Geometric interpretation of the effect of the quadrupole force in the collisions of deformed nuclei

PHYSICAL REVIEW CVOLUME 45, NUMBER 3MARCH 1992 Geome ic in e p e a ion o he e ec o he quad upole o ce in he collisions o de o med nuclei M. V. And es and J.Gomez-Camacho Depa amen o de FAMN, Uni e sidad de Se illa, Facul ad de Fs'sicas, Apa ado 1065, 41080Se illa, Spain M. A. Naga ajan Da esbu y Labo a o y, Wa ing on, WA4 4AD, Uni ed Kingdom (Recei ed 15 Oc obe 1991) The e ec o aquad upole o ce on ase o degene a e s a es o a o a ional band wi h a bi a y spin p ojec ion along he symme y axis Kis s udied. Analy ic exp essions o he eigen alues and eigen ec- o s a e ob ained in e ms o ase o o hogonal polynomials. This is applied o he collision o asphe i- cal nucleus wi h ade o med one in which he coupling o agi en se o o a ional s a es is allowed, ig- no ing exci a ion ene gies. The elas ic S-ma ix, ansi ion ampli udes, and he usion c oss sec ions a e ob ained as aweigh ed a e age o he magni udes co esponding o ase o de ini e o ien a ions o he axis o he de o med nucleus wi h espec o he ela i e coo dina e. Tha weigh ed a e age co esponds o app oxima e he ex eme sudden esul , consis ing o an in eg al o e all he o ien a ions, by agen- e alized Gaussian quad a u e. PACS numbe (s): 24.10.Eq, 24.70.+s, 24.50.+g, 03.65.Nk I. INTRODUCTION The collec i e exci a ions o nuclei play a e y impo - an ole in he eac ion mechanisms. The desc ip ion o hese e ec s equi es he explici inclusion o he exci a- ion by means o coupled-channels calcula ions. Fo he case o o a ional nuclei, ap ope coupled-channels cal- cula ion would equi e he inclusion o many s a es o he o a ional band, making he calcula ion complica ed and ime consuming, and complica ing he in e p e a ion o he esul s. On he o he hand, when he o a ional mo ion can be conside ed slow e sus he ela i e mo ion (sudden app oxima ion) he sca e ing ampli udes can be calcula ed as an in eg al ex ended o all he o ien a ions o he o o o he sca e ing ampli udes calcula ed as i he o ien a ion o he de o med nucleus was ozen du - ing he eac ion, weigh ed wi h he p obabili y densi y ha he g ound s a e has ha o ien a ion [l]. Also, i is known ha when he e ec o he coupling o a es ic ed se o exci ed s a es is conside ed, he exci a ion ene gy is igno ed, he cen i ugal ba ie is sui ably app oxima ed and he coupling o m ac o s ha e he same shape, he coupled-channels sys em can be decoupled, and he sca e ing ampli udes can be exp essed as acombina ion o he ones co esponding o ase o uncoupled eigen- channels [2— 5]. Naga ajan, Balan ekin, and Takigawa [6) con ibu ed o b idge he gap be ween he coupled-channels calcula- ion and he sudden esul demons a ing ha , o a @=0 o a ional band, he coupled-channels e ec co e- sponding o include he o a ional s a es om I=O o I=2%— 2igno ing hei exci a ion ene gies, was equi alen o do aweigh ed a e age o he ampli udes co esponding o No ien a ions. These o ien a ions a e cha ac e ized by he angle 0be ween he symme y axis o he de o med nucleus and he ela i e mo ion, ha akes he alues such ha P2&(cos0) =0. Mo eo e , ha weigh ed a e age