Colorful Electrons

First, we should ask ourselves about the origin of the radiation. Note that the interaction of the electron with the grating can be described by introducing an image charge (of opposite sign). Thus, each electron and its image constitute a dipole which is periodically disturbed by the etchings. In the rest frame of the electron+image, the grating has period given by $d'= d \sqrt{1-\frac{v^{2}}{c^{2}}} \quad \text{(Lorentz contraction)}.$ Therefore, the frequency of the "oscillations" of the dipole formed by the electron+image in its rest frame is $f_{0}=v/d'=\frac{v}{d \sqrt{1-\frac{v^{2}}{c^{2}}}}.$ We can determine the frequency $f$ registered by the detector employing the general formula for the Doppler shift $f= f_{0}\frac{\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c} \cos(\theta)} \Rightarrow \lambda= d(\frac{c}{v}-\cos(\theta)),$ where $\lambda= \frac{c}{f}$ . Solving for $\theta$ , we find the value $\theta=33.27^{\circ}.$ By the way, the emission of visible radiation by electrons passing closely to the surface of a metallic diffraction grating is called the Smith-Purcell effect.