Dynamical restriction for a growing neck due to mass parameters in a dinuclear system

Ž .Nuclear Physics A 671 2000 233–254

www.elsevier.nlrlocaternpe

Dynamical restriction for a growing neck due to massparameters in a dinuclear system

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

a Institut fur Theoretische Physik der Justus-Liebig-UniÕersitat, D-35392 Giessen, Germany¨ ¨b Joint Institute for Nuclear Research, 141980 Dubna, Russiac Institute of Nuclear Physics, 702132 Tashkent, Uzbekistan

Received 16 September 1999; received in revised form 10 December 1999; accepted 16 December 1999

Abstract

Mass parameters for collective variables of a dinuclear system and strongly deformedmononucleus are microscopically formulated with the linear response theory making use of thewidth of single-particle states and the fluctuation–dissipation theorem. For the relative motion ofthe nuclei and for the degree of freedom describing the neck between the nuclei, we calculatemass parameters with basis states of the adiabatic and diabatic two-center shell model. Micro-scopic mass parameters are found larger than the ones obtained with the hydrodynamical modeland give a strong hindrance for a melting of the dinuclear system along the internuclear distanceinto a compound system. Therefore, the dinuclear system lives long enough compared to thereaction time for fusion by nucleon transfer. Consequences of this effect for the complete fusionprocess are discussed. q 2000 Published by Elsevier Science B.V. All rights reserved.

PACS: 25.70.Jj; 24.10.-i; 24.60.-kKeywords: Mass parameters; Complete fusion; Dinuclear system

1. Introduction

The synthesis of transuranium and superheavy elements, the production of superde-formed nuclei and nuclei far from the line of stability stimulate the study of phenomenaof complete fusion in heavy ion collisions at energies less than 15 MeVrnucleon. The

Ž .fusion of heavy ions is characterized by the formation of a dinuclear system DNS atthe initial stage of reaction when a significant part of the kinetic energy is transferredinto excitation energy. In that case one has to describe the evolution of the DNS andstrongly deformed nucleus by collective coordinates Q, namely by the distance R

Ž .between the centers of colliding nuclei or relative elongation l of the system , the mass

0375-9474r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9474 99 00852-0

( )G.G. Adamian et al.rNuclear Physics A 671 2000 233–254234

Ž . Ž .and charge asymmetry degrees of freedom hs A yA rA and h s Z yZ rZ,1 2 Z 1 2Žrespectively, A , A and Z , Z are mass and charge numbers of the nuclei,1 2 1 2

.AsA qA , ZsZ qZ , the parameter ´ of the neck and deformation coordinates of1 2 1 2w xnuclei 1–12 . Then dissipative, conservative and inertial forces for collective variables,

arising from the nucleus–nucleus interaction, are needed to be determined.The present fusion models can be distinguished by the choice of the relevant

collective variables along which fusion mainly occurs. While many models study thew xfusion in R at a practically fixed value of h, the DNS model 9,10,13–19 considers the

DNS evolution in mass asymmetry by nucleon or cluster transfers as the main path tothe compound nucleus. The DNS model assumes basically that the neck degree offreedom is fixed in the evolution in h and the nuclei are hindered to melt together by a

w xvariation in the relative distance. As was shown in Refs. 20–24 , this occurs due to therelatively large inertia of the neck degree of freedom and structural forbiddennesseffects. In this paper we study dynamical restrictions for the growth of the neck in theDNS and suggest proper methods of the calculation of the DNS inertia tensor.

There are various macroscopical and microscopical approaches to calculate the inertiaw x Ž w x.tensor 25 . The macroscopical approaches see, for example, Refs. 6,7,12 are based

on the hydrodynamical model of the nucleus. A calculation of the inertia tensor with aw xtheory for quantum fluid dynamics is suggested in Ref. 26 . By using a random-matrix

model to describe the coupling between a collective nuclear variable and intrinsicdegrees of freedom and with the help of the functional integral approach, mass

w x w xparameters are derived in Ref. 27 . In the linear response theory 28,29 the inertiatensor is found for fissioning nuclei. The microscopical approaches mainly use thecranking type expression and perform calculations in different single-particle bases

w x w xapplying adiabatic 4,5,30 or diabatic 32–34 two-center shell models. Difficulties inthe cranking type calculations arise for collective motions with large amplitudes, forexample, in fusion or fission, due to pseudo-crossings or crossings of levels in thesingle-particle spectrum. Some publications disregard the contributions from the cross-

Ž .ings pseudo-crossings which means a neglect of effects of configuration changes onthe mass parameters during the evolution of the nuclear shape in spite of the fact that the

Ž . w xcollective inertia is strongly influenced by level crossings pseudo-crossings 35,36 . Inorder to overcome this problem, two-body collisions should be regarded which lead to awidth of the single-particle levels and an effective reduction of the level crossing effects.For example, calculations of the nuclear inertia in a generalized cranking model withpairing correlations yielded masses of about one order of magnitude larger than the ones

w xwithout pairing 37 .One of the aims of the present paper is to obtain analytical expressions for mass

parameters using models and methods which include residual interaction effects. InSection 2 the mass parameters are obtained within the linear response theory taking thefluctuation–dissipation theorem and the width of single-particle states into account. Thesame mass parameters are also derived by Fermi’s golden rule and by smoothing thesingle-particle spectrum in the mean-field cranking formula. In Section 3 the mass

Žparameters for the relevant collective variables mass asymmetry, elongation, neck and.deformation parameters of the DNS and strongly deformed nuclear systems are

evaluated in the two-center shell model with adiabatic and diabatic bases, and thecomplete fusion process is discussed. Section 4 contains a short summary.

