Preamble

Dynamics of Light Nuclei Produced in Massive Transfer Reactions

Zi-Han Wang¹, Ya-Ling Zhang²,∗ and Zhao-Qing Feng¹;²†

¹School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China
²State Key Laboratory of Heavy Ion Science and Technology, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China

(Dated: June 15, 2025)

Within the framework of the dinuclear system (DNS) model by implementing cluster transfer into the dissipation process, we systematically investigated the energy spectra and angular distribution of preequilibrium clusters (n, p, d, t, ³He, α, ⁶;⁷Li, ⁸;⁹Be) in massive transfer reactions of ¹²C+²⁰⁹Bi, ¹⁴N+¹⁵⁹Tb, ¹⁴N+¹⁶⁹Tm, ¹⁴N+¹⁸¹Ta, ¹⁴N+¹⁹⁷Au, ¹⁴N+²⁰⁹Bi, and ⁵⁸;⁶⁴;⁷²Ni+¹⁹⁸Pt near Coulomb barrier energies. We find that neutron emission is the most probable compared with charged particles, and the α yields are comparable in magnitude with hydrogen isotopes. The preequilibrium clusters are mainly produced from projectile-like and target-like fragments during dinuclear system evolution. The kinetic energy spectra manifest Boltzmann distributions, and the Coulomb potential influences the structure. The preequilibrium clusters follow the angular distribution of multinucleon transfer fragments.

PACS number(s): 25.70.Hi, 25.70.Lm, 24.60.-k
Keywords: Preequilibrium cluster emission; Massive transfer reaction; Dinuclear system model

I. INTRODUCTION

The cluster structure in an atomic nucleus is a spatially localized subsystem consisting of strongly correlated nucleons with much greater internal binding energy than external ones, which can be treated as a whole without considering its internal structure \cite{1}. In 1968, Ikeda proposed that nuclear cluster states tend to occur in excited states near the cluster threshold energy \cite{2}. In some weakly bound nuclei, the cluster structure is more obvious, and cluster structure is also ubiquitous in light nuclei—for example, the configuration of ⁶Li being composed of an α particle and a deuteron, the 2α structure for ⁸Be, 3α for ¹²C, and 5α for ²⁰Ne \cite{3}. The most convenient way to study cluster structure inside a nucleus is to separate the cluster via pick-up or stripping reactions.

The theoretical explanation of preequilibrium reactions was initially developed through the exciton model. Semi-classical theories were less successful in explaining the angular distribution of emitted particles. The Boltzmann master equation theory was mainly used to calculate energy spectra of particles emitted in nucleon-induced reactions and heavy ion reactions, with details found in review Ref. \cite{4} and its references. On the other hand, emission of preequilibrium clusters in transfer reactions around the Coulomb barrier is also an important physical problem. The emission of preequilibrium clusters is a complex process related not only to the cluster structure of the collision system but also to reaction dynamics. In treating nuclear structure, the cluster state is the overlap of single-particle wave functions. In a nuclear reaction, formation of a preequilibrium particle differs from cluster emission during de-excitation of a composite nucleus, as the pre-equilibrium cluster forms before compound nucleus formation. Its emission continues until composite nucleus formation, and the preequilibrium cluster may be emitted from any fragments during the reaction. Cluster emission provides important information for studying single-particle states or multiparticle correlations of nuclei and serves as a powerful tool for nuclear spectroscopy \cite{4}.

Since multinucleon transfer (MNT) reactions and deep inelastic heavy ion collisions were proposed in the 1970s \cite{5,6,7}, numerous experiments have measured double differential cross sections, angular distributions, and energy distributions of different reaction systems. However, it is worth noting that relatively little research has been conducted on preequilibrium cluster emission in transfer reactions, both experimentally and theoretically. In the 1980s, scientists at RIKEN in Japan and at IMP in China measured preequilibrium cluster emission in transfer reactions of ¹⁴N+¹⁵⁹Tb, ¹⁶⁹Tm, ¹⁸¹Ta, ¹⁹⁷Au, ²⁰⁹Bi \cite{8} and ¹²C+²⁰⁹Bi \cite{9,10}, respectively. Experiments measured angular distributions, kinetic energy spectra, and total production cross sections of emitted particles. As is well known, since the concept of the superheavy stable island was proposed in the 1960s, synthesis of superheavy nuclei has become an important frontier in nuclear physics. In the past few decades, 15 kinds of superheavy elements with Z = 104–118 \cite{11} have been synthesized artificially through hot fusion or cold fusion reactions. However, due to limitations of projectile-target materials and experimental conditions, fusion evaporation reactions are difficult for reaching the next period of the periodic table. With MNT reactions, we can generate many nuclei depending on transfer channels, and with development of separation and detection technology, MNT reactions may be the most promising method for synthesizing unknown superheavy elements. This mechanism has been applied to production of heavy and superheavy isotopes \cite{12,13}.