is p ecisely he combina ion ob ained when he in eg al co esponding o he sudden app oxi- ma ion is app oxima ed by an N-poin Gauss-Legend e quad a u e. The aim o his wo k is o sea ch o asimila esul ha could be applied o a o a ional band o a bi a y E. This is impo an i one is in e es ed in s udying he inhuence o pola iza ion on he eac ion mechanisms, be- cause, i he nucleus is o be pola ized, i s g ound s a e needs o ha e spin di e en om ze o. This pape is o ganized as ollows. In Sec. II we e alua e he ma ix elemen o aquad upole in e ac ion be ween o a ional s a es using he idal spin basis. In Sec. III we map he o a ional s a es in o anew se o s a es cha ac e ized by ase o o hogonal polynomials. In Sec. IV we pe o m an analy ical diagonaliza ion in he new basis. In Sec. Vwe discuss he meaning o eigen- alues and eigen ec o s. In Sec. VI we apply his ea - men o he calcula ion o sca e ing ampli udes. In Sec. VII he ela ion o he geome ical limi is ob ained. Sec- ion VIII is o summa y and conclusions. II. COUPLING POTENTIALS Le us conside he in e ac ion be ween a sphe ical nu- cleus and an axially de o med one. The in e ac ion can be w i en as V( , ')=Vo( )+ V2( )P2( . g), whe e is he ela i e coo dina e and g' s ands o he di ec ion o he symme y axis o he de o med nucleus. In his exp ession we ha e neglec ed spin-o bi e ms and hexadecapole and highe -o de de o ma ion. The ma ix elemen s o he quad upole in e ac ion be ween o a ion- al s a es in he usual coupled-channels basis is gi en by 45 1339 1992 The Ame ican Physical Socie y 1340 M. V. ANDRES, J.GOMEZ-CAMACHO, AND M. A. NAGARAJAN (IKLJI V2( )P~( g)II'K'L'J ) =V2( )5x ~,W(II'LL';2J)LI x&I.o2olI. 'o) &IK20II'K), (2) whe e Iis he in e nal angula momen um o he de- o med nucleus and Kis he p ojec ion along he symme- y axis, ha cha ac e izes he o a ional band. Lis he o bi al angula momen um, and Jis he o al angula momen um. I is known ha his in e ac ion can be pa - ially diagonalized in he idal spin basis [7] (IKMJI V,( )P,( g)II'K'M'J ) =V~( )5M ~5+ KI/I'(IM20II'M )(IK20II'K'), whe e Mis he p ojec ion o he spin Ialong he ela i e coo dina e ( idal spin). No e ha h oughou his pa- pe , Mdeno es he idal spin while Kis he p ojec ion o he spin along he symme y axis o he o o . Bo h mag- ni udes a e conse ed by he coupling po en ial. No e also ha he alue o Jdoes no a Fec he alues o he coupling po en ial in he idal spin basis. So, in wha ol- lows, we will d op he index Jin he cha ac e iza ion o he s a es. The coupling po en ials ha e acommon o m ac o V2( ) and di e en s eng h ac o s. i he exci a ion en- e gy is igno ed o agi en numbe o s a es, he coupling ma ix could be diagonalized nume ically, and ase o eigenchannels be ob ained, om which all he ele an e- ac ion magni udes can be calcula ed [2— S]. Howe e , he eigen alues and eigens a es o he nume ical diagonaliza- ion do no ha e any clea geome ical meaning. The main di icul y in ex ending he analy ic diagonali- za ion [6] o KPO bands is ha each s a e is coupled o ou o he s a es, while o aK=0 band i is only cou- pled o wo o he . Tha makes i di icul o make asui - able unca ion in he s a es