Compton scattering occurs when a photon interacts with an electron by transferring the first energy to the second (inelastic interaction). The wavelength of the photon after the scattering can be calculated by

equation

where

equation=9146

Compton wave length and \theta the angle of deviation of the photon is.

Compton scattering occurs when a photon interacts with a charged particle, in particular with an electron. In the process the photon loses energy and deviates by putting the electron in motion:

image

In the case of Compton scattering, the differential effective section is according to Klein-Nishina

equation

where

equation=9112

is the Thomson total effective section and the

equation=9113

is the normalized energy.

The Compton wavelength is defined by

equation

where h is the Planck constant, m_e mass of the electron and c the speed of light.

Scattering that contributes (in) or describes the abandonment of particles (out) can be plotted as follows:

image

It should be noted that the term collision:

- integrates on all external speeds to those of volume

- includes the likelihood of both speeds leading to scattering simultaneously

- the relative velocity multiplied by the total effective section represents the flow of particles towards the target

The latter can be shown in a simple way through

\Delta v\sigma\sim\displaystyle\frac{dX}{dt}S\sim \displaystyle\frac{dV}{dt}\sim J

The solid angle is defined by

equation

If the differential effective section is taken according to Klein-Nishina

equation=9144

and integrates in the solid angle

equation=9147

the total effective section is obtained

equation

where

equation=9112

is the effective section of Thomson and the

equation=9113

is the normalized energy.

At the limit of small \epsilon\ll1 we have that the total section is

\sigma_{KN}\sim\sigma_T\left(1-2\epsilon+\displaystyle\frac{26}{5}\epsilon^2\ldots\right)

and in the limit \epsilon\gg 1 the total effective section is

\sigma_{KN}\sim\displaystyle\frac{3}{8}\displaystyle\frac{\sigma_T}{\epsilon}\left(\log(2\epsilon)+\displaystyle\frac{1}{2}\right)

The total effective section of Thomson is equal to 2/3 of the surface of a sphere of radius r_0

equation

The radius r_0 corresponds to the classical radius of the electron which is defined as e^2/m_ec ^ 2.

To simplify we introduce the initial energy of the photon E, normalized by m_ec^2

equation

where m_e is the mass of the electron and c the speed of light.

The Klein-Nishina model can be studied in numerical form. This is shown

- the total effective section as a function of photon energy

- the differential section as a function of the angle for the minimum, medium and maximum energies defined

- what would be the total effective section in a one-dimensional system that gives according to the energy transmission or reflection