ST〈TE CYCLOTRON LABORA工ORY 受入 一秘 高工研図書由 PlOMC FUSlON lN ト11≡1AVY lON C◎LL1S1ONS AND Tトl1三 CLUSTER1NG CORRELATl◎N 曳 T◎SH1TAl〈A 1〈AJ1N◎，ト”ROSHl T◎K1ond KEN−1Cト11KU8∪ NOVEM8εR1986 MSUCL−579 Novelnber 1986 Pionic Fusion in Heavy I on Collisions and the Cl us tering CorFelation Tosh i taka Ka J ino National Superconducting Cycl otFon Labo ra to ry )(ichigan State Un ivers i ty East Lansing, NI 48824, V.S .A. and ae De partment of Physics, Tokyo Metropolitan Un iver s i ty Fukazawa 2-1-1, Setagaya-ku, Tokyo 158, Japan an d Hiroshi Toki and Ken-ichi Kubo Department of Physics, Tokyo Metropolitan Un iv er s F ukaz awa 2-1-1, Setagaya-ku, Tokyo 158, Japan t Permanent address i ty ) Be at the 7 4 3 07 subthreshold energy are studied theoretically. Toe cross section Abstract: Pionic fusion reactions He( 4 3He,1T+) Li and He( He, is enhanced largely by the clustering correlation . General trend of targetmass dependence of the A( 3He,1T+)C reaction is also studied. l 1 . Introduction One of the recent topics in pion physics is the pion production in heavy ion collisions at subthreshold energies. Pions more than those expected In the Independent partrcle modell) are observed in heavy-ion collisions2 4) although the beam energy per nucleon is significantly lower than 280 NeV which is the pion production threshold in fFee nucleon-nucleon collision. To explain the phenomena one has to assume a cooperative or a coherent mechanism by which almost all kinetic energy of relative motion of the colliding nuclei is converted into the pion mass . Several recent 5,6) of inclusive pions also have suggested the existence of the measurements collective me9hanism. However, the answer to the question as what kind of collective mechanism really dominates the subthreshold pion production is still unclear both experimentally and theoretically. The first purpose of this aFticle is to propose an interacting cluster model for the pionic fusion. Our picture is that the high energy and high momenturn part of the clustering components in the final nucleus cooperates with the relative motion in the entrance channel to permit coherent pion production. We are therefore interested in the question whether the pionic fusion always shows up the clustering corFelation at the short distance. It is to be noted here that the present model is quite different from the phase-space model with cluster formation which was proposed by Shyam and Knoll f.oF the inclusive pion production. OuF model is based on the micFoscopic cluster model and aims at studying the exclusive pionic fusion reaction . .The pionic fusion process at subthreshold energy provides with a good probe for the study of coherence of pion production since many complicated processes are excluded by identifying the ground state of the final nucleus. There is a small energy left for nuclear excitation. The experimental data 3 He, + are restrlcted to the llght nucleaF systems like3 He( 2 6) 8) Li , 4He(3 He,+ 7) Li9), etc. In this paper we study the two reactions 4 3 +7 4 3 07 He( He,1T ) Li and He( He,1T ) Be' in detail. So far, there have been at least two different theoretical models for studying the exclusive pion production ; one is a semiempirical model of 10) Gelmond and Wilkin which describes the ｦ+_pFoduction in terms of the 3He(p,1T+4 ) He pFocess, and the other is a A-excitation model of Klingenbeck, Dillig and Huber . 1 1 ) Although both models refer the significance of the 4He + d OF 3He + t and Li cluster structure Li nuclei, the harmonic oscillator wave function was practieally used without taking account of the strong clustering correlation of these nuclei. In = = He + t of the final addition , the distoFtion of the entrance channel and the Pauli exclusion principle were neglected. These models have not been extended to heavier systems . The econd purpose of this paper is to extensively estimate the variation of pionic fusion cross sections foF A( 3He,1T+)C with respect to the tanget mass A, which is varied from He to the heavier nuclei C - Ca. Although this estlmate could' be worked out micFoscopically in the same ) Li Feaction, we take here several reasonable 4 3 +7 method as foF the He( He, approximations in order to clearly pick up the target-mass dependence of the cross section. The model and the method for practical calculations are explained briefly in the next section. In sect. 3 we show the calculated results for 4 3 +7 4 3 07 the two reactions He( He,ｦ ) Li and He( He, ) Be, and the Fole of the clustering correlation is discussed. in detail. The target mass dependence of the pionic fusion cross section for A(3He,1T+)C is discussed in sect. 4 and finally, the present study is.summarized in sect 5 3 2 . Nodel 4 3 07 3 +7 Let us consider the pronlc fuslons 4He( He,1T ) Li and He( He,1T ) Be at subthFeshold energy Elab/nucleon = 88.8 MeV.t This energy corresponds to 'ECM = 152.3 l(eV (-- mlT + 12.3 NeV and 17.3 ,(eV for chaFged and neutral pions) . The differential CFOSS Section in the c.m. system Is glven by do E7Mf'vkTr(fv 2 d 8 _Ei 2Vrel 1 (1) k. .k where T)(fv Is the transltlon matrix, Vrel= E + ET Is the relatlve veloclty a m the entrance channel, LD, E E and E7 are the relatlvlstlc energres of a' 1 4 3 7 7 the pion, He, He and Li(oF Be), Ei = Ea + ET, Ef = E +7 a,, and ki and klT denote the relative momenta in the entrance and the exit channels, respectively . The tFansition matrix is defined by ; l TNfv = <klT JfMf H 1 1/2 v >. o (2) As shown In flg. I , Ho Fepresents the pion pFoduction operator, 1 1 /2v> Is 3 4 the initial scatteFing state of the He ,+ He system with channel spin I 12 and its proJection v, and IJfMf> is the final bound stat.e of Li with total spin Jf and its proJection Mf' Expllcit forms of these nuclear states will be defined later . As for the nuclear states , correlated microscopic cluster model wave functionsl2, 13) are adopted 4 < a ;T; I JM > =V 