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

Prediction of synthesis cross-sections of new moscovium isotopes in fusion-evaporation reactions 1 2 corresp first-author † chenpenghui@yzu.edu.cn † chenpenghui@yzu.edu.cn chenpenghui@yzu.edu.cn 1 3 1 4 5 corresp ‡ fengzhq@scut.edu.cn ‡ fengzhq@scut.edu.cn fengzhq@scut.edu.cn School of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China School of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China RIKEN Nishina Center, Wako, Saitama 351-0198, Japan RIKEN Nishina Center, Wako, Saitama 351-0198, Japan College of Electrical, Power and Energy Engineering, Yangzhou University, Yangzhou 225009, China College of Electrical, Power and Energy Engineering, Yangzhou University, Yangzhou 225009, China School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510641, China School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510641, China 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 40,48 Ca bombarding on targets 238 U, 242 Pu, and 243 Am 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 48 Ca + 238 U, 48 Ca + 242 Pu, and 48 Ca + 243 Am 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 278-286 Mc was predicted to be as large as hundreds of pb in the fusion-evaporation reactions of 35,37 Cl + 248 Cf, 38,40 Ar + 247 Bk, 39,41 K + 247 Cm, 40,42,44,46 Ca + 243 Am, 45 Sc + 244 Pu, and 46,48,50 Ti + 237 Np, 51 V + 238 U at some typical excitation energies. Dinuclear system model Superheavy nuclei Complete fusion reactions Production cross-section Introduction 1 Since the "island of stability" of superheavy nuclei was predicted by the shell model in the 1960s[ 1 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 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. 257,258,259 Rf ( Z =104) was discovered simultaneously in Dubna[ 3 3 ] and Berkeley[ 4 4 ] in the reactions of 249 Cf( 12,13 C, 3-4n) 257,258,259 Rf at incident energy E lab = 10.4 MeV/nucleon. 260,261 Db ( Z =105) was discovered simultaneously in Dubna[ 5 5 ] and Berkeley[ 6 6 ] in the reactions of 249 Cf( 15 N,4n) 260 Ds at E lab = 85 MeV and 243 Am( 22 Ne, 4n) 261 Ds at E lab = 114 MeV. 263 Sg ( Z =106) was discovered at Berkeley[ 7 7 ] in the reactions of 249 Cf( 18 O,4n) 263 Sg at E lab = 95 MeV. 286-289 Fl was essentially discovered at Dubna[ 8 8 ] in the reactions of 244 Pu( 48 Ca, 3-6n) 286-289 Fl at E lab = 352.6 MeV. 288 Mc ( Z =115) was essentially discovered at Dubna[ 9 9 ] in the reactions of 243 Am( 48 Ca, 3n) 288 Mc at E lab = 248, 253 MeV. 286-289 Lv ( Z =116) was discovered at Dubna[ 10 10 ] in the reactions of 245 Cm( 48 Ca, xn) 293- x Fl at E lab = 243 MeV. 293-294 Ts ( Z =117) was essentially discovered at Dubna[ 11 11 ] in the reactions of 249 Bk( 48 Ca, 3-4n) 293-294 Fl at E lab = 247, 252 MeV. 294 Og ( Z =118) was essentially discovered at Dubna[ 12 12 ] in the reactions of 249 Cf( 48 Ca, 3n) 294 Og at E lab = 251 MeV. Elements Sg, Bh, Hs, Mt, Ds, Rg, Cn, and Nh were synthesized first in cold-fusion reactions. 259 Sg ( Z =106) was discovered at Dubna[ 13 13 ] in the reactions of 207 Pb( 54 Cr,2n) 259 Sg at E lab = 262 MeV. 262 Bh ( Z =107) was essentially discovered at Gesellschatt Für Schwerionenforschung (GSI)[ 14 14 ] in the reactions of 209 Bi( 54 Cr, 1n) 262 Bh at E lab = 4.85 MeV/u. 263-265 Hs ( Z =108) was synthesized at GSI[ 15 15 ] in the reactions of 208 Pb( 58 Fe, 2n) 265 Hs at E lab = 5.02 MeV/u. 266 Mt ( Z =109) was synthesized at GSI[ 16 16 ] in the reactions of 209 Bi( 58 Fe, 1n) 266 Mt at E lab = 5.15 MeV/u. 269 Ds ( Z =110) was synthesized at GSI[ 17 17 ] in the reactions of 208 Pb( 62 Ni, 1n) 269 Ds at E lab = 311 MeV. 272 Rg ( Z =111) was synthesized at GSI[ 18 18 ] in the reactions of 209 Bi( 64 Ni, 1n) 272 Rg at E lab = 318, 320 MeV. 277 Cn ( Z =112) was synthesized at GSI[ 19 19 ] in the reactions of 208 Pb( 70 Zn, 1n) 277 Cn at E lab = 344 MeV. 278 Nh ( Z =113) was synthesized at RIKEN[ 20 20 ] in the reactions of 209 Bi( 70 Zn, 1n) 278 Nh 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 Table 1 . Chinese superheavy nuclei group synthesized the superheavy isotopes of 258,259 Db[ 21 21 ], 264,265,266 Bh[ 22 22 ], and 271 Ds[ 23 23 ] at the Institute of Modern Physics (IMP) (Lanzhou, China). 1 1 Element 1 1 Isotopes 1 1 Reactions 1 1 Channel 1 1 Lab 1 1 Year 1 1 Ref. 1 1 Rf(104) 1 1 253 Rf 1 1 50 Ti + 204 Pb 1 1 1n 1 1 GSI 1 1 1997 1 1 [ 24 24 ] 1 1 (13) 1 1 267 Rf 1 1 48 Ca + 242 Pu 1 1 α 1 1 Dubna 1 1 2004 1 1 [ 25 25 ] 1 1 Db(105) 1 1 256 Db 1 1 50 Ti + 209 Bi 1 1 3n 1 1 GSI 1 1 2001 1 1 [ 26 26 ] 1 1 (11) 1 1 270 Db 1 1 48 Ca + 249 Bk 1 1 3n α 1 1 Berkeley 1 1 2010 1 1 [ 11 11 ] 1 1 Sg(106) 1 1 258 Sg 1 1 51 V + 209 Bi 1 1 2n 1 1 GSI 1 1 1997 1 1 [ 24 24 ] 1 1 (12) 1 1 271 Sg 1 1 48 Ca + 238 U 1 1 α 1 1 Dubna 1 1 2004 1 1 [ 25 25 ] 1 1 Bh(107) 1 1 260 Bh 1 1 52 Cr + 209 Bi 1 1 α 1 1 Berkeley 1 1 2008 1 1 [ 27 27 ] 1 1 (10) 1 1 274 Bh 1 1 48 Ca + 249 Bk 1 1 3n α 1 1 Dubna 1 1 2010 1 1 [ 11 11 ] 1 1 Hs(108) 1 1 263 Hs 1 1 56 Fe + 208 Pb 1 1 1n 1 1 Berkeley 1 1 2009 1 1 [ 28 28 ] 1 1 (12) 1 1 277 Hs 1 1 48 Ca + 244 Pu 1 1 3n α 1 1 GSI 1 1 2010 1 1 [ 29 29 ] 1 1 Mt(109) 1 1 266 Mt 1 1 58 Fe + 209 Bi 1 1 1n 1 1 GSI 1 1 1982 1 1 [ 30 30 ] 1 1 (7) 1 1 278 Mt 1 1 48 Ti + 249 Bk 1 1 3n α 1 1 Dubna 1 1 2010 1 1 [ 11 11 ] 1 1 Ds(110) 1 1 267 Ds 1 1 59 Co + 209 Bi 1 1 1n 1 1 Berkeley 1 1 1995 1 1 [ 31 31 ] 1 1 (8) 1 1 281 Ds 1 1 48 Ca + 244 Pu 1 1 3n α 1 1 Dubna 1 1 2004 1 1 [ 10 10 ] 1 1 Rg(111) 1 1 272 Rg 1 1 64 Ni + 209 Bi 1 1 1n 1 1 GSI 1 1 1995 1 1 [ 32 32 ] 1 1 (7) 1 1 282 Rg 1 1 48 Ca + 249 Bk 1 1 3n α 1 1 Dubna 1 1 2010 1 1 [ 11 11 ] 1 1 Cn(112) 1 1 277 Cn 1 1 70 Zn + 208 Pb 1 1 1n 1 1 GSI 1 1 1996 1 1 [ 19 19 ] 1 1 (6) 1 1 285 Cn 1 1 48 Ca + 244 Pu 1 1 3n α 1 1 Dubna 1 1 2004 1 1 [ 10 10 ] 1 1 Nh(113) 1 1 278 Nh 1 1 70 Zn + 209 Bi 1 1 1n 1 1 RIKEN 1 1 2004 1 1 [ 33 33 ] 1 1 (6) 1 1 286 Nh 1 1 48 Ca + 249 Bk 1 1 3n α 1 1 Dubna 1 1 2010 1 1 [ 11 11 ] 1 1 FI(114) 1 1 285 FI 1 1 48 Ca + 242 Pu 1 1 5n 1 1 Berkeley 1 1 2010 1 1 [ 8 8 ] 1 1 (5) 1 1 289 FI 1 1 48 Ca + 244 Pu 1 1 3n 1 1 Dubna 1 1 2004 1 1 [ 10 10 ] 1 1 Mc(115) 1 1 287 Mc 1 1 48 Ca + 243 Am 1 1 4n 1 1 Dubna 1 1 2004 1 1 [ 9 9 ] 1 1 (4) 1 1 290 Mc 1 1 48 Ca + 249 Bk 1 1 3n α 1 1 Dubna 1 1 2010 1 1 [ 11 11 ] 1 1 Lv(116) 1 1 290 Lv 1 1 48 Ca + 245 Cm 1 1 3n 1 1 Dubna 1 1 2004 1 1 [ 10 10 ] 1 1 (4) 1 1 293 Lv 1 1 48 Ca + 245 Cm 1 1 1n 1 1 Dubna 1 1 2004 1 1 [ 10 10 ] 1 1 Ts(117) 1 1 293 Ts 1 1 48 Ca + 249 Bk 1 1 4n 1 1 Dubna 1 1 2010 1 1 [ 11 11 ] 1 1 (2) 1 1 294 Ts 1 1 48 Ca + 249 Bk 1 1 3n 1 1 Dubna 1 1 2010 1 1 [ 11 11 ] 1 1 Og(118) 1 1 294 Og 1 1 48 Ca + 249 Cf 1 1 3n 1 1 Dubna 1 1 2006 1 1 [ 12 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 34 ]. Laboratories all over the world such as IMP [ 35 35 ], GSI [ 36 36 , 37 37 ], Dubna[ 38 38 , 39 39 ], RIKEN[ 40 40 - 42 42 40-42 ], and Lawrence Berkeley National Laboratory (LBNL)[ 43 43 , 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 44 - 46 46 44-46 ], the improved quantum molecular dynamics model[ 47 47 - 49 49 47-49 ], a dynamical approach based on Langevin equations[ 50 50 , 51 51 ], and the dinuclear system (DNS) model [ 52 52 - 56 56 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 53 , 52 52 , 56 56 - 62 62 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 1 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 <math display="block" id="M1"><mtable><mtr><mtd><msub><mi>σ</mi><mrow><mtext>ER</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>E</mi><mrow><mtext>c</mtext><mn>.</mn><mtext>m</mtext><mn>.</mn></mrow></msub></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>π</mi><msup><mi>ℏ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>μ</mi><msub><mi>E</mi><mrow><mtext>c</mtext><mn>.</mn><mtext>m</mtext><mn>.</mn></mrow></msub></mrow></mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>J</mi><mo>=</mo><mn>0</mn></mrow><mrow><msub><mi>J</mi><mrow><mtext>max</mtext></mrow></msub></mrow></munderover></mstyle><mo stretchy="false">(</mo><mn>2</mn><mi>J</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>T</mi><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mi>P</mi><mrow><mtext>CN</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><msub><mi>W</mi><mrow><mtext>sur</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> 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 52 ]. The fusion probability P CN ( E c.m. , J ) is the probability to form compound nuclei[ 63 63 , 64 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 65 ]. Capture probability 2 The capture cross-sections of the two colliding partners are expressed as <math display="block" id="M2"><mrow><msub><mi>σ</mi><mrow><mtext>cap</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>π</mi><msup><mi>ℏ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>μ</mi><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>J</mi></munder></mstyle><mrow><mo>(</mo><mrow><mn>2</mn><mi>J</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>T</mi><mrow><mo>(</mo><mrow><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mn>,</mn><mi>J</mi></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> Here, the penetration probability T ( E c.m.,J ) is evaluated by the Hill-Wheeler formula [ 66 66 ] using the barrier distribution function: <math display="block" id="M3"><mtable columnalign="left"><mtr><mtd><mi>T</mi><mrow><mo>(</mo><mrow><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mn>,</mn><mi>J</mi></mrow><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>f</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mrow><mo>{</mo><mrow><mo>−</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mrow><mi>ℏ</mi><mi>ω</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow></mfrac><mrow><mo>[</mo><mrow><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mo>−</mo><mi>B</mi><mo>−</mo><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup><mi>J</mi><mo stretchy="false">(</mo><mi>J</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mi>μ</mi><msubsup><mi>R</mi><mtext>B</mtext><mtext>2</mtext></msubsup><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>]</mo></mrow></mrow><mo>}</mo></mrow></mrow></mfrac><mtext>d</mtext><mi>B</mi><mo>,</mo></mtd></mtr></mtable></math> where <math id="M4"><mrow><mi>ℏ</mi><mi>ω</mi><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo></mrow></math> is the width of the parabolic barrier at position R B ( J ). The normalization constant is with respect to the relation <math id="M5"><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>f</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>B</mi><mo>=</mo><mn>1</mn></mrow></math> . The barrier distribution function is assumed to be in an asymmetric Gaussian form[ 52 52 , 67 67 ]: <math display="block" id="M6"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mi>exp</mi><mo stretchy="false">[</mo><mo>−</mo><mo stretchy="false">(</mo><mfrac><mrow><mi>B</mi><mo>−</mo><msub><mi>B</mi><mtext>m</mtext></msub></mrow><mrow><msub><mtext>Δ</mtext><mn>1</mn></msub></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mi>B</mi><mo><</mo><msub><mi>B</mi><mtext>m</mtext></msub><mn>,</mn></mrow></mtd><mtd><mrow/></mtd></mtr><mtr><mtd><mrow/></mtd><mtd><mrow/></mtd></mtr><mtr><mtd><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mi>exp</mi><mo stretchy="false">[</mo><mo>−</mo><mo stretchy="false">(</mo><mfrac><mrow><mi>B</mi><mo>−</mo><msub><mi>B</mi><mtext>m</mtext></msub></mrow><mrow><msub><mtext>Δ</mtext><mn>2</mn></msub></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mtext> </mtext><mtext> </mtext><mi>B</mi><mo>></mo><msub><mi>B</mi><mtext>m</mtext></msub><mn>,</mn></mrow></mtd><mtd><mrow/></mtd></mtr></mtable></mrow></mrow></mrow></math> where <math id="M7"><mrow><msub><mtext>Δ</mtext><mn>2</mn></msub></mrow></math> = (B 0 -B s )/2, <math id="M8"><mrow><msub><mtext>Δ</mtext><mn>1</mn></msub></mrow></math> = <math id="M9"><mrow><msub><mtext>Δ</mtext><mn>2</mn></msub></mrow></math> -2 MeV, B m =(B 0 +B s )/2, and B 0 and B s are the Coulomb barriers of the side-side collision and saddle-point barriers in dynamical deformations[ 67 67 ]. The nucleus-nucleus interaction potential is expressed as <math display="block" id="M10"><mrow><mi>V</mi><mo stretchy="false">(</mo><mo>{</mo><mi>α</mi><mo>}</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mo>{</mo><mi>α</mi><mo>}</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mo>{</mo><mi>α</mi><mo>}</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mi>V</mi><mrow><mtext>def</mtext></mrow></msub></mrow></math> with <math display="block" id="M11"><mrow><msub><mi>V</mi><mrow><mtext>def</mtext></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>C</mi><mn>1</mn></msub><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>β</mi><mn>1</mn></msub><mo>−</mo><msubsup><mi>β</mi><mn>1</mn><mn>0</mn></msubsup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>C</mi><mn>2</mn></msub><msup><mrow><mrow><mo>(</mo><mrow><msub><mi>β</mi><mn>2</mn></msub><mo>−</mo><msubsup><mi>β</mi><mn>2</mn><mn>0</mn></msubsup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo stretchy="false">)</mo><mn>.