Energy loss distributions of 7 TeV protons axially channeled in the bent 〈1 1 0〉 Si crystal

Nuclear Instruments and Methods in Physics Research B 373 (2016) 10–16

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Energy loss distributions of 7 TeV protons axially channeled in the bent h1 1 0i Si crystal Nace Stojanov a,⇑, Srdjan Petrovic´ b a b

Institute of Physics, Faculty of Natural Sciences and Mathematics, Sts. Cyril and Methodius University, P.O. Box 162, 1000 Skopje, Macedonia Laboratory of Physics (010), Vincˇa Institute of Nuclear Sciences, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia

a r t i c l e

i n f o

Article history: Received 19 October 2015 Received in revised form 12 February 2016 Accepted 16 February 2016

Keywords: Axial channeling Bent crystal Energy loss

a b s t r a c t In this article, the energy loss distributions of relativistic protons axially channeled in the bent h1 1 0i Si crystal are studied. The crystal thickness is equal to 1 mm, which corresponds to the reduced crystal thickness, K, equal to 1.22, whereas the bending angle, a, was varied from 0 to 30 lrad. The proton energy of 7 TeV was chosen in accordance with the concept of using the bent crystals as a tool for selective deflection of the beam halo particles from the LUA9 experiment at LHC. For the continuum interaction potential of the proton and the crystal the Molière’s expression was used and the energy loss of a proton was calculated by applying the trajectory dependent stopping power model. Further, the uncertainness of the scattering angle of the proton caused by its collisions with the electrons of the crystal and the divergence of the proton beam were taken into account. The energy loss distribution of the channeled protons was obtained via the numerical solution of the proton equations of motion in the transverse plane and the computer simulation method. The analysis of the obtained theoretical data shows that the shape of the energy loss distribution strongly depends on the horizontal or vertical direction of the curvature of the crystal. The number of dechanneled protons as a function of the bending angle also strongly depends on the direction of the crystal’s curvature. As a result, the dechanneling rates and ranges, obtained from the Gompertz type sigmoidal fitting functions, have different sets of values for different bending orientations. We have also studied the influence of the proton beam divergence on the energy loss distribution of channeled protons. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction The possibility of deflection of high-energy ion beams by a bent crystal was suggested by Tsyganov [1]. After that, the experimental evidence for channeling deflection of 8.4 GeV protons by the single bent crystal was reported in the Ref. [2]. In that experiment, the fraction of channeled protons was detected by using the measurement of their anomalously low energy loss distribution. In the past, a lot of experimental investigations of various aspects of the axial ion crystal channeling have been reported [3–12]. In parallel, the theoretical methods based on the computer simulations were developed for an accurate explanation of the ion channeling experimental results [13]. Considering the energy loss calculation, in the work of Desalvo and Rosa [14], the calculation was based on the dielectric function theory and the computer simulation method, which took into account the contributions from the valence electrons only. Soon after that, a computer simulation

⇑ Corresponding author. http://dx.doi.org/10.1016/j.nimb.2016.02.038 0168-583X/Ó 2016 Elsevier B.V. All rights reserved.

study of the axially channeled p mesons through a bent Au crystal based on the binary collision model was performed [15]. Improvements of the energy loss model based on the contribution of the valence and core electrons of the atoms led to a better agreement between the experiment and theory [16]. New impulse in the investigation of the energy loss of non-relativistic channeled ions was done by the trajectory dependent energy loss model based on the perturbation theory [17]. In this model, the valence and core electrons were treated equally, while the electron density was averaged along the particle trajectory. The implemented model was later generalized to include the relativistic particles [18,19]. In this work, a detailed study of the energy loss distributions of 7 TeV protons axially channeled through the bent h1 1 0i Si crystal is performed by applying the trajectory dependent stopping power model and the computer simulation method. The proton energy of 7 TeV was chosen in accordance with the concept of using the bent crystals as a tool for particle extraction or selective deflection of the beam halo particles from the LUA9 experiment at LHC [20,21] and the possibility for in-situ calibration of CMS and ATLAS calorimeters by the LHC beam [22]. The aim of our work is to

