††thanks: the authors contributed equally††thanks: the authors contributed
equally

Second institution and/or address

# Microscopic study of deformation and orientation effects in heavy-ion
reactions above Coulomb barrier using the Boltzmann-Uehling-Uhlenbeck model

Yujie Feng1 Huizi Liu1 Yingge Huang1 Fuchang Gu1 Erxi Xiao1 Xin Lei1 Hui Wang1
Jiali Huang1 Long Zhu1,3 Jun Su1,2,3<EMAIL_ADDRESS>Sino-French
Institute of Nuclear Engineering and Technology, Sun Yat-sen University,
Zhuhai 519082, China  China Nuclear Data Center, China Institute of Atomic
Energy, Beijing 102413, China  Guangxi Key Laboratory of Nuclear Physics and
Nuclear Technology, Guangxi Normal University, Guilin 541004, China

###### Abstract

Background

The understanding of the impact of initial deformation and collision
orientation on quasi-fission and fusion-fission reactions remains incomplete.

Purpose

This article aims to explore how the orientation of deformed nuclei influences
quasi-fission and fusion-fission around 1.2 $V_{B}$, employing a
microdynamical method in systems with diverse shapes, namely 24Mg + 178Hf, 34S
+ 168Er, and 48Ti + 154Sm.

Method

Utilizing the Boltzmann-Uehling-Uhlenbeck model, this study investigates
quasi-fission and fusion-fission reactions. The model elucidates micro-dynamic
processes and microscopic observables through the definition of the window and
event-by-event simulations.

Results

The findings reveal that the orientation of deformed nuclei significantly
influences the nucleus-nucleus interaction potential, thereby impacting the
competition between quasi-fission and fusion-fission. Particularly, the
orientation of the deformed target nucleus emerges as the primary factor
affecting this competition. Notably, a higher proportion of fusion-fission
events is observed when the target nucleus is in the belly orientation
compared to the tip. The study also observes that the configuration of the
dinuclear system contributes to fluctuations and dissipation. Collisions with
different orientations result in distinct dinuclear system configurations,
with belly-oriented collisions leading to larger fluctuations between events,
while tip-oriented collisions exhibit smaller fluctuations.

Conclusions

Considering diverse orientations of nuclei with distinct initial deformations,
this study concludes that the orientation of the target nucleus is the key
factor influencing quasi-fission and fusion-fission reactions around 1.2
$V_{B}$.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Quasi-fission (QF) is a pivotal process in heavy-ion reactions, characterized
by the formation of asymmetric fragments Giuliani  _et al._ (2019). Its
significance arises from its competition with fusion, a phenomenon critical to
the understanding of superheavy element synthesis. The propensity of QF to
lead to asymmetric fragment formation poses a challenge to the desired
formation of compound nuclei, thus garnering considerable attention from
researchers in the past few decades Simenel  _et al._ (2021). Consequently, a
thorough investigation into the competition between QF and fusion is
indispensable for optimizing reaction conditions and enhancing the probability
of synthesizing superheavy elements Hinde  _et al._ (2021). Furthermore, the
non-equilibrium nature of QF reactions offers a unique advantage. The
observables of the final state retain crucial information about the entrance
channel conditions, providing valuable insights into the effects of the
initial nuclear configurations Hammerton  _et al._ (2019). This characteristic
makes QF an invaluable tool for studying the intricate interplay of factors
influencing heavy-ion reactions, contributing to our understanding of the
dynamics involved in the synthesis of superheavy elements.

