Prediction of synthesis cross-sections of new moscovium isotopes in fusion-evaporation reactions

NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH

Prediction of synthesis cross-sections of new moscovium isotopes in fusion-evaporation reactions

Peng-Hui Chen，

Hao Wu，

Zu-Xing Yang，

Xiang-Hua Zeng，

Zhao-Qing Feng

Nuclear Science and Techniques

Vol.34, No.1

Article number 7

Published in print Jan 2023

Available online 13 Jan 2023

DOI：10.1007/s41365-022-01157-0

40800

In the framework of the dinuclear system model, the synthesis mechanism of the superheavy nuclides with atomic numbers Z=112, 114, 115 in the reactions of projectiles ^40,48Ca bombarding on targets ²³⁸U, ²⁴²Pu, and ²⁴³Am within a wide interval of incident energy has been investigated systematically. Based on the available experimental excitation functions, the dependence of calculated synthesis cross-sections on collision orientations has been studied thoroughly. The total kinetic energy (TKE) of these collisions with fixed collision orientation shows orientation dependence, which can be used to predict the tendency of kinetic energy diffusion. The TKE is dependent on incident energies, as discussed in this paper. We applied the method based on the Coulomb barrier distribution function in our calculations. This allowed us to approximately consider all the collision orientations from tip-tip to side-side. The calculations of excitation functions of ⁴⁸Ca + ²³⁸U, ⁴⁸Ca + ²⁴²Pu, and ⁴⁸Ca + ²⁴³Am are in good agreement with the available experimental data. The isospin effect of projectiles on production cross-sections of moscovium isotopes and the influence of the entrance channel effect on the synthesis cross-sections of superheavy nuclei are also discussed in this paper. The synthesis cross-section of new moscovium isotopes ^278-286Mc was predicted to be as large as hundreds of pb in the fusion-evaporation reactions of ^35,37Cl + ²⁴⁸Cf, ^38,40Ar + ²⁴⁷Bk, ^39,41K + ²⁴⁷Cm, ^40,42,44,46Ca + ²⁴³Am, ⁴⁵Sc + ²⁴⁴Pu, and ^46,48,50Ti + ²³⁷Np, ⁵¹V + ²³⁸U at some typical excitation energies.

Dinuclear system modelSuperheavy nucleiComplete fusion reactionsProduction cross-section

Introduction

Since the "island of stability" of superheavy nuclei was predicted by the shell model in the 1960s[1], the synthesis of superheavy nuclei has been an exciting frontier field in the laboratories that could provide a unique tool to explore the properties of nuclei and nuclear structure under extremely strong Coulomb force. However, owing to the extremely low production cross-sections, the synthesis of superheavy nuclei in current experiments is time-consuming and costly. Therefore, it is particularly necessary to make reliable theoretical calculations that provide a reasonable reference for experiments. In recent years, synthesizing superheavy elements through low-energy heavy ion collisions near the Coulomb barrier has attracted extensive attention from theorists and experimentalists.

On the experimental side, in the past half century, fifteen superheavy elements characterized by values of Z in the range 104-118 have been synthesized and identified in laboratories all over the world[2]. Generally, superheavy synthesis methods are classified by the excitation energy of compound nuclei as cold fusion and hot fusion, resulting in compound nuclei surviving by emitting 1-2 neutrons and 3-5 neutrons, respectively, against fission. Elements Rf, Db, Sg, Fl, Mc, Lv, Ts, and Og were synthesized first in hot-fusion reactions. ^257,258,259Rf (Z=104) was discovered simultaneously in Dubna[3] and Berkeley[4] in the reactions of ²⁴⁹Cf(^12,13C, 3-4n)^257,258,259Rf at incident energy E_lab = 10.4 MeV/nucleon. ^260,261Db (Z=105) was discovered simultaneously in Dubna[5] and Berkeley[6] in the reactions of ²⁴⁹Cf(¹⁵N,4n)²⁶⁰Ds at E_lab = 85 MeV and ²⁴³Am(²²Ne, 4n)²⁶¹Ds at E_lab = 114 MeV. ²⁶³Sg (Z=106) was discovered at Berkeley[7] in the reactions of ²⁴⁹Cf(¹⁸O,4n)²⁶³Sg at E_lab = 95 MeV. ^286-289Fl was essentially discovered at Dubna[8] in the reactions of ²⁴⁴Pu(⁴⁸Ca, 3-6n)^286-289Fl at E_lab = 352.6 MeV. ²⁸⁸Mc (Z=115) was essentially discovered at Dubna[9] in the reactions of ²⁴³Am(⁴⁸Ca, 3n)²⁸⁸Mc at E_lab = 248, 253 MeV. ^286-289Lv (Z=116) was discovered at Dubna[10] in the reactions of ²⁴⁵Cm(⁴⁸Ca, xn)^293-xFl at E_lab = 243 MeV. ^293-294Ts (Z=117) was essentially discovered at Dubna[11] in the reactions of ²⁴⁹Bk(⁴⁸Ca, 3-4n)^293-294Fl at E_lab = 247, 252 MeV. ²⁹⁴Og (Z=118) was essentially discovered at Dubna[12] in the reactions of ²⁴⁹Cf(⁴⁸Ca, 3n)²⁹⁴Og at E_lab = 251 MeV. Elements Sg, Bh, Hs, Mt, Ds, Rg, Cn, and Nh were synthesized first in cold-fusion reactions. ²⁵⁹Sg (Z=106) was discovered at Dubna[13] in the reactions of ²⁰⁷Pb(⁵⁴Cr,2n)²⁵⁹Sg at E_lab = 262 MeV. ²⁶²Bh (Z=107) was essentially discovered at Gesellschatt Für Schwerionenforschung (GSI)[14] in the reactions of ²⁰⁹Bi(⁵⁴Cr, 1n)²⁶²Bh at E_lab = 4.85 MeV/u. ^263-265Hs (Z=108) was synthesized at GSI[15] in the reactions of ²⁰⁸Pb(⁵⁸Fe, 2n)²⁶⁵Hs at E_lab = 5.02 MeV/u. ²⁶⁶Mt (Z=109) was synthesized at GSI[16] in the reactions of ²⁰⁹Bi(⁵⁸Fe, 1n)²⁶⁶Mt at E_lab = 5.15 MeV/u. ²⁶⁹Ds (Z=110) was synthesized at GSI[17] in the reactions of ²⁰⁸Pb(⁶²Ni, 1n)²⁶⁹Ds at E_lab = 311 MeV. ²⁷²Rg (Z=111) was synthesized at GSI[18] in the reactions of ²⁰⁹Bi(⁶⁴Ni, 1n)²⁷²Rg at E_lab = 318, 320 MeV. ²⁷⁷Cn (Z=112) was synthesized at GSI[19] in the reactions of ²⁰⁸Pb(⁷⁰Zn, 1n)²⁷⁷Cn at E_lab = 344 MeV. ²⁷⁸Nh (Z=113) was synthesized at RIKEN[20] in the reactions of ²⁰⁹Bi(⁷⁰Zn, 1n)²⁷⁸Nh at E_lab = 352.6 MeV. Synthesis information of the most neutron-rich and proton-rich superheavy nuclei with atomic numbers Z=104-118, including elements, isotopes, reactions, channels, laboratories, and year, is provided in Table 1. Chinese superheavy nuclei group synthesized the superheavy isotopes of ^258,259Db[21], ^264,265,266Bh[22], and ²⁷¹Ds[23] at the Institute of Modern Physics (IMP) (Lanzhou, China).

Synthesis information of the most neutron-rich and proton-rich superheavy isotopes with atomic numbers Z=104-118: production reactions, evaporation channel, laboratory, year, and reference.

