Predictions for production of superheavy nuclei with Z=105-112 in hot fusion reactions

Predictions for production of superheavy nuclei with Z =105-112 in hot fusion reactions 1 first-author 1 corresp Corresponding author, zhulong@mail.sysu.edu.cn Corresponding author, zhulong@mail.sysu.edu.cn zhulong@mail.sysu.edu.cn 1 1 1 1 Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai 519082, China The effects of mass asymmetry on the production of superheavy nuclei (SHN), within the dinuclear system (DNS) model, are investigated in this study. It is observed that the fusion probability decreases with decreasing mass asymmetry. A total of 192 possible combinations of projectiles from O to Ti and targets with half-lives longer than 30 days for producing SHN 264 Db, 265 Db, 267 Sg, 268 Bh, 268 Sg, 269 Bh, 271 Hs, 271 Mt, 272 Hs, 272 Mt, 273 Mt, 274 Ds, 275 Ds, 275 Rg, 276 Ds, 276 Rg, 277 Rg, 278 Cn, 279 Cn, and 280 Cn are examined. Further, the optimal combinations and incident energies for synthesizing these nuclei are predicted. Most of the cross sections for production of SHN are larger than 10 pb; therefore, the process can be carried out with the available experimental equipment. DNS model Systematic study Fusion reaction Superheavy nuclei Evaporation residue cross section 1 Introduction 1 In recent years, the synthesis of superheavy elements (SHEs) has been developed considerably, both experimentally and theoretically [ 1 1 - 3 3 1-3 ] and the Periodic Table of elements has been significantly extended. After the first successful synthesis of element 117 in 2010, all the elements in the seventh row of the Periodic Table have been discovered by mankind [ 3 3 - 10 10 3-10 ], which marks a great advancement in man’s cognition of the micro-world and a milestone in the evolution of the synthesis of SHEs. At present, efforts are being made to synthesize superheavy nuclei (SHN) experimentally. The elements from 107 to 113 were synthesized at GSI and RIKEN in cold fusion reactions [ 3 3 - 6 6 3-6 ], and the elements from 114 to 118 were produced at Dubna in hot fusion reactions using a neutron-rich projectile nucleus 48 Ca [ 1 1 , 7 7 - 9 9 7-9 ]. Presently, scientists are interested in the production of SHEs with Z ≥ 119, which will be a tremendous breakthrough in the field of nuclear physics. However, in the synthesis of SHEs, the cross sections decrease with the increase in the charge number of the nuclei, and are close to or even far below the order of pb, which makes the process very difficult. In addition, the production and measurement of the isotopes of already-known SHEs, which have not yet been synthesized, is important, as they can provide information on the trends of the decay properties along the neutron axis approaching the proposed neutron magic number, including the "island of stability". The synthesis of new SHN can greatly extend the map of the nuclides. According to the theory of nuclear structure [ 11 11 - 13 13 11-13 ], there is a double magic SHN, the nuclei in the neighborhood of which are relatively stable. In order to accurately locate the "island of stability", more experimental data needs to be available, therefore, the synthesis of more SHN is evidently important. At present, the main method used in the synthesis of SHN is the heavy-ion fusion evaporation reaction, hence, theoretical study of the process is essential for appropriate performance of the experiments [ 14 14 - 27 27 14-27 ]. Although, the ability to predict the production cross sections and incident energies is limited for theoretical models, it is still beneficial to investigate all the possible combinations systematically and predict the favorable ones for producing SHN [ 28 28 ]. In Fig. 1 Fig. 1 , the blue crosses represent the nuclei that will be investigated in this work. Due to the transition from the cold fusion reactions to hot fusion reactions, the SHN in this region are not