Single-particle levels of nuclei in the vicinity of the doubly magic nuclei 2048 Ca28 and 2856 Ni28

Isotonic and isotopic dependences of single-particle energies of neutron and proton states in 20 ≤ Z ≤ 28 and 24 ≤ N ≤ 32 nuclei are investigated, these energies being determined by matching data on nucleon-stripping and nucleon-pickup reactions on the same nucleus. Regularities of the formation of the spectra of single-particle levels in Z, N = 20, 28 magic nuclei are demonstrated. A distinctive feature is found in the isotonic dependence of the energy of the 1 f _5/2 neutron level, this feature being consistent with the assumption that j _>-j _< interaction is operative in nuclei. The single-particle energies calculated by using the potential of the dispersive optical model are found to be consistent with experimental data within their errors. Original Russian Text ? O.V. Bespalova, I.N. Boboshin, V.V. Varlamov, T.A. Ermakova, B.S. Ishkhanov, E.A. Romanovsky, T.I. Spasskaya, T.P. Timokhina, 2008, published in Yadernaya Fizika, 2008, Vol. 71, No. 1, pp. 37–49.     

Single-particle properties of <Emphasis Type="Italic">N</Emphasis> = 12 to <Emphasis Type="Italic">N</Emphasis> = 20 silicon isotopes within the dispersive optical model

Experimental neutron and proton single-particle energies in N = 12 to N = 20 silicon isotopes and data on neutron and proton scattering by nuclei of the isotope ²⁸Si are analyzed on the basis of the dispersive optical model. Good agreement with available experimental data was attained. The occupation probabilities calculated for the single-particle states in question suggest a parallel-type filling of the 1d and 2s _1/2 neutron states in the isotopes ^{26,28,30,32,34}Si. The single-particle spectra being considered are indicative of the closure of the Z = 14 proton subshell in the isotopes ^30,32,34Si and the N = 20 neutron shell.     

Shell structure fingerprints of tensor interaction

We address the consequences of strong tensor terms in the local energy density functional, resulting from fits to the f _5/2 -f _7/2 splittings in ⁴⁰Ca , ⁴⁸Ca , and ⁵⁶Ni . In this study, we focus on the tensor contribution to the nuclear binding energy. In particular, we show that it exhibits an interesting topological feature closely resembling that of the shell correction. We demonstrate that in the extreme single-particle scenario at spherical shape, the tensor contribution shows tensorial magic numbers equal to N(Z) = 14 , 32, 56, and 90, and that this structure is smeared out due to configuration mixing caused by pairing correlations and migration of proton/neutron sub-shells with neutron/proton shell filling. Based on a specific Skyrme-type functional SLy4_T, we show that the proton tensorial magic numbers shift with increasing neutron excess to Z = 14 , 28, and 50.     

Investigation of special features of the neutron and proton shell structure of the isotopes 90,92,94,96Zr

The neutron and proton single-particle energies and the occupation probabilities for the valence states of the even-even isotopes ^90,92,94,96Zr are determined by matching data on nucleon-stripping and nucleon-pickup reactions on the same nucleus. The data obtained in this way suggest the magicity of the number N = 56 for Z = 40. The single-particle energies of all bound neutron and proton states in the ^90,92,94,96Zr nuclei are described within the experimental errors on the basis of the dispersive optical model.     

The single-particle characteristics of Pb isotopes near the drip lines calculated within the dispersive optical model

The neutron and proton dispersive optical potential for the ²⁰⁸Pb nucleus has been determined for the energy region from–70 to +60 MeV and used to calculate the differential elastic scattering, the total interaction and reaction cross sections, as well as the single-particle characteristics, the neutron and charge densities, rms radii, and the thickness of the nucleus skin. The calculated results are in good agreement with the experimental data. The proton dispersive optical model potential for the spherical and close to spherical Pb isotopes within the neutron and proton drip lines has been obtained by a similar method. The calculation predicts a trend towards the growth of the proton particle-hole gap, which corresponds to Z = 82 shell closure as Z approaches the proton drip line.     

Estimating the occupation probabilities of single-particle orbits in nuclei

Occupation probabilities of neutron and proton single-particle orbits are estimated for a number of spherical and close to spherical nuclei with 20 ≤ Z ≤ 50 and 20 ≤ N ≤ 82 near the Fermi energy. The estimates are made according to the formula of the BCS theory with single-particle energies calculated using the mean-field model with dispersive optical potential. The closeness of the occupation probabilities to 0 and 1 is demonstrated for nuclei with traditional and new magic numbers.     

