The binding energies of super heavy nuclei around the magic nucleus Z = 114, N = 184 was studied through an improved method based on Bethe and Weizsäcker method. This modification has been done by inclusion of some terms to Bethe and Weizsäcker’s initial mass formula. Since these nuclei are deformed in their ground-state, surface and coulomb terms are expanded as a function of deformation parameters. Also, shell effects are considered by including two more terms. Improved formula of the nuclear binding energy consists of seven unknown coefficients that are obtained using known experimental binding energies. The calculated binding energies for some super heavy nuclei are compared with the experimental data of AME 2012’s mass table as well as theoretical results of Zhongzhou Ren and Tiekuang Dong. This comparison indicates that the calculated results using the improved method are well agreed with experimental data than the theoretical results of Zhongzhou Ren and Tiekuang Dong.

Keywords: Binding Energy; Super Heavy Nuclei; Shell Effects, Finite Range Droplet Model

PACS number: 21.10.Dr, 27.90.+b, 21.60.−n

#### Introduction

One of the most important properties of nuclei is its stability that is related directly to its average binding energy (binding energy per nucleon). The nuclear binding energy plays a significant role in study of nuclear mass and its other features such as nuclear stability, decay half-life and nuclear fission. Our knowledge about the decay half-life of super heavy nucleus and the height of fission barrier or its reaction cross-section is also related to the nuclear stability, directly. Therefore, study of the binding energy of nuclei was one of the important issues of nuclear physics and scientists of nuclear physics have spent a lot of time to find it out [1-12]. Be the and Weizsäcker were first scientists that performed their own studies on nuclear mass in 1930s and obtained a useful phenomenological semiempirical relationship for nuclear mass as a sum of its various possible energies [1-2]. Following their studies, Bohr and Wheeler interpreted the energy released from neutron induced fission of 235U (236U compound isotope) using this approach [13]. At present one of the nuclear physics necessity is to find out the mass and binding energies of super heavy nuclei that are produced in heavy ion fusion-fission reactions [4-20]. Various methods have been introduced in order to obtain the binding energies of the nucleus in different region of mass numbers.

Others attempt to study the ground-state properties of nuclei are performed using the finite range droplet model (FRDM), folded Yukawa single particle potential and self-consistent mean field [8-11]. Recently, AdS/CFT correspondence holography model is used to calculate binding energies of light nuclei [20-23]. Considering 1 MeV uncertainty for theoretical calculations of binding energy of super heavy nuclei [24-25], is seems necessary to obtain a more accurate equation to calculate binding energy. In the shell-model representation, those nuclei with closed shell Z or N are called “magic” and also when both Z and N numbers are magic, the nucleus is called “double magic”. So biczewski et al. [26] improved original mass formula to consider closed shell properties of 270Hs, 208Pand 298114 heavy and super heavy isotopes. In addition, Nilsson et al.[27] obtained2×1019 year for the half-life of spontaneous fission of298114superheavy nucleus, which is much higher than the fission halflife of its neighbor isotopes. This means that the fission barrier for this nuclei is significantly different compared to its surrounded nuclei [28]. Therefore it seems necessary to consider this feature as shell effects for super heavy nuclei around 298114. This study attempts to find an improved relationship for binding energies of super heavy nuclei withZ≥105. This paper is organized as follows. In section 2 the improved version of binding energy formalism is presented with seven adjustable parameters. The calculated binding energies for 59 super heavy nuclei calculated using this improved approach are presented and compared with theoretical results of Zhongzhou Ren and TiekuangDong as well as the experimental data of AME 2012 mass table in section 3. Finally, concluding remarks are given in section 4.

#### Definition of Improved Approach

As it is mentioned in previous section, first phenomenological formula that presented for calculating of binding energy in the base of similarity between the liquid drop and nuclear material was proposed by Bethe and Weizsäcker1-3:
$B\left(A,Z\right)={a}_{v}A-{a}_{s}{A}^{\frac{2}{3}}-{a}_{c}{Z}^{2}{A}^{-\frac{1}{3}}-{a}_{a}{\left(\frac{A}{2}-Z\right)}^{2}{A}^{-1}+{a}_{p}\delta {A}^{-\frac{1}{2}}.\left(1\right)$
Where ${a}_{v}$ ،${a}_{s}$ ، ${a}_{c}$ ،${a}_{a}$ and ${a}_{p}$ are volume, surface, coulomb, asymmetry and pairing coefficients, respectively. These parameters can be adjusted using known binding energies of at least five isotopes. δ =1, 0 and -1 are simply agreed for even-even, even-odd and/or odd-even and odd-odd, Z - N nucleus, respectively. Amounts of these coefficients are related to selection of known experimental data of binding energies. Therefore there are many selections of these coefficients. In a study by Zhongzhou Ren and TiekuangDong [29] on heavy nuclei with Z ≥ 90, the following coefficients were obtained without considering the shell effects and deformation
$\left\{\begin{array}{c}{a}_{v}=15.7226MeV\\ {a}_{s}=17.7523MeV\\ {a}_{c}=0.7062MeV\\ {a}_{a}=96.2350MeV\\ {a}_{p}=10.6028MeV\end{array}$
Eq. (1) is improved in advance to consider deformation and shell effects for super heavy nuclei. Because of conservation of volume, the volume of nuclei is not change by deformation. Therefore the portion of volume in binding energy is

