Once we have found the most accurate value of
T
n
o
from the spectrum we can put the energy
levels on an absolute energy scale. We have then determined the energy level structure of the atom
for this set of angular momentum states. The procedure can be carried out using data from emission
spectra. In this case a range of different series will be be provided from the spectrum. It remains to
find the quantum defects for each series of lines and assign values of
l
to each series of levels. From
the discussion above the quantum defects will decrease for increasing
l
.
Figure 10: Plot of quantum defect
δ
(
l
) against Term value, showing effect of choosing
T
n
o
either too
large or too small. The correct value of
T
n
o
yields a horizontal plot.
16

Atomic Physics, P. Ewart
4 Corrections to the Central Field: Spin-Orbit interaction
4 Corrections to the Central Field: Spin-Orbit interaction
The Central Field Approximation gives us a zero-order Hamiltonian
ˆ
H
0
that allows us to solve the
Schr¨
odinger equation and thus find a set of zero-order wavefunctions
ψ
i
. The hope is that we can
treat the residual electrostatic interaction (i.e. the non-central bit of the electron-electron repulsion)
as a small perturbation,
ˆ
H
1
. The change to the energy would be found using the functions
ψ
i
.
The residual electrostatic interaction however isn’t the only perturbation around. Magnetic inter-
actions arise when there are moving charges. Specifically we need to consider the magnetic interaction
between the magnetic moment due to the electron spin and the magnetic field arising from the elec-
tron orbit. This field is due to the motion of the electron in the electric field of the nucleus and the
other electrons. This spin-orbit interaction has an energy described by the perturbation
ˆ
H
2
. The
question is: which is the greater perturbation,
ˆ
H
1
or
ˆ
H
2
?
We may be tempted to assume
ˆ
H
1
>
ˆ
H
2
since electrostatic forces are usually much stronger than
magnetic ones. However by setting up a Central Field we have already dealt with the major part
of the electrostatic interaction. The remaining bit may not be larger than the magnetic spin-orbit
interaction. In many atoms the residual electrostatic interaction,
ˆ
H
1
, does indeed dominate the spin-
orbit. There is, however, a set of atoms where the residual electrostatic repulsion is effectively zero;
the alkali atoms. In the alkalis we have only one electron orbiting outside a spherically symmetric
core. The central field is, in this case, an excellent approximation. The spin-orbit interaction,
ˆ
H
2
will be the largest perturbation – provided there are no external fields present. So we will take the
alkalis e.g. Sodium, as a suitable case for treatment of spin-orbit effects in atoms. You have already
met the spin-orbit effect in atomic hydrogen, so you will be familiar with the quantum mechanics for
calculating the splitting of the energy levels. There are, however, some important differences in the
case of more complex atoms. In any case, we are interested in understanding the physics, not just

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