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My current line of research deals a lot with hydrogen's Lyman-alpha emission and subsequent interactions of the Lyman-alpha photons with the surrounding hydrogen gas. My question is whether (stimulated emission aside) there is any way to predict (or more likely find a probability distribution for) the subsequent propagation direction of a photon after being absorbed and emitted by a neutral atom?

I am also curious to know if there is way to predict (or find a probability distribution for) the propagation direction of a photon after a Compton scattering.

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For Compton scattering, the differential cross section (scattering probability distribution vs. solid angle) is given by the Klein-Nishina formula. A good discussion of Compton scattering, from an experimenter's point of view, can be found in chapter 9 of the now-classic "Experiments in Modern Physics" by Adrian C. Melissinos and Jim Napolitano, recent editions of which are still in print. In the footnotes, Melissinos lists "The Quantum Theory of Radiation", by W. Heitler (a book which is also still in print) as his preferred source for the actual mathematical derivation of the formula.

Not sure about Lyman emission; sorry.

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Probably too late now, but I have to answer this. There is indeed a way to predict the direction of a scattered Lyman $\alpha$ photon. The answer depends on whether the scattering takes place in the core or the wings of the line.

In the core (i.e. closer to the line center than about 3 Doppler widths), we can use the dipole approximation, so the phase function describing the direction of scattering is given by $$W(\theta) \propto 1 + \frac{R}{Q}\cos^2\theta,$$ where $R/Q$ is the degree of polarization for 90º scattering. It turns out (Hamilton 1940) that with a probability $1/3$, this results in an isotropic distribution, while with a probability $2/3$, $$W(\cos\theta) = \frac{7}{16} \left( 1 + \frac{3}{7} \cos^2\theta \right).$$

In the case of wing scattering, Stenflo (1980) found that the scattering behaves like Rayleigh scattering, i.e. it follows a dipole distribution with 100% polarization at 90º, so $$W(\cos\theta) = \frac{3}{8} \left( 1 + \cos^2\theta \right).$$

For a more thorough explanation, I cannot resist referring to chapter 3.2.2 of my thesis "Interpreting Lyman $\alpha$ Radiation from Young, Dusty Galaxies", where you will also find a numerical implementation of the above, as well as a more exact functional form of the transition between the core and line.

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