Fusion of radioactive
Sn with
J. F. Liang, D. Shapira, J. R. Beene, C. J. Gross, R. L. Varner, A. Galindo-Uribarri,
J. Gomez del Campo, P. A. Hausladen, P. E. Mueller, D. W. Stracener
Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
H. Amro, J. J. Kolata
Department of Physics, University of Notre Dame, Notre Dame, IN 46556
J. D. Bierman
Physics Department AD-51, Gonzaga University, Spokane, Washington 99258-0051
A. L. Caraley
Department of Physics, State University of New York at Oswego, Oswego, NY 13126
K. L. Jones
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854
Y. Larochelle
Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37966
W. Loveland, D. Peterson
Department of Chemistry, Oregon State University, Corvallis, Oregon 97331
(Dated: October 28, 2018)
Evaporation residue and fission cross sections of radioactive 132Sn on 64Ni were measured near
the Coulomb barrier. A large sub-barrier fusion enhancement was observed. Coupled-channel
calculations including inelastic excitation of the projectile and target, and neutron transfer are in
good agreement with the measured fusion excitation function. When the change in nuclear size and
shift in barrier height are accounted for, there is no extra fusion enhancement in 132Sn+64Ni with
respect to stable Sn+64Ni. A systematic comparison of evaporation residue cross sections for the
fusion of even 112−124Sn and 132Sn with 64Ni is presented.
PACS numbers: 25.60.-t, 25.60.Pj
I. INTRODUCTION
Fusion of heavy ions has been a topic of interests
for several decades[1]. One motivation is to understand
the reaction mechanisms so that the production yield of
heavy elements can be better estimated by model calcula-
tions. The formation of a compound nucleus is a complex
process. The projectile and target have to be captured
inside the Coulomb barrier and subsequently evolve into
a compact shape. In heavy systems, the dinuclear system
can separate during shape equilibration prior to passing
the saddle point. This quasifission process is considered
the primary cause of fusion hindrance[2, 3, 4].
At energies near and below the Coulomb barrier, the
structure of the participants plays an important role in
influencing the fusion cross section[5, 6, 7]. Sub-barrier
fusion enhancement due to nuclear deformation and in-
elastic excitation has been observed[8, 9, 10, 11, 12].
Coupled-channel calculations have successfully repro-
duced experimental data by including nuclear deforma-
tion and inelastic excitation. Nucleon transfer is another
important channel to be considered[13, 14].
Recently available radioactive ion beams offer the op-
portunity to study fusion under the influence of strong
nucleon transfer reactions. Several theoretical works have
predicted large enhancement of sub-barrier fusion involv-
ing neutron-rich radioactive nuclei[15, 16, 17, 18, 19].
In addition, the compound nucleus produced in such re-
actions is predicted to have a higher survival probabil-
ity and longer lifetimes. This is encouraging for super-
heavy element research. If high-intensity, neutron-rich
radioactive beams become available in the future, new
neutron-rich heavy nuclei may be synthesized with en-
hanced yields. The longer lifetime of new isotopes of
heavy elements would enable the study of their atomic
and chemical properties[20]. However, the current inten-
sity of the radioactive beams is several orders of mag-
nitude lower than that of stable beams. It is thus not
practical to use such beams for heavy element synthesis
experiments, but they do provide excellent opportuni-
ties for studying reaction mechanisms of fusion involving
neutron-rich radioactive nuclei.
Fusion enhancement, with respect to a one-
dimensional barrier penetration model prediction, has
been observed in experiments performed with neutron-
rich radioactive ion beams at sub-barrier energies[21, 22,
23, 24, 25]. For instance, the effect of large neutron excess
on fusion enhancement can be seen in 29,31Al+197Au[23].
http://arxiv.org/abs/0704.0780v2
However, when comparing reactions involving stable iso-
topes of the projectile or target, the fusion excitation
functions are very similar if the change in nuclear sizes is
accounted for.
This paper reports results of fusion excitation func-
tions measured with radioactive 132Sn on 64Ni. The dou-
bly magic (Z=50, N=82) 132Sn has eight neutrons more
than the heaviest stable 124Sn. Its N/Z ratio (1.64) is
larger than that of stable doubly magic nuclei 48Ca (1.4)
and 208Pb (1.54) which are commonly used for heavy
element production[26]. Evaporation residue (ER) and
fission cross sections were measured. The sum of ER and
fission cross sections are taken as the fusion cross section.
The experimental apparatus is described in Sect. II
and data reduction procedures in Sect. III. The results
and comparison with model calculations are presented in
Sect. IV. In Sect. V a comparison of ER and fusion cross
sections with those resulting from stable Sn isotopes on
64Ni is discussed. A summary is given in Sect. VI.
II. EXPERIMENTAL METHODS
The experiment was carried out at the Holifield Ra-
dioactive Ion Beam Facility. A 42 MeV proton beam
produced by the Oak Ridge Isochronous Cyclotron was
used to bombard a uranium carbide target. The fission
fragments were ionized by an electron beam plasma ion
source. The largest yield of mass A=132 fragments was
132Te. Therefore, it was necessary to suppress 132Te.
This was accomplished by introducing sulfur into the ion
source then selecting the mass 164 XS+ molecular ions
from the extracted beam. The 132Te to 132Sn ratio in the
ion beam was found to be suppressed by a large factor
(∼ 7 × 104) compared to that observed with the mass
132 atomic beam. The mass 164 SnS+ beam was con-
verted into a Sn− beam by passing it through a Cs va-
por cell where the molecular ion underwent breakup and
charge exchange[27]. The negatively charged Sn was sub-
sequently injected into the 25 MV electrostatic tandem
accelerator to accelerate the beam to high energies. The
measurement was performed at energies between 453 and
620 MeV. The average beam intensity was 50,000 parti-
cles per second (pps) with a maximum of 72,000 pps. The
ER cross sections measured between 453 and 560 MeV
have been reported previously[24].
The purity of the Sn beam was measured by an ioniza-
tion chamber mounted at zero degrees. Figure 1 displays
the energy loss spectra of a 560 MeV A=132 beam with
and without the sulfur purification. The dashed curves
are the results of fitting the spectrum with Gaussian dis-
tributions to estimate the composition of the beam. In
the upper panel, the beam is primarily 132Te without
sulfur in the ion source. When sulfur was introduced in
the ion source, the beam was 96% 132Sn, as shown in the
lower panel. The small amount of Sb and Te had a neg-
ligible impact on the measurement because their atomic
number is higher. Fusion of the target with these iso-
baric contaminants at sub-barrier energies should have
been suppressed due to the higher Coulomb barriers.
FIG. 1: (Color online) Composition of a 560 MeV mass
A=132 beam measured by the ionization chamber. Top panel:
The mass A=132 beam without purification where Te and Sb
are the major components of the beam. Bottom panel: Sulfur
was introduced into the target ion source and SnS was selected
by the mass separator. The dashed curves are results of fitting
the spectrum with three Gaussian distributions. The isobar
contaminants 132Sb and 132Te were suppressed considerably.
The apparatus for the fusion measurement is shown in
Fig. 2. A thick 64Ni target (1.0 mg/cm2) was used to
compensate for the low beam intensity. Since the com-
pound nucleus decays by particle evaporation and fission,
the evaporation residue (ER) and fission cross sections
were measured. The ERs were detected by the ioniza-
tion chamber at zero degrees and the fission fragments
were detected by an annular double-sided silicon strip
detector.
time−of−flight
Si det
beam defining
Timing
Timing
Timing
target
ionization
chamber
FIG. 2: (Color online) Apparatus for measuring fission and
evaporation residues cross sections induced by low intensity
beams in inverse kinematics.
