Channeling and volume reflection of high-energy charged particles in short bent crystals. Crystal assisted collimation of the accelerator beam halo

Physics Reports 815 (2019) 1–107

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Channeling and volume reflection of high-energy charged particles in short bent crystals. Crystal assisted collimation of the accelerator beam halo W. Scandale a , A.M. Taratin b , a b

∗

CERN, European Organization for Nuclear Research, CH-1211, Geneva 23, Switzerland Joint Institute for Nuclear Research, Joliot-Curie 6, 141980, Dubna, Russia

article

info

Article history: Received 4 July 2018 Received in revised form 16 April 2019 Accepted 26 April 2019 Available online 22 May 2019 Editor: Giulia Zanderighi Keywords: Crystal Channeling Volume reflection Beam deflection Beam collimation

a b s t r a c t The experimental studies of high-energy charged particle deflection due to planar and axial channeling as well as volume reflection and multi volume reflections in short bent crystals at the extracted beams of the CERN Super Proton Synchrotron (SPS) are considered. The experiments on the studies of crystal assisted collimation of the CERN SPS beam halo and the first similar experiment with the CERN Large Hadron Collider (LHC) beam of 6500 GeV/c protons are also considered. © 2019 Elsevier B.V. All rights reserved.

Contents 1. 2.

3.

4.

Introduction............................................................................................................................................................................................. Theoretical description .......................................................................................................................................................................... 2.1. Planar channeling of fast charged particles............................................................................................................................ 2.2. Dechanneling of particles ........................................................................................................................................................ 2.3. Channeling in a bent crystal. Calculation of trajectories ...................................................................................................... 2.4. Dechanneling length reduction. Parabolic potential approximation ................................................................................... 2.5. Beam deflection efficiency by a bent crystal ......................................................................................................................... 2.6. Volume capture of particles into channeling regime ............................................................................................................ 2.7. The trajectory equation of high-energy particles in a bent crystal..................................................................................... 2.8. Volume reflection of high-energy charged particles in a bent crystal................................................................................ 2.9. Axial channeling of fast charged particles.............................................................................................................................. 2.10. Particle deflection due to axial channeling in a bent crystal............................................................................................... 2.11. Calculation of particle trajectories in a bent crystal for axial case ..................................................................................... Experimental setup for studies of beam deflection by short bent crystals .................................................................................... 3.1. Short bent crystals and goniometer ........................................................................................................................................ 3.2. Tracking systems........................................................................................................................................................................ Beam deflection by a bent crystal. Planar orientations ..................................................................................................................... 4.1. Positive particles ........................................................................................................................................................................ 4.1.1. Observation of volume reflection of 400 GeV/c protons in bent silicon crystals ..............................................

∗ Corresponding author. E-mail address: [email protected] (A.M. Taratin). https://doi.org/10.1016/j.physrep.2019.04.003 0370-1573/© 2019 Elsevier B.V. All rights reserved.

2 3 4 5 6 7 7 8 8 8 9 11 12 12 12 13 14 14 14

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4.1.2. 4.1.3.

5.

6.

7. 8. 9.

10.

Volume reflection dependence on the bent crystal curvature ............................................................................. Observation of multiple volume reflection by different planes in one bent silicon crystal for high-energy protons ........................................................................................................................................................................ 4.1.4. Multiple volume reflection of high-energy protons by a sequence of bent silicon crystals ............................ 4.1.5. Possibilities to increase VR and MVR efficiency..................................................................................................... 4.1.6. Planar channeling of protons in short crystals....................................................................................................... 4.1.7. Reduction of inelastic nuclear interactions of protons at planar channeling in crystals .................................. 4.1.8. Electromagnetic dissociation for well channeled heavy ions ............................................................................... 4.1.9. Parametric X-rays produced by protons in bent crystals as a tool to control crystal state ............................. 4.1.10. Beam focusing with a bent crystal .......................................................................................................................... 4.1.11. Reflection of protons by an ultra-thin straight crystal.......................................................................................... 4.2. Negative particles ...................................................................................................................................................................... 4.2.1. Observation of channeling and volume reflection in bent crystals for high-energy negative particles ......... 4.2.2. Observation of multiple volume reflection by different planes in one bent silicon crystal for high-energy negative particles ....................................................................................................................................................... 4.2.3. Measurement of the dechanneling length for high-energy negative pions........................................................ Beam deflection by a bent crystal. Axial orientations ....................................................................................................................... 5.1. Positive particles ........................................................................................................................................................................ 5.1.1. Deflection of high-energy protons through axial channeling along [111] axis.................................................. 5.1.2. High-efficiency deflection of protons through axial channeling along [110] axis ............................................. 5.2. Negative particles ...................................................................................................................................................................... 5.2.1. Deflection of high-energy π − mesons through axial channeling along [111] axis ............................................ 5.2.2. Deflection of high-energy π − mesons in quasi-bound states of doughnut scattering. Volume capture ......... Experimental studies of crystal assisted collimation of the SPS beam halo ................................................................................... 6.1. Observation of the SPS beam halo deflection by a bent crystal .......................................................................................... 6.2. Role of the miscut of a crystal primary collimator ............................................................................................................... 6.3. Leakage reduction in the SPS beam halo collimation with a bent crystal ......................................................................... 6.4. Optimization of the crystal assisted collimation of the SPS beam ...................................................................................... 6.5. Observation of strong leakage reduction in crystal assisted collimation of the SPS beam .............................................. 6.6. Simulation of crystal assisted collimation of the SPS beam................................................................................................. Experimental studies of crystal assisted collimation of the LHC beam halo .................................................................................. Proposal of crystal assisted extraction of the FCC beam halo .......................................................................................................... Possibility of high efficient beam extraction from accelerators with a bent crystal ..................................................................... 9.1. Introduction ................................................................................................................................................................................ 9.2. Simulation studies of the SPS beam extraction with a bent crystal ................................................................................... 9.2.1. Transverse diffusion ................................................................................................................................................... 9.2.2. Parallel orbit bump .................................................................................................................................................... 9.2.3. Optimal orbit bump with angular adjustment ....................................................................................................... Conclusions.............................................................................................................................................................................................. Acknowledgments .................................................................................................................................................................................. References ...............................................................................................................................................................................................

15 19 21 28 32 38 43 44 49 50 52 52 56 58 59 59 59 60 64 64 67 72 72 76 77 79 81 86 91 95 97 97 99 100 100 102 103 105 105

1. Introduction The channeling effect occurs in the passage of fast charged particles through an oriented crystal [1]. If we look at a crystal along the main crystallographic direction, we see a two-dimensional lattice formed by rows of atoms. A fast positive particle entering the crystal at a small angle relative to the crystallographic direction moves along the rows of atoms without closely approaching them, owing to repulsion in correlated small-angle collisions with the row atoms. Planar channeling occurs when the transverse momentum of the incident particle is directed along a main crystallographic plane. In this case, the particle moves oscillating between two adjacent crystal planes. For channeled particles, the yield of processes requiring close collisions with atoms is decreased, and their mean free path is increased. Within the framework of classical mechanics, Lindhard [1] developed a theory of orientation effects of fast charged particles in crystals, which successfully explained the experimental results. The motion of channeled particles is controlled by the electric field of the crystal atoms averaged along the crystallographic axes or planes. In 1976 Tsyganov [2] suggested to use bent crystals for controlling beams of high-energy charged particles. It was assumed that in planar channeling, positive particles would follow the crystal bend up to some critical bend radius determined by the maximum strength Emax of the atomic electric field averaged along the planes Rc =

E eEmax

,

(1)

where E is the particle energy. For the (110) planar channel of a silicon crystal Emax ≃ 6 GV/cm and for 1 TeV protons Rc = 1.6 m.

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Fig. 1. (a) Collimation scheme using a solid state primary collimator–scatterer (SC). (b) Collimation scheme with a bent crystal (BC) as a primary collimator. Halo particles are deflected and directed onto the absorber (TAL — Target Aperture Limitation) far from its edge.

In 1979 the deflection of a 8.4 GeV proton beam by a bent crystal was first observed at the JINR synchrophasotron [3]. At the same time, an experiment on the deflection of 900 MeV electrons was performed in Tomsk [4] and somewhat later at CERN using a beam of secondary particles up to 12 GeV/c [5]. In 1984 a bent crystal was used for the first time to extract a circulating proton beam from the JINR synchrophasotron [6]. The deflection and extraction of a particle beam from a cyclic accelerator by a bent crystal were later studied at IHEP [7], CERN [8–10] and Fermilab [11]. Silicon crystals with a few centimeters length were used. A beam extraction efficiency of 20%–30% was reached. It was shown [9] that there is a considerable contribution to the extraction from particles multiple passed through the crystal. This contribution should increase when a crystal becomes shorter. The extraction efficiency up to 85% was later observed with 2 mm long silicon crystal from the U-70 accelerator [12]. These studies showed that practical applications of crystals for steering particle beams are promising. At the IHEP synchrotron the crystal extraction system functions continuously, which broadens the possibilities for performing experiments, and the setup for splitting the extracted proton beam using bent crystals is used to organize simultaneous studies in several experimental channels. The usage of a short bent crystal as a primary collimator in high-energy hadron colliders such as the LHC is very promising possibility [13]. A multi-stage collimation system is used to absorb the beam halo particles preventing quenches of its superconducting magnets and reducing the collider experiment background. The amplitude increase per turn of halo particles is in general very small (considerably smaller than 1 µm). A primary collimator is a solid state target deflecting halo particles due to Coulomb scattering, therefore increasing their impact parameters with a massive secondary collimator–absorber. However, the impact parameter distribution has still a maximum at the absorber edge. A significant probability exists for these protons to be back scattered in the vacuum pipe and produce losses in the sensitive areas of the accelerator. Tertiary collimators are used to absorb these protons. In principle, backscattering of protons from the absorber should be strongly reduced if a bent crystal is used as a primary collimator. The crystal indeed should deflect particles in channeling states and direct them onto the absorber far from its edge (see Fig. 1). This property could also be used to keep the secondary collimators at larger amplitudes therefore minimizing their beam coupling impedance. The required deflection with a crystal for the LHC beam halo collimation is about 50 µrad [14]. So its length can be 3–5 mm (the critical bend radius Rc ≃ 11 m for 6500 GeV protons). Experiments on the beam halo collimation with short bent crystals have been recently performed at the IHEP synchrotron [12], RHIC [15] and Tevatron [16]. In the last case, the background in the CDF experiment was reduced by using the crystal as a primary collimator by a factor of two with respect to what was obtained in the same conditions with a tungsten primary collimator. Deflection to the side opposite to the crystal bend, a volume reflection, has been predicted in [17]. Volume reflection occurs for high-energy charged particles entering a bent crystal at angles larger than the critical channeling angle near the tangency point with bent crystal planes. This effect was first observed in the experiments [18,19]. Volume reflection was observed also in the collimation experiment at the RHIC [15]. The deflection angle due to volume reflection is small. Besides, strong multiple scattering occurs for volume reflected particles. Therefore, a short bent crystal should be optimal to study the properties of volume reflection. Therefore, experiments with short bent crystals using the extracted and circulating beams of the CERN SPS were started with the purposes to study possibility of the crystal assisted collimation of the LHC beam halo and properties of volume reflection. This review considers the results of these studies for high-energy charged particle deflection due to planar and axial channeling as well as volume reflection and multi volume reflections. The experimental results on the studies of crystal assisted collimation of the CERN SPS beam halo and on the first similar experiment with the CERN Large Hadron Collider (LHC) beam of 6500 GeV/c protons are also considered. 2. Theoretical description Computer simulations of particle passages through bent crystals have been performed for analysis of the experimental results on the studies of channeling and volume reflection of high-energy charged particles in short bent crystals. A short theoretical description of these effects and models for simulations is given in this chapter. More detailed description can be found in [20,21].

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Fig. 2. Averaged potential of the (110) planar channel of a silicon crystal at room temperature in the Moliere approximation for the atomic potential. The dashed line is the parabola approximation for the potential. The coordinate x is calculated from the channel center, l is the channel half-width, xc = l − rc is the maximal oscillation amplitude for stable channeling of particles, rc is the critical approach to the channel walls.

2.1. Planar channeling of fast charged particles The motion of fast charged particles in a crystal at small angles to the atomic planes is described in a first approximation by a continuous planar potential [1], which is periodic in the transverse direction Y¯ (x¯ ) =

∞ ∑

Y1 (x¯ − ndp ) ,

(2)

n=−∞

Y1 (x) = 2π Ndp

∞

∫

√

V ( x2 + r 2 )rdr ,

(3)

0

where Y1 (x) is the potential of a single plane, averaged over the atom locations, assuming that the atoms are uniformly distributed, dp is the distance between planes and N is the average density of atoms, l = dp /2, V(R) is the atomic potential. When the Moliere atomic potential is used, the continuous planar potential for x¯ = (0, dp ) takes the form [22] Y¯ (x¯ ) = 2π Z1 Z2 e2 Ndp

3 ∑ αi cosh [κi (x¯ − l)] , κi sinh (κi l)

(4)

i=1

where Z1 e is the charge of incident particles, Z2 is the atomic number of a crystal, αi , βi are the Moliere potential −1/3 coefficients, κi = βi /a, a is the screening length, a = 0.8853 ao Z2 , ao is the Bohr radius. Fig. 2 shows the (110) planar potential in a silicon crystal obtained by averaging (4) over the thermal vibrations of the atoms at room temperature 20◦ C, u1 = 0.075 Å is the one dimensional vibration amplitude, where Y (x) = Y¯ (x + l) − Y¯ (l), x = x¯ − l. It was shown [1] that the use of classical mechanics is valid for channeling of relativistic particles. The equation of motion of a relativistic particle in a crystal in the direction transverse to the planes dpx

dY (x)

. (5) dt dx The motion of particles along the planes is free. The variation of the relativistic factor can be neglected, and Eq. (5) reduces to the nonrelativistic one =−

mγ x¨ = −Y ′ (x) ,

(6)

that is the transverse energy of the particle is the integral of motion mγ 2 Ex = x˙ + Y (x) = E ∗ θx2 + Y (x) = const . (7) 2 Here x˙ = vθx , θx is the angle of the particle momentum with the plane, E ∗ = pv/2, p and v are the particle momentum and velocity.

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If Ex < Yo , where Yo = Y (l) is the depth of the planar potential well, the transverse motion is bounded. The channeled particle oscillates between two adjacent planes. For Ex > Yo the transverse motion is unbounded. Particles undergoing unbounded motion are called above barrier or quasichanneled. At angles of incidence on the crystal θxo ≥ θc , where the angle

θc =

√

2Yo /pv

(8)

is the critical channeling angle, all particles are quasichanneled. 2.2. Dechanneling of particles In the approximation of a continuous potential, the particle transverse energy does not change with its penetration into a crystal. However, incoherent scattering by nuclei and electrons change the transverse energy of the particles. As a result, channeled particles leave the crystal channels that is dechanneling occurs. The contribution to dechanneling from catastrophic collisions, which lead to scattering at angles larger than the critical channeling angle, is small. Particle dechanneling occurs as a result of multiple scattering on electrons and nuclei with a gradual change of the particle transverse energy. The evolution of the channeled particle density f (Ex , z) due to multiple scattering is described by the Fokker-Planck equation with diffusion coefficients averaged over the electron and nuclear scattering cross-sections and the equilibrium distribution of channeled particles [1]. The equation transforms into a diffusion type, which is in the harmonic approximation for the planar potential

∂f ∂ = ∂z ∂ Ex

( B

∂f ∂ Ex

)

.

(9)

The solution of this equation with diffusion coefficient B(Ex ) = Bo Exα for the boundary and initial conditions f (Exc , z) = 0 ,

(10)

f (Ex , 0) = Fo (Ex ) ,

(11)

where Exc is the critical transverse energy and Fo (Ex ) is the initial distribution of the channeled particles, was obtained in [23]. For the channeled fraction dependence on the penetration depth Fch (z) =

∞ ∑

qn exp(−z /λn ) ,

(12)

n=1

Since λn ∼ 1/µ2n , where µn are the zeros of the Bessel function Jo in the case of α = 1, the channeled fraction is approximately determined by the first term and falls off exponentially. The experimental studies [24] also indicate exponential decrease of the channeled fraction with the penetration depth. Therefore, the dechanneling length can be estimated as S1/e ≈ λ1 = A = E∗⟨

4Exc

µ20,1 A

,

∆ϑx2 ⟩ ∆z

(13)

(14)

where A is the friction coefficient, which gives the average increase of the transverse energy per unit length along z. Lindhard [1] introduced the concept of critical approach to a row or plane of atoms, which limits the region of applicability of the continuous approximation. However, for high energy charged particles the averaged potential approximately determines the behavior also for quasichanneled particles. The concept of the critical approach for high energy particles is used to determine the stability region for channeling. The size of the regions with large nuclear density around the crystal planes (‘‘nuclear corridor’’), where particles undergo strong multiple scattering by nuclei and can fast dechannel, is proportional to the thermal vibration amplitude of atoms u1 . As shown by the experiment [25] a good estimate for silicon is the distance rc = 2.5u1 ≃ a. The dechanneling process consists of two stages. A long stage is due to multiple scattering on the crystal electrons. A fast stage is realized due to multiple scattering on the crystal nuclei for particles with large oscillation amplitudes in the channels xm > xc = l − rc , where l = dp /2 is the channel half width. The critical transverse energy Exc = U(xc ) separates two stages of dechanneling. The mean squared deflection of particles per unit path-length due to multiple scattering on nuclei can be calculated in the Ohtsuki-Kitagawa approximation [26]

∆ϑ 2 ∆ϑ 2 (x) = Pn (x) , ∆z n ∆z R

(15)

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Fig. 3. The mean squared deflection projection of 400 GeV/c protons per unit path length due to multiple scattering on nuclei (1) and electrons (2) in a silicon crystal as a function of the distance to the (110) channel wall.

∆ϑ 2 1 = ∆z R Lr

(

Es pv

)2

,

Pn (x) =

dp (2π u21 )1/2

exp(−x2 /2u21 ) ,

where ∆∆ϑz R is the corresponding value for an amorphous substance, Pn (x) is the transverse distribution of the crystal plane atoms due to thermal vibrations, Es = 21 MeV, and Lr is the radiation length. The mean squared deflection of particles per unit path-length due to multiple scattering on the crystal electrons can be calculated using the proportionality to the energy losses in close collisions [1] 2

me ∆ϑ 2 (x) = 2 ∆z e p

( ) dE ρ (x) − , dz

NZ2

(16)

where −(dE /dz) is the ionization energy losses per unit length and ρ (x) is the electron density in a planar channel. The electron density was determined by the solution of the Poisson equation for the single plane potential in the Moliere approximation with inclusion of the contributions of the two neighboring planes. Fig. 3 shows the mean squared deflection projection of 400 GeV/c protons per unit length due to multiple scattering on the crystal nuclei and electrons as a function of the distance to the channel wall. We see that at the distance smaller than 2.5 u1 from the channel wall, the main contribution to the change of the particle transverse energy occurs from scattering on the atomic nuclei. The expression (16) for the multiple scattering angle on the crystal electrons includes all close collisions with energy transfer up to the maximum Tmax though the upper limit should correspond to energy transfer T < T (θc ) < Tmax (see in [20]). However, the contribution of electrons to the dechanneling process observed in the short crystals is negligible. Therefore, any correction of (16) is not possible to estimate. 2.3. Channeling in a bent crystal. Calculation of trajectories In the crystal bent with the radius Ro the transverse motion of a particle entering the crystal at a small angle to the bent crystal planes in the comoving system rotating around the curvature center with an azimuthal velocity v is governed by the effective potential Ueff (x, Fc ) = U(x) ± Fc x + Uco (Fc ) ,

(17)

where Fc = pv/Ro is the centrifugal force and Uco (Fc ) is a constant, which is chosen such that the effective potential equals zero at the minimum. The sign of the centrifugal term depends on the direction in which the transverse coordinate x is measured; it is minus in the direction of the bend radius. The particle trajectory can be calculated by numerically solving the equation of motion in the effective potential (17) x¨ (t) = −

1

d

mγ dx

Ueff (x, Ro ) .

(18)

The change of the particle longitudinal velocity can be neglected because it is relatively small. Then each step of the time integration of the equation of motion corresponds to the particle traveling a longitudinal distance ∆z = v ∆t in the

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crystal. In the continuous approximation the particle transverse energy Ex is an integral of the motion. So, the integration step ∆t is chosen from the condition that Ex be conserved. The effective potential of the set of bent atomic planes can be calculated using the potential (4) averaged over thermal vibrations of the atoms. After the particles pass a distance ∆zs ≥ ∆z the change of the particle transverse velocity due to multiple scattering on the crystal nuclei and electrons is calculated according to (15) and (16). The step size ∆zs is limited below by the requirement that the theory of multiple scattering be valid. The usual upper limit on ∆zs , from the condition that the particle energy losses be negligible, is always satisfied for heavy high-energy particles in channeling experiments. However, in the planar channels there is very strong dependence of multiple scattering on the transverse positions due to non-uniformity of the nuclei and electron densities. Therefore, the step size should be sufficiently small, ∆zs ≪ λ, where λ is the special period of particle oscillations in the planar channel. 2.4. Dechanneling length reduction. Parabolic potential approximation The critical approach to the channel walls determines the range of stable channeling where slow dechanneling due to multiple scattering on the crystal electrons occurs. The potential of a planar channel can be approximated by a parabola (see Fig, 2) at distances from the channel walls larger than the critical approach U(x) = Uos (x/xc )2 ,

(19)

where Uos = Y (lc ) is the critical transverse energy for stable channeling in a straight crystal. Then the effective potential of a bent crystal can be written as Ueff (x, R) = Uos

(

x − xo

)2

xc

,

xo =

Fc 2Uos

x2c .

(20)

The minimum of the effective potential is shifted to the outer side of the channel by a distance xo , which leads to the corresponding shift of the channeled particle trajectories. Therefore, the critical transverse energy Exc decreased [27]. The dependence of Exc on R and E in a bent crystal can be written as Exc (R) = Uos (1 − Rc /R)2 , Rc (E) = xc

E

Exc (E) = Uos (1 − E /Ec )2 .

∗

Uos

,

Ec∗ (R) = Uos

R xc

(21)

.

The dechanneling length in the first approximation according to (13) should change like Exc (R, E) in a bent crystal S1/e (R) = S1o/e (1 − Rc /R)2 ,

S1/e (E) = S1o/e (E) (1 − E /Ec )2 .

(22)

Here Ec is the critical energy of particles for channeling in a bent crystal. The friction coefficient A (14) decreases with particle energy. Therefore, the dechanneling length in a straight crystal S1o/e increases with energy. The measurements through ionization losses [24] confirmed this behavior; S1o/e was about 10 cm for 200 GeV particles. In a bent crystal the dechanneling length according to (22) should have a maximum at some energy. This behavior predicted in [27] was observed in the experiment [25]. The measurements have been performed through the analysis of the angular distributions of particles. Particles lost from the channeling regime at the depth z in a uniformly bent crystal are deflected by an angle ϑ (z) = z /R. This angular unfolding of the dechanneling process in a bent crystal provides a convenient tool for studying particle dechanneling. 2.5. Beam deflection efficiency by a bent crystal The beam deflection efficiency by a uniformly bent crystal Pd can be estimated as [27] Pd (α, R) = Pc (R) Pch (α, R) ,

(23)

Pch (α, R) = exp[−α R/S1/e (R)] . where Pc (R) is the capture probability of particles into the channeling regime, Pch (α, R) is the probability to pass through the whole crystal in the channeling regime. There are optimal crystal radius R and length L for the beam deflection at a given angle α [27]. The deflection efficiency for a parabolic potential and a beam with uniform angular distribution [28]

(

Pd (α, ρ ) = Pco (1 − ρ )2 exp −

α/θD ρ (1 − ρ )2

)

.

(24)

In this case, the capture probability Pc (R) decreases in a bent crystal the same way as dechanneling length, Pco is its value for a straight crystal. The optimal crystal bend is determined by the condition [28]

α/θD =

2ρ 2 (1 − ρ )2 1 − 3ρ

,

θD = S1o/e /Rc ,

(25)

where ρ = Rc /R and θD is a deflection angle close to the maximum possible one. The optimal bend radius is always larger than 3Rc according to (25).

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2.6. Volume capture of particles into channeling regime When a beam of particles passes a bent crystal there are particles which are captured by bent planar channels in the region tangent to the bent planes. Volume capture in a bent crystal was observed in the experiments [29] by using a beam of 1 GeV protons. The mechanism for the volume capture of particles in a uniformly bent crystal, like in a straight crystal, is multiple scattering. It was shown [30] that this capture mechanism is sufficient for explaining the experimental results [29]. The crystal bend significantly increases the range of orientation angles for which volume capture of particles in the crystal is possible. This range is determined by the crystal bend angle. The capture region is localized due to the crystal bend. The fraction of particles dechanneled and recaptured is small in bent crystals [31]. Experimental studies of volume capture in bent crystals were later performed at IHEP using a beam of 70 GeV protons [32,33]. It was shown that the probability of volume capture is proportional to the crystal bend radius. A simple expression was proposed in [33] for estimating the probability of volume capture. According to the reversibility principle, the probability of a particle transition from the quasi-channeled fraction to the channeled fraction is equal to the probability of particle dechanneling over a length dz, Pdec = dz /Sd . The length of the capture region is proportional to Rθc . The probability of volume capture in this range Pv c ≃

Rθc

R

∼

Sd

.

E 3/2

(26)

This definition of the capture probability can be valid for the crystal bend radii which are not too large and for particle capture into deep levels in the channel potential, when Rθc ≪ Sd . 2.7. The trajectory equation of high-energy particles in a bent crystal Let us consider the passage of quasi-channeled particles through a crystal bent with constant radius in the approximation of continuous potential of atomic planes [17]. For simplicity the particle velocity along the crystal bend axis can be accepted to be zero. Then the motion of particles can be considered in polar coordinates (r , ϕ ). The total energy W and angular momentum M of a particle relative to the center of the field are integrals of the motion W = mγ c 2 + U(r),

(27)

M = mγ r ϕ, ˙

(28)

2

The trajectory equation of particles is found from the integrals of the motion

ϕ (r) = ±M

r −2 dr

∫ U(r))2

[(W −

/

c2

− M 2 /r 2 − m2 c 2 ]1/2

+ ϕo ,

(29)

where ϕo is a constant. Measuring the radial coordinate r from the point with a bend radius Ro , for a crystal size in the radial direction ∆r ≪ Ro and for high-energy particles the equation takes the form [17]

√

ϕ (r) =

E∗

Ro

cos ϑo

∫

r

dr

]1/2 ,

E ∗ sin ϑo + 2E ∗ (r − ro )/Ro − (U(r) − U(ro )) 2

[

ro

(30)

where ϑo is the angle between the particle momentum and the direction of the bent planes at the crystal entrance. Thus the approximation usually used for the description of high-energy particle channeling in a bent crystal was obtained when the bend is taken into account through the constant centrifugal force Fc = 2E ∗ /Ro and the effective potential acting on a particle Ueff (r , Ro ) (see (17)). It is seen that the approximation is valid also for quasi-channeled particles crossing the planes in a bent crystal. Measuring angles by the critical angle ϑc = (Uo /E ∗ )1/2 , the trajectory equation can be presented by a form which is independent of particle energy

ϕ¯ (r , Fc ) =

Fc 1/2 2Uo

∫

r

ro

dr

]1/2 ,

Uo ϑ¯ + Ueff (ro , Fc ) − Ueff (r , Fc )

[

2 o

(31)

where ϕ¯ = ϕ/ϑc , ϑ¯ o = ϑo /ϑc . 2.8. Volume reflection of high-energy charged particles in a bent crystal The trajectories of high-energy quasi-channeled particles in a silicon crystal bent along the (110) planes were studied in [17] using (30) and (31). The case of ϑo = 0 was considered when a quasi-channeled particle starts at the turning point in the effective potential. It was shown that when the crystal bend is far from the critical one, Fc ≪ Fcr (Fcr = eEmax = 5.97 GeV/cm for the (110) planar channels in a silicon crystal), the angular distribution presents a narrow peak at ϑR = −0.8ϑc , that is quasi-channeled particles are deflected opposite to the crystal bend by the field of bent planes (see Fig. 4a). This deflection is doubled when the beam enters the crystal at an angle ϑo > ϑc due to the presence of the two branches of

W. Scandale and A.M. Taratin / Physics Reports 815 (2019) 1–107

9

Fig. 4. Schematic picture of the deflection of high-energy charged particles by the averaged field of atomic planes in a crystal bent with a radius Ro for the case of small crystal bend, Fc ≪ Fcr . (a) ϑo = 0, particles enter a crystal parallel to bent planes. (b) ϑo > ϑc , where ϑc is the critical channeling angle. Here, (1) are channeled particles captured at the entrance (a) or in the crystal volume (b). (2) are quasi-channeled particles, the deflection angle equals θR for (a) and 2 θR for (b).

the trajectory which are symmetric with respect to the turning point in the effective potential (Fig. 4b). The effect was called volume reflection (VR). Negative quasi-channeled particles are also deflected opposite to the crystal bend but the deflection angles are smaller because of the difference in planar potential. For negative particles the gradient of the effective potential is small near its reflecting top (see Fig. 2 in [17]), and they accomplish a longer azimuthal path near turning points than positive particles at the same crystal bend. An explicit expression for the angle of volume reflection in the integral form was obtained in [34]. The numerical solution of the equation of motion in the effective potential described above is a convenient way to study volume reflection of particles in a bent crystal. Fig. 5 shows the trajectory of a quasi-channeled particle in the effective potential for the (110) planes of a silicon crystal bent with the radius Ro = 10 m. The inclination of the planar potential due to the centrifugal term produces the reflecting part. The particle changes the sign of its radial velocity at the turn point x∗ . Fig. 6 shows the dependence of the particle momentum direction on the penetration depth into the crystal. The momentum direction changes abruptly at the turn point of the radial motion. Here, the particle reflection by a bent plane occurs inside the crystal volume. In the considered case, the deflection angle of 400 GeV/c proton due to volume reflection is about 15 µrad. The deflection value depends on the position of the turn point at the reflecting part of the effective potential [35]. Quasi-channeled particles undergo strong multiple scattering crossing the crystal planes. By this reason a considerable broadening of the beam occurs in the condition of volume reflection in a bent crystal, and a small beam fraction is captured in channeling regime. The size of the volume reflection region is the same as for volume capture, Sv r ∼ Rϑc ∼ E 1/2 . The beam broadening over a length Sv r can be estimated assuming that multiple scattering is the same as in an amorphous case

∆ϑ 2 v r = S v r ·

⟨∆ϑ 2 vr ⟩1/2 ∼ E −1/4 . ϑc

∆ϑ 2 ∼ E −3/2 , ∆z R

(32)

The relative angular spread of the particles volume reflected in the crystal decreases with increasing particle energy as E −1/4 . Therefore, volume reflection should be well observed for high-energy particles in short crystals. 2.9. Axial channeling of fast charged particles The motion of fast charged particles in a crystal at small angles to the crystallographic axis is governed by the potential of the lattice of atomic strings averaged along the axis. The averaged potential of a single atomic string is [1] Us (r) =

1 ds

+∞

∫

√

V ( r 2 + z 2 )dz ,

(33)

−∞

where ds is the distance between the string atoms. Using the Doyle–Turner approximation for the electronic scattering factor, the string potential averaged over the thermal vibrations of the crystal atoms takes the form [36] Us (r) = e2

4 2ao ∑

ds

i=1

ai Bi + u2⊥

( exp −

r2 Bi + u2⊥

)

,

(34)

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Fig. 5. The effective potential for the (110) planes of a silicon crystal bent with the radius Ro = 10 m for 400 GeV/c protons. The transverse coordinate x is measured in the direction opposite to the radial one, dp is the planar channel width. Here, ∆Uef is the potential inclination along the channel width, x∗ is the turn point for a particle with the transverse energy Exo , xr is the low coordinate of the reflecting part of the potential.

Fig. 6. The dependence of the momentum direction of 400 GeV/c proton on the penetration depth into a silicon crystal bent along the (110) planes with the radius 10 m. The particle enters the crystal with the angle 25 µrad relative to the plane direction. An abrupt change of the momentum direction occurs due to volume reflection.

√

where ao is the Bohr radius, Bi = bi /4π 2 , ai , bi are the Doyle–Turner approximation coefficients, u⊥ = 2u1 . The axial potential along the [111] axis of a silicon crystal Ua (x, y) calculated by the summation of the closest string potentials is shown in Fig. 7. The angular acceptance of axial channeling (AC) is determined by the critical angle [1]

√ ψ1 =

4Z1 Z2 e2 pv ds

.

