236 
THE COEFFICIENT OF OPACITY 
Dividing this by the energy of a quantum Jiv we obtain the number of 
captures per second 
sn 
, 6477 4 Z 4 e 10 / 
.(163-1), 
3\/3 c 3 h 4 mVv 
where we have inserted the value i/r x = 27T 2 mZ 2 e i /h 2 from (42-62). 
An ionised atom and a free electron will be regarded as a system in 
state 2 in the argument of § 36. The total number of such systems in the 
cubic centimetre is sn', i.e. there are sn' combinations each having a certain 
chance of transformation to a system in state 1. Hence the coefficient 
6 2 i giving the probability of a single system returning to state 1 in unit time 
by the capture process is 
64t7 4 Z% 10 / 
~ 3V3 c 3 hWv 
By (40-63) the atomic absorption coefficient is 
q 2 c 2 6 91 
a = — 
.(163-2). 
q 1 8nv 2 dv ’ 
Also by (45-6) 
q* = p 4 t tVhv, 
since dxdydz is here 1 cu. cm. Again, for a change of V 
Hence 
hdv = mVdV. 
_ m 2 c 2 V j 
a 2q 1 h 2 v 2 21 
3277 4 e 10 mf 
3V3 q^h 6 
Z 4 A 3 (163-3), 
where A = cfv = the wave-length of the radiation absorbed. 
Inserting numerical values we have 
/ 
.(163-4). 
a = -0052 Z 4 A 3 
qi 
The proportionality of a to Z i X 3 agrees with a well-known experimental 
law so that this prediction is confirmed in a highly satisfactory manner. 
To evaluate further the numerical coefficient, consider, for example, the 
ionisation of a A electron. Since the system in state 2 consists of a fully- 
ionised atom and a free electron, our calculation of a applies to an atom 
in state 1 containing just 1 electron in a A orbit. As an approximation 
we shall neglect the interference of the electrons with one another and 
suppose that each A electron in a complete atom gives the absorption 
coefficient (163-4). We have q x = 2 and by (159-3)/ = (|)~ 2 — (f) -2 = - 3 /; 
including a factor 2 to allow for the two A electrons in each atom the 
result 18 a K = 0-0185Z 4 A 3 (163-5).