Experimental Single-particle Energies Research Articles

An ab-initio Gamow shell model approach with a core

Gamow shell model (GSM) is usually performed within the Woods-Saxon (WS) basis in which the WS parameters need to be determined by fitting experimental single-particle energies including their resonance widths. In the multi-shell case, such a fit is difficult due to the lack of experimental data of cross-shell single-particle energies and widths. In this paper, we develop an ab-initio GSM by introducing the Gamow Hartree-Fock (GHF) basis that is obtained using the same interaction as the one used in the construction of the shell-model Hamiltonian. GSM makes use of the complex-momentum Berggren representation, then including resonance and continuum components. Hence, GSM gives a good description of weakly bound and unbound nuclei. Starting from chiral effective field theory and employing many-body perturbation theory (MBPT) (called nondegenerate Qˆ-box folded-diagram renormalization) in the GHF basis, a multi-shell Hamiltonian (sd-pf shells in this work) can be constructed. The single-particle energies and their resonance widths can also been obtained using MBPT. We investigated 23−28O and 23−31F isotopes, for which multi-shell calculations are necessary. Calculations show that continuum effects and the inclusion of the pf shell are important elements to understand the structure of nuclei close to and beyond driplines.

Evolution of nuclear spin-orbit splittings with Skyrme functional SAMi-T

A new Skyrme functional has been developed with tensor term guided by ab initio relativistic Brueckner-Hartree-Fock (RBHF) studies on neutron-proton drops. Instead of extracting information on the tensor force from experimental single-particle energy splittings, the RBHF calculations do not contain beyond mean-field effects such as particle-vibration coupling and therefore the information on the tensor force can be obtained without ambiguities. The new functional gives a good description of nuclear ground-state properties aswell as various giant resonances. The description for the evolution of single-particle energy splittings is also improved by the new functional.

Hartree-Fock Gamow basis from realistic nuclear forces

The development of the radioactive nuclear beams has made it possible to study the exotic nuclei near the drip line. The theoretical description of these nuclei which are far from the valley of stability is still a challenge in nuclear physics. These nuclei exhibit unusual properties such as halo densities and weak bindings. The coupling to the continuum is needed for the theoretical description. The Gamow shell model which extends the traditional shell model to the complex plane, can efficiently include the continuum coupling. This model unifies nuclear structure and reaction properties, and has proven to be a promising tool for the descriptions of weakly bound and unbound nuclei. The starting point of the Gamow shell model is the Berggren completeness where bound, resonant (or Gamow) and scattering states are treated on an equal footing. Employing the Berggren basis, first principle calculations can also be extended to study weak-binding nuclei and nuclear reactions. A big concern is how to obtain the single-particle Berggren basis which is used to construct the Slater determinants in many-body calculations. The single-particle basis in Gamow shell-model calculations are normally constructed from a Woods-Saxon potential with parameters fitted to experimental single-particle energies. However, in a fully microscopic approach, the single-particle basis constructed from the nucleon-nucleon interaction is preferred. In this paper, we present a method to generate the Berggren basis microscopically from realistic nucleon-nucleon forces. The obtained basis is called Hartree-Fock (HF) Gamow basis. We first performed the HF iteration in harmonic oscillator basis, and then the HF potential obtained is analytically continued to the complex- k plane. The continuation is accomplished directly by a basis transformation in which the complex Laguerre polynomials are used. By discretizing the integral of the Schrodinger equation in momentum space, the equation becomes a complex symmetric eigenvalue problem. The HF Gamow basis can be obtained by diagonalizing the complex symmetric matrix. As examples, we apply this method to 4He and 22O with the softened chiral N3LO potential. The single-particle p3/2 resonant state in 4He and d3/2 resonance state in 22O are obtained and the numerical convergence is checked. We also discuss the method to obtain the resonance state from the phase shifts of the real-energy scattering states. With the phase-shift method, the obtained 22O neutron d3/2 resonance state is consistent with the result of the complex-plane calculation. The Gamow basis obtained can be further used in many-body calculations of weakly bound nuclei.

Single-particle properties of N = 12 to N = 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 26,28,30,32,34Si. 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.

Analysis of proton single-particle properties of zinc and germanium isotopes

Experimental proton single-particle energies in the vicinity of the Fermi energy for stable zinc and germanium isotopes are analyzed on the basis the dispersive optical model. The values found for the parameters of the dispersive optical potential are corrected with the aim of matching the total number of protons that is calculated with the aid of the function of Bardeen-Cooper-Schrieffer theory for the occupation probability for single-particle orbits with the charge number Z of the nucleus. The parameters of the dispersive optical potential are extrapolated on the basis of physically motivated arguments to the region of unstable isotopes in which the number N ranges between 34 and 50, and single-particle spectra are predicted by means of calculations with these parameters.

