234 
THE COEFFICIENT OF OPACITY 
undoubtedly contains a great deal of truth, the details are as yet somewhat 
tentative, and, moreover, in its present form, it accounts for only a small 
part of the stellar opacity. 
Comparison with Laboratory Experiments. 
161. If Kramers’ theory were merely a speculation as to the manner 
in which classical laws pass over into quantum laws, it would not greatly 
disturb us to find astronomical results partly discordant with it. But the 
theory has been compared with laboratory experiments and found satis 
factory, so that astronomical conflict with Kramers’ theory is virtually 
a conflict with laboratory experiment. 
In an X ray tube a stream of electrons, all with the same velocity F 
acquired under a known fall of potential, falls on the material of the anti 
cathode. In this case the atoms have their full complement of electrons 
so that there is no opportunity for capture, and only spectrum a can be 
emitted. Within the anticathode the electrons gradually lose their energy 
chiefly by other dissipative causes and only to a small extent by the radia 
tion of spectrum a. The total spectrum is therefore due to electrons with 
all velocities from 0 to F. We can measure the radiation J V dv corre 
sponding to an initial velocity F, and (J v + AJ„) dv corresponding to an 
initial velocity F + AF; then (A J v )dv represents the spectrum emitted 
by the electrons whilst their velocity falls from F + A F to F —after which 
fall they are in a position to radiate J v just as if they were first entering 
the anticathode*. Hence the spectrum due to electrons of the same 
velocity F is of intensity Q v dv, where 
<2„ = AJ„ = ^.-AF (161-1). 
and Q v should be given by (155-42) provided that s represents the number 
of atoms per sq. cm. in a thickness of anticathode such that the average 
electron velocity diminishes by A F in traversing it. 
The best experimental determinations of J v appear to be those of 
H. Kulenkampfff. Due correction has been made for the absorption by 
the anticathode of the radiation emitted within it. Deriving Q v by (161-1), 
these experiments show that the spectrum Q v is of uniform intensity up 
to v 0 and ceases abruptly at v 0 . Further, dJJdv 0 is proportional to Z and 
independent of v 0 , so that dJJdV is proportional to ZV . According to the 
Thomson-Whiddington law the decrease of velocity A F in a sheet contain 
ing a constant number of atoms 5, is proportional to Z/F 3 ; hence by 
* This is only true because the proportion of electrons concerned in radiating 
spectrum a is small; the electrons radiating At7„ have their velocities suddenly 
reduced by a large amount and can take little part in the further radiation J„. 
f Ann. d. PhysiJc, 69, p. 548 (1922).