A C-code for combining a Langevin fission dynamics of hot nuclei with a statistical model including evaporation of light particles and giant dipole γ-quanta

Computer Physics Communications Computer Physics Communications 107 (1997) 223-245

ELSEVIER

A C-code for combining a Langevin fission dynamics of hot nuclei with a statistical model including evaporation of light particles and giant dipole y-quanta I. Gontchar a, L.A. Litnevsky a, R FrObrich b.c a Omsk State Railway Academy, Prospect Marxa 35, 644046 Omsk, Russian Federation b Hahn-Meitner htstitut Berlin. Glienicker Strafle 100, 14109 Berlin, Germany c Fachbereich Physik, Freie Universitiit Berlin, Berlin, Germany

Received 20 August 1997

Abstract

We present a C-code calculating the decay of highly excited and rotating atomic nuclei by fission and light particle (neutrons, protons, a-particles, giant dipole y-quanta) evaporation. The program calculates the distributions of the relevant observables, their averages, variances and correlalions. The most important output parameters are: fission (respectively survival) probabilities, pre-scission neutron and charged-particle multiplicities and spectra, pre-scission gamma-multiplicities and spectra, average energies of these particles, scission lifetime distributions, and excitation energies at scission. For this purpose a dynamical Langevin equation for an overdamped fission process is combined with a statistical model description. In the dynamical part of the model the evaporation of light particles and giant dipole y-quanta is accounted for by coupling a Monte Carlo algorithm to the fission dynamics. In the statistical part of the model a fission width consistent with the dynamical description is used besides the standard way of calculating the light particle emission. The code provides also the possibility to investigate the influence of different nuclear friction form factors on the observable quantities. © 1997 Elsevier Science B.V. PACS: 25.70.Gh; 25.70Jj; 24.60.Ky Keywords: Nuclear fission; l,angevin simulation; Statistical model; Nuclear friction

PROGRAM SUMMARY

Computers: The code has been cleated on a PC, but it is also adaptable to CONVEX or ALPHA workstation

litle tgfprogratn: DESCEND

Operating systems: MS-DOS or UNIX

Catalogue identifier: ADGX

Programming language used: C

Program obtainable from: CPC Program Library, Queen's Univer-

sity of Belfast, N. Ireland Licensing provisions: none

Memory required to execute with typical data: about 8 Mbyte No. of bytes in distributed program, including test data, etc.:

187018

0010o4655/97/$17.00 ~) 1997 Elsevier Science B.V. All rights reserved. Pli S0010-4655 (97)00108-2

224

1. Gontchar et al./Computer Physics Cm~mmnications 107 ~1997) 223-245

Distribution format: uucncoded compressed tar file

Keyw~rds: nuclear fission, Langevin simulations, statistical model, nuclear friction

Nature of physical problem The code calculates distributions of observables characterizing the deexcitation of highly excited and rotating atomic nuclei by fission and light particle and giant dipole 9' evaporation. Averages, variances and correlations of son~ observables are also calculated. The most important of the output parameters are: fission (~spcctively survival) probabilities, pre-scission neutron and chargedparticle multiplicities and spectra~ prc-scission ~, multiplicities and spectra, average energies of these particles, scission lifetime distributions, and excitation energies at scission. The code provides the possibility to investigate the influence of different nuclear friction form factors on the observable quantities.

Method of solution The dynamical evolution of the compound nucleus is simulated numerically by means of a one-dimensional overdamped Langevin equaxion until after a certain delay time 1,t a stationary regime is reached. Afterwards the decay is described by an adequately modified statistical model. An algorithm for light-particle evaporation (p~eutrons. protons, deuterons, a-particles, y-quanta) is coupled to the fi~ion mode by a Monte Carlo procedure thus allowing for emission in discrete steps. Observables are calculated by sampling the corresponding events.

Typical nmning time The running time is defined mostly by the number of necessary trajectories and by the projectile energy. It also depends on the observable the user is interested in. The shortest time during which a meaningful result may be obtained for the fission probability and pre-scission neutron multiplicity is about 2 minutes (on a Pentium processor). If the interesting events are rare ones like giant dipole y-quanta emission, the running time can increase by 2 orders of magnitvde.

LONG WRITE.UP 1. Introduction For a long time the statistical model for fission and light-particle evaporation established by Bohr and Wheeler [ 1 ] and by Weisskopf [ 2] and developed into computer codes by Piihlhofer [ 3], Blann [4] and others has been sufficient to describe the available data for the nuclear fission process. It lasted until the eighties when measurements started to reveal enhanced pre-scission neutron multiplicities as compared to those calculated with a statistical model code. This observation was attributed to the action of nuclear friction which is responsible for a delay of the fission process and therefore more time is available for the emission of neutrons from the fissioning nucleus. We quote here only some general reviews which describe the experimental situation and which also give a survey on the theoretical models for the interpretation of the experiments~ These are the articles of Newton [ 5 ], Hilscher and Rossner [ 6], Hinde [7 ], and the article of Paul and Thoennessen [ 8], where emphasis is put on pre-scission giant dipole ~, emission, which has become an indicator for the action of friction in fission more recently. We mention a number of theoretical papers which introduce friction in connection with the nuclear fission process. The first one is that of Kramers [9], who mentions ihe possibility to treat nuclear fission by a FokkerPlanck equation. Davies et al. [ 10] introduced friction in classical equations of motion in order to describe kinetic energies of fission fragments. A review on the use of multi-dimensional Fokker-Planck equations is Ref. [ I 1 ]. There mainly mass distributions and kinetic energy distributions of fission fragments are investigated. Strumberger et al. [ 12] have combined a Fokker-Planck description of the fission process with rate equations describing the emission of light particles, and analyzed data for pre-scission light particle multiplicities. An alternative to a Fokker-Planck description is the use of Langevin equations. Investigations of fission within this framework started with the work of Abe et al. [ 13 ]. Multi-dimensional Langevin calculations without and with taking into account light particle emission have been performed by different groups. In Refs. [ 14-17] two-dimensional (elongation and constriction) Langevin equations are solved without an algorithm for particle evaporation, whereas in Refs. [ 18,19] particle emission is taken into account by solving rate equations for

L Gontchar et al./Computer Physics Communications 107 (1997) 223-245

225

the emission proc,,ss simultaneously with the Langevin equations. Here particle emission is described as a continuous process. In these papers only symmetric fission is treated. First steps for adding the mass asymmetry degree of freedom are reported in Ref. [201. Tile multi-dimensional Langevin calculations are very computer time consuming. Therefore one has been looking for simplifications. In a series of papers, which are reviewed in Refs. [21,22], a model has been deveiiloped which is simple enough to allow systematic calculations of most of the observables measured in connection with fission of hot nuclei, but which is still realistic enough to describe the essential physics correctly. It turns out to be sufficient to work with a one-dimensional overdamped Langevin dynamics (friction in fission is found to be quite strong in all phenomenological investigations, see e.g. [6] ), and to switch over after a certain delay time to a station~y regime by an adequately modified (i.e. matched to the dynamics) statistical model. This paper presents the explanation of the computer code of this Combined Dynamical and Statistical Model (CDSM). In the following section the model is explained in connection with a flow diagram of the computer code, whose subroutines are explained in the subsequent section. The code is published, first of all, in order to enable experimental groups to analyze experiments on the decay of hot and rotating nuclei.

