Coupling to breakup channels using a transformed harmonic oscillator basis

Coupling o b eakup channels using a ans o med ha monic oscilla o basis A. M. Mo o,1,3 J. M. A ias,1J. Go ´mez-Camacho,1I. Ma el,2F. Pe ´ ez-Be nal,2,*R. C espo,3and F. Nunes4 1Depa amen o de FAMN, Uni e sidad de Se illa, Apa ado 1065, E-41080 Se illa, Spain 2Depa amen o de FAIE, Uni e sidad de Huel a, E-21071 Huel a, Spain 3Depa amen o de Fı ´sica, Ins i u o Supe io Te ´cnico, A . Ro isco Pais, 1049-001 Lisboa, Po ugal 4Uni e sidade Fe nando Pessoa, P ac¸a 9 de Ab il, 4200 Po o, Po ugal 共Recei ed 4 Sep embe 2001; published 12 Decembe 2001兲 The applica ion o a ecen ly p oposed p ocedu e o disc e izing he con inuum o collision p ocesses in ol ing weakly bound nuclei is s udied. In pa icula , he coupling o b eakup s a es in he collision o d ⫹208 Pb a 50 MeV is discussed. Fo illus a i e pu poses, only he s-wa e componen o he bound s a e o he deu e on is conside ed, and he s udy is es ic ed o he case o nuclea s-wa e b eakup. The con inuum disc e iza ion p ocedu e p o ides a basis o ans o med ha monic oscilla o wa e unc ions o accomplish he necessa y calcula ions. App op ia e con e gence o he elas ic and b eakup c oss sec ions wi h inc easing dimension o he basis is epo ed. In addi ion, i is shown ha he esul s ob ained con e ge o hose o a s anda d con inuum disc e ized coupled channels calcula ion, wi h he ad an age ha he con e gence o he me hod is de e mined by only one pa ame e , namely he dimension o he basis. DOI: 10.1103/PhysRe C.65.011602 PACS numbe 共s兲: 24.10.Eq, 25.10.⫹s, 25.45.De, 25.60.Gc The de elopmen o adioac i e beam acili ies has shed ligh on a a ie y o new nuclea s uc u e p oblems 关1兴 which include neu on and p o on ich nuclei close o he d ip lines, and in pa icula halo nuclei. Gi en ha hese a e weakly bound sys ems, hei modeling should necessa ily in- clude in some way he con inuum pa o he spec um. This becomes essen ial when s udying nuclea collisions, whe e he inclusion o he coupling o b eakup channels is no only necessa y o he desc ip ion o p ocesses leading o ag- men a ion, bu also when dealing wi h elas ic sca e ing o ans e eac ions. Reac ion calcula ions whe e b eakup couplings a e in- cluded ha e he addi ional complica ion ha b eakup s a es a e no squa e-no malizable. This p oblem is gene ally su - moun ed eplacing he s a es in he con inuum by a ini e se o no malized s a es, and p o ing ha he calcula ion o sca - e ing obse ables in he ini e basis con e ges when inc eas- ing he numbe o s a es conside ed, o when modi ying he pa ame e s which de ine he no malized s a es. Se e al p ocedu es ha e been de eloped o ob ain a ini e basis o no malized s a es which desc ibes he con inuum. One o he mos widely used o hese app oaches is he me hod o con inuum disc e iza ion coupled channels 共CDCC兲关2兴. I consis s in disc e izing he con inuum by means o aking ixed in e als, o bins, o k alues. Each bin is cha ac e ized by a single adial wa e unc ion, which is ob ained as an a e age o he con inuum wa e unc ions o e he bin. In his a e aged adial wa e unc ion, he oscilla- ions o he di e en componen s end o cancel beyond a ce ain dis ance, and so he bin adial wa e unc ion becomes no malizable. Fo mally, he CDCC me hod equi es he so- lu ion o he Sch o ¨dinge equa ion o all 共o , a leas , many兲 ene gies in he con inuum as a p e ious equi emen o cal- cula e he bin wa