INTERPLAY BETWEEN COMPOUND AND FRAGMENT ASPECTS
IN NUCLEAR FISSION AND HEAVY-ION REACTIONS
A. Iwamoto 1, T. Ichikawa2 and P. Möller3
1
ASRC, Japan Atomic Energy Agency, Japan
2
3
YITP, Kyoto University, Japan
Los Alamos National Laboratory, USA
Abstract: The scission point in nuclear fission plays a special role where one-body system
changes to two-body system. Inverse of this situation is realized in heavy-ion fusion reaction
where two-body system changes to one body system. Among several peculiar phenomena
expected to occur during this change, we focus our attention to the behavior of compound and
fragments shell effects. Some aspects of the interplay between compound and fragments shell
effect are discussed related to the topics of the fission valleys in the potential energy surface of
actinide nuclei and the fusion-like trajectory found in the cold fusion reaction leading to
superheavy nuclei.
1. Introduction
At some stage of nuclear fission or heavy-ion nuclear fusion, a transition from a one-body
system to two-body system occurs. A simple nuclear model like the mean-field approach faces
difficulties to pass through the critical point of this change. For example, a problem of spurious
center-of-motion happens where the spurious motions in one-body system and in two-body
system should be subtracted to obtain the physically plausible connection between them [1]. Not
only the spurious motion but also physically meaningful zero-point motions of one-body and
two-body systems should be connected smoothly. For these problems a unique method does not
exist and one has to refer to some models. There are also other types of connection problem
between one-body and two-body systems. For examples, nuclear congruence energy [2], and the
Wigner and A0 terms in the mass formula [3] are different in the parent and fragment nuclei and
some prescription is necessary to connect them. A consideration on the property of the constanttemperature level density formula at scission point leads to a special repartition of excitation
energy between fission fragments [4].
In this paper, we are concerned with the connection problem in shell structure. This
problem is important because in fission and heavy-ion fusion reactions, it is common to discuss
the role played by the fragments’ shell near scission point or touching point of target and
projectile, whereas the shell structure of one-body system as a whole is used in mean-field
approach. Our concern is to understand the interplay between shell structures of one-body
system and shell structure of fragments’.
It should be kept in mind that there happens another kind of “sudden” change when onebody system changes to two-body system and vise versa. It can occur in classical liquid-drop
approach [5] and in quantum mean-field approach [6] where the neck radius smaller than a given
value cannot exist as a well-defined shape because the system is never stable against separation
into two fragments with respect to a variable that is not a member of the constraint variables.
Therefore we should be careful about the parameterization of nuclear (or single-particle
potential) shape near the scission point because there is a possibility to miss the physically
important instability in this approach. To check how important is a sudden change at scission, as
has been utilized in random neck rapture model [7], it is necessary to develop a full dynamical
model and we will not touch this problem in this talk.
2. Fission valleys seen in the potential energy of actinide nuclei
We briefly summarize our recent paper on this subject [8]. We use a three quadratic
parameterization to calculate the potential
energy in macroscopic-microscopic method.
This parameterization has a merit that even
the shape near the outer saddle point of
fission and beyond can be described in this
5-parameter family naturally.
Since our
concern is the transition from one-body-like
to two-body-like single-particle potentials
and vise versa, use of this parameterization
has merit. This parameterization has
another merit of ability to naturally describe
the nascent fragments in fission potential
energy surface, which aspect is fully utilized
in the discussion given in the next section.
Fig. 1. Five-parameter shape parameterization
used in the potential energy calculations, taken
from [8].
A
schematic
picture
of
the
shape
parameterization is given in Fig. 1 and as is
shown, we use more than five million mesh
points , which is necessary to find all important minima and saddle points.
After identifying the ground state and second minimum, we come to the important process of
barrier determination. To obtain the barrier, we first define the entrée point and exit point, and
seek for a path that connects these two points with minimum height of the highest energy point.
This point is defined as the saddle point. After locating the lowest saddle, we determine the
next-lowest saddle. The actual method used in our calculation is described in [8], where we
named our methods immersion method and dam method.
After locating the saddles, we
determine the fission valleys that connect these minima and saddles.
A typical calculated result is given in Fig.2, where the fission potential energy surface of
232
Th is given. There exist two fission valleys, one is reflection-asymmetric and the other is
symmetric as are depicted in the figure. Both of these fission valleys show typical triple-humped
barrier structure, and the outermost barrier of the asymmetric valley is the dominating barrier and
the middle is dominating for the symmetric valley. The asymmetric-dominating barrier is lower
by about 2 MeV by reflection asymmetric deformation and this nucleus fissions asymmetrically
when excitation energy is not high. Very special aspect is that we can obtain the separating ridge
between these fission valleys at least in outermost region that is terminated at some internucleus
distance where the ridge and one of the valleys merge.
