# Isotopic trends of quasifission and fusion-fission in the reactions
48Ca+239,244Pu

Lu Guo1,2<EMAIL_ADDRESS>Caiwan Shen3 Chong Yu4 Zhenji Wu4 1School of
Nuclear Science and Technology, University of Chinese Academy of Sciences,
Beijing 100049, China  2Institute of Theoretical Physics, Chinese Academy of
Sciences, Beijing 100190, China  3School of Science, Huzhou University, Huzhou
313000, China  4School of Physics, University of Chinese Academy of Sciences,
Beijing 100049, China

###### Abstract

Background: Quasifission and fusion-fission are primary mechanisms to prevent
the production of superheavy elements. The recent experimental measurements
reveal that the fusion-evaporation cross section in the $3n$ reaction channel
of 48Ca+239Pu is 50 times lower than using 244Pu as target nucleus [Phys. Rev.
C 92, 034609 (2015)]. However, the precise mechanisms of this remarkable
isotopic dependence are not well understood.

Purpose: To understand the experimental observation of the rapid decrease of
stability of superheavy nuclei as the neutron number decreases, the
theoretical studies of quasifission and fusion-fission in connection with
experimental production for $Z$=114 flerovium isotopes are required to
investigate the possible differences in reaction mechanisms induced by these
two targets.

Methods: We propose an approach called TDHF+HIVAP to take into account both
the evolution of dinuclear system and the deexcitation of compound nucleus,
which combines the microscopic time-dependent Hartree-Fock (TDHF) method for
the fusion and quasifission dynamics with the statistical evaporation model
HIVAP for fusion-fission dynamics.

Results: Fusion is observed for both reactions 48Ca+239,244Pu with the side
orientation the deformed target nucleus, while quasifission dynamics is
observed for the tip orientation. The nuclear contact times, masses and
charges as well as the kinetic energies of the fragments, and the mass-angle
distribution strongly depend on the colliding energy, impact parameter, and
deformation orientation. The quantum shell effect displays a crucial role in
both the quasifission and the fusion-fission processes. The quasifission is
considerably reduced and the survival probability is enhanced around one order
of magnitude in the reaction using 244Pu target as compared to the 239Pu case.

Conclusions: The studies by using TDHF+HIVAP method well account for the
experimental observations and the present method clearly shows its
applicability in the reaction mechanisms of quasifission and fusion-fission
dynamics. The experimental and theoretical results encourage the use of
neutron-rich targets for the production of new superheavy elements.

##  I Introduction

The creation of superheavy elements (SHEs) is one of the most challenging
research topics in nuclear physics. SHEs up to proton number $Z=118$ have been
experimentally produced in fusion-evaporation reactions, either using 208Pb
(or 209Bi) as the target in cold fusion, or 48Ca-induced hot fusion colliding
with the actinide nuclei. However, the SHEs produced so far in the experiments
are far from the long-lived stability island, which was predicted by many
theoretical approaches Bender  _et al._ (2001); Cwiok (2005); Pei  _et al._
(2005), locating at the neutron magic number $N=184$ and proton magic number
$Z=114-126$ as a result of new shell closures. The existence of the stability
island is supported by the observed increase of stability of heavier isotopes
approaching to the predicted magic number $N=184$ Oganessian  _et al._ (2010);
Ellison  _et al._ (2010); Kozulin  _et al._ (2014a).

The entrance channel dynamics is critical for SHE formation. In heavy-ion
collisions, capture reaction produces a dinuclear system. The shape evolution
of this initial fragile dinucleus (either fusion or quasifission) is
determined by the dynamical dissipation from the collective kinetic energy to
internal degrees of freedom. The dinucleus system may evolve to an
equilibrated compound nucleus in fusion process. Alternatively it can also
break apart, characterized by the massive nucleon transfer and contact time
longer than 5 zs. This process is known as quasifission (QF) Bock  _et al._
(1982); Tōke  _et al._ (1985); Shen  _et al._ (1987). For light and medium-
mass systems typically with $Z_{p}Z_{t}<1600$, the compound nucleus will be
formed once the projectile is captured by the target. For heavy systems
producing SHE, however, the fusion probability is dramatically reduced by the
quasifission. Furthermore, even if the compound nucleus survives against
quasifission, the fusion-fission (FF) has very large probability to happen due
to its excitation. Hence, the production cross section forming SHE is
substantially reduced by the quasifission and fusion-fission processes.

