A microscopic calculation of potentials and inertia parameters for heavy-ion collisions

Nuclear Physics A339 (1980) 336-352 ; © North-Holland Publbhing Co., Anuterdmx Not to be reproduced by photoprint or microfam without written pamiuion from the publisher

A MICROSCOPIC CALCULATION OF POTENTIALS AND INERTIA PARAMETERS FOR HEAVY-ION COLLISIONS H . FLOCARD t, P. H . HEENEN tt and D. VAUTHERIN Division de Physique Théorique t, Institut de Physique Nucléaire, 91406 Orsay Cedex, France Received 29 October 1979 Abstract : Within the adiabatic time-dependent Hartree-Fock formalism, we compute the potential V(R) and the inertia parameter M(R) corresponding to the symmetric heavy-ion collisions ' 2C+' 2C and `60+ 160 . We find that the mass M(R) exhibits very sharp peaks . These peaks are shown to provide a plausible mechanism to explain the oceurence of quasi-moledular resonances.

1 . Introduction The present knowledge of the several components of the Bohr hamiltonian suitable for a heavy-ion collision appears rather uneven . We can now compute the collective potentials by . several methods ranging from phenomenological to self-consistent microscopic 1-6). In the last years our understanding of the dissipation phenomena and their interaction with the collective variables has significantly improved') . In contrast the structure of the collective kinetic energy has been relatively overlooked. As an example, most ofthe calculations assume that the value ofthe mass parameters associated with the separation distance of two colliding ions remains constant, and equal to the reduced mass value. The question of the consistency of this assumption with the other terms ofthe Bohr hamiltonian is also generally neglected. In this work we have calculated the potential and the mass parameter appropriate for a heavy-ion collision within the adiabatic time-dependent Hartree-Fock method (ATDHF). We have found unexpected sharp peaks in the collective mass as a function of the interdistance and investigated the relevance of these structures in the problem of quasi molecular resonances. The ATDHF formalism relies on the assumption that collective motion remains slow compared with uncoherent nucleonic motion. It becomes then possible to transform the time-dependent Hartree-Fock (TDHF) lagrangian into a form similar to the Bohr collective hamiltonian 8-11 ). Thus by restricting one's scope to the study t Present address : Nuclear Science Division, Lawrence Berkeley Laboratory, Berkeley California 94720 ; USA. tt Chercheur qualifié FNRS . Permanent address : Physique Théorique et Mathématique, Université Libre de Bruxelles, CP 229 Bruxelles, Belgique . t Laboratoire associé su CNRS . 336

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of slow collective motions (for heavy-ion collisions an estimate could be 1 MeV per nucleon above the Coulomb barrier in the c.m . frame) one gains a connection with the standard formalism in terms of a hamiltonian quadratic in the velocities and as a consequence one can derive quantal information on the dynamics . One therefore sees the advantage ofATDHF over TDHF . This contrasts with the ordinary TDHF from which one mainly extracts classical information on the behavior of the system. Over less fundamental methods ATDHF has the advantage ofproviding us with a microscopic and unified derivation of the collective potential and kinetic energy . As will be seen in the next sections this removes some of the ambiguities related with a specific choice of the collective variable . In this work we concentrate on the dynamics of the most important variable for a heavy-ion collision: their interdistance. For this reason we give in sect . 2 a short derivation of the well known ATDHF results for a single collective variable. We describe in sect. 3 a numerical method to solve the ATDHF equations. This method uses a 3-dimensional mesh in coordinate space and is therefore well suited for the study ofcollision processes. We also avoid the difficultproblem ofinfinite summations that generally occur in the calculation of collective masses . In sect. 4 we present the potentials and mass parameters for two light heavy-ion systems 12C+ 12C and 16 0 + 160 and discuss some of the consequences on molecular resonances. 2. Adiabatic time-dependent Hartree-Fork along a fixed path The adiabatic time-dependent Hartree-Fock approximation has been derived by several authors e-11). The equations define in principle .both the collective path and the dynamics. They are however rather complicated and up to now calculations have been restricted to dynamical problems for which one fixes a priori the collective path 12 .13) . Then the equations take the much simpler form ofa constrained HartreeFock problem 12). In this section we present a derivation of the ATDHF equations in this particular case without making the usual detour via the full ATDHF formalism [see also sect. 6 of ref. 9)] . Our starting point will be the time-dependent Hartree-Fork action and lagrangian = 2 = ~ .2 ./ -cPdt <«(t)lihd/dt-RI, (1) tt

