Model solution for volume reflection of relativistic particles in a bent crystal

Abstract

For volume reflection process in a bent crystal, exact analytic expressions for positively-and negatively-charged particle trajectories are obtained within a model of parabolic continuous potential in each interplanar interval, with the neglect of incoherent multiple scattering. In the limit of the crystal bending radius greatly exceeding the critical value, asymptotic formulas are obtained for the particle mean de?ection angle in units of Lindhards critical angle, and for the final beam profile. Volume re?ection of negatively charged particles is shown to contain effects of rainbow scattering and orbiting, whereas with positively charged particles none of these effects arise within the given model. The model predictions are compared with experimental results and numerical simulations. Estimates of the volume re?ection mean angle and the final beam profile robustness under multiple scattering are performed.

Nuclear interactions at volume reflection: Perturbative treatment

“…N > 1, it was shown in [9] that for the case of positively charged particles in a purely harmonic interplanar continuous potential, the outcoming particle angular distribution looks as [see Eq. (72) of [9] ] [20] …”

“…This approximation must work better than OðN À1 Þ, presumably as OðN À2 Þ (cf. [9]). However, one should not forget about physical corrections due the approximation of thin planes and the pure continuous potential in itself, which can be actually more significant.…”

“…In a pure continuous potential of a bent crystal, there exists an effect of orbiting (see [9] and references therein), arising due to a retention of the particles on the round top of a potential barrier, which turns as the crystal bends, and it leads to an exponential tail in the angular distribution of the volume-reflected particles, towards the side of the crystal bending. This manifested itself already in Eq.…”

“…The physical origin of the reflection effect in a bent crystal is understood to be due to the asymmetry of the continuous potential in the area where the angles of atomic plane crossing by the particle become comparable to c . Particle dynamics in the pure continuous potential of a uniformly bent crystal can be relatively easily calculated analytically, and the reflected beam angular distribution be evaluated as a function of the beam energy and the crystal bending radius [8,9].…”

“…Then, the problem boils down to evaluation of a sum of inverse plane-crossing angles, and averaging thereof over the impact parameters of particles in the initial beam. In fact, based on the transverse energy conservation law, and even without resorting to model approximations for the interplanar continuous potential (such as those employed in [9]), the entire procedure can be accomplished in closed analytic form, rewarding the simplified approach. The results obtained in this framework were partially published in [15].…”