2. The model in connection with a flow diagram for the computer code The model is described at great length in Refs. [21,22]. Therefore we give here only the formulas necessary to understar,~d the code. For this purpose it is very instructive to consult the flow diagram of Fig. 1. As discussed above, the model consists of a dynamical and statistical branches. In both of them particle emission is taken into account with Monte Carlo procedures which allows particle emission to take part in a discrete way. In order to be able to identify formulas and equations we add after the formulas the names of the files and functions in which they appear. Files, functions and the relevant variables and parameters are described in more detail in the following section; often we write the notation for variables and parameters as used in the program in brackets. For each trajectory simulating the fission motion an angular momentum L = hi is sampled from the spin distribution 2zr 2l+ I o'(1) = k--T l + e x p [ ( l - l c ) / S l ]

(1)

describing the fusion process. The parameters lc (L_crit) and 81 (L_diffus) can be provided by the user in the input file using his own model. If this is not the case a parametrization for lc and 81 is used which reproduces to a certain extent the dynamical results of the surface friction model [ 23 ] for fusion of two nuclei with masses and charges ( A p , Z p ) and (Ar, Zr) forming the compound system (A = Ae + Ar, Z = Zp + Zr). The quantity lc scales as

i, = v / ' A p , m r ~ a . (a//3 + A//3) * ( 0 . 3 3 + 0 . 2 0 5 . ~

)

,

(2)

when 0 < Ecru- Vc < 120 MeV; and when E c m - ]/c :> 120 MeV the term in the last bracket is put equal to 2.5. For the barrier Vc the following ansatz is used: Vc = 5c3 * ZPZP/(AIp/3 + AI/3 + 1.6),

with c3 = 0.7053 MeV. The diffuseness 81 is found to scale as

(3)

226

L Gontchar et al./Computer Physics Communications 107 (1997) 223-245

ev.acor.

re,iaaes

N~+I

A-Av

N

Z-Zv E'- lh-ev

L~

N

L-Lv

T go,~)

Y

qffi

m

i

/

fiuitm Nf+l

Z

<

N

L~ q<

<

L~

A-Av Z-Zv

[-,

E'-s,-e,

L-

N~ C~

~(o,l)

N

r,/r~ ]

'tn,'rq,

,

.

°

1

N

Fig. 1. Flow diagram of the calculationalprocedureof CDSM.

81={(ApAr)3/2*lO-5*[l.5+O.O2*(Ecm-Vc-lO)] (ApAT)3/2*IO-5*[I.5-O.O4*(E~m-Vc-IO)]

for Ecru > Vc + 10, for Ecm < Vc + 10.

(4)

The spin distribution is calculated in the function in the file . The trajectory with the particular angular momentum L ig started at the ground state position qgs of the entropy $(qgs, E~ot, A, Z, L). The quantity q is the dimensionless fission coordinate defined as half of the distance between the centers of masses of the future fission fragments normalized to the radius of the compound system, which is characterized by its mass and charge numbers, A and Z. The total initial excitation energy is given by E~ot = Ecm "t" Q, where Ecru = ElabAT/(Ar + Ap) is the center of mass energy of the fusing heavy-ion system and Q is the fusion Q-value. The quantities Elab and Q appear in the input file . LD where Mp If not specified otherwise by the user of the code, Q is calculated by Q = Mp q- Mr- MeN, and Mr are the masses of projectile and target, which include shell corrections; they are provided by the

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227

file , in which, when available, experimental values are used, otherwise they are taken from the macroscopic-microscopic theoretical calculations of Ref. [24]. The compound nucleus masses M LD are calculated from the liquid-drop model in the file . The entropy is defined by the Fermi-gas expression

S(q,E~*ot, A,Z,L ) = 2v/a(q,,A)[E~ot - V(q,A,Z,L) ] ,

(5)

and the temperature by

T(q'E~*°t'A'Z'L) =

.t

S( q' E~ot'A' Z' L ) 2a(q,A) "

(6)

The potential energy V(q, A, Z, L) is given by the liquid-drop model expression [25]

V(A,Z,L,q)=a2[I_k(N-Z) A

2] A2/3[Bs(q) - I] +c3-~-~3[Bc(q) Z2 - 1] +crL2A-5/3Br(q).

(7)

Here we have dropped terms which do not depend on the deformation coordinate q. The parameter~ !, Eq. (7) are specified to be [25] 2

a2 = 17.9439 M e V ,

c3 = 0.7053 MeV,

mo= 0.01044 MeV(zs)2/fm 2

k = 1.7826,

Cr- 4rnor~ MeV, (8)

ro = 1.2249 fm

= 10 -21 S. These parameters can be changed in the header-file . Our calculations are based on the c, h, te parametrization [26] for the surface, Coulomb, and rotational energy terms, Bs(q),Bc(q), and B,.(q), which depend on the deformation coordinate q. Only symmetric fission ( a = 0) is considered. For the overdamped motion the system follows the bottom of the fission valley which then characterizes a one-dimensional potential in terms of the fission coordinate. It turns out that the bottom of the fission valley of any individual nucleus is very close to the sequence of the saddle points of different nuclei. As long as hsd is a single-valued function of qsd, one can parametrize Bs as a function of q in

1 zs

the tbrm [27] Bs =

I + 0 . 4 ( 6 4 / 9 ) ( q - 0.375) 2, 0.983 + 0 . 4 3 9 ( q - 0.375),

if q < 0.452, if q > 0.452.

(9)

These equations are used if a parameter keyl in the input file is equal to 1. For allowing a more accurate but time consuming calculation (which according to our experience practically does not influence the final results at not too high excitation energies) of nuclei for oblate shapes (q < 0.375), one may use keyl nonequal to 1. In this case Bs is calculated according to In(x/c-3/2

Bs = I +

i+c

-3/2 ] 1

for q < 0.375.

(10)

x/c-3/2_ 1c'--3/"S J 2"-c

In order to find Bc we use the following set of equations. First, we need a relation between the fissility parameter X and the saddle point position q~, which is X = 0.05 In[0.875(qsa - 0.375) -~ - 1 ] + 0.74.

(Il)

Secondly, we use an approximation [ 28 ] for the fission barrier Bf as function of the fissility X,

Bf f 0.2599 - 0.2151X - 0.1643X 2 - 0.0673X3, Essp = ~. 0-7259Y3 - 0-3302y4 +0"6387]'5 + 7"8727y6 - 12"0061y7'

if X < 0.6, if X > 0.6.

(12)

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L Gontchar et al./Computer Physics Communications 107 (1997) 223-245

Here, Y = 1 - X and Essp is the surface energy of a spherical nucleus with fissility X. Then B¢ is given by

Bc=fl+(l-Bs+Bf/Ess~)/2X 1 - 0.2-~(q - 0.375)"

forq>0.452, for q < 0.452.

(13)

In case of key 1 nonequal to 1, Bc is calculated for oblate shapes according to arctan Vtc-3/9- - 1 Bc =

c

for q < 0.375.

(14)

/c:3/2 - i

The quantity Br(q) involved in Eq. (7) is proportional to the inverse rigid body moment of inertia and reads in the (c, h)-parametrization

Br = Jl'( t

if J± < JIJ and

Br = j~i

in all other cases,

q > 0.375,

J.I. =c2{ 1 + c-3 +4Bsh[ 2c3 + (4/15)Bshc3 -- 1]/35}/2, JIJ -'- c2{¢-3 "1-4Bsh[ (4/15)BshC 3 -- 11/35}.

(15)

The nuclear shape function Bsh (c, h) and the collective fission coordinate q(c, h) as function of c and h can be found in Ref. [28],

B~(c,h) = 2 h + ( c - 1 ) / 2 , q(c,h) = -~c(l + ~BshC3) •

(16)

The one-dimensional parametrization should break down for lighter nuclei because hsd(qsa), qsa(X) and h~(csa) in this case cease to be single-valued functions. Therefore the parametrization works only if X > 0.6. If one tries to calculate lighter systems with a fissility X < 0.6, the calculation is terminated and a corresponding message appears. To complete the input for the entropy we have now to specify the level density parameter a(q, A). For an adequate description of the fission process, a coordinate-dependent level density parameter is as important as the potential energy surface. The smooth part of the level density parameter as function of q has the following form [ 291"

a(q,A) = ~lA + ~t2Ae/3Bs(q) ,

(17)

where the dimensionless function Bs(q) was defined in Eq. (9). Following Ref. [ 30] we prefer among the different possibilities the weakest coordinate dependence which is consistent with the data. This corresponds to the Woods-Saxon single-particle potential [ 30,31 ] t = 0.073 MeV-I

and

fi2 = 0.095 MeV-t.

(18)

The user can also choose another set of coefficients according to Ref. [32], al = 0.0685 MeV -t

and

~2 = 4ill MeV -I,

(19)

or a deformation-independent level density, al -- l/did MeV -i

and

a2 =0.0 MeV - l .