To produce the new SHEs and heavier isotopes of known SHE experimentally, the
optimal target-projectile combination and bombarding energy should be chosen
to have the highest residue cross sections for the desired SHE. Experimental
measurements indicate an increase of residue cross sections for the reactions
involving more neutron-rich nuclei Dragojević  _et al._ (2008); Oganessian
_et al._ (2013); Utyonkov  _et al._ (2015). From the theoretical point of
view, the enhancement of residue cross section in the reactions with neutron-
rich target could arise from the decrease of QF and FF probability. However,
the precise mechanisms, especially for the interplay between the QF and FF
process, are not well understood. The dependence of QF on the neutron richness
of compound nucleus is supported by the recent experimental measurements of
mass-angle distributions Itkis  _et al._ (2011); du Rietz  _et al._ (2011);
Lin  _et al._ (2012); Wakhle  _et al._ (2014). In particular, the recent
studies of QF show a great promise for a deep insight of reaction mechanism.
Meanwhile QF dynamics may be affected by many variables, e.g., collision
energy Hinde  _et al._ (2008); Nishio  _et al._ (2012), deformation and
orientation Hinde  _et al._ (1996); Prasad  _et al._ (2016) , shell structure
Kozulin  _et al._ (2014b); Prasad  _et al._ (2015) of the colliding nuclei,
and neutron richness of compound nuclei Hammerton  _et al._ (2015). Besides
the competition between fusion and quasifission, the competition between the
neutron emission and fission of the excited compound nucleus is also crucial
for the synthesis of SHEs. To understand the complex interplay of these
dynamical processes, it is required to carry out the theoretical studies.

Various theoretical models, including both the macroscopic models Shen  _et
al._ (2002); Feng  _et al._ (2009); Shen  _et al._ (2011); Shen and Shen
(2014); Zhu  _et al._ (2014, 2016); Wang  _et al._ (2016, 2017) and the
microscopic approaches Simenel (2012); Nakatsukasa  _et al._ (2016); Wang  _et
al._ (2002); Wen  _et al._ (2013, 2014), have been developed to account for
the experimental observations. Although the macroscopic model may well
reproduce the experimental data, a need for external parameters and a lack of
dynamical effect restrict its predictive power for reactions where no
experimental data are available. In present study, we combine the microscopic
time-dependent Hartree-Fock (TDHF) approach with the statistical evaporation
model HIVAP to investigate the effect of neutron richness on the QF and FF
dynamics. TDHF approach provides a profound understanding of nuclear dynamics,
as seen from the recent applications in fusion Simenel  _et al._ (2004); Umar
and Oberacker (2006); Guo and Nakatsukasa (2012); Umar  _et al._ (2012);
Simenel  _et al._ (2013); Umar  _et al._ (2014); Washiyama (2015); Godbey  _et
al._ (2017); Simenel  _et al._ (2017); Schuetrumpf and Nazarewicz (2017); Guo
_et al._ (2018), quasifission Golabek and Simenel (2009); Oberacker  _et al._
(2014); Umar  _et al._ (2015, 2016); Yu and Guo (2017), transfer reaction
Washiyama  _et al._ (2009); Simenel (2010); Scamps and Lacroix (2013);
Sekizawa and Yabana (2013); Wang and Guo (2016); Sekizawa and Yabana (2016);
Scamps  _et al._ (2017); Sekizawa (2017), fission Simenel and Umar (2014);
Scamps  _et al._ (2015); Goddard  _et al._ (2015, 2016); Bulgac  _et al._
(2016); Tanimura  _et al._ (2017), deep inelastic collisions Maruhn  _et al._
(2006); Guo  _et al._ (2007, 2008); Dai  _et al._ (2014a, b); Stevenson  _et
al._ (2016); Guo  _et al._ (2017); Shi and Guo (2017); Umar  _et al._ (2017),
and resonances dynamics Maruhn  _et al._ (2005); Nakatsukasa and Yabana
(2005); Umar and Oberacker (2005); Reinhard  _et al._ (2007); Simenel and
Chomaz (2009); Fracasso  _et al._ (2012). The statistical evaporation model
HIVAP Shen  _et al._ (2002, 2008) is adopted to take into account the
deexcitation process including both fission and particle evaporation. We will
show that the fission barrier of compound nucleus, dominated mainly by the
quantum shell effect, plays significant effect on the fusion-fission process
in the present systems.

This article is organized as follows. In Sec. II TDHF approach with Skyrme
energy functional and fusion-evaporation dynamics are briefly recalled.
Section III presents the theoretical analysis of the influence of neutron rich
target in the QF and FF dynamics for the reactions 48Ca+239,244Pu. A summary
is given in Sec. IV.