synthesized yet. In this study, the effect of mass asymmetry on the fusion probability and cross sections of the reactions 26 Mg + 248 Cm, 36 S + 238 U, 48 Ca + 226 Ra, which lead to the same compound nucleus 274 HS*, are investigated first. In view of the long duration of the experiments and production rate, combinations of stable projectiles and targets with half-lives longer than 30 days are considered to be suitable. All possible combinations of projectiles from O to Ti and targets with half-lives longer than 30 days for producing SHN 264 Db, 265 Db, 267 Sg, 268 Bh, 268 Sg, 269 Bh, 271 Hs, 271 Mt, 272 Hs, 272 Mt, 273 Mt, 274 Ds, 275 Ds, 275 Rg, 276 Ds, 276 Rg, 277 Rg, 278 Cn, 279 Cn, and 280 Cn are examined and some optimal combinations and incident energies for synthesizing these nuclei are suggested. The article is organized as follows: In Sec. 2, a description of the DNS model is given; The Sec. 3 details an analysis of the results of calculation and a prediction of the optimal combinations for synthesizing several SHN in the gap region; The Sec. 4 presents the summary and the prospects. 2 Theoretical description 1 According to the concept of the DNS model, the complete fusion reaction is described as a diffusion process. All the nucleons of the projectile are transferred to the target and form a compound nucleus. This process is accompanied by the dissipation of energy and angular momentum. The evaporation residue (ER) cross section of the superheavy nucleus under the incident energy E c.m. in the center-of-mass frame can be written as [ 21 21 , 24 24 , 29 29 - 34 34 29-34 ]: <math display="block" id="M1"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>σ</mi><mrow><mtext>ER</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></mrow></mtd><mtd columnalign="left"><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><mtext>.m</mtext><mtext>.</mtext></mrow></msub></mrow></mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>J</mi><mo>=</mo><mn>0</mn></mrow><mi>∞</mi></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><mi>J</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><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><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><mi>J</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> where, T ( E c.m. , J ) is the probability that the collision system overcomes the Coulomb barrier and forms the dinuclear system [ 35 35 ]. P CN ( E c.m. , J ) is the fusion probability. W sur ( E c.m. , J ) is the survival probability of the compound nucleus. The fusion process described as a diffusion process, can be understood by numerically solving the master equations. The time evolution of the distribution probability function P ( Z 1 , N 1 , t ) for fragment 1 with the mass number A 1 and excitation energy E 1 at time t is described by the following master equations [ 17 17 , 24 24 ]: <math display="block" id="M2"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext>d</mtext><mi>P</mi><mo stretchy="false">(</mo><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msubsup><mi>Z</mi><mn>1</mn><mo>'</mo></msubsup></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><msubsup><mi>Z</mi><mn>1</mn><mo>'</mo></msubsup><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><mo stretchy="false">(</mo><msubsup><mi>Z</mi><mn>1</mn><mo>'</mo></msubsup><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><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><mi>P</mi><mo stretchy="false">(</mo><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><msubsup><mi>N</mi><mn>1</mn><mo>'</mo></msubsup></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><msubsup><mi>N</mi><mn>1</mn><mo>'</mo></msubsup></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><mo