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

PHYSICAL REVIEW CVOLUME 45, NUMBER 3MARCH 1992 Geometric interpretation of the effect of the quadrupole force in the collisions of deformed nuclei M. V. Andres and J.Gomez-Camacho Departamento de FAMN, Universidad de Sevilla, Facultad de Fs'sicas, Apartado 1065, 41080Sevilla, Spain M. A. Nagarajan Daresbury Laboratory, Warrington, WA4 4AD, United Kingdom (Received 15 October 1991) The effect of aquadrupole force on aset of degenerate states of arotational band with arbitrary spin projection along the symmetry axis Kis studied. Analytic expressions for the eigenvalues and eigenvectors are obtained in terms of aset of orthogonal polynomials. This is applied to the collision of aspherical nucleus with adeformed one in which the coupling to agiven set of rotational states is allowed, ignoring excitation energies. The elastic S-matrix, transition amplitudes, and the fusion cross sections are obtained as aweighted average of the magnitudes corresponding to aset of definite orientations of the axis of the deformed nucleus with respect to the relative coordinate. That weighted average corresponds to approximate the extreme sudden result, consisting of an integral over all the orientations, by ageneralized Gaussian quadrature. PACS number(s): 24.10.Eq, 24.70.+s, 24.50.+g, 03.65.Nk I. INTRODUCTION The collective excitations of nuclei play avery important role in the reaction mechanisms. The description of these effects requires the explicit inclusion of the excitation by means of coupled-channels calculations. For the case of rotational nuclei, aproper coupled-channels calculation would require the inclusion of many states of the rotational band, making the calculation complicated and time consuming, and complicating the interpretation of the results. On the other hand, when the rotational motion can be considered slow versus the relative motion (sudden approximation) the scattering amplitudes can be calculated as an integral extended to all the orientations of the rotor of the scattering amplitudes calculated as if the orientation of the deformed nucleus was frozen during the reaction, weighted with the probability density that the ground state has that orientation [l]. Also, it is known that when the effect of the coupling to arestricted set of excited states is considered, the excitation energy is ignored, the centrifugal barrier is suitably approximated and the coupling form factors have the same shape, the coupled-channels system can be decoupled, and the scattering amplitudes can be expressed as acombination of the ones corresponding to aset of uncoupled eigenchannels [2— 5]. Nagarajan, Balantekin, and Takigawa [6) contributed to bridge the gap between the coupled-channels calculation and the sudden result demonstrating that, for a @=0 rotational band, the coupled-channels effect corresponding to include the rotational states from I=O to I=2%— 2ignoring their excitation energies, was equivalent to do aweighted average of the amplitudes corresponding to Norientations. These orientations are characterized by the angle 0between the symmetry axis of the deformed nucleus and the relative motion, that takes the values such that P2&(cos0) =0. Moreover, that weighted average is precisely the combination obtained when the integral corresponding to the sudden approximation is approximated by an N-point Gauss-Legendre quadrature. The aim of this work is to search for asimilar result that could be applied to arotational band of arbitrary E. This is important if one is interested in studying the inhuence of polarization on the reaction mechanisms, because, if the nucleus is to be polarized, its ground state needs to have spin different from zero. This paper is organized as follows. In Sec. II we evaluate the matrix element of aquadrupole interaction between rotational states using the tidal spin basis. In Sec. III we map the rotational states into anew set of states characterized by aset of orthogonal polynomials. In Sec. IV we perform an analytical diagonalization in the new basis. In Sec. Vwe discuss the meaning of eigenvalues and eigenvectors. In Sec. VI we apply this treatment to the calculation of scattering amplitudes. In Sec. VII the relation to the geometrical limit is obtained. Section VIII is for summary and conclusions. II. COUPLING POTENTIALS Let us consider the interaction between a spherical nucleus and an axially deformed one. The interaction can be written as V(r, f')=Vo(r)+ V2(r)P2(r. g), where ris the relative coordinate and g' stands for the direction of the symmetry axis of the deformed nucleus. In this expression we have neglected spin-orbit terms and hexadecapole and higher-order deformation. The matrix elements of the quadrupole interaction between rotational states in the usual coupled-channels basis is given by 45 1339 1992 The American Physical Society 1340 M. V. ANDRES, J.GOMEZ-CAMACHO, AND M. A. NAGARAJAN (IKLJI V2(r)P~(r g)II'K'L'J ) =V2(r)5x ~,W(II'LL';2J)LI x&I.o2olI. 