( )G.G. Adamian et al.rNuclear Physics A 671 2000 233–254 235

2. Microscopical inertia

2.1. DeriÕation from collectiÕe response function

Let us consider a nuclear system described by a single collective coordinate Q andŽ .intrinsic single-particle coordinates x with the conjugated momentum p and assumei i

w xthe following effective Hamiltonian 28 :

ˆ ˆ ˆH x , p ,Q sH x , p ,Q q QyQ F x , p ,QŽ . Ž . Ž . Ž .i i i i 0 0 i i 0

2 ˆE H x , p ,QŽ .i i21q QyQ 1Ž . Ž .02 2¦ ;E QQ ,T0 0

The shape of the nuclear mean field is changed with the collective coordinate Q thatŽ .introduces the coupling between Q and the nucleonic degrees of freedom. Eq. 1 is

obtained by expanding the Hamiltonian to second order in the vicinity of Q and by0

approximating the second-order term in a kind of unperturbed limit. The coupling termbetween the collective and intrinsic motion is proportional to the first-order in d QsQ

ˆyQ with an operator F given by the derivative of the mean field with respect to Q in0

the neighborhood of Q . In this way global motion is described within a locally0

harmonic approximation.w xBy applying the linear response theory 28,38 , the Fourier transform of the collective

response function is defined as

x vŽ .x v s , 2Ž . Ž .coll 1qkx vŽ .

Ž .where x v is the Fourier transform of the response function for intrinsic motion andmeasures how, at some given Q and temperature T , the nucleonic degrees of freedom0 0

ˆreact to the coupling term Fd Q. The coupling constant k is written in the form

2 2ˆE H x , p ,Q E E Q,SŽ . Ž .i i 0y1yk s s qx vs0 sC 0 qx 0Ž . Ž . Ž .2 2¦ ;E Q E Q QQ ,T 00 0

3Ž .Ž . Ž .with x 0 and C 0 being the static response and stiffness, respectively. Since the

constant k is entirely determined by quasi-static properties, it is no surprise that E is theinternal energy at a given entropy S or the free energy at a given temperature T . The0 0

Ž .structure of Eq. 2 reflects self-consistency between the treatment of collective andmicroscopic dynamics. It expresses the response of the system of interacting nucleons interms of the response of the individual nucleons.

The local motion in the Q variable is described in terms of the collective responsefunction x . A short calculation leads to the following approximation for the masscoll

w xcoefficient 28,29,38,39 :y1 22 21 E x v C 0 g 0Ž . Ž . Ž .Ž .coll crMsy s 1q M q , 4Ž .2 2 ž /x 0 x 02k Ev Ž . Ž .

vs0


where21 E x vŽ .

crM s 5Ž .22 Evvs0

is the inertia in the zero-frequency limit of the second derivative of the intrinsic responsefunction. M cr can be shown to be similar to the one of the cranking model. For many

Ž . Ž .applications, the value of C 0 rx 0 is much less than unity. The additional term2Ž . Ž . Ž . Ž .g 0 rx 0 in Eq. 4 gives a positive contribution to M where g 0 is the friction

coefficient defined byXXEx v Ex v 1Ž . Ž .

XXg 0 syi s s c 0 . 6Ž . Ž . Ž .

Ev Ev 2Tvs0 vs0 0

XXŽ .The dissipative part of the response function x v is connected with the dissipativeXXŽ .part of the correlation function c v through the fluctuation–dissipation theorem:

1 "vXX XX

x v s tanh c v . 7Ž . Ž . Ž .ž /" 2T0

XXŽ .The c v has a singularity of d-function type at vs0:

cXX

v s2pc 0d v qcXX

v 8Ž . Ž . Ž . Ž .RXXŽ .with c v being regular at vs0. In the case of an independent particle model weR

have2XX < <c v sp " F n e 1yn e d "vye qd "vqe . 9Ž . Ž . Ž . Ž .Ž . Ž .Ý jk j k k j k j

j,k

Ž .Here, e se ye is the difference of single-particle energies, n e are the occupationk j k j jˆ ˆ² < < :numbers and F s j F k the single-particle matrix elements of the operator F. Atjk

jsk and vs0 we find the contributions from the diagonal matrix elements:2

E n e E eŽ . k20 < <c s F n e 1yn e sT . 10Ž . Ž . Ž .Ý Ýk k k k 0 ž /E e E Qesekk k

Ž .The last part in 10 was derived with a Fermi distribution for the occupation numbers,which is characterized by the temperature T . The value of T does not effectively go to0 0

zero with decreasing excitation energy because each single-particle level has a width dueto the two-body interaction. Indeed at zero excitation energy the distribution of theoccupation numbers deviates from a step function at least due to pairing correlations. If

Ž . Ž .we replace the d-functions in Eq. 8 or 9 , we have to apply a Lorentzian functionswith the double single-particle width 2 G where G is the single-particle width, because

w x"v is the transition energy between two single-particle states 28 . Therefore, weŽ . w ŽŽ .2 2 .xsubstitute the d-functions in Eq. 9 by Gr p "v"e qG . Then, using Eqs.k j

Ž . Ž .6 – 10 , we can write the friction coefficient in the following form:

g 0 sg diag 0 qg non -diag 0 , 11Ž . Ž . Ž . Ž .where

2" E n e E eŽ . kdiagg 0 s . 12Ž . Ž .Ý ž /G E e E Qesekk


diagŽ .For smaller temperatures, T -2 MeV, which are of interest here, g 0 is much0non -diagŽ . w xlarger than g 0 28 . The static response is found as

q`XXdv x vŽ .

x 0 s limŽ . Hp vy iee™0 y`

"vXXtanh c vŽ .ž /q` dv 2T0 diag non -diags lim sx 0 qx 0 , 13Ž . Ž . Ž .H

"p vy iee™0 y`

where2

E n e E eŽ . kdiagx 0 s . 14Ž . Ž .Ý ž /E e E Qesekk

diagŽ . non -diagŽ . diagŽ . non -diagŽ .With realistic assumptions g 0 4g 0 and x 0 4x 0 and ne-Ž . Ž . Ž .glecting C 0 rx 0 , we can divide the mass parameter 4 as

