Threshold anomaly in non-central forces

Phymcs Le e s B 300 (1993) 303-307 No h-Holland PHYSICS LETTERS B Th eshold anomaly in non-cen al o ces J. G6mez-Camacho and M.A. Naga ajan ' Depa amen o de Fiswa A 6mwa. Molecula y Nuclea . Un e sMad de Se lla. 4pdo 1065. E-41OSO Se lla. Spa e Recewed 9 No embe 1992 The beha mu o he h eshold anomaly o non-cen al po en ials, wh,ch accoun o collec we excl a ,ons m hea y-~on colh- s~ons. ~s in es iga ed. I ~s shown hai he non-cen al po en mls should exhibi an ene g5 dependence a ene gies m he icini y o he Coulomb ba ie Th~s ene gy dependence is, howe e , di e en om ha o he elas kc op ical po enual, occu ing a lowe ene gies I u he shown hai he e a e co ec mns o he admonal collec wc model such ha . he ans~ mn po en ial ~s exp essed as he de i a i e o he op ical po en ial, he co esponding de o ma ion leng h will be complex and ene gy-dependen Simple model calculauons a e p esen ed Du mg ecen yea s, e idence o he ene gy-depen- dence o he eal ( he h eshold anomaly) and ima- gina y pa s o nucleus-nucleus op ical po en ials has been ound by ca e ul analyses o he elas xc sca e - ing a ene gies m he cml y o he Coulomb ba e [ 1-5] This ene gy dependence has been a ibu ed o he couphng o non-clas c channels o he elas ic channel n his ene gy egion [6.7]. The h eshold anomaly has also been ela ed o he apid mc ease o he su ace imagina y po en ial as he ene gy s m- c eased abo e he Coulomb ba ie and he conse- quen co ec ion o he eal po en ial h ough a dis- pe sion ela ion [8-10]. The op ical po en ial can hus be w i en as U(E)= V o+AI'(E) + H(E), ( I ) whe e go is an ene gy-independen po en ial (dou- ble- olded po en ial, o example), I,I'(E) s he m- agma y po en ial and he dispe si e eal po enual AI'(E) s de ined as AV(E)= P I4'(E') ~dE', (2) whe e P deno es a p mclpal alue m eg al. Gene ahzlng his o he case o a se o s ong-cou- pled channels, i was sugges ed by Sa chle [ 1 1 ] ha Pe manen add ess SERC Da esbu y Labo a o 3., Da esbu s, Wa mg on WA4 4AD, UK each elemen o he po en ial ma ix, which de ines he coupled channels equa ion, should be expec ed o exh b a h eshold anomaly. In such a case, he en- e gy dependence o he di e en diagonal and o -di- agonal elemen s o he po en al ma ix will be a ib- u ed o couphngs o o he channels no mcluded m he subse o coupled channels. Howe e . a p io i, he e s no eason o all he e ms o he po en ial ma ix o ha e he same ene gy dependence. Sa chle [ 1 1 ] a gued ha , m he case o collec i e exci a ions, one would expec a simila ene gy dependence m he coupling m e ac lon ( ansi ion po en ial) and he elas ic op ical po en al, hough one could expec a shi m he h eshold o he coupling po en ial by ap- p ox ma el,, hal o he exci a ion ene gy. He u he concludes ha one uses an op ical po en ial ha s consis en wi h he dispe sion ela on bu s ill needs an ene gy-dependen de o ma ion leng h, hen ei he he op ical model i sel o he nuclea s uc u e model. o bo h, a e madequa e The exci a ion unc ions o he i s exci ed 3 - s a e o 2°8Pb by ~60 a ene gies a ound he Coulomb ba - e ha e been measu ed and analysed [ 12,13] and mo e ecen ly de ailed angula d s bu ions o he same sys em we e measu ed a ene