(35)

For 400 GeV/c protons ψ1 = 20.7µrad for the [111] axial direction in a silicon crystal. Two types of states are realized for particles under axial channeling. The first one called ‘‘hyperchanneling’’ is the confinement of particles bounded up with a single axial channel. The axial potential well is in the string position for

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Fig. 7. The axial potential along the [111] axis of a silicon crystal Ua (x, y) calculated by the summation of the closest string potentials in the Doyle–Turner approximation for the electronic scattering factor.

negative particles. Their bound states are realized through the elliptical orbits precessing around the string. The axial potential wells are formed by neighboring atomic strings for positive particles. In this case, the potential barrier is rather low, only about 1 eV for the [111] axial channels. Therefore, only very small fraction of the positive particles can be captured into the bound states. The other particle states under axial channeling are unbound. Particles move in the lattice of atomic strings changing the transverse momentum direction in collisions with the strings. For particles with angles smaller than ψ1 , a uniform distribution of the transverse momentum directions is reached after traversing a distance λ called equalization length [1] 4

. (36) π 2 Nads ψ1 For 400 GeV/c protons λ = 43 µm for the [111] axial direction in a silicon crystal. The process of scattering by atomic strings that leads to such a distribution is called doughnut scattering. For incident angles larger than ψ1 , multiple scattering λ(ψ1 ) =

of particles by atomic strings leads to a partial equalization of the transverse momentum directions resulting in an asymmetric angular distribution in the form of an arc around the axis direction. 2.10. Particle deflection due to axial channeling in a bent crystal Deflection of axially channeled particles is possible not only due to the bound states of hyperchanneling but also due to the unbound states of doughnut scattering regime. The condition for particle deflection due to doughnut scattering was first formulated in [8] k1 =

λ Rψ1

<1.

(37)

This condition requires that the equalization length should be smaller than the length along which the string direction changes by ψ1 . Thus, the crystal curvature is limited. However, pioneering experiments [8,37] did not observe the beam deflection due to AC for positive particles. Some deflection effect has been observed [37] only for the negative 200 GeV/c beam. The sufficient condition for the particle deflection due to doughnut scattering was formulated in [38] k = k1 k2 < 1 ,

k2 =

L Rψ1

.

(38)

The limitation of the crystal bend angle α = L/R has been added by the use of k2 . The condition (38) can be presented by the other way [38]

λL ψ¯2 = 2 < ψ12 .

(39) R That is the average square of the particle deflection angle due to multiple scattering by the atomic strings at the exit from a bent crystal should be smaller than the square of the critical angle.

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Fig. 8. (a) Schematic picture of a silicon strip crystal with anticlastic curvature along the (110) planes. (b) Cross-sectional view of the crystal. Dimensions are expressed in millimeters. θv and θh are the orientation angles of the [111] axis relative to the beam direction.

2.11. Calculation of particle trajectories in a bent crystal for axial case Let the crystal axis be directed along Z, and the plane of the crystal bend be XZ. The system of equations, describing the transverse motion of particles along the crystal axis in the comoving system has the form x¨ (t) = − y¨ (t) = −

1

(

mγ 1 ∂

∂ Ua (x, y) + Fc ∂x

mγ ∂ y

)

,

(40)

Ua (x, y) .

(41)

The change of the particle transverse velocity due to multiple scattering on the crystal nuclei and electrons is calculated according to (15) and (16) with two-dimensional distribution of nuclei and electrons. The transverse distribution of the string atoms due to thermal vibrations has the form Pn (r) =

rs2 2u21

( exp −

r2 2u21

)

,

(42)

where rs is the effective radius of the symmetry cell along the axis. The electron density of a single string averaged over the thermal vibrations of the crystal atoms can be found using the Doyle–Turner approximation for the X-ray scattering factor [36]

ρs (r) =

4 1 ∑

π ds

i=1

ai Bi + u2⊥

( exp −

r2 Bi + u2⊥

) +

c

1

ds π u2⊥

( exp −

r2 u2⊥

)

,

(43)

where Bi = bi /4π 2 , ai , bi and c are the Doyle–Turner approximation coefficients. The electron density of the symmetry cell ρa (x, y) is found by the summation of the closest string densities. 3. Experimental setup for studies of beam deflection by short bent crystals 3.1. Short bent crystals and goniometer Two methods were used to produce short bent silicon crystals with length of few millimeters and smaller. The first method developed in IHEP [12] was realized for so called strip (ST) crystals. The typical strip dimensions used in the experiments with the extracted beams of the CERN SPS are 70 × 2 × 0.5 mm3 (length × width × thickness). In most cases the large strip faces were parallel to the (110) crystallographic planes with their side faces parallel to the (111) planes. The crystal was placed in a vertical position in such a way that the beam entered the crystal through its side face nearly aligned with the beam direction (see Fig. 8). The crystal was mechanically bent (primary bending) along its length by a holder. The anticlastic curvature induced along the strip width due to primary bending was used for the beam deflection in the horizontal plane. Production of silicon crystals through anisotropic chemical etching developed by Ferrara group [39] allowed obtaining the silicon strips with ultra-flat surface and with negligible change of the crystal quality. Fig. 9 shows schematically the bending device used for the strip crystals.

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Fig. 9. Schematic drawing of the bending device for a strip crystal.

Fig. 10. Schematic picture of a silicon crystal with quasimosaic bending of the (111) silicon planes normal to the crystal plate induced by the primary bending with radius ρ . Radius ρ ′ corresponds to the anticlastic curvature.

The second method uses the quasimosaicity effect, which can produce bending of the crystal planes normal to the large face of the crystal plate when it is bent. In the case of silicon crystals, the quasimosaic bending realizes for the (111) crystallographic planes normal to the crystal plate (see Fig. 10) as it was shown in [40]. Fig. 11 shows schematically the quasimosaic (QM) bent crystal with its bending device. The bending device with a bent crystal is installed on the goniometer. The goniometer consists of four high precision motion units, two linear and two angular. With the linear motions, the crystals were positioned with respect to the beam center with an accuracy of several micrometers within a range of about 10 cm. One of the angular stages (rotational) produces the crystal rotation around the vertical axis to align the crystal plane with the beam axis with an accuracy of 2 µrad. Another angular stage (cradle) produces the rotation around the horizontal axis to align the crystallographic axis with the beam. To increase its mechanical stability, the goniometer was placed on a precisely machined granite table. The goniometer is remotely controlled via a PC and position information is continuously acquired by the data acquisition system of the experiment. The laser system has been produced, which allows performing a preliminary alignment of a crystal relative to the beam direction within a range of 100 µrad. 3.2. Tracking systems High-precision tracking of individual particle trajectories before and after the interaction with crystals was performed by means of silicon microstrip detectors (only for the first experiment [41] a position-sensitive gas chamber was also used to find fast channeling using the beam with higher intensity up to 107 per pulse). Two tracking systems were used in the experiments. The first tracking system (TS1) consisted of four double-sided silicon microstrip detectors 41 × 72 × 0.3 mm3 (see Fig. 12). They have 110 µm and 208 µm readout pitch with 8.5 µm and 30 µm special resolution for the p- and n-sides,

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Fig. 11. Schematic drawing of the bending device for a quasimosaic crystal.

Fig. 12. Tracking system TS1 of the UA9 experiments at the extracted beams of the CERN SPS. Si1–Si4 are the double-sided silicon microstrip detectors, g is the goniometer.

Fig. 13. Tracking system TS2 of the UA9 experiments at the extracted beams of the CERN SPS. S1–S5 planes consist of two sensors measuring two orthogonal coordinates. The S4 plane is rotated by 45◦ . G is the goniometer and T is the scintillation trigger.

respectively [42]. The finer p-side strips were used to measure coordinates along the horizontal direction of the beam deflection. The incident angle of a particle is reconstructed with two first planes Si1 and Si2. Then the hit projection onto the crystal was calculated. The outgoing angle of the particle behind the crystal was reconstructed with this hit projection and either of the Si3 or Si4 detector. The deflection angle acquired by the particle in the crystal was calculated from the difference between the outgoing and incident angles. The estimated resolution of the deflection angles was about 3 µrad. The operation of the tracking system was possible with up to 3 kHz trigger rate. The second tracking system (TS2) used in the UA9 experiments with the extracted beams of the CERN SPS consisted of ten planes of silicon microstrip sensors, arranged as five pairs each measuring two orthogonal coordinates (see Fig. 13) [43]. The sensors are single-sided silicon strip detectors. They are 320 µm thick and consist of 639 strips with a 60 µm readout pitch and 7 µm special resolution. The active area of the XY plane, where the two sensors overlap, is 38 × 38 mm2 . The upstream section of the telescope for the measurement of incoming tracks is formed by planes S1 and S2 while outgoing tracks are measured using planes S3, S4 and S5. The upstream and downstream sections both have lever arms of about 10 m. Plane S4 is a rotated XY plane (by 45◦ ), used in order to resolve ambiguities in reconstruction from multiple outgoing tracks. Events were triggered on the coincidence of signals from a pair of plastic scintillators placed downstream of the telescope. The angular resolution for the incoming angles has been estimated by simulation to be about 2.8 µrad. The measured resolutions of the deflection angles were 5.2 µrad in both horizontal and vertical directions. The data acquisition rate was up to 7 kHz. 4. Beam deflection by a bent crystal. Planar orientations 4.1. Positive particles 4.1.1. Observation of volume reflection of 400 GeV/c protons in bent silicon crystals The experiment on the observation of volume reflection [41,44] was carried out with a 400 GeV/c proton beam at the external H8 beam line of the CERN SPS. The primary beam intensity has been reduced to about 104 particles per second to allow single particle tracking. The tracking system of this experiment consists of two independent silicon microstrip

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Fig. 14. Beam intensity recorded by the silicon micro-strip detectors as a function of the horizontal deflection angle (x-axis) and the crystal orientation (y-axis). Six regions can be distinguished: (1) and (6) amorphous mode; (2) channeling; (3) dechanneling, (4) volume reflection; (5) volume capture.

setups one of which is described above. A fast parallel-plate gas chamber which can withstand a high beam intensity up to 107 per second was used for a preliminary alignment of the crystals to find channeling orientations. For this purpose, the beam was switched to the high intensity mode and then returned to the low intensity one when the channeling orientations were found. The beam line was adjusted to provide a divergence of (8 ± 1) µrad, smaller than the critical channeling angle. The beam spot size was about 1 mm which is close to the strip crystal thickness. Few strip and quasimosaic crystals with length up to 3 mm were used. After the crystal preliminary alignment with the gas chamber the angular scans were performed with every crystal to measure the deflections of particles at different crystal orientations using the silicon microstrip detectors. Fig. 14 shows the beam intensity recorded by the detectors as a function of the horizontal deflection angle and the crystal orientation for the strip crystal ST4. The crystal was bent at an angle of 162 µrad along the (110) crystallographic planes. The crystal length in the beam direction is 3 mm. Six regions can be distinguished for different crystal orientations. In regions (1) and (6) there is no the beam deflection. The beam interacts with the crystal as with amorphous substance. In region (2) the deflection angles are around the crystal bend angle. This region is formed by particles which passed the whole crystal length in the channeling regime. The channeled fraction value is about 56% from the whole beam incident onto the crystal. Region (3) is due to particles which were lost from the bent channels — dechanneled due to multiple scattering by the crystal nuclei and electrons. Region (4) is formed due to volume reflection (VR) of particles. The deflection occurs to the side opposite to the crystal bend and the deflection maximum is about 13.9 µrad. The deflection efficiency due to volume reflection is larger than 95% in this case. The extension of region (4) is comparable with the crystal bend angle which is much wider than for channeling. Region (5) corresponds to particles which are captured into the channeling regime due to multiple scattering by the crystal nuclei and electrons at an intermediate point inside the crystal volume (volume capture), hence undergoing a lower deflection. Thus, volume reflection exhibits better performance compared to channeling in terms of efficiency and angular acceptance. However, the deflection angle due to volume reflection is small, its value depends on the particle momentum, θvr < 1.5θc . 4.1.2. Volume reflection dependence on the bent crystal curvature The volume reflection parameters – the mean deflection angle θv r , its RMS deviation σv r and the efficiency Pv r – as a function of the bend radius R of a (110) silicon strip crystal were studied in the experiment [45]. Results are compared with Monte Carlo (MC) simulation and analytical calculations. It was shown already in [17] that in a bent crystal the distribution of the particle deflection angles due to VR is broadened if the centrifugal force Fc acting on the particles increases, which means that for particles of a given energy E a smaller crystal radius leads to a smaller θv r and a larger σv r . Such a change of the VR parameters becomes clear when the effective potential is considered for the different crystal radii (Fig. 15). VR occurs near the turning points of quasi-channeled particles in the effective potential of a bent crystal, where the particles radial velocity changes its sign. For a large crystal radius, that is R ≫ Rc , where the critical radius for 400 Gev/c protons Rc = 0.68 m, the turning points of all particles are in a narrow region near the inner wall of a planar channel (Fig. 15a), where the electric field

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W. Scandale and A.M. Taratin / Physics Reports 815 (2019) 1–107

Fig. 15. The effective potentials for the (110) planes of a silicon crystal bent with a radius R = 35.71 m (a) and 3.76 m (b) for 400 GeV/c protons. The coordinate x is measured in the direction opposite to the radial one; dp = 1.92 Å is the planar channel width and it is the distance between the two walls of each potential well. The value ∆Uef = (pv/R)dp is the potential inclination along the channel. The trajectory of a particle being reflected with a turning point in the fourth channel is shown. The position of the corresponding reflecting plane is shown by a vertical dashed line at x = 4dp . The thicker lines show the turning point region of the potential inner wall where particles can be reflected.

of the crystal plane is close to its maximal value. Therefore, a larger part of these particles get deflections close to the maximal one. The turning region of the effective potential increases for smaller crystal radii (Fig. 15b). Therefore, the spread of the particles deflections increases and their average value decreases. As experimentally observed [41,44], most of the incident particles are deflected due to VR to a direction opposite to the crystal bend. However, a small part of them is captured into the channeling regime along the tangency area (volume capture, VC), once their transverse momentum has been reduced by multiple scattering with the crystal nuclei and electrons. Volume capture is a competing process and it limits the VR efficiency, that is Pv r (R) = 1 − Pv c (R). For the bend radii R ≫ Rc , the estimation for VC probability has been suggested, Pv c ∼ Rθc /Le [32]. A 70 × 2 × 0.5 mm3 silicon crystal strip obtained from a wafer by means of anisotropic chemical etching [39] was used. Its largest faces are parallel to the (110) crystallographic planes and it was bent along its height (see Fig. 8). The

W. Scandale and A.M. Taratin / Physics Reports 815 (2019) 1–107

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Fig. 16. The deflection angle distribution of the 400 GeV/c proton beam due to VR in the (110) silicon crystal bent with R = 35.71 m (a) and 3.76 m (b). The parameters of the Gaussian fit are the following: the mean deflection angle θv r = 14.08 µrad (a) and 8.68 µrad (b); the RMS deviation σvr = 5.70 µrad (a) and 7.28 µrad (b). The hatched histogram area with angles smaller than θvr + 3σvr defines the VR efficiency Pvr .

anticlastic curvature acquired along the crystal width was used for the beam deflection as it was first suggested in [12] and was changed by changing the principal bending radius of the crystal itself. The crystal was positioned on a high precision goniometer that allowed to align the crystal with respect to the beam. When the crystallographic planes are far from the beam axis direction, particles experience the usual multiple Coulomb scattering in the crystal as in amorphous substance. In this case, the deflection angle distribution has a Gaussian shape with RMS deviation σam which has been measured to be (5.59 ± 0.11) µrad. The crystal has been aligned at the maximal channeling position and, in the assumption of a uniform bending, the crystal bend radius R has been obtained from the channeling deflection angle θch as R = L/θch where L is the crystal length along the beam direction (2 mm). The values for R obtained in this way are listed in Table 1. The smallest bending radius, R = 2.41 m, is about three times greater than Rc . The definitions of the VR parameters are illustrated in Fig. 16, where the angular distributions of protons for two bend radii (8.7 m and 36.62 m) are shown. A Gaussian fit to the deflection angle distributions gives the mean θv r and its RMS deviation σv r . The value of θv r + 3σv r determines the boundary between the volume-reflected portion of the beam and the non-reflected beam, which is due to the volume capture of protons, and allows to compute the VR inefficiency ε . Finally,

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W. Scandale and A.M. Taratin / Physics Reports 815 (2019) 1–107 Table 1 VR parameters for the different crystal bend radii R. R (m)

θvr (µrad)

σ¯ vr (µrad)

2.41 3.76 4.49 8.7 20.85 35.71

5.43 ± 0.10 ± 0.64 8.68 ± 0.14 ± 1.19 9.89 ± 0.09 ± 0.68 12.48 ± 0.08 ± 0.45 13.90 ± 0.16 ± 0.49 14.08 ± 0.11 ± 0.41

6.39 4.50 3.80 1.76 1.68 1.10

± ± ± ± ± ±

0.08 0.12 0.08 0.07 0.15 0.11

ε (%) ± ± ± ± ± ±

0.47 0.42 0.33 0.17 0.22 0.09

1.56 2.07 2.64 3.98 5.52 6.22

± ± ± ± ± ±

0.11 0.21 0.16 0.19 0.31 0.38

± ± ± ± ± ±

0.1 0.8 0.4 0.2 0.9 1.5

Fig. 17. The dependencies of the VR parameters on the crystal curvature and the bending radius R: (a) the deflection angle θv r (dots) and its RMS deviation due to the potential scattering σ¯ v r (squares), (b) the VR inefficiency ε . The solid and open symbols represent the experimental and the MC simulation data, respectively. The curves in (a) show the results of the analytical approach.

σ¯ vr =

√

2 describes the angular spread due to the potential scattering of protons in the crystal. The results are σv2r − σam

shown in Table 1. Fig. 17 shows the experimental dependencies of θv r and σ¯ v r on the crystal curvature and the dependence of ε on the crystal bend radius R. In the measured range the mean deflection angle θv r decreases approximately linearly as the curvature increases. This is similar to the dependence of the critical channeling angle in case of moderate crystal bending [27,46] (θc (R) ∼ 1 − Rc /R). On the contrary, the RMS deviation σ¯ v r increases approximately linearly within the

W. Scandale and A.M. Taratin / Physics Reports 815 (2019) 1–107

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Fig. 18. Angular space around the [111] axis of a silicon crystal. Only main crystal planes are shown. Simplified picture of a proton angle evolution due to MVR in the crystal bent along the vertical (110)v plane is also shown (by blue) in the comoving coordinate system. Volume reflections of the particle occur in the tangency areas with the planes and its momentum direction changes stepwise by the angle θv r in the direction normal to the crystal planes.

measured range of crystal curvatures. This increase is caused by the increase in the spread of the electric field strength near the turning points in the effective potential. The reflection inefficiency ε increases with R since the VC probability increases. The results of a simulation based on the model of planar channeling in a bent crystal described above and the results obtained by using the explicit expression for the VR deflection angle [34] are both shown in Fig. 17. The silicon atom potential obtained through the experimental X-ray scattering factors [47] was used in both cases. Taking into account the observed dependences of the VR parameters on R, an optimal bending radius for a short silicon crystal for beam deflection due to VR is found to be about 10Rc . This value maximizes the product of the deflection angle times the reflection efficiency times the angular acceptance. The increase of R gives larger θv r but decreases the efficiency and the angular acceptance. Therefore, to obtain the same acceptance, the crystal length should be increased but this generates more particle losses in inelastic interactions with the crystal nuclei. 4.1.3. Observation of multiple volume reflection by different planes in one bent silicon crystal for high-energy protons Only volume reflection of particles by bent crystallographic planes normal to the crystal bend plane was investigated in [18,19,41,44,45]. However, VR of particles by the skew planes having acute angles relative to the bend plane was also registered in our experiments with 400 GeV/c protons. It was predicted in [48] that particles entering a bent crystal with a small angle with respect to a crystal axis can be deflected due to the set of volume reflections by different planes crossing the axis. This effect of multiple volume reflections in one crystal (MVR OC) should increase the deflection angles of particles by several times. The experiment [49] gave the first observation of multiple volume reflection for 400 GeV/c protons in a bent silicon crystal when its [111] axis is oriented at a small angle with the beam. The possibility of subsequent reflections of particles by bent planes inside one crystal can be understood by considering the angular space near the bent crystal axis. Fig. 18 shows the main planes, (110) and (112), passing through the [111] axis of a silicon crystal. It shows also in a simplified way in the comoving system the angle evolution of the particle performing MVR. In the comoving coordinate system, which rotates with the particle velocity along the arc of the radius R around the crystal curvature center, the crystal plane orientation is unchanged. The crystal used in the experiment was bent along the vertical (110)v plane, that is the crystal bend plane is coincident with the horizontal (112)h plane. The crystal bend angle α determines the angular interval for changing the particle momentum direction relative to the crystal planes during the particle passage through the crystal. The angular distances between the crystal planes reduce quickly when we approach the axis, in our case along the vertical (110)v plane. Therefore, the angular interval α determined by the crystal bend can cover more and more crystal planes crossed the axis. So, when a particle crosses the crystal its momentum can be subsequently tangential to the crystal planes with different angles χ to the crystal bend plane. In the tangency area with all considered planes volume reflection of the particle occurs. As a result the particle momentum direction changes stepwise by the angle θv r in the direction normal to the plane (see Fig. 18). So, the particle reflection from a skew crystal plane with the inclination angle χ gives it both the horizontal projection of the deflection θx = θvr sinχ and the vertical one θy = θvr cosχ . The horizontal projections of the VR deflections from all crystal planes are directed to the side opposite to the crystal bend. They are summed and the overall deflection is increased several times. In contrast the vertical projections of the VR deflections from the skew planes symmetrically disposed around the vertical plane have opposite signs and cancel each other. So, when the particle trajectory is symmetric in the angular space of the comoving system it does not receive any vertical angular deflection.

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It should be noted that the bend radii of the skew planes are larger than the radius R for the (110)v plane, Rsk (χ ) = R/sinχ . For the (110)sk planes Rsk (30◦ ) = 2R. The VR deflection angle θv r is increased with increasing the crystal bend radius according to [45]. Therefore, for the (110)sk planes it should be larger than for the (110)v plane. However, for the considered bend radius R = 11.43 m the increase is not large, ∆θv r ≈ 5%. The (112) planes are more weak than the (110) planes. The VR deflection angle by the (112) planes should be θv r (112) ≈ 0.75 θv r (110). The particle momentum direction at the crystal entrance shown by the point (θxo , θyo ) in Fig. 18 should satisfy some conditions, which are optimal to observe the MVR effect [48]. The tangency point of the incident particle momentum with the vertical (110)v bent plane should be in the middle of the crystal, that is ∗ = 0.5 α , θxo

(44)

to realize the contributions from the skew planes both on the right and on the left from the vertical crystal plane. At a given angle θxo the vertical angle of the particle momentum θyo determines the number of the skew planes participating in the MVR. Volume reflection occurs from all planes with the inclination angles χ to the crystal bend plane

χ > χo = arctg(θyo /θxo ) ,

(45)

when the condition (44) is satisfied. So, the condition ∗ ∗ θyo = 0.5 θxo ,

χo = 26.6◦

(46)

realizes the contributions of the main (110)sk skew planes, for which χ = 30 and 150 , and all others with χo < χ < 180◦ − χo . The experimental setup I described above was used. A 70 × 4 × 0.5 mm3 silicon strip (height × width × thickness) with the largest faces parallel to the (110) planes and with the side faces parallel to the (111) planes was placed in a vertical position. The beam entered the crystal through its side face. The measured RMS deviation values for the horizontal and vertical divergence of the beam were σx = (11 ± 0.06) µrad and σy = (9.13 ± 0.03) µrad, respectively. The scan of the horizontal orientation angles of the crystal θh was first performed. It allowed finding the angular p position of the vertical (110) planes θh , which is characterized by the maximal deflection efficiency of protons in channeling regime. The deflection angle of channeled protons was measured to be about 350 µrad. The deflection angle is determined by the crystal bend angle α , therefore α = 350 µrad. This gives for the average value of the anticlastic bend radius R = w/α = 11.43 m. Then the scan of the vertical orientation angles of the crystal θv allowed finding the angular position of the [111] axis θva . For the MVR observation the crystal orientation was selected to realize the incident beam direction relative to the ∗ ∗ ∗ [111] crystal axis with θxo = 170 µrad ≈ 0.5 α and θyo = 0.5 θxo = 85 µrad according to (44) and (46). Fig. 19a shows the intensity distribution of the proton beam passed through the crystal in the horizontal and vertical deflection angles. Let us note that the deflection due to volume reflections was written as a positive one for the experimental data in contrast with Fig. 18. The horizontal projection of the distribution is shown in Fig. 19b. Gaussian fit gives the value θm = (66.53 ± 0.27) µrad for the maximum position. This angle is about five times larger than the VR deflection angle of protons by the vertical (110)v plane, whose value θv r = (13.35 ± 0.17) µrad was registered for the beam incidence vr with the angle θyo ≈ 1.3 mrad. So, the additional deflection of protons due to volume reflections by the skew planes is a value of about 4θv r . The (110)sk and (112)sk planes shown in Fig. 18 give approximately a half of this value and the other skew planes, which are more weak and not shown, give a second half. The deflection efficiency to the side opposite to the crystal bend is Pd (θx > 0) = (83.86 ± 0.26)%. The distribution tail to the bend side is caused by the volume capture of protons into the channeling regime when they pass the tangency areas with different crystal planes. A small shift of the maximum position in the vertical direction (see Fig. 19a) can be explained by an incomplete compensation of the contributions from the symmetric skew planes. Fig. 20 shows the dependence of the MVR deflection angle value θmv r , the distribution maximum position, on the vertical angle of the incident beam θyo relative to the axis direction. The angle θyo changes the number of the skew planes, ∗ ∗ at which VR of protons occurs. The deflection angle has a maximal value at the angle θyo = θyo = 85 µrad. For θyo < θyo , reducing the angles of protons with the [111] atomic strings forming the planes destroys the validity condition of the planar potential approach and consequently the stability of the particle trajectories along the planes. This reduces the VR deflection angles. So, this validity condition should be added to the optimum conditions for MVR (44) and (46) ◦

√ θyo ≫ ψ1 ,

ψ1 =

4Z1 Z2 e2 pv d

,

◦

(47)

where ψ1 is the critical angle for axial channeling along the [111] axis, Z1 and Z2 are the atomic numbers of the incident particle and the crystal atom, d is the interatomic spacing in the string. For 400 GeV/c protons ψ1 = 20.7 µrad. So, for ∗ ∗ the considered crystal orientation the condition (47) is satisfied, θyo ≈ 4 ψ1 . With increasing θyo in the range θyo > θyo the angular interval of the crystal bend α includes less and less number of the skew planes. So, at large orientation angles with the axis, for our case at θyo > 1 mrad, we approach the case of a single VR from the vertical plane when the deflection angle of protons θd = θv r .

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Fig. 19. (a) Intensity distribution of the proton beam in the horizontal and vertical deflection angles acquired due to multiple volume reflections from different planes in one bent silicon crystal around the [111] axis. Maximum position is shown by dot-dashed line. (b) The horizontal projection of the distribution. The distribution part with θx > 0 (about 84%) is hatched. Gaussian fit gives for the maximum position θmv r ≈ 66.5 µrad.

Fig. 21 shows the MVR deflection angle distribution of protons for a parallel incident beam with the crystal parameters and its orientation as in the experiment, which was obtained by simulation (1). The model of atomic string lattice described above with the atomic potential and electron density obtained in the Doyle–Turner approximation for the atomic scattering factors was used for simulation. The distribution looks like the experimental one (Fig. 19b). Gaussian fit gives the value θm = (69.77 ± 0.1) µrad for the distribution maximum. The calculated efficiency of the deflection to the side opposite to the bend Pd (θx > 0) = (86.65 ± 0.11) %. For comparison, the calculated distribution of deflection angles for the single VR by the vertical (110)v plane is also presented here (2); its maximum value θv r = (14.06 ± 0.03) µrad. The experiment [49] showed that the additional contribution of particle reflections from the skew planes allows increasing the deflection angle about five times. The efficiency of one side deflection due to MVR OC effect is higher than 80%. The MVR process in one bent crystal enriches possibility of crystal deflectors for steering the beams of high-energy charged particles, although it requires using a two axis goniometer. 4.1.4. Multiple volume reflection of high-energy protons by a sequence of bent silicon crystals The deflection of high-energy particles due to volume reflection can be increased by means of multiple volume reflections (MVR) realized in a sequence of bent crystals. The MVR deflection angle should be proportional to the reflection numbers. However, the MVR inefficiency ε should accumulate similarly. Really, the beam deflection efficiency due to multiple volume reflections in N crystals can be estimated in the first order approximation of independent events as Pmr (N) = PvNr = (1 − ε )N ≈ 1 − N ε .

(48)

Fig. 22 shows schematic picture of the sequence of two bent crystals with a particle (1) having volume reflections in both crystals near the tangency points with bent planes. The tangency point in the second crystal becomes closer to the crystal entrance face.

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Fig. 20. Dependence of the MVR deflection angle θmv r on the vertical angle θyo of the incident beam relative to the [111] axis. Dot-dashed line shows the deflection angle for a single VR from the (110)v plane, θv r ≈ 13.35 µrad.

Fig. 21. MVR deflection angle distribution of protons (1) received by simulation. Histogram 2 shows the distribution for a single reflection by the (110)v plane.

Sequence of separate crystals. Few experiments [50,51] were performed to study the MVR process with different numbers of crystals. Double volume reflection of 400 GeV/c protons by a sequence of two bent QM crystals was first observed [50]. Then the experiments with larger number of the sequence crystals were performed. Fig. 23 shows the beam intensity of 400 GeV/c protons passed through the aligned sequence of five bent QM crystals as a function of the horizontal deflection angle in different angular positions of a goniometer [51]. The goniometer changes the orientation of the whole crystal sequence. Besides, each crystal was supplied with a piezo-element for the mutual alignment of the sequence crystals. The crystal bend angles were about 110 µrad. The channeled particles form a maximum around this angle, which is clearly seen in Fig. 23. Volume reflection of protons with deflections opposite to the crystal bend occurs in all crystals of the aligned sequence in some range of the goniometer orientations. The MVR deflection value is about 53 µrad which is about 5 times larger than for a single reflection. The MVR deflection efficiency is about 90%. So, the deflection inefficiency is 10% which is also about 5 times larger than for a single reflection. The process of the mutual alignment of separate bent crystals is slow and proportional to the crystal number in the sequence.

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Fig. 22. Schematic picture of the sequence of two bent crystals with the particle trajectories. 1 — a particle having volume reflections in both crystals near the tangency points with bent planes. 2 — a particle volume captured in the first crystal and then volume reflected in the second one (for a simplicity the trajectory part after VC is shown by red as a prolongation of the trajectory before VC). θo is the crystal orientation angle, θx is the deflection angle of the particle 1, θv r is the VR deflection angle.

Fig. 23. The beam intensity of 400 GeV/c protons passed through the aligned sequence of five bent QM crystals as a function of the horizontal deflection angle in different angular positions of a goniometer.

Sequence of strip crystals with common bases. MVR assisted by volume capture. The sequence of strip crystals can have the common bases when it is produced from one plate — multi-strip (MST). By this reason the sequence strips for the initial unbent state have the same orientation. In the experiment [52], the sequence of crystals was realized from a 72 × 26.33 × 0.3 mm3 silicon plate with the largest faces parallel to the (110) planes fabricated according to the technologies [39]. The 71 × 0.97 mm2 windows, which were cut in the plate through the same interval, formed the sequence of fourteen strips with the width W=0.98 mm connected by the common bases. The multi-strip crystal was placed in a vertical position in a specially designed holder [53] (see Fig. 24). The beam entered the crystal through its side face. The mechanical bending along the crystal height produced the anticlastic curvature for each strip along the beam direction, which was used for deflection of protons in the horizontal plane due to one of possible mechanisms (channeling, single or multiple volume reflection). The bend angle measured with the interferometer had about the same value for all strips, α = (258 ÷ 276) µrad. The RMS deviation for the horizontal angular distribution of the incident beam was σx = (10.86 ± 0.01) µrad. The scan of the horizontal orientation angles of the crystal deflector θo was performed. Fig. 25 shows the intensity distribution

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Fig. 24. The photograph of the multi strip deflector. All fourteen strips are the parts of one silicon plate. A small anticlastic bend along the width of each strip is produced by a primary bend along their length.

