NEW RESULTS FROM DISSIPATIVE DIABATIC DYNAMICS AND NUCLEAR ELASTOPLASTICITY

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NEW RESULTS FROM DISSIPATIVE DIABATIC DYNAMICS AND NUCLEAR ELASTOPLASTICITY

W. Nörenberg

To cite this version:

W. Nörenberg. NEW RESULTS FROM DISSIPATIVE DIABATIC DYNAMICS AND NU- CLEAR ELASTOPLASTICITY. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-215-C2-224.

�10.1051/jphyscol:1987231�. �jpa-00226498�

JOURNAL DE PHYSIQUE

Colloque C2, supplhment au n o 6, Tome 48, juin 1987

NEW RESULTS FROM DISSIPATIVE DIABATIC DYNAMICS AND NUCLEAR ELASTOPLASTICITY

G e s e l l s c h a f t fiir S c h w e r i o n e n f o r s c h u n g (GSI).

D-6100 D a r m s t a d t 11, F.R.G.

a n d

I n s t i t u t fiir K e r n p h y s i k , T e c h n i s c h e H o c h s c h u l e D a r m s t a d t , D-6100 D a r m s t a d t , F.R.G.

1. INTRODUCTION

Large-amplitude collective nuclear motion as observed in the fission process [ I ] and i n low-energy nucleus-nucleus collisions [2-41 has usually been discussed within the framework of adiabatic single-particle motion [5]. For a single collective variable q the corresponding equation of motion reads

where the mass parameter B(q), the friction coefficient S(q) and the adiabatic potential V(q) are defined within the shell model. Due to the large changes of adiabatic states as functions of the collective variable, the collective mass parameter becomes strongly fluctuating and large compared to its irrotational value [6] which contrasts the assumption of irrotational flow i n the original liquid-drop model as introduced by N. Bohr [7]. A further modification of the liquid-drop model are shell corrections to the static liquid-drop energy of the adiabatic po- tential [a]. Finally, because of the coupling of the intrinsic degrees of freedom to the chang- ing mean field, the nuclear system does not remain i n the adiabatic ground state. The excitation of higher levels leads to damping of the collective motion which readily is taken into account by some viscosity of nuclear matter and gives rise to the friction term 54 i n the equation of motion [1,9]. The microscopic structure of the friction coefficient has been studied together with fluctuations in the collective variable within microscopic transport the- ories based on the random-matrix approach [10,11], the one-body dissipation model [12,13]

and the linear response [14].

The adiabatic approximation for the single-particle motion in a time-dependent mean field i s however restricted to very small collective velocities [5,15]. Already for collective kinetic energies larger than * 0.05 MeV per nucleon a diabatic approximation is more realistic which is defined by scaling the wave functions with an irrotational flow imposed by the time-dependence of the nuclear surface 116-181 and is supported by time-dependent Hartree-Fock (TDHF) calculations [19]. This coherent coupling of single-particle motion to the deformation (shape) degrees of freedom is - for small amplitudes - well known from isoscalar giant vibrations, for example giant quadrupole vibrations [20] which also have been consid- ered to be relevant for dissipation i n nucleus-nucleus collisions [21]. On the basis of general diabatic single-particle motion a transport theory has been formulated [22] which results in the following modifications of the collective equation of motion and is referred to as dissipative diabatic dynamics [16,22,23].

(i) The fluctuating adiabatic mass parameter is replaced by its smooth irrotational value, B

-' Birrot.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987231

C2-2 16 JOURNAL DE PHYSIQUE

(ii) The adiabatic potential with its shell corrections is replaced by its value at the temper- ature T = ,/&* where &* denotes the total excitation energy per nucleon. For temperatures larger than 2 to 3 MeV this potential becomes close to the liquid-drop potential.

