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Spectrum Functions

Absorption coefficient and Cross-sections (XS), Transmittance, Radiance and Absorption function

The monochromatic  absorption line coefficient Ka_jm for a single molecule per unit volume and for a single spectral line corresponding to a transition between levels (i) and (m) is calculated as a product of the line intensity Sjm and the normalized line profile function Φ [ in 1/cm-1 units] at a given pressure P and temperature T:

Ka_jm (WN, T, P) = Sjm (T) · Φ(WNjm, WN, T, P) , [1/(molecule·cm-2 )] (1)

Here Sjm  is the spectral line intensity in HITRAN units [cm-1/(molecule·cm-2)] often reduced to [cm/molecule], WN is the current wave number [cm-1] and WNjm is the wave number of the line center.

The  absorption coefficient Ka for the spectrum of a single molecule per unit volume is obtained by the summation over all transitions j -> m included in the line list

Ka (WN, T, P) = ∑jm Ka_jm (WN, T, P) [1/(molecule·cm-2 )] (2)

In case of a gas containing various molecular species the partial absorption coefficients are weighted with the mixing ratios of these species accounting also for their isotopic abundances.
Many literature sources also use the absorption coefficient KNa “normalized” ( multiplied) by the number N [ cm-3] of absorbing molecules per unit gas volume. This quantity is often denoted as α 

α = KNa (WN, T, P) = Ka (WN, T, P) · N [cm-1] (3)

For an ideal gas N = P/kT [cm-3], where k is the Boltzmann’s constant.

The dimensionless transmittance function is given by

TR(WN, T, P, l) = exp(-KNa(WN, T, P) · l) , (4)

where l is the optical path length.

The dimensionless absorption function is given by

AF(WN, T, P, l) = 1 - TR(WN, T, P, l) , (5)

The Radiance function

L(WN, T, P, l) = AF(WN, T, P, l) · LBB(WN, T) ,  (6)

where AF(WN, T, P, l) is an absorption function, а LBB(WN, T) is the radiance of the black body [erg·c-1·cm-2·Hz-1] which is given by equation

LBB(WN, T) = 2·π·h·c·WN3 / (exp(h·c·WN/(k·T)) - 1) ,  (6')

where h is the Planck's constant, c is the speed of light, k is Boltzmann's constant, T is temperature in Kelvin, and WN is wavenumber in cm-1.

The information system calculates the transmittance, absorption function, and radiance spectra for a fixed path length in a homogeneous medium.

Experimental spectra are often published in the form of absorption cross-section

XS = - ln (TR) / N·l [cm2/ molecule]  (7)

as deduced from observed transmittance TR. The latter definition is given in HITRAN XS-units.

On the theoretical standpoint this quantity corresponds to the absorption coefficient for the spectrum of a single molecule per unit volume (2). Here the wave-number-, temperature- and pressure-dependence of theoretical cross-sections adapted to HITRAN units is computed as

XS(WN, T, P) = ∑jm i Sjm (T) ·ni · Φ(WNjm, WN, T, P) [cm2/ molecule] , (8)

where ni are mixing ratios of the contributing molecular species.

Another relevant quantity often used in the literature is the absorbance A defined as the logarithm of the transmittance

A = - log10 ( TR)                                     (9)

For example the PNNL library (https://secure2.pnl.gov/nsd/nsd.nsf/Welcome) provides the absorbance A for a sample concentration of one part-permillion
(ppm) over an optical path length of one meter (m) at a temperature of 296 Kelvin (K) in units [ ppm-1·m-1].