230 
THE COEFFICIENT OF OPACITY 
another, and some perhaps will escape without radiating. It is in accord 
ance with the general idea of the Correspondence Principle that the 
statistical average of these quantum radiations will amount to the radiation 
given by the classical theory. To put it another way, the Fourier terms in 
the classical radiation are interpreted, not as representing actual radiation 
of that frequency from an accelerated electron, but as probabilities of 
radiation of that frequency. With a large number of electrons it makes 
no difference whether each electron radiates of a quantum or has a 
chance of radiating a whole quantum; so the classical theory should 
give the total radiation of a large number of electrons correctly. 
Turning to the spectrum (/3), the new point arises that the electron 
after capture must be in one of the quantised orbits within the atom, so 
that its final negative energy must have one of a discrete series of values. 
Thus the electrons can emit only frequencies forming a discrete series, 
v K , v L , given by 
hv K = |mF 2 + ip K , hv L = fmF 2 + rfs L , hv M = V 2 + ifj M , (159-2), 
where — i Pk> ~ m> •••> are the energies of the K, L, M, ..., orbits 
in which it can find a resting-place. 
Thus the quantum theory predicts a line spectrum whilst the classical 
theory predicts a continuous spectrum. 
In discussing the theory of “weights of states” in § 48 we have seen 
that as the periodicity becomes more perfect the weight of each unit cell 
becomes more and more strongly concentrated into the single quantised 
orbit contained in that cell. We are scarcely going beyond this principle 
if we suppose that the captured electrons which in the absence of periodicity 
would have been distributed over the cell, are, when periodicity is present, 
to be found concentrated on the quantised orbit which has drained the 
weight of the cell. If the electron had been captured by a very complicated 
system in which the orbits had little or no periodicity so that no quantisa 
tion occurred, we should have had no reason to anticipate a breakdown of, 
the treatment adopted for spectrum a. It is therefore likely that we may 
apply the same principle to spectrum /3, but with the addition that the 
classical radiation corresponding to each cell is heaped up into a single line 
corresponding to the quantised orbit in that cell. 
If it is a question of capture of an electron by an isolated nucleus, the 
dividing lines of the cells are presumably as follows. If ^ (= — j/^) is the 
energy in a one-quantum orbit the energy in an ^-quantum orbit is 
Hence 
t 1 Tr ” 1 ' J1 ' J 1 '