At the collision of charged particles bunches in colliders one bunch is deflected by the electromagnetic field of the opposite bunch. This deflection is described by the coherent kick that is the average deflection of particles in the deflected bunch. The averaging is calculated taking into account the particles distribution in the bunch. This coherent kick leads to orbit shift relative to the design orbit. In the text the elliptical distribution is considered. It is well-known in this case the coherent kick can be calculated using Bassetti-Erskine formulae [1,2]. Nevertheless, these formulae were derived for zero crossing angle between momentums of colliding bunches, whereas the real collision geometries (for example, at LHC [3]) often involve non-zero crossing angles. Of course, the problem can be resolved with numerical sumulations and simplifications reducing the problem to the known case of counter-parallel beams collision [4,5]. Nevertheless, it is of interest to find the direct generalization of Bassetti-Erskine formulae to the case of crossing angle. In this text the procedure of deriving Bassetti-Erskine formulae [1,6,7] is extended to this case. The important simplification is the assumption on the small single-particle momentum kick relative to its initial momentum. Through the text the bunch 1 is the source of the field, bunch 2 is deflected by the field generated by the bunch 1. Correspondingly, the index 1 refers to quantities related to bunch 1 and the index 2 is related to the quantities characterizing the bunch 2. All particles in bunch 1 have the same initial momentum ~p1 and all particles in bunch 2 have the same initial momentum ~ p2. The accelerator frame at the interaction point is: the longitudinal axis Z, horizontal transverse axis X and vertical transverse axis Y. The longitudinal momentum of particles in bunch 1, ~ p1z is directed along Z and the longitudinal momentum of particles in bunch 2, ~ p2z is opposite to Z. Projections of particles’ momentum onto X are positive for both bunches. The angle between ~ p1 and Z is θ. Bunches move symmetrically relative to X, so the angle between ~ p1 and −~ p2 (crossing angle) is α = 2θ, the angle between −~ p2 and Z is θ. The geometry in the accelerator frame is shown in Fig. 1. The preferable frame to consider the interaction of colliding bunches is the rest frame of bunch 1 because in this frame only electric field exists and, untill elliptic bunches in the accelerator frame are considered, the bunch 1 remains elliptic in its rest frame. The longitudinal axis of the rest frame is Z′1, the horizontal transverse axis is X ′ 1, the vertical axis is Y′1. The last coincides with Y. The angle between Z ′ 1 and −~ p′2 is α. Primed quantities correspond to the rest frame of bunch 1, α 6= α due to relativistic effects. Indeed, considering LHC conditions where protons (the rest mass

The coherent deflection of elliptic bunch with the Gaussian charge distribution in the field of an opposite bunch with the Gaussian bunch distribution is considered. The generalization of Bassetti-Erskine formulae is derived taking into account the angle between momenta of bunches.

A theoretical model is developed for the heating and deformation of a thin target at the passage of short electron bunches through it at the energies of modern free electron lasers, and the corresponding simulation is carried out. It is demonstrated that under these conditions, the target can undergo overheating or be irreversibly deformed, which makes it difficult to use such targets for the diagnostics of free electron beams used in modern lasers.

The problem of muon channeling in a standing wave lattice formed by electromagnetic waves is considered. The charged particles motion in a specific electromagnetic field can undergo the features similar to those well-known at the particles channeling in crystals. In this work the muon beam phase space evolution at the wave channeling is evaluated to analyze the feasibility of a beam shaping. The shaping examples have been considered.

The problem of coherent parametric X-radiation (PXR) by the electron bunch is considered. The PXR by different electrons could be coherent at small bunch dimensions. Conditions for this kind of radiation – coherent PXR (CPXR) – were considered. Features of CPXR by the bunch as well as by the train of bunches are demonstrated.

The thin target could be used for beam diagnostics by means the radiation that is induced by interaction of beam particles with target matter. The electron beams used in modern applications (as, for example, modern FELs) have very large brightness, small emittance as well as very short bunch length. For example, the bunch length of XFEL is about of 25 um at bunch charge order of 1 nC and with electrons energy of 17.5 GeV. The passage of this powerful short bunches could damage the target or even completely destroy it. In the presented work the train of such bunches passages through the target is investigated. It is shown the target works in extreme regime close to phase transition temperature.

