REVISTA ~lEXICANA ¡>E FíSICA.U SI ~I'I,E)IE:"\T() 2. X)-lJl

Step-by-step ded"atioll 01'lIJe Iluclear !'usioll radius

A. (iÚlllc/.-C.unadHIlkllllrIafllemo del Acf!emdm: Inslitufo Naciol/al de Inl'CSligaciolles Nllcletlres

Apanado I",sial/S-I02? IISOI ¡\/cxico, n.F. Mnicol'-I/1lI i1: a rIl( '-(lg (a'se' n 'it Ir n: 11//(/11/./11.\".

• T. Udagaw<IDeptlrIlJU'1ll (~IPhysics, nI(' lInil'crsiiY (~/'Ie'rtls al Allstin

r\//sril/. TC.m.\' 78712, USA

RCI.:ihidoel 19 de enero de 19tJX:aceplado d 9 de ahri I de I<JYX

OCTUBRE IlJ9X

A detailrd ~I('p-hy-stcp dc!crminalion of Ihc oplical potclltial (\ \'10') fu~ion radill~ paranll'ler "'" i~ rarried out lmm ;¡ simullaneOu!\ l'.llculali(lnut d~I~lír ~raHcring, lolal reaction ami fmiolllTO!\S sl.'ctions. An approarh CtlllSI.\tl.'lHWilh Ihe dírl.'l."Ilearliolllhl.'ory is lJ.\l.'din Ihe calClIlatiolls.II is found Ih;lt fOI a simlll!aneous good lil lo (he experimenta] elastic sralll.'ring. l(lla] Il.'al."limlamI fusillll tlala. Ihe fllsion radius pamm.:ter1"1-- ~h(lllld lake a \'aluc (:::::::1.1 fm) Iargcr than Ihall'Ollllllonly assullll.'d in othl.'r tl1eoríl.'s.

A.'I'I'Il'ords: Nuclear reaclions il1\'o]\'ing fcV.'-llucleon syslellls: polari¡;lIion ph~l1(lmen;l in rC<lclioJlS

El par<Ímetro dI.' fusión nuclear "/.' de un potenciallÍplico 11',., es delnmin;ldo paso a paso medianIl' una leoría consislente con la Icorí~lde reacnones nudeares directas. Se enclIelHra que para una detcrrninación 'iiJl1ull~ílleade la\ \l'lTiol1cS clic<lces de tlispcrsión el;ística. d~reacción lowl y de t'usilÍn es necesario que r,.. lome valores (::::::1..-1fm) qlle son mayores a aquellos supuestos en otras leorías.

/)ncril'(OH'S: Reaccioncs Illldearcs enlre sislernas de pocos nudeont.'s: fellónll'no de polari/aciún en reacciones

I'ACS: 25.IO+s: 24.70.+s: 24.IO.Eq: 27.30.+1

1. Introduction

111recenl years, hcavy ion induced reacliolls aroulld lileCoulomb harrier energ)' have been gi\'cll IlIlH...h atlcllliol1.There are ¡¡ flulIlhcr of vcry inlcresting observal ions thal havellIo1ivaled Ihis interesl, sllch as lhc wcll known lhrcsholdanoll1aly 11. 2), 01' Ihe vcry inlcrcsting cnhancel11ent observedolllhe fusion cross seclioll (1F [;q hclow Ihe Coulomb barricr1'Ilcrgy and ils currelation lo Ihe tOlal uirccI reaction cross sec-ti(ln 1.1], Scveral theories have becn proposcd It) aCl.'oulIt rol'lhesc clTccls, among lhcm are lhe harril'r pcnelration models(HP~1), ho\\'cver il is \\-'e11knowll that the BPi\-1 has had proh-lL'II1Sto jil silllultaneollsiy lhe fusion cross scclioll above ¡¡1ll1hL'low Ihe Coulomb harricr encrgy, In muer lo rCI11O\'l..'lhisdi flicu It)' SUlIle Illodificaliolls have heell done, ror instanl'c inlhL' sllb~hilrricr regioll Ihe usual approach has hecll lo kecpwilh lhc BPM, hUI lhe effccl 01"low-encrgy surf¡¡ce vihralionshas hcen cOllsidcrcd [51, Olhers lake inlo accounl thc clfcc(01' sl<llic dcformalions (GI or neck formalion (7J hut ncverihe-less, slill none BPM lheory can givc sarisfac(ory alld silllul!a-I1l'OUSlils lo fusion helow ano ahove Ihe Coulomh clle!"gy andsilllllltancous good lits of simple e1astic or inelaslic dala. Ontlw otiler hand some CC-caklllalions [8-10] Ir)' (o cxplainlhc ohscrvcd cnhanceml~nl of the fusion excitalion fUllctiollin Ihe sub-barricr rcgion by cOllsidering Ihe coupling 01' tl1l.'ground slalc to SOIllC rcactioll l'hallllcls likc inelaslic L'xl.'ila-lilllls or particle lransfcL However, as in the BP!\I case, al-Ihollgh SOI1lCof lhcsc CC-approachcs can give good l"esulls

ror Ihe rUSiOll excilalion fUllclion, none can givc salisfacloryand "iIllLJIt~IIlL'(lllSgood IIts lo claSlic ami some dirccl reaclionpnlCcsscs.