Element	Isotopes	Reactions	Channel	Lab	Year	Ref.
Rf(104)	²⁵³ Rf	⁵⁰ Ti + ²⁰⁴ Pb	1n	GSI	1997	[24]
(13)	²⁶⁷Rf	⁴⁸Ca + ²⁴²Pu	α	Dubna	2004	[25]
Db(105)	²⁵⁶Db	⁵⁰Ti + ²⁰⁹Bi	3n	GSI	2001	[26]
(11)	²⁷⁰Db	⁴⁸Ca + ²⁴⁹Bk	3nα	Berkeley	2010	[11]
Sg(106)	²⁵⁸Sg	⁵¹V + ²⁰⁹Bi	2n	GSI	1997	[24]
(12)	²⁷¹Sg	⁴⁸Ca + ²³⁸U	α	Dubna	2004	[25]
Bh(107)	²⁶⁰Bh	⁵²Cr + ²⁰⁹Bi	α	Berkeley	2008	[27]
(10)	²⁷⁴Bh	⁴⁸Ca + ²⁴⁹Bk	3nα	Dubna	2010	[11]
Hs(108)	²⁶³Hs	⁵⁶Fe + ²⁰⁸Pb	1n	Berkeley	2009	[28]
(12)	²⁷⁷Hs	⁴⁸Ca + ²⁴⁴Pu	3nα	GSI	2010	[29]
Mt(109)	²⁶⁶Mt	⁵⁸Fe + ²⁰⁹Bi	1n	GSI	1982	[30]
(7)	²⁷⁸Mt	⁴⁸Ti + ²⁴⁹Bk	3nα	Dubna	2010	[11]
Ds(110)	²⁶⁷Ds	⁵⁹Co + ²⁰⁹Bi	1n	Berkeley	1995	[31]
(8)	²⁸¹Ds	⁴⁸Ca + ²⁴⁴Pu	3nα	Dubna	2004	[10]
Rg(111)	²⁷²Rg	⁶⁴Ni + ²⁰⁹Bi	1n	GSI	1995	[32]
(7)	²⁸²Rg	⁴⁸Ca + ²⁴⁹Bk	3nα	Dubna	2010	[11]
Cn(112)	²⁷⁷Cn	⁷⁰Zn + ²⁰⁸Pb	1n	GSI	1996	[19]
(6)	²⁸⁵Cn	⁴⁸Ca + ²⁴⁴Pu	3nα	Dubna	2004	[10]
Nh(113)	²⁷⁸Nh	⁷⁰Zn + ²⁰⁹Bi	1n	RIKEN	2004	[33]
(6)	²⁸⁶Nh	⁴⁸Ca + ²⁴⁹Bk	3nα	Dubna	2010	[11]
FI(114)	²⁸⁵FI	⁴⁸Ca + ²⁴²Pu	5n	Berkeley	2010	[8]
(5)	²⁸⁹FI	⁴⁸Ca + ²⁴⁴Pu	3n	Dubna	2004	[10]
Mc(115)	²⁸⁷Mc	⁴⁸Ca + ²⁴³Am	4n	Dubna	2004	[9]
(4)	²⁹⁰Mc	⁴⁸Ca + ²⁴⁹Bk	3nα	Dubna	2010	[11]
Lv(116)	²⁹⁰Lv	⁴⁸Ca + ²⁴⁵Cm	3n	Dubna	2004	[10]
(4)	²⁹³Lv	⁴⁸Ca + ²⁴⁵Cm	1n	Dubna	2004	[10]
Ts(117)	²⁹³Ts	⁴⁸Ca + ²⁴⁹Bk	4n	Dubna	2010	[11]
(2)	²⁹⁴Ts	⁴⁸Ca + ²⁴⁹Bk	3n	Dubna	2010	[11]
Og(118)	²⁹⁴Og	⁴⁸Ca + ²⁴⁹Cf	3n	Dubna	2006	[12]

The mechanism of fusion-evaporation (F.E.) cannot easily reach the next new period in the periodic table of elements because of the limited available combinations of projectile-target. With the development of suitable separation and detection techniques, the multinucleon transfer (MNT) mechanism might be the most promising method to synthesize unknown superheavy elements. This mechanism has been applied to produce massive heavy and superheavy isotopes[34]. Laboratories all over the world such as IMP [35], GSI [36, 37], Dubna[38, 39], RIKEN[40-42], and Lawrence Berkeley National Laboratory (LBNL)[43, 43] are focused on synthesizing new superheavy elements and their isotopes. From the chart of nuclei, in the superheavy region, there are substantial isotopes of superheavy elements that are still unknown. One of the objectives of the present study was to predict the production cross-sections of moscovium isotopes in F.E. reactions based on different combinations of projectile-target.

On the theoretical side, to describe the production mechanism of superheavy nuclei, some theoretical models were built, for example, the time-dependent Hartree-Fock model [44-46], the improved quantum molecular dynamics model[47-49], a dynamical approach based on Langevin equations[50, 51], and the dinuclear system (DNS) model [52-56]. The calculations resulting from these theoretical models are in good agreement with the available experimental data, which have their own features. In this study, the DNS model has been applied. This model has some advantages such as better consideration of the shell effect, dynamical deformation, fission, quasi-fission, deep-inelastic and odd-even effects, and high calculation efficiency. In previous studies[53, 52, 56-62], the DNS model accurately reproduced the available experimental results and predicted the synthesis production cross-sections of superheavy elements and exotic heavy nuclei in the mechanisms of F.E. and MNT reactions.

In this study, we investigated the dependence of the evaporation residue cross-sections on collision orientations and the influence of entrance channel effect on the evaporation residue cross-sections. We propose a Gaussian-like barrier distribution function for treating the problem of collision orientation dependence. The article is organized as follows. Section 2 briefly describes the DNS model. Calculated results and discussions are presented in Sect. 3. Section 4 concludes the paper with a summary.

Model Description

Initially, the DNS concept was proposed to describe the deep-inelastic reaction mechanism, which is a molecular-like configuration of two colliding partners, keeping their own individuality in the collision process. The DNS model has been widely used to describe F.E. and multinucleon transfer reactions. The complete fusion evaporation reaction can be described in terms of three processes. First, the colliding partners overcome the Coulomb barrier to form the composite system. Second, the kinetic energy and angular momentum dissipate into the composite system to enable the nucleon transfer between the touching colliding partners. Finally, all the nucleons are transferred from projectile nuclei to the target nuclei, which could form the compound nuclei with excitation energy and angular momentum. The highly excited compound nuclei will be de-excited by evaporation of the light particles (i.e., neutrons, γ-rays, and light-charged particles) or fission. Based on the DNS model, the evaporation residual cross-sections of superheavy nuclei can be expressed as $\begin{matrix} σ_{ER} (E_{c . m .}) = \frac{π ℏ^{2}}{2 μ E_{c . m .}} \sum_{J = 0}^{J_{max}} (2 J + 1) T (E_{c .m .}, J) \\ P_{CN} (E_{c .m .}, J) W_{sur} (E_{c .m .}, J) \end{matrix}$ (1) where the penetration probability T(E_c.m., J) is the probability that the collision system overcomes the Coulomb barrier, which is calculated using the empirical coupling channel model[52]. The fusion probability PCN(E_c.m., J) is the probability to form compound nuclei[63, 64]. The survival probability W_sur is the probability that the highly excited compound nuclei survive by evaporating light particles against fission. The maximal angular momentum is set as J_max = 30-50 because the fission barrier for the superheavy nuclei may vanish at high spin [65].

