bei48482_FM

260 Appendix to Chapter 7

for allowed transitions here is

Selection rule l    1

which is illustrated by the transitions shown. The principal quantum number ncan

change by any amount.

To indicate some of the detail that is omitted in a simple diagram of this kind, the

detailed structures of the n2 and n3 levels are pictured. Not only are all sub-

states of the same nand different jseparated in energy, but the same is true of states

of the same nand jbut with different l. The latter effect is most marked for states of

small nand l, and was first established in 1947 in the “Lamb shift” of the 2^2 S 1  2 state

relative to the 2^2 P 1  2 state. The various separations conspire to split the H spectral

line (n 3 →n2) into seven closely spaced components.

Sodium

The sodium atom has a single 3selectron outside closed inner shells, and so if we

assume that the 10 electrons in its inner core completely shield 10 eof nuclear charge

(which is not quite true), the outer electron is acted upon by an effective nuclear charge

of ejust as in the hydrogen atom. Hence we expect, as a first approximation, that

the energy levels of sodium will be the same as those of hydrogen except that the

lowest one will correspond to n3 instead of n1 because of the exclusion principle.

Figure 7.23 is the energy-level diagram for sodium. By comparison with the hydrogen

levels also shown, there is indeed agreement for the states of highest l, that is, for the

states of highest angular momentum.

To understand the reason for the discrepancies at lower values of l, we need only

refer to Fig. 6.11 to see how the probability for finding the electron in a hydrogen

atom varies with distance from the nucleus. The smaller the value of lfor a given n,

the closer the electron gets to the nucleus on occasion. Although the sodium wave

functions are not identical with those of hydrogen, their general behavior is similar.

Accordingly we expect the outer electron in a sodium atom to penetrate the core of

inner electrons most often when it is in an sstate, less often when it is in a pstate,

still less often when it is in a dstate, and so on. The less shielded an outer electron

is from the full nuclear charge, the greater the average force acting on it, and the

smaller (that is, the more negative) its total energy. For this reason the states of small

l in sodium are displaced downward from their equivalents in hydrogen, as in

Fig. 7.23, and there are pronounced differences in energy between states of the same

nbut different l.

Helium

A single electron is responsible for the energy levels of both hydrogen and sodium.

However, there are two 1selectrons in the ground state of helium, and coupling affects

the properties and behavior of the helium atom. These are the selection rules for allowed

transitions under LScoupling:

L0,    1

LSselection rules J0,  1

S 0

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