4.1 Introduction

Absorption of terrestrial thermal infra-red radiation by molecules in the atmosphere depends on their concentration, type and whether they are heteronu-clear (e.g. CO2, H2O, CH4, O3, NH3) or homonuclear (e.g. N2 and O2), that is whether or not they have an electrical dipole that can interact with infrared radiation. The atmosphere consists of mainly homonuclear molecules and hence it is the trace molecules, which strongly absorb infra-red radiation emitted by the Earth's surface to space, that determine the strength of the greenhouse effect. The absorption of the heteronuclear molecules arises mainly through bound-bound transitions between quantized rotational levels that are affected by the vibration of the molecules. We thus speak of vibrational-rotational lines that form molecular bands of spectral lines that absorb infra-red radiation, particularly between 5 ¡m and 100 ¡m, the spectral region where the Earth primarily emits blackbody radiation.

Absorption by rotational spectral lines through bound-bound transitions depends crucially on the broadening of the bound levels. There are primarily three line-broadening mechanisms; natural broadening that arises from the Heisenberg Uncertainty Principle, collisional broadening and Doppler broadening. The most important line broadening in planetary atmospheres arises through collisions that depend on the atmospheric pressure. The broadening mechanisms allow photons that have energies greater or less than the difference in energy between the levels of the bound-bound transition to be absorbed. This enhances the ability of the molecule to absorb photons over a larger spectral interval with the result that the greenhouse effect becomes stronger as the pressure of the planetary atmosphere increases. This effect enhances the greenhouse effects on Venus (surface pressure of about 100 bar) and Earth (1 bar) compared to the weak greenhouse effect on Mars (7 mbar).

The rate of photon absorption by an atom (or molecule) resulting in the excitation of a bound electron from a lower to an upper energy level depends on the strength of the upwards electronic transition that is given by the Einstein coefficient for absorption, a quantity intrinsic to the particular transition and that is usually measured in spectroscopy laboratories. The absorption also depends

FIG. 4.1. Coherent scattering of a photon with frequency v12 by absorption and re-emission between discrete atomic energy levels.

FIG. 4.1. Coherent scattering of a photon with frequency v12 by absorption and re-emission between discrete atomic energy levels.

on the broadening of the bound energy levels, as this determines the available energy states that an electron can be excited to. Emission, on the other hand, not only depends on the available energy states of the upper level of the transition but also on the population of atoms that have electrons in this excited state. In thermodynamic equilibrium, TE, a system is closed and all processes are in equilibrium, the population of the excited state is given by the Boltzmann distribution, which depends on the atmospheric temperature. The radiation field and the thermodynamic state of the atmosphere can be closely coupled through collisions that excite and de-excite atoms and so govern their populations at each level. When collisions control the populations of the energy levels in a particular part of an atmosphere we have only local thermodynamic equilibrium, LTE, as the system is open to radiation loss. When collisions become infrequent then there is a decoupling between the radiation field and the thermodynamic state of the atmosphere and emission is determined by the radiation field itself, and we have no local thermodynamic equilibrium, or Non-LTE.

4.2.1.1 Natural line broadening Atomic transitions between discrete energy levels, Ei and E2 result in the absorption and emission of photons of energy exactly equal to the difference in energy between the two levels, as shown in Fig. 4.1, where

According to the Heisenberg Uncertainty Principle, an atom (or molecule) has a definite lifetime, At, at an energy level with uncertainty, AE, in its energy, given by

where h = h/2n, and h is the Planck constant. The lifetime of an atom in its ground state is large with the result that this natural energy-level broadening is negligible. The first excited state has a smaller lifetime with the result that the energy-level broadening is significant and measurable (Fig. 4.2). The probability density (absorption profile) that an atom is excited to the upper level of the transition by absorbing a photon of energy corresponding to a wave number w, where w = v/c, is given by the Lorentz profile

FlG. 4.2. Broadening of the energy of the upper subordinate level of a bound-bound transition arising from the ground state.

which is depicted in Fig. 4.3. The natural broadening parameter, bN corresponds to the half-width of the Lorentz profile at half-maximum height, and the profile is centred at wo, the spectral line centre. The absorption profile is normalized according to

4.2.1.2 Collisional line broadening Collisions between the absorbing atom and other atoms decrease the lifetime of the absorbing atom in the excited state with the result that the upper energy level is significantly broadened. The absorption profile is Lorentzian and so it is the same as that arising from natural line broadening. However, the absorption profile half-width parameter, bc, is very large compared to that for natural broadening and is both pressure and temperature depended, having a typical dependence where p and T are the atmospheric pressure and temperature, respectively, while po and To are the reference pressure and temperature of the measurement bCo. We may combine the natural and collisional half-widths into one Lorentzian profile half-width bL = bN + bc, on the reasonable assumption that the two broadening processes are independent. For example, at T=300 K, p=1 bar, bL « 0.1 cm-1 for rotational lines of CO2 with w located near 1000 cm-1. If the transition is between subordinate levels that are broadened then it can be shown that the absorption profile is Lorentzian with half-width equal to the sum of the half-widths of the two levels of the transition.