2.1

Capture probability

The capture cross-sections of the two colliding partners are expressed as $σ_{cap} (E_{c .m .}) = \frac{π ℏ^{2}}{2 μ E_{c .m .}} \sum_{J} (2 J + 1) T (E_{c .m .}, J) .$ (2) Here, the penetration probability T(E_c.m.,J) is evaluated by the Hill-Wheeler formula [66] using the barrier distribution function: $\begin{array}{l} T (E_{c .m .}, J) = \int f (B) \\ \frac{1}{1 + \exp {- \frac{2 π}{ℏ ω (J)} [E_{c .m .} - B - \frac{ℏ^{2} J (J + 1)}{2 μ R_{B}^{2} (J)}]}} d B, \end{array}$ (3) where $ℏ ω (J)$ is the width of the parabolic barrier at position R_B(J). The normalization constant is with respect to the relation $\int f (B) d B = 1$ . The barrier distribution function is assumed to be in an asymmetric Gaussian form[52, 67]: $f (B) = {\begin{matrix} \frac{1}{N} \exp [- (\frac{B - B_{m}}{Δ_{1}})] B < B_{m}, \\ \frac{1}{N} \exp [- (\frac{B - B_{m}}{Δ_{2}})] B > B_{m}, \end{matrix}$ (4) where $Δ_{2}$ = (B₀-B_s)/2, $Δ_{1}$ = $Δ_{2}$ -2 MeV, B_m=(B₀+B_s)/2, and B₀ and B_s are the Coulomb barriers of the side-side collision and saddle-point barriers in dynamical deformations[67]. The nucleus-nucleus interaction potential is expressed as $V ({α}) = V_{C} ({α}) + V_{N} ({α}) + V_{def}$ (5) with $V_{def} = \frac{1}{2} C_{1} {(β_{1} - β_{1}^{0})}^{2} + \frac{1}{2} C_{2} {(β_{2} - β_{2}^{0})}^{2}) .$ where 1 and 2 represent the projectile and target, respectively; R=R₁+R₂+s and s are the distances between the center and surface of the projectile and target, respectively; R₁ and R₂ are the radii of the projectile and target, respectively; $β_{1 (2)}^{0}$ represents the static deformation of the projectile-target; β₁₍₂₎ represents the adjustable quadrupole deformation, which is varied to find the minimal V({α}); and {α} stands for ${R, β_{1}, β_{1}, β_{2}, β_{1}, β_{2}}$ . To reduce the number of deformation variables, we assume that the deformation energy of the colliding system is proportional to its mass[67], that is, $C_{1} β_{1}^{2} / C_{2} β_{2}^{2} = A_{1} / A_{2}$ . Thus, only one deformation parameter, β=β₁+β₂, is required. The stiffness parameters C_i(i=1,2) are calculated using the liquid-drop model[68] through the following parameterization formula: $C_{i} = (λ - 1) [(λ - 1) R_{i}^{2} σ - \frac{3}{2 π} \frac{Z^{2} e^{2}}{R_{i} (2 λ + 1)}]$ (6) where Ri is the radius of the spheroidal nucleus given by Ri=1.18 $A_{i}^{1/3}$ (i=1,2). In this study, the quadrupole deformation was taken into account (λ=2). Note that σ is the coefficient of surface tension that fits $4 π_{i}^{2}$ σ= $a_{s} A_{i}^{2 / 3}$ , where as = 18.32 MeV is the surface energy. The nuclear potential is calculated using the double-folding method [63-65]: $\begin{matrix} V_{N} = C_{0} {\frac{F_{in} - F_{ex}}{ρ_{0}} [\int ρ_{1}^{2} (r) ρ_{2} (r - R) d r \\ + \int ρ_{1} (r) ρ_{2}^{2} (r - R) d R] + F_{e x} \int ρ_{1} (r) ρ_{2} (r - R) d r} \end{matrix}$ (7) where $F_{in(ex)} = f_{in(ex)} + f_{in(ex)}^{,} \frac{N_{1} - Z_{1}}{A_{1}} \frac{N_{2} - Z_{2}}{A_{2}} .$ Note the dependence on the nuclear density and orientation of the deformed colliding partners. We set the following parameter values in our calculations: C₀ = 300 MeV fm³, f_in = 0.09, f_ex=-2.59, $f_{i n}^{,}$ = 0.42, $f_{e x}^{,}$ = 0.54, and ρ₀ = 0.16 fm^-3. The Woods-Saxon density distribution is expressed as $ρ_{1} (r) = \frac{ρ_{0}}{1 + \exp ((r - ℜ_{1} (θ_{1})) / a_{1})}$ (8) and $ρ_{2} (r - R) = \frac{ρ_{0}}{1 + \exp [(| r - R | - ℜ_{2} (θ_{2})) / a_{2}]}$ (9) where $ℜ_{i} (θ_{i}) (i = 1,2)$ denotes the surface radii of the nuclei given by $ℜ_{i} (θ_{i})$ = $ℜ [1 + β_{i} Y_{20} (θ_{i})]$ , where Ri is the spheroidal nuclei radius and ai is the surface diffuseness coefficient, which was set as 0.55 fm in our calculations. The Coulomb potential was derived by Wong’s formula as follows [69]: $\begin{array}{l} V_{C} ({α}) = \frac{Z_{1} Z_{2} e^{2}}{r} \\ + {(\frac{9}{20 π})}^{1 / 2} (\frac{Z_{1} Z_{2} e^{2}}{r^{3}}) \sum_{i = 1}^{2} R_{i}^{2} β_{i} P_{2} (\cos θ_{i}) \\ + (\frac{3}{7 π}) (\frac{Z_{1} Z_{2} e^{2}}{r^{3}}) \sum_{i = 1}^{2} R_{i}^{2} {(β_{i} P_{2} \cos θ_{i})}^{2} \end{array}$ (10) where θi, βi, Ri, and P₂(cosθi) are the angle between the symmetry axis of the deformed projectile-target and collision axis, quadrupole deformation, radius of the projectile-target, and Legendre polynomial, respectively. Wong’s formula is in good agreement with the double-folding method.