t,

where R is the effective many-body hamiltonian and I0(t)> the most general Slater determinant. The set of Slater determinants {di(t), t1 5 t 5 t2} which render the action extremal is a solution of the TDHF equations 9.14). In order to introduce the adiabatic approximation one makes use of a geometrical property valid for any Slater determinant 1°). Under rather general conditions 19) a Slater determinant can be decomposed in its odd and even parts against time reversal I0(t)> = OWl0o(t)>, (2)

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where J(t) is a time-even hermitian operator and 10o(t)i a time-even Slater determinant . The ATDHF equations. along a fixed path can be deduced from the variation of the action si with respect to changes of 10(t)i restricted by the following conditions : . (i) 1 0o(t)i must be of the form 10o(R(t))i where {I0o(R)>, Rl 5 R 5 R2} is the family of time-even Slater determinants which determines the path . Although with slight changes the derivation that follows is valid for any path I(Po(R)>, we shall restrict our discussion to the particular case where a fixed time-even operator Q exists such that I0o(R)> is the solution ofthe constrained Hartree-Fock problem with a constraint AR)¢ : [fi(po ) -Â(R)Q, PO(R)] = 0.

(3)

In eq. (3) PO(R) is the one-body density matrix associated with 10o(R)i. In ordinary space representation its matrix elements read
=

Tói(r')(Poi(r),

(4)

where {Toi(r), i = 1, . . ., A} are a set of orthonormal single-particle wave functions from which I0o> can be built. The operator h(ß o) is the Hartree-Fock hamiltonian associated with the hamiltonian ft and the density Ao . (ii) The operator ,4(t) should be small so as to allow the truncation of the exponential contained in eq. (2). The significance of "small" within the ATDHF context is discussed in ref. 'I) subsect. 6.1. The adiabatic action is obtained by expanding to 2nd order in ,I the quantity zdt
is equal to

t= dt«ole - 'elh(d/dt)e' R l(koi Jt, i

f tt

dt< 4 PoIe - '2RG(R(t))e'1 I0oi,

where R = .dR/dt and the operator a(R) is defined in terms of 100(R)> as B(R) = ifi[OAolaR, po].

(6)

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The adiabatic action becomes then sa0', = £dtY. = £ dtI <15 o(R(t))> .

(8)

Let us fast study the variation of this action with respect to 1. We note that the action does not depend on X . The Euler equations are then 0Ylâx = 0 with fixed R and R. In other words we require that the expectation value of A- AR(R) should be extremal for any time-odd variation of the function 10> around . 100(R)> . For the particular case of a path defined by eq . (3) and in the limit R --., 0 this is equivalent to the following doubly constrained Hartree-Fock problem.

[fi(p) - Â(R)¢ - RÊ(R), i«R, R)]

=

0,

(9)

where p(R,R) is the density associated with the state I4P(R, R)) solution of (9). Indeed we note that the operators A(R)Q and "(R) are respectively time even . and time odd so that the time-even and time-odd parts of (9) decouple in the limit R -> 0 and become respectively identical with (3) and ôylôx = 0. Let us now derive the equation ofmotion for the variable R(t). Rather than working out explicitly the Euler-Lagrange equation 8Y/8R(t) = 0 we shall use a simpler method based on the conservation of energy . Since .19 is independent oftimewe know from Noether's theorem ") that the energy E = Tr(x0Y/ôX~+RaY/ôR-Y = <+<
(10)

is a constant of motion . Let us now notice that eq. (9) implies that the quantity K =

= «ole -'I Ae 'R1,poi -

(11) .

is the second-order variation of the energy when adding an external field AP(R). Therefore K can be written as l ') K = M(R)R2,

(12)

where M(R) is the static polarizability with respect to the operator P(r). Explicitly M(R) is given by z ô M(R) = âR. «R, R)I 1 (R)IoP(R, R)>IR=o = ôR z < O(R, R)Iftld)(R, R)ilii=o, . (13) where 0(R, R) is the Hartree-Fock ground state ofthe hamiltonian t - 4R)¢ - RA3(R} The energy E can thus be written in the standard form of Bohr's hamiltonian E = ZM(R)RZ + *-(R)

(14)

r(R) = <,Po(R)Iftj0o(R)i, ,

(15)

with

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and the equation of motion for the variable R is dt

{ZM(R)R Z +Y(R)} = 0.