The ERs were identified by the time-of-flight measured
with the microchannel plate timing detector located in
front of the ionization chamber and by energy loss in the
ionization chamber. The two microchannel plate timing
detectors located before the target were used to monitor
the beam intensity and to provide the timing reference
for the time-of-flight measurement. The microchannel
plate timing detector in front of the ionization chamber
was position sensitive and was used to monitor the beam
position. It was located 200 mm from the target and had
a 25 mm diameter Mylar foil. The ionization chamber
was filled with CF4 gas so that it could function at rates
up to 50,000 pps. Higher beam intensities occurred in
some of the fission measurements, requiring the ioniza-
tion chamber to be turned off. The data acquisition was
triggered by either the beam signal rate down scaled by a
factor of 1000, the coincidence of the delayed beam signal
and ER signal, or the silicon detector signal. A 350 MeV
Au beam that resembled ERs was measured by the ion-
ization chamber to calibrate the energy loss spectrum.
The ER cross section was obtained by taking the ratio of
the ER yield to the target thickness and the integrated
beam particles in the ionization chamber. A detailed de-
scription of the ER measurement technique used in this
experiment can be found in Ref. [28].
The annular double-sided silicon strip detector (Mi-
cron Semiconductor Design S2) was located 42 mm from
the target. It had 48 concentric strips on one side and 16
pie-shaped sectors on the other side. The inner diame-
ter was 35 mm and the outer diameter was 70 mm. The
thickness of the detector was 300 µm. The detection an-
gles spanned 15.6◦ to 39.6◦. The fission fragments were
identified by requiring a coincidence of events in the Si
detector and by the folding angle distributions of the de-
tected particles.
III. DATA REDUCTION PROCEDURES
A. Evaporation residues
Since this was an inverse kinematics reaction, the ERs
recoiled in the forward direction in a narrow cone. The
apparatus was designed to have high efficiency for detect-
ing ERs. The efficiency of the apparatus was estimated
by Monte Carlo simulations. The angular distribution of
the ERs was generated by statistical model calculations
using the code PACE2[29]. The input parameters for the
statistical model calculations will be discussed later in
this paper. The calculated efficiency for the lowest bom-
barding energy is 93±1%. It increases as the reaction
energy increases and reaches 98±1% at the highest en-
ergy.
A relatively thick target was used in this experiment.
The beam lost approximately 40 MeV after passing
through the target (13 MeV in the center of mass). For
this reason, the measured cross section is an average of
the contributions from the beam interacting throughout
the thickness of the target. The variation of ER cross
sections is not very large at energies above the Coulomb
barrier because the shape of the excitation function is al-
most flat. Therefore, the measured cross section is close
to that would be measured at an energy corresponding to
the middle of the target. However, at energies below the
barrier the ER cross section falls off exponentially. The
cross section near the entrance of the target has more
weight than that near the exit. Smooth curves fitting
the excitation function in this rapidly varying region were
used to determine the reaction energy associated with the
measured cross section.
An iterative method was used to determine the effec-
tive reaction energy for the thick target measurement .
First, the measured cross sections and the beam energies
calculated at the middle of the target were fitted by a
tensioned spline[30] where the smoothness of the curve
could be adjusted. The resulting curve was then used to
calculate the thick target cross section for each measure-
ment, according to
dE/dx
where σ(E) is the curve generated by the spline fit,
dE/dx is the stopping power of 132Sn in 64Ni, and ρ is the
target thickness. The integration limits were the energies
of the beam at the exit of the target and at the entrance
of the target. The energy, Ei, corresponding to the cross
section, σi, was obtained by interpolation using the fit-
ted curve. This set of energies was used as the input for
the next iteration of the fit. The result converged very
quickly. After five iterations, the energies differed from
the previous iteration energies by less than 0.2 MeV. The
validity of this method was checked by generating data
from a known function such as the Wong formula [31]
and folding in the effects of target thickness.
Comparing to the cross-section-weighted-average
method described in Ref. [28], the differences in energies
determined by these two methods are not noticeable
at high energies because the excitation function is
fairly flat. However, at energies below the barrier, the
energy determined by the cross-section-weighted-average
method is larger than that determined by the method
described above and disagrees with the measurement
in Ref. [32], as can be seen in Fig. 3. Furthermore,
it is found that using data generated from a known
function the effective energy obtained by the cross-
section-weighted-average method is shifted to too high
an energy in the exponential falloff region.
The uncertainty of the energy determination was esti-
mated by comparison with the method using the cross
section weighted average. The average uncertainty of
the effective reaction energy is 2.3 MeV in the region
where the excitation function is almost flat and increases
to 3.9 MeV in the exponential fall off region. The uncer-
tainty is larger, 5.8 MeV, for the lowest energy data point
because an extrapolation is required for calculating the
thick target cross section and the extrapolation region
is influenced by the location of the next higher energy
point.
To verify our measurement technique, the ER cross
sections for 124Sn+64Ni in inverse kinematics were mea-
sured and compared to those published by Freeman et al.
measured with a thin target[32]. It is noted that some
of our measurements were performed at energies differ-
ent from those of Ref. [32]. The comparison is shown in
Fig. 3. Our data (open triangles) are in good agreement
with those measured by Freeman et al. [32] (filled stars).
FIG. 3: (Color online) Comparison of ER cross sections for
64Ni+124Sn measured in this work and by Freeman et al.[32]
(filled stars). The filled circles are for energies determined
by the method described in Ref. [28] and the open triangles
are for energies determined by the method described in this
paper.
The solid circles are for energy determined by the cross-
section-weighted-average method described in Ref. [28].
B. Fission
Fission fragments were identified by requiring a coin-
cidence of two particles detected by the pie-shaped sec-
tors of the Si strip detector on either side of the beam.
Figure 4(a) and (b) present two-dimensional histograms
of particle energy and strip number of the Si detector
for coincident events taken from 560 MeV and 620 MeV
132Sn+64Ni, respectively. They were compared to the
kinematics calculation displayed in Fig. 4(c) and (d)
where the fission fragments, elastically scattered Sn and
Ni are shown by the solid, dash-dotted and dotted curves,
respectively. The angular range of the Si strip detector
is between the two vertical dashed lines. The elastically
scattered Ni and Sn appear in the upper right hand cor-
ner and center of the histogram, respectively. The fission
events are located in the gated area.
The folding angle distributions of the fragments were
used to distinguish fission from other reactions, such as
deep inelastic reactions. Since there are two solutions
for the kinematics of the inverse reaction, as shown in
Fig. 4(c) and (d), the fragment angular correlation is not
as simple as that in normal kinematics. Monte Carlo
simulations were performed to provide guidance. It was
assumed that only fusion-fission results from a full mo-
mentum transfer. The width of the mass distribution was
taken from the 58Ni+124Sn measurement[33]. The width
of the mass distribution was varied to estimate the uncer-
(d)(c)
Strip No.
0 10 20 30 40 50 60
Strip No.
0 10 20 30 40 50 60
FIG. 4: (Color online) (a) and (b) Two dimensional his-
tograms of energy and strip number for coincident events from
560 and 620 MeV 132Sn+64Ni, respectively, measured by the
annular double-sided silicon strip detector. The gated area
shows events from fission and other reactions. (c) and (d)
Kinematics of energy as a function of scattering angle for 560
and 620 MeV 132Sn+64Ni, respectively, elastic scattering and
fission fragments.The dash-dotted and dotted curves are for
the elastically scattered Sn and Ni, respectively whereas the
solid curve is for the fission fragments. The angular range of
the Si strip detector is between the two vertical dashed lines.
tainty of the simulation. The transition state model[34]
was used to predict the fission fragment angular distribu-
tion. In Fig. 5 the simulated fission fragment folding an-
gle distributions for 550 MeV 124Te+64Ni are compared
with a stable beam test measurement. The folding angle
distributions for one of the fragments detected in strip 2
(16.2◦), strip 22 (27.7◦), and strip 41 (36.8◦) are shown.
The gap in the spectra at strip 14, 30, 44, 46, and 47 are
malfunctioning strips in the detector.
