Incoherent bremsstrahlung in flat and bent crystal

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.

💡 Research Summary

The paper investigates the incoherent component of bremsstrahlung emitted by relativistic electrons traversing crystalline media, focusing on how the intensity of this radiation depends on crystal orientation and curvature. While the coherent part of the spectrum arises from the periodic lattice potential and is well known to be highly orientation‑dependent, the incoherent part—originating from thermal vibrations of atoms—has traditionally been assumed to resemble that of an amorphous target and to be essentially isotropic. Recent measurements at the Mainz Microtron (MAMI) have, however, revealed pronounced angular variations in the incoherent bremsstrahlung yield, suggesting that the electron beam’s spatial redistribution inside the crystal plays a decisive role.

To address this, the authors apply a Monte‑Carlo simulation framework they previously developed for incoherent bremsstrahlung. The model incorporates several essential physical ingredients: (1) a Gaussian description of atomic thermal displacements, allowing realistic temperature‑dependent lattice disorder; (2) a hybrid quantum‑classical treatment of electron‑atom scattering, using the exact differential cross‑section for high‑energy bremsstrahlung combined with classical trajectory integration; (3) explicit handling of channeling, dechanneling, and over‑barrier motion, which governs how electrons are guided or scattered by the planar or axial potentials of the crystal; and (4) a geometric representation of bent crystals, where the lattice planes follow a curvature radius R, thereby inducing additional focusing or defocusing of the electron flux.

Simulations were performed for both flat (planar) silicon crystals and for crystals bent to radii ranging from a few centimeters down to sub‑millimeter scales. For each configuration, the authors tracked large ensembles of electron trajectories, recorded the instantaneous scattering angles, and accumulated the emitted photon spectra. The resulting incoherent bremsstrahlung intensity was then compared with the experimental angular distributions measured at MAMI for electron energies of several hundred MeV.

The key findings are as follows. First, when the incident beam is aligned closely with a major crystallographic plane (e.g., the (110) plane), a substantial fraction of electrons become trapped in channeling states. In this regime the local electron density near the atomic rows increases, leading to a higher probability of incoherent scattering events. Consequently, the incoherent bremsstrahlung intensity exhibits a sharp peak as a function of the beam‑crystal angle. Slight deviations from perfect alignment cause rapid dechanneling, reducing the local density and producing a pronounced drop in intensity. This angular dependence reproduces the experimental data with high fidelity, confirming that the observed “orientation effect” is not a modification of the intrinsic incoherent cross‑section but a consequence of flux redistribution.

Second, temperature scans show that raising the lattice temperature (and thus the amplitude of thermal vibrations) modestly lowers the overall incoherent yield, as expected from the increased Debye‑Waller factor, yet the shape of the angular dependence remains essentially unchanged. This indicates that the orientation effect is robust against thermal smearing.

Third, for bent crystals the curvature introduces an additional transverse force on the electrons. When the curvature radius is comparable to the critical radius for channeling, the beam is focused toward the inner side of the bend, enhancing the local electron density and thereby amplifying the incoherent emission in that region. Conversely, on the outer side the density is reduced, leading to a suppression of the incoherent component. The simulations capture this asymmetric enhancement and match the measured intensity profiles for various bending radii.

Overall, the study demonstrates that incoherent bremsstrahlung in crystals is highly sensitive to the spatial distribution of the electron beam, which is governed by channeling dynamics and crystal geometry. The authors’ simulation tool provides a quantitative bridge between microscopic electron trajectories and macroscopic radiation observables, enabling accurate predictions for experimental setups that employ oriented or bent crystals as radiators. The results have practical implications for the design of crystal‑based gamma‑ray sources, beam diagnostics, and for interpreting background radiation in high‑energy particle experiments where crystalline targets are used.