Nuclear interactions and multiple coulomb scattering at volume reflection

For volume reflection of charged particles governed by the continuous potential of atomic planes in a bent crystal, we calculate the probability of a nuclear interaction. It is found to differ from the corresponding probability in an amorphous target by an amount proportional to the crystal bending...

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Дата:	2012
Автор:	Bondarenco, M.V.
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Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2012
Назва видання:	Вопросы атомной науки и техники
Теми:	Section A. Quantum Field Theory
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Цитувати:	Nuclear interactions and multiple coulomb scattering at volume reflection / M.V. Bondarenco // Вопросы атомной науки и техники. — 2012. — № 1. — С. 59-63. — Бібліогр.: 12 назв. — англ.

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spelling	irk-123456789-1069832016-10-11T03:02:10Z Nuclear interactions and multiple coulomb scattering at volume reflection Bondarenco, M.V. Section A. Quantum Field Theory For volume reflection of charged particles governed by the continuous potential of atomic planes in a bent crystal, we calculate the probability of a nuclear interaction. It is found to differ from the corresponding probability in an amorphous target by an amount proportional to the crystal bending radius and the particle mean deflection angle, independently of the shape of the interplanar continuous potential. That result is also applied to the description of the final beam angular divergence owing to the multiple Coulomb scattering. The theoretical predictions are compared with the results of recent experiments. Для объемного отражения заряженных частиц, управляемого непрерывным потенциалом атомных плоскостей в изогнутом кристалле, мы вычисляем вероятность ядерного взаимодействия. Найдено, что последняя отличается от соответствующей величины в аморфной мишени на величину, пропорциональную радиусу изгиба кристалла и среднему углу отклонения частиц, независимо от конкретной формы межплоскостного непрерывного потенциала. Данный результат также применяется для описания угловой расходимости, приобретенной пучком за счет многократного кулоновского рассеяния в мишени. Теоретические предсказания сравниваются с результатами недавних экспериментов. Для об'ємного відбиття заряджених частинок, зумовленого неперервним потенціалом атомних площин в зігнутому кристалі, ми обчислюємо вірогідність ядерного зіткнення. Знайдено, що остання відрізняється від відповідної величини в аморфній мішені на величину, пропорційну радіусу згину кристалу та середньому куту відхилення частинок, незалежно від конкретної форми міжплощинного неперервного потенціалу. Даний результат також застосовується для опису кутового розходження набутого пучком за рахунок багаторазового кулонівського розсіяння в мішені. Теоретичні передбачення порівнюються з результатами недавніх експериментів. 2012 Article Nuclear interactions and multiple coulomb scattering at volume reflection / M.V. Bondarenco // Вопросы атомной науки и техники. — 2012. — № 1. — С. 59-63. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 61.85.+p, 29.27.-a, 45.10.-b http://dspace.nbuv.gov.ua/handle/123456789/106983 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Section A. Quantum Field Theory Section A. Quantum Field Theory
spellingShingle	Section A. Quantum Field Theory Section A. Quantum Field Theory Bondarenco, M.V. Nuclear interactions and multiple coulomb scattering at volume reflection Вопросы атомной науки и техники
description	For volume reflection of charged particles governed by the continuous potential of atomic planes in a bent crystal, we calculate the probability of a nuclear interaction. It is found to differ from the corresponding probability in an amorphous target by an amount proportional to the crystal bending radius and the particle mean deflection angle, independently of the shape of the interplanar continuous potential. That result is also applied to the description of the final beam angular divergence owing to the multiple Coulomb scattering. The theoretical predictions are compared with the results of recent experiments.
format	Article
author	Bondarenco, M.V.
author_facet	Bondarenco, M.V.
author_sort	Bondarenco, M.V.