4!3! AaT{[ca( a) c(1/2)(gl) Q iLY(L)(r)]( ) XJL(r)}, tThe corresponding (3) beam energy E lab =266.5 MeV was used in the experiment of Bimbot et al 9) 5 where A T Is the antisyrnrnetrizer of nucleons and ccB(;a) and c (; ) are the T T 4 inteFnal wave functions of He and He (or trlton) respectively, X (r)=X (r)iLY(L)(r) Is the mtercluster relatrve wave functron whlch rs a . JL JL solution of the RGM equation of motionl4) J l,)RGM(+ 'I WD(?)6(; ; )} XJL(rt) drt , = E N RGM (r +, r') X JL(r') ' dFf , where H (r F,' .. ) and NRGM (; ;,) are well known Hamiltonian and normalization kernels. The Hamiltonian kernel consists of two parts , the kinetic energy of nucleons with the c.m. energy subtracted and the interaction energy. As for the nucleon-nucleon effective inteFaction, the modified-Hasegawa-Nagata ( central part) and the Nagata (spin-orbit paFt) forcesl5) aFe used . Coulomb force is taken into account to examine the 4 3 +7 effect of charge sy!nrnetry breaking in the two reactions He( He,1T ) Li and 4He(3He,1T 07) Be. In addltlon to the Feal Interactlon we take account of a local imaginary potential iWD(r) between the two clusters. WD(r) has the same radial dependence as the real paFt of the nuclear direct folding potential and the depth is set equal to WD(O) = -25 MeV for the scattering 7 states and WD(O) O for the bound (Jf 3/2 and 1/2 ) states of Li and 7Be. WD(O) = -25 NeV is a reasonable value obtained in the DWBA analysis 16)although there are observed data only for lo for .the (3He,t) reactions energy region (E CM 40 MeV). The internal wave functions of the constituent clusters c and c aFe assumed to have the highest spatial a T sym:netFles (Os)4[4] and (Os)3[3] with Gaussian radial dependence. Different 6 4 3 ◎sci1ユa七◎r parame七ers B and B are ad◎pted for He and He （◎r 七ri七◎n）， α τ ・ep…七・1y．B・thp…m・七・…ユ・・…七i的肋・…i・七i…1・七・bi1i七y’ c◎nd it i◎ns and 七he wave fuηctions repr◎duce 七he ◎bserved charge radi i and 12，13） bエnding energies very we1ユ． In ◎rder ヒ◎ cユarify 七he r◎1e ◎f 七he cユus七ering c◎rre1a七ion ◎n the 7 pi◎n ic fus ion， we c◎ns ider an◎七her wave func七ion f◎r Li assuming 七he she1ユ ・。・・1・・・・・・・・・・…（・・）4（・・）3［・・1・・・・・・・・・・・・・…ll・1・・…㎝・・…。・・ 一2 7 se七 equaユ 七◎ O．32 fm s◎ 七ha七 七he ◎bserved charge rad ius of Li i s Pepr◎duced by this e◎nfigura七i◎n． N◎te 七hat the cluster wave functi◎n （3） is equivaユent 七◎ 七he she1ユ modeユ wave func七i◎n ωhen 七he c◎rreユa七ion be七ween the 七w◎ c1usters is c◎mple七eユy s凹itched ◎ff in 七he l imi七 ◎f B ◆ B and B “ α 7 τ 37・Thesheユ1㎜◎d・1・…∼・・七i㎝p・mi七・㎝1y七heF・mi㎜◎七ionina harm◎n ic ◎sc iユ1a七◎p weユ1 w i七h◎u七 any ◎七her c一◎rre！a七i◎ns． The in i七iaユ sca七七er ing wave wi七h channeユ spin S ： 1／2 in eq． （2） is c◎nstruc七ed by superp◎sing 七he clus七er wave func七i◎ns （3） as “ ＜ζ …； r11／2 v〉 α 11 ・Σ州／2L・1（LO1／2・lJ・）〈ξζ引洲〉． （5） α τ L，J I・七hi・・・…七h・i・七…1・就・…1・乞i・・囎・・f…七i㎝XJL（・）i・…1・七i・・ ◎f eq． （4） sa七isfying 七he scat七ering b◎undary c◎ndi七i◎n． F◎「七h・pi㎝p「・d・・七i㎝・p…七g・HO…ユy肋…nt「ibuti◎nofsingle nucユe◎n pr◎cess is c◎nsidered． Aユ七h◎ugh 七here are severa1 f◎pms ◎f 七he ¶NN 17） vertex ， we ad◎pted pseudovec七◎r c◎upl ing 七erm which g ives a Gaユiユ6an invaria皿七 f◎r㎜ フ =m J H V41Tf 7 )6(;1T n'{(1i= 7 ) ;n o c nc } n.$(?1T)d;1T' (6) n= 1 Here .o and Tn are spin and isospin operators of nu leons, $(; ) Is the plon n * pion and nucleon . The coupling field, and r and rn are the coordinates of f2= . Slnce we work In the c.m. system of the constant is taken to be O 08 pion and 7Li (or 7Be), the following equation was used in eq. (6); ｦn = (1+=r)V -u nC' co*C )(N 7MN where , and (7) are derivatives with respect to the Felatrve n ITC nC coordinates between pion and nucleon , pion and Li (or Be), and nucleon and 7 7Li (or Be) , Fespectively. Recoil term was neglected in the ealculation. It is to be noted that the p-wave ooupling term of two nucleon process as displayed in fig. 2 is included automatically in the present calculation because both of the initial and final state correlations between nucleons are fully taken into account by the adopted clusteF wave functions . The remaining two-body terms, i.e. the s-wave coupling and A-isobar intermediate coupling terms , aFe neglected here . The created pions at subthreshold energies have a long mean free path XIT 5 fm and, in addi_tion, the nuclear systems to be consideFed here are 12He3He 40ca) . 'It is therefore relatively light (3 He4 +3He, + '+ C,... expected that the pions created in the nuclear interio? are not stFongly affected by the distortions like reabsorption effect. Plane wave solution is assumed for the pion wave function. Inserting eqs. (3), (5) and (6) into eq. (2), we can express the transition matrix TMfv in the following form; 8 = TM fV ( ) A A+ ( Mf v ) - ( Mf V ) A /4 x Y(X) 2Li+1 / L i'Ji (2ｦ)3/2 J x (k ) (Li O 1/2 v l Ji v) (Ji v X Mf v I J M ) ff f Li 1/2 J. 1 Xl} JiJfA { < 1/21 I o (1) l 1 1/2><Lf l lO (o) I ILi>, (8) Lf 112 Jf 0=A:!:i (a) in the ?educed matFix element consists of impulse where the opeFatoF O and recoil terms defined by O( o ) (o) (o) = O Imp . +0Fec O( o ) V41Tf ! /2 l i mp m 1 IJ , ( I u) 7 =7MN ) ( iklT) Fec (o) (pn) ' n= 1 IT O(o) _ V lTf {V21} 4ｦioJo(k Pn) Y o( oo lo I ko) (u¥ M / o (oololAo) N 7 4 (o)' iAJX(k Pn) [Y(X)( n) Q (1)pn] (9) n=1 In eqs . (8) and (9), a puFe LS-coupling scheme was assurned in the cluster model wave functions and the following equations were used for the pion f ield ; <k 1$(;1T)i0> c exp{ I IT'(;1T C)} <klTIHol0>= mlT ･{(1i n 1 )(-i n ) - 7MN NN 9 Vpn} {V21 } exp {-iklT'Pn} * , , ( Io) where the coefficient /2(1) is for positive (neutral) pions, and p =rnn C is the coordinate of the nucleon from the c.m. of 7Li. Since two clusteFs He and He (or He and triton) have different oscillator paFameteFs BG! and B from each other, the Pauli operator A makes it Very difficult to al calculate the reduced matFlx element <Lfl 10 1 1 Li>. Several practlcal 12,18,19) techniques have been exploited in OFdeF to Femove spurious c.m. motion from the :natrix element, and we used those techniques in the present calculat ion . Finally, we give a conHnent on the nucleon-nucleon and intercluster correlation. Since an adopted effective nuclear force has a soft-core as shown in fig. 3, the short range correlation between nucleons is taken into account in the constructed cluster wave function (3). In paFticular the inteFcluster corFelation between 4 He and 3He (or 4He and triton) is maximally taken into conside ation in the present model. In fact, the 7 observed electromagnetic form factors of Li are explained very well even at high tFansfer momentum up to 3 fm 1 by the present wave function . 