EN ERGY LOSS DIS TRI BU TIONS OF 7 TeV PRO TONS CHAN NELED IN A BENT SIL I CON CRYS TALS

Transcription

1 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 013, Vol. 8, No. 1, pp EN ERGY LOSS DIS TRI BU TIONS OF 7 TeV PRO TONS CHAN NELED IN A BENT SIL I CON CRYS TALS by Nace STOJANOV 1*, Srdjan M. PETROVI], and Nebojša B. NEŠKOVI] 1 Institute of Physics, Faculty of Natural Sciences and Maematics, Sts. Cyril and Meodius University, Skopje, Macedonia Laboratory of Physics, Vin~a In sti tute of Nu clear Sciencies, Uni ver sity of Bel grade, Belgrade, Serbia Sci en tific pa per DOI: 10.98/NTRP/ S The energy loss distributions of relativistic protons axially channeled rough e bent <100> Si crys tals, wi e con stant cur va ture ra dius, R = 50 m, are stud ied here. The pro - ton en ergy is 7 TeV and e ick ness of e crys tal is var ied from 1 mm to 5 mm, which cor re - sponds to e re duced crys tal ick ness, L, from.1 to 10.6, re spec tively. The pro ton en ergy was cho sen in ac cor dance wi e large hadron collider pro ject, at e Eu ro pean Or ga ni za - tion for Nu clear Re search, in Geneva, Swit zer land. The en ergy loss dis tri bu tions of e chan - neled pro tons were gen er ated by e com puter sim u la tion meod us ing e nu mer i cal so lu - tion of e pro ton equa tions of mo tion in e trans verse plane. Dis per sion of e pro ton scat ter ing an gle caused by its col li sions wi e crys tal s elec trons was taken into ac count. Key words: rel a tiv is tic chan nel ing, bent crys tal, en ergy loss dis tri bu tion INTRODUCTION It is well known at e en ergy loss of an ion dur ing its pas sage rough e crys tal de pends on its orientation wi respect to e crystallographic axis or planes [1]. When high en ergy charged ion is chan - neled, eir mo tion is mostly re stricted in e re gion wi lower elec tron den sity, and, ac cord ingly, its en - ergy loss is mostly de ter mined from e dis tant col li - sions wi e crys tal s elec trons. In e case of ran - dom ori en ta tion of e ion beam wi re spect to crys tal, e ion en ergy loss is de ter mined from bo close and dis tant col li sions wi e crys tal s elec trons []. There fore, e ran dom (av er age) ion en ergy loss is larger an e chan nel ing one. The chan nel ing ef fects in e straight and bent crystals have been studied in details eoretically [], ex per i men tally [3, 4] and us ing me ods of e com - puter sim u la tions [5, 6]. It has been also shown at e reduced crystal ickness L can clas sify e an gu lar dis tri bu tions of ax i ally chan neled ions in e straight or bent crys tals [7-9]. The re duced crys tal ick ness is de fined by e ex pres sion: Lf.Ln, where f is e fre - quency of trans verse mo tion of ion mov ing close to e atomic strings, L e crys tal ick ness, and n e ion ve loc ity. Fre quency f is de ter mined from e Tay - * Corresponding auor; nacestoj@gmail.com lor ex pan sion of e con tin uum po ten tial of e crys tal in e vi cin ity of e chan nel axis [9, 10]. The val ues of L = 0, 0.5, 1, cor re spond to e be gin nings of e an gu lar dis tri bu tion cy cles [10]. In is work, e en ergy loss dis tri bu tions of rel a - tiv is tic pro tons chan neled in e bent <100> Si crys tal, wi e con stant cur va ture ra dius, R = 50 m, are stud - ied. The pro ton en ergy is 7 TeV and e ick ness of e crys tal is var ied from 1 to 5 mm, which cor re - sponds to e re duced crys tal ick nesses from.1 to 10.6, re spec tively. The pro ton en ergy was cho sen in ac cor dance wi e Large Had ron Collider pro ject, at e European Organization for Nuclear Research, in Geneva, Switzerland. THEORY The sys tem un der con sid er ation is an ion which en ters a bent sil i con crys tal at small an gles wi re spect to some of e main crys tal lo graphic axis. As a re sult of e ion-crys tal in ter ac tion, which con sists of ion-atomic strings in ter ac tions, e cen trif u gal force due to e crys tal cur va ture act ing on e ion, and e ion in ter ac - tion wi e crys tal s elec trons, e (av er age) transversal ki netic en ergy of ion is in creas ing as e ion passes rough e crys tal, mean ing at (av er age) an gle of ion ve loc ity vec tor wi re spect to e crys tal axis is in creas ing. When is an gle is larger an e crit i cal an -