o he band included in he calcula ion, and ye ob ain analy ic exp essions o eigen- alues and eigen ec o s. An excep ion o his a e he K=—, 'bands, because hey can be conside ed as K=O bands o which aj=—, 'pa icle is coupled. Fo hese ana- ly ic exp essions o eigen alues and eigen ec o s can be ound. III. MAPPING To simpli y he complica ed coupling s uc u e o a EWO band we wi11 in oduce ase o s a es ob ained ac - ing wi h he P2( .g) ope a o o e he g ound s a e o he band. I1&=N, (P2lo& — lo &&0IP, lo&), n— 1 In &=N„P,ln — 1& —gli&&ilP, ln — » i=0 (4) These s a es do no ha e, in gene al, good angula momen um I. Explici exp essions o o e lap (nlIKM ) in he case K=— ', a e gi en in Table I(M =— , ') and Table II (M=—, '). The s a es In )a e he combina ions o s a es IIKM) ha a e mos ele an o he quad upole cou- pling. %e expec ha he coupled-channels e Fec s due o he o a ional s a es IIKM) om I=K o I=K+2N would be e y simila o conside he s a es In) om n=0 o n=¹In bo h cases, e ms o o de %+1in J'2 a e igno ed: No e ha coupling s uc u e is simpli ied, because each o he s a es In )is only coupled o In +1) and ln — 1). This way o gene a ing he ln )s a es is known as he Lanczos me hod [8], which allows one o ob ain some eigen alues and eigens a es qui e accu a ely om only pa o he ull ma ix. Al hough he s a es In)ha e acomplica ed expansion in e ms o s a es o good angula momen um, hey ha e a e y appealing exp ession when exp essed as asupe po- si ion o s a es co esponding o de ini e o ien a ions o he o o . The s a e IO) can be w i en as [9] = (2K+1)/16m' Jdn[2P~~(n)InK ) +(— i)2~gF, (n)lnK)], (S) whe e x=cosP and N(x) is ano maliza ion ac o . The s a es lxKM) ha e he same pa i y as he s a es o he o a ional band, and sa is y whe e InK )is as a e ec o o he de o med nucleus co esponding o an o ien a ion gi en by he Eule angles n=(a,p, y) o he o o wi h espec o an ex e nal coo - dina e sys em wi h he zaxis along he ela i e coo di- na e, and ap ojec ion EC o he angula momen um along he symme y axis. InK )is he ime e e sed s a e. We can de ine he s a e lxEM )as lxKM) =N(x) Jdady[2)xM(n)InK) +(— 1)' 2) (n)lnK )], (6) (nII ) TABLE I. Mapping coe icien o K=—, ',M=—. TABLE II. Mapping coe icien s o K=— ', ,M=—, '. &o ((I &2 4465 7755 (o '-', GEOMETRIC INTERPRETATION OF THE EFFECT OF THE. ..134i TAHI.EIII. %eigh unc ions and o hogonal polynomials. I( =—M=— 27 2K=— ',M= — ' m(x) (1+3x )/4 1 Q—, , '2 (15x'— 7) (133x — 126x +17) (3— 3x )/4 I Q—' ,(Sx'— 1) Q64 (21x — 14x'+ 1) I.-.lxKM &=MI.KM &, (Ig') lxKM &=KlxKM &, P2( .g)lxKM &=P2(x)lxKM&, ' (x'K'M'lxKM &=5x x5M ~5(x' x) .— (7) (8) (10) polynomials a e he e en Legend e polynomials. The an- aly ic diagonaliza ion and he geome ic in e p e a ion ob ained in [6] o he K=0 band elied on he o ho- gonali y p ope ies o he Legend e polynomials. Thus, we a e in he si ua ion o applying simila echniques o ou case. They can be in e p e ed as s a es o he o o in which he axis o he o o and he ela i e coo dina e o m a ixed angle p, bu hey a e a e aged o e all he Eule an- gles aand yso ha Mand Ea e good quan um num- be s. In e ms o hese s a es, he g ound-s a e wa e unc ion is jus IV. ANALYTIC DIAGONALIZATION We will demons a e ha he (unno malized) s a e I(()&= gQ„( p)l n=0 (15) lO&=lKKM&= dx w(x)' lxKM&, whe e w(x) =[[dhM(p)] +[d — xM(p)] }. 