##  II Theoretical framework

In TDHF approach the many-body wave function $\Psi(\mathbf{r},t)$ is
approximated as a single Slater determinant composed by the single-particle
states $\phi_{\mathrm{\lambda}}(\mathbf{r},t)$

$\Psi(\mathbf{r},t)=\frac{1}{\sqrt{N!}}\text{det}\\{\phi_{\mathrm{\lambda}}(\mathbf{r},t)\\},$
(1)

and this form is kept at all times in the dynamical evolution. This
approximation leads to the omission of two-body nucleon-nucleon correlations.
By taking the variation of time-dependent action

$S=\int_{t_{1}}^{t_{2}}dt\langle\Psi(\mathbf{r},t)|H-i\hbar\partial_{t}|\Psi(\mathbf{r},t)\rangle$
(2)

with respect to the single-particle states, one may obtain a set of nonlinear
coupled TDHF equations in the multidimensional spacetime phase space

$i\hbar\frac{\partial}{\partial_{t}}\phi_{\mathrm{\lambda}}(\mathbf{r},t)=h\phi_{\mathrm{\lambda}}(\mathbf{r},t),$
(3)

where $h$ is the HF single-particle Hamiltonian. It describes the time
evolution of the single-particle wave functions in a mean field. The set of
nonlinear TDHF equations have been solved on three-dimensional coordinate
space without any symmetry restrictions and with much more accurate numerical
methods.

Most TDHF calculations have been done with Skyrme effective interaction Skyrme
(1956). It is natural to represent the Skyrme force with the energy density
functional (EDF), in which the total energy of the system

$E=\int
d^{3}r\mathcal{H}(\rho,\tau,{\mathbf{j}},{\mathbf{s}},{\mathbf{T}},J;{\mathbf{r}})$
(4)

is expressed as an integral of the energy functional. The number density
$\rho$, kinetic density $\tau$, current density ${\mathbf{j}}$, spin density
${\mathbf{s}}$, spin-kinetic density ${\mathbf{T}}$, and spin-current
pseudotensor density $J$ are obtained as a sum over single-particle wave
functions. The Skyrme EDF is then expressed as

$\displaystyle\begin{split}\mathcal{H}&=\mathcal{H}_{0}+\sum_{\mathrm{t=0,1}}\Bigg{\\{}A_{\mathrm{t}}^{\mathrm{s}}\mathbf{s}_{\mathrm{t}}^{2}+A_{\mathrm{t}}^{\Delta{s}}\mathbf{s}_{\mathrm{t}}\cdot\Delta\mathbf{s}_{\mathrm{t}}\\\
&+A_{\mathrm{t}}^{\mathrm{T}}\bigg{(}\mathbf{s}_{\mathrm{t}}\cdot\mathbf{T}_{\mathrm{t}}-\sum_{\mu,\nu=x}^{z}J_{\mathrm{{t},\mu\nu}}J_{\mathrm{{t},\mu\nu}}\bigg{)}\Bigg{\\}},\end{split}$
(5)

where $\mathcal{H}_{0}$ is the simplified Skyrme functional used in Sky3D code
Maruhn  _et al._ (2014) and most TDHF calculations. For the definition of
coupling constants $A$, see Ref. Lesinski  _et al._ (2007). Due to the
computational complexity, various approximations to Skyrme EDF have been
employed in TDHF calculations, which restrict the number of degrees of freedom
accessible during a collision, and hence the nature and degree of dissipation
dynamics. For instance, the inclusion of spin-orbit interaction Umar  _et al._
(1986) solved an early conflict between TDHF predictions and experimental
observation, and turned out to play an important role in fusion and
dissipation dynamics Maruhn  _et al._ (2006); Dai  _et al._ (2014a). The time-
odd terms appearing in Eq. (5) are also shown to be non-negligible in heavy-
ion collisions Umar and Oberacker (2006). Our HF and TDHF codes contain all of
the time-even and time-odd terms in the energy functional and don’t impose the
time-reversal invariance, which allow us to compute directly the odd system,
such as those studied here, without resorting to the filling approximation. As
pointed out in Refs. Lesinski  _et al._ (2007); Stevenson  _et al._ (2016),
the terms containing the gradient of spin density may cause the spin
instability both in nuclear structure and reaction studies, so we set
$A_{\mathrm{t}}^{\Delta{\mathbf{s}}}=0$ in our calculations.

