Bound-state energy of double magic number plus one nucleon nuclei with relativistic mean-field approach

Transcription

1 Pamana J. Phys. (207) 88: 2 DOI 0.007/s c Indian Academy of Sciences Bound-state enegy of double magic numbe plus one nucleon nuclei with elativistic mean-field appoach M MOUSAVI and M R SHOJAEI Depatment of Physics, Shahood Univesity of Technology, P.O. Box , Shahood, Ian Coesponding autho. nuclea.physics2020@gmail.com MS eceived 3 Septembe 205; evised June 206; accepted 7 June 206; published online 3 Januay 207 Abstact. In this wok, we have obtained enegy levels and chage adius fo the β-stability line nucleus, in elativistic shell model. In this model, we consideed a close shell fo each nucleus containing double magic numbe and a single nucleon enegy level. Hee we have taken 4 Ca with a single neuton in the 40 Ca coe as an illustative example. Then we have selected the Eckat plus Hulthen potentials fo inteaction between the coe and the single nucleon. By using paametic Nikifoov Uvaov (PNU) method, we have calculated the enegy values and wave function. Finally, we have calculated the chage adius fo 7 O, 4 Ca, 49 Ca and 57 Ni. Ou esults ae in ageement with expeimental values and hence this model can be applied fo simila nuclei. Keywods. method. PACS No. Shell model; Diac equation; Eckat potential; Hulthen potential; paametic Nikifoov Uvaov 2.0.D. Intoduction Some static popeties of the nucleus, such as enegy levels and chage adius, ae useful fo descibing the stuctue of the nucleus. The nuclea chage adius plays a key ole in studying the chaacteistics of the nucleus and testing theoetical models of the nuclei as well as in studying astophysics and atomic physics. The calcium isotopes have eceived much attention due to its ich expeimental esults in binding enegy, density distibution, single-paticle enegy, adius, etc. It is useful to calculate these quantities to test micoscopic theoy by futue expeiments ]. The study of nuclei unde exteme conditions has always been a necessity to undestand the nuclea foces. As ealy as 934, Elsasse 2] noticed the existence of special numbes of neutons and potons which confe a paticulaly stable configuation to the coesponding nuclei. In analogy with atomic electons, he coelated these numbes with closed shells in a model of noninteacting nucleons occupying enegy levels geneated by a potential well. Moe than a decade late, the study of shell stuctue egained inteest though the eview of M Goeppet-Maye which has a lage quantity of pecise expeimental data, which wee pointes to the existence of closed shells at numbes 8, 20, 50, 82 and 26 3]. Undestanding the evolution of shell stuctue fom the valley of stability to neuton-ich extemes epesents a key challenge in nuclea stuctue. With a closed poton shell, the 7 O, 4 Ca, 49 Ca and 57 Ni isotopes povide an ideal egion to investigate shell fomation and evolution in medium mass nuclei fom nuclea foces 4,5]. We conside 7 O, 4 Ca, 49 Ca and 57 Ni isotopes. As these isotopes have double magic numbe with a single neuton on top of the close coe, these isotopes ae in the egion of the β-stability line nucleus. The fist step of the shell model to study the enegy levels of nuclei is to conside the nucleon nucleon potentials, which epoduce the nucleon nucleon scatteing data and the popeties of the nuclei 6,7]. Thee ae many nucleon nucleon potentials such as Katze potential 8,9], Woods Saxon potential 0,], Scaf potential 2], Hatmann potential 3,4], Rosen Mose potential 5,6], Hulthen potential 7] and Eckat potential 8,9] that epoduce these data. The Eckat potential which has been studied by many eseaches is one of the most impotant exponentialtype potential in physics and chemical physics 20,2].

