Formation of Heavy and Superheavy
Neutron Rich Nuclei
1
Valery Zagrebaev
∗
and Walter Greiner
†
∗
Flerov Laboratory of Nuclear Reaction, JINR, Dubna, 141980, Moscow region, Russia
†
Frankfurt Institute for Advanced Studies, J.W. Goethe-Universität, Frankfurt, Germany
Abstract. A new way is proposed to discover and examine unknown neutron-rich heavy and
superheavy nuclei at the “north-east” part of the nuclear map. The present limits of the upper part of
the nuclear map is very close to stability while the unexplored area of heavy neutron-rich nuclides
to the east of the stability line (also those located along the neutron closed shell N=126) is extremely
important for nuclear astrophysics investigations and for the understanding of the r-process of
astrophysical nucleo-genesis. A novel idea is proposed for the production of these nuclei via low-
energy multi-nucleon transfer reactions using a gain given by the shell effects. This finding may spur
new studies at heavy ion facilities and should have significant impact for future experiments. The
use of the multi-nucleon transfer reactions in low-energy collisions of heavy actinide nuclei gives
us also the only chance to approach the “island of stability” of superheavy elements. A possibility
for a three-body decay (heavy ternary fission) is predicted for superheavy nuclei.
Keywords: heavy neutron rich nuclei; multi-nucleon transfer; ternary fission.
PACS: 25.70.Hi, 25.70.Jj
MOTIVATION
During the last years, the progress in the investigation of exotic nuclei has been impres-
sive. Nowadays, nuclei far from stability are accessible for experimental study in almost
any region of the chart of the nuclides. The only exception is the northeast part. The
present limits of the chart of the nuclides in the region of heavy neutron-rich nuclei is
still very close to stability although this region is extremely interesting for nuclear struc-
ture and nuclear astrophysics investigations, see Fig. 1. Study of the structural properties
of nuclei along the neutron shell N=126 would contribute to the present discussion of the
quenching of shell effects in nuclei with large neutron excess. Moreover, this region is
extremely interesting in astrophysics, in particular, for the production of heavy elements
in stellar nucleosynthesis through the r-process (the last “waiting point”). According to
a recent report by the National Research Council of the National Academy of Sciences
(USA), the origin of heavy elements from iron to uranium remains one of the greatest
unanswered questions of modern physics (see, for example, [1]) and it is likely to remain
a hot research topic for the years to come.
Usually, new (neutron and proton rich) isotopes located far from the stability line
are obtained in the fragmentation processes, fission of heavy nuclei and in low-energy
fusion reactions. The first two methods are extensively used today for production of new
isotopes in the light and medium mass region including those which are close to the drip
1
AIP Conference Proceedings, 1224 (2010) 259
lines. Because of the “curvature” of the stability line, in the fusion reactions of stable
nuclei, we may produce only proton rich isotopes of heavy elements. That is why we
cannot reach the center of the “island of stability” (Z∼114, N∼184) in the SH mass
region and have almost no information about neutron-rich isotopes of heavy elements
(there are 18 known neutron-rich isotopes of Cs, for example, and only 4 of Pt).
The SH elements produced in the “cold” fusion reactions based on the closed shell
target nuclei of lead and bismuth are situated along the proton drip line being very
neutron-deficient with a short half-life [2, 3]. The cross sections for SH element pro-
duction in more asymmetric (and “hotter”) fusion reactions of
48
Ca with actinide targets
were found to be much larger [4]. These reactions lead to more neutron-rich SH nuclei
with much longer half-lives. However, they are still far from the center of the predicted
island of stability formed by the neutron shell around N=184 (these are the
48
Ca induced
fusion reactions which confirm an existence of this island of stability). Moreover, cal-
ifornium is the heaviest actinide that can be used as a target material in this method.
In this connection, other ways for the production of SH elements with Z>118 and also
neutron-rich isotopes of SH nuclei in the region of the island of stability should be
searched for [5].
FIGURE 1. Top part of the nuclear map. The r-process path is shown schematically. In the inset the
abundance of the elements is shown.
