Langevin fluctuation-dissipation dynamics of hot nuclei: Prescission neutron multiplicities and fission probabilities

Langevin

fluctuation-dissipation

Prescission

neutron

dynamics

multiplicities

of hot nuclei:

and fission probabilities

Received 11 September (Revised 24 November

1992 1992)

Abstract:

We combine a dynamical (Langevin) and statistical description of heavy-ion induced fission in a model which also includes the evaporation of light particles (n, p, a, d, giant-dipole -y’s), and apply it systematically to the data. In the dynamical and statistical parts of our approach important modifications - as compared with the conventional approach - are introduced. In both parts the crucial role is played by the entropy, which is constructed from a deformation-dependent leveldensity parameter, and a dependence on the scission point is introduced in the statistical-model part. The new model turns out to be fairly universal (no input parameter is readjusted for a pa~i~ular system) and reproduces satisfactorily experimental prescission neutron mu~tipI~citi~s and fission p~obab~~it~es for a wide range of systems and excitation energies. We extract info~ation on the deformation dependence of the lever-density parameter, the nuclear friction coefficient, and on the time scale of the fission process.

1. Introduction The

statistical

model

of nuclear

fission

originally

developed

by

Bohr

and

Wheeler ‘) was far a long time sufficient to describe the observed effects. This model assumes thermal equilibrium in all relevant degrees of freedom of the nucleus and that the fission decay rate depends on a particular transition state (the saddle ~oi~t~~ for review articles see refs. 7.3)_For the ~aI~~~ation of fission and ~igbt-panicle decay rates of hot nuclei t e input of the standard model consists in the fission barrier height, the particle binding energies, their transmission coefficients, and the level density parameter and a constant ratio ~+/a, of the level densities at the saddle and at the ground state are used as independent parameters. It turned out, however, in the last years that in particular for describing the rise of prescission neutron multiplicities with increasing bombarding energy observed Correspondence to: Dr. P. FrSbrich, Bereich Schwerionenphysik, Berlin 39, Germany. ’ Also Fachbereich Physik at the Freie Universitiit Berlin. ’ Permanent address: Railway Engineering Institute, Marxa Federation. ~375-9~7~/93/$0~.00

C@1993 - Elsevier

Science

Publis

Hahn-Meitner

prospekt

Institut Berlin, D-1000

35, 644046

ers B.V. All rights reserved

Omsk,

Russian

282

I? Friibrich et al. / Fluctuation-dissipation

in heavy-ion induced fission reactions the standard statistical model fails, see the very illustrative fig. 6 of ref. “). It was realized that one has to include dynamical effects in the description of fission of hot nuclei, in particular one has to introduce the concept of nuclear friction, for reviews see refs. 5,6). Although a number of qualitative theoretical investigations have been published on this subject, e.g. reviewed in ref. ‘), there have been only a few attempts to confront theory 8V9)with data for prescission neutron multiplicities, there have been no attempts to compare theory simultaneously with fission probabilities. In ref. “) and partly in ref. “) theory was compared with prescission neutron multiplicity data of ref. lo) which have been remeasured and corrected to have higher values in ref. ‘I). This means that the values of the reduced friction parameter of about p = 5 x 10” s-’ of ref. “) and of /3 = (6-8) x 102’ ss’ of ref. ‘) should be somewhat increased. Comparing these values with the result p = 20 x 10” s-’ of ref. “) and with /? = 3 x 10” ss’ extracted in ref. 13), and taking also the conclusion of ref. 14) into account, that prescission chargedparticle multiplicities can be described by the standard statistical model (no friction), one is faced with a quite controversial situation. One of the theoretical approaches is to use Langevin equations for a dynamical treatment of nuclear fission. There the main activities have been concerned with investigations whether the stationary limit of Kramers 15) is reached ‘6m’x) after a certain delay time or to calculate the mean value and the width of the total kinetic energy (TKE) distribution of the fission fragments ‘9m2’). Only recently 22) also the correlation between the neutron multiplicities and the TKE distribution has been calculated by coupling particle evaporation to two-dimensional Langevin equations. It is, however, technically difficult to follow Langevin trajectories for arbitrarily long times, and this is also not necessary for physics reasons, because a quasistationary flow is established after a delay time td, i.e. one enters after this time the validity regime of a statistical model. Therefore we have proposed in ref. 13) to combine a Langevin with a statistical-model description. This was done for a one-dimensional model with overdamped motion and allowed us not only to calculate prescission neutron multiplicities but also fission probabilities and (H.I., xn) cross sections as function of the bombarding energy. Later we realized 23,24)that in order to make this combined treatment of dynamics and thermodynamics in a consistent way we have to make important modifications to both the conventional Langevin and the statistical-model approach of ref. 13). Consistency is achieved ‘3,24) by using as the crucial quantity the entropy in the new model. In ref. *‘) we have already reported on some selected results with the new approach. In the present paper we give a comprehensive derivation of the new model and then report on extensive applications to experimental data. In sect. 2 we repeat shortly the derivation of the important equations of the consistent modified model for matching the Langevin dynamics to a statistical-model description and explain the logical sequence of the actual calculations. In sect. 3 we discuss our choice for the coordinate-dependent level density and the reduced

i? Friibrich et al. / Fluctuation-dissipation

283

friction parameter, and explain the input of the model in more detail. show the results of calculations for systems with fissilities X in 0.637 c X s 0.834 in comparison with experimental results. We display excitation functions of prescission neutron multiplicities (n,,,) and abilities PFor survival probabilities 1 - Pf, respectively. We discuss also We finish in sect. 6 with some conclusions and an outlook.

