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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