stretchy="false">(</mo><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msubsup><mi>N</mi><mn>1</mn><mo>'</mo></msubsup><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><msub><mi>d</mi><mrow><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msubsup><mi>N</mi><mn>1</mn><mo>'</mo></msubsup></mrow></msub><mi>P</mi><mo stretchy="false">(</mo><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo stretchy="false">[</mo><msubsup><mi>Λ</mi><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><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>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><mrow><mtext>fis</mtext></mrow></msubsup><mo stretchy="false">(</mo><mi>Θ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mn>,</mn><mi>t</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> The potential energy surface (PES) of the DNS in the fusion process is defined as: <math display="block" id="M3"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>U</mi><mo stretchy="false">(</mo><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>2</mn></msub><mn>,</mn><msub><mi>N</mi><mn>2</mn></msub><mn>,</mn></mrow></mtd><mtd columnalign="left"><mrow><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mtext>B</mtext></msub><mo stretchy="false">(</mo><msub><mi>Z</mi><mn>1</mn></msub><mn>,</mn><msub><mi>N</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>E</mi><mtext>B</mtext></msub><mo stretchy="false">(</mo><msub><mi>Z</mi><mn>2</mn></msub><mn>,</mn><msub><mi>N</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>−</mo><msub><mi>E</mi><mtext>B</mtext></msub><mo stretchy="false">(</mo><mi>Z</mi><mn>,</mn><mi>N</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>V</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where Z = Z 1 + Z 2 and N = N 1 + N 2 . E B ( Z i , N i ) and E B ( Z , N ) are the binding energies of the fragment i and the compound nucleus, respectively. V C and V N are the Coulomb potential and nuclear potential, respectively [ 24 24 ]. After the capture process, the DNS is formed. For the formation of the compound nucleus (CN), the projectile should overcome the inner fusion barrier B fus , defined as the difference between U BG , the highest point called Businaro–Gallone (B.G.) point on the left of U ( ≤ta ), and the potential at the entrance. Hence, summing up the probability on the left side of the B.G. point, the fusion probability of the DNS is given by [ 17 17 , 28 28 , 36 36 ], <math display="block" id="M4"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>P</mi><mrow><mtext>CN</mtext></mrow></msub><mo stretchy="false">(</mo><mi>J</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><msub><mi>Z</mi><mn>1</mn></msub><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><msub><mi>N</mi><mn>1</mn></msub><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>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 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><mtext> </mtext><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> The compound nucleus formed in the fusion reaction is unstable because of its high excitation energy and usually cools down by emitting γ -rays, evaporating particles (neutrons, protons, α and other light charged particles) and fission [ 37 37 - 39 39 37-39 ]. This process is described by a statistical model. Subsequent to the fission of the composite nucleus, the survival probability of the SHN after evaporation of x neutrons (considering only neutron evaporation and fission), can be written as: <math display="block" id="M5"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><msub><mi>W</mi><mrow><mtext>sur</mtext></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><mtext>CN</mtext></mrow><mo>*</mo></msubsup><mn>,</mn><mi>x</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>CN</mtext></mrow><mo>*</mo></msubsup><mn>,</mn><mi>x</mi><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><mo>×</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mstyle