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We present the results of an experimental study of multiple scattering of positively charged high energy particles in bent samples of monocrystalline silicon. This work confirms the recently discovered effect of a strong reduction in the rms multiple scattering angle of particles channeled in the silicon (111) plane. The effect is observed in the plane orthogonal to the bending plane. We show in detail the influence of angular constraints on the magnitude of the effect. Comparison of the multiple scattering process at different energies indicates a violation of the law of inverse proportionality of the rms angle of channeled particles with energy. By increasing the statistics, we have improved the results of multiple scattering measurements for particles moving, but not channeled, in silicon crystals., arXiv admin note: text overlap with arXiv:1910.00250 minor fixes

The UA9 setup, installed in the Super Proton Synchrotron (SPS) at CERN, was exploited for a proof of principle of the double-crystal scenario, proposed to measure the electric and the magnetic moments of short-lived baryons in a high-energy hadron collider, such as the Large Hadron Collider (LHC). Linear and angular actuators were used to position the crystals and establish the required beam configuration. Timepix detectors and high-sensitivity Beam Loss Monitors were exploited to observe the deflected beams. Linear and angular scans allowed exploring the particle interactions with the two crystals and recording their efficiency. The measured values of the beam trajectories, profiles and of the channeling efficiency agree with the results of a Monte-Carlo simulation., 25 pages, 13 figures

Strong reduction of multiple scattering for channeled particles has been observed in an experiment on the deflection of a 180 GeV/c π+-meson beam by bent silicon crystals. The RMS deflections due to multiple scattering for the channeled particles were about six times smaller than for non-channeled ones. It was shown that the approach suggested recently for the description of multiple scattering for channeled particles using the experimental data for random crystal orientations gives fair agreement with the experiment. Keywords: Crystal, Channeling, Multiple scattering, Dechanneling, Beam deflection

Recently studies are started on the application of volume reflection of particles in crystals for the steering beams (for extraction and collimation of a circulating beam in accelerators). Volume reflection is more efficient than channeling, but requires amplification of the deflection angle by applying multicrystals. The report discusses two new applications of multicrystals made like multistrip structures: (i) the property of effective deflection of particle beam was used to protect the septum magnets of the U-70 in the process of extraction of the proton beam with energy of 50 GeV; (ii) the possibility of generation of gamma radiation was studied in the secondary electron beam with energy of 7 GeV. In both cases, promising preliminary data were obtained.

Inelastic nuclear interaction probability of 400 GeV/c protons interacting with bent silicon crystals was investigated, in particular for both types of crystals installed at the CERN Large Hadron Collider for beam collimation purposes. In comparison to amorphous scattering interaction, in planar channeling this probability is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim 36\%$$\end{document}∼36% for the quasi-mosaic type (planes (111)), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim 27\%$$\end{document}∼27% for the strip type (planes (110)). Moreover, the absolute inelastic nuclear interaction probability in the axial channeling orientation, along the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle 110\rangle $$\end{document}⟨110⟩ axis, was estimated for the first time, finding a value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.6\%$$\end{document}0.6% for a crystal 2 mm long along the beam direction, with a bending angle of 55 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu $$\end{document}μrad. This value is more than two times lower with respect to the planar channeling orientation of the same crystal, and increases with the vertical angular misalignment. Finally, the correlation between the inelastic nuclear interaction probability in the planar channeling and the silicon crystal curvature is reported.

The problem of creation of high energy neutrino beams on the basis of modern and future circular proton accelerators with the help of traditional technology seems to be expensive and difficult. Because of this, we propose the solution of this problem based on the usage of focusing bend single crystals. In the paper we demonstrate the possibilities of acceptance and focusing of a pion beam with the help of a crystal optical lens system. As an illustration of these features the calculated neutrino fluxes for energy of circulating proton beam equal to 6.5 TeV are presented., Comment: 16 pages, 6 figures, Latex

International audience; This paper is devoted to an experimental study of focusing and defocusing positively charged particle beams with the help of specially bent single crystals. Four crystals have been fabricated for this purpose. The studies have been performed at the CERN SPS in 400 GeV/c proton and 180 GeV/c pion beams. The results of measurements of beam envelopes are presented. The rms size of the horizontal profile at the focus was 5–8 times smaller than at the exit of the crystals. The measured focal lengths were 4–21 m. The results of measurements are in good agreement with calculations. Possible applications of focusing crystals in present and future high energy accelerators are discussed.

