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1 ELASTIC PROTON SCATTERING ON С AND С NUCLEI IN THE DIFFRACTION THEORY М.А. Zhusupov, Е.Т. Ibaeva, R.S. Kabatayeva - Al-Faab Kazakh Natonal Unvesty, - Insttute of Nuclea Physcs Almaty, Kazakhstan ABSTRACT. Thee s a calculaton of the dfeental coss sectons of poton scatteng on С and С nucle at enegy of GeV wth the shell model wave functons n the famewok of the Glaube theoy n the study. The sngle, double and tple collsons have been taken nto account n the multple scatteng opeato. The ole of each tem of the sees and the ntefeence n the dfeental coss secton have been estmated. It s shown that fo a descpton of the coss sectons n a wde angle/momentum tansfe ange t s necessay to consde not only the fst, but the hghe scatteng odes. INTRODUCTION A study of non-stable, neuton- o poton-ch sotopes put befoe the nuclea physcs the fundamental questons of detemnaton of the nucleon stablty boundaes, nuclea shells evoluton, synthess of supe-heavy elements n acceleatos and cosmc objects. The phenomena unknown untl now ae obseved: nhomogenety of neuton and poton denstes (halo), new defomatons aeas and new types of collectve exctatons at low eneges (soft dpole esonance), beakngs n nuclea shells occupaton, appeang of new magc numbes and othe. The cabon sotopes ae ntensvely studed what s concened wth both the wde dstbuton of the stable C sotope and the obtanng of beams of non-stable,6,7,9,, С sotopes at both the aveage (fom to ev pe nucleon [ ]) and the hgh ( 7 ev pe nucleon [6, 7]) eneges at the facltes n GANIL, NSCL SU, RIKEN and othe. Recently the cabon tagets have been used fo gettng of tenso-polazed deuteons as a esult of the effect called the nuclea spn dchosm. The fst expement was caed out n Cologne at the tandem acceleato whch uses the cabon taget bombaded by deuteons [8]. In Sant-Petesbug Insttute fo Nuclea Physcs n Gatchna and n the Nuclea Cente n Sakley the dfeental coss sectons wee measued at enegy E GeV fo the numbe of nucle 9 Ве,, В,, С o o and othe n the angle ange > θ > [9]. The calculatons of the dfeental coss sectons wee caed out n the famewok of the Glaube dfacton scatteng theoy [] whch epoduces them well fo the fowad angles (untl the second dfacton mnmum (θ~ )). At the fagment-sepaato FRS n GSI one obtaned the beams of -9 С sotopes and the coss sectons of scatteng of potons and С sotope on these sotopes at the enegy of E 9 ev pe nucleon wee measued fo the fst tme. The coss sectons wth the poton taget show the shap gowth wth an ncease of the neutons numbe n the pojectle patcle []. In pape [] the dlatatonal momentum dstbutons of the, С fagments fom the С and the С fagments fom the С wee measued at 8 ev pe nucleon at the cycloton n Rken Pojectle Fagment Sepaato n RIKEN Rng Cycloton Faclty. An analyss of these momentum dstbutons and the coss sectons of eactons of one- o two-neutons stppng was caed out n the Glaube model basng on the coe plus neuton model. An analyss of the coss sectons of - С + р eacton at eneges fom to 8 ev n the famewok of the Glaube theoy was caed out n the pape []. The wave functons of the cabon sotopes ae geneated on the base of the smple model of aveage feld. The evaluaton of contbutons of the neuton and poton coss sectons nto the elastc р + С - scatteng shows that the nuclea suface gves the man contbuton nto the coss secton ( σ R ) what s especally notced at the low eneges (, ev) []. In the elastc р + С - scatteng when enegy changes fom to 8 ev the value of σ vaes fom 8 to 9 mb. Dscussng the R