(iii) The markovian friction term is replaced by a retarded friction force

46 (t)

-

where the non-locality of K is determined by the equilibration within the intrinsic degrees of freedom. For a simple relaxation ansatz the integral kernel becomes

K(t,t') = C exp [-(t-t')lzintr(t')]

where sintr(t) is the equilibration time. According to a Fermi-gas estimate it i s given by [24]

which is inversely proportional to the time-dependent excitation energy. The retardation i n the dissipation process leads to the property of elastoplasticity which is well known for amorphous m terials like glycerine, resin and glass [25]. For nuclei we recover i n the elastic

li

Y

w ) the markovian friction force with 5 = C T . ~ ~ ~ .

As indicated by the name dissipative diabahc dynamics, the theory is based on two ele- ments. The diabatic single-particle motion approximately describes the coherent quantum- mechanical coupling between collective and intrinsic degrees of freedom while dissipative collisions, essentially due to residual two-body interactions, are responsible for the intrinsic equilibration of the system. This approach has been shown i n [lB] to be applicable to nucleus-nucleus collisions in the energy range of 0.3 to 3 MeV per nucleon above the Coulomb barrier which corresponds to bombarding energies above barrier i n the range

for collisions between equal nuclei Ap 2 At.

It is remarkable that the dissipative diabatic dynamics is formally quite close to the ori- ginal liquid-drop model of N. Bohr with irrotational-flow mass parameter and liquid-drop model potential. The essential difference is the non-markovian dissipation (3), (4) which al- lows to describe the elastoplastic property of nuclear matter including giant vibrations and purely frictional motion as limits for fast and slow collective motion. The similarity of diabatic single-particle motion to TDHF [I91 suggests to consider the dissipative diabatic approach as a shell-model version of extended TDHF [26].

I n the following I want to present new results about dissipative diabatic dynamics and nuclear elastoplasticity, in particular on a self-consistent diabatic formulation [27], on first numerical calculations of dissipative diabatic dynamics in two collective degrees of freedom 1281, on quasi-elastic recoil in central nucleus-nucleus collisions [29], on the diabatic hin- drance of fusion reactions [30] and on the diabatic emission of nucleons i n central nucleus- nucleus collisions [31]

2. SELFCONSISTENT DlABATlC FORMULATION [27]

Our approach to dissipative collective nuclear motion is based on the coupled selfconsistent equations for the one- and two-body density matrices. Collective variables are introduced via diabatic shell-model states which satisfy a scaling condition such that one- body coupling terms proportional to the collective velocity vanish. The introduction of the scaling diabatic basis enables us to separate out the smooth part from the selfconsistent single-particle motion. This formulation spans a bridge between the ETDHF theories [26] and extended time-dependent shell-model approaches like the one introduced as dissipative diabatic dynamics [23].

In the derivation of the two-body collision term we apply random properties of the two- body interaction matrix elements but i n addition a time-smoothing procedure which elimi- nates residual fluctuations. Within a linear approximation to the time dependence of the single-particle energies the result resembles the familiar Boltzmann-type collision term, however with a time-smoothed energy conserving F-function, i.e.

=

. ^x

Pya a P r 6 a P r 6 t

denote the smooth two-body interaction matrix elements and n = I - n the the

!::&?EX%

I n addition to the particle number this collision term conserves energy all by itself such that the time-smoothed correlation energy remains constant.

From a variational principle an equation for the collective variables is obtained which is formally the same as in the time-dependent shell-model approach [22], i.e.

d/dt

x ¹

rn rnrn a

The 1.h.s. results from the collective kinetic energy with the irrotational mass tensor B

WTS

(EL

As compared to [22], an additional selfconsistency condition i s required for the validity of this equation, i.e. the one-body couplings need to be negligible as compared to the two- body couplings. Essentially, this is a condition on the choice of the collective variables. The collective equations of motion together with the collision integral for the occupation probabilities describe the elastic response from the single-particle motion like i n TDHF and, simultaneously, the dissipation due to two-body collisions.

Qualitatively, the dissipation process described by (6) and (8) is understood as follows.