Neutron shell structure of 58,60,62,64Ni nuclei and its study within a mean-field model with dispersive optical-model potential

Experimental single-particle energies and occupation probabilities for neutron states near the Fermi energy in 58,60,62,64Ni nuclei were obtained from joint evaluation of the data on nucleon stripping and pickup reactions on the same nucleus. Degeneracy of the 2p 3/2,1/2 and 1f 5/2 subshells was demonstrated. The resulting data were analyzed within a mean-field model with dispersive optical-model potential. Good agreement was obtained between the calculated and experimental single-particle energies of the subshells.

Calculation of single-particle energies in the 38 96 Sr nucleus within the mean field model with the dispersive optical potential

Evaluated and experimental neutron single-particle energies of the 88, 94, 96Sr and 90, 96, 98Zr nuclei near the Fermi energy are compared. A possibility of formation of a considerable particle-hole gap in the 94, 96Sr typical of traditional magic nuclei is investigated. Agreement is obtained between the single-particle energies calculated within the mean field model with the dispersive optical potential and the evaluated energies for the 94, 96Sr nuclei.

Analysis of the new data on single-particle structure of nuclei from the middle of the 1f–2p shell within the dispersive optical model

On the basis of the most reliable experimental information about the single-particle energies of neutron and proton states in 2048Ca28, 2250Ti28, 2452Cr28, and 2654Fe28 nuclei, single-particle energies in mirror 2848Ni20, 2850Ni22, 2852Ni24 and 2854Ni26 nuclei have been evaluated. The evaluated single-particle energies are compared with the energies calculated within the dispersive optical model. Good agreement between the calculated and experimental single-particle energies is demonstrated.

The T=2 mirrors 36Ca and 36S: A test for isospin symmetry of shell gaps at the driplines

Abstract The first excited 2 + state of 36Ca has been identified by its γ-decay, exploiting the two-step fragmentation technique at the FRS-RISING setup at GSI. This is the heaviest T z = − 2 nucleus in the Segre chart in which a γ-decay of an excited state has been observed. A stable beam of 40Ca at 420 A MeV impinged on a primary 9Be target. Out of the secondary beam of fragmentation products, 37Ca was separated by the FRS and struck on a second 9Be target at the final focus of the FRS. The energy for the 2 1 + decay of 36Ca was determined to be 3015(16) keV, which is 276 keV lower than in its T = 2 mirror 36S. This mirror energy difference (MED) is discussed in the framework of shell model calculations using a 16O core, the sd shell isospin symmetric interaction USD and experimental single-particle energies from 17O and 17F. The results show that the MED within the sd shell provide a sensitive test for the evolution of the N, Z = 14 , 16 subshell gaps towards the driplines. Especially the N, Z = 16 gap is determined by Thomas–Ehrman shift in the A = 17 , T = 1 / 2 isospin doublet, while Coulomb effects are found to have marginal influence.

Study of Λ hypernuclei in the quark mean-field model

We extend the quark mean-field model to the study of Λ hypernuclei. Without adjusting parameters, the properties of Λ hypernuclei can be described reasonably well. The small spin–orbit splittings for Λ in hypernuclei are achieved, while the Λ single-particle energies in the present model are slightly underestimated as compared with the experimental values. About 3% deviation from the quark model prediction for the ω– Λ couplings is required in order to reproduce the experimental single-particle energies.

G-matrix calculations in finite hypernuclei.

g-matrix calculations have been performed for $_{\mathrm{\ensuremath{\Lambda}}}^{5}\mathrm{He}$, $_{\mathrm{\ensuremath{\Lambda}}}^{17}\mathrm{O}$, and $_{\mathrm{\ensuremath{\Lambda}}}^{41}\mathrm{Ca}$ by solving the \ensuremath{\Lambda}-\ensuremath{\Sigma} coupled channels Bethe-Goldstone equation. The potentials employed were the Nijmegen model D and Nijmegen soft core. Calculations were performed with various choices for the Pauli-forbidden space and the energy denominator. The Nijmegen model D does a reasonable job of reproducing most experimental single-particle energies, but the calculated ground-state binding energies vary smoothly from being underbound in the $_{\mathrm{\ensuremath{\Lambda}}}^{5}\mathrm{He}$ to overbound in $_{\mathrm{\ensuremath{\Lambda}}}^{41}\mathrm{Ca}$. The Nijmegen soft core underbinds all single \ensuremath{\Lambda} states, but gives level spacings which agree with experimental values.