Study of preequilibrium cluster emission in MNT reactions is not only significant for understanding cluster structure of collision systems but also for exploring cluster formation mechanisms, kinetic information of reaction processes, and nuclear astrophysical processes. The High Intensity Accelerator Facility (HIAF) built in Huizhou, China, has a large energy range and wide species of particle beams \cite{14}, providing an excellent experimental platform for studying nuclear cluster structures and cluster emission.

In this work, we systematically investigate preequilibrium cluster emission in transfer reactions. The article is organized as follows. In Section II we give a brief description of the DNS model for describing preequilibrium cluster production. In Section III, production cross sections, kinetic energy spectra, and angular distributions of preequilibrium clusters are analyzed and discussed. Summary and perspectives are presented in Section IV.

II. BRIEF DESCRIPTION OF THE MODEL

The dinuclear system (DNS) model was first proposed by Volkov \cite{15} to describe deep inelastic heavy ion collisions. Adamian et al. applied the DNS model to fusion evaporation reactions in competition with quasifission to study synthesis of superheavy nuclei for the first time \cite{16,17,18}. The Lanzhou nuclear physics group has further developed the DNS model \cite{19,20,21,22}, e.g., introducing the barrier distribution function in the capture process, considering effects of quasifission and fission in the fusion stage, and using statistical evaporation theory and the Bohr-Wheeler formula to calculate survival probability of superheavy nuclei. The DNS model has been widely used to study production cross sections, quasifission, fusion dynamics, etc., in synthesis of superheavy nuclei based on fusion evaporation (FE) reactions and multinucleon transfer (MNT) reactions \cite{23,24,25}.

With the DNS model, we have calculated temporal evolution, kinetic energy spectra, and angular distributions of pre-equilibrium clusters in transfer reactions with incident energy near the Coulomb barrier. Compared to our previous work \cite{26}, we have introduced cluster transfer into the master equation of the DNS model, and Coulomb force is considered in the preequilibrium cluster emission process. The pre-equilibrium particle forms before compound nucleus formation, and its emission continues until composite nucleus formation. The cross section of pre-equilibrium particle emission (ν = n, p, d, t, ³He, α, ⁶;⁷Li and ⁸;⁹Be) is defined as

$$
\sigma_{\nu}(E_k, \theta, t) = \sum_{J=0}^{J_{\max}} \sum_{Z_1=Z_{\nu}}^{Z_{\max}} \sum_{N_1=N_{\nu}}^{N_{\max}} (2J+1) f(B) \times T(E_{c.m.}, J, B) P(Z_1, N_1, E_1(E_{c.m.}, J), t, B) \times P_{\nu}(Z_{\nu}, N_{\nu}, E_k) dB.
$$

Here, $E_k$ and $\theta$ are the kinetic energy and emission angle of particles when ejected from projectile-like or target-like fragments, $t$ is the reaction process time. The reduced de Broglie wavelength is $\lambda = \hbar/\sqrt{2\mu E_{c.m.}}$, and $P(Z_1, N_1, E_1(E_{c.m.}, J), t, B)$ denotes the realization probability of the DNS fragment $(Z_1, N_1)$. $P_{\nu}(Z_{\nu}, N_{\nu}, E_k)$ is the emission probability of pre-equilibrium particles. $E_1$ is the excitation energy for fragment $(Z_1, N_1)$, associated with center-of-mass energy $E_{c.m.}$ and incident angular momentum $J$. The maximal angular momentum $J_{\max}$ is taken as the grazing collision of two colliding nuclei. The DNS fragments $(Z_1, N_1)$ range from the light one $(Z_{\nu}, N_{\nu})$ to the composite system $(Z_{\max}, N_{\max})$, with $Z_{\max} = Z_T + Z_P$ and $N_{\max} = N_T + N_P$ being the total proton and neutron numbers, respectively.

