Pockets in heavy ion potentials for the deformed systems Th + Ta, U + Th and Th + Th

Nuclear Physics A466 (1987) 439-444 North-Holland, Amsterdam

POCKETS

IN HEAVY SYSTEMS

M. RASHDAN’,

ION POTENTIALS Th+Ta,

Amand

U+Th

FAESSLER,

FOR THE DEFORMED

AND Th+Th*

M. ISMAIL’

and

W. WADIA

Institut fir Theoretische Physik, Universitiit Tiibingen, D-7400 Tiibingen, West Germany Received 3 November (Revised 5 December

1986 1986)

Using a realistic nucleon-nucleon interaction the optical potentials for the strongly deformed systems Th + Ta, U + Th and Tb + Th are calculated at E&A = 6 MeV and for different orientations. Pockets exist in the total real potentials (nuclear plus Coulomb potentials) at surface contact. It has been found that the deepest pockets for the U+Th and Th+Th systems exist when the two colliding nuclei are touching with orientation angle p - 45” while for the Th + Ta system the deepest pocket occurs at p = 90” (equator touching).

Abstract:

1. Introduction TO explain the discrete lines of positrons and electrons in the collisions between heavy nuclei with combined charge 2, = Z, + Z, > 163 at bombarding energies near the Coulomb barrier rm3) one has in some works assumed that the colliding nuclei stick together for times of the order of lo-l9 - 1O-2os, long compared to time scale of collisions assuming Rutherford trajectories. A possible mechanism for producing such time delays is the existence of a minimum or a “pocket” in the heavy-ion (HI) potential at surface contact “). The notion of a pocket is supported to some extent by theoretical models. For example, Seiwert, Greiner and Pinkston ‘) proposed a phenomenological model for HI potentials, equivalent to the double-folding model for large separations and to the liquid drop model in the limit of spherical composite systems which could exhibit the existence of pockets in the HI potential. Pb + Pb, Pb + U and U + U systems.

Their calculations

are performed

for the

Faessler et al. 6-8) proposed a method for calculating the real and imaginary part of the optical potential between heavy nuclei starting from a realistic nucleonnucleon (NN) interaction (Reid soft-core potentials). They consider the collision of two heavy nuclei as a collision of two infinite nuclear matter distributions flowing through each other. They solve the Bethe-Goldstone equation for this situation. In momentum space, the system can be characterized by two overlapping Fermi spheres l This work is supported partly by the Atomic Energy Establishment of Egypt and the International Bureau of Kernforschungsanlage in Jiilich, West Germany and the GSI Darmstadt. ’ Permanent address: Department of Mathematics and Theoretical Physics, Atomic Energy Establishment, Cairo, Egypt. ’ Physics Department, Cairo University, Cairo, Egypt.

0375-9474/87/%03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

440

M. RaskdaR et al. / Pockets in keauy-ion potentials

separated by the relative momentum of the two heavy ions. The resulting reaction matrix is complex because the energy denominator can have a pole due to the non-sphericity of the two overlapping Fermi spheres. This reaction matrix allows to calculate a complex energy density functional which is used to calculate the real and imaginary parts of the HI optical potential. This approach is used to study the energy and deformation dependence of the optical potentials of the Pb+ Pb and U+ U systems *). Pockets are predicted in the total real potentials (nuclear plus Coulomb potentials) with depths depending strongly on nuclear orientations. The aim of the present work is to extend these caIculations to the strongly deformed systems Th + Ta, U + Th and Th + Th. The calculations are performed at the laboratory energy E;,,/A ~6 MeV. In sect. 2 we briefly discuss the method used for calculating the HI potentials and in sect. 3 we present our results and discuss them. 2. Calculation

of the HI potentials

2.1. THE OPTICAL POTENTIAL

In this subsection we briefly outline the method used in our calculation of the HI optical potential since the method is fully discussed elsewhere ‘-‘). The optical potential between two colliding nuclei is defined as the difference between their energy at finite distance R and at infinity 6-8), i.e. V&-G, R) = EK,

W - EK,

00).

