Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 43–49, 2002

43

Nuclear Shapes in Complex Fission Phenomena
D. N. Poenaru,∗,a,b,c W. Greiner,b Y. Nagame,c and R. A. Gherghescua,baNational Institute of Physics & Nuclear Engineering, PO Box MG-6, RO-76900 Bucharest-Magurele,
Romaniab
Institut für Theoretische Physik der J. W. Goethe Universität, Pf 111932, D-60054 Frankfurt am Main,
GermanycJapan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan
Received: November 13, 2001; In Final Form: November 13, 2001
We present a method for finding very general reflection asymmetric equilibrium (saddle-point) nuclear shapes, assolutions of an integro-differential equation, without giving a shape parametrization. By introducing phenomenological shell corrections one obtains minima of deformation energy for binary fission of parent nuclei 238 U, 232,228 Th
at a non-zero mass asymmetry, leading to the same mass number of the heavy fragment A1 = 125. Applications forfission into more than two and three fragments are also illustrated.

1. Introduction
The permanent distortion from a sphere of a given nucleus
in its ground state or the change of deformation during a nuclear process (e.g. fission or fusion) may be described in termsof collective coordinates. One has to choose a function witha number of parameters as low as possible, which in the sametime allows to determine the most important shapes of the nuclear surface. There are many such parametrizations describedin the literature (see various chapters of the books1, 2 and references therein). The surface equation, determined by a set of
deformation coordinates, is frequently used to calculate the potential energy surfaces (PES) which in turn can be applied to findthe nuclear deformations and fission barriers, to explain shapeisomers, to obtain indications about mass asymmetry in fission,
to calculate half-lives against various decay modes or to study
multidimensional tunneling,3 to extend the nuclear chart in theregion of superheavy nuclei, etc.
Since the early days of development of nuclear fission, theshapes during the deformation process from one parent nucleus
to the final fragments, have been intensively studied either statically or dynamically.6, 7 Within a liquid drop model (LDM) all
nuclear shapes in the ground-state are spherical and the fission
fragment mass distributions are symmetric, in contrast to reality. Permanent nuclear deformations and fission fragment mass
asymmetry can be explained by combining the collective (liquid drop-like) and single particle properties in the frameworkof macroscopic-microscopic Strutinsky’s method. By using thetwo center shell model8 to describe the single-particle states,
one can follow the shell structure all the way from the original nucleus, over the potential barriers, up to the final stage ofseparated fragments. Particularly important points on a potential energy surface are those corresponding to the ground-state,9saddle-point(s),10, 11 and scission point.12, 13
The progress in understanding the connection between scission configurations and the binary fission mechanisms and mass
yield was recently reviewed.14 A systematic analysis of a large
body of experimental data (see also Reference 15) on total kinetic energy (TKE) of fission fragments confirmed the existenceof two fission modes in the region of actinides: one with low
TKE components and elongated scission shapes leading to asymmetric mass yield, and the other one with high TKE components and compact shapes connected to a shell-influenced asymmetric yield. Two kinds of fission paths were assumed on thisbasis: with lower barrier going through reflection asymmetric compact configurations, and with higher barrier and sym∗Corresponding author. E-mail: poenaru@ifin.nipne.ro. FAX: +401-457-4440.