4(12) is an eigens a e o he ope a o P2( g) in he subspace gene a ed by [ln &,n=O,N}, co esponding o he eigen- alue P~(x&), i x& is aze o o QN+, .Fo ha , i should be no iced ha he Q„,as o hogonal polynomials, sa is y a ecu sion ela ion o ha can be w i en as (c . [10]) I is s aigh o wa d o see ha , in gene al, In &= dx w(x)'~'g„(x')lxKM&, (13) xQ„(x )=a„„, Q„+,(x )+a„„Q„(x) +a„„, Q„,(x ), (16) whe e Q„ is apolynomial o o de n ha sa is ies he o hogonali y condi ion (nlm &= dx w(x)Q„(x )Q (x )=5„. (14) Using his equa ion and he ac ha Qo =1, one can gen- e a e he polynomials. The explici exp essions o w(x), go, g„and Q2 a e gi en in Table III, o K=—, 'and 2' 2' No e ha , o aK=0band, he s a e ln &has good an- gula momen um I=2n. The co esponding o hogonal I +b„„, g„,(x ). Using ha ela ion, i is s aigh o wa d o see ha (17) whe e ao, =0. The coeScien s in he ecu sion ela ion depend on he no maliza ion o he polynomials. In ou case, he polynomials a e no malized o 1[see Eq. (14)],i can be seen [10] ha a„„+,=a„+,„Simila ly, one can w i e P2 (x)g„(x')=b„„+1Q„+1(x ')+b„„Q„(x ') NN gQ.(3') 2(x)Q.(x')= XQ.(3') 2(3)g (x )+bNN+1[QN(3 )QN+1(x )QN(x )QN 1(+3')] ' n=0 (18) Using hese esul s, we can w i e P,( g)lg&=P, (x&)lg& +bNN+1[QN(xp)IN+1& — QN+1(xp)IN &] . Thus, i he ope a o is es ic ed o he subspace gene a - ed by [ln &,n=O,N} he e m p opo ional o lN+1& cancels. I x& a e aken as he ze os o QN+„ he e m p opo ional o lN & anishes, and we a e le wi h he e- sul we wan ed o demons a e. No e ha QN+, ,as an o hogonal polynomial, has N+1ze os in he in e al (0,1), which would co espond o N+1eigen alues and eigens a es o P2. The eigen al- ues o P2 a e shown in Tables IV and V. Using he Ch is o el-Da boux o mula, one ge s (ply&= yg„(, ')g„( ', )=0 n=0 i x& and x& a e di e en ze os o QN+, .Thus, we con6 m ha he eigens a es a e o hogonal. Finally, he eigens a es can be no malized so ha 1342 M. V. ANDRES, J.GOMEZ-CAMACHO, AND M. A. NAGARAJAN 45 TABLE IV. Eigen alues o he P2 ope a o o K=2,M=2. /=4 N=O N=1 N=2 N=3 N=4 0.2000 — 0.2556 — 0.3871 — 0.4367 — 0.4599 0.6766 0.2303 — 0.0312 — 0.1811 0.8294 0.4984 0.2385 0.8953 0.6512 0.9294 n=O (n~y) =Q„(xp) gQ(xy) i=0 V. INTERPRETATION OF THE EIGENVALUES AND EIGENSTATES (21) The alues o ~ „(13)~ ep esen he p obabili y densi y o ha ing an angle Pbe ween he axis o he o o and he ela i e coo dina e. They a e plo ed in Figs. 1and 2 o K=—, ',n=0,1,2. They do no p esen any p e e ed o ien a ion, and esemble he quali a i e beha io o he modulus squa e o sphe ical ha monics. I we use he equi alen exp ession o he eigens a es I should be no iced ha he s a es ~n ) o m acom- ple e basis o all he combina ions o s a es ~xKM ) ha a e e en in x. Thus, as a e Q—, '(~xKM)+~ — xKM)), ha co espond o an axially symme ic de o med nu- cleus whose symme y axis o ms a ixed angle wi h he ela i e coo dina e, can be expanded in e ms o he s a es ~n )as Q—, '(~xKM )+~xKM ))=— &2w (x)gQ„(x )~n). n=0 (22) I his expansion is unca ed up o n=X, we will ha e he s a e in he subspace gene a ed by he basis [~n),n=O,N] ha esembles mos closely o as a e wi h a ixed o ien a ion. This unca ion is mo e accu a e o he alues o xso ha Q~+~(x )=0. Thus, we can in e p e he eigens a es o P2( g) in asubspace gene a - ed by I~0) .