Recently it was shown that charged particles motion in the field of standing electromagnetic wave can undergo the features similar to the particles channeling in crystals. When a charged particle enters the channels formed by electromagnetic standing waves at a small angle to the node (anti-node) planes its motion represents namely the oscillations between two neighboring planes. The phenomenon is mostly known as channeling in a lattice of the standing waves. Obviously, this effect can be used to handle beams in accelerator physics, more general, for the beam shaping with the specific properties. The advantage of the plane wave channeling is the absence of inelastic scattering that takes place in a crystal. The possibility to re-distribute the current density of particles in the beam by means of the laser standing wave is demonstrated.

Trajectories of relativistic positrons moving in the electric field of oriented crystal near its surface have been simulated. A positron beam enters the field of the crystal at a small angle to crystallographic planes. Thus, planar channeling conditions are satisfied. The miscut surface of a crystal has a specific shape of a step sequence of terraces. It has been shown that such a surface can deflect a noticeable fraction of the beam from the crystal surface by means of quasichanneling. The possibility of the experimental observation of this phenomenon has been analyzed.

The motion of relativistic nuclei which move near to a crystal surface was considered. The nuclei within a beam penetrate the crystal field at the small angle to the crystallographic planes, so the average field approximation and channeling phenomenology are valid. The crystal surface has a specific terrace-like form. It was shown the significant beam fraction could be deflected outwards from the surface due to quasichanneling.

The presented program was designed to simulate the passage of relativistic nuclei through a bent crystal. Namely, the input data is related to a nuclei beam. The nuclei move into the crystal under planar channeling and quasichanneling conditions. The program realizes the numerical algorithm to evaluate the trajectory of nucleus in the bent crystal. The program output is formed by the projectile motion data including the angular distribution of nuclei behind the crystal. The program could be useful to simulate the particle tracking at the accelerator facilities used the crystal collimation systems. The code has been written on C++ and designed for the multiprocessor systems (clusters).Program summaryProgram title: NSBC (Nuclei Scattering by Bent Crystal)Licensing provisions: Standard CPC license, http://cps.cs.qub.ac.uk/license/license.htmlProgramming language: C++ (g++, icc compilers)Computer: multiprocessor systems (clusters)Operating system: any OS based on LINUX; the program was tested under Novell SLES 10Has the code been vectorized or parallelized?: Yes. The code contains MPI directivesRAM: about 1 MB per processorNumber of processors: >1Supplementary material: the user manual readme.pdf, utility to generate the beam of particles Beam_Generator.exe, the pdf presentation that is commented in the Sample A.Classification: 7.10, 11.10External routines: MPI library for GNU C++, Intel C++ compilersRunning time:In general, the running time T depends on the number of both processors N and particles P hitting the crystal, as well as on the crystal thickness. It can be estimated by the ratio T[min] = 3 . 10(-5) . alpha[mu rad]. PAN 1) for the 2.66 GHz processors, where alpha(R) is the crystal bending angle. In our tests the simulations were performed for a few thousands of particles into the crystal up to several mm thickness. The number of the 2.66 GHz processors used counted up to 30. The running time of about 5 min was registered at above mentioned conditions.Nature of problem:Here we deal with planar channeling of fast particles in a bent crystal. The channeled projectile moves along bent planes being in such a way deflected at large angles from the initial direction of motion. This effect is recognized as accelerator techniques to shape the beam. Another attractive phenomenon is known as the volume reflection of quasichanneled projectiles. Volume reflected projectiles can also be deflected at essential angles. In general, channeled and reflected particles are deflected in opposite directions and the initial beam is split into two beams. Hence, there is the practical interest to model the beam tracking in a bent crystal, to obtain the characteristic angles of deflection, to estimate the number of particles, which can be effectively deflected at large angles.Solution method:Initially the beam of relativistic nuclei hitting the bent crystal is considered. The velocity of a particle is defined by two components. The component along the beam direction is relativistic, while the transverse component is nonrelativistic. The particle trajectory in a crystal is defined by the continuous potential of bent planes. Hence, to obtain the trajectory the classical equation of motion is solved numerically. The initial position of a nucleus in the channel is suggested to be random that can be obtained from the uniform distribution. To take into account the multiple scattering of projectiles on crystal both electrons and nuclei the corrections to the trajectory is introduced from time to time. Finally, at the projectile fly-out from the crystal one can obtain the transverse velocity as well as the deflection angle.Restrictions:As known the theory of the channeling effect implies the critical Lindhard angle theta(L). Channeling takes place when the angle theta(0) between the bent planes and the velocity of a particle at the crystal entrance face undergoes the condition vertical bar theta(0)vertical bar < theta(L). The quasichanneling appears when vertical bar theta(0)vertical bar exceeds the value theta(L) but remains close to this value. Thus, it is not recommended to input large values of the crystal orientation angle vertical bar theta(c)vertical bar which defines the range of angles theta(0) (see in Section 2). Nevertheless, in our simulations we found the program gives correct results in the broad range of crystal orientation angles, for example, - 18 theta(L) <= theta(c) <= 4 theta(L), for 400 GeV protons. Characteristic values of critical angle theta(L) are about 10 mu rad for the energy of projectile about 100 GeV/uamu.(C) 2013 Elsevier B.V. All rights reserved.