'o) &IK20II'K), (2) where Iis the internal angular momentum of the deformed nucleus and Kis the projection along the symmetry axis, that characterizes the rotational band. Lis the orbital angular momentum, and Jis the total angular momentum. It is known that this interaction can be partially diagonalized in the tidal spin basis [7] (IKMJI V,(r)P,(r g)II'K'M'J ) =V~(r)5M ~5+ KI/I'(IM20II'M )(IK20II'K'), where Mis the projection of the spin Ialong the relative coordinate r(tidal spin). Note that throughout this paper, Mdenotes the tidal spin while Kis the projection of the spin along the symmetry axis of the rotor. Both magnitudes are conserved by the coupling potential. Note also that the value of Jdoes not afFect the values of the coupling potential in the tidal spin basis. So, in what follows, we will drop the index Jin the characterization of the states. The coupling potentials have acommon form factor V2(r) and different strength factors. if the excitation energy is ignored for agiven number of states, the coupling matrix could be diagonalized numerically, and aset of eigenchannels be obtained, from which all the relevant reaction magnitudes can be calculated [2— S]. However, the eigenvalues and eigenstates of the numerical diagonalization do not have any clear geometrical meaning. The main difficulty in extending the analytic diagonalization [6] to KPO bands is that each state is coupled to four other states, while for aK=0 band it is only coupled to two other. That makes it difficult to make asuitable truncation in the states of the band included in the calculation, and yet obtain analytic expressions for eigenvalues and eigenvectors. An exception to this are the K=—, 'bands, because they can be considered as K=O bands to which aj=—, 'particle is coupled. For these analytic expressions for eigenvalues and eigenvectors can be found. III. MAPPING To simplify the complicated coupling structure of a EWO band we wi11 introduce aset of states obtained acting with the P2(r.g) operator over the ground state of the band. I1&=N, (P2lo& — lo &&0IP, lo&), n— 1 In &=N„P,ln — 1& —gli&&ilP, ln — » i=0 (4) These states do not have, in general, good angular momentum I. Explicit expressions for overlap (nlIKM ) in the case K=— ', are given in Table I(M =— , ') and Table II (M=—, '). The states In )are the combinations of states IIKM) that are most relevant for the quadrupole coupling. %e expect that the coupled-channels efFects due to the rotational states IIKM) from I=K to I=K+2N would be very similar to consider the states In)from n=0to n=¹In both cases, terms of order %+1in J'2 are ignored: Note that coupling structure is simplified, because each of the states In )is only coupled to In +1) and ln — 1). This way of generating the ln )states is known as the Lanczos method [8], which allows one to obtain some eigenvalues and eigenstates quite accurately from only part of the full matrix. Although the states In)have acomplicated expansion in terms of states of good angular momentum, they have avery appealing expression when expressed as asuperposition of states corresponding to definite orientations of the rotor. The state IO) can be written as [9] =v (2K+1)/16m' Jdn[2P~~(n)InK ) +(— i)2~gF, (n)lnK)], (S) where x=cosP and N(x) is anormalization factor. The states lxKM) have the same parity as the states of the rotational band, and satisfy where InK )is astate vector of the deformed nucleus corresponding to an orientation given by the Euler angles n=(a,p, y) of the rotor with respect to an external coordinate system with the zaxis along the relative coordinate, and aprojection EC of the angular momentum along the symmetry axis. InK )is the time reversed state. We can define the state lxEM )as lxKM) =N(x) Jdady[2)xM(n)InK) +(— 1)' 2) (n)lnK )], (6) (nII ) TABLE I. Mapping coefficien for K=—, ',M=—. TABLE II. Mapping coefficients for K=— ', ,M=—, '. &o ((I &2 4465 7755 (of v'-', GEOMETRIC INTERPRETATION OF THE EFFECT OF THE. ..134i TAHI.EIII. %eight functions and orthogonal 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(r.g)lxKM &=P2(x)lxKM&, ' (x'K'M'lxKM &=5x x5M ~5(x' x) .