4.2.1.3 Doppler line broadening Line broadening also arises owing to the random thermal motions of atoms that result in Doppler shifts in photon frequency as seen by the absorbing atom. A photon of frequency v travelling in direction

FlG. 4.3. Normalized Voigt, Lorentz and Doppler profiles of spectral lines, for a = 1, as functions of the dimensionless wave number parameter u, see eqn (4.12).

n relative to an atom with velocity component v, in the direction of the photon, appears to have a frequency that is Doppler shifted by

If v is positive, the atom is approaching the photon and the frequency of the photon is increased (blueshifted) in the rest frame of the atom. The photon frequency is redshifted if the atom is receding.

For an atmosphere in thermodynamic equilibrium, the velocity distribution of the atoms is Maxwellian and the probability of an atom having a component within dv of v, in any given direction, is

ay/n and arises from a Boltzmann distribution of particle kinetic energies, where a represents the most probable speed (in 3D) of the atoms given by

where m is the mass of the atom (or molecule), M its molecular weight (g mole-1), and R = kNA = 8.314 x 107 erg K 1 mole 1 is the universal gas constant, where k = 1.3806 x 10-16 erg K-1 is Boltzmann's constant and Na =

6.022 x 1023 mole 1 is Avogadro's number. The velocity distribution is thus Gaussian with standard deviation equal to a/a/2.

An atom that absorbs a photon through transitions between two discrete states with energy difference hvo can now absorb photons at other frequencies besides vo. If we transform to wave number then we can convert the atom's velocity v to a Doppler shift in wavenumber via v = c(w — wo)/wo (4.9)

and the velocity distribution to a probability density for absorption, or Gaussian absorption profile e-((w-Wo)/7D)2

YdV n with standard deviation equal to 7d/v/2, where 7d = awQ/c is the Doppler shift corresponding to the most probable velocity, while the Doppler half width bD = 7d (ln2)1^2 and is thus given by n)

and depends solely on the atmospheric temperature. For rotational lines of CO2 we have seen that, for T = 300 K at one atmosphere, the collisional broadening half-width is about 0.1 cm-1, while for Doppler broadening of rotational lines with = 1000 cm-1 the half-width is only 0.001 cm-1. Thus, not only are the absorption profiles of collisionally broadened lines wider than those of Doppler-broadened lines within the troposphere, they also have Lorentzian profiles with significant absorption for frequencies far from the line centre vo. In the stratosphere, where the pressure is low, Doppler line broadening dominates over collisional.

4.2.1.4 Mixed line broadening We have seen that collisional line broadening is significant at high pressures, while Doppler line broadening is significant at high temperatures. At intermediate conditions, we have mixed collisional and Doppler line broadening and an associated absorption profile in terms of the Voigt function, H (a, u)

with the normalization condition

where a = &l /yd is the mixing parameter and rT/ ^ a fœ e-y2dy and u = (w — wq)/yd- In terms of u we also have a/n v2 + a2

When a ^ 1 the Voigt profile becomes Doppler, while for a > 1 it becomes Lorentzian. The Voigt profile is similar to the Doppler profile for u < 2, referred to as the Doppler core, and has Lorentzian-type wings outside this region, as shown in Fig. 4.3, for a = 1.

For an atmosphere in thermodynamic equilibrium, the populations of atoms (or molecules) in each energy level is given by the Boltzmann distribution n. n. e hvi,r/kT

where ni r is the number of atoms per unit volume (number density), in level r, ni corresponds to the total population of atoms type i, qir is the statistical weight of level r arising, for example, from the degeneracy (different quantum states with the same energy) of angular momentum. The partition function u is given by

For a particular type of atom, the ratio of the populations in level r and the ground state is given by

There are three radiative processes, as shown in Fig. 4.4, that control the populations of the upper (u) and lower (l) level of a bound-bound transition of energy difference hv0

1. Spontaneous emission of a photon with de-excitation from the upper level to the lower at a rate Aui per second.

2. Absorption of a photon at a rate BluJa per second with excitation from the lower level to the upper.

3. Stimulated emission at a rate of Bul Je per second with de-excitation from the upper level to the lower level.

The coefficients Aul, Blu and Bui, are the Einstein coefficients for the above radiative processes, and Ja is the total effective mean radiance for absorption given by

where J is the mean radiance, defined by eqn (3.8), and we have introduced the absorption profile, $>(v), to allow for the probability of absorption of photons with frequency v, away from the line centre of the transition, vo, due to line-broadening mechanisms. Similarly, the total effective mean radiance for stimulated emission is given by

where ^ is the stimulated emission profile, which for unpolarized radiation is identical to the spontaneous emission profile (Dirac 1958, Oxenius 1965). In thermodynamic equilibrium, TE, all processes are balanced and the radiance Iv is given by the Planck function Bv, the radiance of a blackbody. The radiation field is then isotropic and hence Iv = Jv and hence Jv = Bv. If n\ and nu are the total populations of atoms in the lower and upper state, respectively, then the total number of atoms that are radiatively excited to the upper level per second per unit volume of atmosphere is given by dt with

