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 groundstate, 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 halflife and nuclear
fission. Our knowledge about the decay halflife of super heavy nucleus
and the height of fission barrier or its reaction crosssection 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 [112]. 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 [12]. 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 fusionfission reactions [420].
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 groundstate properties of nuclei are
performed using the finite range droplet model (FRDM), folded Yukawa
single particle potential and selfconsistent mean field [811]. Recently,
AdS/CFT correspondence holography model is used to calculate binding
energies of light nuclei [2023]. Considering 1 MeV uncertainty for
theoretical calculations of binding energy of super heavy nuclei [2425],
is seems necessary to obtain a more accurate equation to calculate binding
energy. In the shellmodel 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 halflife 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
Weizsäcker13:
$$B\left(A,Z\right)={a}_{v}A{a}_{s}{A}^{\frac{2}{3}}{a}_{c}{Z}^{2}{A}^{\frac{1}{3}}{a}_{a}{\left(\frac{A}{2}Z\right)}^{2}{A}^{1}+{a}_{p}\delta {A}^{\frac{1}{2}}.\left(1\right)$$
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 eveneven, evenodd and/or oddeven and oddodd, 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
$$\{\begin{array}{c}{a}_{v}=15.7226MeV\\ {a}_{s}=17.7523MeV\\ {a}_{c}=0.7062MeV\\ {a}_{a}=96.2350MeV\\ {a}_{p}=10.6028MeV\end{array}$$
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
$${B}_{V}={a}_{v}A.\text{}\left(2\right)$$
Considering of deformation on surface energy up to second order convert
it to [31,32]
$${B}_{s}={a}_{s}{A}^{\frac{2}{3}}\left(1+\frac{2}{5}{a}_{2}{}^{2}\frac{4}{105}{a}_{2}{}^{3}+\dots \right).\text{}(3)$$
The effects of deformation also have an impact on Coulomb energy as
well. Thus, by considering deformation, Coulomb energy is rewritten as
[33],
$${B}_{c}={a}_{c}{Z}^{2}{A}^{}{}^{\frac{1}{3}}\left(1\frac{1}{5}{a}_{2}{}^{2}\frac{4}{105}{a}_{2}{}^{3}+\dots \right).\text{}\left(4\right)\text{}$$
The asymmetry energy not affected by deformation, so we have,
$${B}_{a}={a}_{a}{\left(\frac{A}{2}Z\right)}^{2}{A}^{1}.\text{}\left(5\right)$$
Also, the pairing energy not changed considerably by deformation,
$${B}_{p}={a}_{p}\delta {A}^{\frac{1}{2}}.\text{}\left(6\right)$$
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:
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
$$\begin{array}{l}B\left(A,Z\right)={a}_{v}A{a}_{s}{A}^{\frac{2}{3}}\left(1+\frac{2}{5}{a}_{2}{}^{2}\frac{4}{105}{a}_{2}{}^{3}+\dots \right)\\ {a}_{c}{Z}^{2}{A}^{}{}^{\frac{1}{3}}\left(1\frac{1}{5}{a}_{2}{}^{2}\frac{4}{105}{a}_{2}{}^{3}+\dots \right)\\ {a}_{a}{\left(\frac{A}{2}Z\right)}^{2}{A}^{1}{a}_{p}\delta {A}^{\frac{1}{2}}\\ +{a}_{6}\left1\frac{298}{A}\right{a}_{7}\left1\frac{184}{N}\right.\text{}(8)\end{array}$$
Following coefficients are obtained for this formula through fitting method with experimental data,
$$\{\begin{array}{c}{a}_{v}=15.6446MeV\\ {a}_{s}=16.9970MeV\\ {a}_{c}=0.71197MeV\\ {a}_{a}=96.6732MeV\\ {a}_{6}=55.26028MeV\\ {a}_{7}=57.9814MeV.\end{array}$$
And ${a}_{p}=3$
MeVis agreed for pairing energy coefficient. Also, the following values were used for δ:
$$\text{\delta}=\{\begin{array}{c}4.22Zeven,Neven\\ 1Zeven,Nodd\\ 0Zodd,Neven\\ 2.66Zodd,Nodd\end{array}$$
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,
$$<\sigma >=\frac{{{\displaystyle \sum}}_{i=1}^{59}\left{B}_{Expi}{B}_{Cali}\right}{59}=0.524MeV\text{}\left(9\right)$$
$$\sqrt{{\sigma}^{2}}={\left(\frac{{{\displaystyle \sum}}_{i=1}^{59}{\left({B}_{Expi}{B}_{Cali}\right)}^{2}}{59}\right)}^{\frac{1}{2}}=0.597MeV\left(10\right)$$
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.
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 
1990.730051 
1991.825 

