## FANDOM

157 Pages

In the classical Compton scattering, the probability of a photon to scatter in a certain direction depends only on the scattered photon and electron. However, since photons are bosons, the probability of scattering in a certain direction increases with the number of photons already moving in that direction. Let $N \left( \mathbf{k} \right)$ be the distribution function of photons, such that the number density of photons (photons per unit volume) is given by $n = \int \frac{d^3 k}{\left( 2 \pi \right)^3} N \left( \mathbf{k} \right)$, and $\mathbf{k}$ is the wave - number (and the momentum of the photons). Let the distribution functions of the electrons be $f \left( \mathbf{p} \right)$, where $\mathbf{p}$ is the momentum of the electron. Let $w \left(\mathbf{p}, \mathbf{k}, \mathbf{k}' \right)$ be the probability of scattering an electron with momentum $\mathbf{p}$ and a photon of momentum $\mathbf{k}$ to a state where the momentum of the photon is $\mathbf{k}'$ (and the momentum of the electron is $\mathbf{p} + \mathbf{k} - \mathbf{k}'$). The rate equation for the photons is given by

$\frac{\partial N}{\partial t} = \int d^3 p \int \frac{d^3 k}{\left( 2 \pi \right )^3} f \left( \mathbf{p} \right ) N \left( \mathbf{k}' \right ) \left[ 1 + N \left( \mathbf{k} \right ) \right ] w \left(\mathbf{p}, \mathbf{k}', \mathbf{k} \right ) -$

$\int d^3 p \int \frac{d^3 k}{\left( 2 \pi \right )^3} f \left( \mathbf{p} \right ) N \left( \mathbf{k} \right ) \left[ 1 + N \left( \mathbf{k}' \right ) \right ] w \left(\mathbf{p}, \mathbf{k}, \mathbf{k}' \right )$

If the electron does not recoil, then the incoming and outgoing photons have the momenta, so detailed balance reads

$w \left(\mathbf{p}, \mathbf{k}, \mathbf{k}' \right) = w \left(\mathbf{p}, \mathbf{k}', \mathbf{k} \right)$

Substituting into the rate equation eliminates the extra terms (due to symmetry) and only the regular scattering is left.

However, if recoil is taken into account, then detailed balance reads

$w \left(\mathbf{p}, \mathbf{k}, \mathbf{k}' \right) = w \left(\mathbf{p} - \Delta \mathbf{k}, \mathbf{k}', \mathbf{k} \right)$

where $\Delta \mathbf{k} = \mathbf{k}' - \mathbf{k}$. Substituting this into the equation, assuming $k \ll p$ and applying a Taylor expansion yields

$\frac{\partial N}{\partial t} = \int d^3 p \int \frac{d^3 k}{\left( 2 \pi \right )^3} f \left( \mathbf{p} \right ) \left[ N \left( \mathbf{k}' \right ) - N \left(\mathbf{k} \right ) \right ] w \left(\mathbf{p}, \mathbf{k}, \mathbf{k}' \right ) +$

$\int d^3 p \int \frac{d^3 k}{\left( 2 \pi \right )^3} \Delta \mathbf{k} \cdot \frac{\partial f}{\partial \mathbf{p}} N \left( \mathbf{k}' \right ) N \left(\mathbf{k} \right ) w \left(\mathbf{p}, \mathbf{k}, \mathbf{k}' \right )$

The first integral is just the classical Compton scattering. The second term is a correction due to recoil and induced emission.