The Monte Carlo simulated folding angle distributions
for fission are shown in the middle panels of Fig. 5 and
compared to those of measurements shown in the left
panels. For one of the fragments detected at forward
angles, strip 2 for example, the predicted angular dis-
tribution of the other fragment is similar to that of the
measurement. Most of these events are considered as re-
sulting from fission. For one of the fragments detected
near the middle part of the detector, strip 22 for instance,
there are differences between measurement and simula-
tion in the shapes of the angular distributions of the other
fragment. It is predicted that the other fission fragment
is distributed around strip 40. The measured distribution
spreads to more forward angles. For one of the fragments
detected at the backward angles, the yield of the other
fragment is predicted to be small and they are equally
FIG. 5: (Color online) Left panels: Folding angle distributions
for 550 MeV 124Te+64Ni for one of the fragments detected at
16.2◦ (strip 2), 27.7◦ (strip 22), and 36.8◦ (strip 41) by the
annular double-sided silicon strip detector. The elastic scat-
tering events are excluded. The dotted and dashed histograms
are the results of fitting the data with simulated fission and
deep inelastic collisions with Q=–20 MeV, respectively (see
text). Middle panels: Results of Monte Carlo simulations for
fission events. Right panels: Results of Monte Carlo simula-
tions for deep inelastic scattering events. The solid curves are
for reaction Q value of –10 MeV, the dashed curves are for
Q=–20 MeV, and the dotted curves are for Q=–40 MeV.
distributed between the middle part of the detector and
the outer edge of the detector. But the measured events
appear in the middle part of the detector. There are no
events in the region where fission events are expected.
These differences are attributed to the contribution from
other reaction mechanisms, most likely deep inelastic col-
lisions.
An attempt was made to simulate these deep inelastic
collision events. It was assumed that the mass of these
products were projectile- and target-like and the angular
distribution at forward angles followed a 1/sin(θ) depen-
dence. The right panels of Fig. 5 show the results of sim-
ulations performed for reaction Q values of –10 (solid),
–20 (dashed), and –40 MeV (dotted). It can be seen that
the overlap of fission and deep inelastic collisions becomes
larger at more backward angles. At strip 41 (36.8◦), deep
inelastic collisions account for all the events.
The relative contribution of fission and deep inelas-
tic collisions were obtained by fitting the simulated fold-
ing angle distributions to the measured distributions for
all the detector strips using the CERN library program
MINUIT[35]. In the fits, the normalization coefficients for
the simulated distributions were the only two variable
parameters. The results of the fits are shown in the left
panels of Fig. 5 by the dotted and dashed histograms for
fission and deep inelastic collisions with Q=–20 MeV, re-
spectively. The number of fission events in the measured
distributions were taken as the summed events in each
strip multiplied by the relative contribution of fission.
The folding angle distributions for 132Sn+64Ni are
shown in Fig. 6. Due to the low statistics, it was not prac-
tical to extract the fission events by fitting the folding
angle distributions. As an alternative, the fission events
were extracted by setting gates on the folding angle dis-
tributions using the simulated distributions as references.
This gating method was also tested with the 124Te+64Ni
measurement. The fission cross sections obtained by the
fitting method and the gating method agreed within 10%.
FIG. 6: (Color online) Left panels: Folding angle distributions
for 560 MeV 132Sn+64Ni for one of the fragments detected at
16.2◦ (strip 2), 27.7◦ (strip 22), and 36.8◦ (strip 41) by the
annular double-sided silicon strip detector. The elastic scat-
tering events are excluded. Middle panels: Results of Monte
Carlo simulations for fission events. Right panels: Results of
Monte Carlo simulations for deep inelastic scattering events.
The solid curves are for reaction Q value of –10 MeV, and the
dotted curves are for Q=–40 MeV.
The Monte Carlo simulation was also employed to cal-
culate the coincidence efficiency of the detector. The effi-
ciency increased from 5.7±0.9% at 530 MeV to 7.6±0.8%
at 620 MeV bombarding energy.
In the present work, the dynamic range of the ampli-
fiers was not sufficiently large resulting in the distortion
of the high energy signals. In the future, new amplifiers
that are more suitable for measuring the energy of fission
fragments will be used so that the mass ratio of reac-
tion products can be obtained to help distinguish fission
events from other reaction channels.
The formation of a compound nucleus depends on
whether the interacting nuclei are captured inside the
fusion barrier and whether the dinuclear system can sub-
sequently evolve into a compact shape. Quasifission oc-
curs when the dinuclear system fails to cross the saddle
level density parameter (a) A/8 MeV−1
af/an 1.04
diffuseness of spin distribution (∆l) 4 h̄
fission barrier Sierk[45]
TABLE I: Input parameters for statistical model calculations.
point to reach shape equilibrium. Since the beam inten-
sity was several orders of magnitude lower than that of
stable beams and the reaction was in inverse kinemat-
ics, making separation of fusion-fission and quasifission
very difficult, there was no attempt to distinguish quasi-
fission from fusion-fission in this work. Furthermore, the
experimental results are compared to barrier penetration
models which describe the capture process, making it un-
necessary to separate these two processes.
IV. COMPARISON WITH MODEL
CALCULATIONS
A. Statistical model
The compound nucleus formed in 132Sn+64Ni decays
by particle evaporation and fission. Statistical models
have successfully described compound nucleus decay for
a wide range of fusion reactions. The measured ER and
fission cross sections are compared with the predictions of
the statistical model code PACE2[29]. The input param-
eters were obtained by simultaneously fitting the data
from stable Sn on 64Ni[32, 36] and the measured fusion
cross sections[36] were used for the calculations. Fig-
ure 7(a), (b), and (c) displays the comparison of calcula-
tions and data for 112,118,124Sn+64Ni, respectively. The
calculations reproduce the measurements well except for
the ER cross sections of 112Sn+64Ni. Table I lists the in-
put parameters for the calculations. Without adjusting
the parameters, calculations for 132Sn+64Ni were per-
formed. The results are shown in Fig. 7(d). Very good
agreement between the calculation and the data can be
seen.
It is noted that some of the parameters used in our
calculations are different from those used by Lesko et
al. [36]. In their calculations, the code CASCADE[37] was
used. The mass of the nuclei in the decay chain was
calculated using the Myers droplet model[38]. The dif-
fuseness of the spin distribution was ∆l = 15h̄ and the
ratio of level density at the saddle point to the ground
state, af/an, was set to 1.0. In this work, a compilation
of measured masses[39], ∆l = 4h̄, and af/an = 1.04 were
used.
FIG. 7: (Color online) Comparison of measured ER (filled
circles) and fission (open circles) cross sections with statistical
model calculations. The solid and the dotted curves are the
calculated fission and ER cross sections, respectively, using
the measured fusion cross sections as input. (a) 64Ni+112Sn,
(b) 64Ni+118Sn, (c) 64Ni+124Sn (Freeman et al.[32]), and (d)
132Sn+64Ni (this work)
B. Coupled-channel calculation
In general, sub-barrier fusion enhancement can be de-
scribed by coupled-channel calculations. The fusion cross
section of 132Sn+64Ni, the sum of ER and fission cross
sections, is compared with coupled-channel calculations
using the code CCFULL[40]. The interaction potential
(V◦=82.46 MeV, r◦=1.18 fm, and a=0.691 fm) was taken
from the systematics of Broglia and Winther[41]. The
result of the calculations are compared with the data
in Fig. 8. The dotted curve is the prediction of a one-
dimensional barrier penetration model and it can be seen
that it substantially underpredicts the sub-barrier cross
sections. The coupled-channel calculation including in-
elastic excitation of 64Ni to the first 2+ and 3− states
and 132Sn to the first 2+ state is shown by the dashed
curve. The transition matrix elements, B(Eλ), of 64Ni
were obtained from Ref. [42, 43] and the B(E2) of 132Sn
was obtained from a recent measurement by Varner et
al.[44]. This calculation overpredicts the data at energies
near the barrier and underpredicts the data well below
the barrier.
The neutron transfer reactions have positive Q values
for transferring two to six neutrons from 132Sn to 64Ni.
Since there is no neutron transfer data available for this
reaction, the transfer coupling form factor is unknown.