displaystyle="true"><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>x</mi></munderover></mstyle><mrow><mo>[</mo><mrow><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><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><mo>+</mo><msub><mi>Γ</mi><mtext>f</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></mfrac></mrow><mo>]</mo></mrow><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where J and <math id="M6"><mrow><msubsup><mi>E</mi><mrow><mtext>CN</mtext></mrow><mo>*</mo></msubsup></mrow></math> are the spin and excitation energies of the compound nucleus, respectively. The relationship between the excitation energy and incident energy in the center-of-mass frame is <math id="M7"><mrow><msubsup><mi>E</mi><mrow><mi>C</mi><mi>N</mi></mrow><mo>*</mo></msubsup><mo>=</mo><msub><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow></msub><mo>+</mo><mi>Q</mi></mrow></math> . The nuclear ground state masses are taken from [ 40 40 ]. <math id="M8"><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></math> and <math id="M9"><mrow><msub><mi>Γ</mi><mtext>f</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></math> are the widths of the i th neutron evaporation and fission, respectively. The solution is described later. <math id="M10"><mrow><msubsup><mi>E</mi><mi>i</mi><mo>*</mo></msubsup></mrow></math> is the excitation energy of the compound nucleus before evaporation of the i th neutron, which satisfies the following relation: <math id="M11"><mrow><msubsup><mi>E</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mo>*</mo></msubsup><mo>=</mo><msubsup><mi>E</mi><mi>i</mi><mo>*</mo></msubsup><mo>−</mo><msubsup><mi>B</mi><mi>i</mi><mi>n</mi></msubsup><mi>n</mi><mo>−</mo><mn>2</mn><msub><mi>T</mi><mi>i</mi></msub></mrow></math> . <math id="M12"><mrow><msubsup><mi>B</mi><mi>i</mi><mi>n</mi></msubsup></mrow></math> is the evaporation energy of the i th neutron. T i is the temperature before evaporation of the i th neutron, given by the relation <math id="M13"><mrow><msubsup><mi>E</mi><mi>i</mi><mo>*</mo></msubsup><mo>=</mo><mi>a</mi><msubsup><mi>T</mi><mi>i</mi><mn>2</mn></msubsup><mo>−</mo><msub><mi>T</mi><mi>i</mi></msub></mrow></math> . The realization probability for neutron evaporation <math id="M14"><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>E</mi><mrow><mtext>CN</mtext></mrow><mo>*</mo></msubsup><mn>,</mn><mi>x</mi><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo></mrow></math> can be obtained from Ref. [ 34 34 ]. According to Weisskopf’s theory of evaporation, the evaporation width of the particle ν can be written as [ 41 41 ]: <math display="block" id="M15"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>Γ</mi><mi>ν</mi></msub></mrow></mtd><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><msup><mi>E</mi><mo>*</mo></msup><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mn>2</mn><msub><mi>s</mi><mi>ν</mi></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>m</mi><mi>ν</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><mo>×</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mrow><msub><mi>I</mi><mi>ν</mi></msub></mrow></msub></mrow></mstyle><mi>ε</mi><mi>ρ</mi><mo stretchy="false">(</mo><msup><mi>E</mi><mo>*</mo></msup><mo>−</mo><msub><mi>B</mi><mi>ν</mi></msub><mo>−</mo><mi>ε</mi><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><msub><mi>σ</mi><mrow><mtext>inv</mtext></mrow></msub><mo stretchy="false">(</mo><mi>ε</mi><mo stretchy="false">)</mo><mtext> </mtext><mtext>d</mtext><mi>ε</mi><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where s ν is the spin, m ν is the reduced mass relative to the remaining nucleus, and B ν is the binding energy, of the evaporating particle ν . <math id="M16"><mrow><msub><mi>I</mi><mi>ν</mi></msub><mo>=</mo><mo stretchy="false">[</mo><mn>0,</mn><msup><mi>E</mi><mo>*</mo></msup><mo>−</mo><msub><mi>B</mi><mi>ν</mi></msub><mo>−</mo><mi>δ</mi><mo>−</mo><mfrac><mn>1</mn><mi>a</mi></mfrac><mo stretchy="false">]</mo></mrow></math> . The