Highly deformed rotating nuclei and superheavy elements
in the fusionlike deformation valley
G. Royer and C. Bonilla
Laboratoire Subatech, 4 rue A. Kastler, 44072 Nantes Cedex 03, France
(royer@subatech.in2p3.fr)
R. A. Gherghescu
Horia Hulubei - National Institute for Nuclear Physics and Engineering,
P. O. Box MG-6, RO-76900 Bucharest, Romania
Abstract
The potential barriers governing the evolution of rotating highly deformed nuclei and of
superheavy nuclear systems have been calculated within a generalized liquid drop model including a
proximity energy term and the asymmetric two center shell model and the Strutinsky method or
algebraic approximations. The 40Ca, 44Ti, 48Cr, 56Ni and 126Ba nuclei have been particularly
investigated. Besides the normally deformed ground state, minima able to lodge super and highly
deformed states appear progressively with increasing angular momenta. The different possible
entrance channels leading to the superheavy elements are compared while, for the exit channel, the
predictions for the -decay half-lives are given.
1. Introduction
The formation of super and highly deformed rotating nuclei as well as superheavy
elements is possible only via fusion reactions and the potential barriers encountered by the
nuclear system in the entrance channel play a main role. In this path the saddle-point
corresponds to two separated but close nuclei maintained in unstable equilibrium by the
balance between the attractive nuclear proximity forces and the repulsive Coulomb forces.
Later on, a bridge of matter takes form between the two still almost spherical nuclei. Finally
the nuclear system undergoes fusion, quasi-fission or fission while emitting some nucleons.
In the present study these potential barriers have been determined using compact
quasimolecular shapes and a macromicroscopic energy. The macroscopic part is obtained
from a generalized liquid drop model including adiabatically the proximity effects. The shell
effects are calculated within the asymmetric two-center shell model and the Strutinsky method
or a simpler algebraic approach derived from the droplet model.
2. Generalized Liquid Drop Model
The GLDM includes an accurate nuclear radius, the mass and charge asymmetry and a
proximity energy term. This last additional term takes into account the effects of the surface
tension forces between surfaces in regard in a neck or a gap without frozen density
approximation. The proximity function is effectively integrated in the neck and, consequently,
the proximity energy depends explicitly on the adopted nuclear shape. The selected shape
sequence derived from lemniscatoids simulates quasimolecular shapes. It allows with only
two variables to describe smoothly the rapid formation of a deep neck and its filling while
keeping almost spherical ends (Fig. 1). It has been previously demonstrated that the
combination of this selected shape sequence and potential energy is able to reproduce most of
the fusion, fission, alpha and cluster emission data [1-4].
Fig. 1. Two joined lemniscatoids simulating the fusionlike fission path.
3. Highly deformed rotating nuclei
In a previous study [5] the rotating Zr, Ce, Dy and Hg nuclei have been particularly
studied. The main characteristics of the observed superdeformed states such as excitation
energy, quadrupole moment, moment of inertia and spin range have been roughly reproduced.
Here, firstly, predictions relative to the 126Ba nuclear system are given [6]. The
64
Ni(64Ni,2n)126Ba reaction is experimentally under investigation. The calculated potential
barriers (Fig. 2.) are scission barriers preventing the separation of the fragments. The barrier
top corresponds to two separated spheres maintained in unstable equilibrium by the balance
between the attractive nuclear forces and the repulsive Coulomb ones. Macroscopically, with
increasing angular momenta a relative plateau appears due to the proximity energy which
introduces an inflection in the potential energy curve. Above around 50  a macroscopic
highly deformed minimum appears at the bottom of the potential barrier. This potential pocket
is removed by the centrifugal forces only for high angular momenta in reason of the high
curvature and high values of the moment of inertia at the barrier top. This is a specificity of
this deformation path to make compatible a high stability of rotating nuclei with low values of
the fission barrier heights.
