arXiv:2005.04138v2 [physics.acc-ph] 13 May 2020 rsai oeta hc cso rjcie fa in- an If projectile. a angle elec- on cident periodic acts a which create planes potential in trostatic or (axes) strings in ther orientation. crystal strongly the depends [1]). on crystals Ref. with energies in high of charg reviews of interaction of that well-established collection is it a Nowadays, example, 1950s for the since (see, studied high- been passage has the particles on charged energy structure crystal a of influence The Introduction 1 the at facility. are collected (MAMI) calculations MIctrotron data of MAinzer experimental results the The with detail. compared in emitted radiation analysed the elec- are of of distribution inci- angle spectral the deflection and to the the trons respect in in Distribution with one beam. rotation planar dent crystal the the to of from regime course transition channelling the crys- to axial devoted (111) the is silicon attention oriented for Special bent tal. phenomena in channelling MeV the 855 of and analysis modelling computational numerical accurate for applied is ics Abstract date Accepted: / date Received: Russia 194021, fmn tmccnrso h lcrsai ed sa As field. electrostatic the of centres action atomic correlated many a by of rather but motion by atoms the not individual then governed from enough is small projectile is high-energy a plane) of a (or axis an to 6 5 .V Haurylavets V. V. bent silicon experiment oriented versus in theory MeV emission crystal: 855 radiation of and simulations propagation atomistic Explorer MBN b a Mazzolari 1 5 3 2 4 wl eisre yteeditor) the by No. inserted manuscript be C (will J. Phys. Eur. .V Solov’yov V. A. , ..Iff hsclTcnclIsiueo h usa Acad Russian the of Institute Physical-Technical Ioffe A.F. B eerhCne,Atnöeale3 rnfr mMi 6 Main am Frankfurt 3, Altenhöferallee Center, Research MBN orsodn uhr [email protected] author: Corresponding eerhIsiuefrNcerPolm,Blrsa State Belarusian Problems, Nuclear for Institute Research nvriàdgiSuid iao i et e edn,7 2 7, Perdono, del Festa Italy Via 44122, Milano, Ferrara di 1, Studi Saragat degli Via Università Ferrara, di Sezione INFN iatmnod iiaeSinedlaTra nvriàdi Università Terra, della Scienze e Fisica di Dipartimento -al [email protected] e-mail: tsm rsa retto h tm ragdei- arranged atoms the orientation crystal some At h ehdo eaiitcmlclrdynam- molecular relativistic of method The 2 θ .Romagnoni M. , fapoetl’ oetmwt respect with momentum projectile’s a of b,5,6 a,1 .Leukovich A. , 2,3 .Guidi V. , 1 .Sytov A. , 2,4 .B Sushko B. G. , nvriy oriky tet 1 is 200 Belarus 220030, Minsk 11, street, Bobruiskaya University, err i aaa ,Fraa410 Italy 44100, Ferrara 1, Saragat Via Ferrara m fSine,Pltcnceky t.2,S Petersburg St 26, str. Polytechnicheskaya Sciences, of emy ed 2 12Mln,Italy Milano, 0122 .Bandiera L. , 48 Germany 0438, rjcieeprecstecanligmto fits depth the if exceed motion not channelling does energy the transverse experiences projectile a formulated. was of atoms lattice model and pro- important energetic jectiles of The interaction the planes. for potential and to continuum orientation axes relative crystal the the on depends through particles strongly charged crystal of propagation a demon- that has strated [10] study theoretical comprehensive hard’s ilsi retdcytl scle channelling called is crystals oriented in ticles planes. or strings atomic of the in vicinity move close inter-atomic projectiles the charged small into negatively steered at while are region, repulsive they is that so charged field distances positively electrostatic For the direction. projectiles planar) (or along motion axial guided the a experiences projectile the result, 1 ieto 1] Namely, [10]. direction θ by given value (critical) we h atcesvelocity ’s the tween aeil r aoue n ulrts o hc h chan- Refs. the e.g., see, which investigated, such for also of been fullerites, has examples motion and nelling The nanotubes direction. are when random than materials angle any scattering along multiple value the lower moving of much square has mean projectile the “p a of provides which which along material moving structured sages”, any for principle, in odto a efruae ntrso h angle the of This terms well. in formulated potential be can electrostatic condition interplanar or axial hneln ffc a edsusdntol o rsasbut crystals for only not discussed be can effect Channelling L = aigo hsmdli a endmntae that demonstrated been has it model this on Basing hnmnno uddmto fcagdpar- charged of motion guided a of phenomenon A s 2 pv U 5 .V Korol V. A. , 0 2 A. , θ utntece h maximum the exceed not must v n h xa rplanar or axial the and U 1 0 Lind- . fthe of , θ [2–9] be- (1) as- , 2 where p = mγv is the projectile momentum with γ = OZ 1' OZ ε/mc2 being the relativistic Lorentz factor. For an ultra- 1' 2' 3' relativistic projectile one can substitute the product pv 2' with the energy ε.