In this work the operator Q used in eqs. (3) and (9) is the quadrupole moment. There are several advantages in this choice. For a system of well separated ions, the constraint on Q is equivalent to a constraint on the distance between the centers of mass of the ions. For compact shapes of the fused system we also know that the dominant collective mode is the quadrupole one. In addition it is known that the extrema of the potential curves V do not depend on the particular choice of the constraining operator. This means that the top of the heavy-ion barrier and any other extrema occuring along the path will be described correctly. From the derivation above it is clear that the exact definition of the quantity R which labels the collective path plays no role as long as I0o(R)i, D(R), *(R) and M(R) are calculated in a consistent way. To conform with general habit we selected a variable R which behaves asymptotically as the distance between the centers of mass of the ions. It is defined from . the expectation value ofthe quadrupole moment in the following implicit way iAR2 +Q1 +Q 2 (16) = < Oo(R)IQI0o(R)>,

where A is the total number ofnucleons and Q1 and Q2 the HF quadrupole moments of the two ions measured along the collision axis. As a last remark one sees from eq. (13) that the mass M(R) will be positive if the energy increases for a variation of the wave function leaving the . time-even part I45o(R)i unchanged. This is much less restrictive than" asking for stability of the RPA matrix for any point of the path. As an example, in the present case, the RPA matrix for ft is unstable while the RPA matrix for ft -)i.¢ is stable . The stability with. respect to time-odd variations along the path is a property of the effectice nucleonnucleon interaction and examples of some pathological forces may be found in ref. 18). 3. Calculations with the BKN force For computational simplicity we have used the effective nucleon-nucleon interaction of Bonche et al. (BKN) [ref. 19 )]. It gives the HF energy of the nucleus as E = T+ dr{étoP2(r)+ í tsP3(r)}+ V drdr, e-h-r Ir - rIla J 0J x P(r)P(rj+*e2 ('drdr' P(r)Ar, J Ir-r'I

(17)

where T is the kinetic energy and P(r) =
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parameters to, t3, -Yo and a are taken from ref. 19). The corrections due to Coulomb exchange are not included. Also we did not include c.m. corrections of the energy since these should not be performed within ATDHF (in fact subtracting a term from the hamiltonian would lead to a wrong value of the translational mass). There is no spin-orbit interaction and we neglect the isospin degree of freedom so that the degeneracy of each single-particle orbit is four. In order to reduce computational problems we also assumed spatial symmetries . Taking the origin at the c.m. of the total system we have imposed symmetry with respect to the three planes x = 0, y = 0 and z = 0 to each wave function. The Hartree-Fock equations (3) are particularly simple when the BKN force is used. For a nucleus of mass A in the external local field A(R)Q, the single-particle wave functions To,(r) satisfy a set of local Schrödinger equations {_ 2m

d +U(r)- .Z(R)¢(r) op jr) = eiop o Är). . }

(18)

The Hartree-Fock potential U is given by

+ e2 (19) . dr'p(r7 Voe- ' 41r-r j . Ir - r l/a Notice that the Hartree-Fock field (19) depends on the total density p(r) only . As a consequence the ATDHF expressions for the mass parameter (13) reduce to the cranking formula"), Indeed when we add in. eqs. (18) and (19) the small external field" in order to obtain the ATDHF eq. (9) the changes induced in p and therefore in U(r) will be second order in R t. Therefore to calculate the polarizability with respect to D it is. legitimate to replace in eq. (9) the Hartree-Fock field U(r) by its value U0(r) computed with the density AO(r) constructed with the solutions of eq . (18) . In other words it is sufficient to solve the following set of Schródinger equations 2 l U(r) = átop(r)+ -6t3p2(r)+

{- 2m

d + Uo(r)-A(R)¢(r)-R~ } ~p,(r) = e,~Pt(r).

