Optical Spectrum Analyzers Chapter 14: Appendix
Rev D, June 16, 2022 Page 44
Chapter 14 Appendix
14.1. Calculating Absorption Cross Sections
The ThorSpectra software calculates absorption cross sections from the HITRAN line-by-line references by
separating each line into a line strength and a line shape:
𝜎
𝜈
𝑆
𝑇
𝑔𝜈𝜈
𝛿
Here, 𝑆
𝑇 is the temperature-dependent strength of the line, 𝑔 is a function describing the shape of the line,
𝜈
is the center wavenumber of the line, and 𝛿 is the shift in pressure.
The line strength, 𝑆
𝑇, is dependent on temperature because the temperature determines how strongly the
upper and lower energy levels are populated.
𝑆
𝑇
𝑆
exp
ℎ𝑐
𝑘
𝐸
1
𝑇
1
𝑇
𝑇
𝑇
1exp
ℎ𝑐𝜈
𝑇
1exp
ℎ𝑐𝜈
𝑇
Here, 𝐸
is the energy of the lower energy level, 𝑚 represents the temperature dependence of the partition
function, 𝑇 is the given temperature and 𝑇
is the reference temperature.
The line shape is affected by two effects: Doppler broadening – caused by random molecular motion – and
pressure broadening – caused by collisions between molecules. Doppler broadening leads to a Gaussian line
shape with half-width at half-max given by:
𝛼
𝜈
𝑐
2𝑘
𝑇
𝑚
where 𝑚 is the molecular mass. In contrast, pressure broadening leads to a Lorentzian line shape with half-
width at half-max given by:
𝛼
𝑇
𝑇
𝑃
𝑃
𝛾
1𝑞𝛾
𝑞
where 𝛾
is the broadening caused by collisions with other species, 𝛾
is the broadening caused by molecules
of the same species, and 𝑞 is the mixing ratio of the gas. The exponent 𝑛 is transition-dependent and given by
the line-by-line reference.
The combination of Doppler broadening and pressure broadening can be calculated by a convolution between
the Gaussian and Lorentzian line shapes, and produces a profile known as the Voigt line shape.
In many cases, pressure broadening has a significantly greater effect than Doppler broadening. If the half-width
at half-max of the Gaussian line shape is less than 1% of the half-width at half-max of the Lorentzian line shape,
the software approximates the line shape by using only the Lorentzian. Otherwise, the full convolution is
calculated.
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