2.1 The Capture Cross Section of Binary System Overcoming the Coulomb Barrier

In the capture stage, the collision system overcomes the Coulomb barrier to form a composite system. The capture cross section is given by

$$
\sigma_{\text{cap}}(E_{c.m.}) = \pi \lambda^2 \sum_{J=0}^{J_{\max}} (2J+1) \int f(B) T(E_{c.m.}, J, B) dB,
$$

where $T(E_{c.m.}, J, B)$ is the penetration probability over barrier $B$. For light and medium systems, $T(E_{c.m.}, J, B)$ is calculated by the well-known Hill-Wheeler formula \cite{27}

$$
T(E_{c.m.}, J, B) = \left{1 + \exp\left[-\frac{2\pi}{\hbar\omega(J)}\left(E_{c.m.} - B - \frac{\hbar^2 J(J+1)}{2\mu R_B^2(J)}\right)\right]\right}^{-1},
$$

with $\hbar\omega(J)$ being the width of the parabolic barrier at $R_B(J)$. Due to multidimensional quantum penetration effects, the Coulomb barrier exhibits a distribution. By introducing a barrier distribution function based on the original Hill-Wheeler formula \cite{27}, the penetration probability can be written as

$$
T(E_{c.m.}, J) = \int f(B) \left{1 + \exp\left[-\frac{2\pi}{\hbar\omega(J)}\left(E_{c.m.} - B - \frac{\hbar^2 J(J+1)}{2\mu R_B^2(J)}\right)\right]\right} dB.
$$

For heavy systems, the collision system does not form a potential energy pocket after overcoming the Coulomb barrier, so $T(E_{c.m.}, J)$ is calculated by the classical trajectory method:

$$
T(E_{c.m.}, J) =
\begin{cases}
0, & E_{c.m.} < B + J(J+1)\hbar^2/(2\mu R_C^2) \
1, & E_{c.m.} > B + J(J+1)\hbar^2/(2\mu R_C^2)
\end{cases}.
$$

The reduced mass is $\mu = m_n A_P A_T/(A_P + A_T)$ with $m_n$, $A_P$ and $A_T$ being the nucleon mass and mass numbers of projectile and target nuclei, respectively. $R_C$ denotes the Coulomb radius, with $R_C = r_{0c} \times (A_P^{1/3} + A_T^{1/3})$ and $r_{0c} = 1.4 \sim 1.5$ fm.

The barrier distribution function is Gaussian \cite{21,28}:

$$
f(B) = \frac{1}{\sqrt{\pi}\Delta} \exp\left[-\left(\frac{B - B_m}{\Delta}\right)^2\right].
$$

The normalization constant satisfies $\int f(B) dB = 1$. The quantities $B_m$ and $\Delta$ are evaluated by $B_m = (B_C + B_S)/2$ and $\Delta = (B_C - B_S)/2$, respectively. $B_C$ is the Coulomb barrier at waist-to-waist orientation and $B_S$ is the minimum barrier by varying quadrupole deformation of colliding partners. Here we take $B_S$ as the Coulomb barrier at tip-to-tip orientation.

2.2 The Nucleon and Cluster Transfer Dynamics

In the nucleon transfer process, the distribution probability of DNS fragments is obtained by numerically solving a set of master equations \cite{29}. Fragment $(Z_1, N_1)$ has proton number $Z_1$, neutron number $N_1$, internal excitation energy $E_1$, and quadrupole deformation $\beta_1$. The time evolution equation of its distribution probability is described by

$$
\frac{dP(Z_1, N_1, E_1, \beta_1, B, t)}{dt} = \sum_{Z_1', N_1'} \left[W_{Z_1', N_1' \to Z_1, N_1}(t) \times d_{Z_1', N_1'} P(Z_1', N_1', E_1', \beta_1', B, t) - W_{Z_1, N_1 \to Z_1', N_1'}(t) \times d_{Z_1, N_1} P(Z_1, N_1, E_1, \beta_1, B, t)\right].
$$

In this equation, $W_{Z_1, N_1 \to Z_1', N_1'}$ is the mean transition probability from channel $(Z_1, N_1, E_1, \beta_1)$ to $(Z_1', N_1', E_1', \beta_1')$. The quantity $d_{Z_1, N_1}$ indicates the microscopic dimension corresponding to macroscopic state $(Z_1, N_1, E_1, \beta_1)$. In this process, transfer of nucleons or clusters satisfies the relationships $Z_1' = Z_1 \pm Z_{\nu}$ and $N_1' = N_1 \pm N_{\nu}$, each representing transfer of a neutron, proton, deuteron, tritium, ³He, or α particle. Note that we ignored quasifission of DNS and fission of heavy fragments in the dissipation process. The initial probabilities of projectile and target nuclei are set to $P(Z_{\text{proj}}, N_{\text{proj}}, E_1 = 0, t = 0) = P(Z_{\text{targ}}, N_{\text{targ}}, E_1 = 0, t = 0) = 0.5$. The nucleon transfer process satisfies the unitary condition $\sum_{Z_1, N_1} P(Z_1, N_1, E_1, t) = 1$.