First computer experiment results on proton beam deflection by the crystal miscut surface are presented. The phenomenology of proton channeling and quasichanneling has been applied to describe new features of the beam deflection. The analysis predicts efficient beam deflection by the acute crystal end due to repelling miscut potential.

The particle loss in the volume of a bent crystal when relativistic ions pass through the bent crystal under channeling and quasichanneling conditions is simulated. Multiple passage of projectiles through an idealized crystal collimator is considered. The crystal orientation dependence for a particle loss function is obtained.

The motion of relativistic nuclei through an oriented bent crystal was considered on the basis of a developed computer code. To get the angular distributions of projectiles behind the crystals and to estimate the influence of multiple scattering on beam deflection, a detailed study of the beam dynamics due to the processes of channeling, dechanneling and quasichanneling of the particles is presented. The analysis of the beam redistribution in a crystal was performed simulating the main features of the scattering of relativistic Pb ions and protons in the field of bent crystal planes. The comparison of simulated data with experimental ones has been also carried out.

Inelastic nuclear interactions for relativistic channeled and quasi-channeled protons in a bent crystal are theoretically studied. Multiple passage of projectiles through experimental setup was in details analyzed. Paying attention to the features observed, simulation results have been compared with known experimental data.

Penetration of relativistic protons into bent crystals at small angles with respect to the bent crystallographic planes has been evaluated within continuous potential approximation. Namely, in this paper the numerical solution of the equation of motion for channeled and quasi-channeled relativistic protons is presented. Proton trajectories under the conditions of both channeling and volume reflection were simulated. The angular distributions of outgoing beam protons are calculated with the parameters of recent CERN experiments. The rather good agreement with experimental data is achieved.

Resonant coherent excitation (RCE) of relativistic hydrogen-like ions is investigated by computer simulations methods. The suggested theoretical model is applied to the simulations of recent experiments on RCE of 390 MeV/u Ar17+ ions under (220) planar channeling in a Si crystal performed by T.Azuma et al at HIMAC (Tokyo). Theoretical results are in a good agreement with these experimental data and clearly show the appearance of the doublet structure of RCE peaks. The simulations are also extended to greater ion energies in order to predict the new RCE features at the future accelerator facility FAIR GSI and as an example, RCE of 11 GeV/u U91+ ions is considered in detail.

Presently new experiments on channeling radiation by both 150 MeV electrons and 400 MeV positrons are planned at the LNF facilities. These experiments require the preliminary theoretical treatment. The channeling radiation spectrum is formed by transitions between bound energy levels of transverse motion of a channeled particle. The intensity of channeling radiation lines depends on populations of bound energy levels. These populations change during the projectile motion in a crystal that, in turn, influences the channeling radiation intensities. In this manuscript we present theoretical model and computer simulations to investigate the bound energy spectra of planar-channeled electrons and positrons as well as to obtain the populations of bound states. Solving the kinetic equations we explore the dynamics of bound state populations.

The planar channeling of ultra-relativistic heavy hydrogen-like ions is considered. In the rest frame of a channeled ion the magnetic and electric fields act on the orbital electron. The influence of these fields leads to the Stark and Zeeman effects of energy levels of orbital electron, and thus changes the conditions of resonant coherent excitation of an ion. In this work the theory of combined Stark–Zeeman effect (when the influence of electric and magnetic fields is comparable) is developed. Also the theoretical model of resonant coherent excitation in this case is suggested.

The computer model for the resonant coherent excitation of heavy relativistic ions under planar channeling in crystals taking into account the fine structure of the energy levels of the orbital electron and the ion ionization from both the ground and first excited state is presented. The model has been used to explain the experiments carried out under planar channeling of 390 MeV/n 17+Ar ions. Reasonably good agreement for the calculated and experimental data has been obtained.

The effect of a fine structure of the orbital electron energy levels of an Ar17+ ion on the resonant coherent excitation under planar channeling has been investigated by computer simulation technique. The obtained resonance curves are characterized by two closely situated peaks due to the transitions of an electron from the ground to excited states corresponding to the different components of a fine structure of the first excited state which differ in the value of the total electron momentum (1/2 or 3/2).