The influence of entrance channel effects on reaction dynamics near the
Coulomb barrier has captivated the attention of both experimental and
theoretical researchers. In the experimental realm, extensive investigations
have been undertaken to discern entrance channel conditions favoring QF.
Noteworthy factors under scrutiny include mass asymmetry Berriman  _et al._
(2001), fissionability of compound nuclei Lin  _et al._ (2012), reaction
energy Back (1985), magicity Simenel  _et al._ (2012a), and neutron richness
Adamian  _et al._ (2000). In the theoretical domain, the macroscopic dinuclear
system model has been pivotal in revealing the impact of orientation effects
on the distribution of the nucleus-nucleus interaction potential Li  _et al._
(2005). Furthermore, dynamic simulations using microscopic models such as
time-dependent Hartree-Fock (TDHF) and stochastic mean-field (SMF) have
provided valuable insights into reaction dynamics under various initial
conditions Sekizawa (2017); Sekizawa and Yabana (2016); Zheng  _et al._
(2018); Ayik  _et al._ (2019a, b). The microscopic ImQMD model, in particular,
has contributed significantly to the exploration of the effects of initial
deformation and orientation on QF probability, as evidenced by studies such as
Zhao  _et al._ (2013); Wang  _et al._ (2015). These theoretical frameworks not
only shed light on the intricate interplay of entrance channel parameters but
also offer a nuanced understanding of the underlying mechanisms governing QF.

In recent decades, researchers have delved into the intricate interplay of
initial nuclear deformation and orientation in QF and fusion-fission (FF)
reactions. The seminal work in 1995 by Hinde et al. demonstrated a correlation
between nuclear deformation, orientation, and the competition of QF and FF
Hinde  _et al._ (1995). Subsequent investigations by S. Mitsuoka et al. in
2002 brought attention to the role of an ’extra push’ in tip collisions, where
additional kinetic energy was required for fusion. This extra energy was
associated with the elongated contact configuration and the fission saddle-
point configuration. In contrast, belly collisions required less of this
’extra push’ for fusion reactions Mitsuoka  _et al._ (2002). A pivotal
experiment conducted by Knyazheva et al. in 2007 underscored the crucial role
of target nucleus deformation throughout the entire reaction Knyazheva  _et
al._ (2007). Comparing experimental data on QF with theoretical models has
been instrumental in testing the predictive capabilities of these models and
probing the dynamical processes involved. The TDHF calculations by Simenel et
al. in 2012 revealed that tip collisions predominantly result in the QF
component Simenel  _et al._ (2012b). Subsequent the TDHF calculations by
Seikizawa et al. indicated that collisions with different orientations
influence neck nucleon transfer in the dinuclear system, thereby affecting
reactions near the Coulomb barrier Sekizawa (2017). Recent TDHF calculations
have extended this understanding, highlighting the impact of the initial
orientation of a deformed nucleus on intermediate configurations and the
dynamical trajectory of QF reactions McGlynn and Simenel (2023).

While significant progress has been made, the specific dynamical factors
governing the competition between QF and FF are not yet fully elucidated.
Further investigation is needed to uncover the nuanced interactions
influencing this competition and refine our understanding of the underlying
mechanisms in heavy-ion reactions. In our prior study, we introduced and
applied the Boltzmann-Uehling-Uhlenbeck (BUU) model to examine reactions
involving nuclei near the Coulomb barrier. Our findings demonstrated
substantial agreement with experimental data, as detailed in Feng  _et al._
(2023). Building on this work, our current research extends the inquiry to
explore the nuanced influence of orientation and deformation in the dynamical
mechanisms governing the competition between QF and FF reactions around 1.2
times the Coulomb barrier energy ($V_{B}$). The paper is organized as follows.
In Sec. II, we briefly describe the theoretical method. In Sec. III, we
present both results and discussions. Finally, the conclusions and overviews
are given in Sec. IV.

##  II Theoretical framework

The development of successive approximations for the original global N-body
problem is based on the concept of the BBGKY hierarchy Durand  _et al._
(2000). By selectively excluding specific terms in the first equation of the
hierarchy using the Hartree-Fock approximation and ignoring the two body
correlation, we can derive the TDHF equation, which is a time-dependent
formulation of the Hartree-Fock method Dirac (1930):

$i\frac{\partial\hat{\rho}}{\partial
t}=\left[\hat{h}\left[\hat{\rho}\right],\hat{\rho}\right],$ (1)

where $\hat{\rho}$ represents the one-body density matrix, while $\hat{h}$
corresponds to the mean-field Hamiltonian. By making the assumption of
molecular chaos, it is possible to solve the second equation of the hierarchy
and subsequently express the first equation of the hierarchy as the quantum
Boltzmann equation Reinhardt  _et al._ (1985); Reinhard  _et al._ (1986),

$i\frac{\partial\hat{\rho}}{\partial
t}=\left[\hat{h}\left[\hat{\rho}\right],\hat{\rho}\right]+\hat{I}\left[\hat{\rho}\right],$
(2)

where $\hat{I}$ represents the collision term and comes from the solution of
the second equation of the hierarchy.

