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  • The goal of the present study

    2018-10-24

    The goal of the present study was to theoretically analyze the channeling process of high-energy positrons in straight and bent C(110) diamond crystals. Channeling parameters and radiation spectra for a positron beam with the q vd oph Е = 855 MeV and for various crystal bending radii were calculated.
    Modeling the channeling process Molecular dynamics implemented through the MBN Explorer [2] software package was used to obtain 3D particle tracks in a crystalline medium. The standard molecular dynamic algorithm, however, was enhanced by two additional features [1] pertaining to the channeling problem. Firstly, motions of high-energy particles were described by the relativistic equations of motion. Secondly, dynamic modeling of the crystalline medium was performed during the step-by-step track simulation. Only the key points of this procedure are explained below. A more detailed description can be found in Ref. [1]. In classical mechanics the motion of an ultra-relativistic particle with the velocity v, the charge q and the mass m in an external electrostatic field E(r) is described by the equation where is the relativistic momentum, and , is the so-called Lorentz factor, ε is the particle energy, с is the speed of light. The Eq. (1) is solved for t ≥ 0 and the initial conditions for the particle entering the crystal: r(0) = r0 и v(0) = v0. The electrostatic field is calculated as the potential gradient where the electrostatic potential U(r) is regarded as the sum of atomic potentials where R is the radius vector of the j-th atom, is the atomic potential in the Moliere approximation [34]. The sum in Eq. (2) is calculated formally for all atoms of the crystal. However, given the rapid decrease in atomic potential over distances from the nucleus (the Thomas–Fermi radius is used to estimate the average atomic radius), it is convenient to introduce the cutoff radius ρmax beyond which the contribution of the potential is negligible. For the arbitrary position of the particle r, the sum in Eq. (2) may be limited to the number of atoms inside a sphere of radius ρmax, which significantly accelerates the computations. The coordinates of atoms inside the sphere ρmax are calculated using the following algorithm [1]. Let us discuss the channeling inside a crystalline plane with Miller indices (klm). In the first step the crystalline lattice is constructed in a spatial box of volume × × whose linear dimensions are chosen randomly. Let the axis z be oriented along the direction of the beam of incident particles and parallel to the (klm) plane, so that the y axis is perpendicular to this plane. The radius vectors of the (j = 1, 2, …, N) lattice points are generated for a Bravais cell with predetermined translational vectors. Once the lattice points have been defined, the radius vectors of the atomic nuclei are found with respect to the average displacement Δ from the point positions due to temperature fluctuations. The Cartesian components (α = x, y, z) of the displacement are described by a normal distribution: where is the mean-square amplitude of the thermal vibrations of atoms. The values of for various crystals at room temperature are listed in [35]. The solution of Eq. (1) with the above-mentioned software starts for the time t = 0 when a particle (a positron) enters the crystal at z = 0. The initial values x0, y0 of the transverse coordinates are chosen at random in the intervals Δx = 2d, Δy = d, where d is the interplanar spacing. The initial velocity v0 = (v000) is predominantly oriented along the z axis, i.e. the following condition is satisfied
    The perpendicular components of the initial velocity v00 are chosen at crystal entrance with respect to the beam direction and divergence. The following technique is used to obtain the track in a crystal of a finite length L. A particle moves inside the modeled box, interacting with the atoms within the cutoff sphere. Once the distance traveled by the particle reaches the boundaries of the cutoff sphere l ∼ ρmax , a new spatial box of the same size as the previous one is generated, with its center approximately coinciding with the coordinates of the particle\'s location. To ensure that the effect of the qE forces is continuous and consistent, the position of the atoms in the area where the old and the new boxes intersect does not change. For the rest of the new box the position of the atoms is generated anew by the above-described procedure.