Considering of deformation on surface energy up to second order convert it to [31,32]

The effects of deformation also have an impact on Coulomb energy as well. Thus, by considering deformation, Coulomb energy is rewritten as [33],

The asymmetry energy not affected by deformation, so we have,

Also, the pairing energy not changed considerably by deformation,

Considerable difference between theoretical and experimental data for super heavy nuclei around closed shell Z=114 and N=184 indicate that the theoretical formula of binding energy should be revise [30]. Studies on fission barrier [34] indicate that the fission barriers high for nuclei around 298114 are growth considerably than other super heavy nuclei in this region.

Figure 1 clearly verifies this criterion. Therefore we revised the theoretical formula of binding energy by including two terms to emphasis on the shell effects as:

Figure 1: The fission barrier energy, in MeV as a function of mass number (A), for nuclei in the valley of stability. The smooth curve is the results of liquid-drop model. The irregular dashed curve is calculated from Meyers and Swiatecki mass formula that shows shell effects clearly. Nuclei with N ~ 50 (A ~ 90) should require the greatest amount of energy for their disintegration, lighter and heavier elements being more easily disrupted into comparable fragments [28].

By including these new terms and effects of deformation, the improved version of Bethe and Weizsäcker original equation for calculating binding energy of super heavy nuclei around Z=114 and N=184 is rewritten as

Following coefficients are obtained for this formula through fitting method with experimental data,
$\left\{\begin{array}{c}{a}_{v}=15.6446MeV\\ {a}_{s}=16.9970MeV\\ {a}_{c}=0.71197MeV\\ {a}_{a}=96.6732MeV\\ {a}_{6}=55.26028MeV\\ {a}_{7}=57.9814MeV.\end{array}$
And MeVis agreed for pairing energy coefficient. Also, the following values were used for δ:
$\text{δ}=\left\{\begin{array}{c}4.22Z-even,N-even\\ 1Z-even,N-odd\\ 0Z-odd,N-even\\ -2.66Z-odd,N-odd\end{array}$
This improved formula has been used to calculate the binding energies of 59 super heavy nuclei.

#### Numerical Results and Discussion

The calculated results for binding energies of 59 super heavy nuclei along with the results of Zhongzhou Ren and Tiekuang Dong and the experimental data [12]. Are presented in Table 1. This table shows that the calculated binding energies through this approach have been improved over the calculated results of Zhongzhou Ren and Tiekuang Dong and the obtained results are agreed well with the experimental data.

Average deviation and root mean square deviation of binding energies were calculated to indicate agreements between our results and experimental data than other theoretical results as,

$\sqrt{{\sigma }^{2}}={\left(\frac{{\sum }_{i=1}^{59}{\left({B}_{Expi}-{B}_{Cali}\right)}^{2}}{59}\right)}^{\frac{1}{2}}=0.597MeV\left(10\right)$

Figure 2 : shows the calculated results for 59 nuclei along with experimental results [12]. In figure 3 the average binding energies for 59 nuclei are compared with experimental data [12]. The deviations between experimental data and calculated binding energies for 59nuclei are presented in Figure 4. As it can be seen from these figures, the results of this approach are compatible with experimental data than theoretical results of Zhongzhou Ren and Tiekuang Dong.