The shell effects generate a normally deformed minimum close to the sphere. It
corresponds to the slightly deformed ground state (  =0.26). It survives till around 60  . With
increasing angular momenta the shell effects create a second superdeformed minimum in the
macroscopic plateau, in the 46 to 80  range. It becomes the lowest one for 53  . Its moment
of inertia evolves from 5223..75  2 .MeV 1 to 53.813..89  2 .MeV 1 , the quadrupole moment from
0.09
7.7 12.4 eb to 8.6 20.9 eb ,  from 0.4900..08
11 to 0.54 0.05 and the excitation energy from 23.5
MeV to 65 MeV. The limits of the uncertainty range correspond to the characteristics of the
deformed nucleus lodging in the same potential pocket and having an energy of around 0.5
MeV above the minimum energy. A third highly deformed minimum exists from 55  to
around 100  . It becomes the lowest minimum above 70  . Its moment of inertia varies from
66.2 35..54 to 84.855..44  2 .MeV 1 , the quadrupole moment from 14.5 12..44 to 21.7 22 eb,  from
0.03
0.7800..04
09 to 0.96 0.04 and the excitation energy from 36.5 MeV to 83.5 MeV.
Superdeformed rotational bands have been observed also recently in 40Ca [7]. The high
spin states were populated via the reaction 28Si(20Ne,2  )40Ca with an effective beam energy
of 80 MeV. The measured quadrupole moment for transition between states with spin from 16
to 2  is 1.80 00..39
29 eb ; the excitation energy varying from 22.1 to 5.6 MeV; the moment of
inertia being around 8  2 .MeV 1 .
The calculated barriers [8] are displayed in fig. 3. At l=0 there is a plateau but not deep
second minimum. The first minimum around the sphere disappears at the highest angular
momenta. With increasing angular momentum, a second highly deformed minimum appears
and becomes the lowest one also around 17  . Its precise location is due to the shell effects
but the underlying macroscopic energy plays also an important role, particularly the inflection
in the curve due to the proximity energy. The calculated quadrupole moment for transition
between states with spin from 16 to 2  varies then from 2.300..54 eb at 16  to 1.8 00..36 eb at 2
 . The moment of inertia evolves from 9.4 10..08 to 8.310..16  2 .MeV 1 and the excitation energy
from 24.3 to 9.0 MeV. These results agree with the recent experimental data except for the
excitation energy at 2  which is too large.
Some data have been extracted for 44Ti from light charged particle-heavy ion coincident
measurements in the 16O+28Si reaction [9]. The involved highest excitation energy has led to a
value of 35  for the critical angular momentum and an effective moment of inertia of around
11.8  2 .MeV 1 . The macromicroscopic l-dependent potential barriers are shown in fig. 4. A
superdeformed shallow minimum appears only at 21  with a moment of inertia of
10.6 12  2 .MeV 1 and for an excitation energy of 34.5 MeV. It progressively moves to more
external positions. The critical angular momentum is around 36  . Then the moment of inertia
is 13.0 10..76  2 .MeV 1 and the excitation energy 70.5 MeV. These results agree roughly with
the experimental data.
The l-dependent potential barriers governing the 28Si+28Si reaction are given in Fig. 4.
Experimentally, resonances at excitation energies of 70 MeV and spin 42  have been seen.
For an angular momentum of 37  a highly deformed configuration having a moment of
inertia of around 15  2 .MeV 1 has been observed. At low excitation energies two rotational
bands have been detected. One band corresponds to transition between states with spin from
17 to 9  the excitation energy varying from 19.5 to 10.9 MeV (14.5 MeV for 13  ); the
moment of inertia being around 11.5  2 .MeV 1 and the quadrupole moment roughly 2 eb.
Within our approach, besides the spherical ground state we obtain a second minimum for
angular momenta varying from 12 to 50  . For l = 42  its excitation energy is 69.4 MeV
while at l=37  the moment of inertia is 17.811..66  2 .MeV 1 . For spin from 17 to 12  , the
excitation energy varies from 23.5 MeV to 18.0 MeV; the moment of inertia from 14.111..21 to
13.510..39  2 .MeV 1 , and the quadrupole moment from 2.800..67 to 2.500..76 eb. At low spins our
results seem to slightly overestimate the excitation energy and the deformation, besides the
macromicroscopic second well appears at a too high spin. Nevertheless there is no
fundamental disagreement and the existence of hyperdeformed configurations is naturally
accounted for.
Finally the macromicroscopic potential barriers governing the evolution of the 12C+36Ar,
16
O+32S, 20Ne+28Si and 24Mg+24Mg nuclear systems leading to the 48Cr nucleus are displayed
in the lower part of Fig. 4. The persistence of a slightly deformed minimum in all cases at low
angular momentum suggests a non spherical ground state of 48Cr. A very deformed minimum
appears at high spins.
Fig. 2. Sum of the macroscopic (upper part) and macromicroscopic (lower part) symmetric
deformation and rotational energies for 126Ba as functions of the distance r between the mass
centres and the angular momentum (  unit).
Fig. 3. Sum of the macromicroscopic
deformation and rotational energies of the
20
Ne+20Ne nuclear system as functions of
the distance r and the angular momentum.
The vertical dashed line indicates the
transition between one-body and twobody shapes.
====
Fig. 4. Same as Fig. 2 but for the
24
Mg+24Mg nuclear systems.
16
O+28Si,
28
Si+28Si,
12
C+36Ar,
16
O+32S,
20
Ne+28Si and
Angular momenta around 30  seems to be the most favourable value range for
stabilizing super and highly deformed states in this well. For the symmetric reaction, the
behaviour is somewhat different and two minima are visible from roughly 20 to 35 . As the
angular momentum increases, pockets are removed by centrifugal forces, the moment of
inertia varying from 8.7  2 .MeV 1 at 5  to 11 2 .MeV 1 at 35  for the inner well and from
9.1 2 .MeV 1 at 20  to 20.0  2 .MeV 1 at 35  for the external potential well. The excitation
energy varies from 37 MeV at 20  to 61 MeV at 35  for the outer pocket. Experimentally,
in 24Mg+24Mg scattering tentative spins of 36, 38 and 40  have been assigned [10] to three
resonant states at saddle-point energies from 63 to 70 MeV. Our calculated excitation energy
and spin range agree also roughly with these experimental data.
4. On the superheavy elements
The synthesis of very heavy elements has apparently strongly advanced recently using
cold (Zn on Pb [11]) and warm(Ca on U, Pu and Cm [12-14]) fusion reactions. The observed
decay mode is mainly the  emission. The analysis of the experimental data is discussed [15].
Recently [16] in a search for the production of element 112 in the 231 MeV 48Ca+238U
reaction no any events have been observed and an upper limit cross section of 1.8 pb for
evaporation residue-alpha events has been proposed.
Within our approach in the same deformation path these reactions have been recently
investigated [17]. Potential barriers against fusion via the cold reactions 70Zn and 82Se on 208Pb
and warm fusion reactions 48Ca on 238U and 248Cm are displayed in Fig. 5. Different
hypotheses have been assumed. The dashed line corresponds to the pure macroscopic
potential energy. In ordinary fusion studies, it is only that barrier which is taken into account.
The dashed-dotted line incorporates the shell corrections calculated empirically from the
Droplet Model. The solid line is the sum of the GLDM macroscopic energy and of the shell
effects calculated within the asymmetric two-center shell model and adjusted to reproduce the
experimental or estimated (from the Thomas-Fermi model) Q value with a corrected factor
beginning at the contact point. This supposes that the nuclear system has enough time to relax
and built its shells and pairs and then that all the microscopic components contribute to the
total energy. The first external top corresponds always to two separated sphere configurations;
the attractive nuclear forces compensating for the repulsive Coulomb forces. At the contact
point the first external top of the barrier is already passed.
In the cold fusion reactions a wide potential pocket mainly due to the proximity energy
appears at large deformations. From the Zn on Pb reaction, double hump fusion barriers
appear and the inner peak is the highest for the heaviest systems. In the deep minimum
between the two maxima incomplete fusion and fast fission processes may develop since the
neck between the two nuclei is formed and exchanges of nucleons occur. The remaining
excitation energy of the composite system is crucial to decide between complete fusion and
fast fission. It depends on the pre or post equilibrium nature of the evaporation process of the
excess neutron.
In the warm fusion reactions, due to the asymmetry there is no double hump barriers. The
barrier against reseparation being high and wide the system descends toward a quasi-spherical
shape but with an excitation energy of more than 30 MeV if one assumes a full relaxation.
The emission of several neutrons or even an  particle is energetically possible.
The dependence on the value of the next proton magic number is shown on Fig. 6 for the
cold 86Kr + 208Pb fusion reaction and warm 50Ti + 248Cm one. The dashed curve corresponds to
the pure macroscopic barrier and the dashed double dotted line takes into account the exact Q
value. The solid curve corresponds to a proton magic number of 114, the dotted line to 120
and the dashed-dotted one to 126. The height of the first inner peak relatively to the sphere
energy which governs the choice of the beam energy and the stability of the eventual
compound nucleus increases with the proximity of the magic number. Nevertheless the main
features of the barriers are the same which underline the importance of the macroscopic
energy and particularly the proximity energy and the hypothesis of a full relaxation of the
system contained in the exact Q value assumption.
Recently, other reactions have been proposed to form these superheavy elements using
radioactive ion beams or more symmetric reactions [18-19]. The macroscopic fusion barriers
standing before some of these systems leading to the 270110 and 302120 nuclei are displayed in
Fig. 7. It seems that it will be difficult to reach the ground state via the almost symmetric
combinations. Furthermore it has been shown [1] that dynamic fusion barriers significantly
higher than the static ones appear for Z1Z 2  2100  100. With increasing asymmetry the
excitation energy increases strongly and it seems possible for some reactions to reach the
external pocket and, later on, quasispherical configurations with little excitation energy using
tunneling effects through the barriers. This does not prove the stability of the formed system.
For the exit channel all the symmetric and asymmetric fission and  emission barriers must
be investigated.
To determine these  emission barriers the  decay has been viewed [3] as a very
asymmetric spontaneous fission within the generalized liquid drop model. The difference
between the experimental and theoretical Q  value has also been added at the sphere energy
with a linear attenuation factor vanishing at the rupture point. Within such an unified fission
model, the decay constant of the parent nucleus is simply defined as    0 P . There is no
adjustable preformation factor. The assault frequency  0 has been chosen as 10 20 s 1 . The
barrier penetrability P has been calculated within the general form of the action integral. The
predicted  decay half-lives agree with the experimental data in the whole mass range and
also for the known heaviest elements [3, 16]. Analytic formulas have been proposed. For the
even-even nuclei, the  decay half-lives may be calculated using
1.5864 Z
log10 T1/ 2 (s)  25.31  1.1629A1/ 6 Z 
.
Q
For the heaviest elements the theoretical Q  value given by the Thomas-Fermi model has
been selected since it reproduces correctly the mass decrements from Fermium to Z = 112.
The table gives the predictions for the superheavy elements. If such nuclei exist, their halflives vary from microseconds to some days. Generally, for a given element, the half-lives
increase with the neutron number. Curiously some nuclei (319,320126 and 317124) have a very
low Q  value (within the Thomas-Fermi model) and consequently a very high  decay halflife. The calculation of the half-lives against symmetric and asymmetric fission is another
challenge.
Fig. 5. Fusion barriers versus the mass-centre distance r for the 70Zn and 82Se on 208Pb cold
reactions and 48Ca on 238U and 248Cm warm reactions.
Fig. 6. Fusion barriers versus r for the 86Kr + 208Pb and 50Ti + 248Cm reactions.
Fig. 7. Macroscopic potential barriers for reactions leading to the
The contact point is indicated by a vertical bar.
