Multidimensional Stochastic Approach to the Fission...

378

Physics of Particles and Nuclei, Vol. 36, No. 4, 2005, pp. 378–426. Translated from Fizika Elementarnykh Chastits i Atomnogo Yadra, Vol. 36, No. 4, 2005, pp. 712–800.Original Russian Text Copyright © 2005 by Adeev, Karpov, Nadtochii, Vanin.English Translation Copyright © 2005 by Pleiades Publishing, Inc.

CONTENTS

INTRODUCTION 3791. MODEL 3801.1. Shape Parametrization

and Collective Coordinates 3801.2. Equations of Motion 3831.3. Initial and Final Conditions 3831.4. Conservative Force:

The Level-Density Parameter 3861.5. Inertia and Friction Tensors 3891.6. Statistical Model of Excited-Nucleus Decay 3912. MASS–ENERGY DISTRIBUTION (MED)

OF FISSION FRAGMENTS 3932.1. Two-Dimensional MEDs

of Fission Fragments 3942.2. One-Dimensional Mass and Energy

Distributions of Fragments 3952.3. Mechanisms of Nuclear Viscosity

and MEDs of Fission Fragments 3972.4. Correlation Dependences

of MED Parameters 3993. NEUTRON MULTIPLICITIES

AND FISSION TIMES 3993.1. Introduction 3993.2. Prescission Neutron Multiplicities

and Fission Times 400

3.3. Dependence of Prescission Neutron Multiplicity on the Mass and Kinetic Energy of Fragments 400

3.4. Postscission and Total Neutron Multiplicities 4033.5. Fission Probability 4074. CHARGE DISTRIBUTION

OF FISSION FRAGMENTS 4084.1. Features of the Model Describing

Charge Fluctuations 4094.1.1. Potential energy 4094.1.2. Inertial parameter of the charge mode 4104.1.3. Friction parameter of the charge mode 4104.2. Relaxation Times of the Charge Mode:

The Statistical Limit at the Scission Point 4114.3. Variance of the Charge Distribution 4134.4. Determination of the Two-Body Viscosity

Coefficient from Study of Charge ModeFluctuations 414

5. ANGULAR DISTRIBUTION (AD) OF FISSION FRAGMENTS 415

5.1. Transition-State Model 4175.1.1. The effect of prescission neutron evaporation

on AD anisotropy 4185.1.2. The effect of model dimension on AD 4195.2.

K

-Mode Relaxation Time 420CONCLUSIONS 421REFERENCES 422

Dedicated to the memory of Petr Alekseevich Cherdantsev

Multidimensional Stochastic Approach to the Fission Dynamics of Excited Nuclei

G. D. Adeev*, A. V. Karpov**, P. N. Nadtochii*, and D. V. Vanin**

* Omsk State University, pr. Mira 55, Omsk, 644077 Russia** Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia

Abstract

—The purpose of this review is to discuss the results obtained over the last five years from calculationsof the mass–energy, charge, and angular distributions of fragments formed during the fission of excited nuclei.The calculations are performed within the stochastic approach to fission dynamics, which is based on a multi-dimensional Langevin equation, and carried out for a wide range of fissility parameters and excitation energiesof compound nuclei. A temperature-dependent finite-range liquid-drop model, taking into account a diffusenuclear surface, is used in a consistent way to calculate the potential energy and level-density parameter. Inorder to describe the dissipation of collective motion, a modified one-body mechanism of nuclear viscosityincluding a reduction coefficient of the contribution made by the “wall” formula is used. The evaporation oflight prescission particles is taken into account on the basis of a statistical model combined with Langevindynamics. Calculations performed in multidimensional Langevin dynamics satisfactorily reproduce all theparameters of experimentally observed distributions of fission fragments and the prefission neutron multiplicityas a function of compound-nucleus parameters. In order to attain a simultaneous reproduction of the mass–energy distribution of fission fragments and the prefission neutron multiplicity, the reduction coefficient needsto be half or less of the total one-body viscosity. In this review, problems that need to be solved to facilitatefurther development of the multidimensional stochastic approach to fission dynamics are discussed.

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MULTIDIMENSIONAL STOCHASTIC APPROACH TO THE FISSION DYNAMICS 379

INTRODUCTION

Stochastic methods are widely used in the naturalsciences (physics, chemistry, astronomy, and biology)and in technical applications of radiophysics and quan-tum optics [1]. Interest in random fluctuations, as well astheir description via stochastic methods, has greatlyincreased in the last two decades, as is reflected by [2–4]and the references therein.

Since the mid-1980s, when deep inelastic transferreactions were first discovered [5], stochastic methodshave also been widely used in nuclear physics. It shouldbe noted that the use of stochastic equations in nuclearphysics began with Kramers’ classical study [6]. Usingan approach that he named the diffusion model, Kram-ers proposed describing nuclear fission with a smallnumber of degrees of freedom, which would then inter-act with a thermostat formed by all the other single-par-ticle degrees of freedom. Under these circumstances,the collective-variable dynamics becomes similar toBrownian-particle dynamics, since the collective sub-system energy varies only slightly during one act ofinteraction with a single-particle subsystem. The motionin such a physical model is adequately described by theFokker–Planck equation (FPE), introduced for the dis-tribution function of collective coordinates and theirconjugate momenta, or by physically equivalent Lan-gevin equations.

Using the analogy between nuclear-fission dynam-ics and Brownian-particle motion, Kramers calculatedthe diffusion rate of Brownian particles, which are ini-tially located in a potential well, through a potentialbarrier separating the initial and final states of a system.With this approach, Kramers refined the Bohr andWheeler equation [7] for fission width obtained oneyear earlier. The Kramers’ refining factor takes intoaccount the effect of nuclear viscosity on the fissionrate (the fission width). It is interesting to note that,although much research has been devoted to the prob-lem of calculating the barrier-reaction rate (see, forexample, [8, 9]), this problem still remains unsolved. Inthe general case, the problem has a multidimensionalnature and involves complex relief of the potential-energy surface and deformation dependence of trans-port coefficients in the FPE and Langevin equations. Itis also concerned with calculating the induced-fissionwidth, which can be considered, at the present time, asan open problem (for more details, see Section 1.6 ofthis review).

The FPE-based stochastic approach has been suc-cessfully applied to many problems related to collectivenuclear dynamics: theories of deep inelastic transfers[10], induced fission [11–13], and description ofprescission neutron multiplicity [14]. Nevertheless, inrecent years, preference has been given to the Langevinequations, since an exact solution to the FPE is possibleonly for a limited number of low-dimensional modelcases [11] and generally requires the use of variousapproximations. The Langevin equations, in contrast,

can be solved on the basis of numerical methods andwithout additional simplifications for a multidimen-sional case. However, even using the Langevin equa-tions involves serious difficulties at the current level ofcomputer engineering. In order to describe a large num-ber of experimentally observed fission characteristics,it is necessary to introduce a possibly very large num-ber of collective coordinates. The introduction of eachnew coordinate considerably increases the volume ofcalculations. Therefore, it is natural that one-dimen-sional Langevin calculations are performed first and,only then, two-dimensional calculations. One-dimen-sional calculations make it possible to calculate the fis-sion probability and multiplicities of evaporatingprescission particles. Two-dimensional models providethe additional possibility of calculating either the frag-ment-mass distribution corresponding to the most prob-able kinetic energy of fragments or the energy distribu-tion corresponding to a preset ratio between fragmentmasses.

The experimentally observed two-dimensionalmass–energy distribution cannot be obtained in termsof either one-dimensional or two-dimensional Lan-gevin calculations. In this case, it is necessary to haveat least three collective coordinates. For a simultaneousdescription of the charge distribution, it is inevitablynecessary to introduce a fourth collective coordinate soas to determine the charge distribution between frag-ments.

Results from the first estimative three-dimensionalcalculations of the parameters of the distribution of fis-sion fragments over kinetic energy and the meanprescission neutron multiplicity [15] were publishedonly in 1995. Since 2000–2001, the results of system-atic four-dimensional [16] and three-dimensional [17–24] Langevin calculations have begun appear regularlyin publications.

At the present time, the theoretical basis of the sto-chastic approach to collective nuclear dynamics basedon the Langevin equations and the results of one-dimensional and two-dimensional calculations are ade-quately described in the published reviews of Abe

et al.

[25] and Fröbrich and Gontchar [26, 27], and in thebook by Fröbrich and Lipperheide [28]. For this reason,we mostly omit the fundamentals of the stochasticapproach in this review and focus our attention on clar-ification and discussion of the most important resultsobtained in last five years within the multidimensionalstochastic approach. We place particular emphasis onthe specific problems and difficulties resulting from themultidimensionality of the approach developed andused in the calculations.

The stochastic approach to the description of large-amplitude collective nuclear motion is dynamic. There-fore, the model intended for realization of this approachin calculations of experimentally observed values inev-itably involves the following development stages [29]:the choice of parametrization for the nuclear shape and

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collective coordinates, which are considered as classi-cal generalized coordinates satisfying the Fokker–Planck stochastic equations of motion or the physicallyequivalent Langevin equations. Phenomenologicalequations of motion introduced in this way are com-pletely specified by their transport coefficients: the con-servative force and the inertia, friction, and diffusiontensors.

Integration of the equations of motion in the Lan-gevin description determines the stochastic trajectoryof a fissile system in the collective-coordinate space. Itis necessary to consider the evaporation of light parti-cles (neutrons,

γ

quanta, and light charged particles)along each Langevin trajectory. Thus, a statisticalmodel of particle evaporation should be combined withdynamics, i.e., with integration of the equations ofmotion. Naturally, when considering the evolution of afissile system, it is necessary, as in the case of anydynamic problem, to impose conditions that initiate andaccomplish the system’s evolution.

The construction of collective nuclear dynamicswithin a microscopic approach [30, 31] is, on a techni-cal level, extremely difficult to achieve at the presenttime; therefore, in most studies, the transport coeffi-cients of the Langevin equations are determined withinmacroscopic models and approaches, which are knownto viable in relation to fission theory, especially forexcited nuclei. The first attempts to extend thisapproach to the case of low-energy fission are nowbeing made [21]. By extending the stochastic approachto this energy range, it becomes possible to explain theobserved multimodality of fission at low excitationenergies.

In Section 1, we describe the model on which all thefundamental results obtained within the three-dimen-sional Langevin calculations, performed mainly by ourgroup, are based.

In Section 2, we present and analyze results fromcalculations of two-dimensional and one-dimensionalmass–energy distributions of fission fragments as afunction of the fissility parameter and the excitationenergy of a compound nucleus.

In Section 3, we present and analyze results fromcalculations of the mean prescission neutron multiplic-ity and fission times. The dependences of prescissionneutron multiplicity on the mass and the kinetic energyof fragments are also discussed.

In Sections 4 and 5, we present results on the chargeand angular distributions of fission fragments obtainedwithin the three-dimensional Langevin approach.

Finally, we present a summary of our main conclu-sions and discuss the prospects for future application ofthe multidimensional stochastic approach for describ-ing the dynamics of nuclear fusion and fission reac-tions.

This work is a continuation of investigations into thecharacteristics of excited-nuclei fission in terms of the

stochastic approach, whose stages were initiallydescribed in [13, 32].

1. MODEL

1.1. Shape Parametrization and Collective Coordinates

The study of fission reactions is conventionally lim-ited by axially symmetric shapes. In this case, a nuclearshape can be described in cylindrical coordinates by theprofile function

ρ

s

(

z

), whose rotation around the sym-metry axis decides the nuclear surface. The most fre-quently used parameterizations are the

c

,

h

,

α

[33],Trentalange [34], Cassini ovaloid [35–37], and two-center parametrizations [38, 39].

The use of a particular nuclear-shape parametriza-tion is closely connected to the problem of choosingcollective coordinates. All the possible collective coor-dinates can be conditionally divided into coordinatesdescribing a nuclear shape and coordinates setting col-lective degrees of freedom not associated with nuclear-shape variation. For an adequate description of nuclearfission, it is necessary that the chosen parametrizationinvolves at least three parameters and allows introduc-tion of the following shape coordinates: nuclear elonga-tion, a coordinate determining neck evolution in anuclear shape, and a mirror-asymmetry coordinate.Such a minimal set of collective coordinates permitscalculation of the two-dimensional mass–energy distri-butions (MEDs) of fission fragments. Among the col-lective coordinates not associated with nuclear shape,we mention, for example, the charge-asymmetry coor-dinate specifying the charge distribution betweenformed fragments.

A successful choice of collective coordinates pro-vides for the convenience of the dynamic-calculationprocedure and, frequently, the accuracy of the obtainedresults. In this context, we now discuss the introductionof collective shape coordinates using the well-known

c

,

h

,

α

parametrization [33]. In [33, 40], it was shownthat, in terms of this parametrization, it is possible toaccurately reproduce the characteristics of nuclear sad-dle configurations obtained on the basis of variationalcalculations [41, 42] in the sharp-edge liquid-dropmodel.

This parametrization has also been used for staticcalculations performed with the Strutinsky shell-cor-rection method [33], dynamic calculations of MEDs inthe diffusion model [13, 43], and for calculating a largenumber of different fission characteristics in the Lan-gevin approach [18, 19, 44–48]. This parametrizationsets a three-parameter family of shapes and is thereforereasonably convenient for performing three-dimen-sional Langevin calculations. The parameter

c

describes nuclear elongation (the nuclear length, inunits of the initial-sphere radius

R

0

, is equal to 2

c

), theparameter

h

defines neck-thickness variation for a givenelongation, and the coordinate

α

sets the ratio between


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the masses of future fragments. In the chosen shapeparametrization, the nuclear-surface equation is written as

(1)

where

ρ

s

is the polar radius and

z

is the coordinate alongthe nuclear symmetry axis. The factor

B

is expressed bynuclear-shape parameters (

c

,

h

,

α

) [33]:

(2)

The parameter

A

s

is determined from the conditionof conservation of nuclear volume and, for continuousnuclear shapes, has the form

(3)

Nuclear shapes in the

c

,

h

,

α

parametrization areenclosed within

z

min

= –

c

and

z

max

=

c

. If the function

(

z

) vanishes within these limits only at

z

=

±

c

, wehave continuous nuclear shapes. If there are two moreroots at the interval [

z

min

,

z

max

], such shapes are inter-preted as being discontinuous. Other cases (with an oddnumber of roots within [

z

min

,

z

max

]) cannot be consid-ered as nuclear shapes and are frequently referred to asforbidden or “nonphysical” shapes (for a more detaileddiscussion of such shapes, see [49]).

We now introduce the neck concept. The coordinate

z

for which the function (

z

) reaches a minimum isdefined as the neck coordinate

z

N

. The neck coordinateis determined from the condition

(4)

which results in an algebraic equation. If there are threereal roots lying within [

z

min

,

z

max

]), one of them (a min-imum) is the neck coordinate

z

N

and other two (max-ima) are coordinates of the thickest cross sections of acompound-nuclear shape. These maxima determine thecoordinates of formed fragments.

When choosing the collective shape coordinates, thefollowing conditions should be taken into account:

(i) Collective coordinates are functions of thenuclear-shape parameters. Therefore, the simpler theform of these functions, the more convenient it is to usevarious coordinates. By simplicity, we also refer to thepossibility of finding analytical inverse functions for

ρs2

z( ) = c

2z

2–( ) As Bz

2/c

2 αz/c+ +( ), if B 0≥

c2

z2

–( ) As αz/c+( ) Bcz2( ), if Bexp 0,<

B 2hc 1–

2-----------.+=

As =

c3– B

5---, if B– 0≥

43--- B

Bc3( )exp 1 1

2Bc3

------------+ πBc

3– erf Bc

3–( )+

-------------------------------------------------------------------------------------------------------,–

if B 0.<

ρs2

ρs2

∂ρs2

∂z--------- 0,=

the dependence of the shape parameters on the collec-tive coordinates.

(ii) The mesh for dynamic calculations shouldinvolve the largest possible variety of nuclear shapesthat can be given by the chosen parametrization and noforbidden shapes. In addition, if nuclear fission is stud-ied from the fusion instant to the neck rupture, it isdesirable to exclude unconsidered discontinuousnuclear shapes.

(iii) The choice of collective coordinates is closelyrelated to boundary conditions at the mesh edges. In theavailable publications, only one boundary condition,specifically, the rupture condition, has been discussed.However, if there is an unsuccessful choice of collec-tive coordinates or of their variation limits at the mesh,a Brownian particle can hit the boundaries, which isequivalent to introducing physically unjustified infiniteforces restricting nuclear evolution. In order to preventsuch an event, the mesh boundaries should be “inacces-sible,” either due to high values of potential energy (incomparison with the total nuclear excitation energy) ordue to the behavior of inertial and friction coefficients.

Usually, collective coordinates are chosen by twomethods:

(1) Physical values describing nuclear shape, forexample, the spacing R between the mass centers of frag-ments (elongation coordinate), neck thickness rN, andratio of the difference between the masses of formedfragments and the total nuclear mass ηA (the mass-asym-metry coordinate)

(5)

where AR and AL are the mass numbers of the formedfragments (here and below, the subscripts R and L desig-nate right-hand and left-hand fragments, respectively).

In addition to the fact that the coordinates (R, rN, ηA)have a clear physical meaning, there is a further advan-tage to be gained in their use: All the allowed values ofthe coordinates ηA introduced by Strutinsky lie in theinterval –1 to 1 (in [50], the definition of the coordinateηA differs from that given in Eq. (5) by a factor of two),and the minimal value of the coordinate rN = 0. Thismakes such a choice of collective coordinates success-ful from the standpoint of the second and third condi-tions outlined above.

We tested the above choice of collective coordinatesusing well-known Lawrence forms [51]. The profilefunction was taken to be

(6)

where we considered not only symmetric nucleus con-figurations (b = 0 and d = 0) but also asymmetric ones.Thus, we have five unknown coefficients (a, b, c, d, and e)in Eq. (6).

ηA

AR AL–AR AL+-------------------

3

4R03

--------- ρs2

z( ) zd

zN

zmax

∫ ρs2

z( ) zd

zmin

zN

∫–

,= =

ρs2

z( ) az4

bz3

cz2

dz e,+ + + +=

382

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ADEEV et al.

The neck concept, connecting two future fragments,has a physical meaning only under severe nuclear-shape deformation [37]. Therefore, it is also necessaryto extend this concept to weakly deformed shapes so asto be able to define the collective coordinates R and ηA

for every nuclear configuration, including those thathave no pronounced bridge. In this case, the neck alsohas a coordinate defined by Eq. (4) and located at a

point where the function (z) has only an extremum(a maximum) at the interval [zmin, zmax]. Thus, the neck

is always at = 0.

Then, assuming that the reference point coincideswith the neck coordinate zN, we find that the coefficient

d is equal to zero [ = 0 ⇒ d = 0]. Thus,there remain only four unknown coefficients (a, b, c,and e) in Eq. (6). The coefficient e unambiguouslydefines the neck thickness for z = zN = 0. As was men-tioned above, we can choose it in the form of an inde-pendent parameter determining nuclear shape (we

denote this coordinate as ; i.e., e = ). In addition,by imposing a constant-volume condition (in anapproximation of nuclear-matter incompressibility)and using two free parameters (elongation and asym-metry), we can determine the coefficients a, b, and c.For this purpose, we use Eq. (5) and the following equa-tions:

(7)

Because the neck position coincides with the originof the coordinates (zN = 0), many calculations usingEqs. (5) and (7) can be performed analytically. Natu-rally, nonlinear equations require the use of numericalmethods to find the coefficients a, b, and c. In this study,we used the iteration method.

By applying the definition of the coordinate R andthe condition of conservation of nucleus volume, it ispossible to show how the maximal values of the coordi-

nates rN and R are related: = /R. In this context,the most convenient coordinates from a technical pointof view (the possibility of constructing a rectangular

mesh) seem to be (R, ηA, g = ). Thus, the limits onvariation of the collective coordinates are R ∈ [0, ∞],ηA ∈ [–1, 1], and g ∈ [0, 1].

We can ascribe the use of multipole nuclear-densitymoments [52] to the same method of choosing collec-tive parameters. It should be noted that (R, ηA) are thefirst two collective coordinates for large R in thisscheme for introducing the coordinates.

ρs2

∂ρs2/∂z( )z zN=

∂ρs2/∂z( )z zN 0= =

rN2

rN2

V43---πR0

3, R

zρs2

z( ) zd

zN

zmax

∫

ρs2

z( ) zd

zmin

zN

∫----------------------------

zρs2

z( ) zd

zmin

zN

∫

ρs2

z( ) zd

zN

zmax

∫----------------------------.–= =

rN2

R03

rN2

R

(2) Collective coordinates are nuclear-shape param-eters. These particular collective coordinates are conve-nient from the standpoint of satisfying the first condi-tion mentioned above and frequently extremely incon-venient according to the second and third conditions.Below, we explain this tendency and propose our ownmethod of introducing the collective coordinates usingthe c, h, α parametrization.

First of all, we discuss the problem of forbiddenshapes. It should be noted that forbidden shapes existonly for α ≠ 0. In addition, for each c and h, it is possi-ble to find αmax such that nuclear shapes lie only in theregion of |α| ≤ αmax and forbidden shapes lie in theremaining region.

We note that the ultimate values of the parameter ηA(±1) are attained when the mass of one of the fragmentsis equal to zero. This condition can be written as

(8)

Condition (8) means that the minimum (the neck) isat one of the extreme points (zmin or zmax) of a nuclearshape. Using Eq. (8) and taking into account that the

function (z) vanishes at the extreme points, weobtain, for the c, h, α parametrization,

(9)

It can clearly be seen that αmax essentially dependson c and h. It is precisely this dependency that createsthe problem of constructing a mesh with the use of theshape parameter α as the mass-asymmetry coordinate.

The new parameter α', which was introduced in[18, 49], is related to α by the scale transformation

(10)

In part, this introduction solved the problem underconsideration: for |α' | ≤ 1, there are no forbiddenshapes. In addition, the dependence of = αmaxc3

on c and h is much weaker than the similar dependenceof αmax. However, not all the possible asymmetries aregiven by a fall in the parametrization in the region |α'| ≤ 1.In particular, it becomes impossible to consider fissiondynamics from the entrance reaction channel whenthere is a large difference between the masses of the tar-get nucleus and an impinging ion.

We propose a new method of introducing the mass-asymmetry coordinate:

(11)

For such a choice, the problem of forbidden shapesis completely solved: all the possible asymmetric

∂ρs2

z( )∂z

----------------

z zmin zmax( )=0.=

ρs2

αmax

As B+( ), B 0≥As, B 0.<

=

α' αc3.=

αmax'

q3 = α

αmax---------- =

α/ As B+( ), B 0≥α/As, B 0.<



nuclear shapes (for given c and h) are enclosed within|q3 | ≤ 1.

It is convenient to choose the collective parameterresponsible for formation of the neck in a nuclear shapeso that the condition of zero neck thickness is satisfiedfor the same (or nearly the same) value of this param-eter. In the case of α = 0, zero neck thickness isattained for

(12)

From Eq. (12), it follows that the value hsc essen-tially varies depending on c. We introduce the collectiveneck coordinate in the following form:

(13)

The coordinate q2 has the following properties: ifq2 = 0, h = –3/2 (which guarantees reasonably high val-ues of potential energy for eliminating a Brownian par-ticle hit in this area); if q2 = 1, the neck thickness isequal to zero for symmetric shapes. For asymmetricnuclear shapes, the neck thickness vanishes for some-what smaller, but close to unity, values of q2.

