Question

A narrow beam of electrons passes immediately over the surface of a metallic mirror with a diffraction grating with period d = 2.0 μm inscribed on it. The electrons move with velocity v, comparable to c, at right angles to the lines of the grating. The trajectory of the electrons can be seen in the form of a strip, whose colouring depends on the observation angle θ (Fig. 5.38). Interpret this phenomenon. Find the wavelength of the radiation observed at an angle θ = 45°.

Answer 1

An electron moving in front of a metal mirror sees an image charge of equal and opposite type. The two together constitute a dipole. Let us look at the problem in the rest frame of the electron. In this frame the grating period is Lorenz contracted to

Because the metal has etchings the dipole moment of electron-image pair is periodically disturbed with a period d'/v

The corresponding frequency is v/d' which is also the proper frequency of radiation emitted. Due to Doppler effect the frequency observed at an angle θ is