Dynamic effective potential for α-particle bound and quasibound states

Dynamic e ec i e po en ial o a -pa icle bound and quasibound s a es C. H. Dasso,1R. J. Lio a,2and M. Lozano3 1The Niels Boh Ins i u e, Uni e si y o Copenhagen, Blegdams ej 17, DK-2100 Copenhagen O”, Denma k 2Royal Ins i u e o Technology, F esca i a ¨gen 24, S-10405 S ockholm, Sweden and No di a, Blegdams ej 17, DK-2100 Copenhagen O”, Denma k 3Depa amen o de Fı ´sica A o ´mica, Molecula y Nuclea Uni e sidad de Se illa, AP. 1065, E-41080 Se illa, Spain ~Recei ed 21 No embe 1995! We exploi analy ic p ope ies o he op ical po en ial o elas ic sca e ing o a pa icles on nuclei o ex ac in o ma ion on he e ec i e in e ac ion ha should be used o desc ibe he mo ion o a clus e o wo neu ons and wo p o ons bound o a nuclea sys em. This p esc ip ion sol es long-s anding ambigui ies in he o mal- isms used o he s udy o a decay. @S0556-2813~96!01308-8# PACS numbe ~s!: 24.10.H , 21.60.Gx, 23.60.1e, 27.80.1w Alpha decay was one o he i s nuclea phenomena ob- se ed in na u e. No su p isingly, a emp s o desc ibe his p ocess go back o he ea lies days o quan um mechanics. Pe haps he mos anspa en o mula ion was ha o Gamow @1#, who ecognized he possibili ies o quan al unneling o explain he many o de s o magni ude spanned by he expe i- men ally measu ed li e imes. As he gene al knowledge o nuclea s uc u e imp o ed, howe e , his simple pic u e an in o oubles. These p ima ily had o do wi h he need o unde s and mic oscopically he o ma ion inside o he mo he nucleus o an alpha pa icle as a clus e o wo p o- ons and wo neu ons. This goal in oduced all kinds o di - icul ies in he p oblem, anging om complex coupling schemes down o he ea men o he con inuum equi ed o sa is y he app op ia e bounda y condi ions. Especially no e- wo hy a e ques ions aised abou he iabili y o a scheme ha equi es he wa e unc ion o he alpha pa icle no only a he nuclea su ace, bu in he in e io as well @2#. The ese a ions cen e he e mos ly on he inhibi ing ole o he Pauli p inciple, which, ine i ably, en e ed in o he pic u e as soon as he alpha pa icle was hough o in e ms o i s cons i uen nucleons. In his pape we e u n o his ques ion ollowing a pa h ha aims o bypass hese di icul ies. To his end we ely on he wo k by Bell and Squi es @3#~c . also Re . @4#! who in es iga ed he analy ic p ope ies o he complex nucleon- nucleus in e ac ions. In hei con ibu ion hey cla i ied he in ima e connec ion be ween he op ical po en ial o sca e - ing s a es and he e ec i e in e ac ion ha binds he nucle- ons o he co e. In ac , hey es ablished he concep ual iden- i y o hese quan i ies, o he ex en ha he la e is none o he han he analy ic con inua ion o he o me in o he nega i e ~o , mo e p ecisely, bound-s a e!ene gy domain. When Bell and Squi es p esen ed hei esul s, much was al eady known abou he binding po en ial o single nucle- ons. This in o ma ion had been mainly ex ac ed om he analysis o nuclea sys ems ei he di ec ly a ailable in na u e o a i icially p oduced. Thus i could be a gued ha hei wo k was mos ly o academic in e es . Thei conclusions a e, howe e , po en ially much mo e ele an in he case o alpha pa icles. Michel e al. @5#and Delba e al. @6#made signi i- can s eps in he unde s anding o he op ical po en ial o he sca e ing