Geant4 Simulations for the Radon Electric Dipole Moment Search at

energy levels are separated according to the value of J. The 5s orbital does not split,
as shown in Figure 2.4, since l = 0. The energy difference between the 2 S 1/2 and
2 P 1/2 levels coresponds roughly to a 794.8 nm wavelength and is called the D 1 line.
The D 2 line coresponds to the energy difference between the 2 S 1/2 and 2 P 3/2 levels
and has a wavelength of about 780 nm.
The next correction to H o is due to the interaction between the magnetic moment
of the electron and the nuclear magnetic moment. These splittings are called the
hyperfine structure and are given by the following Hamiltonian,
H hf = Ze2 g N 1
2mM N c 2 4π S·[−I∇ 21 ]
r +∇(I·∇)1 , (2.7)
r
where g N is the nuclear g-factor. These levels split according to the total angular
momentum F. 85 Rb has a nuclear spin of I = 5/2, thus the allowed values of F are
5/2−1/2 = 2 and 5/2+1/2 = 3, as shown in Figure 2.4. These splittings are on the
order of 10 −6 eV.
The final splitting is due to the Zeeman effect, also shown in the above simplified
example. The splitting occurs in the presence of a weak external magnetic field (B).
Classically, the magnetic moment of a particle with charge q and angular momentum
L is given by,
µ = q
2mc L . (2.8)
Extending this result into quantum mechanics, the magnetic moment due to the total
electronic angular momentum is given by,
µ J = −g J
e
2mc J , (2.9)
and the magnetic moment of the atom is given by,
µ F = −g F
e
2mc F , (2.10)
21