(20)

To decide between the different options one has to specify the parameter key in . In the case when key > 0 it plays a role of did.

I. Gontchar et aLIComputer Physics Communications 107 (1997) 223-245

229

Having specified the initial conditions the fission process is propagated by a Langevin equation which reads in discretized form

[T~q, dS(q)] [d(T(q)~]

qn+l = qn + [ I~-(-qS-M

dq

II

7" + A -~q \ fi-(qS-M J J ~ + 11

7[Lfi-(-~M T(q) ]J

!1

7"iv,,.

(21 )

Here 7" is the Langevin time step, and wn is a Gaussian distributed random number with variance 2o The Langevin equation occurs in the file ; the time step ~r is called H in the input file , the quantity Wn is calculated in . The quantity T ( q ) is the nuclear temperature, and S(q) is the entropy. M is the inertia parameter which drops out of the overdamped equation. However, because it is usual to deal with the reduced friction parameter fi(q), which is the friction parameter r/divided by the inertia parameter M, we specify the mass parameter by the mass of the compound system (we do not take the reduced mass because our fission coordinate is half of the relative distance). The parameter h allows us to distinguish between the different possibilities to discretize the Langevin equation. These are called interpretations in the literature. In the analysis of the experiments on fission of hot nuclei discussed in the reviews [21,22] and in the papers quoted therein, the Itf-interpretation (,~ = O) [33] has been used exclusively. The program allows us also to use other interpretations, namely that of Stratonovich [34] (,~ = 1/2), or an interpretation which is consistent with the kinetic form of the Smoiucnowski equation of Ref. [ 35 ] ( ,;t = i ). A discussion of the consequences of the use of these interpretations can be found in Refs. [36,37]. The parameter for the choice of one of the interpretations is the parameter key3 in . Now the damping coefficient fi(q) has to be specified. Most of the analysis in Refs. [21,22] is done with a phenomenological universal friction which means that one is using the same set of parameters for all systems; we call this a Standard Parameter g~t (SPS) and the corresponding friction the SPS-friction. The SPS-friction is a special case of a friction which is constant at compact shapes until the necking in starts at qneck = 0.6 and then changes linearly until the scission point at q~c = !.2,

floq fiSPS(q) =

if q <

qneck,

fl~ _- qneck) ,60
(22)

The SPS-friction is weak for compact shapes, fi0q = 2 z s - l ; after the necking in is starting (at qneck - - 0.6), the friction is assumed to increase linearly up to the value of fi~c = 30 zs -! at scission (at q~c " 1.2). In the program fiOq is denoted as BETA0q and flu as betasc. The SPS-friction form factor is shown in Fig. 2 together with the form factor of the One-Body Dissipation (OBD) developed in Ref. [38], for the use of which there is an analytical fit formula in the program. This formula has been developed in Ref. [39] as approximation to the result of the numerical calculation of Ref. [40]. It reads 15/q0"43.4. I -- 10.5q 0"9 -Jr"q2

fiOBD(q) =

32-- 32.21q

if q > 0 . 3 8 , if q < 0.38.

(23)

The third possibility to choose is a form factor which depends on the temperature T (so-called Temperature Dependent Friction, TDF) in the following form:

fir(T) =

t or flor[ ! + ko ( T - Ti ) ] flor[l +ko(Tl - T o ) ]

i f,T < 1"o, if To <_ T < Tl , ifTi _
(24)

230

L Gontchar et al./Computer Physics Communications 107 (1997) 223-245

50

.... I O0 MeV

40

.....

. ....

. . . . . . . . . . . . . . . . . . . . . .

--'°

%30

MeV

"./ !.."....

20

:1

i/

""10 0

/. ." ~ -

li I,

0.0

---

/ ........../:.~.~ .......

7

!

.140 MeV

"

i

0.4

, /

I

- :

."

............

0.8

"!............

1 .2

1 .6

q Fig. 2. The damping coefficient versus deformation for the SPS (solid line), for the OBD (dotted line) in our approximation (Eq. (23)) and for the TDF at different excitation energies for 224Th (dashed and dash-dotted lines) for zero angular momentum. Note that for SIS and OBD the damping coefficient does not depend on the nucleus.

In the program the k0 is denoted as K0, To as TO, Ti as TI, and fl0r as BETAOT. Due to the deformation dependence of the intrinsic excitation energy, the TDF becomes also deformation-dependent. Examples of the coordinate dependence of the TDF are shown in Fig. 2 for different values of the excitation energy. There is also a possibility to generate a deformation- and temperature-dependent damping coefficient multiplying/3ses(q) o r / 3 O B D ( q ) by fir(T). The key parameter for the choice of the type of friction is key2 in . As long as we are staying in the dynamical branch of the model, i.e. as long as we are following the Langevin equation, we allow for the emission of light particles and giant dipole yts in the following way. We calculate the decay widths for neutron and y emission at each Langevin time step r. Then the emission of a particle is allowed by asking along the trajectory at each time step 7. whether a random number ~: is less than the ratio of the Langevin time step r to the decay time 7'dec = h / ( l ' n + / " y ) : ~ < 7./7.dec(0 <_ ~: _< !). 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 F,,/Fpart, where/'part = ~-'~,,I',, and (v = n, p, a, d, 9'). This procedure simulates the law of radioactive decay tbr the different particles. The widths for particle and giant dipole 9' decay are given by the following formulas for which we take the parametrizations due to Blann [41 ]. The emission width of a particle of kind v is given by E° -B,,

Fv=(2sv+ l)a.2li2pc(E, )

f

devpR(E*-e,v)~vO'inv(~,,),

(25)

0

where sp is the spin of the emitted particle v, and my its reduced mass with respect to the residual nucleus. The level densities of the compound and residual nuclei are denoted by pc(E*) and pr(E* - ev) • Because the level density occurs in the nominator and in the denominator pre-exponential factors are not important. The formula p~,, ( 2 L + I) exp (S)

(26)

is used, where S is the entropy. The intrinsic excitation energy is E*, and B,, are the liquid-drop binding energies according to Ref. [ 25], e is the kinetic energy of the emitted particle. The inverse cross sections are given by [41 ]

{ rr (I O'inv(~,) = with

0

for

> v,,,

for ev < V~,,

(27)

L Gontchar et aL/Computer Physics Communications 107 (1997) 223-245

3.4

r,, = 1.21 [ ( A - A~ ) 1/3 + A~/3 ] + ~

8~.,,

231

(28)

where A,, is the mass number of the emitted particle 1,, = n, p, d, a. The barriers for the charged particles are [41] Vv =

( Z - Z,, ) Z,, K,, R~, + 1.6 '

(29)

with K,, = 1.32 for a and deuteron, and 1.15 for proton. The parameter iwMAX in decides to account (iwMAX= 17) or not to account (iwMAX= 1) for the deformation dependence in the emission of charged particles. In the first case the corresponding decay widths are multiplied by the dimensionless area of the nuclear surface Bs(q) defined in Eq. (9) and the Coulomb barriers by the term Bc(q) of Eq. (13) and deformation-dependent binding energies B,, (q) are used. For the emission of giant dipole y-quanta we take the formula given by Lynn [42], J+ !