2 2 Page 2 of 5 Pamana J. Phys. (207) 88: 2 In this wok, we use shell model to calculate the enegy levels and chage adius fo 7 O, 4 Ca, 49 Ca and 57 Ni isotopes. We conside 4 Caasasingleneuton and 40 Ca coe. As these isotopes have one neuton out of the coe, we utilize the elativistic Diac equation to investigate them. These isotopes can be consideed as a single paticle. We apply the modified Eckat plus Hulthen potentials between the coe and a single paticle because these potentials ae impotant nuclea potentials fo descibing the inteaction between the single nucleon and the whole nuclei. Now that the NN potential is selected, the next step is to solve the Diac equation fo the nuclei unde investigation. Seveal numeical and analytical methods have been used to solve the poblem of the Diac equation by using ealistic nuclea potentials. Some of these methods ae: the CRCGV 2], the NCSM 22], the EIHH 23], supesymmetic quantum mechanics 24 26], asymptotic iteation method (AIM) 27,28], factoization method 29,30], Laplace tansfom appoach 3], GPS method 32,33], the path integal method 34 36] and the Nikifoov Uvaov (NU) method 37 39]. We use the paametic Nikifoov Uvaov (PNU) method to solve the Diac equation. The oganization of this pape is as follows: in 2, the PNU method is eviewed. In 3 enegy spectum fo the isotopes is pesented. Conclusion is given in Review of paametic Nikifoov Uvaov method The paametic fom of the NU method takes the fom 37 39] d 2 ds 2 + c c 2 s d s( c 3 s) ds + ( p 2s 2 ] +p s p 0 ) s 2 ( c 3 s) 2 n (s)=0. () The enegy equation and wave function espectively ae obtained fom nc 2 (2n + )c 5 + (2n + )( c 9 + c 3 c8 ) + n(n )c 3 + c 7 + 2c 3 c c 8 c 9 = 0, (2) n,k (s) = N n,k s c 2 ( c 3 s) c 3 P (c 0,c ) n ( 2c 3 s), (3) whee c 4 = 2 ( c ), c 5 = 2 (c 2 2c 3 ), c 6 = c p 2, c 7 = 2c 4 c 5 p, c 8 = c4 2 + p 0, c 9 = c 3 (c 7 + c 3 c 8 ) + c 6, c 0 = c + 2c c 8 >, c = c 2c c 3 c9 >, c 3 = 0, c 2 = c 4 + c 8 > 0, c 3 = c 4 + c 3 ( c 9 c 5 )>0, c 3 = 0. (4) The values of the coefficients c i (i = 4, 5,...,3) depend on p i (i = 0,, 2) and p i depend on enegy so the enegy levels depended on the coefficients c i and we could obtain the enegy levels by solving eq. (2) numeically. 3. Enegy spectum fo the isotopes The wave function fo the Diac equation can be calculated as n,k(,θ,φ)= Fn,k()Y l ] ig n,k()y l, (5) whee F n,k() and G n,k() ae uppe and lowe components, Y l and Y l ae the spheical hamonic functions. n is the adial quantum numbe and m is the pojection of the angula momentum on the z-axis. The obital angula momentum quantum numbes l and l epesent the spin and pseudospin quantum numbes. Unde the condition of spin symmety, i.e. () = 0, the uppe component Diac equation can be witten as 40] ( d2 k(k + ) + d2 2 + h 2 c 2 Mc2 + E] Mc 2 E + ]) () F n,k() = 0. (6) The modified Eckat plus Hulthen potential is defined as 7 9] e 2α V()= v 0 cosech 2 (α) + v ( e 2α ), (7) whee the paametes v 0 and v ae eal paametes, which ae stength paametes, and the paamete α is elated to the ange of the potential. Unde the condition of spin symmety, sum of the potentials V()and S() can be witten as e 2α () = 8v0 ( e 2α ) 2 + 2v e 2α ( e 2α ). (8)