It i." Ihl.' PurPOSL' 01' Ihl' prcsenl work lo show, Ihat il ispossihle [o givc a silllu!tallcolls description of fusion, clasticallll IOtal I'c<lclion noss sl'clion dala by an approach wilhinIhe framcwork 01' dirccl reaclion Iheory. This Illeans thal atany slagc 01' Ihe calcllJatiollS. we slick lo Ihe principie lhallhe oplical pOlential lo he lIsed, musl he Ihal which tlts lheelastil.' sC<lllning dala. In olher wonls. we \\'anl to make ilpos"ihJc [o descrihe Ihe fusioll process in a \Va)' thal is con-sistenl with clastil' sGl!tering, In our approach, Ihe ahsorplivepan lit (Ir an (Ip[ical p(lIcnlial U is spli(teo inlo two parls, lt'r:.lI1d Jl'nu, Ihe lúnlll'r responsihle for.fusion reaclions amllhelatler ror dirl't:1 reaclioll (DR¡ ahsorption processes. The fu-SiOll eross Sl'Clion (1r: is then ohlained frol1llhat porlion 01"lhe(olal I"caclioll eros ....sectioll due solely to Irr:. On lhe othahand, Ihe dirl'Cl rcaelion lTOSS scclion is given in lcrms of\t"v/(, Of l'ourse, the IOlal reactiollcross sectioll is lhe sumof hOlh, fusioll <lnd direct I"cal.'lion cross sections, Since Iritself rellwins llJll1lodilied. Ihe l'usion cross seclioll Gr: can heL'akulalcd illulnsislency wilh e!astic scallcring and other rc-aClion prm:e,"'sL's. I'ossihly Ihe simplesl way (() producc thisseparalion 01' lüsioll i1nd dirccl reaclion ahsorplion proccsscsis lo t.:hoos~ II'/-. = 11" for /' :s R F ami 1\ 'f.' = O rOl"" > n/."whele [hc fusiotl radills UI" is UI" = rp( ..t:/3 + Ai/:l), IIis thell reasoll;lhle (o CXPIXl that Ihe corrcsponding diffusnessparilltlelers 01' lile potclllials 1fr amI 1VDU rcl1ect this cUI-olT

:\. (iÓ.\lEZ-CA\I:\CHll :\\I)T t Il:\li,.\\\"'.\

[lerc, tllL' IUlll'litlll l.',. I = tI, l. h. 1/ dcnotes inlernal slates,(,';,+) is tllL' tlpliL'lllllHltkl (¡I"ecn IUtll'lion in lile challllcll) and(> 11Il';¡Il"inlcgralion in alll'onrdinales except thc fl-channelcoordinale rll. :\ rigorous dni\'alioll nf Eqs. () ;l1ll1 (6) isfOllnd in Re!. [.,. \\'ilh thlo'''l' con"ideralinns is lhell possihlclO \\Tite Illlo'lllla[ fu,ioll crtlSS "n.tioll as

11i.' illlpl1rlalll ltl ntlll' lhal. 1I1l'l[i,lllrtcd wm'e \;,' ) llpplo'aringin Eqs. (2) al1t[ (3) is l'x<lclly Ihe "lII1C. lhat is. in Eq. (3) \VeI 1 1 le, 1 . 1\' h' I {+I.ltllHll rce;l eu ,lle \" )y lI"1n~ "l-'. ut Inste;ll. \" IS

Il1al calcuJalcd \\illl thl' rul! pOlL'1l1ia[ 11." as in Eq, (J l. In a...illlilar fa,hion. lhc ahsorplioll due lo direct rl'actiolls is oh-taincd in Il'l"lllS 01 the potl'llli;¡ll{.".1Jf( = 1\." - II.IIJ", thalIS.

(h)

(5)

(-1)('III' I l")\" "./lU \"

.,.-"

". ('I¡l +-1111' l'I,l/T')-" /, /,¡'

111'<1 iT

111'"

fT 1, 1

\'ext. \\'Ill'll lu ...ion pn)(l'ctlin~ Ihr(lll~h Ihc l'1;lSlk channcl i"Ilol 'llllicil'1I1 lo tlllllL' dat,l. IllSilln Ihrough a channel ¡, wllichi" rC;lchl'd hy ,1dirú'l rl';lctioli II( ,\.11)/, has lo he l'onsidl'l"l'd.ThereJúrl'. fLlrlhl'l' sp[itting of Illl' fusiol1 pOll'ntial 1\',.. is lll'C-L's,ary. lllat is 1\ .l-' = 11"".1"+ 11'/>./".Now ir lhe wave functionin tillo'l'hantlel h i, dl'llotlo'd hy 11';,t). Ihe corresponding crosssl'ction (1,. I Ina} 11L'\\Tillen hy,

, IIr¡ •.f t1ll'or•.tieal deseription

rhl' Sdu{idingl'r L'quation for elaslic scatlering fmm an opti-l.al ['olelltial ClIl he \\"rltlen as,

.t •.••.•lIll1[11101l.1I i, Inund lha!. in the proccss of silllu[lalll'OIl ...JyIl1lill~ 11lL'lu illlL lola[ I"L',IClioll,IIH\c[a"lir noss "erliolls. lilcIll',illll I',ldiu p,U.,lIl1ClL'r,.[-"of Ihc ah.'iorpling \"O\UlllC fu"ion["lllL'IHial 11., 11;Is1(1ha\"l' a \"aluc or aroulld 1.4 hll. whichl' .1 \:tlLlC Ltrgcr Ihan Ihal assulIlcd hcrorc in any BI'\I m('( '-;'pproarh. \Ve \\.i[1 sho\\' Ihe above conclusions in dClail1111'Ihe rC:lrli(ln 11,(}+~w':Pb.Thi" lluclear systclll has Iwcn Sl'-kl'IL'd "inrc il has hl'l'n ;¡ ver)' \Vell 'illldied systelll, h011Jl'X-[ll'ríllll'l\lall} ,llld Il1l'(lrl'lirally Splo'ilKing 1111, and therdoreII Illflors;l \l'ry ~tl\ld exalllplc lo lesl l~ll' "alidil)' of OU! :IP-[lnl,ldl. \\'e hegin llllo' litling analysis ;11lhlo' hOlllharding l'n-,'l'g} nl:-iO ~lc\'. ;llld lalc!" ,,¡H)\\' 1l1allhc COllSCCllcnccs fOlllld;11Stl.\k\' ,lpp[} In se\'cra[ hnmhardingclllo'rgics.

