Volume reflection of a proton beam in a bent crystal

Volume reflection predicted in the mid-1980s by Taratin and Vorobiev has been observed for the first time in the interactions of a 70 GeV proton beam with a short bent crystal. Incident protons deviate from convex atomic planes in the bulk of the crystal as a result of coherent interaction with bent lattice around the tangency point of particle trajectory with a curved atomic plane. The deflection angle 2theta(R) was found to be (39.5+/-2.0) microrad, or (1.65+/-0.08)theta(c) in terms of the critical angle for channeling. The process has a large probability with respect to channeling and takes place in the angular range equal to the bend angle of atomic planes. It could possibly open new fields of application of crystals in high-energy particle beam optics.     

High-efficiency deflection of high-energy protons through axial channeling in a bent crystal

Beam deflection due to axial channeling in a silicon crystal bent along the 111 axis was observed with 400 GeV/c protons at the CERN Super Proton Synchrotron. The condition for doughnut scattering of protons by the atomic strings of the crystal was attained. Such a condition allowed one to observe a beam deflection of 50 murad with about 30% efficiency. The contribution of hyperchanneled states of protons to the observed beam deflection was less than 2% according to simulation results.     

Volume reflection of 1-GeV protons by a bent silicon crystal

The volume reflection of 1-GeV protons by a bent crystal has been observed. The crystal is made of single crystal silicon. The (111) atomic planes are bent owing to the elastic quasimosaicity effect, which makes it possible to reduce the crystal length for a beam to 30 μm. It is found that the probability of the reflection effect is higher than the probability of the channeling effect (0.71 vs. 0.63), and the deflection angle of the protons reflected inside the crystal is equal to 1.39 ± 0.04 in terms of the critical angle for channeling under the conditions of the experiment (170 μrad). The width of the reflected peak is equal to 1.76 ± 0.04 in the same units. The protons that are not involved in channeling at the angular position of maximum channeling undergo volume reflection and are deflected in the direction opposite to the channeled beam by the angle 1.01 ± 0.05 in terms of the critical angle for channeling. The width of the reflected peak is equal to 1.94 ± 0.08 in the same units. Original Russian Text ? Yu.M. Ivanov, N.F. Bondar’, Yu.A. Gavrikov, A.S. Denisov, A.V. Zhelamkov, V.G. Ivochkin, S.V. Kos’yanenko, L.P. Lapina, A.A. Petrunin, V.V. Skorobogatov, V.M. Suvorov, A.I. Shchetkovsky, A.M. Taratin, W. Scandale, 2006, published in Pis’ma v Zhurnal éksperimental’noĭ i Teoreticheskoĭ Fiziki, 2006, Vol. 84, No. 7, pp. 445–450.     

Deflection of MeV protons by an unbent half-wavelength silicon crystal

The interaction of a 2?MeV proton beam with an ultrathin unbent Si crystal was studied through simulation and experiment. Crystal thickness along the beam was set at 92?nm, i.e., at half the oscillation wavelength of the protons in the crystal under planar channeling condition. As the nominal beam direction is inclined by less than the critical angle for planar channeling with respect to the crystal planes, under-barrier particles undergo half an oscillation and exit the crystal with the reversal of the transverse momenta; i.e., the protons are "mirrored" by the crystal planes. Over-barrier particles suffer deflection, too, to a direction opposite that of mirroring with a dynamics similar to that of volume reflection in a bent crystal. On the strength of such coherent interactions, charged particle beams can be efficiently steered through an ultrathin unbent crystal by the same physical processes as for thicker bent crystals.     

Chaotic behavior of channeling particles

Channeling describes the collimated motion of energetic charged particles along the lattice plane or axis in a crystal. The energetic particles are steered through the channels formed by strings of atomic constituents in the lattice. In the case of planar channeling, the motion of a charged particle between the atomic planes can be periodic or quasiperiodic, such as a simple oscillatory motion in the transverse direction. In practice, however, the periodic motion of the channeling particles can be accompanied by an irregular, chaotic behavior. In this paper, the Moliere potential, which is considered as a good analytical approximation for the interaction of channeling particles with the rows of atoms in the lattice, is used to simulate the channeling behavior of positively charged particles in a tungsten (100) crystal plane. By appropriate selection of channeling parameters, such as the projectile energy E(0) and incident angle psi(0), the transition of channeling particles from regular to chaotic motion is demonstrated. It is argued that the fine structures that appear in the angular scan channeling experiments are due to the particles' chaotic motion.     