(20)

Now for a given value of R the field stays constant instead of being determined selfconsistently . By applying first-order perturbation theory to eq. (20) we find that the ATDHF mass parameter M(R) reduces to the cranking mass M(R) = 2ft2 ~ t=1

J> .t

I
(21)

where ipo, and e, are defined by eq. (18). The fact that ATDHF leads to the cranking result is a special feature of the BKN force. It would not be the case for more general interactions e.g. for the Skyrme force' 3, 21).

f From eq . ('9) one sees that the change in the operator density ß, = ß'-ßu is linear in R . However the diagonal elements = óp(r) are zero because A is time-odd and the vector jr) is tie-even.

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342

The Hartree-Fock equations (18) and (19) and the one-body Schr6dinger equations (20) were solved in configuration space. Each orbit was discretized on a threedimensional rectangular mesh. The mesh size is Ax = 0.8 fin for carbon-carbon collisions and 1.0 fin for oxygen-oxygen. This is sufficient to ensure a precision on the potential curve Y(R) better than100 keV. The number ofmesh points is 16x 1.1 x 11 for carbon-carbon and 13 x 11 x 11 for oxygen . Because of the symmetries imposed on the system this corresponds to a 24 x 16 x 16 fin box for the 12C+ 12 C system and a 24 x 20 x 20 fin box for the 160+16 0 system . The Hartree-Fock equations were solved by means of the imaginary time method [ref. 22)]. The basic idea is to follow the time evolution of an arbitrary Slater determinant with a TDHF code using an imaginary time step. At large times the solution eventually converges to the HF ground state. . The operator A has been evaluated in the following separable form A

= ih E {Inix(Poil-I(Poixnt1}, i=1

where

InAR)>

is given by In A R)>

a

A

_IToi> - E ITo;>< . aR ,= 1

(22)

(23)

The derivatives occuring in the definition of Ini> have been constructed from the values of the Poi at R and Rf 0.05 fin or R f 0.1 fin. This step guarantees an overlap between Imoi> and ajToi>/aR less than 10 -4. As we shall discuss in the next section the mass parameter M(R) exhibits pronounced and sometimes sharp structures . In order to describe them accurately we had to solve eqs. (18) and (20) for numerous values of R . As an example the potential *^(R) for the 160+ 160 system has been calculated for 30 different points between the minimum and the top of the Coulomb barrier. A good check of the numerical solution ofeq. (25) is provided by the equality of the masses calculated from eqs. (13). In practice we have found agreement within a few percent. Another good check of the . numerical accuracy is provided by the value of M(R) at large R. Indeed within ATDHF, M(R) should take the reduced mass value in this limit io).1 In practice we have found M = 6.03 in carbon-carbon at R = 9 fin and M = 8 .05 in oxygen-oxygen at R = 11 fin. Let us emphasize that our calculation is free of the convergence difficulties due to continuum states which are encountered when using the Inglis formula (21), as was done in ref. S). In addition by solving the HF equation in coordinate space rather than by an expansion. on a basis we avoid the cumbersome problem ofthe optimization of the basis parameter as a function of R . We had only to check the stability of the results against a decrease of the mesh step Ax. Furthermore because we used the samemesh in all calculations the derivatives occuring in Ini> are obtained in a straightforward manner in contrast with calculations on bases which must use different bases for different values of R.

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34 3

4. Results We have studied two symmetric systems 12 C+ 12C and 160+160 . With the BKN interaction the HF ground state of 12 C is deformed (oblate) so that the potential *'(R) and mass M(R) depend on the relative orientation of the colliding nuclei. In this work we investigated two extreme cases. In the first configuration the symmetry axes of the fragments are aligned along the collision axis and the total system is axially symmetric. In the second case the symmetry axes of the two 12 C are parallel to each otherand perpendicular to the collision axis. The nucleus 160 being spherical, only one configuration needs to be examined TAHiLE

I

Number of nucleons corresponding to a given set px, pr p. of parities with respect to the x = 0, y = 0 and z = 0 planes, for the axial and triaxial configurations of the 12 C+ 12C system and for the 160+ 160 system ; the collision axis is the x-axis Px

P,

P.