Thl' 11L'xl"l'l"lioll. ¡,l' .. Scc\. 2, i...dcvOlcd to a \'c!"y hridprcsclllalillll of OUI"I"orlllalislll I J:2I. In Sú'l. 3. \vc hegin 1l1L'lk'I;likd step-hy slCP calcLllation 01"Ihe I"usioll radius paralll-,'IL'r, tirsl asslIllIing for il tl1e \.aluc nonnaly gi\'en in ,Ollll'BP\t- tU"CC-l"alrulations. that is =::::: 1.0 rm. thcn, \\"e c;lI-l"lllalC lhe olller oplica\ POlclllial parameters hy a \ ~-;¡Jl;¡ly ...i....¡lnl) lilling lllL'l'Iaslic scaltering data. Nexl. \Ve sho\\" hy ll1i1K-in,!! "Il'P~hY-SICP l'hangcs in Ihlo'oplical pOlential partllllcters.\\'hy l!le fusioll radiu" I"F has lo he he dlallgcd ir silllull:llle-1111"tilling" 10 nol ol1ly clastic hut rearlioll and fusioll noss'l'l"linn" arc 1{1hc ohlained. In Ser!. 4. \Ve apply the hnal oj1~Ill.:l1 pl)1L'nlial pilraml'lers found in Slo'CI.3. to L'l1crgks othe!"111;111SO \k\'. 111;1Ii.... \\'L' sludy lhl' energy dej1lo'I1Jencl' 01"thl'"Il-cilll'd f'nl. 7. Sel.tioll:' is dcdicaled 10 lina[ conclusions.

\d1l'rc \Vc dcnoll' hv (f Ihe incidenl challlle!. hv \~I+)Ihe dis-'. .lorlL'd \\'a\'e !"ullction amI hy el = \:1 - ill"" (he opticallo'las-lí,. pOll'ntial. hl'ing 1:, <tmll\.u the real and imaginar)' parls rl'-"fll'L'ti\l'[}. Thell, a ... it is \\'cll KIlO\\'I1.Ihe total rcactioll \:ro......'l'l'liOJl can hl' L'\prl'sscd as.

(8),< R,,. > RF,{

11',.11.

11', 1(1)

lhal is. il is lllHll..'rslood thal all 01" lhe ahsorplion 1>.;fu-...•Ion rm dislalllT.' "ma!ler lhall ;¡ crilical radius Ur, whereUF = /"1"(/1: ':: + ..l~/:\), hcing ,.", an adju ...•lahll' parameter\••..hose valuL' is lo hc dctel"lllilled. 11 \ViII he secn in the nextscct icm ( \ '.! ~l'akulalitl1ls lo that Ibis ClIl-(lll assumpti(lll is sup-POltl'd hy ,mal[ dillU"lllo'S' p<Lr:UlIt..'llo'rSal-' and tl¡IIl for 1hl..'pOll'llliab ti., dlld tl."u Thc real pan uf lhe optica\ pOlen-tial \. of l' = \. - 11r (lhe ,uhscript (/ or ¡, is Olllillo'd asexp\ailll'd I;lllcr) is asslIllled lo blo'nI' lhe fonn,

(T l. = (1., 1 t- z= (T¡" 1".

As said in the ínlrodllctioll, llll' simplesl choice 10 splil lile101;¡\ ahsorpti(ln prt1cess jnto ,l'parale fusion and direL'1 rcac-tions is lo introduce a cut-ofr as",ulllption, thal is. wc chooseIr;.¡.-, rm 1= (1, (1 such lh:H.

(.1)

( 1)

'.'.,. ({<jlll' II{+I)\./ '1. F (/(11 I

/Tu =111'" 1'1'"

\\ hlo'l'e ¡',¡ is tllL' rl'lalivlo' \'l'lodty hd\\'Cl'n targct amI projl'C-li!L'. The imaginar) pOlentia\ ¡{",' is splitted into (\\0 pan ..../.t'., 1\" == 11"".1 + I{'u.uu. where \l', ¡.., responsihle fOl' fu~"iol1 and II.uu 1'01"direcl rcaclions. Ikncc, lhe fusioll nos s"l'l'lion /TI' pron'cding Ihrough Ihe lo'lastic channcl (/ is givl'n'llk'iv in ll,:rm...01' lhlo' cxpl'ctation valuc 01' 1\"r1J" respL'cl lo

'. (». 1 1 . I l'Illl' 11L'¡Orll,:d\\':I\C \" In 11l' lo'astle c la!llle ,1.(' ..

" . 8 /' .,-" L(2'" + 1)-,- 111"(1')1-11",,,1"(1') dI'.',~ l., /:1'" . \ '(,)

\il+ I'XP (,ro)

(~)

SITI'-B )'-STEI' 1)EI{ 1VATIC)NC11:.1'111':NlTLI~AI{ I-LJSI()~ IL\I l]lIS X7

T¡\HI.I. 1. ()ptiCill pO[l'ntial pilriUl1l'lCrS lN'd in tlll' prl'~l'l1! sllldy :lhlll.'! \\'Ilh l'tl1Tl'Sp(llldin~ \ 2'V,.llll'S

l'otl'[]li;d 1'\11. P(I1. 2 Po!. 3 Pul .¡ 1'0L 5 PIIL () POI. 7¡;-".\kV) I :ih: 7 156:7 173;\) 1(1):7 1.13:h 55:0 156:71",,( fllll J.17t) 1 171) 1 17t) 1 17(} 1 17(} 1 17t) 1.17\)II"llm) O,5~'¡ O.5S'¡ O,5:-;'¡ (l5~d 0.5S4 ().(13~ O,5S4

11 ;I'fI(:-'kV) 0.040-1 (1.O775 00775 n,II"¡4 0,7702 "¡t)2() DAS!I"¡, f{( flll) 15 15 1.5 15 IA2h 15 1AlJ71I,,¡,.{lm) 0.7X6 07X6 {l,7Xh 0.7:-;(1 ()4.Q (U)765 O.2XIl';,)-(~kVl I O () 10.0 100 I (), (l 10.0 1(l, () X.9" 1-"1/1111 10 10 1.0 In 10 1.0 1401rI ¡. (1 JI)) O,2S 0.25 (J.25 025 (l 25 () 25 O. 1.11!¡"lmh) 3(1...1 34.(1 4X.2 .\1).5 12.3 O,I() 35.7I! /((mh 1 7X.(1 I 1.1. l 127.2 15J,"'¡ 12:-;.3 126.7 120.1

2().() JI.S 4.29 2.XI O.X(1 (l,()J O.-C

\\ Ill'J\'.I'o = (r - Ro) //10. 01' cnurse 111l'Coulomh polelllial\; "111 has [o he addeu. On lhe otller hand. lhe absOl'plioJl po-IL'IlIial II'{-" and 1\ 'nu 01' Ir ha\'L' a \"O!UIllL'.tnd surfal'c fOl"lns,/.1' ..