MsM diag qM non -diag . 15Ž .ˆThe contribution of the diagonal matrix elements of F to M are

2 2diag 2g 0 " E n e E eŽ . Ž .Ž . kdiagM s s . 16Ž .Ýdiag 2 ž /E e E Qx 0 GŽ . esekk

If the single-particle widths are properly taken into account, the non-diagonal contribu-w xtions to the inertia are 30,31

< < 2X XF n e yn eŽ . Ž .k k k knon -diag cr 2M sM s" . 17Ž .Ý 2 2XXX e yee qG k kk kk/k

The main contribution to M is the diagonal part M diag because it dominates forcollective variables which are responsible for changes of the nuclear shape of the systemw x diag non -diag35,36,40 . Note that the calculation of M is simpler than M . For the case thatthe pairing residual interaction is regarded and only diagonal matrix elements in the

Ž . Žcranking formula are taken into account, Eq. 16 was obtained with GsD D is the. w xpairing gap in Refs. 1,37 . Starting with an equation for the single-particle density

matrix extended with an approximate incorporation of particle collisions in the relax-w x Ž .ation time approach, the authors of Refs. 41,42 derived an expression similar to 16

Ž .with Gs"rt , t is the relaxation time but with a negative sign. This negative signarises from the fact that the condition of self-consistency between collective and

w xnucleonic dynamics was disregarded in 41,42 which is important for a correctw xcalculation of the mass parameters. It was stressed in 28,43 that within the linear

response theory the diagonal component of the friction parameter originates from the‘‘heat pole’’ of the correlation function and vanishes when the system is ergodic. As

w xshown in Ref. 44 , the well-necked DNS-type configurations are not ergodic and stableagainst chaos. Even at zero excitation energy the level crossings at the Fermi surface

w xlead to considerable mass flow 35,36,40 and the diagonal component of the correlationŽ .function or mass parameter does not vanish.


Ž Ž ..Besides the mass and friction coefficients, the diffusion coefficients D k,ls Q, Pk lˆmust also have a component diagonal in the matrix elements of F because they are

connected with correlation functions. For example, the diffusion coefficient in momen-tum is defined as

1 XXD s c vs0 sT g 0 .Ž . Ž .P P 02

2.2. DeriÕation from Fermi’s golden rule

ˆ ˆŽ . Ž . Ž Ž ..By setting H s QyQ F x , p ,Q as perturbation see Eq. 1 , the decay rate ofI 0 i i 0< : < :a collective state n with energy E to the collective state m with energy E is givenn m

in lowest order according to Fermi’s golden rule by

w n™mq"vŽ .2p

2 2ˆ<² < < : < <² < < : <s m QyQ n d "v "v F 0 d E yE y"v r "v .Ž . Ž . Ž .H0 n m qs"

18Ž .

Here, the integral is taken over the final states of the intrinsic system with the densityw xr . With the fluctuation–dissipation theorem for small temperatures we have 45qs

2 "v2ˆ<² < < : <"v F 0 r "v d "v s R v dv 19Ž . Ž . Ž . Ž .qs

p 2

Ž . XXŽ .with the relaxation function R v sx v rv. The half-decay width is obtained fromŽ .Eq. 18 as

G s" w n™mq"v . 20Ž . Ž .Ýnm

Using the properties of the response function and a Taylor expansion

xXX

vŽ .R v sŽ .

v

XX XX21 Ex v 1 E x vŽ . Ž .XX 2s x vs0 q vq v q . . . , 21Ž . Ž .2v Ev 2 Evvs0 vs0

Ž .and the standard formula for mass M E /En m

2 y1"2<² < < : < w xMs m QyQ n E yE 22Ž .Ý 0 n mž /2 m

Ž . Ž .we obtain by setting GsG with Eqs. 18 – 22n

XX" Ex v "Ž .

Ms s g 0 . 23Ž . Ž .G Ev Gvs0

Ž . Ž .High temperatures in 19 effectively lead to a temperature dependence of G in 23 .Ž . Ž .Since g 0 in Eq. 11 contains the terms with the diagonal matrix elements of the

ˆ diag Ž .operator F, the mass parameter M also has the diagonal component M 16 . So, the


contributions to the mass parameter can be again classified as those with diagonal andnon-diagonal matrix elements, respectively.

Ž Ž ..That the mass parameter is proportional to the friction coefficient see Eq. 23 , hasan analogy in the hydrodynamic model. For multipole moments l of the nucleus with0

w xl )1, the following ratio in the limit of an irrotational flow was derived in 460

3 A2r3 1irr irrM rg s ,l l0 0 8p 2l q1 l y1 r nŽ . Ž .0 0 0

where n is the coefficient of the two-body viscosity and r s1.2 fm.0

2.3. DeriÕation from the mean-field cranking formula

Using the single-particle spectrum and the corresponding wave functions, one canobtain the mass parameter with the cranking formula

2E n e yn eŽ . Ž .a bcr 2 < <M s" a b . 24Ž .Ý ¦ ;E Q e yeb aa/b

In reality the Hamiltonian of the system contains a residual two-body interactionbetween the nucleons in addition to the mean field. The residual coupling distributes thestrength of single-particle states over more complicated states. This spectral smoothing

Ž .causes the effect that the sum over a and b appearing in 24 also includes diagonalterms with asb. Let us prove this statement.