</mn></mrow></math> where 1 and 2 represent the projectile and target, respectively; R = R 1 + R 2 + s and s are the distances between the center and surface of the projectile and target, respectively; R 1 and R 2 are the radii of the projectile and target, respectively; <math id="M12"><mrow><msubsup><mi>β</mi><mrow><mn>1</mn><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>0</mn></msubsup></mrow></math> represents the static deformation of the projectile-target; β 1(2) represents the adjustable quadrupole deformation, which is varied to find the minimal V ({α}); and {α} stands for <math id="M13"><mrow><mo>{</mo><mi>R</mi><mn>,</mn><msub><mi>β</mi><mn>1</mn></msub><mn>,</mn><msub><mi>β</mi><mn>1</mn></msub><mn>,</mn><msub><mi>β</mi><mn>2</mn></msub><mn>,</mn><msub><mi>β</mi><mn>1</mn></msub><mn>,</mn><msub><mi>β</mi><mn>2</mn></msub><mo>}</mo></mrow></math> . To reduce the number of deformation variables, we assume that the deformation energy of the colliding system is proportional to its mass[ 67 67 ], that is, <math id="M14"><mrow><msub><mi>C</mi><mn>1</mn></msub><msubsup><mi>β</mi><mn>1</mn><mn>2</mn></msubsup><mo>/</mo><msub><mi>C</mi><mn>2</mn></msub><msubsup><mi>β</mi><mn>2</mn><mn>2</mn></msubsup><mo>=</mo><msub><mi>A</mi><mn>1</mn></msub><mo>/</mo><msub><mi>A</mi><mn>2</mn></msub></mrow></math> . Thus, only one deformation parameter, β = β 1 + β 2 , is required. The stiffness parameters C i (i=1,2) are calculated using the liquid-drop model[ 68 68 ] through the following parameterization formula: <math display="block" id="M15"><mrow><msub><mi>C</mi><mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mi>λ</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mrow><mo>[</mo><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msubsup><mi>R</mi><mi>i</mi><mn>2</mn></msubsup><mi>σ</mi><mo>−</mo><mfrac><mn>3</mn><mrow><mn>2</mn><mi>π</mi></mrow></mfrac><mfrac><mrow><msup><mi>Z</mi><mn>2</mn></msup><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><msub><mi>R</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mn>2</mn><mi>λ</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>]</mo></mrow></mrow></math> where R i is the radius of the spheroidal nucleus given by R i =1.18 <math id="M16"><mrow><msubsup><mi>A</mi><mtext>i</mtext><mrow><mtext>1/3</mtext></mrow></msubsup></mrow></math> ( 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 <math id="M17"><mrow><mn>4</mn><msubsup><mi>π</mi><mi>i</mi><mn>2</mn></msubsup></mrow></math> σ = <math id="M18"><mrow><msub><mi>a</mi><mi>s</mi></msub><msubsup><mi>A</mi><mi>i</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></math> , where a s = 18.32 MeV is the surface energy. The nuclear potential is calculated using the double-folding method [ 63 63 - 65 65 63-65 ]: <math display="block" id="M19"><mtable><mtr><mtd><msub><mi>V</mi><mtext>N</mtext></msub><mo>=</mo><msub><mi>C</mi><mn>0</mn></msub><mrow><mo>{</mo><mrow><mfrac><mrow><msub><mi>F</mi><mrow><mtext>in</mtext></mrow></msub><mo>−</mo><msub><mi>F</mi><mrow><mtext>ex</mtext></mrow></msub></mrow><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow></mfrac><mrow><mo>[</mo><mrow><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msubsup><mi>ρ</mi><mn>1</mn><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><msub><mi>ρ</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo>−</mo><mi>R</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>r</mi></mrow></mrow></mrow></mrow></mtd></mtr><mtr><mtd><mo>+</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msub><mi>ρ</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mrow><mrow><mrow><mrow><msubsup><mi>ρ</mi><mn>2</mn><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo>−</mo><mi>R</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>R</mi></mrow><mo>]</mo></mrow><mo>+</mo><msub><mi>F</mi><mrow><mtext>e</mtext><mi>x</mi></mrow></msub><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msub><mi>ρ</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><msub><mi>ρ</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo>−</mo><mi>R</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>r</mi></mrow><mo>}</mo></mrow></mtd></mtr></mtable></math> where <math display="block" id="M20"><mrow><msub><mi>F</mi><mrow><mtext>in(ex)</mtext></mrow></msub><mo>=</mo><msub><mi>f</mi><mrow><mtext>in(ex)</mtext></mrow></msub><mo>+</mo><msubsup><mi>f</mi><mrow><mtext>in(ex)</mtext></mrow><mn>,</mn></msubsup><mfrac><mrow><msub><mi>N</mi><mn>1</mn></msub><mo>−</mo><msub><mi>Z</mi><mn>1</mn></msub></mrow><mrow><msub><mi>A</mi><mn>1</mn></msub></mrow></mfrac><mfrac><mrow><msub><mi>N</mi><mn>2</mn></msub><mo>−</mo><msub><mi>Z</mi><mn>2</mn></msub></mrow><mrow><msub><mi>A</mi><mn>2</mn></msub></mrow></mfrac><mn>.</mn></mrow></math> Note the dependence on the nuclear density and orientation of the deformed colliding partners. We set the following parameter values in our calculations: C 0 = 300 MeV fm 3 , f in = 0.09, f ex =-2.59, <math id="M21"><mrow><msubsup><mi>f</mi><mrow><mtext>i</mtext><mi>n</mi></mrow><mn>,</mn></msubsup></mrow></math> = 0.42, <math id="M22"><mrow><msubsup><mi>f</mi><mrow><mtext>e</mtext><mi>x</mi></mrow><mn>,</mn></msubsup></mrow></math> = 0.54, and ρ 0 = 0.16 fm -3 . The Woods-Saxon density distribution is expressed as <math display="block" id="M23"><mrow><msub><mi>ρ</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>−</mo><msub><mi>ℜ</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mo>/</mo><msub><mi>a</mi><mn>1</mn></msub></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math> and <math display="block" id="M24"><mrow><msub><mi>ρ</mi><mn>2</mn></msub><mrow><mo>(</mo><mrow><mi>r</mi><mo>−</mo><mi>R</mi></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><msub><mi>ρ</mi><mn>0</mn></msub></mrow><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mrow><mo>[</mo><mrow><mrow><mo>(</mo><mrow><mrow><mo>|</mo><mrow><mi>r</mi><mo>−</mo><mi>R</mi></mrow><mo>|</mo></mrow><mo>−</mo><msub><mi>ℜ</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow><mo>/</mo><msub><mi>a</mi><mn>2</mn></msub></mrow><mo>]</mo></mrow></mrow></mfrac></mrow></math> where <math id="M25"><mrow><msub><mi>ℜ</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1,2</mn><mo stretchy="false">)</mo></mrow></math> denotes the surface radii of the nuclei given by <math id="M26"><mrow><msub><mi>ℜ</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></math> = <math id="M27"><mrow><mi>ℜ</mi><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><msub><mi>β</mi><mi>i</mi></msub><msub><mi>Y</mi><mrow><mn>20</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>θ</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow></mrow></math> , where R i is the spheroidal nuclei radius and a i 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 69 ]: <math display="block" id="M28"><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mo>{</mo><mi>α</mi><mo>}</mo><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>Z</mi><mn>1</mn></msub><msub><mi>Z</mi><mn>2</mn></msub><msup><mi>e</mi><mn>2</mn></msup></mrow><mi>r</mi></mfrac></mtd></mtr><mtr><mtd><mo>+</mo><msup><mrow><mo>(</mo><mrow><mfrac><mn>9</mn><mrow><mn>20</mn><mi>π</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>Z</mi><mn>1</mn></msub><msub><mi>Z</mi><mn>2</mn></msub><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><msup><mi>r</mi><mn>3</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>2</mn></munderover></mstyle><msubsup><mi>R</mi><mi>i</mi><mn>2</mn></msubsup><msub><mi>β</mi><mtext>i</mtext></msub><msub><mi>P</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mi>cos</mi><msub><mi>θ</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>+</mo><mrow><mo>(</mo><mrow><mfrac><mn>3</mn><mrow><mn>7</mn><mi>π</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>Z</mi><mn>1</mn></msub><msub><mi>Z</mi><mn>2</mn></msub><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><msup><mi>r</mi><mn>3</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>2</mn></munderover></mstyle><msubsup><mi>R</mi><mi>i</mi><mn>2</mn></msubsup><msup><mrow><mo>(</mo><mrow><msub><mi>β</mi><mi>i</mi></msub><msub><mi>P</mi><mn>2</mn></msub><mi>cos</mi><msub><mi>θ</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd></mtr></mtable></math> where θ i , β i , R i , and P 2 (cos θ i ) 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. Fusion probability 2 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 ( Z 1 , N 1 , E 1 , t ) contains the proton number, neutron numbers of Z 1 and N 1 , and internal excitation energy of E 1 for a given fragment A 1 . The master equation is [ 64 64 , 57 57 , 70 70 ] <math display="block" id="M29"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mtext>d</mtext><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi></mrow><mo>)</mo></mrow></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><msup><mi>Z</mi><mo>′</mo></msup><mn>1</mn></msub></mrow></munder></mstyle><msub><mi>W</mi><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mo>;</mo><msub><msup><mi>Z</mi><mo>′</mo></msup><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mi>d</mi><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub></mrow></msub><mi>P</mi><mrow><mo>(</mo><mrow><msub><msup><mi>Z</mi><mo>′</mo></msup><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><msup><mi>E</mi><mo>′</mo></msup><mn>1</mn></msub><mn>,</mn><mi>t</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>d</mi><mrow><msub><msup><mi>Z</mi><mo>′</mo></msup><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub></mrow></msub><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi></mrow><mo>)</mo></mrow><mo stretchy="false">]</mo><mo>+</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><msup><mi>N</mi><mo>′</mo></msup><mn>1</mn></msub></mrow></munder></mstyle><msub><mi>W</mi><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mo>;</mo><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><msup><mi>N</mi><mo>′</mo></msup><mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><msub><mi>d</mi><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub></mrow></msub><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><msup><mi>N</mi><mo>′</mo></msup><mn>1</mn></msub><mn>,</mn><msub><msup><mi>E</mi><mo>′</mo></msup><mn>1</mn></msub><mn>,</mn><mi>t</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>d</mi><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><msup><mi>N</mi><mo>′</mo></msup><mn>1</mn></msub></mrow></msub><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi></mrow><mo>)</mo></mrow><mo