The effect of multiple volume reflections in one crystal was observed in each of several bent silicon strips for 400 GeV/c protons. This considerably increased the particle deflections. Some particles were also deflected due to channeling in one of the subsequent strips. As a result, the incident beam was strongly spread because of opposite directions of the deflections. A modified multi-strip deflector produced by periodic grooves on the surface of a thick silicon plate was used for these measurements. This technique provides perfect mutual alignment between crystal strips. Such multi-strip deflector may be effective for collider beam halo collimation and a study is planned at the CERN SPS circulating beam.

Channeling is the confinement of the trajectory of a charged particle in a crystalline solid. Positively charged particles channeled between crystal planes oscillate with a certain oscillation length , which depends on particle energy. A crystal whose thickness is half the oscillation length for planar channeling may act as a mirror for charged particles. If the incident angle of the particle trajectory with the crystal plane is less than the critical angle for channeling, under-barrier particles undergo half an oscillation and exit the crystal with the reversal of their transverse momentum, i.e., the particles are “mirrored” by the crystal planes. Unlike the traditional scheme relying on millimeter-long curved crystals, particle mirroring enables beam steering in high-energy accelerators via interactions with micrometer-thin straight crystal. The main advantage of mirroring is the interaction with a minimal amount of material along the beam, thereby decreasing unwanted incoherent nuclear interactions. The effectiveness of the mirror effect for ultrarelativistic positive particles has been experimentally proven at external lines of CERN-SPS. The mirroring effect in a 26.5-μm-thick Si crystal has been studied in the experiment with a narrow beam of 400 GeV/ c protons at the CERN-SPS. The reflection efficiency for a quasi-parallel beam is larger than 80%.

We studied the dechanneling length of 150 GeV / c π − interacting with a short bent silicon crystal. Dechanneling length measures the rate and the strength of incoherent interactions of channeled particles in a crystal. The mechanism of dechanneling of negatively charged particles has been elucidated through simulation and experiment. It was found that the dechanneling length for negative particles is comparable to the nuclear dechanneling length for positive charges. Indeed, dechanneling of negative particles occurs as a result of incoherent interactions with the nuclei because the trajectories of such particles always intersect atomic planes, explaining the lower channeling efficiency for such particles. Obtained results can be useful for the design of crystals for manipulating high-energy negative particle beams through channeling.

A deflection efficiency of about 61 % was observed for 400 GeV/c protons due to channeling, most strongly along the 〈110〉 axis of a bent silicon crystal. It is comparable with the deflection efficiency in planar channeling and considerably larger than in the case of the 〈111〉 axis. The measured probability of inelastic nuclear interactions of protons in channeling along the 〈110〉 axis is only about 10 % of its amorphous level whereas in channeling along the (110) planes it is about 25 % . High efficiency deflection and small beam losses make this axial orientation of a silicon crystal a useful tool for the beam steering of high energy charged particles.

Two high-accuracy goniometers equipped with two bent silicon crystals were installed in the betatron cleaning insertion of the CERN Large Hadron Collider (LHC) during its long shutdown. First beam tests were recently performed at the LHC with 450 GeV/c and 6500 GeV/c stored proton beams to investigate the feasibility of beam halo collimation assisted by bent crystals. For the first time channeling of 6500 GeV/c protons was observed in a particle accelerator. A strong reduction of beam losses due to nuclear inelastic interactions in the aligned crystal in comparison with its amorphous orientation was detected. The loss reduction value was about 24. Thus, the results show that deflection of particles by a bent crystal due to channeling is effective for this record particle energy., peer-reviewed

The paper devoted to investigation of volume reflection and channeling processes of ultrarela- tivistic positive charged particles moving in germanium single crystals. We demonstrate that the choice of atomic potential on the basis of Hartree-Fock method and correct choice of Debye tem- perature allow us to describe the above mentioned processes in a good agreement with the recent experiments. Moreover, the presented in the paper universal form of equations for volume reflection gives true description of the process at a wide range of particle energies. Standing on this study we make predictions for mean angle reflection (as a function of bending radius) of positive and negative particles for germanium (110) and (111) crystallographic planes., Comment: 5 pages, 2 figures, PACS number: 41.60.-m

The efficiency of the beam extraction and collimation systems in circular accelerators with the use of the channeling effect in a bent crystal is determined. The dependences of the extraction efficiency on the geometrical parameters of the crystal (its length, thickness, and bending angle), the azimuthal location of the system components, and the offset of the septum or scraper are presented. The influence of crystal imperfections (amorphous layers, misorientation, and torsion) on the efficiency of the systems is considered, and their tolerances are proposed. It is shown that an extraction efficiency of >95% can be attained over a wide energy range from 2 GeV to 7 TeV by optimizing the crystal parameters and the positions of the system components., 8 pages, 9 figures

The possibilities of the extraction and collimation of a circulating beam by a new method due to the reflection of particles in crystals with axial orientation were experimentally investigated in the Fall-2010 run at the U_70 synchrotron. Such crystals have positive features, because the axial potential is five times larger than the planar potential. It has been shown that the collimation efficiency can reach 90% due to axial effects in the crystal. Losses of the circulating beam on a collimator have been reduced by several times; this makes it possible to suppress the muon jet near the steel collimator of the circulating beam.

