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Home , Magic number (physics)

New Magicity of Light Nuclei

C. Samanta* and S. Adhikari*

Physics Department, Virginia Commonwealth University, Richmond, Virginia 23284-2000

A new mass formula capable of explaining the binding energies of almost all the known from Li to Bi is prescribed. In addition to identifying the new magic number at number N=16 (Z=7-9), pseudo-magic numbers at N= 14 (Z= 7-10), Z=14 (N=13- 19), and at N=6 (Z=3-8), the formula accounts for the loss of magicity for nuclei with N=8 (Z=4) and N=20 (Z=12-17). The redefinition of the neutron drip line resulting from this formula further allows us to predict the existence of 26O,31F, 32Ne, 35Na, 38Mg, 41Al as bound nuclei and 28O as unbound.

PACS numbers: 21.10.Dr, 21.10.Pc, 25.60.Dz, 27.30.+t

The origin of the unusual Similarly, the density stability of nuclei with numbers distribution in magic atomic clusters is 2, 8, 20, 28, 50, 82 and 126, commonly more localized than those that are less referred to as “magic numbers”, was stable. explained more than half a century ago Recently, atomic clusters have to be due to nuclear shell structure [1-3]. been found whose stabilities are not The discovery in 1984 that similar magic ruled by the electronic shell closure. For numbers also appear in small clusters of example, Na8Mg cluster [6] which Na-atoms [4] was at first surprising since contains 10 valence should not - the nature of force holding atomic be magic, but it is. Similarly, Al 13 and clusters and nuclei are fundamentally Al13K [7], which contains 40 electrons different. In analogy with the stability of each, are magic while Al13Cu which also magic nuclei, the magic numbers in Na- contains 40 electrons is not [8]. The clusters were explained due to electronic same phenomena also appear in nuclei. shell closure. However, due to the Recently the N=16 has different nature of the potentials in been found to be a new magic number in atomic and nuclear domain, the two sets only a few neutron-rich nuclei [9]. The of magic numbers do not exactly central question then arises: why do coincide, namely, the magic numbers in nuclei and clusters deviate from the the Na-clusters are 2, 8, 20, 40, 58, magic numbers prescribed by the shell 92…[4,5] whereas, the nuclear magic structure and is there a convenient way numbers are 2, 8, 20, 28, 50, 82...[3]. to predict the occurrence of the Nevertheless, these two unlikely anomalous magic clusters /nuclei. While fields have found many commonalities such analysis may be easier in atomic such as, giant dipole resonance (which clusters because of the interaction being was well known in ) is well known, similar analysis in nuclei now found to exist in free electron poses some difficulties. atomic cluster [5]. The root-mean-square There exists in literature mass (rms) radii of magic nuclei are relatively formulae [3,10-12] to examine the small compared to other nearby nuclei. stability of nuclei with varying neutron and numbers. However, these extra stability of certain nuclei. This led formulations are more suitable for to the discovery of a new magic number medium to heavy mass nuclei. No mass at N=16 [9] but, the exact Z region could formula so far exists that can account for not be clearly established as the the exotic properties of the light nuclei corresponding cross section data yielded near the dripline. We have critically ambigious results. An anomalous examined the available experimental behaviour in the of very data [13] from Li to Bi, which has enabled us to prescribe a mass formula 50 Li (z = 3) capable of explaining the binding 40 energies of almost all the known 30 20 isotopes from Li to Bi. From a B.E. (MeV) exp comparison with the experimental one- 10 new B-W neutron (Sn) or, one-proton (Sp) 0 separation energies, we can identify the 0246810 140 C (z = 6) locations of new magicity or, loss of it. 120 The redefinition of the dripline resulting 100 from this formula further allows us to 80 B.E. (MeV) B.E. 60 40 predict the existence of several bound exp 20 new B-W nuclei beyond the conventional neutron 0 drip line. 0 5 10 15 20