Z
126
A
Q
Log[T]
124
A
Q
Log[T]
122
A
Q
Log[T]
120
A
Q
Log[T]
118
A
Q
Log[T]
117
A
Q
Log[T]
116
A
Q
Log[T]
115
A
Q
Log[T]
114
A
Q
Log[T]
312
16.78
-10.51
306
16.57
-10.6
300
14.23
-7.24
294
13.53
-6.41
291
12.79
-4.73
289
12.20
-4.12
286
11.65
-3.54
284
10.56
-0.02
283
9.79
1.38
313
16.22
-8.83
307
16.29
-9.37
301
14.14
-6.28
295
13.59
-5.75
292
12.59
-5.09
290
12.14
-3.47
287
11.52
-2.52
285
10.55
-0.74
284
9.64
1.10
314
15.95
-9.29
308
16.39
-10.37
302
14.3
-7.4
296
13.92
-7.17
293
12.49
-4.14
291
11.94
-3.58
288
11.55
-3.35
286
10.45
0.26
285
9.55
2.07
315
15.58
-7.86
309
16.74
-10.06
303
14.96
-7.75
297
13.78
-6.15
294
12.51
-4.96
292
11.93
-3.00
289
11.50
-2.51
287
10.48
-0.59
286
9.61
1.16
316
15.29
-8.26
310
16.58
-10.69
304
15.06
-8.74
298
13.58
-6.57
295
12.42
-4.02
293
11.91
-3.55
290
11.34
-2.89
288
10.34
0.54
287
9.53
2.10
317
15.0
-6.92
311
15.75
-8.61
305
15.16
-8.11
299
13.33
-5.32
296
12.52
-5.02
294
11.90
-2.96
291
11.33
-2.14
289
10.24
0.03
288
9.39
1.80
318
14.71
-7.3
312
15.54
-9.13
306
15.16
-8.95
300
13.63
-6.71
297
12.34
-3.88
295
11.80
-3.34
292
11.03
-2.16
290
10.15
1.07
289
9.08
3.49
270
319
6.71
18.6
313
14.91
-7.27
307
15.83
-9.21
301
13.91
-6.45
298
12.73
-5.49
296
11.59
-2.23
293
11.15
-1.74
291
9.88
1.03
290
8.73
3.95
110 and
320
6.89
16.7
314
14.49
-7.39
308
15.69
-9.83
302
13.95
-7.34
299
12.87
-5.03
297
11.97
-3.76
294
11.19
-2.60
292
9.75
2.28
291
8.66
4.90
302
120 nuclei.
321
10.9
1.94
315
14.23
-6.10
309
15.29
-8.39
303
14.07
-6.78
300
12.94
-5.96
298
12.16
-3.65
295
11.06
-1.55
293
9.69
1.56
292
8.47
4.85
322
12.16
-2.16
316
13.76
-6.08
310
15.35
-9.33
304
14.06
-7.57
301
13.05
-5.42
299
12.25
-4.41
296
11.33
-2.97
294
9.46
3.20
293
8.46
5.58
Table. Predicted Log10[T1/2(s)] for the superheavy elements versus the charge and mass of the
mother nucleus and Q.
323
12.01
-0.96
317
7.1
15.53
311
14.5
-7.09
305
14.72
-7.94
302
13.07
-6.25
300
12.35
-4.13
297
11.38
-2.36
295
9.87
0.99
294
8.83
3.52
5. Conclusion
The potential barriers standing in the fusionlike deformation path for rotating medium
mass nuclei and for the heaviest elements have been determined within a macromicroscopic
energy determined from a generalized liquid drop model and an asymmetric two center shell
model or simpler algebraic expressions.
For most of the nuclei, at low angular momenta the shell effects generate a minimum
close to the sphere which lodges the normally deformed states. It disappears progressively
with increasing angular momenta while the proximity energy and microscopic contributions
create a second minimum where superdeformed states may survive. It becomes progressively
the lowest one at intermediate spins. At still higher angular momenta, the minimum moves
towards the foot of the external scission barrier leading to highly deformed quasimolecular
states. The calculated characteristics roughly agree with the recent observed data on
superdeformed and perhaps highly deformed bands in these nuclei.
For the superheavy elements the cold fusion reactions take place in a double hump path.
Incomplete fusion events may appear between the two peaks. An open question is to know
whether at these large deformations the nucleon shells can take form before investigating a
peculiar exit channel. The moment of the one neutron emission determines the nature of the
reaction. For the heaviest systems and relatively symmetric reactions, the inner barrier is the
highest. In the warm fusion reactions, there is no double hump barriers and the system
descends automatically toward a quasispherical shape but with an excitation energy which is
very high. The rapid emission of several neutrons or even an alpha particle is possible. So, it
is important to be sure that the observed events are not incomplete fusion events. The alpha
decay half-lives of these systems have been calculated within this GLDM adjusted to
reproduce the experimental or the Thomas-Fermi model Q value. The agreement with
experimental data is correct. Analytic formulas have been also proposed.
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