Channeling in a short bent silicon crystal was investigated at the CERN SPS using 400-GeV/c protons with an angular spread much narrower than the critical channeling angle. Particle dechanneling due to multiple scattering on the atomic nuclei of the crystal was observed and its dechanneling length was measured to be about 1.5 mm. For a crystal with length comparable to such dechanneling length, an efficiency of 83.4% was recorded, which is close to the maximum value expected for a parallel beam and exceeds the previously known limitation of deflection efficiency for long crystals.

The staff of the Institute of High-Energy Physics together with researchers from Russian and foreign scientific centers have discovered a new physical phenomenon – reflection of fast charge particles in curved crystals. This article presents the first experimental results using multiple reflections of protons in silicon crystals to increase the beam collimation efficiency in the U-70 accelerator. The experimental data are compared with theoretical models. The technology developed using reflecting crystals can be used to form oppositely propagating beams in accelerators and has prospects for extracting low-emittance beams on ~1 GeV accelerators for electronuclear facilities and in medicine and industry.

The paper is devoted to the study of volume reflection and volume capture of high energy particles moving in planar fields of bent single crystals. The influence of volume capture on the process of volume reflection was considered analytically. Relations describing various distributions of particles involved in the process, the probability of volume capture and the behavior of channeling and dechanneling fractions of a beam were obtained. Results of the study will be useful in the realization of multicrystal devices for collimation and extraction of beams on modern and future accelerators.

W. Scandale a,b,e, F. Andrisani a, G. Arduini a, M. Butcher a, F. Cerutti a, M. Garattini a, S. Gilardoni a, A. Lechner a, R. Losito a, A. Masi a, A. Mereghetti a, E. Metral a, D. Mirarchi a,j, S. Montesano a, S. Redaelli a, R. Rossi a,e, P. Schoofs a, G. Smirnov a, E. Bagli c, L. Bandiera c, S. Baricordi c, P. Dalpiaz c, G. Germogli c, V. Guidi c, A. Mazzolari c, D. Vincenzi c, G. Claps d, S. Dabagov d,k,l, D. Hampai d, F. Murtas d, G. Cavoto e, F. Iacoangeli e, L. Ludovici e, R. Santacesaria e, P. Valente e, F. Galluccio f, A.G. Afonin g, Yu.A. Chesnokov g, A.A. Durumg, V.A. Maisheev g, Yu.E. Sandomirskiy g, A.A. Yanovich g, A.D. Kovalenko h, A.M. Taratin h,∗, Yu.A. Gavrikov i, Yu.M. Ivanov i, L.P. Lapina i, J. Fulcher j, G. Hall j, M. Pesaresi j, M. Raymond j

In ideal two-stage collimation systems, the secondary collimator–absorber should have its length sufficient to exclude practically the exit of halo particles with large impact parameters. In the UA9 experiments on the crystal assisted collimation of the SPS beam a 60 cm long tungsten bar is used as a secondary collimator–absorber which is insufficient for the full absorption of the halo protons. Multi-turn simulation studies of the collimation allowed to select the position for the beam loss monitor downstream the collimation area where the contribution of particles deflected by the crystal in channeling regime but emerging from the secondary collimator–absorber is considerably reduced. This allowed observation of a strong leakage reduction of halo protons from the SPS beam collimation area, thereby approaching the case with an ideal absorber.

IHEP Protvino has pioneered the wide practical use of bent crystals as optical elements in high energy beams for beam extraction and deflection on permanent basis since 1989. In the course of IHEP experiments, crystal channeling has been developed into efficient instrument for particle steering at accelerators, working in predictable, reliable manner with beams of very high intensity over years. Crystal systems extract 70 GeV protons from IHEP main ring with efficiency of 85% at intensity of 10 12 , basing on multi-pass mechanism of channeling proposed theoretically and realized experimentally at IHEP. Today, six locations on the IHEP 70 GeV main ring of the accelerator facility are equipped by crystal extraction systems, serving mostly for routine applications rather than for research and allowing a simultaneous run of several particle physics experiments, thus significantly enriching the IHEP physics program. The long successful history of large-scale crystal exploitation at IHEP should help to incorporate channeling crystals into accelerator systems worldwide in order to create unique systems for beam delivery. We report recent results from the research and exploitation of crystal extraction systems at IHEP.

