The Volume Capture in Structures with Variable Curvature

It is known that the symmetry of straight or bent crystal with a constant curvature does not allow the volume capture of particles in the channeling motion in Hamiltonian mechanics. In quantum mechanics, however, there is a possibility of the volume capture due to an exponentially small tunneling through the potential barrier, but we will not consider this effect here. We will also not touch the incoherent scattering which is the major cause of the volume capture [?,?] in straight and circular bent crystals with constant curvature as well as the normal dechanneling. Our aim is to consider the volume capture, as a pure classical effect, which becomes possible due to curvature variation. In some works [?,?] it is called the 'gradient capture', and it was pointed out first in the computer simulation [?]. Here we describe some general features and relations of volume capture as a classical effect of relativistic electrodynamics starting from classical potential scattering. As a physical phenomenon, the channeling can be considered as a finite motion in one or two directions (oscillations or more complicate finite motion) and free motion in third, generally speaking, curvilinear direction [?]. Such motion can be imposed by a force field in a neighborhood of smooth curve or plane in 3D space. The curvature of the curve or plane surface can also play the role of a force field. If the electric force in a neighborhood of a plane with zero curvature is applied in one direction, it is not enough to localize a particle near the surface. From this point of view, the simple reflection of particles from a flat potential (U(x) < 0 if x > 0 and U(x) = 0 if x < 0, Fig. 1 (A)) is not a channeling, because there is no finite motion in direction perpendicular to the plane. But if the potential is confined in a circular cylinder, (U(ρ) < 0 if ρ < R and U(ρ) = 0 if ρ > R) or sphere, the classical consecutive reflection looks like oscillation in a neighborhood near the potential wall and can be called the 'surface channeling' (see Fig. 1 (B)).

However, if the initial coordinates of the particle is located outside the cylindrical well, any trajectory is a symmetrically refracted on entering and leaving the potential well (see e.g. [?], §14,18). Thus the scattering trajectories of a particle by a circular cylindrical or spherical potential does not include the channeling trajectories. This is not the case, if the particles are scattered by a potential with variable curvature, as it will be shown below.

We shall first consider the deflection of a relativistic particle of mass m and energy E crossing the boundary of a uniform field U 1 which occupies the right side of a curve y = Y (x), Fig. 1 (C), with the curvature k(s), where s is the natural parameter. The path of a particle is refracted at the angles Θ 0 and Θ 2 to the tangent lines, respectively at point A and B. The tangent continuity of a particle’s momentum gives the relativistic relations between pair of angles Θ 0 , Θ 1 and Θ 2 , Θ 3 :

where the

is the square of Lindhard’s angle. In deriving ( 1),( 2), we made one assumption, |U 1 | « m ≤ E, which is true for relativistic as well as nonrelativistic particles. When the curvature k of boundary is constant (circular symmetry), the equality Θ 1 = Θ 2 holds, and the equations ( 1),( 2) give Θ 0 = Θ 3 . Here is no volume capture. Another situation happens when the curvature at the second point of crossing B is less than curvature at the point A. In this case Θ 2 becomes smaller than Θ 1 , but Θ 3 is always smaller than Θ 2 if the potential U 1 is negative inside the boundary. When the curvature at B continuously decreases, the condition Θ 3 = 0 takes place and at this point the second eq.

(2) gives:

).

(4)

The particle undergoes total internal refraction at point B, if Θ 2 < Θ 2cr . It will be captured in channeling mode if the curvature remains the same or decreases further along the boundary of potential. The integral curvature between points A and B defines the relation between the angles Θ 1 and Θ 2 :

In this form it generalizes the way of calculating the angle between two tangent lines to the two separate points on the smooth curve. The the radius vectors of the centers of curvatures are tangent to the evolute of the smooth curve (see Fig. 1). The evolute curve can not be smooth, in general.

One of the most important applications of the formulae derived above is to reverse the scattering problem and put the following question. What would be the incident angle Θ 0 , if the internal glancing angle Θ 2 ≤ Θ 2cr ? From (1), we can receive

Using the critical angle Θ 2cr (4) and the relation ( 5), we obtain an expression for critical angle Θ 1cr :

Now, the critical angle Θ 0cr becomes

For an infinite motion, such as that considered here, if we increase the impact parameter d relatively the edge of the potential (see Fig. 2), the angle Θ 0 will vary from 0 to Θ 0cr . Inside this angle the relativistic particles will be

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