Thus, in our opinion, the collective coordinates q =(q1 = c, q2, q3) are optimal when using the c, h, αparametrization. We use these coordinates with the fol-lowing variation ranges: q1 ∈ [0.5, 4.5], q2 ∈ [0, 1], andq3 ∈ [–1, 1]. When studying the charge mode, it is con-venient to additionally introduce a charge-asymmetrycoordinate in the form of q4 = ηZ = (ZR – ZL)/(ZR + ZL),where ZR and ZL are the charges of the formed right-hand and left-hand fragments.

1.2. Equations of Motion

In the stochastic approach [6, 25, 53], the evolutionof collective degrees of freedom of a fissile nucleus isconsidered by analogy with the motion of a Brownianparticle placed in a thermostat formed by all the othernuclear degrees of freedom. In calculations, a set ofLangevin equations is usually used. For the case of Ncollective coordinates, this set, in the difference form,is written as

(14)

Here, qi is the set of collective coordinates, pi are themomenta conjugate to qi , mij (||µij || = ||mij ||–1) is the iner-tia tensor, γij is the friction tensor, Ki is the conservativeforce, θijξ j is the random force, θij is the random-force

h hsc5

2c3

--------1 c–

4-----------.+= =

q2h 3/2+

hsc 3/2+--------------------.=

pin 1+( )

pin( ) 1

2--- p j

n( )pk

n( ) ∂µ jk q( )∂qi

-------------------

n( )Ki

n( ) q( )––=

-– γ ij

n( ) q( )µ jkn( ) q( ) pk

n( )

τ θij

n( )ξ jn( ) τ,+

qin 1+( )

qin( ) 1

2---µij

n( ) q( ) p jn( )

p jn 1+( )

+( )τ.+=

amplitude, and ξ j is a random variable with the follow-ing statistical properties:

(15)

The superscript n in Eqs. (14) and (15) indicates thatthe corresponding value is calculated at the instant oftime tn = nτ, where τ is the time step in the integrationof the Langevin equations. The angular brackets inEq. (15) denote averaging over the statistical ensemble.In Eqs. (14) and (15), the repeating indices assumesummation from 1 to N. The stochastic Langevin trajec-tory characterized by the shapes that a nucleus acquiresduring fission is obtained by numerically solving set ofEqs. (14) in the collective-coordinate space.

Random-force amplitudes are related to the diffu-sion tensor Dij in the following way:

(16)

In turn, the diffusion tensor satisfies the Einsteinrelation

(17)

From these equations, we found the random-forceamplitudes.

The thermostat temperature T used in calculationscan be determined within the Fermi-gas model:

(18)

Here, Eint is the excitation energy of the internal degreesof freedom of a compound nucleus (the internal energy)and a(q) is the level-density parameter, whose explicitform is discussed in Section 1.4.

When a nucleus moves to the rupture surface alongan entire stochastic Langevin trajectory in the collec-tive-coordinate space, it is necessary to verify that thelaw of conservation of energy is satified:

(19)

Here, E* is the total excitation energy of a compoundnucleus determined from the impinging-ion energy andthe difference between the masses of colliding nucleiand the compound system in the entrance reaction

channel, Ecoll(q, p) = is the kinetic energy

of collective motion of a nucleus, and Eevap(t) is thenuclear excitation energy carried away by particles thatevaporate at the instant of time t.

1.3. Initial and Final Conditions

In the most general case, the initial collective coor-dinates q0, momenta p0, and total compound-nucleus

ξin( )⟨ ⟩ 0,=

ξi

n1( )ξ j

n2( )⟨ ⟩ 2δijδn1n2

.=

Dij θikθkj.=

Dij Tγ ij.=

T Eint/a q( )( )1/2.=

E* Eint Ecoll q p,( ) V q( ) Eevap t( ).+ + +=

12---µij q( ) pi p j

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ADEEV et al.

momentum I can be sampled by the Neumann methodusing the generating function

(20)

The function σfus(I) describes the initial momentumdistribution of compound nuclei and is frequentlyapproximated by the expression

(21)

where k2 = 2µEl.s /2, µ is the reduced mass of theimpinging-ion–target system, and T(I) is the penetrabil-ity coefficient, which is defined within a model repro-ducing experimental fusion cross sections. When per-forming Langevin calculations, the difference betweenthe total nuclear spin I = |l + st + sp | (sp and st are theimpinging-ion and target-nucleus eigenmomenta,respectively) and the angular momentum l is usuallydisregarded. In this case, Eq. (22) describes the initialangular-momentum distribution of nuclei.

In our calculations, the following form of the func-tion T(I) was used:

(22)

The parameters Ic and δI were determined using themethod described in [26], where expressions approxi-mating the results of dynamic calculations [54] weregiven according to the surface-friction model [55] fortwo-ion fusion.

The problem of initial spin distribution has beenwidely discussed in the literature (frequently in the con-text of studying the angular distributions of fission frag-ments); however, it remains an open problem in modernnuclear physics. The majority of distributions containtwo or more varied parameters. It is difficult to deter-mine these parameters because direct experimental datacan only be derived from the fusion cross section, andonly one parameter can be fixed in this way. The otherparameters must be determined from certain theoreticalmodel representations. The angular distribution (AD)of fragments is affected by the second moment of thespin ⟨I2⟩ distribution of nuclei. It is obvious that varioustheoretical approximations can give values of ⟨I2⟩ thatdiffer dramatically from each other, whereas they allcorrectly reproduce the fusion cross section. The onlysuitable approximation for one-parameter partial fusioncross sections is the triangular distribution (the variedparameter is the maximal momentum at which fusiontakes place). However, the triangular distribution can beconsidered only as a zero approximation of a real situ-ation.

When studying the fission of excited compoundnuclei, the initial values of the collective coordinatesare frequently chosen for use as the compound-nucleus

P q0 p0 I t = 0, , ,( ) P q0 p0,( )σfus I( ).∼

σfus I( ) 2πk

2------ 2I 1+( )T I( ),=

T I( ) 11 I Ic–( )/δI[ ]exp+------------------------------------------------.=

ground (spherical) state. In this case, the initial momen-tum distribution is chosen as the equilibrium value.Then,

(23)

where qgs are the coordinates of the nuclear groundstate.

The choice of initial conditions in the form ofEqs. (20)–(23) means that we begin the Langevin cal-culations with a completely statistically equilibriumstate of a compound nucleus for a fixed initial excita-tion energy. It should be noted that such a choice of ini-tial conditions can be considered only as an approxima-tion to a real and more complicated situation. Fusion–fission reactions can be divided into two stages: a fastpre-equilibrium stage and a slower stage of decay of aresidual statistically equilibrium compound nucleus.Their thermalization phase can be described in theintranuclear-cascade model [56]. After the thermaliza-tion stage is accomplished, compound nuclei areformed as a result of fluctuations in the cascade. Alltheir characteristics have a wide distribution: the num-ber of protons and neutrons, excitation energy, and lin-ear or angular momenta. The intranuclear-cascademodel is the only model in which it is possible to takeinto account these fluctuations and to determine thetotal distribution of the parameters describing a com-pound nucleus. Consideration of the pre-equilibriumreaction stage is necessary for comparison of the calcu-lated fission characteristics at high excitation energiesE* > 150–200 MeV. At energies E* < 100 MeV, theconcept of a statistically equilibrium nucleus is a quitesatisfactory approximation of the initial conditions forsimulating fission dynamics.

The most natural choice of initial conditions can beobtained from consideration of the dynamics in theentrance reaction channel. Initial conditions chosen inthe spherical momentum–equilibrium nuclear state areunsuitable for describing the quasifission process anddeep inelastic transfer reactions. It should be empha-sized that the conventional concept of a compoundnucleus becomes a rough idealization of the real andcomplicated situation that exists for large angularmomenta.

Now, we discuss the final conditions under whichthe simulation of compound-nucleus evolution isaccomplished in our calculations. There are two suchconditions: the formation of an evaporation residue andnuclear fission into fragments.

Evaporation residue is detected when, as a result oflight-particle and γ-quanta emission, the nuclear excita-tion energy is reduced to the values Eint + Ecoll(q, p) <min(Bf , Bn), where Bf and Bn are the fission barrier andthe neutron binding energy, respectively.

P q0 p0,( )

∼V q0 l,( ) Ecoll q0 p0,( )+

T------------------------------------------------------–

δ q0 qgs–( ),exp



When performing calculations in the multidimen-sional Langevin dynamics, the choice of rupture sur-face in the collective-coordinate space is important, as,by its intersection, it is possible to ascertain that a com-pound nucleus has disintegrated into fragments. The rup-ture surface is the locus of discontinuous nuclear config-urations. In the case of N collective coordinates, the rup-ture surface is an N – 1-dimensional hypersurface.

The choice of the rupture condition dramaticallyaffects important fission characteristics such as themean values and variances of the energy distributionsof fission fragments. The cause of this sensitivity of theenergy-distribution parameters is obvious: the kineticenergy of fragments predominantly depends on theenergy of their interaction at the instant of scission.

At present, there is no known unambiguous approachto the choice of rupture criterion. However, it is possibleto note several of the most frequently used criteria:

(i) Equality of the neck thickness to zero. At first,this rupture criterion seems a natural choice. However,it has an essential disadvantage and, therefore, can beconsidered only as a zero approximation of the prob-lem. In fact, if the neck radius is comparable with thenucleon size, the description of a nucleus within the liq-uid-drop model becomes meaningless. Usually, the fis-sion of a nucleus into fragments is assumed to occurwith a rather thick neck [33, 41, 57].

(ii) From a physical standpoint, a rupture criterion inwhich a nucleus is disintegrated due to the loss of sta-bility with respect to the neck-thickness variation [13,33] seems to be more attractive. Mathematically, thiscondition can be written as

(24)

If this condition is averaged over the ensemble ofLangevin trajectories, it corresponds to nuclear shapeswith a neck radius of 0.3R0 [33, 41, 44].

(iii) Another physically acceptable rupture criterionis based on the assumption that a nucleus becomesunstable relative to the neck scission at the instant whenthe forces of Coulomb repulsion and nuclear attractionbetween the formed fragments are balanced. In [58], itwas shown that, in the actinide region, this conditioncorresponds to shapes that also have a neck radiusapproximately equal to 0.3R0.

(iv) The most flexible condition is the stochasticnuclear-rupture condition. For example, in [59],nuclear fission into fragments was considered as a fluc-tuation that could occur for arbitrary nuclear shapeswith a neck. The rupture probability was determinedfrom the relation Prup ~ exp(–∆E/T), where ∆E is thechange in the system energy under a deformed-nucleusrupture into two fragments. In [60], it was shown that,if this criterion is used, a rupture under deformationscorresponding to nuclear shapes with rN 0.24R0 ishighly probable.

∂2V

∂q22

---------

q1 q3,

0.=

In view of the above consideration, we consider thatthe fission of a nucleus into fragments occurs for a neckradius equal to 0.3R0, since this condition is in goodagreement with the last three rupture criteria. The equa-tion for the rupture surface can be written as

(25)

where rN is the neck radius corresponding to prescis-sion shapes. The case ρs(zN) = 0 corresponds to the rup-ture condition under which a zero neck radius isattained.

Examples of Langevin trajectories are shown inFigs. 1 and 2. As can be seen from these figures, aspherical nuclear shape is chosen as the initial deforma-tion, and numerical integration of the Langevin equa-tions stops when a nucleus reaches the scission config-uration. The observed quantities were calculated bysimulating the ensemble of Langevin trajectories, withsubsequent statistical averaging over this ensemble.

We now introduce the concept of a mean trajectory.The mean dynamic trajectory is that obtained indynamic Langevin calculations by averaging over anensemble of stochastic trajectories. In this case, theLangevin equations coincide with generalized Hamilto-nian equations [61] because the term responsible forfluctuations (the random force) disappears after suchaveraging. In order to calculate the mean trajectory, theinitial values of the mass-asymmetry coordinate q3 andmomentum p3 are set equal to zero. It is obvious that themean trajectory lies in the plane q3 = 0. In Fig. 3, weshow the potential energy in the coordinates q1 and q2for the case of q3 = 0. The dashed line in this figure rep-

ρs zN( ) rN ,=

–0.3

–0.40.8

h

c

0.3

1.0

0.2

0.1

0

–0.1

–0.2

1.61.2 1.4 1.8 2.0 2.2

5.0

8.0

11.0

2.0

5.0

8.0

14.011.0

17.0

8.0

5.0

2.0

Fig. 1. Example of a stochastic Langevin trajectory in thecollective coordinates (c, h) (α = 0) shown against thepotential-energy background. The numbers at the isolinesspecify values of the potential energy in MeV. The solidthick line corresponds to scission. The trajectory given inthis figure represents a fission event.

11.0

14.0

386


ADEEV et al.

resents the mean trajectory calculated under theassumption of a one-body mechanism of nuclear vis-cosity with ks = 0.25 (see Section 1.5), and the crossesmark the saddle point and nuclear ground state. In addi-tion, this figure shows examples of nuclear shapes inthe c, h, α parametrization. The calculations are per-formed using the 224Th nucleus as an example. Onemore important concept is the mean scission point, i.e.,the point of intersection between the mean dynamic tra-jectory and the rupture surface. The mean scission pointdetermines the mean values of the observed quantities(for example, the mean kinetic energy or average

charge of fission fragments). The spread in scissionconfigurations with respect to the mean scission pointdetermines the variances of the observed quantities (forexample, the width of the energy or charge distributionsof fragments).

1.4. Conservative Force: The Level-Density Parameter

Heated rotating compound nuclei formed in reac-tions with heavy ions represent a thermodynamic sys-tem. As is known, the conservative force operating insuch a system should be determined by part of its ther-modynamic potential (for example, the free energy [62]or entropy [26]).

The expression for conservative force K(q) can bewritten as [26, 29]

(26)

It should be noted that both the given definitions ofconservative force are equivalent. The choice of a cer-tain thermodynamic potential depends on its conve-nience with respect to the calculations, since they arerelated in the following way:

(27)

In the Fermi-gas model, the entropy and level-den-sity parameter can be calculated from the known freeenergy

(28)

Moreover, within the Fermi-gas model, the follow-ing relation is valid:

Ki∂F∂qi

-------

T

– T∂S∂qi

-------

E

.= =

F q T,( ) E* q T,( ) TS.–=

S q T,( ) ∂F q T,( )∂T

----------------------

V

, a q T,( )–S q T,( )

2T------------------.= =

0.8

α'

c

0.4

1.0

0.2

0

–0.2

1.61.2 1.4 1.8 2.0

5.0

8.6

6.8

3.2

3.23.2

8.6

6.8

6.81.4

5.05.0

Fig. 2. The same as in Fig. 1 but in the collective coordinates(c, α') at h = 0. The trajectory shown in this figure representsevaporation residue.

1.00

0.75

0.50

0.25

0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

q2

q1

122.1

87.1

40.4

168

75.4

31.7

25.8

5.4

8.3

2.5

5.4

11.214.2

–0.4–3.3

–9.2

31.7 40.4

52.1

Fig. 3. Potential-energy map in the coordinates q1 and q2 for the 224Th nucleus (q3 = 0) and the corresponding set of nuclear shapes.The crosses denote the ground state and the saddle point. The dashed line is the mean trajectory calculated under the assumption ofthe one-body mechanism of viscosity with ks = 0.25. The numbers at the isolines correspond to the potential energy in MeV.

10.0



(29)

where Eint(q, T) and E*(q, T) are the internal and totalexcitation energy of the system, respectively, and a(q) isthe level-density parameter. The following temperaturedependence of the free energy follows from Eqs. (28)and (29):

(30)

where V(q) is the nuclear potential energy at the tem-perature T = 0 V(q) = F(q, T = 0), and a(q) is thelevel-density parameter of an excited nucleus.

From the first part of Eq. (26), it can be seen that theconservative force in the Fermi-gas model is

(31)

We use this expression for calculating the conserva-tive force involved in the Langevin equations.

The current preference is to calculate the nuclearpotential energy in the liquid-drop model (LDM), tak-ing into account the finite range of nuclear forces anddiffuse nuclear surface [63, 64] with the aid of Sierkparameters [64]. In this model, it is assumed thatnuclear forces have a finite range of action. Fornucleon–nucleon interaction, the Yukava-plus-expo-nential double-folding potential is selected [63]. Its use,as opposed to the simple Yukava potential, in calcula-tions of the potential energy improves description ofexperimental data [64] on the fission barriers and yieldsa more realistic surface energy for strongly deformednuclei [63]. Moreover, for two fragments of nuclearmatter, the interaction potential reaches its lowest valuewhen the fragments are in mutual contact, which fol-lows from the property of saturation of nuclear forces.

In [62], a free-energy formula based on the finite-range LDM was proposed. The nuclear free energy as afunction of the mass number A = N + Z, neutron excess(isotopic number) I = (N – Z)/A, and collective coordi-nates q describing nuclear shape has the form [62]

(32)

where av , as, and ac are the parameters of volume, sur-face, and Coulomb energies, respectively, in the LDM

Eint q T,( ) = E* q T,( ) E* q T = 0,( )– = a q( )T2,

F q T,( ) V q( ) a q( )T2,–=

Ki∂V q( )

∂qi

---------------–∂a q( )

∂qi

--------------T2.+=

F A Z q T L, , , ,( )

= –av 1 kv I2

–( )A as 1 ksI2

–( )Bn q( )A2/3

c0A0

+ +

+ acZ

2

A1/3

---------Bc q( ) ac54--- 3

2π------

2/3 Z

4/3

A1/3

---------–

2L L 1+( )2J⊥ q( )

---------------------------,+

when the diffuse edge is at zero temperature, and kv andks are the corresponding volume and surface parametersof the symmetry energy.

The deformation dependence enters Eq. (32) via thefunctionals Bn(q) and Bc(q) of nuclear and Coulombenergies [64], respectively, and the moment of inertiaJ⊥(q) of a nucleus with respect to an axis perpendicularto the symmetry axis and passing through the nuclearcenter of mass. The latter term in Eq. (32) represents therotational nuclear energy subject to the nuclear-densitydiffusivity [61].

The temperature dependence for the seven coeffi-cients (av , as, kv , ks, r0, a, and ad) involved in Eq. (32)is parametrized in the form [62]

(33)

which can be considered as adequate for T ≤ 4 MeV[65]. Information on the temperature coefficients xi wasobtained in self-consistent microscopic calculations interms of an extended form of the Thomas–Fermimethod and the application of an SkM* interaction,which served as the effective interaction betweennucleons [65, 66]. In [62], the results of these calcula-tions for the Gibbs thermodynamic potential were usedto derive Eq. (32). The values of the fourteen coeffi-cients obtained in [62] are given in Table 1.

The microscopic calculations [65] performed interms of the extended temperature-dependent Thomas–Fermi method showed that Eq. (30) for the free energyF is a reasonably exact approximation for T ≤ 4 MeV.

The level-density parameter a(q) is an excited-nucleus characteristic when considered in terms of theFermi-gas model. In [62, 67], the level-density param-eter was calculated from Eqs. (28) and (32) within thefinite-range LDM and taking into account the nuclearexcitation. In [67], it was shown that, when the nuclearlevel-density parameter a(q) is calculated in thisapproach, it depends only slightly on temperature. Thesame result was obtained in [62] for spherical nuclei.This fact means that it is possible to use the level-den-sity parameter in the finite-range temperature-indepen-dent LDM.

At the same time, the deformation dependence ofthe level-density parameter is frequently represented bya leptodermic expansion [68–71] in the form

(34)

ai T( ) ai T = 0( ) 1 xiT2

–( ),=

a q( ) a1A a2A2/3

Bs q( ).+=

Table 1. Coefficients of Eq. (32)

r0 a ad av kv as ks

ai(0) 1.16 0.68 0.7 16.0 1.911 21.13 2.3

103xi (MeV–2) –0.736 –7.37 –7.37 –3.22 5.61 4.81 –14.79

Note: The first row represents values at zero temperature and the second row gives the thermal coefficients xi.

388


ADEEV et al.

Here, A is the mass number of a fissile nucleus, and thedimensionless factor Bs(q) determines the surface areaof a deformed nucleus in units of the equivalent spher-ical surface, i.e., the surface-energy functional in thesharp-edge LDM [72]. When parametrizing Eq. (34), thetwo sets of coefficients (a1 and a2) proposed byIgnatyuk et al. [69] and recommended by Töke andSwiatecki [71] are most frequently used.

In [67], the deformation dependence of the level-density parameter in the finite-range LDM was approx-imated by Eq. (34). The approximation was carried outfor 70 nuclei along the beta-stability line [73], with themass number Z ranging from 47 to 116. The results ofthe approximation are given in Table 2. In [67], it wasshown that the accuracy of the performed approxima-tion is reasonably high within the entire consideredrange of nuclei.

The coefficient a2 determines the way in which thelevel-density parameter depends on deformation and,therefore, is of importance for statistical and dynamic

models of nuclear fission. The value of this coefficientobtained in [67] is close to the Ignatyuk coefficient a2 =0.095 MeV–1 and over two times less than the value pre-dicted by Töke and Swiatecki (a2 = 0.274 MeV–1). Thesecond coefficient a1 is approximately identical in allthe sets under consideration. As a result, if we use theIgnatyuk coefficients for the level-density parameter,the results of the dynamic Langevin calculations virtu-ally coincide with those obtained in the finite-rangeLDM but noticeably differ from the results obtainedwith the Töke and Swiatecki coefficients.

In [74, 75], the level-density parameter was approx-imated by an expression similar to Eq. (34). The isoto-pic dependence of the level-density parameter and theCoulomb interaction were also taken into account:

(35)

Here, BCoul(q) is a dimensionless functional of Cou-lomb energy in the sharp-edge LDM. In [75], the level-density parameter was estimated for a large number ofspherical even–even nuclei in terms of relativisticmean-field theory and using Eq. (35).

When comparing the results obtained in [67, 74, 75],we can draw the following conclusions. First, in thestudies under consideration, the level-density parame-ter and potential energy were calculated consistently.Clearly, the use of various model representations meansthat we obtain deformation dependences of the level-density parameter that are different from each other.Therefore, when analyzing various nuclear characteris-tics, it is necessary to place special emphasis on theconsistency of the set of parameters in question. In par-ticular, the fission barriers in statistical calculations, theconservative driving force in dynamic calculations, thenucleus temperature, and the level-density parametershould be determined under the same model assump-tions (as occurred in [67, 74, 75]). Second, the level-density parameter obtained in the finite-range LDM canbe approximated by Eq. (35). In [67], it was establishedthat the contribution of the Coulomb term in Eq. (35) isrelatively small, and the values (signs and magnitude)of the coefficients kv and ks strongly depend on the setof nuclei chosen for the approximation. Therefore, itmakes sense to restrict the calculations to only twoterms in expansion (34), which should not greatly affectthe accuracy of the approximation.

In Fig. 4, we show a number of dependences of thelevel-density parameter on the mass number along thebeta-stability line. The results are obtained for thenuclei ground state. It can clearly be seen that, when thelevel-density parameter is calculated with the Ignatyukcoefficients for the Woods–Saxon potential, it agreeswith a(q) obtained in the temperature-dependent LDMto within 10–15% and differs considerably from thedependence predicted by Töke and Swiatecki. Thecurve obtained with the Ignatyuk coefficients also

a q( ) avol 1 kvolI2

+( )A asurf 1 ksurfI2

+( )A2/3

Bs q( )+=

+ aCoulZ2A

1/3–BCoul q( ).