As can be seen from Fig. 1, there is also a gap between the SH nuclei produced in
the “hot” fusion reactions with 48Ca and the continent. This gap does not allow one to
depart from known nuclei when we attribute the charge and mass numbers to the new
SH elements produced in the “hot” fusion reactions. It also hinders one from obtaining
a clear view of the properties of SH nuclei in this region. It is rather difficult to fill this
gap in fusion reactions of stable nuclei.
Here we propose to solve all these problems by production of new heavy neutron-rich
nuclei in the multi-nucleon transfer processes of low-energy damped collisions of heavy
ions. It is well known that in the deep inelastic (or damped) collisions of heavy ions, the
relative motion energy is transformed into the internal excitation of the projectile-like
and target-like reaction fragments which are de-excited by evaporation of light particles
(mostly neutrons). This does not seem to give us a chance for production of neutron-rich
nuclei in such reactions at above-barrier energies. However we found that even at low
collision energies of heavy ions (close to the Coulomb barrier), the cross sections for
the multi-nucleon transfer processes are still rather high. The excitation energy of the
primary reaction fragments produced in these collisions should be not very high, and
after evaporation of a few neutrons, one of the surviving residual nuclei might remain
far from the stability line.
MULTI-NUCLEON TRANSFER REACTIONS
Several models have been proposed and used for the description of mass transfer in deep
inelastic heavy ion collisions, namely, the Focker-Planck [6] and master equations [7],
the Langevin equations [8] and more sophisticated semiclassicall approaches [9, 10].
We employ here the model of low-energy collisions of heavy ions proposed in [11, 12].
This model is based on the Langevin-type dynamical equations of motion. The distance
between the nuclear centers R, dynamic surface deformations
β
1
and
β
2
, and the neutron
and proton asymmetries,
η
N
= (2N −N
CN
)/N
CN
and
η
Z
= (2Z −Z
CN
)/Z
CN
, are the most
relevant degrees of freedom for the description of deep inelastic scattering and fusion-
fission reactions. The use of the neutron and proton asymmetry parameters allows one to
describe properly neutron and proton transfers and obtain the yields of different isotopes
of a given element (including extremely neutron rich ones) [13].
In our approach, the potential energy is calculated within the double-folding proce-
dure (sum of the nucleon-nucleon forces averaged over “frozen” densities of collid-
ing nuclei) at initial (fast or diabatic) reaction stage. After damping of relative mo-
tion kinetic energy and overlapping of the nuclear surfaces (slow reaction stage, adi-
abatic conditions) nucleons have enough time to reach equilibrium distribution within
a volume with normal nuclear density. The corresponding adiabatic potential energy
of the nuclear system is calculated using the extended version of the two-center shell
model [14]. Thus, for the nucleus-nucleus collisions at energies above the Coulomb bar-
rier we need to use a time-dependent potential energy, which after contact gradually
transforms from a diabatic potential energy into an adiabatic one: V (R,
β
,
η
N
,
η
Z
;t) =
V
diab
[1− f (t)]+V
adiab
f (t). Here t is the time of interaction and f (t) is a smoothing func-
tion satisfying the conditions f (t = 0) = 0 and f (t >>
τ
relax
) = 1,
τ
relax
is the adjustable
parameter ∼ 10
−21
s. The diabatic and adiabatic potentials depend on the same variables
and are equal to each others for well separated nuclei. Thus, the total potential energy,
V (R,
β
,
η
N
,
η
Z
;t), provides quite smooth driving forces −
∂
V /
∂
q
i
.
For all the variables, with the exception of the neutron and proton asymmetries, we
use the usual Langevin equations of motion with the inertia parameters,
µ
R
and
µ
β
,
calculated within the Werner-Wheeler approach. For the mass and charge asymmetries
the inertialess Langevin type equations are used (derived from the master equations for
the corresponding distribution functions)
d
η
N
dt
=
2
N
CN
D
(1)
N
+
2
N
CN
√
D
(2)
N
Γ
N
(t),
d
η
Z
dt
=
2
Z
CN
D
(1)
Z
+
2
Z
CN
√
D
(2)
Z
Γ
Z
(t),
(1)
where Γ(t) is the normalized random variable with Gaussian distribution and D
(1)
,
D
(2)
are the transport coefficients. Assuming that sequential nucleon transfers play a
main role in mass rearrangement, i.e. A
′
= A ± 1, we have D
(1)
N,Z
=
λ
N,Z
(A → A + 1) −
λ
N,Z
(A → A−1), D
(2)
N,Z
=
1
2
[
λ
N,Z
(A → A+1)+
λ
N,Z
(A → A−1)], where the macroscopic
transition probability
λ
(±)
N,Z
(A → A
′
= A ± 1) is defined by nuclear level density [6, 7],
λ
(±)
N,Z
=
λ
0
N,Z
√
ρ
(A ± 1)/
ρ
(A) and
λ
0
N,Z
are the neutron and proton transfer rates (note
that the nuclear level density
ρ
∼ exp(2
√
aE
∗
) depends on the excitation energy E
∗
=
E
c.m.
− V (R,
β
1
,
β
2
,
η
N
,
η
Z
) − E
rot
and, thus, on all the degrees of freedom used in the
model). There is no information in literature on a difference between neutron and
proton transfer rates, and for simplicity we assume here that
λ
0
N
=
λ
0
Z
=
λ
0
/2, where
λ
0
is the nucleon transfer rate which was estimated to be ∼ 10
22
s
−1
[6, 7]. We treat
λ
0
as a parameter which should be chosen from appropriate description of available
experimental data on deep inelastic scattering [11, 12].
Nucleon transfer for slightly separated nuclei is also rather probable. This interme-
diate nucleon exchange plays an important role in sub-barrier fusion processes and has
to be taken into account in Eq. (1). It can be treated by using the following final ex-
pression for the transition probability
λ
(±)
N,Z
=
λ
0
N,Z
√
ρ(A±1)
ρ(A)
P
tr
N,Z
(R,
β
, A → A ± 1). Here
P
tr
N,Z
(R,
β
, A → A ± 1) is the probability of one nucleon transfer, which depends on the
distance between the nuclear surfaces. This probability goes exponentially to zero at
R → ∞ and it is equal to unity for overlapping nuclei [15].
The double differential cross-sections of all the processes are calculated as follows
d
2
σ
N,Z
dΩdE
(E,
θ
) =
∫
∞
0
bdb
∆N
N,Z
(b, E,
θ
)
N
tot
(b)
1
sin(
θ
)∆
θ
∆E
.
(2)
Here ∆N
N,Z
(b, E,
θ
) is the number of events at a given impact parameter b in which a
nucleus (N, Z) is formed with kinetic energy in the region (E, E + ∆E) and center-of-
mass outgoing angle in the region (
θ
,
θ
+ ∆
θ
), N
tot
(b) is the total number of simulated
events for a given value of impact parameter. Expression (2) describes the mass, charge,
energy and angular distributions of the primary fragments formed in the binary reaction.
Subsequent de-excitation of these fragments via fission and emission of light particles
and gamma-rays were taken into account explicitly for each event within the statistical
model leading to the final distributions of the reaction products. The sharing of the
excitation energy between the primary fragments is assumed to be proportional to
their masses. For both excited fragments the multi-step decay cascade was analyzed
taking into account a competition between evaporation of neutrons and/or protons and
fission. Note that the model was already applied successfully for description of the
angular, energy and mass distributions of reaction products observed in the deep inelastic
scattering of heavy ions at above barrier energies [11, 12].
PRODUCTION OF NEW HEAVY NEUTRON RICH NUCLEI AT
THE “NORTH-EAST” PART OF NUCLEAR MAP
A novel idea was recently proposed in [13, 16] for the production of the heavy neutron-
rich nuclei (located along the closed neutron shell N = 126) via the multi-nucleon trans-
fer processes of low-energy collisions of heavy ions. It is well known that in the deep
inelastic (damped) collisions of heavy ions the relative motion energy is quickly trans-
formed into internal excitation of the projectile-like and target-like reaction fragments
which are de-excited then by evaporation of light particles (mostly neutrons). This seems
not to give us a chance for production of nuclei with large neutron excess in such reac-
tions. However, if the colliding energy is rather low and the reaction Q-value is not very
high, the formed primary reaction fragments might be not very much excited and will de-
scend to their ground states after evaporation of a few neutrons thus remaining far from
the stability line. The questions are how big is the cross section for the multi-nucleon
transfer reactions at low colliding energies and could these reactions be considered as an
alternative way for the production of exotic nuclei.
