5.1 Fine structure in theLS-coupling scheme 835.1 Fine structure in theLS-coupling scheme
Fine structure arises from the spin–orbit interaction for each of the un-
paired electrons given by the Hamiltonian
Hs−o=β 1 s 1 ·l 1 +β 2 s 2 ·l 2.For atoms with two valence electronsHs−oacts as a perturbation on
the states|LMLSMS〉. In the vector model, this interaction between
the spin and orbital angular momentum causesLandSto change di-
rection, so that neitherLznorSzremains constant; but the total elec-
tronic angular momentumJ=L+S,anditsz-componentJz,areboth
constant because no external torque acts on the atom. We shall now
evaluate the effect of the perturbationHs−oon a term using the vec-
tor model. In the vector-model description of theLS-coupling scheme,
l 1 andl 2 precess aroundL, as shown in Fig. 5.4; the components per-
pendicular to this fixed direction average to zero (over time) so that
only the component of these vectors alongLneeds to be considered,
e.g.l 1 →
{(
l 1 ·L)
/|L|^2
}
L.The time averagel 1 ·Lin the vector model
becomes the expectation value〈l 1 ·L〉in quantum mechanics; also we
have to useL(L+ 1) for the magnitude-squared of the vector. Applying
the same projection procedure to the spins leads to
Hs−o=β 1〈s 1 ·S〉
S(S+1)S·
〈l 1 ·L〉
L(L+1)L+β 2〈s 2 ·S〉
S(S+1)S·
〈l 2 ·L〉
L(L+1)L
=βLSS·L. (5.4)Fig. 5.4In theLS-coupling scheme
the orbital angular momenta of the two
electrons couple to give total angular
momentumL=l 1 +l 2. In the vec-
tor modell 1 andl 2 precess aroundL;
similarly,s 1 ands 2 precess aroundS.
LandSprecess around the total an-
gular momentumJ(but more slowly
than the precession ofl 1 andl 2 around
Lbecause the spin–orbit interaction is
‘weaker’ than the residual electrostatic
interaction).The derivation of this equation by the vector model that argues by
analogy with classical vectors can be fully justified by reference to the
theory of angular momentum. It can be shown that, in the basis|JMJ〉
of the eigenstates of a general angular momentum operatorJand its
componentJz, the matrix elements of any vector operatorVare pro-
portional to those ofJ, i.e.〈JMJ|V|JMJ〉=c〈JMJ|J|JMJ〉.^8 Fig-
(^8) This is particular case of a more
general result called the Wigner–
Eckart theorem which is the corner-
stone of the theory of angular momen-
tum. This powerful theorem also ap-
plies to off-diagonal elements such as
〈JMJ|V
∣∣
JMJ′
〉
, and to more com-
plicated operators such as those for
quadrupole moments. It is used ex-
tensively in advanced atomic physics—
see the ‘Further reading’ section in this
chapter.
ure 5.5 gives a pictorial representation of why it is only the component of
ValongJthat is well defined. We want to apply this result to the case
whereV=l 1 orl 2 in the basis of eigenstates|LML〉, and analogously
for the spins. For〈LML|l 1 |LML〉=c〈LML|L|LML〉the constant
cis determined by taking the dot product of both sides withLto give
c=
〈LML|l 1 ·L|LML〉
〈LML|L·L|LML〉
;
hence
〈LML|l 1 |LML〉=
〈l 1 ·L〉
L(L+1)〈LML|L|LML〉. (5.5)
This is an example of the projection theorem and can also be applied to
l 2 and tos 1 ands 2 in the basis of eigenstates|SMS〉. It is clear that,
for diagonal matrix elements, these quantum mechanical results give the
same result of the vector model.
Fig. 5.5A pictorial representation of
the project theorem for an atom, where
Jdefines the axis of the system.