In fusion-evaporation reaction, the production cross section for the
superheavy evaporation residues (ER) can be defined as

$\sigma_{\rm ER}=\sum_{J=0}^{\infty}\sigma_{\rm cap}(E_{\rm c.m.},J)P_{\rm
fus}(E^{*},J)W_{\rm sur}(E^{*},J),$ (6)

where $\sigma_{\rm cap}$ is the capture cross section for the projectile and
target to come together, $P_{\rm fus}$ is the fusion probability of dinuclear
system against QF, and $W_{\rm sur}$ is the survival probability of compound
nucleus against FF. Fusion occurs when the collective kinetic energy is
entirely converted into the internal excitation of a well-defined compound
nucleus. Traditionally, the fusion cross section is given by

$\sigma_{\rm fus}(E_{\rm c.m.})=\sum_{J=0}^{\infty}\sigma_{\rm cap}(E_{\rm
c.m.},J)P_{\rm fus}(E^{*},J).$ (7)

Since TDHF theory describes the collective motion of fusion dynamics in terms
of semi-classical trajectories, the sub-barrier tunneling of the many-body
wave function can’t be included in TDHF. Consequently, the fusion cross
section can be estimated by the quantum sharp-cutoff formula Bonche  _et al._
(1978)

$\sigma_{\rm fus}(E_{\rm c.m.})=\frac{\pi\hbar^{2}}{2\mu E_{\rm c.m.}}[(l_{\rm
max}+1)^{2}-(l_{\rm min}+1)^{2}],$ (8)

where $\mu$ is the reduced mass of the system and $E_{\rm c.m.}$ the initial
center-of-mass (c.m.) energy. The quantities $l_{\rm max}$ and $l_{\rm min}$
denote the maximum and minimum orbital angular momentum for which fusion
happens.

In the production of superheavy elements, the excited compound nucleus may
undergo the deexcitation process, which is dominated by fission barrier and
neutron evaporation. The survival probability of compound nucleus $W_{\rm
sur}$ in Eq. (6) for $x$ neutron emissions is written as

$W_{\rm sur}(xn)=\prod_{i=1}^{x}P_{in},$ (9)

where $P_{in}$ is the probability of $i$-th neutron emission and given by the
statistical evaporation model Weisskopf (1937); Bohr and Wheeler (1939)

$P_{in}=\frac{\Gamma_{in}}{\Gamma_{in}+\Gamma_{if}},$ (10)

with $\Gamma_{in}$ the $i$-th neutron evaporation width and $\Gamma_{if}$ the
$i$-th fission width. In the numerical calculations of present work, the hivap
code Shen  _et al._ (2002, 2008) is adopted to calculate the survival
probability $W_{\rm sur}$.

##  III results

TDHF approach has recently demonstrated its feasibility and success in fusion
and quasifission dynamics Oberacker  _et al._ (2014); Umar  _et al._ (2015,
2016), in which the theoretical investigations on the reaction mechanism
induced by the different projectiles colliding the actinide target nucleus
account for the experimental observations reasonably. In present work, our
goal is to investigate how the neutron richness in target nucleus affects the
QF and FF dynamics in the production of SHE flerovium with the reactions
48Ca+239,244Pu. We employ the Skyrme SLy5 force Chabanat  _et al._ (1998a, b)
including all of the time-odd terms in the mean-field Hamiltonian in our TDHF
calculations.

In the numerical simulation, first we calculate an accurate static ground
state for the projectile and target nucleus on the symmetry-unrestricted
three-dimensional grid. The numerical coordinate boxes for the static HF wave
functions are chosen as $24\times 24\times 24$ fm3 for 48Ca and $32\times
28\times 32$ fm3 for 239,244Pu, respectively. The nucleus 48Ca shows a
spherical ground state and 239,244Pu exhibits a prolate quadrupole
deformation, which are in agreement with experimental data and other
calculations. The correct description of the initial shape of target and
projectile nucleus is important for the dynamical evolution of heavy-ion
collisions. Second, we apply a boost operator on the static single-particle
wave functions. The nucleus is assumed to move on a pure Coulomb trajectory
until the initial distance so that the initial boost is properly treated in
TDHF evolution. The time propagation is performed using a Taylor-series
expansion up to the sixth order of the unitary mean-field propagator and a
time step of $0.2~{}\mathrm{fm}/c$. For the TDHF dynamical evolution, we use a
numerical box of $60\times 28\times 46$ fm3 and a grid spacing of 1.0 fm. The
reaction is taken in $x$-$z$ plane and along the collision axis $x$. The
initial separation between the two nuclei is set to be 30 fm. The choice of
these parameters assures a good numerical accuracy for all the cases studied
here. The total TDHF energy and particle number are well conserved and shift
less than 0.1 MeV and 0.01 in the dynamical evolution, respectively.

Figure 1: (Color online) Time evolution of the mass density in the side
collision of 48Ca+239Pu at $E_{\mathrm{c.m.}}=204.02$ MeV and $b$=1.5 fm.