A beam line for electrons with energies in the range of 1 to 45 GeV, low contamination of hadrons and muons and high intensity up to 10^6 per accelerator spill at 27 GeV was setup at U70 accelerator in Protvino, Russia. A beam tagging system based on drift chambers with 160 micron resolution was able to measure relative electron beam momentum precisely. The resolution sigma_p p was 0.13% at 45 GeV where multiple scattering is negligible. This test beam setup provided the possibility to study properties of lead tungstate crystals (PbWO_4) for the BTeV experiment at Fermilab.

The results of observation and studies of focusing of 400 GeV/ c proton beam with the help of bent single crystals are presented. Two silicon crystals have been used in the measurements. The focal length of the first and second crystals is found to be 1.48 m and 0.68 m, respectively. The mean square size of the horizontal profile in the focus was 3.1 and 4.3 times as small as at the exit of the crystals.

The possibility for optimization of crystal assisted collimation has been studied at the CERN SPS for stored beams of protons and Pb ions with 270 GeV/ c per unit charge. A bent silicon crystal used as a primary collimator deflects halo particles in the channeling regime, directing them into a tungsten absorber. In channeling conditions a strong reduction of off-momentum particle numbers produced in the crystal and absorber, which form collimation leakage, has been observed in the first high dispersion (HD) area downstream. The present study shows that the collimation leakage is minimal for some values of the absorber offset relative to the crystal. The optimal offset value is larger for Pb ions because of their considerably larger ionization losses in the crystal, which cause large increases of particle betatron oscillation amplitudes. The optimal absorber offset allows obtaining maximal efficiency of crystal-assisted collimation.

The carbon ion +6C beam with energy 25 GeV/nucleon was extracted by bent crystal from the U-70 ring. The bent angle of silicon crystal was 85 mrad. About 2×105 particles for 109 circulated ions in the ring were observed in beam line 4a after bent crystal. Geometrical parameters, time structure and ion beam structure were measured. The ability of the bent monocrystal to extract and generate ion beam with necessary parameters for regular usage in physical experiments is shown in the first time.

This report overviews studies accomplished in the U70 proton synchrotron of IHEP-Protvino during the recent two decades. Major attention is paid to a routine application of bent crystals for beam extraction from the machine. It has been confirmed experimentally that efficiency of beam extraction with a crystal deflector of around 85% is well feasible for a proton beam with intensity up to 1012 protons per cycle. Another trend is to use bent crystals for halo collimation in a high energy collider. New promising options emerge for, say, LHC and ILC based on the "volume reflection" effect, which has been discovered recently in machine study runs at U70 of IHEP (50 GeV) and SPS of CERN (400 GeV)., 12 pages, 14 figures

It has been shown that the use of a bent crystal with a variable curvature radius makes it possible to reduce the fraction of dechanneled particles by an order of magnitude. This effect enables the strong reduction of the particle density at the edge of a collimator or the partition of a septum upon the multiturn extraction of a beam from a ring accelerator. In particular, the beam extraction efficiency at the U-70 synchrotron and Large Hadron Collider can be increased to 95 and 99.65%, respectively., Comment: 4 pages, 4 figures

New crystal technique - array of bent strips and a fan-type reflector, based on thin straight plates - have been used for research of extraction and collimation a circulating beam in the U-70 accelerator at the energy 50 GeV and 1.3 GeV. It is shown, that new devices can effectively steer a beam in a wide energy range. For protons with energy 50 GeV efficiency of extraction and collimation about 90 % has been achieved which is record for this method. Reduction of particle losses in 2-3 times was observed also in accelerator at application of different crystals in comparison with the usual one-stage collimation scheme of beam with a steel absorber., Comment: 14 pages, 12 figures

Different kinds of deflection in a silicon crystal bent along the 〈111〉 axis was observed for 150 GeV/ c negative particles, mainly π − 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 . It was observed volume capture of π − 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 − mesons by different bent planes crossing the axis.