12) On the other hand, the short range correlation between nucleons in the intra- 4 3 clusters He, He and triton is not considered here explicitly although the so called clustering correlation is taken into account. However , the assumed Gaussian functions reproduce the observed electromagnetic form factors Qf these constituent nuclei at q fm 1 owing to the use of different oscillator parameters. Even if one uses the correlated wave function instead of the Gaussian function, the form factor is not modified strongly at q ( 3 fm though enhanced veFy much at the momenturn Fegion 13) igher than 3 ftn 1 . As the transfer momentum per nucleon m the subthreshold (Elab/nucleon 88 8 MeV) pronlc fuslon Is about I ftn 1 , we lO presume that the present Gaussian functions good approximation . ll f o r 4He an d 3He (or tr iton ) ar e 4 3 07 3 The 4He(3He IT+)7Li and He( He,1T ) Be Reactlons We fiFst study the 4He(3He +)7Ll reactron at Elab/nucleon 88 8 r4eV for which the obseFved data exist. Since the ground state ( 3/2 ) is not separated expeFimentally from the 0.48 MeV first excited level ( 1/2 ) 9) we show the surn of the two contributions in fig. 4. Solid and dotted eurves are the calculated Fesults given by using respectively the cluster model and 7 the shell model wave functions for Li. The same scatteFing wave functions in the entrance channel were used for the two cases. Partial waves up to Li = 7 are necessary to obtain convergence for the angular distFibu.tion, while the total cross section oan be calculated with paFtial waves up to Li = 3' Note that the higher partial waves (Li 8) aFe not necessary because the conservation law of total spin and parity makes these contFibutions very small i･n the pionic fusion leading to the low lying states. This point is in a remarkable contrast with the usual hadronic process which needs many partial waves . The pionic fusion cross section calculated with the cluster wave function (solid curve in the case of WD = 25 rieV in fig. 4) explains the observed data fairly well . Although the oalculated result slightly underestimates the absolute cross section , this lack is _attributed to the two-body mechanism which has not fully been considered in the present calculation, namely the s-wave and A-intermediate 20 ) It is worthwhile pointing o,It the followl!ng two facts state couplings. from fig. 4-(a) . First, the clusteF model wave function provides much larger cross section by an oFder of magnitude than the shell model wave function . Second, the relative magnitude is not changed much by the distortion effect of the imaginary potential in the entreince channel . Bertsch has first pointed out that a simple colliding Fermi-sphere model for heavy ion collisiops gives only 1/10 - 11100 of the observed inclusive cross sectron at sub-threshold energie I ) OuF result wlth the shell model wave 12 function is consistent with the conclusion. The question is, then, what kind of mechanism enhances the pion,ic fusion cross section when we use the cluster model wave function. To answer this question we first deoompose the tFansition matFix TMfv into four different terms as displayed in fig. 5. Eacri term from fig. 5-(a) to (d) consists of fuFther three different graphs when one explicitly writes up all seven nucleon-lines with the antisymmetrization effect being considered. All four contFibutions to the pionic fusion cross section are shown in fig . 4-(b) for the case with the cluster model wave function. Most of the calculated cross section is exhausted by the (a)-type among them, namely pion is created from "3He-t sidet' . Based on thls fact we approximate the tFansition matrix (2) as Th(fv = <k ; A {c c Xf(?)}lHolAaT{cac3HeXi(?)}>, (11-a) IT aT a T .4 He><Xf(;) IVN exp{ l <c /N 1 ;)> T Ih OT lc3 -7- ｦ ･; } Xi( ' (11-b) >j/N I X;(;) exp{i( T OT 3He i 7 1T)'F}d;, where the coupling scheme of angular momentum and associated summation over the quantum numbers are omitted for simplicity. hoT Is the pion production opeFator acting on the inteFnal variables of 3He cluster whlch Is deflned m Appendix A, and .r is the intercluster relative coordinate. The Feduced width amplitude is defined by V X(;) =N 1/2 (r r -) -' X(r)-' dr.-' (12) J RGM ' 13 The adopted approximation is threefold : First , only th impulse term of Ho is taken into account. Second , the normalization kernel is substituted for antis metFizer A in eq ( 1 1-b) as usuall used in the ortho onalit condition model21) ThlFd the mltlal scatterlng wave /N x (r) at Ech( = 152.3 MeV is approximated by the plane wave in eq. ( 11-c). As the result, the tFansition matrix TMfv is factored out into twp Parts; transition matrix element for 3He* + + t and the Fourier transfoFm of the reduced width amplitude of 7Li with transfer momentum = i 4/7 ' Such a factorization is Justified for 7Li because the 7Li nucleus has a remarkable 4 probability of finding the tFiton and He clusters in the gFound state. This is a well known fact observed in the (7Li,a) and 3H(a,Y)7Li reactions. FIFst let us conslder the FouFrer tFansform. Fig. 6-(a) illustrates the He + t reduced width amplitude of Li in momentum space . At ECM 152 3 NeV the bombarding 3He nucleus carries a momentum ki =3'5 ] 1 whereas the