Ž .Eq. 24 can be rewritten as2

E n e yn eŽ . Ž .1 2cr 2 < <M s" de d e ye de d e ye e e .Ž . Ž .Ý H H1 1 a 2 2 b 1 2¦ ;E Q e ye2 1a/b

25Ž .Next we use the following replacements:

de g e ™ , de g e ™ , 26Ž . Ž . Ž .Ý ÝH H1 1 2 2k k1 2

1 Gd eye ™r e s 27Ž . Ž . Ž .k k 2 22p eye q Gr2Ž . Ž .k

and the approximation2

E22 < < < <D k k f F . 28Ž .1 2 k k¦ ; 1 2E Q

Here, Ds1rg is average energy distance between single-particle states. Then weŽ .express the mass 25 as

2 2 < < 2F n e yn e" G Ž . Ž .k k k k1 2 1 2crM s .Ý2 2 22 2 e ye4p e q Gr2 e q Gr2Ž . Ž . k ka/b ,k ,k a k b k 2 11 2 1 2

29Ž .


Ž .The energy spreading of the single-particle states is taken into account in Eq. 29 . Aline broadening takes place if collisions of particles and holes with the background resultin single-particle and single hole strength functions that are concentrated around theoriginal single-particle energies. The quantity Dr determines the average strengthk

w xfunction for a particle in state k 47 . In the limit e se se , for e fe , i.e. whenk k k a ,b k1 22 Ž .two neighboring levels near the level k are considered, and 8rp f1, Eq. 29 leads to

Ž .Eq. 16 . Thus, diagonal terms in the mass parameters appear because of the finite widthof the single-particle levels due to the residual interaction.

3. Results of calculations

3.1. Adiabatic two-center shell model

Since in fusion and quasifission we deal with strongly elongated systems, theŽ . w xtwo-center shell model TCSM 4,5,20 is appropriate for calculating the potential

energy surface. In the TCSM the nuclear shapes are defined by the following set of theŽ .coordinates: The elongation ls lr 2 R measuring the length l of the system in units0

of the diameter 2 R of the spherical compound nucleus and used to describe the relative0

motion, the mass and charge asymmetry coordinates h and h , respectively, the neckZ

parameter ´sE rEX defined by the ratio of the actual barrier height E to the barrier0 0

height EX of the two-center oscillator, and the deformation parameters b sa rb ,i i i

is1,2, of axially symmetric fragments, defined by the ratio of the semiaxes of thefragments. Since collisions above the Coulomb barrier are discussed, we firstly considerspherical nuclei with b s1 and then analyze the deformation effects.i

In order to calculate the width of the single-particle states, we use the expressionw x28,48

2 21 e ye q p TŽ . Ž .k F 0G s . 30Ž .k 2 2 2G 1q e ye q p T rcŽ . Ž .0 k F 0

Here, e is the Fermi energy. Both parameters G and c are known from experienceF 0w xwith the optical model potential and the effective masses 28 . Their values are in the

following ranges: 0.030 MeVy1 (Gy1 (0.061 MeVy1, 15 MeV(c(30 MeV. For0Ž .small excitations, Eq. 30 is reduced to the expression known in the theory of a Fermi

Ž .liquid. Since each single-particle state has its own width, Eq. 16 is generalized asŽ .Q sl,h,h ,´ ,b ,bi Z 1 2

f E e E ek k kdiag 2M s" . 31Ž .Ýi j 2 E Q E QG i jkk

Ž .For the Fermi occupation numbers n e , the functionk

dn 1 e yek k Fy2f sy s cosh 32Ž .k ž /de 4T 2Tk 0 0

has a bell-like shape with a width T and is peaked at the Fermi energy e .0 F


Various calculations of the mass parameter for the motion in l were carried out withŽ . w xexpressions similar to Eq. 31 , for example, in Refs. 1,30,36 . When the system

adiabatically moves towards the compound nucleus, the value of M approximatelyll

increases by a factor 10–15 in our and other calculations. In this paper we concentrateon the calculation of the mass parameter M for the motion of the neck to test whether´´

the DNS exists sufficient time enough with a relatively small neck. The dependence ofM on ´ is presented in Fig. 1 for the system 110 Pdq110 Pd at lsl s1.6 which´´ t

corresponds to the touching configuration in this symmetric reaction. The obtainedw xvalues of M have the same order of magnitude as in 30 where the pairing´´

correlations were taken into account. The value of M increases by a factor 2.5 when´´

w xthe system falls into the fission-type valley 20 . This increase reflects the decrease ofthe shell correction dU with ´ towards ´™0. Smaller values of dU correspond tolarger masses.

In order to obtain the same TCSM potential energy as in the DNS model for thew xtouching configuration, the neck parameter ´ should be set about 0.75 20 . With this

value of ´ the neck radius and the distance between the centers of the nuclei areapproximately equal to the corresponding quantities in the DNS.

Ž .For the parameter c in Eq. 30 we use the ‘‘standard’’ value 20 MeV since theŽ . y1masses 31 depend only weakly on this parameter. Setting the parameter G s0

Ž . Ž .Fig. 1. Dependence of the mass parameter M lower part and of the shell correction dU upper part on ´´´

for the system 110 Pdq110 Pd at ls1.6. In the calculation of M an excitation energy of 30 MeV of the DNS´´

and adiabatic single-particle states are used. The mass parameter M WW calculated in the Werner–Wheeler´´

approximation is presented by a dashed line in the lower part. Units: m fm2 with msnucleon mass.


y1 Ž . WW0.045 MeV in 31 and comparing our results with M obtained in the Werner–i j

Wheeler approximation for a touching configuration with excitation energy 30 MeVŽ . WW Ž . WW WWT s1.3 MeV , we find M sM , M f 20–30 M , M f0.4M and0 ll ll ´´ ´´ l´ l´

M r M M <1, practically independent of the mass number of the system. There-(l´ ll ´´

fore, we can conclude that the microscopical mass parameter of the neck is much largerthan the one in the Werner–Wheeler approximation and the non-diagonal componentM is small. Since at the touching configuration the slope of the single-particle levels isl´

small and changes slowly with decreasing elongation, the microscopical mass parameterin l is close to its smooth, hydrodynamical value. In contrast, a large amount of internalreorganization occurs at the level crossings with decreasing ´ and leads to a large neckinertia of the initial DNS. So, the value of M exceeds the mass in the hydrodynamical´´

< <model due to large values of E e rE´ . The restriction for the growth of the neck may bekw xunderstood by analysing the single-particle spectrum as a function of ´ 20 . Well

necked-in shapes with large ´ have single-particle spectra with a good shell structure.The levels show a larger number of level crossings with increasing ´ .