11(l1l1l'CCssary [o rOllsidL'r (llhL'r n:actiol1 channcls he:-.ides lheclastic. ¡:m thi-; rcasoll. llll' suhsnipl /1 in '". \\'¡.- or ¡rouhas bl'l'lI tllllilll'd.

11';,1,

1 +l'xp(.r •.). ( 1111J,l. ('akulaliolls ",itlt ",.. ~ 1.0 fm

('O/clI/o/ioll lei/It /'0/. /

Inlhis seL:!ioll. \\'l' will pcrforlll calculaliülls uf c/astic .SGltter-ill~, n:aclioll and fusioll cross scctiolls for thc nuclear systeJllj(;() + :!nsPh al incidenl cnergy of SO f\-1cV. To oL'gin Wilh.,1 \ 2-analysi:-. 01' lhe e!asliL' scaltcring data ,¡Jolle at the ahon:I1lL'ntioned ellergy 1111 \ViII oc carricd out. The fusion radiusr,.. P,ll,lIJlelcr tlf Eq. (10) \ViII be lirst assulllcd to bL' lixcd al1.0 flll as norlll<llly assLJlllcd in other thcoriL's i I.IJ. \Vilh such,111a.~SlllTlpli(lll ;1I1din <Inclfort to lil simultancously not onlycl;¡..,¡ic bul rcaclioll ,1Ild fllsion data, \Ve \\'ill lIlake ehallges in11ll'olher optical potelllial parameters. Ih:\! is. Ihose 01 \ ". Il"}'dlld 1\ uu (l'\L.Cpl I"F l. TIlL'se cllangcs ¡¡re callcd slcp~bY-Sll'P111(ltiij¡L.<ltioll"(11the original potL'lllial. It \vill be Sl'L'1l[haL i[i.~nol possihlL' to lil ,alllhe experilllentalLToss seeliüns unlessa change in l!le fusioll radius paralllclcr r/-, is lIli.ldc. Thal is. ilwill hc pro ved Il1al a silllultancolls lit to L'!¡¡stiL',rcactioll amifllSioll eross seclions is achicved. unly ir Ihe fusiOIl absorp-lion is inL'reasctlllndcr lhe oarrier hy incrcasing: rF lo lA flll.Fin,dly. \Ve foulld lhat ¡úr lile spherieal nuclear systL'1Il tlllllclstudy. fusioll prneceding unly lhmugll lhe elaslic ehannel issuf1irielll 10 gin." a correet lil 10 lhe data, lhis lIIeans tha!. it is

<lnd.

"o ex» (.rl,,!)II-pu(r)=.llIpU[ l.'.

J + I'Xp(.I"/O/I)-

3. Sll'p-by-stl'p ca!Culatiolls

( 111\Vc hq,!in our l'a!Culation hy cOllsidcring tllL' optical POICIl-lial par;lIl1elCTS ohlaillcd in the oplical lIlodclcillculatiol1 101'

I(,() + ~fI'''l'h hy S:lIl'hlL'r ('/ al .. 1I 11. These are listed in col-1I11l1lI 01' Tahll' J lindel' PoI. 1, Notice thal IhL' I'usioll radiuspilrallleler thc: lI-;ed is r!- = J.O frll. In ReL 1-1.lhere arL' hJurseis nI' pílr,UlletL'rs. Ilowevcl', we llave chosen lile SL'I2 sincelilis gi\'l's lllL' hL'SI lit lo n;',xp. The reaclion ;¡nd fusion cross~L'L'(iullSohlai Ilt'd in l!leir ca!cu lalion are 7H.6 mh ami 36.-IlIIorL'~pL'L'li\"l'I:. The resull fOl' ftlsioll ,Igrces very \Vell with lhel'.\jlnillll'1l1al 3(1 ~'kV [11. I,-'¡, whik' lIJe total reaction l'rOSSsecti(Hl is far fnllll lhe IllL'ilsured value J 13 mo 111. 1;j1. ThL'cakulall'd elaslic sca(ferin:~ (..ross seclion is shO\\-'n hy Ihe fuI!line in Fig. I alollg will1 lhe data ¡II, 1;j1.As dearly scell,lill'calculaled n,'1.1S sl<lrls lo fal! off al;lIl angle ahouI lO de-grees largcr than e.\pl'clcd. This indicalL's Ih:1I Ihe ahsorption01' ('lIl. I is \'el'\ wcak. In facI. lhe \ :!-\'aluc nhwined wilh('01. I is 2h.fl. \\'hidl is qtlilL' Iarge. Possihly, lhc lirst naluralSll'P lo I;II-.e i:-.lo inLTCaSL'Ihc ahsorption. In Fig. 2. Irnu asa fUIlClilJn (lj' 1"is L'tHllparúl lo 11'",,,, \lihich is ()htainetl in a(¡ptical Illodcl allalysis of ct¡¡slic sc;lIlering dala [11. 1;j1. AsSL'CI1.11'/ J /( (I") is lluile small I"espect hl 11-0111 (r) in lhe harriclrcgioll 10 ,. ];\ flll. In principie. Ihis weakness in IFu/{is rcspolIsihk for lIJe hall lil to lhe el<lslic scaltcring dala.

