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( A,Z )= a v A a s A 2 3 a c Z 2 A 1 3 a a ( A 2 Z ) 2 A 1 + a p δ A 1 2 .( 1 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGcbWaaeWaa8aabaWdbiaadgeacaGGSaGaamOwaaGaayjkaiaa wMcaaiabg2da9iaadggapaWaaSbaaSqaa8qacaWG2baapaqabaGcpe GaamyqaiabgkHiTiaadggapaWaaSbaaSqaa8qacaWGZbaapaqabaGc peGaamyqa8aadaahaaWcbeqaa8qadaWcaaWdaeaapeGaaGOmaaWdae aapeGaaG4maaaaaaGccqGHsislcaWGHbWdamaaBaaaleaapeGaam4y aaWdaeqaaOWdbiaadQfapaWaaWbaaSqabeaapeGaaGOmaaaakiaadg eapaWaaWbaaSqabeaapeGaeyOeI0YaaSaaa8aabaWdbiaaigdaa8aa baWdbiaaiodaaaaaaOGaeyOeI0Iaamyya8aadaWgaaWcbaWdbiaadg gaa8aabeaak8qadaqadaWdaeaapeWaaSaaa8aabaWdbiaadgeaa8aa baWdbiaaikdaaaGaeyOeI0IaamOwaaGaayjkaiaawMcaa8aadaahaa Wcbeqaa8qacaaIYaaaaOGaamyqa8aadaahaaWcbeqaa8qacqGHsisl caaIXaaaaOGaey4kaSIaamyya8aadaWgaaWcbaWdbiaadchaa8aabe aak8qacqaH0oazcaWGbbWdamaaCaaaleqabaWdbiabgkHiTmaalaaa paqaa8qacaaIXaaapaqaa8qacaaIYaaaaaaakiaac6cadaqadaWdae aapeGaaGymaaGaayjkaiaawMcaaaaa@65ED@
Where a v MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGHbWdamaaBaaaleaapeGaamODaaWdaeqaaaaa@3832@ ، a s MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGHbWdamaaBaaaleaapeGaam4CaaWdaeqaaaaa@382F@ ، a c MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGHbWdamaaBaaaleaapeGaam4yaaWdaeqaaaaa@381F@ ، a a MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGHbWdamaaBaaaleaapeGaamyyaaWdaeqaaaaa@381D@ and a p MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGHbWdamaaBaaaleaapeGaamiCaaWdaeqaaaaa@382C@ 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
{ a v =15.7226MeV a s =17.7523MeV a c =0.7062MeV a a =96.2350MeV a p =10.6028MeV MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGabaWdaeaafaqabeqbbaaaaeaapeGaamyya8aadaWgaaWcbaWd biaadAhaa8aabeaak8qacqGH9aqpcaaIXaGaaGynaiaac6cacaaI3a GaaGOmaiaaikdacaaI2aGaamytaiaadwgacaWGwbaapaqaa8qacaWG HbWdamaaBaaaleaapeGaam4CaaWdaeqaaOWdbiabg2da9iaaigdaca aI3aGaaiOlaiaaiEdacaaI1aGaaGOmaiaaiodacaWGnbGaamyzaiaa dAfaa8aabaWdbiaadggapaWaaSbaaSqaa8qacaWGJbaapaqabaGcpe Gaeyypa0JaaGimaiaac6cacaaI3aGaaGimaiaaiAdacaaIYaGaamyt aiaadwgacaWGwbaapaqaa8qacaWGHbWdamaaBaaaleaapeGaamyyaa WdaeqaaOWdbiabg2da9iaaiMdacaaI2aGaaiOlaiaaikdacaaIZaGa aGynaiaaicdacaWGnbGaamyzaiaadAfaa8aabaWdbiaadggapaWaaS baaSqaa8qacaWGWbaapaqabaGcpeGaeyypa0JaaGymaiaaicdacaGG UaGaaGOnaiaaicdacaaIYaGaaGioaiaad2eacaWGLbGaamOvaaaaai aawUhaaaaa@6E3D@
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.        ( 2 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGcbWdamaaBaaaleaapeGaamOvaaWdaeqaaOWdbiabg2da9iaa dggapaWaaSbaaSqaa8qacaWG2baapaqabaGcpeGaamyqaiaac6caca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiamaa bmaapaqaa8qacaaIYaaacaGLOaGaayzkaaaaaa@445B@