+ + + + -

+ + + + -

+ + + + -

12C+ 12C

4 4 4 4 4 4 0 0

axial

12C+ 12C

triaxial

8 8 4 4 0 0 0 0

16 0+ 16 0 8 8 4 4 4 4 0 0

Because we impose three plane symmetries, the wave functions of the total system can be sorted in 2 3 blocks . In table 1 we indicate the occupation numbers for the three systems. In the calculation of a given set of curves -r(R) and M(R) the number of wave functions contained in each block is fixed by the corresponding value in the desired asymptotic channel . This prescription amounts thus to a sudden approximation for the occupation numbers. On the other hand, since we determine the collective path by solving the constrained Hartree-Fuck equations (18) and (19), we assume adiabacity with respect to variations -which do not alter the symmetry properties of the individual wave functions. The underlying physical assumption is that the time scale for a change in the symmetry properties of the system is longer than the reaction time which is itself longer than the time scale needed for a readjustment of the single-particle wave functions. The last inequality is supported by the fact that an important part of 'the rearrangement effects that occur during a collision is due to the Pauli principle. Self-consistent potentials ~(R) for 160+ 160 have been calculated by several authors with the Skyrme interaction 34); for the axial 12C+12C system with the Skyrme 6) and the Gogny interaction s) t. The HF potential with a Skyrme force for t

Ref. a) also contains information on the most external part of the axial "C+ "C potential.

44

H. FLOCARD et al .

Fig. 1 . Collective potential and mass parameter for the axial configuration of the 12C+ 12C system .

E-Em

MeV

12C

+ 12 C lriax 101

10

60

-,10~

440

420

Fig. 2. Collective potential and mass parameter for the triaxial configuration of the 12C+ 12C system.

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345

RFm

0

Fig . 3. Collective potential and mass parameter for the 160+160 system.

the triaxial "C+ t 2C system is given in ref. 6). The corresponding results with the BKN force shown in the upper parts of figs . 1-3 are very similar to those obtained in the. above references . The positions and heights of the Coulomb barriers are also consistent with elastic scattering data. As noted in refs. 23,6) in the 12C+ 12 C system the triaxial Coulomb barrier is smaller than the axial one and occurs at a larger radius (R = 9 fm compared to R = 6 fm). In our calculation the potential of the triaxial system shows a minimum around R = 6 fm corresponding to a 15 MeV excited state of 24Mg. As in ref. 6), we find that the corresponding nuclear density is very similar to that of two touching nucleL The minimum in the potential curve of the axial system corresponds to a 1 MeV excited state of 24Mg. (the HF ground state of 24Mg is triaxial) and a very compact shape. The lower parts of figs . 1-3 show the behavior of the mass M(R) for the same systems. The most striking phenomenon is the appearance of very sharp peaks. In the case of 12C+ 12C the maximal ratio of the mass parameter to the reduced mass value reaches 10 while it does not exceed 2.5 for 160+160. The peaks happen for values of R located between the interdistances for the minimum of the potential and for the Coulomb barrier. The reduced mass value expected for large interdistances is in fact already reached at the top of the Coulomb barrier. At the minimum of the potential the mass is about two thirds of the asymptotic mass. From this point on, it increases steadily as the interdistance of the ions decreases. Let us now investigate the meaning of the previous peaks. From the cranking

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H. FLOCARD et al.

formula (21) one notices that an increase of the mass can be either due to a rapid change in the structure of the wave functions (numerator increases) or a crossing or quasi-crossing of single-particle levels (denominator vanishes). One can rule out the second possibility since in the region where the peaks occur, there is no unoccupied level which comes near an occupied one. This is particularly evident for the most external peak located in the vicinity of the Coulomb barrier. Indeed from the upper part 160 + 160 R-7 .75 fm

R.- 6 .35 fm

fm

Fig. 4. Contour lines of the density integrated over the coordinate normal to the scattering plane for the 16 0+ 16 0 system at R = 7.75 and 6.35 fm .

offig. 4 one sees that the density ofthe 160+160 system at this point corresponds to two barely touching fragments and we know from two-center potentials studies, e.g. ref. 2s), that the single-particle spectrum in this case is almost that of two-unperturbed 160 nuclei. Therefore the structured the functions M(R) is almost entirely due to a variation ofthe numerators in eq.. (21). As a check we computed as a function of R the quantity