gies om abo e o below he Coulomb ba ie [14]. These angula dis ibu ions we e analyzed and he esul s hin ed a a poss b h y ha he ansi ion po en ial may ha e an 0370-2693/93/$ 06 00 © 1993 Else ie Science Pubhshe s B V All igh s ese ed 303 Volume 300, numbe 4 PHYSICS LETTERS B 18 Feb ua y 1993 ene gy dependence which is di e en om ha o he elas ic opucal po en ml [ 14 ]. We p esen he e a sxmple model which shows ha , o collec i e exci a mns, he cen al and couphng po en mls, al hough consis en sepa a ely wi h dis- pe sion ela ions, could exhibi di e en ene gy de- pendences No e ha coupling po en mls include he case o conen auon po en ials o he case o elas ic sca e ing o nuclm wq h 1>/I Le us conside he usual collec i e model dese lp mn o inelas ic exc~- a mn. The op ical po en ml ~_" ~s conside ed as a unc mn o he collec i e a iable 6, which is a mea- su e o he change m dis ance o he su aces o he colhdmg nuclm. The eal and mlagma y pa s o he po en ial can be expanded m a Taylo se ies m pow- e s o (~, and o he i s o de md. dl(~ dll. I(,J)=l-~7 " , I~'(6)=~- d o. (3) I he po en mls a e exponen ial in he egmn o m- e es , hen l'(d)=V(l+~/a), '(,5)=H'(l+d/a), (4) whe e a is he di useness pa ame e , whlch has been assumed o be he same o he eal and imagina y po en ials. No e ha , when exp essing he ~magma y po enual as a unc ion o & we a e imphc~ l~ assum- ing ha he collec we a iable changes slow b, com- pa ed wHh he p ocesses ha con ibu e o he ~ma- gma y po en ml Tha implies also ha he ene gy o he ba ie B will be a unc ion old. App oxima ely. we can w i e B(d)..-B( I-~/R) . (5) whe e B is he heigh o he unde o med bame and R ~s he co esponding adms. To explo e he ~mph- ca ons o he e ec o ~ on bo h he geome .~ o he po en ml as well as on he bame , we conside a ac- onzable o m o I '(E) H'(I..') = 14"l-'(. ). (`6) whe e I " is a unc mn o he adms while F(x) ex- p esses he ene gy dependence m e ms o he pa ana- e e x= (B-E)~.1. F(x) ~s assumed o be a Fe mi unc mn F(x)= [ 1 + exp(. ) ]-~. The pa ame e J is a measu e o he ene gy ange o e wh~ch he ,magl- na y po en ml inc eases. Acco ding o eqs (4) and (5), bo h W and "(x) a e dependen on he collec- i e a iable 6. Wi h he ansa z o I '(E) o eq. (6), he dispe - si e po enual AI '(E) o eq. ( I ) becomes aXI'(E) = I 'G(. ), (7) whe e G (. ) is ela ed o F(x) h ough he dispe sion ela mn G(.x)=- p i ---=-F(V)dy. (8) 7/' I'--.V -x The op ical po en ial can be w i en as /,,'(E. d)= I'(E. d)+11 '(E, d). (9) To i s o de in d, one ob ains I'(E, d)= I'o+ l '(/(. ) d + - [ I'+ I 'G(x)+ I 'eG'(. )I, (10) a W(E. 6) = J l.(x) d + - [ liT(x)+ I iF'(. )] , (l 1 ) a whe e we ha e in oduced he pa ame e c = aB/RA and used eq. (4). In hese equauons, he diagonal po- en ml U0(E) s he e m independen on d and he ansmon po enual _',(E) is he e m linea in ~. We can wine h~s exphcl l~ as Uo(E) = I ~, + l!'G(x) +ll!'F(x). ( 12 ) ,( E) d(dU°(E)a ) =- ---+a- ~l'(G'(. )+d:'(x)) . (13) The i s e m on he igh o eq. (13) ~s he usual p esc ip ion o he collec i e model whe e he an- smon po en ml ~s ela ed o he de i a i e o he cen- al po emml. The new ea u e o his model is he second e m on he igh which a ises om he shi o he ba ie due o de o mauon. Eqs ( 12 ) and ( 13 ) sugges ha , while he cen al e ms will exhibi he h eshold anomaly a ound an ene gy B, he non-cen- al e ms will exhibi i a ound an ene gy B( 1 -a/ R ). Besides, all he non-cen al e ms will exhibi he h eshold anomaly a he same ene gy, indepen- den ly o he mul ~pola l y o he couphng s eng h 304 Volume 300, numbe 4 PHYSICS LETTERS B 18 Feb ua y 1993 (p o ided ha a i s o de ea men s alid) I is o m e es o poin ou ha exac ly he same exp essions ( 12 ) and ( 13 ) a e ob amcd o he ccn- al and anss on po en als i one conside s he adi- aba ic hml when he coupled equa ons can be de- coupled m o elgenchannels. In he case o including he g ound s a e and a one-phonon s a e, he wo c~- gcnchannels desc ibe he sca e ing by po en ials o adix R + i, wi h he co esponding ba ie s B( I g i/ R). The ansi ion po en ial is hal he di e ence o he wo cxgenchannel po en ials and up o o de d yields exac ly eq. ( 13 ). I one s dl de ines he coupling po en ial by _ dU(E) U (E)=-6¢ ~- " (14) he a io o he c ec l e o he s a ic de o ma ion leng h o he po en ial becomes complex and ene gy dependen : G' (A) " - I/~" (X) R(E)= =l-c l)~/W+G(x)+ F(x)" (15) Eq. (15) shows sepa a ely he con lbu ons o he e ec i e dc o ma on o he s a ic de o ma ion and o he dispe si e e ec due o he change in he ba - le ene gies. R (E) acqui es an i magma y pa in he VlCln y o he ba ie , o he o de o e W/4I'(B). The eal pa o R (E) has an ene gy dependence a ound he bame ha is consis en w h he magma y pa h ough dispe sion ela ions. The size o he co ec ion o he con en ional model depends c uc ally on Ihc pa ame e e, ha is de e - mined p ima ily by he ange A o e which he ma- glna y po en ial ises ( o E=B-A, he imagina y po en ial is 27%, while o E=B+A, i is 73% o he maximum alue). Depending on he alue o ~, one ob ains di e en shapes o he ene gy depcndcnce o he imagina y pa o he ansi ion po en ial. Fo < 1, he ~magina y ansmon po en ial ises mono- omcally o a cons an alue 6d W/d . Fo ~ > 1. one obse es a cha ac e is ic maximum in he imagina y ans on po en ial jus abo e he ba ie , and hcn a g adual dec ease o he alue dd W/d a la ge cne - gies. Wc llus a e his in igs. 1 and 2, whe e wc show bo h he elas ic (cen al) and ansi ion po en ials o he inelas ic sca e ing o ~60 on z°8pb o he i s 3- s a e o he a ge . Calcula mns ha e been done ak- ing B=80 MeV, a=0 6 m and R= 12 m. Two al- .... I Fig I Cen al ( ull hne) and non-cen al (dashed hne) po en- als e sus he ene gy o A=2 5 MeV The eal pa s a e he up- pe hnes, and he imagina y pa s a e he lowe hnes ~,.? ?() 1(I¢'. I .'0 I 1~ l (g!W, : F~g 2 Sameas ig llo A=455MeV. ues o d. which a c J=4.55 MeV, aken om e . [9], and A= 2.50 MeV, ha e been used. In bo h cases, he op ical po en ial is aken as 1.5 + 11.2 MeV o E= 150 MeV. Bo h calcula ions a e consis en wi h he em- pi ical da a o he op ical po en ial o 160+2°8Pb. These yield alues o e o 0 88 and 1.6 espec i ely. The ansi ion po en ial is no malized so ha i co- incides wi h he cen al po en ial a high ene gies. I is clea ly seen ha he ene gy dependence o he an- si ion and cen al po en ials a e di e en in an en- e gy egion a ound he Coulomb ba ie . E en hough om igs. 1 and 2 i may appea ha ~< I, he