2 3 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 013, Vol. 8, No. 1, pp gle for chan nel ing [1] y c = (4Z 1 Z e /pud) 1/, where Z 1 and Z are e atomic num bers of ion and at oms of e crystal, respectively, e is e el e men tary charge, p e ion (relativistic) impulse and d e dis tance be tween e at oms in e atomic string; e ion is treated as be ing dechanneled. In e cal cu la tion of e ion en ergy loss, we as - sume at e con tin uum in ter ac tion po ten tial of e ion and e i- atomic string of e crys tal is given by e Lindhard s ex pres sion [1] Z Z e Ca 1 s U i ln 1 (1) d p i where p i is e dis tance be tween e ion and e i- atomic string, a s = [9 /18] (Z Z a e screen ing ra dius, a 0 e Bohr ra dius, and C e screen ing con stant. The er mal vi bra tions of e crys - tal at oms are taken into ac count via ex pres sion [10] U i ( x, y) U ( x, y) s i U ( x, y) i D () where U i is e con tin uum po ten tial of e i- atomic string wi e er mal vi bra tions of e at oms ne - glected, xx + yy, x and y are e trans verse com po - nents of e ion po si tion, and s is e one-di men - sional ermal vibration amplitude of crystal atoms. The con tin uum po ten tial of e crys tal, U, is e sum of e con tin uum po ten tials of e atomic strings. For e rel a tiv is tic ion stop ping power we use ex pres sion [11, 1] de 4 Z 1 e 4 p meg n ne ln b dz m ev we (3) where w e = (4e n e /m e ) 1/ and n e = DU /4,m e is e elec tron mass, b = n/c, where c is e speed of light, g b w e e an gu lar fre quency of e elec tron gas os cil la tions of e crys tal in duced by e ion, and n e = n e (x, y) is e (av er age) elec tron gas den sity. The mean-square an gu lar de vi a tion of e ion due to its col li sions wi e elec trons of e crys tal and e ion beam di ver gence be fore its in ter ac tion wi e crys tal are taken into ac count [11]. The en ergy loss strag gling ef fect is ne glected [13, 14]. Due to e ac tion of e cen trif u gal force on e chan neled ion, e ef fec tive con tin uum in ter ac tion po - ten tial of e ion and e crys tal is given by [9] p U eff ( x, y) U ( x, y) u R y (4) We chose e co-or di na tion sys tem in which e centrifugal force is directed toward y axis. The en ergy loss dis tri bu tion of e chan neled ions was ob tained via e nu mer i cal so lu tion of e ion equa tions of mo tion in e trans verse plane and e com puter sim u la tion meod [10, 11]. Fig ure 1. En ergy loss dis tri bu tions of 7 TeV pro tons chan neled rough e bent crys tals (R = 50 m) for e re duced crys tal ick nesses equal to (a) L =.5, (b) L = 3.5, (c) L = 5.0, and (d) L = 10.5

3 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 013, Vol. 8, No. 1, pp Fig ure. Pro tons in e im pact pa ram e ter plane at gen er ate e en ergy loss dis tri bu tions given in fig. 1, for e re duced crys tal ick nesses equal to (a) L =.5, (b) L = 3.5, (c) L = 5.0, and (d) L = 10.5 RESULTS AND DISCUSSIONS As we have al ready men tioned, we shall an a lyze here e en ergy loss dis tri bu tions of rel a tiv is tic pro - tons ax i ally chan neled in e bent <100> Si crys tal, wi e con stant ra dius of cur va ture ra dius R = 50 m. The pro ton en ergy is 7 TeV, e ick ness of e crys - tal, L, var ies from 1 mm to 5 mm, which cor re sponds to e re duced crys tal ick nesses, L, from.1 to The dis tance be tween e at oms in e atomic string is nm [15]. The one-di men sional er mal vi - bra tion am pli tude of sil i con at oms is nm [16]. The num ber of