# Channeling Radiation

## General Properties

Relativistic electrons traversing a single crystal in the vicinity of a crystal axis or plane populate different transverse eigenstates in the continuum potential of that axis or plane. Figurately speaking, they temporarily move through the crystal like along a channel simultaneously fullfilling transverse oscillations around the leading axis or plane. The periodic motion of an electric charge in its rest system results in the emission of electromagnetic dipole radiation

In the framework of quantum mechanics, the frequency of the radiation ω_{0} is determined by the difference E_{f} - E_{i} of the energy levels involved.

The Lorentz transformation to the laboratory system and the consideration of the Doppler effect at a moving radiation source shifts the emitted frequency ω_{0} to an observed maximum value ω = 2γ^{2}ω_{0} which, typically for conventional crystals, low lying states and the electron energies avaiable at ELBE, represents X-ray photons in the energy range from 5 keV up to about 100 keV. Channeling radiation (CR) is emitted into a narrow forward cone θ = [0;1/γ] where γ denotes the relativistic factor.

Energy tuning of CR can be performed continuously by variation of the electron energy E_{e} applying the approximate scaling law E_{x} ~ γ^{3/2}. For large-scale extrapolations the exponent has been found to be a little larger, e.g. for planar electron CR from (1,1,0) diamond and the most intense transition it amounts to about 23/14.

CR is quasi-monochromatic (ΔE_{x}/E_{x} = 5 - 10 %) due to life time effects, the influence of the periodic crystal structure, of multiple scattering, of the electrons beam divergence and of the infinite detector aperture via the relation E_{x}(Θ) ~ E_{x}(Θ=0)/(1+γ^{2}Θ^{2}) where Θ denotes the mean angle of abservation with respect to the direction of the electron beam.

The yield of CR increases with the electron energy nearly like ~γ^{5/2}. As for thicker crystals the population of transverse energy levels is governed by multible scattering of the electrons in the target the dependence of the yield on the thickness d of the source crystal scales like ~d^{1/2}. On the other hand, self absorption of CR in the crystal (μ ~ 1/ω^{α} where μ is the linear absorption coefficient and α > 1) has to be considered. The optimum production rate for some special CR line, therefore, will depend as on E_{e}, E_{x}, d and μ as on the divergence of the electron beam.

Since the flux of CR photons as well as the energy losses of the electrons in the crystal are directly proportional to the electron beam current further limitations may arise from the thermal stability of the source crystal. The largest production rate of quasi-monochromatic CR reached up to now at a low-energy electron accelerator has been obtained by the use of a diamond crystal(~10^{10} photons/s, S-DALINAC at TU Darmstadt).