Channeling of charged particles is accompanied by Detector the emission of Channelling Radiation (ChR) [11]. This specific type of electromagnetic radiation arises due to the oscillatory transverse motion of the projectile (so- called, channelling oscillations). The (quasi-)periodicity of the particle’s trajectory leads to the enhancement of the spectral intensity of radiation at frequencies ω ∼ 2 2γ Ωch (here Ωch stands for the frequency of chan- nelling oscillations). In this frequency range, the inten- sity of ChR greatly exceeds (by more than an order of magnitude) that emitted by the same projectile in an 1 2 3 1 2 amorphous medium (see, e.g., [12]). a) b) As for now, a number of theoretical and experimen- Fig. 1 Schematic representation of the beam-crystal orien- tal studies of the channelling phenomena (the motion tation for the study of a) channelling efficiency, b) volume re- and the radiation emission) in oriented crystals have flection and volume capture effects in a bent crystal. In both panels, the incident electron beam enters the crystal along been carried out (see, e.g., a review [13]). the direction OZ. Interaction with the crystal deflects the The channelling phenomena occur in both straight electrons. A detector, located behind the crystal, allows one and bent crystals. In the latter case, a channelling par- to determine the distribution of electrons with respect to the deflection angle β (shown as thick solid curves 1′ − 2′ − 3′ on ticle experiences the action of a centrifugal force Fcf = the top of the panels). In the crystal, the electrons experience pv/R ≈ ε/R (where R is the bending radius) [14] in ad- two basic types of motion: over-barrier motion (straight seg- dition to that of a crystalline field. A stable channelling ments in the trajectories) and the channelling motion (wavy motion is possible if Fcf is less than the maximum trans- segments). verse force U ′ due to the crystalline field. This leads In panel a) the beam enters the crystal along the tangent to max the crystal planes. The angular distribution of the initially to the restriction on the allowed values of the bending over-barrier electrons at the entrance, label "1", is centered ′ radius, R < Rc ≈ ε/Umax. Since its prediction [14] and along the OZ direction, β = 0. The electrons captured into the experimental confirmation [15], the idea of deflecting channelling mode at the entrance follow the crystal bending high-energy beams of charged particles by means bent and thus are deflected by larger angles. Some of these elec- trons, which channel through a part of the crystal and then crystals has attracted a lot of attention [13, 16]. The ex- dechannel (label "2"), contribute to the central part 2′ of the periments have been carried out with ultra-relativistic angular distribution. Those particles that channel through , ions, , electrons, π−-mesons [17–21]. the whole crystal of length L (label "3") contribute to the maximum 3′ of the distribution at β = R/L. Experimentally, it is more advantageous to study Panel b) corresponds to the case when the angle between the the interaction of charged particles with a bent crys- incident beam and the tangent exceeds the Lindhard angle. tal rather than with a straight one. The former case As a result, most of the particles propagate in the over-barrier provides the opportunity to analyze the channelling ef- mode starting from the entrance. In this case, due to the crys- tal bending a particle can experience a volume reflection or a ficiency by separating the channelling particles from the volume capture as marked by circles in the trajectories 1 and over-barrier particles, i.e. those which travel across the 2, respectively. The angular distribution of electrons provides crystal planes (axes) at the angle larger than the Lind- the quantitative description of these events. hard critical angle. Figure 1 a) shows a sketch of the experimental set- up which can be used to study the channelling effi- 1′−2′−3′ on the top of the panel). Namely, the distribu- ciency by measuring the deflection angle of the incom- tion of the electrons that move in the over-barrier mode ing particles (electrons) due to the interaction with a from the entrance to the exit points (see schematic tra- bent crystal. The direction OZ of an incident electron jectory 1) is centered along the OZ direction, i.e. at beam is aligned with the tangent to the crystal planes β = 0. The particles that channel through the whole at the crystal entrance. Placing a detector behind the crystal of length L (trajectory 3) give rise to the max- crystal one can measure the distribution of electrons imum centered around β = R/L (marked as 3′). Com- with respect to the deflection angle β. Different modes paring the areas below these two maxima one can quan- of the electrons motion contribute to different parts of tify the channelling efficiency, i.e. the relative number of the angular distribution (shown with a thick solid curve the particles channelled through the whole crystal. Fi- 3 nally, the electrons captured into the channelling mode valid only if the incident angle of the projectile with re- at the entrance but dechannelled somewhere inside the spect to the chosen plane (or axis) does not exceed sev- crystal (trajectory 2) experience deflection by the angle eral Lindhard’s critical angles θL. This scheme has been within the interval 0 <β