metric scission configurations. The invariance of scission deformation β was demonstrated16 and new formulas replacing
Viola systematics were given. The deformation was definedby β = D(A1 , A2 )/D0 (A1 , A2 ) where D = Z1 Z2 e2 /T KE(A1 , A2 )is the distance between charge centers of the fragments and D0
is the distance between centers of two touching spherical fragfm. For the first time it was experments with radii Ri = 1.17A1/3
iimentally found17 that scissioning nuclei with a given elongation
lead to a given shape of the fragment mass yield curve. By including into analysis the region of heaviest nuclei with measured
TKE (heavy Fm, Bk, No, Cf, Lr, and Rf isotopes) the existenceof three types of shape elongations was clearly shown: LDM
governed mass symmetric elongated shape (β  1.65); mass
asymmetric deformation (β  1.53), and shell-influenced mass
symmetric deformation (β  1.43).
Important achievements have been also reached in anotherexperimental field including rare phenomena, where new characteristics of fission process18 and new decay modes19 (α and
10
Be accompanied cold fission of 252 Cf, double fine structure
and triple fine structure in binary and ternary fission, respectively) were discovered during the last years by using triple-γ coincidences in a large array of Germanium Compton-suppressed
detectors (the famous GAMMASPHERE). The increased sensitivity of such measurements could allow to detect in the futureother rare phenomena, as multicluster accompanied fission20 andto continue to search for new kinds of quasimolecular states.21
In a statical approach, the equilibrium nuclear shapes are usually obtained by minimizing the energy functional on a certainclass of trial functions representing the surface equation. The
simplest shape one can imagine, still allowing to describe essential features of a cluster decay process is the spherical one.2
Even for a particle-accompanied fission one can imagine21 asingle-deformation parametrization based on spherical shapes.
The purpose of this paper is to present a method allowing toobtain a very general reflection asymmetric saddle-point shape
as a solution of an integro-differential equation without a shapeparametrization apriori introduced. This equation was derived
by minimizing the potential energy with constraints (constantvolume and given deformation parameter). The method allows
to obtain straightforwardly the axially-symmetric surface shapefor which the liquid drop energy, ELDM = Es + EC , is minimum.
By adding the shell corrections δE to the LDM deformation energy, Edef = ELDM + δE, we succeeded to obtain minima at a
finite value of the mass-asymmetry parameter. We use a phenomenological shell correction. Results for binary fission ofparent nuclei 238 U and 232,228 Th are presented. The minima of
the saddle-point energy occur at nonzero mass asymmetry parameters which happens to lead to the same mass number of thec 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 6/30/2002

44

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

heavy fragment, A1 = 125, for all fissioning nuclei mentioned
above. Applications for fission into more than two and threefragments are also illustrated.
2. Saddle PointLet us consider a nuclear system with a shape specified bya set of n generalized coordinates {qi }. For an equilibrium(ground-state or saddle-point) shape22 the deformation energy
E = E(q1 , q2 , ..., qn ) has an extremum, defined by ∂E/∂qi = 0,(i = 1, 2, ..., n). In a LDM the ground state, characterized by thelowest minimum of the potential energy, always corresponds toa spherical shape. One may define a fission valley on the PES ina multidimensional space of deformation parameters, as a conditional minimum ∂E/∂qk = 0, (i=1, 2, ..., k-1, k+1, ..., n) withthe constraint qk = q0k for different values q0k . The maximum
value on this minimum energy determines the saddle-point position, at which all eigenvalues of the symmetric curvature matrix
Ki j = ∂2 E(q)/(∂qi ∂q j ) have a positive sign, except one. For two
deformation coordinates23 at the saddle point
∂2 E 
 ∂2 E ∂q21∂q1 ∂q2  < 0 .
(1)




∂2 E 
 ∂2 E∂q2 ∂q2∂q22
The simplest LDM potential energy vs. one deformation parameter along the fission path is a smooth curve with a minimum at the ground state and a maximum at the saddle point.
Then it decreases continuously going through the scission point,
down to the self energy of the fragments at infinite separation
distance. By representing this quantity for a heavy nucleus vs.
the mass asymmetry coordinate in a transverse direction at thesaddle point we also get a smooth curve with a minimum between two maxima, called Businaro-Gallone mountains. The
saddle point is lying on the bottom of the valley separating thesemountains.
A conditional saddle point is defined by imposing one or several constraints. Of particular interest in fission is the constraintof determined mass asymmetry, η = η0 . If η is one of the generalized coordinates the problem is simplified because one takes
η = constant. In general η depends on q, and the generalizationof the variational equation to the equilibrium with constraints
leads to∂η∂E
+λ= 0, i = 1, 2, ..., n
(2)
∂qi
∂qi
where λ is a Lagrange multiplier.
There is no need to consider any reflection asymmetry or nonaxiality in the calculation of saddle point shapes within LDM,
because the energy increases in the presence of both kinds ofdeviations from symmetry. The parametrization of Legendre
polynomial expansion with even-order deformation parameters
α2n up to n = 18 was used.4 For low fissility, X, the saddle point
shapes are very similar to two tangent spheres, hence it is moredifficult to be described with a small number of deformation
coordinates. By increasing fissility a neck develops between thetwo symmetrical fragments. The length increases up to X = 0.67
and in the same time the neck radius becomes larger. The best
accuracy was obtained at larger fissilities, close to X = 1, forwhich the saddle point shapes are not very different from a single sphere.