mass table is obtained from Ref. [ 40 40 ]. δ = 0, Δ, 2Δ for odd-odd, odd-even, and even-even nuclei, <math id="M17"><mrow><mi>Δ</mi><mo>=</mo><mn>11</mn><mo>/</mo><msqrt><mi>A</mi></msqrt></mrow></math> MeV. σ inv is the inverse reaction cross section for a particle ν with channel energy ε [ 42 42 ]. For the neutron, it is given by <math id="M18"><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></mrow></math> . <math id="M19"><mrow><msub><mi>R</mi><mi>ν</mi></msub><mo>=</mo><mn>1.16</mn><mo stretchy="false">[</mo><mroot><mrow><mi>A</mi><mo>−</mo><mn>1</mn></mrow><mn>3</mn></mroot><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></math> . The width of fission is given by the Bohr-Wheeler formula [ 43 43 ]: <math display="block" id="M20"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mi>Γ</mi></mtd><mtd columnalign="left"><mrow><msub><mrow/><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><mo>×</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mstyle displaystyle="true"><mrow><msub><mo>∫</mo><mrow><msub><mi>I</mi><mtext>f</mtext></msub></mrow></msub></mrow></mstyle><mfrac><mrow><msub><mi>ρ</mi><mi>f</mi></msub><mo stretchy="false">(</mo><msup><mi>E</mi><mo>*</mo></msup><mo>−</mo><msub><mi>B</mi><mtext>f</mtext></msub><mo>−</mo><mi>ε</mi><mn>,</mn><mi>J</mi><mo stretchy="false">)</mo><mtext> </mtext><mi>d</mi><mi>ε</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>e</mi><mi>x</mi><mi>p</mi><mo stretchy="false">[</mo><mo>−</mo><mn>2</mn><mi>π</mi><mo stretchy="false">(</mo><msup><mi>E</mi><mo>*</mo></msup><mo>−</mo><msub><mi>B</mi><mtext>f</mtext></msub><mo>−</mo><mi>ε</mi><mo stretchy="false">)</mo><mo>/</mo><mi>ℏ</mi><mi>ω</mi><mo stretchy="false">]</mo></mrow></mfrac><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> Here <math id="M21"><mrow><msub><mi>I</mi><mtext>f</mtext></msub><mo>=</mo><mo stretchy="false">[</mo><mn>0,</mn><msup><mi>E</mi><mo>*</mo></msup><mo>−</mo><msub><mi>B</mi><mtext>f</mtext></msub><mo>−</mo><mi>δ</mi><mo>−</mo><mfrac><mn>1</mn><mrow><msub><mi>a</mi><mtext>f</mtext></msub></mrow></mfrac><mo stretchy="false">]</mo></mrow></math> . B f ( E * , J ) is the fission barrier, which is of the form <math display="block" id="M22"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><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><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>e</mi><mi>x</mi><mi>p</mi><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mrow><msup><mi>E</mi><mo>*</mo></msup></mrow><mrow><msub><mi>E</mi><mtext>D</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> Here, E D =20 MeV is the shell damping energy. <math id="M23"><mrow><msubsup><mi>B</mi><mtext>f</mtext><mtext>M</mtext></msubsup></mrow></math> is the shell correction energy which is taken from Ref. [ 40 40 ]. After obtaining the widths of neutron evaporation and fission, the final W sur can be obtained from the Eq. (5). 3 Results and discussion 1 In order to investigate qualitatively the effects of mass asymmetry on the fusion probability, the driving potential, which is the minimum value of PES for combinations with the same mass asymmetry, is shown in Fig. 2 Fig. 2 , as a function of mass asymmetry for the reactions 26 Mg + 248 Cm, 36 S + 238 U, and 48 Ca + 226 Ra, leading to the same compound nucleus 274 Hs*. Although the systems evolve in both the proton and neutron degrees of freedom, the mass asymmetry can still be a good collective degree for estimating the fusion probability. The B.G. point is shown in this figure. The blue arrow indicates the inner fusion barrier B fus for the reaction 48 Ca + 226 Ra. The inner fusion barrier plays a main role in the competition between quasifission and complete fusion. According to the results of the calculation, the inner fusion barriers for the reactions 36 S + 238 U and 48 Ca + 226 Ra are B fus, S = 3.6 MeV and B fus, Ca = 8.6 MeV, respectively. It is observed that