FIGURE 2. (Left panel) Schematic picture for preferable proton transfer reactions in low-energy col-
lisions of
136
Xe with
208
Pb. Black rectangles indicate stable nuclei. (Right panel) Landscape of the total
cross section d
2
σ
/dZdN (microbarns, numbers near the curves) for the production of heavy fragments in
collisions of
136
Xe with
208
Pb at E
c.m.
= 450 MeV.
For the production of heavy neutron rich nuclei located along the neutron closed shell
N = 126 we proposed to explore the multi-nucleon transfer reactions in low-energy
collisions of
136
Xe with
208
Pb. The idea is to use the stabilizing effect of the closed
neutron shells in both nuclei, N = 82 and N = 126, respectively (see the left panel of
Fig. 2). The proton transfer from lead to xenon might be rather favorable here because
the light fragments formed in such a process are well bound (stable nuclei) and the
reaction Q-values are almost zero. For example, even for a transfer of 6 protons in the
reaction
136
Xe +
208
Pb →
142
Nd +
202
Os the Q-value is equal to -8.3 MeV.
The landscape of the calculated cross sections for the yield of the different reaction
fragments in low-energy collision of
136
Xe with
208
Pb is shown in the right panel of
Fig. 2, whereas the cross sections for production of final (after a few neutron evapora-
tion) heavy neutron-rich nuclei in this reaction at the incident energy E
c.m.
= 450 MeV,
which is very close to the Coulomb barrier (Bass barrier for this combination is about
434 MeV), is shown in Fig. 3.
Thus, we may conclude that the low-energy multi-nucleon transfer reactions can be
really used for the production of heavy neutron rich nuclei. The estimated yields of
neutron-rich nuclei are found to be rather high in such reactions (much larger than in
high-energy proton-removal nuclear reactions [17], see Fig. 3b) and several tens of new
nuclides can be produced, for example, in the near-barrier collision of
136
Xe with
208
Pb.
This finding may spur new studies at heavy-ion facilities and should have significant
impact on future experiments. Similar reactions with uranium and thorium targets may
be used for the production of new neutron rich isotopes with Z ≥ 82. Accelerated neutron
rich fission fragments (which hardly may be useful for the synthesis of superheavy
nuclei in fusion reactions due to low cross sections [5]) look especially promising for
production of new heavy neutron rich isotopes in low-energy multi-nucleon transfer
processes. In the
132
Sn+
208
Pb reaction, for example, the nuclei
202
Os
N=126
(6 protons
transferred) and
200
W
N=126
(8 protons transferred) are produced with the Q-values of +4
and -3 MeV, correspondingly, which should significantly increases the cross sections.
Note, that a possibility for the production of new heavy isotopes in the multi-nucleon
transfer reactions with neutron-rich calcium and xenon beams at much higher energies
(at which the shell effects do not play any role) has been discussed also in [18] within
the semiclassical approach.
FIGURE 3. (a) Cross sections for production of heavy neutron-rich nuclei in collisions of
136
Xe with
208
Pb at E
c.m.
= 450 MeV. Open circles indicate unknown isotopes. (b) Yield of nuclei with neutron closed
shell N = 126. The dashed curve shows the yield of the nuclei in high energy proton removal process [17].
Cross sections of one microbarn are quite reachable at the available experimental se-
tups. However the identification of new heavy nuclei obtained in the multi-nucleon trans-
fer reactions is a rather complicated problem. Most of these nuclei undergo
β
−
decay.
The atomic mass could be determined by the time-of-flight technique rather accurately.
The identification of the atomic number of the heavy nucleus is more difficult. The same
is true for the determination of its half-life, which is the most important property of the
nuclei in the region of N ∼ 126 (last waiting point in the r-process). In principle, it could
be done by the registration of the electron cascade in the
β
−
decay chain in coincidence
with the gamma-rays of the daughter nuclei. Anyhow, the synthesis and study of these
nuclei (important for many reasons) is a challenge for low-energy nuclear physics now
and in forthcoming years.
