94 TheLS-coupling scheme
is not small compared to the energy gap between the configurat-
ions—this is common in atoms with complex electronic structure.
(b) Thejj-coupling scheme is a better approximation thanLS-coupling
or Russell–Saunders coupling when the spin–orbit interaction is
greater than the residual electrostatic interaction.
(c) The Paschen–Back effect arises when the interaction with an exter-
nal magnetic field is stronger than the spin–orbit interaction (with
the internal field). This condition is difficult to achieve except for
atoms with a low atomic number and hence small fine structure.
Similar physics arises in the study of the Zeeman effect on hyperfine
structure where the transition between the low-field and high-field
regimes occurs at values of the magnetic field that are easily acces-
sible in experiments (see Section 6.3).Further reading
The mathematical methods that describe the way in which angular
momenta couple together form the backbone of the theory of atomic
structure. In this chapter the quantum mechanical operators have been
treated by analogy with classical vectors (the vector model) and the
Wigner–Eckart theorem was mentioned to justify the projection theo-
rem. Graduate-level texts give a more comprehensive discussion of the
quantum theory of angular momentum, e.g. Cowan (1981), Brink and
Satchler (1993) and Sobelman (1996).Exercises
(5.1)Description of theLS-coupling scheme
Explain what is meant by the central-field approx-
imation and show how it leads to the concept of
electron configurations. Explain how perturba-
tions arising from (a) the residual electrostatic in-
teractions, and (b) the magnetic spin–orbit inter-
actions, modify the structure of an isolated multi-
electron configuration in theLS-coupling limit.(5.2)Fine structure in theLS-coupling scheme
Show from eqn 5.4 that theJlevels of the^3 Pterm
in the 3s4p configuration have a separation given
by eqn 5.8 withβLS=β4p/2(whereβ4ps·lis the
spin–orbit interaction of the 4p-electron).
(5.3)TheLS-coupling scheme and the interval rule in
calcium
Write down the ground configuration of calcium
(Z= 20). The line at 610 nm in the spectrumof neutral calcium consists of three components
at relative positions 0, 106 and 158 (in units of
cm−^1 ). Identify the terms and levels involved in
these transitions.
The spectrum also contains a multiplet of six lines
with wavenumbers 5019, 5033, 5055, 5125, 5139
and 5177 (in units of cm−^1 ). Identify the terms
and levels involved. Draw a diagram of the rel-
evant energy levels and the transitions between
them. What further experiment could be carried
out to check the assignment of quantum numbers?
(5.4)TheLS-coupling scheme in zinc
The ground configuration of zinc is 4s^2 .The
seven lowest energy levels of zinc are 0, 32 311,
32 501, 32 890, 46 745, 53 672 and 55 789 (in units
of cm−^1 ). Sketch an energy-level diagram that
shows these levels with appropriate quantum num-
bers. What evidence do these levels provide that