~N)] as he combina ion o s a es ha esemble mos o s a es o ade ini e o ien a ion. This in- e p e a ion is s eng hened by he ac ha he eigen al- ues o P2( g) in he unca ed subspace coincide wi h hose co esponding o he s a es o de ini e o ien a ion ~x&KM ), ha a e P2(x&). To illus a e his, we can exp ess he s a es (n )as a combina ion o s a es co esponding o an angle Pbe- ween he axis o he o o and he ela i e coo dina e: /P) = dP ~(P)/PKM &(24) we ind ha he alues o ~ &(p)~(see Figs. 3and 4) ha e signi ican alues a o ien a ions close o he angles p~ such ha hei cosines gi e he ze os o QN+ ~(x )&» ge s bigge , he unc ion ~ &(p)~becomes a 5 unc ion. VI. APPLICATION TO SCATTERING IN THE SUDDEN LIMIT The in e ac ion be ween he nuclei conse es bo h he idal spin Mand he p ojec ion o he spin along he sym- me y axis o he o o E. Howe e , ha is no ue o he u11 Hamil onian. The cen i ugal e m o he ela i e mo ion changes M, while he Co iolis e m o he in e nal Hamil onian o he o o changes E. Howe e , he Co iolis e m can be neglec ed o no e y high in e nal angula momen um I, and he cen i ugal e m can be subs i u ed by an a e age alue (isocen i ugal app oxi- ma ion [7]) o hea y-ion collisions. No e ha , when he g ound-s a e angula momen um I=0 and one is discuss- ing elas ic sca e ing, he isocen i ugal app oxima ion is equi alen o igno ing he Co iolis o ce o he ela i e mo ion [2]. The isocen i ugal app oxima ion implies ha Mis conse ed in he sca e ing p ocess, while neglec ing he Co iolis o ce o he in e nal deg ees o eedom implies ha Kis conse ed. Any magni ude ela ed o he sca e ing o wo pa i- cles can be ob ained in e ms o he Sma ix. This, in TABLE V. Eigen alues o he P2 ope a o o K=~,M=~. N=0 N=1 N=2 N=3 N=4 — 0.2000 — 0.3780 — 0.4343 — 0.4590 — 0.4720 0.3780 0.0252 — 0.1574 — 0.2606 0.6399 0.3187 0.1008 0.7683 0.5068 0.8393 45 GEOMETRIC INTERPRETATION OF THE EFFECT OF THE. ..1343 1.01.0 0.80.8 0.40.4 0.20.2 0.0~/6 P(»d) 0.00 P( ad) FIG. l. ~ „(P)~' s he angle P o K=— ', ,M=2. The solid line co esponds o n=0, he dashed line o n=1,and he do - dashed line o n=2. FIG. 2. ~ „(p)~2 s he angle p o K=z,M= z. Same n«a- ion as Fig. 1. gene al, will be a unc ion o he o al angula momen um J, he incoming and ou going o bi al angula momen a L,L he incoming and ou going in insic angula mo- men a I,I', and hei p ojec ion along he symme y axis K,K'. When he Co iolis e m can be igno ed, he Sma- ix is diagonal in K. Besides, when idal o ces dom- ina e, and he isocen i ugal app oxima ion can be done [7], he Sma ix elemen s can be w i en in e ms o he idal spin Sma ices as ~J ~L'I'K', LIK =5~x Ph g(LOIM~JM)(L'OI'M~JM)SI I M (25) n, n'=0 (I'KM ~n')S, (n~IKM ).