2.2

Fusion probability

The composite system is formed after the capture process in which the dissipation of kinetic energy and angular momentum takes place to activate the transfer of nucleons in the touching configuration of the projectile target that results in mass probability diffusion. The mass probability of the formed fragments was evaluated by solving a set of master equations. The term of mass probability P(Z₁,N₁,E₁,t) contains the proton number, neutron numbers of Z₁ and N₁, and internal excitation energy of E₁ for a given fragment A₁. The master equation is [64, 57, 70] $\begin{array}{l} \frac{d P (Z_{1}, N_{1}, E_{1}, t)}{d t} = \\ \sum_{{Z^{'}}_{1}} W_{Z_{1}, N_{1}; {Z^{'}}_{1}, N_{1}} (t) [d_{Z_{1}, N_{1}} P ({Z^{'}}_{1}, N_{1}, {E^{'}}_{1}, t) \\ - d_{{Z^{'}}_{1}, N_{1}} P (Z_{1}, N_{1}, E_{1}, t)] + \\ \sum_{{N^{'}}_{1}} W_{Z_{1}, N_{1}; Z_{1}, {N^{'}}_{1}} (t) [d_{Z_{1}, N_{1}} P (Z_{1}, {N^{'}}_{1}, {E^{'}}_{1}, t) \\ - d_{Z_{1}, {N^{'}}_{1}} P (Z_{1}, N_{1}, E_{1}, t)] - \\ [Λ_{A_{1}, E_{1}, t}^{qf} (Θ) + Λ_{A_{1}, E_{1}, t}^{fis} (Θ)] P (Z_{1}, N_{1}, E_{1}, t) . \end{array}$ (11) where $W_{Z_{1}, N_{1}, {Z^{'}}_{1}, N_{1}}$ ( $W_{Z_{1}, {N^{'}}_{1}, Z_{1}, N_{1}}$ ) denotes the mean transition probability from the channel ( $Z_{1}, N_{1}, E_{1}$ ) to ( ${Z^{'}}_{1}, N_{1}, {E^{'}}_{1}$ ) [or ( $Z_{1}, N_{1}, E_{1}$ ) to ( $Z_{1}, {N^{'}}_{1}, {E^{'}}_{1}$ )]; $d_{Z_{1}, N_{1}}$ denotes the microscopic dimension corresponding to the macroscopic state ( $Z_{1}, N_{1}, E_{1}$ ). The sum contains all possible numbers of protons and neutrons for fragment ( ${Z^{'}}_{1}$ , ${N^{'}}_{1}$ ). However, only one nucleon transfer at one time is assumed in the model with relations ${Z^{'}}_{1}$ = Z₁ ± 1, and ${N^{'}}_{1}$ = N₁ ± 1. The excitation energy E₁ is the local excitation energy $� # x 03 B {5;}_{1}^{*}$ for fragment ( ${Z^{'}}_{1}$ , ${N^{'}}_{1}$ ), which is derived by the dissipation of the relative motion along with the potentical energy surface (PES) of the DNS [71]. The time of the dissipation process is evaluated by the parameterization classical deflection function [72]. The motion of nucleons in the interaction potential is governed by the single-particle Hamiltonian: $H (t) = H_{0} (t) + V (t)$ (12) where the total single-particle energy and interaction potential are $H_{0} (t) = \sum_{K} \sum_{ν_{K}} ε_{ν_{K}} (t) α_{ν_{K}}^{+} (t) α_{ν_{K}} (t)$ (13) $\begin{matrix} V (t) = \sum_{K, K^{'}} \sum_{α_{K^{'}} β_{K^{'}}} u_{α_{K^{'}} β_{K^{'}}} α_{α_{K}}^{+} (t) α_{β_{K}} (t) \\ = \sum_{K, K^{'}} V_{K, K^{'}} (t) . \end{matrix}$ (14) where ενK and $u_{α_{K}, β_{K^{'}}}$ represent the single-particle energies and interaction matrix elements, respectively, in which the single-particle state is defined as center of colliding nuclei assumed to be orthogonal in the overlapping region. The annihilation and creation operators are time-dependent. The single-particle matrix elements are parameterized as $\begin{array}{l} u_{α_{K}, β_{K^{'}}} = U_{K, K^{'}} (t) \\ \times {\exp [- \frac{1}{2} {(\frac{ε_{α_{K}} (t) - ε_{β_{K}} (t)}{Δ_{K, K^{'}} (t)})}^{2}] - δ_{α_{K}, β_{K^{'}}}} \end{array}$ (15) where $U_{K, K^{'}} (t)$ and $δ_{α_{K}, β_{K^{'}}} (t)$ are described in Ref.[73]. The proton transition probability is microscopically derived from $\begin{array}{l} W_{Z_{1}, N_{1}; Z_{1}^{'}, N_{1}} = \frac{τ_{mem} (Z_{1}, N_{1}, E_{1}; Z_{1}^{'}, N_{1}, E_{1}^{'})}{d_{Z_{1}, N_{1}} d_{Z_{1}^{'}, N_{1}} ℏ^{2}} \\ \times \sum_{i i^{'}} {| 〈 Z_{1}^{'}, N_{1}, E_{1}^{'}, i^{'} 〈 V 〉 Z_{1}, N_{1}, E_{1}, i 〉 |}^{2} . \end{array}$ (16) The neutron transition probability has a similar formula. The memory time and interaction elements V are described in Ref.[63].

The evolution of the DNS along distance R leads to quasi-fission. The decay probability of quasi-fission is calculated based on the one-dimensional Kramers equation as [73, 74] $\begin{matrix} Λ_{A_{1}, E_{1}, t}^{qf} (Θ) = \frac{ω}{2 π ω^{B_{qf}}} [\sqrt{{(\frac{Γ}{2 ℏ})}^{2} + {(ω^{B_{qf}})}^{2}} - \frac{Γ}{2 ℏ}] \\ \times \exp [- \frac{B_{qf} (A_{1})}{Θ (A_{1}, E_{1}, t)}] \end{matrix}$ (17) where $B_{qf} (A_{1})$ is the quasi-fission barrier; ω and $ω^{B_{qf}}$ are the frequencies of the harmonic oscillator approximation at the bottom and top of the interaction potential pocket, which are constants expressed as $ℏ ω^{B_{qf}}$ = 2.0 MeV and $ℏ ω$ = 3.0 MeV in this study; Γ = 2.8 MeV is the quantity characterizing the average double width of the single-particle state. The local temperature is expressed using the Fermi gas model, i.e., $Θ = (ε^{*} / (A {/ 12))}^{1 / 2}$ . In the nuclear collision process, heavy fragments might lead to fission; the fission probability is calculated by the Kramers formula: $\begin{matrix} Λ^{f i s}_{A_{1} {,E}_{1},t} (Θ) = \frac{ω_{g . s}}{2 π ω_{f}} [\sqrt{{(\frac{Γ_{0}}{2 ℏ})}^{2} + {(ω_{f})}^{2}} - \frac{Γ_{0}}{2 ℏ}] \\ \times \exp [- \frac{B_{f} (A_{1})}{Θ (A_{1}, E_{1}, t)}], \end{matrix}$ (18) where ω_g.s. and ω_f are the frequencies of the oscillators approximating the fission-path potential at the ground state and top of the fission barrier for fragment A₁, respectively, which were set as $ℏ ω_{g .s .}$ = $ℏ ω_{f}$ = 1.0 MeV and Γ₀ = 2 MeV. The fission barrier is calculated by the macroscopic part plus the shell correction energy. In the relaxation process of the relative motion, the DNS is excited by the dissipation of the relative kinetic energy and angular momentum. The excited composite system opens a valence space Δ εK in fragment K (K = 1, 2) that has a symmetrical distribution around the Fermi surface. The nucleons in the valence space are actively enabled to be excited and transferred. The average of these quantities is performed in the valence space: $Δ ε_{K} = \sqrt{\frac{4 ε_{K}^{*}}{g_{K}}}, ε_{K}^{*} = ε^{*} \frac{A_{K}}{A}, g_{K} = A_{K} / 12,$ (19) where $ε^{*}$ is the local excitation energy of the DNS, which provides the excitation energy for the mean transition probability. There are NK = $g_{K} Δ ε_{K}$ valence states and mK = N_K/2 valence nucleons in the valence space ΔεK, leading to dimensions $d (m_{1}, m_{2}) = (\begin{array}{l} N_{1} \\ m_{1} \end{array}) (\begin{array}{l} N_{2} \\ m_{2} \end{array}) .$ (20) The local excitation energy is expressed as $ε^{*} = E_{x} - (U_{d r} (A_{1}, A_{2}) - U_{d r} (A_{P}, A_{T})),$ (21) where $U_{d r} (A_{1}, A_{2})$ and $U_{d r} (A_{P}, A_{T})$ are the driving potentials of fragments A₁, A₂ and A_P, A_T, respectively. The detailed calculations of the driving potentials are given by Eq. 22. The excitation energy E_x of the composite system is converted from the relative kinetic energy dissipation[64]. The PES of the DNS is expressed as $\begin{array}{l} U_{d r} (A_{1}, A_{2}; J, θ_{1}, θ_{2}) = B_{1} (N_{1}, Z_{1}, β_{1}) + B_{2} (N_{2}, Z_{2}, β_{2}) \\ - B_{C N} (N, Z, β) + U_{C} (Z_{1}, Z_{2}, β_{1}, β_{2}, R, θ_{1}, θ_{2}) \\ + U_{N} (Z_{1}, Z_{2}, β_{1}, β_{2}, R, θ_{1}, θ_{2}), \end{array}$ (22) where Bi (i = 1, 2) and B_CN are the negative binding energies of fragment Ai and compound nucleus A = A₁+A₂, respectively, where the shell and pairing corrections are reasonably included; β_i represents the quadrupole deformations of binary fragments; θ_i denotes collision orientations; and U_C and U_N are derived from Eqs. 10 and 7, respectively.

