Compton scattering

Scattering X-rays from electrons

Compton scattering occurs when high-energy light, such as X-rays, scatters from electrons and emerges with a longer wavelength. Classical wave theory could explain some scattering phenomena, but it could not explain the precise wavelength shift observed by Arthur Compton.

The key discovery was that the wavelength shift depends on scattering angle, not on the material in the way classical theory expected. This suggested that light was behaving like particles with energy and momentum colliding with electrons.

Photon energy and momentum

A photon of frequency $f$ has energy

$E=hf$

Using $c=flambda$, this can be written as

E=rac{hc}{lambda}

Special relativity says a massless particle satisfies

$E=pc$

so photon momentum is

p=rac{E}{c}=rac{h}{lambda}

This momentum is essential for understanding Compton scattering.

Collision picture

In the simplest model, an incoming photon strikes an electron initially at rest. After the collision, the photon scatters at angle $heta$ with lower energy and longer wavelength. The electron recoils with kinetic energy and momentum.

Energy and momentum are conserved together. The photon loses some energy to the electron, so its frequency decreases and its wavelength increases.

Compton shift formula

The observed wavelength change is

Deltalambda=lambda'-lambda=rac{h}{m_ec}(1-cos heta)

The quantity

lambda_C=rac{h}{m_ec}

is the Compton wavelength of the electron. Numerically, it is about

$lambda_Capprox2.43 imes10^{-12} m$

Thus

$Deltalambda=lambda_C(1-cos heta)$

Angle dependence

If $heta=0^circ$, the photon is not deflected, and

$Deltalambda=0$

If $heta=90^circ$, then

$Deltalambda=lambda_C$

If $heta=180^circ$, backscattering gives the maximum shift:

$Deltalambda=2lambda_C$

This angle dependence strongly supports the photon collision model.

Why visible light usually shows tiny shifts

The Compton wavelength of the electron is much smaller than visible wavelengths. For visible light, the fractional wavelength shift is usually tiny. X-rays have wavelengths comparable enough to make the effect measurable.

This is why Compton's experiments used X-rays rather than ordinary visible light.

Classical and quantum contrast

Classical electromagnetic waves carry energy and momentum too, but Compton scattering shows energy-momentum transfer in discrete photon events. The outgoing light contains shifted photons whose wavelengths follow relativistic collision kinematics.

This strengthened the case that light has particle-like properties, complementing the photoelectric effect.

The big idea

Compton scattering demonstrated that photons carry momentum $p=h/lambda$. Treating X-ray scattering as a relativistic collision between a photon and an electron explains the wavelength shift Deltalambda=rac{h}{m_ec}(1-cos heta). Light is not merely a continuous classical wave; it exchanges energy and momentum in quantum units.