Potential energy of heavy nuclear system inlow-energy fusion-fission processes

A. V. Karpov∗, V. I. Zagrebaev∗, Y. Aritomo∗, M. A. Naumenko∗ andW. Greiner†

∗Flerov Laboratory of Nuclear Reactions, JINR, Dubna, Moscow region, Russia†Frankfurt Institute for Advanced Studies, J. W. Goethe-Universität Frankfurt am Main, Germany

Abstract. The problem of description of low-energy nuclear dynamics and derivation of multi-dimensional potential energy surface depending on several collective degrees of freedom is dis-cussed. Multi-dimensional adiabatic potential is constructed basing on extended version of the two-center shell model. It has correct asymptotic value and height of the Coulomb barrier in the entrancechannel (fusion) and appropriate behavior in the exit one, giving required mass and energy distribu-tions of reaction products and fission fragments. Explicit time-dependence of the driving potentialwas introduced in order to take into account difference of diabatic and adiabatic regimes of motionof nuclear system at above-barrier energies and also difference of nuclear shapes in fusion and fis-sion channels (neck formation). Derived driving potential is proposed to be used for unified analysisof the processes of deep-inelastic scattering, fusion and fission at low-energy collisions of heavyions.

Keywords: adiabatic driving potential; fusion-fission dynamicsPACS: 25.70.Jj; 24.10.-i

INTRODUCTION

At the present time the interest in the nucleus-nucleus collisions at near-barrier energieshas increased substantially. It is stimulated in part by recent successful experiments onsynthesis of new superheavy elements and by investigations of the hypothetic island ofstability [1]. On the other hand, analysis of dynamics of these processes is very usefulfor understanding of the reaction mechanisms (in particular, the role of shell effects andthe quasifission processes [2]) and also for evaluation of the fundamental characteristicsof nuclear matter, e.g., the nuclear viscosity. Moreover, investigations of the collisionsat deep subbarrier energies are of a special interest for astrophysics.

The analysis of near-barrier nucleus-nucleus collisions shows that the main reactionchannels here are deep-inelastic scattering [3] and quasifission [2, 4]. In particular, thequasifission processes decrease the probability of fusion of heavy nuclei appreciably.Moreover, it is often difficult to distinguish the quasifission processes from the deep-inelastic collisions and also from the normal fission of compound nucleus.

Due to this competition and substantial overlapping of all the channels of the reactionswith heavy ions a unified dynamical approach for the simultaneous description of all thepossible processes is needed. It should be noted that at the present time these processesare usually analyzed separately within different models. There are several problems inthe development and application of the unified model. First of all, the unified approach todescription of strongly coupled channels implies using the degrees of freedom common

V Vadiab =/ diab

Rcontact

r, fm

pote

ntial

energ

yM

eV

,

2010

100

200 V Vadiab = diab

Diabatic way

Adiabatic way

A1 A2

A1

CN

A2

248Cm +

48Ca

FIGURE 1. The potential energy of the system 48Ca+248Cm for fast (the dashed curve) and slow (thesolid curve) collisions.

for all the channels. The choice of the degrees of freedom playing the most importantrole for all the stages of the process is quite a serious problem. The number of thedegrees of freedom should be small enough for solving dynamical equations for thisset of variables. However, for a very small number of the variables it is impossible todescribe fusion, fission, quasifission, and deep-inelastic scattering simultaneously. Themost relevant degrees of freedom, in our opinion, are: elongation of mononucleus r(or distance between mass centers of two separated nuclei), dynamical deformations �β ,mass asymmetry of the system η , and relative orientations of target and projectile Ω1,Ω2 (in the entrance channel).

Then, we need a unified potential energy which depends on the chosen collectivecoordinates and governs the whole process of fusion-fission. The constructed potentialenergy must have correct asymptotic behavior in the region of separated nuclei and bein agreement with the experimental data on fusion and fission barriers and ground statemasses. Only such a potential energy allows us to perform simultaneous realistic analysisof the deep-inelastic processes, quasifission and fusion-fission. Finally, equations ofmotion along with the necessary initial and boundary conditions have to be formulatedto perform numerical analysis of the reactions being studied.