E

i=1 J>,1

( I 1No i >l2 = I< Poj a DR . ,= 1

and checked that it behaves like M(R). The peaks can also be due to a discontinuity in the path as would occur in the case of a first-order transition from one collective valley to an .other. We know that such a transition would induce a slope discontinuity in the potential curve and a discontinuity in all other observables except the quadrupole moment which corresponds to our constraint. From the consideration of

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347

single-particle energies which show no sign of a discontinuity, we can also rule out this possibility. It is thus very likely that the rapid change of the mass for some interdistances tells us that the adequate metric along the collective path is sometimes very different from the one we chose a priori by defining the interdistance R with eq. (16). One can note that M(R) would also exhibit peaks if we had defined R as twice the distance to the origin of the c.m. of the nucleons with x > 0. Thus, more than an incorrect geometrical definition of R, the peaks occuring in M(R) are the signature of a dynamical effect . The most external peak is found for an interdistance slightly smaller than that of the Coulomb barrier. It corresponds to the point where the violation of the Pauli principle that would result from the overlap ofunperturbed wavefunctions becomes so strong that a drastic rearrangement ofthe orbitals becomes necessary. Such a rearrangement necessarily requires a finite amount of time during which the interdistance remains in the vicinity of the peak . The associated decrease of the speed R is achieved by an increase of the mass (the kinetic energy ZMRZ is almost constant for small variations of R since *~(R) is smooth). We shall now show that each peak can be assigned to specific subsets of singleparticle wave functions . From figs. 1-3 we note that M(R) exhibits two peaks for 160 + 16 0 and triaxial " C+ 1ZC and only one for axial "C+ "C. If we call Ox the collision axis the lpx orbits of each ion are filled in the first two systems and not in the axial 12C + 12C system. These wavefunctions have themaximal extension along Ox so that they are the first to interact with the wave functions ofthe colliding nucleus. Our results are thus consistent with the assignment of the most external peak, to the rearrangement of Ipx orbits and the internal peak to the 1py, 1p. and is orbits. The mass for the axial 12C+ "C system shows only the internal peak . One can also note that the internal peak is broader, probably because the rearrangement of the corresponding wave functions is more'gradually induced via the self-consistent deformation of the Hartree-Fock field. The structures of M(R) are smaller for 16 0+ 160 than they are for "C+ "C. This is an indication that single-particle effects become less pronounced. with increasing mass ofthe nuclei Another formulation would be that the interdependence of the orbits via the mean field increases with the mass. For heavy systems it is thus reasonable to expect a mass M(R) without peaks, exhibiting only a large bump slightly inside the Coulomb barrier followed by a broad minimum at shorter interdistances. Calculations for the 4°Ca+ 4° Ca system are now under way to check this hypothesis. Such a behavior ofthe mass parameter must affect the quasi molecular resonances. In this work we investigate this influence for the 160+160 system only. Indeed as was shown in ref. 6), the two potential and mass curves obtained for 12 C+ 12 C must be used within a formalism that describes both the elastic channel and the single and double excitation of the 2+ state of the 12C nuclei This implies that one must solve a set of coupled equations, where these three channels are included . The 160+160 case is free ofthis problem and appears more suitable for a first investigation .

H. FLOCARD et al .

348

Fig. 5. Eigenvalues of the collective Schrádinger .equation, calculated with the reduced mass (solid line) and the adiabatic mass M(R) (dashed line) for the ' 60+' 60 system . The angular momentum scale is linear in L(L+1).

In fig. 5 we compare the spectra of the collective Schródinger equations fi2 V -

2M(R) 2M 6

V+ ~r(R) T(R) = E97(R),

. d + *^(R)J T(R) = ET(R),

(24)

(25)

where Mo = 8m is the reduced mass value. The previous equations have been solved by standard techniques 24,25). The spectrum obtained with .a constant mass exhibits two bands of resonances, the lowest one being almost purely rotational . It agrees quite well with the results ofresonating group and generator coordinate calculations. Such calculations use oscillator wave functions with a fixed parameter, and include angular momentum projection and a proper antisymmetry ofthe total wave function. Results obtained with the Volkov force 26) exhibit two bands quite close to ours, while only one band, very similar to our second band, is obtained with the Brink and Boeker force 2') . This similarity indicates that an exact projection on the relative angular momentum is not essential for the 160 + 160 system, and that the rearrangement effects included in eq. (25) are mainly due to the Pauli principle. When the

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349

effective mass M(R) is used low angular momentum states . are nearly unchanged. However states with large angular momenta are significantly lower and the rotational structure of the first band breaks down. This seems to confirm that the structure in the mass parameter M(R) is due to a self-consistent rearrangement of the individual wave functions, which is not included in resonating group calculations .

