Rotational Raman scattering in the Earth's atmosphere ... - SRON

72 Chapter 3
where λ is the wavelength of the incoming light and λ ′ represents the corresponding wavelength of
the scattered light. f(T,J) is the fractional population in the initial state and can be approximated by
[Joiner et al., 1995]
f(T,J) = 1 [
g(J) [2J+1] exp
N f
− E(J)
kT
]
, (3.74)
where g is the nuclear spin statistical weight factor, k is the Boltzmann’s constant, T is the temperature,
and E(J) = hcBJ(J +1) is the rotational energy (h is Planck’s constant, c is the speed of
light, and B is the rotational constant). The coefficient N f can be determined from the normalization
condition
∑
f(T,J) = 1. (3.75)
J
b(J → J ′ ) represents the Placzek-Teller coefficient for the transition J → J ′ . For simple linear
molecules these coefficients are given by
b(J →J+2) = 3(J+1)(J+2)
2(2J+1)(2J+3) , (3.76)
for Stokes lines and
b(J →J −2) =
3J(J −1)
2(2J −1)(2J+3) , (3.77)
for anti-Stokes lines. The change in wavelength ∆λ=λ ′ −λ due to rotational Raman scattering can be
calculated easily from the energy difference between the initial and final state, ∆E = E(J ′ )−E(J).
The scattering phase matrix P, defined with respect to the plane of scattering, has the same structure
Table 3.1: Parameters A, B, C, and normalization constant N of scattering phase function (3.78) for
Rayleigh, Cabannes and rotational Raman scattering.
A B C N
Rayleigh 3(45+ε) 30(9−ε) 36ε (180+40ε)
Cabannes 3(180+ε) 30(36−ε) 36ε 40(18+ε)
rot. Raman 3 −30 36 40
for Rayleigh, Cabannes and rotational Raman scattering, viz.
⎛
⎞
A(1+cos 2 Θ) + C −A(1−cos 2 Θ) 0 0
P(Θ) =
1 −A(1−cos 2 Θ) A(1+cos 2 Θ) 0 0
⎜
⎟
N ⎝ 0 0 2A cos Θ 0 ⎠
0 0 0 B cos Θ
(3.78)