Dinuclear system in diabatic two-center shell model approach

ELSEVIER Nuclear Physics A 652 (1999) 61-70

NUCLEAR PHYSICS A

www.elsevier.nl/locate/npe

Dinuclear system in diabatic two-center shell model approach

A. Diaz-Torres a, N.V. Antonenko a,b, W. Scheid a

a lnstitutf~ir Theoretische Physik der Justus-Liebig- Universitiit, D-35392 Giessen, Germany b Joint Institute for Nuclear Research, 141980 Dubna, Russia

Received 11 January 1999; revised 23 March 1999; accepted 6 April 1999

Abstract

Diabatic potentials as a function of the elongation and neck parameter for various symmetric heavy-ion systems are studied within the generalized two-center shell model. In the calculations the maximum symmetry method is applied. The diabatic potentials show hindrances for the motion to a smaller relative distance and for the growth of the neck. They are similar to nucleus-nucleus potentials used in the dinuclear system model of fusion which is able to describe the experimental data. The dependence of the diabatic potential on temperature and mass asymmetry is discussed. @ 1999 Elsevier Science B.V. All rights reserved.

PACS: 25.70.Jj; 24.10.-i; 25.70.-z Keywords: Complete fusion; Diabatic two-center shell model; Dinuclear system; Superheavy nuclei; Diabatic potentials

1. Introduction

The synthesis of transuranium and superheavy elements [ 1,2], the production of superdeformed nuclei and nuclei far from the line of stability stimulate the study of fusion processes in heavy-ion collisions at low energies (< 15 MeV/nucleon). Existing theoretical models can be distinguished by the choice of the relevant collective variables in which the fusion occurs. The relative distance between the centers of nuclei R (or elongation of system) and the neck degree of freedom play an essential role in the macroscopic dynamical model [3] which is based on an adiabatic approach to the fusion. Without regarding the competition between complete fusion and quasifission, this model overestimates the fusion cross section of heavy nuclei [4]. Attempts were made to improve this model by including thermal fluctuations and the competition between complete fusion and quasifission in [5 ], but an agreement with the experimental data

0375-9474/99/$ - see front matter (~) 1999 Elsevier Science B.V. All rights reserved. PII S0375-9474(99 )00148-7

62 A. Diaz-Torres et al./Nuclear Physics A 652 (1999) 61-70

was not achieved for the reactions treated in this study. Corrections to these results

could arise from nuclear structure effects in collisions with energies slightly above the Coulomb barrier. In a recent paper [6], the shell effects were incorporated by using

the two-center shell model (TCSM) [7]. The study of the dynamics of fusion within the adiabatic TCSM overestimates the fusion probabilities for most symmetric and near

symmetric reactions. Moreover, the isotopic dependence of the fusion probability resulted incorrectly. Therefore, hindrances for the growth of neck and for the motion to smaller

values of R should be assumed to explain the experimental data. This hindrance allows the dinuclear system (DNS) to keep a relatively small neck over a time comparable with the reaction time. Then the fusion proceeds as a motion of the DNS in mass asymmetry. Such a hindrance of the motion to smaller R is explicitly assumed in the

DNS model [8] which describes the experimental fusion data quite well. The nuclei

in the DNS could be hindered to melt together in R due to diabatic effects [9-13] or a specific behavior of the mass parameters. In a diabatic description the nucleons do not occupy the lowest free single-particle levels as in the adiabatic case, but remain in

the diabatic levels during a collective motion of the nuclear system. As a result, the diabatic potential energy surface is raised with respect to the adiabatic potential energy surface and new potential barriers for collective variables may appear. The values of

these barriers can also be estimated by calculations of the structure forbiddenness of fusion [9].

In this paper we study the diabatic potentials for symmetric heavy nuclear systems as a function of the elongation ,~ and the neck coordinate e using the method of maximum

symmetry [ 1 1] within the generalized TCSM [7]. In the previous calculations [ 12] a simplified version of the TCSM was used and the neck coordinate was fixed at e = 1. We will compare our results with nucleus-nucleus potentials obtained by using a double folding procedure for the nuclear interaction. The calculations are performed for the symmetric systems 9°Zr+9°Zr, 96Zrff-96Zr, l°°Mo+l°°Mo, l~°Pd+t I°Pd, 13°Xeq-13°Xe and

136Xeq-136Xe. The applicability of the diabatic method will be discussed for collisions

near the Coulomb barrier.