o alpha pa icles by ligh e sys ems, such as oxy- gen and calcium. Mic oscopic es ima ion o he s a ic po en- ial be ween an a pa icle and a nuclea co e can also be con empla ed by olding p ocedu es; see, o ins ance, Re s. @7, 8#. Unce ain ies emain, howe e , abou he couplings ha should be used o accoun o he mo ion o he ou - nucleon clus e in a quasibound s a e o a hea ie nucleus ha e en ually a decays. In ac , o calcula e mic oscopically he eno maliza ion o he ba e in e ac ion ha gi es ise o he ac ual ene gy dependence o he a -nucleus po en ial un- de hese ci cums ances is a a he o midable ask. One can inspec , o ins ance, in Re . @9#, he nume ous diag ams ha ha e o be aken in o accoun o cons uc in a pe u ba i e app oach he co ela ion and pola iza ion e ms ha dynami- cally ‘‘d ess’’ he mass ope a o o o dina y nucleons. Needless o say, he si ua ion would ge e en mo e compli- ca ed in he case o alpha pa icles. The p ospec s a e as ly imp o ed, howe e , i one akes a phenomenological app oach in which he ene gy depen- dence o he op ical po en ial is used. Indeed, an ex apola- ion o he empi ically de e mined couplings o nega i e en- e gies, i easible, would yield a binding po en ial whe e Pauli blocking, su ace e ec s, and o he ele an co ec ions a e p ope ly included o all pe u ba i e o de s. I is no necessa y o ‘‘p opose’’ such measu emen s since expe imen al s udies o asymp o ically ee s a es ha e been a ailable o a long ime. In ac , one can simply go back o he li e a u e and sea ch o exis ing op ical-model i s o he elas ic c oss sec ions o a pa icles on nuclei. We shall hence o h ocus ou a en ion only on he eal pa Vo he in e ac ion, as i is expec ed ha his componen will ha e a smoo he ansi ion om posi i e o nega i e ene gies.1We show in Table I a collec ion o op ical model pa ame e s o 1E ec i e in e ac ions ge con ibu ions ela ed o nonlocali y in space and ime. I is well known ha he ene gy dependence o he eal pa o he la e , DV, has a coun e pa in he imagina y pa W. The connec ion be ween hese wo unc ional dependences o hea y ions has been success ully app oached in e ms o dispe sion ela ions ~see, o ins ance, Re . @10#!. The cha ac e is ic shape o W(E) o E'0, howe e , is no sui able o ex apola ion. PHYSICAL REVIEW C SEPTEMBER 1996VOLUME 54, NUMBER 3 54 0556-2813/96/54~3!/1217~4!/$10.00 1217 © 1996 The Ame ican Physical Socie y he speci ic eac ion a 1208Pb. I may be somewha puzzling a i s o see he di e si y o dep hs, adii, and di useness alues ha such a compila ion e eals. One mus , howe e , ecognize ha he numbe s ep esen comple ely independen e o s whe e no a emp was made o pu in e idence an o e all ene gy dependence o he in e ac ion. I is clea ha ou i s ask is hen o ep ocess he in o - ma ion con ained in Table I, so as o es ablish a poin o con ac be ween he di e en measu emen s. The e a e se - e al ways o inco po a e he ene gy dependence o he in e - ac ion. We choose, o ou p esen pu pose, o de ine a com- mon Woods-Saxon geome y V~ !52 V ¯ 0 11exp@~ 2R!/a#,~1! whe e R5 0~2081/3141/3!, and o exhibi he dynamical con- en o he e ec i e in e ac ion in e ms o an ene gy- dependen s eng h V ¯ 05V ¯ 0(E). Fo he adius and di use- ness pa ame e s, we ake 050.98 m, and a50.80 m. This choice is inspi ed by he con enien pa ame iza ion o he ene gy dependence o he in e ac ion gi en in @11#, namely, V0~E!5~182.1641.3!1~20.24860.047!E,~2! which is alid o E.80 MeV. Compa ing po en ials a a s ong-in e ac ion adius ~in m! s51.07~2081/3 141/3)12.722aln@112~Ec.m.2Eb!