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

Introduction 7
energy:
E rot (v,J) = hcB v J(J+1) − hcD v J 2 (J+1) 2 . (1.2)
Here, h is Plank’s constant, c is the speed of light in vacuum, B is the rotational constant, and D is
the centrifugal distortion constant of the specific molecule. The first term in Eq. (1.2) describes how
the rotational energy increases for increasing angular momentum (J). The second term corrects the
energy levels for the stretching of the molecular bond due to the centrifugal force that acts on the
nuclei in the rotating molecule. The values of B v and D v are characteristic for each molecular species
and are given for the ground vibrational state (v = 0): B 0 = 1.99 cm −1 , D 0 = 5.76 × 10 −6 cm −1 for
N 2 and B 0 = 1.44 cm −1 , D 0 = 4.85 × 10 −6 cm −1 for O 2 [Penney et al., 1974].
1.3.2 Elastic and inelastic scattering by air molecules
To describe the interaction between light and molecules we use the concept of photons. Scattering of
light by molecules involves two steps. First, an incident photon brings a molecule to a virtual excited
energy state by absorbing the energy of this photon. Then, the molecule falls back to a lower energy
state by releasing a photon (see Fig. 1.3). The energy of a photon is
E photon = hcν , (1.3)
where ν =1/λ is the wavenumber. Here, E photon equals the energy difference of two molecular states.
According to quantum theory each molecular transition process should obey the quantum law
J →J ′ =J ± 1, where J is the total angular momentum quantum number of the initial state and J ′ is
the quantum number of the final state of the absorbing molecule. Therefore, for a two step process, the
net change in the total angular momentum quantum number becomes ∆J= −2, 0, +2. In addition, the
vibrational energy of the molecule might change, i.e. ∆v= −1, 0, +1. Changes in rotational energy
are allowed only if the molecule has an anisotropic polarizability. Changes in vibrational energy are
allowed only when the polarizability of the molecule changes as the molecule swells and contracts
[Long, 1977, Atkins and de Paula, 2002]. Both criteria hold for the N 2 and the O 2 molecule which
means that in principle both vibrational and rotational Raman scattering need to be considered.
Scattering processes that involve no net exchange of energy (∆J = 0; ∆v = 0) between the
molecule and the photons correspond to elastic scattering, whereas scattering processes that involve
an exchange of rotational and/or vibrational energy are inelastic. Elastic scattering by air molecules
is referred to as Cabannes scattering (Fig. 1.3a). It is worth to mention that the specific naming of
the various spectral components is not always that clear in the literature. In this thesis we adopt
the terminology that was proposed by Young [1982]. The light that is inelastically scattered due to
(rotational-)vibrational Raman scattering (∆v = +1; ∆J = 0, ±2) represents only a small fraction
of the total inelastically scattered light, i.e. approximately 1/30th [Burrows et al., 1996]. Therefore,
it is sufficient to consider pure rotational Raman scattering only (∆v = 0; ∆J = ±2), which is also
assumed in the remainder of this thesis. In future work, however, it might be worthwhile to include
rotational-vibrational Raman scattering in the radiative transfer models 3 .
3 Vibrational Raman scattering by N 2 and O 2 gives rise to “ghosts” of Fraunhofer lines at longer wavelengths. This