2 advantages and dsadvantages of the poton and cabon tagets the authos [] concluded that the poton taget could pobe the suface of the neuton-ch nucle bette than the С taget, especally at low eneges. The pove fo the extended stuctue of the gound and the fst excted states of C nucleus s pesented n pape [] when studyng the elastc magnet electon scatteng on the,7,9 С sotopes. It s shown that one can detemne the obtal of the last nucleon fom the elastc magnet fom-facto. The fom-facto at the lage values n the momenta space s defned by the densty at the small dstances n the coodnate space and vce vesa. The fom-factos at the low and ntemedate momentum tansfes fo all the,7,9 С sotopes ae smla to each othe when the last neuton occupes the same obtal (let admt that s ). Ths means that the asymptotcs of the wave functons defned by the last neuton ae also close to each othe. At the same tme thee s an explct dfeence between the fom-factos fo the same nucleus when the last neuton occupes the dfeent obtals ( s o d ). The total eacton coss sectons of cabon sotopes wth N 6 6 n collsons wth the poton tagets n the enegy ange ev wee systematcally studed []. The coe plus neuton model was chosen fo the odd-nucleon nucle. The empcal fomulas wee suggested whch ae used fo the pedctons of unknown total eacton coss sectons. A paametzaton of the NN-ampltude of scatteng wth one and two gaussods ae dscussed. It s shown that the standad paametzaton of the ampltude by one gaussod gves nealy the same numecal esults fo the coss sectons of the eacton that the two gaussods do that s why t s enough to use one. In pape [] the pa coelatons n the odd sotopes of cabon and the nfluence of Paul pncple account n nteactons lke patcle plus coe n the C and Be nucle ae consdeed. In the two appoaches BCS (Badn-Cupe-Shf) and PBCS (pojectle BCS) the enegy specta and sngle-patcle spectoscopc factos of the low-lyng states of,,7,9 C sotopes wee calculated and a compason wth expemental data was caed out. The esults show that the pa nteacton plays an mpotant ole n the nuclea stuctue of the heavy cabon sotopes due to the chaactestcs: the small bndng enegy and the systematc decease of the bndng enegy at addng of neutons (to ev) when tanstng fom the C to the 9 C. The stable α - lnea chan stuctue of the C sotopes s studed n the mcoscopc cluste α + n - model [-7]. Two excted otatonal sees wee found, whee thee α-cluste stuctues of the С sotope wee obtaned []. The stuctue of / ± states n the C nea the α + n - theshold (E x. ev) ae studed n the complete α + n othogonal states model (ОСМ) [6]. In addton to the α + n - model the effect of decay of one of the α-clustes s taken nto account. The nuclea denstes of the low-lyng states ae calculated [7]. In the shell model the stuctues of the gound states of the С and С nucle ae dfeent [8]. If the gound state of the С nucleus has the shell confguaton (s) (p) 9 : J π, T / -, / >, then the С nucleus n the gound state s (s) (p) s / : J π, T / +, / >. The stuctue of the С nucleus n the gound state s close to the one of the excted state n the С nucleus wth quantum numbes J π / + at Е ev and has the shell stuctue (s) (p) 8 s / : J π, T / +, / >. The objectve of the pesent study s the mcoscopc calculaton and the analyss of the dfeental coss sectons of elastc scatteng of potons on С and С nucle n the famewok of the Glaube theoy wth the wave functons n the shell model at enegy of GeV, and also a study of the ntefeence effects n the Glaube multple scatteng opeato. BRIEF FORALIS In the Glaube dfacton scatteng theoy the matx elements of р С - scatteng ae calculated at enegy of GeV. The analogous calculaton fo the р С - scatteng was caed out n ou pape [9]. The nput paametes of the theoy ae the wave functons and the paametes of the poton-nucleon ampltude extacted usually fom ndependent expements.

3 The wave functon n the shell model fo the С nucleus s pesented by the shell 9 confguaton ( s ) ( p) n the gound state [8]. Let us epesent the shell wave functon n a fom 9,... ) (s ) ( p) (,... ) (,... ), (), f ( nlm nl m whee n l m quantum numbes of the coespondng shell ( n, l, m ; n, l, m, ± ). Each functon s a poduct of sngle-patcle one (,,...) ( ). nlm nlm The matx element (ampltude) k v J J J J d ρ exp( qρ) f Ω, () π hee q s a momentum tansfe n a eacton: q k k, () kk, ncomng and outgong momenta of the ncdent and escape poton. In case of elastc scatteng k k and the momentum q equals: θ q k sn, k ε m, () The opeato of multple scatteng: Ω ω ρ ρ ω ω ω + ω ω ω...( ) ωω... ω. ( ( )) τ τ η τ τ η () Lmtng to tple collsons, substtutng the opeato nto the fomula () and havng ntegated t wth espect to d ρ, dq,..., dq, one obtans % % % % k () () () ΩΩ Ω +Ω π π q π q f () q ω f τ f... k ωω k + < τ k (6) % % % % ωωω % % τ η < τ< η The «tlde» sgn ove the opeatos means the ntegatons done. % ω exp( qρ ), (7) q ωωτ % % exp ( ρ + ρτ) δ( τ) < τ < τ ρ, (8) q % ω % ωω% τ η exp ( ρ + ρτ + ρη) δ ( ρ ρτ) δ ρη ( ρ + ρτ). τ (9) < τ< η < The matx element () n the appoach of tple scatteng: () () () +, () whee