/Eb#, whe e Ebis he Coulomb ba ie @12#, we ha e ex ac ed om he op ical- model pa ame e s lis ed in Table I he po en ial dep hs V ¯ 0(E). These a e plo ed in Fig. 1; eassu ingly, all poin s now all in a a he o de ed pa e n. The s aigh lines co e- spond o he pa ame iza ion ~2!, allowing o he unce ain y in he coe icien s. Fo he many da a poin s a E'20 MeV, we ha e op ed o show hei span in V ¯ 0, which i s qui e well wi hin he end exhibi ed a o he bomba ding ene gies. The choice o 208Pb as a a ge nucleus was gea ed o he e en ual in es iga ion o he adioac i e a decay o 212Po. The sepa a ion ene gy o an alpha pa icle om his nucleus is abou 8 MeV. I ollows om he sys ema ics displayed in Fig. 1 ha he a -208Pb po en ial in his ene gy ange can be app oxima ed by TABLE I. Real po en ial dep h V0, adius pa ame e 0(AT 1/3), and di useness a o a Woods-Saxon geome y as a unc ion o bomba ding ene gy Ein he elas ic sca e ing o a pa icles by 208Pb. Each ow gi es esul s o op ical-po en ial i s om di e en expe imen s, as quo ed in he co esponding e e ences. E~MeV!V0~MeV! 0~ m!a~ m!Re . 16.0 35.0 1.550 0.570 @22# 18.0 35.0 1.550 0.570 @23# 19.0 96.4 1.376 0.625 @24# 19.5 35.0 1.550 0.570 @27# 20.0 96.4 1.376 0.625 @24# 20.0 174.0 1.470 0.470 @27# 20.0 35.0 1.550 0.570 @23# 21.0 32.0 1.550 0.570 @27# 22.0 96.4 1.376 0.625 @24# 22.0 35.0 1.550 0.570 @27# 22.0 94.6 1.364 0.650 @24# 22.0 117.5 1.298 0.700 @24# 22.0 86.9 1.449 0.550 @24# 22.0 159.7 1.387 0.560 @24# 22.0 110.9 1.464 0.500 @24# 22.0 35.0 1.518 0.571 @23# 22.0 92.5 1.384 0.625 @24# 22.0 100.4 1.444 0.542 @24# 22.0 90.3 1.411 0.600 @24# 22.0 200.2 1.390 0.529 @24# 22.5 33.0 1.550 0.570 @27# 23.5 117.5 1.298 0.700 @25# 23.5 86.9 1.449 0.550 @25# 23.5 92.5 1.384 0.625 @25# 23.5 110.9 1.464 0.500 @25# 24.5 37.0 1.550 0.570 @27# 25.5 35.0 1.550 0.570 @23# 25.5 41.0 1.550 0.570 @27# 27.0 98.4 1.371 0.621 @22# 96.0 89.3 1.350 0.710 @26# 104.0 60.0 1.392 0.656 @27# 129.0 89.3 1.350 0.710 @26# 139.0 110.0 1.315 0.705 @27# 139.0 155.0 1.282 0.677 @27# 139.0 200.0 1.261 0.657 @27# 166.0 119.0 1.260 0.740 @27# 172.0 155.0 1.282 0.677 @26# 218.0 119.9 1.260 0.740 @26# 340.0 67.2 1.325 0.800 @28# 480.0 63.0 1.300 0.950 @28# FIG. 1. Po en ial dep h V ¯ 0(E) ex ac ed om he expe imen s quo ed in Table I and con e ed o a common Woods-Saxon geom- e y wi h adius pa ame e 050.98 m and di useness a50.80 m. The solid line gi es he ene gy dependence sugges ed in Re . @1#, while he dashed lines gi e he expe imen al ma gins de ined by he e o ba s. 1218 54 C. H. DASSO, R. J. LIOTTA, AND M. LOZANO V a -Pb~ !5186 11exp@~ 27.36!