -kn 

 *i2^22*23) 1 00) = ( -kn kn 



hihzhzl \0 1/ \ hz -kzi 