4 k q %, () () ( ) f ω π k q % % τ, () () ( ) f ωω π < τ k q % % % τ η. () () ( ) f ωωω π < τ< η The matx element of sngle scatteng: k ( q ) ( ) ( ) ( % % ) ( τ) () m τ ω + ωτ m π τ τ τ whee () () () s () p N + 9 N, () () s k k ( q ) ( ) % ω ( ) ( ) ω d π %, (6) π k k q % % ω d, (7) () p ( ) m( ) ω m( ) m( ) π π N ( ) ( ), N ( ) ( ) m m. (8) () s () s a matx elements of scatteng on nucleons of s-shell ( tems), p on nucleons of р-shell (9 tems), N, N nomalzaton of wave functons. A calculaton of these matx elements wth the wave functons n the shell model n the sphecal coodnates system wth change of the two-dmensonal ρ vecto fo the theedmensonal vecto and an expanson of the expesson exp( q ρ ) n a sees by Bessel J λ + functons and sphecal Y λμ ( Ω q ) hamoncs gve one the followng esult: π () s ( ) q N R ( ) J d q, (9) π q N B q m m Y () p ( ) m π λ+ λ λμ q λμm 9 ( ) ( ) λ λ ( ) ( ) ( ) λμ Ω, q whee ()

5 The adal wave functons ( λ ) B R ( ) J λ+ d. () R C exp, C, () π R C exp, C C. () Hee s connected wth the oscllatoy paamete h ω by the elaton h ( hc) (.97 ev cm). Fo the p-shell nucle h ω.8 ev []. mω mc hω 9 ev ev The matx element of double scatteng k q () ( ) ( ) m( τ) π τ < τ % ωω% + % ω % ω + % ω % ω ( ) ( ) () τ τ m τ τ τ 6 τ () ss () sp τ + () pp () 6 N N, () π q q f () ss ( ) ( ) % ωω% τ( ) k < τ π q q f τ δ τ d k < τ ( ) exp ( + ) ( ), (6) () sp π q f ( ) % ω( ) m( ) % ωm ( ) k (7) () pp π q f m( ) % ωω% τm ( ) k < τ (8) π q q f m( ) exp ( + τ) δ ( τ) d k < τ () whee ss (), sp (), pp matx elements of collsons of poton wth the nucleon on (s)-, (s, p)-, (p)-shells. Let us denote С ( ) π q f k. Applyng the same technque as at calculatng the sngle scatteng matx element one gets fnally

6 () ss π () ( ) ( ) / ( ) q q С R J q d () pp () ( λ ) q λμm 6. (9) π 6 С B Fλμ ( Ω q ), () whee () ( λ ) B R ( ) J λ+ d () F ( Ω ) Y ( Ω ) Y ( Ω ) Y ( Ω ) dω Y ( Ω ) * λμ q m m λμ λμ q (λ + λ mλμ m Yλμ ( Ωq ) π λμm The matx element of tple scatteng k A () (,... ) m (,... ) Ω (,... ) m (,... ) π () % d () The opeato of tple scatteng % % % % Such a epesentaton of the () Ω ωωω τ η < τ< η () % ω % ωω% + % ωω% % ω + % ω % ωω% + % ωωω % %. τ η τ η τ η τ η < τ< η < τ η τ< η < τ< η ~ () Ω dfeental coss sectons fom e-scatteng on the dfeent shells opeato allows one to calculate the contbuton nto the () whee () sss + () ssp + () spp + 8 () ppp, () () sss π q π ( ) f % ωωω % % τ η k < τ< η k q q f f ( ) % ωωω % % ( ) τ η < τ< η () ssp π q π ( ) f % ωω% τ % ωη k < τ η k q q f f ( ) ( ) % ωω% % ω ( ) ( ), m τ η m < τ η (6) (7)

7 7 () spp π q π ( ) f % ω % ω % τωη k τ< η k q q f f ( ) ( ) % ω % ω % ω ( ) ( ), m τ η m τ< η () ppp π q π ( ) 8 f % ωωω % % τ η 8 k < τ< η k q q f f ( ) % ωωω % % ( ) m τ η m < τ< η (8) (9) The uppe lette ndces denote shells. The coeffcents befoe the ampltudes show how many tple e-scattengs the poton undegoes wth the nucleons fom dfeent shells. Afte the vaables sepaaton and ntegaton n the sphecal coodnates system one gets the fnal expessons fo the tple collsons matx elements: () sss D exp J d, () () ssp ( ) C D ( q) H, π 7 exp J λ+ d, ( ) π H k f q, q λ D ( ) (λ+ ) λ λμm m Y ( Ω ) () spp λμmm C C D H, π λ m D ( ) ( ) (λ+ ) L L L λl λμmm LL Ω λμ q ( ) m m L L m L λμl m Yλμ ( q ) exp Jλ+ d, C C D H, π () ppp λ m m m ( ) N λ + D ( ) ( ) L L L λμmm LL KNK N (K + ) K K K m m L L m L m m KN KN m KN λ L K L K N Y Ω J q d ( ) λμ λμ ( q) exp λ+ ( ), 8 C D H. π C, C nomalzaton coeffcents of the adal wave functons, fomulas (), (6). 6 () () ()