E*

E*

F, = ~-~ f d~ PC ( E* - e, l ) /=lS-II o

pc(E*,J)

~ 3

f d~ pc ( E* - 8 ) f ( e )

f(8) _ pc(E*)

(30)

o

with 4 1 + x e 2 NZ FGe.4 f ( e ) = 3zr mnc 2 hc ,4 (FGe.)2 + (~2

_ E2)2

'

(31)

with x = 0.75 according to Ref. [43], and EG and FG are the position and width of the Giant Dipole Resonance (GDR). The exchange force parameter K can be changed in . One can take into account the parallel and perpendicular components of the energies of the giant dipole modes Ec [I= EII/A !/3 and Ec _L= E x / A !/3. In most calculations of Refs. [ 21,22] Ell = Ex = 80 and FG = 5 MeV is used. After each emission act of a particle of kind p the kinetic energy 8,, of the emitted particle is calculated by a hit and miss Monte Carlo procedure, using the integrand of the formula for the corresponding decay width as weight function. This is done in . Then the intrinsic energy, the entropy, and the temperature in the Langevin equation are recalculated (this is done in ) and the dynamics is continued. The loss of angular momentum is taken into account by assuming that a neutron carries away l h, a proton 1 h, an a-particle 2h, a deuteron 2h, and a y-quantum lb. A dynamical trajectory either reaches the scission point, in this case it is counted as a fission event; or if the intrinsic energy E~s = Etot - V(qgs) for a trajectory still inside the saddle (q < qsd) reaches a value E~s < min(Bn, Bf) (Bn is the neutron binding energy and B/ is the fission barrier of the potential energy) the event is counted as evaporation residue one. The code does not calculate the further decay of the evaporation residues. If the Langevin trajectory has not fissioned and has not been counted as evaporation residue event when the time t = ta is reached, where ta is a delay time after which it is guaranteed that fission becomes stationary, and the difference in the entropies at saddle and ground state is above a certain value of the entropy SSTAT, S(qg~ ) - S(qsd ) > SSTAT,

(32)

the statistical branch of the model is entered. If Eq. (32) is not fulfilled the dynamical calculation is continued. The parameters ta (TauD) and SSTAT (Sstat) are specified in , recommended values are (to be sure of an independence of the results on ta) ta = 100 zs and SSTAT= 2. When entering the statistical branch we calculate the decay widths F,, for particle emission as above and the fission width F f = l~Rf according to

L Gontchar et aL/Computer Physics Communications 107 (1997) 223-245

232

rf=h\flgs]

\-ff'~J

;"" 2,rM

exp[S(qsd) - S ( q g s ) ] 2

{

I +erf

[

(qsc--qsd)

"

,

(33)

where (A = 0, 1/2, 1) correspond to the It6, Stratonovich and K-interpretation. In this expression the various qu~tities are taken at the ground state (gs), at the saddle (sd) or at the scission point (so). K e n we use a standard Monte Carlo cascade procedure where the kind of decay is now selected with the weights F.,-/Ftot with (i = fission, n, p, d, a, 7) and/'tot = E i El. This procedure 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 continve the cascade until the intrinsic energy is E~s < rain(B,,, B f). In this case we count the event as evaporation residue event. In the case where the Monte Carlo procedure leads to a fission event this is counted also. All physical quantities are calculated by counting numbers of the corresponding events: number of evaporation residue events, number of fission events, number of fission chances, number of particles of each kind emitted during fission, etc. One can do a pure statistical model calculation by switching off the dynamical branch. For this purpose one has to specify the parameters STATM > 0 together with TauD=0 and Sstat=0 in .

3. The program The code consist5 of nine C files and one header file. It needs four input files. The code produces four output files. There is a special history file where one line is attached alter each run of the code. Below a list of files with short descriptions is presented.

3.1. File This is the main file containing a set of functions including the function main(). These functions perform the Langevin simulations, the statistical calculations, etc. int Number() - Calculates the number of the run. void ChemSymbol() - Creates the chemical symbols of target, projectile, and compound nucleus. void Deslnfo() - Writes the input parameters to the file. void Stemp(FILE *filename,char transfer) - Outputs the number, date, and title to a file. char Qvalue() - Calculates the fusion Q-value of the reaction. void Barrier( ) - Calculates the liquid-drop model fission barriers according to Eq. (12). char SpinDistr( ) - Generates the compound nuclei spin distribution according to Eq. ( I ). void Zerolni() - Initializes the variables whose initial values must be equal to zero. void Definition() - Initializes the variables whose initial values must not be equal to zero. void ParticleEmission( ) - Recalculates the excitation energy, spin, mass and charge number of a nucleus alter each particle emission act. void Scission() - Accumulates information about the nuclei at the scission point: their excitation energies, angular momenta, number of neutrons emitted etc. void EvapResidue() - Accumulates information about the survived nuclei. void Summary() - Calculates the final output quantities.

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void Friction() - Prepares the form factors of coordinate- and temperature dependencies of the damping coefficient. void Temperature( ) - Prepares the temperature-dependent form factor of the diffusion coefficient. char CheckPot( ) - Checks number of maxima and minima of the potential energy and of the entropy. char CheckAI! (char WhatCheck) - Checks whether the excitation energy is reasonable and whether the isotopic table is expired.

3.2. File This file is responsible for reading and checking the input information. void DeslNP() - Reads the input information from input file . char lnpt:tCheck( ) - Checks correctness of the data in the files and .

3.3. File This file contains as the only function void DesOUT() and is responsible for the printout of the output information to the files , , , .

3.4. File This file contains as the only function void INFORM(char scr_inf) and prints the actual information to the screen when the program is running, o

3.5. File This file contains as the only function float POTENTIAL() and calculates the potential e~ergies and the entropies, finds their extreme points and calculates the corresponding quasistationary fission rates. This file calculates also the driving force for the Langevin equation (21).

3.6. File This file contain'; as the only function int STAT_MODEL() and calculates the widths tbr emission of the neutrons, charged-particles and ),-quanta. This file also makes a decision whether a particle is emitted or not and generates the kinetic energies of the emitted particles.

3.7. File This file provides the random numbers. float URANDI () - Generates the uniform random numbers. float GAUS() - Generates the Gaussian random numbers with variance 1. In order to generate the Gaussian random numbers the Polar Marsaglia Method is used, see Ref. [44].

3.8. File This file contains as the only function float ERR_FUN() and provides the error-function by means of the rational approximation 7.1.25 of Refo [451.

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L Gontchar et al./Computer Physics Communications 107 (1997) 223-245

3.9. File This file contains as the only function void DESBAR( ) and calculates the heights of the liquid-drop fission barriers according to the approximate Cohen, Plasil, and Swiatecki routine (Ref. [46]) for comparison with those generated by the file .

3.10. Header file This file contains the instructions of the preprocessor, definition of some constants and the description of the global variables. Before compiling the program one must point out in the first line of the header file the path to the files , , and . Below the constants which may be changed by the user are listed. E_max - Maximum value of the excitation energy needed for the calculations, in MeV. T_max - The number of elements in the arrays which are respo~sible for the temperature dependence. iMEV - The number of steps per 1 MeV in the energy-dependent arrays in the . NE - The number of steps per 1 MeV in the output arrays corresponding to the energy spectra of the emitted particles, NE should be less or equal to iMEV. P_max - The maximum value of the energy of the emitted particles. NLM

-

The maximum number of neutrons to be emitted.

ZLM - The maximum number of protons to be emitted. r0 - Size parameter of the liquid-drop model, in fro. al - Volume energy parameter of the liquid-drop model, in MeV. a2 - Surface energy parameter of the liquid-drop model, in MeV. c3 - Coulomb energy parameter of the liquid-drop model, in MeV. kap - Isospin parameter of the liquid-drop model. qsc

-

The scission point deformation.

elev - Defines the strength of the odd-even term in the mass formula. q_neck - The value of the c a l l , t i r e coordinate "q" along the fission valley at which the necking in starts. iwMAX - A key parameter: if iwMAX = i the deformation dependence of the binding energies and Coulomb barriers are discarded, if iwMAX = 17 the deformation dependence is accounted for.

3.11. lnput file This is the main input-data file. The parameters of the target and projectile, lab energy, number of trajectories, etc., are defined here. There are also several key parameters. Below we present an instruction of the use of and a list of its parameters. Important! The first line must not exceed 79 characters! This line may contain any text information. AP - Projectile mass number. AT - Target mass number. ZP ZT

-

-

Projectile charge number Target charge number.