created pion takes off a small amount of momentum k = O .3 fm 1 VeFy large momentum q = ki 4/7k = 3 '3 fm 1 is therefoFe transferred into 7Li. Let us presume here that the intercluster Felative motion between 4 triton and He receives the whole transfer momentum. Tae dominance of fig. _ 5-(a) agrees with this idea. The cluster model wave function has a high _ momentum component in the relative motion much more than the shell model wave function, as cleaFly shown in fig. 6-(a). This is the first reason why the pion production oross section is enhanced veFy much with the cluster ' 4 model wave function. We expanded the reduced width amplitude of the He + t cluster model wave function in the complete set of harmonic oscillator functions with B = 0.32 fm 2 It is found in fig '6-(b) that the high 14 ㎜◎㎜en七um co㎜p◎nen七 is ◎rigina七ing fr◎㎜ 七he higher n◎daユ aηd high energy comp◎nent ◎f 七he in七ercユus七er reユaセive m◎セi◎n which is in七r◎duced by ヒhe 4 ． ・h・・tdi・七・・・・・…e1・七i・hb・t鵬・・七砒◎P釦dH・・Thi・・・・…y in七eresting finding in addi七ion 七◎ 七he fae七 七ha七 七he higher noda1 c◎mp◎nen七s c◎heren七1y enhance 七he 1◎w m◎㎜entu㎜ c◎mp◎nen七 ◎f 乞he wave func七i◎n as is− a皿n◎unced very ◎f七en in the s七udy ◎f セhe c1us七er ing phen◎㎜ena． I七 is a1s◎ in七erest ing 七◎ see t；he s i㎜ilar resu1七 in 七he quark cユuster m◎de1 calcu1ation 22） 七ha七 七he shor七 range c◎rre1a七i◎n be七ween nucユeons m◎difies 七he high m◎鵬n七㎜co㎜p㎝en七s◎f the eユec七r◎magnetic f◎m fac七◎rs◎f de此er㎝． The second s◎urce ◎f 七he enhancemen七 ◎f 七he pi◎nic fusi◎n cross sec七i◎n is 七he first facセ◎r iη eq． （11■c）， i．e． むhe ma七r ix eユemen七 ◎f is◎vec七◎r M1 3 3 ＋ m◎de f◎r 七he He “ H ＋ π 七rans i七i◎n． 工七 is 七〇 be n◎ted here 七ha七 七he nuc1e◎ns in 七he 七r i七〇n c1us七er have very 1arge ㎜o㎜en七u㎜ in c◎mparis◎n wi七h 7 七h◎se in 七he she11 mode1 wave func七ion ◎f Li because 七r i七〇n has sma1ユer ． 7 nucユear s1ze than Li． Theref◎re， 七he firs七 七er㎜ in eq． （11＿c） aユso increases 七he pi◎n ic fus i◎n cr◎ss sec七i◎n ωhen 七he cユus七er m◎deユ wave func七ion is used as 七he fina1 st；a七e． Fig． 7 sh◎ws 七he enhance㎜en七． Since we neg1ect 七he Pauユi princ ipユe in eq． （11・c）， the in七ernaユ degrees of 3 4 freed◎m of 七he セw◎ c1u−s七ers H and He are c◎㎜p1ete1y separated fr◎m the i・te㏄1uste・・eユa乞i・e㎜◎七i◎・・工ηt肚・apP・◎・i叫i㎝，肋efi・・t七e㎜i・eq・ 一1 （11−c）represen七s七he Ml ma七rix eユe㎜e帥砒h七ransfer m◎ment㎜k・O．3伽 ． ¶ ・1 H◎wever， in fact， the wh◎ユe 七ransfer ㎜◎mentum q：3．3 f血 is shared by aユ1 parセicipating nuc1e◎ns ◎n acc◎un七 ◎f an interp1ay between 亨he intra・c1uster aj＝ld in七ercユus七er degrees ◎f fr・eed◎m by ㎜eans ◎f 七he Pauユi pr incip1e： The − shared m◎㎜entum per nucユe◎n bec◎㎜es near1y 1．1 fm ． A七 七his m◎㎜en七um 七he Ml f◎r㎜ fac七◎r ◎f 七he 乞r it◎n cユuster i s much 1arger 七han．tha七 expec七ed in 七he she11 m◎de1 wave funcちion． 十 〇 In fig． 8 we c◎mpare 七he ・π produc七i◎n cr◎ss sec七i◎n Ni七h 七ha七 ◎f ¶ a七 七he same kenima七ical condi七i◎n E ： 152．3 MeV ◎f 七he inciden七 channeユ． The CM 15 ratio is very close to I /2 . This is a consequence of the charge symmetry 4 3 +7 4 He,ｦ -3 O) Be. 7 between the two reactions He( He,ｦ ) Li and He( Quantitatively , however, there is a small deviation fFom I /2 in th e calculated ratlon because of the small mass difference between IT + and 16 IT o 4. Tapget.Mass Dependence of Pionic Fusion Cross Section for the He, (3 +¥/ Reaotion Let us assume that the pionic fusion process A(3 He, +)C is dominated by the slmllar meehanrsm to the 4He(3He,It+)7Li reaction whatever target nuclei 4 we may use instead of He. We then adopt the same approxlmatlon as eq ( 1 1-c) in order to see the general trend of taFget-mass dependence of the cross section; do EA+3 ' klT 1 d 81T2Ei Vrel 2 x ex i -> / icA : Xf(?) - ).? d;12 (13) l<cTlhoTlc3 P{ ( i A+3 ｦ } ' f , v He where we assurne the triton + A structuFe of final nucleus C as will be defined soon later. This approximation gives substantially a good result 4 3 +7 within a factor of three for the He( He,1T ) Li(3/2 +1/2 ) reaction; 11.4 nb/sr at OcmO' at Elab/nucleon 88 8 NeV Is predrcted by uslng the cluster model wave function for V Xf( ) in eq. (13). This number 1 1 .4 nb/sr is to be compared with 31.5 nb/sr which was obtained in the RGM calculation in the case of WD= 25 rieV (see fig. 4-(a)). As for target, we consider only 4N- nuclei, namely 4He, 1 C, 2 1O,6 Ne. 20 . . and 40ca, because It Is one of the essential ansatzs for the appFoximation ( 13) that the nucleus A has no spinisospin structure. First, we estimate the cross section, adopting the harmonic oscillatot shell model wave function for the composite nucleus C: The ground state wave function is approximated by the lowest shell model configuration with 17 the highest patial permutation sy!rrnetry [4N3] . This wave function is equlvalent to the cluster-coupling wave functlon tp(A+t) V A( ;), AAt{cA(;A) ct(;t) Xf(;)} (14) where the mternal wave function of the target nucleus cA rs descFibed by the SU(3) shell model wave function with oscillator parameter B which is the same as the trlton wave function ct (see table I ) . The (Os)3[3] configuration is assumed for triton. The intercluster relative wave function xf( ) also is described by the shell model wave function with the 3A B For the 7Li 15N 19F and 43Sc nuclei the oscillator parameter A+3 ' ' ' ' constructed wave functions ( 14) correspond to the unique SU(3) Fepresentation ( LL) as listed in table I . This fact makes it easy to calculate the eigen values of normalization kernel by means of several technlques24) as Fegards the SU(3) propertres of the normallzatlon kernel As the Fesult, the reduced width amplitude VN Xf (?) in eq. ( 13), which was defined by eq. ( 12), ' turns out to be simply the harmonic oscillator wave N functioh multiplied by the square Foot of the eigen value n(AIA) Iabelled by (Au) and the total quanta N of valence nucleons; V N I 12 Xf(?)