To formulate this mass formula Ne (z = 10) we started with the Bethe-Weizsacker 200 (B-W) mass formula [3], which gives a 160 reasonable description of the binding B.E. (MeV) 120 exp new B-W energy of medium to heavy mass nuclei. 80 The B-W formula for the binding energy 510152025 Neutron Number (N) of a nucleus of A and proton number Z is, Fig 1 Binding Energy vs. N, calculated with 2/3 1/3 B-W and new formula B(A, Z)= av A – as A – ac Z (Z-1) /A 2 neutron-rich Na isotopes around N=20 – asym (A-2Z) /A + δ (1) was also observed and it was attributed to a loss of magicity [9,14]. Since the B- where, av =15.85 MeV, as =18.34 MeV, W mass formula is inadequate in the low ac=0.71 MeV and asym =23.21 MeV. The -1/2 mass region, it cannot be used to identify pairing energy term δ =+ap A for even -1/2 such deviations. Most importantly, a N-even Z , -ap A for odd N-odd Z, clear evidence of particle stability in 31F and 0 for odd A nuclei and, ap =12 MeV. 30 This mass formula not only (as opposed to instability in F) was found [15], but according to the B-W underestimates the binding energies of 30,31 the light nuclei, it also predicts too large formula both F should be unstable. It a pairing energy for such nuclei (Fig.1, is thus imperative to formulate a mass 2). equation which can help to identify the It is pertinent to note that a mere new magicity or, its loss and, give a survey of the one-neutron separation proper limit of the neutron-drip line. energies of the available nuclei reveals A careful study reveals that a marked deviation between the experimental binding energy and the the magic number show distinct prediction of the B-W mass formula deviation from the calculations. occurs in light nuclei, specially when the neutron proton asymmetry is large. This indicates a more complicated 25 O (Z = 8) dependence of 20 on the neutron-proton number arising 15 exp 10 new from a major change in shape and size 5 Sn (MeV) (like, halo/skin) of the nucleus near the 0 -5 5101520 neutron dripline [16]. Recently, near 15 N≈Z the role of the neutron-proton Ne (Z = 10) interaction and its consequences for p-n 10 exp new pairing has been investigated [17]. A 5 steep decrease of the isoscalar p-n Sn (MeV) 0 10 15 20 pairing energy was suggested with -5   increasing N-Z . A detailed and 15 Mg (Z = 12) systematic search carried out to fit all the 10 exp isotopes from Li to Bi also supports that B-W 5 new

and, to an additional term in the Sn (MeV) 0 12 17 22 27 existing mass formula and a redefinition -5 of the pairing term. The binding energy Neutron Number N of any nucleus of mass A is then defined as, Fig 2 One - neutron vs. N and calculation with formula 2/3 1/3 B(A, Z)new=av A – as A –ac Z(Z-1)/A 2 ∆ δ – asym (A-2Z) /A + (N,Z) + new, Just prior to the magic number, (2) the experimental Sn values are where, ∆(N, Z)= N - 4/3 Z Nk Z e-z/3, significantly higher and, right after it; - z / c δnew = (1 – e ) δ, k = 0.45, and c = the Sn suddenly drops to a lower value. 6.0/Ln 2, (other constants remaining the Sometimes, the Sn value does not drop same). drastically after being high and, we call that particular nucleon number as While almost all the elements pseudo-magic, as it reflects pseudo shell from Li to Bi can be explained by this closure due to rearrangements of shells unique equation (2), binding energies of that changes both position and width of some very neutron deficient light nuclei the shell gaps. In Fig. 2 the known magic are slightly underestimated (Fig.1). number at N=20 (Z=10) can be clearly The power of this mass formula identified. The magicity at N=20 is is its uniqueness in explaining not only found to disappear in the Z= 12 -17 the binding energies but also the one- or, region. Magicity at N=8 also disappears two- neutron and proton- separation at Z=4 (Fig. 3). energies. Fig. 2 shows a comparison of The Sn values from the new 26 28 the Sn data and theoretical calculations formula predicts O and O as bound for Z= 9 and 10. As the latter does not nuclei with very small binding energies incorporate the nuclear shell effect, the (Fig. 2). However, the two-neutron experimental separation energies near separation energy(S2n) is negative for magicity has moved down to N=16 from N=20. Thus, the new formula helps to