Angular distributions of channeled pions and protons up to 250 GeV/c

Channeling phenomena are observed for positive particles of momentum up to 250 GeV/c in a germanium crystal. The polar angular distributions of the channeled particles are compared with theoretical predictions based on a diffusion model. The results indicate that at high particle energy there may be additional mechanisms besides those operative at low energy leading to dechanneling of the particles. In spite of this, channeling effects are observed for particles incident at up to several times the critical angle, in contrast with the results from low energy channeling. Statistical equilibrium in the azimuthal angular distribution has also been observed at all measured beam momenta to about twice the calculated channeling critical angle. The breakdown of statistical equilibrium for the 2 cm crystal used occurs at an incident angle 2–3 times smaller than predicted theoretically.     

Observation of channeling and volume reflection in bent crystals for high-energy negative particles

Deflection due to planar channeling and volume reflection in short bent silicon crystals was observed for the first time for 150 GeV/c negative particles, π⁻ mesons, at one of the secondary beams of the CERN SPS. The deflection efficiency was about 30% for channeling and higher than 80% for volume reflection. Volume reflection occurs, in spite of the attractive character of the forces acting between the particles and the crystal planes, in a wide angular range of the crystal orientations determined by the crystal bend angle.     

First observation of multiple volume reflection by different planes in one bent silicon crystal for high-energy protons

Multiple volume reflection by different planes in a bent silicon crystal with its 〈111〉 axis orientation close to the beam direction was observed for the first time for 400 GeV/c protons at the CERN SPS. The proton beam was deflected to the side opposite to the crystal bend by an angle of about 67 μrad, which is five times larger than in a single volume reflection by the (110) bent planes. The registered efficiency of one side deflection was about 84%. It was shown that multiple volume reflection transforms to a single volume reflection when the orientation angle of the 〈111〉 axis relative to the beam direction is increased.     

A study of small impact parameter ion channeling effects in thin crystals

We have recorded channeling patterns produced by 1–2 MeV protons aligned with ?1 1 1? axes in 55 nm thick silicon crystals which exhibit characteristic angular structure for deflection angles up to and beyond the axial critical angle, ψ _a . Such large angular deflections are produced by ions incident on atomic strings with small impact parameters, resulting in trajectories which pass through several radial rings of atomic strings before exiting the thin crystal. Each ring may focus, steer or scatter the channeled ions in the transverse direction and the resulting characteristic angular structure beyond 0.6ψ _a at different depths can be related to peaks and troughs in the nuclear encounter probability. Such “radial focusing” underlies other axial channeling phenomena in thin crystals including planar channeling of small impact parameter trajectories, peaks around the azimuthal distribution at small tilts and large shoulders in the nuclear encounter probability at tilts beyond ψ _a .     

Influence of incoherent scattering on stochastic deflection of high-energy negative particle beams in bent crystals

An investigation on stochastic deflection of high-energy negatively charged particles in a bent crystal was carried out. On the basis of analytical calculation and numerical simulation it was shown that there is a maximum angle at which most of the beam is deflected. The existence of a maximum, which is taken in the correspondence of the optimal radius of curvature, is a novelty with respect to the case of positively charged particles, for which the deflection angle can be freely increased by increasing the crystal length. This difference has to be ascribed to the stronger contribution of incoherent scattering affecting the dynamics of negative particles that move closer to atomic nuclei and electrons. We therefore identified the ideal parameters for the exploitation of axial confinement for negatively charged particle beam manipulation in future high-energy accelerators, e.g., ILC or muon colliders.     