Thus, the coupled-channel calculation including transfer
and inelastic excitation was performed with one effec-
FIG. 8: (Color online) Comparison of 132Sn+64Ni fusion
data (filled circles) with a one-dimensional barrier penetra-
tion model calculation (dotted curve). The coupled-channel
calculation including inelastic excitation of the projectile and
target is shown by the dashed curve and the calculation in-
cluding inelastic excitation and neutron transfer is shown by
the solid curve.
tive transfer channel using the Q value for two-neutron
transfer. The coupling constant was adjusted to fit the
data. The calculation with the coupling constant set to
0.48 is shown by the solid curve. It reproduces the data
very well except for the lowest energy data point which
has large uncertainties in energy and in cross section. A
better treatment of the transfer channels based on exper-
imental transfer data would help improve understanding
of the influence of transfer on fusion. Experimental neu-
tron transfer data on 132Sn+64Ni in the future would be
very useful.
V. DISCUSSION
The ER cross section can be described by
σER = πλ
(2l+ 1)σl,
where λ is the de Broglie wave length, lc the maximum
angular momentum for ER formation and σl the partial
cross section. The reduced ER cross sections for 64Ni
on stable-even Sn isotopes[32] are compared with that
for 132Sn+64Ni in Fig. 9. The reduced ER cross sec-
tion is defined as the ER cross section divided by the
kinematic factor πλ2. It can be seen that the ER cross
sections saturate at high energies as fission becomes a
significant fraction of the fusion cross section. In addi-
tion, the saturation value increases as the neutron excess
in Sn increases. This is consistent with the fact that the
fission barrier height increases for the more neutron-rich
compound nuclei.
FIG. 9: (Color online) The reduced ER cross section as a
function of the center of mass energy for 64Ni on stable even
Sn isotopes[32] and radioactive 132Sn.
In Fig. 10, the measured reduced ER cross sections for
Ni+Sn as a function of the calculated average mass of the
ERs, predicted by PACE2, are presented. In the same re-
action, the higher mass ERs are produced at lower beam
energies because of the lower excitation energies of the
compound nucleus. As the neutron excess in the com-
pound nucleus increases, neutron evaporation becomes
the dominant decay channel. The PACE2 calculation pre-
dicts that a compound nucleus made with Sn isotopes of
mass number greater than 120 decays essentially 100%
by neutron evaporation and Pt isotopes are the primary
ERs. The mass of the compound nucleus is different
when it is produced with different Sn isotopes. How-
ever, it can be seen that Pt of a particular mass can
be produced with different Sn isotopes if different num-
bers of neutrons are evaporated. The reaction with a
more neutron-rich Sn produces the same Pt isotope at a
higher rate. With 132Sn as the projectile, the ERs are so
neutron-rich that they cannot be produced by stable Sn
induced reactions. This suggests that it may be benefi-
cial to use neutron-rich radioactive ion beams to produce
new isotopes of heavy elements.
The fusion excitation functions of 64Ni on stable even
Sn isotopes[36] are compared with that of 132Sn+64Ni in
Fig. 11. In order to remove the effects of the difference
in nuclear sizes, the cross section is divided by πR2 with
R=1.2(A
t ) fm, where Ap (At) is the mass num-
ber of the projectile (target). The reaction energy in the
center of mass is divided by the barrier height predicted
by the Bass model[46]. It can be seen that the fusion of
132Sn and 64Ni is not enhanced with respect to the stable-
even Sn isotopes when the difference in nuclear sizes is
FIG. 10: (Color online) The reduced ER cross section as a
function of the calculated average mass of ERs predicted by
PACE2[29] for 64Ni on stable even Sn isotopes[36] and radioac-
tive 132Sn.
considered.
FIG. 11: (Color online) Comparison of fusion excitation func-
tions for 64Ni on stable even Sn isotopes[36] and radioactive
132Sn. The change in nuclear sizes are corrected by factor-
ing out the area and the Bass barrier height[46] in the cross
section and energy, respectively.
The lowest energy data point has large uncertainties.
The cross section seems enhanced comparing to the sta-
ble beam measurements in Fig. 9 and Fig. 11. A more
pronounced enhancement appears when the data point
is compared to our coupled-channel calculations (Fig. 8)
and to a time-dependent Hartree-Fock calculation[47].
To further explore if fusion is enhanced at this low energy
region, we plan to repeat the measurement with an im-
proved apparatus where the thickness of the Mylar foil in
the microchannel plate timing detector located in front
of the ionization chamber will be reduced. This will al-
low a better separation of the energy loss signals from
ERs and scattered beams in the ionization chamber at
low bombarding energies.
The Q values for transferring two to six neutrons from
132Sn to 64Ni are positive. It is necessary to include
neutron transfer in coupled-channel calculations to re-
produce experimental results. As the neutron excess in
the Ni isotopes decreases, the number of neutron transfer
channels with positive Q values increases for 132Sn+Ni.
In 132Sn+58Ni, the Q values for transferring one to six-
teen neutrons from 132Sn to 58Ni are positive and range
from 1.7 to 17.4 MeV. A large sub-barrier fusion enhance-
ment due to the coupling to neutron transfer is expected
to occur in 132Sn+58Ni. An experiment to measure the
fusion excitation function of 132Sn on 58Ni is in prepara-
tion.
Although 132Sn is unstable, its neutron separation en-
ergy is 7.3 MeV. This is not very low compared to stable
nuclei. The sub-barrier fusion enhancement observed in
132Sn+64Ni with respect to stable Sn nuclei can be ac-
counted for by the change in nuclear sizes. No extra en-
hancement was found. However, an increased ER yield
at energies above the barrier was observed as compared
to stable Sn. As the shell closure is crossed, the binding
energy for 133Sn decreases by a factor of two. The nu-
clear surface of 133Sn and even more neutron-rich Sn may
be more diffused. The number of neutron transfer chan-
nels with positive Q values increases by a factor of two
or more. Larger sub-barrier fusion enhancement beyond
the nuclear size effect may be expected.
VI. SUMMARY
Neutron-rich radioactive 132Sn beams were incident on
a 64Ni target to measure fusion cross sections near the
Coulomb barrier. With an average intensity of 5×104
pps beams and a high efficiency apparatus for ER detec-
tion, the uncertainty of the measured ER cross section is
small and comparable to that achieved in stable beam ex-
periments. The efficiency for fission fragment detection
was low but the detector had a very fine granularity. By
requiring a coincident detection of the fission fragments
and performing folding angle distribution analysis, fission
events were identified. The excitation functions of ER
and fission can be described by statistical model calcula-
tions using parameters that simultaneously fit the stable
even Sn isotopes on 64Ni fusion data. A large sub-barrier
fusion enhancement with respect to a one-dimensional
barrier penetration model prediction was observed. The
enhancement is attributed to the coupling of the projec-
tile and target inelastic excitation and neutron transfer.
The reduced ER cross sections at energies above the bar-
rier are larger for the 132Sn induced reaction than those
induced by stable Sn nuclei, as expected from the higher
fission barrier of the more neutron-rich compound nu-
cleus. For a specific mass of ER, reactions with a more
neutron-rich Sn have higher cross sections. When the
fusion excitation functions are compared on a reduced
scale, where the effects of nuclear size and barrier height
are factored out, no extra fusion enhancement is observed
in 132Sn+64Ni with respect to stable Sn induced fusion.
The fusion cross section measured at the lowest energy
seems to be enhanced. Experiments to investigate this
with an improved apparatus is planned.
VII. ACKNOWLEDGMENT
We would like to thank D. J. Hinde for helpful and
stimulating discussions. We wish to thank the HRIBF
staff for providing excellent radioactive beams and tech-
nical support. Research at the Oak Ridge National
Laboratory is supported by the U.S. Department of
Energy under contract DE-AC05-00OR22725 with UT-
Battelle, LLC. W.L. and D.P. are supported by the the
U.S. Department of Energy under grant no. DE-FG06-
97ER41026.
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  Evaporation residue and fission cross sections of radioactive $^{132}$Sn on
$^{64}$Ni were measured near the Coulomb barrier. A large sub-barrier fusion
enhancement was observed. Coupled-channel calculations including inelastic
excitation of the projectile and target, and neutron transfer are in good
agreement with the measured fusion excitation function. When the change in
nuclear size and shift in barrier height are accounted for, there is no extra
fusion enhancement in $^{132}$Sn+$^{64}$Ni with respect to stable Sn+$^{64}$Ni.