Poenaru
spect to spherical shape (superscript 0 ), the deformation energy
is defined as
Edef (α) − E 0 = (Es − Es0 ) + (EC − EC0 ) + δE − δE 0
= Es0 [Bs − 1 + 2X(BC − 1)] + δE − δE 0

where Es0 = as (1 − κI 2 )A2/3 and EC0 = aC Z 2 A−1/3 are energies
corresponding to spherical shape and I = (N − Z)/A. The relative surface and Coulomb energies Bs = Es /Es0 , BC = EC /EC0
and the shell correction δE(α) are functions of the nuclear
shape. The dependence on the neutron and proton numbers
is contained in Es0 , the fissility parameter X = EC0 /(2Es0 ) =[3Z 2 e2 /(5R0 )]/2[as (1 − κI 2 )A2/3 ], as well as in shell correctionof the spherical nucleus δE 0 . From a fit to experimental data on
nuclear masses, quadrupole moments, and fission barriers, thefollowing values of the parameters have been obtained:24 as =17.9439 MeV, κ = 1.7826, aC = 3e2 /(5r0 ) = 0.7053 MeV. The
radius of spherical nucleus is R0 = r0 A1/3 with r0 = 1.2249 fm,
and e2 = 1.44 MeV·fm is the square of electron charge. Calculation of δE will be outlined below.
The corresponding variational problem can be reduced to thefollowing equationρρ − ρ2 − [λ1 + λ2 |z| + 10XVs (z, ρ)]ρ(1 + ρ2 )3/2 − 1 = 0 (4)where ρ = dρ/dz, ρ = d 2 ρ/dz2 , and Vs is the Coulomb potential on the nuclear surface. We used the following relationships
for the principal radii of curvature R1 = τρ, R2 −1 = −ρ /τ3 ,in which τ2 = 1 + ρ2 . In eq 4 λ1 and λ2 are Lagrange multipliers corresponding to the constraints of volume conservation(or given mass asymmetry if the volume is conserved in each“half” of the nucleus) and determined value of deformation parameter α, defined as the distance between centers of mass ofthe fragments lying at the left hand side and right hand side of
the plane z = 0, respectively: α = |zcL | + |zcR |. This definition allows to reach all intermediate stages of deformation from one
parent nucleus to two fragments by a continuous variation of its
value. The position of separation plane, z = 0, is given by thecondition: (dρ/dz)z=0 = 0. Lengths are given in units of R0 ,Coulomb potential in units of Ze/R0 , and energy in units of thesurface energy Es0 . One can calculate for every value of α thedeformation energy Edef (α). The particular value αs for which
dEdef (αs )/dα = 0 corresponds to the extremum, i.e. the shapefunction describes the saddle point (or the ground state), andthe unconditional extremum of the energy is the fission barrier.
It can also be obtained by taking λ2 = 0 in eq 4. The other surfaces (for α = αs ) are extrema only with condition α = constant.
In this way one can compute the deformation energy function of
two variables: elongation and mass asymmetry.
The Coulomb potential on the surface depends on the function ρ(z), hence eq 4 is an integro-differential one, as Vs is expressed by an integral on the nuclear volume. One has indeed,
for the general case of non-axial symmetry in cylindrical coordinates (ρ, ϕ, z)Z
Z 2π 
ρe z2
z ∂ρ2
EC =
dz
ρ2 −
V dϕ
(5)
5 z1
2 ∂z
0
where ρ = ρ(z, ϕ) is the nuclear surface equation, z1 and z2 are
the intersections of the surface with Oz axis, and V is the electrostatic potential on the nuclear surface.
For axial-symmetry the shape-dependent, dimensionless surface term is proportional to the surface area:

3. The Model
We are looking for a nuclear surface equation with axial symmetry around the z axis, expressed as ρ = ρ(z) in cylindrical coordinates, which minimizes the potential energy of deformation
with two constraints: volume conservation, and given deformation parameter, α, assumed to be an adiabatic variable. With re-

(3)

Es
d2
Bs = 0 =
2
Es

Z+1
−1

1
y +
4
2



dy2
dx

2 1/2dx(6)where y = y(x) is the surface equation in cylindrical coordinates with −1, +1 intercepts on the symmetry axis, and d =Nuclear Shapes in Complex Fission Phenomena

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002(z2 − z1 )/2R0 is the seminuclear length in units of R0 . Similarly,
for the Coulomb energy25 one has

Z+1 Z+1
5d
dx dx F(x, x ) ,
Bc =
8π
5

−1

(7)

−1

F(x, x ) = a−1
ρ {yy1 [(K − 2D)/3]
 2


3
dy1 dy222
 2
× 2(y + y1 ) − (x − x ) + (x − x )
−
+K
2
dx
dx





x − x dy2
x − x dy21
× y2 y21 /3 + y2 −} . (8)
y21 −
2 dx
2 dx


K, K are the complete elliptic integrals of the 1st and 2nd kindK(k) =

Zπ/2
Zπ/2(1 − k2 sin2t)−1/2 dt; K  (k) = (1 − k2 sin2 t)1/2 dt
0

0

(9)
and a2ρ = (y + y1 )2 + (x − x )2 , k2 = 4yy1 /a2ρ , D = (K − K  )/k2 .
We add the shell correction δE(α) to the LDM energy. The
phenomenological model26 is adapted after Reference 24. At agiven deformation we calculate the volumes of fragments andthe corresponding numbers of nucleons Zi (α), Ni (α) (i = 1, 2),
proportional to the volume of each fragment. Then we add foreach fragment the contribution of protons and neutrons
δE(α) = ∑ δEi (α) = ∑ [ δE pi (α) + δEni (α) ]
i

(10)

i

given by
δE pi = Cs(Zi);
whereδEni = Cs(Ni )s(Z) = F(Z)/Z −2/3 − cZ 1/3

and similar equation for s(N).

 5/3
5/3
3 Ni − Ni−1
5/3
5/3
(n − Ni−1) − n + Ni−1
F(n) =
5 Ni − Ni−1

(11)
(12)

(13)

where n ∈ (Ni−1 , Ni ) is the current number of protons (Z) or neutrons (N) and Ni−1 , Ni are the nearest magic numbers. The parameters c = 0.2, C = 6.2 MeV were determined by fit to experimental masses and deformations. The dependence on deformation α is given by Reference 27Li (α)C
δE(α) =[ s(Ni) + s(Zi ) ]
(14)
2 ∑
Ri
i
where Li (α) are the lengths of fragments along the symmetryaxis. During the deformation process, the variation of separation distance between centers, α, induces the variation of thegeometrical quantities and of the corresponding nucleon numbers. Each time a proton or neutron number reaches a magicvalue, the correction energy passes through a minimum, and it
has a maximum at midshell.
The integration method used to solve eq 4 is based on the
weak dependence of Coulomb energy on the nuclear shape. It is
invariant under subtraction from Vs of a linear function because
λ1 and λ2 are arbitrary constants. The extremal surface depends
on the quantity with which the Coulomb potential on the nuclear
surface differs from the function λ1 + λ2 |z|, where the constants
λ1 , λ2 could be chosen in a way to minimize this difference. In
the next iteration one uses the solution ρ(z) previously determined.
The following boundary conditions have to be fulfilledρ(z1 ) = ρ(z2 ) = 0 ,(15)

lim dρ(z)/dz = ∞ ;z→z1

lim dρ(z)/dz = −∞

z→z2

45
(16)

where z1 and z2 are the intercepts with z axis at the two tips.
Equation 16 are called transversality conditions. For reflection
symmetric shapes z1 = −z2 = −z p , hence one can consider only
positive values of z in the range (0, z p ). In order to get rid of
singularities in eq 16 it is convenient to introduce a new function
u(v) substituting ρ(z)u(v) = A2 ρ2 (z(v))(17)

