5.2 Thejj-coupling scheme 85Energy (eV)1Fine structureGross structure
Residual electrostatic energyHHHeHeNaNaMgMgCsCsHgHg10 10011010210 −^110 −^210 −^310 −^410 −^5Fig. 5.7A plot of typical energies as a
function of the atomic numberZ(on
logarithmic scales). A characteristic
energy for the gross structure is taken
as the energy required to excite an elec-
tron from the ground state to the first
excited state—this is less than the ion-
ization energy but has a similar varia-
tion withZ. The residual electrostatic
interaction is the singlet–triplet separa-
tion of the lowest excited configuration
in some atoms with two valence elec-
trons. The fine structure is the split-
ting of the lowest p configuration. For
all cases the plotted energies are fairly
close to the maximum for that type
of structure in neutral atoms—higher-
lying configurations have smaller val-
ues.electrostatic interaction. WhenHs−oacts directly on a configuration it
causes thelandsof each individual electron to be coupled together to
givej 1 =l 1 +s 1 andj 2 =l 2 +s 2 ; in the vector model this corresponds
tolandsprecessing aroundjindependently of the other electrons.
In thisjj-coupling scheme each valence electron acts on its own, as
in alkali atoms. For an sp configuration the s-electron can only have
j 1 =1/2 and the p-electron hasj 2 =1/2or3/2; so there are two
levels, denoted by (j 1 ,j 2 )=(1/ 2 , 1 /2) and (1/ 2 , 3 /2). The residual
electrostatic interaction acts as a perturbation on thejj-coupled levels;
it causes the angular momenta of the electrons to be coupled to give
total angular momentumJ=j 1 +j 2 (as illustrated in Fig. 5.8). Since
there is no external torque on the atom,MJis also a good quantum
number. For an sp configuration there are pairs ofJlevels for each of the
two originaljj-coupled levels, e.g. (j 1 ,j 2 )J=(1/ 2 , 1 /2) 0 ,(1/ 2 , 1 /2) 1
and (1/ 2 , 3 /2) 1 ,(1/ 2 , 3 /2) 2. This doublet structure, shown in Fig. 5.10,
contrasts with the singlets and triplets in theLS-coupling scheme.