Fig. 25. The intensity distribution of the 400 GeV/c proton beam crossed the fourteen strip silicon deflector in the deflection angles of particles θx at the different goniometer positions θo . The signs 1, 2 and 14 show the positions of the deflection maxima due to channeling of protons in the corresponding strips. The horizontal arrows with the length α show the corresponding VR acceptance areas of the strips. The vertical arrow shows the position with the subsequent reflections of protons in eleven strips.

of the beam passed the crystal sequence in the deflection angles of particles θx at the different angular positions of the goniometer θo . Only particles hitting the crystal with horizontal angles |θxo | < 10 µrad around the beam axis direction were selected. At the very beginning (right) and at the end (left) of the scan the mean deflection angle equals zero due to scattering of particles in the crystal deflector as in an amorphous substance. In the middle area of the scan the proton deflection maxima visible at θx > 0 occur due to channeling in one of the strips. On the other hand, the deflections with θx < 0 happen due to single or multiple volume reflections of protons in the sequence of strips. Let us note that only one deflection maximum due to channeling should be seen if all the strips have the same orientation and bend angle. Our crystal deflector has some spread of the strip orientations. Therefore, a few maxima due to channeling are seen in Fig. 25. The first maximum (marked by 1) on the right side of the scan at the crystal orientation angle θo = θo1 was formed due to channeling of protons in the first strip. The deflection maximum due to channeling of protons in the second strip at the orientation angle θo2 is marked by 2. The horizontal arrows on the left from θo1 and θo2 show the VR acceptance regions for the first and second strips. The deflection maxima due to channeling of protons in the next twelve strips are positioned at the goniometer positions θon on the left from θo2 with small dividing distances. The corresponding VR

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Fig. 26. The distribution of protons in the horizontal deflection angles θx for the case with the multiple volume reflection of particles in the eleven strips (for the goniometer position shown by the arrow in Fig. 25). The Gaussian fit determines the deflection angle θmr and its RMS deviation σmr . The angle θb = θmr + 3σmr . The hatched area with θx < θb defines the MVR efficiency.

acceptance regions for all these strips are overlapped. Therefore, starting from θo2 the number of subsequent reflections of protons in the sequence of strips continuously increases, which is visible through the gradual increase of the MVR deflection angles. On the left from the end of the VR acceptance of the second strip there is a small plateau of θx with eleven subsequent reflections of protons in the strips 3–13. The deflection maximum 14 due to channeling of protons in the fourteenth strip is positioned on the left far from the others. The orientations of the strips 1 and 14 were a little far from the orientations of the other strips because they are at the edges where some deformations in the support elements can arise. Fig. 26 shows the distribution of protons in the horizontal deflection angles θx for the goniometer position marked in Fig. 25 by the vertical arrow where the multiple volume reflection of particles occurs in the eleven strips. The Gaussian fit gives for the maximum and the RMS values θmr = (−110.65 ± 0.16) µrad and σmr = (19.69 ± 0.15) µrad, respectively. With the boundary angle for the maximum θb = θmr + 3σmr the beam deflection efficiency as a whole due to subsequent volume reflections of protons in the eleven strips Pmr (θx < θb ) = (88 ± 0.22)%. The efficiency of one side deflection Pmr (θx < 0) = (94.25 ± 0.14)%. The inefficiency of single reflection obtained in the simulation ε = 1.91%. Therefore, according to the first order approximation (1) the efficiency of eleven reflections should be about 79%. So, the MVR efficiency observed in the experiment is considerably larger than the estimate according to (48). The deflection of protons by the same multi-strip crystal for the conditions of the experiment was studied by simulation based on the model described above. Fig. 27 shows the distribution of the deflection angles of protons for the same orientation of the multi-strip deflector as in Fig. 26 when the subsequent reflections of protons occur in the eleven strips 3–13. The distribution looks like the experimental one. The maximum and the RMS values of the Gaussian fit are θmr = (−110.73 ± 0.17) µrad and σmr = (20.19 ± 0.14) µrad, respectively. The MVR deflection angle and its RMS spread are in good agreement with the experiment. The deflection efficiency Pmr (θx < θb ) = (91.2 ± 0.18)%, which is also close to the experimental one. The history of every particle can be registered in simulation. The hatched area shows the deflection angle distribution for particles, which had been volume captured at least in one of the strips. Most of these particles were fast dechanneled and had the tangent points with the bent planes in the next strips. So, they also participate in the process of multiple volume reflections in the strip sequence. Their contribution in the deflection efficiency is (11.74 ± 0.2)%. Fig. 22 helps to understand the situation. The beam part marked 1, which was only considered for the estimate (48), makes two subsequent reflections with the mean angle θv r passing two bent crystals and obtains the mean deflection angle 2θv r . A smaller beam part 2 is volume captured in the first crystal but fast dechanneled and has a tangent point with the bent planes in the second crystal. Here many of the particles 2 are reflected obtaining the mean deflection angle θvr . So, particles volume captured in one of the sequence crystals also participate in the multiple volume reflection but they obtain a smaller deflection. Experimental results [52] have shown a high efficient deflection of protons due to multiple reflections in the sequence of bent silicon strips. A large increase of the MVR efficiency in comparison with a first order estimate occurs due to the participation of particles volume captured in one of the sequence crystals in the multi reflection process.

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Fig. 27. The same as in Fig. 26 obtained by simulation. The hatched area shows the contribution of particles, which had been volume captured at least in one of the strips.

Surface multi-strip sequence. Observation of MVR OC in few subsequent strips. When a common bending device is used for the multi-strips the edge strips are not sufficiently well aligned to contribute to multiple volume reflections. In the experiment [54], a new multi-strip deflector bent due to internal stresses created by the grooves on the surface of a thick crystal plate was studied. A schematic diagram of the crystal deflector and its photograph are shown in Fig. 28. The deflector was produced from a 70 × 15 × 5 mm3 silicon plate. The large faces of the crystal plate were parallel to the (111) crystal planes, while the entry face was normal to the [110] axis. The depth of the triangular grooves was about 1.1 mm. The bending of 2 mm long separate strips, which are formed between the grooves, was produced by deformation of the surface layers due to the Twyman effect [55]. Because of the thick unbent base of the crystal deflector, mutual alignment of the surface strips, both angular and spatial, is significantly better than with the use of a bending device. The experiment was performed with 400 GeV/c protons at the H8 external beam line of the CERN SPS. The measured angular divergence in both horizontal and vertical planes of the incident beam was about 10 µrad. The scheme of the crystal alignment by the goniometer is shown in Fig. 29. In the first stage a scan of horizontal orientation angles ϕx of the crystal deflector was performed. Fig. 30a shows the beam intensity distribution behind the crystal as a function of the particle deflection angles θx at different horizontal angles ϕx of the goniometer. Only particles hitting the crystal near its surface in the range 0 < x < 200 µm were selected because the bend of the strip layers fast decreases with increasing distance from the surface. At the beginning (left) as well as at the end of the angular scan the mean deflection angle equals zero due to scattering of particles in the crystal deflector as in an amorphous substance. For deflector orientations near ϕx = 0 incident particles are deflected by angles of about 200 µrad due to channeling. There are no other maxima with the same deflection which means that all strips have about the same orientation. So, we may conclude that the method used for the MST production really provides a good mutual alignment of the strips. With increasing ϕx the condition for volume reflection initially appears in the first strip and then in the subsequent ones. For this reason, the deflection angle due to VR increases and reaches the maximum value of 5 θv r when VR occurs sequentially in all five strips. Fig. 31a shows the deflection angle distribution of protons for the goniometer position marked by the arrow in Fig. 30a where multiple volume reflection occurs in all five strips (histogram 1). The efficiency of one side MVR deflection with θx < 0 is about 90%. The MVR maximum position is about 62 µrad, which agrees well with the theoretical prediction for this five strip deflector θmr = 5 θv r = 5 × 13 = 65 µrad [34]. In the second stage the vertical orientation angle ϕy was scanned to find the angular area close to the axis direction. This scan was performed with the horizontal orientation angle fixed in the MVR position marked by the arrow in Fig. 30a. Fig. 30b shows the beam intensity distribution observed behind the crystal as a function of the deflection angles θx at different vertical angles ϕy of the goniometer. The intensity distribution is symmetric relative to ϕy = 0 where the [110] axis direction is parallel to the beam direction. The particle deflection angles increase when the crystal axis direction becomes close to the beam direction. Fig. 31b shows the deflection angle distribution for the crystal orientation marked by arrow 1 in Fig. 30b when the beam broadening is maximal. The RMS proton deflection is about 80 µrad, which is more than 3 times larger than for the amorphous orientation. The deflections in the VR direction increase because the MVR OC effect is realized in a few strips,

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Fig. 28. (a) Schematic representation of bent multi-strips produced by the periodic grooves on the thick crystal surface. (1) bent crystallographic planes, (2) rough surfaces of the grooves, (3) a particle deflected due to channeling, (4) a particle reflected by bent planes. (b) Photograph of the silicon crystal plate with the periodic grooves.

Fig. 29. Scheme of the multi-strip crystal installation relative to the beam. The entrance crystal face is normal to the [110] axis, whose direction is close to the beam direction. The (111) planes are parallel to the strip surface. They are bent due to the grooves. ϕx and ϕy are the horizontal and vertical angles of the crystal orientation to align the (111) planes and the [110] axis with the beam direction, respectively.

which considerably increases the effective reflection angle in all of these strips. Because of these MVR OC deflections in a few previous strips some particles enter the channeling acceptance area and are deflected by the bent planar channels. The resulting deflections of these protons towards the bend side are about α − ϕx .

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Fig. 30. The intensity distribution of the 400 GeV/c proton beam passed through the five-strip silicon deflector in the deflection angles of particles

θx : (a) at the different horizontal angles of the goniometer ϕx . The arrow shows the goniometer position for which volume reflection of protons is realized in all strips. (b) At the different vertical angles of the goniometer ϕy , with ϕx fixed in the position shown by the arrow in (a). The arrows 1–2 are at the goniometer positions for which different mechanisms of particle deflection occur.

Fig. 31c shows the deflection angle distribution for the crystal orientation when the [110] axis direction coincides with the beam direction (shown by arrow 2 in Fig. 30b). In this case, particles undergo multiple potential scattering by the [110] atomic strings in a few first strips (doughnut scattering [8,38,56], it will be described below). This gives them some vertical momentum, which is sufficient to experience the action of both the (111) vertical and skew planes. Only then the MVR OC effect can occur for these particles in a few subsequent strips. As a result the mean particle deflection is about 57 µrad. The experiment [54] showed that the bent strips produced by the periodic grooves on the crystal surface really have good mutual alignment, which allows to achieve volume reflection of incident particles in all strips. It was also shown that the effect of multiple volume reflections of particles in one crystal (MVR OC) can be realized in few subsequent strips. This gives very large broadening of the beam. 4.1.5. Possibilities to increase VR and MVR efficiency Optimal crystal sequence. In the surface multi-strip sequence considered above its strips have about the same orientations. They are approximately parallel. In this case we have parallel Sequence of Volume Reflectors (SVR) [35]. The location of the volume reflection area along the crystal length will be different in each crystal of a parallel sequence. It will become closer and closer to the entry face (see Fig. 32a). In this case, the maximum deflection angle, which a particle can achieve due to multiple reflections in the full crystal sequence, approaches the crystal bend angle. We can optimize the SVR [35] if the crystals will be inclined to each other by the angle of volume reflection θv r (see Fig. 32b). In this case, each crystal is aligned to the propagated beam direction, since the change of direction due to

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Fig. 31. The horizontal deflection angle distributions of 400 GeV/c protons. (a) Histogram 1 for the case shown by the arrow in Fig. 30a when particle volume reflections are realized in all five strips, histogram 2 in the case of multiple scattering for amorphous crystal orientation. (b) For the case shown by the arrow 1 in Fig. 30b when multiple volume reflections in one crystal are realized in the few first strips and deflection due to channeling in the subsequent one. (c) For the case shown by the arrow 2 in Fig. 30b when the [110] axis is aligned with the beam.

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Fig. 32. Schematic picture of the Sequence of two bent crystals where Volume Reflection of particles occurs (SVR). (a) A parallel sequence, the locations of the tangency points are different for the crystals of the sequence. (b) An optimal sequence with the inclination angle θv r between the crystals. Here the tangency points are near the middle for both crystals. The value of θv r is selected to be equal to the deflection angles θ1 , θ2 . That is θ1 = θ2 = θv r .

Fig. 33. The angular distribution of 7 TeV protons passed through a sequence of five silicon crystals uniformly bent along (110) planes with the radius of curvature 100 m, crystal length 1.2 mm and bend angle 12 µrad. The crystals are inclined to each other by the deflection angle in a single reflection θv r = 3 µrad. The hatched distribution part with angles θ < θv r + 3σv r (where θv r and σv r are the mean and RMS deviation of the Gaussian fit for the reflection peak) determines the efficiency of the beam deflection as a whole.

the volume reflection in a previous crystal is compensated for. In such an optimal sequence, the number N of crystals determines the maximum deflection angle due to reflections, θd = N θv r . In this case, the crystal bend angle can be smaller than for a parallel sequence and shorter crystals can be used. For 7 TeV protons in a (110) silicon crystal θc = 2.55 µrad and Rc = 11.89 m. The bend radius of a crystal, for an optimal volume reflection, is a little larger than 100 m. For such bend radii, it is impossible to use effectively a parallel SVR, which requires large bend angles and crystal lengths. In [35] the optimal SVR for 7 TeV protons was considered by simulation. The parameters of this optimal SVR are the following: the crystal bend radius R = 100 m, the bend angle 12 µrad and the crystal length L = 1.2 mm. The deflection due to VR in such a crystal θv r is about 3 µrad. This value determines the inclination angle between two neighboring crystals. The deflection efficiency for such sequence of five crystals is about 93% with the deflection angle 15 µrad (see Fig. 33). VR Efficiency increase in a bent crystal with increasing curvature. The VR efficiency is limited by the concurrent process of volume capture (VC) of particles into channeling states. The probability of particle capture on the top levels in the potential well of a bent channel is determined by the multiple scattering on the crystal nuclei and is of the order of 1% even for the LHC energies of protons. In the sequence of reflections, VC causes an appearance of the deflected beam tail

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Fig. 34. The effective potential of a silicon crystal bent along (110) planes with R = 175 m for 7 TeV protons. Here ∆Uef is the potential inclination along the channel width dp . The vertical arrow shows the transition of an above-barrier particle with the initial transverse energy Exo on the inner level of the channel — the process of volume capture. ∆v c denotes a maximum value of the capture depth.

stretching to the crystal bend side. It is clearly seen in Fig. 33. It was shown by simulation [57] that the VR efficiency can be increased if we will use the crystal with increasing curvature. Fig. 34 shows the effective potential of the silicon crystal bent with radius Ro along (110) planes. The crystal bend produces the potential inclination, whose value at the channel width dp equals ∆Uef = dp pv/Ro . The tangency point of the particle momentum direction with bent crystal planes corresponds to the turn point of particles in the effective potential where the transverse momentum direction changes to the reverse one. A particle crosses the crystal planes at an angle, which is decreased when approaching the turn point. This increases multiple scattering of a particle on the atomic nuclei. As a result, a particle can reduce its transverse momentum and performs a transition on the transverse energy level, which is lower than the height of the potential barrier separating two neighboring bent channels. The capture of a particle into channeling regime occurs therefore inside the crystal (volume capture). The vertical arrow in Fig. 34 shows such a transition. The simulation results show that a volume capture for 7 TeV protons can occur only at the entrance or exit from the channel where their transverse momentum changes sign. The wavelength λ of particle oscillations in the channel equals about 200 µm. The capture depth ∆v c (shown in Fig. 34) does not exceed 1 eV. If the curvature increases along the crystal length fast enough for the potential well depth to decrease by 1 eV per 200 µm, particles will not be captured into the channeling regime. Really, a particle lowering at the inner level of the channel near its outer (left) wall will not be kept because the potential barrier becomes lower than this level when the particle returns after reflection by the right channel wall. Thus, the rate of the potential depth decrease equal to a = ∆Uo /λ = 5 eV/mm is sufficient to avoid VC of 7 TeV protons. Let the inclination of the effective potential ∆Uef for protons increases linearly with the crystal depth z pv R(z)

dp =

pv Ro

dp + az ,

(49)

then for the crystal curvature χ we obtain

χ (z) = χo +

a pv dp

z.

(50)

Fig. 35 shows the dependence of the deflection efficiency for 7 TeV protons on the number of reflections for two different optimal SVRs. Here the efficiency of the beam deflection as a whole (like shown by hatching in Fig. 33) is considered. For the crystals with the increasing curvature at a = 5 eV/mm (curve 2) the deflection efficiency is noticeably higher than for the crystals with a constant curvature (1) and achieves 97% after five reflections. The deflection efficiency determined as a beam fraction deflected to the side of reflection from the initial beam direction is even greater, P(θ < 0) = 98.34%.

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Fig. 35. The dependence of the deflection efficiency of 7 TeV protons on the number of the sequential bent crystals with constant (1) and increasing (2) curvature (on the number of reflections). The change rate of the potential inclination for the curve 2 is a = 5 eV/mm.

4.1.6. Planar channeling of protons in short crystals Deflection efficiency maximum. Observation of nuclear dechanneling. High-energy charged particles entering the crystal with angles relative to the crystal planes smaller than the critical channeling angle θc = (2Uo /pv )1/2 , where p, v are the particle momentum and velocity and Uo the well depth of the crystal potential averaged along the planes, can be captured into the channeling regime [1]. Channeled particles move through a crystal oscillating between two neighboring planes. The averaged planar potential gives an approximate description of channeling, in which the transverse energy of particles is the motion integral. Collisions with atomic electrons and nuclei of the crystal change the transverse energies of particles and as a result they leave the channels (dechanneling). The average square of the angle of particle multiple scattering on the crystal electrons (MSE) and nuclei (MSN) is proportional to their local density [1,26]. The atomic nuclei density is quickly √ reduced with the distance x from the planes according to a Gaussian distribution Pn (x) ∼ exp(−x2 /u2⊥ ), where u⊥ = 2u1 and u1 is the amplitude of the thermal vibrations of the crystal atoms. The amplitude u1 determines the ‘‘nuclear corridor’’ width where particles undergo a strong MSN. It is much smaller than the channel width dp for the main crystal planes. For instance, for the (110) silicon channels at the room temperature one has 6u1 /dp = 0.23. In the central areas of the planar channels particles undergo only scattering on the crystal electrons. The average square of the MSE angle is considerably smaller than for MSN. The critical approach distance to the crystal planes rc (u1 ) is used to determine the boundary of the area of the stable channeling states with the particle oscillation amplitudes x¯ m ≤ x¯ mc = dp /2 − rc (the coordinate x¯ is measured from the channel center x¯ = x − dp /2). Particles leave the stable channeling states through multiple scattering on the crystal electrons. The process has the exponential character, Nch (z) ∼ exp(−z /Le ), where Le is the ‘‘electronic’’ dechanneling length due to MSE. Particles with the large oscillation amplitudes x¯ m > x¯ mc quickly leave the bound states through a strong MSN near the channel walls. This process can be also characterized by ‘‘nuclear dechanneling length’’ Ln ≪ Le [35]. So, the dechanneling process has two stages. In the first stage particles leave the stable states due to MSE and then they leave the unstable bound states with x¯ m > x¯ mc mainly due to MSN, therefore the total dechanneling length Ld = Le + Ln . In all previous measurements with high-energy charged particles [8,24,25,58,59] the crystals with length L ≫ Ln were used. The measured value Ld , which characterizes the channeled fraction reduction with the beam penetration depth into the crystal, gives the electronic dechanneling length because Ld = Le + Ln ≈ Le . For 400 GeV/c protons in straight (110) silicon crystal Le should be about 20 cm according to the extrapolation of the available data [24]. Bent crystals can deflect high-energy charged particles being in channeling states [2]. The crystal bend gives the angular unfolding of the dechanneling process because particles dechanneled at the crystal depth l are deflected by the angle θ = l/R, where R is the crystal bend radius. This was used to measure the electronic dechanneling length in [8,25]. The experimental data [25] have shown that a good approximation for the critical approach distance is rc = 2.5u1 . The crystal bend reduces the dechanneling length of particles mainly due to the decrease of the potential well depth but when R ≫ Rc , where Rc is the critical bend radius [2], the dechanneling length is about the same as in the straight crystal. The effective planar potential Uef (see Fig. 36), which governs the transverse particle motion, has a full depth of Uob (R) that depends on the crystal radius of curvature R. Particles with an initial transverse energy Exo not exceeding Uob (R) are captured into the bound states with the planar channels. In the first order approximation, the deflection efficiency value

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Fig. 36. The effective potentials for the (110) planes of a silicon crystal bent with the radius R = 38 m. The coordinate x is measured in the direction opposite to the radial one, dp = 1.92 Å is the channel width. Uob = 21.7 eV is the depth of the planar potential well, rc = 2.5u1 = 0.1875 Å is the critical approach distance. Exc = Uef (rc ) = 13.52 eV is the critical transverse energy for stable channeling states.

Pd of particles by a bent crystal is the product of the capture efficiency Pc into the bound states and the probability Pch to keep particles in the states during the whole crystal length Exc (R)

∫

Pd (R) = exp(−L/Le (R))

∫

f (Exo ) dExo + exp(−L/Ln (R))

0

Uob (R)

f (Exo ) dExo ,

(51)

Exc (R)

where f (Exo ) is the distribution of the initial transverse energy of particles at the crystal entrance, Exc (R) = Uef (rc , R) is the critical transverse energy for the stable channeling states. In long crystals, with length L ≫ Ln , only particles captured into the stable channeling states with the initial transverse energies Exo < Exc can be deflected. Probability to be in the stable channeling states is limited by dechanneling of particles due to MSE. Therefore, a good approximation for the deflection efficiency in this case is given by the first term of (51) Exc (R)

∫

f (Exo ) dExo .

Pd (R) = Pch · Pc (Exo < Exc ) = exp(−L/Le (R))

(52)

0

This also implies that the deflection efficiency is smaller than the capture efficiency of particles into the stable channeling states Pd < Pc (Exo < Exc ) .

(53)

The value of Pc is maximal for a parallel beam aligned with the crystal planes. Its upper limit is realized in a straight crystal and, in the case of (110) silicon channels, is Pc = 1 − 2rc /dp = 0.805 .

(54)

In the experiment [60], a 70 × 1.94 × 0.5 mm3 silicon strip crystal with the largest faces parallel to the (110) crystallographic planes was used to study the deflection of 400 GeV/c protons. Its length along the beam L = 1.94 mm is much smaller than Se . Therefore, the fast dechanneling stage due to MSN was detected and a record value of the deflection efficiency Pd was measured, which surpasses the limitation for long crystals shown in (54). The proton beam had the RMS values of the horizontal and vertical angular divergence of σx = (9.27 ± 0.06) µrad and σy = (5.24 ± 0.03) µrad, respectively. An angular scan was performed and the optimal orientation was selected, which gives the maximum of the deflected beam fraction. Fig. 37a and 37b show, in linear and semi-logarithmic scale respectively, the distribution of the particle deflection angles at the optimal crystal orientation for the incident beam fraction with horizontal and vertical angles in the range |θxo |, |θyo | < 5 µrad. A Gaussian fit of the right peak provides the mean value θd = (50.5 ± 0.1) µrad and the RMS deviation σd = (5.67 ± 0.04) µrad of the beam fraction deflected by channeling. In the assumption of a uniform bending, the anticlastic bend radius is R = L/θd = 38 m. The fraction of particles deflected by angles greater than θd − 3 σd (hatched area in Fig. 37a) determines the deflection efficiency Pd . For the considered case Pd = (75.2 ± 0.7stat ± 0.5syst )%.

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Fig. 37. The distribution of deflection angles for 400-GeV/c protons in the silicon crystal bent along (110) planes, the crystal length is 1.94 mm. Only particles hitting the crystal with the horizontal and vertical angles |θxo |, |θyo | < 5 µrad were selected. (a) The deflected fraction 76.6% is hatched. (b) Logarithmic scale along Y axis. The exponential fit, which gives the nuclear dechanneling length, is shown by the line between the two maxima.

The peak on the left side in Fig. 37a and 37b is due to particles, which were not captured into the channeling states at the crystal entrance. They were deflected in the opposite direction due to volume reflection [35]. Particles with deflection angles between the two maxima in Fig. 37a and 37b are the dechanneled ones, which were lost due to the MSN. Using the relation l = Rθ between the deflection angle θ and crystal length l traversed by a particle before the dechanneling event, the exponential fit of the area of dechanneling (see the line in Fig. 37b) gives the value of the nuclear dechanneling length Ln = (1.53 ± 0.35stat ± 0.20syst ) mm. The simulation results based on the model described above, in which the average square of multiple scattering angle on the crystal nuclei is proportional to the density of nuclei [26] θ¯n2 ∼ Pn (x), gives a close value Ln = 1.5 mm. The deflection efficiency as a function of the incident angle of particles was studied by selecting different angular fractions of the incident beam. The fractions of particles with horizontal incident directions inside contiguous angular windows each of 2 µrad width were selected. Fig. 38 shows the measured deflection efficiency values (blue squares interconnected by segments) for each beam fraction as a function of the window center position. The maximum value of the deflection efficiency corresponding to the optimal choice of the incoming particle directions is Pd = (83.4 ± 1.6stat ± 0.9syst )%. Such a value is much larger than the upper limit value for long crystals (54). The simulation results are shown in Fig. 38 as circles interconnected by segments. The agreement of simulation and experimental results is rather good in a wide range of incident angles, around the incoming beam axis. The selected angular window width of 2 µrad is much smaller than the critical channeling angle θcb = 10.4 µrad. For this reason, the observed deflection efficiency is close to its maximum value for a parallel beam. The experiment [60] showed that the deflection efficiency limit of higher than 80% for a nearly parallel beam predicted by theory in a single passage through a short crystal is really achievable. A fast stage of particle dechanneling due to multiple scattering on the atomic nuclei has been observed. Dependence of nuclear dechanneling length on incident angle of protons. The analysis of different beam fractions for the crystal orientation optimal for channeling from the experiment [61] allowed to observe the reduction of the nuclear dechanneling length with an increase of the orientation angle of the considered beam fraction relative to the planes at the crystal entrance [62]. A 70 × 1.94 × 0.5 mm3 silicon strip crystal with the largest faces parallel to the (110) crystallographic planes was used. The crystal was bent along the length L=1.94 mm with the bend angle α = 189 µrad. That is its bend radius R=10.26 m was much smaller than in the experiment [60]. The beam of 400 GeV/c protons had the RMS value of the horizontal angular divergence of σx = (9.34 ± 0.06) µrad. An angular scan was performed and the optimal orientation was found, which gives the maximum of the deflected beam fraction.

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Fig. 38. The deflection efficiency for a narrow beam fraction, which is inside an angular window of 2 µrad width, as a function of the window center position. The maximum value of the efficiency is (83.4 ± 1.6stat ± 0.9syst )%. Circles indicate the simulation results.

Fig. 39. The intensity distribution of 400 GeV/c protons passed through the bent silicon crystal when its orientation is optimal for channeling in the deflection angles θx as a function of incidence angle θin of particles relative to the (110) planes at the crystal entrance.

Fig. 39 shows the intensity distribution of protons passed through the bent silicon crystal when its orientation is optimal for channeling in the deflection angles θx as a function of incidence angle θin of particles relative to the (110) plane at the crystal entrance. The beam divergence value allows to observe the different interaction mechanisms with the crystal for the different beam fractions. The fraction 1 was not deflected and consists of the particles passed through the crystal as in the amorphous substance. The particles which passed the full length of the crystal in the channeling regime and deflected by the bend angle represent the fraction 2. The fraction 3 consists of the particles deflected to the side opposite to the crystal bend due to volume reflection. The dechanneled particles, which are found in the angular interval between the initial direction and that of the channeled fraction, represent the fraction 4. Let us consider the different beam fractions with the incident angles θin ± ∆θin , where ∆θin = 1.75 µrad. Fig. 40a shows the deflection angle distribution of protons for the optimal case when θin = 0. The peak on the right consists of the particles passed through the whole crystal in the channeling regime. The peak has a well visible central part because the considered fraction of the incident beam consists of particles with all transverse energies satisfying the channeling conditions including the smallest ones. The maximum of the peak is at an angle θch = 189 µrad and its position corresponds to the crystal bend angle θch = α . The part of particles deflected by the angles larger than θch − 3σch (the boundary is shown by the dot-dashed line), where σch is the RMS deviation of a Gaussian fit of the peak, determines

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Fig. 40. The distributions of deflection angles for 400 GeV/c protons in the silicon crystal bent along (110) planes for the beam fractions with the incident angles θin ± ∆θin , where ∆θin = 1.75 µrad. (a) For θin = 0, (b) for θin = 8.75 µrad. The channeling peak is on the right. The distribution of dechanneled particles is located between two dot-dashed lines. The solid line shows the exponential fit, which determines the dechanneling length.

the deflection (channeling) efficiency, it is Pd =77%. The peak on the left is formed by the particles, which were not captured into the channeling regime. The peak is shifted to the side opposite to the crystal bend by the angle of θv r due to volume reflection. The peak boundary at θv r + 3σv r , where σv r is the RMS deviation of a Gaussian fit of this peak, is shown by the dot-dashed line. The particles with deflection angles within the angular interval indicated in Fig. 40 by two dot-dashed lines are the dechanneled ones Ndc . The particle deflection angle is determined by the distance S passed in the channeling regime, θxs = S /R. The total number of particles in the channeling peak and the dechanneling region represents those particles captured into the channeling regime at the crystal entrance Nch (0). The dechanneling probability is defined as the ratio Pdc = Ndc /Nch (0). For the case under consideration, Pdc = 7.2%. The solid line shows an exponential fit of the central dechanneling region, which gives the dechanneling length value, Sn = (1.38 ± 0.24) mm. Fig. 40b shows the distribution of deflection angles for the beam fraction with an incident angle θin = 8.75 µrad, which is close to the critical channeling angle, θcb (R) = 9.8 µrad, for the considered bend radius R. In this case, the channeling peak is additionally shifted because of the angle θin relative to the planes at the crystal entrance. The upper part of the peak becomes more flat because the number of particles with small oscillation amplitudes decreases (the simulations actually show the two-headed peak, which is also just visible here). The dechanneling probability becomes significantly higher, Pdc = 23.5% and the dechanneling length becomes smaller, Sn = (0.81 ± 0.09) mm. Fig. 41 shows the experimental dependence of the dechanneling probability of 400 GeV/c protons on the incident angle of the beam fraction θin . The dechanneling probability increases monotonously with increasing θin . The dependence becomes close to linear for large orientation angles. The dependence obtained by simulation is shown by the dotdashed line here. There is good agreement with the experiment; the discrepancy is not larger than 10%. Fig. 42 shows

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Fig. 41. The experimental dependence of the dechanneling probability of 400 GeV/c protons in the bent silicon crystal on the incident angle of the beam fraction θin (filled circles). The simulation results are shown by the empty circles and dot-dashed line.

Fig. 42. The dependence of the dechanneling length of 400 GeV/c protons in the bent silicon crystal on the incident angle of the beam fraction θin .

the dependence of the dechanneling length of protons on the incident angle θin . The dechanneling length decreases approximately by a factor of two in the angular range considered. Fig. 43 shows the calculated particle distributions in the transverse energy Ex at the crystal entrance (1) and exit (2) for the same angles of the beam fraction orientation as in Fig. 40. Two dot-dashed lines show the range of transverse energies at which channeled particles can enter the nuclear corridor of the channels. The initial distribution of particles in this range is approximately uniform for θin = 0 (Fig. 43a). In the case of θin = 8.75 µrad the initial distribution of particles in the nuclear dechanneling range of Ex is strongly non-uniform (Fig. 43b). The particle density with the transverse energies close to the potential well depth value is maximal. The distributions of particle transverse energies at the crystal exit clearly show that dechanneling occurs only from the nuclear corridor range. Therefore, when the beam fraction enters the crystal at an angle relative to the planes close to the critical one, the population of the upper part of the nuclear dechanneling range is maximal and those particles determine the dechanneling length value observed in the experiment.

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Fig. 43. The calculated particle distributions in the transverse energy Ex at the crystal entrance (1) and exit (2) for the different beam fractions with the incident angles θin ± ∆θin , where ∆θin = 1.75 µrad. (a) For θin = 0, (b) for θin = 8.75 µrad.