e unc ions. Thus a p ac ical CDCC calcu- la ion equi es o ix he maximum k alue conside ed and he in e al ⌬ko he bins, so ha he numbe o bins is gi en by M⫽kmax /⌬k. To demons a e con e gence in a CDCC calcula ion one has o show ha he sca e ing mag- ni udes a e no modi ied wi h inc easing maximum ene gy (kmax) o dec easing he in e al ⌬k. Howe e , when ⌬kis made smalle , he adius a which he bin wa e unc ions anish becomes la ge , inc easing he ange o he coupling po en ial. As a consequence, CDCC calcula ions a e ypically e y ime consuming. In addi ion, as bo h ⌬kand kmax pa- ame ize he basis, demons a ing con e gence is by no means i ial 共see 关3兴 o a ecen s udy o he con e gence wi hin CDCC兲. Despi e hese di icul ies, he me hod has been success ully applied o a la ge numbe o nuclea eac- ions and is one o he mos eliable app oaches o eac ions in ol ing bina y composi e sys ems. We ha e ecen ly p oposed he use o a ans o med ha - monic oscilla o 共THO兲basis as an al e na i e o desc ibe he e ec o he con inuum 关4,5兴. In Re . 关4兴some applica ions o simple one-dimensional p oblems a e p esen ed and i is shown ha , as he numbe o s a es in he THO basis in- c eases, he eigens a es appea densely packed close o he b eakup h eshold, al hough some o hem lie a highe ene - gies. Besides, i is shown ha global s uc u e magni udes ela ed o he coupling o he con inuum, such as sum ules, a e e y accu a ely desc ibed using ela i ely small THO ba- sis s a es. In Re . 关5兴 his s udy was ex ended o he mo e ealis ic case o he deu e on, showing simila esul s o he one-dimensional case. In he p esen wo k, we show ha he same ideas can be use ul in he s udy o nuclea collisions in ol ing weakly bound nuclei. In he case o a h ee-dimensional p oblem unde cen al o ces, he ele an pa o he wa e unc ion is he adial one, ha can be w i en as R( )⫽ ␾ ( )/ . In he ollowing when e e ing o he wa e unc ion we always mean he adial unc ion ␾ ( ). The basic idea o he me hod p esen ed in Re . 关4兴is o de ine a local scale ans o ma ion 关6兴which con e s he g ound s a e wa e unc ion o a bound nucleus *P esen add ess: Cen e o Theo e ical Physics, Yale Uni e si y, New Ha en, CT 06520-8120. RAPID COMMUNICATIONS PHYSICAL REVIEW C, VOLUME 65, 011602 0556-2813/2001/65共1兲/011602共5兲/$20.00 ©2001 The Ame ican Physical Socie y65 011602-1 ␾ b( ) in o a ha monic oscilla o wa e unc ion ␾ 0 HO(s). The unc ion s( ), which de ines he local scale ans o ma ion, is gi en by 冕 0 兩 ␾ b共 ⬘兲 兩 2d ⬘⫽e 共s兲⫺2sexp共⫺s2兲 冑 ␲ .共1兲 Nex , one gene a es a se o o hogonal wa e unc ions mul- iplying he g ound s a e wa e unc ion by he app op ia e polynomials Pn„s( )…, ␾ n THO共 兲⫽Pn„s共 兲… ␾ b共 兲,共2兲 such ha he s a e wi h n⫽0 coincides wi h he g ound s a e, and he s a es wi h n⬎0 desc ibe he con inuum, o o he bound s a es i hey exis . F om he h ee-dimensional HO wa e unc ions 关7兴 Pn共s兲⫽ ␲ 1/4 2兺 k⫽0 N ␣ nks2k,共3兲 ␣ nk⫽共⫺1兲k k! 冑 2共n!兲⌫共n⫹3/2兲 共n⫺k兲!⌫共k⫹3/2兲.