— (7) (8) (10) polynomials are the even Legendre polynomials. The analytic diagonalization and the geometric interpretation obtained in [6] for the K=0 band relied on the orthogonality properties of the Legendre polynomials. Thus, we are in the situation of applying similar techniques for our case. They can be interpreted as states of the rotor in which the axis of the rotor and the relative coordinate form a fixed angle p, but they are averaged over all the Euler angles aand yso that Mand Eare good quantum numbers. In terms of these states, the ground-state wave function is just IV. ANALYTIC DIAGONALIZATION We will demonstrate that the (unnormalized) state I(()&= gQ„( p)l n=0 (15) lO&=lKKM&= fdx w(x)' lxKM&, where w(x) =[[dhM(p)] +[d — xM(p)] }. 4(12) is an eigenstate of the operator P2(r g) in the subspace generated by [ln &,n=O,N}, corresponding to the eigenvalue P~(x&), if x& is azero of QN+, .For that, it should be noticed that the Q„,as orthogonal polynomials, satisfy arecursion relation for that can be written as (cf. [10]) It is straightforward to see that, in general, In &= fdx w(x)'~'g„(x')lxKM&, (13) xQ„(x )=a„„, Q„+,(x )+a„„Q„(x) +a„„, Q„,(x ), (16) where Q„ is apolynomial of order nthat satisfies the orthogonality condition (nlm &= fdx w(x)Q„(x )Q (x )=5„. (14) Using this equation and the fact that Qo =1, one can generate the polynomials. The explicit expressions for w(x), go, g„and Q2 are given in Table III, for K=—, 'and 2' 2' Note that, for aK=0band, the state ln &has good angular momentum I=2n. The corresponding orthogonal I +b„„, g„,(x ). Using that relation, it is straightforward to see that (17) where ao, =0. The coeScients in the recursion relation depend on the normalization of the polynomials. In our case, the polynomials are normalized to 1[see Eq. (14)],it can be seen [10] that a„„+,=a„+,„Similarly, one can write 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 these results, we can write P,(r g)lg&=P, (x&)lg& +bNN+1[QN(xp)IN+1& — QN+1(xp)IN &] . Thus, if the operator is restricted to the subspace generated by [ln &,n=O,N} the term proportional to lN+1& cancels. If x& are taken as the zeros of QN+„ the term proportional to lN &vanishes, and we are left with the result we wanted to demonstrate. Note that QN+, ,as an orthogonal polynomial, has N+1zeros in the interval (0,1), which would correspond to N+1eigenvalues and eigenstates ofP2. The eigenvalues of P2 are shown in Tables IV and V. Using the Christoffel-Darboux formula, one gets (ply&= yg„(, ')g„( ', )=0 n=0 if x& and x& are different zeros of QN+, .Thus, we con6rm that the eigenstates are orthogonal. Finally, the eigenstates can be normalized so that 1342 M. V. ANDRES, J.GOMEZ-CAMACHO, AND M. A. NAGARAJAN 45 TABLE IV. Eigenvalues of the P2 operator for 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 values of ~f„(13)~represent the probability density of having an angle Pbetween the axis of the rotor and the relative coordinate. They are plotted in Figs. 1and 2for K=—, ',n=0,1,2. They do not present any preferred orientation, and resemble the qualitative behavior of the modulus square of spherical harmonics. If we use the equivalent expression for the eigenstates It should be noticed that the states ~n )form acomplete basis of all the combinations of states ~xKM )that are even in x. Thus, astate Q—, '(~xKM)+~ — xKM)), that correspond to an axially symmetric deformed nucleus whose symmetry axis forms afixed angle with the relative coordinate, can be expanded in terms of the states ~n )as Q—, '(~xKM )+~xKM ))=— &2w (x)gQ„(x )~n). n=0 (22) If this expansion is truncated up to n=X, we will have the state in the subspace generated by the basis [~n),n=O,N] that resembles most closely to astate with afixed orientation. This truncation is more accurate for the values of xso that Q~+~(x )=0. Thus, we can interpret the eigenstates of P2(r g) in asubspace generated by I~0) .