THE COEFFICIENT OF OPACITY 
241 
of upper limits. We shall therefore attempt to calculate more closely the 
line emission from electrons in high quantum orbits. 
By the correspondence principle there is continuity between free 
electrons and electrons in high quantum orbits; Kramers’ theory of 
emission from free electrons can therefore be pushed beyond the zero 
mark so as to include electrons of small negative energy. The investigation 
of § 155 refers strictly to parabolic orbits, but we may use it for eccentricities 
rather less than 1 as we have already used it for eccentricities rather 
greater than 1. By (155-1) and (155-2) Q v is given as a function of the 
angular momentum maV. In quantised orbits the angular momentum is 
given by the second quantum number n', so that we have the equivalence 
Consider a particular orbit (n, n', n"). The number of electrons per atom 
in this state is K/n'HT 
where — K/n 2 is the energy and B is expressed in terms of the density o- 0 
of the free electrons by (46-2), viz. 
The period of the orbit is n 3 h/2K. Hence the number of pericentron 
passages per atom per second is 
Keeping n' fixed we sum this for the values of n belonging to the high 
quantum orbits under consideration, say from n 0 to oo, where n 0 > n'. 
Replacing the summation by integration, the number of “encounters” 
(i.e. perihelion passages) becomes 
To each value of n' there correspond (n' + 1) values of n". We replace 
(n' + 1) by n' (thereby introducing an error not greater than a factor 2) 
and write the number of encounters per atom per second for all orbits in 
a range n' to n' + dn' equal to 
mcrV = n'hj^v 
(166-5). 
B- 
(166-6). 
By (155-1) and (166-5) the corresponding emission per atom is 
E 
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