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tal channel the average angle of the transverse ion’s velocity vector with respect to the crystal axis increases. When this angle is larger than the critical angle for axial channeling, wc, the ion is treated as dechanneled and disregarded [23]. The critical angle is given by the
1=2 expression, wc ¼ 4Z 1 Z 2 e2 =ptd , where Z1 and Z2 are atomic numbers of the ion and the crystal, respectively, e is the elementary charge, p and t are the ion relativistic impulse and speed, respectively, and d is the distance between the atoms in the atomic string. The continuum interaction potential of the ion and the i-th atomic string of the crystal was obtained from the Molière’s expression for the ion–atom interaction potential [24,25]:

U i ðr i Þ ¼

       2Z 1 Z 2 e2 6r i 1:2r i 0:3r i þ 0:55K 0 þ 0:35K 0 ; 0:1K 0 d aTF aTF aTF ð1Þ

Fig. 1. The normalized local electron density averaged along the z direction for the h1 1 0i axial channel of the silicon crystal.

contribute to better understanding of the physics of steering of 7 TeV protons through the bent crystal, with respect to the energy loss problem.

where aTF ¼ 0:8853a0 ðZ 2 Þ1=3 is the screening radius, a0 is the Bohr radius, ri is the distance between the ion and the i-th atomic string and K0 is zero-order modified Bessel function of the second kind. The thermal vibrations of the crystal atoms were taken into account via the expression [25]:

U th i ðx; yÞ ¼ U i ðx; yÞ þ

r2th 2

DU i ðx; yÞ;

ð2Þ

2. Theory We investigate the ion’s motion along the h1 1 0i axial channel of silicon crystal. The z axis coincides with the channel axis, whereas x and y axes lie in the transversal plane, which is perpendicular to the z axis. When an ion enters the crystal at small angle with respect to its channel axis the channeling effect may occur for the straight and bent crystal [1]. In general, due to the ion’s interaction with crystal’s electrons during its motion through the crys-

where Ui is the continuum interaction potential of the i-th atomic string, rth is the one-dimensional amplitude of the thermal vibration of crystal atoms and D = oxx + oyy is the Laplace differential operator. The continuum interaction potential of the crystal is the sum of the continuum interaction potentials of the atomic strings, P th U th ¼ Ui . The (electronic) energy loss of the ion was calculated using the expression for the trajectory dependent stopping power [19,26]:

Fig. 2. The energy loss distributions of 7 TeV protons axially channeled through the h1 1 0i silicon crystal, which is bent in the horizontal direction. The thicknesses of the crystal is L = 1.0 mm, and the bending angle, (a) a = 6 lrad, (b) a = 10 lrad and (c) a = 15 lrad. The low energy loss peak is designated by the blue color, the medium energy transition area is designated by the green color and the high energy loss tail is designated by the red color. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 3. The energy loss distributions of 7 TeV protons axially channeled through the h1 1 0i silicon crystal, which is bent in the vertical direction. The thicknesses of the crystal, L = 1.0 mm, and the bending angle, (a) a = 6 lrad, (b) a = 10 lrad and (c) a = 15 lrad. The low energy loss peak is designated by the blue color, the medium energy loss peak is designated by the green color and the high energy loss tail is designated by the red color. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

dE D ¼  dz 2b2

(

 ) 2me c2 b2 c2 T max T max 2 ; ln  b  d þ qe ðx; yÞ ln  I I 2me c2 c2 ð3Þ r2e me c2 Z 21 hni,

where D ¼ 4p re is the classical radius of the electron, c is the speed of light, hni ¼ NZ 2 is the the averaged electron density, N is the crystal atom density, the maximal energy transfer of a single electron is given by T max ¼ 2me c2 b2 c2 =½1 þ 2cme =mi þ ðme =mi Þ2 , mi is the ion rest mass and the relativistic parameters are b = t/c and c2 = (1  b2)1. The density effect correction [27] hxe Þ  1, I is the crystal atom is dðpÞ ¼ 4:605 logðp=mi cÞ  2 lnðI= mean ionization potential, xe ¼ ð4pne =me Þ1=2 is the angular frequency of oscillations of the electron gas of the crystal induced by the ion and qe ðx; yÞ ¼ ne ðx; yÞ=hni is the normalized local electron density. The local electron density ne(x, y) averaged along the z direction is calculated by using the Poisson equation [28]:

DU th ðx; yÞ ¼ 4p½Q e ðx; yÞ þ Q n ðx; yÞ;