A qualitative description of the transverse motion of axial channeled particles in thin crystals

The qualitative features of the angular distribution of particles emergent from thin ～ 1μ, uniform single crystals of silicon and germanium for incidence at small angles to axial directions are discussed in terms of the continuum picture of particle channeling. Blocking of axially channeled particles in the transverse plane is demonstrated. Implications and possibilities arising from observation of non-equilibrium of the transverse momentum vector are discussed. The use of the radial spreading of the angular distribution to investigate the average inter-atomic potential distribution in the plane transverse to the axial direction is explored and perturbations due to multiple scattering in surface films or inside the crystal and to beam divergence are qualitatively considered.     

Possibility of deflecting relativistic-ion beams by bent single crystals

The propagation of high-energy ions through a bent single crystal near a crystallographic axis is investigated. The results of these investigations reveal that a new mechanism of ion-beam deflection is possible in this case. This mechanism is based on multiple ion scattering by atomic strings in a bent crystal. The results obtained from a computer simulation of the effect are also presented. The effect is shown to depend strongly on the particle charge.     

Relativistic-electron scattering on an atomic string of a crystal at ultrasmall angles of particle incidence on a string

The classical and quantum scattering of fast electrons on an atomic string of a crystal is considered at angles of particle incidence on the string that are much smaller than the critical angle of axial channeling. The investigation was performed within the simplest approximation of the continuous atomic-string potential in the form of a cutoff Coulomb potential. For this case, the azimuthal scattering of particles at an angle exceeding 180° in the plane orthogonal to the string axis is shown to be possible for all impact-parameter values. It is demonstrated that, in particle scattering on a string, an effect can occur that is similar to the Ramsauer-Townsend effect, which consists in a considerable reduction of the total cross section for slow-electron scattering on atoms.     

Comparison of the efficiency of the deflection of proton beams with various energies by a bent crystal with the use of planar channeling and the stochastic deflection mechanism

The efficiency of the deflection of 50-, 15-, and 1.3-GeV proton beams by means of planar channeling in a bent silicon crystal has been compared to that by means of the stochastic mechanism of the deflection of charged particles by the bent crystal. The deflection of protons at single passage through the crystal has been simulated. The results of the experiment on the deflection of a circulating beam at the U-70 accelerator (Institute for High Energy Physics, Protvino, Moscow region) are presented. It has been shown that the efficiency of the stochastic deflection mechanism increases with a decrease in the energy, whereas the efficiency of the planar channeling for deflection decreases.     

Deflection of a 100-MeV positron beam by repeated reflections in thin crystals

The phenomenon of the deflection of a charged particle beam due to channeling in a bent crystal is thoroughly investigated and successfully applied for the extraction of the beam in high-energy accelerators, at the energies of about 10 GeV and higher. However, a big practical interest lies in the task of bending and extracting charged particles with energies below 1 GeV, for example, for the production of ultrastable beams of low emittance for medical and biological applications. That is why a novel crystal technique, namely thin straight crystal targets, is investigated in this article, using crystals as elements for extraction and collimation of the circulating beam in a ring accelerator. The advantages of reflection in straight crystals in comparison with bent crystal channeling consist in the small length of straight crystals along the beam that reduces the amount of nuclear interactions and improves the background. Experimental results were obtained for the bending of a 100 MeV positron beam with using five sequential straight crystals.     

Deflection of high-energy negative particles in a bent crystal through axial channeling and multiple volume reflection stimulated by doughnut scattering

Different kinds of deflection in a silicon crystal bent along the 〈111〉 axis was observed for 150 GeV/c   negative particles, mainly π⁻${\pi }^{-}$ mesons, at one of the secondary beams of the CERN SPS. The whole beam was deflected to one side in quasi-bound states of doughnut scattering (DSB) by atomic strings with the efficiency (95.4 ± 0.2)% and with the peak position close to the bend crystal angle, α=185 μrad$\alpha =185\text{μrad}$. It was observed volume capture of π⁻${\pi }^{-}$ mesons into the DSB states with a probability higher than 7%. A beam deflection opposite to the crystal bend was observed for some orientations of the crystal axis due to doughnut scattering and subsequent multiple volume reflections of π⁻${\pi }^{-}$ mesons by different bent planes crossing the axis.     