2. Diabatic potential for symmetric systems

2.1. Genera l cons idera t ions

The TCSM is the most suitable microscopical model for the DNS. The usual parametrisation [7] of the two-center potential consists of the relative distance between the centers (or elongation), the mass (charge) asymmetry 71 = ( A l - A 2 ) / ( A 1 + A2), the deformations of the fragments /~i and the neck coordinate e. The elongation ~ = l / ( 2 R o )

measures the length 1 of the system in units of the diameter 2Ro of the spherical compound nucleus. This variable can be used to describe the relative motion. The transition of the nucleons through the neck is described by the mass asymmetry 77. The neck parameter e -- E o / E t is defined by the ratio of the actual barrier height Eo to the barrier

A. Diaz-Torres et al./Nuclear Physics A 652 (1999) 61-70 63

height E r of the two-center oscillator. The deformations 1~i = a i /b i of axial symmetric

fragments are defined by the ratio of their semiaxes. The neck grows with decreasing e.

Within the TCSM two methods for the construction of diabatic states were devel- oped [ 11 ]. The method of maximum overlap is based on the construction of crossings of diabatic levels where pseudo-crossings occur in the adiabatic TCSM. The construction of the diabatic states as a function of the relative distance starts from the separated

nuclei where adiabatic and diabatic states coincide. Adiabatic state at a smaller relative

distance is obtained from a linear combination of a few adiabatic states which maxi-

mizes the overlap with a diabatic state at somewhat larger relative distance. Using the so

constructed states, one is able to obtain step by step the diabatic states for smaller relative distances. The method of maximum symmetry eliminates the symmetry-violating parts H r from the total Hamiltonian H of the TCSM. Then the diabatic states are the eigenstates of the difference H - H I, which is taken as the diabatic TCSM-Hamiltonian in the case of equal nuclei (symmetric system). For slightly asymmetric systems, the diabatic states are defined by the expansion of the asymptotic states in terms of the

eigenstates of the symmetric system. Diabatic levels obtained with this method agree with those from the maximum overlap procedure [ 12]. In the following calculations we use the method of maximum symmetry because it turns out to be numerically easier to

handle.

2.2. Diabatic potential

The total diabatic potential is defined as

Vdiab(q) : Vadiab(q) -q- AVdiab(q), ( 1 )

where the set of collective coordinates of the system is denoted by q. Vadiab is calculated within the TCSM using the liquid drop energy, the Strutinsky prescription for the shell correction and the proximity nuclear potential to improve the adiabatic energy for large

elongations [6]. The diabatic contribution AVdiab is expressed as

AVdiab (q ) : Z ~3daiab (q) " nadiab -- 2 .~ eaadiab.tq)n,~- adiab-tq)-

o~ ot

= Z sadiab (q) " {'nadiab __ t/aadiab-tq))- -J- Z " adiab / - \ ( s d i a b ( q ) r t c t I, tt) -- ~aadiab'~q)" )

Ol Or

~ / 3 d i a b ( q ) . diab adiab ~ ~£..~ ot tnot -not t q ) ) , (2)

ot

where the contribution from the second term with (~diab(q) -- ~a adiab/~x~,t4) )x is assumed to

be negligible because the adiabatic and diabatic single-particle levels differ only in the area of the pseudo-crossings of adiabatic states [ 12]. The occupation probabilities "~-diab

are determined by the configuration of the separated nuclei. The adiabatic occupation probabilities _adiab vary with q according to the ground-state configuration where only / t o t

the lowest levels are occupied. The diabatic levels _diab ~ are classified by the quantum

64 A. Diaz-Torres et aL /Nuclear Physics A 652 (1999) 61-70

numbers a = Jz, lz, Sz, np, nz of the eigenstates of the diabatic Hamiltonian. We only used the diagonal elements of the symmetry-violating parts of the generalized TCSM Hamiltonian [7,10]. The influence of the neck parameter ~ is taken into account by means of the diagonal contribution of the difference (see Ref. [7] )