atomic strings is 36, i. e. we took into ac count e atomic strings ly ing on e ree near est square co or di na tion lines. The crit i cal an gle for e chan nel ing is equal to 4.6 mrad. The ini tial po si tions of e ions in e transversal plane were cho sen uni formly wiin e crys tal chan nel. The ini tial num ber of e ions was,019,41. The ions whose ini tial po si tions lie wiin e screen ing ra dius around e atomic strings de fin ing e chan nel were treated as back scat tered and dis re garded. The pro ton beam di ver gence was taken to be 0.1y c = 0.46 mrad. Fig ures 1(a)-(d) show e en ergy loss dis tri bu - tions of e chan neled pro tons for e re duced crys tal ick nesses: L =.5, 3.5, 5.0, and 10.5, re spec tively, which cor re spond to e crys tal ick nesses: L = 1., 1.6,.3, and 4.9 mm, re spec tively. Fig ure 1(a) shows e en ergy loss dis tri bu tion for L =.5 char ac ter ized by two peaks, e low en ergy one which is nar rower and lo cated at DE 1 = 0.7 MeV and e me dium en ergy one, which is broader and lo cated at

4 34 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 013, Vol. 8, No. 1, pp e re duced crys tal ick ness, L, from.1 to 10.6, re - spectively. The proton impact parameters generating e en ergy loss dis tri bu tions are shown. The ob served low en ergy peaks in e en ergy loss dis tri bu tions orig i - nate from e sickle like ar eas pass ing rough e chan nel centre while e high en ergy tails are gen er - ated by e pro tons be ing around e atomic strings. The calculated dechanneling curve can be fit ted wi an ex po nen tial func tion, char ac ter ized by e dechanneling range L d = 0.9, which cor re sponds to e crys tal ick ness of 0.4 mm. Fig ure 3. De pend ence of num ber of e dechanneled pro tons on e re duced crys tal ick ness DE = 0.38 MeV, and e high en ergy tail. In e en ergy loss distribution for L = 3.5, which is shown in fig. (b), one can clearly ob serve at e cor re spond ing me dium en ergy peak is di min ish ing, while e low en ergy peak remains, located at DE = 0.36 MeV. The en ergy loss dis - tributions for L = 5.0 and 10.5 con firm is trend, i. e. e low en ergy peaks re main, lo cated at DE = 0.51 MeV and DE = 1.08 MeV, for L = 5.0 and 10.5, re spec tively, while, for L = 10.5, one can also ob serve small and broad high en ergy peak orig i nat ing from e cor re - spond ing high en ergy tail. Fig ures (a)-(d) show pro tons in e im pact pa - ram e ter plane at gen er ate e en ergy loss dis tri bu - tions for e re duced crys tal ick nesses: L =.5, 3.5, 5.0, and 10.5, re spec tively. It is clear from e ses fig - ures at e low en ergy peaks in e en ergy loss dis tri - bu tions are gen er ated by e pro tons orig i nat ing from e sickle like ar eas pass ing rough e chan nel cen - tre, cor re spond ing to e low crys tal elec tronic den - sity, while e high en ergy tails are gen er ated by e pro tons be ing around e atomic strings, cor re spond - ing to e high crys tal elec tronic den sity. The dechanneling func tion of pro tons, rep re - sent ing e de pend ence of e dechanneled pro tons num ber in e crys tal on its dep (re duced crys tal ick ness) is shown in fig. 3. The anal y sis shows at it ' can be fit ted wi an ex po nen tial func tion: N d /N 0 = 1 exp( x/l d ), where N d is e num ber of dechanneled ' pro tons, N 0 e ini tial num ber of pro tons wi out e backscattered ones, and L d e fit ting pa ram e ter, which rep re sents e dechanneling range, while 1/L d is e dechanneling rate. It fol lows at L d = 0.9, which cor re sponds to e crys tal ick ness of 0.4 mm. CONCLUSIONS We have pre sented here in de tails e evo lu tion of e en ergy loss dis tri bu tion of 7 TeV pro tons chan - neled rough e bent <100> Si crys tals, wi e con - stant cur va ture ra dius, R = 50 m, while e crys tal ick ness var ies from 1 mm to 5 mm, cor re spond ing to