valence electron in such atoms, selection rules are dictated by the allowed
changes in and m:
1
m0, 1The values of the jquantum number depend on the values of the and m
quantum numbers. In these cases, no selection rule depends on j(although it
is recognized that the value of the jquantum number is dictated by the re-
spective values ofand m).
Example 15.4
In his pioneering investigations around 1814 into the spectrum of the sun,
Joseph von Fraunhofer labeled an intense yellow line of the spectrum with
the letter D. It was shown that this emission was due to the sodium atom (in-
dicating, by the way, the presence of sodium in the sun), so this was eventu-
ally termed the “sodium D line.” Later work showed that under high resolu-
tion the sodium D line is actually a pair of closely spaced lines, separated by
6 Å. Assuming that in the lower electronic state of the sodium atom the va-
lence electron has the quantum numbers (n,,j) (3, 0,^12 ), what are the pos-
sible quantum numbers for the upper electronic state?Solution
Since there is no selection rule for n, we cannot say which principal quan-
tum number can be assigned to the upper electronic state. (It is actually due
to an n3 to n4 transition.) However, we can use the selection rules for
and mto determine what the quantum numbers for the upper electronic
state must be for intense—and presumably allowed—transitions. Since
0 for the lower state, the upper state must have 1 (since 1 is not
possible). This means that the upper electronic state must be a porbital. Since
the mquantum number for the lower electronic state must be 0 ( 0
means that mmust be 0), then the upper electronic state must have an m
of either 1, 0, or 1. Such states are degenerate unless a magnetic field is pres-
ent. However, for a single electron,s^12 , so from the upper state’s value of
we can determine that the possible jvalues are ^12 or ^32 . There are, then, two
possible combinations of quantum numbers for the upper state: (n,,j)
(n,1,^12 ) or (n,,j) (n,1,^32 ). The different quantum numbers—j, in par-
ticular—imply that there will be a slightly different energy for the two upper
states. This is the reason there are two closely spaced lines in the spectrum
of Na.Although this is a relatively simple example, it points out a key factor in the
understanding of electronic spectra of atoms: the fact that orbital and spin
angular momenta interact, or couple.Coupling is even more important in the
understanding of electronic spectra of atoms that have more than one electron
in their valence subshell, because now the orbital and spin angular momenta
of different electrons can couple with each other.This makes the spectra po-
tentially more confusing. Luckily, there is a procedure for formalizing the cou-
pling possibilities between more than one electron in a valence subshell. That
will be considered in the next section.
15.4 Angular Momenta: Orbital and Spin 525