The analysis in [62] showed that the dechanneling probability increases with an increase of the incident angle of particles relative to the planes which can be explained by the increase of the particle population in the whole range of nuclear dechanneling. Moreover, the observation of the dechanneling length reduction provides evidence of an increase of the particle population at the top part of this nuclear dechanneling range. 4.1.7. Reduction of inelastic nuclear interactions of protons at planar channeling in crystals Reduction of the processes requiring close collisions of fast charged particles with atoms in the aligned crystals was observed in the first experiments on channeling (see review [63]). A few experiments on the study of nuclear interactions of high energy protons and heavy nuclei in the aligned crystals have been performed at the external accelerator beams [64–67]. Nuclear interactions of high-energy protons in germanium crystals aligned by the [110] axis with the beam had been studied in [64,65]. A considerable decrease of the nuclear interaction probability was registered in the aligned crystals. Recently, the reduced interaction probability has been observed for 33 TeV/c Pb nuclei in a bent silicon crystal aligned by the (110) planes with the beam [66]. A strong suppression by a factor of more than 20 has been observed in [67] for nuclear-charge changing interactions of 18 TeV/c In ions channeled through a bent Si crystal. The atomic density changes along the particle trajectory in the aligned crystal. Thus the nuclear interaction probability changes as well. The atomic density N(x) is quickly reduced with the distance x from the planes according to a Gaussian distribution N(x) = Nam · Pn (x) ,

Pn (x) = √

dp 2π u21

( exp

−

x2 2u21

)

,

(55)

there u1 is the amplitude of thermal vibrations of the crystal atoms, for a silicon crystal at a room temperature u1 = 0.075 Å, dp is the planar channel width. The density at the plane positions is ten times larger than the average density

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Fig. 44. The dependence of the atomic density averaged along the particle trajectory in the (110) silicon crystal on the particle transverse energy.

Fig. 45. Experimental layout at the external beam H8 of the CERN SPS. Here Si1–Si4 are the silicon microstrip detectors, g is the goniometer with a bent crystal. S1 and S2 are the scintillation detectors.

Nam . The ‘‘nuclear corridor’’ width, where the atomic nuclei of the plane are concentrated, is much smaller than the channel width, for the (110) Si dp = 1.92 Å and 6u1 /dp = 0.23. Fig. 44 shows the dependence of the atomic density averaged along the particle trajectory in the (110) silicon crystal on the particle transverse energy Ex . The maximum density at Ex = Uo is three times larger than the average one. For stable channeled states with small transverse energies the atomic density along the trajectories is very small or equals zero. For above-barrier particles with Ex > Uo the atomic density averaged along the trajectory approaches the average one with increasing Ex . For volume reflection of particles in a bent crystal the atomic density averaged along the particle trajectory in a tangency area is significantly larger than Nam . However, the density approaches the average one when the distance of particles from the tangency point is increased. In the experiment [61], the probability of inelastic nuclear interactions in the silicon crystal with the length 1.94 mm bent along the (110) planes has been studied with 400 GeV/c protons at the H8 beam line of the CERN SPS. The probabilities for the crystal orientations optimal for channeling and volume reflection were measured. The probability of inelastic nuclear interactions of protons in a short crystal at its amorphous orientation is Pin ≈ σin Nam L .

(56)

The cross-section of inelastic nuclear interactions of 400 GeV/c protons with silicon nuclei in the Glauber approach calculated according to [68] σin = 0.506 b, the atomic density in a silicon crystal Nam = 0.05 × 1024 cm−3 . Therefore, for the crystal with length L = 1.94 mm the probability Pin = 0.49%. The experimental setup was mainly the same as described above, see Fig. 45. Four microstrip silicon detectors, two upstream and two downstream of the crystal, were used to detect the particle trajectories with an angular resolution of about 3 µrad. Two large scintillation detectors with transverse dimensions 100 × 100 mm2 were placed 60 cm downstream the crystal on both sides from the primary proton beam to register secondary particles generated in inelastic nuclear interactions of protons in the crystal. The distance between the scintillation detectors was 10 mm. So, the angle of the inner edge of both scintillation detectors from the crystal location was θed = 8.33 mrad. It is sufficient to exclude the contributions of primary protons scattered due to elastic nuclear interactions, because θel ≪ θed . A 70 × 1.94 × 0.5 mm3 silicon strip crystal with the largest faces parallel to the (110) crystallographic planes was bent along its length and placed vertically, so that the anticlastic bending induced along the crystal width (1.94 mm) was used

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Fig. 46. The deflection angle distributions of 400 GeV/c protons by the silicon crystal bent along the (110) planes. Only particles with the incident angles |θxo | < θcut = 1.5 µrad were considered. (a) In a linear scale, (b) in a logarithmic scale. Here the boundaries of the dechanneling area are marked by the solid circles.

to deflect particles in the horizontal plane (see Fig. 8). The optimal crystal orientation, which gives the maximum of the deflected beam fraction, was found. The RMS deviation value measured for the horizontal angular distribution of the incident beam was σx = (13.368 ± 0.003) µrad. It is larger than the critical channeling angle, θc ≈ 10 µrad. However, the beam part deflected due to channeling in the crystal can be considerably increased by decreasing the angular size of the incident beam when only particles with the incident angles |θxo | < θcut are considered. Fig. 46 shows the deflection angle distribution of protons observed with the cutting angle θcut = 1.5 µrad. The Gaussian fit in Fig. 46a gives the deflection angle value θxm = (189 ± 0.02) µrad, which equals the crystal bend angle α . The deflected beam fraction Pd = (72.5 ± 0.117)% is hatched. Nuclear interactions occur mainly with undeflected particles either not captured into the channeling regime at the crystal entrance or dechanneled during the crystal passage. Fig. 47 shows the dependence of the undeflected beam part Pnd = 1 − Pd on the cutting angle of the incident beam. It should be mentioned that some instability of the angular position of the goniometer (as large as a few microradians) observed over a long period of measurements and the crystal torsion (different crystal orientations along its height) decreased the deflection efficiency. The analysis of the amplitude spectra of the scintillation detectors registering secondary particles from the crystal allowed to determine the amplitude discrimination thresholds Ab , which cut the intrinsic detector background. Secondary particles were registered also in the case when the crystal was removed from the beam. They were generated by inelastic nuclear interactions of protons upstream the crystal and form the experimental background. The coincidence of the events with the amplitude A > Ab registered by the left and right scintillation detectors was used to subtract efficiently the experimental background including δ -electrons. The frequency of the inelastic nuclear interactions of protons in the crystal

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Fig. 47. The dependence of the undeflected part of protons on the cutting angle θcut of the incident beam. The curve is given to guide the eye.

Fig. 48. The dependencies of the inelastic nuclear interaction frequency of protons on the cutting angle value of the incident beam for the amorphous orientation (1), the aligned crystal (2) and for the case without the crystal in the beam — the experimental background (3). The curves are given to guide the eye.

was defined as Fin = N12 (A > Ab )/No ,

(57)

where N12 (A > Ab ) is the number of the coincidence events in S1 and S2 with amplitudes A > Ab , No is the number of particles with |θxo | < θcut , which hit the crystal. Let us note that the registered number of the coincidence events is not equal to the number of all inelastic interactions of protons in the crystal. The part of all events generated charged particles, which hit both the detectors, the coincidence frequency F12 = 0.655 ± 0.005, was determined by simulation using the FRITIOF model [69]. Fig. 48 shows the measured dependencies of the inelastic nuclear interaction frequency of protons on the cutting angle value θcut of the incident beam for the amorphous orientation (1) and for the aligned crystal (2), as well as without the crystal in the beam that is the experimental background (3). The interaction frequencies without the crystal and with the crystal in its amorphous orientation are practically constant. Whereas the frequency registered in the aligned crystal is

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Fig. 49. The dependencies of the inelastic nuclear interaction probability of protons in the crystal on the value of the cutting angle θcut of the incident beam for the amorphous orientation (1), for the case of volume reflection (2) and in the aligned crystal (3). The dependence (4) was obtained by simulation for the aligned crystal. The curves are given to guide the eye.

considerably smaller than for its amorphous orientation and it decreases with decreasing θcut due to the increase of the fraction of well channeled protons, which do not experience close collisions with the crystal atoms. The probability of inelastic nuclear interactions of protons in the crystal was calculated by subtraction of the experimental background Fin (BG) and taking into account the coincidence frequency F12 estimated by the simulation above Pin = (Fin − Fin (BG))/F12 .

(58)

Fig. 49 shows the dependences of the inelastic nuclear interaction probability of protons in the crystal on the value of θcut in the cases of the amorphous orientation (1) and volume reflection (2) as well as in the aligned crystal (3). The measured am probability value for the amorphous orientation is Pin = (0.505±0.005)%, which is in good agreement with the theoretical estimation made above using (56). For the symmetric case of volume reflection (2) considered in the experiment when the tangency area of the particle momentums with the bent planes is in the middle of the crystal length and practically all particles pass the whole crystal in above-barrier states the probability is 3%–4% larger than for the amorphous orientation. In the aligned crystal for the smallest angular width of the incident beam the probability is more than 3.5 times smaller am than Pin . The particle number in the incident beam was about 5 × 106 . Therefore, the statistical errors are small. The errors shown in the figures are systematic. The inelastic nuclear interaction probability of protons in the aligned crystal has been also studied by simulation using the model described above. The calculated dependence is shown by the curve 4. There is a significant discrepancy with the experiment for small values of θcut because the goniometer instability and the crystal torsion were not taken into account in the simulation. However, for large values of θcut when the angular size of the incident beam is larger than the angular parameters of the goniometer instability and the crystal torsion the agreement with the experiment is good. The simulation allows to register the state of particles before their inelastic interactions with the crystal nuclei. According to the simulation the probability of inelastic interactions for channeled protons is (3 ÷ 4)% of the probability for the amorphous orientation. A possible contribution of channeled protons to the inelastic interactions registered in the aligned crystal has been estimated [61]. Particles, which passed the whole crystal in channeling or above-barrier states, form the maxima at θx = α or θx = θvr , respectively, where θvr is the deflection angle due to volume reflection (see Fig. 46a). Dechanneled particles between two maxima were in channeling states only some part of the crystal length, which is determined by their deflection angle — θx /α . These particles are better seen with using a logarithmic scale (Fig. 46b). Here the Gaussian fit for the left maximum allows to separate volume reflected and dechanneled particles. The total number of particles passed through the whole crystal in above-barrier states taking into account the contributions of dechanneled particles can be determined as Nnch =

i1 ∑ i=1

ni +

i2 ∑ i1

yi +

i2 ∑ i1

(ni − yi )(1 − θxi /α )

,

(59)

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Fig. 50. The dependence of the inelastic nuclear interaction probability in the aligned crystal on the non-channeled part of the beam. The probability am as a ratio to the value for the amorphous orientation is presented, Pin /Pin . The curve is given to guide the eye. The dashed line is a hypothetic am linear dependence of Pin /Pin = Pnch .

where ni is the particle number of the ith histogram bin, yi is the Gaussian fit value in the middle of the ith bin θxi . The boundaries (i1 , i2 ) of the dechanneling area are shown by the solid circles in Fig. 46b. Two first members of (59) determine the beam part, which was not captured into the channeling regime at the crystal entrance and passed the whole crystal in above-barrier states. The third member determines the contribution of dechanneled particles. Fig. 50 shows the inelastic interaction probability of protons measured in the aligned crystal as a function of the beam fraction passed the crystal in above-barrier states, Pnch = Nnch /No (solid circles). The different circles correspond to the different angular cuts of the incident beam θcut . The probability is shown as a ratio to its value for the amorphous am orientation P¯ in = Pin /Pin . The dependence shown by a dashed line, P¯ in = Pnch , is a hypothetic one when the probability is the same as in the amorphous case for all above-barrier protons and the contribution from channeled protons is absent. The probability values measured in the experiment are larger than the hypothetic ones. The difference is about 8% for the large angular sizes θcut of the incident beam. A half of this difference, about 4%, is due to the fact that the interaction probability for above-barrier protons is larger than for the amorphous orientation (see 2 in Fig. 49). The remaining difference, about 4% for the large values of θcut , is due to the contribution of channeled protons, which is in good agreement with the value predicted by the simulation. The experiment [61] has shown that the probability of inelastic nuclear interactions of high-energy protons in the crystal depends strongly on its orientation. For the orientation optimal for volume reflection the probability is a few percents larger than for the amorphous one. The probability is significantly smaller in the aligned crystal because well channeled protons move through the crystal far from the crystallographic planes where the atomic nuclei are concentrated. The contribution of inelastic interactions from channeled protons was estimated to be about 3%–4% of the probability for the amorphous orientation. 4.1.8. Electromagnetic dissociation for well channeled heavy ions Channeling properties of high energy ions in crystals are invariant with respect to pz = p/Z1 , where p is the ion momentum and Z1 is the ion charge. This should lead to the invariance of the deflection efficiency of ions by a bent crystal with respect to pz . However, the cross-section of nuclear interactions and the electromagnetic dissociation (ED) as well as ionization energy losses significantly increase for heavy nuclei. So, the cross-section of inelastic nuclear interactions for Pb nuclei with 270 GeV/c per charge in silicon crystal is σh = 4.3 b [70], which is about 10 times larger than for protons. Besides, already for this energy of Pb nuclei the ED cross-section is comparable with the nuclear one, σed = 1.37 b. Thus, the attenuation length for Pb nuclei in the crystal primary collimator non-aligned with the beam in the experiment [70] was about 3.5 cm. Nuclear interactions can occur in the aligned crystal only for channeled particles with large oscillation amplitudes in the channels, which enter the nuclear corridors. In this case the probability of nuclear interactions still may be calculated using the nuclear cross-section as Ph = nσh L, where n is the atomic density and L is the crystal length. However, the density n should be averaged along the trajectories and ensemble of the beam particles.

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The cross-section conception assumes a uniform distribution of the impact parameters of incident particles with the target nuclei. For nuclear interactions in the aligned crystal the impact parameter distribution is close to uniform for very small parameters involved. However, the ED processes of high energy ions in the aligned crystals are possible for sufficiently large impact parameters comparable with the channel width [71] (see below). Therefore, the assumption about the uniformity of the impact parameter distribution is not already valid and the cross-section conception for the ED processes of high energy ions in the aligned crystals is not applicable. In this case the ED probability should be straight calculated. In [72] the ED probability of deuterons with energy up to 5 GeV in the aligned tungsten crystal was determined along the real calculated trajectories using the equivalent photon method. In the coordinate system connected with an incident particle the charge of the target nuclei provides a short electromagnetic pulse, which can be presented by the virtual photon flow. The minimal energy of photons exciting the giant dipole resonance of Pb nuclei with the emission of one neutron is En1 = 7.4 MeV. A maximal energy of the virtual photons in the collisions with the impact parameter b can be estimated according to [73] as Eγ m = γ h ¯ c /b ,

(60)

where γ is the relativistic factor of particles. Eγ m reaches the value of En1 for Pb nuclei with pz = 270 GeV/c (γ = 114.3) at b = 0.03Å. Thus, in this case the electromagnetic dissociation can occur only for a small fraction of well channeled nuclei with trajectories approaching closely to the nuclear corridors. On the contrary, at the LHC energies for Pb nuclei with pz = 7 TeV/c (γ = 2963) the ED is possible even for channeled particles moving along the center of the (110) channels in silicon crystals when the impact parameter b = dp /2 = 0.96 Å, where dp is the channel width. The ED probability in the nuclei collision with the impact parameter b is calculated using the equivalent photon approach as [73] ∞

∫

dEγ

N(Eγ , b) σ (Eγ )

P(b) =

Eγ

En1

,

(61)

where N(Eγ , b) is the number of the equivalent photons incident on the target per unit area, σ (Eγ ) is the photo-dissociation cross-section of Pb nuclei. The expression taking into account the screening of target nuclei by atomic electrons [74] was used for the equivalent photon number. The experimental data for the photo-dissociation cross-section of Pb nuclei were used. Fig. 51a shows the calculated dependence of the ED probability of Pb nuclei due to collisions with Si nuclei per 1 mm path on the momentum pz for the idealized case of the particle motion along the center of the (110) silicon channel (3.1 × 106 Si nuclei per 1 mm are on the right and on the left side). The ED probability for the given impact parameter (61) should be averaged by the thermal vibrations of the crystal atoms (xa , ya ) and by the vertical coordinates of the particle entrance points yo

¯ = P(x)

1

2

2π

d

u21

d/2

∫

∫

3u1

0

∫

3u1

−3u1

( x2 + y 2 ) a a

√

P( (x − xa )2 + (y − ya )2 ) exp −

dya

dy

−3u1

2u21

dxa ,

(62)

as well as along the channeled particle trajectory with the impact parameter to the crystal plane xo in the channel potential u(x)

˜ o) = P(x

∫

dp /2

√ x0

¯ P(x)dx u(xo ) − u(x)

/∫

dp /2

x0

√

dx u(xo ) − u(x)

.

(63)

Fig. 51b shows the calculated dependence of the ED probability of Pb nuclei averaged according to (62) and (63) on the impact parameter xo to the (110) planes of a silicon crystal in the case of parallel incidence. The ED probability per 1 mm is larger than 10−4 for particles entering the crystal closely to the nuclear corridor when xo is close to 3u1 value. The ED probability per 1 mm averaged by the particle ensemble (xo ) for a parallel beam Ped = 3 × 10−5 . In a bent crystal the effective planar potential transforms and the particle trajectory shifts to the outer channel wall. The well channeled fraction is reduced. Besides, the contribution to the ED of Pb nuclei from the interactions with the Si nuclei of the inner channel wall is reduced. However, the ED probability per 1 mm averaged by the ensemble of well channeled particles remains close to the value for the straight crystal, Ped = 2.8 × 10−5 . Thus, it was shown in [71] that the losses of the well channeled fraction of 7 TeV/c Pb ions due to ED can be of the order of 0.01% for silicon crystals with the length of few millimeters. A visible difference can appear only when a bent crystal is used for the extraction or collimation of the accelerator beam. The increasing losses of heavy nuclei in the crystal due to nuclear interactions and electromagnetic dissociation can reduce possibility of their multiple passages through the crystal. 4.1.9. Parametric X-rays produced by protons in bent crystals as a tool to control crystal state Parametric X-ray radiation (PXR) is emitted by a fast charged particle in a crystal due to diffraction of its virtualphoton field on crystallographic planes. PXR has been theoretically predicted [75–77] and then observed and studied using electron beams of different energies [78,79]. The first experiment with the aim of PXR observation from heavy charged particles was carried out on the 70 GeV proton beam at IHEP [80]. Recently, PXR has been successfully observed from 5 GeV protons and 2.2 GeV/u carbon nuclei in a silicon crystal on the external beams of the Nuclotron at LHE JINR [81,82].

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Fig. 51. (a) The calculated dependence of the ED probability of Pb nuclei due to collisions with Si nuclei per 1 mm path on the momentum pz for the idealized case of the particle motion along the center of the (110) silicon channel. (b) The dependence of the averaged ED probability of Pb nuclei on the impact parameter xo to the (110) planes of a silicon crystal in the case of parallel incidence.

The characteristics of PXR are determined by the particle velocity v and they are independent of the charge sign and mass of the particle. The energy of PXR photons generated by a particle in a crystal at the angle θ relative to its velocity is determined as [75] E n = nh ¯c

⃗ g⃗ · β , √ 1 − εo β cos θ

(64)

where n is the diffraction order, g⃗ is the reciprocal lattice vector, β = v/c and εo is the constant part of the medium permittivity (εo ≈ 1 for X-rays). The experimental test of this formula was performed in [83]. The PXR photon energies generated on the fixed crystal planes may be calculated using the following detailed expression of the formula E n = nh ¯ ω1 = n

2π h ¯c d

1−

√

β sin θB , εo β cos θD cos θy

(65)

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Fig. 52. A schematic view of the layout of the experiment for the observation of parametric radiation produced by protons in bent silicon crystals BC in the horizontal plane. S1 -S5 are the silicon microstrip detectors, Sc is the scintillation trigger. The experiment details: (a) for quasi-mosaic crystal, (b) for strip crystal. The beam enters the crystal parallel to the (111) and (110) deflecting planes for (a) and (b) cases, respectively. The crystal bend is small and is not shown. PXR photons are produced due to diffraction of virtual photons of the proton field on the (110) and (100) planes for (a) and (b) cases, respectively. Photon angles θH are measured from the direction θD of the PXR maximum.

where d is the interplanar distance, θB is the orientation angle of the crystal planes with respect to the particle momentum, θD and θy are the radiation detection angles in the diffraction plane formed by g⃗ and β⃗ and in the √ normal plane, respectively. The direction of the PXR maximum is determined by the angle θD = 2θB and θy (ω) = γ −2 + (ωp /ω)2 , where γ is the relativistic Lorentz factor of the particles and h ¯ ωp = 31 eV is the plasmon energy in silicon. The PXR generated by channeled particles in bent crystals has been considered in [84] where the possibility of the PXR focusing was discussed. The application of PXR for online diagnostics of the beam and the bent crystal state was proposed in [85]. Recently [86], it was suggested to use PXR to control the state of the crystal deflector during its use for beam extraction and collimation in high energy accelerators, which is slowly changed because of defects produced by irradiation. The experiment [87] on the PXR observation from 400 GeV/c protons in bent silicon crystals in the collimation geometry was carried out at the H8 external beam of the CERN SPS. The experimental layout in the horizontal plane is shown in Fig. 52. The figure parts (a) and (b) are for the experiments with quasi-mosaic (QM) and strip (ST) crystals, respectively. The beam entered the crystal in the collimation geometry such that it is parallel to the deflecting planes, which are the (111) and (110) crystallographic planes for the QM and ST crystals, respectively. The divergence of the incident beam measured with the detector telescope was characterized by the RMS deviations σx = 10.7 µrad and σy = 7.6 µrad. The scintillation detector Sc downstream of the silicon telescopes was used as a trigger. It registered the number of protons in the beam No with an accuracy of about 10%. The average cycle time of the SPS during the measurements was about 45 s with the pulse duration 10–11 s and the average number of particles per spill about (1.3 ± 0.1) × 106 . The PXR generated due to diffraction of virtual photons of proton field on the (110) and (100) crystallographic planes (‘‘radiating planes’’) in the case of (a) and (b) shown in Fig. 52, respectively, was registered by the X-ray detector D. The inclination angles of these radiating planes relative to the deflecting ones and hence with the beam were θB = 35.26◦ (a) and 45◦ (b). The X-ray detector was placed close to the PXR maximum direction, θD = 2θB , the detection angles θD were 70.25◦ and 90◦ in the case (a) and (b), respectively. So, the real detector position in the case (a) was shifted a little from the Bragg angle θD = 70.52◦ .

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Fig. 53. X-ray spectrum recorded by the semiconductor detector during the irradiation of the QM crystal by the proton beam after background subtraction. Blue lines show Gaussian fits to the PXR peaks. Numbers in the plots correspond to the diffraction order for the peaks. Table 2 PXR characteristics for QM crystal. Reflex

M (keV)

Nγ

Iγ (10−7 )

Iγ 1 /Iγ n

Iγ (10−7 ) Theory

Iγ 1 /Iγ n Theory

1 2 3

5.58 ± 0.01 11.18 ± 0.01 16.70 ± 0.06

420 ± 20 230 ± 16 47 ± 8

6.9 ± 0.8 3.8 ± 0.4 0.77 ± 0.12

1 1.8 ± 0.3 9 ± 2

4.35 3.33 0.69

1 1.3 6.3

The semiconductor silicon detector D used for detection of PXR photons is 380 µm thick with a sensitive surface area of about 13 mm2 and with a 25 µm beryllium window. The distance between the crystal and detector was changed in the range Sd = (95 ÷ 127) mm. The detector D was placed in a cavity inside a lead brick to reduce the radiation background. The crystal was seen by the detector through a collimating window of 5 mm diameter and 50 mm length. The semiconductor detector was calibrated by using the lines of characteristic radiation excited in a copper target by the proton beam (see [81]). The energy resolution of the detector for the 8.046 keV Kα line was equal to about 250 eV. The detector efficiency is close to 100% for photons with energies (4 ÷ 8) keV and it reduces with further increase of photon energy. The efficiency estimated using the photoabsorption cross-section in silicon was used to compare the experimental results with the PXR theory. PXR Generated in QM crystal deflector. The QM crystal 40 × 30 × 2 mm3 (Height × Width × Thickness) used for the PXR measurements in the case (a) was mechanically bent along its height and placed with its thickness parallel to the beam. The QM curvature produced along the thickness and hence along the (111) planes with bend radius R = 17.4 m was used for beam deflection in the horizontal plane. The energies of PXR photons of different diffraction order n in case (a) calculated according to (65) are equal to En = nE1 with E1 = 5.59 keV. Fig. 53 shows the measured radiation spectrum after background subtraction. The PXR peaks were fitted by Gaussians. The fit parameters, which determine the peak center M, are presented in Table 2. The peak positions are in excellent agreement with the PXR energies En determined by (65). The number of photons Nγ n registered in the peaks are given in Table 2. The total number of protons in the beam No and the beam fraction which passed through the crystal during the PXR measurements, Fcr = 0.77 ± 0.02, allowed to determine the PXR yields per proton Iγ . The PXR yields and their ratio Iγ 1 /Iγ n are also presented in Table 2. The errors of Iγ values are mainly caused by the errors of Nγ and by the error of No , which was estimated to be about 10%. The total registered yield of PXR is larger than 10−6 photon/p. More than half of the detected photons are the first diffraction order ones because the detector is sufficiently close and soft photon absorption in the air is small. PXR Generated in ST crystal deflector. The strip crystal 70 × 5 × 2 mm3 (Height × Width × Thickness) with its largest faces parallel to the (110) planes was used for the PXR measurements in the case (b) and was mechanically bent along its height. The anticlastic bending produced along the crystal width, which had R = 33.3 m, was used for beam deflection in the horizontal plane. The energies of PXR photons in case (b) calculated according to (65) are equal to En = nE1 with E1 = 6.46 keV. Fig. 54 shows the spectrum observed in this case after background subtraction. Only two peaks are visible here. They were also

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Fig. 54. PXR spectrum produced by protons in the ST crystal with the background subtraction. Blue lines show Gaussian fits to the PXR peaks. Table 3 PXR characteristics for ST crystal. Reflex

M (keV)

Nγ

Iγ (10−7 )

Iγ 1 / Iγ n

Iγ (10−7 ) Theory

Iγ 1 /Iγ n Theory

1 2

6.46 ± 0.01 12.85 ± 0.02

179 ± 13 101 ± 10

3.4 ± 0.4 1.9 ± 0.3

1 1.8 ± 0.3

1.88 0.73

1 2.57

fitted by Gaussians and Table 3 presents the measured PXR parameters. The peak positions are also in good agreement with the PXR energies En calculated according to (65). The PXR yield is more than two times smaller than for the QM crystal because the (100) planes have a lower atomic density and hence a lower density of electrons which scatter the virtual photons of the proton field. Calculations of PXR. The calculations of PXR characteristics from 400 GeV/c protons in the silicon crystal deflectors for the experimental conditions have been performed according to the PXR kinematic theory [88]. The absorption of photons in the crystal, air and beryllium window of the detector was considered. The dependence of the detector efficiency on the photon energy was taken into account. It was calculated using the photoabsorption cross-section in silicon. The calculated PXR yields for different diffraction orders as well as the ratios Iγ 1 /Iγ i are presented in Tables 2 and 3. The agreement of the calculated PXR yields for second and third diffraction orders with the experimental values is good for the QM crystal (Table 2). The discrepancy of about 35% for the yield of photons with E1 is explained by their small attenuation length in silicon, which makes the calculation results very sensitive to the beam distribution across the crystal. The PXR yield discrepancy is larger for the ST crystal, which might be caused by a larger error in determining the number of protons that crossed the crystal for this measurement. Conclusions. The experiment [87] allowed to observe parametric X-ray radiation from 400 GeV/c protons in bent silicon crystals aligned with the beam (in a collimation geometry). The radiation was emitted in a direction normal or close to normal to the beam. A few peaks in the X-ray spectra measured by the semiconductor silicon detector correspond well to the PXR photon energies of different diffraction orders En . The total yield of photons was about 10−6 per proton for the quasi-mosaic crystal and a little smaller for the strip crystals. The intensity of PXR emitted from halo protons in a primary crystal collimator of a high-intensity circular accelerator should be sufficient to control its structure state. It should be noted that the PXR yield in the crystal inside the accelerator pipe will be larger than the values observed in our experiment due to vacuum conditions. Furthermore, halo protons will pass through the crystal collimator very close to its surface in the case of perfect alignment. Therefore, PXR attenuation in the crystal will also be suppressed. Both these circumstances will considerably increase the yield of E1 photons. In the case of poor crystal alignment the distribution of halo protons across the crystal collimator becomes close to uniform with a width of a few millimeters. For these orientations the relative yield of higher order photons of PXR increases. Thus, the analysis of PXR spectra recorded from halo protons crossing a primary crystal collimator gives an additional possibility to control its orientation. It should be noted that the radiation hardness of the PXR detector was not considered in [87].

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Fig. 55. (a) Geometry of a straight crystal with a skew-cut back face. (b) Schematic picture of a beam focusing using a bent crystal with a skew-cut back face.

Fig. 56. Schematic picture showing operating principle of focusing with a bent crystal.

4.1.10. Beam focusing with a bent crystal It was shown in the experiment [89] that a bent crystal can focus a beam of high-energy positive particle. A bent crystal with a skew-cut back face deflects particles with different incidence coordinates x by different angles. Therefore, their trajectories will intersect at some distance, the focal length F , behind the crystal, Fig. 55. For the efficient beam focusing, the particle deflection angles should be proportional to their incidence coordinate x, θ = x/F (Fig. 56). First experiment on focusing of 400 GeV/c protons at the H8 beam line of the CERN SPS [90] has been performed with two silicon strip crystals bent along the (111) planes by the angle of 0.3 mrad and 0.6 mrad, respectively. The crystals were skew-cut along the whole length of 3 mm, Fig. 57. Because of the thickness change along the length, the crystal curvature was changed along the beam. Therefore, the proportionality between the deflection angles and the incidence coordinates of the beam particles was a little disturbed. The experimental results showed that the focal length of the crystals is about 1 m. The RMS beam size of the beam passed through the skew-cut part of the crystals in the focus was reduced by factors of about 3 and 4, respectively. A new idea to focus a beam over a short distance was proposed in [91]. It was suggested to use a parallel-side silicon plate whose side faces have the inclination angle β = D/L with respect to some crystallographic planes, where D and L are the crystal thickness and width, respectively (see Fig. 58). That is the crystal plane direction coincides with the direction of the plate diagonal. Under uniform bending of the plate, particles will be focused at a point. The experiment [92] using this new idea was performed with 180 GeV/c pions at the H8 beam line of the CERN SPS. A 0.5 mm thick silicon crystal strip with a width of 10 mm was used. The strip was cut so that the angle between its side faces and the (111) crystallographic planes was about D/L = 50 mrad. The strip was uniformly bent along its width by the angle α = 3.5 mrad due to anticlastic curvature. The expected focal length F = D/α was about 15 cm.

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Fig. 57. A strip crystal for a beam focusing bent along its width that is along the (111) planes due to anticlastic curvature. The crystal is skew-cut along its width. The cross-section of the straight crystal is shown at the top.

Fig. 58. Scheme of a beam focusing by a bent crystal plate whose crystallographic plane direction coincides with the direction of the plate diagonal.

A scan of horizontal orientation angles of the crystal was performed and the channeling orientation was found. Fig. 59 shows the obtained dependence of the horizontal deflection angles of particles θx on their incidence coordinate x. It is clearly seen that the deflection angle for channeled particles (2) is a linear function of their incidence coordinate x. The incident beam divergence was about 30 µrad which is larger than the critical channeling angle θc = 16.7 µrad. Therefore, the channeled fraction of the beam, marked in the figure as area (2), consisted of about 10% of the incident beam. Fig. 60 shows the reconstructed profiles of the channeled fraction straight behind the crystal and at the focal point, which was found at the distance of 15.5 cm. The RMS beam sizes are 125 µm and about 8 µm for these profiles, respectively. That is the beam compression factor is about 15. It should be noted that the beam size under ideal focusing conditions is determined by the critical channeling angle and is equal to F × θc ≈ 3 µm for the considered case. The larger value obtained in the experiment is partly explained by the angular resolution of the tracking system. Thus the experiment [92] demonstrated the possibility of focusing a channeled beam fraction with a bent crystal over a very short distance of ∼10 cm. 4.1.11. Reflection of protons by an ultra-thin straight crystal The crystal bend produces the conditions for reflection of high energy positive particles by the potential of a single plane inside the crystal volume (volume reflection). The conditions are fulfilled about for all particles of the incident beam. Therefore, the volume reflection efficiency can be very high, up to 98%.