By solving a set of master equations, the probability of all possible formed fragments is obtained. The hindrance in the fusion process is named inner fusion barrier, B_fus, which is defined by the difference from the injection position to the B.G. point. In the DNS model, these fragments overcome the inner barrier that is considered to lead to fusion. Therefore, the fusion probability is evaluated by adding all the fragments that could penetrate the inner fusion barrier: $P_{CN} (E_{c .m .}, J, B) = \sum_{Z = 1}^{Z_{BG}} \sum_{N = 1}^{N_{BG}} P (N_{1}, Z_{1}, E_{1} (J), τ_{int} (J)) .$ (23) Here, the interaction time τ_int(E_c.m., J, B) is obtained from the deflection function method [71]. We calculated the fusion probability as $P_{CN} (E_{c .m .}, J) = \int f (B) P_{CN} (E_{c .m .}, J, B) d B .$ (24) The Coulomb barrier distribution function f(B) is given by Eq. 4. Therefore, the fusion cross-section is expressed as $σ_{fus} (E_{c .m .}) = σ_{cap} (E_{c .m .}) P_{CN} (E_{c .m .}, J) .$ (25)

2.3

Survival probability

The compound nuclei are formed by all the transfers of nucleons from projectile nuclei to target nuclei that have a few excitation energies. The excited compound nuclei are extremely unstable and can be de-excited by evaporating γ-rays, neutrons, protons, α, etc., against fission. The survival probability of the channel x-th neutron, y-th proton, and z-th α is expressed as [75, 57, 70] $\begin{array}{l} W_{sur} (E_{C N}^{*}, x, y, z, J) = P (E_{C N}^{*}, x, y, z, J) \\ \times \prod_{i = 1}^{x} \frac{Γ_{n} (E_{i}^{*}, J)}{Γ_{tot} (E_{i}^{*}, J)} \prod_{j = 1}^{y} \frac{Γ_{p} (E_{j}^{*}, J)}{Γ_{tot} (E_{i}^{*}, J)} \prod_{k = 1}^{z} \frac{Γ_{α} (E_{k}^{*}, J)}{Γ_{tot} (E_{k}^{*}, J)} \end{array}$ (26) where $E_{CN}^{*}$ and J denote the excitation energy and spin of the excited nucleus, respectively. The total width $Γ_{tot}$ is the sum of the partial widths of particle evaporation, γ-rays, and fission. The excitation energy $E_{s}^{*}$ before evaporating the s-th particles is evaluated by $E_{s + 1}^{*} = E_{s}^{*} - B_{i}^{n} - B_{j}^{p} - B_{k}^{α} - 2 T_{s},$ (27) with initial condition $E_{i}^{*}$ = $E_{C N}^{*}$ and s=i+j+k; $B_{i}^{n}$ , $B_{j}^{p}$ , and $B_{k}^{α}$ denote the separation energies of the i-th neutron, j-th proton, and k-th alpha, respectively. The nuclear temperature Ti is defined by $E_{i}^{*} = α T_{i}^{2} - T_{i}$ with level density a. The decay width of the γ-rays and the particle decay were evaluated with a method similar to that reported in Ref. [75]. We set $E^{*} - B_{v} - δ - δ_{n}$ to the term $ϱ$ .

The widths of particles decay are evaluated using the Weisskopf evaporation theory as $\begin{array}{l} Γ_{v} (E^{*}, J) = (2 s_{v} + 1) \frac{m_{v}}{π^{2} ℏ^{2} ρ (E^{*}, J)} \\ \times \int_{0}^{ϱ - \frac{1}{a}} ε ρ (ϱ + δ - E_{rot} - ε, J) σ_{inv} (ε) d ε . \end{array}$ (28) Here, sv, mv, and Bv are the spin, mass, and binding energy of the particle, respectively. The pairing correction energy δ was set to be $12 / \sqrt{A}$ , 0, and $- 12 / \sqrt{A}$ for even-even, even-odd, and odd-odd nuclei, respectively. The inverse cross-section is expressed as $σ_{inv} = π R_{ν}^{2} T (ν)$ . The penetration probability was set to 1 for neutrons and $T (ν) = (1 + \exp (π (V_{C} (ν) - ε {) /))}^{- 1}$ for charged particles with $ℏ ω = 5$ and 8 MeV for proton and α, respectively. The Coulomb barrier of the emitting charge particles and daughter nuclei is expressed as $V_{C} = \frac{(Z_{C N} - i) Z_{i} e^{2}}{r_{i} (A_{C N - i}^{1 / 3} + A_{i}^{1 / 3})} .$ (29) In this study, we set proton emitting r_p = 1.7 fm and α emitting α = 1.75 fm; for further information, please refer to [76]. The fission width was calculated using the Bohr-Wheeler formula, as in Ref.[63, 64]. We set $E^{*} - B_{f} - E_{r o t} - δ - δ_{f}$ to the term κ. $Γ_{f} (E^{*}, J) = \frac{1}{2 π ρ_{f} (E^{*}, J)} \int_{0}^{κ - \frac{1}{α_{f}}} \frac{ρ_{f} (κ - ε + δ, J) d ε}{1 + exp [- 2 π (κ - ε + δ + δ_{f}) / ℏ ω]}$ (30) For heavy fragments, the fission width is usually set as α 2.2 MeV [77], and δ_f denotes the pairing correction for the fission barrier. The fission barrier is divided into microscopic and macroscopic parts: $B_{f} (E^{*}, J) = B_{f}^{LD} + B_{f}^{M} (E^{*} = 0, J) \exp (- E^{*} / E_{D}),$ (31) where the macroscopic part is derived from the liquid-drop model: $B_{f}^{LD} = {\begin{array}{l} 0.38 (0.75 - x) E_{S0}, & (1 / 3 < x < 2 / 3) \\ 0.83 {(1 - x)}^{3} E_{S0}, & (2 / 3 < x < 1) \end{array}$ (32) with $x = \frac{E_{C0}}{2 E_{S0}} .$ (33) Here, E_C0 and E_S0 are the Coulomb energy and surface energy of the spherical nuclear, respectively, which could be taken from the Myers-Swiatecki formula: $E_{S0} = 17.944 [1 - 1.7826 {(\frac{N - Z}{A})}^{2}] A^{2 / 3} MeV$ (34) and $E_{C0} = 0.7053 \frac{Z^{2}}{A^{1 / 3}} MeV .$ (35) The microcosmic shell correction energy was taken from [78]. The shell-damping energy was $E_{D} = \frac{5.48 A^{1 / 3}}{1 + 1.3 A^{- 1 / 3}} MeV$ (36) or $E_{D} = 0.4 A^{4 / 3} / a MeV$ (37) where a is the energy level density parameter. The fission level density was set as a_f = 1.1a. The moments of inertia of fission compound nuclei at ground state (gs) and saddle point (sd) configurations are expressed as $ζ_{gs(sd)} = k \times \frac{2}{5} m r^{2} (1 + β_{2}^{gs(sd)} / 3) .$ (38) Here, k = 0.4 is the correction factor of the rigid body and β₂ is the quadrupole deformation taken from Ref. [78]; $β_{2}^{sd}$ = $β_{2}^{gs}$ + 0.2 was the quadrupole deformation at the saddle point calculated by the relativistic mean field theory. Based on the Fermi gas model, the energy level density could be expressed as [78] $\begin{matrix} ρ (E^{*}, J) = K_{coll} \times \frac{2 J + 1}{24 \sqrt{2} σ^{3} a^{1 / 4} {(E^{*} - δ)}^{5 / 4}} \\ \times \exp [2 \sqrt{a (E^{*} - δ)} - \frac{{(J + 1 / 2)}^{2}}{2 σ^{2}}] \end{matrix}$ (39) where $σ^{2} = 6 {\bar{m}}^{2} \sqrt{a (E^{*} - δ)} / π^{2}$ and $\bar{m} \approx 0.24 A^{2 / 3}$ ; K_coll is the collective enhancement factor, which contains the rotational and vibration effects. The level density parameter was set as a=A/12, a_f=1.1a for the fission-level density parameter.

