Download Zeeman Effect

Zeeman Effect
The Zeeman effect is the splitting of a spectral line into several components in the
presence of a static magnetic field. It is analogous to the Stark effect, the splitting of a
spectral line into several components in the presence of an electric field. The Zeeman
effect is very important in applications such as nuclear magnetic resonance
spectroscopy, electron spin resonance spectroscopy, magnetic resonance imaging
(MRI) and Mössbauer spectroscopy.
In most atoms, there exist several electronic configurations that have the same energy,
so that transitions between different pairs of configurations correspond to a single
spectral line.
The presence of a magnetic field breaks the degeneracy, since it interacts in a different
way with electrons with different quantum numbers, slightly modifying their energies.
The result is that, where there were several configurations with the same energy, now
there are different energies, which give rise to several very close spectral lines.
Without a magnetic field, configurations a, b and c have the same energy, as do d, e
and f. The presence of a magnetic field splits the energy levels. A line produced by a
transition from a, b or c to d, e or f now will be several lines between different
combinations of a, b, c and d, e, f. Not all transitions will be possible, as regulated by
the transition rules.
Since the distance between the Zeeman sub-levels is proportional to the magnetic
field, this effect is used by astronomers to measure the magnetic field of the Sun and
other stars.
There is also an anomalous Zeeman effect that appears on transitions where the net
spin of the electrons is not 0, the number of Zeeman sub-levels being even instead of
odd if there's an uneven number of electrons involved. It was called "anomalous"
because the electron spin had not yet been discovered, and so there was no good
explanation for it at the time that Zeeman observed the effect.
If the magnetic field strength is too high, the effect is no longer linear; at even higher
field strength, electron coupling is disturbed and the spectral lines rearrange. This is
called the Paschen-Back effect.
The total Hamiltonian of an atom in a magnetic field is
H = H0 + HM,
where H0 is the unperturbed Hamiltonian of the atom, and HM is perturbation due to
the magnetic field:
,
where is the magnetic moment of the atom. The magnetic moment consists of the
electronic and nuclear parts, however, the latter is many orders of magnitude smaller
and will be neglected further on. Therefore,
,
where μB is the Bohr magneton, is the total electronic angular momentum, and g is
the g-factor. The operator of the magnetic moment of an electron is a sum of the
contributions of the orbital angular momentum and the spin angular momentum
with each multiplied by the appropriate gyromagnetic ratio:
,
,
where gl = 1 or
(the latter is called the anomalous gyromagnetic
ratio; the deviation of the value from 2 is due to the relativistic effects). In the case of
the LS coupling, one can sum over all electrons in the atom:
,
where and are the total orbital momentum and spin of the atom, and averaging is
done over a state with a given value of the total angular momentum.
If the interaction term VM is small (less than the fine structure), it can be treated as a
perturbation; this is the Zeeman effect proper. In the Paschen-Back effect, described
below, VM exceeds the LS coupling significantly (but is still small compared to H0). In
ultrastrong magnetic fields, the magnetic-field interaction may exceed H0, in which
case the atom can no longer exist in its normal meaning, and one talks about Landau
levels instead. There are, of course, intermediate cases which are more complex than
these limit cases.
If the spin-orbit interaction dominates over the effect of the external magnetic field,
and
are not separately conserved, only the total angular momentum
is. The spin and orbital angular momentum vectors can be thought of as
precessing about the (fixed) total angular momentum vector
. The (time)"averaged" spin vector is then the projection of the spin onto the direction of :
.
and for the (time-)"averaged" orbital vector:
.
Thus,
.
Using
and squaring both sides, we get
,
and: using
and squaring both sides, we get