THE ZEEMAN EFFECT 
xv. 4] 
213 
and triplets the final result is a normal Zeeman triplet, in accord 
with the experimental observations of Paschen and Back. 
To correlate the experimental results as regards multiplicities 
it is necessary to introduce the quantum number j, the “ inner 
quantum number.” Thus a spectral term is, as we have seen, 
associated with a total quantum number n, an azimuthal num 
ber k, and an inner quantum number/ For S, P, D . . . terms, 
k has the values 1, 2, 3 . . . and normally only those transitions 
occur for which k changes by unity. According to “ the selec 
tion principle for inner quantum numbers ” the only transitions 
possible are determined by 
Thus, in this case, the transition A j = o is not forbidden. 
Sommerfeld has assigned values for j corresponding to the 
S, P, D terms, and to various values of the multiplicity r. In 
his scheme j is integral for odd multiplets, but for even multi- 
plets is an integer plus one-half. "In a magnetic field each nkj 
term splits up into 2j + 1 equidistant components ; and the 
separation of these from the original position expressed in wave 
numbers is given by 
Av = mgAv n 
15:9 
Here Av n i s the normal Larmor separation (0H/471:m 0 c), m the 
magnetic quantum number which takes the values j, j — 1, 
j — 2, j — 3 . . . — j for each j term (and hence is integral 
for odd multiplets, and half-integral for even multiplets), and g 
the Landé ‘ splitting factor ’ which may be written as 
The expression then covers completely the observations on the 
Zeeman effect in a weak magnetic field for ordinary spectra.” 
We see, then, that the splitting factor of Lande is an empirical 
number which determines the change in the frequency of a term 
due to a magnetic field. If we substitute Sommerfeld’s values 
(pp. 65-6) for r and k, viz. - = s + £, k = l + 1, in the above 
equation, we find 
g = f+ - 
which reduces to 
{( s + j) — (¿+ |)}{( s + I) + [l + I)} 
2j{j + I) 
g = I + 
j(j + 1) + s(s + I) - 1(1 + 1) 
2j{j + I) 
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