stretchy="false">]</mo><mo>−</mo></mtd></mtr><mtr><mtd><mrow><mo>[</mo><mrow><msubsup><mi>Λ</mi><mrow><msub><mi>A</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi></mrow><mrow><mtext>qf</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>Θ</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>Λ</mi><mrow><msub><mi>A</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi></mrow><mrow><mtext>fis</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>Θ</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mi>P</mi><mrow><mo>(</mo><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi></mrow><mo>)</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> where <math id="M30"><mrow><msub><mi>W</mi><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><msup><mi>Z</mi><mo>′</mo></msup><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub></mrow></msub></mrow></math> ( <math id="M31"><mrow><msub><mi>W</mi><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><msup><mi>N</mi><mo>′</mo></msup><mn>1</mn></msub><mn>,</mn><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub></mrow></msub></mrow></math> ) denotes the mean transition probability from the channel ( <math id="M32"><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub></mrow></math> ) to ( <math id="M33"><mrow><msub><msup><mi>Z</mi><mo>′</mo></msup><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><msup><mi>E</mi><mo>′</mo></msup><mn>1</mn></msub></mrow></math> ) [or ( <math id="M34"><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub></mrow></math> ) to ( <math id="M35"><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><msup><mi>N</mi><mo>′</mo></msup><mn>1</mn></msub><mn>,</mn><msub><msup><mi>E</mi><mo>′</mo></msup><mn>1</mn></msub></mrow></math> )]; <math id="M36"><mrow><msub><mi>d</mi><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub></mrow></msub></mrow></math> denotes the microscopic dimension corresponding to the macroscopic state ( <math id="M37"><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub></mrow></math> ). The sum contains all possible numbers of protons and neutrons for fragment ( <math id="M38"><mrow><msub><msup><mi>Z</mi><mo>′</mo></msup><mn>1</mn></msub></mrow></math> , <math id="M39"><mrow><msub><msup><mi>N</mi><mo>′</mo></msup><mn>1</mn></msub></mrow></math> ). However, only one nucleon transfer at one time is assumed in the model with relations <math id="M40"><mrow><msub><msup><mi>Z</mi><mo>′</mo></msup><mn>1</mn></msub></mrow></math> = Z 1 ± 1, and <math id="M41"><mrow><msub><msup><mi>N</mi><mo>′</mo></msup><mn>1</mn></msub></mrow></math> = N 1 ± 1. The excitation energy E 1 is the local excitation energy <math id="M42"><mrow><mi>�</mi><mo>#</mo><mi>x</mi><mn>03</mn><mi>B</mi><msubsup><mrow><mn>5</mn><mo>;</mo></mrow><mn>1</mn><mo>*</mo></msubsup></mrow></math> for fragment ( <math id="M43"><mrow><msub><msup><mi>Z</mi><mo>′</mo></msup><mn>1</mn></msub></mrow></math> , <math id="M44"><mrow><msub><msup><mi>N</mi><mo>′</mo></msup><mn>1</mn></msub></mrow></math> ), which is derived by the dissipation of the relative motion along with the potentical energy surface (PES) of the DNS [ 71 71 ]. The time of the dissipation process is evaluated by the parameterization classical deflection function [ 72 72 ]. The motion of nucleons in the interaction potential is governed by the single-particle Hamiltonian: <math display="block" id="M45"><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>H</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>V</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></math> where the total single-particle energy and interaction potential are <math display="block" id="M46"><mrow><msub><mi>H</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>K</mi></munder></mstyle><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>ν</mi><mi>K</mi></msub></mrow></munder></mstyle><msub><mi>ε</mi><mrow><msub><mi>ν</mi><mi>K</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msubsup><mi>α</mi><mrow><msub><mi>ν</mi><mi>K</mi></msub></mrow><mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><msub><mi>α</mi><mrow><msub><mi>ν</mi><mi>K</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></math> <math display="block" id="M47"><mtable><mtr><mtd><mi>V</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>K</mi><mn>,</mn><msup><mi>K</mi><mo>'</mo></msup></mrow></munder></mstyle><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msub><mi>α</mi><msup><mi>K</mi><mo>′</mo></msup></msub><msub><mi>β</mi><msup><mi>K</mi><mo>′</mo></msup></msub></mrow></munder><mrow><msub><mi>u</mi><mrow><msub><mi>α</mi><msup><mi>K</mi><mo>′</mo></msup></msub><msub><mi>β</mi><msup><mi>K</mi><mo>′</mo></msup></msub></mrow></msub></mrow></mstyle><msubsup><mi>α</mi><mrow><msub><mi>α</mi><mi>K</mi></msub></mrow><mo>+</mo></msubsup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msub><mi>α</mi><mrow><msub><mi>β</mi><mi>K</mi></msub></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>K</mi><mn>,</mn><msup><mi>K</mi><mo>′</mo></msup></mrow></munder></mstyle><msub><mi>V</mi><mrow><mi>K</mi><mn>,</mn><msup><mi>K</mi><mo>′</mo></msup></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> where ε νK and <math id="M48"><mrow><msub><mi>u</mi><mrow><msub><mi>α</mi><mi>K</mi></msub><mn>,</mn><msub><mi>β</mi><msup><mi>K</mi><mo>′</mo></msup></msub></mrow></msub></mrow></math> 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 <math display="block" id="M49"><mtable columnalign="left"><mtr><mtd><msub><mi>u</mi><mrow><msub><mi>α</mi><mi>K</mi></msub><mn>,</mn><msub><mi>β</mi><msup><mi>K</mi><mo>′</mo></msup></msub></mrow></msub><mo>=</mo><msub><mi>U</mi><mrow><mi>K</mi><mn>,</mn><msup><mi>K</mi><mo>′</mo></msup></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><mrow><mo>{</mo><mrow><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>ε</mi><mrow><msub><mi>α</mi><mtext>K</mtext></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>ε</mi><mrow><msub><mi>β</mi><mtext>K</mtext></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mtext>Δ</mtext><mrow><mi>K</mi><mn>,</mn><msup><mi>K</mi><mo>′</mo></msup></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow><mo>−</mo><msub><mi>δ</mi><mrow><msub><mi>α</mi><mi>K</mi></msub><mn>,</mn><msub><mi>β</mi><msup><mi>K</mi><mo>′</mo></msup></msub></mrow></msub></mrow><mo>}</mo></mrow></mtd></mtr></mtable></math> where <math id="M50"><mrow><msub><mi>U</mi><mrow><mi>K</mi><mn>,</mn><msup><mi>K</mi><mo>′</mo></msup></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></math> and <math id="M51"><mrow><msub><mi>δ</mi><mrow><msub><mi>α</mi><mi>K</mi></msub><mn>,</mn><msub><mi>β</mi><msup><mi>K</mi><mo>′</mo></msup></msub></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></math> are described in Ref.[ 73 73 ]. The proton transition probability is microscopically derived from <math display="block" id="M52"><mtable columnalign="left"><mtr><mtd><msub><mi>W</mi><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mo>;</mo><msubsup><mi>Z</mi><mn>1</mn><mo>'</mo></msubsup><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>τ</mi><mrow><mtext>mem</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub><mo>;</mo><msubsup><mi>Z</mi><mn>1</mn><mo>'</mo></msubsup><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msubsup><mi>E</mi><mn>1</mn><mo>'</mo></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msub><mi>d</mi><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub></mrow></msub><msub><mi>d</mi><mrow><msubsup><mi>Z</mi><mn>1</mn><mo>'</mo></msubsup><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub></mrow></msub><msup><mi>ℏ</mi><mn>2</mn></msup></mrow></mfrac></mtd></mtr><mtr><mtd><mo>×</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>i</mi><msup><mi>i</mi><mo>'</mo></msup></mrow></munder></mstyle><msup><mrow><mo>|</mo><mrow><mrow><mo>〈</mo><mrow><msubsup><mi>Z</mi><mn>1</mn><mo>'</mo></msubsup><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msubsup><mi>E</mi><mn>1</mn><mo>'</mo></msubsup><mn>,</mn><msup><mi>i</mi><mo>'</mo></msup><mrow><mo>〈</mo><mi>V</mi><mo>〉</mo></mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub><mn>,</mn><mi>i</mi></mrow><mo>〉</mo></mrow></mrow><mo>|</mo></mrow><mn>2</mn></msup><mn>.</mn></mtd></mtr></mtable></math> The neutron transition probability has a similar formula. The memory time and interaction elements V are described in Ref.[ 63 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 73 , 74 74 ] <math display="block" id="M53"><mtable><mtr><mtd><msubsup><mtext>Λ</mtext><mrow><msub><mi>A</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi></mrow><mrow><mtext>qf</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>Θ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>ω</mi><mrow><mn>2</mn><mi>π</mi><msup><mi>ω</mi><mrow><msub><mi>B</mi><mrow><mtext>qf</mtext></mrow></msub></mrow></msup></mrow></mfrac><mrow><mo>[</mo><mrow><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>Γ</mi><mrow><mn>2</mn><mi>ℏ</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>ω</mi><mrow><msub><mi>B</mi><mrow><mtext>qf</mtext></mrow></msub></mrow></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt><mo>−</mo><mfrac><mi>Γ</mi><mrow><mn>2</mn><mi>ℏ</mi></mrow></mfrac></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>×</mo><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mfrac><mrow><msub><mi>B</mi><mrow><mtext>qf</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mi>Θ</mi><mrow><mo>(</mo><mrow><msub><mi>A</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow><mo>]</mo></mrow></mtd></mtr></mtable></math> where <math id="M54"><mrow><msub><mi>B</mi><mrow><mtext>qf</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></math> is the quasi-fission barrier; ω and <math id="M55"><mrow><msup><mi>ω</mi><mrow><msub><mtext>B</mtext><mrow><mtext>qf</mtext></mrow></msub></mrow></msup></mrow></math> are the frequencies of the harmonic oscillator approximation at the bottom and top of the interaction potential pocket, which are constants expressed as <math id="M56"><mrow><mi>ℏ</mi><msup><mi>ω</mi><mrow><msub><mi>B</mi><mrow><mtext>qf</mtext></mrow></msub></mrow></msup></mrow></math> = 2.0 MeV and <math id="M57"><mrow><mi>ℏ</mi><mi>ω</mi></mrow></math> = 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., <math id="M58"><mrow><mi>Θ</mi><mo>=</mo><mo stretchy="false">(</mo><msup><mi>ε</mi><mo>*</mo></msup><mo>/</mo><mo stretchy="false">(</mo><mi>A</mi><msup><mrow><mo>/</mo><mn>12</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math> . In the nuclear collision process, heavy fragments might lead to fission; the fission probability is calculated by the Kramers formula: <math display="block" id="M59"><mtable><mtr><mtd><msup><mtext>Λ</mtext><mrow><mtext>f</mtext><mi>i</mi><mi>s</mi></mrow></msup><msub><mrow/><mrow><msub><mtext>A</mtext><mtext>1</mtext></msub><msub><mrow><mtext>,E</mtext></mrow><mtext>1</mtext></msub><mtext>,t</mtext></mrow></msub><mo stretchy="false">(</mo><mi>Θ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>ω</mi><mrow><mtext>g</mtext><mn>.</mn><mi>s</mi></mrow></msub></mrow><mrow><mn>2</mn><mi>π</mi><msub><mi>ω</mi><mtext>f</mtext></msub></mrow></mfrac><mrow><mo>[</mo><mrow><msqrt><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>Γ</mi><mn>0</mn></msub></mrow><mrow><mn>2</mn><mi>ℏ</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><msub><mi>ω</mi><mtext>f</mtext></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo>−</mo><mfrac><mrow><msub><mi>Γ</mi><mn>0</mn></msub></mrow><mrow><mn>2</mn><mi>ℏ</mi></mrow></mfrac></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mo>×</mo><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mfrac><mrow><msub><mi>B</mi><mtext>f</mtext></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>]</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> 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 A 1 , respectively, which were set as <math id="M60"><mrow><mi>ℏ</mi><msub><mi>ω</mi><mrow><mtext>g</mtext><mtext>.s</mtext><mtext>.