={ } B)' Xu)(F･3A unl 'A+3 ( 15) where unl Is the harmonlc osclllator wave functlon wlth Fadlal node n and orbltal angular nomentum I For the 23Na nucleus, the cluster coupllng wave 18 6 func tion ( 14) contains sever l SU(3) representations of (80) x (60) = (14-2k k) , where 20Ne is described by the conf iguration (Os)4(Op) 12( Is ,od)4[45 J (80). Extending the definition (15) to this k=3 sys tem , we define the reduced width amplitude as v xf(;) ; { <(80)o,(60)21 l(xv)2>2 N= 14 1 1 12 n(Xu) I Au - 23B), 60 x u (r' (16) 2d ' where <(80)O (60)21 1 (Au)2> is the Feduced Wigner coefficient of the SU(3) N= 1 4 gFoup The elgen values n(Xu) of normallzatlon kernel are calculated m terms of the techniques formulated by HoFiuchi24) and FuJlwara and 25 ) . For the heavier nuclei 27Al 31P 35Cl and 39K, one can define Horiuchi the reduced width amplitude in the same way as eq. ( 16). However, it is not easy to calculate the eigen values of normalization kernel for these systems because there are many possible SU(3) representations resulting from the coupllng (X'u' ) x (60) where (X'u') is the SU(3) representation of the 28 32S and36Ar. We, therefore, assume constituent 4N-nuclei A' =24Mg, Si, the f¥mction V Xf(?) n= -1 12 = n unl * . 3A (r ' A+3 B), ( 17) 0.3, The adopted value of n might be Feasonable N=6 eigen values of no rmal izat ion kernel s are n( 60 ) =0.3462 and foF the A' + triton systems be caus e th e 19 N=9 _ O' 1636 for the triton N=7 _ 0.5049 for the tFiton + 16O system'and n(90) n(70) 40 + Ca system. Harmonic oscillator parameter B is deteFmined so that the observed charge radius of the composite nucleus C is reproduced by the adopted configuration for each nucleus . They are listed in table I . Table 2 shows the pFedicted variation of the oross section for 3 + A( He,1T )C at forward angle (ecm=0') at Elab/nucleon 88 8 MeV whlch was obtained by using the shell model wave functions . We can find at least three interesting features of the pionic fusion cross section in this table. Flrst the cross section for the Feaction leading to 15N decreases veFy 7 much in comparison with that for Li (as 0.92 nb/sr * 0.016 nb/sr) although the beam energy per nucleon is common and the momentum distribution of 7 15 nucleons also is expected to be almost the same: Both Li and N are p-shell nuclei and the observed charge radii indicate almost the same oscillator parameter B=0 .32 fln 2 This decrease becomes even more Femarkable when we compaFe the predicted cross section O .016 nb/sr for 12 3 + 15N with 1 1 4 nb/sF for 4He(3He IT+)7Li which was obtained by using the cluster model wave function instead of the shell model wave 7 function for Li. It is advantageous to work in the c.m, system in order to understand the laFge difference of the cross section between the two reactions. Available beam energies in the c.m. system are i52.3 MeV for 4He(3He ｦ+)7Ll and 213 2 MeV for 12c(3He IT+)15N Consequently the tFansfer momentums are quite different fFom each other, i .e. q=3.3 ftn 1 and 4 .3 fm 1 for the Feactions leading to 7Li and15N, respectively. In the present model, the intercluster relative motion between triton and partner nucleus Feceives the whole tFansfer momentu 1. Therefore, the overlapping of the intercluster relative wave functions of the initial and final states becomes smaller for the 12c(3He,ｦ+)15N reaction than that foF the 4He(3He,1T+)7Li reaction 'due to the gFeater momentum mismatch for the former reaction. This 20 is the' reason why the cross section for the 12.C(3He,1T+)15N reaction decreases very much. We have an experimental data' verifying the above discussion 27 ) for the 12c(3He, +)15N quantitatively . The measured cross section reaction at subthpeshold energy' Elab/nucleon=78.3 eV is 102(+50,-36) pb/sr at el =20'. The present model predicts 70.3 pb/sr at e = O' at the same ab eneFgy, and the angulaF dependence of the differential CFOSS section is very 4 3 +7 weak as for the He( He,1T ) Li reaction. Second, Iet us examine the variation of the cross section for' the Feactions leading to the sd-shell nuclei. Since Elab/nucleon = 88.8 MeV corresponds to Eem=220-240 MeV for these Feactions , the transfer momentum does not differ very much from each other; q = 4.5-4 .9 fm 1 on such conditions, relatively larger cross sections are pFedicted for the Ne( He,ｦ ) Na and 24Mg( He, ) Al reactions. Nucleons in Na and Al move faster than those in the otheF sd-shell nuclei because these nuclei have the oscillator parameters B=0.31 frn 2 and 0.30 frn 2 whlle the otheFS have B=0.27-0.29 fm . This gives rise to an appearance of the faster Felative motion between triton and Ne (or triton and Mg) . Namely, the probability of finding tFiton and 20Ne (OF triton and24Mg) at the high momentum q = 4 .5L4.9 fm 1 is larger than that of finding triton and the partneF nucleus in the other sd-shell nuclei. Hence, the cross section for the above two reactions become laFger than the other reactions . The third remaFkable fact suggested by table 2 is the periodic variation of the cross section reflecting the shell effect. Although the cross section decreases gFadually as the mass of the final nucleus becomes large up to the end of each she 1, it incFeases again at the beginning of the next shell The Increase of the cross sectlon for 40ca(3He IT+)43Sc Is a t ypical example. This is due to the fact that the wave function of 43Sc has 21 high momentum component much more than the light.er nuclear systems because 40 the intercluster relative wave function between triton and Ca has a larger radial node as shown in table I . According to the shell model calculation 28), the intensity of the a_ssumed SU(3) shell nodel state (Os)4(Op)12(Is Od)24(Of)3[4103] (Xu)=(90) is only 18 in the Fealistic wave function of Sc . If only the triton + Ca configuration could be responsible for the pionic fusion reaotion at subthreshold energy as appFoximated in the pFesent study, the predicted cross section 0.56 nb/sr should be multiplied by O. 18. The resulting CFOSS Section 0.101 nb/sr is 3 