z(v) = z p − v/A

(18)

where


dz/dv
= −1/A, u = du/dv = 2A ρ(dρ/dz)dz/dv = −2Aρρ ,√ρ = u/A, u2 = 4uρ2 , 1 + ρ2 = u2 /(4u) + 1, u = d 2 u/dv2 =d(u )/dv = −2A[dρ(z(v))/dv]dρ/dz − 2Aρd(ρ(z(v))/dz =−2A(dρ/dz)(dz/dv)(dρ/dz) − 2Aρ(d 2 ρ/dz2 )dz/dv = 2ρ2 +2ρρ . By substituting into eq 4 one has

5XVs λ1 + λ2 z p
1 2

+
u −2−
u +
u
2A
4A

λ2 v
− 2 (4u + u2 )3/2 = 0.
(19)
4A
2

Now we can introduce a linear function of v by adding andsubtracting a + bv to 5XVs /2A and define Vsd as deviation of
Coulomb potential at the nuclear surface from a linear function
of v
5X
Vsd =
Vs − a − vb
(20)
2A
in which the linear term may be considered an external potentialof deformation
5Xa=Vs (v = 0) ,
(21)
2A


5X
b=
Vs (v = v p ) − a /v p .
(22)
2AConsequently one has
λ1 + λ2 z p
1 2+au − 2 −
u +
u
4A



λ2
+v b −+V(4u + u2 )3/2
sd
4A2

= 0.

(23)

As one can see from eq 23, there are new constants A, z p related
to eq 17, besides the previous ones λ1 and λ2 . Nevertheless thesolution of eq 23 is not dependent on each parameter; important
are the linear coefficients in v of the binomial term within parentheses. By equating with 1 the coefficient of v, one can establish
the following link between parameter A and the Lagrange multiplier λ2
A2 = λ2 /4(b − 1) .
(24)
In this way u(v) is to be determined by equation1u = 2 + [u2 + (v − d +Vsd )(4u + u2 )3/2 ]u(25)

containing a single parameter d. From eq 15, 17 one can deduceat the limit(26)u(0) = 0,u (0) = 1/dand eq 16 is satisfied if z p = v p /A is obtained fromu (v pn ) = 0 .

(27)

The subscript n was introduced as a consequence of the fact thatthe number of points v pn (depending on d and other parameters),
satisfying eq 27 is larger than unity. In other words the function u(v) has the multiplicity of extremes and the subscript n46J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

Poenaru

counts the number of extremes. To various values of n correspond branches of solution of eq 25, describing different classesof shapes. Thus for n = 1 there is no neck, for n = 2 there is one
neck, n = 3 gives two necks, etc. In order to solve eq 25 one
starts with given values of parameters d and n. For reflection
symmetric shapes dL = dR and nL = nR . Although parameter A
is not present in eq 25 we have to know it in order to obtain the
shape function 17. From the volume conservation one has
A=

 Z v
3 pn
u(v)dv
2 0

1/3

.

Z0

Z vp
3
ρ2 (z)dz = A−4(v p − v)u(v)dv
2
z1
0
z1
(29)
depends on d. From α(d), one can obtain the inverse functiond = d(α).
For reflection asymmetrical shapes we need to introduce another constraint: the asymmetry parameter, η, defined by|z|ρ2 (z)dz

Z z2

η=

ML − MRA1 − A2=
ML + MR
A1 + A2(30)

should remain constant during variation of the shape function
ρ(z). Consequently eq 25 should be written differently for left
hand side and right hand side. Now dL is different from dR , andso are the parameters AL and AR . They have to fulfil matchingconditions ρL (z = 0) = ρR (z = 0) hence
1/2
u1/2
L (v p )/AL = uR (v p )/AR .

