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).