20 Be (Z = 4) resolve the previous ambiguity and 15 exp ascertains magicity at N=16 for Z= 7- 9. 10 new This is supported by the cross section

Sn (MeV) 5 data also [9]. 0 For Z= 7-10, one neutron 45678910 -5 separation energy is found to be large at Neutron No. N N=14 (Fig.2) and, one proton separation energy is large at Z=14 for N=13-19 Figure 3 One-neutron separation energy vs. N, (Fig.5). In the above region, the nucleon and calculation with new formula 28O, making it unbound (Fig. 4). So far these isotopes of are not found (N = 14) 20

15 new exp 10

Sp ( Mev) 5

0 10 11 12 13 14 15 16 17 18 30 Proton No. Z Z = 8

25 Figure 5 One-proton separation energy vs. Z 20 and calculation with new formula exp 15 new old

S2n (MeV) 10 number N=14 can be called as, pseudo- magic. The magicity at N=16 was 5

0 30 (a) Even N - Odd Z 25 8 121620 20 -5 15 Tz = -3/2

10 Tz=-1/2 Sn (MeV) Neutron No. N Tz=1/2 5 Tz=3/2

0 Figure 4 Two-neutron separation energy 0 5 10 15 20 25 vs. N and calculations with old and new Neutron Number (N) 25 formulae B (Z = 5) 20 (b)

and experiments [15,18] put upper limits 15 exp to their cross sections to ~ 0.7pb and 10 B-W new 0.2pb respectively. 5 Sn (MeV) 0 3579111315 28 -5 Absence of the doubly magic O -1 0 (Z=8, N=20) nucleus is an interesting Neutro n Number (N) phenomenon as it indicates a significant change in near the Figure 6 One-neutrn separation energy (exp) neutron drip line. Infact, we find that in vs. N; (a) for different Tz and (b) calculations with new formula Nitrogen, Oxygen and Fluorine the predicted to arise from the lowering of the 2S1/2 state below the 1d3/2 [9]. But, the pseudo-magicity at N=14 can occur 15 P (Z = 15) if the 1d5/2 remains below 2S1/2 and the 12

spin-orbit splitting for the d-state is ) 9 exp new large. 6

A region of extra stability is Sn (MeV 3

found at N= 6 for proton numbers Z= 3- 0 15 20 25 30 35 9. The plot of Sn vs. N for different -3 isospin Tz (Fig. 6) shows a clear rise in 15

Sn value for neutron number N=6. The 12 S (Z = 16) ) r.m.s. matter radii [19] for all the nuclei 9 9 10 11 12 13 14 exp ( Li, Be, B, C, N, O) show a 6 new Sn (MeV Sn distinct bunching at a value ∼2.3 fm 3 (Fig. 7) indicating their tight binding. 0 15 20 25 30 35 The nucleus 15F is proton unstable. -3 18 3.2 15 Cl (Z = 17) Li 12 Be ) exp 3 B 9 new C 6