共4兲 Finally, one diagonalizes he Hamil onian ma ix using a i- ni e basis o THO s a es: he esul ing posi i e ene gy eigen- s a es a e aken as con inuum wa e unc ions. He ea e N deno es he numbe o con inuum s a es ob ained a e diago- naliza ion, hus he THO basis dimension is N⫹1. The aim o his wo k is o in es iga e he adequacy o he THO basis o desc ibing he e ec o coupling o b eakup s a es in nuclea eac ions. Fo his pu pose, we ha e consid- e ed a es case whe e he coupling o he con inuum is el- e an : he elas ic sca e ing and b eakup o d⫹208 Pb a 50 MeV. In o de o s udy his p ocess wo ing edien s a e needed: he p o on-neu on in e ac ion and he in e ac ion o he p ojec ile cons i uen s wi h he a ge . A his s age, we do no in end o do a ull calcula ion o he e ec s o deu- e on b eakup no compa e wi h he expe imen al da a bu o show ha he THO disc e iza ion me hod is an adequa e he- o e ical ool o he modeling o eac ions in ol ing weakly- bound nuclei. As an illus a ion, we conside he e a simple heo e ical scena io: he deu e on is aken o be a pu e ss a e, he e ec o Coulomb b eakup is neglec ed, and he nuclea in e ac ion o he p o on and neu on wi h he a ge induce he coupling o s-wa es b eakup s a es only. Wi hin hese es ic ions, we pe o m a con e gence s udy o he THO me hod and compa e wi h he CDCC me hod. The ealis ic case, whe e he dcomponen is inco po a ed in he g ound s a e o he deu e on, and bo h Coulomb and nuclea b eakup o l⫽0 s a es a e coupled in, will be p esen ed in a mo e ex ensi e publica ion. Conce ning he s uc u e o he p ojec ile, we desc ibe he p o on-neu on in e ac ion by means o a Poeschl-Telle po- en ial, V共 兲⫽⫺D1 cosh2共 ␣ 兲,共5兲 wi h dep h 共D兲and ange ( ␣ ) adjus ed o ep oduce he deu- e on binding ene gy and mean squa e adius: D ⫽102.85 MeV and ␣ ⫽0.9407 m⫺1. A ep esen a ion o his po en ial is shown in he inse o Fig. 1. The use o his po en ial p o ides an analy ic g ound s a e wa e unc ion o he deu e on, which is con enien , al hough no essen ial, o he calcula ions in he THO basis. The in e ac ion o p o on and neu on wi h he a ge is aken om he Becche i- G eenless pa ame iza ion 关8兴, e alua ed a hal o he deu- e on inciden ene gy.As men ioned abo e, only he coupling o s-wa e b eakup s a es is conside ed. Once he g ound s a e adial wa e unc ion is ob ained by sol ing he Sch o ¨dinge equa ion wi h he app op ia e po en- ial, Eq. 共5兲in ou case, he unc ion de ining he local scale ans o ma ion is ob ained by di ec in eg a ion o Eq. 共1兲. Wi h hese wo ing edien s he THO wa e unc ions can be gene a ed by using Eq. 共2兲. In Fig. 1 we p esen he THO basis unc ions, om n⫽0 on⫽5. The wa e unc ion wi h n⫽0 is he exac wa e unc ion co esponding o he only bound s a e o he deu e on. I is obse ed ha , as he num- be o nodes 共n兲inc eases, he wa e unc ions ex end o la ge dis ances, bu hey all show a smoo h beha io and decay exponen ially. In p inciple he THO basis is in ini e bu one can unca e i o a ini e dimension and s udy he con e gence o di e en obse ables as a unc ion o his dimension. In he p esen case, he in e nal deu e on Hamil- onian is diagonalized in a unca ed basis and p o ides a ini e se o no malized eigens a es which desc ibe app oxi- ma ely he con inuum. As in he CDCC, he p ocedu e is use ul only i he numbe o con inuum s a es necessa y o p o ide a good app oxima ion o he con e ged esul is small enough. I is wo h o s ess ha uniquely he n⫽0 basis wa e unc ion is an eigens a e o he Hamil onian, whe eas he n⫽0 basis s a es a e no . Consequen ly he s a e n⫽0 is decoupled om he o he s a es included in he basis. In Fig. 2 he eigens a es o he Hamil onian o he case o a unca ed basis o six s a es (N⫽5) a e plo ed. I can be FIG. 1. Radial THO wa e unc ions, ␾ n THO( ), wi h n⫽0 on ⫽5, ob ained by means o he local scale ans o ma ion 共1兲. The deu e on g ound s a e wa e unc ion (n⫽0) is he solu ion o he Poeschl-Telle po en ial