Table 2. Values of the coefficients a1 and a2 obtained in [67,71, 69]

a1 (MeV–1) a2 (MeV–1)

[67] 0.0598 0.1218

[69] 0.073 0.095

[71] 0.0685 0.274

4

20 40

a, MeV–1

A

12

100

26242220181614

1086

260 80 120 140 160 180 200 220 240

Fig. 4. Level-density parameter as a function of the massnumber along the beta-stability line for the nuclear groundstate. The solid curve is the level-density parameter in thefinite-range LDM, the dashed line is the level-densityparameter with the Ignatyuk coefficients [69], and thedashed and dotted line is the level-density parameter withthe Töke and Swiatecki coefficients [71]. The squares showthe dependence of the level-density parameter determinedwithin the extended Tomas–Fermi method involving theSkyrme effective interaction SkM* [66], the circles indicatethe results of consistent calculations in relativistic mean-field theory [75]. The figure is taken from [67].



almost coincides with that for the dependence calcu-lated in terms of relativistic mean-field theory [75].Because, in [62], the temperature coefficients for thefinite-range LDM were obtained from an approxima-tion of the results of calculations within the extendedThomas–Fermi method and using the Skyrme effectiveinteraction SkM* [65, 66], the curves corresponding tothem on the plot superimpose onto each other.

1.5. Inertia and Friction Tensors

The so-called transport coefficients, i.e., the (iner-tial) mass and friction parameters, are an importantcomponent of dynamic models. As numerous calcula-tions have shown, they generally define the character ofthe motion of a fissile system and directly influenceboth the MED parameters and the fission times, as wellas the multiplicities of prescission and postscission par-ticles. Thus, calculation of the transport coefficients isextremely important and, probably, a key factor in real-izing a dynamic simulation.

The inertia and friction tensors are frequently calcu-lated within the hydrodynamic approximation forincompressible vortex-free fluids. In this case, the mech-anism of nuclear viscosity is a two-body one. In thehydrodynamic approximation, the Navier–Stokes equa-tions for a viscous medium are usually solved on thebasis of the Werner–Wheeler approximation [61, 76],which makes it possible to obtain reasonably simpleexpressions for the inertia and friction tensors. TheWerner–Wheeler approximation is based on a represen-tation in which a fluid flows in the form of cylindricallayers, with the fluid particles being unable to escapefrom their layer during motion. The accuracy of theWerner–Wheeler approximation in relation to calcula-tion of the transport coefficients was investigated in[77, 78]. When calculating the mass tensor, it wasfound that the Werner–Wheeler approximation pro-vides very high accuracy for the elongation degree offreedom. For example, when describing the nuclearshape using the Lawrence parametrization [51], mass-tensor components calculated from the elongationcoordinate in the Werner–Wheeler approximation vir-tually coincide with the exact solution to the corre-sponding hydrodynamic Neumann problem [79],which was obtained by a method based on potential the-ory [80, 81]. For the Cassini ovaloid parametrization,there is a small distinction between the solutions onlyfor shapes with an almost zero neck radius [77]. At thesame time, for the collective coordinate determiningevolution of the bridge in a nuclear shape, the differ-ence between the solutions is more significant and canreach, for example, 10% in the case of the Lawrenceparametrization [79]. In [61], it was shown that the dif-ference between mass parameters determined by solv-ing the corresponding Neumann problem and with theWerner–Wheeler method increased proportionally tothe multipolarity of the vibrations near a sphericalshape.

For the friction tensor, the Werner–Wheeler approx-imation of the collective coordinate associated withelongation is in good agreement with the resultobtained for the exact solution to the correspondingNeumann problem [79], as in the case of the mass ten-sor. At the same time, for the coordinate related tobridge evolution, the Werner–Wheeler approximationfor the Lawrence parametrization results in a 30–40%overestimation of the corresponding friction-tensorcomponents.

In [79], an expression taking into account finitenuclear sizes was obtained for the two-body mecha-nism of nuclear viscosity. This expression differs fromthat obtained in [61] for an infinite medium by the dis-sipative-force work in displacing a fluid along thenuclear surface. The results of the calculations [79]showed that, when finite nuclear sizes are taken intoaccount, much smaller values of the friction-tensorcomponents are obtained for the two-body mechanismof nuclear viscosity than when the expressions given in[61] are applied.

In addition to the two-body mechanism of nuclearviscosity, the one-body mechanism [82–84] can also beused in calculations of the friction tensor. In this mech-anism of nuclear viscosity, the authors take into accountthe fact that a nucleus is a fermion system involvingPauli blocking, which forbids nucleon scattering intooccupied states. This circumstance imposes restrictionson the mean free path of particles (it increases up to thesize of the system itself); therefore, the role of two-par-ticle collisions decreases. Nucleons are kept within anucleus due to the presence of a mean field; i.e., parti-cles move almost freely within the nuclear shape andelastically hit only a “wall” simulating the nuclear sur-face, which itself has a certain velocity because themean field in which nucleons move depends on theposition of these nucleons, i.e., on the nuclear collectivecoordinates.

In [82–84], two formulas were obtained: The first,named the “wall” formula, describes the dissipation ofnuclear shapes without a neck, and the second, calledthe “wall-plus-window” formula, was introduced forstrongly extended nuclear shapes under conditions inwhich it is possible to detect formed fragments con-nected by a neck. A quantum consideration of one-bodydissipation showed [85] that the nuclear viscosityamounts to only about 10% of the values calculatedusing the wall formula [82, 84], although the functionaldependence of viscosity on nuclear shape is adequatelyreproduced by this formula. In this context, Nix andSierk proposed a modified variant of one-body dissipa-tion that led to the surface-plus-window formula. In thiscase, the contribution from the wall formula to the dis-sipation is reduced by almost four times. The reductioncoefficient ks was found from an analysis of the experi-mental width of giant resonances and amounted to ks =0.27. From a comparison of the calculated average val-ues of the kinetic energy of fission fragments with the

390


ADEEV et al.

experimental data, it was found [86] that ks lies within0.2 < ks < 0.5. The friction tensor corresponding to thesurface-plus-window formula can be written in cylin-drical coordinates as

(36)

where DR and DL are the positions of the centers ofmass of future fragments and ks is the reduction coeffi-cient for the contribution made by the wall formula. Ifthe coefficient ks = 1 corresponds to the total one-body

γ ijsw 1

2---ρmv

∂R∂qi

------- ∂R∂q j

--------∆σ 329------ 1

∆σ-------

∂V1

∂qi

---------∂V1

∂q j

---------+

=

+ ks π∂ρs

2

∂qi

---------∂ρs

2

∂z---------

∂DL

∂qi

----------+ ∂ρs

2

∂q j

---------∂ρs

2

∂z---------

∂DL

∂q j

----------+

zmin

zN

∫

× ρs2 1

2---

∂ρs2

∂z---------

2

+

12---–

dz

+ π∂ρs

2

∂qi

---------∂ρs

2

∂z---------

∂DR

∂qi

----------+ ∂ρs

2

∂q j

---------∂ρs

2

∂z---------

∂DR

∂q j

----------+

zN

zmax

∫

× ρs2 1

2---

∂ρs2

∂z---------

2

+

12---–

dz

,

viscosity, then Eq. (36) corresponds to the wall-plus-window formula.

The two terms in square brackets in Eq. (36) corre-spond to the wall formula for the left-hand and right-hand fragments. We denote the friction-tensor compo-

nents calculated from the wall formula as . For neck-less nuclear shapes, the friction tensor is usually calcu-lated from the wall formula, and the surface-plus-win-dow formula is applied to strongly deformed shapeswith a thin neck. In order to describe the dissipation inan intermediate case, expressions of the type γij =

+ (1 – f(rN)) are usually used. Thechoice of the function f(rN) is quite arbitrary. As a rule,it is chosen so that it varies smoothly from f(0) = 0 tof(1) = 1. However, this choice is ambiguous [10, 87–89]and can affect the calculated characteristics [84].

In Fig. 5, we show components of the inertia andfriction tensors calculated along the bottom of a fissionvalley under the assumption of one-body and two-bodymechanisms of nuclear viscosity. As can be seen fromthis figure, the deformation dependence of variouscomponents of the friction tensor calculated with dif-ferent mechanisms of viscosity essentially differ fromeach other. This brings about, in the scission for differ-ent mechanisms of viscosity, a different character in afissile system’s motion. It should be emphasized that itis difficult to compare these components of the inertiaand friction tensors with the results of other studies.Even when using the same parametrization to describenuclear shape, the qualitative behavior of the compo-nents depends on the chosen collective coordinates. Forexample, we used the c, h, α parametrization todescribe nuclear shape and chose q1, q2, and q3 as thecollective coordinates, as they are convenient for use indynamic calculations. Therefore, the components of theinertia and friction tensors shown in Fig. 5 do not coin-cide with the tensor components from other studies inwhich the c, h, α parametrization was used.

The reduced friction coefficient β = γ/m is an impor-tant characteristic of the fission process and is fre-quently used during its analysis. In Fig. 6, we show thecomponent = calculated using the one-body and two-body mechanisms of nuclear viscosity.As our calculations show, this component generallydefines the length of time and character of the motionof a fissile nucleus to scission, although the coordinateq2 is not explicitly related to the elongation coordinate.From this figure, it can be seen that amount to sev-

eral units of 1021 s–1 at ks = 0.25. Moreover, their orderof magnitude is close to that of values calculated usingthe two-body mechanism of nuclear viscosity with ν0 =2 × 10–23 MeV s fm–3, but the deformation dependenceis different.

In [82, 90–92], it was shown that the assumptionsused in derivation of the wall formula do not apply

γ ijw

ksγ ijw

f rN( ) γ ijsw

βq2q2γ q2q2

/mq2q2

βq2q2

10

5

0

20

10

0200150100500

–50

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

γq2q2

γq1q2

γq3q3

γq3q3

γq2q2

mq1q1

mq3q3

mq1q2

mq2q2

γq1q2

γq1q1

γq1q1

(a)

(b)

(c)

mij,

10–

40 M

eV s

2γ ij

, 10–

19 M

eV s

q1

Fig. 5. Transport coefficients as functions of the coordinateq1 (h = α = 0): (a) the inertia tensor; (b) the friction tensorunder the assumption of the two-body mechanism ofnuclear viscosity ν0 = 2 × 10–23 MeV s fm–3, and (c) thefriction tensor under the one-body mechanism of viscositywith ks = 0.25. The calculations are performed for the 224Thnucleus.



under certain conditions. For example, the assumptionof the randomness of the particle motion inside anuclear shape can not be applied to the motion of parti-cles in integrable potentials such as an ellipsoidal rect-angular well. In this case, the particle motion inside thenuclear shape is ordered. For this reason, the dissipa-tion predicted by the wall formula is considerably over-estimated. In this context, the authors of [93] proposeda deformation-dependent reduction coefficient in thewall formula reflecting the degree of particle motionrandomization. However, it had already been shown in[82] that, even in integrable potentials, it is possible toobtain values close to those given by the wall formulaif the particle motion is randomized. In [79], it was alsoshown that including collisions between particles in theconsideration results in randomization of the particlemotion, whose degree depends on the temperature anddensity of states near the Fermi energy.

In [94], the reduction coefficient of the contributionfrom the wall formula was defined through the measureof chaos in the single-particle motion of nucleonsinside a nuclear shape. The measure of chaos (the ran-domness of nucleon motion) depends on the nuclearshape in question, i.e., on the collective coordinates.The degree of randomness was found to vary from zeroto one during evolution of a nucleus from the sphericalto scission configurations. Thus, the reduction coeffi-cient explicitly depends on deformation, and thisdependence is calculated on the basis of the generalprinciples of chaos theory [95] instead of being intro-duced, as was done in [26, 27], ad hoc to explain exper-imental data. The first application of such an approachto describing fission widths, prescission neutron multi-plicities, and cross sections for evaporation residuesappeared to be reasonably successful, although the cal-culations were carried out in the one-dimensionalmodel. It would be both interesting and desirable toextend this approach to the multidimensional case.

An important aspect of studying fusion–fissionreactions in terms of the stochastic approach is consid-eration of the temperature dependence of kinetic coef-ficients. The different theoretical approaches used tocalculate kinetic coefficients apply various approxima-tions and parameters, which, however, are determinedwith insufficient accuracy. For this reason, the obtainedtemperature dependences of the friction tensor differdramatically. For example, when the wall formula isderived with the approximations used in [82], it pre-dicts a friction coefficient that is almost temperature-independent. The expression for the wall formula alsocan be derived within linear-response theory [79, 90],but, in this case, the temperature dependence of the fric-tion coefficient is such that the friction increases withtemperature proportionally to T2. In turn, the two-bodyviscosity decreases with temperature in accordancewith 1/T2 [96]. When the temperature dependence ofthe friction was experimentally investigated [97–99], itwas found that the friction coefficient increases withtemperature.

The kinetic coefficients can also be calculatedwithin microscopic approaches [79, 90, 100, 101].Their use makes it possible to remove many of theapproximations that are inevitably applied in calcula-tions of the kinetic coefficients within macroscopicmodels. Nevertheless, performing these calculationswithin microscopic approaches is extremely cumber-some. In addition, the kinetic coefficients obtained inthese calculations and their temperature and deforma-tion dependences depend on the method used, the set ofconstants, etc., and tend to differ considerably. There-fore, the kinetic coefficients most frequently used indynamic simulation of reactions involving heavy ionsare calculated within macroscopic models.

1.6. Statistical Model of Excited-Nucleus Decay

The decay of excited nuclei can be consideredwithin the statistical fission model proposed by Bohrand Wheeler [7] and Weisskopf [102] and further devel-oped in numerous studies. However, in its conventionalform [103], the statistical model is not able to describeexperimental data on variances of mass and energy dis-tributions in the heavy-nuclei range with the parameterZ2/A > 32 [104]. This drawback is ultimately associatedwith the fact that the model does not take into accountthe dynamics of the fission process.

When performing dynamic calculations, the statisti-cal model is usually used for describing light-particleemission, which both accompanies fission and com-petes with it. A combination of the dynamic and statis-tical models makes it possible to bring the theoreticaldescription of nuclear decay and the real process closertogether. Such a combination was used for the first time

4

10

8

6

2

0.8 1.0 1.2 1.4 1.6 1.8 2.0

βq2q2, 1021 s–1

12

q1

Fig. 6. Reduced viscosity coefficient = /

calculated at h = α = 0 for the 224Th nucleus. The solid anddotted curves correspond to the one-body mechanism ofnuclear viscosity with ks = 0.25 and 0.5, respectively. The dot-ted curve was obtained under the assumption of the two-bodymechanism of viscosity with ν0 = 2 × 10–23 MeV s fm–3.

βq2q2γq2q2

mq2q2

392


ADEEV et al.

in [14], where the standard FPE was supplemented witha term describing the continuous-mode emission oflight particles. In spite of the fact that, in this study, theauthors restricted the analysis to only one collectivecoordinate and continuous-mode particle emission,they succeeded in reproducing the principal effects pro-duced by the influence of particle emission on thenuclear-fission dynamics. Later, another schemeincluding the statistical particle-emission model indynamic calculations was proposed in [105]. In thiscase, a reduced Langevin equation was considered, andits numerical integration took into account discrete par-ticle emission. Both the continuous [25] and the dis-crete [18, 19, 26] methods of taking into account parti-cle emission are now widely used.

Within the statistical model, the probability of emis-sion of a certain particle is specified by the nuclear-decay width in the corresponding channel. The decaywidths are expressed according to the density ofnuclear excited states, which, in turn, depends on theexcitation energy, angular momentum, compound-nucleus deformation, and its nucleonic composition.

As a rule, the expression for the nuclear level den-sity is chosen within the Fermi-gas model and has thefollowing form [70, 106]:

(37)

Here, A is the mass number, Eint is the excitation energy,and I is the spin. The values of Kvib and Krot (the collec-tive and rotational enhancements of the level density)are given by the expressions [107]

(38)

In these expressions, the presence of temperature-related attenuation of collective enhancements of the leveldensity is taken into account in accordance with [107].The critical energies Ecr = 40 MeV and ∆E = 10 MeV are

assumed to remain constant. The expressions for

and can be written as

(39)

(40)

The importance of including collective enhance-ments when calculating the level density, as well as

ρ E I q, ,( ) 2I 1+( )a q( )1/2

12------------------------------------

2

2J⊥ q( )-----------------

3/2

=

× 2 a q( )E( )1/2[ ]exp

E2

-------------------------------------------K rot E( )Kvib E( ).

K rot vib( ) E( ) K rot vib( )0

1–[ ] f E( ) 1,+=

f E( ) 1E Ecr–

∆E----------------

exp+1–

.=

K rot0

Kvib0

Kvib0 1, for deformed nuclei

0.0555A2/3

T4/3( ), for spherical nuclei,exp

=

K rot0 1, for spherical nuclei

J⊥T , for deformed nuclei.

=

their attenuations under an increase in temperature, wasnoted, for example, in [107–109]. Nevertheless, collec-tive enhancements were seldom taken into account dur-ing the development of the Langevin models. In [67], itwas shown that the inclusion of collective enhance-ments produces a substantial effect on the calculatedvalues of quantities such as the mean prescission neu-tron multiplicity and the fission probability. Becausethese characteristics are frequently used to obtain infor-mation about, for example, nuclear viscosity, it is nec-essary to place special emphasis on the accuracy of thelevel-density calculation when performing such ananalysis.

It is well-known that, for deformations that remainclose to a spherical nuclear shape, it is possible to intro-duce only a vibrational enhancement of the level den-sity, while rotational enhancement is absent due to thesymmetry effects. In contrast, for strongly deformednuclei, it is possible to take into account only a rota-tional enhancement of the level density, while thevibrational enhancement proves to be 10–100 timeslower and is usually disregarded. However, for theintermediate region of deformations, it still remainsunclear how to pass from vibrational to rotationalenhancements as the deformation increases [107]. Inthis context, an estimation was made in [107] showedthat, for quadrupole deformations with β2 < 0.15, it ispossible to consider a nucleus to be reasonably close toa spherical deformation and that, correspondingly,there is only vibrational enhancement of the level den-sity in this deformation region. For violent deforma-tions, the rotational enhancement sharply increases tothe values given by Eq. (40) and becomes predominantin this deformation region.

The partial widths of nuclear decay with the emis-sion of a particle j ( j = n, p, d, t, 3He, and α particles)can be expressed [106] through the single-particle leveldensity ρj of a residual nucleus and the cross section

for the absorption of this particle by the residualnucleus:

(41)

Here, gj , mj , Bj , and Vj are the statistical factors result-ing from the spin of a particle, its mass, the compound-nucleus binding energy, and the Coulomb barrier,

respectively, and and are the internal excita-tion energy of the initial and residual nuclei.

The radiation widths of γ-quanta emission are calcu-lated from the equation

σinvj( )

Γ j

g jm j

π( )2-------------- 1

ρ0 Eint0( )( )

-------------------=

× Eσinvj( )

E( )ρ j Eintj( )

B j– E–( )E E.d

V j

Eintj( )

B j–

∫

Eint0( )

Eintj( )



(42)

where σγ(E) is the inverse cross section for dipole pho-toabsorption. However, it should be noted that, for acorrect description of particle emission when numeri-cally simulating fission within the Langevin models, itis necessary to take into account the deformationdependence of statistical-model parameters such as theemission barriers, binding energies of particles, andcross sections for inverse processes [21, 110]. Thisnecessity is caused by the fact that a compound nucleusmost often arises in the region of violent deformations,which fundamentally differ from a spherical state.

The statistical model can be used in dynamic calcu-lations related to description of not only light-particleemission but also nuclear fission. With a decrease in thefissility parameter and/or the excitation energy of acompound nucleus, the mean fission time very quicklyincreases and the fissility decreases. Therefore, due tothe long calculation time needed, the use of a purelydynamic approach becomes virtually impossible for fis-sile systems with low fissility. If we instead employ astatistical branch of dynamic calculations, where notonly light-particle evaporation but also fission isdescribed within the statistical model (this approachwas proposed for the first time in [105]), it becomespossible to remove this disadvantage. Such an approachcan be used if the following conditions are met: (i) asystem is in the ground-state region before the ridge,(ii) the time of calculations exceeds a certain time t >tstat, and (iii) the ratio between the nuclear temperatureand a fission barrier is T/Bf < 0.2. The parameter tstat isselected so that it is certain to satisfy the condition thatthe particle flux through the fission barrier attains itsquasisteady value.

Under the statistical branch of calculations, the fis-sion width can be found in the multidimensional case[11]. Because, as was mentioned above, a nucleus is athermodynamic system, the quasisteady fission widthshould be calculated using thermodynamic potentials.Below, we give an expression for the quasisteady value ofthe fission width in the case of the free-energy potential:

(43)

Here, (qgs) = , (qsd) =

, and ωK is the Kramers fre-

quency, which is a unique positive root of the equation

Γγ = 1

πc( )2----------------- 1

ρ0 Eint0( )

( )------------------ Eσγ E( )ργ Eint

0( )E–( )E

2E,d

0

Eint0( )

∫

Γ f = ωK

det Ωij2 qgs( )

det Ωij2 qsd( )

-----------------------------

12---

F qsd( ) F qgs( )–( )/T–( ).exp

Ωij2 µik qgs( ) ∂2

F q( )∂qk∂q j

-----------------

q qgs=Ωij

2

µik qsd( ) ∂2F q( )

∂qk∂q j

-----------------

q qsd=

(44)

Here, E is the unit matrix, and the coordinates qsd andqgs determine the saddle-point and ground-state posi-tions, respectively. In the statistical branch of the pro-gram, the probability of system deexcitation due toemission of a particle or fission is calculated using theMonte Carlo method, with the probabilities of each ofthese events being proportional to the partial decaywidths Γj and Γf . If the system has to be disintegrated,we transfer back from the statistical branch to thedynamic simulation, but, in this case, the calculationproceeds from the ridge surface and the return of a par-ticle to the ground-state region is blocked.

However, it is necessary to note that the choice ofthe free-energy potential F(q) for calculating the qua-sisteady fission width is ambiguous. Strictly speaking,Eq. (43) is an approximation of fission. Because fissionis not an isothermal process, the accuracy of Eq. (43)depends, correspondingly, on how constant the temper-ature remains during nuclear fission. Because fission(occurring between evaporations of particles) proceedswith a constant total excitation energy, the use of theentropy instead of the free energy is more justifiedwhen calculating the quasisteady value of the fissionwidth. Further discussion of this question and results ofdynamic calculations using the entropy can be found,for example, in [26, 27]. However, in our opinion, theuse of the entropy for calculating the fission width isnot the best solution, since only the behavior of thepotential energy near the ground state and at the saddlepoint is taken into account in Eq. (43) and not thepotential-energy landscape. In this sense, a more con-sistent method of fission-width calculation is based onthe concept of a mean first passage time [111, 112] orthe alternative concept of mean last passage time [113].Such a method has been successfully applied for a longtime in other fields of physics [8]. Nevertheless, untilnow, it has been developed only for the nongeneral caseof system motion in an overdamped mode. Therefore,the development and improvement of this method iscurrently one of the important problems to be solved infission physics.

2. MASS–ENERGY DISTRIBUTION (MED) OF FISSION FRAGMENTS

Mass–energy distributions (MEDs) of fission frag-ments are conventionally used as a source of informa-tion about fission dynamics. They were fully investi-gated for the first time by Nix and Swiatecki in theirzero-viscosity dynamic model [114]. Within the frame-work of this model, the authors succeeded in describingMED parameters for fission fragments of light fissilenuclei with Z2/A ≤ 31. For heavier nuclei, the zero-vis-cosity model [114] regularly results in low values of thevariances of mass and energy distributions.

det E 2πωK/( )22πωK/( )µik qsd( )γ kj qsd( )+(

+ Ωij2 qsd( ) ) 0.=

394


ADEEV et al.