ð4Þ

where Q e ðx; yÞ ¼ ene ðx; yÞ is the electron electric charge, Q n ðx; yÞ ¼ Z 2 enn ðx; yÞ is the nuclei electric charge and nn(x, y) is the local nuclei density. For the h1 1 0i axially oriented silicon crystals under the consideration here, the local nuclei density is given by the equation [37]:

nn ðx; yÞ ¼

d 8pr2th

  r2 exp  2 ;

rth

ð5Þ

where d is the distance between the atoms in the h1 1 0i axial direction of silicon and r(x, y) is position of the ion in the transversal plane. For the mean-square angular deviation of the ion, dX2e =dz, caused by its interaction with the crystal’s electrons we used the expression [28]:

" # dX2e me D 2me c2 b2 c2 2 ln ¼ 2  b  d qe ðx; yÞ; dz p 2b2 I

ð6Þ

where me is the electron mass and p is the relativistic momentum of the ion. For the mean-square deviations of the transversal components of the scattering angle of channeled ion caused by its interaction with the crystal’s electrons we used the expressions pffiffiffi Xx ¼ Xy  Xe = 2. In planar or axially bent crystals, in principle, ions can be dechanneled by the following mechanisms, the multiple scattering from crystal’s electrons, the multiple scattering from crystal’s nuclei and the bending dechanneling mechanism [19,29]. However, in this work, due to the fact that the scattering with crystal’s nuclei for the h1 1 0i Si channel, which is by far the largest channel in silicon, is not probable – the area around the atomic strings defined by the screening radius is very much smaller than the area of the channel, the multiple scattering from the crystal’s nuclei was neglected in the calculation. However, it should be noted that taking into account the crystal geometry reasoning, the probability for ion’s multiple scattering with crystal’s nuclei depends on the crystal channel or a plane and generally is higher for a planar with respect to an axial crystal bending. The bending dechanneling mechanism occurs due to the fact that the curvature of radius, R, of the bent crystal induces an additional centrifugal term, U cf ¼ ptq=R, in the ion–atom interaction potential, where q is the transversal coordinate equal to ±x or ±y depending on the sigh and vertical or horizontal direction of the crystal curvature, respectively [19,30]. Therefore, assuming the positive direction of the curvature in the cases under the consideration, the effective continuum interaction potential that ion ‘‘sees” is given by the following expression: th U th eff ðx; yÞ ¼ U ðx; yÞ 

ptq ; R

ð7Þ

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Fig. 4. Protons in the impact parameter plane that generate the corresponding energy loss distributions given in Fig. 2, for the bending angle (a) a = 6 lrad, (b) a = 10 lrad and (c) a = 15 lrad. They are designated by the blue, green and red colors in accordance with Fig. 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

where q = x or q = y, for the vertical or horizontal crystal bending, respectively. The energy loss distribution of axially channeled protons and its dechanneling function were obtained via the numerical solution of the ion’s equations of motion in the transverse plane by using the Runge–Kutta method of the fourth order and the computer simulation method [30]. 3. Results and discussions We shall analyze here in details the energy loss distributions of 7 TeV protons axially channeled through the bent h1 1 0i Si crystal. The critical angle for axial channeling is wc ¼ 5:5 lrad. The thickness of the crystal, L = 1 mm, and the bending angle, a, is varied from 0 to 30 lrad, corresponding to the bending radius of the crystal, R, from 0 to 33.33 m (the bending curvature, j = 1/R, from 0 to 0.03 m1). The number of atomic strings was 36, i.e. we took into account the atomic strings lying on the three nearest square coordination lines. The average frequency of transverse motion of protons moving close to the channel axis was determined from the second order terms of the Taylor expansion of the continuum potential of the crystal in the vicinity of the channel axis [25,31]. The one-dimensional thermal vibration amplitude of silicon atoms is 0.00744 nm [32]. The proton beam divergence, before its interac-