(26) The coe icien s (nIKM )can be ob ained in as aigh - o wa d (bu labo ious) way om he de ini ion o he s a es ~n ). Some o hem a e p esen ed in Tables Iand II.Now i one uses he basis ~(() ) ha diagonalizes he in- e ac ion, i will also diagonalize he Sma ix, and one ge s N+1 S„„=y(n' ( )S (xy)(y~n ). /=1 The Sma ix S(x&) is he one ob ained om aone- channel op ical model calcula ion wi h he po en ial whe e Mis he idal spin, L=(L+L')/2, and Ph is a phase ac o in ol ing Coulomb phase shi s. Now, i we conside he coupling o he o a ional s a es om I=K o K+2N, igno ing he exci a ion ene gies, we can use he mapping discussed p e iously o ge 2.0 V~( )= Vo( )+ V2( )P2(x&) . The elas ic Sma ix o he g ound s a e is jus N+1 SLY y~MKSL( (I5 =1 whe e (28) (29) 1.5 2.0 1.0 0.5 0.00 / /X pp /ipe e= e- /37T/6 P(»d) FIG. 3. &(P)~ s he angle P o K= 3,M= —'. The solid line co esponds o /=0, he dashed line o P= I, and he do - dashed line o /=2. 1.0 0.5 0.00 /6 P( ad) FIG. 4. ~ &(P)~ s he angle II o K= 3,M= —.Same no a- ion as Fig. 3. 1344 M. V. ANDRES, J.GOMEZ-CAMACHO, AND M. A. NAGARAJAN TABLE VI. Weigh ac o s o E=— ', ,I=— ', . N=O N=1 N=2 N=3 N=4 1.0000 0.5113 0.3145 0.2238 0.1733 0.4887 0.4120 0.2973 0.2218 0.2735 0.3068 0.2587 0.1721 0.2286 0.1176 N MK 22 i=0 (30) A~,~(8)=Xd,M M AM(8)dK (32) The usion c oss sec ions, de ined as he c oss sec ion ha does no appea in he channels included explici ly in he calcula ion, o agi en alue he angula momen um J and o he idal spin Mcan be calcula ed in e ms o hose o he uncoupled eigenchannels and one ge s N+1 JM MK J~ /=1 (33) No e ha he usion c oss sec ions o agi en p ojec ion o he spin o he p ojec ile along he beam di ec ion m can be ela ed o he usion c oss sec ions o agi en idal spin Musing idal symme y [11] 6JM OK J K, m(34) m, M No e ha g&w4, =1. The weigh ac o s w& a e shown in Tables VI and VII. As he weigh ac o s do no depend upon L, he sum o ob ain he sca e ing am- pli udes as a unc ion o he sca e ing angle o agi en idal spin can be pe o med, and one ge s N+1 Az~z(8) =gw& A(x&,8) .(31) /=1 No e ha idal symme y leads o he ac ha he ansi- ion ampli udes co esponding o p ojec ions m, m'o he spin along he beam di ec ion a e gi en by ha he momen o ine ia o he o o is e y la ge, and so he o ien a ion o he o o is ixed du ing he sca e - ing p ocess, he sca e ing ampli udes will be gi en in e ms o he a e age o e all he o ien a ions, weigh ed wi h he p obabili y densi y ha hey occu in he g ound s a e [1]: Sz~= Jdx w(x)S (x), (35) Az z(8)= dx w(x) A(x,8), 1 oP= dx w(x)o (x) .(37) — 1 (36) Le P(x) be ase o polynomials ha a e o hogonal wi h espec o he weigh unc ion w(x). One can ap- p oxima e he in eg als as ollows: 1N'+ 1 Jdx w(x) (x)= gA~ (x~), (38) — 11 N' gP(x&)P (x) L~(x)= gP(x~) (40) Using he o hogonali y o P(x) one ge s N' whe e x& a e he ze os o he polinomial PM+, (x), and A&a e weigh s gi en by he exp ession A&= Jdx w(x)L&(x) .(39) — 1 L&(x) is he Lag ange mul iplie unc ion [12] ha can be w i en as A~= gP(x~) (41) whe e OJ is he classical sca e ing angle ha co esponds o an angula momen um J. VII. RELATION TO THE GEOMETRICAL LIMIT I he exci a ion ene gy o all he s a es o he o a ion- al band can be igno ed, which is equi alen o assuming m=0 The unca ion e o is o he o de o he 2N'+ 2de i a- i e o (x) In he case ha w(x) and (x) a e e en unc ions o x, only he polynomials Q„(x )a e ele an , and one can w i e he exp ession TABLE VII. Weigh ac o s o I( =2,M=—, '. N=O N=1 N=2 N=3 N=4 1.0000 0.7646 0.5916 0.4779 0.3995 0.2354 0.3325 0.3380 0.3167 0.0759 0.1532 0.1911 0.0309 0.0779 0.0147 GEOMETRIC INTERPRETATION OF THE EFFECT OF THE. .. 