H i ( e ) - H j ( ~ = 1) - ( e - 1)mow~z2(1 + c z + d z 2) (3) 2

where the coefficients c and d are determined by requiring that the potential and its derivative are continuous with respect to z at z = 0. For z < 0 and z > 0, the oscillator frequencies Wz must be determined numerically from the assumption of volume-

conservation. With the method suggested we can find the diabatic levels close to the adiabatic levels. Differences take place only near the crossing points. In contrast to Ref. [ 12] we can consider the diabatic effects for any neck parameter ~ which yields a better shape parametrisation of the DNS.

3. Results and discussion

We show results of calculations for various symmetric systems. The total diabatic potential is studied as a function of the relative distance R (or elongation A) and the neck parameter e. The diabatic contribution AVdiab as a function of the elongation is presented for the reaction l°°Mo+~°°Mo in Fig. 1. The nuclei are considered as spherical with e = 0.74 which supplies realistic shapes of the DNS for ,~ = 1.5-1.6. AVdiab consists of contributions from neutrons and protons. The diabatic contribution increases with decreasing h or R because many diabatic levels cross the Fermi level. In general, the diabatic contribution increases with the mass number A of the system because of the larger number of level crossings. The diabatic contribution AVdiab of many symmetric systems selected along the line of beta stability shows diabatic shell-structure

effects [9,14]. The role of these effects is demonstrated in Fig. 2a by the diabatic

I ' I ' I " f ' I " I

300 ~ 10OM ° + lOOM °

\ >~ 200 .._ ~ - ~ t o t a l

"-.. ~ ......... neutrons

"-'.~ ~ 100 " ' , , , ,'i'i i i i ~ " pr°t°ns "

0 "~'--':'--'~" -~-- I I I I I I

1.1 1.2 1.3 1.4 1.5 1.6

Fig. 1. Diabatic contribution AVdiab as a function of elongation A for the system t°°Mo+l°°Mo (solid line). The neutron and proton diabatic contributions are shown by dotted and dashed lines, respectively. The nuclei are considered spherical with e = 0.74.


50

> 0 "~ 60

*~ 40

> 20

,-, 500

25o

I " I ' I ' I I n

n ~ . - 90Zr+90Zr ' " , , , . . . . . . . . p ~ Z r + 96Zr

a)

" " , .

b) I , I , I , I , I

1.3 1.4 1.5 9, 1.6 1.7 ' I " I ' I ' ! ' I

.... ......... 136Xe + 136Xe

1.1 1.2 1.3 1.4 1.5 1.6 1.7

Fig. 2. a) Diabatic contributions of protons and neutrons for the systems 9°Zr+9°Zr (solid lines) and 96Zr+96Zr (dotted lines). Diabatic potentials for the systems b) 9°Zr+9°Zr (solid line) and 9621"+962I" (dotted line) and c)13°Xe+~3°Xe (solid line) and 136Xe+136Xe (dotted line). The nuclei are assumed as spherical with e = 0.74.

contributions of neutrons and protons for the systems 9°Zr+9°Zr and 96Zr+96Zr. While

the diabatic contributions of the protons are nearly the same in both the systems, the

diabatic contributions of the neutrons are quite different. As a result, the total diabatic potential is more repulsive in the case of 9°Zr+9°Zr (Fig. 2b). Since the diabatic effects

are small near the touching of the nuclei, they are not important for the potential of the

DNS as a function of the mass asymmetry. The diabatic potential hinders the fusion in

much stronger than in the mass asymmetry degree of freedom [8].

Further diabatic potentials are presented in Figs. 2c and 3 for the systems 13°Xe+13°Xe, 136Xe-l-136Xe, l°°Mo+l°°Mo and l l°pd+ll°pd. The neck parameter is fixed at e = 0.74

and the deformation parameters are set to zero. With the exception of the potentials with

the Xe isotopes (Fig. 2c) the diabatic potentials for all these systems have a pocket near

to the touching configuration ( a = 1.58) in which the DNS could stand some time and

evolve in mass asymmetry. For smaller elongations, the diabatic potential is strongly

repulsive in all symmetric systems. The diabatic potential is similar to the one calculated with the phenomenological

double folding potential. In Fig. 3 we compare the diabatic potential of the system l l °pd+l l °pd with the phenomenological double folding potential. A discrepancy with the

adiabatic potential is seen for small elongations where the phenomenological potential

is more repulsive than the diabatic one. The discrepancy becomes smaller if we take


;>

~5 ;>

160

140

120

100

80

60

40

20

1.2 .... .......... 7

• . . • .- ........... . .... . -. . . . . .