(1)

Here K, is the average relative momentum per nucleon which is related to the laboratory energy of the projectile by the relation

(2) where A, is the mass number of the projectile and m is the nucleon mass (939 MeV). The total energy E is obtained from the energy density functional ~(r; rC,, R) according to E(K,,

R) =

d3rK(r; K,, R) ,

(3)

where the energy density functional K is defined as ‘) KCr; K,, R) =g

r(Ar),

Ar),

K)+ r&(r),

pp(r), G)

(4)

where (h/2m)T and w are the kinetic energy density and the complex potential energy density, respectively, of the total system. These densities are calculated in refs. 6-8) in momentum space for two non-, partially- and totally-overlapping Fermi spheres using a generalized Iocai density approximation 6-s). pT and pp appearing in eq. (4) are the matter density dist~butions of the target and projectile, respectively.

M. Rashdan 2.2. THE COULOMB

441

et al. / Pockets in heavy-ion potentials

POTENTIAL

The Coulomb potential is taken into account as the double-folded

potential (5)

where p$ and p$ are the charge density distributions of the projectile and target, respectively. The six-dimensional integral (5) is solved numerically using a momentum space method which is fully discussed in refs. 9-‘o). 3. Results and discussion Eqs. (l)-(4) are used the Coulomb potential. of the individual nuclei normalization and that way.

to calculate the HI optical potential and eq. (5) to calculate We assumed that both the charge and matter distributions can be described by the Fermi form 8*9)with the appropriate the nuclear radius parameter C is parametrized in the usual C(O)= ~0~~+~*0~20~~~+~,~~0~~~1~

(6)

The values of the parameters are determined experimentally in ref. ‘l) for U and Th nuclei and in ref. 12) for the Ta nucleus. Fig. la, lb and lc show the total real potential (nuclear+Coulomb), the real and the imaginary parts of the optical potential, respectively, for the Th+Ta system at Elab/A = 6 MeV and at three different orientation angles /I = 90”, 45”, 0”. Figs. 2a, 2b and 3a, 3b show the total real and nuclear potentials for the systems U+Th and Th+Th, respectively. The imaginary parts of the optical potential for these two systems are found to depend on orientation in a similar way like that indicated by fig. (Ic) for the Th + Ta system. As for the real potentials the differences are more significant. As we see from figs. (2a) and (3a), the behaviour of the total potentials for the two systems U + Th and Th + Th are similar. In both cases the deepest pocket occurs at the orientation angle p = 45” while for the Th + Ta system the total potential behaves differently. In this case, the deepest pocket occurs at p = 90” (equator touching) as shown in fig. la. Another noticeable difference is at /J = 0”, where a small pocket appears in the total potential of the Th + Ta system while none appears for the other two U+Th and Th+ Th systems. These different trends of the total potentials is mainly due to the different behaviour of the nuclear potentials as seen by fig. lb in comparison with figs. 2b and 3b. These figures indicate that at p = 90” and p = O”,the nuclear potential for the system Th + Ta is more attractive (especially in the case of /3 = 90”) at surface contact than for the other two systems U+ Th and Th + Th. In the case of p = 45” we have an opposite relative behaviour. These different behaviours of the nuclear potentials, at the surface contact, can be explained as due to different hexadecapole deformations. Both U and Th nuclei have a positive hexadecapole deformation. In this case the prolate shape widens in the middle “)

442

M. ~ashdan

: :

% E

I\

et al. / Pockets in heavy-ion potentials

Th + Ta

E,JA

= 6Me'

p=90" -.- p=45'

Ikd, , , , , , , , ,

5aJ 10

18

14 DISTANCE

Rlfml

Fig. la. The total real potential (the real part of the optical potential plus the Coulomb potential) of the Th+Ta system at E&,/A = 6 MeV. The solid, dot-dashed and dashed curves correspond to orientation angles j3 = 90*, /3 = 45” and p = 0”, respectively. The orientation angle is defined as the angle of the symmetry axis of each nucleus relative to the connection of the two nuclei. Both nuclei are oriented in the same way. DISTANCE

R [fml

DISTANCE

R lfml

^_

2 r

Th + Ta

L

g-10-

E&A

&

p-90" -.- pzl+5" ____p: 0"

I

3

--zo-

1:6MeV

1

,,3 Fig. lb, c. The real and imagina~ parts of the optical potential of the Th+Ta system at E,,JA and at o~entation angles fi = 90”, 45”, 0”.