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235

Qval - Fusion Q-value. If Qval = 0 it is ~Iculated by the program using the experimental masses of projectile and target provided by the file and the liquid-drop mass of the compound nucleus calculated in ; provided Qval ~ 0 this value is used in the calculations. Lneu, Lpro, Lalp, Ldeu, Lgam - The angular momenta removed from the compound nucleus by a single emission act of a neutron, proton, alpha-particle, deuteron, and y-quantum. Lsm - When Lsm< 0 it does not influence the calculations; when Lsm is positive the cascade calculations are skipped for L-values from 0 up to Lsm; the recommended value is L s m - 0 . kappa - Correction of the gamma strength function for the exchange forces. E_II, W_GDR..II - Position and width of the parallel GDR. E_I_, W_GDR_I_ - Position and width for the perpendicular GDR. Vcpfit, Vcafit, Vedfit - Factors enabling to correct the heights of the Coulomb barriers for the emission of charged particles, recommended values are !. Ntra_Start_ - Initial number of trajectories; it is close to the total number of the trajectories. Elab - Lab energy of the projectile. L_crit, L_diffus - Parameters of the initial compound nucleus spin distribution provided by the user; if both are put to be negative the standard program routine is used in order to calculate L_crit and L_diffus. L_n'lax, L_min - Borders of the L-window for which the calculations are performed; recommended values are L_max = 100, L_min = 0. Sstat - Controls the switching from the dynamical to the statistical branch of the program; recommended value is Sstat = 2. BETA0q The value of the damping coefficient at the ground state in case of SPS-type friction, in zs -! . -

betasc - The value of the damping coefficient at the scission point in case of SPS-type friction, in zs -I . BETAOT - The value of the damping coefficient at the temperature TO in case of TDF friction, in zs -I . K0 - The slope coefficient of the temperature dependence of the damping coefficient, in MeV -~ . TO - The value of the temperature at which the linear temperature dependence starts, in MeV. Tl - The value of the temperature at which the linear temperature dependence stops, in MeV. H - The time step of the dynamical simulations, in zs; it must be adjusted until a sensitivity to it in the fission probability Pf and the pre-scission neutron multiplicity Neu disappears. The recommended starting value is H=0.3. key - The level density key parameter; if it is negative the prescription of Ref. [31] is used, if it is zero the prescription of Ref. [ 32 ] is used, if positive it is used as the denominator in Eq. (20). STATM - If STATM is positive only statistical calculations are performed; otherwise the ,dynamical simulations are performed. Important! Positive value of STATM gives meaningful results only together with TauD.,,:0 and Sstat = 0! OPNL - If OPNL is positive its value is equal to Ihe number of trajectories to be generate6 for each partial wave in the region L_max - L_min; in this case the Ntra_Start_, L_crit, L_diffus do not play a~y role. If OPNL < 0 it is ignored. TauD - Duration of the dynamical simulations, in zs; must be adjusted until a sensitivity to it in the fission probability Pf and the pre-scission neutron multiplicity Neu disappears; the recommended starting value is TauD = 30. YRAY - If YRAY is positive gamma-quanta are emitted, otherwise not.

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I Gontct, ar et al,/Computer Physics Communications 107 (1997) 223-245

key l - If key l = 1 the harmonic approximation for the potential in the region of the oblate shapes is used, otherwise the precise formula for the ellipsoidal shapes is used; in the last case the time step has to be reduced by an order of magnitude and the CPU time increases by an order of magnitude in order to obtain accurate results; the recommended value is key l = 1; if there are numerous messages of the type of iMS_imS = - 1 during the program is running the key ~ 1 must be used. k e y 2 - If key2 =0 the SPS-type friction is used (see Eq. (22)), if key2 = 1 the OBD is used (see Eq. (23)), if key2=2 the TDF is used (see Eq. (24)), if key2=3 the product of TDF and SPS-type friction is used, if key2 =4 the product of TDF and OBD is used. key3 - If key3 = 1 the It6 interpretation of the Langevin equation (corresponding to the analysis presented in Refs. [ 21,22] ) is active, if key3 = 2 the Stratonovich interpretation is used, if key3 = 3 the K-interpretation is used.

3.12. Input file Provides the number of the actual run of the program.

3.13. Input file This file contains experimental and theoretical masses of about 4 700 nuclides.

3.14. Input file This file contains the q (center-of-mass distance) and c (elongation) arrays. It must not be changed.

3.15. Output file This is the main output file. After each run of the code a new line is attached to it. This line is about 1 200 characters long and contains about 100 output quantities which are mainly the first and second moments of the various distributions. Below a list of the output quantities with a short description is presented. In the brackets the values of the parameters are given corresponding to the input-output example. i w M E M - If iwMEM- 0 no coordinate dependence of the binding energies and Coulomb barriers was accounted for; if iwMEM-6 the calculation was performed with the deformation dependence of the binding energies and Coulomb barriers. AR AT, ZP, ZT, Qval - See . KOF---Ntra_Start_ - See . Elab - See .

Eexc - Initial total excitation energy, in MeV. L_crit, dtL=L_diffus, L_max, L.min, Sstat, BETA0q, BETAOT, K0, TO, TI, H - See . mass - Total mass of the compound nucleus, in MeV zs 2 / f m 2. BfL0 - Height of the fission barrier of the compound nucleus at zero angular momentum according to the routine. B~

- Height of the fission barrier of the compound nucleus at L_crit angular momentum according to the

routine. aCN - Single-particle level density of the compound nucleus. betasc, Lneu, Lpro, Lalp, Ldeu, Lgam, key - See .

l. Gontchar et al./Computer Physics Communications 107 (1997) 223-245

237

Lsm, E_II, W_GDRdl - See . Lf - The average angular momentum of nuclei at the scission point. Lmf - The average initial angular momentum of those nuclei which finally fissioned. Lfvar - The variance of the angular momentum of nuclei at the scission point. Lmfvar - The variance of the initial angular momentum of those nuclei which finally fissioned. Pf0 - The first chance fission probability in percent, percentage of the fission events in which not a single pre-fission (i.e. pre-saddle) neutron has been emitted. kappa - See . ibadAZ - Number of bad events in which the isotopic table of the binding energies has been expired. If bad events appear one should enhance NLM and ZLM in . STATM, OPNL - See . elev - See . CFcs - Complete fusion cross section, in mb. TauD, YRAY - See . Be_0, Be_I - The neutron binding energy in the compound nucleus and that which appears after the emission of one neutron. Bldm_0, Bldm_l - The height of the fission barrier at zero angular momentum in the compound nucleus and that which appears after the emission of one neutron, obtained by means of approximate formula (12). Ntra - The actual number of trajectories. Ntis - The number of fissioned trajectories. P f - The fission probability, in percent. ePf

-

The relative statistical error of the fission probability, in percent.

Eva - The survival probability, in percent. Ntot, Ptot, Atot, Dtot, Gtot - The total number of the emitted neutrons, protons, a-particles, deuterons, y-quanta. Npre, Ppre, Apre, Dpre, Gpre - The number of the neutrons, protons, a-particles, deuterons, y-quanta emitted from the fissioned trajectories. Neu - The pre-scission neutron multiplicity, i.e. the average of the distribution of fission events with respect to the number of emitted neutrons. V2npre - The variance of the distribution of fission events with respect to the number of emitted neutrons. Bn_0 - The neutron binding energy in the compound nucleus. Neugs - The pre-fission neutron multiplicity, i.e. the average of the distribution of fission events with respect to the number of neutrons emitted before the saddle point is reached. dNeu - The absolute statistical error of the pre-scission neutron multiplicity. NeuSS - The ground-to-saddle neutron multiplicity. Avenpre - The average value of the kinetic energy of the pre-scission neutrons, in MeV. V2enpre - The variance of the kinetic energy of the pre-scission neutrons, in MeV 2. Pro, V2ppre, Bp0, Progs, dPro, ProSS, Aveppre, V2eppre - The same as the previous eight quantities but for the protons. Alp, V2apre, Ba0, Alpgs, dAIp, AIpSS, eAlp, Aveapre, V2eapre - The same as the previous eight quantities but for the a-particles.