the injection point of the reaction 26 Mg + 248 Cm is located on the left side of the B.G. point, therefore, the fusion probability for 26 Mg + 248 Cm would be much larger than that of the other two reactions. Figure 3 Figure 3 presents the fusion probabilities as a function of the excitation energy of the compound nucleus in the reactions 26 Mg + 248 Cm, 36 S + 238 U, and 48 Ca + 226 Ra. It can be seen from the calculated results that with the same excitation energy, the fusion probability decreases with decreasing mass asymmetry. This is because a near symmetric system can overcome a high inner fusion barrier for forming a compound nucleus by nucleon transfer. The fusion probability for the reaction 26 Mg + 248 Cm, is close to 1 and much larger than that of the other two reactions. It can also be seen that the fusion probabilities for all the reactions increase with increasing energy. Figure 4 Figure 4 shows the excitation functions for the reactions 26 Mg + 248 Cm, 36 S + 238 U, and 48 Ca + 226 Ra. The available experimental data [ 44 44 - 46 46 44-46 ], which is indicated by points, is also shown. The curves show the results of calculation for different evaporation channels. It can be seen that the calculations duly authenticate the experimental data. The excitation functions for the reactions 22 Ne + 254 Cf, 26 Mg + 250 Cm, and 48 Ca + 228 Ra are shown in Fig. 5 Fig. 5 . All the reactions lead to the same compound nucleus, 276 Hs*. It can be seen that the cross sections decrease with decreasing mass asymmetry. However, it cannot be presumed that the reaction with larger value of mass asymmetry is better for producing SHN. For systems with larger values of mass asymmetry, the reactions are usually very hot. Therefore, in the de-excitation process, many neutrons evaporate. This actually limits the neutron richness of the synthesized SHN. Moreover, due to the limitation of the target, the projectile cannot be very light. In addition, the mass asymmetry not only affects the fusion probability, but also has an impact on the probability of the capture and survival stages. Therefore, in this study, all the possible combinations are examined systematically, with stable projectiles of Z =8-22 and targets with half-lives longer than 30 days, for synthesizing the unknown SHN, as shown in Fig. 1 Fig. 1 . The cross sections of different projectile-target combinations are compared to obtain the optimal ones for synthesizing the unknown SHN, as shown in Figs. 6 Figs. 6 , 7 7 , 8 8 , 9 9 and 10 10 . It can be seen that for producing most of these nuclei, the maximal ER cross sections are larger than 10 pb. For example, for producing the 268 Sg, the ER cross section is close to 1 nb in the reaction 18 O + 254 Es → 268 Sg + 4n with an incident energy of 82.87 MeV. Moreover, for producing the 264 Db, 265 Db, 267 Sg, 268 Bh, 269 Bh, 271 Hs, 271 Mt, 272 Hs, 272 Mt, 273 Mt, 274 Ds, 275Rg, 275 Ds, 276 Ds, 276 Rg, 277 Rg, the most promising reactions are 18 O + 249 Bk → 264 Db + 3n (12 pb at 79.36 MeV), 19 F + 250 Cm → 265 Db + 4n (5.6 pb at 89.55 MeV), 17 O + 254 Cf → 267 Sg + 4n (82 pb at 82.01 MeV), 18 O + 254 Es → 268 Bh + 4n (160 pb at 84.22 MeV), 18 O + 255 Es → 269 Bh + 4n (770 pb at 84 MeV), 18 O + 257 Fm → 271 Hs + 4n (520 pb at 84.56 MeV), 22 Ne + 252 Es → 271 Mt + 3n (190 pb at 99.25 MeV), 22 Ne + 254 Cf → 272 Hs + 4n (190 pb at 101.87 MeV), 32 Si + 243 Am → 272 Mt + 3n (51 pb at 134.9 MeV), 22 Ne + 255 Es → 273 Mt + 4n (140 pb at 103.52 MeV), 21 Ne + 257 Fm → 274 Ds + 4n (23 pb at 104.13 MeV), 30 Si + 248 Bk → 275 Rg + 3n (20 pb at 137.64 MeV), 22 Ne + 257 Fm → 275 Ds + 4n (20 pb at 104.92 MeV), 22 Ne + 257 Fm → 276 Ds + 3n (9.2 pb at 99.92 MeV), 32 Si + 247 Bk → 276 Rg + 3n (10 pb at 139.94 MeV), and 32 Si + 248 Bk → 277 Rg + 3n (18 pb at 139.31 MeV), respectively. However, it can be seen that for synthesis of 278-280 Cn the ER cross sections are approximately 1 pb, therefore, the reaction is still feasible with the available experimental equipment. The maximal evaporation cross sections, optimal incident energies, and corresponding evaporation channels are deduced and presented clearly in Table. 