In the quantum framework, dealing with the collision term for finite fermion
systems can be extremely challenging Reinhard and Suraud (2015); Dinh  _et
al._ (2022). Therefore, in the field of heavy-ion physics, the Wigner
representation is widely utilized, which resulting in the classical Boltzmann
equation.

$\left(\frac{\partial}{\partial
t}+\frac{\mathbf{p}}{m}\cdot\nabla_{r}-\nabla_{r}U(f)\cdot\nabla_{p}\right)f(\mathbf{r},\mathbf{p},t)=I(f).$
(3)

here, $f$ represents the single-particle density in the phase space,
encompassing the degrees of freedom of coordinate $\mathbf{r}$ and momentum
$\mathbf{p}$. The left-hand side of the equation describes the evolution of
the single-particle density, which is influenced by the mean-field potential
$U$. This potential incorporates both the nuclear and Coulomb interactions.
The term on the right-hand side, i.e. $I(f)$, denotes the collision term of
the Uehling-Uhlenbeck form, in which the Pauli principle is considered.

In the numerical simulation of Eq.(3), the parallel-ensemble method is
employed, where Ntp systems are simulated simultaneously to obtain a smooth
density distribution. In the literature, this approach is referred to as the
BUU Bertsch and Gupta (1988), Vlasov-Uehling-Uhlenbeck Aichelin and Bertsch
(1985); Kruse  _et al._ (1985), or Landau-Vlasov models Grégoire  _et al._
(1987).

The mean-field potential $U$ in Eq.(3) includes the Coulomb and nuclear terms.
The Coulomb interaction between protons is described by simple point charges
and calculated in each parallel-ensemble. The specific form and parameters of
the potential are detailed in the previous publication Feng  _et al._ (2023).
The nuclear potential parameters are derived from fitting the main features of
the equation of state (EOS) for infinite nuclear matter, including saturation
density, binding energy, incompressibility coefficient, symmetry energy and
other characteristics.

At intermediate energies in heavy-ion collisions, nucleon-nucleon (NN)
scattering is prevalent, and Pauli blocking in NN scattering is crucial for a
proper quantum treatment, preventing a classical phase space distribution.
However, when the incident energy is near the Coulomb barrier, the scenario
undergoes a shift. In this energy regime, NN scattering becomes less frequent.
Consequently, the NN collision term $I(f)$ in Eq.(3) is implemented using the
Phase-Space-Density Constraint (PSDC) method. In the PSDC method, the phase
space occupation probability is computed by accumulating the contributions of
test particles in the BUU frame, as illustrated in the equation below.

$\overline{f}(\mathbf{r}_{i},\mathbf{p}_{i})=\frac{h^{3}}{2N_{tp}V_{p}}\sum_{j}^{V_{p}}\delta(\tau_{i},\tau_{j})\rho_{ij}$
(4)

where Vp is the accumulating volume in the momentum, $\tau_{i}$ and $\tau_{j}$
are the isospin of the $i$th and $j$th test particles, $\delta$ function is
for select the test particles with the same isospin, $\rho_{ij}$ is the
density contribution of $j$th test particles in the position of the $i$th test
particles. Since the spin degree of freedom is not considered in the model,
there is a number 2 in the denominator.

The PSDC method involves monitoring the phase-space occupation probability
during time evolution. NN scatterings between neighboring nucleons are
conducted to ensure that the phase-space occupation does not exceed 1 while
minimizing the loss of reaction events. Momentum exchanges are selectively
performed for nucleon pairs in which one nucleon has
$\overline{f}(\mathbf{r}_{i},\mathbf{p}_{i})$ less than 0.3, and the other
nucleon has $\overline{f}(\mathbf{r}_{j},\mathbf{p}_{j})$ greater than 1.05.
The choice of 0.3 and 1.05 as empirical parameters is specific to this
treatment.

