Classical theory of Compton scattering: Assessing the validity of the Dirac-Lorentz equation

The Dirac-Lorentz equation describes the dynamics of a classical point charge in an electromagnetic field, accounting for radiative effects in a manifestly covariant and gauge-invariant manner. The validity of this equation is assessed by direct comparison between the Dirac-Lorentz dynamics of an electron subjected to a plane wave in vacuum and the well-known recoil associated with Compton scattering. In the small recoil limit, the classical Dirac-Lorentz is shown to yield the correct momentum transfer. For larger values of the recoil, the quantum scale appears explicitly, and the classical Dirac-Lorentz equation does not properly model this situation, as shown by deriving an exact analytical solution for a monochromatic plane wave of wave number $k_0$ to any order in $k_0{r}_0$, where $r_0$ is the classical electron radius.

Figure 1

(Color) Interaction between an electron initially at rest with an incident photon, propagating along the $z$ axis, with momentum $ħk_0\widehat{\mathbf{z}}$; after the event, the probability distribution for the scattered photons is given by a dipole radiation pattern, which results in an average null momentum for the scattered radiation: the electron recoil is then equal to $⟨m_{0}c{\mathbf{u}}^{+}⟩=ħk_0\widehat{\mathbf{z}}$, on average.

Figure 2

(Color) Top: Compton scattering differential cross section in the Thomson limit $(k_0{ƛ}_c⪡1)$. Bottom: Compton scattering differential cross section for $k_0{ƛ}_c=0.5$. Note that the scales are different: $\sigma (k_0{ƛ}_c=0.5)∕\sigma (k_0{ƛ}_c=0)=0.51167$.

Figure 3

(Color) Comparison between the average axial electron recoil from Compton scattering [red, Eq. (64)] and the Dirac-Lorentz momentum transfer [blue, Eq. (51)]; the dashed line corresponds to a regime where the perturbation in $\epsilon =k_0{r}_0=\alpha \xi <1$ is no longer valid. In both cases, the momentum transfer $\Delta u_z$ is normalized to $\frac{2}{3}{k}_0{r}_0{\int }_{-\infty }^{+\infty }{\mathbf{A}}_{\perp }^2\left(\varphi \right)d\varphi$.