We start from an equilibrium distribution for the occupation probabilities (for example in a nucleus-nucleus collision from the ground-state distributions) and a given collective kinetic energy. Via the diabatic single-particle motion the collective kinetic energy is primarily transformed into a conservative potential. The corresponding force is described by the r.h.s.

of (8). In the limiting case of constant occupation probabilities, no dissipation arises from this term. However, two-body collisions try to establish a new equilibrium distribution for the oc- cupation probabilities, and thus decrease the coherence of the diabatic single-particle mo- tion. This process is time irreversible and leads to dissipation i n the collective motion. With the relaxation ansatz for the occupation probabilities a retarded friction force has been de- rived from the r.h.s. of (8) and shown to be responsible for elastoplastic properties of nuclei [23,28-311.

The selfconsistent diabatic approach gives improvements in the shell-model approach to dissipative nuclear dynamics and further insight into the mechanism of two-body collisions i n ETDHF theories. In applications of this approach one has to take the mean energy con- servation carefully into account. Modifications of the time-smoothed collision term are ex- pected to arise from very fast collective motions and for excitation energies larger than approximately three MeV per nucleon.

C2-218 J O U R N A L D E PHYSIQUE

3. TRAJECTORY CALCULATIONS 3.1. Two-dimensional dynamics [28]

We approximate the collision term (6) by the relaxation equation

where the intrinsic equilibration time as given by (4), becomes time dependent via the changing excitation energy. The equilibrium values Fi, are Fermi functions with chemical potential p and temperature T determined from the conservation of particle number and en- ergy. Equations (8) and (9) form a set of coupled equations which are all local in time, and hence markovian. However, if we eliminate the intrinsic variables n,(t) by the formal inte- gration of eq. (9) we obtain the non-markovian equation of motion

for the collective variables. In the harmonic approximation the integral kernel is given by

and i s referred to as the elastoplasticity tensor. The stiffness tensor is defined by

and thus does not depend explicitly on t. For ~ . ~ ~ ~ - t - t ' , i.e. for the elastic limit, Knm(t,tf) be- comes the stiffness tensor Cnm = Knm (t,t8=t\. For small amplitudes the corresponding vibrations can be identified w ~ t h isoscalar giant vibrations [16,20,22,23]. For slow motion where 4,(t') Z hm(t).we find the frictional limit with the friction tensor tnm given by f dt' Knm(t,tJ). It is interest~ng to note that the values of the friction tensor 5, are quite large, even larger than the values obtained from the one-body dissipation model~cf.[l6]).

The equilibrium force

F, =

-1

a

may be approximated by the derivative of the adiabatic potential which is smoothed accord- ing to the finite temperature. For sufficiently large temperatures this adiabatic potential should be close to the liquid- drop energy, i.e. the sum of surface plus Coulomb energies.

For the numerical treatment of central nucleus-nucleus collisions within the diabatic ap- proach an axially symmetric two-center shell model has been introduced i n I171 together with numerical methods for the construction of diabatic states. We have restricted the col- lective degrees of freedom to the distance 5 between the two potential centers (for conven- ience we refer to 5 as elongation parameter) and the deformation parameter 8 which is defined by the ratio wZ/o of the oscillator constants along and perpendicular to the sym- metry axis, and hence as$mptotically measures the (equal) deformations of the nuclei.

The stiffness tensor which determines the elastoplastic behaviour shows effects from the single-particle level structure at small temperatures T < 2 MeV. In general, non-diagonal elements of the stiffness tensor are large, and therefore strongly affect the collective motion.

Results of trajectory calculations for central collisions of 40Ca

+

lations which are increasingly damped with increasing incident energy. An example is given in Fig. 1.

F&J Results from two-dimensional trajectory calculations for OOZr

+

MeVIu above barrier:

(a) The trajectory shows that the two degrees of freedom 6 and 5 are strongly coupled. Initially a prolate deformation evolves i n contrast to frequent expecta- tions that the initial diabatic repulsion leads to oblate deformations of the colliding nuclei.