Significant success was attained in relation todescription of the MED characteristics of fission frag-ments and understanding of the role of nuclear dissipa-tion within the diffusion model based on the multidi-mensional Fokker–Planck equation for the distributionfunction of collective variables [13, 43]. The importantachievement of the diffusion model is that it can offeran explanation for the sharp increase in the variances ofmass and energy distributions that is observed as theparameter Z2/A increases. However, the model still hasa number of disadvantages. Its main drawback consistsin the fact that it uses approximate methods for solvingthe Fokker–Planck equation and the fluctuations of col-lective variables are taken into account in the form of anaverage. In [45], the variances of energy distributionscalculated from the diffusion and two-dimensionalLangevin models were compared. As a result, the error,which is due to the approximate nature of the methodsused for solving the Fokker–Planck equation, was esti-mated as being equal to about 30%. In the diffusionmodel and the Nix model with zero viscosity, com-pound-nucleus deexcitation due to the emission ofprescission particles, which significantly affects theevolution of the collective degrees of freedom and, also,the parameters of the considered distributions of fissionfragments were disregarded.

The stochastic approach based on the set of three-dimensional Langevin equations makes it possible tostudy the MEDs of fission fragments in full. In thischapter, we present the results of regular (in a wideinterval of fissility parameters and excitation energies)application of the three-dimensional Langevin equa-tions to studying the MED characteristics of fissionfragments.

2.1. Two-Dimensional MEDs of Fission Fragments

The method used for calculation of the observedcharacteristics is based on the concept of a rupture sur-

face. It is assumed that the distribution of collectivevariables and their conjugate momenta are formed dur-ing the descent of system from a conditional saddlepoint to a certain scission configuration at which aninstantaneous rupture of the nuclear neck takes placewithout any change in elongation. The further evolutionof the collective variables produces only a minor effecton the formed distributions of the reaction products.This method has previously been widely used whenstudying the MED within the framework of the diffu-sion model.

When calculating the energy-distribution parame-ters, it was assumed that the total fission-fragmentkinetic energy EK is the sum of the Coulomb repulsionenergy VC, the nuclear attraction energy Vn of the futurefragments, and the kinetic energy of their relativemotion (the prescission kinetic energy Eps). All theterms in this sum (EK = VC + Vn + Eps) were calculatedat the instant of scission. The calculated formulas for VC

and Vn in the LDM, taking into account the finite-rangenuclear forces and diffuse nuclear surface, are listed inthe appendix to [60].

In Fig. 7, calculated fission-fragment MEDs for the224Th compound nucleus are shown at three impinging-ion energies. It can be seen that the MED of the frag-ments broadens with an increase in the excitationenergy. It should also be noted that the shape of the con-tour lines of the distributions qualitatively correspondsto the experimentally observed picture. It is more con-venient to quantitatively compare the calculated char-acteristics of the two-dimensional MED of fragmentswith those observed experimentally in terms of theparameters of one-dimensional mass and energy distri-butions and the correlation dependences of the MEDparameters.

200

190

180

170

160

150

EK, MeV

Elab = 82.8 MeV Elab = 130 MeV Elab = 185 MeV

80 100 120 140 80 100 120 140 80 100 120 140M, amu

0.08

0.160.24

0.32

0.06

0.120.18

0.240.12

0.04

0.08

Fig. 7. Mass–energy distributions of the fission fragments obtained from the reaction 16O + 208Pb 224Th at three values ofimpinging-ion energy. The calculation was performed under the assumption of the one-body mechanism of viscosity with ks = 0.25.



2.2. One-Dimensional Mass and Energy Distributions of Fragments

One-dimensional mass and energy distributions canbe obtained by integration of two-dimensional MEDsover the corresponding parameter. They are character-ized by single-peak curves at high excitation energiesand are usually, both in experimental investigations andtheoretical calculations, approximated by Gaussiancurves with average values and variances, which are theprincipal characteristics of these distributions. As arule, results are discussed in terms of these values.Sometimes, a third γ3 and fourth γ4 moment (asymme-try and excess) of the mass and energy distributions arealso involved. They characterize the difference betweenthese distributions and normal ones and are defined as

(45)

(46)

where X is assumed to be either the kinetic energy EK ofthe fragments or the fragment mass M. For a Gaussiandistribution, γ3 = γ4 = 0.

As is shown by the performed calculations, the firstand second moments of the mass and energy distribu-tions are sensitive to the viscosity used in the calcula-tions and the character of the descent of trajectoriesfrom the saddle to the scission. The parameters of theenergy distribution are also very sensitive to the choiceof rupture condition [58, 84].

The energy-distribution parameters have been inves-tigated for various nuclei in a large number of experi-mental [115–117] and theoretical [44, 45, 118–120]studies. From an analysis of the available experimentaldata, it was found that ⟨EK⟩ is virtually independent ofboth angular momentum and excitation energy [115].In addition, it was shown in [115, 116] that ⟨EK⟩ is nota linear function of the parameter Z2/A1/3, as followsfrom the Viola systematics [121], and has a break atZ2/A1/3 900. The break is observed if only the resultson strongly heated nuclei are selected from the experi-mental data and low-energy and spontaneous fissionstrongly influenced by shell effects, as well as quasifis-sion reactions, are excluded. The systematics proposedin [115, 116] has the form

(47)

According to the Viola systematics [121], ⟨EK⟩ isgiven by the expression

. (48)

In Fig. 8, we show the mean kinetic energy of frag-ments as a function of the parameter Z2/A, which wasobtained in three-dimensional Langevin calculations

γ 3 X X⟨ ⟩–( )3⟨ ⟩ /σX3,=

γ 4 X X⟨ ⟩–( )4⟨ ⟩ /σX4

3,–=

EK⟨ ⟩

= 0.104Z

2/A

1/324.3 MeV Z

2/A

1/3900>( )+

0.131Z2/A

1/3 MeV Z

2/A

1/3 ≤ 900( ).

EK⟨ ⟩ 0.1189Z2/A

1/37.3 MeV+=

with a modified variant of the one-body viscosity. FromFig. 8, it can be seen that the calculated values of ⟨EK⟩agree well with the experimental data and lie closer tothe Viola systematics. It is necessary to emphasize thatthe values we calculated for ⟨EK⟩ [19, 122, 123]decrease with the reduction coefficient ks. A similarresult was obtained in two-dimensional calculations[86] with a neck radius equal to zero taken as the rup-ture condition. In [86], the best description of the exper-imental data was attained with ks approximately equalto 0.3. The values obtained in three-dimensional calcu-lations of ⟨EK⟩ do not appear to be as sensitive to ks;therefore, it is difficult to make conclusions about thevalue of viscosity based on them. The fact that ⟨EK⟩actually ceases to depend on ks is associated with theinclusion of the evolution of a nucleus in the mass-asymmetry coordinate of the three-dimensional calcu-lations, in contrast to that of the two-dimensional calcu-lations [86]. ⟨EK⟩ is found by integration of ⟨EK(M)⟩over M. In [25], it is shown that, in the case of high vis-cosity, when the descent from the saddle to the scissionis slow and Eps can be assumed to be equal to zero, themean kinetic energy ⟨EK⟩3D of fragments obtained in thethree-dimensional calculations is related to the meankinetic energy ⟨EK⟩2D of fragments obtained in the two-dimensional calculations in the following way:

(49)EK⟨ ⟩3D 1 σηA

2–( ) EK⟨ ⟩2D.=

50

800

E, MeV

Z2/A1/3

100

1800

220

200

180

160

140

120

80

60

1600140012001000600

Fig. 8. Dependence of ⟨EK⟩ on the parameter Z2/A1/3. Theopen triangles show the experimental data, the closed trian-gles indicate the results of theoretical calculations with ks =0.25, and the closed circles represent ⟨Eps⟩. The solid line

shows the following systematics [115, 116]: 0.104Z2/A1/3 +24.3 MeV for the region of Z2/A1/3 > 900 and 0.131Z2/A1/3

for the region of Z2/A1/3 ≤ 900. The dashed curve shows theViola systematics [121]: ⟨EK⟩ = 0.1189Z2/A1/3 + 7.3 MeV.

396


ADEEV et al.

Here, is the variance of the distribution over themass-asymmetry coordinate ηA at the instant of scis-

sion. As ks increases, decreases. Correspondingly,the expression in brackets increases, while, in contrast,⟨EK⟩2D decreases. Thus, the values ⟨EK⟩3D become lessdependent on ks than ⟨EK⟩2D. As can be seen from Fig. 8,the assumption of the smallness of Eps in calculationswith the one-body viscosity is thoroughly justifiedbecause, even for the heaviest nuclei, Eps does notexceed 10 MeV. In a wide range of variations of theparameter Z2/A, Eps contributes less than 2% to EK.

In Fig. 9, we show the variance of the mass distribu-tion as a function of the parameter Z2/A. As can be seenfrom this figure, it is possible to reproduce the sharp

increase observed in the experimental values of inthe heavy-nuclei range within the stochastic approachand using the modified variant of the one-body viscos-ity. The results of calculations with ks = 1 [19, 122, 123]virtually coincide with those from the statistical model

[103]. The explanation for the increase in as ks

decreases is as follows: On the one hand, the rigidity ofthe potential energy with respect to the mass-asymme-try coordinate during the descent from the saddle to thescission steadily increases; correspondingly, the massdistribution also narrows. On the other hand, the systemretains a “memory” of the former large width of the dis-tribution because the descent proceeds in a finite time.The faster the descent, the larger the “stored” variances.The rate of descent generally depends on the viscosity,

σηA

2

σηA

2

σM2

σM2

whose value depends on the coefficient ks. A detaileddiscussion of this mechanism of formation of the massdistribution can be found in [13, 43].

Calculated values of for different ks are shownin Fig. 10. It can be seen that the calculations in thethree-dimensional Langevin dynamics with ks ~ 0.1–0.25 make it possible to adequately reproduce theincrease in the experimental variances with Z2/A. Theresults of calculations for ks = 1 virtually coincide withthose performed in the dynamic zero-viscosity model[114, 124]. As the performed calculations show, theinclusion of the third collective coordinate (the mass-asymmetry coordinate) results in a substantial rise in

in comparison with the calculations executed intwo-dimensional models [118, 120] for symmetric fis-sion. This result agrees with qualitative estimates [25]that predict an increase in the energy-distribution widthobtained in two-dimensional calculations of symmetricfission and taking into account fluctuations in the mass-asymmetry coordinate.

In addition, it is particularly important to note thatthe energy-distribution parameters are sensitive to thechoice of parametrization of fissile-nucleus shape andthe rupture condition. For example, in [25], within thetwo-dimensional Langevin calculations, ⟨EK⟩ and were found to be approximately 15% higher for thetwo-center parametrization [38, 39] with a fixed neckparameter than for a parametrization based on Leg-endre polynomials [125]. A similar conclusion isreached from comparison of the results given in [44]

σEK

2

σEK

2

σEK

100

026

σ2M, amu2

300

30 444240383634322824

200

400

500

600

700

800

Z2/A

22

Fig. 9. Variance of the mass distribution as a function of theparameter Z2/A. The closed squares denote the experimen-tal data and the open squares, the results of theoretical cal-culations with ks = 0.25.

26Z2/A

100

0

300

30 4442403836343228

200

400

500

600

700

22 24

σ2EK, MeV2

Fig. 10. Variance of the energy distribution as a function ofthe parameter Z2/A. The closed squares denote the experi-mental data and the open squares and circles, the results oftheoretical calculations with ks = 0.25 and 0.1, respectively.



and [118, 120], in which the calculations were carriedout using the two-body mechanism of nuclear viscositywith approximately identical two-body viscosity coef-ficients ν0. However, the authors of [44] used the c, h, αparametrization [33] and chose a rupture condition witha finite-thickness neck, while the authors of [118, 120]used the Cassini ovaloid parametrization [37] and arupture condition with a zero neck radius. In these stud-

ies, the results of calculations of proved to be verydifferent for close fissile nuclei with almost equal exci-tation energies.

The asymmetry (γ3) and excess (γ4) of mass andenergy distributions have also been studied experimen-tally. In the case of the fission of highly excited nuclei,the condition γ3 = γ4 = 0 is satisfied with high accuracyfor the mass distribution. The same result is alsoobtained in theoretical calculations [18, 19, 122].Experimental investigations [126, 127] into the depen-dences of γ3 and γ4 of the energy distributions of com-pound nuclei from Os to U at various excitation ener-gies show that the energy distributions are character-ized by small and constant values of the coefficientsγ3 –0.1 and γ4 0. The reproduction of these valuesof γ3 and γ4 in theoretical calculations remains to beattained since, as was noted above, the energy-distribu-tion parameters appear to be extremely sensitive to anumber of features of the theoretical calculations, suchas the rupture condition, viscosity value, and nuclear-shape parametrization. In this context, it is necessary tomention [60], where the influence of the rupture condi-tion on the energy-distribution parameters was studiedunder the c, h, α parametrization. It was found thatcalculated γ3 and γ4 move somewhat closer to the exper-imental data if we use a probabilistic simulation of thescission (rN = 0.3R0) instead of the rupture conditionwith a fixed neck thickness. In addition, the use of theprobabilistic rupture condition results in smaller values

of EK and in an increase in . Thus, the energy-dis-tribution parameters can be used not only for determin-ing the viscosity of nuclear matter during fission butalso for investigating the process of disintegration of anucleus into fragments.

From comparison of the variances calculated forvarious ks with the experimental data, it is possible toconclude that, in calculations, it is necessary to use ks ~0.25–0.5 in order to obtain a reproduction of the exper-imental variances in the range of the lightest nuclei con-sidered in this study and ks ~ 0.1 for heavier nuclei.

The statistical model of fission [103] and thedynamic zero-viscosity model [114, 124] cannot evenqualitatively describe the sharp increase in the experi-

mental values of and found for heavier fissile

nuclei in this range of variation of the parameter Z2/A.Significant progress has been achieved in relation to

σEK

2

σEK

2

σEK

2 σM2

description of the dependences of and on thefissility parameter within the diffusion model [13, 43].However, the diffusion-model calculations were per-formed without taking into account the evaporation oflight prescission particles. This effect has a strong influ-ence on the MED parameters [118, 120], as the evapo-rating particles carry away a significant part of the exci-tation energy, and, correspondingly, the variances ofboth the mass and the energy distributions decrease.

Recently, MEDs have been calculated at low excita-tion energies. In [21], the results of two-dimensionalLangevin calculations of the mass distribution werepublished for the 227Pa compound nucleus at low exci-tation energies. Moreover, the shell corrections weretaken into account when calculating the potentialenergy, level-density parameter, and the deformationdependence of the penetrability coefficient in determin-ing the particle-emission widths. In these studies, mul-timodal fission was obtained in theoretical calculations.In addition, the mass-distribution shape appeared to beextremely sensitive to the nuclear viscosity (to thevalue of the coefficient ks). Therefore, the authors of[21] concluded that the mass-distribution parametersare more sensitive to viscosity at low excitation ener-gies than to prescission particle multiplicity. In addi-tion, the authors of [21] stressed the importance of tak-ing into account the temperature dependence of thenuclear viscosity.

2.3. Mechanisms of Nuclear Viscosity and MEDs of Fission Fragments

As was noted above, when describing the dissipa-tion properties of nuclear matter, the one-body and two-body classical mechanisms of nuclear viscosity areconventionally used. The one-body mechanism is,undoubtedly, physically more justified. At the sametime, calculations within the diffusion model [13, 43]executed with the two-body viscosity showed goodqualitative and quantitative agreement with the experi-mental data on MED parameters. Therefore, compari-son of the results obtained using both mechanisms ofviscosity within the three-dimensional Langevinapproach is of interest. In [128], the MED parametersof fragments were calculated in the three-dimensionalLangevin model for the reaction

12C + 232Th 244Cm (Elab = 97 MeV).

We chose this reaction because we previously stud-ied it within the framework of the three-dimensionalapproach and with the one-body viscosity. In [129], themass distribution, both with one-body and two-bodyviscosity, was calculated for this reaction within thetwo-dimensional Langevin approach.

First, we discuss the qualitative side of the compar-ison of the two mechanisms of nuclear viscosity. In thecase of the one-body mechanism of nuclear viscosity(ks = 0.25), the authors of [18] obtained the following

σEK

2 σM2

398


ADEEV et al.

coordinates of the mean scission points: csc = 2.1 andhsc = –0.07 for the 206Po nucleus, and csc = 2.2 and hsc =–0.11 for the 260Rf nucleus. It is well known that thetwo-body mechanism of viscosity results in more pro-late scission shapes in comparison with the one-bodymechanism. The coordinates of the mean scissionpoint, csc = 2.5 and hsc = –0.25, calculated in [128] agreewell with those previously obtained within the diffu-sion model under the assumption of two-body viscosity(ν0 = 3.5 × 10–23 MeV s fm–3). We also note that theposition of the mean scission point weakly varies withthe viscosity coefficient (ks or ν0) for both mechanismsof nuclear viscosity. This dependence is especiallyweak in the case of one-body viscosity.

As the calculations in [128] showed, the use of two-body viscosity, in contrast to one-body viscosity, resultsin nearly Gaussian energy distributions for all the two-body viscosity coefficients ν0. In this respect, the two-body mechanism of viscosity gives the best qualitativeagreement with the experimental data. It should benoted that the mass and mass–energy distributions arealso in qualitatively good agreement with the experi-mental data.

Now, we discuss quantitative characteristics. Calcu-lated MED parameters and ⟨npre⟩ are listed in Table 3 fora number of two-body viscosity coefficients. This tableshows that satisfactory description of the mean prescis-sion neutron multiplicity ⟨npre⟩ is attained for the two-bodyviscosity coefficient ν0 = (10–25) × 10–23 MeV s fm–3.The same conclusion was also reached in [129]:to describe the experimental data on ⟨npre⟩, it is neces-sary to increase the two-body viscosity coefficient toν0 = 25 × 10–23 MeV s fm–3. At the same time, the dataon ⟨EK⟩ are reproduced at ν0 3.5 × 10–23 MeV s fm–3.

Regarding the variances, the variance of the massdistribution is underestimated by 20%, whereas thevariance of the energy distribution differs from theexperimental value nearly by a factor of 2.

The behavior of the variance of the mass distributionas a function of the two-body viscosity coefficient can

σM2

be explained as follows. There are two opposite tenden-cies: on the one hand, the diffusion-tensor componentsincrease with viscosity, which leads to an increase inthe fluctuations in the mass-asymmetry coordinate and,

hence, to an increase in ; on the other hand, the fis-sion time and, consequently, the prescission neutronemission increases. As a result, the excitation energy

decreases as well as . From Table 3, it can be seenthat, for ν0 < 5 × 10–23 MeV s fm–3, an increase in thefluctuations with ν0 is of crucial importance and, forν0 > 10 × 10–23 MeV s fm–3, the behavior of the variancedepends on the second coefficient.

The mechanism of formation of the energy distribu-tion is more complicated and depends on several fac-tors. It follows from our calculations that an essentialdifference between the two-body and one-body viscos-ities is the strong dependence of the mean total kineticenergy on the two-body viscosity coefficient, whereas⟨EK⟩ is almost independent of ks for the one-body mech-anism of viscosity. As was mentioned above, ⟨EK⟩ isfound by summing the average energy ⟨VC + Vn⟩ of theinteraction between fragments and the average energy⟨Eps⟩ of their relative motion at the instant of scission.The calculations showed that ⟨VC + Vn⟩, both for theone-body and two-body mechanisms of viscosity,weakly depends on the value of viscosity because it isgenerally determined by the coordinates of the meanscission point. In contrast, the behavior of ⟨Eps⟩ is verydifferent for the two mechanisms of viscosity. For thisreaction and the excitation energy, the characteristicvalues of the one-body viscosity are ⟨Eps⟩ = 4–7 MeV,which provides a constant mean kinetic energy of fis-sion fragments in addition to a weak dependence of⟨VC + Vn⟩ on the dissipative force. Under the assump-tion of the two-body mechanism of dissipation, themean prescission energy varies from ⟨Eps⟩ = 22 MeVfor ν0 = 2 × 10–23 MeV s fm–3 to ⟨Eps⟩ = 4 MeV for ν0 =25 × 10–23 MeV s fm–3. From Table 3, it can be seen that⟨EK⟩ varies by the same value as ⟨Eps⟩.

In [58, 61], the values of the coefficient ν0 allowingthe authors to describe experimentally observed ⟨EK⟩ werefound to be equal to ν0 = 0.9 ± 0.3 × 10–23 MeV s fm–3 [61]and ν0 = 1.9 ± 0.6 × 10–23 MeV s fm–3 [58] for a widerange of fissility parameters. Study of the MED inthe diffusion model [13] gives the following estimatefor the two-body viscosity coefficient: ν0 =1.5 ± 0.5 × 10–23 MeV s fm–3. We obtained ν0 = 3.5 ×10–23 MeV s fm–3, for which the calculated energies ⟨EK⟩agree well with the experimental data but prove to be atleast 1.5 times higher than the previously obtained esti-mates of this value. A similar relation was also obtainedin [130]. As in [120, 130], we failed to obtain a value ofthe two-body viscosity coefficient at which the meanneutron multiplicity and ⟨EK⟩ would both be adequatelydescribed. The performed three-dimensional Langevincalculations for the two types of viscosity made it pos-sible to conclude that experimentally observed fission

σM2

σM2

Table 3. MED parameters of fission fragments and meanprescission neutron multiplicities calculated under theassumption of the two-body mechanism of viscosity in com-parison with the experimental data [116]

ν0(10–23 MeV s fm–3) (amu2) (MeV2)

⟨EK⟩(MeV)

⟨npre⟩

2 233 122 185 0.8

3.5 270 148 176 1.1

5 304 144 172 1.4

10 302 127 169 2.5

25 261 101 166 3.8

Experiment [116] 366 259 178 3.0

σM2 σEK

2



characteristics such as the mean prescission neutronmultiplicity, mean kinetic energy of fission fragments,and variances of the mass and energy distributions canbe more satisfactorily reproduced within the one-bodymechanism of nuclear viscosity. A similar conclusionwas reached concerning the description of ⟨EK⟩ and⟨npre⟩ in [130].

2.4. Correlation Dependences of MED Parameters

The correlation dependences of the MED parame-ters of fission fragments carry additional informationabout the dynamics of the descent of a fissile system toits last stage immediately before the scission, for exam-

ple, the dependences ⟨EK(M)⟩ and (M) of the meankinetic energy and the variance of the energy distribu-tion on the fragment mass, respectively, as well as the

dependence (EK) of the variance of the mass distri-bution on the kinetic energy. In particular, the depen-dence of the scission configuration on the fragmentmass is directly reflected in the correlation of MEDparameters of fragments.

In the first approximation, the dependence ⟨EK(M)⟩can be described by the parabolic law [131]

(50)

The dependence of the mean kinetic energy on thefragment mass generally reflects the dependence of theCoulomb energy on the distance between the centers ofmass of future fragments at the instant of scission. Wenote that an expression similar to Eq. (50) for β = 1 fol-lows from the Nix dynamic model [114] with zero vis-cosity and corresponds to the simplest assumption onthe independence of the spacing between the centers ofmass of fragments. From the experimental data [131],it follows that β < 1 and depends both on the fissilityparameter and the excitation energy of a compoundnucleus at the energy E* > 20 MeV for nuclei lighterthan 213At. The values of β calculated in our three-dimensional Langevin model vary from β = 0.7 for198Pb to β = 0.9 for 248Cf. For 186Os, the calculated valueof β = 0.9. In the next approximation, it is necessary totake into account the terms containing (1 – 2M/A)4 inEq. (50). The analysis in [132] showed that the experi-mentally observed dependences ⟨EK(M)⟩ actually con-tain the terms (1 – 2M/A)4.