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Fig. 59. The dependence of the horizontal deflection angles of 180 GeV/c pions θx on their coordinate x at the crystal entrance. Region 1 is formed by non-channeled particles, 2 by channeled particles, 3 by dechanneled particles.

Fig. 60. The channeled beam profiles straight behind the crystal (a) and at the focal point (b). The tail of the last profile is due to dechanneled particles.

The reflection may be realized also in a straight crystal. A channeled particle performs the oscillatory motion in a planar channel reflecting by the channel walls and its transverse energy Ex is conserved in the first approximation. When a particle enters a crystal in the channel center with the angle θo < θc it is captured into channeling regime and the maximum change of its direction is observed through the half of the oscillation wavelength λ, θR = 2θo . So, the maximal deflection can be close to θR = 2θc when θo = θc . However, very small beam fraction will be reflected for this large θo . So, if the crystal length will be equal to S = nλ/2, where n is the integer number, it will be possible to observe the channeled particle reflection. However, the oscillation wavelength λ depends upon Ex because the planar potential is not a harmonic one. By this reason if the number n will be large it will not be possible really to observe the beam reflection due to the phase mixing. The best result is realized when n = 1. The ultrathin crystal with its length S = λ/4 was suggested first in [93] to increase the impact parameters of the collider beam halo protons with a bent crystal to overthrow them through some imperfect layer at the bent crystal surface. A quarter wavelength crystal works as an efficient ultrathin scatterer. A half wavelength crystal – a crystal mirror – has been suggested for the same purpose in [94]. The mirroring effect was first demonstrated in [95] for non-relativistic protons of 2 MeV. In the experiment [96] the reflection of 400 GeV/c protons by an ultrathin straight crystal of a half wavelength has been observed. A silicon crystal plate with thickness of 500 µm oriented along the (100) planar channels was used for an ultrathin target production. The oscillation wavelength for 400 GeV/c protons in the (100) planar channels in the harmonic

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Fig. 61. Schematic view of the crystal with an ultrathin central area. The crystallographic directions are shown by arrows and the (001) planes by dashed lines. The beam mirrored by the planes is shown by the thick arrows.

approximation λ = 57 µm and the critical channeling angle θc = 7.6 µrad. The thickness of the central plate area 4 × 4 mm2 square was reduced via anisotropic etching up to 26.5 µm measured with infrared interferometer, which is a little smaller than λ/2 (see Fig. 61). The beam of 400 GeV/c protons enters the central ultrathin crystal area. Pre-alignment of the crystal was made using a laser system. An angular scan of the (100) crystal plane orientations with respect to the beam was then performed to find the mirroring effect. Fig. 62c shows the deflection angle distribution of protons as a function of the incident beam direction with respect to the (100) plane direction. The mirroring effect is observed within the whole angular range of channeling, that is 2θc . The deflection direction changes with the incident beam direction. The mirroring effect is observed for channeled particles in regions (1). The regions (2) correspond to the deflections experienced by over-barrier particles. In regions (3) all incident particles are in over-barrier states and the beam center is not shifted. Fig. 62a shows the results of simulation for a parallel beam. The reflected beam is more narrow in this case in comparison with the experiment. The simulation results taking into account the resolutions for the incident and deflection angles are shown in Fig. 62b. These results are close to the experimental ones. Thus, the experiment [96] confirmed the existence of the mirroring effect for high energy protons in the half wavelength crystals. The simulations without taking into account the angular resolutions of the tracing system allow estimating the deflection efficiency due to the mirroring effect. In the considered case, for the incident beam with the RMS angular size σ = 0.5 µrad the efficiency of one side deflection ε = N(∆θ > 0)/Ntot = 85.5%. It is smaller than for volume reflection in a bent crystal. Besides, the angular acceptance for the mirroring effect is small. The main advantage of mirroring is the interaction with a minimal amount of material along the beam. 4.2. Negative particles 4.2.1. Observation of channeling and volume reflection in bent crystals for high-energy negative particles For negative particles the planar potential is attractive. When the particle transverse energy is smaller than the depth of the planar potential well it moves along the trajectory winding on the crystal plane. As a result, the probability of close collisions with the crystal atoms is high for negative particles in the channeling states. Therefore, the dechanneling length for negative particles is much shorter than for positive ones. For this reason the deflection of negative particles by bent crystals has not been observed up to the experiments in CERN described in this section [97]. Only a broadening of the angular distributions towards the crystal bend side has been observed in the previous experiments [36,37]. Afterwards, the deflections due to channeling and volume reflection have been demonstrated for high energy electrons [98–103]. Negative particles can be also deflected due to volume reflection as it was shown in [17]. Fig. 63 shows the trajectory difference for positive and negative particles with the same momentum at volume reflection. The tangent point and the trajectory parts with a small transverse velocity are near the channel centers where the electric field is minimal. Therefore, these trajectory parts are longer for negative particles. This reduces the VR deflection angle. Short crystals with a length smaller than 1 mm allowed observing the deflection of high energy negative particles in bent crystals due to channeling and volume reflection in [97]. The experiment has been done with 150 GeV/c π − mesons at the H4 beam line of the SPS. The divergence of the beam incident on to the crystal measured with the detector telescope was characterized by the RMS deviations σx = (34.4 ± 0.06) µrad and σy = (28.2 ± 0.04) µrad in the horizontal and vertical planes, respectively. Fig. 64 shows the intensity distribution of the beam crossed the quasi-mosaic crystal (QM2) in the deflection angles of particles at the different angular positions of the goniometer. The crystal was bent along the (111) planes and has a length of 0.84 mm. The crystal bend angle is about 65 µrad. Only particles hitting the crystal with a horizontal angle |θxo | < 10 µrad were selected. The picture is very similar to the one obtained in the experiment with 400 GeV/c protons (see Fig. 14). Therefore, the different areas in the figure are marked in the same way. Moving from the right to the left of the figure we pass first the crystal orientations where a considerable fraction of the incident beam is captured into the bound states with the (111) planes and deflected by the crystal bend angle (channeled particles) (2). The wider area (4), whose width is determined by the crystal bend angle, is the area of volume reflection.

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Fig. 62. The deflection angle distribution of 400 GeV/c protons as a function of the incident beam direction with respect to the (001) planes of a half wavelength Si crystal. (a) Simulation for a parallel beam. (b) Simulation taking into account the angular resolution. (c) The experimental results.

Fig. 65a shows the distribution of particle deflection angles for the crystal orientation optimal for channeling. The right peak was generated by particles deflected by the crystal in channeling states. Particles, which were not captured into the channeling states and were deflected to the side opposite to the crystal bend due to volume reflection, generate the left peak. The beam fraction between the two peaks outside the fits determines the dechanneled fraction of the beam (about 14%). The Gaussian fit of the channeled fraction gives the mean deflection angle θd = (63.24 ± 0.24) µrad. The deflection efficiency determined by the beam fraction under the fit (hatched area) is Pd = (30.24 ± 0.38)%. Fig. 65b shows the distribution of the deflection angles of particles for the crystal orientation in the middle of the VR area. The Gaussian fit gives the peak position, the VR deflection angle, θv r = (−14.64 ± 0.12) µrad and its RMS deviation σvr = (10.06 ± 0.11) µrad. The VR deflection angle is about 0.8 θc . It is smaller than for positive particles as predicted in [17]. The VR efficiency can be determined as the beam fraction under the Gaussian fit with angles θ < θv r + 3σv r . This gives Pv r = (82.74 ± 0.28)%. The large distribution tail on the bend side indicates that the VC probability is considerably higher for negative particles. They undergo strong multiple scattering on the atomic nuclei passing the potential well center where they have large transverse velocities. Therefore, the same multiple scattering angle gives a larger increase of the transverse energy than for the case with positive particles increasing the VC probability. However, most of them

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Fig. 63. The calculated trajectories of 150 GeV/c π + meson (a) and π − meson (b) at volume reflection in a silicon crystal bent along the (110) planes near the turning area in the comoving coordinate system, which rotates around the crystal curvature center with the particle velocity. The coordinate X is measured in the direction opposite to the radial one. Solid horizontal lines show the crystal planes.

Fig. 64. The intensity distribution of the 150 GeV/c π − meson beam crossed the (111) bent silicon crystal (QM2) in the deflection angles of particles at the different angular positions of the goniometer. The following areas are indicated: (1) and (6) amorphous scattering, (2) deflection due to channeling, (3) dechanneling, (4) volume reflection, (5) deflection of volume captured particles.

is quickly dechanneled because of the same strong multiple scattering. Nevertheless, the VR inefficiency due to VC approaches 20% for 150 GeV/c π − mesons, which is about one order of magnitude higher than for 400 GeV/c protons. Fig. 66 shows the distribution of the deflection angles of 150 GeV/c π − mesons for the strip silicon crystal (ST10) when the optimal orientation for channeling occurs. The crystal with length along the beam T = 0.98 mm was bent along the (110) planes. Only particles hitting the crystal with |θxo | < 5 µrad were selected. The Gaussian fits for channeled and reflected fractions are overlapped here because of a smaller bend angle of the crystal. The fit of the channeled fraction gives the mean deflection angle θd = (40.54 ± 0.36) µrad. The deflection efficiency defined as the beam fraction under the fit (hatched area) is Pd = (28.81 ± 0.47)%.

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Fig. 65. The deflection angle distribution of π − mesons crossed the (111) bent silicon crystal (QM2) for the crystal orientations optimal for channeling (a) and volume reflection (b). The curves show the Gaussian fits of the peaks. The hatched areas show the deflected fractions of the beam due to channeling (a) and VR (b).

From the distribution of the particle deflection angles for the VR crystal orientation the VR deflection angle for the strip crystal was determined to be θv r = (11.53 ± 0.23) µrad. For negative particles, in contrast to the positive ones, the VR deflection angle is smaller for the (110) planes than for the (111) ones, θv r = 0.66 θc . The same definition as used above gives for the VR efficiency Pv r = (76.75 ± 0.32)%. The VR efficiency is smaller than for the QM2 crystal because the VC probability of particles is higher due to the larger bend radius. The parameter values for channeling and volume reflection of π − mesons obtained for the (111) and (110) crystals in the experiment and in our simulation are listed in Table 4. The agreement between the experimental and simulation results is rather good. However, the values of the VR deflection angles obtained in the simulation using the Moliere approximation of the atomic potential are 10%–15% larger than the measured ones, which is similar to what we observed with 400 GeV/c protons. The angle values calculated using the explicit expression [34] and the atomic potential obtained through the experimental X-ray scattering factors are closer to the measured values (see Table 4). Thus, the experimental results [97] have shown that the sufficiently stable bound states of high-energy negative particles with the bent crystal planes exist and lead to the deflection of particles. Besides, they have shown that volume reflection in bent crystals occurs also for negative particles in spite of the attractive character of the forces acting between the particles and the crystal planes.

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Fig. 66. The deflection angle distribution of π − mesons crossed the (110) bent silicon crystal (ST10) for the crystal orientations optimal for channeling. Table 4 Channeling and VR parameters for 150 GeV/c pi− mesons. Crystal (111) Si experiment simulation theory (110) Si experiment simulation theory

ϑc (µrad)

R (m)

ϑvr (µrad)

Pv r (%)

Pd (%)

12.92 ± 0.09

14.64 ± 0.12 16.6 ± 0.07 14.28

82.74 ± 0.28 78 ± 0.13

30.24 ± 0.38 30.11 ± 0.15

22.79 ± 0.22

11.53 ± 0.23 12.84 ± 0.11 11.81

76.75 ± 0.32 74.77 ± 0.14

28.81 ± 0.47 28.67 ± 0.14

18.34

17.39

The experimental studies on the deflection of electrons with energies up to 20 GeV [98–102] have been performed with ultra short silicon crystals (with lengths smaller than 0.1 mm along the beam). The observed deflection efficiency was about 20% for channeling and up to more than 80% for VR [100]. The VR deflection angle of 150 GeV/c π − mesons has been considerably increased due to sequential volume reflections realized in the multi-strip silicon deflector; comparative results for protons and π − mesons are given in [104]. 4.2.2. Observation of multiple volume reflection by different planes in one bent silicon crystal for high-energy negative particles The effect of multiple volume reflections in one crystal (MVR OC) predicted in [48] has been observed in the experiment [49] with 400 GeV/c protons. The effect of volume reflection in a bent crystal was predicted [17] and then observed [97] also for high energy negative particles. It was natural to wait that the deflection due to the set of volume reflections by different planes crossing the axis, the MVR OC effect, can be realized also for negative particles. The MVR OC effect in a bent silicon crystal has been really observed for 150 GeV/c π − mesons at the H4 beam line of the CERN SPS in the experiment [105]. As it follows from the optimal conditions for the MVR OC realization near the [111] axis when a silicon crystal is bent along its vertical (110) plane (see (44)–(47)) the crystal bend angle α should be large to satisfy the relations: α = 2θxo , θxo = 2θyo , θyo ≫ ψ1 . The last one is the stability condition for particle trajectories along the crystal planes formed with the [111] atomic strings. The critical angle for axial channeling along the [111] axis for 150 GeV/c π − mesons ψ1 = 33.81 µrad. In the experiment [105] a 70 × 2 × 0.5 mm3 silicon strip with the largest faces parallel to the (110) planes and with the side faces parallel to the (111) planes was placed in a vertical position. The beam entered the crystal through its side face. The anticlastic curvature produced along the crystal width W = 2 mm due to its mechanical bending along the length was used for the beam deflection in the horizontal plane. The bend angle was preliminarily measured in the experiment on the 400 GeV/c proton beam deflection, α = (702 ± 0.4) µrad. The divergence of the beam of π − mesons incident on to the crystal measured with the detector telescope was characterized by the RMS deviations σx = (26.04 ± 0.05) µrad and σy = (25.66 ± 0.04) µrad in the horizontal and vertical planes, respectively. The scan of the horizontal orientation angles of the crystal θh was first performed. It allowed p finding the angular position of the vertical (110) planes θh by observing the deflection of π − mesons in channeling regime.

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Fig. 67. (a) Intensity distribution of the 150 GeV/c π − meson beam in the horizontal and vertical deflection angles acquired due to multiple volume reflections from different planes crossing the [111] axis in one bent silicon crystal. Maximum position is shown by dot-dashed line. (b) The horizontal projection of the distribution. The distribution part with θx < 0 (about 65%) is hatched. Gaussian fit gives for the maximum position θmvr ≈ −47.65 µrad.

Then the scan of the vertical orientation angles of the crystal θv allowed finding the angular position of the [111] axis θva . p The values of θh , θva determine the center of the coordinate system shown in Fig. 18. The crystal orientation in this system ∗ ∗ was selected to realize the incident beam direction close to the optimal one with θxo = 346 µrad and θyo = 168 µrad. The ∗ stability condition for planar trajectories of particles is also fulfilled in this case, θyo ≈ 5 ψ1 . Fig. 67a shows the intensity distribution of the π − meson beam passed through the crystal in the horizontal and vertical deflection angles. The horizontal projection of the distribution is shown in Fig. 67b. Gaussian fit gives the value θm = (47.65 ± 0.31) µrad for the maximum position. This angle is 4.6 times larger than the VR deflection angle of π − mesons by the vertical (110)v plane, whose value θvr = (10.36 ± 0.32) µrad was measured for the beam incidence vr ∗ ≫ θyo . So, the additional deflection of π − mesons due to volume reflections by the skew planes is with the angle θyo a value approaching 4 θv r . The (110)sk and (112)sk planes give approximately a half of this value and the other skew planes, which are more weak and not shown in Fig. 18, give a second half. The deflection efficiency to the side opposite to the crystal bend is Pd (θx < 0) = (64.73 ± 0.31)%. The efficiency is significantly smaller than observed for 400 GeV/c

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Fig. 68. Deflection angle distributions of 150 GeV/c π − mesons by a bent silicon crystal observed in the case of MVR OC near the [111] axis (1) and in the case of a single volume reflection by the (110) vertical planes (2).

protons [49] because of a larger probability of volume capture of π − mesons into the channeling regime when they pass the tangency areas with different crystal planes discussed in [97]. A larger VC probability causes also a more visible tail of π − mesons with small vertical deflection angles volume captured by the vertical (110) planes. Most of them are quickly dechanneled because of the strong multiple scattering by the atomic nuclei. Fig. 68 gives the comparison of the distributions of horizontal deflection angles of π − mesons for the case of MVR near the axis and for the case of a single VR by the vertical (110)v plane far from the axis. The deflection angle distribution for MVR OC obtained by simulation looks like the experimental one with the maximum at θm = (42.83 ± 0.48) µrad. The value of one side deflection efficiency Pd (θx < 0) = (61 ± 0.13)%. So, the agreement with the experiment is sufficiently good. The experimental results [105] have shown that the effect of multiple volume reflections from different crystallographic planes inside one bent crystal exists also for high-energy negative particles, π − mesons, in spite of the attractive character of the forces acting between the particles and the crystal planes. The additional contribution of particle reflections from the skew planes allows increasing the deflection angle in a few times like in the case with high energy protons. 4.2.3. Measurement of the dechanneling length for high-energy negative pions For negative particles the planar potential is attractive and all of them in channeling regime move along the trajectories oscillating around the crystal plane passing through the region with high density of atomic nuclei. Fig. 69 shows the atomic density averaged along trajectories for 150 GeV/c π + and π − mesons entering a straight silicon crystal parallel to the (110) planes as a function of the impact parameter with respect to the crystal plane. The atomic density is considerably larger for negative particles in the whole range of impact parameters. Multiple scattering by atomic nuclei is the main mechanism of dechanneling for the whole channeled fraction of negative particles. Whereas for positive particles this mechanism works only for a large amplitude channeled fraction. In the experiment [106] the dechanneling length for 150 GeV/c π − mesons has been measured in a silicon strip crystal of size 70 × 1.91 × 0.9 mm3 bent along the largest faces, which are parallel to the (110) planes. The crystal was tested first with a 400 GeV/c proton beam at the H8 beam line of the CERN SPS to compensate the initial crystal torsion, which inevitably comes out once the crystal has been mounted on the holder. The deflection angle for channeled protons was θd = 99.6 µrad. Therefore, the crystal bend radius is about 19 m. The dechanneling length measured as in [60] was Ln = (1.23 ± 0.05) mm. Then the crystal was used for the experiment with 150 GeV/c π − mesons. Fig. 70 shows the distribution of deflection angles for 150 GeV/c π − mesons after the passage through the aligned crystal. By fitting the dechanneled part of the distribution between the two maxima with the exponential function, it was found the dechanneling length value Ln = (0.93 ± 0.05) mm. The simulation result for the dechanneling length Ln = (0.95 ± 0.03) mm is in good agreement with the experiment. Thus, the experiment [106] showed that the dechanneling length observed in short crystals has the same order of magnitude for positive and negative pions with close values of their momentum. However, there is a significant difference. The observed dechanneling length Ln describes the reduction of the whole channeled fraction in the case of negative particles. Whereas for the beam of positive particles Ln describes only the reduction of a large amplitude channeled fraction.

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Fig. 69. Atomic density averaged along particle trajectories entering a straight silicon crystal parallel to the (110) planes as a function of the impact parameter with respect to the crystal plane.

Fig. 70. The distribution of deflection angles for 150 GeV/c π − mesons passed through the bent silicon crystal with its orientation optimal for channeling. Only particles hitting the crystal within an interval ±5 µrad are selected.

5. Beam deflection by a bent crystal. Axial orientations 5.1. Positive particles 5.1.1. Deflection of high-energy protons through axial channeling along [111] axis As a particle traverses the crystal with a small angle ψo to one of the main crystallographic axes, the regime of axial channeling (AC) is realized [1]. In this case the particle motion is governed by the potential of the lattice of atomic strings averaged along the axis [1]. The angular acceptance of AC demands the alignment of the crystal axis with the particle momentum, ψo ≤ ψ1 , where ψ1 is the critical angle for axial channeling (see in Section 2). Two types of states, ‘‘hyperchanneling’’ and ‘‘doughnut scattering’’, are realized for particles under axial channeling. The first one is the confinement of particles bounded up with a single axial channel. However, the potential barrier separating the axial channels formed by neighboring atomic strings is rather low; thus only a small fraction of the particles can be captured into hyper-channeling states even for perfect alignment of the crystal axis with the beam. For silicon crystal, such potential barrier is about 6 eV for the [110] direction and only 1 eV for the [111] direction. Particles with a transverse energy higher than the potential barrier wander in the lattice of atomic strings changing the transverse momentum direction in collisions with the strings. For particles with an angle smaller than ψ1 , a uniform distribution of the transverse momentum directions is reached after traversing a distance λ (equalization length) [1]. The process of scattering by atomic strings that leads to such a distribution is called doughnut scattering. Indeed, AC in a bent crystal would represent an alternative to planar channeling for particle steering. The deflection of positively charged particles axially channeled in a bent crystal was first shown in [107] by simulation using a binarycollision model. The observed wide spectrum of deflection angles from zero to the full bend angle of the crystal was caused by channeled particles wandering among bent axial channels.

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With some limitations on the crystal bending, doughnut scattering does not significantly smear out the trajectories transversely, i.e., the ensemble of channeled particles proceeds along the bent axes and undergoes deflection by the crystal bend angle. Thus, deflection of axially channeled particles is possible not only due to the bound states of hyperchanneling but also due to the unbound states of doughnut scattering regime. Indeed, the mechanism of doughnut scattering would allow one to steer a considerable part of the beam. The condition for particle deflection in a bent crystal due to doughnut scattering was first formulated in [8]. However, pioneering experiments [8,37] actually did not observe the beam deflection due to AC. Such experiments mainly highlighted a strong feed-in of particles into skew planar channels stretched to the side opposite to that of bending. Indeed, a sufficient condition for deflection of particles through doughnut scattering in AC was formulated in [38] (see in Section 2). This condition for the deflection of protons due to doughnut scattering was not fulfilled in [8,37]. The deflection of particles by a bent silicon crystal due to doughnut scattering has been first observed in the experiment [56] with 400 GeV/c protons at the H8 beam line of the CERN SPS. A 70 × 2 × 0.5 mm3 silicon strip with the largest faces parallel to the (110) crystallographic planes and with its side faces parallel to the (111) planes was used. The crystal was placed in a vertical position in such a way that the proton beam entered the crystal through its side face nearly parallel to the large faces (see Fig. 8). Thus, the [111] axis direction, which is normal to the side crystal faces, became nearly aligned with the beam direction. The anticlastic curvature produced by the bending device along the crystal width was used for beam deflection in the horizontal plane. The bending angle was α = 50 µrad and the corresponding bending radius R = 40 m. The critical angle for AC of 400 GeV/c protons along [111] direction is ψ1 = 20.7 µrad (i.e., about twice the critical angle for planar channeling), the equalization length λ = 43 µm, and k = 0.126. Thus, the condition (38) was satisfied. The divergence of the beam of protons incident on to the crystal measured with the detector telescope was characterized by the RMS deviations σx = (9.27 ± 0.06) µrad and σy = (5.24 ± 0.03) µrad in the horizontal and vertical planes, respectively. After the installation, an angular scan of the crystal around the vertical axis (θv ) was done to orient the (110) planes parallel to the beam direction. Then, an angular scan around the horizontal axis (θh ) was performed to reach the best possible alignment of the [111] axis with the beam. Fig. 71 shows beam intensity distributions as a function of the particle deflection in the horizontal (θx ) and vertical directions (θy ) due to the interaction with the crystal at three different values of θv . When the beam direction is far from the axial direction the beam is undeflected, resulting in a spot, not shown in the figure, centered in (θx ; θy ) = (0; 0) with standard deviations close to the beam divergence quoted above. As θv approached the alignment condition, the distribution in Fig. 71a was observed. A large fraction of the particles was still undeflected though part of the beam was deflected due to scattering by atomic strings and feeding into skew planar channels. The bottom spot is more intense than the top one because of non perfect alignment of the crystal with respect to the beam in the direction of θh . By increasing θv , the distribution is further modified as shown in Fig. 71b. Here only a small fraction of the beam remained undeflected. Axial channeling along the [111] axis occurred and deflected the beam by the crystal bending angle, which was about 50 µrad. A fraction of the initial axially-channeled particles leaked into skew planar channels because the [111] axis is the intersection of several planes. Scattering with [111] atomic strings may accidentally direct a particle parallel to one of the skew planes; then the crystal bending strengthens the stability of this planar motion. Deflection of particles due to channeling by the strongest (110) skew channels was clearly observed as two tails departing from the axial spot. The best alignment condition is shown in Fig. 71c. The undeflected part of the beam has completely disappeared and the spot corresponding to axial channeling is the most populated by beam particles. A ‘‘swallow-tail’’ profile was clearly observed, formed by deflection of particles due to axial channeling (rightmost spot) and channeling in skew planar channels (two symmetric spots on the left), as predicted by simulation [108]. In contrast to planar channeling, there is no beam near the incident beam direction in the distribution. Fig. 71d shows the angular distribution for 400 GeV/c protons deflected at a perfect alignment with the [111] axis of a bent silicon crystal simulated for the same conditions of the experiment. The simulation results allowed one to estimate the contribution of hyperchanneled particles to the distribution peak near the full bending angle. This contribution is smaller than 2% even for perfect alignment of the crystal axis with the beam. Curve 2 in Fig. 72 shows the horizontal projection of the experimental distribution for almost perfect axial alignment (Fig. 71c) in comparison with the unperturbed beam profile (curve 1). Deflection efficiency due to AC can be conservatively estimated through the fraction of particles whose horizontal deflection angles exceed 40 µrad. The particles with smaller deflection angles were dechanneled or captured by the skew channels with high probability. Deflection efficiency for AC was estimated to be about 30%, i.e., doughnut scattering allowed to significantly enhance the number of deflected particles in comparison with the hyperchanneling contribution. In summary, deflection of high-energy protons via axial channeling in a bent crystal, mainly driven by the mechanism of doughnut scattering, was clearly observed in [56]. Hyperchanneling provides only a small contribution ∼ 1% to the deflection efficiency. 5.1.2. High-efficiency deflection of protons through axial channeling along [110] axis Strong feed-in of protons into the (110) skew planes was observed in the experiment [56] on the deflection of 400 GeV/c protons by a silicon crystal bent along the [111] axis. This reduces the deflection efficiency due to doughnut scattering

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Fig. 71. Beam intensity distribution as a function of the horizontal and vertical deflections at some orientation angles (θv ) of the [111] axis with respect to the incident beam direction: (a) θv = −40 µrad, (b) θv = −15 µrad, (c) θv = 0 µrad. The beam particles shown in the figure are only 2 2 those hitting the entry face of the crystal and satisfying condition (θxo + θyo ) < (5 µrad)2 , where θxo and θyo are the horizontal and vertical angles of the incident particle, respectively. (d) Simulation results for the experimental conditions in the case of perfect alignment of the [111] axis with the beam direction, for comparison with the experimental case (c).

Fig. 72. Projection on the horizontal plane of the unperturbed beam profile (curve 1) and of the distribution shown in Fig. 71c (curve 2). Each distribution is normalized to unity. The figure highlights that most of the particles in the incoming beam were shifted apart by significant angles through interaction with the crystal.

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Fig. 73. The distributions of 400 GeV/c protons in the horizontal and vertical deflection angles after the passage through the silicon crystals with their orientations optimal for axial channeling, (a) along the [111] axis, (c) along the [110] axis. (b) and (d) are the simulation results for the corresponding conditions.

(DS) regime. The reduction of the DS fraction because of the feed-in process can be characterized by some length, which was called the relaxation length in [109]. It reduces when the crystal bend radius reduces. The simulation results in [110] for the deflection of high energy protons by a silicon crystal bent along the [110] axis showed that the probability of close collisions of protons with atoms is strongly reduced in DS regime and the number of particles in skew planes is around 15% only. The [110] axis is the strongest one in a silicon crystal, the critical angle for axial channeling is larger than for the [111] axis, ψ1 = 22.9 µrad for 400 GeV/c protons. The high-efficient deflection of 400 GeV/c protons by a silicon crystal bent along the [110] axis was studied in the experiment [111]. Two bent silicon crystals were used. Both the crystals have ∼ 2 mm length and are bent along the (110) plane with the bend angle about 60 µrad. However, the bent axes parallel to the beam direction are different, being [111] and [110], respectively. As a consequence comparison of the AC properties for two different axes was possible. Both the crystals were aligned first by the (110) planes with the beam to observe the beam deflection due to planar channeling. Then their axes were aligned with the beam. Fig. 73 shows the distributions of 400 GeV/c protons in the horizontal and vertical deflection angles after the interaction with the crystals under the conditions optimal for axial channeling. Only particles with the incident angles to the axis direction θo smaller than 3 µrad were selected. Fig. 74 shows the distributions of protons in the horizontal and vertical deflection angles separately. The distributions in the vertical deflection angles are symmetric about the vertical plane for both of the crystals. The distribution of the horizontal deflection angles has a single maximum with the position close to the bend crystal angle for the [110] crystal. Whereas for the [111] crystal the distribution has two maxima generated due to axial channeling in DS regime and due to channeling in skew planar channels. The deflection efficiency due to axial channeling was evaluated as the fraction of particles in the circle centered on the position of the corresponding deflection maximum. The deflection efficiency was 22 ± 2% for the [111] crystal and 61 ± 2% for the [110] crystal.

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Fig. 74. The distributions of 400 GeV/c protons in the horizontal (a), (c) and vertical (b), (d) deflection angles after the passage through the [111] crystal (a), (b) and through the [110] crystal (c), (d).

The tails due to the feed-in process by the skew planes are weak for the [110] crystal. As it was described above, the feed-in process starts when scattering by the atomic strings accidentally directs a particle transverse momentum nearly parallel to one of the main skew planes. Then this planar motion of the particle can be stabilized due to the crystal bend. This stabilization is less efficient for the [110] crystal because its (111) skew planes have larger angle, about 55◦ , with respect to the plane of the crystal bending. Besides, the deflection angles due to channeling in the (111) skew planes are closer to the crystal bend angle. The secondary particles produced due to the inelastic nuclear interactions (INI) of protons with the crystal atoms were measured by two pairs of scintillation detectors installed behind the crystal. The INI event frequency was measured for both of the crystals for the axial and planar as well as for amorphous orientations. Fig. 75 shows the INI frequency in the crystals with √ the orientations optimal for axial channeling as a function of the angular range radius θom around the 2 + θ 2 < θ . The frequency values are normalized to the value for the amorphous orientation. When θxo om yo the angular size of the incident beam θom increases the beam fraction in axial channeling regime decreases leading to the increase of the average INI frequency. For a narrow beam fraction with θom = 5 µrad the INI frequencies under AC regime

AC orientation

are 0.3 and 0.1 for the [111] and [110] crystal, respectively. That is the beam losses are reduced by 10 times for axial channeling of protons along the [110] axis. The simulation performed with Geant4 takes into account the whole experimental setup and the contribution of multiple scattering of protons in the beam line elements. The simulation results are also presented in Figs. 73 and 74. There is good agreement between these simulation results and the experiment. The simulations performed for the ideal conditions without taking into account the experimental setup resolutions show that the deflection efficiency for the considered crystal with the [110] axial orientation may reach about 80% for a narrow incident beam with θom = 3 µrad.

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Fig. 75. The INI frequency of 400 GeV/c protons in the crystals with the orientations optimal for axial channeling as a function of the angular range radius θom around the AC orientation. Curve 2 for the [111] crystal, curve 3 for the [110] crystal. Curve 1 for the crystal orientation optimal for the planar channeling along the (110) planes. The values are normalized to the INI frequency for the amorphous crystal orientation.

The experiment [111] showed that the deflection efficiency of protons due to the [110] axial channeling in a bent crystal can be comparable with the deflection efficiency due to planar channeling. The measured probability of inelastic nuclear interactions of protons in channeling along the [110] axis is only about 10% of its amorphous level. 5.2. Negative particles 5.2.1. Deflection of high-energy π − mesons through axial channeling along [111] axis For negative particles the atomic string potential in crystals is attractive and the bound states of the particles with the atomic strings can be realized. The deflection possibility for relativistic negative particles in the bound states with atomic strings in a bent crystal had been shown by simulation in [112] using the model of binary collisions of particles with the crystal atoms. However, doughnut scattering by atomic strings occurs also for negative particles in crystals. The mechanism of doughnut scattering should allow the deflection of negative particles by a bent crystal for its axial orientation. The sufficient conditions for this deflection mechanism discussed above (see (39) in Section 2) are the same as for positive particles. In the experiment [113] 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. A 70 × 0.98 × 0.5 mm3 silicon strip (length × width × thickness) with the largest faces parallel to the (110) planes and with the side faces, which are 70 × 0.5 mm2 , parallel to the (111) planes was used in the experiment. The crystal bending and its installation were the same as described above. The deflection angles of particles were measured with resolution of 8 µrad, which is limited by the multiple scattering in the detectors and the air. The crystal bend angle α according to the measurement of the deflection angle of particles at the planar orientation of the crystal was about 43 µrad that is the bend radius R = 22.79 m. For 150 GeV/c π − mesons the critical angle of channeling along the [111] axis is ψ1 = 33.8 µrad, and the equalization length λ = 26.3 µm that gives ψ 2 = 0.043 ψ12 . Thus, the condition (39) for π − mesons deflection due to doughnut scattering was satisfied in this experiment. The divergence of the beam incident on to the crystal measured with the detector telescope was characterized by the RMS deviations σx = (34.4 ± 0.06) µrad and σy = (28.2 ± 0.04) µrad in the horizontal and vertical planes, respectively. After the crystal installation its [111] axis was aligned with the beam. Fig. 76 shows the beam intensity distribution in the deflection angles of particles in the horizontal θx and vertical θy planes for the different orientation angles θv of the [111] crystal axis with the beam direction. The narrow fraction 2 2 1/2 of the incident beam with the angles (θxo + θyo ) ≤ 10 µrad was selected. When the crystal axis becomes close to the beam direction particles are governed by the averaged potential of atomic strings. They undergo multiple scattering by the atomic strings. As a result the transverse momentums of particles are partly randomized. We can see an arc shape of the particle distribution for the orientation angle of the [111] crystal axis θv = −40 µrad (a). The arc radii are determined by the particle angles to the axis direction at the crystal entrance |θv | and exit |θv | + α . For the orientation angle θv = −20 µrad (b), the distribution shape becomes close to a circle, which is shifted to the bend side. However, the maximum of the beam intensity continues to be near the incident beam direction. For the nearly perfect alignment at θv = 0 (c), the whole beam is deflected by the bend angle. In the contrary to the case with positive particles [56] there is no any leakage of π − mesons into skew planar channels intersecting the [111] axis because the planar motion of negative particles having small angles with atomic strings of the planes cannot be stabilized. Really, the impact parameters of

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Fig. 76. Beam intensity distribution in the deflection angles of 150 GeV/c π − mesons in the horizontal θx and vertical θy planes for the different orientation angles θv of the [111] crystal axis with respect to the beam: (a) θv = −40 µrad, (b) θv = −20 µrad and (c) θv = 0. The narrow fraction 2 2 1 /2 of the incident beam with the angles (θxo + θyo ) ≤ 10 µrad was selected. (d) The distribution obtained by simulation for the perfect alignment with the axis. The bend crystal angle α = 43 µrad is shown by the dotted line in (c) and (d).

π − mesons in subsequent collisions with the atomic strings of skew planes fast decrease due to the attractive character of forces. Therefore, they undergo strong scattering in close collisions with the atomic strings by the angle larger than the critical one for planar channeling and leave the planar channel. Fig. 77 shows the horizontal projection of the two-dimensional distribution for the nearly perfect alignment presented in Fig. 76c. The distribution maximum position corresponds to the crystal bend angle, α = 43 µrad. The distribution width FWHM= (55 ± 2) µrad. The crystal considerably broadens the beam due to multiple potential scattering of particles by the atomic strings, which leads to the randomization of their transverse momentum, and due to strong incoherent multiple scattering on the atomic nuclei. The efficiency of the beam deflection to one side from the initial direction P(θx > 0) = (90.6 ± 0.5)%. Fig. 76d shows the two-dimensional intensity distribution of the beam crossed the crystal at a perfect alignment, which was obtained by simulation for the experimental conditions. There is a satisfactory agreement with the experiment. The deflection angle distribution is more narrow, FWHM= 45 µrad, and P(θx > 0) = 96.5%. It should be noted that the experimental resolution of deflection angles was not taken into account in these simulations. The simulation allows knowing the state of every particle during the passage through the crystal. Fig. 78 shows the dependence of the particle fraction in the bound states with the atomic strings, with the transverse energies Ex < U(Rs ), where U(Rs ) = 99.4 eV is the potential value at the distance equal to the string radius Rs (half the distance between the strings), on the penetration depth into the crystal. The capture efficiency of particles into the bound states for the experimental conditions is about 42%. However, the particles fast leave the bound states due to strong multiple scattering on the atomic nuclei. The contribution of hyperchanneled particles, which have crossed the whole crystal length in the bound states, is about 0.55% (the curve 1). However, incoherent multiple scattering causes also the reverse process of the intense recapture of particles into the bound states. The fraction of particles recaptured in the considered crystal layer is shown by the curve 2. The recapture process occurs along the whole crystal length. The total number of particles in the bound states with the atomic strings increases due to the recapture (the curve 3). The beam fraction deflected at the bend angle in the bound states is about 15%.

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Fig. 77. Horizontal projection of the beam intensity distribution shown in Fig. 76(c). The dotted line shows the crystal bend angle value α = 43 µrad.

Fig. 78. Fraction of particles in the bound states with the [111] atomic strings as a function of the penetration depth into the crystal (curve 3). Curve 1 is for the hyperchanneled fraction. Curve 2 is for the fraction recaptured into the bound states in the considered crystal layer. The dependencies were obtained by simulation for the perfect alignment of the [111] axis with the beam.

Fig. 79 shows the trajectory of one of the particles, which underwent a few recaptures crossing the crystal. The particle was captured into the bound state at the crystal entrance. It fast left this state and was recaptured by the atomic string next to the neighboring one. The precessing elliptical orbit indicates the bound state of the particle. The recapture occurred the second time by the atomic string closest to the first one before the particle left the crystal. The experiment [113] allowed observing the deflection of high energy negative particles at the axial orientation of a bent silicon crystal. The deflection mainly occurs due to doughnut scattering of particles by the atomic strings. Besides, it was shown by simulation that strong incoherent multiple scattering of particles on the atomic nuclei causes not only fast dechanneling but also a high probability of the reverse capture of particles into the bound states with the atomic strings. So, the bound states of particles should also give a contribution of about 15% in the observed beam deflection.

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Fig. 79. The trajectory in the transverse space calculated for the particle, which was captured into the bound state with the atomic string located at (X , Y ) = (0, 0) at the crystal entrance and then two times was recaptured by the neighboring atomic strings. The bound states are seen by the precessing elliptical orbits. Red dots indicate the [111] axes positions and the blue one — the position of the entrance point of the particle.

5.2.2. Deflection of high-energy π − mesons in quasi-bound states of doughnut scattering. Volume capture High energy charged particles performing doughnut scattering at the axial orientation of a bent crystal can be deflected as it was shown for protons [56,111] and π − mesons [113]. The particles deflecting in this case can be considered as quasibound with the axis direction in contrary to the real bound states with a single atomic string. This regime was called ‘‘Doughnut Scattering Bound’’ (DSB) in [114]. The motion regime for particles performing multiple scattering by atomic strings but do not following to the crystal bend was called ‘‘Doughnut Scattering Unbound’’ (DSUB). Let us note some peculiarities, which are important for understanding the DSUB regime. The particle momentum randomization center is determined by the angular position of the axis therefore it shifts with the penetration of a particle into a bent crystal. Besides, the transverse momentum of a particle, which determines the randomization radius at the given crystal depth, is changed due to Coulomb multiple scattering and doughnut scattering (DSUB) in the previous crystal layers. The experiment [114] on the deflection of negative particles, 150 GeV/c π − mesons, by a bent silicon crystal oriented along the [111] axis was done with the considerably larger bend angle α = 185 µrad = 5.5 ψ1 . This allowed to observe the dependence of the particle fraction deflected in the quasi-bound states with the bent axis direction, in the DSB regime, on the crystal orientation. It was observed volume capture of π − mesons into the DSB regime. Besides, the beam deflection to the side 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 the bent planes crossing the axis. A 70 × 8 × 0.3 mm3 silicon strip (length × width × thickness) with the largest faces parallel to the (110) planes and with the side faces, which are 70 × 0.3 mm2 , parallel to the (111) planes was used in the experiment. The crystal bending and its installation were the same as described above. However, here θh and θv are the horizontal and vertical angles of the crystal inclination with the beam, respectively, as opposed to the notations used in Fig. 8. Because of a small planar dechanneling length for negative particles the crystal has been preliminarily tested using 400 GeV/c proton beam. The crystal bend angle measured through the deflection of protons by the (110) bent planar channels was α = (185±0.4) µrad. The condition (39) for the particle deflection due to doughnut scattering is fulfilled for this case, ψ 2 = 0.098 ψ12 . The measured divergence of the incident beam of π − mesons was characterized by the RMS deviations σx = σy = (26.5 ± 0.2) µrad. After the installation the [111] axis of the crystal was aligned with the beam. Fig. 80a shows the beam intensity distribution in the deflection angles of 150 GeV/c π − mesons in the horizontal θx and vertical θy planes for the perfect alignment of the [111] axis of a bent silicon crystal with the beam, θh = θv = 0, when the fraction of particles deflected with the horizontal angles around α (shown by the dashed line) is maximum. The 2 2 1/2 + θyo ) ≤ 10 µrad was selected. Fig. 80b shows the horizontal narrow fraction of the incident beam with the angles (θxo projection of this distribution by the hatched histogram. The distribution maximum is at the angle θx < α . The deflection efficiency to the bend side is Pd (θx > 0) = (95.4 ± 0.2)%. The histogram 2 shows the distribution obtained by simulation for the experimental condition. The one side deflection efficiency obtained in the simulation Pd (θx > 0) = (98.5 ± 0.1)%. The maximum position of the calculated distribution is about the crystal bend angle. The reduction of the deflection angles in the experimental distribution maximum may be partly explained by the crystal torsion, which leads to different orientations of the crystal axes along the vertical direction. For comparison the histogram 3 shows the deflection angle

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Fig. 80. (a) Beam intensity distribution in the deflection angles of 150 GeV/c π − mesons in the horizontal θx and vertical θy planes for the perfect 2 2 1 /2 alignment of the [111] axis of a bent silicon crystal with the beam. The narrow fraction of the incident beam with the angles (θxo + θyo ) ≤ 10 µrad was selected. The vertical dashed line shows the crystal bend angle value α = 185 µrad. (b) The horizontal projection of the experimental distribution (1). The histogram 2 is the simulation result for the same condition. The histogram 3 shows the distribution for the straight crystal obtained by simulation.

distribution of π − mesons calculated for the straight crystal with the same length. The distribution has a Gaussian shape centered at θx = 0 with the RMS deviation value σ = (28.65 ± 0.24) µrad, which is larger than the Coulomb multiple scattering angle θo = 24 µrad for the amorphous orientation due to the contribution of doughnut scattering by the atomic strings of the crystal. The interesting peculiarity of the beam deflection by a bent crystal in the case of its axial orientation exists. When there is a small vertical inclination of the crystal bend plane, where the bent atomic strings are located, relative to the incident beam plane, θv ≤ ψ1 , the beam particles obtain also some vertical deflection to be on the bend plane. For particles in the quasi-bound regime of doughnut scattering the distribution maximum position of the vertical deflection angles is at θy = θv as in the case presented in Fig. 81. Fig. 82 shows the distributions of horizontal deflection angles of π − mesons for the crystal orientation angles θh = 24 µrad (1) and -24 µrad (2) obtained by simulation. The distributions have well seen maximum generated by particles deflected in the quasi-bound regime DSB. The maximum positions are at θxd = α + θh (these values are shown by the dashed lines). The distribution tail to the side of the initial beam direction increases in comparison with the perfect

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Fig. 81. (a) The same as in Fig. 80a for the vertical crystal inclination with θv = 35 µrad. (b) The vertical projection of the distribution (a). The mean value of Gaussian fit m = (35.2 ± 0.7) µrad.

alignment of the crystal and it acquires a convex shape. Let us note that the crystal-beam mutual orientation is always changed by the crystal inclination in the experiment. It is sometimes more convenient to talk about the beam incident angles, which are θxo = −θh and θyo = −θv . Fig. 83 shows the dependence on the beam incident angle θxo for the beam fraction of π − mesons deflected by the angles around θxd with |θx − θxd | < 32 µrad obtained in the experiment (1) and by simulation (2). The dependence difference may be partly explained by the torsion of the crystal used in the experiment. It should be noted that there are also particles in the unbound regime DSUB in the considered interval of the horizontal deflection angles. However, the main contribution is due to the quasi-bound fraction; curve (3) for the DSB fraction was obtained by simulation. All particles in the DSB regime with the incident angles |θxo | > ψ1 were captured in the crystal volume due to Coulomb multiple scattering by the crystal nuclei and due to doughnut scattering by atomic strings. Possibility for volume capture of π − mesons into the DSB regime remains along the whole arc of the bent crystal as it is seen for the positive values of θxo in Fig. 83. The corresponding volume capture probability is larger than 7%. Fig. 84 shows the deflection angle distributions of π − mesons for different crystal orientations, which are far from the perfect alignment. For θh = 50 µrad (a) the distribution is formed due to unbound states of doughnut scattering

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Fig. 82. The distributions of horizontal deflection angles of π − mesons for the crystal orientation angles θh = 24 µrad (1) and −24 µrad (2) obtained by simulation. The dashed lines show the deflection angle values θxd = α + θh .

Fig. 83. The dependence on the beam incident angle θxo for the beam fraction of π − mesons deflected by the angles around θxd with |θx − θxd | < 32 µrad obtained in the experiment (1) and by simulation (2). Curve 3 shows the quasi-bound deflected fraction of π − mesons with ((θx − θxd )2 + θy2 )1/2 < ψ1 obtained by simulation.

of π − mesons in the condition when the crystal axis direction becomes farther from the incident beam direction with increasing the penetration depth of particles into the crystal. The distribution maximum is close to θx = 0. The partial randomization of the transverse momentum of particles leads to the arc shape distribution with the average radius larger than θh but smaller than α/2. The distribution tail stretches significantly farther than the bend angle α . For the crystal orientation angle θh ≈ −ψ1 (b) the particle fraction in the DSB regime is well seen through the maximum on the right. In this case, the deflection occurs due to the volume capture of particles into the DSB regime as it was discussed above. For the crystal orientation angle θh ≈ −α/2 (c) the tangency area of the incident beam with the bent axes is about the middle of the crystal where the initial negative angular position of the axes changes to the positive one. Let us remember again that the axis position is the center of the particle momentum randomization in the DSUB regime. The deflection angle distribution is very wide and approximately centered at θx = 0 in this case. The hatched histogram shows the distribution obtained for the amorphous crystal orientation far from the axes and plane directions (the distributions are

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Fig. 84. The horizontal deflection angle distributions of π − mesons for different crystal orientations far from the perfect alignment. (a) θh = 50 µrad, (b) θh = −32 µrad, (c) θh = −89 µrad, (d) θh = −129 µrad. The dashed line in (b) shows the value of the deflection θxd = α + θh and in (d) it shows the maximum position θx = −88 µrad. The hatched histogram shows the distribution for the amorphous orientation of the crystal in (c) and the distribution obtained by simulation for the same condition in (d).

not equally normalized). Doughnut scattering of particles by bent atomic strings for the crystal orientation with θh ≈ −α/2 increases the beam broadening about three times in comparison with the amorphous orientation. The effect of the beam deflection unexpected for the axial crystal orientation was observed for the angles θh < −α/2. The tangency areas of the initial momentum directions of particles are closer to the crystal exit in this case. For the case presented in Fig. 84d θh = −129 µrad. The distribution maximum is on the side opposite to the bend at θx = −88 µrad. This deflection is larger than it was observed in our experiment for 150 GeV/c π − mesons in the conditions of multiple volume reflections in one crystal (MVR OC) by bent planes crossing the [111] axis [105], which was about θmv r = 50 µrad. The hatched histogram shows the distribution obtained by simulation for the experimental conditions. The agreement is very well because the crystal torsion should not be very important for the regime of unbound doughnut scattering of particles realizing in this case. Fig. 85 shows the beam intensity distributions in the deflection angles obtained by simulation for the case presented in Fig. 84d at the different penetration depths of the beam into the crystal. The axis direction at the crystal entrance coincides with the center of the circle with the radius R = θxo shown in Fig. 85a for the depth Z = 2 mm. Particles at this depth are partly randomized along the arc with some radius, which is smaller than θxo because of the crystal bend. Many particles obtain large vertical momentums due to this randomization. They obtain also negative horizontal deflections. Therefore, the tangency areas of their momentums with bent planes crossing the axis become closer to the crystal entrance than for the initial beam direction. These changes of transverse momentums of particles relative to the bent axis inside the crystal due to doughnut scattering lead to the realization of the MVR OC for a large part of the beam. This forms the distribution maximum at θx < θmv r (see Fig. 85b–d) because the MVR deflection is added to the negative deflections already obtained due to doughnut scattering of these particles in the previous crystal layers.

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Fig. 85. The beam intensity distributions in the deflection angles obtained by simulation for the case presented in Fig. 84d at the different penetration depths of the beam into the crystal: 2 mm (a), 4 mm (b), 6 mm (c) and 8 mm (d). The circle shown by dashed line in (a) has the center at θx = θh and the radius R = |θh |. The transverse momentum randomization of π − mesons occurs along this circle in the straight crystal. The distribution maximum position is shown by the dashed line in (d).

Fig. 86 shows for comparison the deflection angle distribution of π − mesons for the straight crystal of the same length with the same orientation angle, θh = −129 µrad, obtained by simulation. This is the arc distribution around the fixed center determined by the crystal axis direction, which coincides with the center of the circle of the radius R = |θh | shown by the dashed line in Fig. 86a. The horizontal projection of the distribution is shown in Fig. 86b. It has only a very small negative shift of its maximum and can be compared with the corresponding distribution for the bent crystal (Fig. 84d). The experiment [114] has shown possibility to deflect high energy negative particles in the quasi-bound regime of doughnut scattering DSB by the angles significantly larger than the critical angle ψ1 . The dependence of the beam fraction deflected in the DSB regime on the crystal orientation was obtained. The large bend angle allowed to observe the volume capture of π − mesons into the DSB regime, whose probability was larger than 7%. Particles in the DSB regime move along the bent atomic strings. Therefore, at the vertical inclination of the bend plane θv they obtain also the corresponding vertical deflection θy = θv . The beam deflection opposite to the crystal bend for large orientation angles was observed, which was stimulated by doughnut scattering of π − mesons in the previous crystal layers. 6. Experimental studies of crystal assisted collimation of the SPS beam halo 6.1. Observation of the SPS beam halo deflection by a bent crystal The main goal of the UA9 experiment is to investigate the beam halo collimation assisted by a bent crystal at the CERN-SPS. Fig. 87 shows the schematic layout of the UA9 experiment. The collimation system is made of two stations

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Fig. 86. (a) The deflection angle distribution of π − mesons after passage through the straight crystal of the same length with the same orientation angle, θh = −129 µrad, obtained by simulation. The randomization circle is shown like in Fig. 85a. (b) The horizontal projection of the distribution shown in (a).

whose relative horizontal betatron phase advance is close to 90 degrees. Both of them are installed at SPS azimuths with large value of the horizontal beta function, upstream of the quadrupoles QF518 and QF520 (marked as QF1 and QF2 in Fig. 87), respectively. The first station contains bent crystals alternatively used as a primary collimator. The two silicon crystals, C1 and C2, of 2 mm long with bend angles of about 150 µrad and 140 µrad, respectively, were used in the first experiment [115]. They were installed in a tank on two in-vacuum goniometers. The crystal C1 was a strip crystal bent along the (110) equidistant planes. The crystal C2 was a quasi-mosaic crystal bent along (111) non-equidistant planes. The crystal orientation and horizontal position relative to the closed orbit were controlled by the goniometers. The BLMs developed for LHC [116] with low noise and high sensitivity and the scintillation telescopes [117] were used as beam loss monitors. The second station contains a 60 cm long tungsten absorber (TAL) used as a secondary collimator, its radial and vertical dimensions are 2 and 5 cm, respectively. The TAL is installed in a tank upstream of the quadrupole QF2, where the angular deflection of protons by the crystal is transformed into a large horizontal displacement.

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Fig. 87. The simplified layout of the UA9 experiment. The crystal primary collimator C is located upstream the quadrupole QF518 (QF1). The TAL acting as a secondary collimator–absorber is upstream the quadrupole QF 520 (QF2). The scraper SC is used as a target in a high dispersion HD area to measure off-momentum particles produced in the collimation process. BLM, BLM1 and BLM2 are the beam loss monitors. Table 5 Relevant accelerator parameters. Parameter

C1

COL

MED

TAL

βx (m) σx (mm) ∆µx (2π )

81.953 0.980 0

24.134 0.532 0.1621

28.211 0.575 0.1781

87.665 1.014 0.2445

A MEDIPIX-type detector (MED) with 256 × 256 square pixels [118] is installed in a Roman pot 20 m upstream of the TAL. The pixel size is 55 µm. The Roman pot can bring the detector into the deflected beam. The LHC collimator prototype (COL) with two 1 m long graphite jaws is located about 2 m upstream the MEDIPIX. Both jaws of the COL are moveable. The inner jaw was moved to intercept the deflected beam. During the tests [115], the SPS proton beam was accelerated and stored at 120 GeV/c with a typical transverse normalized emittance ε ∗ = 1.5 µm · rad and nominal betatron tunes QH = 26.13 and QV = 26.18. The relevant accelerator parameters at the azimuths of the UA9 elements are listed in Table 5, where βx is the value of the horizontal beta-function, σx is the RMS value of the horizontal beam size and ∆µx is the horizontal phase advance between the elements. In most of the cases, the beam was made of a single bunch with few 109 up to 1011 protons and the diffusion speed was governed by the multiple scattering of circulating protons with the residual gas in the beam pipe. When the crystal was positioned in the periphery of the circulating beam, the beam lifetime was varying from a few minutes up to 10 h (depending on the presence or the absence of a transverse beam excitation) and the number of particles hitting the crystal could be varied from 10 to 103 protons per turn. The amplitude growth of particle oscillations per turn in average was smaller than 0.1 nm in the absence of transverse excitation. At the beginning of each run the LHC collimator was centered with respect to the closed orbit of the beam. The collimator half gap X1/2 determined the reference beam envelope. After that, all the UA9 movable elements (crystals, MEDIPIX and TAL) were put just at the edge of the LHC collimator shadow that is at the same initial distance (in σx ) from the orbit, using the readings of the closest beam loss monitor downstream. After the alignment of the transverse positions, all the elements with the exception of one of the crystals and the TAL were moved back to the garage positions at large distances (more than 30 σx ) from the orbit. The crystal in use was then moved closer to the beam, while the TAL was additionally moved away from the orbit. In this configuration the TAL intercepts the particles deflected by the crystal in channeling states at the same turn, still allowing a free passage to the particles deflected by multiple scattering in the crystal, therefore providing them additional probability of channeling in the following turns. In these positions, we started scanning the horizontal angles of the crystal using the goniometer. In Fig. 88, curve 1 shows the dependence of the BLM1 counts on the angular position of the crystal C1, during the goniometer scan. The BLM1 registers secondary particles generated due to inelastic nuclear interactions in the crystal. The UA9 elements were aligned with the COL half gap X1/2 = 2.2 mm (about 4σx ). The alignment positions of the crystal and the TAL were 4.05 mm and 4.19 mm, respectively. For the measurements the crystal and the TAL were moved. Their operational positions became XC 1 = 3.55 mm and XTAL = 5.69 mm for the crystal and the TAL, respectively. Curve 2 shows the result of a simulation performed as described in [119]. The agreement between the experiment and simulation results is fair. The angular dependence and its width are well reproduced. The dot-dashed horizontal line in Fig. 88 shows the loss count level for the crystal orientations far from the main crystallographic axes and planes when it works as an amorphous substance (‘‘amorphous’’ orientations). The number of particle passages through the crystal before they either reach the TAL aperture or are lost due to inelastic interactions is determined by the angular kick values due to multiple Coulomb scattering in the crystal. A minimum appears in the loss count at the goniometer angle of about −1700 µrad (channeling orientation). For this orientation the fraction of beam halo protons deflected by the crystal in channeling states avoiding inelastic nuclear interactions is maximal: the beam loss decreases by five times with respect to the amorphous orientation of the crystal in the considered case. There is a wide area of a significant beam loss reduction (‘‘shoulder’’) on the right of the minimum observed also in [15,120], which is due to volume reflection (VR) of halo particles in the crystal. The angular kick of protons due to volume reflection is θv r = 22 µrad whereas the RMS multiple scattering angle in the crystal is θms = 10 µrad. Therefore, in volume reflection mode the particles perform a smaller number of passages through the crystal to reach the TAL aperture than for the amorphous orientations of the crystal. This reduces the total number of inelastic interactions in the crystal. There is a second minimum in the expected dependence represented by curve 2. This minimum is at an

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Fig. 88. (1) The dependence of the BLM1 count on the angular position of the crystal C1, (2) The dependence of the number of inelastic nuclear interactions of protons in the crystal on its orientation angle obtained by simulation. The dot-dashed line shows the level of the beam losses for the amorphous orientation of the crystal.

angular distance from the channeling orientation about equal to the bend angle (150 µrad). Its explanation is that for these specific orientations, the whole VR region of the crystal is on the same side relative to the beam envelope direction. Therefore, angular kicks due to VR always increase the oscillation amplitudes of particles and they more quickly reach the TAL. The experimental scan (curve 1) presented here has no second minimum and the minimum due to the channeling is not as deep as the theoretical one (curve 2). This can be caused by some crystal torsion producing different orientations of the crystal planes along the vertical direction (the torsion of these first crystals installed in the SPS can be about 10 µrad/mm). In this case, the different vertical parts of the crystal deflect halo particles in a different way and their effect is averaged. Instabilities of the goniometer (mainly its vibrations) or of the circulating beam can also cause a smoothing down of the main and the second minimum. To investigate the distribution of the deflected beam, the crystal orientation was fixed close to the beam loss minimum detected by the angular scan. The LHC prototype collimator (COL) was used to scan horizontal positions near the circulating beam. During this motion the inner jaw of the COL eventually intercepted the beam halo deflected by the crystal in channeling states and the BLM detector downstream the COL registered secondary particles generated due to inelastic nuclear interactions of protons in the COL. Fig. 89 shows the dependence of the BLM signal on the COL position. The BLM signal value is proportional to the number of protons intercepted by the COL. The BLM signal quickly increases when the COL intercepts the beam halo deflected in channeling states by the crystal. Then the signal growth becomes slow when the COL intercepts the tail of protons dechanneled in the crystal. Then a sharp growth of the BLM signal occurs because the COL reaches the TAL aperture and touches the secondary multi-turn halo produced by multiple scattering of protons in the crystal. The COL scan provides the integral of the deflected halo distribution. The dashed line presented in Fig. 89 shows a fit of the experimental data with an error function. The fit center is at am = (−7.82 ± 0.007) mm and its RMS value σa = (0.51 ± 0.005) mm. Since an error function is the integral of a Gaussian, am and σa are the parameters of the Gaussian, which fits the position and width of the beam deflected in channeling states. The angular kick produced by the crystal, which gives the registered position am of the deflected beam from the beam orbit at the COL location, should be equal to θk = −173 µrad. The deflection efficiency of halo protons in channeling states can be estimated as the ratio of the error function plateau value to the maximum value at the scan edge (see the right vertical axis in Fig. 89). For the considered case, the estimated value of the channeling efficiency is Pch = (75 ± 4)%. The measured value of the angular kick produced by the crystal was larger than the crystal bend angle α , which is about 150 µrad. Our simulation demonstrates that this effect as well as a lower channeling efficiency can be explained by a non-perfect alignment of the crystal with the beam halo. Fig. 90 shows the visual image of the beam deflected with the crystal C1 (a) and its horizontal projection (b) obtained with the MEDIPIX detector. The experimental results [115] have proven that the larger part of the beam halo can be directed with a bent crystal onto the absorber far from its edge. Besides, the nuclear loss rate in the aligned crystal registered in the experiment was reduced by up to five times with respect to the crystal amorphous orientation.

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Fig. 89. The dependence of the BLM signal on the horizontal position of the inner jaw of the COL during its scan from the beam periphery towards the orbit (acoll = 0) when it intercepts the beam deflected by the crystal 1. The error function fit is shown by the dot-dashed line. Ncoll is proportional to the number of secondary halo protons hitting the collimator in its current position; Ncry is proportional to the total number of the secondary halo particles.

Fig. 90. The image of the beam deflected with the crystal 1 (a) and its horizontal projection (b) obtained with the Medipix detector. Table 6 Crystal parameters. Crystal

Length (mm)

Bend angle (µrad)

Torsion (µrad/mm)

Miscut angle (µrad)

C3 C4

2.1 2.0

165 176

1 0.6 - 1.0

90 200

6.2. Role of the miscut of a crystal primary collimator A new goniometer constructed and produced by IHEP with two new low torsion bent crystals C3 and C4 was installed in the upstream position to the old tank with the crystals C1 and C2 . This new goniometer has an angular accuracy of about ±10 µrad, which is four times better than for the goniometers supporting the crystals C1 and C2. The parameters of C3 (quasi-mosaic) and C4 (strip) crystals are presented in Table 6. The measurements [121] with the quasi-mosaic crystal C3 for the SPS beam of 120 GeV/c protons showed the dependence of beam losses in the crystal on its orientation similar to the dependence observed for the strip crystal C1. The beam loss reduction in the crystal aligned for channeling was close, Rbl (C 3) = 5.8. The crystals C3 and C4 have low torsion values, which excluded the torsion role in the deflection efficiency reduction, but sufficiently large miscut angles θm between the crystal surface and the crystallographic planes. The problem of a crystal miscut was already considered in the first experiments on the SPS beam extraction with a bent crystal [9]. A right orientation relative to the incident beam was determined for a bent crystal with a miscut. This right orientation was applied for all bent crystals used in the UA9 experiment on the SPS beam collimation. Fig. 91 shows the situations when

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Fig. 91. The crystal primary collimator with the miscut angle θm = 0 (a) and θm ̸ = 0 (b). For (a) the first hits of halo particles occur with the entrance face AC, where particles are captured in channeling and deflected. For (b) the first hits occur with the side face AB, where particles pass the crystal as amorphous substance and can have inelastic interactions with the crystal nuclei.

the crystal primary collimator has θm = 0 (a) and θm ̸ = 0 (b). In both cases the crystal planes at its entrance face AC are parallel to the beam envelope (the dashed line touching the crystal on the left). The crystal touches the beam envelope at the point A of its entrance face in the case (a) and at the point B of its exit face in the case (b). As a result, a particle with the same impact parameter b enters the crystal from its entrance face and is captured into channeling in the case (a) and enters through its side face with a large angle to the bent planes and traverses the crystal acting as an amorphous material in the case (b). In the last case a particle acquiring some amplitude increase due to multiple scattering can enter the crystal next time during the following turns through its entrance face AC and be captured into channeling. However, some particles entering through the side face at the first passage will be lost due to inelastic interactions. These losses are additional in comparison with the case (a). Thus, the beam loss reduction in channeling Rbl for the real crystal with a miscut angle should be smaller. The decrease of Rbl due to the crystal miscut depends on the ratio between the impact parameter values in the first hits and the projection value of the crystal side face ∆ (see Fig. 91). In our case ∆ = R(cos(θm − α ) − cos(θm )) = 0.224 µm, where R and α are the crystal bend radius and angle (see Table 6). In the experiment [70] the time required to repopulate the halo scraped by the TAL near the crystal distance from the orbit was measured. This gave the average rate of the oscillation amplitude growth as λ = 2.2 µm/s. According to our simulation the average impact parameter of particles with the crystal at this rate is b = 0.28 µm (it is comparable with the crystal side face ∆) and the crystal miscut more than doubles the beam losses at perfect alignment. The beam loss reduction obtained by simulation for C4 taking into account its miscut became closer to the experimental value [70]. 6.3. Leakage reduction in the SPS beam halo collimation with a bent crystal The performance of the existing collimation system is estimated using the beam loss map around the ring measured by the distributed system of BLMs. However, for the purpose of study and optimization of a new crystal-assisted collimation system, it is better to have a sensitive and fully controlled instrument for measurement of the leakage, because the sensitivity of the SPS BLMs around the ring is insufficient. Off-momentum particles with δ = p/po − 1 ̸ = 0 (p and po are momenta of a circulating particle and of the synchronous particle, respectively) produced in the crystal or in the absorber and escaping from the collimation area have large displacements from the orbit at high dispersion azimuths, xδ = Dx δ , where Dx is the value of the dispersion function. A new station was installed in the first high dispersion (HD) area downstream the absorber TAL for an optimal detection of the off-momentum halo escaping from the collimation area. The station consists of a beam scraper SC, a movable Cherenkov detector and the beam loss monitor BLM2. The low noise and high sensitivity BLM developed for the LHC and the scintillation telescope installed in the first high dispersion (HD) area downstream the TAL absorber are used for

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Fig. 92. Beam of 270 GeV/c protons. Curves (1) are the dependencies of beam losses observed in the crystal (a) and in the HD area target (b) on the angular position of the crystal C4 normalized to its value for the amorphous orientation of the crystal (dot-dashed line). (2) The dependence of the number of inelastic nuclear interactions of protons in the crystal on its orientation angle obtained by simulation with taking into account the crystal miscut.

leakage measurement in the UA9 experiment. Off-momentum particles produced in the collimation area have here the first possibility to hit the pipe or a target at the beam periphery. The experiments [70] on crystal collimation were performed with the SPS beam of 270 GeV/c protons. It should be noted that the crystal channeling characteristics were changed by changing the particle energy. The critical channeling and volume reflection angles, θc , θv r ∼ p−1/2 , decreased by a factor of 1.5 and the angle of multiple Coulomb scattering θms ∼ p−1 decreased by a factor of 2.25. Decreasing particle scattering should require a larger number of passages through the non-oriented crystal to reach the TAL aperture. This causes growth of the beam losses and consequently the loss reduction Rbl should increase. In Fig. 92 curves 1 show the dependencies of beam losses observed in the crystal (a) and in the HD area target (b) on the angular position of C4 for the SPS beam of 270 GeV/c protons. The crystal and the TAL collimation positions were XC 4 = 5.6 mm (9 σx ) and XTAL = 8.92 mm (13.5 σx ). The gap between them at the TAL azimuth was therefore Xoff = 2.89

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mm (4.4 σx ). It is larger than for the case considered above with 120 GeV/c protons. Here the measured beam loss reduction in the aligned crystal Rbl = 20. This Rbl value is considerably larger than for 120 GeV/c protons, which is probably caused by the larger gap and by smaller angular kicks of multiple scattering. Both these factors lead to a beam loss increase for the amorphous crystal orientation, which increases Rbl . The second minimum in the VR region caused by the amplitude drift due to volume reflections is clearly seen in the experimental dependence. Simulation (curve 2 in Fig. 92a) well describes the loss dependence character in the crystal and its level in the VR region. The duralumin scraper of a 10 cm length was used as the HD area target, its radial and vertical dimensions are 2 and 5 cm, respectively. It was put far from the orbit, Xsc = 41 mm, to avoid disturbance of the collimation. The crystal angular position corresponding to the beam loss minimum in the scraper (Fig. 92b) is the same as for the crystal. Here the beam loss reduction Rbl (HD) = 6. The loss reduction in the VR area is visible but considerably smaller than for the case of 120 GeV/c protons. A possible reason is more frequent scattering of protons from the TAL back to the beam because of smaller impact parameters caused by smaller angular kicks acquired in the crystal, that is the TAL contribution to the off-momentum halo generation is larger. Experiments on the crystal collimation of the SPS stored beam of Pb ions with 270 GeV/c per charge were also carried out. The crystal channeling characteristics are determined by the ratio of pz = p/Z and they are the same for protons and Pb ions with the same pz . However, their interactions with a crystal are considerably different. As noted in [121] the difference is caused by considerably larger ionization losses, which are proportional to Z 2 . Their average value in a 2 mm long silicon crystal for the considered energy is about 7 GeV. This causes a strong orbit shift and, as a consequence, a strong increase of oscillation amplitudes. Three passages of Pb ions through the crystal cause their debunching. Besides, the beam attenuation cross-section in a silicon crystal for Pb ions is more than 10 times larger than for protons σtot = σh + σed = 4.35 + 1.37 = 5.72 b, where σh and σed are the contributions from inelastic nuclear interactions and electromagnetic dissociation, respectively [68,122,123]. The attenuation length is about 3.5 cm, therefore about 5.6% of Pb ions will be lost in the passage through the non-aligned crystal. This should decrease the contribution of multiple passages of ions through the crystal in the extraction efficiency. Curves 1 in Fig. 93 show the dependencies of beam losses in the crystal (a) and in the HD area target (b) on the crystal orientation for the SPS beam of Pb ions with 270 GeV/c per charge. The collimation positions of the crystal and the TAL were XC 4 = 3.1 mm (3.4 σx ) and XTAL = 6.9 mm (7 σx ). The gap between them at the TAL azimuth was Xoff = 3.56 mm (3.6 σx ). The roman pot used as a target was installed at the distance XRP = 35.7 mm. A remarkable similarity of the beam loss dependencies in the crystal and in the RP is observed. The loss reduction at the channeling orientation is larger than 7 in both cases. This is evidence that the TAL contribution to the formation of the off-momentum halo is negligible because practically all Pb ions should be lost in the TAL due to inelastic nuclear interactions. Curves 2 show the simulation results. They describe sufficiently well the experiment near the channeling minimum. The stored multi-bunch beam (train of 48 bunches spaced by 25 ns) has been set up for 270 GeV/c protons to provide a possibility for the beam loss measurements in regions of the SPS ring far from the crystal where the sensitivity of the BLMs is not sufficient for the single bunch beam. Fig. 94 shows the beam losses detected by the BLMs located near the main quadrupoles spaced by 32 m in the SPS sector 6, which is 1 km downstream of the crystal, for two crystal orientations corresponding to channeling and amorphous modes. The losses for the channeling orientation are few times smaller than for the amorphous one. Thus, the experiment [70] showed that the aligned crystal really can reduce the collimation leakage. 6.4. Optimization of the crystal assisted collimation of the SPS beam The collimation leakage is formed during the interactions of the beam halo particles with primary and secondary collimators. The usage of a bent crystal as a primary collimator can decrease the leakage formed both in the crystal and in the secondary collimator–absorber. A bent crystal should be first aligned with the collider beam halo to be an efficient primary collimator. However, not all halo particles which hit the aligned crystal are deflected at the first passage. They can be lost due to inelastic nuclear interactions inside the crystal or will hit the absorber edge if its distance from the orbit is about the same as for the crystal. The possibility of extra passages of halo particles through the crystal, to be deflected in channeling regime during subsequent turns, may be realized when the absorber is placed with some offset relative to the crystal. The dependence of the collimation leakage upon the offset value between the TAL absorber and the crystal has been studied in the experiment [124]. The SPS stored beam of Pb ions with pz = 270 GeV/c was used in the experiment [124]. There were 10 bunches in the beam separated by 200 ns, with an average number of about 1.1 × 108 ions per bunch. The two-sided collimator COL was centered relative to the closed orbit with the half gap X1/2 = 1.735 mm (about 4.2 σx ). The alignment positions relative to the reference beam envelope determined by the value X1/2 were found for the crystal and the absorber as well as for the scraper. Then the crystal and the TAL were placed at a distance XC 4 = 2.77 mm (3.55 σx ) and XTAL = 12.8 mm (16.34 σx ) from the orbit, respectively, so the TAL offset relative to the crystal was Xoff = 10 mm. Under these conditions, an angular scan was made and the dependence of beam losses in the crystal on its angular position, similar to the one shown in Fig. 93a, was obtained. As a result, the best alignment of the crystal relative to the beam envelope direction was found at which the losses have a minimal value due to a maximal fraction of channeled halo particles.

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Fig. 93. Beam of Pb ions with 270 GeV/c per charge. The same as in Fig. 92.

For this channeling orientation of the crystal the measurements of the beam losses in the crystal with BLM1 , and in the HD area with BLM2 , were made for different distances of the TAL from the orbit, that is for different values of its offset relative to the crystal. A 10 cm long tungsten scraper SC, its radial and vertical dimensions are 2 and 5 cm, respectively, used as a target for off-momentum particles in the HD area was placed at a distance XSC = 25.4 mm from the orbit to increase the beam losses detected with BLM2 . Fig. 95 shows the dependence of beam losses observed in the crystal (a), and in the HD scraper (b), on the TAL offset value. The losses in the crystal monotonically increase by a factor of 3.5 when the offset increases from 1 mm to 10 mm. Conversely, losses in the scraper have a minimal value for offsets of 3–5 mm. They increase by a factor of about 3 for the smallest and largest Xoff . This beam loss behavior observed in the HD area means that the number of the off-momentum particles produced both in the crystal and in the TAL (the collimation leakage) has a minimum in this range of offset values. The behavior is caused by the existence of two sources of collimation leakage in the crystal and in the TAL edge. The SPS beam halo collimation for these experimental conditions was studied by simulation. Fig. 96 shows the calculated distributions of the impact parameters of Pb ions with the TAL for Xoff = 1 mm (1) and 10 mm (2). There is a sufficiently large maximum near the TAL edge for Xoff = 1 mm. Therefore, in this case, a large number of off-momentum particles could be produced by the TAL and scattered back into the beam. Conversely, for the large offset value Xoff = 10

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Fig. 94. The beam loss map in the SPS sector 6 for the amorphous and channeling (the histogram filled by blue) orientations of the crystal.

mm the TAL edge population is small. Fig. 97 shows the calculated dependence on the TAL offset value of the number of inelastic nuclear interactions of Pb ions in the crystal (a) and for the number of particles hitting the TAL edge with the impact parameters smaller than 0.5 mm (b). Our simulation confirms that the number of particles with a large momentum reduction produced in the TAL should decrease, whereas that produced in the crystal should increase, with increase of the TAL offset. From the observed dependence of beam losses in the HD area (Fig. 95b) we may conclude that collimation leakage occurs mainly in the TAL edge for small offset values, while for large ones it originates in the crystal. As a result, the leakage has a minimal value for some offset. Similar experiments were performed with an SPS beam of 270 GeV/c protons. Fig. 98 shows the observed beam loss dependence in the crystal (a) and in the scraper (b) on the TAL offset value. The experimental points have large errors in this case due to large beam instabilities. Beam losses in the crystal increase considerably faster for large offsets than is the case with Pb ions. Losses in the scraper begin to increase after Xoff = 2.8 mm where they have a minimal value. In this case the measurement was not performed for a small offset of about 1 mm but our other experiments always showed an increase in this range of Xoff . Therefore, we may conclude that for protons the collimation leakage also has a minimum but for Xoff = 2 − 3 mm. Thus, the optimal TAL offset is larger for Pb ions than for protons. The crystal channeling characteristics are the same for protons and Pb ions with the same momentum per charge pz . The difference is caused by considerably larger ionization losses of Pb ions, which are proportional to Z 2 and whose average value in the crystal is about 7 GeV. The momentum change shifts the particle orbit and, as a consequence, a considerable increase of its betatron oscillation amplitude occurs. This large amplitude increase by Pb ions requires a larger TAL offset to allow subsequent passages through the crystal after their first unsuccessful ones. Thus, the experiment [124] showed that the crystal assisted collimation of a circulating beam halo can reach maximal efficiency with usage of the optimal absorber offset. 6.5. Observation of strong leakage reduction in crystal assisted collimation of the SPS beam In the experiments [70,115,121,124] on the SPS beam halo collimation assisted by a bent crystal, the collimation leakage was measured by the monitor BLM2 installed in the first high dispersion (HD) area downstream of the collimator– absorber where off-momentum particles produced in the collimation area have the first possibility to hit the beam pipe. A considerable reduction of the collimation leakage was always observed for the channeling orientations of all tested crystals. For the case of an SPS beam of Pb ions with 270 GeV/c momentum per unit charge, the loss reduction observed in the HD area by BLM2 was practically the same as in the crystal because the probability of backscattering from the tungsten absorber is very small for Pb ions due to their high probability of nuclear interactions. In the case of protons, the beam loss reduction detected by BLM2 was always smaller than detected by BLM1 in the crystal because of the contribution of particles emerging from the absorber [70]. In the UA9 experiments, a 60 cm long tungsten bar is used as a secondary collimator–absorber. It is insufficient for the full absorption of the halo protons. The nuclear inelastic cross-section for 270 GeV/c protons in tungsten

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Fig. 95. Beam of Pb ions with 270 GeV/c per charge. The dependence of beam losses observed in the aligned crystal (a) and in the HD scraper (b) on the TAL offset value. The losses are normalized to its minimal value.

σin = 1.725 b [125] and the interaction length Sin = 9.18 cm corresponds to the attenuation probability of the proton beam Pin = exp(−L/Sin ) = 1.45 × 10−3 . Besides, protons after losing a small part of their momentum by diffractive scattering can also remain in the beam. Protons deflected by a crystal deeply into the absorber but emerging from it with some momentum loss give a large contribution to the beam losses measured by the monitor BLM2 . Thus, the imperfect absorption of halo protons in our collimator–absorber leads to an underestimation of the efficiency which is achievable with a crystal assisted collimation system. The situation may be considerably improved already with a 1 m long tungsten absorber, Pin = 1.86 × 10−5 . Multi-turn simulation of the crystal assisted collimation of the SPS beam halo with a SixTrack code and real beam pipe aperture [14] allowed finding the azimuth in the first high dispersion area where the loss reduction for the crystal channeling orientation should be considerably larger than that observed in the position of BLM2 . New beam loss monitor BLM3 has therefore been installed at this azimuth downstream the BLM2 location. The experiment [126] where this new

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Fig. 96. The calculated distributions of the impact parameters of Pb ions with the TAL for offset values relative to the aligned crystal Xoff = 1 mm (1) and 10 mm (2). Table 7 Relevant accelerator parameters. Parameter

C1

TAL

QF3

βx (m) σx (mm) ∆µx (2π )

90.945 0.9047 0 −0.857

87.660 0.888 0.2405 −6.8 × 10−4

107.023 0.981 0.4909 3.772

Dx (m)

BLM3 was used allowed observing the collimation leakage reduction which was larger than the beam loss reduction in the crystal. Second new goniometer constructed and produced by IHEP with two new bent crystals C1 and C2 was installed instead of the old tank. The silicon strip crystal C1 with the bend angle of 165 µrad was used as a primary collimator. The crystal miscut angle between the crystal surface and the (110) crystallographic planes is about 10 µrad, which is much smaller than for the crystals used in our earlier experiments. This feature helps to reduce the particle losses at the crystal channeling orientation. Fig. 99 shows the dispersion function change in the collimation area and in the first HD area downstream of the absorber. The positions of the monitors BLM2 and BLM3 are close to the first and the second dispersion maximums, respectively. Additionally, it is shown that the betatron phase advance between the absorber TAL and the monitor BLM2 equals approximately 90◦ . It is very important that protons strongly scattered in the TAL should acquire a maximal betatron deviation from the orbit near BLM2 . The targets limiting the accelerator aperture installed in the HD area were not used in this experiment. The beam loss monitors BLM2 and BLM3 detected secondary particles generated by protons in the pipe walls. The SPS stored beam of 270 GeV/c protons consisting of 12 bunches with a total intensity of 1.3 × 1012 protons was used in this experiment. The relevant accelerator parameters at the azimuths of some UA9 elements are listed in Table 7. After the alignment the crystal C1 and the absorber TAL were placed at a distance XC 1 = 4 mm (4.5 σx ) and XTAL = 7 mm (7.94 σx ) from the orbit, respectively. The TAL offset relative to the crystal was about 3 mm. Under these conditions a scan of the horizontal angular positions of the crystal was performed. Fig. 100a shows the dependence of beam losses in the crystal observed with BLM1 (curve 1). The left minimum corresponds to the crystal orientation optimal for channeling. The beam loss reduction behind the crystal in channeling is Rbl (1) = 11.8. Curve 2 shows the dependence of nuclear inelastic interaction number in the crystal on its orientation obtained by simulation of the collimation process with the detailed calculation of particle trajectories in the crystal as described in [119]. The simulation for a given particle was finished when it hits the TAL or when a nuclear inelastic interaction occurred in the crystal. The calculated dependence describes well both the width and the shape of the experimental dependence but gives smaller values of the beam losses for the channeling as well as for the VR orientations of the crystal. Curve 3 shows the multi-turn simulation results obtained by using the SixTrack code to transport particles in the SPS. The process of diffractive scattering of protons in the crystal and TAL was taken into account. The interaction of

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Fig. 97. The calculated dependence on the TAL offset value of the relative number of inelastic nuclear interactions of Pb ions with 270 GeV/c per charge in the crystal (a) and for the relative number of ions hitting the TAL edge with the impact parameters smaller than 0.5 mm (b).

protons with the crystal was considered using approximations for different processes described in [14]. In this case, the calculated width of the angular dependence is smaller than in the experiment because of the approximations used for the description of the proton interactions with the crystal. However, the loss value in the VR region is in better agreement with the experiment. The beam loss reduction observed in the HD area with the monitor BLM2 was Rbl (2) = 8.3, which is smaller than behind the crystal as in the previous experiments. Fig. 100b shows the beam loss dependence on the crystal orientation observed in the new position with monitor BLM3 . The angular dependence is the same as behind the crystal but the beam loss reduction in channeling is considerably larger, Rbl (3) = 18.1. The reduction values of the proton losses on the beam pipe obtained by the simulation with the SixTrack code are 9.1 and 20.1 upstream BLM2 and BLM3 , respectively. The observation of the large reduction of collimation leakage with BLM3 was possible because the number of particles deflected by the crystal in channeling regime deeply inside the TAL but emerging from it considerably reduced on their way to BLM3 .

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Fig. 98. Beam of 270 GeV/c protons. The same as in Fig. 95.

The particles were lost at the beam pipe near the first dispersion maximum where the betatron phase advance from the TAL exit is about 90◦ . Actually, the RMS deflection due to multiple Coulomb scattering in the TAL taking into account nuclear elastic scattering, is large, θms = 0.742 mrad, and the average ionization losses are estimated as δ = −7.62 × 10−3 . The betatron amplitudes for protons deflected by the crystal in the channeling regime are about 15 mm at the TAL entrance face. At the TAL exit, protons deflected through θms due to multiple scattering will have an amplitude of Xm = 68.6 mm. Their amplitude near BLM2 will be Xm = 75 mm. Besides, the average shift for these particles due to high dispersion here is Xδ = δ Dx = −28.75 mm. The horizontal and vertical dimensions of the SPS beam pipe in the quadrupoles QFs are 76 mm and 19.15 mm, respectively. The simulations show that a larger fraction of particles deflected by the crystal but which avoided absorption in the TAL should be lost in this part of the pipe. Thus, the situation observed in the experiment [126] was close to the ideal case when full absorption of particles with large impact parameters occurs in the secondary collimator.

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Fig. 99. The dependence of the dispersion function on the distance along the collimation area and the first HD area downstream of the TAL absorber.

6.6. Simulation of crystal assisted collimation of the SPS beam Model for simulation of the UA9 crystal-assisted collimation experiment at the CERN-SPS and some simulation results are presented in [119]. Four azimuths in the SPS ring were considered in the simulations of collimation experiments. Linear 6-dimensional transfer matrices M(6,6) were used to transport particles between the azimuths. The azimuths correspond to the locations of the crystal primary collimator, the secondary collimator–absorber TAL, some target in the first high dispersion area downstream the absorber and the accelerating cavity. Far off-momentum particles of the beam halo produced in the crystal and the absorber were detected at the HD azimuth. The momentum of particles was changed by the accelerating voltage at the azimuth of the RF cavity

δ=δ+

eV Eo

sin(π + 2π hl/Co ) ,

(66)

where δ = p/po − 1 is the deviation of the particle momentum p from its synchronous value po , V and h are the amplitude and harmonic number of the accelerating voltage, l is the longitudinal deviation of a particle in a bunch, Co is the orbit length. The history of every particle was finished in the inelastic interaction with nuclei in the crystal or when it hits the absorber. The crystal and absorber are located in the inner side of the SPS orbit in the UA9 experiment. The simulation for the real conditions of the experiment [70] on the collimation of the SPS beam of 120 GeV/c protons using the silicon strip crystal C4 bent along the (110) planes was considered in [119]. The initial distributions of halo particles in the transverse and longitudinal coordinates were generated at the crystal azimuth. First of all the longitudinal coordinates (δ, l) were determined from the Gaussian distributions. The RMS deviation from the bunch center was measured in the experiment [70], σl = 0.18 m. The RMS deviation of the particle momentum was determined using the theoretical relation between the values of l and δ , σδ = 0.6 × 10−3 . Initially, each particle trajectory should pass very close to the crystal without touching it. For this reason, the betatron oscillation amplitude Xm of a particle was chosen to satisfy the condition Xm + d = |XC 4 | − |Dx δm | ,

(67)

where δm is the amplitude value of δ in synchrotron oscillations. The value of d = 30 µm was used. The betatron phases of particles were randomly and uniformly distributed in the interval (0,2π ). Fig. 101a shows the initial trajectory of a synchronous particle, δ = 0, in the horizontal phase space (dot-dashed line). The oscillation amplitudes of the beam halo particles increase due to scattering onto the residual gas and other reasons. As a result halo particles begin to hit the crystal at the phases of betatron oscillations near ϕ = π . The first hit trajectory is shown by solid line in red. In the UA9 experiment the time required to repopulate the halo scraped by the TAL near the crystal distance from the orbit was measured. This gave the average rate value of the oscillation amplitude growth, λ = 2.2 µm/s. That is the average amplitude increase per turn was smaller than 1 nm. The situation shown in Fig. 101a for the synchronous particle is an idealization. All halo particles really have some momentum deviation δ , which leads to the additional trajectory shift Xδ = Dx δ . The synchrotron oscillations of the particle

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Fig. 100. Beam of 270 GeV/c protons. Curves (1) show the dependence of beam losses observed in the crystal (a) and in the HD area with the BLM3 monitor (b) on the angular position of the crystal C1 normalized to its value for amorphous orientation of the crystal (dot-dashed line). Curves (2) and (3) show the dependence of the number of nuclear inelastic interactions of protons in the crystal on its orientation angle obtained by simulation according to [119] and [14], respectively.

momentum produce the oscillations of the particle trajectory near the orbit, Fig. 101b. At the crystal azimuth, the maximal shift of the trajectory from the orbit for δm = σδ is Xδ m = 0.47 mm. This value is considerably larger than the increase of the betatron oscillation amplitude Xm during the synchrotron oscillation period Ts , which is Ts ≈ 150 To for δm = σδ , where To is the revolution time. Due to the orbit change at the crystal azimuth, induced by the synchrotron oscillations, the first hit with the crystal always occurs when a particle has the synchrotron oscillation phase near (δm ,0) and the betatron oscillation phase near ϕ = π . The impact parameters become considerably larger than the average amplitude increase per turn. Their values result from the interplay of the diffusion rate and the particle oscillations in the longitudinal and horizontal directions. Fig. 102 shows the impact parameters and angles of particles at the first hits with the crystal, which were obtained by simulation with the average diffusion rate λ = 2.2 µm/s and λ = 8.8 µm/s. The average impact parameter is b = 0.28 µm in the case with λ = 2.2 µm/s. The width of the impact angle distribution is FWHH≈ 2 µrad, which is much smaller than the critical channeling angle θc . Therefore, a high channeling efficiency was possible and really observed. The average

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Fig. 101. Schematic picture. a) For a synchronous particle, δ = 0. The initial trajectory (dot-dashed) and the first hit trajectory with the oscillation amplitude sufficient to touch the crystal (solid). The position of the crystal is shown. (b) For a particle with δ ̸ =0. The particle trajectories shifted from the orbit towards the crystal for the momentum deviation δ = δm and to the opposite side for δ = −δm during the synchrotron oscillations of the particle momentum.

impact parameter of halo particles is comparable with the width of the non-effective crystal cross-section ∆ because of the C4 crystal miscut. As it was already mentioned above, this means that a considerable part of the beam halo enters the crystal side face at the first hits and explains the loss increase of about 140% in the aligned crystal C4 obtained in the simulation. The distributions of the impact parameters and angles of particles at the first hits with the crystal become wider for a larger value of λ. It was considered in [119] how the miscut of the crystal can change its optimal parameters for the particle extraction from the circulating beam of the accelerator. The optimal length L and corresponding radius R exist for the deflection of particles with a given energy by a fixed angle α [27]. The deflection efficiency Pd (α, R) is determined by the probability for particle capture into the channeling regime Pc and the probability for the particles to pass through the whole crystal in the channeling regime Pch (see in Section 2).

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Fig. 102. The distributions of impact parameters (a) and angles (b) at the first hits with the crystal for the SPS beam of 120 GeV/c protons obtained by simulation for the average rate of the oscillation amplitude growth λ = 2.2 µm/s (1) and 8.8 µm/s (2).

When a bent crystal is used for the particle extraction from the circulating accelerator beam the deflection (extraction) efficiency increases due to the contribution of multiple passages (n > 1) of particles through the crystal [127,128] Pex (α, R) = Pex,1 (α, R) + Pex,n>1 (α, R) ,

(68)

where the extraction efficiency at the first passage Pex,1 (α, R) ≡ Pd (α, R). For particles uncaptured into the channeling regime the crystal length growth increases the losses due to inelastic nuclear interactions and the multiple Coulomb scattering in the crystal reducing the probability of their capture into the channeling regime at the subsequent passages. Both these circumstances decrease the contribution of multiple passages of particles through the crystal to the extraction efficiency. Therefore, the optimal crystal length for the particle extraction from the circulating accelerator beam is smaller than for the single passage. Fig. 103 shows the calculated dependences on the crystal length for the extraction efficiency of particles from the circulating beam of the SPS in the conditions of the collimation experiment [70] at the first passage (a) and for the total efficiency (b). For the perfect crystal (curves 1) the extraction efficiency behavior is the same as described above. The efficiency at the first passage is maximal for L = 1 mm (Fig. 103a) whereas the total extraction efficiency has a maximum for a smaller crystal length L = 0.4 mm. For the crystal with a miscut angle θm ̸ = 0 the additional parameter depending

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Fig. 103. The dependences on the crystal length for the extraction efficiency of 120 GeV/c halo protons from the circulating beam of the SPS at the first passage through the crystal (a) and for the total efficiency (b). Curves 1 for a perfect crystal and curves 2 for the crystal with a miscut angle θm = 200 µrad.

on the crystal length appears. The value of the non-effective crystal cross-section ∆ is proportional to the crystal radius and consequently its length L. The extraction efficiency dependences on the crystal length for the crystal with the miscut angle θm = 200 µrad are shown by curves 2 in Fig. 103. The efficiency at the first passages Pex,1 decreases more quickly with increasing the crystal length than in the case of a perfect crystal. This causes also a stronger dependence for the total extraction efficiency. The optimal length of the crystal with the miscut for the beam halo extraction is the same as for the deflection in a single passage, Lopt = 0.4 mm. The loss of the extraction efficiency when we use the crystal length L > Lopt is about two times larger than for a perfect crystal in the considered conditions. The simulation results show that there is no visible dependence of the extraction efficiency of halo particles from the SPS circulating beam on the transverse dimension (thickness) of the crystal because the impact parameters of particles with the crystal are very small. There is only a small increase of about 10% for the particle losses in the crystal when its thickness increases from 0.1 mm up to 1 mm. The simulations of the crystal collimation of the SPS beam halo [119] sufficiently well describe the experimental results. The maximal discrepancy observed for the beam loss reduction at channeling can be considerably reduced if the simulation results will be averaged upon the crystal orientations near the perfect one according to the angular accuracy

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Table 8 Relevant goniometer parameters. Angular range (mrad)

Angular resolution (µrad)

Linear range (mm)

Linear resolution (µm)

10

0.1

40

5

Table 9 Parameters of the ST crystal after its production. Length (mm)

Bend angle (µrad)

Torsion (µrad/mm)

Miscut angle (µrad)

4.1

52

<1

6

of the goniometers. The simulations allowed to explain the effect of the miscut angle between the crystal planes and its surface on the beam loss value in the crystal and the collimation efficiency. 7. Experimental studies of crystal assisted collimation of the LHC beam halo The experiment UA9 studying crystal assisted collimation at the CERN Super Proton Synchrotron (SPS) [70,115,121, 124,126] showed strong reduction of the collimation leakage when the crystal deflector is in channeling conditions [126]. Perfect alignment of the crystal to have deflection of halo particles due to channeling was always obtained quickly by using information from the beam loss monitors (BLM) installed downstream of the crystal. These positive results allowed to start studies with crystals at the LHC. For the (110) planar channels of a silicon crystal at a room temperature the critical channeling angle for 6500 GeV/c protons θc = 2.6 µrad. This imposes challenging requirements to the angular control of the crystals that has to be in the sub-microradian angular resolution range. Two piezo-goniometers with bent silicon crystals for horizontal and vertical collimation of the LHC beam have been installed according to the recommendations [14] in the betatron cleaning insertion of Ring 1 of the LHC in 2014. The present setup for studying crystal assisted collimation was designed with a minimum impact on the LHC collimation system. Layout and the crystal parameters were chosen such that existing secondary collimators can be used to intercept the channeled beams [14]. Goniometers which satisfied the high requirements of sub-microradian angular resolution were developed [129]. Some relevant goniometer parameters are presented in Table 8. An important feature of the goniometer design is its complete transparency for the normal LHC operations. This is ensured by a movable segment of the beam pipe that masks the crystal and the goniometer itself. It is remotely retracted only during the special collimation tests, to allow the crystal insertion. The goniometers were mounted on standard collimation supports using the same fast plug-in technology, which ensures fast handling of the object in the tunnel. The value of the crystal bend angle was chosen for reasons of obtaining the maximal impact parameters of deflected halo particles with the collimator–absorbers while ensuring that the deflected halo should remain at a safe distance from the beam pipe [14]. The bend angle of the crystal for the LHC beam collimation was selected to be α = 50 µrad from these considerations. This bend angle value α can be realized with different crystal length L and consequently with different bend radius R, L = α R. However, the bend radius should be considerably larger than the critical one [2], Rc . For protons with 6500 GeV/c momentum, Rc = 11 m in the (110) silicon channels. The channeling efficiency of protons for the crystal with a given bend angle α , considering the possibility of their multiple passages through the crystal, is maximal when its bend radius R is in the interval (3 ÷ 10)Rc . The length of the LHC crystals was chosen to be L = 4 mm, that is their bend radius R = 80 m ≈ 7Rc . A silicon strip (ST) crystal bent along the (110) planes due to anticlastic deformation was installed for the LHC beam collimation in the horizontal direction. A QM crystal bent along the (111) planes due to the quasi-mosaic effect was installed for the collimation in the vertical plane. The bending devices of both the crystals were made from titanium to reduce possible electron emission from them when the LHC proton bunches pass the azimuths of their location. Table 9 reports the main parameters for the ST crystal at the manufacturing stage. The studies described below show that the bend angle of the ST crystal increased to 65 µrad in comparison with its design value. Fig. 104 shows the horizontal projection of the trajectory of a halo particle deflected by the strip crystal due to channeling at the bend angle α = 65 µrad (solid line): (a) for the beam injection with 450 GeV/c, (b) for the maximum momentum 6500 GeV/c. A dashed line shows the beam envelope at 5.4 σx . This value corresponds to the operational position of the crystals in the first measurements at the LHC [130]. The vertical lines show the longitudinal positions of primary collimators (TCP) and secondary collimators made from CFC (TCSG) as well as the shower–absorber collimators made from a tungsten heavy alloy (TCLA). TCSG and TCLA collimators were placed at about 7 σx and 10 σx , respectively, in the injection case and at about 8 σx and 14 σx , respectively, in the top momentum case. The positions of the beam loss monitors BLM1 and BLM2 for measuring the losses in the crystal and in the first horizontal TCSG1 downstream the crystal, respectively, are also shown. The relevant accelerator parameters at the azimuths of the crystal and the collimators are listed in Table 10, where βx is the horizontal beta-function, σx is the RMS value of the horizontal beam size (for the beam of 6500 GeV/c protons with the RMS normalized emittance ε ∗ = 3.5 µm rad), xim is the impact parameter with the

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Fig. 104. The horizontal projection of the trajectory of a halo particle deflected by the crystal due to channeling at the bend angle α = 65 µrad (solid line up to the first horizontal TCSG1 and then by dot-dashed line to show the trajectory propagation in the case without the collimators): (a) for the beam injection with 450 GeV/c, (b) for the maximum momentum of 6500 GeV/c. Dashed lines show the beam envelope for the crystal distance from the orbit. Vertical lines show the positions of the collimators TCSG and TCLA and the primary collimators TCP (they are not visible for the injection case because they were shifted far from the orbit for the measurements). The positions of BLM1 and BLM2 for measuring the losses in the crystal and in the first horizontal TCSG1 behind the crystal, respectively, are shown schematically. Table 10 Relevant accelerator parameters. Parameter

BC

TCSG1

TCSG2

TCLA1

TCLA2

TCLA3

βx (m) σx (mm)

342.98 0.416

xim (mm) ∆µx (2π )

0

141.22 0.267 3.2 0.0443

338.96 0.414 16 0.3166

161.71 0.286 8 0.3425

66.20 0.183 2.5 0.3979

64.12 0.180 −0.2 0.4456

collimators for a particle deflected by the crystal with 65 µrad, and ∆µx is the horizontal phase advance between the crystal and collimators, TCSG1 and TCSG2 are two horizontal collimators behind the crystal. A single bunch with 1011 protons was injected in the first run on the LHC collimation studies with bent crystals. At the beginning all collimators of IR7 were placed at their standard injection settings: the primary collimators (TCPs) at 5.7 σx , the secondary collimators (TCSGs) at 6.7 σx and the absorbers (TCLAs) at 10 σx . The crystal was aligned precisely to the circulating beam and set at the TCP opening. Then the crystal was moved by 0.5 mm (about 0.3 σx ) towards the beam orbit to become the primary collimator. In this position angular scans with the crystal were performed with the collimators settings listed in Table 11. As mentioned above, well-channeled particles do not experience nuclear interactions, therefore the channeling orientation of the crystal may be found through the beam loss reduction in the crystal. This loss reduction was indeed observed with the BLM downstream the crystal — BLM1 . Three angular scans were made and in all of them the crystal orientation for channeling was about the same. Two first scans were made with a goniometer rotation speed of 0.5 µrad/s when all collimators upstream the crystal were in their standard injection positions. One scan was performed with 1 µrad/s rotation speed when all collimators upstream of the crystal were retracted.