title	Nuclear interactions and multiple coulomb scattering at volume reflection
title_short	Nuclear interactions and multiple coulomb scattering at volume reflection
title_full	Nuclear interactions and multiple coulomb scattering at volume reflection
title_fullStr	Nuclear interactions and multiple coulomb scattering at volume reflection
title_full_unstemmed	Nuclear interactions and multiple coulomb scattering at volume reflection
title_sort	nuclear interactions and multiple coulomb scattering at volume reflection
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2012
topic_facet	Section A. Quantum Field Theory
url	http://dspace.nbuv.gov.ua/handle/123456789/106983
citation_txt	Nuclear interactions and multiple coulomb scattering at volume reflection / M.V. Bondarenco // Вопросы атомной науки и техники. — 2012. — № 1. — С. 59-63. — Бібліогр.: 12 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT bondarencomv nuclearinteractionsandmultiplecoulombscatteringatvolumereflection
first_indexed	2025-07-07T19:17:07Z
last_indexed	2025-07-07T19:17:07Z
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fulltext	NUCLEAR INTERACTIONS AND MULTIPLE COULOMB SCATTERING AT VOLUME REFLECTION M.V. Bondarenco∗ National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine (Received November 1, 2011) For volume reflection of charged particles governed by the continuous potential of atomic planes in a bent crystal, we calculate the probability of a nuclear interaction. It is found to differ from the corresponding probability in an amorphous target by an amount proportional to the crystal bending radius and the particle mean deflection angle, independently of the shape of the interplanar continuous potential. That result is also applied to the description of the final beam angular divergence owing to the multiple Coulomb scattering. The theoretical predictions are compared with the results of recent experiments. PACS: 61.85.+p, 29.27.-a, 45.10.-b 1. INTRODUCTION Volume reflection [1] is a phenomenon when a fast charged particle reflects from a family of curved atomic planes of an oriented bent crystal to side op- posite to that of the crystal bending. It is considered to be a promising mechanism of local beam steering at high energy particle accelerators. The transverse direction asymmetry in the volume reflection effect originates from the asymmetry of the continuous po- tential of bent atomic planes, especially in the area where the angles of atomic plane crossing by the par- ticle become comparable to the Lindhard’s critical angle θc = √ 2V0/E, with V0 the interplanar contin- uous potential well depth, and E � V0 the particle energy. The extent of the volume reflection area is estimated as ∼ Rθc, where R is the crystal bending radius [2]. Although the origin of the volume reflection ef- fect is due to the coherent potential scattering, in a real crystal one must also take into account incoher- ent Coulomb scattering on individual atomic nuclei at close interactions with them. The condition for the incoherent scattering not to spoil the volume reflec- tion effect is the smallness of the multiple Coulomb scattering r.m.s. angle accumulated along the whole traversed crystal compared to the mean volume re- flection angle. That condition permits usage for the volume reflection experiments of the crystals of thick- ness 1÷ 2 mm, by ∼ 10 times exceeding the essential volume reflection region extent.1 However, in other respects, for instance for evaluation of the outcom- ing beam angular divergence, the multiple Coulomb scattering is crucial. Another manifestation of the nuclear interactions at high energy are the multiple hadron production events, which can be registered in downstream detectors (beam loss monitors). Outside the volume reflection area, the parti- cle motion becomes highly over-barrier and straight- ens out even relative to the active atomic planes. Thereat, the rate of a fast particle scattering on atomic nuclei must approach that in an amorphous medium (see Sec. 2). In a thick-crystal limit, the number of close nuclear interactions in the whole crystal will be dominated by the pre- and post- volume reflection areas, and will become about equal to that in an amorphous target of same material and thickness. But there remains a finite difference gener- ated in the volume reflection region, which carries in- formation about the volume reflection dynamics and kinetics, and usually is sufficiently sizeable to man- ifest itself in the experimental data. In the present article we calculate that difference, and compare the result with the recent related measurements. For a more detailed discussion of the involved problems see [3]. 