Detector smaller than the size of the crystal. The corresponding standard deviations of the beam divergences of σh′ = 70 Ionization Chamber and σv′ = 30 µrad, are much smaller than Lindhard’s critical angle θL = 0.217 mrad for a 855 MeV electron the Si(111) planar channel.

Bent Pb Shilding crystal Magnet Calorimeter 2.2 The crystal used in the experiment Beam d=40mm

8.7 m A bent silicon single crystal of thickness 30.5 µm along the beam direction was used as a target. The crystal was Fig. 2 Schematic representation of the experimental setup bent using a mechanical holder that provided a uniform at the MAMI facility. The collimated electron beam prop- bending of the (111) planes with the bending radius agates through the oriented bent crystal. On the exit, the electrons are deflected by a magnet system towards the de- R = 33.5 mm. The bending curvature was manifested as tector to measure the deflection angle distribution. The pho- a result of the quasi-mosaic effect. More details on the ton beam from the target is collimated and directed to the bending procedure and characterization can be found calorimeter where the photons energy is measured. For more in [59]. The crystal was mounted on a high-precision details see, e.g., Ref. [57, 58]. goniometer with three degrees of freedom which allowed for a precise rotation of the target [57]. The orientation Experimental setup used to measure the radiation of the axes and planes in crystal was verified by means emission spectra and the distribution of electrons with of high resolution X-ray diffraction [60]. 5

2.3 Crystal orientation Y

In the experiment, the crystal was first oriented by aligning the crystalline axis h112i with the incident elec- tron beam. Then, the crystal was rotated around the h111i axis to the position in which the beam is directed along the crystal plane (111). The choice of the (111) planes is due to the fact that they are most effective ones for reflecting negatively charged particles. Beam Z

X 3 Case studies

Fig. 3 Orientation of the crystal axes and the beam direc- In our simulations we used a bent crystal with the pa- tion along the planar direction (111)¯ used for studying the rameters (thickness, bending radius) close to those used transient effects from axial to planar channelling. in the experiments. The simulations were carried out for a 855 MeV electron beam targeting the crystal at the di- axis θ, rad rections corresponding to those probed experimentally h112i 0 as well as at the direction that were not explored. It h123i 0.190126 has allowed us to compare the simulated dependences h134i 0.281035 with the experimentally measured data and to study h145i 0.333473 h156i 0.367422 the evolution of the angular distributions of deflected h167i 0.391144 electrons and of the emission spectra with variation in h178i 0.408638 the relative orientation of the crystal and the beam. h189i 0.422064 In what follows we analyze in detail the transition Table 1 Angles between the h112i and h1n(n + 1)i (n = from the axial to the planar mode and compare the 1,..., 8) axial directions in a silicon crystal. results with available experimental data presented in Refs. [20, 21]. The atomistic approach implemented in MBN Ex- assumed with constant bending radius R = 33.5 mm plorer allows one to simulate the trajectories of charged lying in the (v0,y) plane. particles entering a crystal along an arbitrary direction. The case θ =0 corresponds to the axial channelling We take this advantage and study the transition from along the h112i axis. Transition to the planar chan- electron capture by the crystalline axis h112i to the pla- nelling regime can be realized by increasing the values nar channelling mode in the bent (111) plane at differ- of θ. In our simulations this has been implemented by ent rotation angles. The crystal was oriented from the means of the following two routines. h112i axis along the beam direction to the channelling Firstly, to move away from the axial channelling position in the (111) plane. along h112i we have considered the values of θ corre- sponding to the axes with higher Miller indices such as h123i, h134i, ..., h1n(n + 1)i up to n = 9. The limit 3.1 Crystal orientation used in the case studies n ≫ 1 corresponds to the planar channelling regime. The data presented in Table 1 indicates that for all Let us describe the geometries of the beam-crystal ori- n considered the corresponding values of θ are in the entation which were used in the case studies. range of hundreds of mrad. The first case study refers to the analysis of the Another option to monitor the transition from the change in the channelling properties in the course of axial channelling to the planar is by considering the transition from the axial to the planar channelling regime. values of θ within the range [0,θmax] where θmax to be Figure 3 illustrates the beam-crystal orientation in this chosen much larger that Lindhard’s critical angle for case. The y-axis is directed along the h111¯i crystallo- the h112i axis but much smaller than 0.190 rad, see graphic axis which is normal to the (111)¯ plane. The z- Table 1. axis is aligned with the h112i crystallographic direction. We note that the experimental data presented in At the entrance, the beam velocity v0, being tangent to Refs. [20, 21] refer, as stated, to the planar channelling the (111)¯ plane, is directed at the angle θ to the y-axis. regime. However, the corresponding value of θ is not For given θ the uniform bending of the (111)¯ plane is quoted. 6

