2 Concepts in Magnetism
57
2.4.2 Spin–Orbit Interaction and Crystal Fields
Now that we have the possibility of some partially filled d-levels, there can be magnetism. Next let us turn to the spin–orbit interaction which has the form
λ ˆ
S · ˆ
L ,
(2.47)
so that the ˆ
S operator acts on the spin part of the wave function, while the ˆ
L operator
acts on the spatial part of the wave function. If the states are approximately atomic
states, and the spin–orbit interaction acts as a perturbation, one can focus on the
λS
z L
z part and note that
ˆ
L
z
= −i
∂
∂φ
,
(2.48)
which has eigenfunctions e
imφ , i.e.
ˆ
L
z e
imφ
= −i
∂
∂φ
e
imφ
= me
imφ
.
(2.49)
Now the crystal field is a real potential which is due to electrostatic fields from
neighbouring ions. The eigenfunctions of the crystal field cannot be proportional to
e
imφ because we require real solutions. (Recall the problem of particle in a box where
the solutions are real and take the form cos kx or sin kx, but not e
ikx , but of course
we can make real functions by making linear combinations such as e
ikx
± e
−ikx and
making a wave function out of something proportional to that.) To make a crystal
field state we must, therefore, look for linear combinations of eigenfunctions such
as
|ψ = e
imφ
± e
−imφ
.
(2.50)
This kind of state though will automatically have zero angular momentum along the
z-direction because it is made up of an equal contribution of a state with L
z
= m
and L
z
= −m. In fact, this idea works for all directions and
ˆ
L = 0 .
(2.51)
For example, if l = 1, there are three states with m = 1, 0, −1 with wave functions given by the spherical harmonics Y 1m (θ, φ), and are therefore proportional to
sin θ e
iφ , cos θ and sin θ e
−iφ , respectively. We could write these states as |1, |0 and
| − 1. However, these are not the famous p-orbitals familiar from chemistry books.
These arise in the formation of chemical bonds due to (real) electrostatic effects and
must, therefore, be linear combinations of exactly the kind we are talking about.
Thus the p-orbitals that line up along the x-, y- and z-directions [see Fig. 2.10a] are
the zero-angular-momentum linear combinations given below:
57
2.4.2 Spin–Orbit Interaction and Crystal Fields
Now that we have the possibility of some partially filled d-levels, there can be magnetism. Next let us turn to the spin–orbit interaction which has the form
λ ˆ
S · ˆ
L ,
(2.47)
so that the ˆ
S operator acts on the spin part of the wave function, while the ˆ
L operator
acts on the spatial part of the wave function. If the states are approximately atomic
states, and the spin–orbit interaction acts as a perturbation, one can focus on the
λS
z L
z part and note that
ˆ
L
z
= −i
∂
∂φ
,
(2.48)
which has eigenfunctions e
imφ , i.e.
ˆ
L
z e
imφ
= −i
∂
∂φ
e
imφ
= me
imφ
.
(2.49)
Now the crystal field is a real potential which is due to electrostatic fields from
neighbouring ions. The eigenfunctions of the crystal field cannot be proportional to
e
imφ because we require real solutions. (Recall the problem of particle in a box where
the solutions are real and take the form cos kx or sin kx, but not e
ikx , but of course
we can make real functions by making linear combinations such as e
ikx
± e
−ikx and
making a wave function out of something proportional to that.) To make a crystal
field state we must, therefore, look for linear combinations of eigenfunctions such
as
|ψ = e
imφ
± e
−imφ
.
(2.50)
This kind of state though will automatically have zero angular momentum along the
z-direction because it is made up of an equal contribution of a state with L
z
= m
and L
z
= −m. In fact, this idea works for all directions and
ˆ
L = 0 .
(2.51)
For example, if l = 1, there are three states with m = 1, 0, −1 with wave functions given by the spherical harmonics Y 1m (θ, φ), and are therefore proportional to
sin θ e
iφ , cos θ and sin θ e
−iφ , respectively. We could write these states as |1, |0 and
| − 1. However, these are not the famous p-orbitals familiar from chemistry books.
These arise in the formation of chemical bonds due to (real) electrostatic effects and
must, therefore, be linear combinations of exactly the kind we are talking about.
Thus the p-orbitals that line up along the x-, y- and z-directions [see Fig. 2.10a] are
the zero-angular-momentum linear combinations given below: