- Email: [email protected]
Fission modes in the compound nucleus 238Np
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A 594 (1995) 45-56
Fission modes in the compound nucleus 238Np P. S i e g l e r a'I, E - J . H a m b s c h a, S. O b e r s t e d t c, J.P. T h e o b a l d b a JRC-Institute for Reference Materials and Measurements IRMM, Retieseweg, B-2440 Geel, Belgium b Technische Hochschule Darmstadt, D-64289 Darmstadt, Germany c SCK, CEN, Boeretang 200, B-2400 MoL Belgium Received 7 March 1995; revised 19 July 1995
Abstract
Potential energy calculations in conjunction with the multi-modal random neck-rupture model of Brosa, Grogmann and Milller (BGM-model) indicate the existence of four fission modes in the compound nucleus 23SNp. For the first time the splitting of the standard mode into three submodes could be demonstrated theoretically. The existence of a third standard mode in fission has already been proposed for heavier nuclei, where a normegligible yield of more asymmetric fission fragments is experimentally observed. The calculated fission fragment properties for 23SNp are discussed and compared to experimental results.
1. Introduction
The idea of multi-modal fission was announced already more than 40 years ago. In 1951, Turkevich and Niday postulated the existence of two fission modes for the interpretation of the mass distribution of 232Th(n, f) [ 1 ]. Britt et al. used this approach for their analysis of the fission yields from light actinides [2]. However, this picture could not successfully be applied to heavier actinides like, e.g., 235U and thus failed as a general description of fission throughout the whole actinide region. Some years ago Brosa, Grogmann and MUller introduced a new concept of multimodal random neck-rupture (BGM-model) to explain the experimentally studied fission fragment properties from 213At to 258Fm. In this frame, fission modes as minimized trajectories in the potential energy landscape of the fissioning nucleus between the l Present address: Japan Atomic Energy Research Institute (JAERI), Tokai-mura, Naka-gun, Ibaraki-ken 319-11, Japan. 0375-9474/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDI 03 7 5 - 9 4 7 4 ( 9 5 ) 0 0 3 6 4 - 9
46
P. Siegler et al./Nuclear Physics A 594 (1995) 45-56
saddle and the scission point have been connected to the random neck-rupture model, which is applied to the pre-scission configuration of the nucleus [ 3 ]. The first successful application of the BGM-model has been demonstrated by Knitter et al. for the fissioning system 236U [4]. For the first time, the superposition of three predicted fission modes, namely, the super-long and two standard modes satisfactorily explained the dip of about 22 MeV in the average total kinetic energy (TKE) at symmetric fission fragment masses. Recently, the BGM-model has been applied to the compound system 238Np. This nucleus was chosen in the framework of an experiment which was performed to study the fission fragment properties of neutron induced fission of 237Np as a function of the incident neutron energy, A comparison of the theoretical predictions and experimental results will be given in the following.
2. Computational method A complete description of the BGM-model and the computational method can be found in Ref. [ 3 ]. At IRMM, the program package to calculate fission fragment properties within the BGM-model has been implemented [5] and considerably improved concerning I/O routines and data presentation [6,7]. However, for the sake of clarity the essential features are recalled here. The potential energy landscape has been calculated in a five-dimensional parameter space of the generalized Lawrence shapes [8] in cylindrical coordinates p and ~" according to N p2(~-) = (12 -- r E) E a n n--0
(~ - Z) 2 ,
(1)
where l denotes the half-length of the nucleus. In order to use shape parameters with a geometric meaning, r, c and s have been introduced besides l and z. Their distinct meaning is obvious from Fig. 1. Using N = 4 in Eq. (1) and considering volume conservation as well as the definition of the neck curvature, all parameters an are successively defined by one set (l, r, z, c, s). A detailed description of the parameters an is given in Ref. [3]. In this macroscopic-microscopic approach, the shell and pairing corrections of the Strutinsky type are included. The liquid drop part was calculated according to MyersSwiatecki [9,10], and a Woods-Saxon type single-particle potential was used to perform the shell corrections. For details we refer to Ref. [3]. In this five-dimensional potential energy landscape fission modes are defined as minimized trajectories between ground state and scission. In the frame of the BGM-model, scission may occur if one of the following conditions is satisfied: either the ratio between the half-length l of the nucleus and its neck radius r fulfills the Rayleigh criterion [ 11 ], 1 = ar,
(2)
P Siegler et aL/Nuclear Physics A 594 (1995) 45-56
47
P
2I
=-
Fig. 1. Generalized Lawrence representation of the nuclear shape. Here, 1 denotes the nuclear half-length, r the neck radius, z the location of the neck and a measure of the mass asymmetry, c the neck curvature in units of the spherical radius of the compound nucleus and s the position of the nuclear center-of-mass. with a = 11/2 as derived in Ref. [3], or the neck radius r becomes smaller than one nucleonic radius. The complete model calculations separate into two parts, namely the fission mode search and the successive derivation of the fission fragment properties from the calculated scission configuration. In the first part, potential energy landscapes are calculated in a two-dimensional subspace, mainly in the (l, r) plane with a fixed value for the asymmetry parameter z, and displayed as a contour plot in the upper part of Fig. 2. There the potential energy is shown for a symmetric configuration, i.e., z = 0. Within this contour plot, evidence already exists for the symmetric super-long mode given by the full line. However, the other modes are not well pronounced because of their asymmetry. For that reason many similar datasets have to be calculated for different values of z and compared to each other in order to find starting parameters and to fix the different fission modes. Since this standard method is quite cumbersome, a better scanning procedure in the threedimensional subspace (l, r, z), which directly uses the Rayleigh criterion of Eq. (2) to connect I and r at scission, has been developed [6]. The resulting potential energy landscape is plotted in the lower part of Fig. 2 as a contour map in the ( r ( 1 ) , z ) plane. In that way three minima, corresponding to three well-separated fission modes, are immediately visibly indicated by asterisks ( . ) . In subsequent steps, this cut perpendicular to the (l, r) plane is shifted towards the ground state by introducing an offset ( on l, l = a ( r n - ~ / 2 ) - sc,
(3)
where rn = r0 + ~:/2 [6,7]. The simultaneous shift of the r-interval is necessary in order to keep all calculations in a region of reasonable combinations of the parameters l and r. The factor 1/2, however, is chosen arbitrarily according to the characteristic shape of these potential energy landscapes in the (l, r) plane. Based on a sequence of similar
48
P Siegler et al./Nuclear Physics A 594 (1995) 45-56 I
.
.
.
.
I
.
.
.
z
.
I
=
'
'
'
0.0 fin
6
2
10
15
20
l(fm)
4
2 N
2.5
3.0
3.5
r (fro) Fig. 2. Potential energydisplayedas a contourplot for 238Np.Upperpart: in the (l, r) plane for a symmetrical shape (z = 0); lower part: in the (r(l), z) plane where l is calculated accordingto Eq. (2). plots, the search of fission modes is considerably accelerated. The final results obtained for each fission mode at scission are the elongation of the nucleus, e.g., the distance of the center-of-charges of the future fragments, the mean mass division, which is determined by the neck position z, and the fission-barrier heights. Additionally, the mean value for the total kinetic energy (TKE) is defined by the shape of the nucleus, mainly by its length 21. In order to obtain the characteristic properties of the fission fragment distributions, the width of the mass as well as of the TKE distribution, the dynamics at scission have to be considered. These dynamics may be expressed in terms of two instabilities in the motion around scission, namely, the shift instability and the capillarity or Rayleigh instability. The concept is roughly as follows: on a flat neck (close to scission) small strangulations may happen everywhere, but most probably at the position z, where the neck is thinnest. It has been demonstrated in Ref. [3] and references mentioned there, that such a tiny dent can shift on a flat neck as in an unstable motion along the (-axis.
R Siegleret aL/Nuclear PhysicsA 594 (1995)45-56
49
Finally, the capillarity instability leads to rupture provided that the Rayleigh criterion (see Eq. (2)) is fulfilled. The probability w((r) for rupture at a position (r, which produces a fragment pair (A c< p2((r), Acn - A), is given by a Boltzmann distribution, W(~"r) O( exp { - 2 ~ y 0 [p2((r) - p2(z)] } / T ,
(4)
where T = (8Es/Acn) V2 is the nuclear temperature, with E s the intrinsic excitation energy, both given in MeV and y0, the surface tension coefficient. The respective width of the mass distribution O'A follows then to be proportional to x ~ . The width of the TKE distribution, o'TKZ, is assumed to be due to fluctuations in length, Al, as well as in pre-scission kinetic energy. The distinct expression of o'Tr,E was derived in Ref. [3] from a standard Langevin equation. The force is taken to be constant and depends on A U, the energy difference between the top of the outer barrier and scission, and on Al, which is the increase of l between the outer barrier and the scission point. The random Langevin force has a constant strength incorporating damping as well as the nuclear temperature T as defined above. According to the solution of the Langevin equation O'TKE is a function of Al/l, T ( E s) and AU, where AU is partly transformed to the pre-scission kinetic energy and partly dissipated into the intrinsic excitation energy E s. Furthermore, from the deformation energy from the shape of the nascent fragments as well as from the assumption of the mass-equal share of the intrinsic excitation energy it is possible to calculate the neutron multiplicity. However, this part will not be considered in the following.
3. Results and discussion
The results on the fission mode calculation of the compound nucleus 23SNp are displayed in Fig. 3. Since each mode passes through the five-dimensional parameter space (l, r, z, c, s), the projections are shown on the (l, r), (l, z) and (r, z) planes for better visualization. The curvature c and the center-of-mass position s have not been shown, because they are not essential for the further discussion. The potential energy of the deformed nucleus, defined as excess of the ground-state binding energy Egs over the binding energy Epot of the deformed nucleus, is given as a function of I. Four modes are visible in these projections. A striking feature is the fact that the so-called "standard path", given by the dotted line, is splitted into three submodes. Starting from the ground state (gs) the common fission path exhibits the lowest barrier. In the second minimum, a first bifurcation takes place. The super-long mode branches off from the standard modes and reaches its second fission barrier, which is considerably higher than that of the standard mode. The super-long mode ends per definition in the minimum of the potential energy at a length of 21 fm (see Fig. 3d). After the second minimum the remaining standard mode reaches its second barrier, and shortly behind a second bifurcation takes place. Like in other actinides, the well-known standard I and standard II modes are appearing, but for the first time the standard HI
P. Siegler et al./Nuclear Physics A 594 (1995) 45-56
50
4P
(,..~
(a)
m,~
3
6
t~ ~andmrd
Ill -
-
~
~
-
~
N
21 = l l r
.
:"...,
1
----
Itattdltrd
II
q
~.-•
i
.
,
.
i
I
i
ltand~trd
%11 ,
,
15
10
11 l
,
,
,
20
1 (fro)
'
i
. . . .
. . . . . . II~ndard
Ill -~o/
...-'
"'~
/
~,
llltandard
I
stmndlurd
.....~
.
I -'3~'~
..-.....................
(,:)
..
0
:~
•
['~
I
,'
,r,
~
..£'
"-,
2~mln
i
.
,
,
*',
Sill
long
(d)
\
I -
standard
-1o
- ,,p super
. . . .
..... ,{'4".,,..."',. . . . ,,t.,,,d,,,-dIt,
:/ " " " " . . "..,,,~ ' \ ~ ~ ,~.: -
;
./
II - - ~
i
i
l0
3
20
15
10
I (fin)
--"',
super-long
"e
-20
i
,
I
I0
r
(fm)
,
,
,
,
/
. . . .
15
1
,
,
20
l(fm)
Fig. 3, Fission mode picture for the compound nucleus 238Np. Different projections of the multi-dimensional dataset are shown: (a) in the ( l , r ) plane; (b) in the (l, z ) plane; (c) in the ( r , z ) plane; (d) in the ( l , Egs - Epot) plane, gs stands for ground state and 2nd min for second minimum.
mode could be identified at higher mass asymmetry, i.e., larger z-values. The scission point of standard I and II is marked by the end of each mode. A peculiarity of the standard path is the fact that standard I and standard II overlap in both the projection onto the (l, r) and (l, Egs - Epot) plane. They could only be discriminated by their asymmetry parameter z. For standard III more particularities exist. First of all, the potential energy is increasing again and a third fission barrier develops. The second curiosity is the early ending of this fission mode at a neck radius r with more than twice the radius of a nucleon. Nevertheless, the Rayleigh criterion (see Eq. ( 2 ) ) for the neck instability is fulfilled and, thus, neck-rupture is possible (see Fig. 3a, full line). As a consequence, this mode might contribute to possible fission configurations. In Fig. 3c the location of the neck z, which is an indication for the mass asymmetry, is plotted as a function of the length parameter I. At scission the z-values are quite specific and give the characteristic mean mass split for each fission mode. In Fig. 4, the energy Egs - Epot and the neck radius r are shown as functions of the parameter D, which is defined as the distance of the center-of-masses of the future fragments. This presentation has been introduced in Ref. [6] in order to describe the evolution of fission in a way which is independent from the underlying shape parameterization. As is obvious from the upper part of Fig. 4, the well-known shape of the
P. Siegler et al./Nuclear Physics A 594 (1995) 45-56
51
/ ,
10
-.; t" ;'"t4"
cD
"~.--- Standard Ill
o 0
g~
standard
-10
'~ - - I -~',
Super-long
I U
standard
11 --''-......
"4..._S/
-20 I
....
1'0 ....
1'5 ....
....
l
I
I
,o
D (fro) ,,,,l,,,,i,,,,1,,,,i,,,,
6
•?.. _ • :.,.. T--- super-long ~ 4
"~'~'~- ~ - standard Ill s t a n d a ~ "
standard
10
15
II --"..~.. 20
25
D (fm) Fig. 4. Fission modes for the compound nucleus 238Np; upper part Ee,s - Elmt and lower part r as functions of the center-of-mass distance D.
double-humped fission barrier is obtained in this representation. Additionally, we see that the (r, D) representation (lower part of Fig. 4) allows to uniquely determine the bifurcation points at least for the standard 11I and super-long modes. The obtained shape parameters at scission for each fission mode as well as the corresponding barrier parameters are used to predict the following observables quantitatively as described in the previous chapter: the mean mass of the heavy fragment A, its width O'A, the mean total kinetic energy TKE and the corresponding width O'TKE. In Table 1 the predictions are listed together with the shape parameters (l, r, z) at the scission point of the different modes calculated for zero incident energy. Due to the generally limited precision of the shell corrections, absolute binding energies and locations in the potential energy landscape cannot be more accurate than 1 MeV and 0.5 fm, respectively. Therefore, all predictions may not be seen to be more accurate than TKE-4- 5 MeV, A + 3 ainu and o'A + 25% [3]. In the following, the theoretical predictions have been compared to results of a fit
P. Siegler et al./Nuclear Physics A 594 (1995) 45-56
52
Table 1 Parameters of the scission point configurations of the different fission modes calculated for zero incident energy (En = 0 MeV) Mode
standard I standard II standard III super-long
B [MeV]
l [fm]
r [fm]
z lfm]
TKE [MeV]
O-TKE [MeV]
A
trA
7.5 7.5 7.4 10.9
15.5 16.8 16.1 21.0
1.8 1.2 2.8 2.0
0.3 0.7 2.2 0.2
189.8 178.6 173.6 155.0
9.7 9.6 6.3 10.1
135.4 139.1 153,0 119,0
3.4 5.5 3.8 13.0
to the two-dimensional experimental yield distribution as a function of mass and TKE, based on a description given in Ref. [ 13]: Y(A,TKE) = E
Y~(A)Y~(TKE),
(5)
i
with Yi(A) the part for the mass distribution and Y/(TKE) the part for the TKE distribution of the fission mode labeled with the index i. In the case of e38Np, the total fission fragment mass distribution can be considered as a superposition of four Gaussian distributions, one for each theoretically predicted fission mode:
Yi(a)-w/2Wio.2,iexp
(
20.2A,i
j.
(6)
The TKE distribution, however, cannot be approximated by superposition of simple Gaussian distributions, since it is rather skewed and exhibits a sharp cutoff at higher energies defined by the Q-value and a low-energy tailing. A best-suited function to describe the TKE distribution was introduced in Ref. [ 13] :
( 200
Yi(TKE) = ~,-~---~j hi (2(dmax, i - drain,i) × exp \ dd~,;
L ddec,i
(dmax,i - dmin,i)2~ -L~e--~,i ,I '
(7)
where the label i again refers to one distinct fission mode. In Eq. (7), L is defined as L = D - dmin. The distance D is calculated considering only Coulomb-interaction between the fragments, D -
ZlZhe 2 TKE
(8)
The parameters zl and zh are the nuclear charges of the light and heavy fragments assuming an unchanged charge distribution, respectively. The free parameters are dmax, drain and ddec, which have to be adjusted during the fitting process with the following physical meaning:
P. Siegler et aL/Nuclear Physics A 594 (1995) 45-56 I
'
'
'
[
,
,
,
]
,
,
'
I
'
'
'
53
I
200
>
° !
180
160
I
140
80
i
i
i I 140
120
tO0
, I 160
Fragment Mass
' 200i
. . . .
I
80
100
'
'
'
I
'
'
'
i
'
I
'
180
160i
140
120
140
I i i 160
Fragment Mass Fig. 5. U p p e r part: C o m p a r i s o n
o f the e x p e r i m e n t a l distributions w i t h the theoretical fit for 237Np(n, f) at an
i n c i d e n t n e u t r o n e n e r g y o f En = 5.5 MeV, d i s p l a y e d as a c o n t o u r plot o f f r a g m e n t m a s s v e r s u s T K E . L o w e r part: P o s i t i o n a n d w i d t h o f the four fission m o d e s i n d i c a t e d b y a c o n t o u r line at a relative h e i g h t o f 5 0 % o f the m a x i m u m
y i e l d for e a c h m o d e .
• dmax is the charge distance at the yield maximum of the distribution, • drain is the smallest possible charge distance corresponding to an upper limit of the
TKE, • ddec describes the exponential decrease of the yield with increasing distance D.
The parameter h approximates the height of the distribution, because 200 MeV is about the TKE of the fission of the actinides (see the factor (200/TKE) 2 in Eq. ( 7 ) ) . A comparison of the experimental distribution for 237Np(n, f) at an incident neutron energy of En = 5.5 MeV with the fitted result is given in the upper part of Fig. 5 as a contour plot. This incident neutron energy has been chosen because of the good statistics of the experiment especially in the symmetric fragment mass region. The good agreement between the experiment (thin full line) and the fit (thick full line) is evident. In the lower part of Fig. 5 the position and the width of the four fission modes are indicated
P. Siegler et al./Nuclear Physics A 594 (1995) 45-56
54
Table 2 Comparison of the theoretical parameters of the fission modes calculated for incident energy En = 5.5 MeV with results from the fit to the experimental data at En = 5.5 MeV. The values indicated by * have been adopted from the theoretical predictions Mode
B [MeV]
TKE [MeV]
O'TKE [MeV]
A
o'a
Theory
standard I standard II standard 1II super-long
7.5 7.5 7.4 10.9
189.8 4- 5.0 178.6 4- 5.0 173.64-5.0 155.0 4- 5.0
10.3 4- 1.0 10.1 4- h 0 7.04-0.7 9.4 4- 0.9
135.4 4- 3.0 139.1 4- 3.0 153.04-3.0 119.0 4- 3.0
3.9 4- 1.0 5.9 -4- 1.5 4.94-1.2 13.5 4- 3.4
Experiment
standard I standard II standard III super-long
6.1 6.1 6.1 10.3
186.04-0.1 170.04-0.1 152.74-0.4 160.9 4- 0.2
7.24-0.1 9.14-0.1 6.64-0.3 8.5 4- 0.1
134.74-0.1 140.3+0.1 153.04-3.0" 119.0 4- 3.0*
3.74-0.1 6.44-0.1 4.94-1.2" 13.5 4- 3.4*
by a contour line at a relative height of 50% of the maximum yield for each mode. A comparison of the parameters from the fit with their theoretical predictions calculated for 5.5 MeV incident neutron energy is given in Table 2. Only the parameters indicated with an asterisk (*) have been kept fixed in the fit-procedure and their values have been adopted from the theoretical predictions. This was necessary because of the very small abundance of the standard III and super-long modes. Omitting these modes, however, resulted in a worse fitting result. All experimentally achieved parameters are in reasonable agreement with the theoretical predictions, except the TKE of the standard III mode. A possible explanation is that the standard III mode could not be followed theoretically up to the real scission point, resulting in a rather short configuration and therefore a higher TKE. In order to demonstrate the good agreement between fit and experiment the different moments of the three-dimensional mass-TKE distribution have been compared in Fig. 6 to the fit-results. In the upper part of Fig. 6, the four distributions describing the superlong (SL), standard I (S1), standard II ($2) and standard III ($3) mode are plotted as dashed lines for the mass yield and for the TKE as a function of mass. Not only these moments of the experimental distribution are very well reproduced, the dispersion o- and the dissymmetry of the TKE distribution in the lower part of Fig. 6 are very well reproduced, also. In all cases, the sum of the superposition is represented by the full line. It should be emphasized that fitting the two-dimensional yield distribution as a function of mass and TKE is essential for the reliability of the fit-parameters. If, however, only the mass distribution averaged over all TKEresolution of the experiment and the initial start values for the fit-procedure. Concerning the good agreement between the fit and the experimental data, it is also evident that the super-long and standard III modes must be introduced to the fit-procedure in order to get a better overall agreement. A similar conclusion has been drawn for the fissioning system 239pu(n, f), where a detailed fission mode calculation does not exist
P. Siegler et aL/Nuclear Physics A 594 (1995) 45-56
55
SI
]
10 o ~
N ~
1fl0
I0"
>"~°-~ /' lO-a
" . . . . . . . . . . I10
100
120
Fragment
140
i I 0
160
130
140
Fragment
Mass
150
160
Mass
14
I
L7.
121)
130
140
Fragment
150
Mass
160
120
130
140
Fragment
150
180
Mass
Fig. 6. Comparison of the experimental mass, TKE, o-(TKE) and dissymmetry distributions with the theoretical fit for 237Np(n, f) at an incident neutron energy of En = 5.5 MeV.
yet [ 14]. The authors have also compared their experimental yield distribution as a function of mass and TKE to a fit taking into account two asymmetric modes (standard I and standard II) and found a surplus yield in the mass region 150 < A < 160, compared to the restricted fit with only standard I and standard II, implying an additional fission mode (presumably standard III). The inclusion of this mode gave a much better agreement between the fit-results and the experimental distributions. For 237Np(n, f), the dip in the TKE for symmetric masses, as already seen for 235U(n, f) [4], is also very well reproduced by the fit-procedure. This again is due to the superposition of the super-long mode and the standard modes and the lower TKE of the super-long mode compared to the standard modes.
4. Conclusions
With potential energy calculations in the frame of the multi-modal random neckrupture model (BGM-model), we were able to follow the fission modes of the compound nucleus 238Np from ground state to scission. For the first time, the standard III fission mode could be identified from the bifurcation point to a position in the five-dimensional parameter space, where the Rayleigh criterion justifies scission. An experiment on neutron induced fission of 237Np(n, f) as a function of the incident neutron
56
P Siegler et al./Nuclear Physics A 594 (1995) 45-56
energy confirmed the existence of this third standard mode besides the standard I, II and the super-long fission modes. The reasonable agreement between theoretical predictions and experimental data is quite promising. This agreement has been found over a wide range of excitation energies, which will be reported elsewhere [ 15].
Acknowledgements We are indebted to U. Brosa for the transfer of his programs to perform the mode search and fission fragment property calculations. Also many fruitful hints on how to use his code are appreciated. ES. wants to express his gratitude to the Commission of the EU for his fellowship and the IRMM for the excellent and pleasant working conditions.
References [ 1] [2] [3] [4] [5 ] 16] [7] [81 19] [ 10] [ 11] [12] [ 13]
A. Turkevichand J.B. Niday, Phys. Rev. 84 (1951) 52. H.C. Britt, M.E. Wegner and J.C. Gursky,Phys. Rev. 129 (1963) 2239. U. Brosa, S. GroBmannand A. MUller,Phys. Reports 197 (1990) 167. H.-H. Knitter, F.-J. Hambsch,C. Budtz-J¢rgensenand J.P. Theobald,Z. Naturforsch. 42a (1987) 786. U. Brosa, private communication( 1991). S. Oberstedt, IRMM internal report GE/R/VG/77/93 (1993). P. Siegler, IRMM internal report GE/R/VG/84/94 (1994). J.N.P. Lawrence, Phys. Rev. 139 (1965) B1227. W.D. Myers and W.J. Swiatecki, Nucl. Phys. 81 (1966) 1. W.D. Myers and W.J. Swiatecki, Acta Fys. 36 (1967) 343. J.W. Rayleigh, Proc. London Math. Soc. X (1878) 4. S. Gro6mann,U. Brosa and A. Miiller,Nucl. Phys. A 481 (1988) 340. U. Brosa and H.-H. Knitter, Proc. XVlllth Int. Symp. on Nuclear physics - physics and chemistry of fission, Gaussig, Germany, 1988, eds. H. M~Lrtenand D. Seeliger, ZfK-732,p. 145. [ 141 P. Schillebeeckx,C. Wagemans,A.J. Deruytter and R. Barthelemy,Nucl. Phys. A 545 (1992) 623. 115] P. Siegler,E-J. Hambschand J.P. Theobald,to be published.
Nuclear Physics A 594 (1995) 45-56
Fission modes in the compound nucleus 238Np P. S i e g l e r a'I, E - J . H a m b s c h a, S. O b e r s t e d t c, J.P. T h e o b a l d b a JRC-Institute for Reference Materials and Measurements IRMM, Retieseweg, B-2440 Geel, Belgium b Technische Hochschule Darmstadt, D-64289 Darmstadt, Germany c SCK, CEN, Boeretang 200, B-2400 MoL Belgium Received 7 March 1995; revised 19 July 1995
Abstract
Potential energy calculations in conjunction with the multi-modal random neck-rupture model of Brosa, Grogmann and Milller (BGM-model) indicate the existence of four fission modes in the compound nucleus 23SNp. For the first time the splitting of the standard mode into three submodes could be demonstrated theoretically. The existence of a third standard mode in fission has already been proposed for heavier nuclei, where a normegligible yield of more asymmetric fission fragments is experimentally observed. The calculated fission fragment properties for 23SNp are discussed and compared to experimental results.
1. Introduction
The idea of multi-modal fission was announced already more than 40 years ago. In 1951, Turkevich and Niday postulated the existence of two fission modes for the interpretation of the mass distribution of 232Th(n, f) [ 1 ]. Britt et al. used this approach for their analysis of the fission yields from light actinides [2]. However, this picture could not successfully be applied to heavier actinides like, e.g., 235U and thus failed as a general description of fission throughout the whole actinide region. Some years ago Brosa, Grogmann and MUller introduced a new concept of multimodal random neck-rupture (BGM-model) to explain the experimentally studied fission fragment properties from 213At to 258Fm. In this frame, fission modes as minimized trajectories in the potential energy landscape of the fissioning nucleus between the l Present address: Japan Atomic Energy Research Institute (JAERI), Tokai-mura, Naka-gun, Ibaraki-ken 319-11, Japan. 0375-9474/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDI 03 7 5 - 9 4 7 4 ( 9 5 ) 0 0 3 6 4 - 9
46
P. Siegler et al./Nuclear Physics A 594 (1995) 45-56
saddle and the scission point have been connected to the random neck-rupture model, which is applied to the pre-scission configuration of the nucleus [ 3 ]. The first successful application of the BGM-model has been demonstrated by Knitter et al. for the fissioning system 236U [4]. For the first time, the superposition of three predicted fission modes, namely, the super-long and two standard modes satisfactorily explained the dip of about 22 MeV in the average total kinetic energy (TKE) at symmetric fission fragment masses. Recently, the BGM-model has been applied to the compound system 238Np. This nucleus was chosen in the framework of an experiment which was performed to study the fission fragment properties of neutron induced fission of 237Np as a function of the incident neutron energy, A comparison of the theoretical predictions and experimental results will be given in the following.
2. Computational method A complete description of the BGM-model and the computational method can be found in Ref. [ 3 ]. At IRMM, the program package to calculate fission fragment properties within the BGM-model has been implemented [5] and considerably improved concerning I/O routines and data presentation [6,7]. However, for the sake of clarity the essential features are recalled here. The potential energy landscape has been calculated in a five-dimensional parameter space of the generalized Lawrence shapes [8] in cylindrical coordinates p and ~" according to N p2(~-) = (12 -- r E) E a n n--0
(~ - Z) 2 ,
(1)
where l denotes the half-length of the nucleus. In order to use shape parameters with a geometric meaning, r, c and s have been introduced besides l and z. Their distinct meaning is obvious from Fig. 1. Using N = 4 in Eq. (1) and considering volume conservation as well as the definition of the neck curvature, all parameters an are successively defined by one set (l, r, z, c, s). A detailed description of the parameters an is given in Ref. [3]. In this macroscopic-microscopic approach, the shell and pairing corrections of the Strutinsky type are included. The liquid drop part was calculated according to MyersSwiatecki [9,10], and a Woods-Saxon type single-particle potential was used to perform the shell corrections. For details we refer to Ref. [3]. In this five-dimensional potential energy landscape fission modes are defined as minimized trajectories between ground state and scission. In the frame of the BGM-model, scission may occur if one of the following conditions is satisfied: either the ratio between the half-length l of the nucleus and its neck radius r fulfills the Rayleigh criterion [ 11 ], 1 = ar,
(2)
P Siegler et aL/Nuclear Physics A 594 (1995) 45-56
47
P
2I
=-
Fig. 1. Generalized Lawrence representation of the nuclear shape. Here, 1 denotes the nuclear half-length, r the neck radius, z the location of the neck and a measure of the mass asymmetry, c the neck curvature in units of the spherical radius of the compound nucleus and s the position of the nuclear center-of-mass. with a = 11/2 as derived in Ref. [3], or the neck radius r becomes smaller than one nucleonic radius. The complete model calculations separate into two parts, namely the fission mode search and the successive derivation of the fission fragment properties from the calculated scission configuration. In the first part, potential energy landscapes are calculated in a two-dimensional subspace, mainly in the (l, r) plane with a fixed value for the asymmetry parameter z, and displayed as a contour plot in the upper part of Fig. 2. There the potential energy is shown for a symmetric configuration, i.e., z = 0. Within this contour plot, evidence already exists for the symmetric super-long mode given by the full line. However, the other modes are not well pronounced because of their asymmetry. For that reason many similar datasets have to be calculated for different values of z and compared to each other in order to find starting parameters and to fix the different fission modes. Since this standard method is quite cumbersome, a better scanning procedure in the threedimensional subspace (l, r, z), which directly uses the Rayleigh criterion of Eq. (2) to connect I and r at scission, has been developed [6]. The resulting potential energy landscape is plotted in the lower part of Fig. 2 as a contour map in the ( r ( 1 ) , z ) plane. In that way three minima, corresponding to three well-separated fission modes, are immediately visibly indicated by asterisks ( . ) . In subsequent steps, this cut perpendicular to the (l, r) plane is shifted towards the ground state by introducing an offset ( on l, l = a ( r n - ~ / 2 ) - sc,
(3)
where rn = r0 + ~:/2 [6,7]. The simultaneous shift of the r-interval is necessary in order to keep all calculations in a region of reasonable combinations of the parameters l and r. The factor 1/2, however, is chosen arbitrarily according to the characteristic shape of these potential energy landscapes in the (l, r) plane. Based on a sequence of similar
48
P Siegler et al./Nuclear Physics A 594 (1995) 45-56 I
.
.
.
.
I
.
.
.
z
.
I
=
'
'
'
0.0 fin
6
2
10
15
20
l(fm)
4
2 N
2.5
3.0
3.5
r (fro) Fig. 2. Potential energydisplayedas a contourplot for 238Np.Upperpart: in the (l, r) plane for a symmetrical shape (z = 0); lower part: in the (r(l), z) plane where l is calculated accordingto Eq. (2). plots, the search of fission modes is considerably accelerated. The final results obtained for each fission mode at scission are the elongation of the nucleus, e.g., the distance of the center-of-charges of the future fragments, the mean mass division, which is determined by the neck position z, and the fission-barrier heights. Additionally, the mean value for the total kinetic energy (TKE) is defined by the shape of the nucleus, mainly by its length 21. In order to obtain the characteristic properties of the fission fragment distributions, the width of the mass as well as of the TKE distribution, the dynamics at scission have to be considered. These dynamics may be expressed in terms of two instabilities in the motion around scission, namely, the shift instability and the capillarity or Rayleigh instability. The concept is roughly as follows: on a flat neck (close to scission) small strangulations may happen everywhere, but most probably at the position z, where the neck is thinnest. It has been demonstrated in Ref. [3] and references mentioned there, that such a tiny dent can shift on a flat neck as in an unstable motion along the (-axis.
R Siegleret aL/Nuclear PhysicsA 594 (1995)45-56
49
Finally, the capillarity instability leads to rupture provided that the Rayleigh criterion (see Eq. (2)) is fulfilled. The probability w((r) for rupture at a position (r, which produces a fragment pair (A c< p2((r), Acn - A), is given by a Boltzmann distribution, W(~"r) O( exp { - 2 ~ y 0 [p2((r) - p2(z)] } / T ,
(4)
where T = (8Es/Acn) V2 is the nuclear temperature, with E s the intrinsic excitation energy, both given in MeV and y0, the surface tension coefficient. The respective width of the mass distribution O'A follows then to be proportional to x ~ . The width of the TKE distribution, o'TKZ, is assumed to be due to fluctuations in length, Al, as well as in pre-scission kinetic energy. The distinct expression of o'Tr,E was derived in Ref. [3] from a standard Langevin equation. The force is taken to be constant and depends on A U, the energy difference between the top of the outer barrier and scission, and on Al, which is the increase of l between the outer barrier and the scission point. The random Langevin force has a constant strength incorporating damping as well as the nuclear temperature T as defined above. According to the solution of the Langevin equation O'TKE is a function of Al/l, T ( E s) and AU, where AU is partly transformed to the pre-scission kinetic energy and partly dissipated into the intrinsic excitation energy E s. Furthermore, from the deformation energy from the shape of the nascent fragments as well as from the assumption of the mass-equal share of the intrinsic excitation energy it is possible to calculate the neutron multiplicity. However, this part will not be considered in the following.
3. Results and discussion
The results on the fission mode calculation of the compound nucleus 23SNp are displayed in Fig. 3. Since each mode passes through the five-dimensional parameter space (l, r, z, c, s), the projections are shown on the (l, r), (l, z) and (r, z) planes for better visualization. The curvature c and the center-of-mass position s have not been shown, because they are not essential for the further discussion. The potential energy of the deformed nucleus, defined as excess of the ground-state binding energy Egs over the binding energy Epot of the deformed nucleus, is given as a function of I. Four modes are visible in these projections. A striking feature is the fact that the so-called "standard path", given by the dotted line, is splitted into three submodes. Starting from the ground state (gs) the common fission path exhibits the lowest barrier. In the second minimum, a first bifurcation takes place. The super-long mode branches off from the standard modes and reaches its second fission barrier, which is considerably higher than that of the standard mode. The super-long mode ends per definition in the minimum of the potential energy at a length of 21 fm (see Fig. 3d). After the second minimum the remaining standard mode reaches its second barrier, and shortly behind a second bifurcation takes place. Like in other actinides, the well-known standard I and standard II modes are appearing, but for the first time the standard HI
P. Siegler et al./Nuclear Physics A 594 (1995) 45-56
50
4P
(,..~
(a)
m,~
3
6
t~ ~andmrd
Ill -
-
~
~
-
~
N
21 = l l r
.
:"...,
1
----
Itattdltrd
II
q
~.-•
i
.
,
.
i
I
i
ltand~trd
%11 ,
,
15
10
11 l
,
,
,
20
1 (fro)
'
i
. . . .
. . . . . . II~ndard
Ill -~o/
...-'
"'~
/
~,
llltandard
I
stmndlurd
.....~
.
I -'3~'~
..-.....................
(,:)
..
0
:~
•
['~
I
,'
,r,
~
..£'
"-,
2~mln
i
.
,
,
*',
Sill
long
(d)
\
I -
standard
-1o
- ,,p super
. . . .
..... ,{'4".,,..."',. . . . ,,t.,,,d,,,-dIt,
:/ " " " " . . "..,,,~ ' \ ~ ~ ,~.: -
;
./
II - - ~
i
i
l0
3
20
15
10
I (fin)
--"',
super-long
"e
-20
i
,
I
I0
r
(fm)
,
,
,
,
/
. . . .
15
1
,
,
20
l(fm)
Fig. 3, Fission mode picture for the compound nucleus 238Np. Different projections of the multi-dimensional dataset are shown: (a) in the ( l , r ) plane; (b) in the (l, z ) plane; (c) in the ( r , z ) plane; (d) in the ( l , Egs - Epot) plane, gs stands for ground state and 2nd min for second minimum.
mode could be identified at higher mass asymmetry, i.e., larger z-values. The scission point of standard I and II is marked by the end of each mode. A peculiarity of the standard path is the fact that standard I and standard II overlap in both the projection onto the (l, r) and (l, Egs - Epot) plane. They could only be discriminated by their asymmetry parameter z. For standard III more particularities exist. First of all, the potential energy is increasing again and a third fission barrier develops. The second curiosity is the early ending of this fission mode at a neck radius r with more than twice the radius of a nucleon. Nevertheless, the Rayleigh criterion (see Eq. ( 2 ) ) for the neck instability is fulfilled and, thus, neck-rupture is possible (see Fig. 3a, full line). As a consequence, this mode might contribute to possible fission configurations. In Fig. 3c the location of the neck z, which is an indication for the mass asymmetry, is plotted as a function of the length parameter I. At scission the z-values are quite specific and give the characteristic mean mass split for each fission mode. In Fig. 4, the energy Egs - Epot and the neck radius r are shown as functions of the parameter D, which is defined as the distance of the center-of-masses of the future fragments. This presentation has been introduced in Ref. [6] in order to describe the evolution of fission in a way which is independent from the underlying shape parameterization. As is obvious from the upper part of Fig. 4, the well-known shape of the
P. Siegler et al./Nuclear Physics A 594 (1995) 45-56
51
/ ,
10
-.; t" ;'"t4"
cD
"~.--- Standard Ill
o 0
g~
standard
-10
'~ - - I -~',
Super-long
I U
standard
11 --''-......
