0198506961.pdf

102 Hyperfine structure and isotope shift

Table 6.1Comparison of fine and hyperfine structures.

Fine structure in the Hyperfine structure in the

LS-coupling scheme IJ-coupling scheme

Interaction βL·S AI·J

Total angular momentum J=L+SF=I+J

Eigenstates |LSJMJ〉|IJFMF〉

Energy,E β 2 {J(J+1)−L(L+1)−S(S+1)} A 2 {F(F+1)−I(I+1)−J(J+1)}

EJ−EJ− 1 =βJ EF−EF− 1 =AF

Interval rule (ifE

s−o<Ere)(ifA^ ∆EQuadrupole)

That exercise also shows how this rule can be used to deduceF and

(^12) This interval rule for magnetic dipole hence the nuclear spinIfrom a given hyperfine structure. 12
hyperfine structure can be disrupted by
the quadrupole interaction. Some nu-
clei are not spherical and their charge
distribution has a quadrupole moment
that interacts with the gradient of the
electric field at the nucleus. This elec-
tric quadrupole interaction turns out to
have an energy comparable to the inter-
action of the magnetic dipole moment
μIwithBe. Nuclei, and atoms, do not
have static electric dipole moments (for
states of definite parity).
The hyperfine-structure constantA(n, l, j) is smaller forl>0than
forl=0andthesamen. Exact calculation shows that the hyperfine-
structure constants of the hydrogenic levels np^2 P 1 / 2 and ns^2 S 1 / 2 are
in the ratio
A(n^2 P 1 / 2 )
A(n^2 S 1 / 2 )

=

1

3

. (6.15)

This ratio is smaller in the alkalis, e.g.∼ 1 /10 in the examples below,

because the closed shells of electrons screen the nuclear charge more

effectively for p-electrons than for s-electrons.

6.1.4 Comparison of hyperfine and fine structures

The analogy between hyperfine and fine structures is summarised in

Table 6.1.

For fine structure in the alkalis we found the Land ́e formula (eqn 4.13)

EFS∼

Zi^2 Zo^2

(n∗)^3

α^2 hcR∞. (6.16)

TheZ^4 scaling for a hydrogenic system is reduced toEFS∝Z^2 for neu-

tral atoms since the effective outer atomic number isZo=1,andZi∼Z

gives a reasonable approximation in the inner region. Applying similar

considerations to the hyperfine structure shows that the dependence on

Z^3 in eqn 6.10 reduces to

EHFS∼

ZiZo^2

(n∗)^3

me

Mp

α^2 hcR∞. (6.17)

The mass ratio arises fromμN/μB=me/Mp. Hyperfine structure scales

asZ, whereas fine structure scales asZ^2 ;thusEHFSvaries much less

thanEFS, as the following comparison of the splittings for Na and Cs

shows.