However, the width of a spectral line is insignificant compared to the width of the Planck function and hence we can assume that Bv is constant over the width

of the spectral line and so Ja = BVo. In a similar way we can obtain the total number of atoms undergoing radiative de-excitation from the upper level dn

where

In TE, collisions are effective in maintaining complete redistribution of the populations of atoms in the upper state and so preserve the line-broadening distribution which is that given by the absorption profile, hence = and Je = BVo. Further, in TE there must be a balance between the radiation processes so that dnex/dt = dndex/dt and hence niBiuBv0 = nu(Aul + BUB0). (4.26)

On rearranging we obtain

niBiu/nuBui - 1

which can only be true for the Planck function if

on using the TE Boltzmann distribution for nu/ni. These are known as the Einstein relations that are independent of temperature and pressure and so they hold whether or not we have TE.

The line absorption coefficient kv, defined in Chapter 3, can now be written in terms of the absorption profile and the Einstein coefficient for absorption

where

and k0 is the frequency integrated line absorption coefficient, also known as line strength or line intensity. We can rewrite the line absorption coefficient to include a correction that takes into account stimulated emission, regarded as negative absorption. We can do this since stimulated emission results in photons that have the same direction as the stimulating photons. On the other hand, spontaneous emission takes place isotropically. Thus we have

Under TE we have complete redistribution and so 4>(v) = 4>(v) and we can use the Einstein relations plus the Boltzmann distribution to obtain

This can be expressed also in terms of the oscillator strength of the transition, f\u that is related to B\u via hvoB\u/4n = ne2f\u/mec, (4.34)

where me is the rest mass of the electron. Thus, given the absorption profile and either the Einstein coefficient or the oscillator strength we can calculate the line absorption coefficient and hence the line optical depth if we know the absorber amount. The optical depth can now be written as

if we can assume that the absorption coefficient is constant in the layer that we have absorber amount or path X (cm), kv then has units of cm-1, as $ has units of Hz-1 and n\ has units of cm-3. The line strength k0 then has units of cm-1Hz, in frequency units. We note that as an integral in wavenumber, the line strength has units of cm-2 .

In the absence of reflection, the transmissivity, or transmission coefficient, tv, of the atmospheric layer, is related to its absorptivity, av, through tv = e-Tv = 1 - av. (4.36)

We can now generally define the line absorptance, or line equivalent width, A0, in terms of line strength K0, for a line centred at v0, by

where tv (a,X) = K0$(a,v)X, where a is the mixing parameter for the Voigt profile. We note that frequency can only vary between zero and infinity. As we have seen in §3.5.4, an atmospheric layer of uniform temperature emits thermal radiation according to evBv where the emissivity of the layer is given by ev = 1 - e-Tv (4.38)

based on Kirchoff's Law that states that the absorption coefficient, kv is equal to the emission coefficient, ev. Thus, the spectral line emissivity, ev, is equal to the spectral line absorptivity, av, for the atmospheric layer. For a blackbody, we have that av = eu = 1.

For rotational lines, it is convenient to work with wavenumber, w = v/c in units of cm-1, and so we can now define the mean absorptivity, ao, over the spectral line via

J Au where Aw is the integration interval of the spectral line. Thus ao is dimension-less, while the line absorptance has units of cm-1. We can also convert to the dimensionless wave number parameter u = (w — w0)/yD and so obtain

if the line is symmetric about line centre, wo. A useful dimensionless form of the equivalent width is w(a,X) = Ao(a,X)/2yd.

The curve of growth gives the variation of the line absorptance, or equivalent width, as a function of absorber amount, X. If we introduce the dimensionless absorber-amount parameter y, defined by v = (4.41)

YDV n where the line strength, Ko, is in cm-2 (integrated in wave number units), X is in cm and yd is in cm-1, then in terms of dimensionless quantities, the curve of growth is given by w(a,y) versus y.

4.2.4.1 Limiting forms Let us denote wD, wL and wV, the dimensionless equivalent width for Doppler, Lorentz and Voigt absorption profiles, respectively. If a ^ 0, or y ^ 0, then wV ^ wD. When a > 1 or y > 10/a then wV ^ wL. For y < 10 the Doppler equivalent width can be computed from (Struve and Elvey, 1934)

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