('11101101;011 \I';1h l'tll. '2

According to 111L'cOllclusiullS madc JUSI aho\'L'. \Ve decicJed lolIlolliry 11Tlfr :-'0 as lo reproduce 111l'cxpL'rimCnlal aRo In the

ss A. GÚ;"'lEZ-CAI\I"\CHO Ai\[) T 1I1J,\(j;\\\,\

The (aelllla!. Ihe ea!clllated L'!;¡slie noss scction with POI. I isIar~e rOl"litr~l' al1~k'" can hl' "ollll'whal rl'lIledil'd h) inen:as-il1~ lhe ahsorpliílll lIndl'r Ihl' harril'r hy incn:asing thl' mag-nillllie of tlll' sudan; ahsorptive part ()f thl' Illlcll'ar opticalpotl'nlial 1I'}I,/!. Ihis was achic\'ed wilh POI. 2. Here lhe pa-rameler \;1 wdl be ea!culatel! t'rtllll Ihe \ 2-analysis by keer-ing all the nthcr parallle[ers lhe same as in POlo 2 (see Tahle 1).Thl' resultanll'la"tic cross s('ctioll is shown hy Ihe dashed Hneof roigo l. As Sl'cn. thc dala as lx'cn IItted for largc angles,hO\vevcr tlll' lil is slilllll1salisl";:lclory. Thc reaclion amI fusionnos s scc!iolls now hecolllc larger Ihan with Pot. 2. The \ 2-\'aluc has hccn rl'duccd 10 4.2l),

lO/Clllolion 1\';1" /'01. 3

Co/culat1OIl \I'lr" Hlr . .¡

hUI howc\'L'r. Ihe lit 10 lhe c1aslic scallcring prcdiclion hc-comes nol JllllCh bcllcr thanlhc ohtaincd wilh PoI. l .- ...•::----,

.•••••....,".,'.,"\'.,

'.1,.\\\,

'\\

\,,\.

'':..:

Pot.11'0t.2

- . - . - 1'0t.4

I:I( ;lIRE I . Angular díslrihlllions of elastil' scattcring cross sc{,:liolls(l<l!itls (o RUlhcrford) (lf !l,O + 20HPh al E1ab ::= 80 MeY.

0(C.M.)" •• " "" ." •••

\Ve dccilkd no\\' lo \'ary \ () ami 1\ 'jjf{ silllu!taneously. keep-ill~ Ihe rcsl of Illl' paramctcrs lixed as in Pot. 3, The resul-tan! l'laslic scallering ca!Culatioll is graphcd in Fig, 1 hy thedOI-dashed line. The \ :.!-value becollles now 2.X l. \\'hich islo\\"ef Ihan with PoI. 3. Howl'\'L'r. (hefe is an imporlalll dis-creranc)' \\.'ilh Illl' data. for angles VIl < (-)< I:W degrees.lila! is in tlll' shlllllder region \vhere Ihe experimcnlal ralioda /alllllli relllains practically unil)' and our prcdictilln fallsoff loo rapidly, This lIIeans lhal Ihl' ansorrtion in the tail 01'lFf)u(r) is nmv too s¡rong. (1H has increased signilicanlly.while (1¡. is l'IOSl' to ¡he data.

Cll!clllatiOI/ \\';1" /'01. 5

Thc stnlng alN Ilpl ion in the I¡¡il \111\',) /( (r) can he laken caretlr by 1lI0dit'ying ils gl'ollletry. in particular OVil' flere, inlhe\ :.!-anal)'si\. (I/>/I ami "UIl werl' "aried. llevertheless. we no-liced Ihal Ihe resultant \ :.!-valllc increased respecI to formervallles, Therdore. \\'c included \ íl and \r2J1{ in the \ :.!-filling.No\\'. Ihe \ :.!_P;U.ullclcr is redllced lo Ihe very small vallle(l.X6. The reactioll cross seclion (1 u hl'COlllCS 12l':.3 mh whichis ahovc hUl c1osl' lo the cxperimental 113 mh, Ho\Vever, <Inul1expecIl'd fcature 01' ¡his calculation is thal the calcul •.lled(11" is (Hl\\" 12..~ lIlb which is less Ihan hall' the expecled 36IIlh. The prcdicled clastic cross seclion. shO\vn in Fig. 3 is uplo now lhe hest lit 10 Ihe dala.

By starting with the paramelers of PoI. 4. another \ :!-littingwas done. \ ;l, 11" of Ihe real POIl'(llial ano lrgu amI tLOIl oí"Ihe surtiu.:e polell!ial werc allo\\'cd lo vary. The \ 2_value he-came ver)' small {UL'. Figurc ..l shO\vs the c3lculaled claslicscattering cross \l'Clion. As Sl'cn, the lil is no\\' very goml.(Tu = I:Yi,llllh \\'hieh is close lo I U mh. hui still Ihl' proh-Icm is lhalllle l'usioll cross seclioll (T,.. is again very smalllhalis, 0.1 \) IIIh. TlJis is dlle 10 Ihe Vl'ry large value for ilT)f{ (see

Cu/cllla/ion \I'ilh /'01. 6""""90,001

11 '2

R (fm)

FIl;tJRE 2. \\'(,.) of Ihe potcntial Pul. comparcd 10 (he optical

llllldcl C<llnilalioll of Re!". 11.

lhird i.:olumn nr Tahle 1. Ihe results of lhe \ :!-analysis is Pf('-....l,lItl'i.L whcrc al1 lile othcr paramelcrs have heell kcpt IheS:llllC as in POI. l. As s('en, the l1'}ju-valllc has almosl bl:cnd\luhkd ami (he \ :..!-valuc has hren rcduccd to II.X. \Ve can"eL' Ihal (T/I has heen fitlcd vcry \Vell. (JF is sfil! closc lo (J~~xp

STEP-BY-STEP DERIVATIC)N ()F THE NI.'CL[:\[{ FUSil l:'\ RAI ltUS

•

1'01.5

..o o

~o"

1'01.7

..o o

~O."

Fa;LlR!: J. Thc samc as Hg.1 cxccpt tha! lhe solid line fepn.'SL'Il1S

lhe rhC(lfclic:t1 cross scclioll with lhe potencial POI. 5.

F](;IIRE 5. The S:lmc as Fig. 1 cxccpt that lhe solid [ine represen!slhe thcon:tical eross seclioll "'¡lh the POlclllÍnl POlo 7.