In the present paper the unified multidimensional potential energy is developed basingon extended variant of the two-center shell model. For the fast nucleus-nucleus collisionsthe first non-equilibrium (diabatic) stage of the reaction is taken into account. The cor-responding diabatic potential energy is calculated within the double-folding procedure.

ADIABATIC AND DIABATIC REGIMES. TIME-EVOLUTION OFPOTENTIAL ENERGY

Let us describe adiabatic and diabatic limit regimes of motion of nuclear system. Fig-ure 1 explains their difference. The interaction potential of two separated nuclei can becalculated quite easily. But after overcoming the Coulomb barrier there are two differentregimes of further evolution of the system. The diabatic regime (or so-called regime of“frozen nuclei”) realizes if the approaching speed of two nuclei is fast and comparablewith nucleons speed in the nuclei. In this case after the contact the nuclei try to penetrateinto each other, which results in doubling of nuclear density and, finally, in appearance

of repulsive core in the potential energy preventing from the density doubling [5, 6]. Inthe case of slow near-barrier collisions the system has enough time to change its shapeand single particles levels in order to keep nuclear density constant (adiabatic condition).We see (Fig. 1) that the adiabatic potential energy noticeably differs from the diabaticone after passing the contact point. At the same time, they have to coincide for wellseparated nuclei.

In order to construct the potential energy for the analysis of reaction at energysubstantially above the Coulomb barrier (in the region above 10 MeV/nucleon) it isnecessary to start from the nonequilibrium diabatic regime as an initial stage andto consider a transition to the equilibrium adiabatic one. This transition to equilib-rium nucleon distribution and to adiabatic regime is rather fast. The characteristictime for the relaxation process is estimated [7, 8] to be τrelax ∼ 10−21 s. The relax-ation time is a parameter of the model which can be determined from the analysisof enormous experimental data on the deep-inelastic scattering of nuclei. Thus, at en-ergies above 10 MeV/nucleon the following expression for the time-dependent poten-

tial energy can be used: Vfus−fis(r,�β ,η;AP,ZP,AT ,ZT ;τ) = Vdiab · exp(− τ

τrelax

)+Vadiab ·[

1− exp(− τ

τrelax

)], where τ is the time of interaction of the nuclei. Note that for slow

near-barrier collisions the first diabatic term in this expression plays a minor role andthe potential energy is purely adiabatic.

DIABATIC POTENTIAL ENERGY

We use the following definition of the diabatic potential energyVdiab(A,Z;r,�β1,Ω1,�β2,Ω2,η):

Vdiab = V12 +M(A1,Z1;�β1)+M(A2,Z2;�β2)−M(AT ,ZT ;�β g.s.T )−M(AP,ZP;�β g.s.

P ). (1)

Here V12(A1,Z1,A2,Z2;r,�β1,Ω1,�β2,Ω2) is the interaction energy of the nu-clei, M(A1,2,Z1,2) are the masses of future fragments, and the constant valueM(AT ,ZT ) + M(AP,ZP) (the sum of the ground state masses of target and projectile)determines zero value of the potential energy in the entrance channel at infinite distancebetween the nuclei. It is also easy to see that in the channels with mass rearrangementthe value of Vdiab at infinite distance equals to the Q-value of the reaction.

We apply double-folding procedure to calculate the interaction energy V12. It consistsin summation of the effective nucleon-nucleon interaction (see, e.g., [9]). In this ap-proach the effects of deformation and orientation are taken into account automatically.According to the folding procedure the interaction energy of two nuclei is given by

V12(r;�β1,Ω1,�β2,Ω2) =∫V1

ρ1(r1)∫V2

ρ2(r2)vNN(r12)d3r1d3r2, (2)

where vNN(r12 = r+ r2 − r1) is the effective nucleon-nucleon interaction and ρi(ri) arethe density distributions of nuclear matter in the nuclei (i = 1,2). The nuclear density

is usually parametrized by the Fermi-type function ρ(r) = ρ0

[1+ exp

(r−R(Ωr)

a

)]−1,

where R(Ωr) is the distance to the nuclear surface (Ωr are the spherical coordinatesof r), and the value ρ0 is determined from the condition

∫ρid3r = Ai. There are two

independent parameters in this formula: the diffuseness of the nuclear density a and thenuclear radius parameter r0.