The time-dependence of the neck parameter calculated with the microscopical andWerner–Wheeler mass formulas are compared in the upper part of Fig. 2. The lower

Fig. 2. Upper part: Time-dependence of the neck parameter ´ in the system 96 Zrq96Zr calculated withŽ . Ž .microscopical solid curve and Werner–Wheeler dashed curve mass parameters. Lower part: Trajectories in

Ž . 136 136 Ž .the l,´ -plane calculated for the system Xeq Xe with microscopical mass parameters solid curve andŽ .with Werner–Wheeler mass parameters dashed curve . The end points of the solid and dashed curves in the

drawing are at time ts2=10y21s and ts2=10y22 s, respectively.


Ž .part of Fig. 2 shows trajectories in the ´ ,l -plane, calculated with microscopic andWerner–Wheeler masses. An adiabatic potential energy surface is used in all thesecalculations. Since there are no suitable barriers at smaller values of l and ´ in theadiabatic potential which hinder a growth of the neck, the neck parameter and systemlength decrease steadily to smaller values, faster in the case with the Werner–Wheelermasses and much slower with the microscopical masses. The experiments on fusion ofheavy nuclei can not be explained as a melting with increasing neck together with a

w xdecreasing l in an adiabatic potential 20 . It seems that there is an intermediatesituation between the adiabatic and diabatic limits. The study of the transition betweendiabatic and adiabatic regimes gives a potential energy surface which contains quite high

w xbarriers for the motion to smaller l and ´ 22,49 . Therefore, the dynamical calculationswith the adiabatic potential energy show a maximal possible growth of the neck. Sincethe moment of inertia is large in heavy nuclear systems, we disregard the dependence ofthe potential energy on the angular momentum in dynamical calculations. Indeed, in

Ž .fusion reactions with massive nuclei only low angular momenta - 20 " contribute tothe evaporation residue cross sections. The isotopic composition of fusioning nucleiforming a DNS is chosen by the condition of an NrZ equilibrium in the system with hZ

w xfollowing the value of h 13–19 . In processes developing in shorter times, thecoordinate h could be considered as an independent collective variable.Z

As result of Fig. 2 we find that the microscopical mass parameters keep the systemnear the entrance configuration for a sufficient long time comparable with the time ofreaction even in an adiabatic potential. Therefore, this situation justifies the assumption

w xof a fixed neck as we assume it in the DNS model 9,10,13–19 . When the DNSconfiguration exists a long time, then thermal fluctuations in the mass asymmetrycoordinate play the essential role in the fusion process. Indeed, these fluctuations are

w xresponsible for the fusion in the DNS model 9,10,13–19 . Thus, the dynamicalrestriction for the growth of the neck can be caused partly by a large microscopical massparameter for the neck motion, partly by the potential energy surface intermediatebetween diabatic and adiabatic limits.

In the adiabatic two-center shell model the mass parameter M slightly increases´´

with the deformation parameters b of the two nuclei in the symmetric systemi110 Pdq110 Pd as shown in Fig. 3. The small variation of M is due to the shell´´

structure. Therefore, the relatively large value of M is a general result for collisions of´´

both spherical and deformed nuclei. Fig. 4 shows the mass parameter M at the´´

touching configuration of symmetric systems hs0 with b s1 as a function of thei

mass number A. The mass M increases with the mass of the system. It slightly´´

decreases with increasing mass asymmetry of the DNS.

3.2. Diabatic two-center shell model

Ž .Let us now consider the calculation of the mass parameters 31 with the diabaticsingle-particle energies obtained with the method of maximum-symmetry in the diabatic

w xtwo-center shell model 33,34,49 . In the diabatic motion the nucleons do not occupy thelowest single-particle states as in the adiabatic case, but remain in their diabatic states.In comparison to the calculations with the adiabatic single-particle levels, the numericalprocedure with diabatic single-particle levels is much easier because each level has a


Fig. 3. Mass parameter M as a function of deformation b calculated for the system 110 Pdq110 Pd at the´´

touching configuration with excitation energy 30 MeV and adiabatic single-particle states. Units: m fm2 withmsnucleon mass.

complete set of quantum numbers and the drawing of the levels as a function of acollective variable is simpler and unique.

Due to the non-zero widths of the single-particle states, the distribution of single-par-Ž .ticle strength over more complicated states is a Lorentzian distribution r e ask

Ž . Ž . w xintroduced in Eq. 27 instead of the d-function d eye 28 . The occupation numberk˜Ž .n e of a state k with energy e and the corresponding value of f are obtained,˜ k k k

Ž .Fig. 4. Mass parameter M as a function of the mass number A of symmetric systems hs0 calculated with´´

adiabatic single-particle states for b s1, ls1.6, and ´ s0.75. Units: m fm2 with msnucleon mass.i


Ž . Ž .respectively, from functions n e and dn e rde calculated with zero widths of thelevels

n e s n e r e de, 33Ž . Ž . Ž . Ž .˜ Hk k

dn eŽ .f̃ sy r e de. 34Ž . Ž .Hk kde

The Lorentzian distribution increases the diffuseness of the Fermi distribution. TheFermi distribution which is given at the touching configuration of the nuclei in the DNS,is destroyed if the further motion of the system runs diabatically. To treat the diabatic

