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A New Mass Formula and Magic Number At 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 isotopes from Li to Bi is prescribed. In addition to identifying the new magic number at neutron 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 electron density stability of nuclei with nucleon 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 electrons 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 neutron number 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 nuclear physics) 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 proton 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 binding energy 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 atomic number 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 nuclear binding energy 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, leads 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 separation energy 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.
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