ske ched in he inse , wi h pa ame e s ha ep oduce he deu e on binding ene gy 共ma ked wi h dashed line兲 and i s oo -mean-squa e adius. RAPID COMMUNICATIONS A. M. MORO e al. PHYSICAL REVIEW C 65 011602 011602-2 no ed ha he ange o hese ␾ k( ) wa e unc ions is de e - mined by he ange o he las s a e included in he THO basis. All eigen unc ions o he Hamil onian p esen a smoo h beha io and a e no malized. In Fig. 3 we p esen he ene gies o he deu e on Hamil- onian diagonalized in a THO basis as a unc ion o he basis dimension. One can e i y ha he ene gy densi y o hese s a es is highe close o he b eakup h eshold. An a ac i e ea u e o his me hod is ha he only e- qui emen o cons uc he THO basis is sol ing he Sch o ¨- dinge equa ion o he g ound s a e, ei he analy ically o nume ically. The THO basis is hen ob ained by means o Eqs. 共1兲and 共2兲. As he con inuum wa e unc ions a e ob- ained h ough he Hamil onian diagonaliza ion, hei calcu- la ion does no equi e he in eg a ion o he Sch o ¨dinge equa ion. Fu he mo e, he sca e ing calcula ion is equi a- len o a s anda d coupled channels calcula ion wi h bound s a es, whose in e nal ene gies and wa e unc ions a e gi en by he diagonaliza ion o he deu e on Hamil onian in his THO basis. The only disc e e pa ame e o be changed in o de o in es iga e con e gence is he numbe o s a es o be included in he basis. Using his inpu , we sol e he coupled channels calcula ions using he compu e code FRESCO 关9兴. In Table I we p esen he esul s o a s anda d CDCC calcula ion o he o al b eakup c oss sec ion, as well as he a e age inal exci a ion ene gy and he a e age in e se exci- a ion ene gy, using in bo h cases he b eakup c oss sec ions as weigh s. We ha e kep ixed he maximum exci a ion en- e gy a 30 MeV and ha e done calcula ions a ying he num- be o con inuum bins and, acco dingly, he bin size ⌬k. In Table II we p esen he same obse ables as in Table I, bu now using he THO me hod. I is ound ha he con e - gence wi h espec o he numbe o s a es is e y sa is ac- o y. Taking he esul s o N⫽10 as e e ence, we ind ha he THO calcula ions wi h ou , six, and eigh con inuum s a es gi e simila esul s. E en he THO calcula ion wi h jus wo con inuum s a es gi es as easonable an es ima e as he co esponding CDCC. An unusual ea u e o he THO me hod is ha he diago- naliza ion o he in e nal Hamil onian in he THO basis may gi e ise o closed channels, i.e., s a es wi h exci a ion ene - gies la ge han he sca e ing ene gy. Al hough hese s a es do no con ibu e di ec ly o he ou going lux, hey migh be signi ican ly exci ed in he p oximi y o he a ge , hus a - FIG. 2. Radial eigen unc ions o he Po ¨schl-Telle Hamil onian, ␾ k( ), wi h k⫽0 ok⫽5, ob ained by diagonalizing he Poeschl- Telle Hamil onian in a six-dimensional THO basis. FIG. 3. G ound s a e and con inuum ene gies ob ained upon diagonalizing he deu e on Hamil onian as a unc ion o he dimen- sion o he THO basis (N⫹1). No e ha o N⬎5 he las s a e is ou o scale. TABLE I. Con e gence o he b eakup c oss sec ion ( ␴ bu), a - e age exci a ion ene gy ( 具 ⑀ ⫺ ⑀ 0 典 ), and a e age in e se exci a ion ene gy 关 具 ( ⑀ ⫺ ⑀ 0)⫺1 典 兴in he CDCC calcula ion, wi h espec o he numbe o con inuum bins (M). All calcula ions we e pe o med keeping kmax⫽0.85 m⫺1共co esponding o a maximum exci a ion ene gy o ⑀ max⫽30 MeV) and using a di e en numbe o con- inuum bins. M ␴ bu 具 ⑀ ⫺ ⑀ 0 典具 ( ⑀ ⫺ ⑀ 0)⫺1 典 共mb兲共MeV兲(MeV⫺1) 2 86.8 6.20 