Non-empirical pairing energy density functional

We perform systematic calculations of pairing gaps in semi-magic nuclei across the nuclear chart using the Energy Density Functional method and a non-empirical pairing functional derived, without further approximation, at lowest order in the two-nucleon vacuum interaction, including the Coulomb force. The correlated single-particle motion is accounted for by the SLy4 semi-empirical functional. Rather unexpectedly, both neutron and proton pairing gaps thus generated are systematically close to experimental data. Such a result further suggests that missing effects, i.e. higher partial waves of the NN interaction, the NNN interaction and the coupling to collective fluctuations, provide an overall contribution that is sub-leading as for generating pairing gaps in nuclei. We find that including the Coulomb interaction is essential as it reduces proton pairing gaps by up to 40%.     

Microscopic three-body force for asymmetric nuclear matter

Brueckner calculations including a microscopic three-body force have been extended to isospin-asymmetric nuclear matter. The effects of the three-body force on the equation of state and on the single-particle properties of nuclear matter are discussed with a view to possible applications in nuclear physics and astrophysics. It is shown that, even in the presence of the three-body force, the empirical parabolic law of the energy per nucleon vs. isospin asymmetry β = (N - Z)/A is fulfilled in the whole asymmetry range 0≤β≤1 up to high densities. The three-body force provides a strong enhancement of the symmetry energy which increases with density in good agreement with the predictions of relativistic approaches. The Lane's assumption that proton and neutron mean fields linearly vary vs. the isospin parameter is violated at high density due to the three-body force, while the momentum dependence of the mean fields turns out to be only weakly affected. Consequently, a linear isospin split of the neutron and proton effective masses is found for both cases with and without the three-body force. The isospin effects on multifragmentation events and collective flows in heavy-ion collisions are briefly discussed along with the conditions for direct URCA processes to occur in the neutron star cooling. Received: 18 February 2002 / Accepted: 16 May 2002     

Evolution of proton shells in 20 ⩽ Z ⩽ 28 and 20 ⩽ N ⩽ 50 nuclei and dispersive optical model

Experimental single-particle proton energies in spherical and nearly spherical 20 ? Z ? 28 medium-mass nuclei and their counterparts evaluated with the aid of data formirror nuclei were analyzed on the basis of the dispersive optical model. The parameters of the dispersive optical potential were extrapolated to the region of unstable nuclei, and the values obtained in this way were then used to predict the single-particle proton energies in the 20 ? N ? 50 nuclei under study. The evolution of the particle-hole energy gap was traced, and features peculiar to single-particle spectra of magic and nonmagic nuclei were revealed by comparing single-particle energies with proton-separation energies.     

Neutron single-particle characteristics of Ag isotopes in the dispersive optical model

The neutron dispersive optical potential for Ag isotopes is constructed for a wide range of variation in the N number. Good agreement with the experimental data on the scattering of neutrons by ¹⁰⁷Аg isotopes, and on single-particle energies and the probabilities of subshell occupation near the Fermi energy, is obtained for stable isotopes ^{107, 109}Ag using the dispersive optical model. Calculations that predict the evolution of the neutron single-particle energies up to the boundary of neutron stability are performed. It is shown that new magic number N = 56 (for Z = 40) disappears upon moving from Zr to Sn in an isotonic chain with N = 56, due to rapid deepening of level 1g _7/2.     

New double-magic nucleus <Superscript>96</Superscript>Zr and conditions for existence of new magic nuclei

New information about energies and occupation probabilities of neutron and proton single-particle (hole) subshells in even—even nuclei was obtained in previous studies applying the method of putting into correspondence all available data on one-nucleon pickup and stripping reactions. The most important and interesting resultwas identification of several nuclei as new magic ones. Namely, itwas found that a filling of the neutron 2d _5/2 subshell in ⁹⁶Zr makes this nuclide a magic one. Moreover, changes in proton subshell energies with increasing neutron number N are accompanied by an increasing energy gap between closed 2p _1/2 and empty 1g _9/2 subshells. Thus, the neutron number N = 56 appears to be a magic one if the proton number Z is equal to 40. The proton number Z = 40 manifests properties of the magic number in ⁹⁶Zr. Therefore, ⁹⁶Zr was identified as a new double-magic nucleus. Further investigations revealed that the energy of the first 2⁺ state E(2 ₁ ⁺ ) in ⁹⁶Zr is much higher than that in the neighboring isotopes and isotones, whereas the ratio E(4 ₁ ⁺ )/E(2 ₁ ⁺ ) and the quadrupole deformation parameter β2 are, vice versa, clearly lower. Moreover, the A dependence of the neutron separation energy B(n) in Zr isotopes has an irregularity at N = 56 which is typical of magic nuclei. As a result of these investigations, it was found that, near the Fermi energy, there are two closed subshells with the same (and large) angular momentum j = 5/2 (viz. π1f _5/2 and ν2d _5/2). We call this situation the jj connection. The magic numbers under discussion (Z = 40 and N = 56) are achieved at the points where both subshells are closed, and in addition, the closed subshell with j = 1/2, π2p _1/2, occurs above the proton π1f _5/2 subshell. This looks like a result of some additional attractive proton-neutron interaction. It was found that application of this scheme (jj connection) to other subshells reveals several other new magic nuclei: ⁵⁴Ca (closed π1d _3/2 and ν2p _3/2 together with closed ν2p _1/2), ³⁰S and ³⁰Si (closed π1d _5/2 and ν1d _5/2 together with closed (π/ν)2s _1/2), and ¹⁴O and ¹⁴C (closed π1p _3/2 and ν1p _3/2 together with closed ν2p _1/2). The text was submitted by the authors in English.     