We have done such calculations for actinide nuclei and others in the region Z=78 to Z=125.
When we limit ourselves to typical light actinide nuclei, the structure of potential energy surface
is quite similar to what is shown in Fig. 2, i.e., there is always two fission valleys, one is
asymmetric and the other is symmetric, which are separated by the ridge line. As mass becomes
heavier, the ridge structure becomes
less prominent but the system still
tends
to
fission
asymmetrically.
Therefore, we might say generally that
for light actinide, the bimodal nature is
seen
prominently
in
the
fission
potential energy surface. By bimodal,
we mean that two parallel (not
sequential), one is symmetric and the
other is asymmetric, fission barriers of
different deformations appear where in
Fig. 2. Fission barriers for 232Th corresponding
to different fission modes as a function of
quadrupole moment Q2, taken from [8].
between a ridge that separates the two.
Evidence of the bimodal structure is
seen in the fission excitation function, that has two different thresholds for 227Ra [9].
It should be kept in mind that multi-modal structure is often discussed in the treatment of the
multi-channel fission analysis [7]. Normally in this treatment, however, the channel is defined in
terms of scission configurations, which are different from the fission barrier. Scission point
inherits the difficulty that the mathematical definition of it is not clear. This feature is in sharp
contrast with the saddle where even in multi-dimensional space, the mathematical definition of it
is clear. Another problem in scission model is that it cannot explain the threshold effect [9]. In
our consideration based on the adiabatic assumption, scission points are located somewhere in
the fission valleys. As we stated before, there are only two important valleys in the light actinide
cases, which tells that there are no more than two scission points of physical importance. In this
respect, the bimodal structure seen in our model seems not to be compatible with the multimodal
analysis [7] where there are at least two mass-asymmetric modes, standard I and standard II, in
addition to mass-symmetric mode. We cannot eliminate, however, the existence of two-body
channels that contribute to the mass distribution, because our calculation is based on one-body
system.
We have to develop the theory that treats the one-body and two-body channels
simultaneously to study the possible coexistence of two different kinds of channels.
Normally in multichannel analysis [7], the mass distributions are fitted with several kinds of
Gaussian distributions. It should be kept in mind, however, that there is no a priori reason that
the mass distribution of one mode has Gaussian shape, because the underlying potential that
governs the width of the mass distribution is not restricted to the quadratic shape. The discussion
referring the Gaussian shape to the mass distribution based on the central limit theory seems
hardly appropriate and therefore, the analysis that decomposes the experimental mass
distribution into several Gaussian shapes is not a priori justified. It seems more natural to
assume that the deviation from a single Gaussian distribution can be attributed to the nonquadratic form of the underlying potential governing the mass distribution.
3. Fusion-like trajectory found in the cold fusion reaction leading to superheavy nuclei
Cold fusion reactions have been used quite extensively [11], together with hot fusion
reactions, to produce superheavy elements. In cold fusion, the typical excitation energies of the
compound nuclei are quite low and as a result, the main reaction channel leading to the
formation of SHE is 1n or 2n. Therefore, the static potential energy surface calculated for zero
excitation energy as is done here is a good approximation and is suitable to analyze some aspects
of the cold fusion reaction.
We have calculated the fission potential energy surfaces of 8 even-even superheavy nuclei
carefully [10]. We found that there is always a normal fission valley originating at the inner
fission saddle. There is a signature of outer saddle for light SHE nuclei but as masses become
heavier, the outer saddle is no more prominent. Along this fission valley the nucleus takes
mostly reflection symmetric (in some cases, small degree of mass asymmetric) shape.
In
addition to this kind of normal fission valley, however, there appears another kind of valley with
quite large mass asymmetry that almost corresponds to the mass asymmetry of the incident
channels. This latter kind of valley seems to correspond to the fusion valley with respect to mass
ratio. However, the deformations of the fragments have typical feature that although the Pb- or
Bi-like targets are almost spherical, the projectile-like fragments are highly deformed to prolate
shape with deformation of about ε=0.4. In order to understand the feature of this fusion-likevalley, we concentrate on one typical system, namely a compound nucleus of Darmstadtium
272
Ds (Z=110). This nucleus was synthesized in the reaction 64Ni + 208Pb –> 272Ds at GSI [11].
In Fig.3, we show the potential-energy surface of this compound nucleus
272
Ds as a function
of the quadrupole moment q2 which is dimensionless and is given in unit of 3ZR02/4π(e2b) where
Z is the charge number and R0 is the nuclear radius. This figure is essentially the same as Fig.7
of [12] except for the definition of the quadrupole moment. Shown by the lowest solid curve is
the normal fission valley on which the shape is almost mass symmetric. The triangles and
squares on this curve are the saddle points and the minima obtained by water immersion method.