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

Fig. 3. Diabatic potentials for the systems l°°Mo+l°°Mo (dotted line) and l l °Pd+H°Pd (solid line). The phenomenological double folding potential for the system H°Pd+ll°Pd is shown by the dashed-dotted line. The discrepancy between this potential and the diabatic one becomes smaller if, by starting at the minimum of the pocket, the neck parameter e is diminished with decreasing A (dashed line for l l °pd+l l °Pd) .

a decrease of e with decreasing elongation into account. This is demonstrated for the system l l ° p d + l l ° p d in Fig. 4 where the diabatic potential is shown as a function of e

for constant values of ,~. The minimum of the diabatic potential moves to smaller values

of e with decreasing values of A (or R). Fig. 5 shows diabatic potentials for the systems 9°Zr+9°Zr and 96Zrq-96Zr as a function

of e for ~ = 1.54 near to the minimum of the pocket in ,L The diabatic contribution is smaller as a function of e for the 96Zrq-96Zr system than for the 9°Zr+9°Zr system

( lower part of Fig. 5) . The reason for this can be seen by an analysis of the single- particle spectra (Fig. 6) . In the case of 9°Zr+9°Zr, more diabatic levels with larger

50

t_, 40 > 0.,)

3 0

20 ;>

10

0 0 . 0

I I I I I

llOpd + llOpd

. " , . . )~ = 1.34

" ' - , . . . . . . . . . . . . . . . , " " "

I I I I I

0.2 0.4 0.6 0.8 1.0

E

Fig. 4. Diabatic potentials as a function of e for the system Ju)Pd+ll°Pd at a = 1.34 and 1.56.

A. Diaz-Torres et aL /Nuclear Physics A 652 (1999) 61-70 67

100 . . . . . .

.-. 80 ~ 9OZr + 9OZr

2O 1 1 : 1 : 1 : 1 " I

80

--- 60

N 40

> ~ 20

! I I I I I

0.0 0.2 0.4 0.6 0.8 1.0 E

Fig. 5. The same as in Fig. 4 for the systems 90Zr+90Zr and 96Zr+96Zr for A = 1.54, which is near to the

minimum of the pocket in A (upper part). The diabatic contributions for these systems are presented as a

function of e in the lower part.

slopes cross the Fermi level at larger values of e. From Figs. 1 and 5 it follows that the

diabatic contributions are larger with respect to the relative coordinate than with respect to the neck coordinate. This is explained by the number of crossings of diabatic states in the variation of both coordinates. In most systems considered the diabatic potential has a minimum as a function of e around e = 0.65-0.85. Since the mass parameter in e has

been shown to be large [6], it strongly hinders the growth of the neck even for a small energy gain from the diabatic contributions during a variation of the neck coordinate.

In the upper part of Fig. 7 the diabatic contributions for the system 22°u(ll°pd+ ll0Pd)

are shown for r /= 0 and 0.5 at e = 0.74. For an asymmetric clusterization of 22°U, the

diabatic hindrance for the motion to smaller values of a is smaller than for the symmetric configuration and the quasifission barrier of the pocket in the diabatic potential is larger.