!. Gontchar et al./Computer Physics Communications i07 (1997) 223-245

238

Deu, V2dpre, Bd0, Deugs, dDeu, DeuSS, Aved, V2edpre - The same as the previous eight quantities but for the deuterons. Gampre, V2gpre, eGam, Gamgs, dGam, GamSS, Aveg, V2egpre - The same as the previous eight quantities ~ t for the y-quanta, except eGam which is the relative statistical error of Gampre, in percent. x317

-

A long-lifetime fission component: the percentage of fission events occurring after 3 x 104 zs.

ex317 - Relative statistical error of the previous quantity, in percent. xl6 - A second long-lifetime fission component: the percentage of fission events occurring after 105 zs. exl6 - Relative statistical error of the previous quantity, in percent. tf_10_21 - The average fission, lifetime, in zs. V 2 t f - The variance of the fission lifetime, in zs 2. Hop - Percentage of trajectories passing through the statistical branch. Tsc - The average temperature at the scission point. E_sc - The average intrinsic excitation energy at the scission point. V2E_sc - The variance of the intrinsic excitation energy at the scission point, in MeV 2. B~M0 - The height of the fission barrier of the compound nucleus obtained by at zero angular momentum. Erelease - The potential energy difference between the ground state and the scission point, Kc - Vgs, at zero angular momentum. BVLmf - The height of the fission barrier of the compound nucleus obtained by at the angular momentum Lmf. BVLf - The height of the fission barrier of the compound nucleus obtained by at the angular momentum Lf. Nfdyn - The number of fission events occurring in the dynamical branch. Ganlpre_9 - Multiplicity of the pre-scission y-quanta with energies higher than 9 MeV. GamER_9 - Multiplicity of the y-quanta with energies higher than 9 MeV emitted in coincidence with evaporation residues. XCN - Fissility parameter of the compound nucleus.

3.16. Output file This file provides information on the fission barriers of the compound nucleus for different angular momenta: Bfmy - The barriers generated by the file (i.e. those which are actually used in the calculations). BfCPS - The barriers obtained by means of the Cohen, Plasil and Swiatecki routine (see Ref. [46] ). Sgs_Ssd - The difference between the entropies at the ground state and at the saddle point. iSgs - Deformation of the compound nucleus at the ground state of the entropy x 100. iSsd

-

Deformation of the compound nucleus at the saddle point of the entropy × 100.

iVgs - Deformation of the compound nucleus at the ground state of the potential energy × 100. iVsd - Deformation of the compound nucleus at the saddle point of the potential energy × 100.

L Gontchar et al./Computer Physics Communications 107 (1997) 223-245

239

3.1Z Output file This file contains various output quantities differential with respect to angular momentum calculated for the compound nucleus: L -- The angular momentum. sigcfL - The complete fusion cross section, in mb. B_n_CN - The neutron binding energy. BV_CN - The height of the potential energy fission barrier, in MeV. BS_CN - The height of the entropy fission barrier. aSsd - The level density parameter at the saddle point of the entropy. ER - Number of evaporation residue events for each partial wave divided by the total number of evaporation residue events times 100. Fi - Percentage of the fission events from each partial wave normalized to be 100 after summing up through all the partial waves presented. iVgs - See . Vgs - The value of the potential energy at the ground state. iVsd - See . Vsd - The value of the potential energy at the saddle point. Vsc - The value of the potential energy at the scission point. a l a n - The ratio of the value of the level density parameter at the saddle point of the entropy to that at the ground state.

3.18. Output file This file contains distributions of different observables: Queue distributions for fission - 1"he distribution of the number of the step que_f in a deexcitation chain for fissioned trajectories. If for instance for que_f= ! Npre=0.1, this means that a neutron is emitted in the first step of the de.excitation chain in 10% of the cases; if for que_f=2 Apre=0.3, this means that an a-particle in 30% of the cases is emitted in the second step. Queue distributions for survival - The distribution of the number of the step que_ER in a deexcitation chain for survived trajectories. Time distributions for fission - The time distribution of the emission acts for fissioned trajectories and for fission itself. The lg_time is the upper border of the proper bin. In particular, the first row contains events occurring during a time shorter than 1 zs. The lg_time is the Iog(time/zs). Evaporation residues - The N.Z evaporation residue distribution, in percent. Fission events/chances - The N.Z fission event distribution, in percent. Fission chances - gamma correlations - The N_Z ),-quanta chart for fissioned trajectories, in number of events. Prescission particle spectra - Energy spectra of pre-scission particles, En is the lower border ~,f the energy bin, in MeV. Deformation dependencies and distributions - Defomlation (q) dependencies of neutron (B_n), proton (B_p), a-partic}e (B_a), deuteron (B_d) bindi.ng energies, and of surface (Bs) and Coutemb (Bc) form factors. q, Npre, Ppre, Apre, Dpre, Gpre - Distributions of the pre-scission particles with respect to the deformation at

240

L Gontchar et ai./Con::',:cer Ptg"sh's Communicaeions 107 (1997) 223-245

which they have been emitted, each normalized to 100%. Deformation dependencies of: VCN0 - The compound nucleus potential energy at zero angular momentum, in MeV. SCN0- ~e

compound nucleus entropy at zero angular momentum.

fvCN0 - The driving force as minus the derivative of the potential with respect to the coordinate. ffCN0 - The driving force as the derivative of the Helmholtz free energy with respect to the coordinate. Npre_integr - Integral of Npre over the detbrmation; at qse it becomes equal to the pre-scission neutron multiplicity. VCNL_min - The compound nucleus potential energy at the minimum angular momentum, in MeV. SCNLmin - The entropy of the compound nucleus at the minimum angular momentum.

3.19. File

This is a history file, updated. After each run the code stores here the values of basic input parameters and keys. Values of some constants defined by the preprocessor instructions are stored here too. These constants are attached to the tale of the main input file .

4. Special features of the program

-

In the following we list some special features the user has to pay attention to, when running the Betbre compiling the program one must point out in the first line of the header file the path , , and . After each change in the file the program should be compiled again. The first line of the input file must not exceed 79 characters. A positive value of STATM gives meaningful results only together with TauD = 0 and Sstat = 0 in pure statistical model calculations. For a value of fl < 2 zs-I all results are not reliable because the approximation of overdamped which the model is based is expected to break down.

program. to the files

performing motion on

References

[ I ! N. Bohr, J.A. Wheeler, The mechanismof nuclear fission, Phys. Rev. 56 (1939) 426-450. 121 J.M. Blatt, V.E Wesskopf, Theoretical Nuclear Physics (New York, London, 1952). 13l E Piihlhofer, On the interpretation of evaporation residue mass distributions in heavy-ion induced fusion reactions, Nucl. Phys. A 280 (1977) 267-234. [41 M. Blann, M. Beckerman, Statistical model for nuclei at high excitation and angular momenta: some new considerations, Nucleonika 23 (1978) 1-34. 151 LO. Newton, Nuclear fission induced by heavy ions, Soy. J. Part. Nucl. 21 (1990) 349-383. [61 D. Hilscher, H. Rossner, Dynamicsof nuclear fission, Ann. Phys. Ft. 17 (1992) 471-552. [ 7 i D.J. Hindu, Neutron emission as a clock and thermometer to probe the dynamics of fusion-fission and quasi-fission, Nucl. Phys. A 553 (1993) 255-270c. 181 P. Paul, M. Thoennes~n, Fission time scales from giant dipole resonances, Ann. Rev. Part. Nucl. Sc. 44 (1994) 65-92. 191 H.A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (1940) 284-304.