8 8 RESULTS ANALYSIS The calculaton of the dfeental coss sectons of potons elastc scatteng on С and С nucle at enegy of GeV n the Glaube theoy has been done. Sngle-, double and tplecollsons have been taken nto account n the opeato of multple scatteng. In fgue the dfeental coss sectons of elastc р С-scatteng at Е. GeV ae shown wth account of sngle- (dash-dot), double- (dashed), tple- (dots) collsons and the total coss secton (sold cuve). The expemental data ae povded fom the pape [9]. It s shown n the pctues, that at small angles the sngle scatteng domnates, as the angle nceases the sngle scatteng deceases shaply, t s equated to the double scatteng, and the last one defnes the coss sectons behavo at ntemedate angles. Wth the futhe ncease of the angle when the double collsons coss sectons decease, the tple collsons begn to play a ole, the last ones descbe the behavo of the coss sectons. In the ponts of equalty of the patal coss sectons the mnmums appea snce the multple scatteng sees s an altenatng-sgn one. At the enegy of GeV the domnatng contbuton of the sngle collsons s obseved untl angles only, the double one untl, the tple one ove. All odes ae pesent n the total dfeental coss secton. The calculaton descbes the behavo of the dfeental coss sectons coectly; some dscepances ae n the mnmums snce ths aea s moe sensble to the stuctue of the wave functon and to the nteacton dynamcs. dσ/dω, mb/s θ, deg Fgue. Contbuton nto the dfeental coss sectons of р С-scatteng: sngle- (dash-dot), double- (dash), tple- (dots) collsons and the sum (sold cuve) at Е. GeV

9 9 dσ/dω, mb/s - - θ, deg Fgue. The same as n fgue fo the р С-scatteng In fgue thee s a contbuton of the seveal fst odes of scatteng nto the dfeental coss sectons of р С-scatteng. Each subsequent tem of the sees () gves a contbuton of one ode less than the pevous one. As t was multply ponted out n the lteatue (begnnng fom the []), the tems of the multple scatteng sees decease nonunomly wth the scatteng angle ncease: the sngle scatteng deceases too shaply and even n the ange of θ ~ angles ts contbuton nto the dfeental coss sectons becomes neglgbly small; as fo the double scatteng, beng fa less than the sngle one at small θ < angles, t begns to domnate n the ange of >θ > angles and gves one the man contbuton nto the dfeental coss sectons; the tple scatteng tem whch s fo two odes less than the sngle one at zeo angle, s equated to the double one at θ ~ angle and gves one the man contbuton nto the coss secton at θ >. In the ange whee the cuves of one-, double- and tple collsons ntesect (.е. whee the dfeental coss sectons ae equated by absolute values), the ntefeence mnmums appea snce the sees () s an altenatng-sgn one and when squang the matx element the coss tems ae subtacted. As t s seen fom the esult of ou calculaton n ode to descbe the behavo of the dfeental coss secton n the wde angle ange t s necessay to account not only sngle, but the hghe odes of scatteng ethe. Ths vey esult demonstates enough fast convegence of the multple scatteng sees (at the gven enegy of GeV pe nucleon), when t s enough to lmt to the tple scatteng snce n the ange of small angles (whee the calculaton by the Glaube theoy s elevant) the odes hghe than the thd one wll not gve notceable contbuton nto the coss secton.