In intermediate-energy heavy-ion collisions, the contribution of surface
energy is negligible and can be disregarded. However, as the collision energy
approaches the Coulomb barrier, the impact of surface energy becomes
pronounced. To account for this, we introduced a surface energy constraint
term into our model. The surface potential is

$U_{s}=g\nabla^{2}\rho,$ (5)

where $g$ is a parameter to adjust the strength of the surface potential. The
value of $g$ chosen in this work is 5 MeV fm5, which is the result obtained
from the previous study Feng  _et al._ (2023). The inclusion of this
constraint term ensures the preservation of the initial shape of the
projectile and target nuclei for an extended duration in the BUU collision
dynamics process, accounting for dissipative kinetic energy.

Figure 1: (a)The setting of coordinate axes during the dynamical evolution of
the dinuclear system involves. (b)The variation of the $\textit{I}(\theta,t)$
value as a function of the cosine value of the rotation angle of the
coordinate axes. (c)The linear density distribution along the collision
symmetry axis.

To investigate the dissipative dynamics in the QF process, it is essential to
establish a window that distinguishes between the projectile and target nuclei
at each time step. Initially, determining the symmetry axis of the dinuclear
system is crucial, satisfying two conditions: passing through the center of
mass of the dinuclear system and minimizing the rotational inertia of all test
particles. The $xz$ coordinate axis, with the center of mass of the dinuclear
system as the origin, is established, as illustrated in Fig.1(a). At this
point, the coordinates of all test particles are represented as ($z$, $x$).
After rotation by an angle $\theta$, the coordinates of the test particles
become ($z^{\prime}$, $x^{\prime}$). They satisfy the relationship:
$\begin{pmatrix}z^{\prime}\\\
x^{\prime}\end{pmatrix}$=$\begin{pmatrix}z\cos\theta+x\sin\theta\\\
x\cos\theta-z\sin\theta\end{pmatrix}$ The $\textit{I}(\theta,t)$ values which
is proportional to the rotational inertia is expressed in the following
relationship:

$I(\theta,t)\propto\int x^{\prime}z^{\prime}\rho(\mathbf{r},t)d\mathbf{r}$ (6)

The relationship between $\textit{I}(\theta,t)$ and $\cos(\theta)$ is depicted
in Fig.1(b). The $z$-axis that satisfies $\textit{I}(\theta,t)=0$ serves as
the symmetry axis of the dinuclear system, meeting the two aforementioned
conditions.

Secondly, we calculate the linear density $\rho_{L}$ of the test particles
along this symmetry axis, as illustrated in Fig.1(c). The $\rho_{L}$
distribution exhibits a double-peak shape, and the valley between the two
peaks identifies the position of the window. A plane perpendicular to the
symmetry axis at this position defines the window of the dinuclear system. By
employing this window, researchers can investigate the temporal variations in
mass, energy, and momentum of the projectile and target nuclei.

Figure 2: The time range computed for the BUU model and GEMINI++ model

Figure 2 illustrates the models employed in this work at different time
scales. The BUU model is utilized to simulate the dynamical process before the
formation of the compound nucleus within a time range of 10000 fm/c,
encompassing QF reactions and fusion. Prior to 10000 fm/c, the collision and
capture of the projectile and target nuclei form a dinuclear system, which can
subsequently undergo either QF or fusion reactions. The BUU model
distinguishes between these two reactions by defining windows and provides the
proportion of fusion and QF reactions through event-by-event simulations.
After 10000 fm/c, the excited compound nuclei formed through fusion reactions
undergo symmetric fission (referred to as FF), simulated using the GEMINI++
model Charity (2010); Mancusi  _et al._ (2010). By combining the BUU and
GEMINI++ models, we can obtain the final-state fragments of the entire
collision reaction, encompassing fragments from both QF and FF reactions.

