Compton scattering is the inelastic scattering of a photon by a charged particle, usually an electron that is initially
at rest or only weakly bound. The key experimental result is that the outgoing photon has a longer wavelength than the
incoming photon, which means it has lower energy after the collision. This wavelength increase cannot be explained by
classical wave scattering alone. It follows naturally when light is treated as a particle carrying both energy and
momentum.
In the standard Compton model, an incoming photon of wavelength \(\lambda\) strikes an electron. After the collision,
the photon leaves at an angle \(\theta\) with a new wavelength \(\lambda'\), while the electron recoils and carries away
kinetic energy. By combining conservation of energy with conservation of momentum, one obtains the famous Compton shift
formula.
Compton wavelength shift.
\[
\begin{aligned}
\Delta \lambda &= \lambda' - \lambda \\
&= \frac{h}{m_e c}\left(1-\cos\theta\right)
\end{aligned}
\]
The constant
Electron Compton wavelength.
\[
\begin{aligned}
\lambda_C &= \frac{h}{m_e c} \\
&\approx 0.002426\ \mathrm{nm}
\end{aligned}
\]
is called the Compton wavelength of the electron. Using it, the scattering formula becomes
Compact form.
\[
\begin{aligned}
\Delta\lambda &= \lambda_C(1-\cos\theta)
\end{aligned}
\]
This equation has two especially important consequences. First, the shift depends on the scattering angle \(\theta\).
Second, the shift does not depend on the incoming wavelength directly. The incident wavelength changes the
initial and final photon energies, but the amount by which the wavelength increases is determined only by the angle and
fundamental constants.
Special cases
If \(\theta = 0^\circ\), then \(\cos\theta = 1\), so
No scattering shift at 0°.
\[
\begin{aligned}
\Delta\lambda &= \lambda_C(1-\cos 0^\circ) \\
&= \lambda_C(1-1) \\
&= 0
\end{aligned}
\]
so the photon wavelength does not change. If \(\theta = 180^\circ\), then \(\cos\theta = -1\), and the shift is
largest:
Maximum single-scattering shift.
\[
\begin{aligned}
\Delta\lambda_{\max} &= \lambda_C(1-\cos 180^\circ) \\
&= \lambda_C(1-(-1)) \\
&= 2\lambda_C
\end{aligned}
\]
This is the greatest wavelength increase possible for a single Compton scattering event from an electron at rest.
Scattered wavelength and energies
Once the shift is known, the scattered wavelength is found by simple addition:
Scattered wavelength.
\[
\begin{aligned}
\lambda' &= \lambda + \Delta\lambda
\end{aligned}
\]
Photon energy is related to wavelength by
Photon energy.
\[
\begin{aligned}
E &= \frac{hc}{\lambda}, \\
E' &= \frac{hc}{\lambda'}
\end{aligned}
\]
Because \(\lambda' > \lambda\), the scattered photon always has lower energy than the incident photon. The difference
between those two energies becomes the kinetic energy of the recoiling electron in the ideal free-electron model.
Electron recoil energy.
\[
\begin{aligned}
K_e &= E - E'
\end{aligned}
\]
This is why Compton scattering is called an inelastic scattering process: the photon does not keep all of
its original energy.
Sample interpretation
Suppose the incident wavelength is \(\lambda = 0.1\,\mathrm{nm}\) and the scattering angle is \(\theta = 90^\circ\). Since
\(\cos 90^\circ = 0\), the shift becomes
Shift at 90°.
\[
\begin{aligned}
\Delta\lambda &= \lambda_C(1-\cos 90^\circ) \\
&= \lambda_C(1-0) \\
&= \lambda_C \\
&\approx 0.002426\ \mathrm{nm}
\end{aligned}
\]
Therefore the scattered wavelength is
Scattered wavelength for the sample.
\[
\begin{aligned}
\lambda' &= \lambda + \Delta\lambda \\
&= 0.1 + 0.002426 \\
&= 0.102426\ \mathrm{nm}
\end{aligned}
\]
The photon loses some energy because its wavelength has increased. In this case, the incident energy is about
\(12.4\,\mathrm{keV}\), the scattered energy is slightly smaller, and the difference goes into electron recoil.
Why Compton scattering mattered
Compton scattering was one of the strongest early confirmations of the particle picture of light. Classical
electromagnetic wave theory could describe scattering of radiation, but it did not predict the observed angle-dependent
wavelength increase. Arthur Compton explained the data by treating the photon as a particle with momentum
\(p = h/\lambda\). Once energy and momentum conservation were applied to a photon-electron collision, the experimental
shift was reproduced exactly.
Today, Compton scattering is important in atomic physics, medical imaging, radiation detection, astrophysics, and X-ray
science. At a more advanced university level, the next step is to study the full angular dependence of scattering
probabilities using the Klein–Nishina formula. However, the wavelength-shift equation remains the cleanest entry point
because it captures the central physics with a small number of quantities: \(\lambda\), \(\theta\), \(\lambda'\), and the
recoil electron energy.
| Concept |
Main relation |
Meaning |
| Electron Compton wavelength |
\(\lambda_C = h/(m_e c)\) |
Fundamental length scale for photon-electron scattering |
| Wavelength shift |
\(\Delta\lambda = \lambda_C(1-\cos\theta)\) |
Angle-dependent increase in wavelength |
| Scattered wavelength |
\(\lambda' = \lambda + \Delta\lambda\) |
Outgoing photon wavelength |
| Photon energy |
\(E = hc/\lambda\), \(E' = hc/\lambda'\) |
Incident and scattered photon energies |
| Recoil energy |
\(K_e = E - E'\) |
Energy transferred to the electron |
| Maximum shift |
\(\Delta\lambda_{\max} = 2\lambda_C\) |
Backscatter limit at \(180^\circ\) |