(31)

The similar condition for derivatives ρ (z) in z = 0, ρL (z = 0) =ρR (z = 0) = 0, is automatically satisfied due to eq 27. The second derivative ρ (z) can have a discontinuity in z = 0 if dL = dR .
The parameters AL and AR are easily expressed in terms of η, if
we write eq 30 as
Z vp
2π
ML =
uL (v)dv ,(32)(1 + η) = πA−3
L
3
0
Z vp
2π
uR (v)dv .
(33)
MR =
(1 − η) = πA−3R
30
We assume that ML + MR is equal to the mass of a sphere with
R = 1. From eq 32, 33 we obtain

AR = (1 − η)

(34)

AR0 ,

(35)

where we introduced notations similar to eq 28
 Zv
3 p
AL0(R0) =u (v)dv
2 0 L(R)dL − dR
0.04
0.06
0.08

η
0.04988
0.075170.09512A1
124.93124.72124.84condition of continuity of the second derivative ρ (z) in z = 0,
which (using eq 25) may be written as
1/2(dL − v pL )u1/2
L (v pL ) = (dR − v pR )uR (v pR ) .

(37)

This equation establishes the relationship between dL and dR .
Unlike the first approach in which dL and dR are free parameters
determining α and η, this time we have a class of shapes with
only one parameter; at a given deformation, the mass asymmetry is perfectly determined. This is a drawback when we intend to study the variation of deformation energy with α and η.
Moreover, these kind of shapes have a low chance to be met inpractice because of higher value of the energy compared to symmetrical shapes with the same parameter d. Generally speaking
the solutions of the variational problem are obtained for independent values (not related by eq 37) of parameters nL = nR , dL
and dR . It should be noted that the energies of the corresponding
shapes will be higher than those of shapes with the same valuesof dL and dR but with nL = nR .
In order to solve the eq 25 we employ the method of successive approximations. In the first iteration one obtains thesolution of differential equation in which Coulomb potential
at the nuclear surface is assumed to be a linear function of z
(or v), i.e. if one assumes initially Vs = 0. After solving theequation in such a manner, one calculates the parameters A,
a, b, which depend on the Coulomb potential and its deviation
Vsd from a linear function. The quantity Vsd obtained in sucha way is introduced in eq 25 and the whole procedure is repeated until the desired accuracy is reached. In every iteration
the equation is solved with the 2nd order Runge-Kutta methodwith constant integration step. The initial value u (v = 0) can befound straightforwardly from eq 25 by removing the indetermination in the point v = 0, u (0) = −2 + (1 − b + g)/2d 2 , whereg = [5X/(2A)][dVs (v)/dv]v=0 . The differential equation is integrated up to the point v = v pn , in which the first derivative u (v pn )vanishes. The number n of extremal values of u(v) (equal to thenumber of necks plus one unit) is an external parameter. The
value v pn was determined by linear interpolation between two
neighbouring points in which u has opposite signs.
4. Results

AL = (1 + η)−1/3 AL0 ,−1/3

Nuclei
238
U
232
Th
228
Th

(28)

After solving the integro-differential equation one can findthe deformation parameter α = zcL + zcR , wherezcL =TABLE 1: Mass number of the heavy fragment corresponding to
the minimum of the saddle-point energies in Figure 4.

1/3

.

(36)

The shape of a nucleus with given mass asymmetry, η, is completely determined by eq 34–36; the quantities uL (v p ) and uR (v p )are obtained as solutions of the differential eq 25. Again, the deformation α is the distance between centers of mass of the left
hand side and right hand side.
There is an alternative way to obtain asymmetric shapes.
When we previously discussed the energy minimization withthe constraint of deformation conservation, we observed that besides symmetrical solutions for which dL = dR and nL = nR (nL(R)is the number of solutions of eq 27 for the left hand side and righthand side), there are also solutions with nL = nR , meaning shapeswith different number of necks on the two sides separated by theplane z = 0. The parameters dL and dR are linked through the