= 6 MeV

\

5 : ,

U + Th E,,blA

= 6 MeV

./,’ ,../

,

600 ?.J

,

,

,

,

,

Il’kTANCE

I

2

R [fm]

-UQ

lb),, , ,

Fig. 2a, b. Same as fig. la, b but for the U+Tb I

I

I

I

,

I

I

system.

,

Th + Th E&A

)

DISTANCE

R

[fml

= 6 MeV

--p=!v -.p = 45* -_-- p = 0’

_--_ p = 0’

650

I ”

&31(a), 10 I

I

I

,

,

15 DISTANCE

I

,

R lfml

‘,

4’

20

Fig. 3a, b. Same as fig. la, b but for the Tb+Tb

system.

444

M. Roan

et al. / Pockefs in heavy-ion potentials

and the contact surfaces in a collision where the two heavy nuclei have both positive hexadecapole deformation are expected to be maximum around B = 45”. On the other hand, the Ta nucleus has a negative hexadecapole deformation which narrows the prolate shape around the middle ‘*) and the maximum contact areas for a collision of the type Th+Ta, where the two nuclei have an opposite hexadecapole deformations, is expected to be around @= 90” (equator touching). We therefore conclude that the real part of the potential of two heavy nuclei can have pockets for special orientations relative to each other. A quantitative difference for forming pockets is found due to the opposite sign of the hexadecapole moment of Ta (p4 < 0) and the heavier nuciei Th and U. This changes the relative orientation at which the overlap of the two nuclei sets in most rapidly during the approach and thus the orientation angle at which the two nuclei have a pocket. References 1) M. Clemente, E. Berdermann, P. Kienle, H. Tsertos, W. Wagner, C. Kozuharov, F. Basch and W. Koenig, Phys. Lett. 137B (1984) 41 2) T. Cowan, H. Backe, M. Beyemann, K. Bethge, H. Bokemeyer, H. Folger, J.S. Greenberg, H. Grien, A. Gruppe, Y. Kido, M. Khiver, D. Schwalm, J. Schweppe, K.E. Stiebing, N. Trautmann and P. Vincent, Phys. Rev. Len. 54 (1985) 1761; T. Cowan et al., Phys. Rev. Lett. 56 (1986) 444 3) 3. Reinhardt, B. Miilier and W. Greiner, Z. Phys. A303 (1981) 173; A. Schafer, J. Reinhardt, 9. Miiller and W. Greiner, 2. Phys. A324 (1986) 243 4) U. Heinz, V. Miiller. J. Reinhardt, 9. Miiller, W. Greiner and W.T. Pinkston, 2. Phys. A316 (1984) 341 5) M. Seiwert, W. Greiner and W.T. Pinkston, J. Phys. Gil (1985) L21 6) A. Faessler, W.H. Dickhoff, M. Trefz and M. Rboades-Brown, Nucl. Phys. A428 (1984) 271~ 7) M. Trefz, A. Faessler, W.H. Dickhoff, Nucl. Phys. A443 (1985) 499 8) M. Ismail, M. Rashdan, A. Faessler, M. Trefz and H.M.M. Mansour, Z. Phys. A323 (1986) 399 9) M.J. Rhoades-Brown, V.E. Oberacker, M. Seiwert and W. Greiner, Z. Phys. A310 (1983) 287 10) M. Rashdan, Ph.D. thesis, Cairo University (1986), Cairo, Egypt 11) T. Cooper et al., Phys. Rev. Cl3 (1976) 1083 12) R.J. Powers, F. Boehm, A. Zehnder, A.R. Kunselman and P. Roberson, Nucl. Phys. A278 (1977) 477