</mtext></mrow></msub></mrow></math> = <math id="M61"><mrow><mi>ℏ</mi><msub><mi>ω</mi><mtext>f</mtext></msub></mrow></math> = 1.0 MeV and Γ 0 = 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 ε K 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: <math display="block" id="M62"><mrow><mtext>Δ</mtext><msub><mi>ε</mi><mi>K</mi></msub><mo>=</mo><msqrt><mrow><mfrac><mrow><mn>4</mn><msubsup><mi>ε</mi><mi>K</mi><mo>*</mo></msubsup></mrow><mrow><msub><mi>g</mi><mi>K</mi></msub></mrow></mfrac></mrow></msqrt><mn>,</mn><mtext> </mtext><msubsup><mi>ε</mi><mi>K</mi><mo>*</mo></msubsup><mo>=</mo><msup><mi>ε</mi><mo>*</mo></msup><mfrac><mrow><msub><mi>A</mi><mi>K</mi></msub></mrow><mi>A</mi></mfrac><mn>,</mn><mtext> </mtext><msub><mi>g</mi><mi>K</mi></msub><mo>=</mo><msub><mi>A</mi><mi>K</mi></msub><mo>/</mo><mn>12,</mn></mrow></math> where <math id="M63"><mrow><msup><mtext>ε</mtext><mo>*</mo></msup></mrow></math> is the local excitation energy of the DNS, which provides the excitation energy for the mean transition probability. There are N K = <math id="M64"><mrow><msub><mi>g</mi><mtext>K</mtext></msub><mtext>Δ</mtext><msub><mi>ε</mi><mi>K</mi></msub></mrow></math> valence states and m K = N K /2 valence nucleons in the valence space Δ ε K , leading to dimensions <math display="block" id="M65"><mrow><mi>d</mi><mo stretchy="false">(</mo><msub><mi>m</mi><mn>1</mn></msub><mn>,</mn><msub><mi>m</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>N</mi><mn>1</mn></msub></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>m</mi><mn>1</mn></msub></mrow></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>N</mi><mn>2</mn></msub></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>m</mi><mn>2</mn></msub></mrow></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mn>.</mn></mrow></math> The local excitation energy is expressed as <math display="block" id="M66"><mrow><msup><mi>ε</mi><mo>*</mo></msup><mo>=</mo><msub><mi>E</mi><mi>x</mi></msub><mo>−</mo><mo stretchy="false">(</mo><msub><mi>U</mi><mrow><mtext>d</mtext><mi>r</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mn>,</mn><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>U</mi><mrow><mtext>d</mtext><mi>r</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>P</mtext></msub><mn>,</mn><msub><mi>A</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where <math id="M67"><mrow><msub><mi>U</mi><mrow><mtext>d</mtext><mi>r</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mn>,</mn><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></math> and <math id="M68"><mrow><msub><mi>U</mi><mrow><mtext>d</mtext><mi>r</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mtext>P</mtext></msub><mn>,</mn><msub><mi>A</mi><mtext>T</mtext></msub><mo stretchy="false">)</mo></mrow></math> are the driving potentials of fragments A 1 , A 2 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 64 ]. The PES of the DNS is expressed as <math display="block" id="M69"><mtable columnalign="left"><mtr><mtd><msub><mi>U</mi><mrow><mtext>d</mtext><mi>r</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mn>,</mn><msub><mi>A</mi><mn>2</mn></msub><mo>;</mo><mi>J</mi><mn>,</mn><msub><mi>θ</mi><mn>1</mn></msub><mn>,</mn><msub><mi>θ</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>B</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>β</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>B</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>N</mi><mn>2</mn></msub><mn>,</mn><msub><mi>Z</mi><mn>2</mn></msub><mn>,</mn><msub><mi>β</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>B</mi><mrow><mtext>C</mtext><mi>N</mi></mrow></msub><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>Z</mi><mn>,</mn><mi>β</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>U</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>Z</mi><mn>2</mn></msub><mn>,</mn><msub><mi>β</mi><mn>1</mn></msub><mn>,</mn><msub><mi>β</mi><mn>2</mn></msub><mn>,</mn><mi>R</mi><mn>,</mn><msub><mi>θ</mi><mn>1</mn></msub><mn>,</mn><msub><mi>θ</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>U</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>Z</mi><mn>2</mn></msub><mn>,</mn><msub><mi>β</mi><mn>1</mn></msub><mn>,</mn><msub><mi>β</mi><mn>2</mn></msub><mn>,</mn><mi>R</mi><mn>,</mn><msub><mi>θ</mi><mn>1</mn></msub><mn>,</mn><msub><mi>θ</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable></math> where B i ( i = 1, 2) and B CN are the negative binding energies of fragment A i and compound nucleus A = A 1 + A 2 , 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: <math display="block" id="M70"><mrow><msub><mi>P</mi><mrow><mtext>CN</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mn>,</mn><mi>J</mi><mn>,</mn><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>Z</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>Z</mi><mrow><mtext>BG</mtext></mrow></msub></mrow></munderover></mstyle><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mrow><mtext>BG</mtext></mrow></msub></mrow></munderover></mstyle><mi>P</mi><mo stretchy="false">(</mo><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>E</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo><mn>,</mn><msub><mi>τ</mi><mrow><mtext>int</mtext></mrow></msub><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Here, the interaction time τ int (E c.m. , J, B) is obtained from the deflection function method [ 71 71 ]. We calculated the fusion probability as <math display="block" id="M71"><mrow><msub><mi>P</mi><mrow><mtext>CN</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mi>f</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><msub><mi>P</mi><mrow><mtext>CN</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mn>,</mn><mi>J</mi><mn>,</mn><mi>B</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>B</mi><mn>.</mn></mrow></math> The Coulomb barrier distribution function f ( B ) is given by Eq. 4. Therefore, the fusion cross-section is expressed as <math display="block" id="M72"><mrow><msub><mi>σ</mi><mrow><mtext>fus</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>σ</mi><mrow><mtext>cap</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mo stretchy="false">)</mo><msub><mi>P</mi><mrow><mtext>CN</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Survival probability 2 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 75 , 57 57 , 70 70 ] <math display="block" id="M73"><mtable columnalign="left"><mtr><mtd><msub><mi>W</mi><mrow><mtext>sur</mtext></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><mtext>C</mtext><mi>N</mi></mrow><mo>*</mo></msubsup><mn>,</mn><mi>x</mi><mn>,</mn><mi>y</mi><mn>,</mn><mi>z</mi><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><mtext>C</mtext><mi>N</mi></mrow><mo>*</mo></msubsup><mn>,</mn><mi>x</mi><mn>,</mn><mi>y</mi><mn>,</mn><mi>z</mi><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>×</mo><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>x</mi></munderover></mstyle><mfrac><mrow><msub><mi>Γ</mi><mtext>n</mtext></msub><mo stretchy="false">(</mo><msubsup><mi>E</mi><mi>i</mi><mo>*</mo></msubsup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>Γ</mi><mrow><mtext>tot</mtext></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>E</mi><mi>i</mi><mo>*</mo></msubsup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo></mrow></mfrac><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>y</mi></munderover></mstyle><mfrac><mrow><msub><mi>Γ</mi><mtext>p</mtext></msub><mo stretchy="false">(</mo><msubsup><mi>E</mi><mi>j</mi><mo>*</mo></msubsup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>Γ</mi><mrow><mtext>tot</mtext></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>E</mi><mi>i</mi><mo>*</mo></msubsup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo></mrow></mfrac><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>z</mi></munderover></mstyle><mfrac><mrow><msub><mi>Γ</mi><mi>α</mi></msub><mo stretchy="false">(</mo><msubsup><mi>E</mi><mi>k</mi><mo>*</mo></msubsup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>Γ</mi><mrow><mtext>tot</mtext></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>E</mi><mi>k</mi><mo>*</mo></msubsup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr></mtable></math> where <math id="M74"><mrow><msubsup><mi>E</mi><mrow><mtext>CN</mtext></mrow><mo>*</mo></msubsup></mrow></math> and J denote the excitation energy and spin of the excited nucleus, respectively. The total width <math id="M75"><mrow><msub><mi>Γ</mi><mrow><mtext>tot</mtext></mrow></msub></mrow></math> is the sum of the partial widths of particle evaporation, γ -rays, and fission. The excitation energy <math id="M76"><mrow><msubsup><mi>E</mi><mi>s</mi><mo>*</mo></msubsup></mrow></math> before evaporating the s -th particles is evaluated by <math display="block" id="M77"><mrow><msubsup><mi>E</mi><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow><mo>*</mo></msubsup><mo>=</mo><msubsup><mi>E</mi><mi>s</mi><mo>*</mo></msubsup><mo>−</mo><msubsup><mi>B</mi><mi>i</mi><mtext>n</mtext></msubsup><mo>−</mo><msubsup><mi>B</mi><mi>j</mi><mtext>p</mtext></msubsup><mo>−</mo><msubsup><mi>B</mi><mi>k</mi><mi>α</mi></msubsup><mo>−</mo><mn>2</mn><msub><mi>T</mi><mi>s</mi></msub><mn>,</mn></mrow></math> with initial condition <math id="M78"><mrow><msubsup><mi>E</mi><mi>i</mi><mo>*</mo></msubsup></mrow></math> = <math id="M79"><mrow><msubsup><mi>E</mi><mrow><mtext>C</mtext><mi>N</mi></mrow><mo>*</mo></msubsup></mrow></math> and s = i + j + k ; <math id="M80"><mrow><msubsup><mi>B</mi><mi>i</mi><mtext>n</mtext></msubsup></mrow></math> , <math id="M81"><mrow><msubsup><mi>B</mi><mi>j</mi><mtext>p</mtext></msubsup></mrow></math> , and <math id="M82"><mrow><msubsup><mi>B</mi><mi>k</mi><mi>α</mi></msubsup></mrow></math> denote the separation energies of the i -th neutron, j -th proton, and k -th alpha, respectively. The nuclear temperature T i is defined by <math id="M83"><mrow><msubsup><mi>E</mi><mi>i</mi><mo>*</mo></msubsup><mo>=</mo><mi>α</mi><msubsup><mi>T</mi><mi>i</mi><mn>2</mn></msubsup><mo>−</mo><msub><mi>T</mi><mi>i</mi></msub></mrow></math> 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 75 ]. We set <math id="M84"><mrow><msup><mi>E</mi><mo>*</mo></msup><mo>−</mo><msub><mi>B</mi><mi>v</mi></msub><mo>−</mo><mi>δ</mi><mo>−</mo><msub><mi>δ</mi><mtext>n</mtext></msub></mrow></math> to the term <math id="M85"><mi>ϱ</mi></math> . The widths of particles decay are evaluated using the Weisskopf evaporation theory as <math display="block" id="M86"><mtable columnalign="left"><mtr><mtd><msub><mi>Γ</mi><mi>v</mi></msub><mo stretchy="false">(</mo><msup><mi>E</mi><mo>*</mo></msup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo><mrow><mn>2</mn><msub><mi>s</mi><mi>v</mi></msub><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mfrac><mrow><msub><mi>m</mi><mi>v</mi></msub></mrow><mrow><msup><mi>π</mi><mn>2</mn></msup><msup><mi>ℏ</mi><mn>2</mn></msup><mi>ρ</mi><mo stretchy="false">(</mo><msup><mi>E</mi><mo>*</mo></msup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo></mrow></mfrac></mtd></mtr><mtr><mtd><mo>×</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>ϱ</mi><mo>−</mo><mfrac><mn>1</mn><mi>a</mi></mfrac></mrow></msubsup></mrow></mstyle><mi>ε</mi><mi>ρ</mi><mrow><mo>(</mo><mrow><mi>ϱ</mi><mo>+</mo><mi>δ</mi><mo>−</mo><msub><mi>E</mi><mrow><mtext>rot</mtext></mrow></msub><mo>−</mo><mi>ε</mi><mn>,</mn><mi>J</mi></mrow><mo>)</mo></mrow><msub><mi>σ</mi><mrow><mtext>inv</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ε</mi><mo stretchy="false">)</mo><mtext>d</mtext><mi>ε</mi><mn>.