Y The beam dimensions are significantly larger than the size of the unit cell of the crystal, therefore, at the crystal entrance the uniform spatial distribution of the beam particles was assumed. In the course of simulations, the crystal structure is generated dynamically following the propagation of the particle. In a bent crystal, the code modifies coordinates of the nodes in newly generated unit cells accounting Z for the finite value of the bending radius R. In the simu- Beam lations, we used R = 33.5 mm and the crystal thickness was set to 30.5 µm. as quoted in Ref. [21]. X

5 Simulation of the electrons angular distribution Fig. 4 Orientation of the crystal axes and the beam direc- tion used in the second case study simulations (see explana- The modification of the particle angular distributions tion in the text). The crystal bending is carried out in the behind the bent crystal (see Figure 1) due to the change ¯ (y,z) plane (i.e., in the (110) crystallographic plane). in the beam-crystal orientation can be studied at the atomistic level of detail by means of MBN Explorer. In The second case study, the geometry of which is il- this paper we analyse this modification during the tran- lustrated by Figure 4, was devoted to the analysis of sition from the axial to the planar channelling regimes the volume reflection and volume capture phenomena. by changing the rotation angle θ, see Figure 3. In the Here, the direction of the incident beam is character- simulations, θ was varied over the broad interval, 0 ≤ ized by two angles: θ defined as above, and the angle α θ . 0.5 rad. This interval includes, in particular, the between v0 and the (111)¯ plane. Bending of the (111)¯ value θ = 95 mrad, for which we present a comparison plane is assumed with bending radius within the (y,z) of the simulation results with the experimental data. plane. To match the experimental conditions indicated in 5.1 Angular distribution in the case of axial Refs. [20, 21], we set θ = 95 mrad in the simulations. channelling The α angle was varied from −0.5 to 1.5 mrad. Setting θ = 0 one directs the incident beam along the h112i axis. This geometry corresponds to the axial chan- 4 Input parameters for simulation nelling. The resulting distribution of electrons with re- spect to the deflection angle is presented in Figure 5, The simulations have been performed by means of the green curve with crosses. The electrostatic field in the MBN Explorer package [45], which supports a number oriented crystal influences the motion of electrons forc- of physical models including, in particular, the atom- ing them to follow the (bent) axial direction. As a re- istic approach based on the relativistic molecular dy- sult, there is a strong asymmetry of the distribution namics to simulate channelling of charged particles in with respect to the incident beam direction (this corre- crystals [37]. The work with MBN Explorer is facili- sponds to the deflection angle equal to zero). This be- tated by a multitask software toolkit, MBN Studio [61], haviour differ notable from a symmetric profile of the which provides tools for designing crystalline systems, distribution obtained for non-oriented (amorphous) sil- preparing the input data for simulation, setting up cal- icon, orange curve with filled circles. culations, monitoring their progress and examining the results. The electrons with energy 855 MeV were used as a 5.2 Evolution of the angular distribution with θ source of initial particles for the simulation. As men- tioned above, the angular divergence, σh′ and σv′, of As mentioned in Sect. 3.1 above, one can increase θ by the beam at MAMI is much smaller than Lindhard’s aligning the incident beam the axial directions h1n(n + angle. Therefore, in the main set of simulations the 1)i along the (111)¯ plane. Gradual increase in the Miller beam divergence was not accounted for, i.e. we used index n leads to the transformation of the electrostatic σh′ = σv′ =0. field acting on the projectile from the axial type to the 7