tion with the crystal [25,30], is taken to be n = 0.1wc = 0.55 lrad. The initial position of the proton in the transversal plane was chosen uniformly within the area of the channel and the initial number of protons was N = 115,307. The proton whose initial position lies within the screening radius around the atomic strings defining the channel was treated as backscattered and disregarded. The number of backscattered protons was Nb = 2620. The mean average ionization potential of the silicon atom was taken to be I = 172.25 eV [33]. Fig. 1 shows the normalized electron density averaged along the z direction, qe, for the h1 1 0i axial silicon crystal channel. It is clear that the small areas close to the atomic strings correspond to the electron density higher than 1 and the large area around the channel axis corresponds to the electron density much smaller than 1. Fig. 2(a)–(c) shows the energy loss distributions for 7 TeV protons axially channeled through the h1 1 0i silicon crystal, which is bent in the positive horizontal direction, for the bending angle, a, equal to 6, 10 and 15 lrad, respectively. The energy loss distribution for a = 6 lrad is characterized by the low energy loss peak, designated by the blue color, located at DE1,2a = 0.774 MeV, a transition area, designated by the green color and the long high energy loss tail, which is designated by the red color. It should be noted that the low energy peak area was determined from its FWHM,

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Fig. 5. Protons in the impact parameter plane that generate the corresponding energy loss distributions given in Fig. 3, for the bending angle (a) a = 6 lrad, (b) a = 10 lrad and (c) a = 15 lrad. They are designated by the blue, green and red colors in accordance with Fig. 3. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

whereas the transition areas and tail energy loss areas are determined taking into account results presented in Fig. 4(a). For the energy loss distributions shown in Fig. 2(b) and (c), one can observe decreasing of the yield of the low energy loss peaks and their shifting toward higher energy loss values as the bending angle increases. The corresponding low energy loss peaks are located at DE1,2b = 0.778 MeV and DE1,2c = 0.782 MeV, respectively.

For the positive vertical bending of the h1 1 0i silicon crystal, Fig. 3(a)–(c) shows the energy loss distributions of 7 TeV protons for the bending angle, a, equal to 6, 10 and 15 lrad, respectively. These cases were analyzed by dividing the energy loss distribution into the low, medium and tail parts taking into account FWHM of the low energy loss peak and results presented in Fig. 5(a). For a = 6 lrad, the low energy loss peak is located at DE1,3a = 0.778 -

Fig. 6. The dechanneling functions for the 7 TeV protons axially channeled through the h1 1 0i silicon crystal, which is bent in the (a) horizontal and (b) vertical directions.

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Fig. 7. The energy loss distributions of 7 TeV protons axially channeled through the h1 1 0i silicon crystal, for the bending angle a = 10 lrad and the beam divergence, n = 2.8 lrad, in the (a) horizontal and (b) vertical directions.

MeV. The yield of the low energy peaks decrease as the bending angle increases and they are located at DE1,3b = 0.78 MeV and DE1,3c = 0.786 MeV, for the bending angles of 10 and 15 lrad, respectively. Further, one can clearly observe the medium energy loss peaks located at DE2,3a = 0.802 MeV, DE2,3b = 0.834 MeV and DE2,3c = 0.862 MeV, respectively. We have performed the analysis of the impact parameters of the protons generating the energy loss distributions presented in Figs. 2 and 3. The results are shown in Figs. 4 and 5, respectively. These figures also show the equipotential contour lines of the corresponding effective continuum interaction potentials (7). Comparison of Figs. 2(a) and 4(a) shows that the low energy loss peak originates from protons with large impact parameter with respect to the atomic strings. These protons create a ‘‘central blue area” with the local electron density lower than 10% of the average electron density of the crystal. This area decreases and moves in the y < 0 direction toward higher electronic density area, as the bending angle increases (see Fig. 4(b) and (c)). Clearly, this movement is due to the action of the centrifugal force. Further, movement of the low energy peak toward higher values, as the bending angle increases, can be explained with the fact that the protons on average ‘‘experience” higher electron density. On the other hand, for the bending angle a = 6 lrad, the high energy loss tail in the energy loss distributions is generated by the protons, designated by the red color, in the areas close to the atomic strings (with small impact parameters). The green transition area is located in between the blue and red areas. This clear distinguish between the blue and red areas were used to define the green area in the impact parameter plane and consequently the corresponding areas in the energy loss distribution, i.e. the blue–green–red division in the energy loss distribution. The relation between the energy loss distribution of the protons and their corresponding impact parameters, when the crystal is bent in the vertical direction, can be done by comparing Figs. 3 and 5, in the same way as in the case of the horizontal crystal bending. For the bending angle a = 6 lrad, the low energy loss peak in the energy loss distribution originates from the corresponding ‘‘central blue area”, shown in Fig. 5(a). It should be noted that this area is rotated by 90° with respect to the corresponding area shown in Fig. 4(a). As the bending angle is increasing, this area is decreasing and moving in the x < 0 direction under the influence of the centrifugal force, which is clearly shown in Fig. 5(b) and (c). In these cases, the medium energy loss area (green area) separates the blue and red areas, corresponding to the large and small proton’s impact parameters with respect to the atomic strings defining the crystal channel. It should be noted that this separation is clearer in comparison with the horizontal crystal bending (see Fig. 4(a)).