Considering of deformation on surface energy up to second order convert it to [31,32]
B s = a s A 2 3 ( 1+ 2 5 a 2 2 4 105 a 2 3 + ).    (3) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGcbWdamaaBaaaleaapeGaam4CaaWdaeqaaOWdbiabg2da9iaa dggapaWaaSbaaSqaa8qacaWGZbaapaqabaGcpeGaamyqa8aadaahaa Wcbeqaa8qadaWcaaWdaeaapeGaaGOmaaWdaeaapeGaaG4maaaaaaGc daqadaWdaeaapeGaaGymaiabgUcaRmaalaaapaqaa8qacaaIYaaapa qaa8qacaaI1aaaaiaadggapaWaaSbaaSqaa8qacaaIYaaapaqabaGc daahaaWcbeqaa8qacaaIYaaaaOGaeyOeI0YaaSaaa8aabaWdbiaais daa8aabaWdbiaaigdacaaIWaGaaGynaaaacaWGHbWdamaaBaaaleaa peGaaGOmaaWdaeqaaOWaaWbaaSqabeaapeGaaG4maaaakiabgUcaRi abgAci8cGaayjkaiaawMcaaiaac6cacaqGGaGaaeiiaiaabccacaqG GaGaaiikaiaaiodacaGGPaaaaa@5577@
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 1 3 ( 1 1 5 a 2 2 4 105 a 2 3 + ).     ( 4 )   MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGcbWdamaaBaaaleaapeGaam4yaaWdaeqaaOWdbiabg2da9iaa dggapaWaaSbaaSqaa8qacaWGJbaapaqabaGcpeGaamOwa8aadaahaa Wcbeqaa8qacaaIYaaaaOGaamyqa8aadaahaaWcbeqaa8qacqGHsisl aaGcpaWaaWbaaSqabeaapeWaaSaaa8aabaWdbiaaigdaa8aabaWdbi aaiodaaaaaaOWaaeWaa8aabaWdbiaaigdacqGHsisldaWcaaWdaeaa peGaaGymaaWdaeaapeGaaGynaaaacaWGHbWdamaaBaaaleaapeGaaG OmaaWdaeqaaOWaaWbaaSqabeaapeGaaGOmaaaakiabgkHiTmaalaaa paqaa8qacaaI0aaapaqaa8qacaaIXaGaaGimaiaaiwdaaaGaamyya8 aadaWgaaWcbaWdbiaaikdaa8aabeaakmaaCaaaleqabaWdbiaaioda aaGccqGHRaWkcqGHMacVaiaawIcacaGLPaaacaGGUaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccadaqadaWdaeaapeGaaGinaaGaayjkaiaa wMcaaSGaaeiiaiaabccaaaa@5AD8@
The asymmetry energy not affected by deformation, so we have,
B a = a a ( A 2 Z ) 2 A 1 .    ( 5 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGcbWdamaaBaaaleaapeGaamyyaaWdaeqaaOWdbiabg2da9iaa dggapaWaaSbaaSqaa8qacaWGHbaapaqabaGcpeWaaeWaa8aabaWdbm aalaaapaqaa8qacaWGbbaapaqaa8qacaaIYaaaaiabgkHiTiaadQfa aiaawIcacaGLPaaapaWaaWbaaSqabeaapeGaaGOmaaaakiaadgeapa WaaWbaaSqabeaapeGaeyOeI0IaaGymaaaakiaac6cacaqGGaGaaeii aiaabccacaqGGaWaaeWaa8aabaWdbiaaiwdaaiaawIcacaGLPaaaaa a@4A1D@
Also, the pairing energy not changed considerably by deformation,
B p = a p δ A 1 2 .           ( 6 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGcbWdamaaBaaaleaapeGaamiCaaWdaeqaaOWdbiabg2da9iaa dggapaWaaSbaaSqaa8qacaWGWbaapaqabaGcpeGaeqiTdqMaamyqa8 aadaahaaWcbeqaa8qacqGHsisldaWcaaWdaeaapeGaaGymaaWdaeaa peGaaGOmaaaaaaGccaGGUaGaaeiiaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccadaqadaWdaeaa peGaaGOnaaGaayjkaiaawMcaaaaa@4B0A@
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].
