34.3 Rayleigh Scattering
Lord Rayleigh calculatedlow energy elastic scattering of lightfrom atoms using classical
electromagnetism. If the energy of the scattered photon is muchless than the energy needed to
excite an atom,ω << ωji, then the cross section may be approximated.
∓ωji
ωji±ω=
∓ωji
ωji(1±ωωji)=∓(1∓
ω
ωji) =∓1 +
ω
ωjidσelas
dΩ=
(
e^2
4 πmc^2) (^2) (
mω
̄h
) 2
∣ ∣ ∣ ∣ ∣ ∣
∑
j[
ωji〈i|ˆǫ′·~x|j〉 〈j|ˆǫ·~x|i〉
ωji−ω−
ωji〈i|ˆǫ·~x|j〉 〈j|ˆǫ′·~x|i〉
ωji+ω]
∣ ∣ ∣ ∣ ∣ ∣
2=
(
e^2
4 πmc^2) (^2) (
mω
̄h
) 2
∣ ∣ ∣ ∣ ∣ ∣
∑
j[(〈i|ˆǫ′·~x|j〉 〈j|ǫˆ·~x|i〉−〈i|ˆǫ·~x|j〉 〈j|ǫˆ′·~x|i〉)+
ω
ωji(〈i|ˆǫ′·~x|j〉 〈j|ˆǫ·~x|i〉+〈i|ˆǫ·~x|j〉 〈j|ˆǫ′·~x|i〉)]∣
∣
∣
∣
2=
(
e^2
4 πmc^2) (^2) (
m
̄h
) 2
ω^4∣ ∣ ∣ ∣ ∣ ∣
∑
j[
1
ωji(〈i|ˆǫ′·~x|j〉 〈j|ˆǫ·~x|i〉+〈i|ˆǫ·~x|j〉 〈j|ˆǫ′·~x|i〉)]
∣ ∣ ∣ ∣ ∣ ∣
2For thecolorless gasses(like the ones in our atmosphere), the first excited state in the UV,so the
scattering of visible light with be proportional toω^4 , which explains why the sky is blue and sunsets
are red. Atoms with intermediate states in the visible will appear to becolored due to the strong
resonances in the scattering.Rayleighgot the same dependence from classical physics.
34.4 Thomson Scattering
If the energy of the scattered photon is much bigger than the binding energy of the atom,ω >>1 eV.
then cross section approaches that forscattering from a free electron, Thomson Scattering.
We still neglect the effect of electron recoil so we should also requirethat ̄hω << mec^2. Start from
the Kramers-Heisenberg formula.
dσ
dΩ=
(
e^2
4 πmc^2) 2 (
ω′
ω)
∣ ∣ ∣ ∣ ∣ ∣
δniˆǫ·ˆǫ′−1
m ̄h∑
j[
〈n|ˆǫ′·~p|j〉〈j|ˆǫ·~p|i〉
ωji−ω+
〈n|ˆǫ·~p|j〉〈j|ˆǫ′·~p|i〉
ωji+ω′]
∣ ∣ ∣ ∣ ∣ ∣
2The ̄hω = ̄hω′ denominators are much larger than 〈n|ˆǫ
′·~p|j〉〈j|ˆǫ·~p|i〉
m which is of the order of the
electron’s kinetic energy, so we can ignore the second two terms. (Even if the intermediate and final
states have unbound electrons, the initial state wave function willkeep these terms small.)
dσ
dΩ=
(
e^2
4 πmc^2) 2
|ˆǫ·ˆǫ′|
2This scatteringcross section is of the order of the classical radius of the electron squared,
and isindependent of the frequencyof the light.
The only dependence is on polarization. This is a good time to take a lookat the meaning of
thepolarization vectorswe’ve been carrying around in the calculation and at the lack of any