Similar to cascade transfer of nucleons \cite{21}, cluster transfer is also described by the single-particle Hamiltonian $H(t) = H_0(t) + V(t)$. Single-particle states are defined with respect to centers of interacting nuclei and assumed orthogonalized in the overlap region, so annihilation and creation operators depend on reaction time. The total single-particle energy is $H_0(t) = \sum_K \varepsilon_K(t) a_K^+(t) a_K(t)$. The interaction potential is $V(t) = \sum_{K, K'} u_{KK'}(t) a_K^+(t) a_{K'}(t) V_{K,K'}(t)$. The cluster formation probability involves cluster structure, cluster potential, cluster binding energy, Mott effect, etc. More improvements are still needed in future work.

The memory time is connected with internal excitation energy \cite{33}. The quantity $\varepsilon_K$ represents single-particle energies, and $u_{KK'}$ is the interaction matrix elements parameterized as

$$
u_{KK'} = \frac{U_{K,K'}(t)}{\varepsilon_K(t) - \varepsilon_{K'}(t)} \Delta_{K,K'}(t) - \delta_{KK'}.
$$

The calculation of $U_{K,K'}(t)$ and $\delta_{KK'}(t)$ has been described in Ref. \cite{30}.

In the relaxation process of relative motion, DNS becomes excited by dissipation of relative kinetic energy and angular momentum. The excited DNS opens a valence space in which valence nucleons have symmetrical distribution around the Fermi surface. Only particles in states within the valence space are actively excited and undergo transfer. Averages on these quantities are performed in the valence space as follows:

$$
\Delta\varepsilon_K = \varepsilon^*, \quad g_K = A_K/12,
$$

where $\varepsilon^*$ is the local excitation energy of DNS fragments, providing excitation energy for the mean transition probability. The number of valence states in the valence space is $N_K = g_K \Delta\varepsilon_K$, with $g_K$ being the single-particle level density around the Fermi surface. The number of valence nucleons is $m_K = N_K/2$. The microscopic dimension for fragment $(Z_K, N_K)$ is evaluated by

$$
d(m_1, m_2) = \frac{(m_1 + m_2)!}{m_1! m_2!}.
$$

The mean transition probability is related to local excitation energy and transfer of nucleons or clusters, and can be microscopically derived from the interaction potential in valence space as

$$
W_{Z_1, N_1 \to Z_1', N_1'}^{\nu} = \frac{1}{d_{Z_1, N_1} d_{Z_1', N_1'}} \left|\langle Z_1', N_1', E_1', i' | V | Z_1, N_1, E_1, i \rangle\right|^2.
$$

$G_{\nu}$ represents spin-isospin statistical factors, and we use the Wigner density approach to identify particle types \cite{31,32}, i.e., $G_{\nu} = 1, 1, 3/8, 1/12, 1/12, 1/96$ for neutron, proton, deuteron, tritium, ³He, and α, respectively. The cluster transition probability is related to memory time $\tau_{\text{mem}}(Z_1, N_1, E_1; Z_1', N_1', E_1')$.

The interaction matrix elements are calculated by

$$
|V_{ii'}|^2 = \omega_{11}(i_1, i_1') + \omega_{22}(i_1, i_1') + \omega_{12}(i_1, i_1') + \omega_{21}(i_1, i_1'),
$$

in which $\omega_{KK'}(i, i_1') = d_{Z_1, N_1} \langle V_{KK'}, V_{KK'}^* \rangle$, with states $i(Z_1, N_1, E_1)$ and $i'$.