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il0l K.T.R. Davies, A.J. Sierk, J.R. Nix, Effect of viscosity on the dynamics of fission, Phys. Rev. C 13 (1976) 2385-2403. l l l l G.D. Adeev, i. Gontchar, V.V. Pashkevich, O.!. Serdyuk, A diffusion model tbr formation of the fission-fragment distributions, Soy. J. Part. Nucl. 19 (1988) 529-1298. !121 E. Strumberger, K. Dietrich, K. Pomorski, A more detailed calculation of particle evaporation and fission of compound nuclei, Nucl. Phys. A 529 (1991) 522-564. 1131 Y. Abe, C. Gregoire, H. Delagrange, Langevin approach to nuclear dissipative dynamics, J. Phys. (France) 47 (1986) C4-329-C4338. i141 G.-R. Tillack, Two-dimensional Langevin approach to nuclear fission dynamics, Phys. Lett. B 278 (1992) 403-406. 1151 I. Gontchar, G.I. Kosenko, N.I. Pischasov, O.!. Serdyuk, Calculation of the moments of the fission-fragment energy distribution by the method of Langevin equations, Yad. Fiz. 55 (1992) 920-928. l l 6 i J. Bao, Y. Zhuo, X. Wu, Systematic studies of fission fragment kinetic energy distributions by l.angevin simulations, Z. Phys. A 352 (1995) 321-325. li71 T. Wada, N. Carjan, Y. Abe, Multi-dimensional Langevin approach to fission dynamics, Nucl. Phys. A 538 (1992) 283-290. 1181 G.-R. Tillack, R. Reif, A. Schiiicke, P. Fr/Sbrich, H.J. Krappe, H.G. Reusch, Light particle emission in the Langevin dynamics of heavy-ion induced fission, Phys. Lett. B 296 ( ! 992) 296-301. 1191 'L Wada, Y. Abe, N. Carjan, One-body dissipation in agreement with prescission neutrons and fragment kinetic energies, Phys. Rev. Let~. 70 (1993) 3538-354 !. 1201 Y. Abe, S. Ayik, P.-G. Reinhard, E. Suraud, On stochastic approaches of nuclear dynamics, Phys. Rep. 275 (1996) 49-196. 1211 i. Gon,'char, [,,angevin fluctuation-dissipation dynamics of fission of excited atomic nuclei, Fiz. Eiem. Chast. At. Yadra 26 (1995) 932-10(,~0. 1221 P. Frfibrich, i. Gontchar, Langevin description of fusion, deep-inelastic collisions and heavy-ion induced fission, Phys. Rep. (1997), in press. 123 ] J. Marten, P. Frfbrich, Langevin description of heavy-ion collisions within the surface friction model, Nucl. Phys. A 545 (1992) 854-870. 1241 P. MOller, W.D. Myers, W.J. Swiatecki, J. Treiner, Nuclear mass formula with a finite-range droplet model and a folded-Yukawa single-particle potential, At. Data & Nucl. Data Tables 39 (1988) 225-233. 1251 W.D. Myers, WJ. Swiatecki, Nuclear masses and deformations, Nucl. Phys. 81 (1966) 1-60; Anomalies in nuclear masses, Ark. Fys. 36 (1967) 343-352. 1261 M. Brack, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M. Strutinsky, C.Y. Wong, Funny hills: the s~'~ell-correctionapproach to nuclear shell effects and its applications to the fission process, Rev. Mod. Phys. 44 (1972) 320.-405. 127l i. Gontchar, P. Frtbnch, N.i. Pischasov, Consistei~i 0~,,iami~:ai an0 stausticai tlescriptio,1 of t~ssion of ho~ auciei, Phys. Rev. C 4? (1993) 2228-2235. 1281 R.W. Hasse, W.D. Myers, Geometrical Relationships of Macroscopic Nuclear Physics (Springer, Berlin, Heidelberg, New York, 1988). 1291 A.V. lgnatyuk, G.N. Smirenkin, M.G. itkis, S.i. Mulgin, V.N. Okolovich, Investigation of the fissility of the prc-actinide nuclei in charged-particle induced reactions, Fiz. El. Chast. At. Yadra 16 (1985) 709-772. 1301 E.M. Rastopchin, Yu.V. Ostapenko, M.I. Svirin, G.N. Smirenkin, Nuclear surface effect on level density and fission probability, ?Cad. Fiz. 49 (!989) 24-32. 1311 A.V. Ignatyuk, M.G. Itkis, V.N. Okolovich, G.N. Smirenkin, A.S. Tishin, Fission of pre-actinide nuclei. Excitation functions for the ~tr, f ) reaction, Yad. Fiz. 21 (1975) 1185-1205. 1321 J. Toke, W.J. Swiatecki, Surface-layer corrections to the level-density formula for a diffusive fermi gas, Nucl. Phys. A 372 ( 1981 ) 141-150. 1331 H. Risken, The Fokker-Pianck equation, 2rid ed. (Springer, Berlin, 1989). 1341 R.L. Stratonovich, Topics in the theory of random noise, Vols. ! and II (Gordon & Breach, New York, 1967). 1351 Yu.L. Klimontovich, Nonlinear Brownian motion, Physics Uspekhi 37 (1994) 737-766. 1361 A.E. Gettinger, i. Gontchar, R.S. Kutananov, L.A. Litnevsky, Fission rate of excited nuclei in presence of deformation dependent friction, Omsk State Railway Academy, preprint (1996). 1371 i. Gontchar, A.E. Gettinger, R.S. Kurmanov, L.A. Litnevsky, The stochastic approach to the fission process - a problem of the interpretation of the Langevin equation with a multiplicative noise, Int. School Seminar on Heavy Ion Physics (Dubna, 1997). J. 1381 Blocki, Y. Boneh, J.R. Nix, J. Randrup, M. Robel, A.J. Sierk, W.J. Swiatecki, One-body dissipation and the super-viscidity of nuclei, Ann. Phys. 113 (1978) 330-386. 1391 I. Gontchar, L.A. Litnevsky, Dependence of nuclear dissipation upon deformation or temperature: analysis of the data using a Langevin-Monte-Carlo approach, Z. Physik A (1997), in press. g401 N. Carjan, Classical (Langevin) and quantum (SchriSdinger) approaches to fission dynamics, Workshop on Open Problems in Hea~,y Ion Reaction Dynamics at Vivitron Energies, CRN, Strasbourg, France, CRN 93-22 (1993) 524-546. M. Blann, Decay of deformed and superdetbrmed nuclei formed in heavy ion reactions, Phys. Rev. C 21 (1980) 1770-1782. 141] .I.E. Lynn, Theory of Neutron Resonance Reactions (Clarendon, Oxford, 1968). 1421

242 {43] [44] [45 ] [46]

/. Gontchar et al. / Computer Physics Communications 107 (1997) 223-245 V.G. Nedorezov, Yu.N. Ranyuk, Fotodelenie Yader za Gigantskim Rezonansom (Naukova Dumka, Kiev, 1989) [in Russian]. R Kloeden, E. Platen, H. Schurz, Numerical Solution of SDE Through Computer Experiment (Springer, Berlin, Heidelberg, 1994). M. Abramovitz, l.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965). S. Cohen, E Plasil, WJ. Swiatecki, Equilibrium configurations of rotating charged or gravitating liquid masses with surface tension, Ann. Phys. (NY) 82 (1974) 557-596.

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TEST RUN OUTPUT

RUN*l: Input file <*****DESCEND***** - - - T h i s l i n e i s d e s i g n e d f o r comments. 79 symbols o n l y . - - - > 19 181 9 73 O. 1 1 2 2 1 -10 AP AT ZP ZT Oval Lneu Lpro L a l p Ldeu L g a m Lsm

.75 kappa

80. E_II

I0000. Ntra_Start_ 2. BETAOq

30. betasc

5. W_GDR_II

80. E_I_

5. W_GDR_I_

1. Vcpfit

1. Vcafit

1. Vcdfit

150. Elab

-14 L_crit

-3.9 L_diffus

I00. L_max

O. L_min

2.00 SsZat

1.172 BETAOT

19.19 KO

1.126 TO

4.00 T1

0.300 H

12. -10. kty(-!,OT~S,+A!key) STATM

0.0 OPNL

0ee=-46.49

Use STATM > 0 only with

1 key1, har_pot,

1 key2, fric.,

NUM

DATE

30. TauD

1 YRAY

TauD=O and with Sstat=O

!!!

3 key3 inter]).