2. The new approach

for combining

the Langevin

dynamics

In sect. 4 we the range of in particular fission probfission times.

with the statistical

model

In this section we derive in a comprehensive way the formulas which have led 73-25) to a consistent combination of the dynamical Langevin with the statistical model. We start with giving arguments for the necessity of modifications of the conventional ‘.“.“~“). We used for instance in ref. ‘j) an overdamped Langevin approaches of refs. equation of the form *_ dt

1

dV

(1)

MP dq

Here the dimensionless quantity q is half the distance between the centers of mass of the future fission fragments divided by the radius of the compound nucleus, V(q) is the potential energy, M the inertia parameter, y the friction coefficient and p = y/M the reduced friction coefficient; /3 and M are treated here as constants. The temperature is denoted by T and r(t) is a fluctuating force with (r(t)) = 0 and (r( r)T(t’)) = 26( t - t’). The fission rate can be calculated from eq. (I) as 1 Rr’t)=(NIDI-

dN,(r) dt

N,(t))

(2)

by counting the number of trajectories Nr( t) which have reached the scission point at time t. N,,, is the total number of trajectories. When switching to the statistical branch of the model we used in ref. 13) the Kramers modified Bohr-Wheeler expression RFHW for the fission rate (overdamped case) RKBWf

WsdWg.r.

j3T

HW

Rf

’

where (4)

Here RFW is the standard statistical Bohr-Wheeler fission-rate the level density of the compound nucleus at the ground state level-density parameter a,), pf is the level density at the saddle by the parameter af), Bf is the fission barrier and E& is the

expression 3), PcN is (characterized by the point (characterized total internal energy

284

I? Friibrich et al. / Fluctuation-dissipation

of the system given by the center-of-mass energy corrected for the Q-value. The quantities w,d and w~.~.are the frequencies at the saddle point and at the ground state. In the long-time limit the rates calculated from eqs. (1) and (3) should approach each other. However, in eq. (1) any information on a coordinate-dependent leveldensity parameter is missing contrary to eq. (3) where the coordinate dependence enters in the difference between a,, and a,, whose ratio is often used as fitting parameter when describing data within the statistical model. On the other hand there is no information on the scission point in eq. (3). So one would expect agreement between the results of eqs. (1) and (3) in the long-time limit only if the coordinate dependence of the level-density parameter and the scission point do not play any role, which is not generally the case. Earlier 16-18) the correspondence between Langevin and Kramers rates were unfortunately checked only for the situation where the scission point is far away from the saddle point and the level-density parameter is coordinate independent. In refs. 23*24)we have removed the inconsistency above and proposed improved versions of both the dynamical and the statistical description of the nuclear fission process. In order to make the present paper selfcontained we repeat the corresponding derivations. It is well established 26,‘7) that in order to describe hot nuclear systems an equation of motion should be applied which is governed by the free energy F. The free energy is related to the coordinate-dependent level-density parameter a(q) by a formula valid for the Fermi-gas model which we use as the most simple example,

F(q, T) = V(q) In this way the information concerning equation of motion which now reads

a(q)T’ .

the level density

(5) is introduced

into

the

T is given by the derivative of the free energy with The driving force K = -(dF/dq)l respect to the fission coordinate at fixed temperature. We would like to stress the fact that it does not matter which thermodynamical quantities are used in the description. In the following we prefer to discuss the situation in terms of the entropy q) which is a function of q alone due to the fact that the total internal S(ES,, energy E$,( q, S), with its natural variables S and q, is constant. The internal energy E$,, of the system is the sum of three terms, the kinetic energy Ekin (which in the overdamped case is zero), the potential energy V(q), and the intrinsic excitation T= energy E”: E,,,* = Eki, + V(q) + E*. Then, using F = E,*,, - TS, the force -dF/dql is given by the temperature times the derivative of the entropy with Tas/aql ET”, respect to the fission coordinate at fixed total internal energy. Because E&is constant we use in the following the total derivative of the entropy with respect to the

The Langevin

equation

then reads

dq -=T dt

S’+

Mp

In the actual ca~c~l~tio~s the entropy is constructed with a particular level-density parameter (see sect. 3) from the Fermi-gas expression SCq) = 2~Q(q)~~~~

- V1(& .

choice of the

(81

We prefer to use the entropy in the following also because, when normatized to its ground-state value, it will turn out in our modified approach to play the role of statistical (- V/ T) which is th e crucial quantity in the decay rate of the conventional model, which we shall modify in the following way. We have derived in refs. 23*‘4) an expression for the statistical decay rate for fission as the inverse of a mean first-passage time [MFPT, for mean first-passage time approaches consult refs. ‘“,““>] which not only includes the coordinate-dependent level-density parameter via the entropy but also contains, contrary to eq. (3), the position of the scission point qsc. Using eq. (7) we repeat shortly the derivation of this formula for the fission decay rate. The overda~~~d Langcv~n equation (7) is (for constant temperatures known to be strictly equivalent, see e.g. ref. “), to the Smoluchowski equation for the distribution function d(q, t) (9) with the Fokker-Planck

operator

We obtain a rate formula from the inverse of a mean ~rst-passage time trr(cf). The latter is calculated from the conditional probability density Prr ( y, t; q, t’) for reaction the point y E ft at t if the trajectory started at I’ at CJE R. Here f2 is the domain of the potential before the nucleus fissions. The probability density which defines the mean first passage time by

dy &(Y, t; q, f’) obeys the Smoluchowski Because we are interested

equation (9) in the variable y for the fission process. in the variable 4 we need the so-catted backward equation

I? Friibrich et al. / Fluctuation-dissipation

286

which acts on the second variable Fokker-Planck operator L+,

q.

