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

36 Chapter 2
2.A Appendix: Optical properties of Rayleigh, Cabannes and Raman
scattering
The strength of Rayleigh scattering is described by the scattering cross section [Bucholtz, 1995, Naus
and Ubachs, 2000]
σ ray,air (ν ′ ) =
∑
χ i σ ray,i (ν ′ ), (2.24)
i=N 2 ,O 2 ,Ar
where ν ′ is the wavenumber of the incoming radiation, i indicates the molecule species, χ i is the
volume mixing ratio (see Table 2.1), and
[
σ ray,i (ν ′ ) = 128π5 [ν ′ ] 4 α 2
3
i(ν ′ ) 1 + 2 ]
9 ε i(ν ′ ) . (2.25)
Here, α i is the average polarizability and ε i is the anisotropic polarizability factor. Values for these
quantities can be derived from Bates [1984] and Peck and Fisher [1964]. The term containing 2ε 9 i(ν)
is attributed to the anisotropic polarizability of molecules [Long, 1977, Young, 1982]. Noble gases
such as Ar show no anisotropy, hence ε Ar = 0.
Rayleigh scattering is only an effective description of scattering by molecules. In reality, the
anisotropic part is distributed to other wavenumbers than the one for the incoming light, due to rotation
and vibration of molecules. In this paper we assume that N 2 and O 2 are simple linear rotors [Sioris,
2001]. For pure rotational Raman scattering only energy transitions J →J ± 0, 2 are allowed, where
J is the rotational angular momentum quantum number.
The ∆J =0 transitions are added to the isotropic scattering component and make up one quarter of
the anisotropic energy [Bhagavantam, 1931, Young, 1982]. The elastic scattering component contains
the isotropic part plus one quarter of the anisotropic part and is described by the Cabannes cross
section
σ cab,air (ν ′ ) =
∑
χ i f cab,i (ν ′ )σ ray,i (ν ′ ), (2.26)
i=N 2 ,O 2 ,Ar
where the elastic fraction [Kattawar et al., 1981, Joiner et al., 1995] is given by
f cab,i (ν ′ ) = 18 + ε i(ν ′ )
18 + 4ε i (ν ′ ) . (2.27)
The rest of the anisotropic part is scattered inelastically due to Raman scattering. The total rotational
Raman cross section for each molecule is given by
σ ramtot,i (ν ′ ) = [1 − f cab,i (ν ′ )]σ ray,i (ν ′ ). (2.28)
The energy contained in the total rotational Raman cross section is distributed to other wavenumbers
according to
σ ram,i (ν;ν ′ ) = 4 [ ν
] 4
3 f i(T,J)b i (J →J ′ ) σramtot,i (ν ′ ), (2.29)
ν ′