## Compton Scattering of X-rays

The Compton formula was published in 1923 in the Physical Review. Compton explained that the X-ray **shift** is caused by particle-like **momentum of photons**. **Compton scattering formula** is the mathematical relationship between the **shift in wavelength** and the scattering angle of the X-rays. In the case of Compton scattering the photon of frequency *f* collides with an electron at rest. Upon collision, the photon bounces off electron, giving up some of its initial energy (given by Planck’s formula E=hf). While the electron gains momentum (mass x velocity), the **photon cannot lower its velocity**. As a result of momentum conservation law, the photon must lower its momentum given by:

So the decrease in photon’s momentum must be translated into **decrease in frequency** (increase in wavelength Δ**λ = λ’ – λ**). The shift of the wavelength increased with scattering angle according to **the Compton formula**:

where **λ** is the initial wavelength of photon **λ’** is the wavelength after scattering, **h **is the Planck constant = 6.626 x 10^{-34} J.s, **m _{e}** is the electron rest mass (0.511 MeV)

**c**is the speed of light

**Θ**is the scattering angle. The minimum change in wavelength (

*λ′*−

*λ*) for the photon occurs when Θ = 0° (cos(Θ)=1) and is at least zero. The maximum change in wavelength (

*λ′*−

*λ*) for the photon occurs when Θ = 180° (cos(Θ)=-1). In this case the photon transfers to the electron as much momentum as possible. The maximum change in wavelength can be derived from Compton formula:

The quantity h/m_{e}c is known as the **Compton wavelength** of the electron and is equal to **2.43×10 ^{−12}**

**m**.

## Interaction of X-rays with Matter

Although a large number of possible interactions are known, there are three key interaction mechanisms with matter. The strength of these interactions depends on the **energy of the X-rays** and the elemental composition of the material, but not much on chemical properties, since the X-ray photon energy is much higher than chemical binding energies. The photoelectric absorbtion dominates **at low-energies of X-rays, **while Compton scattering dominates at higher energies.

**Photoelectric absorption****Compton scattering****Rayleigh scattering**