### Spectrum Functions

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

The monochromatic

*absorption line coefficient*

**K**_{a_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

**S**^{jm }and the normalized line profile function

*[ in 1/cm*

**Φ**^{-1}units] at a given pressure

*and temperature*

**P**

**T:**

**K**_{a_jm }

**(WN, T, P) = S**^{jm}

**(T)****·**

**Φ**

**(WN**^{jm}

*, [1/(molecule·cm*

**, WN, T, P)**^{-2})] (1)

Here

**S**^{jm}

*is the spectral line intensity in HITRAN units [cm*

^{-1}/(molecule

**·**cm

^{-2})] often reduced to [cm/molecule],

*is the current wave number [cm*

**WN**^{-1}] and

**WN**^{jm}is the wave number of the line center.

The

*absorption coefficient*

**K**_{a}

*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

**K**_{a }

**(WN, T, P) = ∑**_{jm}

**K**_{a_jm }

*[1/(molecule·cm*

**(WN, T, P)**^{-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*

**KN**_{a}“normalized” ( multiplied) by the number

*N*[ cm

^{-3}] of absorbing molecules per unit gas volume. This quantity is often denoted as

*α*

**α = KN**_{a }

**(WN, T, P) =**

**K**_{a }

**(WN, T, P)****·**

*[cm*

**N**^{-1}] (3)

For an ideal gas

**N = P/**

**k***[*

**T***cm*

^{-3}], where

*is the Boltzmann’s constant.*

**k**The dimensionless

*transmittance function*is given by

**TR****(**

**WN, T, P, l****) = exp(-**

**KN**_{a}

**(**

**WN, T, P****) ·**

**l****)**, (4)

where

*is the optical path length.*

**l**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) · L**^{BB}

*, (6)*

**(WN, T)**where

*is an absorption function, а*

**AF(WN, T, P, l)**

**L**^{BB}

**(**

**WN, T****)**is the radiance of the black body [erg·c

^{-1}·cm

^{-2}·Hz

^{-1}] which is given by equation

**L**^{BB}

**(WN, T) = 2·**

**π**

**·**

**h**

**·**

**c**

**·WN**^{3}

**/ (exp(**

**h**

**·**

**c**

**·WN/(**

**k***, (6')*

**·T)) - 1)**where

*is the Planck's constant,*

**h***is the speed of light,*

**c***is Boltzmann's constant,*

**k****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**

**·***[cm*

**l**^{2}/ molecule] (7)

as deduced from observed transmittance

*. The latter definition is given in HITRAN XS-units.*

**TR**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}

**S**^{jm}

**(T)****·**

**n**_{i}

**·**

**Φ(WN**^{jm}

*[cm*

**, WN, T, P)**^{2}/ molecule] , (8)

where

*n*

_{i}are mixing ratios of the contributing molecular species.

Another relevant quantity often used in the literature is the

*absorbance*defined as the logarithm of the transmittance

**A***(9)*

**A = - log**_{10}( TR)For example the PNNL library (

*https://secure2.pnl.gov/nsd/nsd.nsf/Welcome*) provides the

*absorbance*for a sample concentration of one part-permillion

**A**(ppm) over an optical path length of one meter (m) at a temperature of 296 Kelvin (K) in units [ ppm

^{-1}·m

^{-1}].

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