2 1.4 <112> axis Planar channeling, experiment not oriented 1.2 <1,1,2> ) ) 3 1.6 3 <1,3,4> 10 10 1 × ×

1.2 0.8

0.8 0.6

0.4 Counts (normalized)( Counts (normalized) ( 0.4 0.2

0 0 -1.5 -1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 -1 -0.5 0 0.5 1 1.5 Deflection angle, mrad Deflection angle, mrad

Fig. 5 Distribution of 855 MeV electrons in the deflection 1.4 angle after passing through axially oriented (θ = 0, see Figure Planar channeling, experiment 1.2 <1,6,7> ) 3) bent silicon crystal. For the sake of comparison, angular 3 <1,9,10> distribution of electrons in the case of a non-oriented crystal 10 1 × (amorphous silicon) is also presented. 0.8

0.6

0.4 Counts (normalized)( 0.2

0 -1 -0.5 0 0.5 1 1.5 planar one. As a result, one can monitor the change Deflection angle, mrad in the profile of the deflection angle distribution. Two panels in Figure 6 show the variation of electron angular Fig. 6 Distribution of 855 MeV electrons in the deflection distribution with respect to angles θ corresponding to angle after passing through bent silicon crystal oriented as shown in Figure 3 at angles θ corresponding to different direc- several directions indicated in Table 1. For the sake of tions h1n(n+1)i, as indicated. The experimental data (which reference we also present the experimentally measured corresponds to some angle within the range 10 ≪ θ < 180 distribution, for which, however, the value of θ has not mrad) attributed to the planar channelling regime are shown been clearly identified. for comparison.

Intensity of the axial electrostatic field is the high- 5.3 Evolution of angular distribution in the region of est for the axial directions with small values of n. In smaller angles θ this case, the projectiles are captured, predominantly, into the axial channelling mode so that their distribu- Another approach to simulate the planar channelling is tion in the deflection angle is notably different from to direct incident beam along the (111)¯ plane at the an- the planar case. This feature is illustrated by the up- gle θ (measured with respect to the h1,1, -1i axis) that per panel in Figure 6 where the calculated distributions (i) is much less than 190 mrad, which corresponds to for channelling along h112i and h134i (n = 1 and 3, the nearest h1,2,3i axial direction, see Table 1, and (ii) respectively) directions are compared with the experi- is well above Lindhard’s critical angle for axial chan- mentally observed one. As n increases, the axial field nelling along h1,1, -1i. intensity decreases and the particle’s motion more and more acquires features of the planar channelling. In the Figure 7 presents angular distributions calculated simulations, it was noticed that starting with n = 6 for several values of θ as indicated. In the upper panel, the angular distributions of electrons becomes virtually the distribution calculated for a small values θ = 0.5 independent on values of n and the profile of the sim- mrad is shifted to the left from that obtained for the ulated distributions reproduces the features of the pla- pure axial channelling (the curve θ =0) and also exhibit nar channelling seen in the experiment: the two maxima strong deviations from the experimental data. For the corresponding to the particles scattered at the entrance incident angles within the interval 0.5 . θ ≤ 4 mrad the and to those which channel through the whole crystal, shape of the simulated dependence experiences strong see lower panel in the figure. It is seen, however, that the modifications but for θ> 4 mrad it stabilizes becoming first maximum in the simulated distributions is slightly close to that of the experimentally measured one, see shifted towards higher scattering angles. the lower panel. 8

1.4 Planar channeling, experiment 1.2 Axis <112>, 0 mrad 1.5 1.2

3 0.5 mrad 10

⋅ 1

1 ) 1 3

0.8 10 0.8 × 0.6 0.5 0.6 0.4

Counts (normalized) 0 ection angle, mrad

0.2 fl 0.4 De

-0.5 Counts (normalized)( 0 0.2 -1 -0.5 0 0.5 1 1.5 Deflection angle, mrad -1 0 0 1 2 3 4 5 6 7 8 1.4 Planar channeling, experiment n, The Miller index of axes <1,1+n,2+n> 1.2 1 mrad