B Shell = a 6 | 1 298 A | a 7 | 1 184 N |.          (7) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaWGcbWdamaaBaaaleaapeGaam4uaiaadIgacaWGLbGaamiBaiaa dYgaa8aabeaak8qacqGH9aqpcaWGHbWdamaaBaaaleaapeGaaGOnaa WdaeqaaOWdbmaaemaapaqaa8qacaaIXaGaeyOeI0YaaSaaa8aabaWd biaaikdacaaI5aGaaGioaaWdaeaapeGaamyqaaaaaiaawEa7caGLiW oacqGHsislcaWGHbWdamaaBaaaleaapeGaaG4naaWdaeqaaOWdbmaa emaapaqaa8qacaaIXaGaeyOeI0YaaSaaa8aabaWdbiaaigdacaaI4a GaaGinaaWdaeaapeGaamOtaaaaaiaawEa7caGLiWoacaGGUaGaaeii aiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaacIcacaaI3aGaaiykaaaa@5B98@

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
B( A,Z )= a v A a s A 2 3 ( 1+ 2 5 a 2 2 4 105 a 2 3 + ) a c Z 2 A 1 3 ( 1 1 5 a 2 2 4 105 a 2 3 + ) a a ( A 2 Z ) 2 A 1 a p δ A 1 2 + a 6 | 1 298 A | a 7 | 1 184 N |.                   (8) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaa aaaaWdbiaadkeadaqadaWdaeaapeGaamyqaiaacYcacaWGAbaacaGL OaGaayzkaaGaeyypa0Jaamyya8aadaWgaaWcbaWdbiaadAhaa8aabe aak8qacaWGbbGaeyOeI0Iaamyya8aadaWgaaWcbaWdbiaadohaa8aa beaak8qacaWGbbWdamaaCaaaleqabaWdbmaalaaapaqaa8qacaaIYa aapaqaa8qacaaIZaaaaaaakmaabmaapaqaa8qacaaIXaGaey4kaSYa aSaaa8aabaWdbiaaikdaa8aabaWdbiaaiwdaaaGaamyya8aadaWgaa WcbaWdbiaaikdaa8aabeaakmaaCaaaleqabaWdbiaaikdaaaGccqGH sisldaWcaaWdaeaapeGaaGinaaWdaeaapeGaaGymaiaaicdacaaI1a aaaiaadggapaWaaSbaaSqaa8qacaaIYaaapaqabaGcdaahaaWcbeqa a8qacaaIZaaaaOGaey4kaSIaeyOjGWlacaGLOaGaayzkaaaabaGaey OeI0Iaamyya8aadaWgaaWcbaWdbiaadogaa8aabeaak8qacaWGAbWd amaaCaaaleqabaWdbiaaikdaaaGccaWGbbWdamaaCaaaleqabaWdbi abgkHiTaaak8aadaahaaWcbeqaa8qadaWcaaWdaeaapeGaaGymaaWd aeaapeGaaG4maaaaaaGcdaqadaWdaeaapeGaaGymaiabgkHiTmaala aapaqaa8qacaaIXaaapaqaa8qacaaI1aaaaiaadggapaWaaSbaaSqa a8qacaaIYaaapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaOGaeyOeI0 YaaSaaa8aabaWdbiaaisdaa8aabaWdbiaaigdacaaIWaGaaGynaaaa caWGHbWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWaaWbaaSqabeaape GaaG4maaaakiabgUcaRiabgAci8cGaayjkaiaawMcaaaqaaiabgkHi TiaadggapaWaaSbaaSqaa8qacaWGHbaapaqabaGcpeWaaeWaa8aaba Wdbmaalaaapaqaa8qacaWGbbaapaqaa8qacaaIYaaaaiabgkHiTiaa