A systematic comparison of evaporation residue cross sections for the fusion of
even $^{112-124}$Sn and $^{132}$Sn with $^{64}$Ni is presented.

Fusion of radioactive
Sn with
J. F. Liang, D. Shapira, J. R. Beene, C. J. Gross, R. L. Varner, A. Galindo-Uribarri,
J. Gomez del Campo, P. A. Hausladen, P. E. Mueller, D. W. Stracener
Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831
H. Amro, J. J. Kolata
Department of Physics, University of Notre Dame, Notre Dame, IN 46556
J. D. Bierman
Physics Department AD-51, Gonzaga University, Spokane, Washington 99258-0051
A. L. Caraley
Department of Physics, State University of New York at Oswego, Oswego, NY 13126
K. L. Jones
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854
Y. Larochelle
Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37966
W. Loveland, D. Peterson
Department of Chemistry, Oregon State University, Corvallis, Oregon 97331
(Dated: October 28, 2018)
Evaporation residue and fission cross sections of radioactive 132Sn on 64Ni were measured near
the Coulomb barrier. A large sub-barrier fusion enhancement was observed. Coupled-channel
calculations including inelastic excitation of the projectile and target, and neutron transfer are in
good agreement with the measured fusion excitation function. When the change in nuclear size and
shift in barrier height are accounted for, there is no extra fusion enhancement in 132Sn+64Ni with
respect to stable Sn+64Ni. A systematic comparison of evaporation residue cross sections for the
fusion of even 112−124Sn and 132Sn with 64Ni is presented.
PACS numbers: 25.60.-t, 25.60.Pj
I. INTRODUCTION
Fusion of heavy ions has been a topic of interests
for several decades[1]. One motivation is to understand
the reaction mechanisms so that the production yield of
heavy elements can be better estimated by model calcula-
tions. The formation of a compound nucleus is a complex
process. The projectile and target have to be captured
inside the Coulomb barrier and subsequently evolve into
a compact shape. In heavy systems, the dinuclear system
can separate during shape equilibration prior to passing
the saddle point. This quasifission process is considered
the primary cause of fusion hindrance[2, 3, 4].
At energies near and below the Coulomb barrier, the
structure of the participants plays an important role in
influencing the fusion cross section[5, 6, 7]. Sub-barrier
fusion enhancement due to nuclear deformation and in-
elastic excitation has been observed[8, 9, 10, 11, 12].
Coupled-channel calculations have successfully repro-
duced experimental data by including nuclear deforma-
tion and inelastic excitation. Nucleon transfer is another
important channel to be considered[13, 14].
Recently available radioactive ion beams offer the op-
portunity to study fusion under the influence of strong
nucleon transfer reactions. Several theoretical works have
predicted large enhancement of sub-barrier fusion involv-
ing neutron-rich radioactive nuclei[15, 16, 17, 18, 19].
In addition, the compound nucleus produced in such re-
actions is predicted to have a higher survival probabil-
ity and longer lifetimes. This is encouraging for super-
heavy element research. If high-intensity, neutron-rich
radioactive beams become available in the future, new
neutron-rich heavy nuclei may be synthesized with en-
hanced yields. The longer lifetime of new isotopes of
heavy elements would enable the study of their atomic
and chemical properties[20]. However, the current inten-
sity of the radioactive beams is several orders of mag-
nitude lower than that of stable beams. It is thus not
practical to use such beams for heavy element synthesis
experiments, but they do provide excellent opportuni-
ties for studying reaction mechanisms of fusion involving
neutron-rich radioactive nuclei.
Fusion enhancement, with respect to a one-
dimensional barrier penetration model prediction, has
been observed in experiments performed with neutron-
rich radioactive ion beams at sub-barrier energies[21, 22,
23, 24, 25]. For instance, the effect of large neutron excess
on fusion enhancement can be seen in 29,31Al+197Au[23].
http://arxiv.org/abs/0704.0780v2
However, when comparing reactions involving stable iso-
topes of the projectile or target, the fusion excitation
functions are very similar if the change in nuclear sizes is
accounted for.
This paper reports results of fusion excitation func-
tions measured with radioactive 132Sn on 64Ni. The dou-
bly magic (Z=50, N=82) 132Sn has eight neutrons more
than the heaviest stable 124Sn. Its N/Z ratio (1.64) is
larger than that of stable doubly magic nuclei 48Ca (1.4)
and 208Pb (1.54) which are commonly used for heavy
element production[26]. Evaporation residue (ER) and
fission cross sections were measured. The sum of ER and
fission cross sections are taken as the fusion cross section.
The experimental apparatus is described in Sect. II
and data reduction procedures in Sect. III. The results
and comparison with model calculations are presented in
Sect. IV. In Sect. V a comparison of ER and fusion cross
sections with those resulting from stable Sn isotopes on
64Ni is discussed. A summary is given in Sect. VI.
II. EXPERIMENTAL METHODS
The experiment was carried out at the Holifield Ra-
dioactive Ion Beam Facility. A 42 MeV proton beam
produced by the Oak Ridge Isochronous Cyclotron was
used to bombard a uranium carbide target. The fission
fragments were ionized by an electron beam plasma ion
source. The largest yield of mass A=132 fragments was
132Te. Therefore, it was necessary to suppress 132Te.
This was accomplished by introducing sulfur into the ion
source then selecting the mass 164 XS+ molecular ions
from the extracted beam. The 132Te to 132Sn ratio in the
ion beam was found to be suppressed by a large factor
(∼ 7 × 104) compared to that observed with the mass
132 atomic beam. The mass 164 SnS+ beam was con-
verted into a Sn− beam by passing it through a Cs va-
por cell where the molecular ion underwent breakup and
charge exchange[27]. The negatively charged Sn was sub-
sequently injected into the 25 MV electrostatic tandem
accelerator to accelerate the beam to high energies. The
measurement was performed at energies between 453 and
620 MeV. The average beam intensity was 50,000 parti-
cles per second (pps) with a maximum of 72,000 pps. The
ER cross sections measured between 453 and 560 MeV
have been reported previously[24].
The purity of the Sn beam was measured by an ioniza-
tion chamber mounted at zero degrees. Figure 1 displays
the energy loss spectra of a 560 MeV A=132 beam with
and without the sulfur purification. The dashed curves
are the results of fitting the spectrum with Gaussian dis-
tributions to estimate the composition of the beam. In
the upper panel, the beam is primarily 132Te without
sulfur in the ion source. When sulfur was introduced in
the ion source, the beam was 96% 132Sn, as shown in the
lower panel. The small amount of Sb and Te had a neg-
ligible impact on the measurement because their atomic
number is higher. Fusion of the target with these iso-
baric contaminants at sub-barrier energies should have
been suppressed due to the higher Coulomb barriers.
FIG. 1: (Color online) Composition of a 560 MeV mass
A=132 beam measured by the ionization chamber. Top panel:
The mass A=132 beam without purification where Te and Sb
are the major components of the beam. Bottom panel: Sulfur
was introduced into the target ion source and SnS was selected
by the mass separator. The dashed curves are results of fitting
the spectrum with three Gaussian distributions. The isobar
contaminants 132Sb and 132Te were suppressed considerably.
The apparatus for the fusion measurement is shown in
Fig. 2. A thick 64Ni target (1.0 mg/cm2) was used to
compensate for the low beam intensity. Since the com-
pound nucleus decays by particle evaporation and fission,
the evaporation residue (ER) and fission cross sections
were measured. The ERs were detected by the ioniza-
tion chamber at zero degrees and the fission fragments
were detected by an annular double-sided silicon strip
detector.
time−of−flight
Si det
beam defining
Timing
Timing
Timing
target
ionization
chamber
FIG. 2: (Color online) Apparatus for measuring fission and
evaporation residues cross sections induced by low intensity
beams in inverse kinematics.
