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Mass Chains of Light Hypernuclei and Separation Energies of Mirror Hypernuclei from BWMH Mass Formula

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Mass Chains of Light Hypernuclei and Separation Energies of Mirror Hypernuclei from BWMH Mass Formula Mass Chains of Light Hypernuclei and Separation Energies of Mirror Hypernuclei from BWMH Mass Formula A. Armat1 and H. Hassanabadi1* 1Physics Department, University of Shahrood, Shahrood, Iran *[email protected] Abstract In this work, by using generalized semi-empirical mass formula, decay chains of light hypernuclei for odd-A and even-A are calculated and the unstable hypernuclei of mass chains are indicated. The separation energy per baryon of the mirror hypernuclei is obtained. Keywords: Hypernuclei, Mass formula, Binding energy, Mass chains, Separation energy PACS numbers: 21.80.+a, 25.80.-e, 21.10.Dr, 32.10.Bi, 12.90.+b 1. Introduction Hypernuclear physics is an important branch of nuclear physics and has attracted lots of attention during the past decades. Many novel phenomena have been discovered in this field and many others are anticipated. Hypernuclei are bound systems of nucleons with one or more strange baryons (hyperons), which contain the strange quark. These systems provide us with a reliable basis to investigate many nuclear and particle systems [1].) The hypernuclei physics has been progressing quite rapidly in both experimental and theoretical backgrounds [2, 3].). Hypernuclei are typically produced in some excited states through hadronic reactions such (K , ) or (,K). Nevertheless, they can reach their ground state through electromagnetic and/or nucleon emission [4]. The study of hypernuclei provides us with a deeper insight into fundamental interactions where particle reactions are not accessible such as the hyperon- nucleon [5-6] and hyperon-hyperon interactions. Information on these interactions is only possible through the study of the production and decay of hypernuclei which makes the field quite vital for related studies [7]. The research on the structure and property of hypernuclei had produced many meaningful results in 1970s. The hypernuclei properties include well depth of -nuclear potential, the shrinkage of hypernuclear size [8-9], hypernuclear spins and life times, hypernuclei spectroscopy[10], excited states of hypernuclei, and binding energy per baryon and the hyperon binding energy [11]. The total nuclear binding energy of any nucleus plays a crucial role in nuclear stability and can be therefore used to examine the single nucleon binding near the drip-line regime. The binding energies in single hypernuclei with in different single-particle states is obtained by Xue et al. with the spherical and the triaxial RMF calculations [12] .The semi empirical mass formula proposed by Bethe-Weizsäcker was generalized for light nuclei (BWM)[13-16] and the more improved version (BWMH) which works well for both strange and nonstrange nuclei was developed recently [17]. We use BWMH to compute decay chains of light Hypernuclei for odd-A and even-A. The unstable hypernuclei of mass chains is shown in Fig. 1 and Fig. 2. Proton, neutron and Lambda hyperon of mirror strange nuclei separation energies and binding energy are reported in Table 1. The separation energies difference is shown in Fig. 3. The binding energy difference and baryons separation energies difference between the number of mirror hypernuclei and mirror nuclei are compared in Figs. 4, 5 and 6. 2. Generalized Semi-Empirical Mass Formula Semiempirical mass formula for hypernuclei is M(A,Z) M(A,zcY ) M B(A,Z) (1) where M(A,Z) and MY denote the hypernuclei mass and hyperon mass, respectively. M(A,zc ) is the nucleus mass M(A,z ) M B(A,z ) (2) ccN B(A,zc ) and B(A,Z) are respectively the binding energies of a nucleus and hypernucleus. A is the total number of baryon and Z is defined as Z zc n Y q Y (3) Y where nY is the number of hyperons of a particular kind in a nucleus, zc denotes the number of protons, qY is the charge number (with proper sign) of hyperon(s) constituting the hypernucleus and MN , the nucleon mass, is MN z c m p nm n (4) where m is the proton mass, m denotes the neutron mass, and n is the number p n of neutrons. We consider the generalized mass formula for non-strange, strange and multiply-strange nuclear systems [17] 2 2 Z(Z 1) a (n z ) 3 sym c A B(A, Z) a A asc 4A a1 (1 exp( )) 3 (1 exp(A ))A 30 A 17 2 3 nYY [0.0335(m ) 26.7 48.7 S A Y 2 2 A/17 aYc {(N z ) (N n) }/{(1 e )A}] (5) where mY is the mass of hyperon in MeV, S is the strange number of hyperon and N shows the total number of hyperons. Parameter a Y is introduced to consider the hyperon number asymmetry with respect to the proton and neutron numbers. The pairing energy is 1/2 12A for even n even zc 1/2 12A for odd n odd zc (6) 0 for odd A We calculated mass chains for A=25 and