Compton scattering is one of the classic demonstrations that light carries both wave-like and particle-like properties.
In this process, an incoming photon scatters from an electron, and the photon emerges with a longer wavelength than before.
That increase in wavelength means the scattered photon has less energy, while the electron takes away the missing energy and momentum.
This effect was crucial in establishing the quantum picture of electromagnetic radiation.
The central formula for the wavelength shift is
\[
\Delta\lambda=\lambda'-\lambda=\frac{h}{m_ec}(1-\cos\theta).
\]
The quantity
\(\dfrac{h}{m_ec}\)
is called the Compton wavelength of the electron, usually written as
\(\lambda_C\).
Numerically,
\[
\lambda_C \approx 2.426\,\text{pm}.
\]
Using this notation, the Compton shift formula becomes
\[
\Delta\lambda=\lambda_C(1-\cos\theta),
\]
where \(\theta\) is the photon scattering angle.
This equation shows immediately that the shift depends only on angle, not on the original wavelength.
At
\(\theta=0^\circ\),
we have
\(\cos\theta=1\),
so the wavelength shift is zero.
At
\(\theta=180^\circ\),
the photon is backscattered and the shift is largest:
\[
\Delta\lambda_{\max}=2\lambda_C.
\]
Once the shift is known, the scattered wavelength is simply
\[
\lambda'=\lambda+\Delta\lambda.
\]
Photon energy is related to wavelength through
\[
E=\frac{hc}{\lambda},
\qquad
E'=\frac{hc}{\lambda'}.
\]
This lets us convert the wavelength shift directly into an energy loss.
Since the scattered wavelength is larger, the scattered photon energy is smaller.
The difference
\(E-E'\)
is the energy transferred to the recoiling electron.
There is also a very useful direct energy formula for the scattered photon:
\[
E'=\frac{E}{1+\left(E/m_ec^2\right)(1-\cos\theta)}.
\]
Here
\(m_ec^2\)
is the electron rest energy, approximately
\(511\,\text{keV}\).
This formula is equivalent to the wavelength-shift equation and is often more convenient in X-ray and gamma-ray problems where photon energies are given directly.
For the sample case of an incident wavelength
\(\lambda=0.1\,\text{nm}\)
and scattering angle
\(\theta=90^\circ\),
the cosine term is
\(\cos90^\circ=0\),
so
\[
\Delta\lambda=\lambda_C(1-0)=\lambda_C\approx 2.426\,\text{pm}.
\]
Since
\(2.426\,\text{pm}=0.002426\,\text{nm}\),
the scattered wavelength becomes
\[
\lambda' = 0.1 + 0.002426 \approx 0.1024\,\text{nm}.
\]
This is exactly the classic Compton-shift scale for X-ray scattering from electrons.
The shift is small compared with the original wavelength, but it is measurable and physically important.
The effect cannot be explained correctly by treating light only as a classical wave.
The wavelength change emerges naturally when one conserves both energy and momentum in a collision between a photon and an electron.
That is why Compton scattering is such a strong piece of evidence for photon momentum and for the particle aspect of radiation.
At a more advanced university level, one can extend the discussion to differential cross sections and the Klein–Nishina formula, which describes the angular distribution and polarization dependence of relativistic photon-electron scattering.
But for an introductory treatment, the most important insight is already contained in the shift formula:
scattering at larger angles produces a larger wavelength increase and therefore a lower scattered photon energy.
\[
\Delta\lambda=\lambda_C(1-\cos\theta),
\qquad
\lambda'=\lambda+\Delta\lambda,
\qquad
E'=\frac{E}{1+\left(E/m_ec^2\right)(1-\cos\theta)}.
\]
These equations connect scattering angle, wavelength shift, and energy transfer in a compact way and explain why Compton scattering is one of the fundamental quantum effects in high-energy photon interactions.
