Experiments with Supersonic Beams as a Source of Cold Atoms

B
Figure 4.3: A pictorial illustration of the angular momentum coupling of L and S
which takes place in the high field limit. The coupling of the spin and orbital angular
momenta from the spin-orbit interaction is broken by the high magnetic field, and
as such L and S precess around the magnetic field independently. As such, the
projections Lz and SZ along B are constant, meaning that mL and mS are good
quantum numbers.
4.1.2 The Paschen-Back Effect
S
In high field, the perturbing Hamiltonian produces a shift which is larger than
the fine structure splitting. In this limit the interaction is known as the Paschen-Back
effect. Because the magnetic interaction is stronger than the spin-orbit coupling, L
and S precess independently around the magnetic field. To correctly calculate the
energy levels as a function of magnetic field, the fine structure Hamiltonian must be
considered a perturbing interaction on top of the shifts induced by the magnetic field.
In this regime, the coupling between L and S is broken, and both precess
independently, making the projections Lz and SZ along B constant. This means that
mL and mS are good quantum numbers. The angular momentum coupling of L and
S in the Paschen-Bach regime is illustrated in figure 4.3. Starting from equation 4.5
the Hamiltonian is now
This results in an energy shift of
L
HB = μBB (gLLz + gSSz) . (4.13)
ΔE = μBB (mL +2mS) . (4.14)
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