﻿472 Messrs. C. G. Darwin and R. H. Fowler 



on 



From the general theorem (9*7) we at once have 



so that such vibrators have just twice as much energy as 

 the line vibrators. 



In exactly the same way we can treat the case of three 

 dimensions. To illustrate the weights we again take the 

 case of 2e and quantize the svstem in x, y, z. There are 

 six alternatives (2e, 0, 0), (0/2e, 0), (0, 6, 2e), (e, €, 0), 

 (e, 0, e), (0, e, e). The general form for re is J(r+ l)'(f + 2). 

 The partition function is now 



= (l-e)" 3 5 

 which leads at once to the expected result 



§ 11. Rotating Molecules. 



Another interesting example to which the calculations at 

 once apply is that part of the specific heat of a gas due to 

 the rotations of the molecules. Various writers * have 

 quantized the motions of a rigid body, and it is found that 

 the system has at most two instead of three periods, so that 

 it is partly degenerate. We may consider for simplicity a 

 diatomic molecule. Then, on account of the small moment 

 of inertia about the line of centres, the third degree of 

 freedom may be omitted altogether — its quantum of energy 

 is too large. A simple calculation then leads to energies of 

 rotation e r given by 



^=*5r 2 > c*d 



where I is the moment of inertia about a transverse axis, 

 which we shall assume to be independent of r. This is a 

 degenerate system, and considerations of the number of cases 

 which occur if it is quantized for the two degrees shows 

 that the weight to be attached is 2r-f 1. This is on the 

 principles suggested by Bohr f with a simplifying modifica- 

 tion; for Bohr had to suppose that certain quantized motions 

 were excluded for other reasons which are not operative 



* Among others, Ehrenfest, Verh. Deutsch. Phys. Ges. xv. p. 451 

 (1913). Epstein, Phys. Zeitschr. xx. p. 289 (1919;. F. Reiche, Ann. 

 der Physik, liv. p. 421 (1917). 



f Loc. cit. p. 26. 