Figure 2: Binding energies of 59 nuclei compared with experimental data and results of other theoretical methods.
Figure 3: Average binding energies of 59 nuclei compared with results of other theoretical methods and experimental data.
Figure 4: The deviations between experimental and calculated binding energies as a function of mass number.
Table 1:Calculated binding energies for 59 super heavy nuclei along with theoretical and experimental data.
 Z N A Elt. 105 164 269 Db 1969.106067 1969.887 7320 7323 105 165 270 Db 1974.623 1974.78 7313 7314 106 165 271 Sg 1979.243519 1979.655 7303 7305 106 166 272 Sg 1985.670635 1985.872 7300 7301 106 167 273 Sg 1991.320192 1990.443 7294 7291 107 163 270 Bh 1970.846677 1971.28 7299 7301 107 164 271 Bh 1978.177785 1977.758 7299 7298 107 165 272 Bh 1982.084731 1982.88 7287 7290 107 166 273 Bh 1988.859482 1989.078 7285 7286 107 167 274 Bh 1994.551196 1994.174 7278 7278 107 168 275 Bh 2001.091304 2000.075 7276 7273 108 166 274 Hs 1992.89976 1993.624 1994.014 7273 7276 7277 108 167 275 Hs 1998.682605 1998.425 1999.468 7268 7267 7271 108 168 276 Hs 2005.513782 2004.864 2006.069 7266 7264 7268 109 166 275 Mt 1994.987672 1995.675 1996.430 7256 7257 7260 109 167 276 Mt 2001.518211 2001.276 2002.078 7252 7251 7254 109 168 277 Mt 2007.988945 2007.696 2008.870 7249 7248 7252 109 169 278 Mt 2013.8358 2012.998 2014.294 7244 7241 7246
 110 165 275 Ds 1990.73 1991.83 7239 7243 110 166 276 Ds 1998.18 1999.07 7240 7243 110 167 277 Ds 2004.35 2004.37 2005.523 7236 7236 7240 110 168 278 Ds 2011.57 2011.61 2012.504 7236 7236 7239 110 169 279 Ds 2017.5 2016.61 2018.118 7231 7228 7233 110 170 280 Ds 2024.48 2023.56 2024.870 7230 7227 7232 111 167 278 Rg 2005.9 2006.6 2007.291 7216 7217 7220 111 168 279 Rg 2013.21 2013.54 2014.463 7216 7217 7220 111 169 280 Rg 2019.44 2019.36 2020.268 7212 7212 7215 111 170 281 Rg 2026.5 2026.29 7212 7211 111 171 282 Rg 2032.5 2031.81 7207 7205 112 167 279 Cn 2008.36 2009.08 2009.896 7198 7201 7204 112 168 280 Cn 2015.95 2016.56 2017.255 7200 7202 7204 112 169 281 Cn 2022.27 2022.36 7197 7197 112 170 282 Cn 2029.62 2029.55 7197 7197 112 171 283 Cn 2035.71 2035.05 7193 7191
 112 172 284 Cn 2042.83 2042.53 7193 7192 113 169 282 Ed 2023.38 2024.2 7175 7178 113 170 283 Ed 2030.82 2031.37 7176 7178 113 171 284 Ed 2037.2 2037.42 7173 7174 113 172 285 Ed 2044.41 2044.59 7173 7174 113 173 286 Ed 2050.56 2050.33 7170 7169 113 174 287 Ed 2057.53 2057.22 7169 7168 114 171 285 Fl 2039.5 2040.03 7156 7158 114 172 286 Fl 2047.08 2047.47 7158 7159 114 173 287 Fl 2053.31 2053.2 7154 7154 114 174 288 Fl 2059.83 2060.64 7152 7155 114 175 289 Fl 2066.58 2066.06 7150 7149 115 173 288 Ef 2054.37 2055.17 7133 7136 115 174 289 Ef 2061.71 2062.3 7134 7136 115 175 290 Ef 2068 2068.28 7131 7132 115 176 291 Ef 2075.12 2075.12 7131 7131 116 173 289 Lv 2056.31 2057.39 7115 7119
 116 174 290 Lv 2063.94 2064.8 7117 7120 116 175 291 Lv 2070.48 2070.76 7115 7116 116 176 292 Lv 2077.71 2078.16 7115 7117 116 177 293 Lv 2084.09 2083.52 7113 7111 117 175 292 Eh 2070.94 2072.03 7092 7096 117 176 293 Eh 2078.42 2079.42 7094 7097 117 177 294 Eh 2084.86 2085.34 7091 7093 118 175 293 Ei 2072.45 2073.56 7073 7077

#### Conclusion

Considering the fact that there is limited experimental information regarding super heavy nuclei, our main objective was to introduce an improved version to calculate accurate binding energies for super heavy nuclei. Two new terms originating from shell effect were included to Bethe and Weizsäcker’s original equation. Moreover, surface and coulomb energies of liquid drop formula are deformed as a function of deformation parameter up to second order. Seven unknown coefficients of the improved equation were obtained through fitting with experimental data. This improved approach is used to calculate the binding energies of 59 super heavy nuclei. Obtained results were compared with theoretical results of Zhongzhou Ren and Tiekuang Dong as well as experimental data. The average deviation between theoretical results and experimental data is obtained equal to 0.524 MeV that is illustrating the acceptable accuracy of new improved approach.