3 Pn(y, f; q, 4

This

equation

= -L+(q)P(y,

is governed

f;

q, t’)

by the adjoint

(12)

)

with

(13) Acting with the adjoint operatorL+ time t’ = 0 at the end we obtain

on eq. (ll),

using eq. (12), and putting

L+(q)b(q) = -1 ,

the initial

(14)

or explicitly s,

dt,(q)+d’t,,W -=-dq

MP

dq'

(15)

T

Introducing w(q) = dtl,/dq the solution of the homogeneous part of the equation is given by w(q) = C(q) exp [-S(q)]. The solution of the inhomogeneous equation is obtained by the method of variation of the constant C(q) and applying the proper boundary conditions for fission: q + -0~ is a reflecting, and q = qsc is an absorbing border. The reciprocal of this solution is interpreted as fission rate RMFPT R

T

MFPTPM

q\c

dy ev [-S(y)1 qgs

-1

I

-x

dz exp [S(z)]

.

(16)

The outer integration goes from the ground state qg.s.to the scission point qsc position. This is a generalization to hot systems of a formula first derived in ref. ‘O) a long time ago. By introducing the entropy in the description of hot nuclei instead of the bare potential we have thus been able to derive a dynamical equation containing the information on the level density and a statistical fission decay-rate formula which includes the position of the scission point. Thus the inconsistencies concerning formulas (1) and (3) used in ref. 13) are removed by replacing them by the physically more correct formulas (7) and (16). We do not use eq. (16) in the applications because it is very time consuming to calculate the double integrals many times when entering the statistical branch of the model. To avoid this we replace eq. (16) by an approximate expression which is derived by quadratic expansions of the entropies in eq. (16) after having extended qg.s.+ --CO. the inner integration to infinity ( y + 00) and setting in the outer integration

P. Friibrich et al. / Fluctuation-dissipation

The resulting

approximate .,

expression

287

for the decay rate is

_

[S(qsd)-S(qg.a.)12(1+erf[(q,,-qsd)~,d~l~~’,

Rap,,--exp 27-43

(17)

where dt exp (-t’) is the error function. The saddle-point and the ground-state positions are defined by the stationary points of the entropy and not, as in the conventional approach, by the potential energy. Also the frequencies cZ_ = JIS~~_ T/ M and &, = Js:‘,TIM are now calculated from the second derivative of the entropy at the stationary points. From this formula the influence of the scission point is clear: If the scission point is far away from the saddle point the error function goes to unity. If the saddle point and scission point coincide the error function is zero, which leads to an enhancement by a factor of two as compared to the situation when the scission point is far away from the saddle point. The fact that the Kramers formula has to be improved if the saddle and scission points are close to each other was of course noticed before, e.g. in ref. “). We should mention that in the Langevin calculations we use a coordinate-dependent temperature, whereas in the approximate formula the temperature at the stationary points should be used. Actual calculations, however, show that there is no significant change in the rates if we use the temperature at the saddle point or at the ground state. The well-known less general expression of Kramers 15) RK for the overdamped case is obtained from eq. (17) if the level density is independent of the coordinate and the scission point is far away from the saddle point and E,*,,s V( qsd) > V( qg.s.): RK=

wg.s.w,d exp ii24

V(q\d) + V(q,J)l Tl,

(18)

where w~.~.= md an CO,,,= $-,( V 1.J M are now related to the curvatures of the potential at the stationary points. By describing now the logical sequence of the actual calculations we explain how the emission of light particles (n, cr, p, d, y) is combined with the fission model described above. As in ref. 13) we start the calculations by calculating the fusion spin distribution, which represents the formation probability of the compound nucleus, with the surface friction model “), which describes systematically fusion excitation functions and simultaneously spin distributions in agreement with experiments. The spin distribution acts as the proper weight function for starting the dynamical Langevin trajectories with eq. (7). As long as we are staying in the dynamical branch of the model we allow for the emission of light particles and giant-dipole y’s in the

288

following

l? Frijbrich et al. / Fluctuation-dissipation

way. We calculate

at each Langevin

time step T the decay widths

r,( v =

n, p, (Y,d, y) by using inverse cross-section formulas as in ref. 33). Then the emission of a particle is allowed by asking along the trajectory at each time step whether a random number 5 is less than the ratio of the Langevin time step T to the particle decay time 7rart: .$<~/r~~~ (Os5~1; rpart=h/rtot, where rtot=C.r,). If this is the case a particle is emitted and we ask for the kind of the particle v by a Monte Carlo selection with the weights T,,/T,,,. This procedure simulates the law for radioactive decay for the different particles. After each emission act of a particle of kind v the energy of the emitted particle is calculated by a hit and miss Monte Carlo procedure and the intrinsic excitation energy, the potential and the temperature in the Langevin calculation are recalculated and the dynamics is continued. The loss of angular momentum is taken into account only in a rough way, by assuming that a neutron carries away lh, a proton lh, an cu-particle 2h, a deuteron 2h and a ylh. These values correspond to an upper estimate of standard statistical-model calculations 34). A dynamical trajectory will either reach the scission point, in this case it is counted as a fission event; or if the intrinsic excitation energy E * = Ez, V(q) for a trajectory still inside the saddle (q < qsd) reaches a value E * < min (B,, B,) (B, is the binding energy of the particle v and B, is the fission barrier) the event is counted as evaporation residue event. If the Langevin trajectory has not fissioned and has not been counted as evaporation residue event when the time t = td is reached and the difference in the entropies at saddle and ground state is above a certain value of the entropy Ss,,,: (S(G)

- S(q,,.))

> &TAT,

(19)

we enter the statistical branch of the model. If eq. (19) is not fulfilled we continue the dynamical calculation. The actual values for the delay time td and the critical value for the entropy SsTAT are td = 100 x lo-” s and SsTAT = 2. The precise value of SSTAT is not important as long as td is large enough. In fig. 1 calculated prescission neutron multiplicities for the system ‘O”Pb at total of the delay time t, and internal energy E,,,* = 150 MeV are shown as functions SSrAT. The values for the calculated neutron multiplicities (their contribution from inside the saddle and from saddle to scission are also displayed) saturate for a delay time of td > 30 x 1O-21 s (fig. la) and for S srAT> 1.5 (fig. lb). In the actual calculations we use td = 100 x lo-” s and SsTAT = 2. This guarantees that the results of all calculations do not depend on these switching parameters to the statistical regime. When entering the statistical branch we calculate the decay widths r, again according to ref. 33) and the fission width r,= hR,,, according to eq. (17) and use a standard Monte Carlo cascade procedure which allows for multiple emissions of light particles and higher-chance fission. After each emission act we again recalculate the intrinsic energy and the angular momentum, and continue the cascade until the intrinsic energy is E” < min (B,, B,). In this case we count the event as evaporation residue.