The effective nucleon-nucleon potential consists of the Coulomb and nuclear parts

vNN = v(N)NN + v(C)

NN . One of the most frequently used nucleon-nucleon potential in thetheory of nuclear reactions is M3Y potential [10, 11, 12]. It is a sum of three Yukawafunctions and consists of direct and exchange parts. The parameters of the M3Y potentialwere fitted well to the experimental data on nucleus scattering. It is also possible toreproduce the fusion barriers using this potential by the appropriate choice of the nuclearradii and diffuseness of the nuclear matter distribution.

At the same time, the M3Y potential leads to a very strong attraction in the region ofoverlapping nuclei, where due to the Pauli principle the repulsion has to appear. One ofsolutions of this problem was suggested in [13]. The idea was to add to the original M3Ypotential a zero-range term that would simulate repulsion. The folding expression withthis extended M3Y potential has five free parameters. In our opinion, it is very difficultor even impossible to make a good systematics of this five parameters basing only onthe fusion barrier data.

Therefore, we prefer to use another nucleon-nucleon potential proposed byA. B. Migdal [14]. The Migdal nucleon-nucleon potential is zero-range density-dependent potential. It has the form

v(N)NN (r1,r2) = C

[Fex +(Fin−Fex)

ρ1(r1)+ρ2(r2)ρ00

]δ (r12) = veff(r1,r2)δ (r12),

Fex(in) = fex(in) ± f ′ex(in). (3)

Here “+” corresponds to the interaction of identical particles (proton-proton or neutron-neutron) and “-” to the proton-neutron interaction. For the fixed value of the constantC=300 MeV fm3 the following values of the amplitudes were recommended in [14]:fin = 0.09; fex =−2.59; f ′in = 0.42; f ′ex = 0.54. The value ρ00 is a mean value of nucleardensities in the centers of the nuclei ρ00 = (ρ01 + ρ02)/2. The potential (3) is definedby the amplitude Fex (“ex” means external) for the interaction of “free” nucleons (i.e.,nucleons from the tails of the nuclear density distributions, where ρ1(r1)+ρ2(r2) � 0);by the amplitude Fin (“in” means internal) for the interaction of a free nucleon with anucleon inside the nucleus (ρ1(r1)+ρ2(r2)� ρ00); and by the value (2Fin−Fex) if bothnucleons are inside the nuclei (double nuclear density region).

The density of nuclear matter ρi is a sum of the proton and neutron densities ρi =ρ(p)

i +ρ(n)i . Then the final expression for V12 is

V12(r;�β1,Ω1,�β2,Ω2) =∫V1

ρ(p)1 (r1)d3r1

∫V2

ρ(p)2 (r2)

e2

r12d3r2 +∫

V1

{[ρ(p)

1 (r1)ρ(p)2 (r1− r)+ρ(n)

1 (r1)ρ(n)2 (r1− r)

]v(+)eff (r1,r1 − r)+[

ρ(p)1 (r1)ρ

(n)2 (r1− r)+ρ(n)

1 (r1)ρ(p)2 (r1− r)

]v(−)eff (r1,r1− r)

}d3r1, (4)

where upper index (“(+)” or “(-)”) in the effective nucleon-nucleon potential correspondsto the sign in the second equation in (3).

charge number

0 10 20 30 40 50 60 70 80 90 1000.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

r 0(p) ,fm

FIGURE 2. Dependence of the parameter r(p)0 on Z. The open squares [15] and circles [16] are the

experimental data. The curve represents our parametrization of the data by expression (5).

