## Compton Scattering

## Compton Scattering Formula

The Compton formula was published in 1923 in the Physical Review. Compton explained that the particle-like momentum of photons causes the X-ray shift. 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. The photon bounces off the electron upon collision, 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 a **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 the 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