</mn></mtd></mtr></mtable></math> Here, s v , m v , and B v are the spin, mass, and binding energy of the particle, respectively. The pairing correction energy δ was set to be <math id="M87"><mrow><mn>12</mn><mo>/</mo><msqrt><mi>A</mi></msqrt></mrow></math> , 0, and <math id="M88"><mrow><mo>−</mo><mn>12</mn><mo>/</mo><msqrt><mi>A</mi></msqrt></mrow></math> for even-even, even-odd, and odd-odd nuclei, respectively. The inverse cross-section is expressed as <math id="M89"><mrow><msub><mi>σ</mi><mrow><mtext>inv</mtext></mrow></msub><mo>=</mo><mi>π</mi><msubsup><mi>R</mi><mi>ν</mi><mn>2</mn></msubsup><mi>T</mi><mo stretchy="false">(</mo><mi>ν</mi><mo stretchy="false">)</mo></mrow></math> . The penetration probability was set to 1 for neutrons and <math id="M90"><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>ν</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">(</mo><msub><mi>V</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mi>ν</mi><mo stretchy="false">)</mo><mo>−</mo><mi>ε</mi><msup><mrow><mo stretchy="false">)</mo><mo>/</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> for charged particles with <math id="M91"><mrow><mi>ℏ</mi><mi>ω</mi><mo>=</mo><mn>5</mn></mrow></math> and 8 MeV for proton and α , respectively. The Coulomb barrier of the emitting charge particles and daughter nuclei is expressed as <math display="block" id="M92"><mrow><msub><mi>V</mi><mtext>C</mtext></msub><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><mrow><msub><mi>Z</mi><mrow><mtext>C</mtext><mi>N</mi></mrow></msub><mo>−</mo><mi>i</mi></mrow><mo>)</mo></mrow><msub><mi>Z</mi><mi>i</mi></msub><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><msub><mi>r</mi><mi>i</mi></msub><mrow><mo>(</mo><mrow><msubsup><mi>A</mi><mrow><mtext>C</mtext><mi>N</mi><mo>−</mo><mi>i</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><mo>+</mo><msubsup><mi>A</mi><mi>i</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow><mo>)</mo></mrow></mrow></mfrac><mo>.</mo></mrow></math> 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 76 ]. The fission width was calculated using the Bohr-Wheeler formula, as in Ref.[ 63 63 , 64 64 ]. We set <math id="M93"><mrow><msup><mi>E</mi><mo>*</mo></msup><mo>−</mo><msub><mi>B</mi><mtext>f</mtext></msub><mo>−</mo><msub><mi>E</mi><mrow><mtext>r</mtext><mi>o</mi><mi>t</mi></mrow></msub><mo>−</mo><mi>δ</mi><mo>−</mo><msub><mi>δ</mi><mtext>f</mtext></msub></mrow></math> to the term κ . <math display="block" id="M94"><mrow><msub><mi>Γ</mi><mtext>f</mtext></msub><mo stretchy="false">(</mo><msup><mi>E</mi><mo>*</mo></msup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>π</mi><msub><mi>ρ</mi><mtext>f</mtext></msub><mo stretchy="false">(</mo><msup><mi>E</mi><mo>*</mo></msup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>κ</mi><mo>−</mo><mfrac><mn>1</mn><mrow><msub><mi>α</mi><mtext>f</mtext></msub></mrow></mfrac></mrow></msubsup></mrow></mstyle><mfrac><mrow><msub><mi>ρ</mi><mtext>f</mtext></msub><mrow><mo>(</mo><mrow><mi>κ</mi><mo>−</mo><mi>ε</mi><mo>+</mo><mi>δ</mi><mn>,</mn><mi>J</mi></mrow><mo>)</mo></mrow><mtext>d</mtext><mi>ε</mi></mrow><mrow><mn>1</mn><mo>+</mo><mtext>exp</mtext><mrow><mo>[</mo><mrow><mo>−</mo><mn>2</mn><mi>π</mi><mrow><mo>(</mo><mrow><mi>κ</mi><mo>−</mo><mi>ε</mi><mo>+</mo><mi>δ</mi><mo>+</mo><msub><mi>δ</mi><mtext>f</mtext></msub></mrow><mo>)</mo></mrow><mo>/</mo><mi>ℏ</mi><mi>ω</mi></mrow><mo>]</mo></mrow></mrow></mfrac></mrow></math> For heavy fragments, the fission width is usually set as α 2.2 MeV [ 77 77 ], and δ f denotes the pairing correction for the fission barrier. The fission barrier is divided into microscopic and macroscopic parts: <math display="block" id="M95"><mrow><msub><mi>B</mi><mtext>f</mtext></msub><mo stretchy="false">(</mo><msup><mi>E</mi><mo>*</mo></msup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>B</mi><mtext>f</mtext><mrow><mtext>LD</mtext></mrow></msubsup><mo>+</mo><msubsup><mi>B</mi><mtext>f</mtext><mtext>M</mtext></msubsup><mo stretchy="false">(</mo><msup><mi>E</mi><mo>*</mo></msup><mo>=</mo><mn>0,</mn><mi>J</mi><mo stretchy="false">)</mo><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><msup><mi>E</mi><mo>*</mo></msup><mo>/</mo><msub><mi>E</mi><mtext>D</mtext></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where the macroscopic part is derived from the liquid-drop model: <math display="block" id="M96"><mrow><msubsup><mi>B</mi><mtext>f</mtext><mrow><mtext>LD</mtext></mrow></msubsup><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>0.38</mn><mrow><mo>(</mo><mrow><mn>0.75</mn><mo>−</mo><mi>x</mi></mrow><mo>)</mo></mrow><msub><mi>E</mi><mrow><mtext>S0</mtext></mrow></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo><</mo><mi>x</mi><mo><</mo><mn>2</mn><mo>/</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>0.83</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><msub><mi>E</mi><mrow><mtext>S0</mtext></mrow></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>/</mo><mn>3</mn><mo><</mo><mi>x</mi><mo><</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math> with <math display="block" id="M97"><mrow><mi>x</mi><mo>=</mo><mfrac><mrow><msub><mi>E</mi><mrow><mtext>C0</mtext></mrow></msub></mrow><mrow><mn>2</mn><msub><mi>E</mi><mrow><mtext>S0</mtext></mrow></msub></mrow></mfrac><mn>.</mn></mrow></math> 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: <math display="block" id="M98"><mrow><msub><mi>E</mi><mrow><mtext>S0</mtext></mrow></msub><mo>=</mo><mn>17.944</mn><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><mn>1.7826</mn><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>N</mi><mo>−</mo><mi>Z</mi></mrow><mi>A</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow><msup><mi>A</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup><mtext> </mtext><mtext>MeV</mtext></mrow></math> and <math display="block" id="M99"><mrow><msub><mi>E</mi><mrow><mtext>C0</mtext></mrow></msub><mo>=</mo><mn>0.7053</mn><mfrac><mrow><msup><mi>Z</mi><mn>2</mn></msup></mrow><mrow><msup><mi>A</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac><mtext> </mtext><mtext>MeV</mtext><mn>.</mn></mrow></math> The microcosmic shell correction energy was taken from [ 78 78 ]. The shell-damping energy was <math display="block" id="M100"><mrow><msub><mi>E</mi><mtext>D</mtext></msub><mo>=</mo><mfrac><mrow><mn>5.48</mn><msup><mi>A</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><mn>1</mn><mo>+</mo><mn>1.3</mn><msup><mi>A</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac><mtext> </mtext><mtext>MeV</mtext></mrow></math> or <math display="block" id="M101"><mrow><msub><mi>E</mi><mtext>D</mtext></msub><mo>=</mo><mn>0.4</mn><msup><mi>A</mi><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>/</mo><mi>a</mi><mtext> </mtext><mtext>MeV</mtext></mrow></math> where a is the energy level density parameter. The fission level density was set as a f = 1.1 a . The moments of inertia of fission compound nuclei at ground state (gs) and saddle point (sd) configurations are expressed as <math display="block" id="M102"><mrow><msub><mi>ζ</mi><mrow><mtext>gs(sd)</mtext></mrow></msub><mo>=</mo><mi>k</mi><mo>×</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><mi>m</mi><msup><mi>r</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msubsup><mi>β</mi><mn>2</mn><mrow><mtext>gs(sd)</mtext></mrow></msubsup><mo>/</mo><mn>3</mn><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Here, k = 0.4 is the correction factor of the rigid body and β 2 is the quadrupole deformation taken from Ref. [ 78 78 ]; <math id="M103"><mrow><msubsup><mi>β</mi><mn>2</mn><mrow><mtext>sd</mtext></mrow></msubsup></mrow></math> = <math id="M104"><mrow><msubsup><mi>β</mi><mn>2</mn><mrow><mtext>gs</mtext></mrow></msubsup></mrow></math> + 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 78 ] <math display="block" id="M105"><mtable><mtr><mtd><mi>ρ</mi><mo stretchy="false">(</mo><msup><mi>E</mi><mo>*</mo></msup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>K</mi><mrow><mtext>coll</mtext></mrow></msub><mo>×</mo><mfrac><mrow><mn>2</mn><mi>J</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>24</mn><msqrt><mn>2</mn></msqrt><msup><mi>σ</mi><mn>3</mn></msup><msup><mi>a</mi><mrow><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>E</mi><mo>*</mo></msup><mo>−</mo><mi>δ</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>5</mn><mo>/</mo><mn>4</mn></mrow></msup></mrow></mfrac></mtd></mtr><mtr><mtd><mo>×</mo><mi>exp</mi><mrow><mo>[</mo><mrow><mn>2</mn><msqrt><mrow><mi>a</mi><mrow><mo>(</mo><mrow><msup><mi>E</mi><mo>*</mo></msup><mo>−</mo><mi>δ</mi></mrow><mo>)</mo></mrow></mrow></msqrt><mo>−</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>J</mi><mo>+</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>σ</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>]</mo></mrow></mtd></mtr></mtable></math> where <math id="M106"><mrow><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mn>6</mn><msup><mover accent="true"><mi>m</mi><mo>¯</mo></mover><mn>2</mn></msup><msqrt><mrow><mi>a</mi><mo stretchy="false">(</mo><msup><mi>E</mi><mo>*</mo></msup><mo>−</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow></msqrt><mo>/</mo><msup><mi>π</mi><mn>2</mn></msup></mrow></math> and <math id="M107"><mrow><mover accent="true"><mi>m</mi><mo>¯</mo></mover><mo>≈</mo><mn>0.24</mn><msup><mi>A</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math> ; 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 =1.1 a 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 <math display="block" id="M108"><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><mtext>CN</mtext></mrow><mo>*</mo></msubsup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><mtext>CN</mtext></mrow><mo>*</mo></msubsup><mo>−</mo><msub><mi>B</mi><mi>s</mi></msub><mo>−</mo><msub><mi>E</mi><mrow><mtext>rot</mtext></mrow></msub><mo>−</mo><mn>2</mn><mi>T</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>σ</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>]</mo></mrow></mrow></math> 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: <math display="block" id="M109"><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><mtext>CN</mtext></mrow><mo>*</mo></msubsup><mn>,</mn><mi>s</mi><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mtext>Δ</mtext><mi>s</mi></msub><mn>,2</mn><mi>s</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mo>−</mo><mi>I</mi><mo stretchy="false">(</mo><msub><mtext>Δ</mtext><mrow><mi>s</mi><mo>+</mo><mn>1</mn></mrow></msub><mn>,2</mn><mi>s</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math> where I and <math id="M110"><mtext>Δ</mtext></math> are given by <math display="block" id="M111"><mrow><mi>I</mi><mo stretchy="false">(</mo><mi>z</mi><mn>,</mn><mi>m</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>m</mi><mo>!