For the calculations we use equal proton and neutron densities in the nucleus center:ρ(p)

0 = ρ(n)0 and different radii of these densities. Thus, we have two free parameters: the

radius of the charge distribution R(p) = r(p)0 A1/3 and the diffuseness a. Parametrization of

r(p)0 can be obtained by fitting the corresponding experimental data [15, 16]. We propose

the following parametrization:

r(p)0 (Z) = 0.94+

32Z2 +200

, (5)

shown in Fig. 2. This expression can be used for the nuclei heavier than carbon. The

ACN

Diffe

rence

inbarr

ier

heig

hts

MeV

,

Diffe

rence

inbarr

ier

positio

ns

fm,(a)

ACN

(b)

0 50 100 150 200 250 300 350-4

-3

-2

-1

0

1

2

3

4

0 50 100 150 200 250 300 350-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

FIGURE 3. Difference between the fusion barrier heights (a) and their positions (b) obtained within thefolding potential with the Migdal forces and the Bass potential ("experimental data"). The calculationswere performed for all the possible combinations of the spherical nuclei 16O, 40Ca, 48Ca, 60Ni, 90Zr,124Sn, 144Sm, 208Pb.

values of the second parameter of the model (the diffuseness of the nuclear matterdistribution a) were fitted in order to describe the experimental fusion barriers forspherical nuclei. Using all the possible combinations of the spherical nuclei 16O, 40Ca,48Ca, 60Ni, 90Zr, 124Sn, 144Sm, 208Pb we obtained the parametrization: a(Z) = 0.734−

150/(Z2 +500

), which is recommended for the calculation of the nucleus-nucleus

folding potentials for nuclei A1,2 ≥ 16.The difference between the calculated and “experimental” (the Bass barriers [17])

fusion barriers is shown in Fig. 3. We reproduce the “experimental” data with accuracyof 2 MeV for the barrier height and 0.3 fm for the barrier position.

192

193

194

195

196

��,.

degr, fm

V,

MeV

( )a

��,.

deg

204

205

206

207

208

r, fm

V,

MeV

( )b

10

12

14

16

150

160

170

180

190

200

r, fm

V,

MeV

( )c

��

1

2=, deg.

11.712.012.3

12.6 12.9-180

-90

0

90180

10.811.1

11.411.7

-180-90

0

90180

-90-60

-300

3060

90

FIGURE 4. Folding potential with the Migdal forces as a function of the relative distance r andvarious orientations of the nuclei 64Zn(β g.s.

2 = 0.22) and 150Nd(β g.s.2 = 0.24). Case (a) corresponds to

θ1 = θ2 = π/4, case (b) – to θ1 = θ2 = π/2, and case (c) – to Δϕ = 0. The relative positions of the nucleiare shown schematically in the upper part of the figure.

Figure 4 shows the dependence of the folding potential with the Migdal forces on thedistance between mass centers and relative orientations for the system 64Zn +150 Nd.Dependence on the azimuthal angle Δϕ is given in Fig. 4 (a) and (b). Case (c) showsthe dependence on the polar orientation (orientation in the reaction plane). The polarangle θ influences the diabatic potential energy significantly while the dependence onthe angle Δϕ is very weak. In the case (a) the value of the fusion barrier changes on thevalue about 2 MeV and in the case (b) the change is even less (about 1 MeV). The barrierposition in the cases (a) and (b) changes insignificantly too. It should be also mentionedthat the diabatic double-folding potential with the Migdal forces has qualitatively correctbehavior for small distances between mass centers of the interacting nuclei (see Fig. 4(c)): the repulsive core appears in the region of overlapping nuclear densities.

ADIABATIC POTENTIAL ENERGY

The adiabatic potential energy is defined as a difference between the mass of the wholenuclear system (the system could be either mononucleus or two separated nuclei) andthe ground state masses of target and projectile

Vadiab(A,Z;r,�β ,η) = M(A,Z;r,�β ,η)−M(AT ,ZT ;�β g.s.T )−M(AP,ZP;�β g.s.

P ). (6)

The last two terms in expression (6) provide a zero value of the adiabatic potential energyin the entrance channel of the reaction for the ground state deformations of the target andprojectile at infinite distance between them.