E_max

T_max

P_max

NLM

ZLM

LMAX

NE

iMEV

RUN*I: Output file NUM DATE SYSTEM 1 : 30. 7.97 : 19 F + 181Ta => 200Pb < COMMENTS > <****.DESCEND***** --- This line is designed for comments. 79 symbols only.---> iwMEM, O,

AP, 19,

AT, 181,

ZP, 9,

ZT, 73,

Oval, -29.81,

KOF, 10000,

Elab, 150.0,

Eexc, 105.9,

L_cri¢, 65.2,

dtL, 4.9,

L_max, 100,

L_min, O,

Sstat, 2.00,

BETAO, 32.000,

KO, 19.19,

TO, 1.13,

T1, 4.00,

H, 0.300,

mass, 107.1,

BfLO, 12.90,

BfLc, 1.56,

aCN, 16.67,

betasc, 30.00,

Lneu, 1,

Lpro, 1,

Lalp, 2,

Ldeu, 2,

Lgam, 1,

key, 12.00,

Lsm, -10,

E_II, 14.41,

W_GDR_II, L f , 5.00, 51.5,

Lmf, 57.3,

Lfvar, 1.31e+02,

Lmfvar, PrO, 1.03e+02, 4.4,

kappa, 0.75,

ibadAZ, O,

STATM, -10,

OPNL, O,

elev, 11,

CFcs, 1219.04,

TauD, 30,

YRAY, I,

Be_O, 8.15,

Be_l, 6.75,

B1 dm_O, 12.31,

B!dm_l, 12.09,

243

244

L Gontchar et aL / Computer Physics Communications 107 (1997) 223-245

Ntra, 9950,

Nfis, 5186,

Pf, 52.12,

ePf, Eva, 1.39e+00, 47.9,

Ntot, 67427,

Ptot, 967,

A¢o¢,

Dt;o¢,

G¢o¢,

1295,

80,

2945,

Npre, 29012,

Ppre, 269,

Apre, 310,

Dpre, 18,

Gpre, 181,

Neu, 5.594,

V2npre, 3.346,

Bn_O, 8.15,

Neugs, 5.241,

dNeu, 0.08,

NeuSS, 0.353,

Avenpre, 3.068,

V2enpre, 5.368,

Pro, V2ppre, BpO, 5.19e-02, 5.15e-02, 5.42,

Progs, dPro, ProSS, 4.76e-02, 3.23e-03, 4.24e-03,

Aveppre, V2eppre, 12.862, 5.702,

Alp, V2apre, BaO, 5.98e-02, 5.62e-02, -4.60,

A1pgs, dAlp, AIpSS, 5.46e-02, 3.40e-03, 5.21e-03,

Aveapre, 23.345,

Deu, V2dpre, BdO, 3.47e-03, 3.46e-03, 11.11,

Deugs, dDeu, DeuSS, Aved, 3.47e-03, 8.18e-04, O.OOe+O0, 14.972,

V2edpre, 7.513,

Gampre, V28pre, eGaN, 3.49e-02, 3.52e-02, 7.6,

GaNgs, dGam, GaNSS, 3.39e-02, 2.65e-03, 9.64e-04,

V2egpre, 19.491,

x317, 16.99,

ex317, 3.37,

x16, 9.89,

ex16, 4.42,

Hop, 81.47,

Tsc, 1.33,

E_sc, V2E_sc, BFLDNO, Erelease, 3.15e+01, 3.32e+02, 1.29c+01, 5.84e+00,

Nfdyn, 1844,

GaNpre_9,GamER_9, XCN 1.33e-02, 1.85e-01, 7.012e-01

Aveg, 7.160,

V2eapre, 4.447,

tf_lO_21, V2¢f, 6.50e+04, 7.68e+10, BVLmf, BVLf, 3.63e+00, 5.25e+00,

RUN*2: Input file <*****DESCEND***** - - - This l i n e is designed for comments. 79 symbols o n l y . - - - >

-10

19

181

9

73

O.

1

1

2

2

1

AP

AT

ZP

ZT

Oval

Lneu

Lpro

Lalp

Ldeu

L g a m Lsm

.75 kappa

80. E_II

10000. Ntra_Start_ 2. BETAOq

30. betasc

5. W_CDR_II

80. E_l_

5. W_GDR_I_

1. Vcpfit

1. Vcafit

150. Elab

-14 L_crit

-3.9 L.diffus

100. L_max

O. 0.00 L _ m i n Sstat

1.172 BETAOT

19.19 KO

1.126 TO

4.00 T1

0.300 H

0.0 OPNL

O. TauD

1 YRAY

12. 10. key(-I,OTkS,+A/key) STATM

0ee--46.49

Use STATM > 0 only with

1 keyl, har_po~,

1 key2, frt¢.,

NUM

TauDfOand with SstatffiO

!~!

3 key3 inZerp.

DATE E_max T_max P_max NLM

ZLM

LMAX NE

iMEV

1. Vcdfi¢

L Gontchar et aL/Computer Physics Communications 107 (1997) 223-245

RUN*2: Output file NUM DATE SYSTEM 1 : 30. 7.97 : 19 F + 1 8 1 T a => 200Pb < COMMENTS > <*****DESCEND***** --- This line is designed for commenCs. 79 symbols only.--->

iwMEM, O,

AP, 19,

AT, 181,

ZP, 9,

ZT, 73,

Qval, -29.81,

KOF, 10000,

Elab, 150.0,

Eexc, 105.9,

L_crit, 65.2,

dtL, 4.9,

L_max, 100,

L_min, O,

Sstat, 0.00,

BETAO, 32.000,

KO, 19.19,

TO, 1.13,

T1, 4.00,

H, 0.300,

mass, 107.1,

BfLO, 12.90,

BfLc, 1.56,

aCN, 16.67,

betasc, 30.00,

Lneu, 1,

Lpro, 1,

Lalp, 2,

Ldeu, 2,

Lgam, 1,

key, 12.00,

Lsm,

E_II,

W_GDR_II, Lf,

-10,

14.41,

5.00,

50.6,

Lmf, 53.5,

Lfvar, 1.95e+02,

Lmfvar, PrO, 1.51e+02, 26.3,

kappa, 0.75,

ibadAZ, O,

STATN, 1C,

OPNL, O,

elev, 11,

CFcs, 1219.04,

TauD, O,

YRAY, 1,

Be_O, 8.15,

Be_l, 6.75,

Bldm_O, Bldm_1, 12.31, 12.09,

NCra, 9950,

Nfis, 6636,

Pf, 66.69,

ePf, Eva, 1.23e+00,33.3,

N¢ot, 45411,

Ptot, 808,

AZot, 1039,

Dtot, 71,

GZot, 1846,

Npre, 18891,

Ppre, 216,

Apre, 225,

Dpre, 22,

Gpre, 74,

Neu, 2.847,

V2npre, 6.539,

Bn_O, 8.15,

Neugs, 2.S~7,

dNeu, 0.05,

NeuSS, 0.000,

Avenpre, 3.386,

V2enpre, 6.281,

Pro, V2ppre, BpO, 3.25e-02, 3.24e-02, 5.42,

Progs, dPro, ProSS, A v e p p r e , V2eppre, 3.25e-02, 2.25e-03, O.OOe+O0, 13.190, 7.087,

Alp, V2apre, BaO, 3.39e-02, 3.28e-02, -4.60,

Aipgs, dA1p, AIp~S, Aveapre, V2eapre, 3.39e-02, 2.26e-03, O.OOe+O0, 23.529, 4.571,

Deu, V2dpre, BdO, 3.32e-03, 3.30e-03, 11.11,

Deugs, dDeu, DeuSS, Aved, 3.32e-03, 7.07e-04, O.OOe+O0, 13.955,

V2edpre, 5.112,

Gampre, V2gpre, eGam, 1.12e-02, 1.16e-02, 7.6,

Gamgs, d G a m , GamSS, Aveg, 1.12e-02, 1.33e-03, O.OOe+O0, 7.284,

V2egpre, 18.541,

x317, 4.51,

ex317, 5.78,

x16, 2.53,

ex16, 7.72,

Hop, 100.00,

Tsc, 1.83,

E_sc, V2E_sc, BFLDMO, Erelease, 5.90e+01, 6.15e+02, 1.29e+01, 5.84e+00,

Nfdyn, O,

Gampre_9,GarnER_9, XCN 4.22e-03, 1.83e-01, 7.012e-01

¢f_10_21, V2tf, 1.19e+04, 7.35e+10, BVLmf, BVLf, 4.44e+00, 5.25e+00,

245