ACKNOWLEDGEMENT S. M. Petrovi} and N. B. Ne{kovi} ac knowl edge e sup port to is work pro vided by e Min is try of Ed u ca tion, Sci ence and Tech no log i cal De vel op ment of e Re pub lic of Ser bia rough e pro ject Phys ics and Chem is try wi Ion Beams, No. III AUTHOR CONTRIBUTIONS The o ret i cal anal y sis was car ried out by N. Stojanov, S. M. Petrovi}, and N. B. Ne{kovi}. The manu script was writ ten by N. Stojanov and S. Petrovi} and e fig ures were pre pared by N. Stojanov. REFERENCES [1] Lindhard, J. K., In flu ence of Crys tal Lat tice on Mo - tion of En er getic Charged Par ti cles, Mat.-Fys. Medd. Dan. Vidensk. Selsk., 34 (1965), 14, pp [] Esbensen, H., Golovchenko, J. A., En ergy Loss of Fast Chan neled Par ti cles, Nu clear Phys ics A, 98 (1978), pp [3] Elishev, A. F., et al., Steer ing of Charged Par ti cle Tra - jec to ries by a Bent Crys tal, Phys. Lett. B, 88 (1979), pp [4] Bak, J., et al., Bend ing of High-En ergy Beams Us ing Axial and Planar Channeling, Phys. Lett. B, 93 (1980), pp [5] Taratin, A. M., et al., Computer Simulation of Deflec - tion Ef fects for Rel a tiv is tic Charged Par ti cles in a Curved Crys tal, Phys. Lett. A, 7 (1979), pp [6] Biryukov, V. M., Com puter Sim u la tion of Crys tal Ex - trac tion of Pro tons from a Large-Hadron-Collider Beam, Phys. Rev. Lett., 13 (1995), pp [7] Krause, H. F., et al., Angular Distribution of Ions Axi - ally Chan neled in a Very in Crys tal: Ex per i men tal and The o ret i cal Re sults, Phys. Rev. A, 49 (1994), pp [8] Mileti}, L., et al., De cay of Zero-De gree Fo cus ing of Channeled Ions, Nucl. Instrum. Me. Phys. Res. B, 115 (1996), pp [9] Stojanov, N., et al., Angular Distributions of Relativ - is tic Ions Chan neled in e Bent <100> Si Crys tals, Nucl. Instrum. Me. Phys. Res. B, 193 (00), pp [10] Petrovi}, S., et al., Angular Distributions of Ions Chan neled in e <100> Si Crys tals, Nucl. Instrum. Me. Phys. Res. B, 193 (00), pp

5 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 013, Vol. 8, No. 1, pp [11] Stojanov, N., et al., An gu lar Dis tri bu tions of 7 TeV Pro tons Ax i ally Chan neled rough e Thin Bent <100> Si Crys tal, Nucl. Instrum. Me. Phys. Res. B, 44 (006), pp [1] Gemmell, D. S., Chan nel ing and Re lated Ef fects in e Mo tion of Charged Par ti cles rough Crys tals, Rev. Mod. Phys., 46 (1974), pp [13] Golovchenko, J. A., et al., Charge State De pend ence of Chan neled Ion En ergy Loss, Phys. Rev., B 3 (1981), pp [14] Petrovi}, S., et al., En ergy Loss of Ions Chan neled in a in Crys tal, Rad. Eff. and Def. in Sol ids, 141 (1997), pp [15] Straumanis, M. E., et al., Lat tice Pa ram e ters, Co ef fi - cients of Ther mal Ex pan sion, and Atomic Weights of Purest Silicon and Germanium, J. Appl. Phys., 3 (195), pp [16] Batterman, B. W., et al., Vi bra tional Am pli tudes in Ger ma nium and Sil i con, Phys. Rev., 17 (196), pp Re ceived on Au gust 3, 01 Ac cepted on Feb ru ary 8, 013 Nace STOJANOV, Sr an M. PETROVI], Neboj{a V. NE[KOVI] RASPODELE ENERGIJSKIH GUBITAKA PROTONA OD 7 TeV KANALISANIH U ZAKRIVQENIM KRISTALIMA SILICIJUMA Izu~avane su energijske raspodele relativisti~kih protona koji su aksijalno kanalisani kroz zakrivqene <100> Si kristale, ~iji radijus krivine iznosi R = 50 m. Energija protona iznosi 7 TeV, a du`ina kristala se mewa od 1 do 5 mm, {to odgovara redukovanoj du`ini kristala od.1 do Energija protona je izabrana u skladu sa projektom velikog hadronskog sudara~a Evropske organizacije za nuklearna istra`ivawa [vajcarska. Energijska raspodela kanalisanih protona dobijena je pomo}u metoda kompjuterske simulacije koriste}i re{ewa jedna~ina kretawa protona u transverzalnoj ravni. Disperzija ugla rasejawa jona koja je posledica wegovih sudara sa elektronima kristala ukqu~ena je u prora~un. Kqu~ne re~i: relativisti~ko kanalisawe, zakrivqen krista, raspodela energijskih gubitaka