We first consider the reaction 48Ca+239Pu at $E_{\mathrm{lab}}$ = 245 MeV
(corresponding to $E_{\mathrm{c.m.}}$ = 204.02 MeV), which is the energy used
in the Dubna experiment Utyonkov  _et al._ (2015). The time evolution of the
mass density is displayed in Fig. 1, in which the symmetry axis of the prolate
deformed nucleus 239Pu is initially set perpendicular to the internuclear
axis. This is the so-called side orientation which leads to the largest
contact time in central collisions Wakhle  _et al._ (2014). We observe that
TDHF calculation predicts fusion for this collision. Our definition for fusion
is that an event has the contact time larger than 35 zs, and furthermore a
mononuclear shape without any neck formation is required. As shown in Fig. 1,
in the early stage of collision, there forms a neck between the two fragments,
and the dinuclear system starts to rotate. With the time evolution, more
nucleons transfer from heavy to light fragment and the neck grows into a
mononuclear shape. This mononuclear system keeps its shape for enough long
time and results in the fusion process.

Figure 2: (Color online) Time evolution of the mass density in the tip
collision of 48Ca+239Pu at $E_{\mathrm{c.m.}}=204.02$ MeV and $b$=2.5 fm.

By contrast, quasifission is observed for the same reaction, but with the tip
orientation and impact parameter $b$=2.5 fm, as shown in Fig. 2. Until the
neck formation, the density evolution is similar as in Fig. 1. After that, due
to the competition between Coulomb repulsion and centrifugal force, the
dinuclear system prolongs and then breaks in two fragments. This is the QF
process which is characterized, as compared to the deep inelastic collision
(DIC), by the long contact time and significant nucleon transfer. In this tip
collision, the contact time is found to be 8.17 zs and roughly 30 nucleons are
transferred from 239Pu to 48Ca. Here the contact time is calculated as the
time interval in which the lowest density in the neck exceeds half of the
nuclear saturation density $\rho_{0}/2=0.08~{}\mathrm{{fm}^{-3}}$ Wakhle  _et
al._ (2014); Oberacker  _et al._ (2014); Umar  _et al._ (2016). The light and
heavy fragments in the exit channel are around 79Ge and 208Pb, respectively.
The heavy fragment lying around $N=82$ magic shell indicates that the quantum
shell effect plays an important role in the QF dynamics. Such a multi-nucleon
transfer is crucial to understand the dissipation dynamics in heavy-ion
collisions.

Figure 3: (Color online) Contact time (a), mass and charge of the heavy
fragments (b) as a function of $E_{\mathrm{c.m.}}/V_{\mathrm{B}}$ for the
central collisions 48Ca+239Pu (open circle) and 48Ca+244Pu (solid circle) with
the tip of 239,244Pu.

We now compare the energy dependence of fusion and quasifission dynamics in
the central collisions 48Ca+239,244Pu. The energy range is chosen as
$E_{\mathrm{c.m.}}/V_{\mathrm{B}}=$1.0–1.2, denoted by the ratio between the
colliding energy $E_{\mathrm{c.m.}}$ and TDHF capture barrier
$V_{\mathrm{B}}$. The TDHF capture barrier is calculated for two extreme
orientations of the deformed targets 239,244Pu (tip and side). For the tip
collision, the barrier is found to be 177 MeV with 239Pu target and 181 MeV
for 244Pu, respectively. The side collision results in a significantly higher
barrier of 199 MeV both for 239,244Pu targets, as expected. TDHF calculations
show that both central collisions 48Ca+239,244Pu with the side of target
nuclei lead to fusion within the above energy range. Certainly, the noncentral
collisions with the side orientation may show quasifission. Hence, we leave
out the results for the side collisions, and the central collisions with the
tip orientation are shown in Fig. 3. For the reaction 48Ca+239Pu (open
circle), we observe that the contact time is within the value of 5–10 zs over
a wide range of energies. But, a dramatically different behavior appears for
48Ca+244Pu (solid circle). At energies close to barrier
($E_{\mathrm{c.m.}}/V_{\mathrm{B}}=1.0-1.06$.), the contact time is larger
than 35 zs and a mononuclear shape is kept, which is considered as fusion
reaction leading to the formation of compound nucleus. As the energy
increases, quasifission is observed, in which the contact time is
systematically larger than the 239Pu case. The long contact time of the
quasifission process with the neutron-rich system relative to the neutron-
deficient system have also been observed in Cr$+$W reactions Hammerton  _et
al._ (2015). In the dynamical evolution of the neutron-rich system, the
elongation of the dinuclear system is much slower and the compact
configuration with mononuclear shape remains much longer, which are expected
to lead to a longer contact time. These results demonstrate that the reaction
with the neutron-rich target favors the fusion. In similar reactions 48Ca+238U
Oberacker  _et al._ (2014) and 48Ca+249Bk Umar  _et al._ (2016), tip
orientations are never found to lead to fusion in TDHF calculations using
SLy4d force Chabanat  _et al._ (1998a, b). This difference may arise from the
choice of different Skyrme interaction, numerical approximations, and the
shell structure of target nuclei. In particular, the influence of Skyrme
interaction in QF should be examined. Since TDHF calculations for reactions
leading to superheavy systems require very long CPU times, the interaction
dependence of QF dynamics will be the subject of future works, which is beyond
the purpose of present study.