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Table 11 Collimator positions in units of RMS beam size.

TCP TCSG TCLA CRY a

Nominal Injection (σx )

Nominal Flat top (σx )

Crystal MD Injection (σx )

Crystal MD Flat top (σx )

5.7 6.7 10 Out

5.5 8.0 14 Out

Out 6.7a 10 5.4

8.0 8.0 14 5.4

TCSGs upstream of the crystal are in out positions.

Fig. 105. The dependence of the beam losses observed with the BLM1 downstream of the crystal (curve 1) for the injection case with 450 GeV/c protons. Curve 2 shows the dependence of the number of inelastic nuclear interactions of protons in the crystal on its orientation angle obtained by simulation.

For the last case, curve 1 in Fig. 105 shows the observed dependence of the BLM1 count on the angular position of the crystal at 450 GeV/c. The dot-dashed line shows the loss level for the crystal orientations far from alignment with the (110) planes, when it works as an amorphous substance (‘‘amorphous’’ orientation). The losses are normalized to the beam flux and the loss value for the amorphous orientation. Curve 2 shows the number of inelastic nuclear interactions of protons in the crystal as a function of the crystal orientation angle obtained by simulations. They were done with the tools described in [119] by adding also the interaction with other collimators relevant for the experimental setup, taking into account ionization losses, multiple Coulomb scattering and nuclear interactions. The deep minimum on the right corresponds to the optimal orientation for channeling. There is a wide area of a beam loss reduction on the left of the minimum, which is due to volume reflections (VR) of halo particles in the crystal. Agreement of the simulation with the experiment is sufficiently good. Some discrepancies near the edges of the beam loss dependence are not yet well understood and will be investigated in new measurements. The loss reduction in channeling is 47.3 and 50.7 from the experiment and simulation, respectively. The loss reduction values in the VR region are also very close. The first horizontal collimator TCSG1 behind the crystal was used to scan horizontal positions across the beam deflected by the crystal when its angular position was fixed close to the beam loss minimum detected in the angular scan. The BLM2 downstream of the collimator registered secondary particles generated by inelastic nuclear interactions of protons in the collimator. Fig. 106 shows the dependence of the BLM2 signal on the collimator position by a dot-dashed line. The BLM2 signal increases when the collimator intercepts the beam halo deflected in channeling states by the crystal. The observed profile is consistent with the presence of a well-defined channeled halo separated from the beam core by a distance that can be determined with optical transport between the two locations after the angular kick of the crystal bend angle value. The loss dependence gives the integral of the deflected beam distribution. The solid line in Fig. 106 shows a fit of the dependence with an error function. The fit center is at xm = 7.9 mm. An error function is the integral of a Gaussian therefore xm is the center of the Gaussian which fits the deflected beam distribution. The detected displacement xm of the deflected beam halo from the beam envelope at the collimator location determines the angular kick produced by the crystal for channeled particles, θm = 65 µrad. This deflection should be equal to the bend crystal angle if the crystal planes at its entrance were really parallel to the beam envelope for the collimator scan.

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Fig. 106. The dependence of the BLM2 signal on the horizontal position of the first TCSG1 behind the crystal during its scan from the beam periphery towards the orbit (X = 0) is shown by dot-dashed line. The crystal angular position was fixed to be in the optimal conditions for channeling. The error function fit is shown by the solid line.

Fig. 107. The dependence of the beam losses observed with the BLM1 downstream of the crystal (curve 1) for the LHC coasting beam of 6500 GeV/c protons. Curves 2 (solid line) and 3 (dotted line) show the dependence of the number of inelastic nuclear interactions of protons in the crystal on its orientation angle obtained by simulation according to [119] and [14], respectively.

The beam test with bent crystals for 6500 GeV/c protons was performed in the next run. In this case, 16 bunches of 109 protons, evenly spaced around the ring, were injected and accelerated to 6500 GeV/c. The collimators were in their nominal positions for these conditions, as in Table 11, the primaries at 5.5 σx , the secondary collimators at 8 σx and absorbers at 14 σx . The crystal was aligned with respect to the primary collimators and then was moved toward the orbit by about 0.05 mm to intercept the primary halo. The angular scan was performed with the goniometer rotation speed of 0.2 µrad/s and with collimator settings shown in Table 11. Individual bunches were excited one at a time to increase beam losses during the measurements. Curve 1 in Fig. 107 shows the dependence of the BLM1 count on the angular position of the crystal. Channeling was clearly observed. It is clearly seen that the loss reduction ‘‘well’’ in the region of channeling and volume reflection has steeper walls than what was observed at 450 GeV/c. This may be explained by a

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strong reduction of multiple Coulomb scattering at higher energy; the results show that a particle cannot come to the channeling region when its incident angle with the crystal planes is larger than 10 µrad. The channeling minimum on the right is narrower because the critical channeling angle is about 4 times smaller than at injection energy. The loss reduction in channeling is about 24. The second minimum on the left in the VR region is clearly seen here. This minimum was also observed in our studies with bent crystals at the SPS. The minimum is at an angular distance from the channeling orientation about equal to the bend angle value, 65 µrad. In this case, the whole VR region is on the same side relative to the beam envelope direction. Therefore, angular kicks due to VR always increase the oscillation amplitudes of particles and they more quickly reach the secondary collimators. Curve 2 shows the dependence of the number of inelastic nuclear interactions on the crystal orientation obtained by simulation according to [119]. The agreement with the experiment is fair. The walls of the loss reduction dependence coincide well with the experimental ones. This means that the crystal bend angle measured by the collimator scan is right (the angular crystal position for this scan was chosen close to the perfect one for channeling). The loss reduction for VR region is the same as in the experiment. However, the loss reduction value for channeling, R = 187, is considerably larger than the experimental one. This difference may be explained by the residual angular instabilities of the goniometer. Multi-turn simulation with a SixTrack code, including other collimators in all ring insertions and a detailed beam aperture model [14], gives also good agreement with the angular scan data (curve 3), although the loss reduction value, R = 140, is also larger than the experimental one. Similar measurements performed with the QM crystal for the LHC proton beam collimation in the vertical plane and measurements with the stored beam of Pb ions for the injection energy also showed a strong beam loss reduction in the aligned crystals. First experiments on the study of the LHC beam halo collimation assisted by bent crystals [130] were successfully performed in machine studies during the LHC run II, at the injection energy of 450 GeV as well as at the record proton beam energy of 6500 GeV. Beam losses due to inelastic nuclear interactions of particles in the aligned crystal were strongly reduced in comparison with the amorphous crystal orientations. This proves that deflection of particles due to channeling in a bent crystal is effective at this record particle energy. The beam loss measurements in the dispersion suppressor region behind the collimation area showed that the collimation leakage was significantly reduced with using a bent crystal as a primary collimator. 8. Proposal of crystal assisted extraction of the FCC beam halo There is a proposal on the design of a 100 TeV proton collider coming from CERN [131]. It was made a suggestion [132] to include in the studies on the FCC project the consideration of possibilities of a natural beam halo extraction by a bent crystal using it as a primary collimator. A possible scheme of the extraction may be similar to the one considered for the SSC [133] and it is shown in Fig. 108. A horizontal dogleg is formed in a long straight section of the FCC where a bent crystal is installed at the beginning behind a magnet at the vertical distance of 6 σy from the closed orbit. Halo particles deflected vertically by a full angle of the crystal bend enter the passive aperture of the Lambertson magnet and thus they are extracted. Other halo particles deflected with smaller angles hit a collimator–absorber installed upstream the Lambertson. In addition to obvious advantages of possessing the extracted beam, the problem of the absorption of a very narrow beam deflected by a bent crystal may be avoided if the crystal assisted extraction of the FCC beam halo will be considered. In the paper [132] the optimal parameters of the crystal deflector with the bend angles of 50 and 100 µrad and the extraction efficiency of a natural halo of 50 TeV proton beam were estimated by simulation. Fig. 109 shows the dependence of the deflection efficiency Pd for 50 TeV protons on the length of a silicon crystal bent along the (110) planes by the angle α = 50 µrad. Maximal efficiency of about 86% is observed for Ldm = 12 cm. For the beam deflection by 100 µrad the optimum crystal length Ldm = 16 cm. The extraction efficiency of particles from a circular accelerator Pex can be higher than the deflection efficiency Pd due to multiple passages of circulating particles through a crystal. Simulation of the natural halo extraction from the FCC has been performed in [132]. Two azimuths in the FCC ring were considered. The azimuths correspond to the locations of a bent silicon crystal and a collimator–absorber. The betatron phase shift between two azimuths is 90◦ to realize a maximal shift of particles deflected by the crystal from the absorber edge. The distances from the closed orbit were Ybc = 6 σy and Ysc = 7 σy for the crystal and absorber, respectively. The RMS beam size is σy = 0.14 mm for the vertical beta-function βy = 500 m and the normalized emittance εn = 2.2 × 10−6 planned for the FCC. The FCC beam halo particles begin hitting the crystal due to a random increase of their betatron oscillation amplitudes according to an exponential diffusive law with λ = 0.1 µm per turn. Linear 6-dimensional transfer matrices M(6,6) were used to transport particles between the azimuths. The dependence of the extraction efficiency Pex of 50 TeV protons from the FCC on the crystal length obtained by simulation is shown in Fig. 109. It was assumed that the (110) crystal planes at its entrance are perfectly aligned with the beam envelope. The maximum extraction efficiency of about 94% is observed for a crystal length of about 3 cm. For the FCC beam halo extraction with the deflection angle α = 100 µrad the optimal crystal length is larger, 6 cm. Fig. 110 shows the dependence of the extraction efficiency on the crystal orientation. The full dependence width is about 3 µrad. An extraction efficiency larger than 90% may be obtained in the angular interval smaller than 1 µrad. That means the goniometer step should be about 0.1 µrad.

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Fig. 108. Possible scheme of the beam halo extraction from the FCC with a bent crystal. (a) The horizontal dogleg with a bent crystal upstream the Lambertson septum. (b) The cross-sectional view at the position of the crystal. A vertical beam halo intercepts crystal. (c) The cross-sectional view in the Lambertson septum. The extracted beam, which is the portion of the vertical beam halo that the bent crystal intercepts and deflects vertically, has been separated from the circulating beam and passes through the Lambertson hole.

Fig. 111 shows the dependence of the beam halo losses of 50 TeV protons due to inelastic nuclear interactions in the silicon crystal on its orientation angle obtained by simulation. The angular range of volume reflection with a considerable reduction of the beam losses is wide and equal to the crystal bend angle. This simplifies the crystal alignment with the beam envelope. Fig. 112 shows the distribution of impact parameters of 50 TeV protons with the collimator–absorber when the crystal bend angle is 100 µrad. The channeling peak is very narrow of about 1 mm. The peak position depends on the betafunction value and is about 50 mm from the absorber edge for the considered case (the absorber edge is located at 7σy from the closed orbit). An absorber position straight before the Lambertson magnet will be optimal. The vertical size of the

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Fig. 109. The dependencies on the length of a silicon crystal bent along the (110) planes by the angle α = 50 µrad for the deflection efficiency Pd and for the extraction efficiency Pex from a collider for 50 TeV protons.

Fig. 110. The dependence on the crystal orientation of the extraction efficiency from a collider for 50 TeV protons, the crystal length L = 3 cm.

collimator should be sufficient to absorb the tail of the deflected particles (hatched part of the distribution in Fig. 112) and its horizontal size should be optimized to realize the collimator cooling. The channeled particles pass by the collimator and are extracted from the FCC through the passive aperture of the Lambertson magnet. The estimations performed in [132] show that the extraction of a natural beam halo from a collider of 50 TeV protons using a bent crystal is possible and may be efficient if a precise goniometer with the angular accuracy of 0.1 µrad can be realized. 9. Possibility of high efficient beam extraction from accelerators with a bent crystal 9.1. Introduction The deflection by a small angle with short crystals can be efficient (about 80%) if the incident beam has a divergence considerably smaller than the critical channeling angle θc [60]. Particles of the diffusive halo surrounding the circulating

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Fig. 111. The dependence of the beam halo losses of 50 TeV protons due to inelastic nuclear interactions in the silicon crystal on its orientation angle, the crystal length L = 3 cm.

Fig. 112. The distribution of impact parameters of 50 TeV protons with the collimator–absorber when the crystal bend angle is 100 µrad, L = 6 cm.

beam of the LHC have a very small angular spread in collisions with a primary collimator. Recently the first successful tests of the crystal deflectors were fulfilled for the LHC beam halo collimation [130]. The contribution of multiple passages through the crystal increases the channeling efficiency up to 95% according to our simulation results. The channeling efficiency is determined by the beam fraction passed through the whole crystal length in channeling regime. The collimation efficiency increases additionally due to dechanneled particles with large deflection angles. A short bent crystal with a small bend angle can be also used for the efficient beam extraction from the accelerators. In this case a crystal deflects particles directing them over the wire of the electrostatic septum or septum magnet of the slow extraction system. Such beam extraction assisted by a bent crystal is wide used in IHEP Protvino [134]. A local orbit bump is used to deliver the accelerated beam onto the crystal. The extraction efficiency from U-70 with a bent crystal reaches 85%. However, the growing orbit bump changes the envelope inclination at the crystal (see Fig. 113). This effect decreases the deflection probability of particles by the crystal. The direct extraction when the beam deflected by a bent crystal exits the accelerator pipe had been studied at the CERN SPS [9]. This requires a large deflection therefore the bend crystal angle was 8.5 mrad. Transverse diffusion of

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′ Fig. 113. Schematic picture. A particle trajectory of the beam edge (solid line) touches the crystal, which is aligned with the beam envelope, θch = Xbc . A particle trajectory of the beam core shifted by the orbit bump is shown by dot-dashed line. Its envelope direction is far from θch .

Table 12 Relevant accelerator parameters. Parameter

C1

TAL

βx (m) σx (mm) ∆µx (2π )

90.945 0.9047 0 3

87.660 0.888 0.2405 6

X a (mm)

a

Here X is the initial distance from the orbit bump.

the circulating beam due to the injection of noise pulses into the transverse damper plates was used to deliver the beam particles onto the crystal. The angular spread of particles at the first hits with the crystal determined by the noise amplitude can be made smaller than the critical channeling angle. The whole beam passes through the crystal at the fixed crystal position relative to the beam orbit. The extraction efficiency can be very high if one will use a short crystal with a small bend angle to deflect the beam particles into the electrostatic septum of the slow extraction system. Possibility of a high efficiency of the SPS beam extraction assisted by a bent crystal using the transverse diffusion or the orbit bump was considered by simulations in [135]. The whole accelerated beam is delivered onto the crystal. The beam fraction passed through the whole crystal length in channeling regime and a small dechanneled fraction with deflection angles sufficient to jump over the septum wire are extracted from the accelerator. The extraction losses are small, 0.5–0.6%, and determined by the inelastic nuclear interactions of protons in the crystal and in the septum wires. 9.2. Simulation studies of the SPS beam extraction with a bent crystal The SPS beam extraction by a bent crystal was studied by simulation of particle trajectories in the crystal and accelerator as it was described in [119]. The simulations were performed for the SPS beam of 270 GeV/c protons for which the critical channeling angle of the (110) silicon channels θc = 13 µrad and the critical radius of the crystal bend Rc = 0.46 m. Four azimuths in the SPS ring were considered, which correspond to the equipment positions of the UA9 crystal collimation experiment. The initial beam which is Gaussian in the six dimensional phase space was given at the azimuth of the crystal 1 (C1). The TAL absorber (see Section 6.1) was at the second azimuth. One can imagine the electrostatic septum for the beam extraction in this position. The third azimuth corresponds to the RF accelerating system position. The fourth azimuth was at the position of the transverse damper plates. The positions of the crystal and the TAL absorber are with large values of the horizontal beta function. The horizontal betatron phase advance between them is close to 90 degrees. The relevant accelerator parameters at these two azimuths are listed in Table 12. The parameters for the crystal 1 are presented in Table 13. Almost the whole accelerated beam with Gaussian distribution is in the range of ±3 σx . The RMS horizontal beam size at the azimuths of the crystal and absorber is close to 1 mm. Therefore, the crystal distance from the orbit was selected to be XBC = 3 mm. The SPS beam collimation studies with a bent crystal [124] showed that the collimation leakage (both

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Bend angle (µrad)

Bend radius (m)

Miscut angle (µrad)

1.87

165

11.33

10

Fig. 114. The distribution of protons in the impact angles for the first hits with the crystal. The case of the transverse diffusion initiated by the Gaussian noise with σd = 0.04 µrad injected to the damper plates of the SPS.

in the crystal and absorber) is minimal with the optimal absorber offset (2–3 mm) relative to the crystal. The absorber offset of 3 mm was selected for this simulation. Thus, the absorber (septum) distance from the orbit with the selected offset value is Xtal = 6 mm. It was assumed that at the beginning of the extraction process, the crystal orientation for channeling θch is always ′ aligned with the beam envelope direction at its position Xbc = (−αx /βx )Xbc , where αx and βx are the horizontal betatron functions, to have maximum probability for particles to be captured into the channeling regime. 9.2.1. Transverse diffusion Transverse diffusion of the circulating beam for the beam delivering onto the crystal is generated due to the injection of noise pulses into the transverse damper plates. At every turn the circulating particles acquire some angular kick when they pass the damper azimuth. The kick is randomly selected from the Gaussian distribution with the RMS value σd . The maximum value of σd for 270 GeV/c protons, which can be realized in the damper is about 0.04 µrad according to [9]. Let us consider the situation realized for this regime with the maximum value of σd . Fig. 114 shows the angular distribution of particles at their first hits with the crystal. The distribution maximum is near the beam envelope direction. The distribution width at its half height is 5.8 µrad. The half width is considerably smaller than θc therefore the channeling efficiency is high, about 95%. Fig. 115 shows the impact parameter distribution of particles with the TAL absorber (septum). Assuming a septum wire diameter of 0.2 mm, the particles which hit the septum (0.2 mm from its edge) are about Nw = 0.4%. Inelastic nuclear interactions in the crystal were occurred for Ncr = 0.2% of the beam particles. The total beam losses of the extraction process in the crystal Ncr and in the septum wire Nw are about 0.6%. Thus, the beam extraction efficiency from the SPS with the bent crystal can be larger than 99%. The distribution of the number of turns performed by particles before the extraction is shown in Fig. 116. The distribution for the considered case with the RMS kick σd = 0.04 µrad (1) is non-uniform with a long tail. About 3 × 106 turns are necessary for the beam extraction. That is the extraction time is about 70 s. This time can be reduced by a factor of 4 using the noise with the RMS kick value σd = 0.08 µrad. The extraction distribution can be made more uniform by changing the kick strength. The distribution (2) in Fig. 116 is for the case when σd linearly increases from 0.02 µrad to 0.08 µrad during 5 × 106 turns. 9.2.2. Parallel orbit bump Let Nex be the number of turns for the beam extraction, which we would like to have. Then at every turn due to the growing orbit bump at the crystal azimuth, particles should be shifted to the crystal by the value ∆Xb = Xbc /Nex . A similar

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Fig. 115. The case of the transverse diffusion. The distribution of protons in the impact parameters with the TAL absorber (extraction septum). (a) linear scale, (b) logarithmic scale. The maximum on the right is the channeled fraction. The narrow dashed part on the left shows particles, which hit the septum wires (0.2 mm thick). The particles between are dechanneled ones, which obtained deflection angles in the crystal sufficient to be extracted.

result can be produced by the corresponding shift of the crystal to the orbit. Let us consider the situation when the beam extraction should be performed for Nex = 106 turns. The revolution time in the SPS is about 23 µs. Thus the extraction time τex = 23 s and the required rate of the bump growing is 0.13 mm/s. The beam envelope direction at the crystal is changed by the value ∆Xb′ = (−αx /βx )∆Xb due to the orbit bump at every turn. Fig. 117 shows the angular distributions of particles at the first hits with the crystal (1) and at the hits before the extraction from the circulated beam (2). The width of the distribution (1) is about 36 µrad because of the situation shown in Fig. 113. The distribution is considerably shifted from the angular range of channeling, which corresponds to the position of the distribution (2). By this reason particles at the first hits cannot be captured into the channeling regime. They come to the angular range of channeling only after few passages through the crystal. The average passage number is larger than 15. The impact parameter distribution of particles with the TAL absorber (septum) is considerably wider in this case (Fig. 118). About 1% of particles hit the septum wire and 8% are lost in the inelastic nuclear interactions in the crystal. The beam extraction efficiency may be about 91% in this case. The impact parameter distribution of particles with the crystal covers its whole thickness. The distribution of particles in the turn number before the extraction is presented

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Fig. 116. The case of the transverse diffusion. The distribution of protons in the number of turns before the extraction. (1) with the RMS kick σd = 0.04 µrad, (2) σd linearly increases from 0.02 µrad to 0.08 µrad for 5 × 106 turns.

Fig. 117. The distributions of protons in the impact angles for the first hits with the crystal (1) and before the extraction (2). The case of the parallel bump with the growing rate of 0.13 mm/s.

in Fig. 119. The distribution has a maximum but it can be made close to a uniform one by some regulation of the growing bump process. 9.2.3. Optimal orbit bump with angular adjustment The situation with the orbit bump for the beam delivering onto the crystal can be considerably improved if the orbit inclination by the value ∆Xb′ can be made simultaneously with the orbit shift. In this case the beam envelope direction at the crystal will be unchanged. It is probably difficult to realize such an optimal bump. Additional investigations are required to clarify this issue. The result will be the same if the crystal will be shifted to the orbit with its simultaneous rotation by the value −∆Xb′ . The required rate of the angular rotation of the crystal is about 3.1 µrad/s. Due to the angular adjustment in the case of the optimal bump, the angular distribution of particles at the first hits with the crystal is very narrow. For a slow bump with Nex = 106 the distribution width is about 1 µrad. Most of the particles are deflected by the crystal at their first hits. The channeling efficiency reaches 96%. The impact parameter distribution with the absorber is about the same as in the transverse diffusion case for the beam delivering onto the crystal (see

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Fig. 118. The case of the parallel bump. The distribution of protons in the impact parameters with the TAL absorber (extraction septum).

Fig. 119. The case of the parallel bump. The distribution of protons in the number of turns before the extraction.

Fig. 115). About 0.25% of the beam particles hit the septum wire and the beam losses in the crystal are about 0.2%. Thus, the total beam losses are smaller than 0.5%, therefore the extraction efficiency can reach a value exceeding 99%. Thus, the simulation studies [135] showed that the main condition for the high efficient beam extraction from the accelerators with a bent crystal can be fulfilled using the transverse diffusion or the optimal orbit bump for delivering the whole circulating beam onto the crystal. Thus, the beam extraction efficiency assisted by a bent crystal can reach 99%. The irradiation of the septum wires during the beam extraction can be considerably reduced. 10. Conclusions The experimental studies of deflection effects for high energy charged particles by short bent crystals with length much smaller than their electron dechanneling length, few millimeters, at the beams of 400 GeV/c protons and secondary particles mainly 150 GeV/c π mesons of the CERN SPS were considered. 1. The dechanneling processes due to multiple scattering by crystal nuclei, which occur in the narrow ranges around the crystal planes, were studied at the planar orientations of the crystals. Multiple scattering by nuclei determines

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2.

3. 4.

5. 6.

7.

8. 9.

10. 11.

12.

13.

the dechanneling rate only for the channeled fraction with large oscillation amplitudes in the case of positive particles, and for the whole channeled fraction in the case of negative particles. The measured nuclear dechanneling length approaches 1.5 mm for 400 GeV/c protons and about 1 mm for 150 GeV/c π − mesons. The deflection efficiency of larger than 80%, which is larger than the theoretical limit for long crystals, was obtained for the beam of 400 GeV/c protons with the angular divergence much smaller than the critical channeling angle. At the same time the beam losses due to inelastic nuclear interactions were reduced up to 5 times. The full deflection by the bend angle due to planar channeling has been observed for negative particles. The characteristics of parametric X-ray radiation generated due to diffraction of virtual photons of proton field on the crystallographic planes in the oriented crystals in the collimation conditions were studied. The radiation is emitted at large angles relative to the beam direction and can be used for monitoring the structure of the crystal collimator. Usage of a bent crystal for effective focusing of channeled fraction of high energy positive particle beams with a short focal length of ∼ 10 cm was shown. The deflection of the proton beam due to reflection of particles by crystallographic planes in an ultrathin straight crystal with its thickness of the half oscillation wavelength of particles in the planar channels has been studied. The deflection angle is close to the critical channeling angle. The deflection of a considerable beam fraction due to multiple coherent scattering by atomic strings, which produces the transverse momentum randomization of particles (doughnut scattering) around the string direction, has been observed at the axial orientations of bent crystals for positive and negative particles. The particles can be considered as quasi-bound with the crystal axes in this case. The deflection efficiency was about 80% with the silicon crystal oriented along the [110] axis for the beam of 400 GeV/c protons. The volume capture of negative particles into this quasi-bound regime with atomic strings in a bent crystal has been observed. Volume reflection (VR) in bent crystals and its dependence on the bend radius has been studied for 400 GeV/c protons. The VR efficiency can reach 98%. The angular acceptance is determined by the bend angle and can be much larger than in the channeling case. The VR deflection angle increases with the curvature reduction but its efficiency decreases because the volume capture of particles into channeling increases. The optimal radius of a bent crystal for the beam deflection due to VR is about 10 Rc , where Rc is the critical radius for channeling. Multiple volume reflection by different planes in one bent crystal (MVR OC) has been studied with 400 GeV/c protons. The deflection angle due to MVR OC is about five times larger than for single VR. Different crystal sequences to realize the sequence of volume reflections have been studied. The sequences allow increasing the deflection angle but the deflection efficiency decreases. The sequence will be optimal if its neighboring crystals have inclination about the VR deflection angle. The production of the crystal sequence with right mutual orientations where the VR deflection occurs in all crystals is a difficult task. The sequence of the crystal reflectors can be more efficient than one crystal in channeling mode if the required deflection is small and should be done in one passage. However, the extraction efficiency of particles from the circulating beam is higher with one short crystal in channeling mode due to multiple passages of particles through the crystal. VR and MVR OC effects in bent crystals have been observed also for negative particles. The deflection angle and efficiency for VR and MVR OC are smaller than for positive particles because the planar potential is attractive for negative particles.

The experiments on the studies of crystal assisted collimation of the CERN SPS beam halo and the first similar experiment with the CERN Large Hadron Collider (LHC) beam of 6500 GeV/c protons were also considered. 1. Perfect alignment of the crystal to achieve deflection of halo particles from the SPS circulating beam due to channeling could be obtained quickly and easily reproducible. The beam losses are strongly reduced for the aligned crystal that is registered by the beam loss monitors (BLM). The loss reduction value depends on the collimation conditions as well as on the beam and crystal characteristics (miscut and torsion). So, if the miscut angle is larger than the crystal bend angle then the beam losses in the oriented crystal can be more than doubled. 2. Considerable reduction of the collimation leakage was observed through the beam loss reduction in the first area with high dispersion behind the collimation area when the crystal was oriented for channeling. 3. The optimal offset distance between the crystal and absorber exists at which the collimation leakage is minimal. The optimal offset value is larger for Pb ions than for protons because of larger ionization losses in the crystal. 4. First experiments with a bent crystal used as a primary collimator at the LHC showed that the crystal orientation perfect for channeling is found quickly through the strong reduction of losses in the crystal. The oriented crystal gives a considerable reduction of the collimation leakage. Simulation studies showed that bent crystals can be used for high efficient beam extraction from accelerators if the whole accelerated beam can be delivered onto the crystal with very small angular divergence.

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Acknowledgments The experimental data discussed here were collected in the CERN accelerator complex by the UA9 Collaboration from 2007 onwards. UA9 is a CERN approved experiment, investigating different deflection mechanisms of high-energy particle beams by short bent crystals, with the purpose of improving the collimation efficiency of hadron colliders, such as the LHC, and the beam extraction performance in circular accelerators, such as the SPS. The SPS Committee (SPSC) recommended the approval of UA9 in April 2008, thanks to the precious support of Robert Aymar, former CERN Director General, Steve Myers, former director of the CERN accelerators and Fernando Ferroni, former president of the CSN1 of the INFN-Italy. Roberto Losito and later Simone Gilardoni, group-leaders of the CERN-EN-STI group, strongly contributed, together with their collaborators, to the construction and the validation of the UA9 experimental equipments and to their integration in the UA9 setups installed in the North Area, in the SPS ring and in the LHC. Simone Montesano and later Marco Garattini secured the technical coherence of the UA9 setups, guaranteeing their adequacy for the experimental runs. Geoff Hall and collaborators provided the telescope tracker used in the SPS North Area to investigate the beam deflection mechanisms by short bent crystals. Earlier versions of the telescope were built by Roberto Battiston and collaborators as well as by Michela Prest and collaborators. Strip crystals bent due to anticlastic curvature by a special holder were invented and developed by Yury Chesnokov and collaborators. ‘‘Quasi-mosaic’’ crystals were developed by Yuri Ivanov, Yury Gavrikov and collaborators. Other crystals were provided by Vincenzo Guidi, Pietro Dal Piaz and collaborators, in particular Andrea Mazzolari. Production of silicon crystals through anisotropic chemical etching developed by Ferrara group allowed obtaining the silicon strips with ultra-flat surface and with negligible change of the crystal surface quality. Gianluigi Arduini proposed the initial experimental layout of UA9 in the SPS. Ralph Assman and later Stefano Redaelli and co-workers provided the conceptual support and the financial effort to perform crystal assisted collimation tests in LHC. Mike Lamont and later Rende Steerenberg and co-workers of the CERN-BE-OP group implemented smooth operational conditions during the experimental tests. Alexander Kovalenko considerably supported the investigation of the crystal assisted extraction concept. Achille Stocchi and Alexei Vorobyev continuously encouraged and supported the UA9 activity in their respective laboratories. We are grateful to Andrea Mazzolari for his help with some figures. We gratefully acknowledge the competence and the professionalism of the entire UA9 Collaboration and the generous effort of the UA9 funding agencies: CERN (CH), INFN (Italy), LAL (France), Imperial College (UK), PNPI (RU), IHEP (RU), JINR (RU). For a limited period of time also SLAC (US), LBNL (US) and FNAL (US) provided an appreciated additional support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

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