~N)] as the combination of states that resemble most to states of adefinite orientation. This interpretation is strengthened by the fact that the eigenvalues of P2(r g) in the truncated subspace coincide with those corresponding to the states of definite orientation ~x&KM ),that are P2(x&). To illustrate this, we can express the states (n )as a combination of states corresponding to an angle Pbetween the axis of the rotor and the relative coordinate: /P) =fdP f~(P)/PKM &(24) we find that the values of ~f&(p)~(see Figs. 3and 4) have significant values at orientations close to the angles p~ such that their cosines give the zeros of QN+ ~(x )&» gets bigger, the function ~f&(p)~becomes a 5 function. VI. APPLICATION TO SCATTERING IN THE SUDDEN LIMIT The interaction between the nuclei conserves both the tidal spin Mand the projection of the spin along the symmetry axis of the rotor E. However, that is not true for the fu11 Hamiltonian. The centrifugal term of the relative motion changes M, while the Coriolis term of the internal Hamiltonian of the rotor changes E. However, the Coriolis term can be neglected for not very high internal angular momentum I, and the centrifugal term can be substituted by an average value (isocentrifugal approximation [7])for heavy-ion collisions. Note that, when the ground-state angular momentum I=0 and one is discussing elastic scattering, the isocentrifugal approximation is equivalent to ignoring the Coriolis force of the relative motion [2]. The isocentrifugal approximation implies that Mis conserved in the scattering process, while neglecting the Coriolis force of the internal degrees of freedom implies that Kis conserved. Any magnitude related to the scattering of two particles can be obtained in terms of the Smatrix. This, in TABLE V. Eigenvalues of the P2 operator for 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(rad) FIG. l. ~f„(P)~' vs the angle Pfor K=— ', ,M=2. The solid line corresponds to n=0, the dashed line to n=1,and the dotdashed line to n=2. FIG. 2. ~f„(p)~2 vs the angle pfor K=z,M= z. Same n«ation as Fig. 1. general, will be afunction of the total angular momentum J, the incoming and outgoing orbital angular momenta L,Lthe incoming and outgoing intrinsic angular momenta I,I', and their projection along the symmetry axis K,K'. When the Coriolis term can be ignored, the Smatrix is diagonal in K. Besides, when tidal forces dominate, and the isocentrifugal approximation can be done [7], the Smatrix elements can be written in terms of the tidal spin Smatrices 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 coefficients (nIKM )can be obtained in astraightforward (but laborious) way from the definition of the states ~n ). Some of them are presented in Tables Iand II.Now if one uses the basis ~(() )that diagonalizes the interaction, it will also diagonalize the Smatrix, and one gets N+1 S„„=y(n' (t )S (xy)(y~n ). /=1 The Smatrix S(x&) is the one obtained from aonechannel optical model calculation with the potential where Mis the tidal spin, L=(L+L')/2, and Ph is a phase factor involving Coulomb phase shifts. Now, if we consider the coupling of the rotational states from I=K to K+2N, ignoring the excitation energies, we can use the mapping discussed previously to get 2.0 V~(r)= Vo(r)+ V2(r)P2(x&) . The elastic Smatrix for the ground state is just N+1 SLY y~MKSL( (I5 =1 where (28) (29) 1.5 2.0 1.0 0.5 0.00 / /X pp /ipe e= evr/37T/6 P(»d) FIG. 3. f&(P)~ vs the angle Pfor K= 3,M= —'. The solid line corresponds to /=0, the dashed line to P= I, and the dotdashed line to /=2. 1.0 0.5 0.00vr/6 P(rad) FIG. 4. ~f&(P)~ vs the angle II for K= 3,M= —.Same notation as Fig. 3. 1344 M. V. ANDRES, J.GOMEZ-CAMACHO, AND M. A. NAGARAJAN TABLE VI. Weight factors for 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 fusion cross sections, defined as the cross section that does not appear in the channels included explicitly in the calculation, for agiven value the angular momentum J and of the tidal spin Mcan be calculated in terms of those of the uncoupled eigenchannels and one gets N+1 JM MK J~ /=1 (33) Note that the fusion cross sections for agiven projection of the spin of the projectile along the beam direction m can be related to the fusion cross sections for agiven tidal spin Musing tidal symmetry [11] 6JM OK J K, m(34) m, M Note that g&w4, =1. The weight factors w& are shown in Tables VI and VII. As the weight factors do not depend upon L, the sum to obtain the scattering amplitudes as afunction of the scattering angle for agiven tidal spin can be performed, and one gets N+1 Az~z(8) =gw& A(x&,8) .