The influence of the impact parameters of channeled protons on the energy loss distributions presented in Figs. 2–5, is connected to the geometry of the silicon crystal channel into consideration. As shown in Figs. 2 and 4, the h1 1 0i axial channel is defined by two pairs of atomic strings located at the left and right sides of the channel being further from the channel center, and by the atomic strings at the top and bottom sides of the channel, being closer to the atomic center. Therefore, in the case of the crystal’s vertical bending (see Fig. 5), the effective potential (7) is stronger than in the case of the crystal’s horizontal bending (see Fig. 4). Thus, one can expect the bending dechanneling process to be more pronounced for the vertical in comparison with the horizontal crystal bending, which will be clearly presented in Fig. 6. We shall now analyze the dechanneling functions of the 7 TeV protons axially channeled through the h1 1 0i Si crystal bent in the horizontal and vertical directions, presented in Fig. 6 (a) and (b), respectively. These functions represent the dependencies of the ratio of the number of the dechanneled and initial protons on the crystal bending angle. The analysis of the data shows that the dechanneling function can be excellently fitted with the Gompertz type dechanneling function: Nd/N0 = (exp [exp (k (a  ac))]  exp [exp (kac)])/(1  exp [exp (kac)]), where Nd is the number of dechanneled protons, N0 = N  Nb is the initial number of protons without the backscattered ones, k is the dechanneling rate and ac is the dechanneling range [30]. According to the theory [34,35], ac corresponds to the inflection point of the Gompertz type dechanneling function, whereas k corresponds to the slope at the inflection point. In the case of horizontal bending of the crystal, the best fitting parameters have the following values ac = 11.97 lrad and k = 0.15 lrad1, whereas for the vertical bending of the crystal, the best fitting parameters are ac = 7.65 lrad and k = 0.32 lrad1. Therefore, the dechanneling process is much more pronounced in the case of the vertical bending. This can be explained with much stronger effective potential (7) in the vertical in comparison with the horizontal bending case (see Figs. 4 and 5). Fig. 7(a) and (b) shows the energy loss distributions for 7 TeV protons axially channeled through the h1 1 0i silicon crystal, for the horizontal and vertical bending cases, respectively, for the same bending angle, a = 10 lrad, and the beam divergence, n = 0.5wc = 2.8 lrad. Comparison of Fig. 7(a) and (b) with Figs. 2 (b) and 3(b), respectively, clearly shows that the yield of channeled protons decreases as the beam divergence increases, whereas, the shape of energy loss distributions have been changed, especially in the case of the horizontal crystal bending (Fig. 7(a)), with a clear appearance of the medium energy loss peak in both of the bending orientations [36]. The analysis of data shows that this is due to the enlargement of protons in the ‘‘green” area. Also, as a result of the increase of proton’s transversal energy since the beam divergence has been increased, the areas in the impact parameter plane that

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generate the medium energy loss peak and the high energy tail have been more spread throughout the channel in comparison with the cases presented in Figs. 4 and 5, for n = 0.1wc.

4. Conclusion We have analyzed in details the energy loss distribution dependence on the bending angle for 7 TeV protons axially channeled through the 1 mm long h1 1 0i Si crystal, for the horizontally and vertically bent crystal cases. The analysis was performed in the impact parameter plane by dividing the areas corresponding to the protons generating the low energy, medium and tail parts of the energy loss distribution. We have shown that the bending angle dependence of the energy loss distribution strongly depends on the bending orientation. It was also shown that the proton dechanneling function, which can be excellently fitted with the Gompertz type dechanneling function, strongly depends on the bending orientation. The obtained values for the horizontally bent crystal are ac = 11.97 lrad and k = 0.15 lrad1, whereas, for the vertically bent crystal are ac = 7.65 lrad and k = 0.32 lrad1. Finally, it was shown, by taking into account larger beam divergence, n = 0.5wc = 2.8 lrad, the appearance of the medium energy loss peak in both the horizontal and vertical crystal bending orientations and explained its origin.