The realization probability of evaporation channels is an important component in the survival probability equation. The realization probability of one particle evaporation is expressed as $P (E_{CN}^{*}, J) = \exp [- \frac{{(E_{CN}^{*} - B_{s} - E_{rot} - 2 T)}^{2}}{2 σ^{2}}]$ (40) where σ is the half-height width of the excitation function of the residual nucleon in the F.E. reactions, which was set as 2.5 MeV in our calculations, and E_rot is the rotation energy. For the multiple neutron evaporation channels (x >1), the realization probability can be derived using the Jackson formula: $P (E_{CN}^{*}, s, J) = I (Δ_{s},2 s - 3) - I (Δ_{s + 1},2 s - 1)$ (41) where I and $Δ$ are given by $I (z, m) = \frac{1}{m!} \int_{0}^{z} u^{m} e^{- u} d u$ (42) $Δ_{s} = \frac{E_{CN}^{*} - \sum_{i = 1}^{s} B_{i}^{v}}{T_{i}}$ (43) where $B_{i}^{v}$ is the separation energy of the evaporation of the i-th particle and s=x+y+z. The spectrum of realization probabilities determines the distribution shape of survival probability in the evaporation channels.

Results and Discussion

In the framework of the DNS model involving all the collision orientations, we calculated the excitation functions of 2n-, 3n-, 4n-, and 5n-evaporation channels for the collisions of ⁴⁸Ca+²⁴³Am, ⁴⁸Ca+²⁴²Pu, and ⁴⁸Ca+²³⁸U, marked by solid olive, dash red, dash-dot blue, and orange short-dash lines, respectively, in Fig. 1. In panel (a), the olive-filled up-triangle, red-filled square, and blue-filled circle represent experimental results of 2n-, 3n-, and 4n-evaporation channels for ⁴⁸Ca+²⁴³Am taken from Ref. [9, 83, 84]. According to Ref. [9], the experiments concerning ⁴⁸Ca+²⁴³Am at incident energies E_lab = 248, 253 MeV were carried out at FLNR, JINR. At E_lab = 248 MeV, three similar decay chains consisting of five consecutive α decays were identified. At E_lab = 253 MeV, the decay properties of these synthesized nuclei are consistent with consecutive α decays originating from the parent isotopes of the new element, Mc, i.e., ²⁸⁷Mc and ²⁸⁸Mc, produced in the 3n- and 4n-evaporation channels with cross-sections of approximately 3 pb and 1 pb, respectively. According to Ref. [83], the cross-section for the 3n-evaporation channel reaches its maximum, σ_3n = ${8.5}_{- 3.7}^{+ 6.4}$ pb, at E^* = 34.0 - 38.3 MeV and decreases with further increase of the excitation energy of the compound nucleus ²⁹¹Mc. At excitation energy, E^* = 44.8 ± 2.3 MeV, not a single event indicating the formation of ²⁸⁸Mc was detected. The upper cross-section limit can thus be set at level σ_3n ≤ 1 pb. At excitation energy in the range of E^* = 31.1 - 36.4 MeV, the cross-sections for the formation of ERs in the 3n- and 2n-evaporation channels were approximately ${3.5}_{- 1.5}^{+ 2.7}$ pb and ${2.5}_{- 1.5}^{+ 2.7}$ pb, respectively. At energies E^* ≤ 36 MeV, which could be expected for the 2n-evaporation product, ²⁸⁹Mc was not detected. The upper cross-section limit can be set at level σ_2n ≤ 3 pb. According to Ref. [84], the cross-sections for the formation of ERs in the 3n- and 2n-evaporation channels are approximately ${3.2}_{- 1.2}^{+ 0.8}$ pb and ${0.3}_{- 0.2}^{+ 0.7}$ pb at energies E^* = 33 MeV, respectively. In Fig. 1 (b), the olive up-triangle, red square, blue circle, and orange down-triangle represent the experimental results of 2n-, 3n-, 4n-, and 5n-evaporation channels for ⁴⁸Ca+²⁴²Pu, respectively, where filled, half-filled, and open symbols represent three experiments for ⁴⁸Ca+²⁴²Pu [8, 79, 80, 86]. According to Ref. [79], a maximum cross-section of ${10.4}_{- 2.1}^{+ 3.5}$ pb was measured for the ²⁴²Pu(⁴⁸Ca, 3n)²⁸⁷Fl reaction. According to Ref. [8], at excitation energy E^* = 50 MeV, the ²⁴²Pu(⁴⁸Ca, 5n)²⁸⁵Fl cross-section was ${0.6}_{- 0.5}^{+ 0.9}$ pb. The no-observation of a 3n-evaporation product led to an upper limit for the ²⁴²Pu(⁴⁸Ca, 3n)²⁸⁷Fl reaction of 1.1 pb. The 3n and 4n cross-section values measured at E^* = 41 MeV were ${3.1}_{- 2.6}^{+ 4.9}$ pb. In Fig. 1 (c), the red square and blue circle stand for the experimental results of 3n- and 4n-evaporation channels for ⁴⁸Ca+²³⁸U, respectively, where filled, half-filled, and open symbols represent three experiments for ⁴⁸Ca+²³⁸U [79, 81-82]. According to Ref. [80], the maximum cross-section values of the xn-evaporation channels for the reaction ²³⁸U(⁴⁸Ca, xn)^286-xCn were measured to be σ_3n = ${2.5}_{- 1.1}^{+ 1.8}$ pb and σ_4n = ${0.6}_{- 0.5}^{+ 1.6}$ pb. At the excitation energies of the compound nucleus E^* = 34.5 MeV, two decay events from ²⁸³Cn were observed, resulting in a cross-section of ${2.0}_{- 1.3}^{+ 2.7}$ pb[82]. The cross-section deduced from all four events was ${0.72}_{- 0.35}^{+ 0.58}$ pb, measured at an excitation energy of 34.6 MeV of the compound nucleus ²⁸⁶Cn[81]. From the above three panels, we can conclude that our calculations are in good agreement with the available experimental excitation functions of the reactions ⁴⁸Ca+²⁴³Am, ⁴⁸Ca+²⁴²Pu, and ⁴⁸Ca+²³⁸U.

Fig. 1

(Color online) The calculated excitation functions of 2n-, 3n-, 4n-, and 5n-evaporation channels for the reactions of ⁴⁸Ca+²⁴³Am, ⁴⁸Ca+²⁴²Pu, and ⁴⁸Ca+²³⁸U are marked by solid olive, dash red, dash-dot blue, and orange short-dash lines, respectively. The experimental measurement results of excitation functions for 2n-, 3n-, 4n-, and 5n-evaporation channels are represented by an up-triangle, square, circle, and down-triangle. Vertical error bars correspond to total uncertainties. Symbols with arrows show upper cross-section limits. Data marked by open, half-closed, and filled symbols are taken from [8, 9, 79-85], respectively.

To investigate the dependence of the production cross-section of superheavy nuclei in F.E. reactions on collision orientation, we exported four configurations of the collision orientations from our calculations for the reaction of ⁴⁸Ca+²⁴³Am as (0°, 0°), (30°, 30°), (60°, 60°) and (90°, 90°), marked by the solid black, red dash, olive dash-dot, and blue short-dash lines, respectively, in Fig. 2. The projectile nuclei ⁴⁸Ca and target nuclei ²⁴³Am have theoretical quadrupole deformation values β_P = 0. and β_T = 0.224, respectively. In Fig. 2, panel (a) shows the distributions of interaction potential energy with respect to the distance between the surfaces of projectile nuclei and target nuclei. The interaction potential V_CN consists of Coulomb potential V_C and nucleus-nucleus potential V_N, which were calculated by Wong’s formula [69] and the double-folding method [87], respectively. The interaction potential energies were increased with the large collision orientations because of the large effective interaction face. Panel (b) displays the distributions of radial kinetic energy with respect to the interaction time. The kinetic energy decreased exponentially with increasing reaction time at the prescribed impact parameter, i.e., L=20 ℏ. These evolutions of kinetic energy reached equilibrium at approximately 2 × 10^-21s. These equilibrium kinetic energies were 225 MeV, 228 MeV, 235MeV, and 239 MeV, corresponding to collision orientations (0°, 0°), (30°, 30°), (60°, 60°), and (90°, 90°), respectively. The kinetic energy dissipated into the internal excitation of the composite system, which correspondingly increased exponentially with the reaction time for the same relaxation time, as illustrated in panel (c). According to Fig. 2, we can conclude that the interaction potential and evolution of kinetic energy and internal excitation energy were highly dependent on the orientations. These were the basic reasons causing the dependence of the final synthesis cross-sections of superheavy nuclei on collision orientations.