1 Table. 1 . It is found that for synthesizing the isotopes 267 Sg and 268 Sg, the target 254 Cf can be used. The 18 O is predicted as a projectile, for producing both the isotopes 268 Bh and 269 Bh. In addition, the combinations of projectile and target are the same for synthesizing the 275 Ds and 276 Ds, but in different evaporation channels. ON: Objective Nucleus BC: Best projectile and target Combination EC: Evaporation Channel <math id="M24"><mrow><msubsup><mi>E</mi><mrow><mtext>CN</mtext></mrow><mo>*</mo></msubsup></mrow></math> is the excitation energy where the largest ER cross section appears in the unit of MeV, <math id="M25"><mrow><msubsup><mi>E</mi><mrow><mtext>c</mtext><mtext>.m</mtext><mtext>.</mtext></mrow><mo>*</mo></msubsup></mrow></math> is the incident energy in the center-of-mass frame corresponding to <math id="M26"><mrow><msubsup><mi>E</mi><mrow><mtext>CN</mtext></mrow><mo>*</mo></msubsup></mrow></math> . σ ER, max is the largest ER cross section in the unit of pb. 1 1 *ON 1 1 2*BC 1 1 2*EC 1 1 E CN * ( E c.m. ) 1 1 2* σ ER, max (pb) 1 1 264 Db 1 1 18 O + 249 Bk 1 1 3n 1 1 36 (79.36) 1 1 1.2 × 10 1 1 1 265 Db 1 1 19 F + 250 Cm 1 1 4n 1 1 44 (89.55) 1 1 5.6 1 1 267 Sg 1 1 17 O + 254 Cf 1 1 4n 1 1 40 (82.01) 1 1 8.2 × 10 1 1 1 268 Sg 1 1 18 O + 254 Cf 1 1 4n 1 1 39 (82.87) 1 1 9.2 × 10 2 1 1 268 Bh 1 1 18 O + 254 Es 1 1 4n 1 1 39 (84.22) 1 1 1.6 × 10 2 1 1 269 Bh 1 1 18 O + 255 Es 1 1 4n 1 1 39 (84.00) 1 1 7.7 × 10 2 1 1 271 Hs 1 1 18 O + 257 Fm 1 1 4n 1 1 38 (84.56) 1 1 5.2 × 10 2 1 1 271 Mt 1 1 22 Ne + 252 Es 1 1 3n 1 1 34 (99.25) 1 1 1.9 × 10 2 1 1 272 Hs 1 1 22 Ne + 254 Cf 1 1 4n 1 1 39 (101.87) 1 1 1.9 × 10 2 1 1 272 Mt 1 1 32 Si + 243 Am 1 1 3n 1 1 32 (134.9) 1 1 5.1 × 10 1 1 1 273 Mt 1 1 22 Ne + 255 Es 1 1 4n 1 1 39 (103.52) 1 1 1.4 × 10 2 1 1 274 Ds 1 1 21 Ne + 257 Fm 1 1 4n 1 1 43 (104.13) 1 1 2.3 × 10 1 1 1 275 Rg 1 1 30 Si + 248 Bk 1 1 3n 1 1 33 (137.64) 1 1 2.0 × 10 1 1 1 275 Ds 1 1 22 Ne + 257 Fm 1 1 4n 1 1 39 (104.92) 1 1 2.0 × 10 1 1 1 276 Ds 1 1 22 Ne + 257 Fm 1 1 3n 1 1 34 (99.92) 1 1 9.2 1 1 276 Rg 1 1 32 Si + 247 Bk 1 1 3n 1 1 32 (139.94) 1 1 1.0 × 10 1 1 1 277 Rg 1 1 32 Si + 248 Bk 1 1 3n 1 1 32 (139.31) 1 1 1.8 × 10 1 1 1 278 Cn 1 1 30 Si + 251 Cf 1 1 3n 1 1 35 (140.57) 1 1 1.7 1 1 279 Cn 1 1 30 Si + 252 Cf 1 1 3n 1 1 36 (140.22) 1 1 1.1 1 1 280 Cn 1 1 30 Si + 254 Cf 1 1 4n 1 1 43 (144.86) 1 1 3.0 4 Summary 1 The production of several SHN was investigated systematically within the dinuclear system model. The mass asymmetry effects were studied and it was observed that the fusion probabilities decreased with decreasing mass asymmetry. Suitable reactions for producing SHN in the gap region, as shown in Fig. 1 Fig. 1 , were predicted by reviewing the stable beam induced hot fusion reactions, and 192 possible combinations with projectiles from O to Ti and targets with half-lives longer than 30 days, for producing SHN 264 Db, 265 Db, 267 Sg, 268 Bh, 268 Sg, 269 Bh, 271 Hs, 271 Mt, 272 Hs, 272 Mt, 273 Mt, 274 Ds, 275 Ds, 275 Rg, 276 Ds, 276 Rg, 277 Rg, 278 Cn, 279 Cn, and 280 Cn were examined. The optimal combinations and incident energies for synthesizing these nuclei were predicted. It was found that the production cross sections, for synthesizing most of these SHN, were larger than 10 pb. The predicted cross section was 920 pb for the production of 268 Sg; hence, the reaction could be carried out with the available experimental equipment. 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(Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2018 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/s41365-018-0501-2 http://dx.doi.org/10.1007/s41365-018-0501-2 . 2018 Nuclear Science and Techniques 11 29