##  III  Results and discussions

reaction system | $\beta_{2}^{\rm(P)}$ | $\beta_{2}^{\rm(T)}$ | shape of projectile nucleus and target nucleus
---|---|---|---
24Mg + 178Hf | 0.393 | 0.278 | prolate nuclei + prolate nuclei
34S + 168Er | \- 0.235 | 0.297 | oblate nuclei + prolate nuclei
48Ti + 154Sm | 0.011 | 0.270 | spherical nuclei + prolate nuclei

Figure 3: The initial shape of nuclei for the three collision systems. The 34S
nucleus is oblate. The 48Ti nucleus is spherical, while other nuclei are all
prolate. Table 1: The $\beta_{2}$ values and their corresponding initial shape
for the three collision systems. If the absolute value of the $\beta_{2}$
value is less than 0.1, the nucleus is approximately spherical. If the
$\beta_{2}$ value is greater than 0.1, the nucleus is prolate. If the
$\beta_{2}$ value is less than - 0.1, the nucleus is oblate.

To investigate the effects of deformation and orientation on the competition
between QF and FF, we selected collision systems with diverse deformations.
This selection aimed to thoroughly examine the influence of initial nuclear
deformations on the competition mechanism between QF and FF. Figure 3 displays
the shapes of the selected collision nuclei in this study. The 48Ti nucleus is
spherical, while the 34S nucleus is oblate, and the remaining nuclei are
prolate. Table I presents the combinations of the three collision systems and
the $\beta_{2}$ values for each nucleus, along with the corresponding shape
combinations for each selected collision system.

Figure 4: Mass distributions of BUU + GEMINI++ calculations (QF + FF) of model
(red line) and experimental data (circles) for (a) 24Mg+178Hf, (b) 34S+168Er,
and (c) 48Ti + 154Sm at around 1.2 $V_{B}$. The incident energy (Elab) and the
excitation energy of 202Po (The compound nucleus synthesized from the three
systems) for each system are indicated. The experiment data and Elab are from
Ref. Rafiei  _et al._ (2008), and the BUU model calculates the excitation
energy. The $x$-axis corresponds to the mass ratio MR = $\frac{m}{m1+m2}$,
where m denotes the fragment mass number, and m1 and m2 represent the mass
numbers of the projectile and target nuclei.

In Figure 4, a comparison is made between experimental data and model
calculations of fragment mass distributions (MDs) for three distinct reaction
systems. The MDs of QF are simulated using the BUU model, while the MDs of FF
are simulated with the GEMINI++ model. For the (a) 24Mg + 178Hf and (b) 34S +
168Er collisions, the model calculations align well with experimental data,
demonstrating good agreement. However, some discrepancies are noted around
M${{}_{R}}\approx$ 0.7 and M${{}_{R}}\approx$ 0.3. These deviations can be
attributed to fluctuations in the BUU model simulations, and the presence of
only a small fraction of fragments in this region. Regarding the 48Ti + 154Sm
collision, the model simulations match the experimental data for fragments in
the $0.4\leq\textit{M}{{}_{R}}\leqslant 0.6$ range. Nevertheless,
discrepancies are observed for fragments in the
$0.3\leq\textit{M}{{}_{R}}\leqslant 0.4$ and $0.6\leq\textit{M}_{R}\leqslant
0.7$ ranges. This is attributed to the smaller mass asymmetry of the 48Ti +
154Sm system compared to the other two systems, leading to predominantly
quasi-elastic and deep inelastic events that are not considered in our
calculations.

Furthermore, the figure reveals that the three reactions form the same
compound nucleus, 202Po, with identical excitation energy. However, the curves
for both experimental and model data differ for different reaction systems,
indicating variations in the MDs of QF for different systems. MDs of QF carry
information about the entrance channel, influenced by factors such as the
initial deformation of the projectile and target nuclei and the mass asymmetry
of the projectile-target system, resulting in different MDs of QF for various
reaction systems.

Figure 5:  (Color online) The dynamic and static nucleus-nucleus (NN)
interaction curves for the 48Ti + 154Sm system, where the dynamic NN
interactions calculation is the average result of simulating 30 events at an
incident energy of Elab = 235 MeV. The calculations in the figure are
performed for central collisions. V represents the NN interaction, and R
represents the center-of-mass distance between the two nuclei. The solid red
lines represent the static NN interaction for the target nuclei in the belly
orientation, while the solid red circle-dotted lines represent the dynamic one
for the target in belly. The dashed black lines represent the static curve for
the target in tip, and the solid red square-dotted lines represent the dynamic
curve for the target in tip.