Ž .case, we use the following function n e for an arbitrary configuration of the systemN

n e s a Q eye yQ eye , 35Ž . Ž . Ž . Ž .Ž .Ý l l lq1ls0

Ž .where Q x is the Heaviside function and e the energy of single-particle state l withl

the occupation number a . Here, the numbers ls0, . . . , N count the single-particle statesl

in the region of the Fermi level. The values e and e are the low and high limits of0 Nq1

the single-particle energies. For lower and higher energies, the occupation numbers areŽ .one and zero, respectively. Therefore, we assume a s1 and a s0 in 35 . The0 N

Ž .derivative dn e rde is expressed asNy1dn eŽ .

y s 1ya d eye q a ya d eye qa d eye .Ž . Ž . Ž . Ž . Ž .Ý1 1 ly1 l l Ny1 Nde ls2

36Ž .˜ Ž .Then we obtain f with 34 as follows:k

Ny1

f̃ s 1ya r e q a ya r e qa r e . 37Ž . Ž . Ž . Ž . Ž . Ž .Ýk 1 k 1 ly1 l k l Ny1 k Nls2

Ž .In the calculations we assume the same average width for each Lorentzian r e . Thek

diabatic occupation numbers a are fixed at the touching configuration of the systeml

using a Fermi distribution for a smaller temperature T ) -T . The value of T ) is0 0 0˜Ž .chosen in such a way that the occupation numbers n e and the values of f , obtained˜ k k

at the touching configuration of the nuclei with the Fermi distribution or the approxi-Ž . Ž .mate expressions 35 – 37 , are nearly equal.

Similar results as in the adiabatic two-center shell model are obtained. The massparameters M and M depend weakly on the mass asymmetry h and on thell ´´

Ž 110 110 .deformations b of the nuclei bsb sb for Pdq Pd at the touching configu-1 2

ration. This result is presented in Fig. 5. One can observe a global minimum of M as all

function of b in the reaction 110 Pdq110 Pd around the ground state deformation of 110 PdŽ .bf1.2 . Since we regard the widths of the single-particle states, the mass parametersdepend more smoothly on the mass asymmetry h than the mean field cranking masses.

w xIn comparison to Ref. 26 where the hydrodynamical-type treatment was used, the shelleffects are here more transparent because all peculiarities of the single-particle spectra

86 134 Ž .are taken into account. Let us consider the fragmentation Krq Ba hs0.2 in thesystem 220 U where the numbers of neutrons in the fragments are close to the magic


numbers 50 and 82, respectively, and the single-particle levels have a smaller slope withrespect to l than in the neighboring combinations. We find a smaller contribution to

diag Ž .M from M and a local minimum in the dependence of M on h Fig. 5 . Forll ll ll

h-0.6, the dependence of mass parameter M on h is consistent with the results ofll

w xRef. 26 . The shape parametrisation of the TCSM is not appropriate for h)0.7.Therefore, we can not calculate M at the moment for such large mass asymmetries´´

w xwhere a strong increase of M with h is expected in accordance with Ref. 26 .´´

The mass parameter M increases about two times with decreasing ´ from 1.0 to´´

0.5 or with increasing neck as shown in Fig. 6 and depends weakly on the mass numberA. In the reactions 110 Pdq110 Pd and 48 Caq172 Hf, which lead to the same compoundnucleus 220 U, the difference between the corresponding values of M becomes larger´´

for small values of ´ . Such a behavior correlates with the fact that the diabaticcontribution to the potential energy grows faster with decreasing ´ in symmetric

w xconfigurations than in asymmetric ones 33,34,49 .For fixed ´s0.75, the mass M grows with decreasing l on average as shown inll

Fig. 6 and is 5 times larger than the reduced mass at lf1.3. The dependence of Mll

on l has fluctuations around an average upwards trend which are more pronounced with

Ž . Ž .Fig. 5. Upper part: Mass parameters M solid line and M dotted line as functions of mass asymmetry hll ´´

at the touching configuration for DNS which leads to the same compound nucleus 220 U. The value of the neckcoordinate is ´ s0.75 and excitation energy of the DNS is 30 MeV. The calculation is done with sphericalnuclei. Lower part: The same as in the upper part, but as a function of deformation b for the reaction110 110 Ž . 2Pdq Pd hs0 . Diabatic single-particle states are used. Units: m fm with msnucleon mass.


Fig. 6. Upper part: Mass parameter M as a function of the neck coordinate ´ at the touching configuration in´´96 96 Ž . 110 110 Ž . 136 136 Ž . 48 172the reactions: Zrq Zr solid line , Pdq Pd dashed line , Xeq Xe dotted line and Caq Hf

Ž .dashed-dotted line . The excitation energy of the DNS in these reactions is 30 MeV. Lower part: The same asin the upper part, but for M as a function of the elongation l. Diabatic single-particle states are used. Units:ll

m fm2 with msnucleon mass.