N (MeV Sn 3 O 2.8 0

-3 15 20 25 30 35

2.6 Neutron Number (N) Rrms (fm) Rrms

2.4 Figure 8 One-neutron separation energy vs. N, and calculations with old and new formula

2.2

2 24681012 This results in unequal shifts in single Neutron Number (N) particle levels. The s-orbital is more influenced than the p and d orbitals as Figure 7 r.m.s. matter radii vs. N for Z = 3-8 the repulsive centrifugal potentials insulates the higher angular momenta from the narrow central region. Movement of single particle levels Another interesting finding is the causes appearance of new magic weakening of the N=28 shell closure in numbers and disappearance of the old Z=15-17 region (Fig. 7). According to ones [5]. Similarly, in nuclei away from Shell model and relativistic mean field the β-stability line, the rearrangement of calculations, this arises from deformed the nuclear shells causes drifting of prolate ground state associated with nuclear magic numbers from the known shape coexistence [18]. to unexpected values. This has been In atomic mixed clusters, the already observed in 11Be for which the heteroatom at the center was found to ground state spin parity is ½+ instead of - cause a narrow depression (or, a local ½ indicating lowering of the 2s1/2 state maximum, depending on the electron below1p1/2[20]. density) near the center of the potential. Unlike the B-W formula, the new [3] K. Heyde, “Basic ideas and concepts formula supports experimentally in nuclear physics”, IOP publication, established stability of 31F (and 1999 30 instability of F) [15]. A survey of Sn [4] W.D.Knight et al., Phys. Rev. Lett., 32 35 38 and S2n suggests Ne, Na, Mg, and 52, 2141 (1984) 41Al as bound nuclei which are beyond [5] C.Yannouleas, P.Jena and the neutron-drip line predicted by the B- S.N.Khanna , Phys. Rev. B 46, 9751 W formula. (1992) In summary a “new” mass [6] M.M.Kappes et al., Phys. Lett. 119, formula capable of explaining binding 11 (1985) energies of almost all the known [7] B.K.Rao, S.N.Khanna and P.Jena, isotopes from Li to Bi is prescribed. The Phys. Rev. B 62, 4666 (2000) calculated binding energies, as well as [8] O.C.Thomas, W.Zheng and the one-neutron separation energies, are K.H.Bowen, J. Chem. Phys. (in press); compared with the experimental data S.N.Khanna et al., J.Chem. Phys. (in and the predictions of the Bethe- press) Weizsacker mass formula. In addition to [9] A.Ozawa et al., Phys. Rev. Lett. identifying the new magic number at 84, 5493 (2000) N=16 (Z=7-9), we suggest pseudo-magic [10] W. D. Myers and W. J. Swiatecki, numbers at N=6 (Z=3-8), N=14 (Z=7- Nucl. Phys. 81, 1 (1966) 10), and Z=14 (N=13-19). The new [11] R.C.Nayak and L.Satpathy, Atomic formula also accounts for the loss of data and Nucl. Data tables 73, 213 magicity for nuclei with N=8 (Z=4) and (1999), L.Sathpathy and R.C.Nayak, J. N=20 (Z=12-17). The redefinition of the Phys. G: Nucl. Part. Phys. 24, 1527 neutron drip line resulting from this (1998); ibid. Phys. Rev. Lett. 51, 1243 formula further allows us to predict the (1983) existence of 26O, 31F, 32Ne, 35Na, 38Mg, [12] S. Liran, A. Marinov, and N. 41Al as bound nuclei which are beyond Zeldes, Phys. Rev. C62, 047301 (2000) the conventional (B-W) neutron dripline. [13] Nuclear wallet cards, January It is now left as a challenge to find more 2000; Brookhaven National Laboratory fundamental basis for this mass formula [14] C. Thibault et al., Phys. Rev. C12, and its origin based upon the basic 644 (1975) nucleon-nucleon interaction. [15] H. Sakurai et al., Phys. Lett. B We would like to thank P.Jena 448, 180 (1999) for stimulating discussions on atomic [16] I. Tanihata, J. Phys. G: Nucl. Part. clusters and, I. Samanta for his help in Phys.22, 157 (1996) computation. [17] G.Ropke et al., Phys. Rev. C 61, 024306 (2000) th References: [18] Z.Dlouhy et al, Proc. of the 9 Int. Conf. On Mechanisms, [1] M.G.Mayer, Phys. Rev. 75, 1969 Ed. Gadioli, Varena, June 5-9, 2000; (1949); ibid, Phys. Rev. 78, 16 (1950) see references therein. [2] D.Haxel, J.H.D.Jensen and [19] A.Ozawa et al., Nucl. Phys. A 608, H.E.Suess, Phys. Rev. 75, 1766 (1949); 63 (1996); see references there in ibid. Z. Physik 128, 295 (1950) [20] M.Fukuda et al., Phys. Lett. B 268, 339 (1991)