432 C. Travis — Behavior of Crystals in Light 



or, dropping the subscript, 



* = ^rw (4) 



2 V being the angle between the optic axes, and ~k a small con- 

 stant. Under the same conditions, (3) may be written 



4, = VkK (5) 



Hence in the figure given by either kind of crystal, the loci of 

 points of equal phase difference A are concentric circles. 

 When A is an integer, the factor sinVA in (1) vanishes ; when A 

 is an odd number of times one-half, sinVA attains its maximum 

 value, which is unity. The dark circles around the axis are 

 the loci of points for which A is equal to n, and the bright 

 ones those of points for which A is equal to n— -J, n being an 

 integer. 



It is evident that the average intensity of the portion of the 

 figure included in a circle whose center is the axis, increases 

 with the size of the circle until this coincides with the first 

 ring of maximum brightness ; after this the average intensity 

 fluctuates, but undergoes no great change. If, then, a crystal 

 section is illuminated by a pencil of angular radius r, the 

 intensity of the light transmitted is zero when r is zero, 

 increases with r until r is equal to the radius of the first bright 

 ring, and is approximately constant as r increases beyond this. 



The only important difference between biaxial and unaxial 

 crystals, in this regard, lies in the difference between the radii 

 of the first bright rings in the two cases. If the birefringence 

 and the thickness of section are the same for both, h (equations 

 (4) and (5)) has roughly the same value for either kind of crys- 

 tal. But the radius of the first bright ring for the biaxial 

 crystal is 



<j>b ~ sinW 

 whereas for the uniaxial crystal it is 



<K = VP" 



and if k is very small (as it commonly is), <£ u is much larger 

 than <£ b . 



This is well shown by comparing sections l mrn thick of 

 aragonite and calcite. It is to be noted that while calcite has 

 the greater birefringence, the first bright ring for this substance 

 is six times as large as that for aragonite. In the calculation,* 

 sodium light is assumed, for which A = '000589 mm . 



* The radii of the circles of maximum and minimum intensity are calcu- 

 lated as they appear after refraction from the crystal into the air. 