Multiple volume reflections of the 400 GeV / c proton beam by the sequence of fourteen bent silicon strips has been studied at the CERN SPS. The sequence is close to be parallel that is the spread of the strip orientation angles is much smaller than their bend angle and eleven strips working coherently in the regime of volume reflections deflected the beam by 110 μrad with the efficiency 88%, which is significantly larger than the estimation based on independent reflections. The mechanism giving the efficiency increase has been studied by simulation. It appears that many particles volume captured in one of the strips take part in volume reflections in the subsequent ones. Such a crystal multi reflector can be successfully used as a primary collimator for the beam halo collimation of high-energy accelerators.

The radiation emitted by $180\ensuremath{-}\mathrm{GeV}∕c$ volume-reflected electrons and positrons impinging on a bent crystal has been measured by the H8RD22 Collaboration on the H8 beamline at the CERN SPS. A dedicated spectrometer has been developed to measure high-energy photon spectra (up to $\ensuremath{\sim}100\phantom{\rule{0.3em}{0ex}}\mathrm{GeV}$) under volume reflection: photon and charged particle beams have been separated by a bending magnet and leptons were detected and tagged by microstrip silicon detectors and a Pb-scintillator sampling calorimeter. A comparison between the experimental and analytical data for the amorphous and volume-reflection cases is presented and the differences are discussed.

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.

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.

Deflection due to axial channeling in a silicon crystal bent along the 〈111〉 axis was observed for 150 GeV/c negative particles, mainly π − mesons, at one of the secondary beams of the CERN SPS. The whole beam was deflected to one side with the efficiency of about 90% and with the peak position at the bend crystal angle α = 43 μrad . The deflection occurs mainly due to doughnut scattering of above-barrier particles by the atomic strings of the crystal. However, due to a high probability of particle recapture into bound states with the atomic strings their contribution to the deflection should be about 15% for our case according to simulation results.

The interactions of 400 GeV protons with different sequences of bent silicon crystals have been investigated at the H8 beam line of the CERN Super Proton Synchrotron. The multiple volume reflection of the proton beam has been studied in detail on a five-crystal reflector measuring an angular beam deflection theta = 52.96 +/- 0.14 microrad. The efficiency was found larger than 80% for an angular acceptance at the reflector entrance of 70 microrad, with a maximal efficiency value of epsilon = 0.90 +/- 0.01 +/- 0.03.

The trend of volume reflection parameters (deflection angle and efficiency) in a bent (110) silicon crystal has been investigated as a function of the crystal curvature with 400 GeV/c protons on the H8 beam line at the CERN Super Proton Synchrotron. This Letter describes the analysis performed at six different curvatures showing that the optimal radius for volume reflection is approximately 10 times greater than the critical radius for channeling. A strong scattering of the beam by the planar potential is also observed for a bend radius close to the critical one.

This paper presents a detailed study of the deflection phenomena of a $400\text{ }\text{ }\mathrm{GeV}/c$ proton beam impinging on a new generation of bent silicon crystals; the tests have been performed at the CERN Super Proton Synchrotron H8 beam line. Channeling and volume reflection angles are measured with an extremely precise goniometer and with high resolution silicon microstrip detectors. Volume reflection has been observed and measured for the first time at this energy, with a single-pass efficiency as large as 98%, in good agreement with the simulation results. This efficiency makes volume reflection a possible candidate for collimation with bent crystals at the CERN Large Hadron Collider.

The doubling of the angle of beam deflection due to volume reflection of protons by a sequence of two bent silicon crystals was experimentally observed at the 400 GeV proton beam of the CERN SPS. A similar sequence of short bent crystals can be used as an efficient primary collimator for the Large Hadron Collider.

A high performance apparatus has been designed and built by the H8-RD22 collaboration for the study of channeling and volume reflection phenomena in the interaction of 400 GeV/c protons with bent silicon crystals, during the 2006 data taking in the external beamline H8 of the CERN SPS. High-quality silicon short crystals were bent by either anticlastic or quasimosaic effects. Alignment with the highly parallel (8 murad divergence) proton beam was guaranteed through a submicroradian goniometric system equipped with both rotational and translational stages. Particle tracking was possible by a series of silicon microstrip detectors with high-resolution and a parallel plate gas chamber, triggered by various scintillating detectors located along the beamline. Experimental observation of volume reflection with 400 GeV/c protons proved true with a deflection angle of (10.4+/-0.5) murad with respect to the unperturbed beam, with a silicon crystal whose (111) planes were parallel to the beam.

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 mu rad with about 30% efficiency. The contribution of hyperchanneled states of protons to the observed beam deflection was less than 2% according to simulation results.