3 Pamana J. Phys. (207) 88: 2 Page 3 of 5 2 Substituting the tansfomation s = exp( 2α) into eq.(5),wefind F (s) + s F (s) + { (E 2 M 2 c 4 ) 4αs 2 h 2 c 2 (E + Mc2 ) h 2 c 2 ] s 8v 0 ( s) 2 + 2v s ( s) } 2 k(k + ) 4α ( s) 2 F(s) = 0. (9) Equation (9) is exactly solvable only fo k = 0. In ode to obtain the analytical solutions of eq. (9), we employ the impoved Pekeis appoximation and eplace the spin obit coupling tem with the expession that is valid fo α 4,42]. k(k + ) 2 4α2 k(k + ) (e 2α ) 2. (0) We can wite eq. (9) as given below: F n,k ( s) (s) + s( s) F n,k (s) + s 2 ( s) 2 p 2 s 2 + p s p 0 ]F n,k (s) = 0, () whee the paametes p 2, p and p 0 ae as follows: p 2 = (E + Mc2 ) 4α 2 h 2 c 2 2v + (E Mc 2 )], p = (E + Mc2 ) 4α 2 h 2 c 2 8v 0 + 2v + 2(E Mc 2 )], p 0 = k(k + ) (E2 M 2 c 4 ) 4α 2 h 2 c 2. (2) Now compaing eq. () with eq. (), we obtain the coefficients c i (i =, 2, 3) as follows: c = c 2 = c 3 =. (3) The values of the coefficients c i (i = 4, 5,...,3) ae also found fom eq. (4) as below: c 4 = 0, c 5 = 2, c 6 = 4 + p 2, c 7 = p, c 8 = p 0, c 9 = p 2 p + p 0 + 4, c 0 = 2 p 0, c = 2 p 2 p + p 0 + 4, c 2 = p 0, c 3 = 2 + p 2 p + p (4) With espect to eq. (2) in the PNU method, we attain the enegy elation as 2 ( p 0 p 2 p +p 0 + ) +(2n+) p 2 p +p (2n + ) ( p 0 + 2p 0 p + n + ) = 0. (5) The gound-state and fist excited enegies of 7 O, 4 Ca, 49 Ca and 57 Ni isotopes ae obtained by using eq. (5). These esults ae compaed with the expeimental data in table 44]. The calculated enegy levels ae in good ageement with expeimental values. Theefoe, the poposed model can well be used to investigate othe simila isotopes. With espect to eq. (3) in PNU method 39,43,45] and eq. (4), the uppe component of the Diac spino is given by eq. (6). F n,k() = N(e 2α ) p 0 ( e 2α p ) 2 p +p P 2 p 0,2 p 2 p +p n ( 2e 2α ), (6) whee N is the nomalization constant. The lowe component of the Diac spino can be calculated fom eq. (6) as G n,k() = h 2 c 2 ( d E + Mc 2 d + k ) F n,k(). (7) The wave function can be calculated by substituting eqs (6) and (7) in eq. (5) as Y l ψ n,k(,θ,φ)=n i d M + E n,k] d + k ] Y l (e 2α ) p 0 ( e 2α ) P 2 p 0,2 p 2 p +p p 2 p +p n ( 2e 2α ). (8) We calculate the chage adius by using eqs (8) and (9) fo 7 O, 4 Ca, 49 Ca and 57 Ni isotopes. ( ψ 2 /2 n,k = ()2 ψ n,k()d 3 ) /2 ψ ()ψ n,k n,k()d 3. (9)

4 2 Page 4 of 5 Pamana J. Phys. (207) 88: 2 Table. The gound-state and the fist excited enegy (MeV) fo the isotopes. Isotope State Ou wok Exp. 7 O E d5/ ] E 2s/ ] 4 Ca E f 7/ ] E 2p3/ ] 49 Ca E 2p3/ ] E f 5/ ] 57 Ni E 2p3/ ] E f 5/ ] Table 2. The chage adius (fm) fo gound-state isotopes. Isotope 2 /2 ou wok (fm) 2 /2 Exp (fm) 7 O ] 4 Ca ] 49 Ca Ni In table 2 we show the calculated chage adii fo the gound-state isotopes, and compaed them with the expeimental data. Fo example, the calculated paametes of the modified potential ae: α (fm ) = 0.027, V 0 = 0.292, V = The obtained chage adii fo the gound-state isotopes ae in good ageement with the expeimental value. The esults show that ou model can be used to investigate othe simila isotopes, because Diac equation specially gives good esult fo single-paticle systems. Diac equation descibes paticles of half-intege spin. So this model cannot conside nuclei with intege spin. 4. Conclusions In this pape, we have calculated some enegy levels of 7 O, 4 Ca, 49 Ca and 57 Ni