We understand that the orientation of deformed nuclei can impact the nucleus-
nucleus interaction potential, subsequently affecting the capture probability
and fusion cross-section of the projectile and target nuclei. Fig. 5 depicts
the static and dynamic interaction potential as a function of distance for the
48Ti + 154Sm system, considering the tip and belly orientations of the target
nucleus. The interaction potential comprises the nucleus interaction potential
$U_{N}$, the Coulomb interaction potential $U_{C}$, and the centrifugal
potential. As these simulations are for central collisions, the centrifugal
potential does not play a role. It’s observed that when the target nucleus is
in the belly orientation, both the static and dynamic barriers are higher
compared to the tip orientation. Moreover, the highest point of the
interaction corresponds to a smaller R compared to the tip orientation. This
indicates that when the target nucleus is in the belly orientation, the
collision results in a deeper contact between the projectile and target
nuclei, leading to a larger $U_{C}$ and consequently a higher barrier.
Additionally, when the target nucleus is in the belly orientation, the static
and dynamic interactions coincide at larger R. However, at closer distances,
the dynamic interaction is lower than the static interaction. This occurs
because, at larger distances, the nucleon density of the projectile and target
nuclei remains largely unchanged, resulting in the coincidence of the static
and dynamic interactions. As the nuclei approach each other, dynamic changes
in nuclear density lead to corresponding changes in dynamic interaction.

When the target nucleus is in the tip orientation, the dynamic and static
interactions coincide at larger R. However, at closer R, the dynamic
interaction surpasses the highest point of the static barrier, and the minimum
distance at which the projectile and target nuclei can be in contact is
smaller. The dynamic interaction for the tip orientation and the belly
orientation of the target nucleus lies between the static interactions for the
tip and belly orientations. This is attributed to the dynamic evolution of the
BUU model, where nuclei cannot maintain their initial deformations and tend to
evolve towards a spherical shape. In conclusion, the initial orientation of
deformed nuclei can influence the distribution of effective nucleus-nucleus
interactions, aligning with conclusions from prior research Hammerton  _et
al._ (2019); Li  _et al._ (2005).

Figure 6:  The proportion of quasi-fission (QF) reaction and fusion-fission
(FF) reaction for 24Mg + 178Hf, 34S + 168Er, and 48Ti + 154Sm system in
central collision at near Coulomb barrier energy, The initial deformation and
orientation for projectile and target nuclei are indicated.

Our calculations reveal that the initial orientation significantly influences
the competition between QF and FF. In Figure 6, we illustrate the proportions
of QF and FF events among the total reaction events in 24Mg + 178Hf, 34S +
168Er, and 48Ti + 154Sm reactions for all possible combinations of
orientations. From the figure, it is evident that the competition between QF
and FF is greatly affected by the orientation of the target nucleus.
Specifically, in the three systems, the proportion of FF events is notably
higher in the belly orientation of the target nuclei compared to the tip
orientation. Another noteworthy theoretical observation is that, as the charge
product in the entrance channel (Z1Z2) increases, the overall proportion of QF
events tends to increase. This phenomenon aligns with previous findings in the
literature Tsang  _et al._ (1983).

Figure 7:  (Color online)Mass distributions of quasi-fission (QF) fragments in
(a) 24Mg+178Hf, (b) 34S+168Er, and (c) 48Ti + 154Sm in central collisions at
near-Coulomb barrier with different orientations.

On the other hand, collision configurations also influence the mass
distributions of QF. Figure 7 illustrates the mass distributions of QF
reactions for (a) 24Mg + 178Hf, (b) 34S + 168Er, and (c) 48Ti + 154Sm under
different orientations. The red dots represent the mass distributions of the
target nucleus in the belly orientation, while the black line represents the
mass distributions of the target nucleus in the tip orientation. It is evident
that the width of the mass distribution is larger in the belly orientation of
the target nucleus compared to the tip orientation. In QF reactions, the width
of the mass distributions can be attributed to the sticking time, referring to
the survival time of the dinuclear system before separating into fragments
Rafiei  _et al._ (2008); Back (2011); Shen  _et al._ (1987). Reactions with
longer sticking times have a higher probability of forming fragments with a
near-equilibrium mass ratio (MR = 0.5), resulting in narrower mass
distributions. Conversely, shorter sticking times of the dinuclear system lead
to wider mass distributions of QF fragments. Therefore, in the case of the
target nucleus in the belly orientation, the sticking time of the dinuclear
system is shorter compared to the tip orientation of the target nucleus. This
factor contributes to influencing the competitive proportion between QF and
FF.