"4..._S/
-20 I
....
1'0 ....
1'5 ....
....
l
I
I
,o
D (fro) ,,,,l,,,,i,,,,1,,,,i,,,,
6
•?.. _ • :.,.. T--- super-long ~ 4
"~'~'~- ~ - standard Ill s t a n d a ~ "
standard
10
15
II --"..~.. 20
25
D (fm) Fig. 4. Fission modes for the compound nucleus 238Np; upper part Ee,s - Elmt and lower part r as functions of the center-of-mass distance D.
double-humped fission barrier is obtained in this representation. Additionally, we see that the (r, D) representation (lower part of Fig. 4) allows to uniquely determine the bifurcation points at least for the standard 11I and super-long modes. The obtained shape parameters at scission for each fission mode as well as the corresponding barrier parameters are used to predict the following observables quantitatively as described in the previous chapter: the mean mass of the heavy fragment A, its width O'A, the mean total kinetic energy TKE and the corresponding width O'TKE. In Table 1 the predictions are listed together with the shape parameters (l, r, z) at the scission point of the different modes calculated for zero incident energy. Due to the generally limited precision of the shell corrections, absolute binding energies and locations in the potential energy landscape cannot be more accurate than 1 MeV and 0.5 fm, respectively. Therefore, all predictions may not be seen to be more accurate than TKE-4- 5 MeV, A + 3 ainu and o'A + 25% [3]. In the following, the theoretical predictions have been compared to results of a fit
P. Siegler et al./Nuclear Physics A 594 (1995) 45-56
52
Table 1 Parameters of the scission point configurations of the different fission modes calculated for zero incident energy (En = 0 MeV) Mode
standard I standard II standard III super-long
B [MeV]
l [fm]
r [fm]
z lfm]
TKE [MeV]
O-TKE [MeV]
A
trA
7.5 7.5 7.4 10.9
15.5 16.8 16.1 21.0
1.8 1.2 2.8 2.0
0.3 0.7 2.2 0.2
189.8 178.6 173.6 155.0
9.7 9.6 6.3 10.1
135.4 139.1 153,0 119,0
3.4 5.5 3.8 13.0
to the two-dimensional experimental yield distribution as a function of mass and TKE, based on a description given in Ref. [ 13]: Y(A,TKE) = E
Y~(A)Y~(TKE),
(5)
i
with Yi(A) the part for the mass distribution and Y/(TKE) the part for the TKE distribution of the fission mode labeled with the index i. In the case of e38Np, the total fission fragment mass distribution can be considered as a superposition of four Gaussian distributions, one for each theoretically predicted fission mode:
Yi(a)-w/2Wio.2,iexp
(
20.2A,i
j.
(6)
The TKE distribution, however, cannot be approximated by superposition of simple Gaussian distributions, since it is rather skewed and exhibits a sharp cutoff at higher energies defined by the Q-value and a low-energy tailing. A best-suited function to describe the TKE distribution was introduced in Ref. [ 13] :
( 200
Yi(TKE) = ~,-~---~j hi (2(dmax, i - drain,i) × exp \ dd~,;
L ddec,i
(dmax,i - dmin,i)2~ -L~e--~,i ,I '
(7)
where the label i again refers to one distinct fission mode. In Eq. (7), L is defined as L = D - dmin. The distance D is calculated considering only Coulomb-interaction between the fragments, D -
ZlZhe 2 TKE
(8)
The parameters zl and zh are the nuclear charges of the light and heavy fragments assuming an unchanged charge distribution, respectively. The free parameters are dmax, drain and ddec, which have to be adjusted during the fitting process with the following physical meaning:
P. Siegler et aL/Nuclear Physics A 594 (1995) 45-56 I
'
'
'
[
,
,
,
]
,
,
'
I
'
'
'
53
I
200
>
° !
180
160
I
140
80
i
i
i I 140
120
tO0
, I 160
Fragment Mass
' 200i
. . . .
I
80
100
'
'
'
I
'
'
'
i
'
I
'
180
160i
140
120
140
I i i 160
Fragment Mass Fig. 5. U p p e r part: C o m p a r i s o n
o f the e x p e r i m e n t a l distributions w i t h the theoretical fit for 237Np(n, f) at an
i n c i d e n t n e u t r o n e n e r g y o f En = 5.5 MeV, d i s p l a y e d as a c o n t o u r plot o f f r a g m e n t m a s s v e r s u s T K E . L o w e r part: P o s i t i o n a n d w i d t h o f the four fission m o d e s i n d i c a t e d b y a c o n t o u r line at a relative h e i g h t o f 5 0 % o f the m a x i m u m
y i e l d for e a c h m o d e .
• dmax is the charge distance at the yield maximum of the distribution, • drain is the smallest possible charge distance corresponding to an upper limit of the
TKE, • ddec describes the exponential decrease of the yield with increasing distance D.
