### 12. Zeeman Effect

The Zeeman effect for "weak" magnetic fields (the anomalous Zeeman effect) is
of special interest because of the importance of Zeeman data in the analysis
and theoretical interpretation of complex spectra. In a weak field, the
*J* value remains a good quantum number although in general a level is
split into magnetic sublevels [3].
The *g* value of such a level may be defined by the expression for the
energy shift of its magnetic sublevel having magnetic quantum number *M*,
which has one of the 2*J* + 1 values, -*J*,
-*J* + 1, ..., *J*:

The magnetic flux density is *B*, and *µ*_{B} is the Bohr
magneton (*µ*_{B} = *e*/2*m*_{e}).
The wavenumber shift Δσ corresponding to this energy shift is

Δσ = *gM *(0.466 86 *B* cm^{-1})
, |
(7) |

with *B* representing the numerical value of the magnetic flux density in
teslas. The quantity in parentheses, the Lorentz unit, is of the order of 1 or
2 cm^{-1} for typical flux densities used to obtain Zeeman-effect
data with classical spectroscopic methods. Accurate data can be obtained with
much smaller fields, of course, by using higher-resolution techniques such as
laser spectroscopy. Most of the *g* values now available for atomic
energy levels were derived by application of the above formula (for each of the
two combining levels) to measurements of optical Zeeman patterns. A single
transverse-Zeeman-effect pattern (two polarizations, resolved components, and
sufficiently complete) can yield the *J* value and the *g* value for
each of the two levels involved.

Neglecting a number of higher-order effects, we can evaluate the *g*
value of a level β*J* belonging to a pure *LS*-coupling term
using the formula

The independence of this expression from any other quantum numbers (represented
by β) such as the configuration, etc., is important. The expression is
derived from vector coupling formulas by assuming a *g* value of unity for
a pure orbital angular momentum and writing the *g* value for a pure
electron spin as *g*_{e} [15]. A
value of 2 for *g*_{e} yields the Landé formula. If the anomalous
magnetic moment of the electron is taken into account, the value of
*g*_{e} is 2.002 319 3. "Schwinger" *g*
values obtained with this more accurate value for *g*_{e} are
given for levels of *SL* terms in
Ref. [8].
The usefulness of *g*_{SLJ} values is enhanced by their relation
to the *g* values in intermediate coupling. In the notation used in
Eq. (4) for the wave function
of a level β*J* in intermediate coupling, the corresponding *g*
value is given by

where the summation is over the same set of quantum numbers as for the wave
function. The *g*_{βJ} value is thus a weighted
average of the Landé *g*_{SLJ} values, the weighting factors being just the
corresponding component percentages.
Formulas for magnetic splitting factors in the
*J*_{1} *J*_{2} and
*J*_{1} *L*_{2} coupling schemes are given in
Refs. [8] and
[15]. Some higher-order effects
that must be included in more accurate Zeeman-effect calculations are treated
by Bethe and Salpeter [4] and by
Wybourne [15], for example. High
precision calculations for helium are given in
Ref. [16].