The ERs were identified by the time-of-flight measured
with the microchannel plate timing detector located in
front of the ionization chamber and by energy loss in the
ionization chamber. The two microchannel plate timing
detectors located before the target were used to monitor
the beam intensity and to provide the timing reference
for the time-of-flight measurement. The microchannel
plate timing detector in front of the ionization chamber
was position sensitive and was used to monitor the beam
position. It was located 200 mm from the target and had
a 25 mm diameter Mylar foil. The ionization chamber
was filled with CF4 gas so that it could function at rates
up to 50,000 pps. Higher beam intensities occurred in
some of the fission measurements, requiring the ioniza-
tion chamber to be turned off. The data acquisition was
triggered by either the beam signal rate down scaled by a
factor of 1000, the coincidence of the delayed beam signal
and ER signal, or the silicon detector signal. A 350 MeV
Au beam that resembled ERs was measured by the ion-
ization chamber to calibrate the energy loss spectrum.
The ER cross section was obtained by taking the ratio of
the ER yield to the target thickness and the integrated
beam particles in the ionization chamber. A detailed de-
scription of the ER measurement technique used in this
experiment can be found in Ref. [28].
The annular double-sided silicon strip detector (Mi-
cron Semiconductor Design S2) was located 42 mm from
the target. It had 48 concentric strips on one side and 16
pie-shaped sectors on the other side. The inner diame-
ter was 35 mm and the outer diameter was 70 mm. The
thickness of the detector was 300 µm. The detection an-
gles spanned 15.6◦ to 39.6◦. The fission fragments were
identified by requiring a coincidence of events in the Si
detector and by the folding angle distributions of the de-
tected particles.
III. DATA REDUCTION PROCEDURES
A. Evaporation residues
Since this was an inverse kinematics reaction, the ERs
recoiled in the forward direction in a narrow cone. The
apparatus was designed to have high efficiency for detect-
ing ERs. The efficiency of the apparatus was estimated
by Monte Carlo simulations. The angular distribution of
the ERs was generated by statistical model calculations
using the code PACE2[29]. The input parameters for the
statistical model calculations will be discussed later in
this paper. The calculated efficiency for the lowest bom-
barding energy is 93±1%. It increases as the reaction
energy increases and reaches 98±1% at the highest en-
ergy.
A relatively thick target was used in this experiment.
The beam lost approximately 40 MeV after passing
through the target (13 MeV in the center of mass). For
this reason, the measured cross section is an average of
the contributions from the beam interacting throughout
the thickness of the target. The variation of ER cross
sections is not very large at energies above the Coulomb
barrier because the shape of the excitation function is al-
most flat. Therefore, the measured cross section is close
to that would be measured at an energy corresponding to
the middle of the target. However, at energies below the
barrier the ER cross section falls off exponentially. The
cross section near the entrance of the target has more
weight than that near the exit. Smooth curves fitting
the excitation function in this rapidly varying region were
used to determine the reaction energy associated with the
measured cross section.
An iterative method was used to determine the effec-
tive reaction energy for the thick target measurement .
First, the measured cross sections and the beam energies
calculated at the middle of the target were fitted by a
tensioned spline[30] where the smoothness of the curve
could be adjusted. The resulting curve was then used to
calculate the thick target cross section for each measure-
ment, according to
dE/dx
where σ(E) is the curve generated by the spline fit,
dE/dx is the stopping power of 132Sn in 64Ni, and ρ is the
target thickness. The integration limits were the energies
of the beam at the exit of the target and at the entrance
of the target. The energy, Ei, corresponding to the cross
section, σi, was obtained by interpolation using the fit-
ted curve. This set of energies was used as the input for
the next iteration of the fit. The result converged very
quickly. After five iterations, the energies differed from
the previous iteration energies by less than 0.2 MeV. The
validity of this method was checked by generating data
from a known function such as the Wong formula [31]
and folding in the effects of target thickness.
Comparing to the cross-section-weighted-average
method described in Ref. [28], the differences in energies
determined by these two methods are not noticeable
at high energies because the excitation function is
fairly flat. However, at energies below the barrier, the
energy determined by the cross-section-weighted-average
method is larger than that determined by the method
described above and disagrees with the measurement
in Ref. [32], as can be seen in Fig. 3. Furthermore,
it is found that using data generated from a known
function the effective energy obtained by the cross-
section-weighted-average method is shifted to too high
an energy in the exponential falloff region.
The uncertainty of the energy determination was esti-
mated by comparison with the method using the cross
section weighted average. The average uncertainty of
the effective reaction energy is 2.3 MeV in the region
where the excitation function is almost flat and increases
to 3.9 MeV in the exponential fall off region. The uncer-
tainty is larger, 5.8 MeV, for the lowest energy data point
because an extrapolation is required for calculating the
thick target cross section and the extrapolation region
is influenced by the location of the next higher energy
point.
To verify our measurement technique, the ER cross
sections for 124Sn+64Ni in inverse kinematics were mea-
sured and compared to those published by Freeman et al.
measured with a thin target[32]. It is noted that some
of our measurements were performed at energies differ-
ent from those of Ref. [32]. The comparison is shown in
Fig. 3. Our data (open triangles) are in good agreement
with those measured by Freeman et al. [32] (filled stars).
FIG. 3: (Color online) Comparison of ER cross sections for
64Ni+124Sn measured in this work and by Freeman et al.[32]
(filled stars). The filled circles are for energies determined
by the method described in Ref. [28] and the open triangles
are for energies determined by the method described in this
paper.
The solid circles are for energy determined by the cross-
section-weighted-average method described in Ref. [28].
B. Fission
Fission fragments were identified by requiring a coin-
cidence of two particles detected by the pie-shaped sec-
tors of the Si strip detector on either side of the beam.
Figure 4(a) and (b) present two-dimensional histograms
of particle energy and strip number of the Si detector
for coincident events taken from 560 MeV and 620 MeV
132Sn+64Ni, respectively. They were compared to the
kinematics calculation displayed in Fig. 4(c) and (d)
where the fission fragments, elastically scattered Sn and
Ni are shown by the solid, dash-dotted and dotted curves,
respectively. The angular range of the Si strip detector
is between the two vertical dashed lines. The elastically
scattered Ni and Sn appear in the upper right hand cor-
ner and center of the histogram, respectively. The fission
events are located in the gated area.
The folding angle distributions of the fragments were
used to distinguish fission from other reactions, such as
deep inelastic reactions. Since there are two solutions
for the kinematics of the inverse reaction, as shown in
Fig. 4(c) and (d), the fragment angular correlation is not
as simple as that in normal kinematics. Monte Carlo
simulations were performed to provide guidance. It was
assumed that only fusion-fission results from a full mo-
mentum transfer. The width of the mass distribution was
taken from the 58Ni+124Sn measurement[33]. The width
of the mass distribution was varied to estimate the uncer-
(d)(c)
Strip No.
0 10 20 30 40 50 60
Strip No.
0 10 20 30 40 50 60
FIG. 4: (Color online) (a) and (b) Two dimensional his-
tograms of energy and strip number for coincident events from
560 and 620 MeV 132Sn+64Ni, respectively, measured by the
annular double-sided silicon strip detector. The gated area
shows events from fission and other reactions. (c) and (d)
Kinematics of energy as a function of scattering angle for 560
and 620 MeV 132Sn+64Ni, respectively, elastic scattering and
fission fragments.The dash-dotted and dotted curves are for
the elastically scattered Sn and Ni, respectively whereas the
solid curve is for the fission fragments. The angular range of
the Si strip detector is between the two vertical dashed lines.
tainty of the simulation. The transition state model[34]
was used to predict the fission fragment angular distribu-
tion. In Fig. 5 the simulated fission fragment folding an-
gle distributions for 550 MeV 124Te+64Ni are compared
with a stable beam test measurement. The folding angle
distributions for one of the fragments detected in strip 2
(16.2◦), strip 22 (27.7◦), and strip 41 (36.8◦) are shown.
The gap in the spectra at strip 14, 30, 44, 46, and 47 are
malfunctioning strips in the detector.
