The description of the electrostatic field

The description of the electrostatic field is done using the Molière approximation [19] of the electrostatic potential of a neutral atom: where Z is the atomic number, the coefficients in the screening function S(ρ) are as follows: (so that ) and . The Thomas–Fermi radius is related to the Bohr radius a0 via
In order to simulate the motion of a particle in the medium we used a dynamic simulation box technique (see Ref. [10]). With this approach the crystalline medium is generated in the close vicinity of the particle taking the predefined unit cell of the crystal and the set of transformations: rotation, displacement and bending. In order to take into account thermal vibration of atoms in grid structure, atoms are randomly shifted from their nodal positions. Each component of the displacement vector is normally distributed with the root-mean-square amplitude which corresponds to the room temperature [20]. In this article, the following directions were chosen for comparison: axes 〈100〉, 〈110〉, 〈111〉 and plane (110). The structure of chloride channels for axial directions is illustrated in Fig. 1. This structure is a result of the Voronoi decomposition of the plane being perpendicular to the beam direction. Coordinates of the projections of atoms are used as seeds of this decomposition for electrons. As for positrons, those with the minimum of the potential energy surface are taken as seeds of the decomposition. This description of channels can be applied to any axial direction and any type of lattice. The axial channels’ structure depends on the projectile charge. For electrons the channel center coincides with the line of Si atoms. For positrons the channel center is situated between the lines of atoms. The projectile was considered to be in a channel if the distance to this channel center is the shortest among the ones to all channels for the given type of projectiles. If the projectile stays in one channel and changes the velocity sign in the X or Y directions four times, then it is considered as captured in a channeling mode. The concept of the crystalline undulator implies a motion of a projectile in a periodically bent crystal. This bending leads to a change of the radiation spectrum and produces undulator radiation in addition to channeling radiation. In simulations, the periodic crystal bending is implemented using a simple transformation of the coordinates of the particle: where d is a bending amplitude, p is an inverse period vector modulus and s is a phase shift.
Numerical results In order to analyze the effect of axial channeling, numerical calculations of trajectories were performed using random sampling with different initial positions of projectiles in the channel and with random positioning of Si atoms in the medium. The average number of trajectories for each case was at a level of 104, which allowed reducing the statistical error of obtained quantitative characteristics to a level of a few percent. In all calculations, the Z axis was aligned with the main direction of channeling (the beam direction). The Z coordinate of the particle is a penetration distance. The used combinations of the Z–Y axes are as follows:
For planar channeling the taken axis directions are as follows:
The beam direction along the Z axis was taken in a (110) plane.
Summary
Acknowledgments
Introduction A crystalline undulator (CU) [1] is a device producing high-energy radiation by propagating charged relativistic particles through a periodically bent oriented crystalline medium. The feasibility of creating such a device has been predicted theoretically [2–4] and now is being studied experimentally [5]. This concept is based on the channeling effect that involves the propagation of projectiles in oriented crystals along crystalline planes or axes. Projectiles in channels interact with crystalline medium and oscillate around the center of the channel. The shape of the channel is responsible for the amplitude and the period of channeling oscillations and the corresponding radiation. Channeling in bent crystals leads to a special type of radiation which depends on the parameters of crystal bending. In recent years two main types of crystalline undulators have been discussed: small-amplitude short-period (SASP) undulators [5–7], and large-amplitude large-period (LALP) undulators [5,6,8,9]. The bending amplitude is considered high if its value is more than the inter-planar distance in the crystal. The bending period is considered large if its value is larger than that of the period of channeling oscillations in the channel. Short-period undulators can be used to generate the radiation with photon energies higher than the characteristic energy of the channeling radiation.