isotopes with modified Eckat plus Hulthen potentials using the PNU method. We obtained the enegy levels and wave functions fo these isotopes. The wave functions satisfy the bounday conditions; also we obtained the chage adii and compaed them with expeimental values. The esults ae in ageement with expeimental values and hence contain impotant physics. Acknowledgements The authos would like to thank the efeee fo the helpful comments and suggestions which have impoved the manuscipt geatly. Refeences ] Z Shuangquan, M Jie and Z Shangui, Sci. China Phys. Mech. Aston. 46, 632 (2003) 2] W Elsasse, J. de Phys. et Rad. 5, 625 (934) 3] M Goeppet-Maye, Phys. Rev. 74, 235 (948) 4] T Baumann, A Spyou and M Thoennessen, Rep. Pog. Phys. 75, (202) 5] O Solin and M-G Poquet, Pog.Pat.Nucl.Phys.6, 602 (2008) 6] G Puddua, Eu. Phys. J. A 3, 63 (2007) 7] E Cauie, Pog.Pat.Nucl.Phys.59, 226 (2007) 8] W C Qiang, Chin. Phys. 3, 575 (2004) 9] W C Qiang, Chin. Phys. 2(0), 054 (2003) 0] H Feizi, M R Shojaei and A A Rajabi, Eu. Phys. J. Plus 27, 4 (202) ] J Y Guo and Z Q Sheng, Phys. Lett. A 338, 90 (2005) 2] X C Zhang, Q W Liu, C S Jia and L Z Wang, Phys. Lett. A 340, 59 (2005) 3] C Y Chen, Phys. Lett. A 339, 283 (2005) 4] A de Souza Duta and M Hott, Phys. Lett. A 356, 25 (2006) 5] L Z Yi, Y F Diao, J Y Liu and C S Jia, Phys. Lett. A 333, 22 (2004) 6] A D Alhaidai, J. Phys. A: Math. Gen. 34, 9827 (200) 7] M R Shojaei and M Mousavi, Adv. High Enegy Phys. 206, Aticle ID , 2 pages (206) 8] C Eckat, Phys. Rev. 35, 303 (930) 9] B J Falaye, Cent. Euo. J. Phys. 0, 960 (202) 20] A P Zhang, W C Qiang and Y W Ling, Chin. Phys. Lett. 26(0), (2006) 2] J Y Guo, X Z Fang and F X Xu, Phys. Rev. A66, (2002) 22] E Hiyama et al, Phys. Rev. Lett. 85, 270 (2000) 23] P Navatil, J P Vay and B R Baet, Phys. Rev.C62, 0543 (2000) 24] N Banea, W Leidemann and G Olandini, Phys. Rev. C67, (2003) 25] C S Jia, P Gao and X L Peng, J. Phys. A: Math. Gen. 39, 7737 (2006) 26] A C Astoga, D J Andez and J Nego, SIGMA 8, 082 (202) 27] H Feizi, A A Rajabi and M R Shojaei, Acta Phys. Polon. B 42, 243 (20) 28] H Ciftci, R L Hall and N Saad, J. Phys. A 36, 807 (2003) 29] O Öze and G Lévai, Rom. J. Phys. 57(3 4), 582 (202) 30] S H Dong, Factoization method in quantum mechanics (Spinge, Dodecht, 2007) 3] I Infeld and T E Hull, Rev. Mod. Phys. 23, 2 (95) 32] A Ada and R Seve, Commun. Theo. Phys. 58, 27 (202) 33] A K Roy, Phys. Lett. A 32, 23 (2004) 34] A K Roy, Int. J. Quant. Chem. 3, 503 (203)

5 Pamana J. Phys. (207) 88: 2 Page 5 of ] J Cai, P Cai and A Inomata, Phys. Rev.A34, 462 (986) 36] A Diaf, A Chouchaoui and R J Lombad, Ann. Phys. 37, 354 (2005) 37] A Diaf and A Chouchaoui, Phys. Sc. 84, (20) 38] C Bekdemi, A Bekdemi and R Seve, Phys. A: Math. Gen. 399, 3455 (2006) 39] M R Shojaei and M Mousavi, Int. J. Phys. Sci. 0(9), (205) 40] W Geine, Relativistic quantum mechanics: Wave equations (Spinge, 2000) 4] E H Hill, Am. J. Phys. 22, 2 (954) 42] A A Rajabi and M Hamzavi, Int. J. Theo. Phys. 7, 7 (203) 43] M R Shojaei and N Roshanbakht, Chin. J. Phys. 53, 2030 (205) 44] G Audi, F G Kondev, M Wang et al, Chin. Phys. C 36, 57 (202) 45] A F Nikifoov and V B Uvaov, Special functions of mathematical physics (Bikhause, Basel, 988) 46] K P Angeli and Mainova, At. Data Nucl. Data Tables 99, 69 (203)