103 



ions, however, the reason for this must obviously be soughl in the lad llial in liie 



other elements the motion of the outer electron will, due to action of the inner 



electrons, not be simply periodic so that for a connection with the ordinary theorv 



of radiation the existence of several series of stationary states is required. Thus, as 



pointed out by Sommerfeld^), a clue to the interpretation of the spectra under 



consideration is offered by his fundamental theory considered in Part I on page 1 7 



of the stationary states of a system consisting of a particle moving in a central field 



of force. While for a simply periodic system the stationary states are characterized 



by the value of one integral positive number they are for this system determined 



by two such numbers 77i and 7)2 of which rii serves to fix the value of an integral 



of the type (15) corresponding to the radial motion and «2 fixes the value of the 



angular momentum of the particle round the centre. Comparing the effect of tlic 



inner electrons with that of a central field of force the potential of which may be 



represented bj^ a series of descending powers of the distance from the nucleus and 



putting 77i + 7J2 = n and 772 = t, Sommerfeld found that it is possible to obtain 



expressions for the energy in the stationary states which for constant r show a 



remarkable general resemblance with the empirical formulae of Rydberg and Ritz 



for fr(n) and which offer a suggestive interpretation of the fact that the empirical 



values of fj(n), looking apart from the eventual complex structure of the lines, can 



for the spectrum of an element generallj^ be arranged in a simple scheme of the 



following form: , , 



yi(l), yi(2), yi(3), 9)1(4), 



92(2), ^2(3), 92(4), 



ysO), 9)3(4), 



9i (4), 



in which (pj-in) approaches to unity for constant r and increasing n as well as for 

 constant 71 and increasing t. It will be observed that from this point of view the 

 structure of the series spectra of the other elements is analogous to the hydrogen 

 spectrum if the fine structure of the hydrogen lines is taken into account, and that the 

 difference consists only in the fact that in the latter case, due to the much smaller devi- 

 ations of the orbit of the electron from a periodic orbit, the functions f\ (n) show dilTe- 

 rences which are much smaller than the corresponding differences for the other spectra. 

 The above general view as regards the origin of the series spectra of the ele- 

 ments is supported in an instructive manner by the considerations of the former 

 section about the probabilities of transitions between the difTerent stationary slates of 

 an atomic system. Thus the displacement of an electron moving in a central field of 

 force will be given by a set of expressions of the same type as that given by (73) on 

 page 68, and we shall therefore assume that for this system only such transitions are 

 possible in which n^ varies by one unit, or what is the same, in which the angular 



momentum of the electron decreases or increases by z—- . This corresponds to the fact 



Z TT 



') A. Sommerfeld, Ber. Akad. München, 1915, p. 425; 191Ü, p. 131. 



14* 



