The bremsstrahlung cross section for relativistic electrons in a crystal is split into the sum of coherent and incoherent parts (the last is due to a thermal motion of atoms in the crystal). Although the spectrum of incoherent radiation in crystal is similar to one in amorphous medium, the incoherent radiation intensity could demonstrate substantial dependence on the crystal orientation due to the electrons' flux redistribution in the crystal. In the present paper we apply our method of the incoherent bremsstrahlung simulation developed earlier to interpretation of some recent experimental results obtained at the Mainz Microtron MAMI.
It is well known (see, e.g. [1,2,3]) that high energy electron beam incident on an oriented single crystal produces the coherent radiation that is due to the spatial periodicity of the lattice atoms, and the incoherent one, that is due to the thermal spread of atoms from their positions of equilibrium in the lattice. For the first look, the incoherent part of radiation is similar to the last in amorphous medium (with Bethe-Heitler spectrum), and do not depend on the crystal orientation in relation to the particles beam.
However, in [4,5] it was paid attention to the fact that some features of the particle’s dynamics in the crystal (channeling effect etc.) could lead to various substantial orientation effects in the hard range of the spectrum, where (for ε ∼ 1 GeV electrons) the incoherent part is predominant. The semi-numerical approach developed in [4,5] was used for interpretation of early experimental data [6].
The ideass of [5] had been referred by the authors of recent experiments [7] to interpret some of their results. In our article we present the results of simulation of the incoherent radiation under the conditions of the experiment [7]. A good agreement with the experimental data confirms the interpretation given in [7].
For the reader convenience, in the next section we outline some theoretical ideas of our approach.
Radiation of relativistic electron in matter develops in a large spatial region along the particle’s momentum. This region is known as the coherence length (or formation length) [1,2] l coh ∼ 2εε ′ /m 2 c 3 ω, where ε is the energy of the initial electron, ω is the radiated photon frequency, ε ′ = ε -ω, m is the electron mass, c is the speed of light. In the large range of radiation frequencies the coherence length could exceed the interatomic distances in crystal:
In this case the effective constant of interaction of the electron with the lattice atoms may be large in comparison with the unit, so we could use the semiclassical description of the radiation process. In the dipole approximation the spectral density of bremsstrahlung is described by the formula [2]
where δ = m 2 c 3 ω/2εε ′ ∼ l -1 coh , and the value
is the Fourier component of the electron’s acceleration in the direction orthogonal to its initial velocity.
The main contribution to the integral in ( 2) is made by the small values of q, q ∼ δ. On the other hand, the characteristic distances on which the electron’s acceleration in the field of the atom in (3) would substantially distinct from zero are equal by the order of magnitude to the atomic radius R. The corresponding time intervals in which the integrand in (3) is distinct from zero are ∆t ∼ R/c. But under the condition (1) in the frequency range of our interest δ ≪ R -1 . So we could present W q in the form [8]
where ϑ n is the two-dimensional electron scattering angle under collision with the n-th atom, t n is the time moment of the collision. Consider now the radiation of the electron incident onto the crystal under small angle ψ to one of its crystallographic axes. It is known [1,2] that averaging of the equation for |W q | 2 over the thermal vibrations of atoms in the lattice leads to the split of this value (and so the radiation intensity) into the sum of two terms describing coherent and incoherent effects in radiation:
where ρ n = ρ(t n )ρ 0 n is the impact parameter of the collision with the n-th atom in its equilibrium position ρ 0 n , ρ(t) is the trajectory of the electron in the plane orthogonal to the crystallographic axis (which could be obtained by numerical integration of the equation of motion), and u n is the thermal shift of the n-th atom from the position of equilibrium. In the range of radiation frequencies for which
where a is the distance between two parallel atomic strings the closest to each other, the incoherent term (6) makes the main contribution into the bremsstrahlung intensity (2). The radiation by the uniform beam of particles is characterized by the radiation efficiency, that is the radiation intensity (2) integrated over impact parameters of the particles’ incidence onto the crystal in the limits of one elementary cell. So, the efficiency is the classical analog of the quantum cross section. In the further consideration we shall compare the radiation efficiency in the crystal to the Bethe-Heitler efficiency of bremsstrahlung in amorphous medium.
For further computational details see [4,5].
When charged particles are incident onto the crystal under small angle θ to one of the atomic planes densely packed with atoms, the channeling phenomenon could takes the place (see, e.g., [2,3]). Under planar channeling the electron moves in the potential well formed by the attractive continuum potential of the atomic plane (see figure 1, left panel). The largest incidence angle, for which the capture into the channel is possible, is called as the critical channeling angle θ c [2,3]. Under θ ≪ θ c the most part of the incident electr
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