As was observed under experimental conditions, the

calculated variances (M) are virtually independent

of the fragment mass. The dependence (EK) alsoqualitatively corresponds to the experiment data (italmost hyperbolically decreases as EK increases). The

dependences (EK) and (M) obtained for fissionfragments can be explained from the shape of sections

σEK

2

σM2

EK M( )⟨ ⟩ EK A/2( )⟨ ⟩ 1 β 1 2MA

--------– 2

– .=

σEK

2

σM2

σM2 σEK

2

of the MED for M = const and EK = const. This problemis considered in detail in [13]. If the energy-distributionwidth is virtually independent of M, the width, andeven the shape, of the mass distribution stronglydepend on EK.

3. NEUTRON MULTIPLICITIESAND FISSION TIMES

3.1. Introduction

Among all the particles that evaporate from anucleus, neutrons play a special role [133] because,during fission, they evaporate in much greater quanti-ties than charged particles or γ quanta. For this reason,we mainly consider results obtained for prescission(evaporation from a compound nucleus) and postscis-sion (evaporation from fission fragments) neutrons inthis review. In the experimental studies of the last 10–15 years, the results obtained in on fission neutronyields, which depend on various compound-nucleusparameters, have been accumulated and systematized[104, 133–135]. Neutrons that evaporate before thecompound-nucleus fission into fragments significantlyaffect the fission process. They reduce the excitationenergy and mass of fissile nuclei and, thus, consider-ably complicate the fission picture. Nevertheless, vari-ous characteristics of prescission neutron multiplicitycontain valuable information about fission, in particu-lar, the mean prescission neutron multiplicity (⟨npre⟩),which represents an original type of “clock” measuringthe fission time. In addition, prefission neutron multi-plicities can be successfully used, together with otherobservable quantities, for determining the importantcharacteristic of the viscosity of nuclear matter [25,45]. It should also be noted that it is not only mean pre-fission neutron multiplicities that are actively studied inexperimental investigations but also their dependences(⟨npre(M)⟩) on the mass and (⟨npre(EK)⟩) fission-fragmentkinetic energy [133, 134, 136]. Formulating a descrip-tion of these dependences within the framework of atheoretical approach for a wide range of nuclei is anextremely complicated problem.

The most complete review of experimental results con-cerning both prescission and postscission neutrons evapo-rating in fusion–fission reactions is given in [115, 133]. Inparticular, it was found in these studies, based on exper-imental investigations for a large number of nuclei, thatthe number of postscission neutrons increases moreslowly than the number of prescission neutrons as theexcitation energy increases. As was noted in [133], thisresult indicates that the fission process is very slow andis accompanied by a significant dissipation of the col-lective energy into internal energy. Consequently,almost all the initial nuclear excitation energy is trans-formed into internal energy and, then, is carried awayby evaporating particles. Theoretical calculations gen-erally corroborate these conclusions. For example, itwas found in [130], using two-dimensional Langevin

400


ADEEV et al.

calculations, that one-body viscosity is more suitablefor reproducing the prescission neutron multiplicitywhile, in the case of the two-body viscosity, it is neces-sary to use an extraordinary high viscosity coefficient.Specifically, it is 5–10 times higher than the coefficientfound in [61, 128] during study of the mean kineticenergy of fission fragments.

3.2. Prescission Neutron Multiplicities and Fission Times

A comparison of the results of calculations of themean prescission neutron multiplicity in the three-dimensional Langevin model [19, 122] for various kswith the experimental data are shown in Fig. 11. In thisfigure, the mean neutron multiplicity is shown sepa-rately for (a) light nuclei with A < 224 and (b) heavynuclei. Along the abscissa axis, we plot TI2, where I =(N – Z)/A. As can be seen from this figure, good agree-ment with the experimental data is attained with ks =0.25 for reasonably light fissile nuclei with A < 224(Fig. 11a). In the case of heavier nuclei (Fig. 11b), it isnecessary to set ks > 0.25 to reproduce calculations ofthe prescission neutron multiplicity. A similar resultwas previously obtained in [137], where it was shown,from studying reactions resulting in the formation ofheavy nuclei (with mass numbers ACN > 260), that it is

necessary to use ks > 4 to reproduce experimental val-ues of ⟨npre⟩. It should be noted that the results of calcu-lations of the prescission neutron multiplicity in manyrespects depend on the parameters used for calculationof nuclear decay width in various channels. For exam-ple, calculation with or without consideration of thecollective enhancements (and their attenuation withtemperature) [107] can strongly influence the level den-sity, and, hence, the nuclear decay width. However,there is no common opinion at present as to whether itis necessary to take into account these enhancementswhen realizing a dynamic simulation of nuclear decay.

Because neutron multiplicity acts as a type of clockfor measuring fission time, it is interesting to comparethe obtained theoretical estimates of fission times τf andthe available experimental data. There are no directexperimental data on fission times. They are conven-tionally derived from prescission neutron multiplicityusing model representations and can differ by an orderof magnitude when using different models and sets ofparameters [115, 133]. Therefore, at present it is notpossible to use fission times for determining the dissi-pative properties of excited nuclei. However, the major-ity of experimental and theoretical estimates of the fis-sion time given in [115, 133] show that itamounts toapproximately 10–20–10–18 s in a wide range of variationof the fissility parameters. As the fissility parameterdecreases, the fission time increases. The results of ourtheoretical investigations [18, 19, 123] agree well withsuch behavior and values of τf . In Fig. 12, we show themean fission time as a function of the parameter Z2/A,which we obtained via Langevin calculations with theviscosity coefficient ks = 0.25. In our calculations, themain contribution in τf is the time spent by a nucleusbefore reaching the saddle point, and the time ofdescent from the saddle point to the scission amounts toless than (5–8) × 10–21 s independently of the fissilityparameter. In this review, we selected reactions inwhich strongly excited nuclei are produced for theoret-ical investigation. For these nuclei, the influence ofshell effects on the decay process can be disregarded,and the contribution of quasifission reactions is muchless than that of fusion–fission reactions.

In a recent study [138], it was found, in one-dimen-sional theoretical calculations, that the dependence ofthe fission time on the parameter Z2/A has a peak atZ2/A 33. However, we observed no such a behaviorof τf in our calculations.

3.3. Dependence of Prescission Neutron Multiplicityon the Mass and Kinetic Energy of Fragments

The dependences of prescission neutron multiplicityon the mass and the kinetic energy of fission fragmentshave been experimentally investigated. These depen-dences provide additional information on fission. In[134], it was shown that many more neutrons are evap-orated in symmetric fission events than in asymmetric

7

0.06

6

5

4

3

2

1

7

6

5

4

3

2

10.07 0.08 0.09 0.100.05

(a)

(b)

npre

T I2, MeV

Fig. 11. Mean prescission neutron multiplicity as a functionof the parameter TI2 for nuclei (a) with A < 224 and (b) A >224. The closed squares show the experimental data. Theopen squares show the results of calculations with ks = 0.25.



ones and that the energy of evaporating neutrons isindependent of fragment masses. In addition, this ten-dency appeared to be almost completely independent ofthe mass of a fissile compound nucleus. As a result ofan analysis of experimental dependences ⟨npre(M)⟩taken from [133], it was established that, for the major-ity of the nuclei under consideration, they can beapproximated with good accuracy by the parabolic law

(51)

where Ms is the fragment mass for symmetric nuclearfission and the coefficient cpre characterizes the depen-dence of prescission neutron multiplicity on the massasymmetry. As was noted in [133, 134], the mechanismof formation of these dependences cannot be foundonly from an analysis of experimental data; rather, acombined detailed investigation within the framework ofboth theoretical models and experimental approaches isrequired. In [19, 122], we succeeded in reproducing theexperimentally observed parabolic dependence⟨npre(M)⟩ by applying the three-dimensional Langevincalculations involving the modified variant of one-bodynuclear viscosity. Examples of these dependences areshown in Figs. 13a and 14 for the 215Fr and 205Fr nuclei.

When explaining the parabolic shape of the depen-dence ⟨npre(M)⟩, the problem of where the neutrons evapo-rate becomes of fundamental importance. In Fig. 15, weshow ⟨npre⟩ as a function of the elongation coordinate cfor the 215Fr and 256Fm nuclei. Our calculations show thata large fraction of the neutrons (as well as other particles)evaporate from the ground-state region. Approximately50–80% of the total number of evaporating particles areemitted before the saddle point, and approximately10% are emitted when passing the saddle configura-tions, independently of the final mass asymmetry.Among all the nuclei under consideration, only in thefission of nuclei heavier than 256Fm does an appreciablefraction of neutrons evaporate at the stage of descentfrom the saddle point to the scission. A similar resultwas previously obtained in [139], where it was shownthat the number of neutrons evaporating during thedescent from the saddle to the scission increases withan increase in excitation energy. Due to the fact that weused the modified variant of one-body nuclear viscositycorresponding to the overdamped mode in our calcula-tions [19, 122], fissile nuclei resided for a significantperiod of time near the ground state before reaching thesaddle-configuration region. Therefore, a large numberof prescission neutrons had time to evaporate from theground-state region before the fissile system reachedthe fission barrier. The kinetic energy of particles evap-orating from the ground-state region is independent ofthe final mass asymmetry of the fission fragmentsbecause the fissile system almost totally “forgets” thedynamic evolution present in the ground-state regionafter overcoming the saddle-configuration region. Howmany particles are emitted by a nucleus from theground-state region is reflected only by the nuclear

npre M( )⟨ ⟩ ns⟨ ⟩ cpre Ms M–( )2,–=

excitation energy. Due to the random movement ofLangevin trajectories in the space of collective coordi-nates, different trajectories reach the ridge surface atdifferent times. Fissile nuclei, which rapidly attain thesaddle-configuration region and, thereafter, evaporate asmall number of neutrons, retain most of their excita-tion energy (a large phase volume is accessible to themduring the descent from the saddle point to the scis-sion); thus, it is possible for them to quickly reach therupture surface and possess a high mass asymmetry. Incontrast, those nuclei for which, during their evolution,a large number of particles evaporate from the ground-state region lose a significant part of their excitationenergy. After passing the ridge surface, for a massasymmetry approximately equal to zero, they are ableonly to descend slowly to the bottom of the liquid-dropfission valley and unable to ascend over the potential-energy surface into the large mass-asymmetry region.For such nuclei, the accessible phase volume is smallerduring the descent from the saddle to the scission, andthe fission time is longer in comparison with that for thenuclei that evaporate a smaller number of neutrons.Thus, it seems that nuclei that evaporate a small numberof neutrons during the formation of the dependence⟨npre(M)⟩ predominantly fall at the edges of the massdistribution. All nuclei evaporating an arbitrarily largenumber of prescission particles and, thus, overcomingthe ridge surface can fall in a mass-asymmetry regionapproximately equal to zero. Therefore, it is found thatthe mean neutron multiplicity is lower at the edges ofthe mass distribution than it is near the zero mass asym-metry. In Fig. 13b, we show the calculated mean timenecessary for fissile nuclei to attain the rupture surfaceas a function of the fragment mass. Figure 13b showsthat the fission times differ by almost a factor of 2 for

20 25

⟨ff⟩, s

Z2/A30

10–17

10–18

10–19

35 40 45

Fig. 12. Mean fission time as a function of the fissilityparameter Z2/A. The dots represent the results of calcula-tions performed with ks = 0.25.

402


ADEEV et al.

symmetric and asymmetric nuclear fission. Thus, weconcluded in [19, 122] that the final mass asymmetryand nuclear fission time depend on the prehistory of thefission’s dynamic evolution in the ground-state region.The smaller the number of particles that evaporate froma nucleus in the ground state region, the higher theprobability of an event in which a nucleus rapidlyreaches the rupture surface for a large mass asymmetry.

It should be noted that the parabolic shape of thedependence ⟨npre(M)⟩ is not characteristic for all thereactions under consideration. For example, in our cal-culation, we found that the calculated coefficients cpreshow a very weak parabolic dependence ⟨npre(M)⟩ forheavy 252Fm and 256Fm nuclei. This result follows fromthe fact that these nuclei have a very high excitationenergy, a small barrier, and are quickly disintegrated.

The evaporation of light prescission particles does notfor long enough to considerably decrease the excitationenergy of these nuclei. Consequently, almost all fissilenuclei have an equally large available phase space dur-ing the descent from the saddle to the scission. In thiscontext, the approximately one order of magnitude dif-ference between the experimental data on cpre for 252Fmand 256Fm remains inexplicable in terms of theoreticalcalculations with ks = 0.25–0.50.

Furthermore, no parabolic dependence ⟨npre(M)⟩ isobserved for during fission of the 205Fr nucleus.Because this nucleus is strongly neutron-deficient, oneneutron, on average, has time to evaporate before thenucleus decays into fragments. Thus, when simulating205Fr decay, all the fissile nuclei are assumed to have anapproximately identical excitation energy (an identical

1.5

0 35Fragment mass, amu

2.5

70 105 140 175 210

2.0

3.0

3.5

4.0

1.0

0

2

4

6

8

10

12

1

3

2

4

5

6

2

4

6

8

10

12(a)

(b)

Y(M

), %

⟨En⟩

, MeV

⟨npr

e⟩⟨t f

⟩ 10–2

0 , sFig. 13. Experimental data and results of calculations for the reaction 18O + 197Au 215Fr (El.s = 158.8 MeV) [134]. (a) Themass distribution: the thick solid line shows the experimental data [134] and the thin solid line and dashed lines show the results ofcalculations with ks = 0.5 and 0.25, respectively. The neutron multiplicity as a function of the fragment mass: the closed circles showthe experimental data [134]; the closed squares and open squares show the results of calculations with ks = 0.25 and 0.5, respec-tively; and the crosses show the results of calculations under the condition that neutron evaporation takes place only during the

descent from the saddle to the scission point ( = 0). (b) The kinetic energy of neutrons as a function of the fragment mass:

the closed circles show the experimental data from [134] and the closed squares show the result of calculations with ks = 0.5. Thetriangles show the fission time calculated with ks = 0.5.

npregs⟨ ⟩



available phase space), since the evaporation of prescis-sion particles is the only factor that can change the exci-tation energy of nuclei. Therefore, there is no parabolicdependence ⟨npre(M)⟩ for 205Fr.

A short time ago, in [140], an attempt to provide atheoretical description of the experimentally observeddependence ⟨npre(M)⟩ was undertaken within the two-dimensional dynamic model based on the classicalEuler–Lagrange equations. In the calculations [140], acombination of one-body and two-body viscosities wasused. It was ascertained that the viscosity may be dra-matically reduced by multiplying the friction-tensorcomponents by exp(–K ∗ α ∗ α), where K = 161 ± 3,depending on the mass-asymmetry coordinate for thereproduction of the experimentally observed depen-dence ⟨npre(M)⟩. At the same time, it follows from thecalculations in [19, 122] that it is feasible to quantita-tively reproduce the dependences ⟨npre(M)⟩ observedunder experimental conditions well for many nucleiwithout introducing additional free parameters in themodel. Unfortunately, the calculations in [140] yieldedno data on the number of neutrons that evaporate beforeattaining the saddle point or during the descent fromthis point to the scission. Therefore, a detailed compar-ison between the results in [19, 122] and [140] is diffi-cult.

The experimentally observed dependence ⟨npre(EK)⟩[136] shows a significant increase with EK. After apply-ing a recalculation procedure to these results that takesinto account recoil effects [134], ⟨npre(EK)⟩ actuallyceases to depend on EK. In our calculations, ⟨npre(EK)⟩and ⟨tf(EK)⟩ appeared to be almost independent of EK

within the calculation error, and only in the region oflow EK was a small decrease in the dependences

⟨npre(EK)⟩ and ⟨tf(EK)⟩ (see Fig. 16) observed. Such adecrease can be explained by the steadily increasingcontribution of events with a large mass asymmetry asEK decreases, which follows from the general characterof the two-dimensional MED. It should also be notedthat these calculations, which were performed using themodified variant of one-body viscosity, reproduce theshape of the dependence ⟨npre(EK)⟩ more satisfactorilythan calculations within the two-dimensional model[120] including two-body viscosity.

3.4. Postscission and Total Neutron Multiplicities

The evaporation of postscission neutrons weaklyaffects the observed fission characteristics, for exam-ple, the mass–energy distribution of fission fragments.However, postscission particles emitted from fissionfragments carry important information about the fissionprocess because the postscission neutron multiplicitiesprovide data on the critical stage of evolution of a fissilenucleus, i.e., on the rupture of a continuous configura-tion into fragments. At the instant of separation of acontinuous shape into fragments, the excitation energypossessed by a compound nucleus before the scission isdistributed between the formed fragments. Thus, thepostscission particle multiplicities include informationabout the instant of scission of a compound nucleus andcharacterize its excitation energy at the instant of fis-sion into fragments. It is also necessary to emphasizethat the correlation dependences of the total neutronmultiplicity on the mass and kinetic energy of fissionfragments can considerably differ from the correlationdependences of prefission neutron multiplicities [133].This difference is defined by the postscission neutronmultiplicity. Therefore, a simultaneous description of

Neutron multiplicity9876543210

205Fr

80 100 120 14060Fragment mass, amu

Fig. 14. Experimental data and results of calculations forthe reaction 36Ar + 169Tm 205Fr (El.s = 205 MeV)[136]. The open symbols represent the experimental datafrom [136] and the closed symbols show the results of cal-culations with ks = 0.5. The squares represent the prescis-sion neutron multiplicity; the circles, the postscission neu-tron multiplicity; and the triangles; the total neutron multi-plicity.

1

1.0

Yn, %

c

7

1.5

6

5

4

3

2

02.0 2.5

256Fm

215Fr

Fig. 15. Calculated prescission neutron yield (as a percent-age of the total number) as a function of the elongationcoordinate c for the 215Fr (solid curve) and 256Fm (dottedcurve) nuclei. The arrows indicate the saddle point position.

404


ADEEV et al.

the correlation dependences of prefission, postscission,and total neutron multiplicities is a serious test for the-oretical models. However, a consistent description ofthese correlation dependences would provide informa-tion on how the fission process proceeds, from the stageof compound-nucleus formation to the separation offragments.

In order to calculate the postscission neutron multi-plicity, it is necessary to know the excitation energy ofthe fragments formed after compound-nucleus fission.In this case, it is usual to assume that the compound-nucleus temperatures before the scission and the frag-ment temperatures immediately after the scission areequal. In addition, it is conventional to apply the law ofconservation of energy, which can be written as [123]

(52)

Here, Qf is the reaction energy, which is calculated asthe difference between the masses of a compound nucleus

Q f Ecoll tsc( ) V qsc( )– EK–+ Edef1( )

Edef2( )

.+=

and the formed fragments; Ecoll(tsc) and V(qsc) are the col-lective-motion energy and potential energy at theinstant of scission of a nucleus into fragments; EK is the

kinetic energy of motion of the fragments; and and

are the deformation energies of the fragments. Thecondition that the system maintains a constant temper-ature before its fission into fragments and the energyconservation law make it possible to determine theinternal excitation energy of the formed fragments andtheir total deformation energy but render it impossibleto calculate the deformation energy for each fragment.Therefore, when describing the shape of fragmentsusing several independent parameters, it is necessary toinclude additional conditions to attain an unambiguousdefinition of the deformation energy of these frag-ments. These conditions are as follows: the condition ofconservation of the lowest moments of the nuclear-den-sity distribution during fission of an initial compound

Edef1( )

Edef2( )

5

50 100EK, MeV

15

150 200 250

10

20

25

30

0

0

2

4

6

8

10

12

2

4

6

8

10

12(a)

(b)

Y(E

K),

%⟨t f

⟩ 10–

20, s

⟨npr

e⟩

40

35

45

0

Fig. 16. Experimental data and results of calculations for the reaction 18O + 197Au 215Fr (El.s = 158.8 MeV) [134]. (a) Theenergy distribution: the thick solid line shows the experimental data from [134], and the thin solid and dashed lines represent theresults of calculations with ks = 0.25 and 0.5, respectively. The neutron multiplicity as a function of the fragment kinetic energy: theclosed circles show the experimental data from [134]; the closed squares and open squares show the results of calculations withks = 0.25 and 0.5, respectively; and (b) the triangles represent the fission time calculated with ks = 0.5.



nucleus [141] or the condition of maximum entropy ofa system consisting of two fragments [142]. Becausethe energy conservation law in the form of Eq. (52) issatisfied for an arbitrary way of distributing the defor-mation energy between fragments, the mean postscis-sion neutron multiplicities are virtually identicaldespite the additional conditions used for finding thedeformation energy of each fragment. As a result, cal-culations [123, 142] performed using various methodsof determining the deformation energies of fragmentslead to approximately identical results on the meanpostscission neutron multiplicities. The use of variousconditions for determining the deformation energies offragments affects only the dependences of postscissionneutrons on the mass and kinetic energy of the frag-ments.

In Figs. 17 and 18, we show the results of calcula-tions of the mean multiplicities for prefission andpostscission neutrons and, in addition, the total neutronmultiplicity depending on the initial excitation energy ofthe compound nucleus for the reaction 18O + 208Pb 224Th (Elab = 130, 110, 90, and 83 MeV).

In these figures, the solid and dotted lines show theapproximation of the prescission and postscission neu-tron multiplicity proposed in [143, 144]. From a com-parison between Figs. 17 and 18, it can be seen that,depending on the value of viscosity, the number of neu-trons evaporating from a compound nucleus and itsfragments is different; moreover, the results of calcula-tions of the total neutron multiplicity are actually inde-

pendent of the viscosity, as can be expected for physicalreasons. With an increase in the viscosity, the numberof particles evaporating from a compound nucleusincreases and its internal excitation energy and, conse-quently, the internal excitation energy of the fragmentsdecreases.

Experimental data and our theoretical calculations[123] performed with the one-body mechanism ofnuclear viscosity show that, as the excitation energyincreases, the mean postscission neutron multiplicityincreases more slowly than the prefission neutron mul-tiplicity. Such a result is valid for ks > 0.5. For ks = 1 (seeFig. 18), the postscission neutron multiplicity dependsvery weakly on the excitation energy and the calculatedvalues of mean postscission neutron multiplicity are ingood agreement with the experimental data. This resultconfirms the conclusion made in [133], on the basis ofan analysis of a large amount of experimental data, that,at the instant of scission, a nucleus is relatively cold;i.e., a significant portion of the excitation energy of acompound nucleus is carried away by prefission neu-trons before compound-nucleus fission into fragments.

Investigation of the correlation dependences ⟨npost(M)⟩and ⟨npost(EK)⟩ of the postscission neutron multiplicityon the mass of fragments and their kinetic energy,respectively, is also of interest. The calculated depen-dences ⟨npost(M)⟩ and ⟨npost(EK)⟩ for the 205Fr nucleus areshown in Figs. 14 and 19.

As can be seen from Fig. 14, ⟨npost(M)⟩ is a virtuallylinear dependence on M. This type of dependence is aconsequence of the fact that the internal energy Eint of acompound nucleus before scission is virtually indepen-dent of the mass of the formed fragments and Eint is dis-tributed between the fragments proportionally to theirmasses during the separation of a compound nucleus.

070

Ecm, MeV90

9

100 110 120

Neutron multiplicity

80

8

7

6

5

4

3

2

1

Fig. 17. Mean prefission (squares) and postscission (circles)neutron multiplicities and the total neutron multiplicity (tri-angles) as functions of the excitation energy for the 224Thnucleus. The open symbols represent the experimental datafrom [196] and the closed symbols represent the results ofcalculations with ks = 0.5. The solid and dotted lines showthe approximations of prescission and postscission neutronmultiplicities proposed in [143, 144].

