Brehmsstrahlung Spectrum of EnergeticβParticles Traversing Single Crystals in Lattice-Directed Trajectories

For lattice-directed trajectories, the Fourier spectrum of the interaction between an energetic ($\gtrsim {10}^5$ eV) $\beta$ 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 ($\frac{\mathrm{sv}}{2a}$). Here $s$ is an integer, $v$ is the velocity of the $\beta$ 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 $\beta$ particle. It is suggested that these frequencies will dominate the brehmsstrahlung spectrum of an energetic $\beta$ 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.