" " te 110

8(C.M.)'" .., " " 8( C M. )

'" '"

FI(;UI{I; -l. Thc same as Fig. 1 cxccpt thallhc solid Jinc rcprl'selltslile thcorclicll cross scction wilh Ihe potcntial PoI. 6.

• •

8( C M. )

Cu/eu/orioll lI'irh /)01. 7

Tahk 1) <lml lhe ve!')' small flnu which sOl11chO\\' produce a\'L'l)' IIIlUSlli.¡] sllape rOl" Ihe surfacc ahsorptivc polcntial.

It should he slrcsscd lha!. in all nf lhe forcgoing calcula-,ion s, non!".'01" Ihe i"lIsion potclltial paramctcrs have heen var-ied. In <In ~ITort lo lil Ihc experimcnlal elastic scaltcring dala.we have becll l"orced lo vary Ihe paramelers 01" ll'"f)/((r) amisOlllclimes ¡hose tlr \ . (r). In l~lCl,in fitting Ihe clastic scalter-ing dala, Ihcre has hCl:n a lentlcncy 01' incrcasing Ihe surfaceahsorplioll al Ihe (osi tlf severcly reducing Ihe ealculaled fu-sitlll eross seelion. II Sl:elllS Ihal in order lo lit simullaneouslyIhe fllsion dala il is nl:cessary lo also var)' Ihe parameters 01"II"I-'(r). In Ihis L',lIl'Ulalloll. we imend lo vary Ihe six parame-lers tlr 1\ '{)u( 1") and 1\ 'l-'( 1"). kceping Ihe paramelel's 01' \' (1')l!la sal11c as in POI. l. In Ihe \ :!-analysis, Ihe (111 and t7D dala\vere also inclllded. Figure 5 shows lhe lit lo claslie sealtcringdala. \vhich is very good. Ihe \ :!-value is 0.45, which is up lono\\' IhL' smallesl value. Thc calculateJ reacllon and fusioneros s seclions are 17/1 = 1 :20.1 mh and aF = 3;').llIlh respec-livel)'. Thel'c are several inlcresting COlTlll1Cnls10 he malleabolll Ihis cakulalion. First 01' all. lhe large value oblainedfor 1',..= 1.4 fm. which incrcasc lhe range wherc fusion lakesplace lIlaking Ihe ahsorption Ululer the barrier stronger. As\\'l' wil1 laIL'!" 'ihllW. Illi'i !":lL'¡ playes ¡¡Iso a cl"lI(.:ial role in 1..'01"-

reelly lilling lhe fusioll eross seclion "unllel' the Coulomb har-riel'. SCl'lllllÍ. Ihe lusion polelllial 11 '{.{,.) is sharply cul off alU,.. = I ..1 (.-t I + ..I:!) 1/:~ fm si !lce Ihe paramclcr f/ ¡.. has a \'ery

..,'"'".,""

1'01.6

<fe" O'"O."

A (;('\lE7.(,A\1:\(!JO "\\111 1 1):\(;,\\\\

¡-----6'1 • 20llPb

~..,'" .., '¡", '._ ..' .'. ,,

•,

1 . \"

." '00 .,']. r (~" 'J '

0.0

\'.-, ',,'eV \

-~,' •.

"

'.'.

1'-1(;11<1~ \11~1I¡.1Idl'trIhlllill11' 1'1L'I."!ll ',.;llll'rLll~ 1."10'"'CI."-ti(ll1~11.111(\'In Rll\hc!hlld¡ !llr Ihc ,~,tCr1l l' ()~: 'l'h;1I \;lri()u~

,'11l'1)21,'S

90 100E tob( '1eV)

90

, <

,o .W, •Wo.W,1',

>,.O2

" Oó -

1'j(,llld /1 rile \'alllC:-;01 lhe l'e,i1and illla~ill,lry pOlCllll<i1....;111 -::: 1"21 1111~l....rUllcliollS (Ir (!le illcidcll! cller~ic:-;.Tlic cill'k.~,Iumn 1(11I1 i, 111l'fuI! II.-\;lllll', whill' lile n(ls~c""are Ihe \.1111!r111111Hlll'lr\llll 11,. TI1l' nll\\' dl.l\\n fOl"\ is l!ll' prctiic(1\ll\ ll(llll

tl1,'dl'Pl't~hlll rl'lallllll

>":;:; 2 O

F!l;¡ In:-; I"Ik' l";lklll.lh'd <IlldC\PC11IllCI1l;dIU,jPll cnl\"; "l'clioll';1' Illlldlnll' nt lile 1!1,I,kllll'[)('1'~~

11311\ IE/)IIF/

1'"' r. )(J-."' - L"I'

rl'sp,'l"li\'ely. \\'11\'1\',\\'l' 11:1\'\"chos,'n L'o = .'):1..-)ivkV. ¡t"uI.li \kV. ~ = l.,,",,\kV lúr 11.(1':);11111L'" ~ ~l(l ~kV and\', 1" \k\' Illl \ (f- l. I"i.~'-ll['l'7 "lhIW'" l!ll' l'all'uJaliOlh

ill el;l"li,. 'l'allLTill~ lw li\L: dillcl'l'lll L'IlCTt!iL'''. :\" ckarl~"1','J);Llllhl' pr\..t1iLllllll' 111111('l'\I'l'rillll'lllal dala \'l'r~ \\l'll. InFi~ ~. lile ,'IlLT~~ lkll,'Il,!L'IlCC 01 IJ1l' lu"illll L'\l'ildlioll l'un\,-

liPll 1'"gl\L'11.A ..•h,' IOlL', ;1 \L'r~ goud lil has hl'l'll "chin'ell

h\.'lo\\ ;11111;lhO\c lhc ('(IlJlolllh h,lITiL'r L'lI('rg~.