</mo></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>z</mi></msubsup></mrow></mstyle><msup><mi>u</mi><mi>m</mi></msup><msup><mi>e</mi><mrow><mo>−</mo><mi>u</mi></mrow></msup><mtext>d</mtext><mi>u</mi></mrow></math> <math display="block" id="M112"><mrow><msub><mtext>Δ</mtext><mi>s</mi></msub><mo>=</mo><mfrac><mrow><msubsup><mi>E</mi><mrow><mtext>CN</mtext></mrow><mo>*</mo></msubsup><mo>−</mo><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>s</mi></msubsup><mrow><msubsup><mi>B</mi><mi>i</mi><mi>v</mi></msubsup></mrow></mstyle></mrow><mrow><msub><mi>T</mi><mi>i</mi></msub></mrow></mfrac></mrow></math> where <math id="M113"><mrow><msubsup><mi>B</mi><mi>i</mi><mi>v</mi></msubsup></mrow></math> 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 1 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 48 Ca+ 243 Am, 48 Ca+ 242 Pu, and 48 Ca+ 238 U, marked by solid olive, dash red, dash-dot blue, and orange short-dash lines, respectively, in Fig. 1 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 48 Ca+ 243 Am taken from Ref. [ 9 9 , 83 83 , 84 84 ]. According to Ref. [ 9 9 ], the experiments concerning 48 Ca+ 243 Am 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., 287 Mc and 288 Mc, produced in the 3n- and 4n-evaporation channels with cross-sections of approximately 3 pb and 1 pb, respectively. According to Ref. [ 83 83 ], the cross-section for the 3n-evaporation channel reaches its maximum, σ 3n = <math id="M114"><mrow><msubsup><mrow><mn>8.5</mn></mrow><mrow><mo>−</mo><mn>3.7</mn></mrow><mrow><mo>+</mo><mn>6.4</mn></mrow></msubsup></mrow></math> pb, at E * = 34.0 - 38.3 MeV and decreases with further increase of the excitation energy of the compound nucleus 291 Mc. At excitation energy, E * = 44.8 ± 2.3 MeV, not a single event indicating the formation of 288 Mc 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 <math id="M115"><mrow><msubsup><mrow><mn>3.5</mn></mrow><mrow><mo>−</mo><mn>1.5</mn></mrow><mrow><mo>+</mo><mn>2.7</mn></mrow></msubsup></mrow></math> pb and <math id="M116"><mrow><msubsup><mrow><mn>2.5</mn></mrow><mrow><mo>−</mo><mn>1.5</mn></mrow><mrow><mo>+</mo><mn>2.7</mn></mrow></msubsup></mrow></math> pb, respectively. At energies E * ≤ 36 MeV, which could be expected for the 2n-evaporation product, 289 Mc was not detected. The upper cross-section limit can be set at level σ 2n ≤ 3 pb. According to Ref. [ 84 84 ], the cross-sections for the formation of ERs in the 3n- and 2n-evaporation channels are approximately <math id="M117"><mrow><msubsup><mrow><mn>3.2</mn></mrow><mrow><mo>−</mo><mn>1.2</mn></mrow><mrow><mo>+</mo><mn>0.8</mn></mrow></msubsup></mrow></math> pb and <math id="M118"><mrow><msubsup><mrow><mn>0.3</mn></mrow><mrow><mo>−</mo><mn>0.2</mn></mrow><mrow><mo>+</mo><mn>0.7</mn></mrow></msubsup></mrow></math> pb at energies E * = 33 MeV, respectively. In Fig. 1 (b) 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 48 Ca+ 242 Pu, respectively, where filled, half-filled, and open symbols represent three experiments for 48 Ca+ 242 Pu [ 8 8 , 79 79 , 80 80 , 86 86 ]. According to Ref. [ 79 79 ], a maximum cross-section of <math id="M119"><mrow><msubsup><mrow><mn>10.4</mn></mrow><mrow><mo>−</mo><mn>2.1</mn></mrow><mrow><mo>+</mo><mn>3.5</mn></mrow></msubsup></mrow></math> pb was measured for the 242 Pu( 48 Ca, 3n) 287 Fl reaction. According to Ref. [ 8 8 ], at excitation energy E * = 50 MeV, the 242 Pu( 48 Ca, 5n) 285 Fl cross-section was <math id="M120"><mrow><msubsup><mrow><mn>0.6</mn></mrow><mrow><mo>−</mo><mn>0.5</mn></mrow><mrow><mo>+</mo><mn>0.9</mn></mrow></msubsup></mrow></math> pb. The no-observation of a 3n-evaporation product led to an upper limit for the 242 Pu( 48 Ca, 3n) 287 Fl reaction of 1.1 pb. The 3n and 4n cross-section values measured at E * = 41 MeV were <math id="M121"><mrow><msubsup><mrow><mn>3.1</mn></mrow><mrow><mo>−</mo><mn>2.6</mn></mrow><mrow><mo>+</mo><mn>4.9</mn></mrow></msubsup></mrow></math> pb. In Fig. 1 (c) Fig. 1 (c) , the red square and blue circle stand for the experimental results of 3n- and 4n-evaporation channels for 48 Ca+ 238 U, respectively, where filled, half-filled, and open symbols represent three experiments for 48 Ca+ 238 U [ 79 79 , 81 81 - 82 82 81-82 ]. According to Ref. [ 80 80 ], the maximum cross-section values of the x n-evaporation channels for the reaction 238 U( 48 Ca, x n) 286- x Cn were measured to be σ 3n = <math id="M122"><mrow><msubsup><mrow><mn>2.5</mn></mrow><mrow><mo>−</mo><mn>1.1</mn></mrow><mrow><mo>+</mo><mn>1.8</mn></mrow></msubsup></mrow></math> pb and σ 4n = <math id="M123"><mrow><msubsup><mrow><mn>0.6</mn></mrow><mrow><mo>−</mo><mn>0.5</mn></mrow><mrow><mo>+</mo><mn>1.6</mn></mrow></msubsup></mrow></math> pb. At the excitation energies of the compound nucleus E * = 34.5 MeV, two decay events from 283 Cn were observed, resulting in a cross-section of <math id="M124"><mrow><msubsup><mrow><mn>2.0</mn></mrow><mrow><mo>−</mo><mn>1.3</mn></mrow><mrow><mo>+</mo><mn>2.7</mn></mrow></msubsup></mrow></math> pb[ 82 82 ]. The cross-section deduced from all four events was <math id="M125"><mrow><msubsup><mrow><mn>0.72</mn></mrow><mrow><mo>−</mo><mn>0.35</mn></mrow><mrow><mo>+</mo><mn>0.58</mn></mrow></msubsup></mrow></math> pb, measured at an excitation energy of 34.6 MeV of the compound nucleus 286 Cn[ 81 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 48 Ca+ 243 Am, 48 Ca+ 242 Pu, and 48 Ca+ 238 U. 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 48 Ca+ 243 Am 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 Fig. 2 . The projectile nuclei 48 Ca and target nuclei 243 Am have theoretical quadrupole deformation values β P = 0. and β T = 0.224, respectively. In Fig. 2 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 69 ] and the double-folding method [ 87 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 -21 s. 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 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. The PES and driving potential (DP) of the reaction 48 Ca+ 243 Am 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 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 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. 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 <math id="M127"><mrow><mi>T</mi><mi>K</mi><mi>E</mi><mo>=</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mo>−</mo><msub><mi>V</mi><mrow><mtext>CN</mtext></mrow></msub><mo>−</mo><msub><mi>Q</mi><mrow><mtext>gg</mtext></mrow></msub><mo>−</mo><msup><mi>E</mi><mo>*</mo></msup></mrow></math> . Fig. 4 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 <math id="M128"><mrow><mi>T</mi><mi>K</mi><mi>E</mi><mo>=</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mo>−</mo><msub><mi>U</mi><mrow><mtext>dr</mtext></mrow></msub><mo>−</mo><msup><mi>E</mi><mo>*</mo></msup></mrow></math> . 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 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. Figure 5 Figure 5 shows the TKE-mass distributions at excitation energies <math id="M129"><mrow><msubsup><mi>E</mi><mrow><mtext>C</mtext><mi>N</mi></mrow><mo>*</mo></msubsup></mrow></math> = 10 MeV, 40 MeV, 70 MeV, and 100 MeV for the tip-tip collisions of 48 Ca+ 243 Am, as illustrated in panels (a), (b), (c), and (d), respectively. Fig. 5 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. 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 48 Ca+ 243 Am 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 83 , 9 9 ]. The calculated excitation functions of 48 Ca+ 243 Am 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 Fig. 6 shows thatthe DNS model involving barrier distributions could reproduce the experimental results relatively well. 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 42 Ca+ 243 Am, 44 Ca+ 243 Am, 46 Ca+ 243 Am, 48 Ca+ 243 Am, 44 Ti+ 237 Np, 46 Ti+ 237 Np, 48 Ti+ 237 Np, and 50 Ti+ 237 Np at excitation energies within the interval E * = 1 - 80 MeV. Fig. 7 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 287-290 Mc. The predictions of maximum cross-sections of the new 281-286 Mc 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 283 Mc as 0.15 nb in the reactions 42 Ca+ 243 Am. 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 281 Mc as 0.2 nb in the reactions 46 Ti+ 237 Np. The new moscovium isotopes of 278-286 Mc 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. 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 35 Cl + 248 Cf ( η = 0.75), 40 Ar + 247 Bk ( η = 0.72), 39 K + 247 Cm ( η = 0.73), 40 Ca + 243 Am ( η = 0.72), 48 Ca + 243 Am ( η = 0.67), 45 Sc + 244 Pu ( η = 0.69), 48 Ti + 237 Np ( η = 0.66), and 51 V + 238 U ( η = 0.65) based on the DNS model, as illustrated in panels (a), (b), (c), (d), (e), (f), (g), and (h), respectively. Fig. 8 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 278-286 Mc 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 247 Cm( 39 K, 3n) 283 Mc. Conclusion 1 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 48 Ca+ 243 Am 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 48 Ca+ 243 Am 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 48 Ca+ 243 Pu and 48 Ca+ 238 U, which reproduced the experimental excitation functions well. Based on the DNS model involving the barrier distribution function, we systematically calculated the reactions of projectiles 42-48 Ca bombarding on target 243 Am and projectiles 42-48 Ca on target 237 Np. 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 283 Mc in the reaction 237 Np( 46 Ti, 2n) 283 Mc. The reactions of 35 Cl+ 248 Cf, 40 Ar+ 247 Bk, 39 K+ 247 Cm, 40 Ca+ 243 Am, 48 Ca+ 243 Am, 45 Sc+ 244 Pu, 48 Ti+ 237 Np, and 51 V+ 238 U 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 278-286 Mc 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 35,37 Cl + 248 Cf, 38,40 Ar + 247 Bk, 39,41 K + 247 Cm, 40,42,44,46 Ca + 243 Am, 45 Sc + 244 Pu, and 46,48,50 Ti + 237 Np, 51 V + 238 U at some typical excitation energies. References 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 http://doi.org/10.1088/0034-4885/78/3/036301 Super-heavy element research M. Thoennessen , Discovery of isotopes of elements with Z ≥ 100 . Atomic Data and Nuclear Data Tables 99 , 312 - 344 ( 2013 ). doi: 10.1016/j.adt.2012.03.003 http://doi.org/10.1016/j.adt.2012.03.003 Discovery of isotopes of elements with Z≥100 G. Heinlein , K. Bächmann , J. Rudolph , in Superheavy Elements , Model experiments with short-lived nuclides for the investigation of the quantitative chemical behaviour of new elements . ( Pergamon , 1978 ), pp. 407 – 414 doi: 10.1016/B978-0-08-022946-1.50051-7 http://doi.org/10.1016/B978-0-08-022946-1.50051-7 Superheavy Elements, Model experiments with short-lived nuclides for the investigation of the quantitative chemical behaviour of new elements A. Ghiorso , M. Nurmia , K. Eskola , et al. , 261rf; new isotope of element 104 . Physics Letters B 32 , 95 - 98 ( 1970 ). doi: 10.1016/0370-2693(70)90595-2 http://doi.org/10.1016/0370-2693(70)90595-2 261rf; new isotope of element 104 G.N. Flerov , Y.T. Oganesyan , Y.V. Lobanov , et al. , Synthesis of element 105 . Sov. At. Energy 29 , 967 ( 1970 ). Synthesis of element 105 A. Ghiorso , M. Nurmia , K. Eskola , et al. , New element hahnium, atomic number 105 . Phys. Rev. Lett . 24 , 1498 - 1503 ( 1970 ). doi: 10.1103/PhysRevLett.24.1498 http://doi.org/10.1103/PhysRevLett.24.1498 New element hahnium, atomic number 105 A. Ghiorso , J.M. Nitschke , J.R. Alonso , et al. , Element 106 . Phys. Rev. Lett . 33 , 1490 - 1493 ( 1974 ). doi: 10.1103/PhysRevLett.33.1490 http://doi.org/10.1103/PhysRevLett.33.1490 Element 106 P.A. Ellison , K.E. Gregorich , J.S. Berryman , et al. , New superheavy element isotopes: 242 Pu( 48 Ca, 5n) 285 114 . Phys. Rev. Lett . 105 , 182701 ( 2010 ). doi: 10.1103/PhysRevLett.105.182701 http://doi.org/10.1103/PhysRevLett.105.182701 New superheavy element isotopes: 242Pu(48 Ca, 5n)285114 Y.T. Oganessian , V.K. Utyonkoy , Y.V. Lobanov , et al. , Experiments on the synthesis of element 115 in the reaction 243 Am( 48 Ca, xn)291- x 115 . Phys. Rev. C 69 , 021601 ( 2004 ). doi: 10.1103/PhysRevC.69.021601 http://doi.org/10.1103/PhysRevC.69.021601 Experiments on the synthesis of element 115 in the reaction 243Am(48 Ca, xn)291-x115 Y.T. Oganessian , V.K. Utyonkov , Y.V. Lobanov , et al. , Measurements of cross sections for the fusion-evaporation reactions 244 Pu( 48 Ca, xn) 292- x 114 and 245 Cm( 48 Ca, xn) 293- x 116 . Phys. Rev. C 69 , 054607 ( 2004 ). doi: 10.1103/PhysRevC.69.054607 http://doi.org/10.1103/PhysRevC.69.054607 Measurements of cross sections for the fusion-evaporation reactions 244 Pu(48Ca, xn)292-x 114 and 245Cm(48Ca, xn)293-x116 Y.T. Oganessian , F.S. Abdullin , P.D. Bailey , et al. , Synthesis of a new element with atomic number Z=117 . Phys. Rev. Lett . 