In the lower panel of Fig. 3 are the corresponding mass and charge of the
heavy fragments. We find that the transferred nucleon number is within a small
variation as a function of energy. For the reaction 48Ca+239Pu, the heavy QF
fragments has the charge $Z_{H}\simeq 81.4-83.9$ and mass $A_{H}\simeq
205.3-211.9$ in this energy range, which is centered in the vicinity of the
doubly magic nucleus 208Pb. The tip collisions clearly favor the production of
heavy fragments near 208Pb at all energies, indicating a strong influence of
the quantum shell effects. A similar effect was observed in TDHF calculations
for the tip collisions using 238U Oberacker  _et al._ (2014) and 249Bk Umar
_et al._ (2016) targets. For the reaction 48Ca+244Pu, the 1–2 more protons and
3–4 more neutrons are transferred on average, as compared to 48Ca+239Pu, due
to the charge equilibrium.

Figure 4: (Color online) Contact time (a), mass and charge of the heavy
fragments (b) as a function of impact parameter both for the tip (open circle)
and side (solid circle) collisions of 48Ca+239Pu at $E_{\mathrm{c.m.}}=204.02$
MeV used in Dubna experiment Utyonkov  _et al._ (2015).

We next concentrate on the impact parameter dependence of fusion and
quasifission dynamics in connection with the experimental observation, in
which the isotopic dependence of measured ER cross section of Flerovium
isotopes has been displayed in Ref. Utyonkov  _et al._ (2015). As stated in
the Abstract of Ref. Utyonkov  _et al._ (2015), the measured ER cross section
in 239Pu(48Ca,$3n$)284FI reaction channel are about 50 times lower than using
244Pu as target nucleus. However, the precise mechanisms of this remarkable
isotopic dependence are not well understood yet. In the following, we will
investigate how the QF and FF affect the production of SHEs.

The collision 48Ca+239Pu at $E_{\mathrm{lab}}$ = 245 MeV ($E_{\mathrm{c.m.}}$
= 204.02 MeV), which is the energy used in the Dubna experiment Utyonkov  _et
al._ (2015), is shown in Fig. 4. For the tip collision (open circle), the
contact time is roughly linear decreasing for the impact parameter $b<3$ fm,
and then form a plateau with small variance until $b$=4.0 fm. After that, a
quick decrease of contact time is observed which corresponds to the inelastic
scattering. No fusion events are predicted for the tip orientation. The side
collision (solid circle) shows a very different behavior as compared to the
tip collision. We find that TDHF predicts fusion for $b<2$ fm, and then a
rapid drop-off within a narrow range of impact parameter 2`-`3 fm. At
$b$=3`-`5 fm the quasielastic collisions happen, characterized by the nearly
zero contact time and identical fragments before and after collision. In Fig.
4(b) are the corresponding mass and charge of the heavy fragments. For the tip
collision, the transferred nucleon is nearly constant in the quasifission
region $b$=0`-`4 fm, producing the typically heavy fragments around 208Pb.
This implies that the magic shell effect is also observed to be crucial as a
function of impact parameter for the tip collisions. For the side collision,
the transferred nucleon is proportional to the contact time. We find that only
collision with the tip of 239Pu produces QF fragments in the magic $Z$=82
region, while collisions with the side are the only ones that may result in
fusion. These findings are consistent with the experimental observation in the
collision with 238U target Wakhle  _et al._ (2014).

Figure 5: (Color online) Contact time (a), mass and charge of the heavy
fragments (b) as a function of impact parameter both for the tip (open circle)
and side (solid circle) collisions of 48Ca+244Pu at $E_{\mathrm{c.m.}}=216.76$
MeV used in GSI experiment Gates  _et al._ (2011).

For comparison, the same observables for the reaction 48Ca+244Pu at energy
$E_{\mathrm{lab}}$ = 259.4 MeV ($E_{\mathrm{c.m.}}$ = 216.76 MeV) used in GSI-
TASCA experiment Gates  _et al._ (2011) are displayed in Fig. 5. For the tip
collision, the contact time behaves similar as in 48Ca+239Pu. The multi-
nucleon transfer from the heavy to light nucleus produces the heavy fragments
with the charge $Z_{H}\simeq 74-82$ and mass $A_{H}\simeq 192-212$. The heavy
QF fragment is again in the vicinity of Z=82 magic shell. For the side
orientation of 244Pu nucleus, TDHF shows fusion for $b<2.5$ fm and
quasifission at $b$=2.5–4 fm, both of which are in a larger angular momentum
window as compared to the 239Pu case, indicating a high possibility for fusion
with 244Pu target.