The middle solid curve with small circles is the special fusion valley corresponding to this
system. As will be explained later this valley is not the fusion valley itself but a kind of fusionlike valley but we will call it fusion
valley for simplicity. The topmost curve
with triangles is the ridge separating the
two valleys, fission valley and fusion
valley. The fusion valley is limited in
between points A and B, outside of
which this valley structure becomes
unclear or some sudden change of the
configuration
happens.
of
the
wave-function
Between A and B we can
define a clear valley structure and there
Fig. 3. Structure in the calculated fivedimensional potential energy surface for
as a function of quadrupole moment q2.
is no sudden change of the wave
272
Ds
Fig. 4. Single-particle levels of proton (left) and neutron (right) along the fusion valley
shown in Fig. 3. Symbol A and B correspond the deformations shown in Fig. 3.
function. We concentrate on the study of this part of fusion valley in the following.
In Fig.4, we plot the proton (left) and neutron (right) Nilsson-type single particle diagram
along the fusion valley of Fig.3. In this figure, left-end point of abscissa is point A and right-end
is point B. When we look at the energy levels near the Fermi level between points A and B, we
see that the level density just above the Fermi energy is lower compared with the level densities
a few MeV above and a few MeV below regions. Actually, corresponding to this observation,
the calculated shell correction energies along fusion valley are negative and large both for
protons and for neutrons. Therefore, this fusion valley is stabilized due to the shell correction
energies.
Next task is to investigate the reason why such low level density region appears along that
special valley. For this purpose, it is of interest to compare the level diagram of Fig.4 with the
Nilsson diagram of isolated
diagrams of
208
208
Pb and
64
Ni. In Fig.5, we show the proton and neutron Nilsson
Pb as a function of deformation ε2. In this figures, the energy gap of Z=82 and
N=126 are clearly seen. If we see the neutron levels, the shell gap N=126 appears for spherical
deformation in the energy rage of -4.5 to -7.5 MeV. This energy range is just the region of a low
level density just above the Fermi level in the neutron level diagram shown in Fig.4. In Fig.6,
Fig. 5. Single-particle levels of proton (left) and neutron (right) as a function of
deformation ε2 for 208Pb, taken from [12].
Fig. 6. Same ad Fig. 5 but for 64Ni, taken from [12].
we show the proton and neutron Nilsson diagrams of 64Ni as a function of deformation ε2. In this
figure, for neutrons of spherical deformation, we see a region of low level density from -6 to -9
MeV, What is more interesting is that at deformation near ε2=0.4, we see a gap again from -7 to 9 MeV, which energy region overlaps with the
208
Pb neutron energy gap and with the low level
density region in the total system shown in Fig.4. Thus it is expected that the reason of low level
density near the Fermi level of total system is its energetical coincidence with the Fermi levels
and low level density regions of spherical
208
Pb and highly prolate-deformed
64
Ni. The same
type of analysis is possible for protons but in this case one has to take care of the long-range
Coulomb force correction, for example, the proton single-particle levels of light fragment in total
system is energetically pushed upward by the Coulomb field of the heavy fragment. After this
correction, the same type of quantitative discussion is possible for proton levels.
The consideration up to this point means that the structure of the single-particle levels of the
total system resembles to those of two fragments with respect to the energies of Fermi levels and
the level densities above the Fermi levels. It suggests that the interaction between two fragments
in the total system is not strong enough to change the Fermi energies and the low level density
regions above the Fermi energies. The ideal and extreme case that assures this situation is when
the single-particle wave functions are well localized to either of the fragments. In this case, the
single-particle levels of the total system are the simple sum of those of two fragments. In this
case, the shell correction energy of the total system is approximately the same as the sum of the
shell correction energies of two fragments, provided the Fermi energies and the averaged Fermi
energies of two fragments differ not much. Therefore, we will check in the following if the
wave-function localization is really achieved or not.
In Fig. 7, we show the localization of the neutron wave function at the point B of Fig. 3. As
the shape in this figure tells, it is well necked in and therefore one expect that the wave function
is well localized. To check it, what we plot in Fig. 7 is the numbers of neutrons localize to the
large and the small fragments as a function of the extent of the localization. For example, the 72
neutrons are localized to heavy fragment more than 90%, and 14 neutrons are localized to heavy
fragment between 80 to 90 %. It means 72 neutrons are localized to light fragment less than
10 %, and 14 neutrons are localized to light fragment between 10 to 20 %. In this figure, we see
that although the localization to the heavy fragment is rather good, the localization to the light
fragment is very poor. There are only 18 neutrons localized more than 50 % to the light
fragment. But if we integrate the total number of neutrons localized to heavy fragment, it is
127.32 and those localized to light fragment is 34.68. These two numbers resemble well to the
neutron number 126 of 208Pb and 36 of 64Ni. It means that although the mass ratio is quite close
to the cluster structure of
208
Pb and
64
Ni, the extent of the wave function localization is quite
Fig. 7. Localization of neutron single-particle wave functions corresponding to
the point B of Fig. 3.