104 , 142502 ( 2010 ). doi: 10.1103/PhysRevLett.104.142502 http://doi.org/10.1103/PhysRevLett.104.142502 Synthesis of a new element with atomic number Z=117 Y.T. Oganessian , V.K. Utyonkov , Y.V. Lobanov , et al. , Synthesis of the isotopes of elements 118 and 116 in the 249 Cf and 245 Cm+ 48 Ca fusion reactions . Phys. Rev. C 74 , 044602 ( 2006 ). doi: 10.1103/PhysRevC.74.044602 http://doi.org/10.1103/PhysRevC.74.044602 Synthesis of the isotopes of elements 118 and 116 in the 249Cf and 245Cm+48Ca fusion reactions Y.T. Oganesyan , Y.P. Tretyakov , A.S. Ilinov , et al. , Synthesis of neutron-deficient isotopes of fermium, kurchatovium, and element 106 . JETP Lett. (USSR) 20 , 265 ( 1975 ). Synthesis of neutron-deficient isotopes of fermium, kurchatovium, and element 106 G. Münzenberg , S. Hofmann , F.P. Heßberger , et al. , Identification of element 107 by α correlation chains . Zeitschrift fur Physik A Hadrons and Nuclei 300 , 107 - 108 ( 1981 ). doi: 10.1007/BF01412623 http://doi.org/10.1007/BF01412623 Identification of element 107 by α correlation chains G. Münzenberg , P. Armbruster , H. Folger , et al. , The identification of element 108 . Zeitschrift für Physik A Atoms and Nuclei 317 , 235 - 236 ( 1984 ). doi: 10.1007/BF01421260 http://doi.org/10.1007/BF01421260 The identification of element 108 G. Münzenberg , P. Armbruster , F.P. Hessberger , et al. , Observation of one correlated α -decay in the reaction58fe on209bi → 267109 . Zeitschrift für Physik A Atoms and Nuclei 309 , 89 - 90 ( 1982 ). Observation of one correlated α-decay in the reaction58fe on209bi →267109 S. Hofmann , V. Ninov , F. Heßberger , et al. , Production and decay of269110 . Zeitschrift für Physik A Hadrons and Nuclei 350 , 277 - 280 ( 1995 ). doi: 10.1007/BF01291181 http://doi.org/10.1007/BF01291181 Production and decay of269110 S. Hofmann , V. Ninov , F. Heßberger , et al. , The new element 111 . Zeitschrift für Physik A Hadrons and Nuclei 350 , 281 - 282 ( 1995 ). doi: 10.1007/BF01291182 http://doi.org/10.1007/BF01291182 The new element 111 JOUR , S. Hofmann , V. Ninov , et al. , The new element 112 . Zeitschrift für Physik A Hadrons and Nuclei 354 ,. doi: 10.1007/BF02769517 http://doi.org/10.1007/BF02769517 The new element 112 K. Morita , K. Morimoto , D. Kaji , et al. , Experiment on the synthesis of element 113 in the reaction 209bi(70zn,n)278113 . Journal of the Physical Society of Japan 73 , 2593 - 2596 ( 2004 ). doi: 10.1143/JPSJ.73.2593 http://doi.org/10.1143/JPSJ.73.2593 Experiment on the synthesis of element 113 in the reaction 209bi(70zn,n)278113 Z. Gan , Z. Qin , H. Fan , et al. , A new alpha-particle - emitting isotope 259 Db . European Physical Journal A 10 , 21 - 25 ( 2001 ). doi: 10.1007/s100500170140 http://doi.org/10.1007/s100500170140 A new alpha-particle - emitting isotope 259Db Q. Z. , W.X. L. , D.H. J. , et al. , Alpha decay properties of 266 Bh . Nuclear Physics Review 23 , 404 ( 2006 ). doi: 10.11804/NuclPhysRev.23.04.404 http://doi.org/10.11804/NuclPhysRev.23.04.404 Alpha decay properties of 266Bh Z.Y. ZHANG , Z.G. GAN , L. MA , et al. , Observation of the superheavy nuclide 271 Ds . Chin. Phys. Lett . 58 ,. doi: 10.1088/0256-307X/29/1/012502 http://doi.org/10.1088/0256-307X/29/1/012502 Observation of the superheavy nuclide 271Ds F.P. Heßberger , S. Hofmann , V. Ninov , et al. , Spontaneous fission and alpha-decay properties of neutron deficient isotopes 257-253 104 and 258 106 . Zeitschrift für Physik A Hadrons and Nuclei 359 , 415 - 425 ( 1997 ). doi: 10.1007/s002180050422 http://doi.org/10.1007/s002180050422 Spontaneous fission and alpha-decay properties of neutron deficient isotopes 257-253104 and 258106 U. Tippawan , S. Pomp , A. Ataç , et al. , Light-ion production in the interaction of 96 MeV neutrons with silicon . Phys. Rev. C 69 , 064609 ( 2004 ). doi: 10.1103/PhysRevC.69.064609 http://doi.org/10.1103/PhysRevC.69.064609 Light-ion production in the interaction of 96 MeV neutrons with silicon JOUR , F. Heßberger , S. Hofmann , et al. , Decay properties of neutron-deficient isotopes 256, 257 Db, 255 Rf, 252, 253 Lr . The European Physical Journal A - Hadrons and Nuclei . http://dx.doi.org/10.1007/s100500170039 doi:10.1007/s100500170039 Decay properties of neutron-deficient isotopes 256, 257Db, 255Rf, 252, 253Lr S.L. Nelson , K.E. Gregorich , I. Dragojević , et al. , Lightest isotope of bh produced via the 209 Bi( 52 Cr, n) 260 reaction . Phys. Rev. Lett . 100 , 022501 ( 2008 ). doi: 10.1103/PhysRevLett.100.022501 http://doi.org/10.1103/PhysRevLett.100.022501 Lightest isotope of bh produced via the 209Bi(52Cr, n)260 reaction I. Dragojević , K.E. Gregorich , C.E. Düllmann , et al. , New isotope 263 Hs . Phys. Rev. C 79 , 011602 ( 2009 ). doi: 10.1103/PhysRevC.79.011602 http://doi.org/10.1103/PhysRevC.79.011602 New isotope 263Hs C.E. Düllmann , M. Schädel , A. Yakushev , et al. , Production and decay of element 114: High cross sections and the new nucleus 277 Hs . Phys. Rev. Lett . 104 , 252701 ( 2010 ). doi: 10.1103/PhysRevLett.104.252701 http://doi.org/10.1103/PhysRevLett.104.252701 Production and decay of element 114: High cross sections and the new nucleus 277Hs G. Münzenberg , P. Armbruster , F. Heßberger , et al. , Observation of one correlated α -decay in the reaction 58 Fe on 209 Bi → 267 109 . Zeitschrift für Physik A Hadrons and Nuclei 309 , 89 - 90 ( 1982 ). Observation of one correlated α-decay in the reaction 58Fe on 209Bi→ 267109 A. Ghiorso , D. Lee , L.P. Somerville , et al. , Evidence for the possible synthesis of element 110 produced by the 59 Co+ 209 Bi reaction . Phys. Rev. C 51 , R2293 - R2297 ( 1995 ). doi: 10.1103/PhysRevC.51.R2293 http://doi.org/10.1103/PhysRevC.51.R2293 Evidence for the possible synthesis of element 110 produced by the 59Co+209Bi reaction JOUR , S. Hofmann , V. Ninov , et al. , The new element 111 . Zeitschrift für Physik A Hadrons and Nuclei 350 ,. doi: 10.1007/BF01291182 http://doi.org/10.1007/BF01291182 The new element 111 K. Morita , K. Morimoto , D. Kaji , et al. , Experiment on the synthesis of element 113 in the reaction 209 Bi( 70 Zn,n) 278 113 . Journal of the Physical Society of Japan 73 , 2593 - 2596 ( 2004 ). doi: 10.1143/jpsj.73.2593 http://doi.org/10.1143/jpsj.73.2593 Experiment on the synthesis of element 113 in the reaction 209Bi(70Zn,n)278113 V.V. Volkov , Deep inelastic transfer reactions — the new type of reactions between complex nuclei . Phys. Rep . 44 , 93 - 157 ( 1978 ). doi: 10.1016/0370-1573(78)90200-4 http://doi.org/10.1016/0370-1573(78)90200-4 Deep inelastic transfer reactions — the new type of reactions between complex nuclei Z. Gan , W. Huang , Z. Zhang , et al. , Results and perspectives for study of heavy and super-heavy nuclei and elements at IMP/CAS . The European Physical Journal A 58 ,. doi: 10.1140/epja/s10050-022-00811-w http://doi.org/10.1140/epja/s10050-022-00811-w Results and perspectives for study of heavy and super-heavy nuclei and elements at IMP/CAS S. Heinz , H. Devaraja , Nucleosynthesis in multinucleon transfer reactions . The European Physical Journal A 58 ,. doi: 10.1140/epja/s10050-022-00771-1 http://doi.org/10.1140/epja/s10050-022-00771-1 Nucleosynthesis in multinucleon transfer reactions M. Block , F. Giacoppo , F.P. Heßberger , et al. , Recent progress in experiments on the heaviest nuclides at SHIP . Nuovo Cimento Rivista Serie 45 , 279 - 323 ( 2022 ). doi: 10.1007/s40766-022-00030-5 http://doi.org/10.1007/s40766-022-00030-5 Recent progress in experiments on the heaviest nuclides at SHIP B. Sergey , S. Dmitriev , N. Kazarinov , et al. , She factory: Cyclotron facility for super heavy elements research. 2020 . doi: 10.18429/JACoW-Cyclotrons2019-THC01 http://doi.org/10.18429/JACoW-Cyclotrons2019-THC01 She factory: Cyclotron facility for super heavy elements research. 2020 G. Adamyan , N. Antonenko , A. Diaz-Torres , et al. , How to extend the chart of nuclides? European Physical Journal A 56 , 47 - 1 ( 2020 ). doi: 10.1140/epja/s10050-020-00046-7 http://doi.org/10.1140/epja/s10050-020-00046-7 How to extend the chart of nuclides? D. Kaji , K. Morimoto , A. Yoneda , et al. , Target for the heaviest element production at riken . Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 590 , 198 - 203 ( 2008 ). Toward the Realization of Target and Stripper Foil Technologies for High-Power Proton and Radioactive Ion Accelerators . doi: 10.1016/j.nima.2008.02.090 http://doi.org/10.1016/j.nima.2008.02.090 Target for the heaviest element production at riken H. Haba , A new period in superheavy-element hunting . Nature Chemistry 11 , 10 - 13 ( 2019 ). doi: 10.1038/s41557-018-0191-8 http://doi.org/10.1038/s41557-018-0191-8 A new period in superheavy-element hunting T. Tanaka , K. Morita , K. Morimoto , et al. , Study of quasielastic barrier distributions as a step towards the synthesis of superheavy elements with hot fusion reactions . Phys. Rev. Lett . 124 , 052502 ( 2020 ). doi: 10.1103/PhysRevLett.124.052502 http://doi.org/10.1103/PhysRevLett.124.052502 Study of quasielastic barrier distributions as a step towards the synthesis of superheavy elements with hot fusion reactions J.L. Pore , J.M. Gates , R. Orford , et al. , Identification of the new isotope 244 Md . Phys. Rev. Lett . 124 , 252502 ( 2020 ). doi: 10.1103/PhysRevLett.124.252502 http://doi.org/10.1103/PhysRevLett.124.252502 Identification of the new isotope 244Md L. Guo , C. Simenel , L. Shi , et al. , The role of tensor force in heavy-ion fusion dynamics . Physics Letters B 782 , 401 - 405 ( 2018 ). doi: 10.1016/j.physletb.2018.05.066 http://doi.org/10.1016/j.physletb.2018.05.066 The role of tensor force in heavy-ion fusion dynamics K. Sekizawa , Tdhf theory and its extensions for the multinucleon transfer reaction: A mini review . Frontiers in Physics 7 ,. doi: 10.3389/fphy.2019.00020 http://doi.org/10.3389/fphy.2019.00020 Tdhf theory and its extensions for the multinucleon transfer reaction: A mini review J. Maruhn , P.G. Reinhard , P. Stevenson , et al. , The tdhf code sky3d . Computer Physics Communications 185 , 2195 - 2216 ( 2014 ). doi: 10.1016/j.cpc.2014.04.008 http://doi.org/10.1016/j.cpc.2014.04.008 The tdhf code sky3d N. Wang , Z. Li , X. Wu , Improved quantum molecular dynamics model and its applications to fusion reaction near barrier . Phys. Rev. C 65 , 064608 ( 2002 ). doi: 10.1103/PhysRevC.65.064608 http://doi.org/10.1103/PhysRevC.65.064608 Improved quantum molecular dynamics model and its applications to fusion reaction near barrier X. Jiang , S. Yan , J.A. Maruhn , Dynamics of energy dissipation in heavy-ion fusion reactions . Phys. Rev. C 88 , 044611 ( 2013 ). doi: 10.1103/PhysRevC.88.044611 http://doi.org/10.1103/PhysRevC.88.044611 Dynamics of energy dissipation in heavy-ion fusion reactions K. Zhao , Z. Li , X. Wu , et al. , Production probability of superheavy fragments at various initial deformations and orientations in the 238 U + 238 U reaction . Phys. Rev. C 88 , 044605 ( 2013 ). doi: 10.1103/PhysRevC.88.044605 http://doi.org/10.1103/PhysRevC.88.044605 Production probability of superheavy fragments at various initial deformations and orientations in the 238U +238U reaction A.V. Karpov , V.V. Saiko , Modeling near-barrier collisions of heavy ions based on a langevin-type approach . Phys. Rev. C 96 , 024618 ( 2017 ). doi: 10.1103/PhysRevC.96.024618 http://doi.org/10.1103/PhysRevC.96.024618 Modeling near-barrier collisions of heavy ions based on a langevin-type approach V.I. Zagrebaev , A.V. Karpov , W. Greiner , Possibilities for synthesis of new isotopes of superheavy elements in fusion reactions . Phys. Rev. C 85 , 014608 ( 2012 ). doi: 10.1103/PhysRevC.85.014608 http://doi.org/10.1103/PhysRevC.85.014608 Possibilities for synthesis of new isotopes of superheavy elements in fusion reactions Z.Q. Feng , G.M. Jin , F. Fu , et al. , Production cross sections of superheavy nuclei based on dinuclear system model . Nuclear Physics A 771 , 50 - 67 ( 2006 ). doi: 10.1016/j.nuclphysa.2006.03.002 http://doi.org/10.1016/j.nuclphysa.2006.03.002 Production cross sections of superheavy nuclei based on dinuclear system model X.J. Bao , Y. Gao , J.Q. Li , et al. , Influence of the nuclear dynamical deformation on production cross sections of superheavy nuclei . Phys. Rev. C 91 , 011603 ( 2015 ). doi: 10.1103/PhysRevC.91.011603 http://doi.org/10.1103/PhysRevC.91.011603 Influence of the nuclear dynamical deformation on production cross sections of superheavy nuclei L. Zhu , W.J. Xie , F.S. Zhang , Production cross sections of superheavy elements Z=119 and 120 in hot fusion reactions . Phys. Rev. C 89 , 024615 ( 2014 ). doi: 10.1103/PhysRevC.89.024615 http://doi.org/10.1103/PhysRevC.89.024615 Production cross sections of superheavy elements Z=119 and 120 in hot fusion reactions G. Adamyan , N. Antonenko , A. Diaz-Torres , et al. , How to extend the chart of nuclides? European Physical Journal A 56 , 47 - 1 ( 2020 ). doi: 10.1140/epja/s10050-020-00046-7 http://doi.org/10.1140/epja/s10050-020-00046-7 How to extend the chart of nuclides? Z.Q. Feng , g.m. no , H. Ming-Hui , et al. , Possible way to synthesize superheavy element Z=117 . Chinese Physics Letters 24 ,. doi: 10.1088/0256-307X/24/9/024 http://doi.org/10.1088/0256-307X/24/9/024 Possible way to synthesize superheavy element Z=117 Z.Q. Feng , G.M. Jin , J.Q. Li , Influence of entrance channels on the formation of superheavy nuclei in massive fusion reactions . Nuclear Physics A 836 , 82 - 90 ( 2010 ). doi: 10.1016/j.nuclphysa.2010.01.244 http://doi.org/10.1016/j.nuclphysa.2010.01.244 Influence of entrance channels on the formation of superheavy nuclei in massive fusion reactions K.X. Cheng , J. Pu , Y.T. Wang , et al. , Non-frozen process of heavy-ion fusion reactions at deep sub-barrier energies . Nuclear Science and Techniques 33 , 132 ( 2022 ). doi: 10.1007/s41365-022-01114-x http://doi.org/10.1007/s41365-022-01114-x Non-frozen process of heavy-ion fusion reactions at deep sub-barrier energies F. Niu , P.H. Chen , Z.Q. Feng , Systematics on production of superheavy nuclei Z = 119-122 in fusion-evaporation reactions . Nuclear Science and Techniques 32 , 103 ( 2021 ). doi: 10.1007/s41365-021-00946-3 http://doi.org/10.1007/s41365-021-00946-3 Systematics on production of superheavy nuclei Z = 119-122 in fusion-evaporation reactions M.T. Jin , S.Y. Xu , G.M. Yang , et al. , Yield of long-lived fission product transmutation using proton-, deuteron-, and alpha particle-induced spallation . Nuclear Science and Techniques 32 , 96 ( 2021 ). doi: 10.1007/s41365-021-00933-8 http://doi.org/10.1007/s41365-021-00933-8 Yield of long-lived fission product transmutation using proton-, deuteron-, and alpha particle-induced spallation Y.Q. Xin , N.N. Ma , J.G. Deng , et al. , Properties of Z=114 super-heavy nuclei . Nuclear Science and Techniques 32 , 55 ( 2021 ). doi: 10.1007/s41365-021-00899-7 http://doi.org/10.1007/s41365-021-00899-7 Properties of Z=114 super-heavy nuclei X.B. Yu , L. Zhu , Z.H. Wu , et al. , Predictions for production of superheavy nuclei with Z = 105-112 in hot fusion reactions . Nuclear Science and Techniques 29 , 154 ( 2018 ). doi: 10.1007/s41365-018-0501-2 http://doi.org/10.1007/s41365-018-0501-2 Predictions for production of superheavy nuclei with Z = 105-112 in hot fusion reactions Z.Q. Feng , G.M. Jin , J.Q. Li , Production of new superheavy Z=108-114 nuclei with 238 U, 244 Pu, and 248,250 Cm targets . Phys. Rev. C 80 , 057601 ( 2009 ). doi: 10.1103/PhysRevC.80.057601 http://doi.org/10.1103/PhysRevC.80.057601 Production of new superheavy Z=108-114 nuclei with 238U, 244Pu, and 248,250Cm targets Z.Q. Feng , G.M. Jin , J.Q. Li , et al. , Formation of superheavy nuclei in cold fusion reactions . Phys. Rev. C 76 , 044606 ( 2007 ). doi: 10.1103/PhysRevC.76.044606 http://doi.org/10.1103/PhysRevC.76.044606 Formation of superheavy nuclei in cold fusion reactions G.G. Adamian , N.V. Antonenko , R.V. Jolos , et al. , Effective nucleus-nucleus potential for calculation of potential energy of a dinuclear system . International Journal of Modern Physics E 05 , 191216 ( 1996 ). Effective nucleus-nucleus potential for calculation of potential energy of a dinuclear system D.L. Hill , J.A. Wheeler , Nuclear constitution and the interpretation of fission phenomena . Phys. Rev . 89 , 1102 - 1145 ( 1953 ). doi: 10.1103/PhysRev.89.1102 http://doi.org/10.1103/PhysRev.89.1102 Nuclear constitution and the interpretation of fission phenomena V.I. Zagrebaev , Y. Aritomo , M.G. Itkis , et al. , Synthesis of superheavy nuclei: How accurately can we describe it and calculate the cross sections? Phys . Rev. C 65 , 014607 ( 2001 ). doi: 10.1103/PhysRevC.65.014607 http://doi.org/10.1103/PhysRevC.65.014607 Synthesis of superheavy nuclei: How accurately can we describe it and calculate the cross sections? Phys W.D. Myers , W.J. Swiatecki , Nuclear masses and deformations . Nuclear Physics 81 , 1 - 60 ( 1966 ). doi: 10.1016/0029-5582(66)90639-0 http://doi.org/10.1016/0029-5582(66)90639-0 Nuclear masses and deformations C.Y. Wong , Interaction barrier in charged-particle nuclear reactions . Phys. Rev. Lett . 31 , 766 - 769 ( 1973 ). doi: 10.1103/PhysRevLett.31.766 http://doi.org/10.1103/PhysRevLett.31.766 Interaction barrier in charged-particle nuclear reactions Z.Q. Feng , G.M. Jin , J.Q. Li , et al. , Production of heavy and superheavy nuclei in massive fusion reactions . Nuclear Physics A 816 , 33 - 51 ( 2009 ). doi: 10.1016/j.nuclphysa.2008.11.003 http://doi.org/10.1016/j.nuclphysa.2008.11.003 Production of heavy and superheavy nuclei in massive fusion reactions J.Q. Li , G. Wolschin , Distribution of the dissipated angular momentum in heavy-ion collisions . Phys. Rev. C 27 , 590 - 601 ( 1983 ). doi: 10.1103/PhysRevC.27.590 http://doi.org/10.1103/PhysRevC.27.590 Distribution of the dissipated angular momentum in heavy-ion collisions J. Li , X. Tang , G. Wolschin , Anticorrelated angular-momentum distributions in heavy-ion collisions . Physics Letters B 105 , 107 - 110 ( 1981 ). doi: 10.1016/0370-2693(81)91000-5 http://doi.org/10.1016/0370-2693(81)91000-5 Anticorrelated angular-momentum distributions in heavy-ion collisions G.G. Adamian , N.V. Antonenko , W. Scheid , Characteristics of quasifission products within the dinuclear system model . Phys. Rev. C 68 , 034601 ( 2003 ). doi: 10.1103/PhysRevC.68.034601 http://doi.org/10.1103/PhysRevC.68.034601 Characteristics of quasifission products within the dinuclear system model P. Grangé , L. Jun-Qing , H.A. Weidenmüller , Induced nuclear fission viewed as a diffusion process: Transients . Phys. Rev. C 27 , 2063 - 2077 ( 1983 ). doi: 10.1103/PhysRevC.27.2063 http://doi.org/10.1103/PhysRevC.27.2063 Induced nuclear fission viewed as a diffusion process: Transients P.H. Chen , Z.Q. Feng , J.Q. Li , et al. , A statistical approach to describe highly excited heavy and superheavy nuclei . Chinese Physics C 40 , 091002 ( 2016 ). doi: 10.1088/1674-1137/40/9/091002 http://doi.org/10.1088/1674-1137/40/9/091002 A statistical approach to describe highly excited heavy and superheavy nuclei A.S. Zubov , G.G. Adamian , N.V. Antonenko , et al. , Competition between evaporation channels in neutron-deficient nuclei . Phys. Rev. C 68 , 014616 ( 2003 ). doi: 10.1103/PhysRevC.68.014616 http://doi.org/10.1103/PhysRevC.68.014616 Competition between evaporation channels in neutron-deficient nuclei A. Zubov , G.G. Adamyan , N. Antonenko , et al. , Survival probabilities of superheavy nuclei based on recent predictions of nuclear properties . Eur. Phys. J. A 23 , 249 - 256 ( 2005 ). doi: 10.1140/epja/i2004-10089-5 http://doi.org/10.1140/epja/i2004-10089-5 Survival probabilities of superheavy nuclei based on recent predictions of nuclear properties P. Moller , J. Nix , W. Myers , et al. , Nuclear ground-state masses and deformations . Atomic Data and Nuclear Data Tables 59 , 185 - 381 ( 1995 ). doi: 10.1006/adnd.1995.1002 http://doi.org/10.1006/adnd.1995.1002 Nuclear ground-state masses and deformations Y.T. Oganessian , V.K. Utyonkov , D. Ibadullayev , et al. , Investigation of 48 Ca-induced reactions with 242 Pu and 238 U targets at the JINR superheavy element factory . Phys. Rev. C 106 , 024612 ( 2022 ). doi: 10.1103/PhysRevC.106.024612 http://doi.org/10.1103/PhysRevC.106.024612 Investigation of 48Ca-induced reactions with 242Pu and 238U targets at the JINR superheavy element factory Y.T. Oganessian , V.K. Utyonkov , Y.V. Lobanov , et al. , Measurements of cross sections and decay properties of the isotopes of elements 112, 114, and 116 produced in the fusion reactions 233,238 U, 242 Pu, and 248 Cm+ 48 Ca . Phys. Rev. C 70 , 064609 ( 2004 ). doi: 10.1103/PhysRevC.70.064609 http://doi.org/10.1103/PhysRevC.70.064609 Measurements of cross sections and decay properties of the isotopes of elements 112, 114, and 116 produced in the fusion reactions 233,238U, 242Pu, and 248Cm+48Ca S. Hofmann , D. Ackermann , S. Antalic , et al. , The reaction 48 Ca + 238 U → 286 112 * studied at the GSI-SHIP . European Physical Journal A 32 , 251 - 260 ( 2007 ). doi: 10.1140/epja/i2007-10373-x http://doi.org/10.1140/epja/i2007-10373-x The reaction 48Ca + 238U → 286112* studied at the GSI-SHIP D. Kaji , K. Morimoto , H. Haba , et al. , Decay measurement of 283 Cn produced in the 238 U( 48 Ca,3n) reaction using GARIS-II . Journal of the Physical Society of Japan 86 , 085001 ( 2017 ). doi: 10.7566/JPSJ.86.085001 http://doi.org/10.7566/JPSJ.86.085001 Decay measurement of 283Cn produced in the 238U(48Ca,3n) reaction using GARIS-II Y.T. Oganessian , F.S. Abdullin , S.N. Dmitriev , et al. , Investigation of the 243 Am+ 48 Ca reaction products previously observed in the experiments on elements 113, 115, and 117 . Phys. Rev. C 87 , 014302 ( 2013 ). doi: 10.1103/PhysRevC.87.014302 http://doi.org/10.1103/PhysRevC.87.014302 Investigation of the 243Am+48Ca reaction products previously observed in the experiments on elements 113, 115, and 117 Y.T. Oganessian , F.S. Abdullin , S.N. Dmitriev , et al. , New insights into the 243 Am+ 48 Ca reaction products previously observed in the experiments on elements 113, 115, and 117 . Phys. Rev. Lett . 108 , 022502 ( 2012 ). doi: 10.1103/PhysRevLett.108.022502 http://doi.org/10.1103/PhysRevLett.108.022502 New insights into the 243Am+48Ca reaction products previously observed in the experiments on elements 113, 115, and 117 Y.T. Oganessian , V.K. Utyonkov , S.N. Dmitriev , et al. , Synthesis of elements 115 and 113 in the reaction 243 Am+ 48 Ca . Phys. Rev. C 72 , 034611 ( 2005 ). doi: 10.1103/PhysRevC.72.034611 http://doi.org/10.1103/PhysRevC.72.034611 Synthesis of elements 115 and 113 in the reaction 243Am+48Ca L. Stavsetra , K.E. Gregorich , J. Dvorak , et al. , Independent verification of element 114 production in the 48 Ca+ 242 Pu reaction . Phys. Rev. Lett . 103 , 132502 ( 2009 ). doi: 10.1103/PhysRevLett.103.132502 http://doi.org/10.1103/PhysRevLett.103.132502 Independent verification of element 114 production in the 48Ca+242Pu reaction I.I. Gontchar , D.J. Hinde , M. Dasgupta , et al. , Double folding nucleus-nucleus potential applied to heavy-ion fusion reactions. Zhangjiang Laboratory, Shanghai 201210 , ChinaPhys. Rev. C 69 , 024610 ( 2004 ). doi: 10.1103/PhysRevC.69.024610 http://doi.org/10.1103/PhysRevC.69.024610 Double folding nucleus-nucleus potential applied to heavy-ion fusion reactions. Zhangjiang Laboratory, Shanghai 201210 All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Peng-Hui Chen, Hao Wu and Zu-Xing Yang. The first draft of the manuscript was written by Peng-Hui Chen and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-022-01157-0 This work was supported by National Natural Science Foundation of China (Nos. 12105241, 12175072), Natural Science Foundation of Jiangsu Province (No. BK20210788), Jiangsu Provincial Double-Innovation Doctoral Program (No. JSSCBS20211013), University Science Research Project of Jiangsu Province (No. 21KJB140026), Lv Yang Jin Feng (No. YZLYJFJH2021YXBS130), and Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences (No. IMPKFKT2021001). 2022-10-18 2022-11-24 2022-11-25 2023-01-13 2023-02-20 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2023 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-022-01157-0 http://dx.doi.org/10.1007/s41365-022-01157-0 . Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. 2023 Nuclear Science and Techniques 01 34