# Equation for amount of laser light scattered/absorbed by an elemental gas dependent on density

I am looking for an equation that can effectively describe the proportion of an incoming laser with wavelength $$\lambda$$ and intensity $$I$$, that is absorbed or scattered by a gas atmosphere (eg Aluminium vapour) at a given density $$\rho$$. I know that Rayleigh scattering would play a role here but I am unsure how the amount of scattering/ absorption depends on the density of the gas that the laser is passing through.

Does such an equation or empirical relationship exist? I would expect one should exist as a higher density means a smaller mean free path for the laser photons and thus more absorption.

• Resonant fluorescence, as in the sodium laser guide star is also in the absorption and scattering mix of effects, depending on what the gas phase species happens to be and the laser wavelength.
– Ed V
Dec 2, 2021 at 2:31

You’re looking for the Beer-Lambert law.

Wavelength dependency is complicated, especially for mixtures, with characteristic peaks and troughs. For instance, “greenhouse gasses” have absorption peaks in the infrared band. You’ll want an empirical attenuation cross section for the sample in question, or a set of empirical cross sections that you can add together.

Density: Attenuation for a given wavelength passing through a given substance is directly proportional to the molar concentration of that substance. Attenuation of light passing through a mixture is the sum of attenuation of the components.

Intensity: Attenuation coefficient is independent of intensity, if you hold temperature constant. Transmitted intensity is directly proportional to incident intensity.

• I am not sure I am looking for that. Given that for many elements, the molar attenuation coefficient is not known, this makes finding the absorbance difficult. I wish to ideally find an equation that focuses on intrinsic physical properties of the gas (like its density). In addition, BL does not take into account scattering which would further reduce the intensity. Dec 2, 2021 at 8:55
• Upvoting. Hard to see how avoiding the absorption coefficients is possible: they are intrinsic to the substances, i.e., fluorine gas is pale yellow, chlorine gas is yellow-green, bromine vapor is red-brown, etc. Even oxygen gas, at high pressure, is pale blue. It is doubtful that gas phase absorbance measurements have been made for most metallic elements, simply because of the obvious experimental difficulties.
– Ed V
Dec 2, 2021 at 13:57
• @tjsmert44 you could in principle program a computer to simulate the quantum mechanics of the photon and molecule interactions and predict the wavelength dependency. There isn’t a simple, material agnostic formula.
– g s
Dec 2, 2021 at 16:30
• Beer-lambert law does include scattering, expressed as a coefficient that adds with absorption. See “attenuation coefficient”
– g s
Dec 2, 2021 at 16:43