PRODUCTION OF SUPERHEAVY ELEMENTS IN COLLISIONS
OF ACTINIDE NUCLEI
The use of multi-nucleon transfer from heavy-ion projectile to an actinide target nucleus
for the production of new nuclear species in the transuranium region has a long history.
The level of 0.1
µ
b was reached for chemically separated Md isotopes [19]. These ex-
periments seem to give not so great chances for production of new SH nuclei. However,
we may expect that the shell structure of the driving potential (deep valleys caused by
the double shell closure Z = 82 and N = 126) might significantly influence the nucleon
rearrangement between primary fragments at low collision energies.
FIGURE 4. Potential energy at contact “nose-to-nose” configuration for the two nuclear systems formed
in
48
Ca+
248
Cm (a) and
232
Th+
250
Cf (b) collisions. The spheroidal deformation is equal to 0.2 for both
cases. The arrows indicate initial configurations and possible clusterization of nuclear systems.
In Fig. 4, the potential energies are shown depending on mass rearrangement at con-
tact configuration of the nuclear systems formed in
48
Ca+
248
Cm and
232
Th+
250
Cf colli-
sions. The lead valley evidently reveals itself in both cases (for
48
Ca+
248
Cm system there
is also a tin valley). In the first case (
48
Ca+
248
Cm), a discharge of the system into the lead
valley (normal or symmetrizing quasi-fission) is the main reaction channel, which de-
creases significantly the probability of CN formation [20]. In collisions of heavy nuclei
(Th+Cf, U+Cm and so on), one may expect that the existence of this valley can no-
tably increase the yield of surviving neutron-rich super-heavy nuclei complementary to
the projectile-like fragments around
208
Pb (“inverse” or anti-symmetrizing quasi-fission
process) [21].
FIGURE 5. Landscape of the cross sections (microbarns, logarithmic scale) for production of primary
fragments in collision of
238
U with
248
Cm at 780 MeV center-of-mass energy (contour lines are drawn over
one order of magnitude). Vertical and horizontal strips indicate the magic proton and neutron numbers.
The calculated cross sections for formation of primary fragments in low-energy colli-
sions of
238
U with
248
Cm target are shown in Fig. 5 by the counter lines in logarithmic
scale. As can be seen, the superheavy nuclei located very close to the center of the is-
land of stability may be produced in this reaction with rather high cross section of one
microbarn. Note one again that this region of the nuclear map cannot be reached in any
fusion reaction with stable projectiles and long-lived targets. Of course, the question
arises whether these excited superheavy primary fragments produced in the transfer re-
actions may survive in competition with fast fission which is a dominated decay channel
for heavy nuclei.
Indeed, the yield of survived SH elements produced in the low-energy collisions of
actinide nuclei is rather low, though the shell effects (see two double magic crossing in
Fig. 5) give us a definite gain as compared to a monotonous exponential decrease of the
cross sections with increasing number of transferred nucleons. In Fig. 6 the calculated
EvR cross sections for production of SH nuclei in damped collisions of
238
U with
248
Cm
at 800 MeV center-of-mass energy are shown along with available experimental data. As
can be seen, really much more neutron rich isotopes of SH nuclei might be produced in
such reactions (new isotopes of Siborgium (Z = 106) are shown in Fig. 6 by the open
circles). Reactions of such kind can be also used to fill the gap between the SH nuclei
produced in the “hot” fusion reactions with
48
Ca and the continent of known nuclei.
FIGURE 6. Yield of primary and survived isotopes of SH nuclei produced in collisions of
238
U with
248
Cm at 800 MeV center-of-mass energy. Experimental data obtained in [19] are also shown. Dashed
line shows the expected locus of reaction cross sections without the shell effects. Open circles at the curve
with Z = 106 indicate unknown isotopes of Siborgium and their positions at the nuclear map (right panel).
TERNARY FISSION OF SUPERHEAVY NUCLEI
The problem of ternary fission has a long history and was studied both experimentally
and theoretically. One of the first calculations of the fission saddle points for three-
cluster nuclear systems was performed in [22], where the “chain” (three fragments along
the line) and triangle configurations were studied within the liquid drop model. Later in
[23] the three-center shell model was developed to describe potential energy of such
configurations. However an overall study of the problem was not performed yet, and
the question about a possibility for three-body clusterization of heavy nuclei remains
unanswered both in theory and in experiment.