Figure 8:  (Color online)The relationship between the center-of-mass distance
R12 of two nuclei and the linear density $\rho_{L}$ at the window for sph.-tip
collisions (black hollow circles) and sph.-belly collisions (red solid
circles) in the 48Ti + 154Sm system at Elab = 235 MeV as a function of
reaction evolution time.

To investigate the microscopic dynamic mechanisms influencing the sticking
time of the dinuclear system, we selected 48Ti + 154Sm, with different
orientation combinations showing the maximum difference in width of mass
distribution. Figure 8 illustrates the evolution of the 48Ti + 154Sm system at
an incident energy of Elab = 235 MeV, comparing two orientations: target
nucleus in the tip orientation (black hollow circles) and target nucleus in
the belly orientation (red solid circles). The figure shows (a) the center-of-
mass distance of the projectile and target nucleus R12 and (b) the linear
density $\rho_{L}$ at the window position as a function of reaction time.

Several distinct points are observable, the turning point of the curves, the
extremum of the curves, and the duration of the curves at the extremum.
Firstly, in Fig. 8 (b), the black line starts to rise at approximately 40
fm/c, indicating the formation of the window around 40 fm/c when the target
nucleus is in the tip orientation. The red line starts to rise at around 100
fm/c, signifying the formation of the window at approximately 100 fm/c when
the target nucleus is in the belly orientation. The sequential formation of
the window is due to the elongated shape of the target nucleus, where in the
tip orientation, the target nucleus comes into contact with the projectile
nucleus earlier.

Secondly, the extremum of the curves in Fig. 8(a) and (b) differ. In Fig.
8(a), the minimum value that the red line can reach is around 7 fm, while the
black line reaches a minimum of around 10 fm. This implies that collisions
with the target nucleus in the belly orientation have a deeper contact with
the projectile nucleus compared to collisions with the target nucleus in the
tip orientation. The extremum of the curves in Fig. 8(b) also confirms this
point, showing that collisions with the target nucleus in the belly
orientation exhibit a higher linear density at the window.

Thirdly, regarding the duration of the extremum, the black line reaches its
extremum at 200 fm/c and remains unchanged until 7000 fm/c, while the red line
reaches its extremum at around 400 fm/c and remains unchanged until
approximately 3500 fm/c. Similarly, the duration shown in Fig. 8(b) follows
the same pattern. This indicates that the dinuclear system maintains a stable
evolution for a longer time in sph.-tip compared to sph.-belly. This
observation also aligns with the results in Fig. 6, where the width of the MD
in QF is related to the sticking time of the dinuclear system, with a wider
mass distribution representing a shorter sticking time.

Figure 9:  (Color online)The mass-angle distributions of quasi-fission (QF)
fragments for central collisions in different orientations of the 34S + 168Er
system at Elab = 174 MeV.

Figure 9 illustrates the mass-angle distributions for the 34S+168Er system in
central collisions at an incident energy of Elab = 174 MeV, considering four
different collision orientations. The previously mentioned mass distribution
are a subset of the mass-angle distribution, which not only provide
information about the sticking time between the projectile and target nuclei
through the fragmentation deflection angle but also offer insights into
fluctuations between events.

Comparing (a) and (b) with (c) and (d) in Fig. 9, three distinct points can be
observed. Firstly, when the target nucleus is in the tip orientation, the
reaction fragments are concentrated around MR = 0.5. On the other hand, when
the target nucleus is in the belly orientation, the two reaction fragments are
distributed on both sides of MR = 0.5. This implies that when the target
nucleus is in the tip orientation, the dinuclear system in the reaction has a
longer survival time compared to when the target nucleus is in the belly
orientation. This indicates a more extensive exchange of nucleons between the
projectile and target nuclei, resulting in a more symmetric MD of the two
fragments after the reaction.