P. Friibrich et al. / Fluctuation-dissipation

V

289

.A&?’ A 0 _...* ‘...A “““” “““’ 1 ()-22 1 O-1 2 1 ()02 1 012 1 (122 td (1 O-21set)

0

-1

0

1

2

a

.

A

A

A

n

.

n

3

4

5

6

&TAT Fig. 1. Calculated prescission neutron multiplicities (filled circles) for ‘“‘Pb at total internal energy 150 MeV are shown as a function of the logarithm of the delay time td(a) and of Ss,,, (b). The ground-state-to-saddle (p.s.) and saddle-to-scission (s.s.) contributions to (n,,,) are also displayed. The curves show saturation for t, > 30 x IO-” s and for SSTAT> 1.5. Calculations are performed for L = 50 2 in (a), and for L = 60 and 1, = 0 for (b). and Ss,,,=

All physical quantities can now be calculated by counting the numbers of the corresponding events. Before entering the discussions of the results of our calculations we continue in sect. 3 with discussing our choices for the further input of the model, in particular for the level density parameter a(q) and the reduced friction parameter /3. 3. The detailed input of the new model It is very instructive

to describe

the way in which we are led to the detailed

input

of the new model. In constructing the entropy (eq. (8)) one has to specify the potential V(q) and the level-density parameter a(q). The potential is calculated from the liquid-drop model 35). We use the c, h, (Y parametrization 36) (with a = 0, symmetric fission), for which it is well known 36) how to calculate our dynamical variable q which is half the distance between the centers of mass of the future fragments divided by the radius of the compound nucleus. We determine the one-dimensional potential of non-rotating nuclei by following the bottom of the fission valley (overdamped motion). A proper rotational term with deformationdependent moment of inertia is added. This is explained in detail in ref. 24). If also the level-density parameter u(q) is specified the input of the model is fixed except for the reduced friction parameter p as the only free parameter.

290

P. ~r~~ri~~ et al. / Ffuctuation-dEssipat~~n

Let us start to discuss the problems with the original model described in ref, 13f. In that paper we have been able to describe simultaneously the excitation functions of the neutron multiplicity and fission probability for the system ‘“F+ ‘*ITa-+ ‘OoPb with a value of /!I = 3 x 102’ s-’ and using Sierk 53) barriers and the level-density parameter a(q) of Tiike and Swiatecki 37). We are also able to do so for the systems “0 + 14’Nd + 158Er and ‘9F+ 232Th ~ 251Es but we have to use different values of the friction coefficient @ = 20 x 10” s-’ and p = 7 x 10” s-l, respectively (see fig. 2). This is not a satisfactory situation because we believe that a physically meaningful reduced friction /3 should be a universal parameter for different systems. This belief is supposed by the fact that the universality of /3 can be shown for example with

bombarding energy (M~V) Fig. 2. Prescission neutron multiplicities ( nprc) and fission probabilities PC for the systems I60 + 14’Nd + zoapb and ‘9~+232n+ 251 “‘Er, 19F+‘8’Ta+ Es (P,= 100%) are calculated using the model of ref. 13) (Langevin fluctuation-dissipation dynamics, LFDD, dots connected by solid lines) and compared to the data. Agreement with the data can be obtained only with different values of the reduced friction /?: “‘Er (0 = 20 x 10”’ SC’), “‘Es (/3= 7 x 10” s-‘) and 2ooPb (/3 = 3 x 10” s-*, see ref. 13)). The symbols for the data for (n,,,) refer to: 0 and 0 (‘80+‘50Sm), ref. 4); 0 and 0 (“Ne+“‘Ta), ref. 4’); V (‘8O+ “?!Sm), ref. 41). The symbols for Pr and (t,) refer to:O, ref. 43j; Cl, ref. *‘); V, ref. 45); Q, ref. 46); shadowed area, ref. ‘I). The dots connected with dotted lines are results of standard statistical model (SSM) ~aIc~la~~o~s.

P.

calculations dimensional

~r~~~c~ et al. f

Fluctuarian-dtss~~pntion

291

in the framework of the Werner-reeler approximation in a twomodel 20). This is demonstrated 38) in fig. 3, where p(q) is shown for

three quite different nuclei calculated as yqYyu/myr,or from ~~~~(rn-‘),,, respectively, where yyv and m+, are the friction and inertia tensors. Observing a non-universality of the value of /3 in the new calculations using ref, I31 has led us to the modifications discussed in sect. 2. When applying the improved model of sect. 2 with the level-density parameter a(g) of ref. 37) the situation changes. Now it is still not possible to reproduce with the same value of /3 neutron multiplicities and fission probabilities. However, one can reproduce the experimental prescission Textron multiplicities (pip,,) with ,B = 20 x 10” s-’ and the fission probabilit~gs PC with a different value for j3 = 3 x 10” s--j for all systems. This is shown in fig. 4 for seven systems with fissilities X ranging between 0.671 and 0.834. However, this is still not a universal description. The value of /3 = 20 x 10” s-* is close to the friction obtained by the wall formula, as can be seen in ref. 5x), w h ere figures for the corresponding friction and mass parameters are displayed. Our result means that in the framework of our model the wall friction is not able to reproduce simultaneously excitation functions of prescission neutron multiplicities and fission probabilities. As the next step we noticed that in order to reproduce simultaneously (n,,,} and P,- with a universal J3 it is ~~ces~~r~ to introduce a weaker coordinate dependence of the level-density parameter a(q) as that of ref. ‘?). We propose the level density found by ~g~a~~~~ ef aI. 39)_A special study jctl supports our choice by favoring also the prescription ofref”“) as compared to ref. ‘37)by dialyzing data for level densities.