Fig. 2

(Color online) Panel (a) shows the interaction potential for the collisions of ⁴⁸Ca+²⁴³Am as a function of distance with different collision angles. The collision orientations (0°, 0°), (30°, 30°), (60°, 60°), and (90°, 90°) correspond to solid black, red dash, olive dash-dot, and short dash lines, respectively. Panel (b) represents the radial kinetic energy decreases along with the reaction time at different collision orientations under an angular momentum L=20 ℏ. Panel (c) shows that the internal excitation energy of the composite system varies with the sticking time for the given angular momentum L=20 ℏ.

The PES and driving potential (DP) of the reaction ⁴⁸Ca+²⁴³Am were calculated by Eq. 22 for the collision orientations of sphere-sphere, (0°, 0°), (30°, 30°), (60°, 60°) and (90°, 90°), as illustrated in Fig. 3. The PES and DP are listed in the upper and lower panels, respectively. Panels (a) and (f) show the PES and DP of the no-deformation of projectile-target nuclei. The minimum trajectories and injection points are attached to the PESs, which are represented by solid black lines and filled black stars. The structure effect is clearly shown in the PESs and DPs by the comparison of no-deformation collision with quadrupole deformation collision. The inner fusion barrier was set as the difference between the injection points and Businaro-Gallone (B.G.) points, which were 8 MeV, 11.5 MeV, 10.5 MeV, 7.1 MeV, and 6 MeV corresponding to collision orientations of no-deformation, (0°, 0°), (30°, 30°), (60°, 60°), and (90°, 90°), respectively. It was found that the inner fusion barrier was highly dependent on the collision orientations, which could reveal the fusion probability directly. The inner fusion barriers were decreased with the increased collision orientation, reaching its minimum at the waist-waist collision. Sketches of collision orientations are shown at the top of Fig. 3. The potential energy of the symmetry field in the PES increased with increasing collision orientations because the corresponding Coulomb force increased as well.

Fig. 3

(Color online) Panels (a), (b), (c), (d), and (e) represent the PES of ⁴⁸Ca+²⁴³Am at collision orientations of sphere-sphere, (0°, 0°), (30°, 30°), (60°, 60°), and (90°, 90°), respectively. Panels (f), (g), (h), (i), and (j) correspond to their collision orientation-based valley trajectories in PES along with the mass asymmetry η with respect to

η = (A_{T} - A_{P}) / (A_{T} + A_{P})

. Their inner barrier value is given by B__BG. The arrow lines represent injection points.

In the collision process, when overcoming the Coulomb barrier, the kinetic energies of the colliding partners rapidly dissipate into the composite system. The probability of projectile and target diffusing along the PES was calculated by solving a set of master equations. The TKE of binary fragments was related to the incident energy, ground-state binding energy, and internal excitation energy as $T K E = E_{c .m .} - V_{CN} - Q_{gg} - E^{*}$ . Fig. 4 presents the TKE-mass distributions for collision orientations of no-deformation, (0°, 0°), (30°, 30°), (60°, 60°), and (90°, 90°) at incident energy E_c.m. = 1.1 × V_B, as shown in panels (a), (b), (c), (d), and (e), respectively. The TKE term could be rewritten as $T K E = E_{c .m .} - U_{dr} - E^{*}$ . The TKE-mass distribution shape is highly dependent on the driving potential. The TKE-mass distribution for no-deformation collision in panel (a) was smoother than others in panels (b), (c), (d), and (e), thereby showing the structure effect in the TKE-mass distribution. The fragments in the black square passing the B.G. points are supposed to lead to fusion. The fusion probability was calculated by summing all the formation probabilities passing B.G. points. Fig. 4 reveals that it is difficult to evaluate the dependence of fusion probability on the collision orientations; the reason is that only one incident energy, E_c.m. = 1.1 × V_B, is shown.

Fig. 4

(Color online) Panels (a), (b), (c), (d), and (e) display the calculations of TKE-mass distribution of the primary fragments in the collisions of ⁴⁸Ca+²⁴³Am at E__c.m. = 1.1× V_B for their collision orientations of sphere-sphere, (0°, 0°), (30°, 30°), (60°, 60°), and (90°, 90°), respectively.

Figure 5 shows the TKE-mass distributions at excitation energies $E_{C N}^{*}$ = 10 MeV, 40 MeV, 70 MeV, and 100 MeV for the tip-tip collisions of ⁴⁸Ca+²⁴³Am, as illustrated in panels (a), (b), (c), and (d), respectively. Fig. 5 shows that the TKE-mass distribution was broader for increasing incident energy. It is evident that the fusion probability increased with larger excitation energy. However, the compound nuclei with large excitation energy could easily lead to fission. The maximum evaporation residue cross-section of the high-excitation compound nuclei was the balance between fusion probability and survival probability.

Fig. 5

(Color online) Panels (a), (b), (c), and (d) show the calculations of TKE-mass distribution of the primary fragments in the head-on collisions of ⁴⁸Ca+²⁴³Am at incident energies corresponding to excitation energies of compound nuclei, that is, 10 MeV, 40 MeV, 70 MeV, and 100 MeV, respectively.

To approximate the real collision process as much as possible, we propose Gaussian-like barrier distributions to consider all the collision orientations. Eq. (4) can be employed for this purpose. The olive solid, red dash, blue dot-dash, and orange short-dash lines represent the calculated excitation function of the 2n-, 3n-, 4n-, and 5n-evaporation channels. The olive-filled up-triangle, red-filled square, and blue-filled circle represent the experimental excitation function of the 2n-, 3n-, and 4n-evaporation channels, respectively. For the reactions ⁴⁸Ca+²⁴³Am at excitation energies within the interval E^* = 20–100 MeV, the excitation functions of the 2n-, 3n-, 4n-, and 5n-evaporation channels were calculated by the DNS model involving the barrier distribution, as shown in panel (a); these functions are in good agreement with the experimental data [83, 9]. The calculated excitation functions of ⁴⁸Ca+²⁴³Am for the collision orientations (0°, 0°), (30°, 30°), (60°, 60°), (90°, 90°), and no-deformation are shown in panels (b), (c), (d), (e), (f), respectively. It was found that the (0°, 0°) collisions underestimate the experimental results. Collisions (30°, 30°) agree with the experimental results relatively well. Collisions (30°, 30°), (60°, 60°), and (90°, 90°) overestimate the experimental data. Fig. 6 shows thatthe DNS model involving barrier distributions could reproduce the experimental results relatively well.

Fig. 6

(Color online) For the collisions of ⁴⁸Ca+²⁴³Am, the panels show the calculations of excitation functions in 2n-, 3n-, 4n-, and 5n-evaporation channels corresponding to solid olive, red dash, blue dash-dot, and orange short-dash lines, respectively. Panels (b), (c), (d), (e), and (f) display the excitation functions at orientations (0°, 0°), (30°, 30°), (60°, 60°), (90°, 90°), and sphere to sphere, respectively. Panel (a) shows the total excitation function when considering all the collision orientations using the method of Gaussian distribution. Experimental data are marked by filled up-triangle, square, circle, and down-triangle symbols, as in [83, 9].