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] [32]

Acknowledgments S.P. acknowledges the support to this work provided by the Ministry of Education and Science and Technological Development of Serbia through project Physics and Chemistry with Ion Beams, No. III 45006.

[33] [34] [35] [36] [37]

E.N. Tsyganov, Fermilab Reports TM-682 and TM-684, 1976. A.F. Elishev et al., Phys. Lett. B 88 (1979) 387. J. Bak et al., Nucl. Phys. B 242 (1984) 1. E. Uggerhöj, Nucl. Instr. Meth. 170 (1980) 105. A. La Ferla et al., Nucl. Instr. Meth. Phys. Res. B 66 (1992) 339. J.F. Dias et al., Nucl. Instr. Meth. Phys. Res. B 148 (1999) 164. G.M. Azevedo et al., Phys. Rev. B 65 (2002) 075203. L. Shao et al., Nucl. Instr. Meth. Phys. Res. B 249 (2006) 51. W. Scandale et al., Phys. Rev. Lett. 101 (2008). V. Guidi et al., Proceedings of PAC09 Vancouver, BC, Canada, WE6RFP092, 2009. M. Motapothula et al., Nucl. Instr. Meth. Phys. Res. B 283 (2012) 29. A. Baurichter et al., Nucl. Instr. Meth. Phys. Res. B 164–165 (2000) 27. D.V. Morgan, Channeling, Wiley, New York, 1973, p. 79. A. Desalvo, R. Rosa, J. Phys. C: Solid State Phys. 10 (1977) 1595. A.M. Taratin et al., Phys. Lett. A 72 (1979) 145. A. Dygo et al., Phys. Lett. A 127 (1987) 281. H. Esbensen, J.A. Golovchenko, Nucl. Phys. A 298 (1978) 382. A.F. Burenkov et al., Phys. Status Solidi B 99 (1) (1980) 417. A.M. Taratin, Phys. Part. Nucl. 29 (1998) 437. V.M. Biryukov, S. Bellucci, Nucl. Instr. Meth. Phys. Res. B 266 (2008) 235. W. Scandale et al., Phys. Lett. B 714 (2012) 231. V.M. Biryukov, S. Bellucci, Nucl. Instr. Meth. Phys. Res. B 252 (2006) 7. J. Lindhard, K. Dan, Vidensk. Selsk. Mat.-Fys. Medd. 34 (1965) 14. D.S. Gemmell, Rev. Mod. Phys. 46 (1974) 129. S. Petrovic´ et al., Nucl. Instr. Meth. Phys. Res. B 193 (2002) 152. V.M. Biryukov, Phys. Rev. E 51 (1995) 3552. R.M. Sternheimer, Phys. Rev. B 24 (1981) 6288. A. Babaev, S.B. Dabagov, Eur. Phys. J. Plus 127 (2012) 62. V.M. Biryukov et al., Crystal Channeling and Its Application at High-Energy Accelerators, Springer-Verlag, 1997. N. Stojanov et al., Nucl. Instr. Meth. Phys. Res. B 302 (2013) 9. N. Stojanov et al., Nucl. Instr. Meth. Phys. Res. B 244 (2006) 457. B.W. Batterman et al., Phys. Rev. 127 (1962) 690; Ya S. Umanskii et al., Fiz. Tverd. Tela 7 (1966) 2958 (Sov. Phys. Solid State 7 (1996) 2399). R.M. Sternheimer et al., Phys. Rev. B 3 (1971) 3681. M. Kokkoris et al., Eur. Phys. J. B 34 (2003) 257. S. Petrovic´ et al., Nucl. Instr. Meth. Phys. Res. B 256 (2007) 177. A.M. Taratin, Nucl. Instr. Meth. Phys. Res. B 119 (1996) 156. M. Kitagawa, Y.H. Ohtsuki, Phys. Rev. B 8 (1973) 3117.