10 p C p C p 6 O p C dσ/dω, mb/s - - θ, deg Fgue. Compason of the dfeental coss sectons of the С, С, С, 6 О nucle at enegy Е, GeV pe nucleon. Expemental data fo С, С ae fom [], fo 6 О fom [] In fgue thee s a demonstaton of the dfeental coss sectons on С, С, С, 6 О nucle at enegy Е. GeV. A compason of the coss sectons shows that wth the gowth of the nucleon numbe the coss secton nceases slghtly at zeo scatteng angle, what eflects the gowth of the mean-squae adus (fom ms R. fm fo С, tll. fm fo 6 О) and the mnmum n the coss secton s shted n the ange of smalle scatteng angles. The locaton and the value of the mnmums dfe sgncantly snce the pctue by the odnate axs s n the logathmc scale. The esult obtaned by us fo the С nucleus does not contadct to the expemental data obtaned fo othe cabon sotopes and fo the 6 О nucleus. CONCLUSIONS Thee s a calculaton technque of the poton elastc scatteng matx element on the С nucleus n the famewok of the Glaube dfacton theoy n the pesent study. A neglect of the small nuclea momenta Q n compason wth the tansfeed one q, taken n the calculaton, allowed us to calculate the ente sees of multple scatteng, and the use of the shell wave functons n the hamonc oscllato bass povded analytcal calculaton of the dynamc ntegals (dependent on the scatteng opeato n the ams of the wave functons). The obtaned calculaton fomula of the matx element pesents a poduct of two multples, one of whch s a sum of the multple scatteng sees, and the othe one stands fo the scatteng on nucleons fom dfeent shells of С nucleus. Thus t became possble to calculate the contbuton nto the dfeental coss sectons fom both the dfeent odes of scatteng and the scattengs on nucleons on the dfeent shells. Havng calculated the contbuton nto the dfeental coss secton fom the fst thee tems of the multple scatteng sees, we showed that the sngle collsons domnate at the vey o small scatteng angles, untl θ <, the double ones n the ange of the second maxmum n the coss secton at > θ >, the tple ones n the ange of the thd maxmum at θ >. In the ntesecton ponts of dfeent scatteng odes the typcal ntefeence mnmums appea n the dfeental coss sectons snce the multple scatteng sees s an altenatng-sgn one and the coss tems ae subtacted fom the total sum. An absence of the expemental data fo the р С-scatteng motvated us to cay out a compason of the dfeental coss sectons of the close cabon С and С sotopes and the 6 О nucle at enegy of GeV, fo the last one data ae avalable. The coss secton fo the р С- R ms

11 scatteng follows the typcal peculates of othes, beng consstent wth them by both the absolute value and the locaton of the mnmums and the maxmums. Refeences. U. Datta Pamank et al., Phys. Lett. B, 6-7 ().. TUNL Nuclea Data Evaluaton Poject avalable: Natonal Nuclea Data Cente, Bookhaven Natonal Laboatoy webpage W. Houch, Y. Susuk, B. Abu-Ibahm, and A.Kohama, Phys. Rev. C7, 67 (7).. B. Abu-Ibahm, W. Houch, A. Kohama, and Y. Susuk, Phys. Rev. C77, 67 (8). 6. Yu. L. Pafenova,. V. Zhukov, and J. S. Vaagen, Phys. Rev. C6, 6 (). 7. R. Khaab, R. Kuma, P. Sngh, H. C. Shama, Physcs of Atomc Nucle, Vol. 7, No., P. 9- (7). 8. H. Seyfath, R. Engels, F. Rathmann et al, Phys. Rev. Lett., (). 9. G.D. Alkhazov, Physcs of Atomc Nucle, Vol., No. P.8- (978).. R.J. Glaube, n Lectues n Theoetcal Physcs (Intescence, New Yok, 99), Vol., p... Tanhata, S. Teashma, R. Kanungo et al., axv:.99v [hep-ph] Nov.. D. Q. Fang, T. Yamaguch, T. Zheng et al., Phys. Rev. C69, 6 ().. Tekuang Dong, Zhongzhou Ren, and Yanqng Guo, Phys. Rev. C76, 6 (7).. A. R. Samana, T. Tautna, F. Kmpotc et al., Nucl. Phys. A79, 6-6 (7).. Fuutach,. Kmua, Phys. Rev. C8, (R) (). 6. Tach Yamada and Yasuo Funak, Phys. Rev. C9, 6 (). 7. T. Yoshda, N. Itagak, T. Otsuka, Phys. Rev. C79(), 8/6 (9). 8. А. V. Boyakna, n Stuctue of р-shell nucle (SU, 969) 6 pages. 9. М. А. Zhusupov et al., Bull. of Russ. Acad. of Scence: Phys., Vol. 7, No 6, pp. 8-8 (9).. V. G. Neudatchn, Yu. F. Smnov, n Nucleon assocatons n lght nucle (М.: Scence. 969).. H. Palevsky et al., Phys. Rev. Lett. 8, No 6. P. (967).. G. D. Alkhazov, V. V. Ansovch, P. E. Volkovtsky, n Dfacton nteacton of hadons wth nucle at hgh eneges (L.: Scence, 99).