90 TheLS-coupling scheme
5.4 Selection rules in theLS-coupling scheme
Table 5.1 gives the selection rules for electric dipole transitions in the
LS-coupling scheme (listed approximately in the order of their strict-
ness). The rule forJreflects the conservation of this quantity and is
strictly obeyed; it incorporates the rule for ∆jin eqn 2.59, but with
the additional restrictionJ=0J′= 0 that affects the levels with
J= 0 that occur in atoms with more than one valence electron. The
rule for ∆MJfollows from that for ∆J: the emission, or absorption, of
a photon cannot change the component along thez-axis by more than
the change in the total atomic angular momentum. (This rule is rele-
vant when the states are resolved, as in the Zeeman effect described in(^15) There is no simple physical ex- the following section.) (^15) The requirement for an overall change in parity
planation of why an MJ =0to
MJ′ = 0 transition does not occur
ifJ =J′; it is related to the sym-
metry of the dipole matrix element
〈γJ MJ=0|r|γ′JMJ=0〉,whereγ
andγ′ represent the other quantum
numbers. The particular case ofJ=
J′=1and∆MJ= 0 is discussed in
Budkeret al. (2003).
and the selection rule for orbital angular momentum were discussed in
Section 2.2. In a configurationn 1 l 1 n 2 l 2 n 3 l 3 ···nxlxonly one electron
changes its value ofl(and may also changen). Therulefor∆Lal-
lows transitions such as 3p4s^3 P 1 –3p4p^3 P 1. The selection rule ∆S=0
arises because the electric dipole operator does not act on spin, as noted
in Chapter 3 on helium; as a consequence, singlets and triplets form
two unconnected sets of energy levels, as shown in Fig. 3.5. Similarly,
the singlet and triplet terms of magnesium shown in Fig. 5.9 could be
rearranged. In the mercury atom, however, transitions with ∆S =1
occur, such as 6s^21 S 0 –6s6p^3 P 1 , that gives a so-called intercombination
(^16) This line comes from the second level line with a wavelength of 254 nm. (^16) This arises because this heavy atom
in the table given in Example 5.1, since
0.254μm= 1/(3. 941 × 106 m−^1 ).
is not accurately described by theLS-coupling scheme and the spin–
orbit interaction mixes some^1 P 1 wavefunction into the wavefunction
for the term that has been labelled^3 P 1 (this being its major compo-
nent). Although not completely forbidden, the rate of this transition is
considerably less than it would be for a fully-allowed transition at the
same wavelength; however, the intercombination line from a mercury
lamp is strong because many of the atoms excited to triplet terms will
decay back to the ground state via this transition (see Fig. 5.9).^17
(^17) Intercombination lines are not ob-
served in magnesium and helium. The
relative strength of the intercombina-
tion lines and allowed transitions are
tabulated in Kuhn (1969).
5.5 The Zeeman effect
The Zeeman effect for atoms with a single valence electron was not
presented in earlier chapters to avoid repetition and that case is covered(^18) Most quantum mechanics texts de- by the general expression derived here for theLS-coupling scheme. 18
scribe the anomalous Zeeman effect for
a single valence electron that applies to
the alkalis and hydrogen.
The atom’s magnetic moment has orbital and spin contributions (see
Blundell 2001, Chapter 2):
μ=−μBL−gsμBS. (5.9)
The interaction of the atom with an external magnetic field is described
byHZE=−μ·B. The expectation value of this Hamiltonian can be
calculated in the basis|LSJMJ〉, provided thatEZEEs−oEre,
i.e. the interaction can be treated as a perturbation to the fine-structure