2. NUCLEAR INTERACTION PROBABILITY AT VOLUME REFLECTION The volume reflection implies particle interaction with the bent crystal in a planar orientation. Thereat, the atomic density in each plane may be re- garded as uniform. Since all the nuclei are located in the planes, the particle crossing of one atomic plane may be regarded as an elementary act of nuclear in- teraction. In the simplest case when all the planes are equivalent (which corresponds to silicon crystal in orientation (110)), the surface density of nuclei in each plane equals natd, where nat is the atomic vol- ume density of the crystal, and d the inter-planar dis- ∗E-mail address: bon@kipt.kharkov.ua 1At an exemplary beam energy E � 400 GeV (CERN SPS), which entails θc ∼ 10−5 rad, and at optimal radius R ∼ 10 m, this longitudinal scale amounts to Rθc ∼ 10−1 mm. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1. Series: Nuclear Physics Investigations (57), p. 59-63. 59 tance. If nuclear concentrations along the planes may be regarded as thin (despite the thermal broadening) and thus crossed by the fast particle at a definite tan- gential angle θ, the probability of any kind of nuclear interaction in one atomic plane amounts to P1 = natσA d sin θ (1) with σA the corresponding cross-section of the par- ticle interaction with a single nucleus. For elastic scattering one has to take the transport cross-section (σA → σtr), while for inelastic interactions – the cor- responding total inelastic cross-section on a Silicon nucleus (σA → σinel). Generalization to a case with a few non-equivalent planes within a period (rele- vant, e.g., for silicon crystal in orientation (111)) is straightforward. Consider first the case of a straight crystal tra- versed by a fast particle in a highly over-barrier regime, when the angle θ between the particle mo- mentum and the planes by far exceeds the critical value, θ � θc. (2) Thereat, θ varies negligibly within the crystal (varθ < θc � θ). Summing up contributions (1) for ≈ L sin θ d crossed planes then yields the total nuclear interac- tion probability: P = P1 L sin θ d = natσAL (highly over-barr. pass.) (3) This value is independent of d and θ, and equals to the corresponding probability in an amorphous (i.e. poly- crystalline) medium – not surprisingly since the uni- form particle flow covers each nucleus with the same density, regardless of the existence of a far atomic or- der in the medium. In that sense, one can as well speak about an “amorphous orientation” of a perfect crystal. Next we consider the case of a bent crystal, to which a particle enters and exits in a highly over- barrier regime, but undergoes volume reflection some- where in the middle of the crystal. In the vicinity of the reflection point, the particle transverse kinetic en- ergy is comparable to the potential energy, and con- dition (2) breaks down, and the “amorphous orien- tation” does not apply. Thereat, the plane crossing angle varies considerably along the particle path, and from plane to plane. Hence, for accurate evaluation of nuclear interaction probabilities (1) at each plane crossing, one needs at least to evaluate the whole trajectory beyond the straight line approximation. Within the volume reflection region, it may be le- gitimate to neglect the multiple Coulomb scattering entirely, and compute the particle trajectory in the pure continuous potential. That will be our approxi- mation in the present article. Provided the crystal is bent uniformly (which is sufficiently credible at present technology level), the trajectory description simplifies in cylindrical coordi- nates, with the bent planes corresponding to surfaces of constant radius relative to some axis far outside the crystal. Thereat, the plane crossing angle sine enter- ing Eq. (1) expresses simply as the time derivative of the particle radial coordinate: sin θ ≈ ṙ/c (4) (we deal with ultra-relativistic particles moving nearly at speed of light c). Inserting (4) to Eq. (1) and summing over all the planes crossed by the par- ticle, we obtain the total inelastic nuclear interaction probability in a bent crystal: P ≈ natσAcd ∑ n 1 ṙn (unif. bent crystal). (5) Since