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
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 1 3 A 17 mmnp z A 1 e (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
S B(A, Z) B(A 1,Z) n hyper 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 2.9675 13.6171 N F 16 126.108 11.6430 10.1279 14.1867 16 122.164 13.6683 7.6983 14.2724 N O 14 103.111 16.2671 5.4296 13.3086 14 106.6460 8.9646 13.1591 13.2202 N C 13 86.8443 16.9444 2.8636 11.8538 13 93.4874 6.4054 14.2838 11.6742 N B 12 69.9000 22.9124 0.2414 12.9367 12 79.2036 3.7986 20.7231 12.6629 N Be 17 127.269 6.1275 20.0058 15.1975 17 110.701 23.1616 1.3940 15.5357 C Ne 16 121.141 5.1611 17.5838 13.3016 16 109.3070 18.5518 0.8122 13.5590 C F 15 115.9800 9.3333 15.6055 13.8859 15 108.495 18.7255 5.3837 14.0600 C O 11 69.6585 20.1494 4.9154 11.7954 11 75.4049 8.0254 17.9866 11.6099 C Be 10 49.5091 22.6204 1.8043 11.7157 10 57.4183 4.9349 20.9723 11.4330 C Li 14 100.3750 6.8874 16.7507 13.6754 14 89.7697 21.0117 2.9253 13.9407 B O 9 47.7047 19.5205 2.3942 10.0849 9 52.4834 17.9092 5.04288 9.8936 B Li 10 57.4186 4.9349 20.9723 11.4330 10 49.5091 22.6204 1.8043 11.7157 Li C 28 222.6820 18.9906 1.05731 17.8877 28 240.9200 7.4615 13.5615 17.6652 P Mg 29 237.3150 14.6326 2.5237 17.9271 29 249.794 8.9253 8.8742 17.7803 P AL 25 187.685 2.2063 21.0474 17.5224 25 210.024 8.3000 16.3107 17.2164 Si Ne 26 204.1470 3.8038 16.4623 17.5141 26 221.4030 9.8900 11.3789 17.2870 Si Na 27 221.6250 17.4778 5.3767 17.5713 27 233.458 11.458 12.0554 17.4215 Si Mg 20 Mg 127.7740 21.7322 0.05130 15.4422 20 N 151.3150 5.4990 18.1333 16.3278 21Mg 149.6440 21.8703 1.9241 16.6475 21O 169.203 7.3579 17.8877 16.3268 22 Mg 166.672 17.0279 3.7274 16.6294 22 F 181.8760 9.1515 12.6728 16.3919 23 Mg 184.376 17.7034 5.5047 16.7063 23 Ne 194.861 10.922 12.9859 16.5498 20 Na 147.7200 19.9975 1.0464 16.0980 20 O 161.8450 13.445 19.9975 15.8544 21 Na 162.945 15.2246 2.8973 13.5470 21F 172.7240 7.9683 10.8792 13.3893 19 Ne 146.674 18.0662 5.5307 15.5879 19 O 155.720 10.242 14.1089 15.4234
Fig.1. Hypernuclei decay chain for A=24
Fig.2 Hypernuclei decay chain for A=25
Fig .3 Proton, neutron and Lambda hyperon of mirror hypernuclei
separation energies difference
Fig .4. Binding energy difference of light mirror hypernuclei and mirror nuclei
Fig .5. Neutron separation energies difference of light mirror hypernuclei
and mirror nuclei
Fig. 6. proton separation energies difference of light mirror hypernuclei
and mirror nuclei