The Monte Carlo simulated folding angle distributions
for fission are shown in the middle panels of Fig. 5 and
compared to those of measurements shown in the left
panels. For one of the fragments detected at forward
angles, strip 2 for example, the predicted angular dis-
tribution of the other fragment is similar to that of the
measurement. Most of these events are considered as re-
sulting from fission. For one of the fragments detected
near the middle part of the detector, strip 22 for instance,
there are differences between measurement and simula-
tion in the shapes of the angular distributions of the other
fragment. It is predicted that the other fission fragment
is distributed around strip 40. The measured distribution
spreads to more forward angles. For one of the fragments
detected at the backward angles, the yield of the other
fragment is predicted to be small and they are equally
FIG. 5: (Color online) Left panels: Folding angle distributions
for 550 MeV 124Te+64Ni for one of the fragments detected at
16.2◦ (strip 2), 27.7◦ (strip 22), and 36.8◦ (strip 41) by the
annular double-sided silicon strip detector. The elastic scat-
tering events are excluded. The dotted and dashed histograms
are the results of fitting the data with simulated fission and
deep inelastic collisions with Q=–20 MeV, respectively (see
text). Middle panels: Results of Monte Carlo simulations for
fission events. Right panels: Results of Monte Carlo simula-
tions for deep inelastic scattering events. The solid curves are
for reaction Q value of –10 MeV, the dashed curves are for
Q=–20 MeV, and the dotted curves are for Q=–40 MeV.
distributed between the middle part of the detector and
the outer edge of the detector. But the measured events
appear in the middle part of the detector. There are no
events in the region where fission events are expected.
These differences are attributed to the contribution from
other reaction mechanisms, most likely deep inelastic col-
lisions.
An attempt was made to simulate these deep inelastic
collision events. It was assumed that the mass of these
products were projectile- and target-like and the angular
distribution at forward angles followed a 1/sin(θ) depen-
dence. The right panels of Fig. 5 show the results of sim-
ulations performed for reaction Q values of –10 (solid),
–20 (dashed), and –40 MeV (dotted). It can be seen that
the overlap of fission and deep inelastic collisions becomes
larger at more backward angles. At strip 41 (36.8◦), deep
inelastic collisions account for all the events.
The relative contribution of fission and deep inelas-
tic collisions were obtained by fitting the simulated fold-
ing angle distributions to the measured distributions for
all the detector strips using the CERN library program
MINUIT[35]. In the fits, the normalization coefficients for
the simulated distributions were the only two variable
parameters. The results of the fits are shown in the left
panels of Fig. 5 by the dotted and dashed histograms for
fission and deep inelastic collisions with Q=–20 MeV, re-
spectively. The number of fission events in the measured
distributions were taken as the summed events in each
strip multiplied by the relative contribution of fission.
The folding angle distributions for 132Sn+64Ni are
shown in Fig. 6. Due to the low statistics, it was not prac-
tical to extract the fission events by fitting the folding
angle distributions. As an alternative, the fission events
were extracted by setting gates on the folding angle dis-
tributions using the simulated distributions as references.
This gating method was also tested with the 124Te+64Ni
measurement. The fission cross sections obtained by the
fitting method and the gating method agreed within 10%.
FIG. 6: (Color online) Left panels: Folding angle distributions
for 560 MeV 132Sn+64Ni for one of the fragments detected at
16.2◦ (strip 2), 27.7◦ (strip 22), and 36.8◦ (strip 41) by the
annular double-sided silicon strip detector. The elastic scat-
tering events are excluded. Middle panels: Results of Monte
Carlo simulations for fission events. Right panels: Results of
Monte Carlo simulations for deep inelastic scattering events.
The solid curves are for reaction Q value of –10 MeV, and the
dotted curves are for Q=–40 MeV.
The Monte Carlo simulation was also employed to cal-
culate the coincidence efficiency of the detector. The effi-
ciency increased from 5.7±0.9% at 530 MeV to 7.6±0.8%
at 620 MeV bombarding energy.
In the present work, the dynamic range of the ampli-
fiers was not sufficiently large resulting in the distortion
of the high energy signals. In the future, new amplifiers
that are more suitable for measuring the energy of fission
fragments will be used so that the mass ratio of reac-
tion products can be obtained to help distinguish fission
events from other reaction channels.
The formation of a compound nucleus depends on
whether the interacting nuclei are captured inside the
fusion barrier and whether the dinuclear system can sub-
sequently evolve into a compact shape. Quasifission oc-
curs when the dinuclear system fails to cross the saddle
level density parameter (a) A/8 MeV−1
af/an 1.04
diffuseness of spin distribution (∆l) 4 h̄
fission barrier Sierk[45]
TABLE I: Input parameters for statistical model calculations.
point to reach shape equilibrium. Since the beam inten-
sity was several orders of magnitude lower than that of
stable beams and the reaction was in inverse kinemat-
ics, making separation of fusion-fission and quasifission
very difficult, there was no attempt to distinguish quasi-
fission from fusion-fission in this work. Furthermore, the
experimental results are compared to barrier penetration
models which describe the capture process, making it un-
necessary to separate these two processes.
IV. COMPARISON WITH MODEL
CALCULATIONS
A. Statistical model
The compound nucleus formed in 132Sn+64Ni decays
by particle evaporation and fission. Statistical models
have successfully described compound nucleus decay for
a wide range of fusion reactions. The measured ER and
fission cross sections are compared with the predictions of
the statistical model code PACE2[29]. The input param-
eters were obtained by simultaneously fitting the data
from stable Sn on 64Ni[32, 36] and the measured fusion
cross sections[36] were used for the calculations. Fig-
ure 7(a), (b), and (c) displays the comparison of calcula-
tions and data for 112,118,124Sn+64Ni, respectively. The
calculations reproduce the measurements well except for
the ER cross sections of 112Sn+64Ni. Table I lists the in-
put parameters for the calculations. Without adjusting
the parameters, calculations for 132Sn+64Ni were per-
formed. The results are shown in Fig. 7(d). Very good
agreement between the calculation and the data can be
seen.
It is noted that some of the parameters used in our
calculations are different from those used by Lesko et
al. [36]. In their calculations, the code CASCADE[37] was
used. The mass of the nuclei in the decay chain was
calculated using the Myers droplet model[38]. The dif-
fuseness of the spin distribution was ∆l = 15h̄ and the
ratio of level density at the saddle point to the ground
state, af/an, was set to 1.0. In this work, a compilation
of measured masses[39], ∆l = 4h̄, and af/an = 1.04 were
used.
FIG. 7: (Color online) Comparison of measured ER (filled
circles) and fission (open circles) cross sections with statistical
model calculations. The solid and the dotted curves are the
calculated fission and ER cross sections, respectively, using
the measured fusion cross sections as input. (a) 64Ni+112Sn,
(b) 64Ni+118Sn, (c) 64Ni+124Sn (Freeman et al.[32]), and (d)
132Sn+64Ni (this work)
B. Coupled-channel calculation
In general, sub-barrier fusion enhancement can be de-
scribed by coupled-channel calculations. The fusion cross
section of 132Sn+64Ni, the sum of ER and fission cross
sections, is compared with coupled-channel calculations
using the code CCFULL[40]. The interaction potential
(V◦=82.46 MeV, r◦=1.18 fm, and a=0.691 fm) was taken
from the systematics of Broglia and Winther[41]. The
result of the calculations are compared with the data
in Fig. 8. The dotted curve is the prediction of a one-
dimensional barrier penetration model and it can be seen
that it substantially underpredicts the sub-barrier cross
sections. The coupled-channel calculation including in-
elastic excitation of 64Ni to the first 2+ and 3− states
and 132Sn to the first 2+ state is shown by the dashed
curve. The transition matrix elements, B(Eλ), of 64Ni
were obtained from Ref. [42, 43] and the B(E2) of 132Sn
was obtained from a recent measurement by Varner et
al.[44]. This calculation overpredicts the data at energies
near the barrier and underpredicts the data well below
the barrier.
The neutron transfer reactions have positive Q values
for transferring two to six neutrons from 132Sn to 64Ni.