3 4.0 mrad 10

⋅ 1 10 mrad

0.8 1.5 1.2

0.6 1

1 3 10 0.4 ⋅

Counts (normalized) 0.8 0.5 0.2 0.6 0 0

-1 -0.5 0 0.5 1 1.5 ection angle, mrad

fl 0.4

Deflection angle, mrad De -0.5 Counts (normalized) 0.2 Fig. 7 Deflection angle distribution of electrons behind the BC oriented as in Figure 3 at different values of angles θ. The -1 0 experimental data (obtained for some angle within the range 0 2 4 6 8 10 10 ≪ θ < 180 mrad) are presented for comparison. Rotation angle, θ(mrad)

Fig. 8 Colour map of distribution in deflection angle for elec- 5.4 Scan over the rotation angle θ trons behind the BC oriented as shown in Figure 3. Upper graph corresponds large θ, comparable to the angles between To obtain a more detailed description on the depen- the h112i direction and the axes h1(n + 1)(n + 2)i with dif- dence of the distribution of electrons in the deflection ferent Miller indexes n (see Table 1 for the θ values). Lower graph shows the scan over smaller values of θ. angle the simulations have been performed scanning over the two ranges of θ indicated in Sects. 5.2 and 5.3. out using the parameters that match the experimental Upper panel in Figure 8 shows an angular scan over conditions [20, 21]. the region of large θ, which includes the axial directions Figure 9 compares the results of simulation with the h1(n + 1)(n + 2)i with the Miller index n from 0 to 8. experimental data for the angular distribution of 855 The results presented indicate that the axial directions MeV electrons incident along the (111)¯ plane (α = 0) corresponding to n ≥ 5 can be equally used to simu- at the entrance of bent silicon crystal. On the whole, late the planar channelling along the (111)¯ plane. Lower there is good agreement between theory and experiment panel in the figure presents the scan over smaller θ val- but there are also some differences. ues, i.e. much less that 190 mrad, which corresponds First, the maximum centered at zero deflection an- to the h123i angular direction. Here, the planar chan- gle, which corresponds to the distribution of electrons nelling can be adequately simulated in the region θ ≥ 4 that become over-barrier at the entrance, in the simu- mrad where the angular distribution remains virtually lated curve is slightly shifted to the right as compared unchanged. to the maximum in the experimentally measured depen- dence. In the course of simulations it has been noticed that the closest agreement in the maxima positions is 5.5 Comparison with the experiment achieved at smaller incident angles 10 ≤ θ < 190 mrad. In this range the simulated angular distribution does The data presented in this section refer to the second not virtually depend on the the incident angle. In con- case study, see Figure 4. The simulations were carried trast to this, for higher incident angles, corresponding 9

1.4 Experiment 1.2 MBN Explorer 1.5 ) 3

10 1 1 ×

1 3

0.8 10 ⋅

0.6 0.5 0.1 0.4 0 Counts (normalized)( ection angle, mrad

0.2 fl De Counts (normalized) 0 -0.5 0.01 -1 -0.5 0 0.5 1 1.5 Deflection angle, mrad -1 -0.5 0 0.5 1 1.5 Fig. 9 Simulated and measured angular distributions of 855 α MeV electrons exiting the oriented bent silicon crystal. At the Rotation angle, (mrad) entrance, the electron beam is aligned with the (111)¯ plane. The experimental geometry is in accordance with Figure 4 Fig. 10 Colour map of the angular distribution of electrons with θ = 95 mrad and α = 0. after interaction with bent silicon crystal as a function of the incident angle α, see Figure 4. The angle θ is set to 95 mrad. to the axial directions h1(n + 1)(n + 2)i with high in- 2 dices n, the deviation from the experiment is more pro- Experiment 1.8 MBN Explorer nounced. This feature indicates that although the field ) 3 1.6 10

intensities of the high-index axes are weak they, never- × 1.4 theless, influence the angular distribution of particles. 1.2 The second difference to be noted is that the peak, 1 centered at the deflection angle about 0.8 mrad, in 0.8 0.6 the simulated dependence is lower that its experimen-