(b,c) The time dependence of the deformation 6 and the quadrupole moment Q2 shows high-frequency oscillations (h m2 S 22 MeV) i n 6 superimposed on the

<-vibration with hw2 7 MeV.

(d) Collective kinetlc energy Ekin, excitation energy E' and total energy

v.

result of dissipation, the mean value of Ekin (E*) decreases (increases). T e oscil- lations reflect the partially elastic response of nuclear matter.

In connection with Fig. 1 I like to point out an interesting effect which might serve as a signature for elastoplasticity of nuclei. In the fusion processes shown i n Fig. 1 we observe large-amplitude quadrupole vibrations. These vibrations give rise to a finite probability for the emission of a gamma quantum. From the semi-classical theory of radiation [32] the number of gammas which are emitted by a source with the electric quadrupole moment (Z/A)eoQ2 oscillating with frequency w and litetime t, is given by

C2-220 JOURNAL DE PHYSIQUE

For the oscillations shown i n Fig. 6c we find for a single collision event Ny Z 2 lodwith energies hs, 2 (20 ... 25) MeV and N

"

Y -

dipole emission is hindered in fusion reactions of equal nuclei such precompound gia,~t- quadrupole gammas around 22 MeV may become detectable.

3.2. Quasi-elastic recoil in central nucleus-nucleus collisions [29]

At present this investigation i s limited to one collective degree of freedom - t h e elongation

<. This parameter denotes the distance between the centers of the potentials in a two-center shell model [I71 and for not very compact configurations roughly coincides with the distance between the centers of masses. The limitation to one collective variable means that we con- sider only spherical shapes of the fragment nuclei (smoothly joined by a neck with a fixed parameter which is appropriate for the contact configuration).

The system '08Pb + '24Sn is chosen to maximize the effect of the elastic response. Indeed, the adiabatic potential has only a very shallow pocket (less than 1 MeV) just inside the con- tact distance and rises almost linearly with decreasing <. Therefore, the elastic response cannot be obscured by the adiabatic potential. The mass parameter shows a smooth de- pendence on the distance and approaches the reduced mass value for large <. The stiffness parameter exhibits shell effects for low temperatures. These effects are smoothed out with increasing temperature and vanish practically for T 1 2 MeV. The results of the trajectory calculations are summarized in Fig. 2.

Fi 2 Interaction t i m e (top) and kinetic energy e a t t h e exit point (bottom) as functions o f the ~ n i t i a l kinetic energy cti, per nucleon above the Coulomb barrier f o r -rintr given b y eq.(4), solid lines, multiplied w i t h 0.5, dash-dotted lines, and multiplied w i t h 2, dashed lines. There exists a certain i n i t i a l energy ( 3 ~ e V / u f o r t h e sglid curves) beyond which t h e t o t a l kinetic energy i s dissipated and the interaction t i m e increases suddenly t o three times larger values.

From the present investigation we can draw the following conclusions which are specific for the elastoplastic properly tested in central nucleus-nucleus collisions.

1. For energies up to roughly 2...3 MeVIu above the Coulomb barrier one can expect a quasi-elastic behaviour in central collisions of heavy systems. Therefore, in an exper- iment Z08Pb on Iz4Sn with inverse kinematics one would observe the recoiling ""Sn-like nuclei with velocities larger than the beam velocity. The critical bombarding energy, however, depends strongly on the intrinsic equilibration time.

2. Beyond the critical bombarding energy- where the exit kinetic energy vanishes, the interaction times are more than three times larger than below.

At the present stage our studies cannot give quantitative predictions because they are restricted to one collective degree of freedom. However, we expect that the qualitative re- sults of the quasi-elastic recoil due to elastoplasticity remain valid also when further essen- tial collective degrees of freedom are included. Work in this direction is in progress.