0.166 4 86.1 5.81 0.219 6 87.0 5.85 0.222 8 84.8 5.94 0.224 10 82.5 6.04 0.223 12 81.8 6.07 0.224 TABLE II. Calcula ed o al b eakup c oss sec ion ( ␴ bu), a e - age exci a ion ene gy ( 具 ⑀ ⫺ ⑀ 0 典 ), and a e age in e se exci a ion en- e gy 关 具 ( ⑀ ⫺ ⑀ 0)⫺1 典 兴, using he THO basis, o di e en choices o he numbe o con inuum s a es (N). The column Ncoup indica es he numbe o con inuum s a es which a e coupled oge he in he coupled channels calcula ion. NN coup ␴ bu 具 ⑀ ⫺ ⑀ 0 典具 ( ⑀ ⫺ ⑀ 0)⫺1 典 共mb兲共MeV兲共MeV⫺1) 2 2 89.2 3.98 0.273 4 3 83.8 5.48 0.231 6 5 84.0 5.46 0.226 8 7 83.5 5.66 0.224 10 8 83.5 5.85 0.220 10 10 82.8 5.87 0.221 RAPID COMMUNICATIONS COUPLING TO BREAKUP CHANNELS USING A . . . PHYSICAL REVIEW C 65 011602 011602-3 ec ing he elas ic and b eakup c oss sec ions. The di ec me hod o sol ing di e en ial equa ions used in FRESCO is nume ically uns able o hese s a es wi h high exci a ion en- e gies. The e o e, in o de o in es iga e he impo ance o hese s a es, R-ma ix coupled channel calcula ions we e pe - o med. We ound ha s a es wi h exci a ion ene gies abo e 30 MeV play a mino ole and, consequen ly, closed chan- nels do no a ec he sca e ing obse ables o his eac ion. Wi h he excep ion o he las ow, he calcula ions p esen ed in Table II a e pe o med excluding he coupling o closed channels. The numbe o s a es which a e coupled in each case is indica ed by he ow labeled Ncoup . In Fig. 4 we compa e he elas ic di e en ial c oss sec ions ob ained wi hin he CDCC app oach ( ⑀ max⫽30 MeV and M⫽10 bins o uni o m wid h in k) and he co esponding THO esul s o N⫽10 共dashed line兲. I is obse ed ha he ag eemen be ween bo h calcula ions is excellen . We also p esen he esul o a minimal THO calcula ion, including only wo s a es 共do ed-dashed line兲. We see ha his e y schema ic calcula ion does indeed desc ibe he e ec o cou- pling o he con inuum easonably well. As an illus a ion o he ele ance o b eakup s a es in he elas ic sca e ing, we ha e also included he calcula ion wi h he olded po en ial, i.e., igno ing he coupling o he b eakup channels. In Fig. 5 he b eakup c oss sec ion as a unc ion o he exci a ion ene gy o he deu e on is depic ed o he CDCC and THO app oaches. In he THO me hod, we p esen esul s wi h wo di e en numbe s o basis s a es, N⫽8共open ci cles兲and N⫽10 共 illed ci cles兲, in o de o show he con- e gence o he esul s. In he CDCC case, i is ob ained di iding he c oss sec ion o each bin by he wid h o he bin. In he THO case, i is ob ained di iding he c oss sec ion o each con inuum s a e, co esponding o an ene gy ⑀ iby an ene gy wid h which is gi en by ( ⑀ i⫹1⫺ ⑀ i⫺1)/2. The consis- ency be ween he THO calcula ions wi h di e en numbe s o s a es and he CDCC calcula ion indica es he alidi y o he THO me hod o calcula e b eakup exci a ion unc ions. A ea u e o he THO me hod is ha i p o ides a de ailed desc ip ion o he con inuum in he low ene gy egion whe e b eakup is mo e ele an . We ha e checked ha , changing he deu e on Hamil onian o dec ease 共inc ease兲 he binding ene gy, hen he b eakup s eng h mo es o lowe 共highe 兲 ene gies, and so do he eigen alues o he Hamil onian in he THO basis. To achie e his in a CDCC calcula ion, one needs o adjus he bin size by hand. In Fig. 6 we show he angula dis ibu ion o he b eakup c oss sec ion. The THO calcula ion wi h N⫽10 s a es is in good ag eemen wi h he CDCC calcula ion, especially a o wa d angles o which he c oss