70Ecm, MeV

90

9

100 110 120


80

8

7

6

5

4

3

2

1

Fig. 18. The same as in Fig. 17. The values are calculatedwith ks = 1.

0

406


ADEEV et al.

As can be seen from Figs. 20 and 21, the internal energyof the fragments considerably exceeds their deforma-tion energy in the case of high viscosity, thus determin-ing the general character of the dependence ⟨npost(M)⟩.

In Figs. 20 and 21, it can be also seen that the defor-mation energy of the fragments nonlinearly depends ontheir mass. Such a type of dependence is mainly speci-fied by the behavior of Qf in Eq. (52), for which tabu-lated values of the nuclear masses [145] were used inthe calculation. For fissile nuclei from 170Yb to 258Fm,the filling of a shell with Z = 50 and N = 82 fundamen-tally affects Qf(M). In this case, a pronounced peakreveals itself at M 132 up to At in the dependenceQf(M). For lighter fissile nuclei, the Qf(M) peak isshifted to M ACN/2. Correspondingly, as can be seenfrom Figs. 20 and 21, the peaks of the deformationenergy can fall for both symmetric fission, as in the caseof 256Fm, and asymmetric fission, as in the case of 215Fr.Thus, the deformation energy of the fragments specifiesthe deviation of the total fragment excitation energy fromlinear dependence on M. A similar result was previouslyobtained in [142] for the reaction 4He + 209Bi 213At(Elab = 45 MeV), where the deformation energy wascalculated using the maximum-entropy condition onthe system of fragments.

It is convenient to approximate the dependence ofthe total neutron multiplicity ⟨ntot(M)⟩, as well as⟨npre(M)⟩, by a parabolic dependence on M with thecoefficient ctot. Naturally, the closer the dependence⟨npost(M)⟩ is to a straight line, the less the coefficient ctotdiffers from cpre. When the dependence ⟨npost(M)⟩ is rep-

resented by a straight line, ctot should be equal to cpre. Asthe calculations show, the difference between ⟨npost(M)⟩and a straight line generally depends on the deforma-tion energy, which mainly depends on Qf(M). There-fore, if Qf(M) peaks for symmetric fission, ctot > cpre;otherwise, ctot < cpre. This effect most strongly manifestsitself in 205Fr fission (see Fig. 14) for which the prefissionneutron multiplicity is actually independent of mass. Inthis case, the total multiplicity has an explicit parabolicdependence on the fragment mass due to the postscissionneutron multiplicity. The dependence ⟨npost(M)⟩ pro-duces the opposite effect in 215Fr fission, reducing thecoefficient ctot in comparison with cpre.

0140

1

EK, MeV

2

3

4

5

6

7

8

9

150 160 170 180 190


Fig. 19. Neutron multiplicities as functions of the fragmentkinetic energy for the 205Fr nucleus calculated with ks = 0.5.The closed squares show the prefission neutron multiplicity,the open circles represent the postscission neutron multi-plicity, and the closed triangles refer to the total neutronmultiplicity.

Fragment mass, amu

Eex, MeV

75 125 150100

120

100

80

60

40

20

0

215Fr

Fig. 20. Fragment excitation energy Eex (closed circles) forthe 215Fr nucleus as a function of the fragment mass. Theopen circles represent the internal excitation energy of acompound nucleus before scission. The open squares andtriangles show the internal excitation energy and deforma-tion energy of the fragments, respectively.

20

50 75

Eex, MeV

Fragment mass, amu

140

100 125 150 175 200

40

60

80

100

120

252Fm

Fig. 21. The same as in Fig. 20 but for the 252Fm nucleus.



The calculated and experimental dependences⟨npost(EK)⟩ for 205Fr are shown in Fig. 19. As can be seenfrom this figure and Table 4, the multiplicities⟨npost(EK)⟩ and ⟨ntot(EK)⟩ decrease with an increase inEK, and such behavior agrees well with the availableexperimental data. The decrease in the multiplicity⟨npost(EK)⟩ as EK increases follows from the law of con-servation of energy in the form of Eq. (52). With anincrease in EK, the total excitation energy of the frag-ments decreases (for a fixed ratio between the fragmentmasses) and, consequently, the mean postscission neu-tron multiplicities decrease as well. The total neutronmultiplicity also decreases with an increase in EKbecause the prefission neutron multiplicity weaklydepends on EK.

3.5. Fission Probability

As was noted above, the evaporation of prescissionparticles reduces the excitation energy and, thus, com-petes with the fission process. If the nuclear excitationenergy proves to be lower than the fission barrier during

particle evaporation, a nucleus becomes an evaporationresidue. Along with the prescission particle multiplici-ties, the cross section for evaporation residues (or fis-sion probability) is an important characteristic, whichcan be compared with experimental data, and dependson both the parameters used in the statistical model ofnuclear decay and the viscosity of nuclear matter.

In dynamic approaches, data on the prescission neu-tron multiplicity and evaporation probabilities are con-ventionally used for deriving information about the fis-sion time and nuclear viscosity. Significant successes indescribing ⟨npre⟩ and the fission probability Pf wereattained in the one-dimensional Langevin model pro-posed by the Gontchar and Fröbrich group [26]. Inorder to attain a simultaneous description of the data on⟨npre⟩ and Pf , it was necessary to introduce the new uni-versal dependence of the reduced friction parameter βon the fission coordinate: β = 2 × 1021 s for compactnuclear shapes (such a low value of the parameter βresults in a high fission probability). From the instant atwhich a neck occurs to the instant of scission, β linearly

Table 4. Characteristics of compound-nucleus fission

CNE*

(MeV)ks

cpre10–4

ctot10–4

10–2 (MeV–1)

⟨npre⟩ ⟨npost⟩ ⟨ntot⟩

162Yb 114 0.25 8 –1 3.2 0.054 1.8 2.6 7

0.5 11 –3 2.7 0.058 2.4 2.3 7

Expt. [134] 12 1 3.5 – 2.45 1.7 5.85172Yb 121 0.25 20 12 3.6 0.038 3.4 3 9.4

0.5 32 8 2.7 0.052 4.5 2.4 9.3

Expt. [134] 14 10 3.6 0.056 4.4 2 8.4205Fr 77 0.25 1.3 8 2.6 0.039 0.4 3.4 7.2

0.5 1.9 11 2.7 0.041 0.9 3.2 7.3

Expt. [136] 0 9.5 1.8 0.046 1.2 – –215Fr 111 0.25 5.1 4.8 4.3 0.037 3 3.2 9.4

0.5 7.1 6 4.1 0.041 4.3 3 10.3

Expt. [134] 6.5 4.4 3.8 0.047 4.1 2.7 9.5256Fm 101 0.25 1.6 6.6 4.3 0.046 2 6.2 15.5

0.5 3.6 6.5 4.1 0.066 3.1 5.8 14.7

Expt. [134] 8.2 5.4 4.1 – 5.1 4.25 13.6252Fm 140 0.25 2.7 2.2 5.4 0.052 2.7 7 16.7

0.5 3.6 4 4.8 0.061 4.0 6.5 17

Expt. [146] 0 2.4 5 – 6.95 3.83 14.6Note: The following are the full forms of the notation used in the table: compound nucleus (CN); compound-nucleus excitation energy

(E*); the reduction coefficient (ks) of the wall-formula contribution; the coefficients (cpre) and (ctot) of the parabolic mass depen-

dence for the prefission and total neutron multiplicities, respectively; the coefficients ( ) and ( ) of the linear kinetic-

energy and mass dependences, respectively, for the postscission neutron multiplicity; the mean prefission, postscission, and totalneutron multiplicities (⟨npre⟩), (⟨npost⟩), and (⟨ntot⟩), respectively.

dnpost

dM-------------

dntot

dEK-----------–

dnpost

dM-------------

dntot

dEK----------–

408


ADEEV et al.

increases to 30 × 1021 s, which provides a good descrip-tion of the mean prescission neutron multiplicity.

Further study of the fission probability within themultidimensional Langevin approaches, together withother observed quantities (in particular, the MEDparameters of fission fragments), is undoubtedly ofinterest.

In [47], the two-dimensional Langevin model wasused to simultaneously calculate three observed quanti-ties: the prescission neutron multiplicity, fission proba-bility, and variance of the mass distribution. In thisstudy, two sets of coefficients were used for the level-density parameter: those of Ignatyuk et al. [69] andthose of Töke and Swiatecki [71]. In addition, thepotential energy was calculated using two variants ofthe liquid-drop model: one with a sharp-edge and theother with a diffuse-edge nuclear surface. The influenceof the choice of LDM variant and coefficients for calcu-lating the level-density parameter on the obtained val-ues of observed quantities was analyzed. Similarly tothe results of [19, 67], it was found in [47] that the useof the level-density parameter proposed by Töke andSwiatecki allowed the authors to better describe thefission probability. In contrast, experimental values ofthe mean prescission neutron multiplicity were moresatisfactorily reproduced with the coefficients ofIgnatyuk et al.

In [19, 123], on the basis of the three-dimensionalLangevin calculations, we showed that the fission prob-ability (the evaporation-residue cross section σER) isextremely sensitive to nuclear viscosity, especially inthe low-energy range where ⟨npre⟩ weakly depends onthe dissipative force. Thus, at low excitation energies,σER is an important experimentally observed character-istic that should be included in investigations of theproblem of nuclear dissipation. A similar conclusion onthe sensitivity of evaporation-residue cross sectionswas made in [147] on the basis of results of one-dimen-sional calculations. Furthermore, in [19, 123], wefound that good quantitative agreement with the exper-imental data on σER was attained at ks = 0.25–0.50.However, calculations executed with the total one-bodyviscosity (ks = 1) considerably overestimate the data onevaporation residues.

It is also of interest to gage the influence of the useof the free-energy thermodynamic potential, instead ofthe potential energy, for determining the conservativeforce. Our calculations [19] showed that replacement ofthe driving potential V(q) by F(q) reduces the prescis-sion neutron multiplicity and the fraction of evapora-tion residues. This result is obtained because the use offree energy makes the fission barrier lower, the meanfission time decrease, and, hence, the mean prescissionneutron multiplicity decrease and the fission probabil-ity increase.

For superheavy nuclei, theoretical investigation ofthe fission probability has gained in importance due torecent experiments on the synthesis and study of vari-

ous characteristics of new superheavy elements. How-ever, the theoretical calculations for reactions resultingin the formation of superheavy elements now face seri-ous problems. Primarily, these problems are associatedwith large uncertainty in relation to the statistical-model parameters in the heavy-nuclei region. The pro-cess of formation of heavy compound nuclei includesvarious stages such as the fusion of heavy ions and theformation and decay of a compound system. Todescribe each stage of the process is rather difficult initself, and the results obtained also strongly depend onthe parameters and approximations used. However, sig-nificant progress has recently been achieved in this lineof research due mainly to the steady development oftheoretical approaches and models such as the coupledchannel method [148, 149] and the Langevin dynamicmodels for description of the fusion of ions and the evo-lution of a compound system [22, 24].

4. CHARGE DISTRIBUTIONOF FISSION FRAGMENTS

The charge mode is related to the transfer of nucle-ons between fragments and strongly differs from thepreviously considered collective modes of describingvariation in nuclear-surface shape. The principal differ-ence between these modes consists in the different ratiobetween the relaxation times of collective and single-particle modes and between the relaxation times of col-lective modes and the mean time τss of descent of a fis-sile nucleus from the saddle to the instant of scission.As is known, the charge degree of freedom is very fastin the sense that the relaxation time of a charge is muchshorter than τss and comparable with the relaxation timeof internal degrees of freedom. The relaxation times ofthe collective modes determining nuclear shape are ofthe same order of magnitude as the descent time [150].

First, such a ratio between the times makes it possi-ble to consider the charge collective coordinate andnuclear-shape coordinates as independent. Second,because the charge mode is fast, the statistical modelshould provide a good description of the properties ofthe charge distribution of fragments. This assumptionhas been confirmed in a number of theoretical andexperimental investigations of the charge distribution inthe low-excitation energy range and makes dynamicstudy of charge fluctuations a valuable source of infor-mation on the properties of nuclear matter, in particular,on nuclear viscosity. Indeed, the experimental situationis already understood: at the instant of scission of anucleus into fragments, an instantaneous statisticallimit [20, 151] is established in the charge mode.Finally, because the characteristic times of an internalsubsystem and the charge degree of freedom are notvery far from each other in the time scale, it is not cer-tain that the Langevin approach (in particular, the Lan-gevin approach used within the Markovian approxima-tion) can be applied to a description of such a fast modeas the charge mode.



4.1. Features of the Model Describing Charge Fluctuations

In order to study the charge distribution of fissionfragments within the Langevin approach, it is necessaryto introduce a coordinate determining the charge parti-tion between fragments. It is convenient to use theparameter ηZ = (ZR – ZL)/(ZR + ZL), where ZR and ZL arethe charges of the right-hand and left-hand fragments,respectively, for this coordinate. The coordinate ηZ hasfrequently been used as the charge coordinate [152, 153]and, in our opinion, is the most convenient. Consider-ation of asymmetric fission is of minor interest from thestandpoint of studying charge fluctuations and, in addi-tion, results in significant complication of the model.For this reason, in [20, 151], we considered the case ofsymmetric fission, which enabled us to introduce justthree collective coordinates: two coordinates todescribe nuclear shape in terms of the c, h, α param-etrization (an elongation c and a neck coordinate h) andthe charge-asymmetry coordinate ηZ. When restrictingthe analysis to the case of symmetric fission (α = 0), itshould be remembered that the model can also be gen-eralized to the case of asymmetric shapes.

In [20, 151], set of three-dimensional LangevinEqs. (14) serves as a set of equations of motion. At thesame time, because the charge mode is a finite mode, itis necessary to use the effective temperature [154],which takes into account quantum vibrations in thecharge coordinate, instead of the temperature T fordetermining random-force amplitude from the coordi-nate ηZ (see Eqs. (16) and (17)):

(53)

Here, ωZ is the frequency of vibrations in ηZ. We notethat, in contrast to the charge distribution, it is unneces-sary to introduce an effective temperature into themass-asymmetry coordinate when studying the massdistribution (at least at T > 1 MeV) due to the smallnessof the vibration frequencies for this collective mode incomparison with the thermostat temperature T [13]. Inthis case (T ωA/2, where ωA is the frequency ofvibrations in the mass coordinate), the effective temper-ature is approximately equal to the temperature T.

It is obvious that considering the charge transferbetween fragments is meaningful only if it is possibleto unambiguously separate one fragment from another.Therefore, in the dynamic calculations described in[20, 151], evolution of the charge degree of freedombegan when a neck appeared in a nucleus, while onlythe coordinates determining nuclear-surface shapeevolved for shapes without a neck. For the 236U nucleus,chosen for the calculations, the neck appears after thenucleus passes the ridge separating the ground-stateregion from the fission valley. Therefore, the questionarises as to whether it is expedient to begin dynamic

TZ*

TZ* ωZ

2----------

ωZ

2T----------

.coth=

T A*

calculations from the ground state instead of, for exam-ple, the ridge line. The reason for this choice of initialconditions is obvious. In [19, 122], we showed that anoverwhelming prescission neutron fraction (more thanhalf) evaporates in the ground-state region before anucleus attains the ridge. Thus, for the correct accountof neutron evaporation, it is necessary to begin dynamiccalculations from the ground state of a compound sys-tem. In addition, when discussing methods of calculat-ing the potential energy and the transport coefficients ofthe charge mode, we assumed that there is a bridge inthe nuclear shape.

4.1.1. Potential energy. When calculating the poten-tial energy within the liquid-drop model, the chargedensity is usually assumed to be constant throughout anucleus. However, it is obvious that, due to the Cou-lomb repulsion of protons, the nucleons in a nucleus areredistributed so that the charge density becomes higherat the periphery and lower inside the nucleus. The sim-plest solution to the problem on the type of charge-den-sity function was proposed in [51, 155], where thecharge density was approximated by a linear functionof the coordinate z (along the symmetry axis). Theproblem of the charge distribution in a nucleus wasinvestigated in detail using variational methods in[32, 156], and two liquid-drop models were used in [32].From the experimental data on the charge distribution[157–159], it is possible to draw the conclusion that thepolarizability of nuclear matter is negligible. Therefore,when performing Langevin calculations of the chargedistribution in [20, 151], we assumed that the chargedensity is different but constant inside each future frag-ment [32, 160]. Such an approximation of the chargedensity is rather rough, but, at the same time, it meantthat we did not have to introduce any additional param-eters (for example, the charge-polarization parameter[51, 155]) and, using reasonably simple formulas, itallowed us to describe the charge partition betweenfragments.

Using the condition of incompressibility of nuclearmatter, it is possible to show that the proton densities

and in the formed fragments are related to thecharge coordinate ηZ in the following way:

(54)

Here, = Z/(4/3π ) and = N/(4/3π ) are thedensities of protons and neutrons, respectively, for auniform charge distribution over the entire nucleus andk = AR/AL is the mass ratio for the formed fragments.From Eqs. (54), the corresponding neutron densities

and are easily expressed.

ρRp ρL

p

ρRp ρ0

p k 1+( )2k

---------------- 1 ηZ+( ),=

ρLp ρ0

p k 1+( )2

---------------- 1 ηZ–( ).=

ρ0p

R03 ρ0

nR0

3

ρRn ρL

n

410


ADEEV et al.

In the sharp-edge LDM taking the charge degree offreedom into account, the potential energy is formed asa sum of the symmetry energy, the Coulomb and sur-face energies, and the energy of the rotation of anucleus as a whole. Calculation of the Coulomb energyis essentially simplified for a homogeneous charge dis-tribution in each fragment and, for the dependence ofthe potential energy on the charge-asymmetry parame-ter ηZ, can be expressed as

(55)

(here and below, q = (c, h)). The mean value of thecharge-asymmetry parameter ⟨ηZ(q)⟩ is the point of a localminimum of the potential energy for these shape parame-ters (c, h). In the case of symmetric fission, ⟨ηZ⟩ = 0. Thecoefficient of rigidity of the potential relative to the param-

eter ηZ is (q) = (∂2V(q, ηZ)/∂ . The expres-sion for the coefficient of rigidity can be found, forexample, in [20, 32, 160, 161]. The coefficient of rigid-ity of the potential on the charge coordinate weaklydepends on the nuclear deformation, since the maincontribution is due to the symmetry energy. In its orderof magnitude, = (7–8) × 103 MeV [20, 151].

4.1.2. Inertial parameter of the charge mode. Animportant problem in nuclear dynamics is calculationof the transport coefficients. For the charge problem,calculation of the transport coefficients is complicatedbecause two collective coordinates describe nuclear-shape variation while the third coordinate (ηZ)describes the charge redistribution in a nucleus and hasquite a different physical nature from the shape coordi-nates. Therefore, different models are necessary whendescribing the charge component and all the other com-ponents of the mass tensor and friction tensor.

We now discuss the method of calculation of thecharge-mode mass parameter . In [161], wefound the mass parameter from the formula

(56)

where ω1 is the frequency of the longitudinal dipoleisovector vibrations obtained from the solution to theHelmholtz equation for charge-density fluctuation[162, 163].

In [164], in the case of a nonviscous-fluid flowthrough a round aperture of radius rN connecting twotouching spherical fragments, the following expressionwas obtained for the charge-mode inertial parameter:

(57)

Here, m is the nucleon mass.

V q ηZ,( ) V q ηZ⟨ ⟩,( )CηZ

q( )2

---------------- ηZ ηZ⟨ ⟩–( )2+=

CηZηZ

2)ηZ ηZ⟨ ⟩=

CηZ

mηZηZ

mηZηZq( )

CηZq( )

ω12 q( )

----------------,=

mηZηZq( ) π

6---r0

3m

Z A2

N---------- 1

rN

-----.=

When deriving Eq. (57), the fact that the neck con-necting the fragments has a nonzero length was com-pletely disregarded. In addition, viscosity plays animportant role in the Langevin approach. In [165], itwas shown that, for the flow of a viscous incompress-ible fluid through a cylindrical neck of radius rN andlength lN, the charge-mode mass parameter is describedby the expression

(58)

where ρ is the nucleon density in a nucleus.A comparison between the results given by Eqs. (56)–

(58) is shown in Fig. 22a for the parameter h = 0 (as theline h = 0 approximately corresponds to the bottom ofthe fission valley [33]). The mass-parameter values forEq. (56) are taken from [161], in which the massparameter is given as a function of the spacing betweenthe centers of mass of future fragments for several fixedvalues of the neck parameter h. The mass-parametervalues given in this figure correspond to a large value ofthe elongation parameter c in the nuclear-deformationregion from the saddle point to the scission. All threedependences show a characteristic sharp increase in theinertial parameter when approaching the scissionpoint. From this figure, it can also be seen that Eqs. (56)and (58) yield close values of while those calcu-lated from Eq. (57) lie much lower.

Equation (56) is the most consistent; however, cal-culation of the frequencies of dipole isovector vibra-tions result in a mathematical problem that is rather dif-ficult to realize. From an analysis of the three equationswritten above, it follows that Eqs. (56) and (58) yieldclose values of the mass parameter in the entireregion of variation of the nuclear deformation parame-ters (c, h) that we are interested in. Therefore, it is rea-sonable to use Eq. (58) in the calculation of the charge-mode mass parameter. To apply this equation, it is nec-essary to know the neck radius rN and the neck length lN

for given c and h. To this end, in the region of deforma-tions with a pronounced neck, a nuclear shape can beapproximated by two spherical fragments connected bya cylindrical neck of radius rN. In this case, the centersof the spheres are the centers of mass of the formingfragments and the sphere radii RR and RL are foundfrom the condition of conservation of nuclear volume.Then, the neck length is lN = R – RR – RL, where R is thespacing between the centers mass of the future frag-ments.

4.1.3. Friction parameter of the charge mode. In thetheoretical studies of the isobaric distribution (see, forexample, [161, 166]), the friction coefficient hasfrequently been assumed to be a coordinate-indepen-dent varied parameter. However, such an assumption isa rough approximation, and it is necessary to take into

mηZηZq( ) m

3πρ----------Z A

2

N----------

lN 2rN+( )

rN2

------------------------,=

mηZηZ

mηZηZ

mηZηZ

γ ηZηZ



account the dependence of the charge-mode frictionparameter on nuclear deformation. The coordinatedependence of the friction parameter was takeninto account in [20, 151] in two ways: under theassumption of the one-body and two-body mechanismsof dissipation. The most simple way to estimate thefriction parameter is in the hydrodynamic model bystudying the energy dissipation in the motion of a vis-cous incompressible fluid along a pipe (a neck connect-ing the forming fragments) of length lN and radius rN. Inthis case, the reduced friction coefficient of the chargemode has the form [150, 151, 165]

(59)

where ν is the kinematic viscosity coefficient. FromEq. (59), it can be seen that, in the hydrodynamicmodel, the charge-mode friction parameter depends on the approach used for calculating the iner-tial parameter . From experimental data on thewidth of giant dipole resonances [167], it was foundthat ν = 13.5 × 1021 fm2 s–1. Conventionally, thedynamic viscosity coefficient is used in nuclear physicsas the two-body viscosity coefficient. Dynamic viscos-ity is related to kinematic viscosity as follows: ν0 = ρ0ν,where ρ0 is the nuclear-matter density. For the parame-ter r0 = 1.22 fm [72], the dynamic viscosity coefficientobtained in [167] amounts to ν0 = 1.8 × 10–23 MeV s fm–3.This coefficient ν0 can be used as the initial value indynamic calculations with the two-body mechanism ofnuclear viscosity, since the dipole isovector vibrationsalong the nuclear symmetry axis are the basic mecha-nism of charge redistribution between the fragments.