1 35 39 still larger than those for the reactions leading to P , Cl and K . In the present estimate, we neglected the distortion effect of pion field and the initial channel was described by plane wave. Although these 4 3 12 3 approxlmations may work well foF Iight nuclear systems like He+ He, C+ He and F He, they might not do for heavieF systems. It was reported by Stachel et al. that the 65 -80 of created pions in N+ Ni collisions might encounter the reabsorption effect in the nuclear interior. The present authors, tneFefore, estimated the reabsorption effect for the nuclear systems discussed here with the use of the mean free path of pions derived fFom the pion-nucleus optical potential29) based on the analysis of 30 ) . NeaFly 35 -55 pions are pion distorted waves by Kaps and Bertsch reabsorbed in the nuclear interior. In addition, we assurned the SU(3) shell model wave functions for the composite nuclei. The adopted configuFations for the 15N, sd-shell and 43 Sc nuclei (in table i) exhaust the intensities and･ 18 28) more than31)8085 50 , 28,32) - , Fespectively, in the realistic wave functions of these nuclei. Our prediction for heavier systems, t erefore, could be considered as upper limit of the pionic fusion CFOSS sect ion . Let us discuss the effect of clustering correlation. It was shown in the previous section that the clustering corFelation enhances the pionic 22 4 3 +7 ftlsion cross section fo the He(_ He,ｦ ) Li reactlon We aFe now mterested in the question whether it shows up in any pionic fusion process . The 16O(3 He, +) 19F Feactlon seems to provide a very good example to answer thls 19 question because there are several low exoited states of F which aFe very strongly populated by the triton-transfer reactions. In fact, acconding to 33) 19 , the excited 3/2 (6.09 the theoretical study of F by Sakuda and Nemoto MeV) state which is the band-head of the rotational levels is a remarkable 16 triton + O clustering state and the higher nodal states of the clustering components ( XO) with X 8 aFe admixed by nearly 30 even in the gFound 1/2+state. The SU(3) shell model conflguratlons wlth the lowest allowed quanta aFe (60) and (70) for the gFound 1/2+ and excited 3/2 states, respectively. We here compare the predicted pionic fusion cross sections which aFe obtained by using the cluster model and shell model wave functions of 1 9F fFom each otheF . The clusteF model wave functions calculated by Sakuda and Nemoto were used in the present calculation . As for the shell model wave function of the excited 3/2 state, the oscillator parameteF 8 was determined so as to give the same charge radius as that given by the cluster model wave functiont since the calculated mean intercluster distance 5 .04 frn of the excited state is larger than 3.95 frn of the ground state, the value of B (fm 2) for the excited state is smaller than that for the 'ground state, as shown in table I . FOF the gFound state of 19F, both of the cluster model and shell model wave functions give the same charge radius as the observed one. t In order to avoid the spurious c.m. component, the charge radius of the excited 3/2 state was calculated in the following equation, corresponding to the clusteF-coupling wave function ( 14) ; 23 V< r2c > = /19 2t 9 82 1 uo 32 8 2 <r > + {<r t >=3249<rhel> 2 where V<F > and /<rl60> are the charge radii of triton and 1 60 , and V<r2 > rel 16 See ref. 33 for is the r ,m.s. intercluster distance between triton and O details. Note that the charge radii of the ground states of al l nuclei in table I were calculated exactly without approximation. 24 In figs. 9-(a) and (b) we compare the trlton + 160 cluster model and harmonic oscillator shell model wave functions for the ground 1/2+ and exci･ted 3/2 states, separately, as m flg 6 (a) As Is well known the clustering component predominates the low q reglon for both of I /2+ and 3/2 states whlle at 1 5 fm 1 ( ･q ( 3.5 ftn I for the gFound 1/2+ state and 2 O frn I ( of ( 4.0 fm 1 for the 3/2 excited state the shell model component domlnates. However, at q=4.51 fm I and 4 53 fm I whlch are the transfer 3 +) reactions leading to the ground 1/2+ and momentums for the 16 O( He, excited 3/2 states, respectively, the clustering component prevails again. Therefore, the pionic fusion cross section may be enhanced as shown in table 2 by the use of the cluster model wave function. + 15 Quite recently, the pion production reaction12C( 3He, ) N has been observed34) at the Indiana University and It was found that the unFesolved excited states of the 15N nucleus in the 6 .5-10.5 MeV excitation energy range are strongly populated by this reaction. From the theoretical viewpo int 31) 12 , there are many triton + C clustering states in the energy range 5-15 rteV . Experirnentally, these excited states are known to have large reduced tFiton-width as obseFved by the tFiton-transfer reactions. High resolution measurement which can separate each excited level is highly requ ired . ' The clustering component prevails over harmonic oscillator shell model at very high momentum region . This should be the case because the short distance correlation between two clusters is fully taken into account in the cluster model wave function. Thus the pionic fusion cross section should be enhanced in most cases, reflecting the clusteFing correlation in the final states . 