By comparing the impact parameter dependence of fusion and quasifission
dynamics between the two reactions (Fig. 4 and Fig. 5), the fusion happens in
the side collision for $b<2.0$ fm with 239Pu target and $b<2.5$ fm with 244Pu.
We further refine the maximum impact parameter for fusion within the precision
of 0.1 fm, and find that the critical value is 1.7 fm and 2.3 fm for 239Pu and
244Pu cases, respectively. The fusion window is about 0.6 fm large in
48Ca+244Pu, indicating that quasifission is considerably reduced as compared
to the 239Pu case. By using the sharp cut-off formula Eq. (8), the ratio of
fusion cross section between 244Pu and 239Pu side collisions is roughly 1.8.
One should note that the average over the orientation for the deformed target
nucleus should be done if one wants to get an accurate value for this ratio.

Figure 6: (Color online) Mass-angle distribution of QF fragments both for the
tip (open circle) and side (solid circle) collisions of 48Ca+239Pu at
$E_{\mathrm{c.m.}}=204.02$ MeV (a) and 48Ca+244Pu at
$E_{\mathrm{c.m.}}=216.76$ MeV (b). The impact parameters (in units of fm) are
written next to each data point.

On the other hand, the fusion-fission of compound nucleus dominated mainly by
the height of fission barrier and neutron evaporation may be sensitive to the
neutron-richness of compound nucleus. This FF process is at a much longer
time-scale than QF and has no memory of the entrance channel, which is beyond
the scope of TDHF studies. In order to calculate the ratio of ER cross
sections between 292Fl ($4n$) and 287Fl ($3n$), besides the ratio of their
fusion cross section, we also need to calculate the ratio of their survival
probability $W_{\rm sur}$. However during the decay, due to the very low
fission barrier $B_{\rm f}$, the fission dominates the decay process and has
several orders higher probability than the neutron emissions, and thus the
$W_{\rm sur}$ after several neutron emissions is several orders smaller than
the fusion cross section Shen  _et al._ (2002). Empirically, 1 MeV larger of
the $B_{\rm f}$ causes one order higher of the survival probability $W_{\rm
sur}$.

Some theoretical investigations indicated that 298Fl could be the next double-
magic nucleus after 208Pb, and then one may expect more stable and larger
$B_{\rm f}$ of 292Fl compared to 287Fl. This is confirmed by several models
predicting $B_{\rm f}$ Möller  _et al._ (2009); Staszczak  _et al._ (2013);
Jachimowicz  _et al._ (2017) and giving the difference of the fission barrier
$\Delta B_{\rm f}$ (= $B_{\rm f}(^{292}$Fl) $-B_{\rm f}(^{287}$Fl)) around
$0.82\sim 1.98$ MeV and averaging at $B_{\rm f}=1.2$ MeV. Another simpler but
frequently used $B_{\rm f}$ is approximated as $B_{\rm f(LD)}-E_{\rm{sh}}$,
where $B_{\rm f(LD)}$ is the liquid drop fission barrier and $E_{\rm sh}$ the
shell correction energy. As an example, by adopting the $B_{\rm f(LD)}$ from
Ref. Dahlinger  _et al._ (1982) and the $E_{\mathrm{sh}}$ from Ref. Wang  _et
al._ (2014), corresponding to $\Delta B_{\rm f}=0.55$ MeV, the $W_{\rm
sur}(4n)$ for 292Fl at $E^{*}=41.7$ MeV is $8.21\times 10^{-8}$, while the
$W_{\rm sur}(3n)$ for 287Fl at $E^{*}=37.7$ MeV is $5.5\times 10^{-9}$, and
hence the ratio of the survival probability for the two reactions is 14.9.
Finally, involving estimated ratio of fusion cross sections, the approximate
ratio of ER cross sections between the two reactions is roughly 27, close to
the ratio the experimental data showing.

Figure 7: (Color online) TKE-mass distribution of QF fragments both for the
tip (open circle) and side (solid circle) collisions of 48Ca+239Pu at
$E_{\mathrm{c.m.}}=204.02$ MeV (a) and 48Ca+244Pu at
$E_{\mathrm{c.m.}}=216.76$ MeV (b). The Viola systematics (solid line) is also
shown.