In the relaxation process of relative motion, DNS becomes excited by dissipation of relative kinetic energy. The local excitation energy is determined by dissipation energy from relative motion and the potential energy surface of DNS \cite{22,24}:

$$
\varepsilon^*(t) = E_{\text{diss}}(t) - \left(U({\alpha}) - U({\alpha_{EN}})\right).
$$

Here $\alpha_{EN} = Z_P, N_P, Z_T, N_T, J, R, \beta_P, \beta_T, \theta_P, \theta_T$ for the projectile-target system. The excitation energy of DNS fragment $(Z_1, N_1)$ is $E_1 = \varepsilon^*(t = \tau_{\text{int}}) A_1/A$. $\tau_{\text{int}}$ denotes interaction time, associated with the reaction system and relative angular momentum, and can be obtained by the deflection function \cite{34}. The energy dissipated into DNS is

$$
E_{\text{diss}}(t) = E_{c.m.} - B - \frac{\langle J(t) \rangle (\langle J(t) \rangle + 1) \hbar^2}{2\zeta_{\text{rel}}} - \langle E_{\text{rad}}(J, t) \rangle.
$$

The radial energy is $\langle E_{\text{rad}}(J, t) \rangle = E_{\text{rad}}(J, 0) \exp(-t/\tau_r)$, with relaxation time of radial motion $\tau_r = 5 \times 10^{-22}$ s and initial radial energy $E_{\text{rad}}(J, 0) = E_{c.m.} - B - J_i(J_i + 1)\hbar^2/(2\zeta_{\text{rel}})$. Dissipation of relative angular momentum is described by

$$
\langle J(t) \rangle = J_{\text{st}} + (J_i - J_{\text{st}}) \exp(-t/\tau_J),
$$

with angular momentum at sticking limit $J_{\text{st}} = J_i \zeta_{\text{rel}}/\zeta_{\text{tot}}$ and relaxation time $\tau_J = 15 \times 10^{-22}$ s. $\zeta_{\text{rel}}$ and $\zeta_{\text{tot}}$ are the relative and total moments of inertia of DNS, respectively. The initial angular momentum is set to $J_i = J$ in Eq. (1). Relaxation times of radial kinetic energy and angular momentum are associated with friction coefficients in binary collisions, with values taken from empirical analysis in deeply inelastic heavy-ion collisions \cite{34,35}.

The potential energy surface (PES) of DNS is evaluated as

$$
U({\alpha}) = B(Z_1, N_1) + B(Z_2, N_2) - B(Z, N) + V({\alpha}),
$$

with the relationship $Z_1 + Z_2 = Z$ and $N_1 + N_2 = N$ \cite{36,37}. The symbol ${\alpha}$ denotes quantities $Z_1, N_1, Z_2, N_2, J, R, \beta_1, \beta_2, \theta_1, \theta_2$. In calculations, the distance $R$ between fragment centers is chosen at the touching configuration where DNS is assumed formed, with $R = r_0 \times (A_1^{1/3} + A_2^{1/3})$ and $r_0 = 1.2 \sim 1.3$ fm.

$B(Z_i, N_i)$ ($i = 1, 2$) and $B(Z, N)$ are negative binding energies of fragment $(Z_i, N_i)$ and compound nucleus $(Z, N)$, respectively. $\beta_i$ represent quadrupole deformations of the two fragments at ground state, and $\theta_i$ ($i = 1, 2$) denote angles between collision orientations and symmetry axes of deformed nuclei. The interaction potential between fragments $(Z_1, N_1)$ and $(Z_2, N_2)$ is derived from

$$
V({\alpha}) = V_C({\alpha}) + V_N({\alpha}) + V_{\text{def}}(t),
$$

where $V_C$ is the Coulomb potential using the Wong formula \cite{38}, $V_N$ is the nucleus-nucleus potential using the double folding potential \cite{39}, and $V_{\text{def}}(t)$ denotes deformation energy of DNS at reaction time $t$:

$$
V_{\text{def}}(t) = C_1(\beta_1 - \beta_T(t))^2 + C_2(\beta_2 - \beta_P(t))^2.
$$

The quantities $C_i$ ($i = 1, 2$) denote stiffness parameters of the nuclear surface, calculated by the liquid drop model \cite{40}. Detailed calculations of $V_{\text{def}}(t)$ can be obtained from Ref. \cite{41} and references therein.

[FIGURE:1] shows the PES in collisions of ¹⁴N+²⁰⁹Bi and ⁶⁴Ni+¹⁹⁸Pt. The zigzag lines are driving potentials estimated by minimal PES values during nucleon transfer. The incident point is denoted by a star symbol.

