I was wondering if there's a simple way to compute the mean free path of UV photons in a optically thick medium with density n.
I've looked up at the literature and found out that the mean free path of Lyman-Werner band photon is ~1 Mpc on cosmological scales but 0.1 pc inside clouds. However, I would like to read an article about that but I can't find any.
If you know specifically what medium, you can look it up in the NIST XCOM database:
http://www.nist.gov/pml/data/xcom/index.cfm
Click on the "Database Search Form" and enter the details you have. Read the introduction text to determine how to use the results, but you will likely only need to multiply what the chart says by the density of the material.
For light traveling through a medium in the direction x,
$$I(x) = I_0 e^{-\alpha x}$$
where $I(x)$ is the intensity of light at position $x$ and $I_0$ is the intensity of light at position $x=0$. $\alpha$ is called "absorption coefficient". The absorption coefficient is related to absorption cross-section by
$$\alpha = n\sigma$$
where $n$ is the number of absorbing "things" (I guess atoms or molecules) per volume and $\sigma$ is the absorption cross-section of a single "thing". The mean free path is--I assume--defined as the mean distance which a photon can travel before getting absorbed. That would be $1/\alpha$.
"1.) I was wondering if there's a simple way to compute the mean free path of UV photons in a optically thick medium with density n."
"2.) I've looked up at the literature and found out that the mean free path of Lyman-Werner band photon is ~1 Mpc on cosmological scales but 0.1 pc inside clouds. However,
- I would like to read an article about that but I can't find any."
1.) The mean free path is the average distance traveled by a moving particle (such as an atom, a molecule, a photon) between successive impacts (collisions), which modify its direction or energy or other particle properties.
Wikipedia: https://en.m.wikipedia.org/wiki/Mean_free_path#Mean_free_path_in_radiography
"In gamma-ray radiography the mean free path of a pencil beam of mono-energetic photons is the average distance a photon travels between collisions with atoms of the target material. It depends on the material and the energy of the photons:
$$\ell =\mu ^{{-1}}=((\mu /\rho )\rho )^{{-1}},$$
where $μ$ is the linear attenuation coefficient, $μ/ρ$ is the mass attenuation coefficient and $ρ$ is the density of the material.
The Mass attenuation coefficient can be looked up or calculated for any material and energy combination using the NIST databases. See: "X-Ray Mass Attenuation Coefficients" or "XCOM: Photon Cross Sections Database".
In X-ray radiography the calculation of the mean free path is more complicated, because photons are not mono-energetic, but have some distribution of energies called a spectrum. As photons move through the target material, they are attenuated with probabilities depending on their energy, as a result their distribution changes in process called spectrum hardening. Because of spectrum hardening, the mean free path of the X-ray spectrum changes with distance.
Sometimes one measures the thickness of a material in the number of mean free paths. Material with the thickness of one mean free path will attenuate 37% (1/e) of photons. This concept is closely related to half-value layer (HVL): a material with a thickness of one HVL will attenuate 50% of photons. A standard x-ray image is a transmission image, an image with negative logarithm of its intensities is sometimes called a number of mean free paths image."
Matlab code to calculate mean free path for various frequencies of the electromagnetic spectrum in various elements is available here.
2.) The mean free path of Lyman-Werner band photons are a more complicated subject due to gravity, distance, redshift and fluctuations of the cloud density (including dust), and thus involve quantum mechanics.
Wikipedia: https://en.wikipedia.org/wiki/Lyman%E2%80%93Werner_photons
"A Lyman-Werner photon is an ultraviolet photon with a photon energy in the range of 11.2 to 13.6 eV, corresponding to the energy range in which the Lyman and Werner absorption bands of molecular hydrogen (H2) are found. A photon in this energy range, with a frequency that coincides with that of one of the lines in the Lyman or Werner bands, can be absorbed by H2, placing the molecule in an excited electronic state. Radiative decay (that is, decay into photons) from this excited state occurs rapidly, with roughly 15% of these decays occurring into the vibrational continuum of the molecule, resulting in its dissociation.1 This two-step photodissociation process, known as the Solomon process, is one of the main mechanisms by which molecular hydrogen is destroyed in the interstellar medium."
A Google book search turns up many books and papers covering the subject, I am not sure why you had trouble locating some articles. Here is a simple and particularly good explanation of "Lyman-Werner Photons and the Solomon Process".
Source: "Light-Cone Effect of Radiation Fields in Cosmological Radiative Transfer Simulations"
"Cosmological radiative transfer simulations are used to study astrophysical processes which occur on a large, cosmic scale. A notable example of such processes is the process of cosmic reionization, in which individual H II regions take up large volumes, of order (∼ 20 comoving Mpc)$^3$ or larger, after ∼ 50% of all the baryons are reionized by astrophysical radiation sources (Furlanetto & Oh 2005; Furlanetto et al. 2006; Iliev et al. 2007; Zahn et al. 2007). The typical radiation sources long after the recombination epoch are stars and quasars, which emit predominantly ultra-violet (UV) and X-ray photons, respectively.
The mean free path of hydrogenionizing (H-ionizing) UV photons is simply the average size of H II regions,$^1$ while the mean free path of Xray photons is much larger than that of UV photons due to the relatively small optical depth (the scattering cross section of UV photons is much larger than that of X-ray photons: e.g., Mesinger et al. 2013; Xu et al. 2014). Therefore, the mean free path of X-ray photons is truly cosmological, surpassing several hundred Mpc when the rest-frame photon energy is larger than a few keV (Xu et al. 2014).
Another example of long mean free paths is the Lyman-Werner (LW) band photons. These photons lie at the energy range of 11 - 13.6 eV, and thus traverse cosmological distances freely until they redshift into hydrogen Lyman resonance lines and are scattered oneutral hydrogen atoms (Haiman et al. 2000; Ahn et al. 2009). Practically all LW band photons are scattered multiple times and reprocessed into lower-energy photons when they traverse the "Lyman-Werner horizon" r$_{LW}$ ∼ 100[21=(1+z)]$^{0.5}$ Mpc, where $z$ is the redshift at which the photons are emitted (Ahn et al. 2009).
The light-cone effect, which denotes the effect related to fields travelling at the speed of light and thus tracing past-time events along the corresponding light cones, is inherent in any physical processes on cosmic scales. Radiation fields and gravitational fields are the obvious examples governed by the light-cone effect: both fields travel at the speed of light (see Hwang et al. 2008 showing that the gravitational field travels at the speed of light in the form of the electric part of the Weyl tensor). The relevant time or length scale over which the light-cone effect matters can be estimated by an effective time-scale $tlc ≡ f=(df=dt)$, where $f$ is any physical quantity relevant to the problem of interest and $df=dt$ is the change rate of $f$.
For astrophysical problems, the time scale may be naively estimated by the lifetime of radiation sources of interest. In contrast, gravitational fields under non-relativistic motion (velocity ≪ $c$) will be corrected from the Newtonian field, which is constructed in an action-at-a-distance manner, only very slightly (about 10$^{−6}$ - 10$^{−4}$ or somewhat larger if cumulative effects are considered) by general relativistic effects including this light-cone effect (Hwang et al. 2008).
Light-cone effect of radiation fields traversing cosmological distances has been usually approximated by a uniform, spatially-averaged quantity in semi-analytical studies (e.g., Haiman et al. 2000 for LW and H-ionizing backgrounds), or calculated by computationally expensive, brute-force methods of ray-tracing with finite speed of light (e.g., Bryan et al. 2014 for H-ionizing background).