25 5 . Summary and Conclusion 4 3 +7 We have calculated-the pionic fusion cposs.section for He( He,1T ) Li 4 3 07 and He( He,ｦ ) Be at subthreshold eneFgy Elab /nucleon = 88.8 MeV in the cluster model. In this model we exactly take account of the strong clustering correlation of the A=7 nuclear systems, the distortion of the entrance channel and the Pauli exclusion principle between nucleons. We have obtained the result that the pionic fusion cross section is enhanced very much by the clustering correlation. We have also found that it originates from the high momentum component of the intercluster relative 4 wave function between triton and He. Assuming that the pion pFoduction is dominated by the high momentum component of the intercluster relative motion, we extended the analysis of pionic fusion to heavier systems and estimated the variation of the pion production cross section . Several kinds of shell effects were clearly 16 +¥1 * predicted. It was found that the cross section for the O(3He,ｦ / 9* reaction also might be enhanced by the clustering correlation. A very simple pion production operator was assumed in the pFesent calculation. There are, however , pFobably important s-wave coupling and A20 ) excitation teFms , which we have neglected here. A quantitatively satisfactory description of the pionic fusion will be fulfilled by considering these terms with the strong clustering correlation being taken into account as in the present study. Although a qualitative aspect- is stressed in this paper , we have learned that the short distance correlation between heavy ions might be observed systematically in the pionic fusion reactions by vaFying the combination of projectile and target. The nuclear cluster model has ever enjoyed a success for relatively low momentum phenomena. For example, the microscopic clusteF model has succ eded in predicting the radiative f¥Ision CFOSS Sections for 4 3 7 4 3 7 He( He,Y) Be and He( H,Y) Li at low energies which affect seveFal 26 astrophysical problens very much. 35-37 ) In the present paper, it wa suggested that the cluster model provides a powerful tool for the study of pionic fusion reactions, too, although these are the dynamical phenomena accompanying with high transfer momentum. The authors wish to thank Profs. H. Ikegami , I. Tanihata, t(. Ishihara, K. Yazaki, K. Shimizu and A. Arima for many valuable discussions . One of the authors (T.K.) also is grateful to Profs. G. Bertsch, A. Brown, W. Benenson and S. Austin for useful oonversations and comuents . Numerical calculations were performed by using VAX-11 at the National SupeFconducting Cyclotron Laboratory, )(ichigan State University and M680H at the Computer Center, UniveFsity of Tokyo and financially supported by Institute for Nuclear Study, University of Tokyo and ReseaFch Center for NucleaF Physics , Osaka UniveFsity . This woFk is also financially supported by the Grant-in-Aid for Scientific Research of Ministry of Education, Science and Culture, Japan . 27 Appendix A The following eq ua t i on for the p ron produc tion operator Ho was used in eq. (11); 8n'{(1+ u) )(_i 7MN mlT n = 1 ) M {hol+S11' r}exp{-i_ k4 pn}exp{-i ' n} Tn . }+{h + . }exp{i-3 oa la r '; ( A- I ) hOT 3exp{-''{A(-i } n IT)+B 1 - Y] nT}Tnl nT } (A-2) {* }= hlT n=1 1-3B8n ' hoa 7 {- }= 4exp{-i hla '{A(-i IT)+B . } }TY{ n } (A-3) -7!S{ on where A=1+=r B=co LD , .nT(.na) is the coordinate of nucleon fFom the c.m. 7)( N ' NN 3 4 of He ( He) and 'V, nl ( na) is the derivative with Fespect to *nT( *;na) ･ Note that one can always divide one-body operator into two parts w i thou t approximation acting respectively on the internal variables of the constituent clusters (like (h h ) and (h h ) oT' IT oa' Ia relative coordinate *F. 28 ) and the inteFcl uster Referenceき 1） G．F． Ber七sch， Phys． Rev．」≡…ユ亘 （1977） 713． 2） W・Benens㎝e七aユ．，Phys．Rev．L嚇．坦（1979）683・ 3） S・Naga㎜iyae七aユ．，Phys．Rev．Le沈．坦（1982）1780． 4） T． J◎hanss◎n e七 a1．， Phys． Rev． Let七． 48 （1982） 732． 5） M． Prakash， P． Braun−Munz1nger and J． S七ache1， Phys． Rev． C33 （1986） 937． 6） J． Sセache1 e七 aユ．， Phys． Rev． C33 （1986） 1420． 7） R． Shyam and J． Kn◎1ユ， Phys． Le七t． 136B （1984） 221； Nucユ． Phys． A426 （1984） 606． 8） Y・LeBomece七a1．，Phys．Rev．Le七七．虹（1981）1870． 9） L． Bi㎜b◎t e七 a1．， Phys． Le七七． 114B （1982） 311． 10） J． Germ◎nd and C． W1ユk1n， Phys． Le七七． 106B （1981） 449． 11） K．K1i㎎enbeck，M．Dil1igandM．Huber，Phys．Rev．Le眈．坦（1981） 1654； M． Huber， K． Kユ1ngenbeck aハd R． Hupke， Nuc1． Phys． A396 （1983） 191c． 12） T． KaJ1n◎， T． Matsuse and A． Ar：Lma， Nuc1． Phys． A413 （1984） 323． 13） T．KaJ1n◎，T．Ma七suse and A．Ar1ma，Nuc1．Phys． A414（1984） 185． 14） J・A・冊eele・，Phys．Rev．里（1937）1083，l107・ 15） SupP1・P・og・The◎・・Phys・幽（1980）P．215・ 16） G． Bruge e七aユ．，Nucユ．Phys． A129（1969） 417． 17） P．Hecki㎎，R．Br◎c㎞am and W．Weise，肋ys．Le眈．72B（1978）432． 18） A・T◎hsaki−Su・uki，P・・9．肺e◎・．Phys．型（1978）1261・ 19） M． Ka㎜i1−lura， SupP1． Pr◎9． Theor． Phys． N◎． 62 （1977） 236． 20） E・ Ose七， H． T◎k i and W． Weise， Phys． Rep．旦；…」（1983） 273・ 21） S・S・i七・・P・・g・Th・…Phy・・卑（1968）893；ibid・旦（1969）705・ 22） M． Oka and K． Yazaki， Nuc1． Phys． A402 （1983） 477． 23） M． Harvey， Advances 1n Nuc1ear Phys1cs Vo1． 1 （ed． by H． Baranger and E． V◎9七， 1968） 67． 24） H． H◎r1uch1， SupP1． Pr◎9． Theor． Phys． N◎． 62 （1977） 90． 25） Y・F・ji胞・♀・・dH・H・汕・hi・P・…Th・…Ph…坦（19gO）895・ 29 26 ) C. Jager, H. Vries and C. Vries, Atomic Data and Nucl. Data Tables 14 _ (i974) 479. 27 ) L. Bimbot et al., Phys. Rev. C30 (1984) 739. 28 ) 29 ) J.B. NcGrory, Phys. Lett. 47B (1973) 481. J.A. Carr, H. McManus and K. Stricker-Bauer, Phys. Rev. C25 ( 1982) 952. 30 ) R. Kaps , preprint; G. Bertsch, private corrmunication. 31 ) Y. Goto and H. Horiuchi, PFog. Theor. Phys. 62 (1979) 662. 32 ) 33 ) B.A. Brown, private communication. T. SakLrda and F. Nemoto, Prog. Theor. Phys. 62 (1979) 1274 and 1606. 34 ) W. Schott et al., Phys. Rev. C34 (1986) 1406. 35 ) 36 ) T. KaJino and A. Arima, Phys. Rev. Lett. 52 (1984) 739. T. Kajino, Nucl . Phys. A460 ( 1986) 559. 37 ) T. KaJino, H. Toki and S. Austin, preprint of Michigan State University ( 1986) , submitted to Astrophys. J. 30 丁自b1e 1． A‘＝lopted ha㎜■＝1nic◎sei1ユa七〇rε㎏1ユmode1旧ve fしInction． d1理ge ◎sc工］ユa七◎r intぼc1uster Su（3）鞭entation◎f S口（3）nep㈱nta七i◎n◎f 紬。。† 因㎜瞼幽・。。戯i㎝雌。。戚・胸ω由讐雌如・㎏ω脳 鬼。（航）・（〆）冷 （〃） （畑） J¶ 7 − Li3122．蜘 0．契 1p 0，32 lp 0，32 lp 0，29 3s 0，26 3P ㌔3／2＋2．｝ 0，31 射 27 ＋ 虹5／23．06 0，30 困 31 ＋ P1／23．20 0，29 3s 35 ＋ ＋ C13／23．30 0，28 困 39 ＋ K3／23．蜘 0，27 射 0．26、 改 1／2 （O．478MさV） 1㍉1／2・2，65 1キ1／2＋2．90 3／2・（6．