A=24. Fig. 1 shows a decay chain for A= 24. The unstable hypernuclei approach stability by converting a neutron into a proton or a proton into a neutron. Fig. 2 shows the mass chains of hypernuclei for A=25. For this A, the pairing term gives two parabolas. For constant A, Eq. (5) represents a parabola of M vs Z. The parabola will be centered about the point where (5) reaches a minimum. We must find the minimum, where M zc 0 1.42 185.68 mm A1 3 1 e A 17 np z (7) cmin 2.84 371.36 A1 3 A(1 e A 17 ) which gives 46.42(A 2z )2 M 36.68A2 3 31.554A cmin m (A z ) m z M 2(1 e A 30 ) minA(1 eA 17 ) n cmin p c min 1.42(1 z n q )( z n q ) 48.7 S ccmin min n ( 26.7 0.0335M ) (8) AA1 3 2 3 Other useful and interesting properties which are often considered are the neutron, proton and hyperon separation energies. The hyperon separation energy SY is the amount of energy required to remove a hyperon from a hypernuclei, or equivalently the difference in binding energies between initial and final hypernuclei states. The generalized hyperon separation energy SY is defined as SY B(A,Z) hyper B(A n Y ,Z) hyper (9) Y where B(A,Z) is the binding energy of a hypernucleus and the separation energies for neutron and proton are Sn B(A, Z) hyper B(A 1,Z) hyper Sp B(A, Z) hyper B(A 1,Z 1) hyper (10) Their respective binding energies provide necessary guidelines for studying the nuclear stability near the drip lines against decay by emission of protons or neutrons. 3. Conclusions The study of hypernuclei is one of the important developing areas of nuclear physics. At present most of the information on hypernuclei has been extracted from (k , )reactions. Major goals of hypernuclear physics are to understand baryon-baryon interactions and the structure of multi-strangeness systems. In this work, We used the values of a 15.777MeV , as 18.34MeV ,ac 0.71MeV , asym 23.21MeV , k 17MeV , c 30MeV and aY 0.02 . By using the generalized semi-empirical mass formula, decay chains of light hypernuclei for odd-A and even-A are calculated. The unstable hypernuclei of decay chains is shown in Fig. 1 and Fig. 2. Proton, neutron and Lambda hyperon of mirror strange nuclei separation energies and binding energy are reported in Table 1. The separation energies difference is shown in Fig. 3. The binding energy difference and baryons separation energies difference between the number of mirror hypernuclei and mirror nuclei are compared in Figs. 4, 5 and 6. References [1] A. Parre˜no Lect. Notes Phys. 724, 141–189 (2007). [2] A. Gal and D. J. Millener, Hyperfine Interact. 210, 77 (2012). [3] C. Rappold et al, Phys. Rev. C 88, 041001 (2013). [4] A. Parreno and A. Ramos, Phys. Rev. C, 56, 1 (1997). [5] Bo-Chao Liu, Nuclear Physics A 00 1–4 (2013). [6] A. Ohnishi et al., Nuclear Physics A 00 1–8 (2013). [7] A. Parre˜no Lect. Notes Phys. 724, 141–189 (2007). [8] Motoba et al., PTP70 189(1983). [9]E. Hiyama, et al.,Phys. Rev. C 59 2351(1999). [10] Hashimoto O and Tamura H, Prog.Part.Nucl.Phys. 57 564 (2006). [11] Lan Mi-Xiang et al , Chinese Phys. Lett. 26 072101 (2009). [12] W. X. Xue, J. M. Yao, K. Hagino, Z. P. Li, H. Mei, and Y. Tanimura, Phys. Rev. C 91, 024327 –(2015). [13] C. Samanta and S. Adhikari , Phys. Rev. C 65, 037301 (2002). [14] César Barbero, Jorge G. Hirsch , Alejandro E. Mariano, Nuclear Physics A 874 (2012). [15] C. C. Barros, Jr. and Y. Hama, Phys. Rev. C, 63, 065203 (2001). [16] Z. Hasan, V. Kumar, and Daksh Lohiya, Proceedings of the DAE Symp. on Nucl. Phys. 57 (2012). [17] C. Samanta, P. Roy Chowdhury and D.N. Basu, J. Phys. G: Nucl. Part. Phys. 32, 363. (2006). TABLE 1: Binding energy and separation energies of mirror hypernuclei B Sn Sp S 4 5.0017 28.7415 2.8134 0.0282 4 4.1072 25.4819 8.3215 0.1183 H He 6 26.9775 7.7054 19.1623 5.2093 6 25.4146 20.0906 19.1623 5.3072 He Li 8 47.4405 9.9172 14.5226 8.3345 8 45.3105 16.1042 7.7827 8.4315 Li Be 10 67.3795 8.3786 14.8961 10.4581 10 64.7431 17.0384 5.7421 10.5523 Be B 12 87.0819 10.9126 11.6769 12.0160 12 83.9807 14.3222 7.8113 12.1072 B C 14 106.646 8.9646 13.1591 13.2202 14 103.111 16.2671 5.4296 13.3086 C N 16 126.108 11.6430 10.1279 14.1867 16 122.164 13.6683 7.6983 14.2724 N O 18 145.478 9.4182 12.0629 14.9846 18 141.143 16.012 5.0835 15.0680 O F 20 164.7560 12.1936 9.0351 15.6582 20 160.047 13.3736 7.4854 15.7394 F Ne 22 183.9390 9.7827 11.2146 16.2370 22 178.871 15.9261 4.7175 16.3162 Ne Na 24 201.724 12.6225 8.1604 16.7415 24 185.479 13.2310 4.5310 16.8189 Na Mg 26 222.00 10.0956 10.4879 17.1867 26 216.2480 15.9049 4.3436 17.2624 Mg Al 28 240.8690 12.9761 7.4103 17.5836 28 234.79 13.1645 6.8969 17.6577 Al Si 30 259.623 10.3665 9.8289 17.9405 30 253.225 15.9101 3.9684 18.0132 Si P 20 151.3150 5.4990 18.1333 16.3919 20 127.7740 21.7322 0.0513 16.6294 N Mg 18 141.6120 8.1967 14.3425 15.2376 18 128.608 17.9062 3.4766 15.4878 N Ne 17 133.415 7.3066 12.2734 13.4426 17 125.131 15.8236
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