1111111'"I.'clitlll. \\'c \\.iJl Iry lo sllow Ihal ealculation" tlt rT

,1'1 ;lIld al \\ilh Ihe pOlenlial PoI. 7, L'<Inhe applicd lo Oll1l'll.lll.rgiL •..•dillcll'nl rW1I1 l!"l'll. = ~()Ml'V. .f() SIH)\\,'Ihi:.;. \Vl'

11:1\'1.'Illilik' 11\,' nI' Ihe dal" ni R('fs. 1I allil 15 and IIltl(!L' a

\ - -;lIlaly"i ..•01 lhi" dala. Thl' Jlar;lln~ll'rS \;1 uf [he n.'al pO(L'n-11:11 ,. 1"). 11"}'1}( 01' 1'./Ju(I") amlllf 01' 1\", are "11I)\\,,,tI 111\;¡I~ in lhc \ -'-lilling oll'la"lil.:, rcaclioll illld lusiotl dala. 1'/

i" Kl'Pl li\cd al I ...t fm, Figure h shows Ihe clIlpirical \.( ¡"');lIId I¡'(rl. \\11L'r~1I'(/~) :;; JI'P/dE) + II.",(E). ohlaill,'d

111lile \ -'-analy\i" ~valllall'd al lhe slrong ah"lH"pli(Hl r;ldiu"',",1 = 1:2.IIIlI.Thcclllpiricalvalul'\olll'(EJand\'(l:'}

Ilav\.' hl'~ll 11liL'dhy lhe I'qu,llion:-..

.J. Eneq!y c1ependelll'e 01"Ihe polential PoI. 7

1\1\ (E) = "[1 +,11 -1 ",1--'1

"Inall v¡¡lul' ((l.13 fm), Ibis supports our original asslllllplioll

(d;[ ....Ilarp ,:lIl olfof Eq, (X). Ollle!" illlporlillll linding i" 111<11

lhl'l<1\al illl;l¡..!illary pOlenliall{. = ¡{"uu + ll'¡. i" re;l''';(lll;lhl~

lilrgL' allhl' ....lt"Ongabsorptioll radius IH." = 1:!.llml.lh¡¡1 i"().~l) \kV. \\lliLh i...•in agrl'l'lllCll1 wilh wh;¡1 \Va" ohsl'n'l'd in

l\l'l. ~. :\" il i" ....l'i ...n in ¡:ig. 5. lilc e!aslil' LTOS"sel'li(l1l i" \.l'r~

Ilil'l'l~ l.l\hl' lo lllL' dala. (.(lI\"it!L'ring Ihe e.\perimclllal cITor

1tlr lllL' rC;ll.lill[l \.:1"0""sl'L"lion rT¡, = 11:\ 1: 1(J mh. Olll ,:;11(11-

1:lIL'd\;lIu,' I ~().I lllh i\ \\"el! wilhin lhc l'\pe!"illlcnlal r¡¡ll~l'.

l!Jl' l.:;Iil'lll"ll'd l'u",ion no:.;" SL'L'lillll ¡jI = :'i."").7 11Ihi:.;{lnl)

(l,' mh :l\vay lrorn lhe e.\perilll~ntal V<tIUl', Tllereloll'. \\e

l.(lltld "a~ Ih;ll.lhe Sillllllt~lllC()U", lit lo Ihe angulardislrihulioll

pi L'la"lic "c;llll:ring, (l1eI"e.:lclioll ami fllSioll nos" 'il'l'tion ..•al

:-;()\II.'V Iw ..•hl'l'll achic\'l'd .

STEP-BY-STEI' J)ERJVATH)N ()F lilE ~u°l.EAR Fl.1SION RAI)llIS 91

5. SUlI1l11ary ami COJlciusioJls

\\'c h,J\"l' done stl'p-hy-stl'J1 calculatiolls 01 L'lastic. rcaclioll;lIld rU."itlllLTOSS scclions tllllhl' syslcllI Ih() + :!U'''Ph al Iheprojeclill' lah L'lll'rgy 01' X() l\kV. \Ve use ;l1l optical pOll'lltiallIlodcllhal is consistenl will1 lhe s(andanl dirl'CI rcaclioll 111c-ory. ll1al ¡s. lhe optical polcl1tial llscd in thL' ca!Culilliolls 01

fcaclion and fusioll eros" sL'ction" is a potclltial lhat fmm Ihehl'gining lIlu ....llit lhe cJaslic sl,,:atlcring data. The hasiL"poiu! nillil' approach is lo shm\' thal, jI is possihlL' lo dcscrihL' Sillllll-lalll'ollsly L'!aslic. rcaclion and lüsioll dala wilhin lhe frallh>

\\'orh: of dirl'c[ I"caclioll theory. By starting wilh thl' opticajpOlclltial raritIlll'tcrs (POI. 1) nI' Ref. ]-1.rOl' lile same nuclear"iVstelll, and in which the 11lSioll radills parametcr /"/,' \Vas a"i-s~lI11cd10 he 1.0 fm, \Ve ha\'L' dOlle slep-hy-step rnodilieations{Ir this pOlenlial (cal1ed PoI. 2 lo POI. 7) \vhich are Slllllm;[-rill'd in Tahle 1. The original potentiall'ol. 1 nf ReL 1-1.rOIl-lains I\\"{)ahsorptioll pOlelllials \lo/-" (volllllle) ami 11°0 (sur-f;ll"t.o')ami the) are used in slleh a \Vayas lo he resp{llIsihle¡in (usioll alld dircct rcae[iolls respectively, \Ve have roJlowl'dIhe "iallll' prl'seription throllgholll. As \Ve \Vl'nl rmm POI. 1 1(1Pll\. 6. Wl' kep' Ihe paralllctl'r /',,' I¡.\el! at 1.0 fm anl! IÍlundIhat tllL' lil (o l'Iaslie scattcring data \Vas gradllall.y imprtlwdhu[ nOl quile satisfacloraly, hO\vevcr, the lit wilh [he POI. () pa-ramelers resulted lIluch hellL'r. Al the same time, as IllL' l'Ias-lie scalerring lit improved, \Ve Iloliccd Ihal the \ :.!-parallleter~radLlall\' deLTeased. the m¡¡gnitllJe 01' ¡rua graJuallv ill-:Tea'icd :lIld the fusion cross ~"icelion gradually dCLTeas~d. It

:\ B,1L'¡:l. B. Bif\\'C':-.. J. Día!, ami J.L. FnrLTo. ,vl/d I'ltl"' ..\-t11}([()l'-l)-l12

.! J.S. Lílky /'/ ul., f'lty.\. 1./'11.lJ 151 (1(}X5) JX!.