The standard macro-microscopic model based on the Strutinsky shell-correctionmethod [18, 19] is usually used for calculation of the total mass:

M(A,Z;r,�β ,η) = Mmac(A,Z;r,�β ,η)+δE(A,Z;r,�β ,η). (7)

Here Mmac is the liquid drop mass which reproduces a smooth part of the dependenceof the mass on deformation and nucleon composition. The second term δE is themicroscopic shell correction which is usually calculated using the Strutinsky shell-correction method. It gives non-smooth behavior due to irregularities in shell structure.

The macroscopic mass Mmac can be calculated in the framework of finite-range liquid-drop model [20, 21, 22] (FRLDM):

MFRLDM (A,Z;r,�β ,η) = MpZ +MnN−av(1− kvI2)A

+ as(1− ksI2)Bn(r,�β ,η)A2/3 +

35

e2Z2

r0A1/3BC(r,�β ,η)

− 34

e2

r0

(9Z4

4π2A

)1/3

+ f (kFrp)Z2

A− ca(N−Z)+a0A

0

+ W

(|I|+

{1/A, Z and N equal and odd0, otherwise

})

+

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

−Δp +

−Δn −δnp, Z and N odd

−Δp, Z odd and N even−Δn, Z even and N odd0, Z and N even

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

−aelZ2.39. (8)

Here the meanings of the terms are the following: the masses of Z protons and N neu-trons; volume energy; nuclear (surface) and Coulomb energies depending on deforma-tion via dimensionless functionals Bn(r,�β ,η) and BC(r,�β ,η); Coulomb exchange cor-rection; proton form-factor correction to the Coulomb energy; charge-asymmetry energy[(N−Z)-term]; A0-term (constant); Wigner energy; average pairing energy; and energyof bound electrons. For calculation of the shell-correction we can apply the well-knowntwo-center shell model (TCSM) proposed in [23, 24, 25].

200

-2

-1

0

1

2

�M

(sd)

MeV

,

250

(a)

mass number0 50 100 150 200

-6

-4

-2

0

2

4

6

rms=1.19 MeV

250

(b)

�M

g.s

.(

)M

eV

,

mass number210 220 230 240 250

rms=0.94 MeV

(c)

-4

-2

0

2

4

6

8

10

0 50 100 150 200

�M

g.s

.(

)M

eV

,

mass number

FIGURE 5. Difference between the experimental and theoretical ground state masses (δM = Mexp −Mth): (a) with parameters recommended in [22]; (b) with parameters obtained in the present work (seeTab. 1). (c) Difference between the experimental and theoretical saddle point masses.

Figure 5 shows difference between the experimental and calculated ground statemasses as a function of the mass number A. In case (a) the difference is obtained withthe original values of the parameters of the macroscopic mass formula suggested byP. Möller et al. [22]. We see that the dependence has a systematic slope. This slope canbe corrected by additional fitting of five constants in the Weizsäcker-type formula (8).The results are shown in Fig. (b) and the values of the fitted parameters are listed in

Table 1. The obtained rms error is 1.19 MeV, which is good enough for our purposes.For these calculations we restricted ourselves by ellipsoidal shapes of the nuclei. Thenext important characteristic of the potential energy landscape is the fission barrierwhich is the difference between the nuclear masses at the saddle point and ground state

Bf = M(sd)−M(g.s.). In Fig. 5 (c) we compare the experimental (B(exp)f +M(exp)(g.s.))

and theoretical saddle point masses. This quantity is reproduced within 2 MeV. Thesaddle point deformations have been calculated in three dimensional deformation space(see next section for details of the degrees of freedom used).

TABLE 1. Parameters of macroscopic mass formula (8)

parameter av (MeV) kv a0(MeV) ca (MeV) W (MeV)

[22] 16.00126 1.92240 2.615 0.10289 30.0present work 16.02590 1.91385 6.711 0.04998 27.276

g.s.

R cont

6 8 10 12 14 16 18

150

160

170

180

190

200

210

Po

ten

tia

le

ne

rgy,

Me

V

r, fm

g.s.

6 8 10 12 14 16 18

150

160

170

180

190

200

210

Po

ten

tia

le

ne

rgy,

Me

V

r, fm

(a) (b)

296116

48 248Ca+ Cm↔

296116

48 248Ca+ Cm↔

R cont

FIGURE 6. The adiabatic potential energy for the system 296116 ↔48 Ca+248 Cm obtained within theextended (solid curve) and standard (dash-dotted curve) version of the macro-microscopic model. Thedashed curve is the diabatic potential energy calculated within the double-folding model.