In Figure 1 we present reflection symmetric nuclear shapes
for binary fission of a nucleus with the fissility parameter X =0.6 (e.g. 170 Yb), obtained for nL = nR = 2 (one neck), dL = dR =1.4, 1.5, 1.7, and 1.91 (for which α = 1.314, 1.644, 2.100, and2.304) and a vanishing mass asymmetry η = 0. The saddle point(maximum value of the conditioned deformation energy minimum) is obtained for dL = 1.91, at which the shape is deformed
and necked-in. The deformation, α (distance between the masscenters of fragments in units of radius of the spherical parent
nucleus, R0 ), was given above.
A comparison between three nuclear shapes at the saddle
point for nuclei with fissilities X = 0.60, 0.70, and 0.82 (corresponding to 170 Yb, 204 Pb, and 252 Cf nuclei lying on the lineof β stability) is presented in Figure 2. One can see how thenecking-in and the elongation are decreasing (α = 2.304, 1.822,
and 1.165) when fissility increases from X = 0.60 to X = 0.82,
in agreement with Reference 4. In the limit X = 1 the saddle
point shape is spherical.
Within LDM a nonzero mass asymmetry parameter (see theshapes from Figure 3) leads to a deformation energy which increases with η as is illustrated in Figure 4, where η is replaced

Nuclear Shapes in Complex Fission Phenomena

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

47

Figure 1. Reflection symmetric nuclear shapes obtained by solving
the integro-differential equation for the following values of parameters:
nL = nR = 2; dL = dR = 1.4, 1.5, 1.7, and 1.91 (for which the deformations in units of R0 are α/R0 = 1.314, 1.644, 2.100, and 2.304) at η = 0.
Binary fission of a nucleus with the fissility X = 0.60 (e.g. 170 Yb).

Figure 3. Nuclear shapes (one reflection symmetric and three reflection asymmetric) obtained by solving the integro-differential equation
for the following values of parameters: nL = nR = 2; dL = 1.40, 1.45,
1.50, and 1.60 keeping dR = 1.40 and binary fissility X = 0.60. The
mass asymmetry η was automatically changed in such a way.

by an almost linear dependent quantity (dL − dR ). The reflection asymmetric shapes plotted in Figure 3, resulted by choosing
the input parameters as follows: nL = nR = 2; dL = 1.40, 1.45,
1.50, and 1.60 while dR = 1.40 was kept constant, and so was
X = 0.60. The increasing deformation energy with mass asymmetry in Figure 4, refers to different values of fissilities, namely
X = 0.769, X = 0.754, and X = 0.758 for 238 U, 232 Th, and 228 Th,
respectively. By adding the shell corrections δE to the LDM
deformation energy, Edef = ELDM + δE, we succeeded to obtainthe minima shown in Figure 4 at a dL − dR of about 0.04,0.06,
and 0.08 for 238 U, 232 Th, and 228 Th nuclei. Interestingly enough,the mass number of the heavy fragment which corresponds tothese minima is the same, namely almost A1 = 125 (more exactly 124.93, 124.72, and 124.84). One has indeed the resultspresented in Table 1.
On the other hand, for experimentally determined mass asymmetry28, 29 the maximum of the fission fragment mass distributions is centered on A1 = 140 in a broad range of mass numbersof parent nuclei. Qualitatively we obtain for the binary cold
fission a similar behaviour. Nevertheless, the numerical discrepancy (125 instead of 140) remains to be explained.
The next result presented in Figure 5 refers to asymmetricalshapes with multiple necks which are obtained for nL = 4 andnR = 2. At a given value of dL = 2.39, 2.44, and 2.47 one has

