Brehmsstrahlung Spectrum of Energetic β Particles Traversing Single Crystals in Lattice-Directed Trajectories

For lattice-directed trajectories, the Fourier spectrum of the interaction between an energetic (10⁵ eV) β particle and the lattice atoms is different from the Fourier spectrum of the interaction with randomly distributed target atoms by significant terms of frequencies near (sv/2a). Here s is an integer, v is the velocity of the β particle parallel to the low-index crystallographic direction, and $2a$ is the interatomic spacing in this low-index direction in the reference frame of the β particle. It is suggested that these frequencies will dominate the brehmsstrahlung spectrum of an energetic β particle traversing a single crystal in a lattice-directed trajectory. This effect is considered in terms of the quantal time-dependent perturbation theory. Its magnitude is estimated using the classical theory of radiation.