Since there is no neutron transfer data available for this
reaction, the transfer coupling form factor is unknown.
Thus, the coupled-channel calculation including transfer
and inelastic excitation was performed with one effec-
FIG. 8: (Color online) Comparison of 132Sn+64Ni fusion
data (filled circles) with a one-dimensional barrier penetra-
tion model calculation (dotted curve). The coupled-channel
calculation including inelastic excitation of the projectile and
target is shown by the dashed curve and the calculation in-
cluding inelastic excitation and neutron transfer is shown by
the solid curve.
tive transfer channel using the Q value for two-neutron
transfer. The coupling constant was adjusted to fit the
data. The calculation with the coupling constant set to
0.48 is shown by the solid curve. It reproduces the data
very well except for the lowest energy data point which
has large uncertainties in energy and in cross section. A
better treatment of the transfer channels based on exper-
imental transfer data would help improve understanding
of the influence of transfer on fusion. Experimental neu-
tron transfer data on 132Sn+64Ni in the future would be
very useful.
V. DISCUSSION
The ER cross section can be described by
σER = πλ
(2l+ 1)σl,
where λ is the de Broglie wave length, lc the maximum
angular momentum for ER formation and σl the partial
cross section. The reduced ER cross sections for 64Ni
on stable-even Sn isotopes[32] are compared with that
for 132Sn+64Ni in Fig. 9. The reduced ER cross sec-
tion is defined as the ER cross section divided by the
kinematic factor πλ2. It can be seen that the ER cross
sections saturate at high energies as fission becomes a
significant fraction of the fusion cross section. In addi-
tion, the saturation value increases as the neutron excess
in Sn increases. This is consistent with the fact that the
fission barrier height increases for the more neutron-rich
compound nuclei.
FIG. 9: (Color online) The reduced ER cross section as a
function of the center of mass energy for 64Ni on stable even
Sn isotopes[32] and radioactive 132Sn.
In Fig. 10, the measured reduced ER cross sections for
Ni+Sn as a function of the calculated average mass of the
ERs, predicted by PACE2, are presented. In the same re-
action, the higher mass ERs are produced at lower beam
energies because of the lower excitation energies of the
compound nucleus. As the neutron excess in the com-
pound nucleus increases, neutron evaporation becomes
the dominant decay channel. The PACE2 calculation pre-
dicts that a compound nucleus made with Sn isotopes of
mass number greater than 120 decays essentially 100%
by neutron evaporation and Pt isotopes are the primary
ERs. The mass of the compound nucleus is different
when it is produced with different Sn isotopes. How-
ever, it can be seen that Pt of a particular mass can
be produced with different Sn isotopes if different num-
bers of neutrons are evaporated. The reaction with a
more neutron-rich Sn produces the same Pt isotope at a
higher rate. With 132Sn as the projectile, the ERs are so
neutron-rich that they cannot be produced by stable Sn
induced reactions. This suggests that it may be benefi-
cial to use neutron-rich radioactive ion beams to produce
new isotopes of heavy elements.
The fusion excitation functions of 64Ni on stable even
Sn isotopes[36] are compared with that of 132Sn+64Ni in
Fig. 11. In order to remove the effects of the difference
in nuclear sizes, the cross section is divided by πR2 with
R=1.2(A
t ) fm, where Ap (At) is the mass num-
ber of the projectile (target). The reaction energy in the
center of mass is divided by the barrier height predicted
by the Bass model[46]. It can be seen that the fusion of
132Sn and 64Ni is not enhanced with respect to the stable-
even Sn isotopes when the difference in nuclear sizes is
FIG. 10: (Color online) The reduced ER cross section as a
function of the calculated average mass of ERs predicted by
PACE2[29] for 64Ni on stable even Sn isotopes[36] and radioac-
tive 132Sn.
considered.
FIG. 11: (Color online) Comparison of fusion excitation func-
tions for 64Ni on stable even Sn isotopes[36] and radioactive
132Sn. The change in nuclear sizes are corrected by factor-
ing out the area and the Bass barrier height[46] in the cross
section and energy, respectively.
The lowest energy data point has large uncertainties.
The cross section seems enhanced comparing to the sta-
ble beam measurements in Fig. 9 and Fig. 11. A more
pronounced enhancement appears when the data point
is compared to our coupled-channel calculations (Fig. 8)
and to a time-dependent Hartree-Fock calculation[47].
To further explore if fusion is enhanced at this low energy
region, we plan to repeat the measurement with an im-
proved apparatus where the thickness of the Mylar foil in
the microchannel plate timing detector located in front
of the ionization chamber will be reduced. This will al-
low a better separation of the energy loss signals from
ERs and scattered beams in the ionization chamber at
low bombarding energies.
The Q values for transferring two to six neutrons from
132Sn to 64Ni are positive. It is necessary to include
neutron transfer in coupled-channel calculations to re-
produce experimental results. As the neutron excess in
the Ni isotopes decreases, the number of neutron transfer
channels with positive Q values increases for 132Sn+Ni.
In 132Sn+58Ni, the Q values for transferring one to six-
teen neutrons from 132Sn to 58Ni are positive and range
from 1.7 to 17.4 MeV. A large sub-barrier fusion enhance-
ment due to the coupling to neutron transfer is expected
to occur in 132Sn+58Ni. An experiment to measure the
fusion excitation function of 132Sn on 58Ni is in prepara-
tion.
Although 132Sn is unstable, its neutron separation en-
ergy is 7.3 MeV. This is not very low compared to stable
nuclei. The sub-barrier fusion enhancement observed in
132Sn+64Ni with respect to stable Sn nuclei can be ac-
counted for by the change in nuclear sizes. No extra en-
hancement was found. However, an increased ER yield
at energies above the barrier was observed as compared
to stable Sn. As the shell closure is crossed, the binding
energy for 133Sn decreases by a factor of two. The nu-
clear surface of 133Sn and even more neutron-rich Sn may
be more diffused. The number of neutron transfer chan-
nels with positive Q values increases by a factor of two
or more. Larger sub-barrier fusion enhancement beyond
the nuclear size effect may be expected.
VI. SUMMARY
Neutron-rich radioactive 132Sn beams were incident on
a 64Ni target to measure fusion cross sections near the
Coulomb barrier. With an average intensity of 5×104
pps beams and a high efficiency apparatus for ER detec-
tion, the uncertainty of the measured ER cross section is
small and comparable to that achieved in stable beam ex-
periments. The efficiency for fission fragment detection
was low but the detector had a very fine granularity. By
requiring a coincident detection of the fission fragments
and performing folding angle distribution analysis, fission
events were identified. The excitation functions of ER
and fission can be described by statistical model calcula-
tions using parameters that simultaneously fit the stable
even Sn isotopes on 64Ni fusion data. A large sub-barrier
fusion enhancement with respect to a one-dimensional
barrier penetration model prediction was observed. The
enhancement is attributed to the coupling of the projec-
tile and target inelastic excitation and neutron transfer.
The reduced ER cross sections at energies above the bar-
rier are larger for the 132Sn induced reaction than those
induced by stable Sn nuclei, as expected from the higher
fission barrier of the more neutron-rich compound nu-
cleus. For a specific mass of ER, reactions with a more
neutron-rich Sn have higher cross sections. When the
fusion excitation functions are compared on a reduced
scale, where the effects of nuclear size and barrier height
are factored out, no extra fusion enhancement is observed
in 132Sn+64Ni with respect to stable Sn induced fusion.
The fusion cross section measured at the lowest energy
seems to be enhanced. Experiments to investigate this
with an improved apparatus is planned.
VII. ACKNOWLEDGMENT
We would like to thank D. J. Hinde for helpful and
stimulating discussions. We wish to thank the HRIBF
staff for providing excellent radioactive beams and tech-
nical support. Research at the Oak Ridge National
Laboratory is supported by the U.S. Department of
Energy under contract DE-AC05-00OR22725 with UT-
Battelle, LLC. W.L. and D.P. are supported by the the
U.S. Department of Energy under grant no. DE-FG06-
97ER41026.
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