Fig . 6 . Collective potentials, including the centrifugal barrier for L = 0 and 12, for the 160+ 160 system, calculated with the reduced mass (solid line) and the adiabatic mass M(R) (dashed line) .

To understand the lowering of large L states when M(R) is used, we compare in fig. 6 the potentials including the centrifugal barriers, vL(R)

=

'r(R) +

WL(R) = 'K(R) +

a 2f L( R+

1)

,

hZ L(L+1) Rz . ' 2M(R)

for L = 12. It may be seen on this figure that the large value of M(R) near R = 6 fm lowers the centrifugal barrier significantly. The minimum in COL is thus deeper than the minimum in vL. A deeper minimum and a larger mass in this region both induce a lowering of the corresponding state. It may also be noted on fig. 6 that. the minimum in coL lies further outside than the minimum in vi. We thus expect the L = 12 wave function to be shifted outwards .

350

H. FLOCARD et al.

0

4

6

8

10

R(fm)

Fig. 7. Radial wave functions of the first L = 12 state obtained with the reduced mass (solid line) and the adiabatic mass M(R) (dashed line) for the 16 0+ 16 0 system .

It can be checked that this is indeed the case on fig. 7, which shows the L = 12 radial wave functions. To make the comparison easier we have normalized both functions to unity inside a sphere of radius Ro = 12 fm and we have used the boundary condition W(R O) = 0. It is seen that the peak in the wave function is sharper and shifted by about 0.7 fin when M(R) is used instead of Mo. An interesting consequence is that, since large angular momentum resonances with M(R) have their wave functions localized far away (about 6.5 fnm), they would presumably persist in the presence of a realistic imaginary potential. The peaks we have obtained in the mass parameter M(R) may thus provide a plausible scheme to explain the molecular resonances which have been observed in some light ion-ion collisions 28 .291, The above results remain qualitatively unchanged if a different quantization of the collective hamiltonian is used. However quantitative changes occur because of the rapid variations in M(R). For instance ifthe collective kinetic energy is quantized as P2/4M+ (114M)P2 instead of P(112M)P as was done in eq. (24), energies in thefirst band become -1 .31, -0.60, 0.96, 3.15, 5.52, 7.57 MeV for L = 0, 2, 4, 6, 8,10 respectively instead of -1.90, -1 .12, 0.60, 3.03, 5.84, 8.71 MeV. 5. Conclusion We have presented a method suitable to calculate inertial parameters for heavyion collisions . It uses the framework of the adiabatic time-dependent Hartree-Fock approximation. At each value of the interdistance R this method requires only the numerical solution ofa set ofHartree-Fock equations. We have shown that in practice,

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calculations are most efficiently performed by using a three-dimensional mesh in coordinate space. Results have been presented for the symmetric heavy-ion collisions t 60 + t 60 and "C+ "C. We have found sharp peaks in the collective mass M(R) in the region where the fragments are touching. This result is in contrast to the assumption of a constant M(R) usually made in phenomenological calculations . These peaks can be assigned to specific single-particle orbits and their height decreases significantly when going from the "C + t ZC system to the 160+160 one. The effect ofthe previous peaks on the quasimolecular resonance band has been worked out for the 160+160 system . We compared the solutions of the collective Schr6dinger equation with either M(R) or with a constant mass equal to the reduced mass M(R) = Mo. We showed that while small angular momentum states are almost unaffected by M(R), a significant lowering of the energy arises for large angular momenta. As a result the L(L + 1) energy dependence of the rotational bands which is observed with a constant mass disappears when M(R) is used and states with large L's are more bound. Thus the peaks in M(R) induce a greater number of sharp quasimolecular resonances. In other words the inclusion ofthe effective mass has a stabilizing effect on the rotational molecular bands. An interesting open question is to know whether these resonances will survive to the addition of a realistic imaginary part to the ion-ion potential. This might happen because the effective mass has a tendency to shift the wave functions towards larger radii where the imaginary potential is generally less effective. It would also be worthwhile studying in more detail the 12C+ 11C system which is more complicated due to the existence of coupled channels in the collective Schródinger .equation 6). Another important problem is to find out whether the peáks in M disappear for collisions of heavier nuclei, which, for these systems, would justify the assumptions of phenomenological models . Work in this direction is now in progress. We are most grateful to M. V6néroni for many stimulating discussions. We also wish to thank D. Baye for valuable comments, and the Service de Calcul de la Faculté des Sciences d'Orsay for extended computing facilities . References