In spite of a rather good agreement with the experimental ground state masses andfission barriers, direct application of the standard macro-microscopic approach and, inparticular, expression (8), to the case of highly deformed mononucleus or two separatednuclei leads to incorrect result. In Fig. 6 (a) the adiabatic potential energy calculatedwithin the standard macro-microscopic model and the diabatic one are shown. As men-tioned above, they have to coincide in the region of well separated nuclei. But in thisregion the standard macro-microscopic approach results in a wrong behavior of the adi-abatic potential energy. In order to understand the main reason of this discrepancy weshould analyze the expression for the macroscopic mass (8). One can see that some ofthe terms in this formula are nonadditive over Z and N numbers. In fact, the only ad-ditive part in this expression is MpZ + MnN − ca(N − Z). In the special case of equalcharge densities in the target, projectile, mononucleus, and then in reaction fragmentsZ1/A1 = Z2/A2, the volume, surface, and Coulomb terms will be also additive (but notin the general case). In the entrance channel the charge densities in the projectile andtarget are very different usually, i.e. ZP/AP �= ZT /AT .

This nonadditivity of (8) (in particular, the difference in the charge densities) results inincorrect description of transition from the ground state mass of the compound nucleusto the masses of two separated fragments. This problem with the constant and Wignerterms was pointed out in [26, 27, 28, 29]. It was suggested there to take into account adeformation dependence of these terms.

In the present paper we propose to use the following procedure. It was shown abovethat the standard macro-microscopic model agrees well with the experimental data onthe ground state masses and fission barriers. On the other hand, the double-foldingmodel reproduces the data on the fusion barriers and the potential energy in the regionof separated nuclei (in this region the diabatic and adiabatic potential energies shouldcoincide). Thus, we propose to use the correct properties of these two potentials and toconstruct the adiabatic potential energy as

Vadiab(A,Z;r,�β ,η) ={[

MFRLDM(A,Z;r,�β ,η)+δETCSM(A,Z;r,�β ,η)]

−[MFRLDM(AP,ZP;�β g.s.

P )+δETCSM(AP,ZP;�β g.s.P )

]−

[MFRLDM(AT ,ZT ;�β g.s.

T )+δETCSM(AT ,ZT ;�β g.s.T )

]}B(r,�β ,η)

+ Vdiab(A,Z;r,�β1,�β2,η)[1−B(r,�β ,η)

]. (9)

The function B(r,�β ,η) defines transition from the properties of two separated nu-clei to those of the mononucleus. The function B(r,�β ,η) is rather arbitrary. We onlyknow that it should be unity for the ground state region of mononucleus and it shouldtend to zero for completely separated nuclei. We use the following expression for it:

B(r,�β ,η) =[1+ exp

(r−Rcont

adiff

)]−2, where Rcont(�β ;A1,A2) is the distance between mass

centers corresponding to the touching or scission point of the nuclei, and adiff is theadjustable parameter. Using the value adiff = 0.5 fm we reproduce the fusion barriers.

We call the new procedure for the calculation of the adiabatic potential energy,defined by expression (9), the extended macro-microscopic approach. An example of theadiabatic potential energy calculated within the extended macro-microscopic approachis shown in Fig. 6 (b). It is seen that this procedure leads to the correct adiabatic potentialenergy which reproduces the ground state properties of mononucleus properly as wellas the fission and fusion barriers and the asymptotic behavior for two separated nuclei.

COLLECTIVE DYNAMICS OF FUSION-FISSION

The choice of collective degrees of freedom for low-energy nuclear dynamics is quitea complicated problem. As mentioned above, the distance between centers of mass ofthe separated nuclei (or elongation of mononucleus), their dynamical deformations, andmass asymmetry play the most important role in the fusion-fission process. However,difference in nuclear shapes for the entrance (fusion) and exit (fission) channels can alsobe very important.

