In spectroscopy, oscillator strength is a dimensionless quantity that expresses the probability of absorption or emission of electromagnetic radiation in transitions between energy levels of an atom or molecule.
An atom or a molecule can absorb light and undergo a transition from one quantum state to another.
The oscillator strength
f
12
of a transition from a lower state
|
1
⟩
to an upper state
|
2
⟩
may be defined by
f
12
=
2
3
m
e
ℏ
2
(
E
2
−
E
1
)
∑
α
=
x
,
y
,
z
|
⟨
1
m
1
|
R
α
|
2
m
2
⟩
|
2
,
where
m
e
is the mass of an electron and
ℏ
is the reduced Planck constant. The quantum states
|
n
⟩
,
n
=
1,2, are assumed to have several degenerate sub-states, which are labeled by
m
n
. "Degenerate" means that they all have the same energy
E
n
. The operator
R
x
is the sum of the x-coordinates
r
i
,
x
of all
N
electrons in the system, etc.:
R
α
=
∑
i
=
1
N
r
i
,
α
.
The oscillator strength is the same for each sub-state
|
n
m
n
⟩
.
To make equations of the previous section applicable to the states belonging to the continuum spectrum, they should be rewritten in terms of matrix elements of the momentum
p
. In absence of magnetic field, the Hamiltonian can be written as
H
=
1
2
m
p
2
+
V
(
r
)
, and calculating a commutator
[
H
,
x
]
in the basis of eigenfunctions of
H
results in the relation between matrix elements
x
n
k
=
−
i
ℏ
/
m
E
n
−
E
k
(
p
x
)
n
k
.
.
Next, calculating matrix elements of a commutator
[
p
x
,
x
]
in the same basis and eliminating matrix elements of
x
, we arrive at
⟨
n
|
[
p
x
,
x
]
|
n
⟩
=
2
i
ℏ
m
∑
k
≠
n
|
⟨
n
|
p
x
|
k
⟩
|
2
E
n
−
E
k
.
Because
[
p
x
,
x
]
=
−
i
ℏ
, the above expression results in a sum rule
∑
k
≠
n
f
n
k
=
1
,
f
n
k
=
−
2
m
|
⟨
n
|
p
x
|
k
⟩
|
2
E
n
−
E
k
,
where
f
n
k
are oscillator strengths for quantum transitions between the states
n
and
k
. This is the Thomas-Reiche-Kuhn sum rule, and the term with
k
=
n
has been omitted because in confined systems such as atoms or molecules the diagonal matrix element
⟨
n
|
p
x
|
n
⟩
=
0
due to the time inversion symmetry of the Hamiltonian
H
. Excluding this term eliminates divergency because of the vanishing denominator.
Sum rule and electron effective mass in crystals
In crystals, energy spectrum of electrons has a band structure
E
n
(
p
)
. Near the minimum of an isotropic energy band, electron energy can be expanded in powers of
p
as
E
n
(
p
)
=
p
2
/
2
m
∗
where
m
∗
is the electron effective mass. It can be shown that it satisfies the equation
2
m
∑
k
≠
n
|
⟨
n
|
p
x
|
k
⟩
|
2
E
k
−
E
n
+
m
m
∗
=
1.
Here the sum runs over all bands with
k
≠
n
. Therefore, the ratio
m
/
m
∗
of the free electron mass
m
to its effective mass
m
∗
in a crystal can be considered as the oscillator strength for the transition of an electron from the quantum state at the bottom of the
n
band into the same state.
