0198506961.pdf

2.3 Fine structure 37

has a magnitude close to one Bohr magneton (μB =e
/ 2 me). The
interaction of the electron’s magnetic moment with the orbital field gives
the Hamiltonian

H=−μ·B

=gsμBs·

mec^2

(

1

er

∂V

∂r

)

l. (2.48)

However, this expression gives energy splittings about twice as large
as observed. The discrepancy comes from theThomas precession—a
relativistic effect that arises because we are calculating the magnetic
field in a frame of reference that is not stationary but rotates as the
electron moves about the nucleus. The effect is taken into account by
replacinggswithgs− 1 1.^46 Finally, we find the spin–orbit interaction,

(^46) This is almost equivalent to using
gs/ 2
1, butgs−1 is more accu-
rate at the level of precision where the
small deviation ofgsfrom 2 is impor-
tant (Haar and Curtis 1987). For fur-
ther discussion of Thomas precession
see Cowan (1981), Eisberg and Resnick
(1985) and Munoz (2001).
including the Thomas precession factor, is^47
(^47) We have derived this classically, e.g.
by using
l=r×mev. However, the
same expression can be obtained from
the fully relativistic Dirac equation for
an electron in a Coulomb potential by
making a low-velocity approximation,
see Sakurai (1967). That quantum me-
chanical approach justifies treatingl
andsas operators.
Hs−o=(gs−1)

^2

2 m^2 ec^2

(

1

r

∂V

∂r

)

s·l. (2.49)

For the Coulomb potential in hydrogen we have

1

r

∂V

∂r

=

e^2 / 4 π 0

r^3

. (2.50)

The expectation value of this Hamiltonian gives an energy change of^48

(^48) Using the approximationgs− 1

1. Es−o=

^2

2 m^2 ec^2

e^2

4 π 0

〈

1

r^3

〉

〈s·l〉. (2.51)

The separation into a product of radial and angular expectation values
follows from the separability of the wavefunction. The integral

〈

1 /r^3

〉

is
given in eqn 2.23. However, we have not yet discussed how to deal with
interactions that have the form of dot products of two angular momenta;
let us start by defining the total angular momentum of the atom as the
sum of its orbital and spin angular momenta, l

s

j

Fig. 2.4The orbital and spin angular

momenta add to give a total atomic an-

gular momentum ofj.

j=l+s. (2.52)

This is a conserved quantity for a system without any external torque
acting on it, e.g. an atom in a field-free region of space. This is true
both in classical and quantum mechanics, but we concentrate on the
classical explanation in this section. The spin–orbit interaction between
landscauses these vectors to change direction, and because their sum
is constrained to be equal tojthey move around as shown in Fig. 2.4.^4949 In this precession aboutjthe magni-
tudes oflandsremain constant. The
magnetic moment (proportional tos)
is not altered in an interaction with
a magnetic field, and because of the
symmetrical form of the interaction in
eqn 2.49, we do not expectlto be-
have any differently. See also Blundell
(2001) and Section 5.1.

Squaring and rearranging eqn 2.52, we find that 2s·l=j^2 −l^2 −s^2.
Hence we can find the expectation value in terms of the known values
for

〈

j^2

〉

,

〈

l^2

〉

and

〈

s^2

〉

as

〈s·l〉=

1

2

{j(j+1)−l(l+1)−s(s+1)}. (2.53)

Thus the spin–orbit interaction produces a shift in energy of

Es−o=

β

2

{j(j+1)−l(l+1)−s(s+1)}, (2.54)