(31) /=1 Note that tidal symmetry leads to the fact that the transition amplitudes corresponding to projections m, m'of the spin along the beam direction are given by that the moment of inertia of the rotor is very large, and so the orientation of the rotor is fixed during the scattering process, the scattering amplitudes will be given in terms of the average over all the orientations, weighted with the probability density that they occur in the ground state [1]: Sz~= Jdx w(x)S (x), (35) Az z(8)=fdx w(x) A(x,8), 1 oP= fdx w(x)o (x) .(37) — 1 (36) Let P(x) be aset of polynomials that are orthogonal with respect to the weight function w(x). One can approximate the integrals as follows: 1N'+ 1 Jdx w(x)f(x)= gA~f(x~), (38) — 11 N' gP(x&)P (x) L~(x)= gP(x~) (40) Using the orthogonality of P(x) one gets N' where x& are the zeros of the polinomial PM+, (x), and A&are weights given by the expression A&= Jdx w(x)L&(x) .(39) — 1 L&(x) is the Lagrange multiplier function [12]that can be written as A~= gP(x~) (41) where OJ is the classical scattering angle that corresponds to an angular momentum J. VII. RELATION TO THE GEOMETRICAL LIMIT If the excitation energy of all the states of the rotational band can be ignored, which is equivalent to assuming m=0 The truncation error is of the order of the 2N'+ 2derivative off(x) In the case that w(x) and f(x) are even functions ofx, only the polynomials Q„(x )are relevant, and one can write the expression TABLE VII. Weight factors for 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 fdx w(x)f(x)= gB&f(x&), — 1P— 1 where x& are the positive zeros of Q~+,(x ), and 1N B~=fdx w(x)L~(x )= gQ„(x~) — 1n=0 (42) (43} The truncation error is of the order of the 4N+4 derivative off(x). Comparing these expressions with the ones obtained before, we conclude that the effect of including the coupling of the states in )from n=0 to N, or the rotational states iIKM )from I=K to K+2N, ignoring the excitation energies, is equivalent to performing an (N+1) point Gaussian integration on the geometrical expressions of these magnitudes. Moreover, the coupling to the states I=K+2N+1 and I=E+2N+2 is going to be important for amagnitude such as the elastic scattering or the reaction cross section when the geometrical expression of that magnitude as afunction of the orientation angle has asignificant contribution of the 4N+4 derivative. states of the rotational band in acoupled-channels calculation will affect significantly the elastic magnitudes (S matrix and transition amplitudes} and the fusion cross sections when the parametric expression of these magnitudes as afunction of the cosine of the orientation angle presents significant contributions of the 4N+4 derivative. As afinal comment, we think that the analytic diagonalization found in this work and in other cases discussed in [6] is not just amathematical curiosity. One should expect to have it for other kinds of collective excitation. An algebraic treatment of the diagonalization of the collective excitation, based on group theory, can be an alternative to the analytic approach presented here, based on the properties of the orthogonal polynomials, and it can provide adeeper understanding of the effect of internal degrees of freedom on the reaction mechanisms. This work was partially supported by the Acciones Integradas HB-196 and the Spanish DGICYT PB89-0636. APPENDIX: COMPARISON OF THE ANALYTIC RESULTS WITH NUMERICAL DIAGONALIZATION VIII. SUMMARY AND CONCLUSIONS The quadrupole coupling in arotational band with KAO can be simplified if the states with definite angular momentum are mapped into aset of states in )obtained acting with the P2 operator on the rotational ground state and orthogonalizing. These states correspond to a superposition of states with definite orientations, weighted with aset of orthogonal polynomials. In general, the states in )do not correspond to states of good angular momentum and hence the truncation of the basis states ~n )is, except for the case of K=0 and IC =—, 'bands, distinct from the truncation in terms of the members of the ground-state rotational band. Nevertheless, the results obtained when truncating in these two different ways are very similar, as it can be seen in the Appendix. The eigenvalues and eigenvectors of the P2 operator in basis defined by the states in )from n=0 