Counts (normalized) ( 0.4 tal counterpart. This peak is due to the electrons that 0.2 channel through the whole length L of a crystal. To 0 make this peak pronounced the value of L should be -1 -0.5 0 0.5 1 1.5 fl comparable with the dechannelling length Ld. The sim- De ection angle, mrad ulations carried out in Ref. [47] by means of the MBN Fig. 11 Angular distribution of 855 MeV electrons with en- Explorer produced the result Ld = 16.6 ± 0.8 µm for ergy 855 MeV after interaction with bent silicon crystal. The 855 MeV electrons channelled in oriented Si(111) crys- simulated dependence was calculated at α = 0.45 mrad and θ = 95 mrad to match the experimental conditions. tal bent with radius R = 33 mm. The crystal thickness along the beam L = 30.5 mm which is very close to the value 33.5 mm used in the experiments [20, 21] was thin The simulated angular distribution is presented in enough to observe the peak of channelled electrons. Figure 11 together with the experimental data. In ac- A set of simulations has been carried out aimed at cordance with illustrative picture b) in Figure 1 and studying modifications in the angular distributions of the corresponding qualitative explanation outlined in electrons after interaction with the crystal as a func- the caption, the angular distribution the main modifi- tion of the incident angle α. The results obtained are cation in the angular distribution, as compared to the presented in Figure 10 which shows a scan over the case α =0, is that the main maximum due to the over- rotation angle α from −0.5 to 1.5 mrad. The pattern barrier particles is shifted to the left. This is a direct of the angular distribution is similar to that observed consequence of the volume reflection events which occur experimentally [21]. in the crystal bulk. The second less powerful maximum seen at the deflection angle of about 0.5 mrad appears as a result of the volume capture events: the electrons 5.6 Volume reflection: Simulation versus experiment are captured in the channelling mode at some point in the bulk where there velocity becomes aligned with the A more detailed study of the impact of the volume re- tangent of the bent plane [22]. flection phenomenon on the angular distribution of elec- The dependences shown indicate that in the sim- trons has been carried out for the incident angle α = ulations the peak due to VC corresponds well to the 0.45 mrad. The second angle θ was set to 95 mrad. experiment, whereas the peak associated with VR is 10

lower and slightly wider than the experimentally mea- 4.5 sured one. 4 Experiment 3.5 0.5 mrad 3

10 1.0 mrad ⋅ 3 6 Radiation emission spectra 1.5 mrad 3.0 mrad 2.5 The transition from the axial to planar channelling regimes E(dN/dE) 2 is revealed also in the changes in spectral distributions of the radiation emitted by projectiles. 1.5 Calculation of emission spectra is provided by a spe- 1 1 2 3 4 5 6 7 8 cial module of MBN Explorer which uses pre-calculated Energy, E(MeV) trajectories as the input data. The spectral intensities of radiation presented below have been calculated from Fig. 13 Radiation spectra by 855 MeV electrons incident along the (111)¯ plane at small angles θ. the same electron trajectories as used in analyzing the angular distributions.

6.1 Evolution of the emission spectra with the incident and, thus, radiates intensively. As θ increases, a pro- angle θ jectile moves across the axes experiencing smaller ac- celeration in the transverse direction. As a result, the intensity decreases (compare the curves corresponding to θ = 0.5, 1.0 and 1.5 mrad). For θ much larger than 5.5 the critical angle for the axial channelling, the spectrum 5 approaches its limit, which corresponds to the planar Experiment 4.5 <1,2,3> channelling regime. In Figure 14 the emission spectra <1,3,4>

3 4

10 <1,5,6> ⋅ 3.5 <1,7,8> 3

E(dN/dE) 2.5 2 1.5 9 1 8 1 2 3 4 5 6 7 8 7 Energy, E(MeV) 6

Fig. 12 Radiation spectra emitted by 855 MeV electrons 5 1 ¯ incident along the (111) plane at different axial directions 4 h1n(n + 1)i as indicated.

Energy, E(MeV) Energy, 3 2 Spectral Intensity (arb. units) 1 Figure 12 shows emission spectra calculated for 855 0.1 MeV electrons entering silicon crystal tangentially to its 0 -0.5 0 0.5 1 1.5 (111)¯ bent plane along several axial directions h1 n (n + α 1)i. These directions correspond to large values of the Rotation angle, (mrad) incident angles measured with respect to the h112i axis, Fig. 14 Colour map for radiation spectra emitted by 855 see Table 1. With n increasing, the averaged axial elec- MeV electrons in bent silicon crystal. The data refer to the tric field decreases leading to the decrease in the ra- incident angle θ = 95 mrad and different rotation angles α. diation intensity. At n = 7 (the h178i axial direction) the spectrum approaches that emitted in the planar channelling regime and becomes virtually insensitive to further increase of n. presented in the form of a scan over angle α varying Similar modification of the spectra occurs when the within the interval from -0.5 to 1.5 mrad as in the ex- incident angle varies within the interval θ ≪ 190 mrad, periment [20]. On total, the colour map corresponds to see Figure 13. In this case, small values of θ correspond that presented in Figure 1(b) of the cited paper. How- to the motion close to the axial direction h112i where ever, to carry out a more detailed comparison, below a projectile experiences action of the strong axial field we consider emission spectra at certain geometries. 11

