50 
QUANTUM THEORY 
Planck’s Law correspondingly shortened. We have here preferred to 
avoid this excursion into an extraneous subject, and in the present 
derivation Boltzmann’s formula is obtained from pure quantum theory. 
By (37-5) 
(7 = ^21 __ ^21 £2 1 
a 21 v V2 a i2 7l v 12 3 
so that — = C — v 12 3 (38-4), 
% 2 q 2 2 
giving the relation between the coefficients of absorption and of spontaneous 
emission. 
Also, considering the atoms in state 2, the emission per atom at 
* temperature T is to the emission per atom at temperature zero in the 
ratio 
6 21 + ®21 ^ ( v 12) T) : 6 2 1 
by (36-3). This is equal to 
CV + '■ C V = (1 (38-5). 
The ratio is greater than unity, so that emission is stimulated by the 
presence of radiation in the field. This stimulated emission is called by 
Einstein negative absorption. 
As an example of this formula consider a radio-active process consisting 
of a simple readjustment of the nucleus of an atom with emission of a 
y ray of frequency v 12 . The effect of raising the temperature is to increase 
the radio-activity in the ratio given by (38-5). The frequency of y rays is so 
high that even a temperature of 10 7 degrees (in the interior of a star) makes 
no appreciable difference to the radio-activity. 
When the atoms are crowded together in dense material, the absorbing 
and emitting power of the individual atom may be to some extent modified 
by the proximity of its neighbours, so that a 12 , a 21 and b 2l are then not 
purely atomic constants. The extent of the interference will depend on 
p and T, and there is a breakdown of the foregoing argument which as 
sumes that axa, etc. are independent of T. This does not in any way affect 
the proof of Planck’s Law, because we have proved that the distribution 
law is the same for diffuse and dense matter; we determine once for all 
the form of the universal function / by considering a diffuse distribution 
which lends itself to simple treatment. But Boltzmann’s formula is 
deduced only for diffuse matter in which the atoms are so far apart as to 
act independently; it becomes inaccurate in dense matter. 39 
39 . The present investigation is not confined to transitions in which 
the atom remains intact. It applies also to transitions in which an electron 
is expelled from or captured by an atom with absorption or emission of