In our opinion, a three-body clusterization might appear in a vicinity of the scission
point of heavy nucleus (more exactly, on the path from the saddle point to scission),
where the shared nucleons ∆A may form a third cluster located between the two heavy
cores a
1
and a
2
. In Fig. 7 schematic view is shown for normal (two-body) and ternary
fission starting from the configuration of the last shape isomeric minimum of CN con-
sisting of two magic tin cores and 36 extra nucleons sharing between the two cores and
moving in the whole volume of the mono-nucleus. In the two-body fission these extra
nucleons pass into one of the fragments with formation of two nuclei in the exit chan-
nel (Sn and Dy in our case, mass-symmetric fission is energetically less favorable here).
However there is a chance for these extra nuclei ∆A to concentrate in the neck region
between the two cores and form finally the third fission fragment (
36
S in our case).
FIGURE 7. Schematic view of the normal (with formation of
130
Sn and
166
Dy nuclei) and heavy ternary
fission of the superheavy nucleus
296
116.
It is clear that there are too many collective degrees of freedom needed for a proper
description of potential energy of a nuclear configuration consisting of three deformed
heavy fragments. Thus, we restricted ourselves by consideration of potential energy,
V (Z
1
, Z
3
, R
12
), of a three-body configuration as a function of three variables, Z
1
, Z
3
and R
12
at fixed deformations of the fragments being in contact (
β
1
=
β
2
=
β
3
= 0.1).
To make the result quite visible we minimized the potential energy over the neutron
numbers of the fragments, N
1
and N
3
.
FIGURE 8. Landscape of the potential energy for three-body clusterization of the nucleus
296
116
formed in collision of
48
Ca with
248
Cm (top) and of giant nuclear system formed in collision of
238
U+
238
U.
We found that for actinide nuclei the potential energy smoothly increases with increas-
ing charge of the third nucleus and no other local deep minima appear on the potential
energy surface. However, the situation changes for heavier transactinide nuclei. With
increasing mass number of heavy nucleus more and more possibilities for its clusteriza-
tion appear. In the top part of Fig. 8 the potential energy is shown for three-body clus-
terization of the nucleus
296
116 formed in collision of
48
Ca with
248
Cm. The two-body
clusterization of this nucleus onto Pb–Se (quasi-fission) and Sn–Dy (normal fission) is
clearly visible in Fig. 8 at Z
3
→ 0. However a possibility for a three-body clusterization
also persists in this nuclear system. A minimum is observed for the Xe–O–Xe configu-
ration and even the combination
130
Sn–
36
S–
130
Sn with formation of two magic clusters
is quite reachable. The energy release (Q-value) for such three-body decay channels is
much higher than for the two-body fission. This fact makes quite possible the observation
of the ternary fission in collisions of medium mass projectiles with actinide targets and,
probably, in spontaneous decay of SH elements. Note, that the experiments, in which the
two-body fission and quasi-fission processes have been studied in collisions of
48
Ca with
248
Cm [20], were not aimed originally on a search of the three-body reaction channels.
This should be done with a new experimental procedure.
Conditions for the three-body fission (quasi-fission) are even better in the giant nu-
clear systems formed in low-energy collisions of actinide nuclei. In this case the shell
effects significantly reduce the potential energy of the three-cluster configurations with
two strongly bound lead-like fragments. In the bottom part of Fig. 8 the landscape of the
potential energy surface is shown for a three-body clusterization of the nuclear system
formed in collision of U+U. Beside the two-body Pb–No clasterization and the local
three-body minimum with formation of light intermediate oxygen-like cluster, the po-
tential energy has the very deep minimum corresponding to the Pb-Ca-Pb–like configu-
ration (or Hg-Cr-Hg) caused by the N=126 and Z=82 nuclear shells. In our opinion, an
existence of this three-body clusterization can be rather easily proved experimentally by
a coincident detection of the two Pb–like fragments in collisions of two uranium nuclei.
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