a- di ional collec i e model, whe e he ansl on po en- 305 Volume 300, numbe 4 PHYSICS LETTERS B 18 Feb ua y 1993 lal is aken as he p oduc o a s a ic de o ma ion leng h and he de a we o he complex po en ial, is alid, his is no s ic ly ue In ig. 3 we show he eal and imagina y pa s o R(E) as a unc ion o he ene gy o he wo cases 3=4.55 McV and 3=2.5 MeV I is seen ha e en o , =4.55 MeV (~< I ), he de o ma ion leng h becomes complex. I is e i- den , o cou se, ha e y high quah 3 inelas ic da a will be necessa y o ex ac his complex de o ma ion leng h a ene gies a ound he Coulomb ba ie The pa ame e e can be unde s ood m a ime-de- penden pic u e, =,5//I has bccn desc ibed as a e- a da ion ime in he ime-dependen p~c u e o he 1magmas.' po en ial [ 15 ]. I is he cha ac e is ic ime o he p ocesses ha con ibu e o he maagma y po- en ial, ha is, o he couphng o s a es no ela ed o he collec we a iable d. aB/R is app oxlma eb equal o I" Classically, he hamil oman o a lb a- Uonal collec i e a iable d is II=P]/2M+Kd2/2- l'cxp(d/a) , (16) whe e M and K a e, espec wel. ., he ine ia pa ame- e and he ha monic cons an o he collec i e a i- able We can de ine he cha ac e is ic ime o collec- i e exci a ion as dl~/d I/T= -- (17) ',~ "7 . :z'6(] --" ~(; 1(;(: : ~;" I 1(' ? I *d, ', : Fig 3 Real (uppe hnes) and imagina y, (lo , e hnes) pa s o lhe auo o he e ec i e and slallc de o ma mn leng hs as a une- I on o he ene gy The ull hnes co espond o 3=2 5 MeV and he dashed hnes o A=4 55 MeV dP,,/d call be ob ained om eq. (16) using Hamil- on's equa ion dP~ dH d I" - dd C + - a exp(:i/a). (18) d The second e m on he RHS domina es o e he i s in he adlabauc limi . On he o he hand, as 6 will ake alues o he o de o a, we can es ima e he alue o P,, as h/a. Thus, T=hR/aB is he cha ac e is ic ime o collec i e exci a ions. Hence he pa ame e = /T is he a ]o o he ~me o p ocesses ha ake away lux om he coupled channels sys em o he mae o he p ocesses ha couple mhm he coupled channels sys em I < 7" hc couphng o non-collec- we channels is as compa ed o he couphng o col- lec we channels, and hen i s e ec on he diagonal and ansi ion po en ials is simila , p oducing a s~m- la ene gy dependence o bo h. I > 7-, he couphng o non-collec i e channels Is slow compa ed o ha o collec i e channels, and so a di e en ene gy de- pendence appea s. No e ha hese condmons a e no incompa ible wi h he assump ion ha he collec i e a iable changes slowly compa ed wi h p ocesses ha con ibu e o he imagma po en ial. The cha ac e - is ic ime o change old can be de ined as dd/d 1/7-- (19) d dd/d can be ob ained om eq. (16) using Hamil- on's equa ion dd _ dll _ l',s . (20) dl dP~ .ll Hence. using he p e ious es ima es o (5 and/~, one ge s T'= 31a'/h. which can be la ge han i he in- e ia pa ame e ~s la ge enough. To conclude, we ha e shown ha he ansi ion po- cn mls o collecm.e exci a ion ha e a di e en en- e gy dependence o he diagonal po en ials. The de- o ma mn leng h o he po en ml becomes complex and ene gy-dependen a ound he Coulomb bame . The size o his e ec depends c ucially on how as he imagina y op ical po en ial ises o i s maximum alue as he ene gy inc eases. 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