Spin-orbit interaction

The **spin-orbit interaction** is an interaction in atomic physics between the magnetic moment of an electron due to its intrinsic spin and the magnetic field it sees from moving in the electric field of the nucleus. It is written using the spin-orbit Hamiltonian,

Δ

H
^

S
O

=

μ
B

ℏ

m
e

e

c
2

1
r

∂
U

∂
r

L
→

.

S
→

{\displaystyle \Delta \hat H\_{SO} = \frac{\mu\_B}{\hbar m\_e e c^2}\frac{1}{r}\frac{\partial U}{\partial r} \vec L . \vec S}

, where

r

{\displaystyle r}

 is the distance between the electron and the nucleus,

U

{\displaystyle U}

 is the central potential of the nucleus' electric field,

L
→

{\displaystyle \vec L}

 is the orbital angular momentum of the electron, and

S
→

{\displaystyle \vec S}

 is the spin of the electron.

This interaction means that, for multiple electrons, there is an energy difference depending on the total angular momentum, explaining a small splitting in the spectral lines that can be observed. This splitting is called the fine structure of the atom.

Since

Δ

H
^

S
O

{\displaystyle \Delta \hat H\_{SO}}

 is typically small, this can usually be treated using perturbation theory.

## Derivation

The energy that an electron has from a magnetic field is

−

1
2

μ
→

.

B
→

{\displaystyle - \frac{1}{2} \vec \mu . \vec B}

. This is equal to the energy of a magnetic dipole moment

μ
→

{\displaystyle \vec \mu}

 in a magnetic field

B

{\displaystyle B}

, multiplied by a factor of a half to account for Thomas precession.

### Magnetic Field

The magnetic field that the electron fields is found by Lorentz transforming the electric field from the nucleus

E

r
^

{\displaystyle E \hat r}

 into the instantaneous rest frame of the electron, giving

# B

−

v
→

×

E
→

c
2

{\displaystyle B = \frac{-\vec v \times \vec E}{c^2}}

. Since the electron's momentum is

v
→

=

m
e

v
→

{\displaystyle \vec v = m\_e \vec v}

, and the electric field from the nucleus is purely radial so

E
→

=

E
r

r
→

{\displaystyle \vec E = \frac{E}{r} \vec r}

, this can be rewritten as

B
→

=

r
→

×

p
→

m
e

c
2

E
r

{\displaystyle \vec B = \frac{\vec r \times \vec p}{m\_e c^2} \frac{E}{r}}

. The orbital angular momentum is defined as

L
→

=

r
→

×

p
→

{\displaystyle \vec L = \vec r \times \vec p}

, so this is

B
→

=

1

m
e

c
2

E
r

L
→

{\displaystyle \vec B = \frac{1}{m\_e c^2} \frac{E}{r} \vec L}

Since the electric field is defined as the negative gradient of the potential, and the potential and electric field are both purely radial,

−
e
# E
−

∂
U

∂
r

{\displaystyle -eE = -\frac{\partial U}{\partial r}}

, so

B
→

=

1

m
e

e

c
2

1
r

∂
U

∂
r

L
→

{\displaystyle \vec B = \frac{1}{m\_e e c^2} \frac{1}{r} \frac{\partial U}{\partial r} \vec L}

.

### Magnetic Moment

The magnetic moment of the electron is found from the spin through

μ
→

S

=
−

g
s

μ
B

ℏ

S
→

{\displaystyle \vec \mu\_S = - g\_s \frac{\mu\_B}{\hbar} \vec S}

, where

g
s

{\displaystyle g\_s}

 is the gyromagnetic factor for spin.

### Energy

By putting both of these terms into the expression for the Hamiltonian, the overall Hamiltonian for the spin-orbit interaction is found to be

Δ

H
^

S
O

=

μ
B

ℏ

m
e

e

c
2

1
r

∂
U

∂
r

L
→

.

S
→

{\displaystyle \Delta \hat H\_{SO} = \frac{\mu\_B}{\hbar m\_e e c^2}\frac{1}{r}\frac{\partial U}{\partial r} \vec L . \vec S}

.

## Solving the Perturbation

In order to determine the size of the energy shift caused by the spin-orbit interaction, one needs a form for the central potential

U

{\displaystyle U}

. For many-electron atoms, this does not exist analytically (similar to the 3-body problem), so it must be determined numerically. The exception to this is in alkali metals, where there is only one outer electron so all of the inner electrons form a spherically symmetric wavefunction.