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

9 Fig. 3. Universality of p: The reduced friction parameter calculated with the two-dimensional model of ref. 20) is shown to behave universally as a function of q for three different systems. Plotted as /3 is y~~/~~~ (open symbols) and ~,,(m-~),, (closed symbols), where r,, and myy are the diagonal elements of the friction and inertia tensors.

40 20

120

60

initial

100

140 20

total excitation

60

100

140

energy (MeV)

Fig. 4. Prescission neutron multiplicities (n,,,) and fission probabilities Pf or survival probabilities 1 - Pf, calculated with the model of sect. 2 using the level-density parameter of Take-Swiatecki 37), are shown in comparison with the data. The symbols for the data for (n,,,) refer to: 0 and (> t2?Si+ “‘Er), ref. “); 0 and U (20Ne + ‘siTa), ref. 4i); A, ref. 47). Th e symbols for P,, 1 - Pf, and (tr) refer to: 0, ref. 43); 0, ref. 48); q, ref. 44); A, ref. 49); shadowed area, ref. 41). Note that the same data in fig. 2 have the same symbols. Systematic agreement with the data for (n,,,) for all systems is obtained with /3 = 20 x 102’ s-l (dash-dotted lines), whereas one needs p = 3 x 102r SC’ (solid lines) to reproduce the measured fission (P,) and survival (1 - Pf) probabilities. Results for /3 = 7 x 10” SC’ are also shown (dashed lines).

The lever-density

parameter

has the form a(q) = a,A + a,A2’3B,(q)

,

WV

where B, is the dimensionless surface area (for a sphere parametrized in ref. 24) by the analytical expression 1+2.844+ B,(q) =

B, = 1) which

we have

for q -C0.452

-0.375)2

{ 0.983 + 0.439 * (q - 0.3759

for q > 0.452 .

The parameters u, and a, are displayed in table 1. The absolute values of level-density parameters (at the ground state) favored by the different authors are shown as a function of the atomic mass in the upper part of fig. 5, whereas the coordinate dependence of the different level-density parameters for a particular mass number A isdemonstrated in the lower part of fig. 5. Using the Ignatyuk level-density parameter we are now able to reproduce both (n,,,) and Pr for light and medium heavy nuclei {X KO.8) reasonably well with p = 2 x 102’ s-’ to a two-body viscosity of vo= 2 x 1O-‘1 (corresponding MeV . s 9fme3). However, we still observe a systematically increasing slight underestimation of (n,,,) with increasing bombarding energy for lighter systems, and for the heaviest system ( 251Es, X = 0.834) (all,,) turns out to be clearly too small by a factor of two. Examples for the corresponding calculations are shown in fig. 6. The reason for the observed facts is that for lighter systems the main part of the neutrons is emitted before the saddle point is reached (at least for not too higb ~~er~ies~~ wbereas for 151Es the situation is completely di rent. Here because of the low fission barrier and the long descent path more th alf of the neutrons are emitted during the descent from saddle to scission (see also the detailed discussion in sect. 4). In order to be able to describe the heavy systems as well as the lighter ones at higher energies we now propose to use an enhanced friction coefficient (in comparison to p = 2 x IO"SC'), but only in that region of deformation where it does not influence the fission probability any more. A ur?iuersal way for doing so is shown in fig. 7, where p is plotted as a function of the deformation coordinate q. From the ground state up to q = 0.6 where the necking in of the fissioning nuclei starts to set in we use a constant /3 = 2 x 10” CL, followed by a linear increase up to a value of p = 30 x 102” s-’ at the scission point at (q = 1.19). oice of this value for p at scission is obtained by calculating the prescission neutron multiplicities for ‘15Es at the energies Elab = IO2 and I37 MeV as function TABLE 1 Parameters Ref. TGke and Swiatecki Ignatyuk et al. 39)

37)

a, and a, a,

a,

0.0685 0.073

0.274 0.095

294

Fig. 5. Level-density ~aramcte~s: In the upper part of the figure different ~eve~-de~s~ty parameters (at the ground state) are shown as function of the mass number A [T&S: ref. 37); Ign: ref. 39)]. In the lower part the coordinate dependence ofthe level-density parameter a(q) = a,A f ~~Az/3~~(~~ of ref. 37) {T&S; a, = 0.0685, Q, = 4a,) is displayed and compared with that of ref. 39) (Ign; u, = 0.073, a, = 0.095) and the values for iA, &A and &A for A = 216.

of psC. This is shown in fig. 8 where also the experimental “) values for (n,,,) are entered. The value of psC = 30 x 10”’ s-l is compatible with the data and is assumed to be universally the same for all other systems at all energies. The systematic confrontation with the data is made in sect. 4. A qualitative physical explanation for the increase of the damping coefficient when the neck appears would be that here the mechanism of friction is beginning to change its nature. The thinner the neck and the more the fragments are separated, the closer one is to a surface friction mechanism “1, which was applied before to describe fusion and deep-inelastic heavy-ion collisions. An estimate of the surface friction coeficient at contact gives even a value larger than 30 (the situation in the entrance channel corresponds to oblate deformations). Finally we add a short discussion concerning the validity of assuming overdamped motion. The crucial point here is whether the quantity q = p/20 is larger than 1 (overdamped motion). Here w = w, where the local curvature C can be estimated from the potential curves of fig. 9; for A4 we use the reduced mass. As an example we show the calculation for the most ~~~~~~ri~~ situations namely for the

295

P Friibrich et al. / Fluctuation-dissipation

6 a)

1W+l 81Ta=SmPb

60 40. 20.

c) 1W+232Th=951Es

0

4

0

0

1 92

O/f&s ._ _* 0/ 70

90

110

150

130

Elab (MeV) (n,,,) and (b) the survival probabilities 1 - Pf Fig. 6. For “‘Pb (a) the prescission neutron multiplicities are compared with calculations with the Ignatyuk 39) level densities and reduced friction values of p = 2 (dashed-dotted lines) and 3 x 102’s-’ (solid lines). The survival probabilities are reproduced with p = 2 but (n,,,) is too small at high energies. (c) The calculated prescission neutron multiplicities (n,,,) are definitely too small for 25’Es( Pf = 100%) with both values of /3. The symbols for the data are the same as in fig. 4.

