E.2.9 (HL)—Compton wavelength shift

Use the Compton equation

The wavelength shift is the scattered wavelength minus the incident wavelength. The electron is treated as initially at rest, and $\theta$ is the photon's scattering angle.

Worked example — $30^\circ$ scattering

Using $h/(m_ec)=2.426\times10^{-12}\,\mathrm{m}$, $\Delta\lambda=(2.426\times10^{-12})(1-\cos30^\circ)=3.25\times10^{-13}\,\mathrm{m}$. The shift is positive and depends on angle, not on the incident wavelength.

Check the limits

For $\theta=0^\circ$, $\Delta\lambda=0$. For back-scattering $\theta=180^\circ$, the shift is maximal at $2h/(m_ec)$. The shift is always non-negative for the usual scattering geometry.

Solve for angle

If $\Delta\lambda$ is given, rearrange to $\cos\theta=1-\frac{m_ec\Delta\lambda}{h}$, then take the inverse cosine and check that the result is physically allowed.

Common trap

Do not use the scattered wavelength itself as $\Delta\lambda$. The equation requires the difference $\lambda_f-\lambda_i$ and the scattering angle.