This means that the evolution of the asymmetric DNS to the compound nucleus is more favored which is supported by the experimental data. The dependence of the diabatic

contribution on the temperature is presented in the lower part of Fig. 7 for the same system. For this calculation we take the occupation probabilities _diab given by the H O t

Fermi distribution at finite temperature for the configuration of the separated nuclei. The initial excitation energy of the system decreases the repulsive character of the diabatic contribution due to smaller occupation numbers of the diabatic states under the Fermi level. If these states are occupied and cross the Fermi level from below, they increase the repulsive contributions of the potential, while occupied states above the Fermi level, crossing it from above with decreasing A, diminish the diabatic contributions. The


54

52

50 > (D

54

52

50

48 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 6. Diabatic neutron-single-particle spectra as a function of e for the systems 9°Zr+9°Zr (upper part) and 96Zr+96Zr (lower part) for a = 1.56, corresponding roughly to the minimum of the pocket. The Fermi level is indicated by triangles.

behavior of the diabatic contributions with deformation of the nuclei corresponds to

calculations of the structure forbiddenness [ 9 ]. The diabatic contributions increase with

prolate deformations and decrease for oblate deformations. However, in order to draw

conclusions about an advantageous fusion with oblate nuclei, the corresponding mass

parameters should first be analysed.

The estimated collective velocities in h and e are large enough to consider diabatic

effects in the DNS. We estimated the minimal excitation energy per nucleon e* for the

applicabil i ty of the diabatic treatment [ 11 ]. The value of e* must fulfill the following

inequality:

2¢rIH=el2 (4) e* ~> (1.4 x 10 -22 MeVs) hlaE,,MOql K,


400

300

200

100

:~ o

~400

300

200

100

0 1.1

~ ' I ' I • I • I ' I

220U(1 lOpd + l l0pd)

\ . . . . . . . . ~ 5

: : ', : 1 : 1 : 1

T = 0 MeV "...~W'............... ......... =

, I , I , I , I , I

1 . 2 1 . 3 1 . 4 1 . 5 1 . 6

Fig. 7. Diabatic contributions as a function of h. for the mass asymmetries r/= 0 (dotted line) and 0.5 (solid line) in the system ll°pd+ll°Pd at e = 0.74 (upper part). The dependence on temperature is shown for 7/= 0 in the lower part. T = 0 MeV: solid line, T = 1 MeV: dotted line.

where x is the mean distance in q between two subsequent crossings. The numerical

factor in (4) is obtained by using the Fermi-gas model for estimating the decay time

due to residual two-body collisions [ 11 ]. From the study of the diabatic and adiabatic

levels of different systems we obtained e* ~> 0.03 MeV for the relative motion in R

and e* ~> 0.07 MeV for the motion in the neck coordinate e. Therefore, the diabatic

effects are already important for relatively small excitation energies of 6-14 MeV. For

the relative motion, the values of the coupling In~l and the derivative ]SEa~/aR] in

the slopes of the crossing levels are 0.17 MeV and 1.52 MeV/fm, respectively. For

the motion in the neck coordinate, we obtained IH,,#I = 0.2 MeV and laE~/3/a~l -- 0.77

MeV. Since in Ref. [ 11 ] the diabatic single-particle spectra were calculated with another

Hamiltonian, a larger minimal value of e* was obtained.

4. Summary

In the present paper we studied diabatic potentials as a function of elongation and neck coordinate for various symmetric heavy systems. The calculations were performed

using the maximum symmetry method with the generalized TCSM. The diabatic effects give rise to hindrances for the growth of the neck and for the motion to smaller


relat ive distances. The diabatic potentials as a funct ion o f the e longat ion are s imilar

to the phenomeno log ica l double folding potentials used in the D N S mode l o f fusion

which descr ibes the exper imenta l data quite well. For the asymmetr ic DNS, the diabatic

hindrance for the mot ion to smal ler e longat ions is smaller than for the symmetr ic DNS

and, therefore, its evolut ion to the c o m p o u n d nucleus is more favored. The temperature

decreases the repuls ive character o f the diabatic potential. The diabatic T C S M supports

the mode l o f fus ion based on the D N S concept , where a hindrance for the mot ion to

smal ler values o f A is assumed. The diabatic effects give a just if icat ion for the use o f

the D N S in heavy- ion col l is ions at low energies.

Acknowledgements

We thank Prof. Yu.M. Tchuv i l ' sky and Dr. G.G. Adamian for fruitful discussions.

A.D.T. is grateful to the D A A D for support. N.V.A. thanks the Alexander von Humbold t -

St i f tung for support. This work was supported in part by DFG.

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Dinuclear system in diabatic two-center shell model approach