00.0

i 0.4

0.8

1.2

Fig. 7. Coordinate dependence of the reduced friction p(q): The of p for different systems according to ref. 38) (see also fig. 2). The present calculations: fi = 2 for q < 0.6 and a linear increase up 0.6~ q < 1.19. The vertical lines represent the universal ground state positions.

1.6

open solid to a (q =

symbols show the line is the ansatz value at scission 0.375) and scission

universality used for the p,,= 30 for point (1.19)

l? Friibrich et al. / Fluctuation-dissipation

296

8

6

I

19F+232Th=>251

Es

- El&, = 102 MeV --&,=I37 MeV ___-------__--

100 50 f3sc (1021 set-1)

150

Fig. 8. Selecting the value of the reduced friction at scission: p,, = 30 x lO*r SC’. Plotted is the calculated (n,,,) for “‘Es for the energies 102 and 137 MeV as function of the reduced friction at scission p,,. Included are also the corresponding data 4). The calculations are compatible with the data for a reduced friction value at scission of p,, = 30 x lo*’ SC’. This value is also used for all other systems and energies.

heaviest system and for angular momentum I = 0. We find a value of 7 = 0.86 for the ground state and 77= 1.23 at the saddle for A = 251 (einsteinium), using the smallest value of /3 = 2 X 102’ SC’. This in fact means that one is at the border line of the validity of an overdamped motion. However, the case I = 0 enters with zero weight in the cross section, and finite Z-values widen the local curvatures of the effective potential, i.e. 7 is increasing. The situation is more in favor for overdamped motion for lighter systems and in the regime where p(q) is larger. Moreover, we have done test calculations with the model in the version of ref. 13) with the full Langevin equation (taking into account the kinetic-energy term), and found that one starts to obtain severe differences from the overdamped case when approaching a value of j? = 1 X 10” s-r. This means that one is allowed to use the overdamped Langevin

equation

with our choice

of parameters.

4. Results of the model and comparison

with experimental

data

Before discussing comparisons of the model with data we show in fig. 9a the potentials, V(q), and minus the entropy times the temperature, -ST, and in fig. 9b as function of q for systems with minus the entropy, -S, and V/T, respectively, different fissilities. The potential and entropy curves are normalized to zero at the ground-state position. The calculations are made with a relatively low excitation energy of 50 MeV and zero angular momentum. One observes in the potential and in the entropy curves the transition from a light system (X = 0.637) with a high barrier and a saddle point close to the scission point to a heavy system (X = 0.834) with a low barrier and a long descent from saddle to scission. This different behavior is important for the later discussions. One can notice in fig. 9 that for the lightest system (highest barrier) the potential curve V/T is below that of -S, whereas the

I? Frijbrich et al. / Fluctuation-dissipation

291 30

10

7

5 -10 >

’

t b)

I.

0.0

.(..

,a._

0.5

1.0

-30

9

Fig. 9. (a) Potentials V(q) (dashed lines) and negative entropies times the temperature, -ST (solid lines), are displayed as function of y for calculations with the Ignatyuk level density, L=O, and I?,*,, = 50 MeV for systems “sW (X = 0.637), “‘Pt (X = 0.671), *“‘Pb (X = 0.701), 2’3Fr (X = 0.743). “%I (X = 0.763) and *“Es (X = 0.834), from top to bottom. (b) For the same systems the negative entropy (the entropy is normalised to be zero at the ground state), -S (solid lines), and the potential divided by the temperature, V(q)/ T (dashed lines), are plotted as function of q.

opposite is true for the heavier systems (lower barriers). That is understandable making an expansion (up to second order in V/E&) in the quantity

by

-s-(v/T)=-2Jz&+;;~+. . . . tot

(Note that -S is positive after normalizing the entropy to zero at the ground state.) For heavy systems (low barriers) the first term on the right-hand side is dominant whereas for light systems (large barriers) the second term can exceed the first one. Because the entropy determines the driving force ( TS’) in the Langevin equation (eq. (7)) and also is the exponent in the fission rate equations (eqs. (16) and (17)) we consider it as the relevant quantity for determining the physics. In order to get some physical feeling for the entropy in describing the fission of hot nuclei one has to note that its negative value at the barrier roughly describes the ratio between the difference in the saddle and ground-state energy to the temperature. Now we compare the results of our model with the data obtained in a large variety of measurements for many systems for a wide range of excitation energies; we have already shown a few results in ref. 25). The universal model (no parameter was readjusted for a particular system) reproduces the experimental excitation functions of the prescission neutron multiplicities (n,,,) (see figs. lOa,c,e,g,i,k,m) and fission probabilities Pr (figs. lOb,d,f) or survival probabilities 1 - Pf (figs. lOh,j,l) for systems in the wide range of fissilities 0.637 s X G 0.834 to the extent shown in the figures. Considering the prescission neutron multiplicities (n,,,) as a function of the total internal energy two regions with different slopes are observed in fig. 10. These slopes are determined by the ground-state-to-saddle (n,.,,) and the saddle-to-scission

4

c)

lo*-

19F+16QTm=>l88Pt

2

d)

cd

(0

10

5

- 8

Q.