7239 
7243 

110 
166 
276 
Ds 
1998.182942 
1999.068 

7240 
7243 

110 
167 
277 
Ds 
2004.353034 
2004.372 
2005.523 
7236 
7236 
7240 
110 
168 
278 
Ds 
2011.565406 
2011.608 
2012.504 
7236 
7236 
7239 
110 
169 
279 
Ds 
2017.501792 
2016.612 
2018.118 
7231 
7228 
7233 
110 
170 
280 
Ds 
2024.478874 
2023.56 
2024.870 
7230 
7227 
7232 
111 
167 
278 
Rg 
2005.903234 
2006.604 
2007.291 
7216 
7217 
7220 
111 
168 
279 
Rg 
2013.206421 
2013.543 
2014.463 
7216 
7217 
7220 
111 
169 
280 
Rg 
2019.435951 
2019.36 
2020.268 
7212 
7212 
7215 
111 
170 
281 
Rg 
2026.502409 
2026.291 

7212 
7211 

111 
171 
282 
Rg 
2032.501171 
2031.81 

7207 
7205 

112 
167 
279 
Cn 
2008.357044 
2009.079 
2009.896 
7198 
7201 
7204 
112 
168 
280 
Cn 
2015.949574 
2016.56 
2017.255 
7200 
7202 
7204 
112 
169 
281 
Cn 
2022.2684 
2022.357 

7197 
7197 

112 
170 
282 
Cn 
2029.622158 
2029.554 

7197 
7197 

112 
171 
283 
Cn 
2035.708877 
2035.053 

7193 
7191 

112 
172 
284 
Cn 
2042.828835 
2042.528 

7193 
7192 

113 
169 
282 
Ed 
2023.380036 
2024.196 

7175 
7178 

113 
170 
283 
Ed 
2030.822932 
2031.374 

7176 
7178 

113 
171 
284 
Ed 
2037.199747 
2037.416 

7173 
7174 

113 
172 
285 
Ed 
2044.4075 
2044.59 

7173 
7174 

113 
173 
286 
Ed 
2050.555048 
2050.334 

7170 
7169 

113 
174 
287 
Ed 
2057.532682 
2057.216 

7169 
7168 

114 
171 
285 
Fl 
2039.5004 
2040.03 

7156 
7158 

114 
172 
286 
Fl 
2047.07736 
2047.474 

7158 
7159 

114 
173 
287 
Fl 
2053.311319 
2053.198 

7154 
7154 

114 
174 
288 
Fl 
2059.8252 
2060.64 

7152 
7155 

114 
175 
289 
Fl 
2066.579426 
2066.061 

7150 
7149 

115 
173 
288 
Ef 
2054.365367 
2055.168 

7133 
7136 

115 
174 
289 
Ef 
2061.711572 
2062.304 

7134 
7136 

115 
175 
290 
Ef 
2068.0048 
2068.28 

7131 
7132 

115 
176 
291 
Ef 
2075.1223 
2075.121 

7131 
7131 

116 
173 
289 
Lv 
2056.310388 
2057.391 

7115 
7119 

116 
174 
290 
Lv 
2063.938 
2064.8 

7117 
7120 

116 
175 
291 
Lv 
2070.481813 
2070.756 

7115 
7116 

116 
176 
292 
Lv 
2077.7132 
2078.164 

7115 
7117 

116 
177 
293 
Lv 
2084.089 
2083.523 

7113 
7111 

117 
175 
292 
Eh 
2070.9419 
2072.032 

7092 
7096 

117 
176 
293 
Eh 
2078.42155 
2079.421 

7094 
7097 

117 
177 
294 
Eh 
2084.8576 
2085.342 

7091 
7093 

118 
175 
293 
Ei 
2072.450485 
2073.561 

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.
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