/0.80#,~3! whe e Vand a e exp essed in MeV and m, espec i ely. The shape o his po en ial is qui e di e en om he one o McFadden and Sa chle @13#, in spi e o he close sa u a ion alue. A ai knowledge o he e ec i e in e ac ion ~3!is a necessa y ing edien o es ima e he ela i e mo ion o he alpha pa icle p io o i s unneling decay om 212Po. Spe- ci ically, i ei he de e mines he wa e unc ion o he alpha pa icle nea he nuclea su ace in he R-ma ix heo y o Teichman and Wigne @14#~which we will apply he e; c . also Re . @15#! o bo h bulk eloci y and ba ie p o ile in Gamow’s. Fo cla i y o p esen a ion, we b ie ly e iew be- low he R-ma ix app oach o decay p ocesses. A dominan ea u e o eac ion p ocesses in which wo nuclei collide a low ene gies o o m a compound nucleus ha subsequen ly disin eg a es is he p esence o esonances. In a ime-independen amewo k, and assuming ha he esonances do no o e lap and a e no close o h esholds, he Sma ix as a unc ion o ene gy can be pa ame ized wi h a many-le el B ei -Wigne o mula Scc8~E!5ei~ d c1 d c 8! F d cc82i( n Gn,c 1/2Gn,c8 1/2 E2 n G ,~4! whe e d cis he phase shi in channel c, he pa ial decay wid h Gn,cis p opo ional o he p obabili y ha he eso- nance labeled by ndecays in o he channel c, and En5Re n and Gn522Im na e he posi ions and wid hs o he eso- nances. Fo isola ed esonances only one e m domina es Eq. ~4!, and he uni a i y o he Sma ix equi es ha i s esidues Gn,c 1/2Gn,c8 1/2 be eal. In ha case each Gn,ccan also be aken o be eal, and (cGn,c5Gn. This si ua ion does no occu e- quen ly ~e.g., among neu on-uns able s a es!, and he physi- cal meaning o he pa ial wid h can become ques ionable because hey a e uly complex quan i ies. Howe e , o he a decay o low-lying s a es o hea y nuclei, his p oblem does no a ise because he alues o Gn/En ange be ween 10210 and 10230. In he R-ma ix heo y one di ides he con igu a ion space o he a 1daugh e -nucleus sys em in o wo egions: an ‘‘in e nal egion’’ V o which he compound s a e is e- s ic ed and an ‘‘ex e nal egion’’ ha encompasses he es o he space. The bounda y be ween he egions should be chosen such ha all nuclea in e ac ions and e ec s induced by he Pauli p inciple a e only non-negligible in he in e nal egion. In he ex e nal egion he mo ion o he agmen s is go e ned by he Coulomb in e ac ion and he daugh e nucleus and he a pa icle beha e like a bina y sys em mo - ing ou wa ds wi h asymp o ic ene gy En. The solu ion inside he olume Vsa is ies a sel -adjoin bounda y condi ion on he su ace So V. In he decay o sphe ical nuclei he su - ace Scan be chosen o be a sphe e o adius ccen e ed a ound he esidual nucleus. The esidues o he Sma ix can hen be compu ed in e ms o quan i ies e alua ed on he su ace S, and one inds @15# ha he li e ime o he decay is gi en by ~ c!5 2P~ c! g 2~ c!,~5! whe e Pis he pene abili y h ough he ba ie and g is he educed wid h ampli ude. We conclude wi h a conc e e calcula ion o he li e ime o a decay o 212Po ha exploi s he p ocedu es ad oca ed he e ~ his quan i y has also been ecen ly s udied in de ail using a di e en app oach o he s ong in e ac ion in Re . @16#!. Fo ou p esen pu pose we ha e used he compu e code GAMOW @17# o sol e he Sch o ¨dinge equa ion o he a -Pb binding po en ial ~3!, imposing ou going bounda y con- di ions o he egula solu ions. The eigen alue co espond- ing o en adial nodes u ns ou o be 8.96 MeV wi h negli- gible imagina y pa , a alue ema kably close o he kine ic ene gy o he emi ed alpha pa icle. We no e ha one could also in p inciple es ima e he li e ime using ha alue o he imagina y pa o he complex ene gy. Howe e , as poin ed ou abo e, he a io be ween he imagina y and eal pa s o he ene gy is so small ha i canno be calcula ed nume i- cally wi h su icien accu acy. Wi h he po