4

3 2

4

3

z it 3

4

2

2

1

Q

0

0

5

20

60

initial

IO0

140

total

180

20

excitation

60

100

energy

140

(MeV)

Fig. 10. Calculated prescission neutron multiplicities (n,,,. (points connected by solid lines) are compared with the data for seven systems (figs. lOa,c,e,g,i,k,m). The ground state-to-saddle (g.s.) and saddle-toscission (s.s.) contributions to (n,,) are also displayed. Calculated fission probabilities Pr (figs. lob, d, f) and survival probabilities 1 - Pf (figs. IOh, j, 1 f, respectively (points connected with solid lines), are also compared with the data. For “‘Es (P,= 100%) we shaw {fig. IOn) calculated mean fission times {points connected by a solid line) in comparison with a time extracted from the data FQF a similar system [Ne + Th, fig. 5b of ref. 4f ); shaded area]. The symbols for the data for (n,,} refer to: 0 and C, CzxSi+ ‘“OEr), ref. “f; 0 and 0 I”Ne + “‘Ta), ref. “1; A, ref. 47); V, V (‘“0 + ‘@Tm, c; ““Of L84W, g; *% i- ‘97A~, i->‘8*‘60+238U, m), ref. 41). The symbols for Pr, 1 - Pr, and (tr) refer to: 0, ref. 431; 0, ref. 4”I; El, ref. 44}; A, ref. “). The symbols for the data common with fig. 4 coincide.

P. Friibrich et al. J Fluctuation-dissipation

299

( ns.J

contributions to ( npre) which are also entered in the figures. For lighter systems with fission barriers higher than or equal to the neutron binding energy and for low bombarding energies ( ns.J is small as compared to (n,.,). Therefore, the low-energy part of (n,,,) is dominated by (n,,). For heavier systems and higher energies the contribution of (n,,,,) increases, and is dominant for the heaviest system with the lowest barrier and the longest descent from saddle to scission. It is interesting to note that for not too light systems the contribution (n,,) seems to saturate with increasing bombarding energy, whereas the ( nS.J contribution increases monotonically with increasing energy; i.e. the observed steady increase of the prescission neutron multiplicities with the bombarding energy is according to the model due to the continuous increase of the saddle-to-scission contribution. Now we want to show that the increase of (n,,,) at lower energies is due to the fact that only a very narrow range of I-values contributes, whereas the saturation of (n,,,) at higher energies has to do with an increasing range of partial waves leading to fission. For that purpose we consider fig. 11, where in the upper panel (figs. lla-c) neutron multiplicities and in the lower panel (figs. lld-f) fission probabilities Pf and also complete fusion spin distributions uck( I) [calculated with the surface friction model “)I are displayed as function of the angular momentum for the ‘“Ff ‘“‘Ta+ ‘O”Pb system at different energies. In the figures also gates are entered (shadowed areas) which represent the range of I-values contributing to fission. These gates are determined by the mean values of the fissioning partial waves (IF) + G, where &f = (/f) - (13’ are the variances of the I-values contributing to fission. We o,bserve that for the lowest energy (figs. 1 la, d) only a narrow I-range contributes, and (n,,) is only weakly l-dependent; whereas with increasing energies (figs. 1 lb, e and figs. 1 lc, f) the range of contributing I-values increases considerably, and (n,,) is going down for I> IBt ( Bf= fission barrier). We discuss the situation more in detail in fig. 12. It was shown previously 34,5’) and we find the same that for a particular partial wave (n,.,.) increases monotonically with increasing excitation energy. The weight of the contribution of each partial wave to (n,,,) is, however, determined by that part of the spin distribution which leads to fission as was discussed above. In fig. 12 (n,.,.) is plotted for different I-values as a function of the total internal energy again for the example ‘“F+ “‘Ta + ‘O”Pb of figs. log, h. For small energies only a few partial waves contribute to (n,,,), with increasing energy more and more angular momenta come into play. In fig. 12 also (n,,,,((lr)))(boxes) are inserted, where (I,) is the mean value of the partial waves contributing to fission, together with the vertical bars which represent those (n,,,) covering

the

region

max ((n:.,.)),

min ((n:.,),

where

(n:.,) = (n,.,.((Q)),

(nf::) =

(n,,((h-) *Gj). For low energies (E& < 80 MeV) the dominant contribution to (n,,) comes from a narrow Ir range; one observes an increase of (n,.,,) with increasing energy. We predict that the slope of our calculated curves in the low-energy regime should be similar to results for light-particle induced fission for not too heavy targets. But

a

m

I

. ,,.,I.

,..

120 80 Et”ot (MeW

160

p~~~ssi~~ neutron muitip~~~jt~es (a_) as fu~~ti~~ of Egl are displayed Fig. 12. reground-state-to-saddle to for different partial waves (I= 15, 23, 35, 40, 60, 70) for the ‘O”Pb system. The boxes ~o~espo~d {~~,~,~{~~~)}, where (&} is the mean value of the angular moments ~ontr~~utin~ to fission; the vertieat bars correspond to the ~ont~butions to (n,.,.) due to the width of the fr dist~bution, see discussion in the text.

21 0 l*

20

13

012345678 Ig(tf/l o-21set)

5

7

9

a m *t-t 4-r

0

nprf3

Fig. 13. In figs. 13a,c,e calculated fission-time distributions (fission yields (in 96) as function of the logarithm of the fission time) and the mean fission times (tr) (vertical lines) are shown for the system ‘OF+ r8’Ta+200Pb at E ,=,,= 85, 105, 13.5 MeV. In figs. 13b,d,f calculated fission yields with respect to the number of prescission neutrons and their mean values {vertical lines) are shown for the same system at the same energies.