The two-center parametrization [25] has been chosen for description of nuclearshapes. It has five free parameters. It is possible, consequently, to define five indepen-dent degrees of freedom determining the shape of the nucleus. We use the following setof them: r – the distance between mass centers; δ1 and δ2 – two ellipsoidal deformationsof the nascent fragments; η = (A2 −A1)/(A2 + A1) – the mass asymmetry parameter(A1 and A2 are the mass numbers of the fragments) and ε – the neck parameter. This

parametrization is quite flexible and gives reasonable shapes for both the fusion andfission processes. However, inclusion of all five degrees of freedom of the two-centerparametrization in the dynamical equations is beyond the present computational pos-sibilities. In order to decrease the number of collective parameters we propose to useone unified dynamical deformation δ instead of two independent δ1 and δ2. Relationbetween δ and δi is given by{

2δ = (δ1−δ (0)1 )+(δ2−δ (0)

2 ),Cδ1(δ1 −δ (0)

1 ) = Cδ2(δ2−δ (0)2 ).

(10)

The deformations δ (0)i provide a minimum of the potential energy (at fixed values of

the other parameters). The first equation in (10) means that zero dynamical deformationcorresponds to the bottom of the potential energy landscape. The second equation comesfrom the condition of equal forces of deformation between two halves of the system.We calculate these forces taking only the first term in liquid-drop expansion of thedeformation energy. The quantities Cδ i are the stiffnesses of the potential energy withrespect to the deformation δi. We apply the liquid drop model for the calculation of Cδ i.

0.2

0.4

0.6

0.8

1.0

ne

ck

pa

ram

ete

r(

)�

3.50.5 1.0 1.5 2.0 2.5 3.0

r/R0

0.4

0.8 3.31.8 2.81.3 2.3 3.8

0.2

0.6

0.8

1.0

r/R0

neck

para

mete

r(

)�

8

4

4

4

0

-10

-20

-30

-40

-50

8

-60

12 -70

16 20

FIGURE 7. The potential energy (a) and the corresponding shapes of nuclei (b) in the coordinates (r,ε)for the system 224Th calculated within FRLDM [20, 21] for η = 0 and δ1 = δ2 = 0. The potential energyis normalized to zero for the spherical compound nucleus. The thick solid curve is the scission line.

Now let us discuss the possibility of approximate consideration of the evolution ofthe neck parameter ε . Figure 7 shows the macroscopic potential energy in coordinates(r,ε) and the map of respective nuclear shapes. Nuclear shapes corresponding to scissionconfigurations in the fission channel have large distance between mass centers and awell pronounced neck. On the contrary, the shapes at the contact point in the fusionchannel are rather compact and almost without neck. These shapes can be describedwell with ε = 1. For the exit (fission) channel the value of the neck parameter shouldbe chosen to minimize the potential energy along the fission path. The value ε ≈ 0.35was recommended in [30] for the fission process. One can see from Fig. 7 that thescission configuration (and, consequently, the kinetic energy of fission fragments) isvery sensitive to the value of ε . The potential energy is almost independent of the neckparameter in the region of separated nuclei. Therefore, the neck parameter keeps theconstant value (ε = 1) in the entrance channel (before contact). The fission process(motion from the ground state along the bottom of the potential energy landscape toscission point and then to infinity) takes place at ε < 1.

It is clear that we should take into account the difference of the entrance and exitchannel shapes. In order to restrict ourselves by the three-dimensional deformation space(r,η,δ ) we propose to consider evolution of the neck parameter as a relaxation processwith the characteristic time τε . Finally, the potential energy for the fusion-fission processVfus−fis(r,�β ,η;AP,ZP,AT ,ZT ;τ) becomes time-dependent and can be written as

Vfus−fis = Vdiab · exp

(− ττrelax

)+Vadiab(ε ,τ) ·

[1− exp

(− ττrelax

)], (11)

where Vadiab(ε ,τ) is the adibatic potential energy which also depends on time and on theneck parameter

Vadiab(ε ,τ) = Vadiab(ε = 1) · exp

(− ττε

)+Vadiab(εout) ·

[1− exp

(− ττε

)], (12)

where the first term corresponds to the entrance channel (ε = 1) and the second – to theexit channel (ε = εout). The characteristic time τε is parameter of the model and shouldbe extracted from the comparison of the experimental data with theoretically calculated.