Chaotic phenomena of charged particles in crystal lattices

In this article, we have applied the methods of chaos theory to channeling phenomena of positive charged particles in crystal lattices. In particular, we studied the transition between two ordered types of motion; i.e., motion parallel to a crystal axis (axial channeling) and to a crystal plane (planar channeling), respectively. The transition between these two regimes turns out to occur through an angular range in which the particle motion is highly disordered and the region of phase space spanned by the particle is much larger than the one swept in the two ordered motions. We have evaluated the maximum Lyapunov exponent with the method put forward by Rosenstein et al. [Physica D 65, 117 (1993)] and by Kantz [Phys. Lett. A 185, 77 (1994)]. Moreover, we estimated the correlation dimension by using the Grassberger-Procaccia method. We found that at the transition the system exhibits a very complex behavior showing an exponential divergence of the trajectories corresponding to a positive Lyapunov exponent and a noninteger value of the correlation dimension. These results turn out to be linked to a physical interpretation. The Lyapunov exponents are in agreement with the model by Akhiezer et al. [Phys. Rep. 203, 289 (1991)], based on the equivalence between the ion motion along the crystal plane described as a "string of strings" and the "kicked" rotator. The nonintegral value of the correlation dimension can be explained by the nonconservation of transverse energy at the transition.     

Rosette motion and string scattering of 20 MeV electrons in MgO single crystals

Analytical estimates and computer simulations were undertaken to perceive the motion of negative particles through a lattice structure, the interaction being classical binary scattering. Three distinct modes of particle motion along atomic strings were found depending on the magnitude of the transverse energy and the angular momentum L of the particle with regard to the string axis. At small and large L increased scattering on the strings as compared with random penetration dominates. At medium L and negative transverse energy (bound state particles in the attractive potential) a rosette motion along the string occurs. In this case small impact parameters to the string atoms are avoided and thus an increased penetrability of the negative particles results. The influence of thermal lattice vibrations on these motions was studied.

Experimentally, the negative particle motion modes manifested themselves in the penetration profiles of 20 MeV electrons through an 8 μm MgO single crystal.     

Multicrystal microundulator

The mode of propagation of relativistic, positively charged particles through a system of mutually oriented and periodically arranged ultrathin crystals whose thicknesses are equal to the half-period of the particle trajectory during planar channeling in a thick crystal is considered. In the case of an incidence angle that is less than the critical channeling angle, a certain fraction of particles is specularly reflected from the atomic planes of the crystal. Therefore, passing through a stack of crystals, a particle moves along quasiundulator trajectories. The characteristics of the radiation of a particle passing through such a “multicrystal microundulator” are found. The radiation spectrum is discrete, and the first-harmonic frequency and the number of harmonics in the spectrum are dependent on the distance between the crystals, the particle energy, and the potential of atomic planes of the crystal. Radiation is concentrated in a narrow cone in the direction of the average velocity of particles and is mainly polarized in a plane that is orthogonal to the atomic planes of the crystal. The microundulator can be composed of separate crystals with micron thicknesses and can be fabricated using modern methods of microlithography and micromechanics with deep, for example, plasmochemical etching of the crystal surface.     

正弦平方势与小振幅近似下的弯晶沟道辐射

在理想情况下和经典力学框架内,引入正弦平方势,把粒子在弯晶中的运动方程化为具有外力矩的摆方程。并对系统的相平面特征进行了数值分析。在小振幅近似下,把粒子运动方程化为具有硬特性的弹簧-振子系统,用Jacobian椭圆函数和椭圆积分解析地给出系统的解和粒子运动周期。讨论了弯晶沟道辐射频率、无量纲偏转角和辐射谱的一般特征。指出利用沟道辐射作为激光的可能性。以正电子在碳单晶中沟道辐射为例进行了具体计算,得到了与其他工作基本一致的结果。