536 G. H. BEAVEN, E. R. HOLIDAY, AND E. A. JOHNSON 



ellipsoid. These are related to the non-oriented value (A') for the same 

 compound by the equation 



A' = {Ax + A^ + At)/S (biaxial case) 



or 



A' = i2Ax + A||)/3 (uniaxial case) 



The calculated value of A' can be used to estimate the thickness of the 

 crystal specimen by applying the Beer-Lambert absorption laws to solution 

 data for the same compound, assuming that the specific absorptivity (a or 

 e) is identical in the crystal and in solution (with identical concentration 

 units) . 



Except in certain favorable cases (e.g., hexamethylbenzene, see below) 

 the absorbing groups in crystals and oriented materials are not all parallel 

 to each other, but may be in arrangements in which the dichroism of one 

 set of parallel groups is neutralized, to a greater or lesser extent, by the 

 opposing dichroism of other sets of parallel groups. Consequently the ob- 

 served dichroic ratio is often much smaller (2 or less) than would be ex- 

 pected from simple theory. As emphasized by Seeds^"' the interpretation 

 that can be given to such low dichroic ratios depends on making a choice 

 between two assumptions. The system under study can be assumed to be 

 partly disoriented, so that the observed low dichroism is taken as some 

 measure of the degree of orientation. Such an assumption is obviously not 

 valid for crystals proper, for which, in any case. X-ray data might be avail- 

 able to indicate that the crystal lattice is unfavorable. For artificially ori- 

 ented materials of natural or synthetic origin, imperfect orientation is 

 much more likely, although errors arising from it can be minimized in 

 microspectrographic procedures by confining the measurements to the 

 specimen areas with the highest birefringence. It may then be assumed 

 that the specimen is fully oriented and that the observed dichroism is a 

 measure of the average angle of tilt of the absorbing groups in the molecule 

 to the principal axis of the specimen. If it is further assumed that the di- 

 chroic ratio for a planar ring is very large and that dichroism in the plane 

 of the ring is small, the variation of D with angle of tilt for some simple 

 model structures can be calculated. 



For the important practical case of a uniaxial fiber with rotational sym- 

 metry, in which the normals to the planes of the absorbing groups lie on a 

 cone of semi-angle 6 generated about the fiber axis, Fraser"^ has shown that 



Dfiber = Sin2 e/l - i Sin« 



"2 R. D. B. Fraser, Ph. D. Thesis, London University, 1951, quoted by Seeds. '"' 