1345 1%+1 dx w(x) (x)= gB& (x&), — 1P— 1 whe e x& a e he posi i e ze os o Q~+,(x ), and 1N B~= dx w(x)L~(x )= gQ„(x~) — 1n=0 (42) (43} The unca ion e o is o he o de o he 4N+4 de i a- i e o (x). Compa ing hese exp essions wi h he ones ob ained be o e, we conclude ha he e ec o including he cou- pling o he s a es in ) om n=0 o N, o he o a ional s a es iIKM ) om I=K o K+2N, igno ing he exci a- ion ene gies, is equi alen o pe o ming an (N+1) poin Gaussian in eg a ion on he geome ical exp es- sions o hese magni udes. Mo eo e , he coupling o he s a es I=K+2N+1 and I=E+2N+2 is going o be impo an o amagni ude such as he elas ic sca e ing o he eac ion c oss sec ion when he geome ical ex- p ession o ha magni ude as a unc ion o he o ien a- ion angle has asigni ican con ibu ion o he 4N+4 de i a i e. s a es o he o a ional band in acoupled-channels calcu- la ion will a ec signi ican ly he elas ic magni udes (S ma ix and ansi ion ampli udes} and he usion c oss sec ions when he pa ame ic exp ession o hese magni- udes as a unc ion o he cosine o he o ien a ion angle p esen s signi ican con ibu ions o he 4N+4 de i a- i e. As a inal commen , we hink ha he analy ic diago- naliza ion ound in his wo k and in o he cases discussed in [6] is no jus ama hema ical cu iosi y. One should expec o ha e i o o he kinds o collec i e exci a ion. An algeb aic ea men o he diagonaliza ion o he col- lec i e exci a ion, based on g oup heo y, can be an al e - na i e o he analy ic app oach p esen ed he e, based on he p ope ies o he o hogonal polynomials, and i can p o ide adeepe unde s anding o he e ec o in e nal deg ees o eedom on he eac ion mechanisms. This wo k was pa ially suppo ed by he Acciones In- eg adas HB-196 and he Spanish DGICYT PB89-0636. APPENDIX: COMPARISON OF THE ANALYTIC RESULTS WITH NUMERICAL DIAGONALIZATION VIII. SUMMARY AND CONCLUSIONS The quad upole coupling in a o a ional band wi h KAO can be simpli ied i he s a es wi h de ini e angula momen um a e mapped in o ase o s a es in )ob ained ac ing wi h he P2 ope a o on he o a ional g ound s a e and o hogonalizing. These s a es co espond o a supe posi ion o s a es wi h de ini e o ien a ions, weigh - ed wi h ase o o hogonal polynomials. In gene al, he s a es in )do no co espond o s a es o good angula momen um and hence he unca ion o he basis s a es ~n )is, excep o he case o K=0 and IC =—, 'bands, dis- inc om he unca ion in e ms o he membe s o he g ound-s a e o a ional band. Ne e heless, he esul s ob ained when unca ing in hese wo di e en ways a e e y simila , as i can be seen in he Appendix. The eigen alues and eigen ec o s o he P2 ope a o in basis de ined by he s a es in ) om n=0 o Na e ob- ained in e ms o he ze os o he o hogonal polynomial o o de N+1. The eigen ec o s can be in e p e ed as he combina ion o s a es in ) om n=0 o N ha mos esemble as a e