The parameter h approximates the height of the distribution, because 200 MeV is about the TKE of the fission of the actinides (see the factor (200/TKE) 2 in Eq. ( 7 ) ) . A comparison of the experimental distribution for 237Np(n, f) at an incident neutron energy of En = 5.5 MeV with the fitted result is given in the upper part of Fig. 5 as a contour plot. This incident neutron energy has been chosen because of the good statistics of the experiment especially in the symmetric fragment mass region. The good agreement between the experiment (thin full line) and the fit (thick full line) is evident. In the lower part of Fig. 5 the position and the width of the four fission modes are indicated
P. Siegler et al./Nuclear Physics A 594 (1995) 45-56
54
Table 2 Comparison of the theoretical parameters of the fission modes calculated for incident energy En = 5.5 MeV with results from the fit to the experimental data at En = 5.5 MeV. The values indicated by * have been adopted from the theoretical predictions Mode
B [MeV]
TKE [MeV]
O'TKE [MeV]
A
o'a
Theory
standard I standard II standard 1II super-long
7.5 7.5 7.4 10.9
189.8 4- 5.0 178.6 4- 5.0 173.64-5.0 155.0 4- 5.0
10.3 4- 1.0 10.1 4- h 0 7.04-0.7 9.4 4- 0.9
135.4 4- 3.0 139.1 4- 3.0 153.04-3.0 119.0 4- 3.0
3.9 4- 1.0 5.9 -4- 1.5 4.94-1.2 13.5 4- 3.4
Experiment
standard I standard II standard III super-long
6.1 6.1 6.1 10.3
186.04-0.1 170.04-0.1 152.74-0.4 160.9 4- 0.2
7.24-0.1 9.14-0.1 6.64-0.3 8.5 4- 0.1
134.74-0.1 140.3+0.1 153.04-3.0" 119.0 4- 3.0*
3.74-0.1 6.44-0.1 4.94-1.2" 13.5 4- 3.4*
by a contour line at a relative height of 50% of the maximum yield for each mode. A comparison of the parameters from the fit with their theoretical predictions calculated for 5.5 MeV incident neutron energy is given in Table 2. Only the parameters indicated with an asterisk (*) have been kept fixed in the fit-procedure and their values have been adopted from the theoretical predictions. This was necessary because of the very small abundance of the standard III and super-long modes. Omitting these modes, however, resulted in a worse fitting result. All experimentally achieved parameters are in reasonable agreement with the theoretical predictions, except the TKE of the standard III mode. A possible explanation is that the standard III mode could not be followed theoretically up to the real scission point, resulting in a rather short configuration and therefore a higher TKE. In order to demonstrate the good agreement between fit and experiment the different moments of the three-dimensional mass-TKE distribution have been compared in Fig. 6 to the fit-results. In the upper part of Fig. 6, the four distributions describing the superlong (SL), standard I (S1), standard II ($2) and standard III ($3) mode are plotted as dashed lines for the mass yield and for the TKE as a function of mass. Not only these moments of the experimental distribution are very well reproduced, the dispersion o- and the dissymmetry of the TKE distribution in the lower part of Fig. 6 are very well reproduced, also. In all cases, the sum of the superposition is represented by the full line. It should be emphasized that fitting the two-dimensional yield distribution as a function of mass and TKE is essential for the reliability of the fit-parameters. If, however, only the mass distribution averaged over all TKEresolution of the experiment and the initial start values for the fit-procedure. Concerning the good agreement between the fit and the experimental data, it is also evident that the super-long and standard III modes must be introduced to the fit-procedure in order to get a better overall agreement. A similar conclusion has been drawn for the fissioning system 239pu(n, f), where a detailed fission mode calculation does not exist
P. Siegler et aL/Nuclear Physics A 594 (1995) 45-56
55
SI
]
10 o ~
N ~
1fl0
I0"
>"~°-~ /' lO-a
" . . . . . . . . . . I10
100
120
Fragment
140
i I 0
160
130
140
Fragment
Mass
150
160
Mass
14
I
L7.
121)
130
140
Fragment
150
Mass
160
120
130
140
Fragment
150
180
Mass
Fig. 6. Comparison of the experimental mass, TKE, o-(TKE) and dissymmetry distributions with the theoretical fit for 237Np(n, f) at an incident neutron energy of En = 5.5 MeV.
yet [ 14]. The authors have also compared their experimental yield distribution as a function of mass and TKE to a fit taking into account two asymmetric modes (standard I and standard II) and found a surplus yield in the mass region 150 < A < 160, compared to the restricted fit with only standard I and standard II, implying an additional fission mode (presumably standard III). The inclusion of this mode gave a much better agreement between the fit-results and the experimental distributions. For 237Np(n, f), the dip in the TKE for symmetric masses, as already seen for 235U(n, f) [4], is also very well reproduced by the fit-procedure. This again is due to the superposition of the super-long mode and the standard modes and the lower TKE of the super-long mode compared to the standard modes.
4. Conclusions
With potential energy calculations in the frame of the multi-modal random neckrupture model (BGM-model), we were able to follow the fission modes of the compound nucleus 238Np from ground state to scission. For the first time, the standard III fission mode could be identified from the bifurcation point to a position in the five-dimensional parameter space, where the Rayleigh criterion justifies scission. An experiment on neutron induced fission of 237Np(n, f) as a function of the incident neutron
56
P Siegler et al./Nuclear Physics A 594 (1995) 45-56
energy confirmed the existence of this third standard mode besides the standard I, II and the super-long fission modes. The reasonable agreement between theoretical predictions and experimental data is quite promising. This agreement has been found over a wide range of excitation energies, which will be reported elsewhere [ 15].
Acknowledgements We are indebted to U. Brosa for the transfer of his programs to perform the mode search and fission fragment property calculations. Also many fruitful hints on how to use his code are appreciated. ES. wants to express his gratitude to the Commission of the EU for his fellowship and the IRMM for the excellent and pleasant working conditions.
References [ 1] [2] [3] [4] [5 ] 16] [7] [81 19] [ 10] [ 11] [12] [ 13]
A. Turkevichand J.B. Niday, Phys. Rev. 84 (1951) 52. H.C. Britt, M.E. Wegner and J.C. Gursky,Phys. Rev. 129 (1963) 2239. U. Brosa, S. GroBmannand A. MUller,Phys. Reports 197 (1990) 167. H.-H. Knitter, F.-J. Hambsch,C. Budtz-J¢rgensenand J.P. Theobald,Z. Naturforsch. 42a (1987) 786. U. Brosa, private communication( 1991). S. Oberstedt, IRMM internal report GE/R/VG/77/93 (1993). P. Siegler, IRMM internal report GE/R/VG/84/94 (1994). J.N.P. Lawrence, Phys. Rev. 139 (1965) B1227. W.D. Myers and W.J. Swiatecki, Nucl. Phys. 81 (1966) 1. W.D. Myers and W.J. Swiatecki, Acta Fys. 36 (1967) 343. J.W. Rayleigh, Proc. London Math. Soc. X (1878) 4. S. Gro6mann,U. Brosa and A. Miiller,Nucl. Phys. A 481 (1988) 340. U. Brosa and H.-H. Knitter, Proc. XVlllth Int. Symp. on Nuclear physics - physics and chemistry of fission, Gaussig, Germany, 1988, eds. H. M~Lrtenand D. Seeliger, ZfK-732,p. 145. [ 141 P. Schillebeeckx,C. Wagemans,A.J. Deruytter and R. Barthelemy,Nucl. Phys. A 545 (1992) 623. 115] P. Siegler,E-J. Hambschand J.P. Theobald,to be published.