the particle motion is supposed to straighten out away from the volume reflection area, there the nuclear interaction rate per unit length must ap- proach that in an amorphous medium. Hence, the difference between the number of nuclear interactions in an oriented crystal and in an “unoriented” crystal may be expressed as ΔP = natσAΔL (6) with the isolated geometrical factor ΔL = lim L→∞ (∑ n cd ṙn − L ) , (7) expected to be finite and independent of L, repre- senting the excess (or deficit) of the target nuclear interaction range. The evaluation of limit (7) and averaging over the angles and impact parameters of particles in the ini- tial beam simplifies under the condition R � Rc, and may be performed without the need to specify the ex- act shape of the inter-planar potential. For positively charged particles, the result reads [3]: 〈ΔL〉 = R 〈χ〉 (1 + O(R2 c/R2) ) (pos. particles), (8) where 〈χ〉 is the correspondingly averaged deflection angle of particles in the crystal. Importantly, 〈χ〉 includes O(Rc/R) corrections (cf. Eq. (17) below), which may be important at practice, but higher order corrections are beyond the accuracy of relation (8). Remarkably, result (8) holds both in orientations (110) and (111). For negatively charged particles, at leading order in Rc/R, when 〈χ〉 ≈ θc, the result is similar to (8): 〈ΔL〉 = −Rθc (1 + O(Rc/R)) (neg. particles). (9) Corrections O(Rc/R) can also be computed, but they are not absorbable into 〈χ〉 and differ for (110) and (111) orientations (see [3]). The principal difference between Eq. (8) and (9) is that for positive particles the nuclear range excess is positive, while for negative particles it is nega- tive (representing a deficit), being of the same or- der in magnitude. That property is natural from 60 the viewpoint that positive particles are repelled from the atomic planes and cross them more tangentially, while negative particles are attracted, crossing the planes quicker. 3. COMPARISON WITH THE EXPERIMENTAL DATA Inelastic nuclear interaction probability. Pre- dictions (8, 9) can be tested against the available ex- perimental data. The most direct check is supposed to be against the results of experiments on inelastic nuclear scattering. At present, there is one such ex- periment, performed at CERN with 400 GeV protons and a L = 2 mm thick silicon crystal at a single value of the crystal bending radius R = 10 m [4]. When the cutting angle (essentially the initial beam collimation angle) was sufficiently large (which ought to correspond to perfect averaging over b or E⊥), the measured relative difference between the number of inelastic nuclear interaction events at vol- ume reflection and in the “amorphous orientation” was about constant, holding on the level ΔP P ≈ (5 ± 2)%. (experim.) (10) For comparison, our prediction, using the experimen- tally determined [7] mean value 〈χ〉exp = 13.35 μrad at the given curvature 1/R = 0.1 m−1 amounts to ΔP P = 〈ΔL〉 L = R 〈χ〉exp L = 6.67%. (theor.) (11) The agreement between (10) and (11) may be re- garded as satisfactory. Impact on the final beam angular divergence. The nuclear interactions also manifest themselves through angular broadening of the final beam due to elastic Coulomb scattering. There is, however, an- other contribution to the broadening, present even in a pure continuous potential, and stemming from the impact parameter dependence of the deflection angle. In fact, the continuous potential contribution is ab- sent in the direction parallel to the planes, but the beam spread parallel to the planes is rarely measured, so in the published experimental data on the beam dispersion in the direction of deflection the amor- phous and continuous potential contributions enter together. Reasonably, we can decompose the kinetics of the particle passage through the crystal into three dis- tinct stages: pure incoherent multiple scattering up- stream the volume reflection region (where the beam acquires Gaussian shape), pure dynamical broaden- ing in the volume reflection region, and pure incoher- ent multiple scattering downstream of it. Denoting by dw/dχ the final angular distribution function, and by dwcoh/dχ the intrinsic volume reflection distribu- tion function, obtained with the neglect of multiple Coulomb scattering, the aggregate angular dispersion σ2 = ∫ ∞ −∞ dχ (χ − 〈χ〉)2 dw dχ (∫ ∞ −∞ dχ dw dχ = 1 ) (12) about the mean value 〈χ〉 = ∫ ∞ −∞ dχχ dw dχ (13) is