sec ions a e mos impo - an . The THO calcula ion wi h N⫽2 s a es p o ides a ea- sonable app oxima ion o he con e ged esul . FIG. 4. Elas ic di e en ial c oss sec ion angula dis ibu ion as a a io o Ru he o d c oss sec ion o he eac ion d⫹Pb a 50 MeV. All calcula ions we e pe o med wi h he Becche i-G eenless p e- sc ip ion o he p o on- a ge and neu on- a ge op ical po en ials. FIG. 5. B eakup c oss sec ion as a unc ion o he p o on- neu on ela i e ene gy in he ou going channel, o he eac ion d ⫹Pb a 50 MeV. The CDCC calcula ion wi h en con inuum bins o uni o m size in k, up o a maximum exci a ion ene gy o ⑀ max ⫽30 MeV is compa ed wi h wo calcula ions using he THO basis wi h N⫽8 and N⫽10 s a es. FIG. 6. B eakup c oss sec ion as a unc ion o he deu e on cen e -o -mass sca e ing angle, o he eac ion d⫹Pb a 50 MeV. The CDCC calcula ion wi h en con inuum bins o uni o m size in k, up o a maximum exci a ion ene gy o ⑀ max⫽30 MeV is com- pa ed wi h wo calcula ions using he THO basis wi h N⫽2 and N⫽10 s a es. RAPID COMMUNICATIONS A. M. MORO e al. PHYSICAL REVIEW C 65 011602 011602-4 In conclusion, we ha e shown in his Rapid Communica- ion ha he con inuum disc e iza ion me hod p oposed in Re . 关4兴can be applied o p oblems o cu en in e es in nuclea physics, in pa icula o he case o collisions in ol - ing weakly bound nuclei. The only p e ious equi emen o he applica ion o he me hod is he knowledge o he g ound s a e wa e unc ion o he nucleus, ei he in an analy ical o nume ical o m. Then, Eqs. 共1兲and 共2兲p o ide he THO basis which can be used o diagonalize he nuclea Hamil- onian. We ha e pe o med a se ies o schema ic calcula ions o he case o he d⫹208 Pb elas ic sca e ing and b eakup a 50 MeV. The compa ison o THO and CDCC esul s has demons a ed he alidi y o he i s me hod and has shown ha he THO disc e iza ion me hod wo ks p ope ly e en hough he basis is unca ed o e y ew s a es. One o he g ea es ad an ages o he THO me hod is ha i au oma i- cally dis ibu es he basis s a es in he ele an ene gy e- gions o he con inuum, o e ing an easy, compu a ionally e icien , al e na i e o he s anda d CDCC me hod. This wo k was suppo ed in pa by he Spanish DGICYT unde P ojec s Nos. PB98-1111 and FPA2000-1592-C03-02 and by he Po uguese Science and Technology Founda ion unde G an No. SAPIENS/36282/99. F.P.B. wishes o hank he inancial suppo p o ided by he Spanish Di eccio ´n Gen- e al de Uni e sidades 共Minis e io de Educacio ´n, Cul u a y Depo e兲 h ough a g an . We acknowledge use ul discus- sions wi h R. C. Johnson and I. J. Thompson. 关1兴Repo o Nuclea Physics Eu opean Collabo a ion Commi ee, edi ed by J. Ve ie e al., 1997. 关2兴N. Aus e n, Y. Ise i, M. Kamimu a, M. Kawai, G. Rawi sche , and M. Yahi o, Phys. Rep. 154, 125 共1987兲. 关3兴R. Piyadasa, M. Kawai, M. Kamimu a, and M. Yahi o, Phys. Re . C 60, 044611 共1999兲. 关4兴F. Pe ´ ez-Be nal, I. Ma el, J. A ias, and J. Go ´mez-Camacho, Phys. Re . A 63, 052111 共2001兲. 关5兴F. Pe ´ ez-Be nal e al. Few Body Sys . 共 o be published兲. 关6兴M. S oi so , J. Dobaczewski, P. Ring, and S. Pi el, Phys. Re . C61, 034311 共2000兲. 关7兴M. Moshinsky and Y. Smi no , The Ha monic Oscilla o in Mode n Physics 共Ha wood Academic, Chu , Swi ze land, 1996兲. 关8兴F. Becche i and G. G eenlees, Phys. Re . 182, 1190 共1969兲. 关9兴I. Thompson, Compu . Phys. Rep. 7, 167 共1988兲. RAPID COMMUNICATIONS COUPLING TO BREAKUP CHANNELS USING A . . . PHYSICAL REVIEW C 65 011602 011602-5