dR = 1.40, 1.43, and 1.46. The corresponding deformation parameters in units of R0 are 1.840, 1.977, and 2.060, respectively.
The total deformation energy in units of Es0 increases from 0.135
to 0.148 and 0.152.
The elongated shapes for ternary and quaternary fissions
are shown in Figures 6 and 7. For shapes with three fragments and two necks (nL = nR = 3, see Figure 6), by increasing dL = dR , from 2.25 to 2.80 and 7.00 the deformation increases from 1.650 to 2.306 and 2.730. In the same time theelongation is initially increased from 5.234 to 5.392 and thendecreased to 5.24; the fragment radii are 0.461/0.814/0.461,
0.592/0.753/0.592, and 0.673/0.659/0.673, leading to decreasing energies in units of Es0 from 0.165 to 0.150 and 0.134.
The last configuration with E/Es0 = 0.134 is not far from a “trueternary-fission” in which the three fragments are almost iden56
56
58
tical: 170
70 Yb → 23 V + 23 V + 24 Cr and the Q value is 83.639 MeV.
One may compare the above E/Es0 value with the touching-point
energy of these spherical fragments (Et − Q)/Es0 = 0.239. It islarger, as expected, because of the finite neck of the shapes in
Figure 6. For α-accompanied fission of 170 Yb with two 83
34 Se
fragments Q = 87.484 MeV is larger and the touching point energy (Et − Q)/Es0 = 0.103 is lower. A lower Q = 70.859 MeV
and higher energy barrier (Et − Q)/Es0 = 0.147 is obtained for10
Be accompanied fission of 170 Yb with 80
33 As fission fragments.
One should stress that we present shapes in Figures 1, 3, 5–7
which are produced for various values of input parameters; only
one of these shapes in every figure corresponds to the saddlepoint. One should not be confused about the unexpected shapewith dL = dR = 2.25 in Figure 6 having a large fragment between two smaller ones; it was produced due to the low value of
dL = dR .

Figure 2. Saddle-point nuclear shapes obtained by solving the integrodifferential equation for the following values of parameters: nL = nR =2; dL = dR = 1.91, 1.55, and 1.38 and the binary fission fissilities X =0.60, 0.70, and 0.82 (for which the deformation α/R0 = 2.304, 1.822,
and 1.165) at η = 0.

Figure 4. Saddle-point deformation energy versus mass asymmetry
parameter for binary fission of 238 U and 232,228 Th nuclei. One can see amonotonous increase within a pure liquid drop model and minima when
the shell corrections are included.

48

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

Poenaru
ceeded to obtain the minima shown in Figure 4 at a finite valueof mass asymmetry for the binary fission of 238 U, 232 Th, and228
Th nuclei. Moreover, the mass number of the heavy fragment
which corresponds to these minima is the same A1  125. A
discrepancy (if any because our result was obtained in the absence of dissipation, hence it refers to cold fission) remains to
be explained in the future.

Figure 5. An alternative way to generate asymmetrical shapes in whichnL = nR illustrated for nL = 4 and nR = 2. At a given value of dL = 2.39,
2.44, and 2.47 one obtains dR = 1.40, 1.43, and 1.46. The binary fissility
X = 0.60.

Acknowledgement. This work was partly supported by theCenter of Excellence IDRANAP under contract ICA1-CT-200070023 with European Commission, Brussels, by Bundesministerium für Bildung und Forschung (BMBF), Bonn, Gesellschaft
für Schwerionenforschung (GSI), Darmstadt, by Japanese Society for the Promotion of Science (JSPS), Tokyo, by the Japan
Atomic Energy Research Institute (JAERI), Tokai, and by the
Ministry of Education and Research, Bucharest. One of us(DNP) would like to acknowledge the hospitality and financial
support received during his research stage in Frankfurt am Main
and Tokai, when the present work was completed.
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Figure 6. Nuclear shapes obtained by solving the integro-differential
equation for nL = nR = 3 and dL = dR = 2.25, 2.80, and 7.00. The binary
fissility X = 0.60.

Figure 7. Nuclear shapes obtained by solving the integro-differential
equation for nL = nR = 4 and dL = dR = 2.30, 2.70, and 4.00. The binary
fissility X = 0.60.

The shapes with four fragments and three necks (nL =nR = 4) can be seen in Figure 7. When dL = dR increases
from 2.30 to 2.70 and 4.00 the deformation takes the values
2.144, 3.136, and 3.233 and the elongations are 6.077, 6.916,
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42
42
43
43
70 Yb → 17 Cl + 17 Cl + 18 Ar + 18 Ar. Again the configuration withaligned spherical fragments in touch is higher in energy:(Et − Q)/Es0 = 0.324. Even more complex shapes can be obtained by further increasing the values of nL = nR .
5. Conclusions
The method of finding the most general axially-symmetricshape at the saddle point without introducing apriori aparametrization (inherently limited due to the finite numberof deformation coordinates), by solving an integro-differential
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