represented as a sum of independent contributions: σ2 = σ2 am(L + 〈ΔL〉) + σ2 coh ≡ σ2 am(L) + σ2 am(R 〈χ〉) + σ2 coh, (14) with σ2 coh = ∫ ∞ −∞ dχ (χ − 〈χ〉)2 dwcoh dχ , (15) and provided σ2 am(T ) is a linear function of T (see Eq. (19) below). The intrinsic angular distribution dwcoh/dχ was evaluated in [5] in the model of harmonic continu- ous potential between (110) silicon crystallographic planes. It has some differences for positive and neg- ative particles, as does the nuclear interaction rate described in Sec. 2. Positively charged particles. For positive particles, at R > 4Rc the angular distribution of the intrinsic vol- ume reflection has approximately rectangular shape (see Eq. (72) of [5], where the calculation was con- ducted in the harmonic approximation for the inter- planar continuous potential)2 dwcoh dχ ≈ Rθc πd Θ ( πd 2Rθc − \|χ − 〈χ〉 \| ) . (16) In Eq. (16) Θ(s) is the Heavyside unit step function, and 〈χ〉 equals3 〈χ〉 ≈ θlim ( 1 − d θ2 cR ) , θlim = π 2 θc. (17) By Eq. (15), the corresponding σcoh is found to be σcoh ≈ π 2 √ 3θc d R , (18) notably being inversely proportional to the crystal bending radius. For σam we may adopt the simple formula of Gaussian diffusion: σam(T ) ≈ E0 E √ T X0 , (19) with X0 the radiation length (X0 ≈ 9.36 cm for sil- icon), and E0 an universal constant. With the ac- count of Rutherford asymptotics of the scattering cross-section, and of the distribution function at large 2In paper [5] the final beam angular distribution was described in terms of dλ/dχ, the differential cross-section. But obvi- ously, dividing that quantity by d, we obtain the normalized probability distribution dwcoh dχ = 1 d dλ dχ , � dχ dwcoh dχ = 1 handled in the present paper. 3Note that d/θ2 c ≈ 2Rc, but when the continuous potential is not exactly harmonic, expression d/θ2 c works better. 61 χ, σ2 defined by Eq. (12) logarithmically diverges, but instead it can be determined from a Gaussian fit to the experimentally measured distribution [9]. In this case, E0 weakly depends on T , and in the range T ∼ 0.2 ÷ 1 mm, practically important for volume reflection, the approximate value for E0 is E0 ≈ 11 MeV (see [3]). Measurements of total σ2 for 400 GeV protons interacting with a (110) silicon crystal were made in CERN experiment [7]. There, in order to get access to the intrinsic volume reflection angular divergence σcoh, the difference σ2 − σ2 am(L) = σ̄2 v.r. (20) was evaluated. Yet, according to Eq. (14), it differs from pure σcoh: σ̄v.r. = √ σ2 − σ2 am = √ σ2 coh + σ2 am(R 〈χ〉). (21) Inserting explicit theoretical expressions (18), (19) into Eq. (21) leads to a non-scaling dependence of the measured quantity σ̄v.r. on R: σ̄v.r. = √ π2 12θ2 c d2 R2 + πθc 2 ( E0 E )2 R − d/θ2 c X0 . (22) The most characteristic feature of function (22) is the existence of a minimum with respect to variation of R. The minimum location is found by differentiating the radicand: R∗(E) = 1 θc 3 √ π 3 X0d2 ( E E0 )2/3 � ( E 38 GeV )7/6 [m]. (23) It may serve to mark the scale of R where the mul- tiple scattering compares with coherent deflection angles. 10 R� 20 30 40 R�m� 1 2 3 4 5 6 7 Σv.r.�Μrad� Subtracted final beam angular width vs. the crystal bending radius, for E = 400 GeV protons in a L = 2 mm silicon crystal. Solid curve: theoretical prediction [Eq. (22)]. Dotted curve: pure σcoh (also compatible with calculation of [10] in a more realistic continuous potential model). Points: experimental data [7] The data of experiment [7] do show a flattening of the R-dependence around R∗ (see the figure), but at greater R there is an indication of further decrease. More experimental points in the region R > R∗ are needed to establish a clear trend. Negatively charged particles. For negative particles the experimental data are yet too scarce to extract a picture of σ̄2 v.r.