dQfaaiaawIcacaGLPaaapaWaaWbaaSqabeaapeGaaGOmaaaakiaadg eapaWaaWbaaSqabeaapeGaeyOeI0IaaGymaaaakiabgkHiTiaadgga paWaaSbaaSqaa8qacaWGWbaapaqabaGcpeGaeqiTdqMaamyqa8aada ahaaWcbeqaa8qacqGHsisldaWcaaWdaeaapeGaaGymaaWdaeaapeGa aGOmaaaaaaaak8aabaWdbiabgUcaRiaadggapaWaaSbaaSqaa8qaca aI2aaapaqabaGcpeWaaqWaa8aabaWdbiaaigdacqGHsisldaWcaaWd aeaapeGaaGOmaiaaiMdacaaI4aaapaqaa8qacaWGbbaaaaGaay5bSl aawIa7aiabgkHiTiaadggapaWaaSbaaSqaa8qacaaI3aaapaqabaGc peWaaqWaa8aabaWdbiaaigdacqGHsisldaWcaaWdaeaapeGaaGymai aaiIdacaaI0aaapaqaa8qacaWGobaaaaGaay5bSlaawIa7aiaac6ca caqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaiikaiaaiIdacaGGPaaaaaa@AC54@
Following coefficients are obtained for this formula through fitting method with experimental data,
{ a v =15.6446MeV a s =16.9970MeV a c =0.71197MeV a a =96.6732MeV a 6 =55.26028MeV a 7 =57.9814MeV. MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGabaWdaeaafaqabeGbbaaaaeaapeGaamyya8aadaWgaaWcbaWd biaadAhaa8aabeaak8qacqGH9aqpcaaIXaGaaGynaiaac6cacaaI2a GaaGinaiaaisdacaaI2aGaamytaiaadwgacaWGwbaapaqaa8qacaWG HbWdamaaBaaaleaapeGaam4CaaWdaeqaaOWdbiabg2da9iaaigdaca aI2aGaaiOlaiaaiMdacaaI5aGaaG4naiaaicdacaWGnbGaamyzaiaa dAfaa8aabaWdbiaadggapaWaaSbaaSqaa8qacaWGJbaapaqabaGcpe Gaeyypa0JaaGimaiaac6cacaaI3aGaaGymaiaaigdacaaI5aGaaG4n aiaad2eacaWGLbGaamOvaaWdaeaapeGaamyya8aadaWgaaWcbaWdbi aadggaa8aabeaak8qacqGH9aqpcaaI5aGaaGOnaiaac6cacaaI2aGa aG4naiaaiodacaaIYaGaamytaiaadwgacaWGwbaapaqaa8qacaWGHb WdamaaBaaaleaapeGaaGOnaaWdaeqaaOWdbiabg2da9iaaiwdacaaI 1aGaaiOlaiaaikdacaaI2aGaaGimaiaaikdacaaI4aGaamytaiaadw gacaWGwbaapaqaa8qacaWGHbWdamaaBaaaleaapeGaaG4naaWdaeqa aOWdbiabg2da9iaaiwdacaaI3aGaaiOlaiaaiMdacaaI4aGaaGymai aaisdacaWGnbGaamyzaiaadAfacaGGUaaaaaGaay5Eaaaaaa@7B5E@
And    a p =3 MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaGGGcGaaiiOaiaadggapaWaaSbaaSqaa8qacaWGWbaapaqabaGc peGaeyypa0JaaG4maaaa@3C50@ MeVis agreed for pairing energy coefficient. Also, the following values were used for δ:
δ={ 4.22Zeven,Neven 1Zeven,Nodd 0Zodd,Neven 2.66Zodd,Nodd MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacaqG0oGaeyypa0Zaaiqaa8aabaqbaeqabqqaaaaabaWdbiaaisda caGGUaGaaGOmaiaaikdacaWGAbGaeyOeI0IaamyzaiaadAhacaWGLb GaamOBaiaacYcacaWGobGaeyOeI0IaamyzaiaadAhacaWGLbGaamOB aaWdaeaapeGaaGymaiaadQfacqGHsislcaWGLbGaamODaiaadwgaca WGUbGaaiilaiaad6eacqGHsislcaWGVbGaamizaiaadsgaa8aabaWd biaaicdacaWGAbGaeyOeI0Iaam4BaiaadsgacaWGKbGaaiilaiaad6 eacqGHsislcaWGLbGaamODaiaadwgacaWGUbaapaqaa8qacqGHsisl caaIYaGaaiOlaiaaiAdacaaI2aGaamOwaiabgkHiTiaad+gacaWGKb GaamizaiaacYcacaWGobGaeyOeI0Iaam4BaiaadsgacaWGKbaaaaGa ay5Eaaaaaa@6D26@
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,
<σ>= i=1 59 | B Expi B Cali | 59 =0.524MeV   ( 9 ) MathType@MTEF@5@5@+= feaagGart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqGH8aapcqaHdpWCcqGH+aGpcqGH9aqpdaWcaaWdaeaapeWaaubm aeqal8aabaWdbiaadMgacqGH9aqpcaaIXaaapaqaa8qacaaI1aGaaG yoaaqdpaqaa8qacqGHris5aaGcdaabdaWdaeaapeGaamOqa8aadaWg aaWcbaWdbiaadweacaWG4bGaamiCaiaadMgaa8aabeaak8qacqGHsi slcaWGcbWdamaaBaaaleaapeGaam4qaiaadggacaWGSbGaamyAaaWd aeqaaaGcpeGaay5bSlaawIa7aaWdaeaapeGaaGynaiaaiMdaaaGaey ypa0JaaGimaiaac6cacaaI1aGaaGOmaiaaisdacaWGnbGaamyzaiaa dAfacaqGGaGaaeiiaiaabccadaqadaWdaeaapeGaaGyoaaGaayjkai aawMcaaaaa@5CAE@
σ 2 = ( i=1 59 ( B Expi B Cali ) 2 59 ) 1 2 =0.597MeV( 10 ) MathType@MTEF@5@5@+= feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaGcaaWdaeaapeGaeq4Wdm3damaaCaaaleqabaWdbiaaikdaaaaa beaakiabg2da9maabmaapaqaa8qadaWcaaWdaeaapeWaaubmaeqal8 aabaWdbiaadMgacqGH9aqpcaaIXaaapaqaa8qacaaI1aGaaGyoaaqd paqaa8qacqGHris5aaGcdaqadaWdaeaapeGaamOqa8aadaWgaaWcba WdbiaadweacaWG4bGaamiCaiaadMgaa8aabeaak8qacqGHsislcaWG cbWdamaaBaaaleaapeGaam4qaiaadggacaWGSbGaamyAaaWdaeqaaa GcpeGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaIYaaaaaGcpaqa a8qacaaI1aGaaGyoaaaaaiaawIcacaGLPaaapaWaaWbaaSqabeaape WaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaaaaOGaeyypa0Ja aGimaiaac6cacaaI1aGaaGyoaiaaiEdacaWGnbGaamyzaiaadAfada qadaWdaeaapeGaaGymaiaaicdaaiaawIcacaGLPaaaaaa@5DF1@

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

1990.730051

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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