1) H. J. Krappe and J. R. Nix, Proc . Third IAEA Symp. on the physics and chemistry of fission, vol. 1 (IAEA, Vienna, 1974) p. 159; J. Blocki, J. Randrup, W. J. Swiatecki and C. F. Tsang, Ann. of Phys . 105 (1977) 427 2) D. M. Brink and F. Stancu, Nucl . Phys . A243 (1975) 175 ; C. NgB, B. Tamain, .M . Beiner, R. J. Lombard, D. Mas and H. H. Deublet, Nucl. Phys . A252 (1975) 237; G. Reidemeister, Nucl . Phys. A197 (1972) 631 3) H. Flocard, Phys. Lett . 49B (1974) 129 4) P. G. Zint and U. Mosel, Phys . Rev. C14 (1976) 1488 5) J. F. Berger and D. Gogny, Nucl. Phys. A333 (1980) 302 6) J. Cugnon, H. Doubre and H. Flocard, Nucl . Phys . A331 (1979) 213

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7) H. A. Weidenmüller, Progress in particle and nuclear physics, ed . D. H. Wilkinson (Pergamon, Oxford, 1979) 8) F. M. H. Villars, Nucl. Phys . A285 (1977) 269 9) D. M. Brink, M. J. Giannoni and M. V6néroni, Nucl . Phys . A258 {1976) 237 10) M. Baranger and M. Vén6roni, Ann. of Phys. 114 (1978) 123 11) K. Goeke and P. G. Reinhard, Ann. of Phys . 112 (1978) 328 12) M. J. Giannoni et al ., Phys . Lett . 65B (1976) 305 13) M. J. Giannoni and P. Quentin, Phys . Rev., in press 14) A. K. Kerman and S. E. Koonin, Ann. of Phys . 100 (1976) 332 15) M. J. Giannoni, J. Math. Phys., in press 16) N. N. Bogoliubov and D. V. Shirkov, Introduction to the theory of quantized fields (Interscience, New York, 1959) p. 19 17) D. J. Thouless and J. G. Valatin, Nucl. Phys. 31 (1962) 211 18) B. D. Chang, Phys. Lett . 56B (1975) 205 ; S. O. Biickman, A. D. Jackson and J. Speth, Phys . Lett. 56B (1975) 209 19) P. Bonche, S. E. Koonin and J. W. Negele, Phys. Rev. C13 (1976) 1226 20) D. R. Inglis, Phys. Rev. 103 (1956) 1786 21) See for instance H. Flocai-d, Nukleonika 24 (1979) 19 22) K. T. R. Davies, H. Flocard, S. J. Krieger and M. S. Weiss, Phys. Rev., in p.-ess 23) H. Charidra and U. Mosel, Nucl. Phys . A298 (1978) 151 24) D. Vautherin and D. M. Brink, Phys. Rev. C5 (1972) 626 25) C. B. Dover and N. V. Giai, Nucl . Phys . A177 (1971) 559 26) T. Ando, K. Ikeda and A. Tohsaki-Suzuki, Prog . Theor. Phys . 59 (1978) 306 27) D. Baye and P. H. Heenen, Nucl . Phys. A276 (1977) 354 28) D. A. Bromley, J. A. Kuehner and E. Ahnquist, Phys. Rev. Lett. 4 (1960) 365 ; Phys . Rev. 123 (1961) 878 29) Proc . Conf. on Nuclear molecular phenomena, Hvar, 1977, ed . N. Cindro