2.3 The Preequilibrium Cluster Emission

Based on the Weisskopf evaporation theory \cite{42,43}, particle decay widths are

$$
\Gamma_{\nu}(E^, J) = \frac{(2s_{\nu} + 1) m_{\nu}}{\pi^2 \hbar^2 \rho(E^, J)} \int_0^{E^ - B_{\nu} - E_{\text{rot}} - V_c} \varepsilon \rho(E^ - B_{\nu} - E_{\text{rot}} - V_c - \varepsilon, J) \sigma_{\text{inv}}(\varepsilon) d\varepsilon,
$$

with $s_{\nu}$, $m_{\nu}$ and $B_{\nu}$ being spin, mass and binding energy of evaporating particles, respectively. The inverse cross section is $\sigma_{\text{inv}} = \pi R_{\nu}^2 T(\nu)$, with radius $R_{\nu} = 1.21(A - A_{\nu})^{1/3} + A_{\nu}^{1/3}$. The penetration probability is $T(\nu) = 1$ for neutrons and $T(\nu) = [1 + \exp(2\pi(V_C(\nu) - \varepsilon)/\hbar\omega)]^{-1}$ for charged particles, with $\hbar\omega = 5$ MeV for hydrogen isotopes and 8 MeV for other charged particles.

It should be mentioned that local equilibrium of DNS is assumed formed, and excitation energy for the $i$-th fragment is associated with local excitation energy with the mass table \cite{44}. The level density is calculated from the Fermi-gas model as

$$
\rho(E^, J) = \frac{2J + 1}{2\sigma^3 a^{1/4} (E^ - \delta)^{5/4}} \exp\left[2\sqrt{a(E^* - \delta)} - \frac{(J + 1/2)^2}{2\sigma^2}\right],
$$

with $\sigma^2 = 6\bar{m}^2 A^{5/3}/\pi^2$ and $\bar{m} \approx 0.24 A^{2/3}$. The pairing correction energy $\delta$ is set to $0$, $\Delta$, and $2\Delta$ for even-even, even-odd and odd-odd nuclei, respectively, where $\Delta = 12/\sqrt{A}$. The level density parameter is related to shell correction energy $E_{\text{sh}}(Z, N)$ and excitation energy $E^*$ of the nucleus as

$$
a(E^, Z, N) = \tilde{a}(A)\left[1 + \frac{E_{\text{sh}}(Z, N) f(E^ - \Delta)}{E^* - \Delta}\right].
$$

The asymptotic Fermi-gas value of the level density parameter at high excitation energy is $\tilde{a}(A) = \alpha A + \beta A^{2/3} b_s$, and the shell damping factor is $f(E^) = 1 - \exp(-\gamma E^)$ with $\gamma = \tilde{a}/(\epsilon A^{4/3})$. Parameters $\alpha$, $\beta$, $b_s$ and $\epsilon$ are taken as 0.114, 0.098, 1.0 and 0.4, respectively \cite{36,37}.

Emission probabilities of pre-equilibrium clusters with kinetic energy $E_k$ are calculated by the uncertainty principle within time step $t \sim t + \Delta t$ via

$$
P_{\nu}(Z_{\nu}, N_{\nu}, E_k) = \frac{\Delta t \Gamma_{\nu}}{\hbar}.
$$

The kinetic energy of pre-equilibrium particles is sampled by Monte Carlo method within energy range $\varepsilon_{\nu} \in (0, E^* - B_{\nu} - E_{\text{rot}} - V_C)$ and $E_k = \varepsilon_{\nu} + V_C$. Here $V_C$ represents the Coulomb force outgoing particles need to overcome, with $V_C = 0$ for neutrons.

The time step is set to $\Delta t = 0.5 \times 10^{-22}$ s for reactions induced by ¹²C and ¹⁴N, but $\Delta t = 0.25 \times 10^{-22}$ s for reactions induced by ⁵⁸;⁶⁴;⁷²Ni isotopes.

Watt spectrum is used for neutron emission \cite{45} and expressed as

$$
\frac{dN_n}{d\varepsilon_n} = C_n \exp\left(-\frac{\varepsilon_n}{T_w}\right) \sinh\left(\sqrt{2\varepsilon_n T_w}\right),
$$

with $T_w = 1.7 \pm 0.1$ MeV and normalization constant $C_n$.

For charged particles, Boltzmann distribution is taken into account as

$$
\frac{dN_{\nu}}{d\varepsilon_{\nu}} = \frac{8\pi \varepsilon_{\nu}}{m_{\nu}} \left(\frac{m_{\nu}}{2\pi T_{\nu}}\right)^{3/2} \exp\left(-\frac{\varepsilon_{\nu}}{T_{\nu}}\right),
$$

with mass $m_{\nu}$ and local temperature $T_{\nu} = \sqrt{E^*/a}$, where $a = A/8$ is the level density parameter.

