The scattering of fast charged particles in a bent crystal has been analyzed in the framework of relativistic classical mechanics. The expressions obtained for the deflection function are in satisfactory agreement with the experimental data for the volume reflection of relativistic protons obtained in [1,2,3]. The features of the scattering of the particles on ring potentials are considered in a wide range of impact parameters.PACS numbers: 61.85.+p, 45.50.Tn In the 1980s, studying the effect of the volume cap-ture of relativistic protons into the channeling regime, Taratin and Vorobiev [4,5] demonstrated the possibility of volume reflection, i.e., the coherent small-angle scattering of particles at angle θ< 2θ L (θ L is the Lindhard critical angle) to the side opposite to the bending of the crystal. Recent experiments reported in [13] confirm the presence of this effect for 1-, 70-, and 400-GeV proton beams in a Si crystal. The conclusions made in [4,5] were based primarily on the numerical simulation. In view of this circumstance, the aim of this work is to derive analytical expressions for the deflection function of relativistic particles. At first sight, the perturbation theory in the potential can be applied at relativistic energies and weak crystal potential [U (r) ≈ 10 − 100]. However, the relativistic generalization of the known classical formula for small-angle scattering in the central field [6],where b is the impact parameter and φ(r) = 2U(r)E p 2 ∞ c 2 , U (r), E, p ∞ are the centrally symmetric continuous potential of bent planes, total energy, and particle momentum at infinity, respectively, is inapplicable for the entire range of impact parameters. Indeed, the above formula is the first nonzero term of the expansion of the classical deflection functionin the power series in the effective interaction potential φ(r). The crystal interaction potential U (r) is the sum of the potentials of individual bent planes concentrically located in the radial direction with period d. It has no singularities (i.e., is bounded in magnitude) and is localized in a narrow ring region at distances R − N d < r < R * JETP Letters, 2008, Vol. 87, No.2, pp. 87 -91 (where the crystal thickness N d << R and N is the number of planes). In this region, U (r) > 0 and U (r) < 0 for the positively and negatively charged scattered particles, respectively. Beyond the ring region, it vanishes rapidly. The perturbation theory in the interaction potential is obviously well applicable if the impact parameter satisfies the inequality b < (R − N d). In this case, the scattering area localized in the potential range is far from the turning point r o determined from the relation b = r o 1 − φ(r o ) and the root singularity of the turning point does not contribute to integral (1). In the general case, it can be shown [7,8] that the condition of the convergence of the power series of φ is a monotonic increase in the function u(r) = r 1 − φ(r) (e.i. u(r) ′ > 0) in the r regions substantial for integral (1). Such a monotonicity is achi...

The mechanisms of the volume reflection of positively and negatively charged relativistic particles in a bent crystal have been analyzed. It has been shown that the empty core effect is significant for the negatively charged particles. The average reflection angle of the negatively charged particles has been determined and the conditions for the observation of the reflection and refraction are discussed. PACS numbers: 13.88.+e, 61.85.+p The experiments reported in [1][2][3] confirmed the effect of the volume reflection of 1-, 70-, and 400-GeV protons in a bent Si crystal, which was revealed by Taratin and Vorobiev [4,5] using the Monte Carlo method, and demonstrated the possibility of its application to collimate accelerated beams [6]. The numerical calculations performed in [4,5,7] also indicated the volume reflection of negatively charged particles, but the corresponding reflection angle is smaller than that for positively charged particles. However, the reflection of the negatively charged particles differs in nature from the reflection of the positively charged particles and requires a more detailed analysis. Indeed, within the framework of classical mechanics, the reflection and scattering of the negatively charged particles in the field of a one-dimensional potential well (see Fig. 1a) are absent, whereas the grazing incidence of the postively charged particles on a one-dimensional barrier, α < θ L (α is the angle between the particle momentum and the boundary of the one-dimensional barrier and θ L is the Lindhard critical angle, see Fig. 1b), is accompanied by the complete reflection. For the case of a centrally symmetric potential, this phenomenon is responsible for the volume reflection of positively charged relativistic particles in a bent crystal. Indeed, the impact parameter b measured from the tangential edge (point T in Fig. 1c) of the centrally symmetric ring barrier with the height U o is given by the expression b = R(1 − cos(α)) and the average reflection angle is specified by the integralHere, the maximum impact parameter, b max = R(1 − cos(θ L )), depends on the critical channeling angle θ L = 2U o E/(p 2 c 2 ) and the potential-barrier radius R. For small angles α, θ L << 1 , b max = Rθ 2 L /2, db = Rα dα, and Eq. (1) provides the average reflection anglē

The peak position, impact-parameter range, and optimal conditions for observing spiral scattering of relativistic particles in a uniformly bent crystal are estimated. The existence of spiral scattering with a square-root singularity is pointed out. In this case, the secondary process of volume capture to the channeling mode is absent and the conditions for observing this effect are most favorable.The phenomenon of spiral scattering occurs due to the appearance of a negative logarithmic singularity of the classical deflection function χ(b) of a particle or light ray for a certain impact parameter b = b s [1] . Resonance scattering is a quantum mechanical analog of spiral scattering [2]. However, resonance scattering includes a wider class of quantum-mechanical phenomena. In particular, it can appear in the scattering of fast particles by a cylindrical well (see, e.g., [3]), whereas classical spiral scattering by such a potential is absent [4].To illustrate this feature and determine the spiralscattering boundaries, let us compare scattering by a cylindrical or spherical potential well of radius R and depth −U 0 and scattering by a well of the same depth and radius, but with a smoothed parabolic edge of width d/2 (d << R) (curves 1 and 2, respectively, in Fig. 1a):(1) 3 1 1 1 1 2 2 2 2 2 FIG. 1: (a) Square-well potential shown by solid curve 1 and a smoothed well given by dashed curve 2. (b) The function u(r) given by Eq. (3) for the (line 1) well, (line 2) smoothed well, and (line 3) barrier. (c) The deflection function α = χ/2. (d) The trajectory of a particle in the square well. (e) The spiral trajectory of a particle in the smoothed square well.The classical deflection functionwhere φ(r) = 2U (r)E/(p 2 ∞ c 2 ) and b, U (r), E, p ∞ are the impact parameter, centrally symmetric potential, total energy, and momentum of a particle at infinity, respectively, can be calculated for both cases, but the absence of spiral scattering for a square well can be seen directly in the plot (curve 1 in Fig. 1b) of the functionwhich has a finite step and a local minimum for r = R. Therefore, the derivative u(R) ′ is indeterminate at this point. Deflection function (2) is expressed in terms of function (3) aswhere, as before, the turning point r 0 is determined from the equation u(r 0 ) = b. In the case of the smoothed potential (curve 2), the function u(r) has a smooth local minimum at the point rmin, at which u ′ (rmin) = 0. If the impact parameter coincides with this minimum, i.e., b s = u(r min ),both the radial velocity and radial acceleration of the particle become zero (for more detail, see [1,5]) and the conditions for spiral scattering are implemented. Here, the function u(r) in the vicinity of the minimum r min is represented in the formwhile the local-minimum point r min determined from the condition u ′ (r min ) = 0 is specified by the equation 1 − φ(r min ) − r min 2 φ(r min ) ′ = 0.This equation holds for any potential; for Eq. (1), it reduces to the simple form 2r 2 min − 3r min + 1 + δ = 0.

The motion of channeled particles is accompanied by the photon emission. This feature can be used for the stimulated generation of high energy photons, but the required density of channeled particles must be very high.

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