09剛ま 43 一 十 Sc7／23．50 4距（oo） 4He（00） 12 C（04） 16 0（00） 16 0（00）’ 牝（80） 24 ㎏（釧） 28． S1（120） 弩（㎎） 36吐く08） 蜘Ca．（00） （30） （30） （01） （60） （70） ㎜並記 ．○ ” ，1 ” 〈90） 十 竿削射蛾・紬ii（爬f・26）・ ’十 鯛帽ユueS． ±池紬お。帥・鉦七伽。16・d晦㎏、㈱。・1キ．（・。。㈱舳。汕．） 讐 3 AdoPted ぼiton wave fしrcti◎n （Os） 工3］ is a Su（3） s◎≡1ユar （00）． A(3He,1T+)C reaction pred icted Table 2 . Pionic fusion cross section for the m eq. ( 13) at ecm=0' at E lab /nucleon='88 . 8 MeV 3 A( He, IT +)C Shel l JIT( cgr . ) 9i d t (nblsr) JIT(C ) do (O' ) d ( nb/sr ) model wave function foF C c 4He 12c 160 - 20Ne 24 Mg 28si 32s 36 Ar 4 O ca Clus te F 7Li 3/2 +1/2 1 5N I 12 1 9F I /2+ 23Na 3 /2+ 0.92 0.016 o . 064 3/2 (6.09 NeV) 0.11 3/2 0.41 o . 62 27Al " 5/2+ 3 1 P I /2 0.28 o . 030 35cl 3/2+ 39K 3/2+ ' 43Sc 7/2 o . 043 0.016 0.56 model wave function for c A c 4He 7Li 160 19F 3/2 1 /2+ + I 12 11 .4 2.3 (6.09 MeV) Figure captions Fig. 1 7L i ( Jf ) 4 3 He * Schematic Fepresentation of the mechanism of the He + + reaction H represents the pion production operator and C + IT ' O denotes the clustering coFrelation containing the Pauli exclusion pr incipl e . P-wave coupling teFm among two nucleon processes of the pion production operator, which is included automatically in the present model caloulation in addition to the single nucleon process (see text) . Fig . 2 Fig . 3 Effective nucleon - nucleon interaction Nagata force) adopted in the pFesent calculation . ( the 3E , modified - Hasegawa - 1E, 3O and 10 represent the triplet even, singlet even, triplet odd and singlet odd channels, respectively. 4 3 Fig.4 (a): Angular distribution of the He + He * 7Li(3/2 + 1/2 ) + IT+ reactron cross sectron at the Incident energy Elab/nucleon 88 8 MeV. Solid and dashed curves are the calculated results obtained by using 7 respectively the cluster model and the shell model wave functions of Li . WD denotes the adopted 'strength of the imaginary part of the optical potential in the entrance channel (see text). Black dots with erFor bars are the observed data fFom ref.9. (b) : Decomposition of the calculated differential cross section in the case of WD = 25 MeV with the cluster wave function (solid curve in Fig.4-(a)) into four different contFibutions displayed in Fig.5-(a) to 5-(d). FouF different terms of the transltron matrix for Fig . 5 7 + Li reaction. An oval box denotes the antisy!runetrizer of nucleons. IT 4 3 Fig.6 7 the 4He + 3He * (a): Reduced width amplitude of the He + H component of Li(3/2 ) in momentum space. Solid and dashed curves are cases foF the cluster model and the shell model wave functions, respectively. q = 3.3 fin 1 is the tFansfer momentum of the 4He + 3He * 7Li + + reaction at Elab/nucleon = 88.8 h(eV. (b): Decomposition of the reduced width amplitude of the cluster model wave function (solid curve in Fig.6-(a)) into harmonic oscillator functions u* ( ) with 8 - O 2 fm 2 Solid and dotted curves *,* q 7 - '3 ' denote Fespectively the positive and negative amplitudes . Note that the Op - amplitude does not vanish bec use the different size parameters Ba = 0.574 = 0.460 fm 2 aFe used for He and 3H nuclel respectrvely Ratio of the squared transition matFix appearing in eq . (11-c) of Fig . 7 3H + 3 the He + IT + process for the cluster model wave function to that for the shell mo del wave function as a function of transfer momenturn. Fig . 8 Compar ison of the calculated angular distribution of - 7Li(3/2 7Be(3/2 + 112 ) + + 1/2 ) + ' 4 t reaction cross section with that of He + at Elab /nucleon optical potential has the stFength WD 4 3 the He + He He * 88.8 )teV . The adopted WD of the - 25 MeV. Fig.9 (a) : Reduced width amplitude of the 16 O + H3 component of the ground state ( 1/2 ) of19F in momentum space. Solld and dashed curves are the same as in Fig.6-(a) ; cluster model and shell model wave functions, respectively . q = 4.5 fm 1 is the transfer momentum of the 16O +3H * 19F + + reactron at E /nncleon 88 8 MeV b ' The same as in Fig.9-(a) for lab = ' ' ( ) ' the 3/2 (6.09 )ieV) excited state of19F. MSu-86-383 FIGURE l 7Li ( Jf ) + 7E / Ho / / / / / / / / / / ((A), -) // _K7r) / c 4He 5 He FIGURE 2 //. / / ll / Ho / / / / MSU-86 - 384 FIGURE 3 1000 3E b . IE , >o ' . 500 t t , ¥ t Lzz t ' o ' , ,b I ,__ 1' l' 1" lp 2 ,. d. s rNN (fm) -250 , .> o 10 t. , 50 ' 30 100 f¥ 'b 'I ' ¥ ' 'I' 1l 1lb ' ,l ･Ib,b, .bl･1-_,1lbl･_.1,-1-1･-1-ll-1-･.-1･-. Lz J 1 -50 2 s rNN (fm) ' LL lur¥L d07d (nblsr) a ELab/N = 88.8 MeV 1 02 4He (SHe, 7c+ ¥/ 71Li312 +v2 101 1-1--1"I,'-d'Q1-..11･,I 11･a･*-_. 1 lbll･l O_ d,1_'_dP .,I , 1 ,,tb ¥ ¥ ¥¥ ¥¥ ¥¥ ¥¥ 1lplPd d 11--'--""I- ¥ ' bl'dPl"I / d ¥ /. 'b'_dl'd' 1 11- 11- Ib ' I tb, 'l' . 1 0 1 1 2 O' CM (deg) 1 8 Oo MSU＿86＿386 FIGuRE 4 b d0；7dΩ‘（nblSr） 2 10 ’WD＝一25 1d1 （O） 11 （b） 10 Oo ．． 0 120◎ ◎。。（d・g） 0 180 MSU - 86-387 FIGURE 5 (a) (b) 7C /1 / / / / + 3H / ll / i 1 3 He 3 He (c) 4 He 7c+ (d) + 4He 7c ¥ ¥ ¥ ¥ ¥ ¥ 4 He 7C 4 He / / + - J q ' f : XJL(q) , MSU-86-388 JL He + 3H =7Li 312 q = 5.5 fm 1 1 / / / ,, / / / / / ¥ ¥ ¥ 1 / ¥ i 101 ¥ l : ¥ l ¥ ¥ ¥ ¥ -2 10 ¥ ¥ ¥ ¥ 103 q (fm 1) 4 MSU-86- 389 FIGURE 6 b 4He + 5H = 7L13/2 q' N XJL(q) lp 1 /,-¥' l 1 0 1 l l l ,, f I V l l l -2 10 l l l , / ¥ I , l ¥ l ¥ ¥¥ ll ¥ ¥ ll ¥ l l t 10 o 1 2 q (fm 1 ) 7 p ¥ / 4 FIGURE7 一・φ。1り。τ1φ・。べ、s、、、肌 1・φ。1h㎡1細、・1；。、岨肌 一 ◎ ⊥ 一 ◎ ’ ．o ヨ∼ ⊥ ） ぴ ζ ω ⊂ I oo ① I ｝ ⑩ O MSU-86-391 FIGURE 8 d /d (nb/sr) E Lab /N = 88.8 MeV 1 02 4 He ( 3He x+ )7LI 312 + v2 101 1 4 He ( 101 Cf 120' CM (deg) 1800 MSU-86-375 FIGURE 9(a) I q . Jj 160 + 3H = 19Fv2 XJL(q) I q = 4.5 fm I I -'b I t It / I l l f f I l l0 I fb / I ll Il / ¥ // ¥ ¥ ¥ / l ll ll ¥ ¥ ¥ tl ll ll ¥ ll ¥ Il l ll I ¥ ¥ ¥ ¥ ¥ ¥ I0 2 ' ¥ L L l0 3 o 2 q (fm 1) MSU-86-374 FIGURE 9 b l q . Jj 160 + 3H = 19F3/a XJL(q) l q = 4.5 fm l l l l / 1 l X I / I I l /*¥ / '¥ l 1 ll tl ll / / ll l i l l I l0 l l / b¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ l0 2 ¥ l ¥ l0 3 o 2 q (fm 1)

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