We use the deflection function method \cite{34,46} to calculate angular distributions of pre-equilibrium particles emitted from DNS fragments as

$$
\Theta(J_i) = \Theta_C(J_i) + \Theta_N(J_i).
$$

The Coulomb deflection is given by the Rutherford function

$$
\Theta_C(J_i) = 2 \arctan\left(\frac{Z_p Z_t e^2}{2 E_{c.m.} b}\right),
$$

with incident energy $E_{c.m.}$ and impact parameter $b$. The nuclear deflection is calculated by

$$
\Theta_N(J_i) = -\beta \frac{\Theta_C^{\text{gr}}(J_i)}{1 + \exp\left(\frac{J_i - J_{\text{gr}}}{\delta}\right)},
$$

where $\Theta_C^{\text{gr}}(J_i)$ is the Coulomb scattering angle at grazing angular momentum $J_{\text{gr}} = 0.22 R_{\text{int}} \sqrt{A_{\text{red}}(E_{c.m.} - V(R_{\text{int}}))}$. $J_i$ is incident angular momentum, $A_{\text{red}}$ is reduced mass of the collision system. The quantity $\eta = Z_1 Z_2 e^2/\hbar v$ is the Sommerfeld parameter, and relative velocity is $v = \sqrt{2(E_{c.m.} - V(R_{\text{int}}))/\mu}$. For the $i$-th DNS fragment, emission angle is determined by

$$
\Theta_i(J_i) = \Theta(J_i) \frac{\xi_i}{\xi_1 + \xi_2},
$$

with moment of inertia $\xi_i$ for the $i$-th fragment.

III. RESULTS AND DISCUSSION

Preequilibrium cluster emission in transfer reactions is very complicated, related not only to collision system structure (e.g., formation factor) but also to dynamic evolution of reaction processes, i.e., dissipation of relative motion and coupling of internal degrees of freedom. Emission of preequilibrium clusters is a non-equilibrium process of time and space evolution, serving as a powerful probe for deeply investigating MNT reaction dynamics.

[FIGURE:2] and [FIGURE:3] show temporal evolution of emission probabilities for n, p, d, t, ³He and α from transfer reactions of ¹⁴N + ¹⁵⁹Tb, ¹⁶⁹Tm, ¹⁸¹Ta and ¹⁹⁷Au at $E_{\text{lab}} = 115$ MeV. Compound nucleus formation occurs on the order of a few hundred zeptoseconds, while preequilibrium process reaction time is about several zeptoseconds. Emission of preequilibrium clusters continues until composite nucleus formation. At reaction beginning, emission probability increases rapidly, reaching maximum at about $20 \sim 40 \times 10^{-22}$ s, then remaining stable or decreasing gradually. Emission probabilities of α and hydrogen isotopes are comparable, with yields about 3–4 orders of magnitude lower than neutrons but much larger than ³He. Local excitation energy of DNS fragments increases with time, and emitted clusters can take away part of energy, which is conducive to formation of compound nucleus with lower excitation energy. Total emission cross sections are obtained by counting temporal evolution of cluster yields, shown in [TABLE:I]. α yields are comparable with proton emission.

[FIGURE:4] shows kinetic energy spectra of light nuclei produced in transfer reactions of ¹⁴N + ¹⁵⁹Tb, ¹⁶⁹Tm, ¹⁸¹Ta at $E_{\text{lab}} = 115$ MeV. Different reactions show similar shape, presenting Boltzmann distribution. Neutron emission is most important. Compared with previous work \cite{26}, we introduced Coulomb barrier correction. Hydrogen isotopes have similar emission probabilities due to same charge amount, with kinetic energy spectrum peak at about 10 MeV. Since α and ³He are more charged, kinetic energy spectra shift to greater energy. Emission probability of α is about three to five orders of magnitude higher than ³He because α has lower separation energy and is more easily emitted from DNS fragments. These results are consistent with experimental data \cite{8,10}.

[FIGURE:5] exhibits kinetic energy spectra of preequilibrium clusters (n, p, d, t, ³He, α, ⁶;⁷Li, ⁸;⁹Be) in transfer reactions induced by ¹²C and ¹⁴N on the same target nucleus ²⁰⁹Bi. Kinetic energy spectra show nuclear structure effects and dynamic characteristics. Available experimental data for α emission from HIRFL for ¹²C+²⁰⁹Bi \cite{10} and from RIKEN for ¹⁴N+²⁰⁹Bi \cite{8} are nicely reproduced with DNS model. Excitation energy of DNS fragments, transition probability, binding energy and separation energy of transferred nucleons (clusters) affect kinetic energy spectra. Emission cross section is mainly related to formation probability and emission probability. In our calculation, assuming clusters already exist in DNS, emission cross sections are mainly determined by emission probabilities. Higher charge of emitted particle means higher Coulomb barrier. Larger separation energy of cluster means smaller decay width and lower emission probability. Kinetic energy spectra are strongly related to Coulomb barriers and excitation energies of composite system.