wi h de ini e o ien a ion o he symme y axis o he o o wi h espec o he ela i e coo dina e. The eigen alues co espond o P2(x), whe e xis he cosine o he o ien a ion angle. The Sma ix, elas ic ansi ion ampli udes, and usion c oss sec ions in he idal spin basis co espond o a weigh ed a e age o hese magni udes co esponding o he o ien a ions ha de ine he eigen alues and eigen ec- o s o P2. The classical sudden esul o hese magni udes co e- spond o an in eg al o e all he o ien a ion angles, weigh ed wi h he p obabili y ha agi en o ien a ion occu s in he g ound s a e. The weigh ed a e age de- sc ibed be o e co esponds o agene alized Gaussian quad a u e o he sudden in eg al exp ession. The e ec o he inclusion o he 2N+1 and 2N+2 AcoR bE— Vb o. =ln 1+exp 2m. 2E %co (A1) The ba ie heigh depends on he o ien a ion o he o- o wi h espec o he ela i e coo dina e as Vb =Vp+ V2P2(cosg). Hence, wi hin he sudden app ox- ima ion, he usion c oss sec ion will be gi en by AcoR b O = gW; 2E E— Vo — V2P2 Xln 1+exp 2w (A2) The coupling ma ix o Eq. (3) can be diagonalized nu- me ically, including he o a ional s a es iIKM) om I=K o I=K+2N. Thus, one ob ains 2N+1 eigen al- ues and eigen ec o s, om which he sca e ing magni- udes can be ob ained. Howe e , he analy ical diagonal- iza ion including he s a es ~n) om n=0 o Ngene a es N+1 eigen alues and eigen ec o s. Despi e his ac , we will show ha he p edic ions o sca e ing magni udes happen o e y simila in bo h cases. When Ko M ake he alue 0, only he alues o I wi h IKe en a e— coupled. The s a es in) coincide wi h he s a es o agi en angula momen um I=E+2n. When Ko M ake he alue —, ', o he 2N+1 eigens a es o P2, Na e o hogonal o he g ound s a e, and hence do no a ec he eac ion mechanism. The emaining N+1 coincide wi h he eigens a es in he basis in) and ha e he same eigen alues. This is due o he ac ha o a- ional s a es wi h K(o M) equal o —, 'can be conside ed as o a ional s a es wi h Ko Mequal o 0 o which a pa icle wi h j=—, 'is coupled. Fo he o he cases, he eigens a es and eigen alues in he iIKM) basis di e om hose in he ~n)basis. Le us conside he Wong o mula o he usion c oss sec ion M. V. ANDRES, J.GOMEZ-CAMACHO, AND M. A. NAGARAJAN TABLE VIII. Enhancemen ac o s o he usion c oss sec ion calcula ed in he ~n)basis and in he ~Il M )basis. Imax 3 2 7 2 11 2 15 2 19 2 F{n) 7.3891 424.29 1098.3 1375.9 1435.2 E«V, F(I) 7.3891 488.36 1150.6 1390.3 1437.4 F(n) 3.0685 4.8265 4.7079 4.6719 4.6989 E=Vo F(I) 3.0685 4.7463 4.7081 4.6784 4.6950 F= gw;exp 2m V2 P2 (A3) When he ene gy coincides wi h he unde o med ba ie Vo whe e P2 is he i h eigen alue o Pz and w, is he squa e o he o e lap o he g ound s a e wi h he i h eigens a e. Le us de6ne an enhancemen ac o Fas he a io o he coupled-channels usion c oss sec ion o he uncoupled c oss sec ions. When he ene gy is well below he ba ie , ha ac o is gi en by N F= gw, ln 1+exp 2m V2 P2 ln2 (A4) In Table VIII we p esen he e alua ion o he ac o F o anucleus wi h K=M =—, ', calcula ed in he basis ~n) and in he basis ~IKM ). We ha e aken 2 V2/ iw =10, ha is easonable o he sys em Na +Pb. 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