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8 Electric and Magnetic Phenomena
The changes in energy of the electronic levels of a multi-electron atom or ion in
the presence of a constant homogeneous applied magnetic field B are governed by
the Zeeman interaction Hamiltonian, H Z , given by
H Z = − μ
(m)
· B,
(8.5.21)
in which μ (m) is the magnetic dipole moment operator of the atom or ion. We
shall approximate the magnetic moment operator by its dominant paramagnetic
component [9] as 5
μ
(m)
= −μ B (2 J − L),
(8.5.22)
in which J is the total angular momentum, L is the total orbital angular momentum
for the multi-electron atom or ion, while μ B , defined by
μ B ≡
¯
h|e|
2m e
,
(8.5.23)
with ¯
h the Planck constant divided by 2π , |e| the magnitude of the electronic charge,
and m e the mass of the electron, is the Bohr magneton. The Bohr magneton is the
fundamental unit of electronic magnetic moments, and has the approximate value 6
9.27401 × 10 −24 J T −1 . A much smaller diamagnetic component of the electronic
magnetic moment is too small to play a role in our present considerations.
As the Zeeman interaction energy is small in comparison with the field-free
electronic energy, we may treat the Zeeman Hamiltonian as a perturbation to
the field-free Hamiltonian, H 0 , with eigenenergies E 0 (α, L, S, J ). According to
perturbation theory, the energy due to the Zeeman interaction can therefore be
expressed in a power series in the magnetic field strength B as
E(α, L, S, J, M J ) = E 0 (α, L, S, J ) + W 1 (L, S, J, M J )B
+ W 2 (L, S, J, M J )B
2
+ · · · ,
(8.5.24)
in which W 1 (L, S, J, M J ) and W 2 (L, S, J, M J ) are known as the first- and secondorder Zeeman energies of the atom or ion in an energy level characterized by
α, L, S, J . The field-free energy E 0 (α, L, S, J ) can be split into two parts as
E 0 (α, L, S, J ) = E 0 (α, L, S) + E so (L, S, J ),
(8.5.25)
5 For a derivation of this form for the magnetic dipole moment of a multi-electron atom see, for
example, pp. 128–129 of [9].
6 That the Bohr magneton is much larger than the nuclear magneton that we encountered earlier
is simply due to the much smaller mass of the electron. For a more precise value of the Bohr
magneton, see the NIST website https://physics.nist.gov/PhysRefData.
8 Electric and Magnetic Phenomena
The changes in energy of the electronic levels of a multi-electron atom or ion in
the presence of a constant homogeneous applied magnetic field B are governed by
the Zeeman interaction Hamiltonian, H Z , given by
H Z = − μ
(m)
· B,
(8.5.21)
in which μ (m) is the magnetic dipole moment operator of the atom or ion. We
shall approximate the magnetic moment operator by its dominant paramagnetic
component [9] as 5
μ
(m)
= −μ B (2 J − L),
(8.5.22)
in which J is the total angular momentum, L is the total orbital angular momentum
for the multi-electron atom or ion, while μ B , defined by
μ B ≡
¯
h|e|
2m e
,
(8.5.23)
with ¯
h the Planck constant divided by 2π , |e| the magnitude of the electronic charge,
and m e the mass of the electron, is the Bohr magneton. The Bohr magneton is the
fundamental unit of electronic magnetic moments, and has the approximate value 6
9.27401 × 10 −24 J T −1 . A much smaller diamagnetic component of the electronic
magnetic moment is too small to play a role in our present considerations.
As the Zeeman interaction energy is small in comparison with the field-free
electronic energy, we may treat the Zeeman Hamiltonian as a perturbation to
the field-free Hamiltonian, H 0 , with eigenenergies E 0 (α, L, S, J ). According to
perturbation theory, the energy due to the Zeeman interaction can therefore be
expressed in a power series in the magnetic field strength B as
E(α, L, S, J, M J ) = E 0 (α, L, S, J ) + W 1 (L, S, J, M J )B
+ W 2 (L, S, J, M J )B
2
+ · · · ,
(8.5.24)
in which W 1 (L, S, J, M J ) and W 2 (L, S, J, M J ) are known as the first- and secondorder Zeeman energies of the atom or ion in an energy level characterized by
α, L, S, J . The field-free energy E 0 (α, L, S, J ) can be split into two parts as
E 0 (α, L, S, J ) = E 0 (α, L, S) + E so (L, S, J ),
(8.5.25)
5 For a derivation of this form for the magnetic dipole moment of a multi-electron atom see, for
example, pp. 128–129 of [9].
6 That the Bohr magneton is much larger than the nuclear magneton that we encountered earlier
is simply due to the much smaller mass of the electron. For a more precise value of the Bohr
magneton, see the NIST website https://physics.nist.gov/PhysRefData.