4 6 8 10 12 14 16 18 20 22

120

140

160

180

200

220

Rcont

R scission

pote

ntialenerg

yM

eV

,

entrance channel shapes

fission channel shapes

r, fm

100

Mo100

Mo

200

Po

FIGURE 8. An example of the adiabatic potential energy in the entrance (solid curve) and fissionchannel (dashed curve) for the reaction 100Mo+100 Mo →200 Po. The calculations have been performedfor zero dynamical deformation δ = 0 and mass asymmetry η = 0.

Figure 8 shows the entrance and exit channel (εout = 0.35) adiabatic potential energyand nuclear shapes for the system 100Mo +100 Mo. Notice the significant differencebetween the positions of the contact point in the entrance channel of the reaction and thescission point in the fission channel. This difference plays a crucial role in the dynamicalcalculations because the energy dissipation and mass transfer occur mainly while thenuclei are in contact.

CONCLUSIONS

Potential energy is a fundamental characteristic determining the statical and dynamicalproperties of heavy nuclear system at low energies. The unified potential energy for thesimultaneous analysis of the deep-inelastic, quasifission and fusion-fission processes isproposed in this paper. The results are summarized in Fig. 9. The initial stage of the

68

1012

1416

18

100

120

140

160

180

200

220

-0.8-0.6

-0.4-0.2

0.00.2

0.40.6

0.8

mass asymmetry

r, fm

po

t.en

erg

y,M

eV

contact point

68

1012

1416

18

100

120

140

160

180

200

220

-0.8-0.6

-0.4-0.2

0.00.2

0.40.6

0.8

mass asymmetry

r, fm

po

t.en

erg

y,M

eV

contact point

ground state

68

1012

1416

18

100

120

140

160

180

200

220

-0.8-0.6

-0.4-0.2

0.00.2

0.40.6

0.8

mass asymmetry

r, fm

po

t.en

erg

y,M

eV

ground state

(a)

(b)

(c)

FIGURE 9. Time evolution of the potential energy for the system 296116 ↔48 Ca+248Cm at zerodynamical deformation δ = 0. (a) The diabatic potential energy calculated using the double-foldingprocedure (the first stage). The entrance-channel (b) and fission channel (c) adiabatic potential energiesobtained within the extended macro-microscopic approach. The white arrows show schematically themost probable reaction channels: deep-inelastic scattering (a); deep-inelastic scattering, quasifission, andfusion (b); and multimodal fission (c).

nucleus-nucleus collision is governed by the diabatic potential energy (see Fig. 9 (a)).The double-folding procedure with the density-dependent Migdal nucleon-nucleon in-teraction is suggested for the calculation of the diabatic potential energy. It reproduceswith a good accuracy the experimental fusion barriers for nuclei heavier than carbon.

We propose to use empirical time-dependent potential energy in order to modeltransition from the nonequilibrium diabatic stage of the reaction to equilibrium adiabaticone. It allows us to analyze nucleus-nucleus collisions at above-barrier energies. Thetransition is treated as a relaxation process (11) with characteristic time τrelax ∼ 10−21 s.The extended macro-microscopic approach is proposed for calculation of the adiabaticpotential energy. It gives correct asymptotic behavior of the potential energy and alsoreproduces the ground state masses, fusion (in entrance channel) and fission barriers incontrast with the standard macro-microscopic approaches. An example of the entrance-channel adiabatic potential energy is shown in Fig. 9 (b).

Time-evolution of the neck parameter is taken into account in a phenomenologicalway. It allows us to consider very elongated nuclear configurations in the exit (fission)channel of the reaction. The corresponding fission-channel adiabatic potential energy isshown in Fig. 9 (c).

Calculation of the proposed driving potential for any nuclear system can be done atthe web-server [31] with a free access.

ACKNOWLEDGEMENTS

One of us (A. V. K.) is grateful to the INTAS for financial support of the presentresearches (Grant No. INTAS 05-109-5058).

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