::. L.e. V;I/. J.I\1. Alcxandl.:r. :lmJ GR. S<llchlcr. 1'11.1'.\. HI'V (JI)

( 19X 1) 373: J.R. BirkenJunu ¡¡IlU J.R. lIui/ellga. ATIII/t. HIT.

XI/d. Hm. Sci. ~U (19XJ) 265.1. .\1. lk'derlll:ln ('/111., 1'11\',\, R/'\'. e 25 (Il)X2) XJ7; K.E. Rdllll.

F.L.II. \Volr:-.. A.~l. v:1lllkn Bcrg. ami \V', Ilcnning. 1'11.1',1 Nf'l'.L('//. 55 (11)X5) 2XO

,J. 1l. 1::-.hel1~cll. SI/el. I.JII\ .\. A .\52 ( 19X 1) I~7.

ti. R.G Slo\.;:-.lad ilndEE (;l'Os:-'. Plly.l'. H{'I: ('2.\ (Il)XJ) 2XI

- C.E. r\guiar. Ve. Barhusa. L.F. CUlIo. and R.Donangc!o.I'III',I.1,('If. n 2lH (19XX) 22.

:". S, LandO\vlll' :md S.e. I'iepn. IJ/¡.\'.I". Hel', e 2lJ (Il)X-t) 1352;s.e. Piep\'r .. \!.J. Rho;ldes-Brown. and S. Landowllc. I'ltl'\.Lr!f. )(,2 B \ J()X5) ~3.

\Vas very .surprising lo I¡nd that \vilh POI. 6, the cakul¡¡led fu-SiOlllTO\S SeC!iOll hecame as 10\l,'as 0.19 1110, a vallle which is\'l'r~ far ImmlllL' e,'pL'rilllental \'aluc J() mil, This rael pointedlo [!tL'l'ollclusioll [ha[ [he ahsorpliollllnder Ihe harrier had 10hL' illlTl';¡'L'd. ThnellllL'. \Ve deL'idL'd 10 go haek 10 Ihe originalpo[cl1\ial PoI. J, amI Ihl\V allo\Ved lo frcel)' vary al] [he pa-rallll'lns llr the illlagi nary po[clllials. iIll'luding /' J' whieh hadh•...'el1 kepl ti\L'd al 1.0 frn. The rL'slIltant potclllial. called PoI.7. adlil'\l'll sillllll[allL'OUs good tj¡.., [o e!astil', reaclioll amI fu-sion d;lla, 1\ no[ahle ral'l 01' POI. 7 is Iha! the fllSioll radiusparalllL'ler /'¡.- or 11.,., resul(ed to he 1....•fm. It i.s very inlcr-eslillt' 1(\ !lole lünheL thal POI. 7 has a very small dilTllsnessparallleler /1 F supp0rling our original assllmplioll thal the fu-sion ptllclltial has a sharp cut nlT nalure IEl]. (X)I. Further. \\'l'(llcll applil'd the Ilil1L'lltial Po!. 7 to othe!' cnergies hy l1Iakinga \ '.'-;ulalysis. ",llL're Ihe valtlcs or \ ;•. 11'}\ a and 11/ .. are \'al"-iL'd a\ fUllctiolls nflhL' incidenl ellergy. The resultanllils \Verl'\'L'r~ good ror tllL'dillcrellll'llergics and [he values 01' real anoilllaginary po[cllliaLs al lhe strong ahsorplion radius salislied[he disl'L'rsioll rcbtioll. This rael runfler SlIpports lhe val¡dilyorille plllL'llli;lI ]J11!. 7.

(hit" L'{HlL'!lIsion i\ Ih¡ll. il has hCL'11pro\'cd lhal ir an opti-L'¡l!mollcl Iheory is USL'din a Silllu!t:lIleOlls cakulalioll of nolonly L'!astic. hut n:.1L'lion alld t'usioll cross seclioll dala, it isnol possible lo ¡ix Ihe rusion radills paralllctcr 1',.' lo a valuclIsL'd in (l!lh.'r ca!L'ul;l[ions. hui in\lead. il has 10 he Illodiliedlo <l \':lIlIc nI' aroulld lA rm

!J. (iR S;lldlk'l. :-'1,r\. ~,lgarajan. JI.. Lillcy. :11](..1 !.J. Tholllp~Oll.,.\1111 I 'It l'.\. IN. U 17N ll9X71 J 10

10. 1I E:-.hcn:-.en and s. L<llldO\vl1l'. /VIIII "h.r.\'. A -t92 (Il)Xl)) -t7J.

J J. F. Videhaek ('1 (/1.. I'h.",\. R('\: r 15 (1 tJ77) 95.+.

l:!. T. l 'dagawa. B.T Killl. amJ T T:llllura, /'I" ..\',Ni'I' (' .\2 (JlJX5)124.

]:). T. 1.1¡]dg<lwaand T T:llJ1Ura. Ph\'.1 1«'1'. e 29 (19X-I.) 1922.

1.1 (j R S:llchkr. \IA ~agarajal1. JS. LiJJey'. and!.J Thlll1lr~on.,.\/{II. 01- I'hn. 17X Ilt)S7) '10

J .~l. E. Vlllg;¡ri~. 1.. (;lod¡in:-.. S.(j. Stl'<ld1l1<ln. amI R. LL'doll\. /'111',1NIT. ('J~(Il)X(JJ 1..1')5

1(;. (" \Iahall\, 11. ~g(l. amI GR. SaH.'hlcr. Nucl. P".".~,A ~9(1()Xh) J5~

RI'\: .\/n. Fr\. ..w S2 (1IJI)X) X5-lJl