Secondly, when the target nucleus is in the tip orientation, QF fragments are
concentrated at 0 degrees and 180 degrees, while for the belly orientation, QF
fragments are concentrated at 30 degrees and 150 degrees.

Thirdly, the distribution of fragments is more concentrated when the target
nucleus is in the tip orientation, whereas it is more dispersed when the
target nucleus is in the belly orientation. This implies that events with the
target nucleus in the tip orientation exhibit less fluctuation between events,
while events with the target nucleus in the belly orientation have larger
fluctuations. Therefore, the dynamical properties of each event and their
uncertainties are greater when the target nucleus is in the belly orientation.
The fluctuations between events are also a contributing factor to the
dependence of the competition between QF and FF on the orientation of the
target nucleus.

Figure 10:  (Color online)The phase space distribution of the 34S + 168Er
system at different reaction times and orientations in central collisions with
an incident energy of Elab = 174 MeV.

Figure 10 shows the phase space distribution of the 34S + 168Er collision
system at different orientations and moments in central collisions. From the
figure, two pieces of information can be inferred related to the fluctuations
between events.

Firstly, at a time of 1000 fm/c, the dinuclear system configuration for the
tip-oriented target nucleus is elongated, while for the belly-oriented target
nucleus, it is triangular. This implies that the triangular configuration of
the dinuclear system is more unstable and will rapidly break apart into two
fragments, resulting in larger uncertainties in the angles and masses of the
emitted fragments.

Secondly, at a time of 10000 fm/c, the dinuclear system with the tip-oriented
target nucleus exhibits minimal rotation relative to the incident direction,
whereas the dinuclear system with the belly-oriented target nucleus undergoes
a certain degree of rotation relative to the incident direction. This aligns
with the previous observation that the triangular configuration of the
dinuclear system is unstable. Therefore, this can explain the observed
behavior in panels (c) and (d) of Figure 6, where fragments obtained from
collisions with the belly orientation of target nuclei exhibit a certain
angular deflection and tend to be mass asymmetric. Additionally, there is a
larger fluctuation between events in this type of collision.

##  IV CONCLUSION

In this study, we employed the Boltzmann-Uehling-Uhlenbeck (BUU) model to
investigate the impact of orientation and deformation in heavy-ion collision
reactions occurring around 1.2 times the Coulomb barrier (1.2 $V_{B}$). We
selected three distinct shape combinations: 24Mg + 178Hf, 34S + 168Er, and
48Ti + 154Sm for our calculations.

Our investigation unveiled that different orientations of deformed nuclei have
a significant influence on the nucleus-nucleus interaction potential.
Specifically, the potential barrier is notably higher when the target nucleus
is in the belly orientation. Furthermore, we established that the orientation
of deformed target nuclei plays a pivotal role in determining the competition
between QF and FF. A higher proportion of FF events is observed when the
target nucleus is in the belly orientation compared to the tip orientation.
Additionally, our findings indicate that the orientation of the deformed
target nucleus affects the width of the mass distributions of QF.
Specifically, the mass distributions exhibit a wider spread when the target
nucleus is in the belly orientation.

These observed phenomena are attributed to underlying micro-mechanisms,
including the sticking time between the projectile and target nuclei and
fluctuations between events in the reaction. Microscopic dynamical simulations
using the BUU model revealed that the depth of contact between the projectile
and target nuclei influences the survival time of the dinuclear system.
Moreover, the configuration of the formed dinuclear system in the reaction is
identified as a micro-mechanism impacting the fluctuations between events.

It should be emphasized that the BUU model can predict reaction dynamics in a
nonempirical manner, as demonstrated in this study. Therefore, conducting
further systematic BUU calculations for various projectile- target
combinations with different orientations, deformations, and entrance channel
conditions has the potential to provide effective reference settings for
determining the initial conditions that influence fusion probabilities and
facilitate the experimental synthesis of new isotopes.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China
under Grants Nos. 11875328 and 12075327, the Central Government Guidance Funds
for Local Scientific and Technological Development, China (No. Guike
ZY22096024), the Key Laboratory of Nuclear Data foundation(JCKY2022201C157).

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