2.8 sities is related to that seen in the angular distribution Experiment of the electrons, Figure 11. 2.6 RADCHARM ++ MBN Explorer 2.4

3 2.2 10 ⋅ 2 7 Conclusions

1.8 E(dN/dE) The atomistic approach implemented in MBN Explorer 1.6 allowed us to monitor in detail the changes in the dis- 1.4 tribution of the deflected electrons and in the emission 1.2 1 2 3 4 5 6 7 8 spectra that occur when the 855 MeV electron beam ¯ Energy, E(MeV) enters thin bent oriented silicon (111) crystal along dif- ferent directions. In particular, we have revealed the Fig. 15 Radiation spectra emitted by 855 MeV electron beam incident on bent silicon crystal at angles θ = 95 mrad ranges of the angle θ between the beam and the h112i and α = 0. axis which correspond to the planar channelling regime. To achieve the planar channelling one can direct the beam either along the axial directions h1n(n+1)i with 6.2 Comparison with the experiment n ≥ 6 (this corresponds to large angles, θ . 0.4 rad) or to vary the angle within the interval 3 < θ ≪ 190 For angles θ = 95 mrad and α = 0 (the channelling mrad, i.e. where the upper boundary stands for the an- regime), a comparison of the spectra obtained in the gle between the h112i and h123i axes. simulations with those measured experimentally is pre- Comparison of the results of simulation with the sented in Figure 15. The dependence calculated by means experimental data has been carried out for the two of the RADCHARM code [42] is presented as well. The geometries of the beam-crystal orientation correspond- simulation is normalized for the experiment at a given ing to (i) the channelling conditions, when the beam orientation of the crystal. We state that for this geome- is aligned with the tangent to the (111)¯ planar direc- try, which corresponds to the planar channelling regime, tion at the entrance, and (ii) the volume reflection and there is a good agreement between the experiment and volume capture regime. We may state that on total theory. there is good agreement between theory and experi- ment. In case (i) and (ii) some the discrepancies have been revealed in the the distribution of electrons. In the

2 channelling regime the experiment goes slightly above Experiment 1.9 the simulated curve in the vicinity of the channelling RADCHARM ++ 1.8 MBN Explorer peak whereas in case (ii) more notable difference is 1.7

3 seen for the peak associated with volume reflection. The

10 1.6 ⋅ emission spectrum simulated for the channelling regime 1.5 agrees perfectly with the experimentally measured one. 1.4 E(dN/dE) 1.3 In case (ii) the simulated data exhibit approximately 1.2 10 per cent excess over the experiment. 1.1 Possible reasons for the discrepancies can be asso- 1 ciated with (i) particular force field (the Moliére one) 1 2 3 4 5 6 7 8 chosen to describe the electron-atom interaction in the Energy, E(MeV) course of simulations and (ii) effects not included into Fig. 16 Radiation spectra emitted by 855 MeV electrons the simulations (e.g., quantum effects in multiple scat- in BC oriented as depicted in Figure 2 at the experimental θ = 95 and α = 0.5 mrad. This geometry corresponds to tering in crystals [62]). We plan to clarify these issues electron volume reflection of the bent (111) crystalline plane. in our future work.

For the geometry, which allows for the VR and VC Acknowledgements regime, a comparison of the simulation results and ex- perimental data is shown in Figure 16. It is seen, that al- We acknowledge support by the European Commis- though the shapes of the simulated and measured spec- sion through the N-LIGHT Project within the H2020- tra are the same, the absolute values differ by about 10 MSCA-RISE-2019 call (GA 872196) and by Deutsche per cent (in the maximum). The difference of the inten- Forschungsgemeinschaft (Project No. 413220201). This 12 work has been also partially supported through the 15. A.F. Elishev et al., Steering of charged particle tra- CSN5-ELIOT experiment and the Grant73-OSCaR project. jectories by a bent crystal. Phys. Lett. B 88, 387- 391 (1979) 16. V.M. Biryukov, Yu.A. Chesnokov, V.I. Kotov. References Crystal Channeling and its Application at High- Energy Accelerators. (Springer Science & Business 1. A.W. Sáenz, H. 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