Based on the DNS model involving barrier distribution, to investigate the dependence of evaporation residue cross-section on the isospin of the projectile, we systematically calculated the reactions of ⁴²Ca+²⁴³Am, ⁴⁴Ca+²⁴³Am, ⁴⁶Ca+²⁴³Am, ⁴⁸Ca+²⁴³Am, ⁴⁴Ti+²³⁷Np, ⁴⁶Ti+²³⁷Np, ⁴⁸Ti+²³⁷Np, and ⁵⁰Ti+²³⁷Np at excitation energies within the interval E^* = 1 - 80 MeV. Fig. 7 shows that the excitation functions of the evaporation residue cross-section are highly dependent on the isospin of the projectile. Regarding the isotopes of Ca-induced reactions, the cross-sections of 2n- and 3n-evaporation channels decreased with the projectile of Ca isotopes with large N/Z, which might be caused by fusion probability. The ratio of σ_3n/σ_2n increased with increasing N/Z, which implies that more-neutron-rich compound nuclei are prone to evaporating more neutrons. The existing moscovium isotopes are ^287-290Mc. The predictions of maximum cross-sections of the new ^281-286Mc were 4 pb, 45 pb, 150 pb, 50 pb, 101 pb, and 30 pb, respectively, in calcium-isotope-induced F.E. reactions. The maximum synthesis cross-section of new moscovium isotopes was ²⁸³Mc as 0.15 nb in the reactions ⁴²Ca+²⁴³Am. Concerning Ti-isotope induced reactions, the 2n-evaporation channel was dominant in the evaporation residue cross-sections. The maximum synthesis cross-section of Mc was ²⁸¹Mc as 0.2 nb in the reactions ⁴⁶Ti+²³⁷Np. The new moscovium isotopes of ^278-286Mc were evaluated as 0.5 pb, 9 pb, 12 pb, 10.5 pb, 150 pb, 11 pb, 100 pb, 10 pb, and 31 pb, respectively, in titanium-isotope-induced F.E. reactions.

Fig. 7

(Color online) The calculations of excitation functions for the collisions of ⁴²Ca+²⁴³Am, ⁴⁴Ca+²⁴³Am, ⁴⁶Ca+²⁴³Am, ⁴⁸Ca+²⁴³Am, ⁴⁴Ti+²³⁷Np, ⁴⁶Ti+²³⁷Np, ⁴⁸Ti+²³⁷Np, and ⁵⁰Ti+²³⁷Np are shown in panels (a), (b), (c), (d), (e), (f), (g), and (h), respectively. The 2n-, 3n-, 4n-, and 5n-evaporation channels correspond to black solid, red dash, blue dash-dot, and olive short-dash lines, respectively

To investigate the influence of the entrance effect on the synthesis cross-section of superheavy moscovium in the F.E. reactions, we systematically calculated the collisions of ³⁵Cl + ²⁴⁸Cf (η = 0.75), ⁴⁰Ar + ²⁴⁷Bk (η = 0.72), ³⁹K + ²⁴⁷Cm (η = 0.73), ⁴⁰Ca + ²⁴³Am (η = 0.72), ⁴⁸Ca + ²⁴³Am (η = 0.67), ⁴⁵Sc + ²⁴⁴Pu (η = 0.69), ⁴⁸Ti + ²³⁷Np (η = 0.66), and ⁵¹V + ²³⁸U (η = 0.65) based on the DNS model, as illustrated in panels (a), (b), (c), (d), (e), (f), (g), and (h), respectively. Fig. 8 shows that the reaction systems with large η are prone to producing large production cross-sections because the large mass asymmetry reactions are in turn prone to fusion. In these calculations, the new moscovium ^278-286Mc was predicted with production cross-section values of 1 pb, 10 pb, 130 pb, 50 pb, 15 pb, 100 pb, 30 pb, 200 pb, 40 pb, respectively. The 2n- or 3n-evaporation residue channels were dominant in the evaporation survival process. The ratio σ_3n/σ_2n illustrates the role of the odd-even effect on the production cross-section of superheavy nuclei. The maximum production cross-section of moscovium isotopes was predicted as 200 pb in the reaction ²⁴⁷Cm(³⁹K, 3n)²⁸³Mc.

Fig. 8

(Color online) The calculations of excitation functions in the collisions of ³⁵Cl+²⁴⁸Cf, ⁴⁰Ar+²⁴⁷Bk, ³⁹K+²⁴⁷Cm, ⁴⁰Ca+²⁴³Am, ⁴⁸Ca+²⁴³Am, ⁴⁵Sc+²⁴⁴Pu, ⁴⁸Ti+²³⁷Np, and ⁵¹V+²³⁸U are shown in panels (a), (b), (c), (d), (e), (f), (g), and (h), respectively. The 2n-, 3n-, 4n-, and 5n-evaporation channels correspond to black solid, red dash, blue dash-dot, and olive short-dash lines, respectively.

Conclusion

As a summary, to simulate the real collision process, we propose a Gaussian-like barrier distribution function used to include all collision orientations. To investigate the dependence of the production cross-section of superheavy isotopes on the collision orientations, we systematically calculated the reactions of ⁴⁸Ca+²⁴³Am at excitation energies within the interval 0-100 MeV for the collision orientations of no-deformation, i.e., (0°, 0°), (30°, 30°), (60°, 60°), and (90°, 90°). In the DNS model, for a given collision orientation, some physical quantities such as interaction potential, radial kinetic energy, internal excitation energy, TKE-mass, PES, DP, and inner fusion barrier were exported to show the influence of collision orientations; the conclusion is that these quantities are highly dependent on the collision orientations. We compared the calculated excitation functions of ⁴⁸Ca+²⁴³Am at some fixed collision orientations with available experimental results. We found that large collision orientations showed an overestimated value compared to experimental data. The collision orientation nearby (30°, 30°) fit the experimental data very well. The barrier-distribution-based excitation function was in good agreement with the experimental data. To test the barrier distribution function, we calculated the reactions of ⁴⁸Ca+²⁴³Pu and ⁴⁸Ca+²³⁸U, which reproduced the experimental excitation functions well. Based on the DNS model involving the barrier distribution function, we systematically calculated the reactions of projectiles ^42-48Ca bombarding on target ²⁴³Am and projectiles ^42-48Ca on target ²³⁷Np. The influence of the isospin of a projectile on the production cross-section was studied. For Ca-induced F.E. reactions, σ_2n and σ_3n were dominant in the evaporation residue cross–sections, which decreased with increasing N/Z in projectiles. The ratio σ_3n/σ_2n increased with increasing N/Z in projectiles, which might be caused by neutron-rich compound nuclei prone to losing neutrons. For Ti-induced F.E. reactions, the maximum cross-section was 150 pb for ²⁸³Mc in the reaction ²³⁷Np(⁴⁶Ti, 2n)²⁸³Mc. The reactions of ³⁵Cl+²⁴⁸Cf, ⁴⁰Ar+²⁴⁷Bk, ³⁹K+²⁴⁷Cm, ⁴⁰Ca+²⁴³Am, ⁴⁸Ca+²⁴³Am, ⁴⁵Sc+²⁴⁴Pu, ⁴⁸Ti+²³⁷Np, and ⁵¹V+²³⁸U were calculated to investigate the entrance channel effect on production cross-sections of superheavy nuclei. Large mass asymmetry systems lead to large production cross-section. We also found that the odd-even effect might play a role in the evaporation residue cross-section. We predicted the new moscovium isotopes ^278-286Mc with maximum cross-sections of 0.5 pb, 9 pb, 12 pb, 10.5 pb, 150 pb, 11 pb, 100 pb, 10 pb, and 31 pb in the collisions of ^35,37Cl + ²⁴⁸Cf, ^38,40Ar + ²⁴⁷Bk, ^39,41K + ²⁴⁷Cm, ^40,42,44,46Ca + ²⁴³Am, ⁴⁵Sc + ²⁴⁴Pu, and ^46,48,50Ti + ²³⁷Np, ⁵¹V + ²³⁸U at some typical excitation energies.

References

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Y.T. Oganessian, V.K. Utyonkov,

Super-heavy element research

. Reports on Progress in Physics 78, 036301 (2015). doi: 10.1088/0034-4885/78/3/036301