[FIGURE:6] compares time evolution and kinetic energy distribution for transfer reaction ⁵⁸Ni+¹⁹⁸Pt at different incident energies of 220 MeV and 240 MeV. The left part shows temporal evolution, right part shows kinetic energy spectra. Kinetic energy is mainly determined by local excitation energy of projectile-like and target-like fragments, with high local excitation energy beneficial for cluster emission. Emission probability at $E_{c.m.} = 240$ MeV is about 2–3 orders of magnitude higher than at $E_{c.m.} = 220$ MeV, indicating emission probability increases with incident energy.

[FIGURE:7] compares transfer reactions bombarding target nucleus ¹⁹⁸Pt with heavier Ni isotopes at $E_{c.m.} = 240$ MeV. Left shows kinetic energy spectra for ⁶⁴Ni+¹⁹⁸Pt, right for ⁷²Ni+¹⁹⁸Pt. Compared to ⁶⁴Ni+¹⁹⁸Pt, ⁷²Ni-induced reaction seems more likely to emit neutrons, while the former is more likely to emit protons. In both systems, proton isotope kinetic energy spectra peak at about 9 MeV, while α kinetic energy spectra peak at about 17 MeV.

Particles emitted in preequilibrium process and from compound nucleus have different kinetic energy and angular distributions \cite{47}. Direct particles primarily emit in the same direction as incident particles with similar energy, while compound process particles emit isotropically in equal amounts forward and backward. Preequilibrium particles tend to emit forward and are generally more energetic than those from composite nucleus.

[FIGURE:8] shows angular distributions of emitted preequilibrium clusters in ⁵⁸;⁶⁴;⁷²Ni+¹⁹⁸Pt at $E_{c.m.} = 240$ MeV. Angular distributions differ for different reaction systems. For the same system, shapes for different clusters are quite similar because different clusters may evaporate from the same excited DNS fragment. Angular distributions are anisotropic, showing similar characteristics as fragment angular distributions in multinucleon transfer reactions \cite{34,46}. In calculations, we ignored values where output was less than $1 \times 10^{-10}$. Under all three reaction systems, angular distributions increase rapidly when center-of-mass angle is about 26°, reaching maximum between 32° and 36°. There exhibits a window of 32°–160° for preequilibrium neutron emission. Study of angular distributions is significant for understanding primary fragment angular distributions in MNT reactions and helpful for experimental measurement planning.

IV. CONCLUSIONS

In summary, within the DNS model framework, we investigated emission mechanisms of preequilibrium clusters in massive transfer reactions near Coulomb barrier energies: temporal evolution, kinetic energy spectra, and angular distributions of n, p, d, t, ³He, α, ⁶;⁷Li, ⁸;⁹Be in collisions of ¹²C+²⁰⁹Bi, ¹⁴N+¹⁵⁹Tb, ¹⁶⁹Tm, ¹⁸¹Ta, ¹⁹⁷Au, ²⁰⁹Bi and ⁵⁸;⁶⁴;⁷²Ni+¹⁹⁸Pt. Cluster transfer and dynamic deformation are coupled to relative dissipation of angular momentum and motion energy in the DNS model.

Emission of preequilibrium clusters strongly depends on incident energy, separation energy and Coulomb barrier from primordial DNS fragments. Yields of hydrogen isotopes and α production have similar magnitude, but are more probable than heavy particles. Kinetic energy spectra manifest differences for charged particles, i.e., α emission has higher kinetic energy than protons. Preequilibrium clusters follow MNT fragment emission angular distributions and are related to nucleon correlations. This reaction mechanism is helpful for investigating cluster structure of atomic nuclei and MNT fragment formation—i.e., yields, shell effects, emission dynamics—which is being planned for forthcoming experiments at HIAF in Huizhou.

ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China (Projects No. 12175072 and No. 12311540139).

∗ Corresponding author: zhangyl@impcas.ac.cn
† Corresponding author: fengzhq@scut.edu.cn

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