The application of the one-body viscosity model forcalculating the friction parameter of the charge degreeof freedom is of particular interest. It should be notedthat the one-body viscosity mechanism has been suc-cessfully applied to the physics of the fission of excitednuclei when studying the width of giant dipole reso-nances [168, 169]. In [20, 151], we obtained the follow-ing expression for the charge-mode friction parameterwhen we assumed the one-body mechanism of viscosity:

(60)

Here, and are the mean proton and neutronvelocities, respectively.

In Fig. 22b, we show the charge-mode frictionparameter as a function of the elongation parameter cfor the two-body (the solid curve) and one-body (thedashed curve) mechanisms of viscosity, respectively.From Eq. (60), it can be seen that the parameter depends only on the neck thickness for the one-body

mechanism; hence, ~ 1/ for rN 0 (i.e.,

γ ηZηZ

βηZηZq( )

γ ηZηZq( )

mηZηZq( )

---------------------6νrN

2------,= =

γ ηZηZ

mηZηZ

γ ηZηZq( ) 4m

9ρ------- AZ

N------- Nv p Zv n+[ ] 1

∆σ-------.=

v p v n

γ ηZηZ

γ ηZηZrN

2

when approaching scission). For the two-body mecha-nism, the inertial parameter calculated from Eq. (58) is

proportional to 1/ when the neck radius approacheszero; hence, the friction parameter behaves according to

1/ at rN 0 for two-body viscosity. From Fig. 22b, itcan be seen that the two-body mechanism of nuclearviscosity yields smaller friction parameters than theone-body mechanism for the deformations characteris-tic at the saddle point. However, sharplyincreases when approaching scission under the assump-tion of two-body viscosity and is twice as high at thescission point than when this parameter is calculatedfrom Eq. (60). We note another important distinctionbetween the two equations for the charge-mode frictionparameter: the presence of a varied parameter in theviscosity ν0 in the equation of two-body viscosity andthe absence of varied parameters in the one-bodymodel.

4.2. Relaxation Times of the Charge Mode: The Statistical Limit at the Scission Point

The charge distribution for the 236U compoundnucleus has been reasonably thoroughly investigatedboth experimentally and theoretically. The isobariccharge distribution for the nuclear fission of 236U wasinvestigated theoretically using the multidimensionalFokker–Planck equation in [161, 166]. However, in

rN2

rN4

γ ηZηZ

γ ηZη Z

(10–

18 M

eV s

)m

η Zη Z

(10–

39 M

eV s

2 )

c

1.0

0.8

0.6

0.4

0.2

08

6

4

2

1.6 1.7 1.8 1.9 2.0 2.1 2.20

(‡)

(b)

h = 0

Fig. 22. (a) Comparison between the dependences of theinertial parameter on the coordinate c at the fixed neckparameter h = 0: the solid curve shows the mass parameter

calculated from Eq. (58) in [165]; the dashed curve

is obtained from Eq. (57) in [164]; and the squares representthe mass parameter calculated from Eq. (56) in [161]. (b)The viscosity parameter of the charge mode as a function ofthe elongation parameter c along the mean trajectory underthe assumption of the two-body mechanism of viscosityν0 = 1.8 × 10–23 MeV s fm–3 (solid curve) and the one-bodymechanism of viscosity (dashed curve).

mηZηZ

412


ADEEV et al.

these studies, the charge-mode friction parameter waschosen independently of the collective coordinates andwas a varied parameter. In [20, 151], we performedthree-dimensional Langevin calculations for the chargedistribution using both the one-body and the two-bodymechanisms of viscosity for the charge mode.

There is a large amount of experimental data onvariances of the charge distribution for the 236U nucleusat low excitation energies. For example, it is well-known

[170, 171] that the charge variance = 0.40 ± 0.05 in235U fission induced by thermal neutrons and is inde-pendent of the excitation energy. Such behavior of the

variance indicates the quantum nature of the forma-tion of the charge distribution at low excitation ener-

gies. It is also known that the experimental data on are described well in this energy range by the followingexpression for the statistical limit at the mean scissionpoint:

(61)

Here, CZ(⟨qsc⟩) = (⟨qsc⟩)/Z2 and ⟨qsc⟩ are the coor-

dinates of the mean scission point. If T ωZ/2, ωZ/2, which explains the constancy of the charge vari-ance at low energies. The dynamic model, which disre-gards shell effects in calculation of the transport coeffi-cients and the potential energy, cannot be used todescribe low-energy fission. Nevertheless, it is possible

to calculate and to compare the resulting valuewith experimental data. For an excitation energy of6.4 MeV (corresponding to 235U thermal-neutron fis-

sion), it was found in [20, 151] that = 0.35, whichagrees well with the experimental data.

We now consider the results of dynamic calcula-tions. Most importantly, we discuss the characteristictimes for the nuclear charge mode. In a system with dis-sipation, the characteristic time is the relaxation time[150] in the corresponding collective mode:

(62)

Here, = + is the generalizeddamping coefficient for the charge mode. In Eq. (62),

the first case ( ≥ /2) corresponds to the mode of

damped vibrations and the case < /2 corre-sponds to the mode of aperiodic damping. The resultsof calculations of are shown in Fig. 23 for both

σZ2

σZ2

σZ2

σZ .st2

Eint( )TZ

* qsc⟨ ⟩( )CZ qsc⟨ ⟩( )------------------------.=

4CηZ

TZ*

σZ .st2

σZ .st2

τηZ

2βηZ

1–, ωηZ

βηZ/2≥

βηZ/2 βηZ

2/4 ωηZ

2–( )

1/2–[ ]

1–, ωηZ

βηZ/2.<

=

βηZ βηZmηZηZ

/mηZηZ

ωηZβηZ

ωηZβηZ

τηZ

mechanisms of viscosity. We now discuss relaxationtimes calculated under the assumption of the two-bodymechanism of nuclear viscosity. It is possible to see thatthe coordinate dependence of is essentially deter-

mined by two-body viscosity. In particular, decreases with an increase in the deformation when thecharge oscillator is in the damping mode (Figs. 23aand 23b) and, in contrast, increases in the over-damped mode (Figs. 23e and 23f]. The case shown inFigs. 23c and 23d corresponds to the damped-vibrationmode at the beginning of charge-mode evolution, andthe system passes into the aperiodic-damping mode atc 2.1 for ν0 = 1.8 × 10–23 MeV s fm–3 and c 1.75 forν0 = 5.7 × 10–23 MeV s fm–3. Furthermore, Fig. 23cshows that the charge-mode relaxation time,

0.4 × 10–21 s, remains virtually constant in the entireinterval of motion of the system from the saddle pointto the scission point under the assumption of one-bodyviscosity, which is not the case for two-body viscosity.This is associated with the fact that the inertial and fric-tion coefficients of the charge mode for the one-body

mechanism of friction are proportional to 1/ . Hence,

the generalized damping coefficient does not varyalong a mean trajectory (the ratio is small

in comparison with ). In the two-body mechanism,this friction coefficient [see Eq. (59)] depends on theneck thickness; therefore, also appreciably dependson the coordinate involving two-body viscosity for thecharge mode.

Another important problem arising in connectionwith the discussion of relaxation times for the chargemode is the applicability of the Langevin equations fordescribing charge fluctuations. The use of the Langevinequations in the Markovian approximation assumesthat the relaxation times τint of the internal degrees offreedom are much shorter than the relaxation times ofthe collective mode under consideration. In [172], itwas shown that τint amounts to about 0.2 × 10–21 s. Wecan consider the Markovian limit to be valid if therelaxation time of the charge degree of freedom is atleast two to three times as long as τint (i.e., > (0.4–

0.6) × 10–21 s). As was noted above, 0.4 × 10–21 sin the case of the one-body mechanism of dissipation,which is at the boundary of the validity range of theMarkovian approximation. In the case of two-body vis-cosity for ν0 < 0.57 × 10–23 MeV s fm–3 and ν0 > 1.8 ×10–23 MeV s fm–3, the Markovian description is valid.The most problematic case from all those consideredabove is the case shown in Fig. 23c. The minimal valuethat can be attained by the charge relaxation time isabout 0.3 × 10–21 s, and the condition of the Markoviancase is not met. However, in the region determining the

τηZ

τηZ

τηZ

τηZ

rN2

βηZ

mηZηZ/mηZηZ

βηZ

τηZ

τηZ

τηZ



parameters of the charge distribution (near the instantof scission), = (0.7–0.9) × 10–21 s. This value is atleast three times higher than τint. Thus, the above argu-ments provide evidence that the charge-distributionparameters calculated from the Langevin equations inthe Markovian approximation are correct. In our opin-ion, the use of the Langevin approach in the Markovianlimit is the first necessary step in solving the problem ofevolution of the charge degree of freedom. However, ifthe Markovian description is unsuitable for studyingcharge fluctuations, it is still possible to use the Lan-gevin equations, but with so-called “delayed” friction(see [173] or review [25]). It is necessary to note thatthe charge mode is interesting in relation to studyingthe memory effects caused by small relaxation times.

4.3. Variance of the Charge Distribution

Studying the behavior of the variance of the chargedistribution with the excitation energy is of great inter-est. Most importantly, it is necessary to compare theresults obtained for both mechanisms of viscosity. As isshown by the calculations, the relaxation times for thecharge coordinate are much shorter than the character-istic times for the other modes associated with a varia-tion in nuclear-surface shape (see, for example, [32]).Because of this circumstance, it is possible to predictthat statistical equilibrium is established for the chargemode not only at low excitation energies but also at

τηZ

high ones. In Fig. 24a, we show the variance of thecharge distribution as a function of the internalnuclear excitation energy at the scission point. The

curve in Fig. 24a shows the variance as a functionof Eint. When plotting this curve, we took into accountthat the neck length is virtually independent of thenuclear deformation, as is shown in our calculationsand the results presented in [174].

On the basis of the data shown in Fig. 24a, it is pos-sible to draw the following conclusions. First, the cal-culated variance of the isobaric distribution shows acharacteristic increase as the excitation energyincreases. Second, the calculations performed with thedifferent mechanisms of viscosity result in identicalvalues of variance within the statistical error associatedwith the limited number (about 104) of Langevin trajec-tories. Thus, we can state that, within the Langevinmodel developed in [20, 151], the variance of thecharge distribution is insensitive to the mechanism ofnuclear viscosity selected. It is noteworthy that chargedistribution was investigated in [175] for the same 232Thfission reaction induced by helium ions in an excita-tion-energy range from 20 to 57 MeV. It was found thatthe charge distribution can be described with a Gauss-ian curve and that the variance is independent of theexcitation energy below 39 MeV. The dynamic calcula-tions performed in [20, 151] at an excitation energy of

39 MeV yield = 0.46, which is in agreement with

σZ .st2

σZ2

Rel

axat

ion

times

τη z(1

0–21

s)

Rel

axat

ion

times

τη z(1

0–21

s)

7

5

3

1

1.0

0.8

0.6

0.4

0.2

1086420

2.0

1.5

1.0

0.5

4

3

2

1

30

20

10

01.6 1.7 1.8 1.9 2.0 2.1 2.2 1.6 1.7 1.8 1.9 2.0 2.1 2.2

v0 = 0.18 v0 = 0.57

v0 = 1.8

v0 = 18

v0 = 5.7

v0 = 57

(a) (b)

(c) (d)

(e) (f)

cc

Fig. 23. Relaxation times for the charge mode as functions of the elongation coordinate c along the mean trajectory. The solid

curves are obtained under the assumption of the two-body mechanism of viscosity in the charge coordinate for various viscositycoefficients. The magnitudes of the viscosity coefficients are indicated in the figures in units of (10–23 MeV s fm–3). The dottedcurve shows the results of the calculation for the one-body mechanism of viscosity. The figure is taken from [20].

τηZ

414


ADEEV et al.

the results given in [175], where = 0.45–0.50 was

obtained for the entire energy range under investiga-tion. Third, as can be seen from Fig. 24a, the variance

obtained in dynamic calculations lies almost at the

statistical-limit curve, which passes through the pre-sented experimental data on the charge variance at theexcitation energies E* = 6.4 and 39 MeV. Therefore, itis possible to conclude that statistical equilibrium for

σZ2

σZ2

the charge coordinate is established both at low andhigh energies.

Figure 24b shows the second derivative

, which includes important informa-tion on the mechanism of formation of the charge dis-tribution as a function of the internal excitation energyat the mean scission point. It can be seen that the energyaxis can be divided into two intervals: Eint < 20 MeVand Eint > 20 MeV. In the first interval,

> 0; i.e., quantum fluctuations playthe predominant role in formation of the charge distri-

bution. In the second interval, < 0 andfluctuations of the charge mode generally have a ther-mal nature.

4.4. Determination of the Two-Body Viscosity Coefficient from Study of Charge Mode Fluctuations

In our discussion on calculation of the friction ten-sor, we mentioned the different behavior of the frictionparameter in relation to the charge coordinate for thetwo mechanisms of viscosity. This difference manifestsitself especially distinctly when a nucleus approachesscission. However, Langevin calculations show that themechanism of nuclear viscosity exerts almost no influ-ence on the charge-distribution width. Therefore, we

should expect a weak dependence of on the onlyfree parameter of the model—the dynamic viscosity ν0.In order to investigate this problem, we calculated thevariance of the charge distribution in a wide range ofν0 = (0.18–57) × 10–23 MeV s fm–3 [20]. The resultsof these calculations are shown in Fig. 25 in the log-

arithmic scale. Figure 25 shows that when is cal-culated dynamically in the interval (0.6 ≤ ν0 ≤ 1.8) ×10–23 MeV s fm–3, it agrees well with the statistical limitat the mean scission point and the experimental data

on . This interval represents an estimate of the two-body viscosity coefficient obtained from studying fluc-tuations of the charge mode.

The coordinate dependence and relaxation times shown in Fig. 23 for all the considered values of

viscosity provide the key to understanding the behavior

of . Statistical equilibrium is established for the

charge mode if is much shorter than the time τss

(τss (5–10) × 10–21 s) taken for descent of a fissile sys-tem from the ridge to the scission. Figure 23b showsthat 0.5 × 10–21 s τss near the scission, which isthe most significant (for the charge-distribution param-eters of fission fragments) deformation region. In thecase shown in Fig. 23c, < 10–21 s for the entiredescent from the saddle to the scission. A sharp

d2σZ .st

2( )/ dEint2( )

d2σZ .st

2( )/ dEint2( )

d2σZ .st

2( )/ dEint2( )

σZ2

σZ2

σZ2

τηZ

σZ2

τηZ

τηZ

τηZ

(d2 σ2 Z

.st)/

(dE

int),

10

–4 M

eV–

2σ2 Z

Eint, MeV

(a)

(b)

0.70

0.65

0.60

0.55

0.50

0.45

0.40

0.35

0.307

6

5

4

3

2

1

0

10 20 4030 50 60 70 80 90 100

Fig. 24. (a) Variance of the isobaric distribution as a

function of the internal excitation energy of a compoundnucleus at the scission point: the squares represent the cal-culations with the two-body mechanism of viscosity for thecharge mode (ν0 = 1.8 × 10–23 MeV s fm–3); and the trian-gles represent the calculations with the one-body mecha-nism of viscosity. The solid curve shows the statistical limitat the mean scission point (see Eq. (61) and comments

related to it). (b) The second derivative

as a function of the internal excitation energy Eint at themean scission point.

σZ2

d2σZ .st

2( )/ dEint2( )

2



increase in the charge-mode relaxation time at the finalstage of the descent cannot lead to a significant devia-tion of the dynamically calculated charge variance fromthe statistical limit. Hence, it is possible to concludethat the behavior and the relaxation times in the

interval (0.6 ≤ ν0 ≤ 1.8) × 10–23 MeV s fm–3 are such thatthe statistical description of the charge variance shouldbe valid. However, the relaxation times are compa-

rable for ν0 < 0.57 × 10–23 MeV s fm–3 (Fig. 23a) andν0 > 1.8 × 10–23 MeV s fm–3 (Figs. 23d–23f) and evenexceed τss. Therefore, no statistical equilibrium isestablished at the instant of scission.

The mechanism determining an increase in thecharge variance for ν0 > 1.8 × 10–23 MeV s fm–3 can becalled a “memory effect.” Its essence can be explainedas follows. A weak coordinate dependence of the rigid-ity factor and an increase in the charge-mode massparameter as the deformation of a fissile nucleusincreases lead to a decrease in the effective temperature

. In turn, this leads to a decrease in charge fluctua-tions during the descent of a nucleus to scission and toa narrowing of the equilibrium charge distribution(determined by the ratio /CZ). In contrast, a fast

increase in the friction parameter results in the

charge degree of freedom being frozen, and nolonger varies in arbitrary way during the further descentof the system towards scission. The charge mode is fro-zen at a high two-body viscosity coefficient ν0 > 1.8 ×10–23 MeV s fm–3, which corresponds to the over-damped mode. In addition, the higher the coefficient ν0,the earlier the charge degree of freedom is frozen and,hence, the broader the charge distribution of fissionfragments. This mechanism qualitatively explains theincrease in charge variance as ν0 increases.

The mechanism of two-body viscosity has previ-ously been widely used to study the mass–energy dis-tribution of fission fragments, and certain conclusionshave been formulated about the value of the frictioncoefficient. In [58, 61], the mean kinetic energy of fis-sion fragments was calculated in a wide interval ofZ2/A1/3, and the following values of this coefficientwere found from fitting the obtained results to experi-mental data: ν0 = (0.9 ± 0.3) × 10–23 MeV s fm–3 [61]and ν0 = (1.9 ± 0.6) × 10–23 MeV s fm–3 [58]. In [13],we investigated the MED of fission fragments with theaid of the multidimensional Fokker–Planck equationand showed that the results of dynamic calculationsagreed most closely with the experimental data for ν0 =(1.5 ± 0.5) × 10–23 MeV s fm–3. In [176], various typesof collective nuclear motion (in particular, the fission ofa nucleus into fragments and giant dipole resonances)were considered. Based on the obtained results, theauthors assumed that the two-body viscosity coefficientwas universal for all types of collective nuclear motion

τηZ

τηZ

CηZ

TZ*

TZ*

γ ηZηZ

σZ2

and had the value ν0 10–23 MeV s fm–3. It is easy tosee that the interval of the coefficient ν0 found by us forthe charge mode agrees well with the former estimates.

In summary, we note once again that, at present,Langevin calculations of the charge distribution areperformed only for symmetric nuclear fission into frag-ments. Study of the mass–charge distribution is of par-ticular interest, and, in order to calculate it, it is neces-sary either to introduce a fourth coordinate (of massasymmetry) into the model or, in the three-dimensionalmodel, to select the mass-asymmetry coordinate, thecharge-asymmetry coordinate, and a third coordinateresponsible for nuclear elongation and, eventually, forfission into fragments as the collective coordinates.

5. ANGULAR DISTRIBUTION (AD) OF FISSION FRAGMENTS

Statistical theory describes the anisotropy of theangular distribution (AD) of fission fragments well formany nucleus-target–impinging-particle combinationsin a wide excitation-energy range. At early the stages ofexperimental research into the ADs of fission frag-ments, reactions in which neutrons, 3He ions, and αparticles were used as impinging particles were consid-ered [157, 177]. The compound nuclei formed in suchreactions have a temperature of about 1 MeV and lowangular momenta. For these reactions, the fission-bar-rier height is much higher than the nuclear temperature,and the conventional saddle-point transition-state(SPTS) model [157, 178, 179] gives a reasonably exactreproduction of the experimental data on the AD anisot-ropy of fragments.

0.50

0.18

σ2Z

ν0, 10–23 MeV s fm–3

0.80

1.8

0.75

0.70

0.65

0.60

0.55

0.450.57 5.7 18 57

Fig. 25. Variance of the charge distribution as a function

of the two-body viscosity coefficient ν0 at the excitationenergy E* = 50 MeV. The solid line shows the statisticallimit at the mean scission point. The vertical dashed–dottedlines restrict the interval of values of the two-body viscositycoefficient obtained in [13, 58, 61, 176]. The figure is takenfrom [20].

σZ2

416


ADEEV et al.

Later, ADs were studied with more massive imping-ing ions of carbon, oxygen, and heavier ions [177]. Thestudy of the ADs of fragments produced in the fissionof heavier nuclei with much higher temperatures andangular momenta became possible. For such systems, itwas found that the standard SPTS model regularly pre-dicted low values of AD anisotropy in comparison withthe experimental data.

The challenges arising in the theoretical descriptionof ADs in reactions with heavy ions can be condition-ally separated into two problems (for more details, see,for example, review [180] and the references therein):

(1) At impinging-ion energies much higher than theCoulomb barrier, theoretical models based on the SPTSmodel usually predict a decrease in AD anisotropy inresponse to an increase in the impinging-ion energy,whereas an increase or the approximate constancy ofthis value is observed under experimental conditions(see, for example, [181]). In this section, we mainlyconsider this problem. Several hypotheses have beenproposed to explaining the high values of AD anisot-ropy observed in this energy range:

(i) The transition state for compound nuclei with ahigh fissility and momentum is located not at the saddlepoint but at a more deformed scission point [182, 183].Thus, the scission-point transition-state (SCTS) modelwas proposed. Great success in describing the AD inreactions with heavy ions was achieved with the ver-sion of the SCTS model developed in [184]. Theauthors of this study proposed a model which moreaccurately (in comparison with the first variants of theSCTS model [182, 183]) took into account the spin(twisting and wriggling) modes of formed fission frag-ments.

(ii) At the same time, it was shown in [185] that theexperimentally observed AD anisotropy cannot bedescribed by either the SPTS or SCTS models. There-fore, it was assumed, in the general case, that the tran-sition state determining the AD of fission fragments islocated somewhere between the saddle point and thescission point.

(iii) Abnormal values of AD anisotropy for a numberof reactions can be explained by the presence of a signif-icant fraction of events, which are not fusion–fissionevents, proceeding without the formation of a compoundnucleus in the total fission cross section. To such reac-tions, we assign quasifission [88, 186–190], fast fission[188, 191, 192], and pre-equilibrium fission [193].

(2) For deformed actinide target nuclei, a sharpincrease in the AD anisotropy is experimentallyobserved at energies close to and below the Coulombbarrier (in reactions such as 16O, 19F + Th, U, Np) whenthe impinging-ion energy is decreased. No suchincrease is observed if the Bi and Pb nuclei, which arelighter and spherical in the ground state, are used as thetarget. This abnormal behavior of the AD anisotropy isassociated either with a sharp increase in the secondmoment of the total momentum distribution of nuclei

[194] (as a result of the Coulomb-barrier influence) orwith an increase in the contribution made by quasifis-sion events [195] (for head-on collisions).

Detailed study of ADs shows that the AD anisotropyis extremely sensitive to an entrance-channel asymme-try α = (At – Ap)/(At + Ap), where At and Ap are themasses of a target nucleus and impinging particle,respectively.

The value of the Businaro–Gallone parameter αBGdefines the validity range of the standard SPTS model:for α > αBG, the theoretical calculations are in accor-dance with the experimental data; for α < αBG, experi-mental methods give considerably higher values ofanisotropy than are predicted by theoretical calcula-tions with the saddle point as the transition state.

However, even in the “safe” range of α > αBG, wherethe SPTS model gives a satisfactory description of ADanisotropy, calculations executed in statistical modelswithout taking into account the dynamic features ofnuclear fission encounter serious difficulties in deter-mining the saddle configuration of a fissile nucleus(temperature and surface shape).