Fig. 8. Same as Fig. 7. but for proton single-particle wave functions.
poor for the light fragment. The total density of the light fragment is composed of so many
numbers of wave functions, each contributing not much. This aspect is quite contradictory to the
ideal image of the cluster wave function.
The same analysis as Fig.7 is given in Fig.8, where the localization of the proton wave
functions at the point B of Fig.3 is shown. Also in this case, the localization of the singleparticle wave functions to the light fragment is very poor, where only 18 protons are localized
more than 50 %. However, when we calculate the total proton localization, we obtain that 83.18
protons are localized to heavy fragment and 26.82 protons are localized to light fragment. Again,
these numbers are quite similar to the proton number 82 of 208Pb and 28 of 64Ni. In proton case
too, the clusterization with respect to the charge ratio is well achieved but with respect to the
single-particle localization of wave functions, the clusterization is not good. For the deformation
corresponding to the point A of Fig. 3, the localization of neutrons and protons are much worth
than at point B as expected and cluster picture is not good at all.
Another interesting aspect is the charge ratio defined by the proton number divided by the
total nucleon number for each fragment. At point B, this charge ratio for heavy fragment is
83.18/210.5=0.3952, whereas for light fragment it is 26.82/61.50=0.4361. For 208Pb, this ratio is
82/208=0.3942 and for 64Ni is 28/64=0.4375. This result shows that the charge ratios are quite
well conserved and therefore, the charge equilibrium is not at all established. What is more
astonishing is that even at inner point A, the charge ratio for the heavy fragment is
89.06/224.30=0.3971 and for light fragment it is 20.94/47.70=0.4390, which are again quite
similar to those of incident
satisfied.
208
Pb and
64
Ni. The stability along this fusion valley is so well
To find the origin of the robustness of the charge ratio on fusion valley is an
interesting future problem.
Although we showed our result only for the special case of
272
Ds, the discussion from Fig. 3
to Fig. 8 applies to most of the cold fusion reaction systems. It means that the Fermi levels of
targets and projectiles fall on similar energy range and the level densities just above the Fermi
levels are commonly low. That is why the special fusion valleys are seen in most of the cold
fusion systems [10]. The single-particle wave functions on these valleys are no more the same as
the original ones, and the spreading out of the original wave functions to the direction of the
reaction partner happens very frequently. However, the structure of the spherical target and
highly deformed projectile can be preserved even when target and projectile overlaps
appreciably due to the coincidence of the Fermi energies and occurrence of the low level density
regions just above the Fermi energy. The existence of this special fusion valley must have
important physical role in the cold fusion reactions.
4. Summary
We have discussed the compound and fragments’ aspect of shell structure in the actinide and
SHE nuclei. In actinide case, the outer fission saddle region has bimodal structure, that is, the
fission barriers and valleys are separated into mass-asymmetric and mass-symmetric
deformations and there exist a ridge line that separate the two valleys. The extension of these
two valleys to the scission direction will give us an idea that there should be two sources of
fission mass distributions, for example, the mass-symmetric and a mass-asymmetric distributions.
The possibility of the third source of special importance is not seen in our calculation, which
somehow contradicts to the multimodal analysis [7]. Further work to seek the one-body and
two-body configurations simultaneously is required to fully answer this problem.
In case of SHE potential energy surface corresponding to the cold fusion synthesis, we find
always a very special fusion-like valley structure at outer part of the potential energy, which is
separated from the normal fission valley by a ridge line. The configuration of the valley has a
structure of original spherical target plus a highly prolate-deformed projectile, although the
single-particle wave functions are rather well mixed up. Because of the coincidence of the Fermi
energy and the low level density area just above the Fermi energy of target and projectile, which
happens for cold fusion reaction systems, the stability of the incident-channel-like structure
survives up to some internucleus distance where the strong overlapping of the matter breaks the
magic-like incident-channel shell structure completely. Up to this distance, the charge ratios of
the incident system are very well preserved, which is a great wonder. The physical consequence
of the existence of such special fusion valley structure together with the relation to the incident
two-body-channel consideration given in [10] should be studied further to explore the subject
completely.
Acknowledgement
This work was carried out under the auspices of the National Nuclear Security Administration of
the U. S. Department of Energy at Los Alamos National Laboratory under ContractNo.\ DEAC52-06NA25396, and a travel grants for P. M.\ to JUSTIPEN (Japan-U. S. Theory Institute for
Physics with Exotic Nuclei) under grant number DE-FG02-06ER41407 (U. Tennessee).
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