The mass-angle distributions (MADs) of QF fragments have been measured in
experiments, which can be used to distinguish the QF from FF events. In
experiments, FF fragments are usually more symmetric than in QF. In Fig. 6, we
show the scattering angle as a function of the mass ratio
$M_{\mathrm{R}}=m_{1}/(m_{1}+m_{2})$, where $m_{1}$ and $m_{2}$ are the masses
of QF fragments, both for the tip (open circle) and side (solid circle)
collisions of 48Ca+239,244Pu. The values of impact parameters are written next
to each data point. The scattering angle is calculated as the sum of the
incoming and outgoing Coulomb scattering angles plus the TDHF scattering angle
in the center-of-mass frame. We observe that for the tip collisions of
48Ca+239Pu the mass ratio of QF events is near $M_{\mathrm{R}}=$0.22–0.27 for
light fragments and $M_{\mathrm{R}}=$0.73–0.78 for heavy fragments. At larger
impact parameter $b$=4.5`-`5 fm for the tip collisions and $b$=2.5`-`5 fm for
the side collision, the light fragments lying around $M_{\mathrm{R}}=$0.16 and
heavy fragments $M_{\mathrm{R}}=$0.84 correspond to the quasielastic and deep-
inelastic reactions. One should note that TDHF is a deterministic theory which
gives only the most probable event instead of a probability distribution of
various events. In the lower panel of Fig. 6, the MADs in 48Ca+244Pu show
similar behavior in comparison to the 48Ca+239Pu case, except the larger mass
ratio due to the more nucleon transfer. The MADs for the side orientation span
a larger angular range in 48Ca+244Pu. The detailed differences can be
recognized by the values of the MADs for the specific impact parameters.

The correlation between mass and total kinetic energy (TKE) of QF fragments is
another key experimental observable to distinguish the quasielastic events
from fully damped events such as QF and FF. In TDHF, the TKE is calculated as
the sum of kinetic energy of the fragments after the separation and the
Coulomb potential energy assuming that the fragments are pointlike charges.
Figure 7 shows that the TKE-mass distribution of QF fragments both for the tip
(open circle) and side (solid circle) collisions of 48Ca+239,244Pu. The Viola
systematics (solid line) Viola  _et al._ (1985); Hinde  _et al._ (1987) is
also shown and has been known in good agreement with the measured TKE of
experimental fission fragments. We observe that the TKE of QF fragments are
distributed around the Viola systematics, indicating that most of the relative
kinetic energy has been dissipated in the collision.

##  IV Summary

We investigate the isotopic dependence of quasifission and fusion-fission in
connection with the experimental production of superheavy flerovium isotopes
for the reactions 48Ca+239,244Pu by using the microscopic TDHF approach and
the statistical evaporation model HIVAP. Quasifission dynamics may be
characterized by many variables, for instance, nuclear contact times, masses
and charges as well as kinetic energies of the fragments, and the mass-angle
distribution. We investigate the dependence of these variables on bombarding
energy, impact parameter, and orientation of the deformed target. Long contact
times associated with fusion are observed in both reactions with a side
orientated target nucleus, whereas only quasifission dynamics is observed for
the tip orientation. For the tip collisions, the quasifission fragments lying
in the vicinity of the magic nucleus 208Pb indicate the importance of the
quantum shell effect. No quantum shell effects are observed in the collisions
with the side of 239,244Pu targets. The calculated mass-angle and TKE-mass
distributions of QF fragments can be used for a direct comparison with the
experimental measurements.

The quasifission in the reaction 48Ca+244Pu is found to be remarkably reduced
as compared to the 48Ca+239Pu case, which leads to the ratio of fusion cross
section between 244Pu and 239Pu side collisions roughly 1.8. The differences
are attributed to the neutron-richness of target nucleus. While considering
the isospin-dependence of the fission barrier, the ratio of survival
probability of 292Fl ($4n$) to 287Fl ($3n$) is about 14.9, and thus the
predicted ratio of their residue cross sections is about 27, close to the
value shown by experimental data. Therefore, our studies explain the
experimental observations and make clear of the precise reaction mechanisms,
which may motivate the experimentalists to produce SHEs with more efficient
target-projectile combinations. The present method TDHF+HIVAP demonstrates its
usefulness for the microscopic reaction mechanism of quasifission and fusion-
fission dynamics.

##  V Acknowledgments

L. G. thanks A. S. Umar for helpful discussions. This work is partly supported
by NSF of China (Grants No. 11175252, 11575189, 11747312, U1732138, 11790325,
11790323), NSFC-JSPS International Cooperation Program (Grant No.
11711540016), and Presidential Fund of UCAS. The computations in present work
have been performed on the High-performance Computing Clusters of SKLTP/ITP-
CAS, Tianhe-1A supercomputer located in the Chinese National Supercomputer
Center in Tianjin, and the C3S2 computing center in Huzhou University.

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