A relativistic mean-field study of magic numbers in light nuclei from neutron to proton drip-lines

In an axially deformed relativistic mean-field calculation of single-particle energy spectra ofN = 8 (Li-Mg) andN = 14,16 (C-Mg) isotonic chain and the one- and two-neutron separation energies of various isotopes of Li-Mg, new magic numbers are found to exist atN = 6 andN = 16 and/orN = 14, which are in addition to theN = 8 andN = 20 magic numbers. In neutron-rich nuclei, the shell gap atN = 6 is larger than atN = 8 and a large gap is observed forN = 16 or 14 for the neutron-rich andN = 14 for proton-rich nuclei. Large shell gaps are also found to exist atN = 14 and 16 orN = 16 alone for nuclei near theβ-stability line. The above results are independent of the parameter sets TM2, NL3 and NL-SH used here. Similarly, new large shell gaps are predicted atZ = 616 and/or 14 for protons.     

A pilgrimage through superheavy valley

We searched for the shell closure proton and neutron numbers in the superheavy region beyond Z = 82 and N = 126 within the framework of non-relativistic Skryme–Hartree–Fock (SHF) with FITZ, SIII, SkMP and SLy4 interactions. We have calculated the average proton pairing gap Δ_p, average neutron pairing gap Δ_n, two-nucleon separation energy S _2q and shell correction energy E _shell for the isotopic chain of Z = 112–126. Based on these observables, Z = 120 with N = 182 is suggested to be the magic numbers in the present approach.     

Single-particle neutron characteristics of Cd isotopes with N of 50 to 82

Experimental data on single-particle neutron energies of Cd isotopes are analyzed within a dispersive optical model. Parameters of the potential are extrapolated to the region of unstable isotopes with a neutron excess. The evolution of calculated single-particle spectra and occupation probabilities of single-particle orbits corresponds to formation of magic features in Cd isotopes with N = 50, 82. The results from calculations agree with the concept that the ¹⁷⁴Cd isotope with N = 126 forms the end of the peninsula at the neutron drip line.     

Non-local mean field effect on nuclei near sub-shell

Evolutions of single-particle energies and Z=64 sub-shell along the isotonic chain of N=82 are investigated in the density dependent relativistic Hartree–Fock (DDRHF) theory in comparison with other commonly used mean field models such as Skyrme HF, Gogny HFB and density dependent relativistic Hartree model (DDRMF). The pairing is treated in the BCS scheme, except for Gogny HFB. It is pointed out that DDRHF reproduces well characteristic features of experimental Z-dependence of both spin–orbit and pseudo-spin–orbit splittings around the sub-shell closure Z=64. Non-local exchange terms of the isoscalar σ and ω couplings play dominant roles in the enhancements of the spin–orbit splitting of proton 2d states, which is the key ingredient to give the Z=64 sub-shell closure properly. On the other hand, the π and ρ tensor contributions for the spin–orbit splitting cancel each other and the net effect becomes rather small. The enhancement of the sub-shell gaps towards Z=64 is studied by the DDRHF, for which the local terms of the scalar and vector meson couplings are found to be important.     

Calculated properties of superheavy nuclei

Ground-state properties of the heaviest nuclei are analyzed within a macroscopic-microscopic approach. The main attention is paid to such properties as deformation, deformation energy, energy of the first rotational state 2⁺ of a nucleus, and the branching ratio of α decay to this 2⁺ state with respect to the decay to the ground state 0⁺. The analysis concerns the problem of experimental confirmation of theoretically predicted deformed shapes of superheavy nuclei situated in the region around the nucleus ²⁷⁰Hs. A large region of even-even nuclei with proton, Z=82–128, and neutron, N=126–190, numbers is considered.