Ž .increasing total mass number A of the system Fig. 6 and at smaller temperaturesy1r2 Ž .T ;A Fig. 7 . The average trend is connected with an increase of the average0

slope of the single-particle levels with decreasing l or ´ and with the enlarged numberof crossings between the diabatic levels. Such a behavior correlates with the diabatic

w xcontributions to the potential energy 33,34,49 . The fluctuations arise from the factor˜ 2 Ž .f rG in Eq. 31 and are related to the changing number of crossings between thek k

w xsingle-particle levels near the Fermi level with decreasing l 49 .The choice of the width is crucial for the value of M diag in both the diabatic and

adiabatic cases within a reasonable variation. If instead of Gy1 s0.045 MeVy1 the0

value of Gy1 is taken as 0.03 MeVy1 at the lower limit, the value of M becomes0 ´´

2.25 times smaller, but remains larger than M WW by about one order of magnitude. For´´

˜ 2higher temperatures, the average width G increases and the function f rG becomesk k

smoother. The mass parameter M depends on temperature T mainly due to the width´´ 0Ž 2 . y4G of the single-particle levels G ;T . The value of M decreases with T . One-k k 0 ´´ 0

w xand two-body interactions 40 contribute to the non-diagonal and diagonal parts of themass parameter M , respectively. Whereas the one-body non-diagonal contribution to´´

the mass is relatively insensitive to the temperature of system, the diagonal two-bodydiagŽcontribution increases strongly with decreasing temperature. So, M T s´´ 0

. diagŽ .1.0 MeV rM T s1.5 MeV ;5.2 which is shown in Fig. 7. We found the largest´´ 0


Fig. 7. Mass parameter M as a function of ´ at the touching configuration in the 110 Pdq110 Pd reaction for´´

Ž . Ž .temperatures T s1 MeV solid line and 1.5 MeV dashed line . Diabatic single-particle states are used.0

Units: m fm2 with msnucleon mass.

contribution of the two-body component to the neck mass parameter already at a quiteŽ .high excitation energy of 30 MeV T s1.3 MeV of the DNS in fusion reactions. For0

comparison, the excitation energies of the initial DNS in Pb- and Bi-based cold fusionŽ .reactions are smaller than 20 MeV T -1.0 MeV . Hot fusion reactions of heavy nuclei0

Ž .have the excitation energies of the initial DNS smaller than 40 MeV T s1.5 MeV and0

a rapid decrease of the survival probability of the compound nucleus with the excitationenergy. Finally, we should stress that the mass parameter in the neck remains alwaysmuch larger in fusion reactions than the mass calculated with the hydrodynamicalWerner–Wheeler approximation.

The mass parameters at the touching configuration are the same in the adiabatic anddiabatic descriptions because the single-particle levels and occupation numbers arepractically the same in these two limits. The diabaticity destroys the initial Fermidistribution of the occupation numbers at the touching configuration and gives rise to

˜ 2fluctuating values of f rG near the Fermi level with decreasing elongation l and neckk kŽ .parameter ´ of the system Fig. 6 . It is known that the mass parameters are smaller in

the diabatic case for a zero width of single-particle levels than in the adiabatic casew x28,32 . We found values for the masses M and M within the diabatic two-centerll ´´

shell model approach which are closer to the ones obtained in adiabatic calculations inthe definite range of collective variables because we took the width G of single-particlek

states into consideration. The finite decay widths G partially destroy the diabatick

motion. As a result, the diabatic distribution of the single-particle occupation numbersapproaches the adiabatic Fermi distribution for fixed values of ´ and l and excitationenergy of the system. Since we showed the similarity of the data obtained with adiabaticand diabatic single-particle states using the same width, we can use diabatic single-par-ticle states in further calculations of mass parameters in order to simplify the numericalprocedure.


3.3. Influence of large mass in neck on the fusion process

Since the mass of the neck degree of freedom is large, the neck parameter can beŽ . w xtaken fixed ´s0.75 during the fusion. In contrast to Ref. 20 , where the fusion

probability through the fission-type valley was considered with a small ´ , the obtainedfusion probabilities P , calculated with fixed large ´ , depend correctly on the initialCN

w xvalue of h in the entrance channel. As in Ref. 20 , the adiabatic treatment of fusion inl yields fusion probabilities P which are considerably overestimated in comparisonCN

to the experimental data. The reason of this overestimate is clearly seen in Fig. 8, wherethe adiabatic potentials for different fragmentations leading to the same compoundnucleus are shown as a function of l at ´s0.75. In contrast to experiment, the

Ž .hindrance of fusion in these potentials is practically absent P f 1 because there isCN

no fusion barrier for the motion to smaller elongations.In order to describe the experimental data, one should consider the structural

w xforbiddenness effects for the motion to smaller l. As shown in Refs. 21,22 , thediabatic potential measures the structural forbiddenness which reflects the action of thePauli principle. The time-dependent transition between a diabatic interaction potential inthe entrance channel and an adiabatic potential developing during the fusion process was

w xinvestigated within the two-center shell model in Ref. 22 . This transition is a slowerŽ .process than the quasifission the decay of the DNS . Therefore, the system has not

enough time for destroying the ‘‘memory’’ on the structural forbiddenness and a largehindrance for the motion to smaller elongations l at fixed neck is obtained. The

Ž .dynamical diabatic potential at the time of the lifetime of the initial DNS Fig. 9 has a

Fig. 8. Adiabatic potentials of different combinations leading to 246 Fm as functions of l for a fixed value of´ s0.75. The crosses denote the touching configurations.


Ž .Fig. 9. Dynamical diabatic potentials ´ s0.75 . The notations are the same as in Fig. 8.

very large fusion barrier in l and, correspondingly, the fusion probability in l isnegligible for combinations leading to 246 Fm. The decay process in l determines mainlythe lifetime of the DNS dependent on the quasifission barrier B l . The DNS lifetimesqf

w xwere calculated with the microscopical masses in contrast to Ref. 22 where we used thehydrodynamical-type masses. With both types of masses the qualitative picture is thesame with the result that the dynamical potential is considerably higher than the

Ž .adiabatic potential for l-l l is value at the touching point . It should be noted thatt t

dynamical potentials were calculated using the minimally possible relaxation time forw xthe transition between diabatic and adiabatic potentials 22 . The calculated energy

thresholds for the complete fusion in the l- and h-channels lead to the conclusion thatthe DNS evolution to the compound nucleus proceeds in the mass asymmetry degree offreedom. For example, the average fusion barriers B) in mass asymmetry are about 10,fus

76 170 Ž . 86 160 Ž .12 and 15 MeV for the reactions Geq Er hs0.4 , Krq Gd hs0.3 and110 136 Ž . w xPdq Xe hs0.1 , respectively 22,33,34 . The fusion barrier in l is about 3–4times larger than the fusion barrier Bh in mass asymmetry. In compound systemsfus

heavier than 246 Fm the difference between the fusion barriers and probabilities in bothchannels is even larger. It demonstrates that a structural forbiddenness exists for a directmotion of the nuclei to smaller internuclear distances.