4. QUASI-STATIC EFFECTS

4.1. Diabatic fusion barriers and fluctuations [30].

Inelastic processes which take out energy from the radial motion, hinder the fusion proc- ess by shifting the observed barrier to higher energies. Such dynamically hindered fusion processes have been found particularly in collisions of symmetric or almost symmetric sys- tems [33-361. Coupled-channels calculations [33,37-381 show that the excitation of intrinsic degrees of freedom can indeed account for the major part of barrier shifts i n Ar- and Kr- induced fusion reactions. Due to transfer and mutual excitations the fusion barrier is no longer a single-valued quantity but becomes a distribution of different values [39] which is characterized by a mean value < 6 > and a standard deviation erg (or variance

s).

For heavy systems the number of intrinsic degrees of freedom is large and therefore it becomes difficult to describe the fusion process by coupled-channels equations. On the basis of classical friction models it is possibe to describe the energy loss during the approach of the nuclei and thereby account for the additional energy which is needed to overcome the fusion barrier [40-431. Gross features of fusion hindrance as function of the product of projectile and target charges as well as mass asymmetry i n the entrance channel are quite well reproduced within these models. However, puzzling isotopic trends opposite to the gross dependence have been reported recently [33,35-361. These findings suggest that shell-structure effects modify the general trend.

In the application of dissipative diabatic dynamics to the fusion problem we consider the diabatic limit. This approximation is reasonable because the excitation energies remain small (< 20 MeV) throughout the transition over (or through) th arrier. For such excitation energies the intrinsic equilibration time 4) are large ( > 2 *lO'Ps) compared to times nec- essary for passing the barrier ( < 10qds, from Heisenberg's uncertainty relation). Thus, keeping the single-particle occupation probabilities fixed during the approach of the nuclei we obtain a diabatic shifl AB and standard deviation og of the fusion barrier. These modifi- cations of the fusion barrier are not the results of Landau-Zener crossings. It is rather the initial occupation probabilities determined by the pairing correlations which give rise to A 6 and oB (cf. Fig. 3). For a schematic model, mean shift and fluctuations are given by the sim- ple analytical formulae

C2-222 JOURNAL DE PHYSIQUE

Diabatic neutron levels (1.h.s.) and nucleus-nucleus potential (r.h.s.) around the barrier for '24Sn

+

Table 1

Experimental [35] and theoretical [30] values for the mean shift AB with respect to the adiabatic barrier B and standard deviation og of the fusion barrier for '"Sn

+

+

where a single degenerate level in each fragment F ( = l and 2) for protons (p) and neutrons(n), respectively, has been assumed asymptotically, i.e. for the separated nuclei.

The quantities NiF, g.F and yiF denote the number of valence nucleons, the asymptotic de- generacy and the wldth of the spreading of the proton (i=p) or neutron (i=n) levels at the

90 Zr + " ~ r l Z 4 ~ n + ^{9 6 ~ r}

" ~ r '"r

aB(MeV) Bad

(MeV)

185 216 217 219

exp.

2f0.6

8.9k1.2 8.8t1.2 7.850.7 AB (MeV)

d i a b .

0 7 . 8 7.4 7.3 e x p

.

-3i2 26kk 22+L 19f4

d i a b .

0 2 7 24 2 1

barrier. The pairing strength GiF is well approximated by 20/AF fpr neutrons and protons.

General expressions are obtained within the projected BCS formal~sm.

Numerical studies have been performed for the fusion barriers of 'OZr

+

+ 96 94 92

Zr. The comparison (cf. Table 1) of experimental [35] and theoretical values shows that for '24Sn

+

+

4.2. Diabatic emission of neutrons and protons in central nucleus-nucleus collisions [31]

Analytical expressions are derived for the double-differential multiplicities with respect to energy and angle for the diabatic emission of neutrons and protons in central nucleus- nucleus collisions. Results are presented for "OCa + 40Ca, 58Ni

+

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