to Nare obtained in terms of the zeros of the orthogonal polynomial of order N+1. The eigenvectors can be interpreted as the combination of states in )from n=0 to Nthat most resemble astate with definite orientation of the symmetry axis of the rotor with respect to the relative coordinate. The eigenvalues correspond to P2(x), where xis the cosine ofthe orientation angle. The Smatrix, elastic transition amplitudes, and fusion cross sections in the tidal spin basis correspond to a weighted average of these magnitudes corresponding to the orientations that define the eigenvalues and eigenvectors ofP2. The classical sudden result for these magnitudes correspond to an integral over all the orientation angles, weighted with the probability that agiven orientation occurs in the ground state. The weighted average described before corresponds to ageneralized Gaussian quadrature ofthe sudden integral expression. The effect of the inclusion of the 2N+1 and 2N+2 AcoR bE— Vb o.f=ln 1+exp 2m. 2E %co (A1) The barrier height depends on the orientation of the rotor with respect to the relative coordinate as Vb =Vp+ V2P2(cosg). Hence, within the sudden approximation, the fusion cross section will be given by AcoR b Of= gW; 2E E— Vo — V2P2 Xln 1+exp 2w (A2) The coupling matrix of Eq. (3) can be diagonalized numerically, including the rotational states iIKM) from I=K to I=K+2N. Thus, one obtains 2N+1 eigenvalues and eigenvectors, from which the scattering magnitudes can be obtained. However, the analytical diagonalization including the states ~n)from n=0to Ngenerates N+1 eigenvalues and eigenvectors. Despite this fact, we will show that the predictions for scattering magnitudes happen to very similar in both cases. When Kor Mtake the value 0, only the values of I with IKeven are— coupled. The states in) coincide with the states of agiven angular momentum I=E+2n. When Kor Mtake the value —, ', of the 2N+1 eigenstates ofP2, Nare orthogonal to the ground state, and hence do not affect the reaction mechanism. The remaining N+1 coincide with the eigenstates in the basis in) and have the same eigenvalues. This is due to the fact that rotational states with K(or M) equal to —, 'can be considered as rotational states with Kor Mequal to 0to which a particle with j=—, 'is coupled. For the other cases, the eigenstates and eigenvalues in the iIKM) basis differ from those in the ~n)basis. Let us consider the Wong formula for the fusion cross section M. V. ANDRES, J.GOMEZ-CAMACHO, AND M. A. NAGARAJAN TABLE VIII. Enhancement factors for the fusion cross section calculated in the ~n)basis and in the ~IltM )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 the energy coincides with the underformed barrier Vo where P2 is the ith eigenvalue of Pz and w, is the square of the overlap of the ground state with the ith eigenstate. Let us de6ne an enhancement factor Fas the ratio of the coupled-channels fusion cross section to the uncoupled cross sections. When the energy is well below the barrier, that factor is given by N F= gw, ln 1+exp 2m V2 P2 ln2 (A4) In Table VIII we present the evaluation of the factor F for anucleus with K=M =—, ', calculated in the basis ~n) and in the basis ~IKM ). We have taken 2trV2/fiw =10, that is reasonable for the system Na +Pb. The agreement is remarkable, and justi6es the mapping performed in Sec. III. [1]D. M. Chase, Phys. Rev. 104, 838 (1956). [2] P. Jacobs and U. Smilansky, Phys. Lett. 127B,313 (1983). [3) C. H. Dasso, S. Landowne, and A. Winther, Nucl. Phys. A405, 381 (1983);A407, 221 (1983);A432, 555 (1985). [4] R. Lindsay and N. Rowley, J.Phys. G10, 805 (1984). [5]M. A. Nagarajan, in Proceedings of the Second La Rabida Summer School on Nuclear Structure and Nuclear Reactions, edited by M. Lozano and G. Madurga (World Scientific, Singapore, 1986),p. 349. [6]M. A. Nagarajan, B.Balantekin, and N. Takigawa, Phys. Rev. C34, 894 (1986). [7]J. G6mez-Camacho and R. C. Johnson, J. Phys. G12, L235 (1986);14, 609 (1988). [8]P. J.Brussard and P. W. M. Glaudemans, Shell Model Ap-- plications in Nuclear Spectroscopy (North-Holland, Amsterdam, 1977),p. 371. [9]D. M. Brink et al.,J.Phys. G13,629 (1987). [10]M. Abramowitz and I.A. Stegun, Handbook ofMathemat ical Functions (Dover, New York, 1965),p. 773. [11]J.G6mez-Camacho, Phys. Lett. B185, 310(1987). [12]F. Scheid, Theory and Problems of Numerica! Analysis (McGraw-Hi11, New York, 1968),p. 125.