Primarily, these difficulties are caused by the factthat the statistical models predict a decrease in the meanprescission neutron multiplicity ⟨npre⟩ in response toincreasing excitation energy [104]. Consequently, theygive an underestimated value of ⟨npre⟩ and, therefore, anoverestimated value of the temperature Tsd of a nucleusat the saddle point at high excitation energies. In addi-tion, the fraction of neutrons evaporating after the sys-tem passes the saddle increases with the excitationenergy and mass number of a fissile nucleus. Therefore,even if the calculations in the statistical model repro-duce the experimental value of the total prescissionneutron multiplicity, the presence of neutrons emittedat the stage of evolution between the saddle and theinstant of scission results in an incorrect value of Tsd. Inthe dynamic models including an evaporation branch,no problems associated with definition of the nuclearconfiguration arise because the nuclear configuration isobtained as the solution to equations of motion. A sec-ondary factor causing difficulties in determination ofthe saddle configuration is that the multidimensionalnature of the fission barrier is disregarded in the statis-tical SPTS models when determining the transition-stateposition, which leads to seriously underestimated calcu-lated values of the AD anisotropy in the range of highexcitation energies (for more details, see Section 5.1.2).In this context, a promising approach would seem to beto actively use multidimensional dynamic models in thetheory of angular distributions of fission fragments.

In the analysis of an AD, it is usually assumed thatfission fragments are emitted in the direction of thenuclear symmetry axis. In this case, the AD is specifiedby three quantum numbers: I, K, and M, where I is thetotal momentum of a compound nucleus, K is the pro-jection of I towards the nuclear symmetry axis, andM is the projection of the total moment in the direction



of the impinging-ion beam. In the case of fusion ofspinless ions, the value of M = 0; hence, the totalmoment I coincides with the orbital momentum l. Then,the AD for fixed values of I and K has the form

(63)

where (θ) is the Wigner rotation function and θ isthe angle between the nuclear symmetry axis and theaxis of the impinging-ion beam. For large values of I,the following expression is valid:

(64)

The angular distribution of fission fragmentsobserved in experiments can be obtained by averagingEq. (63) over the quantum numbers I and K:

(65)

From Eq. (65), it can be seen that, for calculation ofADs, it is necessary to specify the type of distributions[σI] and [ρ(K)] of compound nuclei over I and K,respectively. We described the problems associatedwith calculation of the distribution σI over the totalnuclear momentum in Section 1.3. Therefore, we nowdiscuss the problem of finding the distribution ρ(K).

5.1. Transition-State Model

The transition-state model [157, 178, 179] is con-ventionally used in theoretical analysis of the data onADs of fission fragments. The essence of this modelconsists in the assumption that there is a certain chosen(transition) configuration of a fissile system that deter-mines the AD of the fission fragments. Thus, there aretwo limiting assumptions on the position of the transi-tion state and, correspondingly, two variants of the tran-sition-state model: the saddle-point transition-state(SPTS) model and the scission-point transition-state(SCTS) model.

When developing the classical SPTS model, certainkey assumptions are made: (i) The mean residence timeof a nucleus in the saddle-point region is sufficientlylong for an equilibrium distribution over K to be estab-lished at the saddle point. In other words, the time τgs ofthe motion of the system from the ground state to thesaddle point is much longer than the relaxation time ofthe K degree of freedom (τgs τK). (ii) The mean timeτss of descent of a nucleus from the saddle to the instantof scission is short in comparison with τK. In this case,the K distribution formed at the saddle point is retainedat the scission point. (iii) The type of K distributiondepends on the factor exp(–Erot /T] [179],

W θ I K, ,( ) I 1/2+π

---------------- d0 K,I θ( )

2,=

dM K,I

W θ I K, ,( ) I 1/2+

π---------------- I 1/2+( )2 θsin

2K

2–[ ]

12---–

.

W θ( ) σI ρ K( )W θ I K, ,( ).K I–=

I

∑I 0=

∞

∑=

(66)

Thus, the K equilibrium distribution is

(67)

The parameter K0 determines the width of this dis-tribution:

(68)

Here, T is the temperature of a nucleus in the transitionstate, Jeff is the effective moment of inertia, J|| and J⊥ arethe solid-state moments of inertia of a nucleus withrespect to the symmetry axis and to the axis perpendic-ular the symmetry axis, respectively.

By averaging Eqs. (63) and (64) over ρ(K), weobtain an expression for the angular distribution with afixed I and given K0:

(69)

(70)

Here, J0 is the zero-order Bessel function and p =

. Equation (70) is known as the Halp-

ern–Strutinsky equation [157, 179].If p 1, it is possible to show that the AD anisot-

ropy is given by the approximate relation

(71)

Equation (71) is pictorial and, consequently, conve-nient for revealing qualitative features in the behaviorof the AD anisotropy. Equations (69) or (70) are usedfor quantitative analysis. It is necessary to note that boththese expressions give identical probabilities W(θ, I)[186] to a good level of accuracy.

In addition to the initial spin distribution of com-pound nuclei, it is possible to select two more factorsdetermining angular-distribution anisotropy in theSCTS model: the effective nuclear moment of inertia(or the deformation) in the transition state and the

Erot

2K

2

2J ||------------

2

I2

K2

–( )2J⊥

---------------------------.+=

ρ K( )

K2

2K02

----------–exp

K2

2K02

----------–expK I–=

I

∑----------------------------------------.=

K02 T

2

-----Jeff, Jeff

J ||J⊥

J⊥ J ||–----------------.= =

W θ I,( ) = I 1/2+( )

π---------------------

d0 K,I θ( )

2K

2/2K0

2–( )exp

K I–=

K = I

∑

K2/2K0

2–( )exp

K I–=

K = I

∑-----------------------------------------------------------------------,

W θ I,( ) 2 pπ

------p θsin

2–[ ]J0 p 2θsin–[ ]exp

erf 2 p[ ]------------------------------------------------------------------.

I12---+

2

/ 4K02( )

W 0°( )W 90°( )------------------- 1

I2⟨ ⟩

4K02

----------.+

418


ADEEV et al.

temperature in this state. The following two sectionsare devoted to discussing the new features that arisewhen studying the AD of fission fragments within theLangevin approach in comparison with the statisticalmodels.

5.1.1. The effect of prescission neutron evaporationon AD anisotropy. The nuclear temperature T = Tsd inthe transition state is closely related to the prescissionneutron multiplicity and, therefore, to dissipation of thecollective energy into an internal form. The influence ofprescission neutron evaporation on the calculated frag-ment ADs was studied in [19, 196–200]. An improve-ment in the agreement between the calculated values ofthe AD anisotropy and the experimental data wasobserved in all these studies if we take into accountneutron evaporation. The typical results obtained bothwith and without inclusion of the neutron evaporation

are shown in Fig. 26. As can be seen, with an increasein the excitation energy, the prescission neutron multi-plicity increases, and, hence, the difference between thecurves obtained with and without inclusion of the neu-tron evaporation increases as well.

In [196–198], the temperature in the transition statewas determined under the assumption that all the neu-trons evaporate before a nucleus attains the saddlepoint. In [19, 199, 200], the results of one-dimensional[199, 200] and three-dimensional [19] Langevin calcu-lations were used, and it was concluded that only thepresaddle fraction of neutrons affects the transition-state temperature. An obvious advantage of thedynamic models is that they can be used to determinethe fraction of neutrons that evaporate before attainingthe saddle configuration, which is impossible in purestatistical models. The temperature Tsd of a nucleus atthe saddle point depends on the mean presaddle neutron

multiplicity , i.e., on the fraction of neutrons thatevaporate before the nucleus attains a conditional sad-dle point. The mean prescission neutron multiplicitydepends on the mean nuclear dissipation, while the

ratio /⟨npre⟩ is determined mainly by the coordi-nate dependence of the friction-tensor components.Therefore, ADs can be considered as a source of infor-mation on the magnitude and mechanism of nuclearviscosity. It should be noted that we investigated ADanisotropy as a function of a value of the one-body vis-cosity in [19]. We showed that, although the variation inviscosity affects the ADs under consideration, thischaracteristic is less sensitive to nuclear viscosity in thequantitative sense than the prescission neutron multi-plicity.

As was shown in [19, 199], the ratio /⟨npre⟩increases in response to an increase in the excitationenergy. Therefore, as can be seen from Fig. 26, theangular distributions obtained under the assumptionthat all the neutrons are evaporated before reaching thesaddle point differ appreciably from the results of cal-culations taking into account only the presaddle neu-tron fraction when determining Tsd under the effect ofincreasing excitation energy.

We note that the energy dependence of /⟨npre⟩obtained in [19] qualitatively agrees with the depen-dence obtained in [199] for the same reaction. Theresults of [199] enabled the authors to demonstrate thatquantitative agreement with the experimental data onAD anisotropy for high excitation energies is possibleonly when a significant prescission neutron fractionevaporates during the descent from the saddle point to

the scission point. The predicted ratio /⟨npre⟩ is 3–5 times higher than that obtained in [19] at the sameexcitation energies. However, good quantitative agree-ment was achieved with experimental data in both stud-ies. The disparity can be explained by the fact that dif-

npregs⟨ ⟩

npregs⟨ ⟩

npregs⟨ ⟩

npregs⟨ ⟩

npregs⟨ ⟩

*

*

*

**

**

**

4

3

2

1

W(0°)/W(90°)

208Pb(16O, f )

70 90 110 130Ec.m., MeV

, , exp

TSM(νpreexp)

TSM(νpreexp•(νgs

th / νpreth ))

TSM(νpre = 0)

Fig. 26. Comparison of experimental data on the AD anisot-ropy (closed circles, squares, and triangles) with the resultsof theoretical analysis within the SPTS model. The resultsof the calculations are obtained using various methods oftaking into account evaporation of prescission neutrons: thesolid curve is obtained without taking into account neutronevaporation [196]; the asterisks connected by the dottedcurve correspond to a situation in which all the neutronsevaporate before a nucleus reaches the saddle configuration[196]; the rectangles connected by the dashed and dottedcurve refer to the evaporation occurring at the stage ofdescent of the system from the saddle point to the scissionpoint. The figure is taken from [199].



ferent models and, in particular, a different number ofcollective coordinates were used.

5.1.2. The effect of model dimension on AD. The setof all the accessible transition states is determined bythe potential energy landscape and, hence, by the num-ber of collective coordinates. At the same time, the par-ticular ensemble of transition points strongly dependson the fission dynamics and, consequently, is sensitivevirtually to all the components of the model used: theconservative force, nuclear-viscosity mechanism,method of calculation of the mass tensor, etc. It shouldbe noted that there is only one transition state, the sad-dle point, in the one-dimensional models for each angu-lar momentum, while the entire ensemble of condi-tional saddle points forms the set of transition states inthe multidimensional models. The multidimensionaldynamic models, in comparison with the one-dimen-sional ones, take into account the multidimensionalnature of the fission barrier. This circumstance canstrongly influence the AD anisotropy predicted in mod-els with a different number of collective coordinates.

In [19], we assumed that the model dimension influ-ences the calculated AD anisotropy. In addition, thisinfluence should increase with an increase in thenuclear fissility and excitation energy. In fact, underthese conditions, the fission barrier lowers and therigidity of the potential-energy trough decreases.Hence, a greater number of transition states are allowedfor a fissile nucleus. An increase in the nuclear excita-tion energy also results in an increase in the phase-space volume accessible to a nucleus at the ridge.

In [123], we considered the reaction 16O + 232Th 248Cf in the three-dimensional Langevin model andcompared our results with the results of calculations forthe same reaction [200] obtained in the one-dimen-sional model. In [200], the AD anisotropy was calcu-lated for the reaction 16O + 232Th 248Cf and for cer-tain systems of impinging-ion–target combinationsclose to 16O + 208Pb. When calculating the AD anisot-ropy of the fragments, corrections were made to theemission of presaddle neutrons according to [196]. Theauthors of [200] obtained good agreement with theexperimental data on AD anisotropy for systems of the16O + 208Pb type. In contrast, for the heavier system of16O + 232Th, the calculated values of the anisotropydrastically differ from the experimental data. Thus, thedisagreement between theoretical calculations andexperimental data increases as the excitation energydecreases, which qualitatively agrees with the calcula-tions in [123]. Generally, the results obtained in thethree-dimensional approach are in better agreementwith the experimental data. If the excitation energyincreases, the difference between theoretical and exper-imental values decreases faster than in the one-dimen-sional calculations. In [123], this result was interpretedas being due to the effect of inclusion of several collec-tive coordinates.

In order to study the dependence of the calculatedAD of fragments on the number of collective coordi-nates included in the model in more detail, we calcu-lated the AD anisotropy in the one-dimensional andthree-dimensional Langevin models for two reactions:16O + 208Pb 224Th and 16O + 232Th 248Cf. Torule out the influence of the temperature and angular-momentum variation resulting from prescission parti-cle evaporation, we disregarded evaporation in the cal-culations. Figure 27 shows that, although the ADanisotropy almost coincides in the one-dimensional andthree-dimensional calculations at low excitation ener-gies, the three-dimensional model predicts consider-ably higher values of the AD anisotropy than the one-dimensional model as the excitation energy increases.The greatest difference is found at the highest consid-ered energy E* 150 MeV. It can be seen that the ADanisotropy obtained in the three-dimensional calcula-tions for the reaction resulting in the formation of thelighter compound nucleus 224Th at E* 150 MeV is athird higher than that obtained in the one-dimensionalcalculations. For the reaction resulting in the formationof the heavier compound nucleus 248Cf, this differencereaches almost 40%, which agrees with the assumptionthat the influence of model dimension should be stron-ger for heavier nuclei.

In order to understand why the multidimensional cal-culations give higher values of anisotropy than the one-dimensional ones, we should consider Fig. 28. As can beseen, the saddle point is located at h 0 and α = 0.A weak dependence of the effective moment of inertiaon the mass-asymmetry parameter and a strongerdependence on the parameter h are shown. In this case,the values of Jeff decrease if the parameter h deviatesfrom zero either on the positive or the negative sides ofthe figure. In the multidimensional model for calcula-tion of the AD anisotropy, averaging over the ensembleof transition points takes place, while, in the one-dimensional model, only one transition state, the saddlepoint, is realized. Therefore, the values of K0 obtainedin the three-dimensional calculations are lower on aver-age than in the one-dimensional case both due to anincrease in the effective moment of inertia and adecrease in the transition-state temperature for a devia-tion from the saddle point. Lower values of the param-eter K0 correspond to a narrower AD.

It should be noted that both the calculations in [181]and the one-dimensional calculations described aboveshow a decrease in the AD anisotropy at high excitationenergies, whereas this value is observed to increase inexperiments. In [181], it was shown that the reason forthis decrease in the anisotropy is a sharp increase in Jeff,calculated at the saddle point in the diffuse-edge LDM,if the compound-nucleus spin increases. The authors of[181] therefore concluded that fission events proceed-ing without the formation of a compound nucleusamount to a significant fraction of the total for highmomenta (which are realized at high excitation ener-gies), which results in a desirable decrease in Jeff. How-

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ever, it can be seen from the results shown above thatthe problem can be solved if all the events are consid-ered as true fusion–fission events and the multidimen-sional nature of the fission barrier is taken into account.

5.2. K-Mode Relaxation Time

In recent years, evidence has appeared indicatingthat it is necessary to take into account the dynamic fea-tures of AD formation. In the most general case, the Kmode should be considered as an independent collec-tive coordinate, and its evolution can be studied using,for example, the multidimensional Langevin approach.Such a completely dynamic approach makes it possibleto determine, in the most general form, the desired dis-tribution ρ(K). However, in this case, the problem ofcalculating the conservative force for the K mode, aswell as the inertial and friction parameters, must be

dealt with. From the known dependence of the rota-tional energy on K in Eq. (66), it is easy to determinethe desired conservative-force component. At the sametime, a method for use in calculation of the frictionparameter for the K mode has not been described in pre-vious publications. Therefore, a fully dynamic consid-eration of the evolution of the K degree of freedom isstill difficult.

However, the dynamic aspects of AD formation canbe contained within a characteristic named the K-moderelaxation time τK. In [201], it was proposed that K-mode evolution be considered using the Monte Carlomethod. In [201], compound-nucleus fission was char-acterized by two collective degrees of freedom: thenuclear elongation and K. The initial K distribution wasassumed to be uniform in the nuclear ground state. Ateach step in the process of solving the Langevin equa-tions for the elongation coordinate, there was a varia-tion of K under fulfillment the condition ξ < τ/τK, whereξ is a random number uniformly distributed in the inter-val [0, 1] and τ is the integration step of the solution tothe Langevin equations. A new value of K was sampledbased on the following distribution:

(72)

Here, ∆F is the free-energy increment resulting fromvariation in K. It is necessary to note that the methodused by the authors of [201] is reasonably general in thesense that it allows us to consider the evolution of anarbitrary finite collective mode in a similar way.

The described algorithm was applied in [201] to cal-culating the AD of the fission fragments for the systems16O + 232Th, 16O + 238U, and 16O + 248Cm at impingingoxygen ion energies ranging from 80 to 160 MeV. Froman analysis of the obtained results and experimentaldata, the authors succeeded in deriving a K-mode relax-ation time of τK = 21 × 10–21 s.

P K( ) ∆F/T( ).exp∼

E*, MeV20 40 60 80 100 120 140 160

1.2

1.6

2.0

2.4248Cf

224Th

3-dim

1-dim

3-dim

1-dim

3.6

3.2

2.8

2.4

2.0

1.6

W(0°)/W(90°)

Fig. 27. Angular-distribution anisotropy obtained in one-dimensional (circles) and three-dimensional (squares) Lan-gevin calculations for the one-body mechanism of nuclearviscosity with ks = 0.25 for 224Th and 248Cf nuclei.

1.00.80.60.40.2

0–0.2–0.4–0.6–0.8–1.0

–0.6 –0.4 –0.2 0 0.2 0.4h

α'

0.500.55

0.650.60

0.70

0.800.75

0.850.90

0.951.11.21.2

1.0

1.1

1.0

1.2 1.2 1.2

1.1

1.2

Fig. 28. Effective moment of inertia Jeff/Jsph of a nucleus inunits of the moment of inertia of an equivalent sphericalnucleus on a ridge surface. The calculations are performedin the coordinates (h, α') for a 224Th nucleus with the spinI = 20.

0.95



It is more often assumed that the K-mode relaxationtime is constant. The obtained estimates of τK lie inthe interval (5–8) × 10–21 s [104, 193, 202, 203] to 60 ×10–21 s [204]. We note that the relaxation time τK wasdetermined in [203] within the one-dimensional Lan-gevin model, while, in other studies, the calculationswere performed within statistical models.

At the same time, it follows from [205] that there isa dependence of τK on the nuclear rotation velocity; i.e.,considering the K-mode relaxation time as constant isonly an approximation. Until now, there have been nodynamic calculations performed using this result. Theresults of [205] were applied in an analysis of an ADwithin the framework of the statistical model proposedin [206]. The authors of these studies achieved reason-ably good agreement with the experimental data in theregion of near-barrier and sub-barrier fusion–fission.

In summarizing this subsection, it is necessary tonote that clarification of the role of dynamic coeffi-cients in formation of the ADs of fission fragments isstill at an early stage. The use of the multidimensionalLangevin models, which consider the K mode as anindependent collective degree of freedom, seems to bepromising. In this case, during the calculation of an AD,it is extremely important to consider the reaction fromthe instant of contact of colliding ions.

CONCLUSIONS

In recent years, a stochastic approach based on mul-tidimensional Langevin equations has been widelyused for describing the dynamics of induced fission andheavy-ion fusion. In this review, we showed potential ofthis approach with regard to description of the largenumber of different quantities observed in fusion–fis-sion reactions. In our opinion, the results obtained inrecent years with the use of the multidimensional sto-chastic approach are impressive. In particular, calcula-tions in the three-dimensional stochastic approach nat-urally reproduce the large variances and significantincrease observed in the mass–energy distribution offragments with the compound-nucleus fissility parame-ter. These characteristic pronounced features of the dis-tribution cannot be explained within the framework ofthe fission models conventionally used for analysis offragment distributions and the previously favoured two-dimensional Langevin calculations. In addition, withinthe developed multidimensional stochastic approach, itis possible to satisfactorily describe quantitatively finecorrelation characteristics of the mass–energy distribu-tion of fragments and the correlation of evaporated neu-trons with the mass and kinetic energy of fragments. Aconsistent description of the correlation dependencesfor the prefission, postscission, and total neutron multi-plicities makes it possible to obtain information on thetime taken at all the stages of the fission process fromcompound-nucleus formation to the deexcitation offragments.

Now, there is evidently no doubt that the stochasticapproach based on multidimensional Langevin equa-tions in combination with the evaporation of light par-ticles and gamma quanta provides the most adequatedynamic description of the investigated fusion–fissionreactions. At the same time, considerably poorer agree-ment is observed among researchers with respect to thecomponents representing the physical basis of the sto-chastic approach. As before, it primarily concerns thechoice of the dependence of the friction parameter (thetensor) on the collective coordinates and (or) tempera-tures.

In summary, it would also be desirable to emphasizethat the regular calculations of the mass–energy distri-bution of fragments and prescission neutron multiplici-ties performed within the three-dimensional stochasticapproach have made it possible to unambiguouslychoose, after much discussion, which nuclear-viscositymechanism (two-body or one-body) is realized duringnuclear fission. A simultaneous description of theparameters of the mass–energy distribution of frag-ments and the mean prescission neutron multiplicitiesis attained for the one-body mechanism of viscosity inits modified variant with the reduction coefficient ks =0.25–0.5 for the contribution from the wall formula. Atthe same time, it is necessary to mention that, for moresubstantiated conclusions concerning both the defor-mation and the temperature dependences of nuclearviscosity, more realistic variants have to be used to sta-tistically describe particle emission in dynamic calcula-tions, particularly in relation to the deformation depen-dence of the particle-emission barriers and the bindingenergy. As an example, we note the study of Pomorskiet al. [21, 207] in which the influence of nuclear defor-mation was shown for a partial nuclear-decay width. Inaddition, as supplementary observed quantities, exper-imental data on the emission of charged particles can beemployed for deeper analysis of nuclear dissipativeproperties.

The attenuation of the dissipation relative to thewall-formula predictions for a fissile nucleus with ahighly symmetric configuration (such as a sphere orweakly deformed ellipsoid) is now understood wellboth from the use quantum-mechanical calculationsand from the ideas of chaos theory. However, the abnor-mally sharp increase in the dissipation for stronglydeformed configurations close to scission, which wasintroduced in the calculations of Fröbrich and Gont-char, has yet to receive a proper theoretical explanationdespite the good agreement of their calculations withthe data from numerous experimental studies. There-fore, the character of the mechanism of dissipation real-ized in fission requires both further theoretical and fur-ther experimental study. Finally, in dynamic simulationof a fusion–fission reaction, the calculations shouldbegin from the point of contact between fusing ionsinstead of an arbitrary initial configuration correspond-ing to a statistically equilibrium compound nucleus.The first of such calculations have now been performed.

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ACKNOWLEDGMENTS

We are grateful to A.Ya. Rusanov for his part in numer-ous interesting discussions and useful consultations on theexperimental data, which formed the basis of this review,throughout the entire period in which the review was beingformulated. We also thank H.J. Krappe, P. Fröbrich,K.Kh. Schmidt, M.G. Itkis, V.A. Rubchenya, A.S. Il’inov,G.I. Kosenko, and M.V. Mebel’ for discussing certainchapters of the review, their constructive suggestions,and their attention to this paper.

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