296 BELL SYSTEM TECHNICAL JOURNAL 



with a major fact of experience: the energy-values of the atom in its 

 various "magnetic levels," as I shall hereafter call them, are known 

 from spectroscopy to follow upon one another in a uniform evenly- 

 spaced sequence. This is just what the rule requires: for if M denotes 

 the magnetic moment of the atom and dj the angle which it makes with 

 the 2-direction, the energy due to the field is equal to Mil cos dj, 

 which is MH{mjl^j{j + 1), and this changes by uniform steps as m 

 is changed from each of its permitted values to the next. 



Now recalling the importance of the ratio of magnetic moment to 

 angular momentum, and noticing that Mj^jij + 1) is none other 

 than this ratio, and introducing the g-factor of equation (7), and the 

 precession co of equation (8) , we may write for A U the energy-difference 

 between one magnetic level and the next: 



At/ = g{e/2mc)II = w, (9) 



so that a measurement of the energy-difference or separation of two 

 magnetic levels of an atom gives immediately the value of g for that 

 atom. One sees at once how to make a special test of the value 2 

 assigned to the g-factor of the electron-spin; for when the sodium 

 atom is in any state for which / = 0, it is the spinning valence-electron 

 which contributes the whole of the magnetic moment and the angular 

 momentum of the atom; and when from the spectrum of sodium vapor 

 in the magnetic field the value of AU is determined for these states, 

 it is precisely the value 2(e/2mc)H which is found. 



To get a notion of what actually happens in the general case, it is 

 best to take a sheet of paper and make a graphic composition of the 

 angular momenta. These three — s, I, and j, to denote them by their 

 quantum-numbers — are to be laid down as a triangle having sides of 

 the lengths V5(F+^, V/(/ + 1), and Vj'O' + 1) ; for convenience I 

 drop out the common factor h/lir for the next few lines. The cosines 

 of the three angles are obtained in terms of the sides by applying the 

 well-known trigonometric formula and getting three equations of which 

 here is one, 



sis -f 1) = /(/ + 1) + jO" + 1) - 2V/(/+ l)Vi(j+ 1) cos e,, ,. (10) 



The magnetic moment of the orbital electron-motion is a vector 

 parallel to / and of the length g{el2mc)^l{l -f 1), with unity put as the 

 value of g. The magnetic moment of the electron-spin is a vector 

 parallel to 5 and of the length g(e/2mc)yls{s + 1), with two put as the 

 value of g. Owing to the inequality of these g factors, the resultant 

 of the magnetic moments is not parallel toj. We could easily calcu- 

 late its magnitude and direction, but they are not relevant. It is 