The competition between complete fusion and quasifission is extremely important inthe DNS evolution. The quasifission process delivers products with different h. But themain contributions to the quasifission channel come from the initial or near initial

Žconfigurations which have more or less the same quasifission barriers B in l or in theqf.relative distance R . It was proved in calculations with the multidimensional Fokker–

w xPlanck equation 13–19 that one can use the value B of the initial DNS as a goodqf


approximation for the quasifission barriers even for asymmetric reactions. All these factsallow us to calculate the quasifission rate lKr and the fusion rate lKr for the initial DNSl h

Ž w x.with Kramers-type expressions see discussion in 13–19 :

1 v B2i i2Kr Bi(l s Gr 2" q v yGr 2" exp y . 38Ž . Ž . Ž .i iB ž /i2p Tv 0i

Ž . Ž ) . Ž .Here, B isl,h defines the height of the fusion B sB or quasifission B sBi h fus l qfŽ . Bi Ž .barriers. In Eq. 38 , the frequencies v isl,h of the inverted harmonic oscillatorsi

approximate the potential in the variable i around the tops of the barriers B , and v arei i

the frequencies of the harmonic oscillators approximating the potential in the samevariables around the coordinates of the initial DNS. The local oscillator assumption ofthe potential energy surface is a good approximation here. In the calculations we use the

Ž . Ž .expressions 11 and 12 for the friction coefficients g . The mass parameters M andi i ll

Ž .M calculated for touching configurations with Eqs. 15 are about two times largerhh

w xthan the ones of Ref. 26 and lead to the following values of the frequencies:"v Bl f0.8 MeV, "v Bh f1.4 MeV, "v f1.4 MeV and "v f0.5 MeV for thel h l h

reactions considered.

Fig. 10. Fusion probability P in the reactions leading to 246 Fm with excitation energy 30 MeV as a functionCN

of the mass asymmetry in the entrance channel. The result of the adiabatic treatment of the fusion in l ispresented by the dotted line. The upper limit of the fusion probability in l in the dynamical diabatic treatmentis presented by the dashed line. The fusion probability in the h channel at closed fusion channel in l is

Ž w x.presented by the solid line and is in a good agreement with the experiment see Refs. 13–20 .


w xAs shown in Fig. 10, the fusion probability 13–19

lKrh

P s 39Ž .CN Kr Krl qlh l

strongly increases with mass asymmetry in the entrance channel. The same behavior wasw x 40 206 76 170experimentally established 50 . For the reactions Arq Pb and Geq Er the

values of P yield a good agreement with experimental data from evaporation residueCN

cross sections. Fusion was not observed in the reactions 86 Krq160Gd and 110 Pdq136 XeŽ .because of the small values of P . Since P is defined by the ratio 39 , theCN CN

calculated fusion probabilities depend weakly on the mass parameters. There is more orw xless no difference between fusion probabilities calculated with the masses of Ref. 26

and of the present paper. The fusion probability has nearly the same value for realisticŽ y1 y1 y1.values of the single-particle width 0.030 MeV (G (0.061 MeV . The micro-0

scopical friction and mass parameters give very similar values of the fusion probabilityw x 90 90P in the h-channel as in Refs. 13–19 for the symmetric reactions Zrq Zr,CN

100 Moq100 Mo, 110 Pdq110 Pd, 123,124Snq123,124Sn and 136 Xeq136 Xe as well as for thePb-, Bi- and actinide-based fusion reactions.

4. Summary

Mass parameters for the relevant collective variables of strongly deformed systemsconsisting of two nuclei as the DNS are evaluated in the two-center shell model.Formulas for the masses are derived within the linear response theory by taking thefluctuation–dissipation theorem and the width of single-particle states into account, andby other methods as Fermi’s golden rule and a spectral smoothing in the mean-fieldcranking formula. The obtained mass parameter for the neck degree of freedom is muchlarger than the one obtained in the hydrodynamical model with the Werner–Wheelerapproximation. By applying the microscopical mass parameters we find a relativelystable neck during the time of reaction. In addition, a large energy threshold due to the

w xstructural forbiddenness effect 21,23,24 hinders the motion to proceed to smallerinternuclear distances. We observe a strong correlation between a large mass and a largedegree of the structural forbiddenness for the heavy DNS as a function of thefragmentation. Therefore, the DNS configuration exists a sufficiently long time andthermal fluctuations in the mass asymmetry coordinate could develop and lead to fusionand quasifission which are the essential reactions in the DNS concept. The comparisonof our predicted results in the DNS concept with many experimental data for fusionreactions let us conclude that the DNS is stable against a melting in l and ´ . Thetreatment of the fusion in h shows agreement with the experimental data. Our resultsalso support the existence of long living cluster configurations which appear, for

w xexample, in ternary fission 51 .

Acknowledgements

We thank Profs. H.Hofmann, S.P. Ivanova, F.A. Ivanyuk, R.V. Jolos, V.V. Volkovand Drs. E.A. Cherepanov, A.B. Larionov, A.K. Nasirov for fruitful discussions and


suggestions, and Mrs. N.G. Adamian for her help in preparing the paper. G.G.A. andA.D.-T. are grateful for support of the Alexander von Humboldt-Stiftung and DAAD,respectively. This work was supported in part by DFG and RFBR.

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Dynamical restriction for a growing neck due to mass parameters in a dinuclear system

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