Compton scattering is the inelastic scattering of a photon by an electron that is initially at rest or nearly at rest.
In this process, the photon changes direction and loses part of its energy, while the electron recoils and gains the
transferred energy as kinetic energy. This effect is one of the classic demonstrations that electromagnetic radiation
behaves like a stream of particles carrying both energy and momentum.
The standard wavelength form of the Compton relation is
Compton wavelength shift.
\[
\Delta\lambda=\lambda'-\lambda=\lambda_C(1-\cos\theta),
\qquad
\lambda_C=\frac{h}{m_ec}.
\]
Here, \(\lambda\) is the incident photon wavelength, \(\lambda'\) is the scattered photon wavelength, \(\theta\) is
the scattering angle of the photon, \(h\) is Planck’s constant, \(m_e\) is the electron mass, and \(c\) is the speed
of light. The quantity \(\lambda_C\) is the Compton wavelength of the electron. The shift depends only on the
scattering angle and not directly on the incident wavelength.
Since photon energy and wavelength are related by
\[
E=\frac{hc}{\lambda},
\]
the same physics can be written directly in terms of photon energies. This is often the most convenient form for X-ray
and gamma-ray problems.
Energy form of Compton scattering.
\[
E'=\frac{E}{1+\dfrac{E}{m_ec^2}(1-\cos\theta)}
\]
In this equation, \(E\) is the incident photon energy and \(E'\) is the scattered photon energy. The factor
\(m_ec^2\approx 0.511\ \mathrm{MeV}\) is the electron rest energy, which sets the natural energy scale of the problem.
If the incident photon energy is small compared with \(m_ec^2\), the energy loss is relatively modest. If the photon
energy is comparable to or larger than this scale, the energy transfer can become substantial.
The energy transferred to the electron is simply the difference between initial and final photon energies:
Transferred energy.
\[
\Delta E = E - E'.
\]
In the simplest ideal treatment, this transferred energy becomes the kinetic energy of the recoil electron. Because the
scattered photon energy decreases as the angle increases, the transferred energy increases with angle. The maximum
transfer occurs for backscatter at \(\theta = 180^\circ\).
Maximum energy transfer
At \(\theta = 180^\circ\), the cosine becomes \(-1\), so the denominator is largest:
\[
E'_{\min}=\frac{E}{1+2E/(m_ec^2)}.
\]
Therefore, the maximum electron energy gain is
\[
\Delta E_{\max}=E-E'_{\min}.
\]
This is the largest possible single-event energy transfer for a photon of that incident energy scattering from a free
electron initially at rest.
Sample example: \(E=1\ \mathrm{MeV}\), \(\theta=180^\circ\)
For a 1 MeV photon, first form the dimensionless parameter
\[
\alpha=\frac{E}{m_ec^2}=\frac{1}{0.511}\approx 1.96.
\]
At \(\theta=180^\circ\), we have \(1-\cos\theta=2\). Then
\[
\begin{aligned}
E' &= \frac{1\ \mathrm{MeV}}{1+2(1/0.511)} \\
&\approx 0.204\ \mathrm{MeV}.
\end{aligned}
\]
So the transferred energy is
\[
\Delta E = 1.000 - 0.204 \approx 0.796\ \mathrm{MeV}.
\]
This means a large fraction of the photon’s energy can be transferred to the electron in a backward-scattering event.
The photon still emerges with nonzero energy, but it is much less energetic than before the collision.
Physical interpretation
Compton scattering is especially important for X-rays and gamma rays interacting with matter. At low photon energies,
the wavelength shift and energy loss are relatively small. At higher energies, the effect becomes much more pronounced.
The dependence on angle is also crucial: forward scattering corresponds to small energy transfer, while large angles
correspond to larger energy transfer.
In more advanced university treatments, one often studies the differential cross section using the Klein–Nishina
formula, which describes the angular probability distribution of the scattering process. However, for many practical
calculations, the single-event energy formula used here is the key result.
| Concept |
Main relation |
Meaning |
| Compton wavelength shift |
\(\Delta\lambda=\lambda_C(1-\cos\theta)\) |
Change in photon wavelength after scattering |
| Energy relation |
\(E'=\dfrac{E}{1+\dfrac{E}{m_ec^2}(1-\cos\theta)}\) |
Scattered photon energy |
| Transferred energy |
\(\Delta E = E - E'\) |
Energy gained by the recoil electron |
| Backscatter limit |
\(\theta=180^\circ\) |
Maximum energy transfer |
| Electron rest energy scale |
\(m_ec^2 \approx 0.511\ \mathrm{MeV}\) |
Natural energy scale of the effect |