en ial ~3!and o >9 m, only he unneling h ough he Coulomb ield is ele an . Since he esul ing unc ion P( c) is s ongly dependen on c, one may hink ha i is possible o ob ain p ac ically any alue o he li e- ime by adjus ing he dis ance a which he pene a ion o he alpha clus e occu s. Ac ually, a consis ency check on he heo y is ha ( c) should be a he s a iona y o dis ances a ound he nuclea su ace. The educed wid h ampli ude can be w i en as g ~ c!5 A 2 c 2 m C~ c!,~6! FIG. 2. Li e ime a io be ween heo y and expe imen and pen- e a ion p obabili y as a unc ion o he adius pa ame e c o he a decay o 212Po. No e ha he la e quan i y has been scaled up by a ac o o 1010. The do ed line ~uni y!is gi en as a e e ence. 54 1219DYNAMIC EFFECTIVE POTENTIAL FOR a -PARTICLE... whe e Cis he wa e unc ion o he a pa icle. The exp es- sion o he li e ime quo ed abo e does no ake in o accoun ha only a ac ion o he mo he nucleus wa e unc ion co esponds o he mo ion o an a clus e a ound he daugh- e nucleus. This ac o mus be included o compa e wi h expe imen ally measu ed li e imes. A ecen de ailed mic o- scopic calcula ion @2#shows ha he a clus e spec oscopic ac o is 0.025. We adop his alue, which is also in good ag eemen @18#wi h he a - o ma ion ampli ude ex ac ed om a compa ison be ween he Thomas heo y @15#and ex- pe imen . We no e ha an olde , la ge-scale shell model cal- cula ion @19#p oduced a alue o ;1023 o ha quan i y. In Fig. 2 we p esen he a io be ween he expe imen al and calcula ed li e imes as a unc ion o he pa ame e c. We also show in he igu e he pene abili y Pas a unc ion o dis ance. One can see ha he heo e ical calcula ion ag ees ex emely well wi h he expe imen in a egion close o he nuclea su ace ~ he quali y o his p edic ion may be bes app ecia ed by poin ing ou ha disc epancies be ween heo y and expe imen ega ding he absolu e alues o a -decay wid hs can easily span o de s o magni ude!. No ice ha he li e ime esul s a e indeed p ac ically independen o co e a la ge in e al. While he pene a ion changes by nine o de s o magni ude in he depic ed ange, he calcu- la ed a io emains uni y wi hin, a wo s , a ac o o 2. The success o his simple calcula ion e eals he po en- ial o he ideas in oduced in his con ibu ion. I should be no ed ha he p esc ip ion applies equally well o he in es- iga ion o mo e exo ic emission p ocesses ha ha e also been con empla ed ~c ., e.g., Re . @20#and e e ences quo ed he ein!. These in ol e he decay o hea ie nucleon clus e s such as ca bon, oxygen, e c., which, a he mic oscopic dy- namical le el, would be e en ha de o app oach. The in o - ma ion p esen ly a ailable o he elas ic sca e ing o some o hese sys ems may no be su icien o ex ac a eliable ene gy dependence o he co esponding op ical po en ials. Wi h he use o mode n de ec ion equipmen , howe e , hese measu emen s a e no ime consuming, as a ecen de e mi- na ion o diagonal and o -diagonal ma ix elemen s o he e ec i e in e ac ions has shown @21#. Discussions wi h T. Ve se a e g a e ully acknowledged. This wo k was suppo ed in pa by G an No. DFG Bo 1109, EU Human Capi al and Mobili y p og am G an No. ERBCHRX-CT92-0075, and Spanish CICYT P ojec No. PB92-0663. @1#G. Gamow, Z. Phys. 51, 204 ~1928!. @2#K. Va ga, R. G. 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