(R) behavior, so we restrict ourselves to a few remarks. For negatively charged particles the expression for σcoh(R) differs only by a numerical coefficient (actually, logarithmically dependent on R), but the main ∼ 1/R-dependence remains. But according to Eq. (4), the difference of the amorphous contribution changes sign: Δσ2 am ∝ 〈ΔL〉 < 0. Therefore, the expression for σ̄2 v.r. for negative particles is similar to the radicand of Eq. (22), but with a negative co- efficient at the second term. That implies that for negative particles σ̄2 v.r. turns to zero at some value of R, and becomes negative beyond it. That is the salient feature of the final beam angular distribution for negative particles, which would be interesting to verify experimentally. Secondly, since σcoh for positively and for nega- tively charged particles differ, in general it is not as straightforward to compare the angular broadenings for positive and negative particles, as it was for the rate of inelastic nuclear interactions. However, in the region R > Rc where σcoh gets relatively small, that must become possible. The simplest way of pinning down σcoh, though, is to measure both angular beam divergence components perpendicular and parallel to the family of the active atomic planes. Acknowledgement The author thanks to V. Guidi and A.V. Shchagin for discussions and to A.M. Taratin for useful corre- spondence. References 1. A.M. Taratin and S.A. Vorobiev. Deflection of high-energy charged particles in quasi-channeling states in bent crystals // Nucl. Instrum. Methods. 1987, v. B26, p. 512-521; A.G. Afonin et al. The schemes of proton extrac- tion from IHEP accelerator using bent crystals // Nucl. Instrum. Methods. 2005, v. B234, p. 14-22; V.M. Biryukov et al. Crystal collimation as an option for the large hadron colliders // Nucl. In- strum. Methods. 2005, v. B234, p. 23-30; V. Shiltsev et al. Channeling and volume reflec- tion based crystal collimation of tevatron circu- lating beam halo (T-980) // Proc. of IPAC-2010, Kyoto, Japan, p. 1243-1245. 2. A.M. Taratin and W. Scandale. Volume reflec- tion of high-energy protons in short bent crys- tals // Nucl. Instrum. Methods. 2007, v. B262, p. 340-347. 3. M.V. Bondarenco. Account of Nuclear Scatter- ing at Volume Reflection // arXiv : 1108.0648v1. 4. W. Scandale et al. Probability of inelastic nu- clear interactions of high-energy protons in a bent 62 crystal // Nucl. Instrum. Methods. 2010, v. B268, p. 2655-2659. 5. M.V. Bondarenco. Model solution for volume re- flection of relativistic particles in a bent crystal // Phys. Rev. 2010, v. A82, 042902, 19 p. 6. M.V. Bondarenco. Comments on theory of vol- ume reflection and radiation in bent crystals // Il Nuov. Cim. 2011, v. C34, p. 381-388.; arXiv : 1103.0770. 7. W. Scandale et al. Volume Reflection Depen- dence of 400 GeV/c Protons on the Bent Crys- tal Curvature // Phys. Rev. Lett. 2008, v. 101, 234801, 4 p. 8. W. Scandale et al. Observation of channeling and volume reflection in bent crystals for high- energy negative particles // Phys. Lett. 2009, v. B681, p. 233–236. 9. G.R. Lynch and O.I. Dahl // Nucl. Instrum. Methods. 1991, v. B58, p. 6; V.L. Highland // Nucl. Instr. Methods. 1975, v. 129, p. 497; ibid. 1979, v. 161, p. 171; K. Nakamura et al. (Particle Data Group) // Re- view of Particle Physics. J. Phys. 2010, v. G37, 075021. 10. V.A. Maisheev. Volume reflection of ultrarela- tivistic particles in single crystals // Phys. Rev. ST Accel. Beams. 2007, v. 10, 084701, 11 p. 11. Yu.A. Chesnokov et al. Radiation of photons in process of charge particle volume reflection in bent single crystal // JINST. 2008, v. 3, P020052; A.G. Afonin et al. Investigation of the emission of photons induced in the volume reflection of 10- GeV positrons in a bent silicon single crystal // JETP Lett. 2008, v. 88, p. 414-417; W. Scandale et al. Experimental study of the radiation emitted by 180-GeV/c electrons and positrons volume-reflected in a bent crystal // Phys. Rev. 2009, v. A79, 012903, 9 p.; M.V. Bondarenco. Coherent bremsstrahlung in a bent crystal // Phys. Rev. 2010, v. A81, 052903, 14 p.; Yu.A. Chesnokov et al. Photoproduction of electron-positron pairs in bent single crystals // Phys. Rev. ST Accel. Beams. 2010, v. 13, 070706, 6 p. 12. V. Biryukov. The Theory of the scattering in- duced feeding in in bent crystals // Phys. Lett. 1995, v. A205, p. 340-343; V.M. Biryukov and S. Bellucci. Simulations of volume reflection and capture in bent crystals // Nucl. Instr. Methods. 2008, v. B266, p. 235-241; V.M. Biryukov. The Limits of volume reflection in bent crystals // Nucl. Instr. Methods. 2009, v. B26, p. 2457-2459. �� !�� " �� #� � �� $ � �� "�� $ � �� #� �� %�� &�� '(� � � �)�� ) �� !� � �) �� )� �� (�� )�� )� )�� )�� * �� !� �� )��)�$ �(�� )� �)��)� �+ �� " )� �)#� ) �� )� � � �)�� )�� )� �� + "�� )��!� � � ��$ �� ) �� (�� )�� )� � � �)#� )� %�� ) �� )�$ �� )� �� )�� ,-

Nuclear interactions and multiple coulomb scattering at volume reflection

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