352 BELL SYSTEM TECHNICAL JOURNAL 



and the distribution-functions with respect to v, tv, r, s, and / differ 

 only by the substitution of the appropriate kinetic-energy term for 

 Ku, and (if necessary) of /i or /o or Is for m. 



For the distribution-function with respect to 9, we shall write an 

 equation copied after (5), as follows: 



dN = NCe exp (- Ve/kT) sin Odd 



= NCe exp {MH cos O/kT) sin ddd. (7) 



The constant Ce is to be so adjusted that the integral of dN over 

 the entire range of values of 6 (which extends from to vr) shall be 

 equal to A^. It turns out that 



Ce = a/ie" - e'") = a/2 sinh a (8) 



in terms of the parameter 



a = MII/kT, (9) 



which we shall use often enough to justify the special symbol for it. 

 The factor sin d in equation (7) requires comment. Imagine all the 

 molecular magnets brought together at a point P, and their axes 

 prolonged until these intersect a sphere of unit radius traced around 

 P as center. The locus, upon this sphere, of the points of intersection 

 of lines associated with magnets inclined at angles between 6 and dd 

 to the field is a belt or collar of area 2ir sin ddd. There are dN of 

 these points, and they are distributed over this belt with surface- 

 density dNjlir sin ddd. By making dN proportional to the product 

 of sin d into an exponential function, we make that surface-density, 

 which is the density-in-solid-angle of the directions of the magnetic 

 axes, proportional to the exponential function itself; and this is 

 what is done. 



We proceed to calculate the net magnetic moment of the assemblage 

 of N molecular magnets. Resolving the moment of each, we find 

 M cos d for its component parallel to the field-direction (with the 

 perpendicular component we are not concerned, since the average of 

 its values for all the molecules is obviously zero) . Summing the values 

 of these parallel components for all the molecules, we have: 



7 = r Mcos ddN, (10) 



-I 



the symbol / being used for the sum of the parallel components, since 

 this sum is precisely the intensity of magnetization per unit volume 

 defined near the beginning of this article. Remembering (7) and (8), 

 and performing the integration, we arrive at 



1 