302

P. Friibrich et al. / Fluctuation-dissipation

there are, as for the heavy-ion range. At higher energies

case, also not so many data available

(E& 180

MeV) one observes

a saturation

50) in this energy

of (n,.,,). This is due

to the fact that (see fig. 12) although (n,,((Zr))) is going down with increasing energy a much broader range of partial waves contributes since the variances c:, increase, with an increasing larger weight for contributions of Z-values lower than the mean (If). This we consider to be the reason for the observed saturation of (n,.,,) at higher energies. The model allows also to extract information on the time scales of the fission process. In ref. “) for example the existence of a long life-time fission component is discussed. It is easy to calculate the complete distributions of the fission times. Examples for time distributions are shown in fig. 13 for the system “‘Pb at different energies. In the figure also the mean fission times (tr) are entered. These mean fission times are not of much physical significance because the mean values are far away from the maximum of the time distributions. The fission time distributions are narrower for heavy systems because of the lower barriers and might be characterized by a mean time. Therefore, we have entered in fig. 10n the calculated mean fission times for the 25’Es system, which are compared to times from the analysis of ref. “) for a very similar system (Ne+Th, the extracted range of mean times from fig. 5b of ref. 4’) is shown by the shaded area). The calculated mean times in the 25’Es example are compatible with those extracted from experiment. With the model also the fission yields with respect to the fission chances can be easily calculated. It would be interesting if they could be measured. Predictions for fission yields with respect to the number of emitted neutrons for the system 19F+ 181Ta+200 Pb are made for several energies and are entered in fig. 13.

5. Conclusions

and outlook

A model for the description of heavy-ion induced fission has been developed which is a consistent combination of a dynamical (one-dimensional, overdamped) Langevin treatment with a statistical-model description including the emission of light particles (n, p, (Y,d, y). The driving force of the Langevin equation and the statistical expression for the fission width are governed by the entropy, not by the potential energy as in the conventional approach. The entropy is constructed from the liquid-drop model 35) as input for the potential energy and from a coordinatedependent level-density parameter, for which we favor that proposed in ref. 39). The reduced friction which we have extracted from the data is compatible with a two-body viscosity as long as the shape of the fissioning nucleus is compact, but as soon as the necking in starts to develop a strong increase of the friction coefficient is needed, which might be related to a surface friction mechanism in the late stage of fission, in order to reproduce the observed increase of the prescission neutron multiplicities (n,,,) with increasing bombarding energy. Besides (npre), the universal model (no

l? Friibrich et al. / Fluctuation-dissipation

parameter

is readjusted

for a particular

system)

describes

303

also the general

trend

of

the excitation function of the fission probabilities Pf or survival probabilities 1 - Pf, respectively, for a variety of systems with fissilities in the range of 0.637 G X G 0.834. The model allows also to extract information on fission times. We would like to add a few remarks about the possibility to change the essential input quantities of the model. The use of the liquid-drop model 35) as input for the potential energy V(q) and the choice of the level-density parameter a(q) according to Ignatyuk 39) leads to our particular choice of the universal reduced friction p(q). It is well known that the choice of u(q) and V(q) have strong influence on the fission rates; in the statistical model, for instance, one can obtain results of equal quality by using higher barriers and simultaneously a stronger coordinate dependence of u(q) (larger a’). We have shown that using the liquid-drop potential and Take-Swiatecki 37) level-density parameters (see fig. 4) does not allow to reproduce the data with a universal reduced friction. Using u(q) of ref. 37) together with Sierk ‘3) barriers would make the situation of fig. 4 even worse. This means that there is not much freedom left to change the potential energy and the coordinate dependence of u(q) without losing the universality of p(q). We remark that using a value of p = 20 x 10” s-‘, which corresponds to the wall formula for friction, does within our model not allow to reproduce simultaneously excitation functions for prescission neutron multiplicities and fission probabilities. In our model it is also possible to calculate multiplicities and energy spectra of the emitted charged particles and giant-dipole y’s. This will be pursued in a later publication where we shall try for instance to analyze the data for prescission proton and a-particle multiplicities “,54) and for y-emission I’). We have also made predictions for fission yields with respect to the number of emitted neutrons (fission chances). And we also predict that the slopes of neutron multiplicities as function of the excitation energy for light-particle induced fission for not too heavy targets should be the same (due to the role played by the angular momenta) as for heavy-ion induced fission at low excitation energy. When discussing deviations of the results of the calculations from experimental data one has to keep in mind the fact that the model we employ here is still quite crude. We use only a one-dimensional overdamped Langevin equation for the fission degree of freedom. We have also not tried to investigate shell corrections in constructing the entropy, which might not be very important, because the fissioning systems despite the cooling through light-particle emission seem to have still temperatures of about 1.5 MeV at scission 55). Th e emission of light particles has been treated by calculating the decay widths with inverse cross sections 33) and we have used liquid-drop values for the binding energies of the emitted particles. This certainly can be improved by using transmission coefficients from optical potentials and empirical binding energies. A generalization to Langevin equations of higher dimensions and beyond the overdamped case is certainly possible along the lines of refs. “-I’). In this case

304

P. Friibrich et al. / Fluctuation-dissipation

formulas for a more-dimensional statistical model in the not-overdamped regime have to be developed when a switching to the statistical branch is required. We stress that this switching is necessary if one wants to describe systems with fission probabilities of less than 100%. If it turns out in the future that a non-markovian treatment of fission is necessary, Langevin equations remain to be a useful tool. First investigations in this direction have been reported

in ref. 56).

Also quanta1 corrections to the Langevin approach, which is completely in the framework of classical statistical mechanics, have been already treated ‘8*57).They seem, however, not to be large enough to be detectable in experiments for heavy-ion induced fission. At the moment the present model should be confronted with as many experimental data as possible because the detection of shortcomings of the model in the analysis of data can be a guide for further developments.

1.1. Gontchar acknowledges Hahn-Meitner Institut.

the warm

hospitality

and financial

support

of the

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