Excitation cross sections for electron collisions with sulfur atoms

An analytic atomic independent particle model potential adjusted to experimental single-particle energy levels is used to generate wave functions for the valence and excited states of sulfur. Using these wave functions in conjunction with the Born approximation and LS coupling, we calculate generalized oscillator strengths and integrated cross sections up to 5 keV for various electron impact excitations from the 1s22s22p63s23p4(3P2) ground state.

Information on effective interactions from experimental single-particle energies.

A method is presented for obtaining centroids of the effective nucleon-nucleon interaction directly from measured single-particle energies. An application to nuclei in the mass region A=28 to 56 is provided to illustrate the technique. Uncertainties in the centroids are estimated at 100 keV or less, except in cases of exceptional fragmentation of the single-particle strength. Methods such as the one proposed here can be used to place constraints on realistic interactions calculated from bare nucleon-nucleon potentials, which are known to possess defective centroids in many cases.

Optical oscillator strengths for germanium, tin, and lead

A semiempirical analytical atomic independent-particle-model potential adjusted to experimental single-particle energy levels is used to generate wave functions for the valence and excited states of Ge i, Sn i, and Pb i. Using the Born approximation and LS coupling we calculate optical oscillator strengths for excitations from the ground states 4p2(3P0) of Ge i, 5p2(3P0) of Sn i, and 6p2(3P0) of Pb i. The results are compared to experimental and theoretical data.

Neutron-40Ca mean field between -80 and +80 MeV from a dispersive optical-model analysis.

The n${\mathrm{\ensuremath{-}}}^{40}$Ca complex mean field is derived from a dispersive optical-model analysis of the available experimental cross sections. In this analysis the real part of the mean field contains dispersive contributions which are derived from the imaginary part by means of a dispersion relation. These dispersive contributions must be added to the Hartree-Fock potential which is assumed to have a Woods-Saxon shape, with a depth ${V}_{H}$(E) that depends exponentially upon energy. The input experimental data are 14 differential cross sections in the energy domain (5.3, 40.0 MeV), five polarization cross sections in the domain (9.9, 16.9 MeV), and the total cross section in the domain (2.5, 80 MeV).The resulting optical-model potential is an analytic function of energy. It can thus be extrapolated towards negative energies, where it should be identified with the shell-model potential. This extrapolation yields good agreement with the experimental single-particle energies in the two valence shells of $^{40}\mathrm{Ca}$. The model also predicts the radial shape and the occupation probabilities of the single-particle orbits and the spectroscopic factors of the single-particle excitations. In order to reproduce the experimental energies of the deeply bound 1p and 1s orbits, one must use a linear rather than an exponential energy dependence of ${V}_{H}$(E) at large negative E. It is shown that this is precisely the behavior expected from the fact that the energy dependence of ${V}_{H}$(E) actually represents the nonlocality of the original microscopic Hartree-Fock field. The model also correctly predicts the distribution of the single-particle strength of the 1d(5/2 excitation in $^{39}\mathrm{Ca}$. The calculated distributions of the 1p strength in $^{39}\mathrm{Ca}$ and of the 1f(5/2 strength in $^{41}\mathrm{Ca}$ show that the available experimental information extends over less than half the expected peak, whose energy is thus poorly known experimentally. In the energy domain (2.5, 9 MeV) the predicted total cross section deviates from the experimental data; this reflects the fact that at low energy the calculated cross section is very sensitive to small modifications of the mean field.

Electron impact excitation cross sections for chlorine

An analytic atomic independent particle model potential adjusted to experimental single-particle energy levels is used to generate wave functions for the valence and excited states of chlorine. Using these wave functions in conjunction with the Born approximation and LS coupling, we calculate generalized oscillator strengths and integrated cross sections up to 5 keV for various electron impact excitations from the 1s2 2s2 2p6 3s2 3p5(2P3/2) ground state.

Unified description of the neutron-208Pb mean field between -20 and +165 MeV from the dispersion relation constraint.

The real part of the central neutron${\mathrm{\ensuremath{-}}}^{208}$Pb mean field is the sum of a Hartree-Fock component plus a dispersive component. In keeping with theoretical expectations, the Hartree-Fock field is assumed to have a Woods-Saxon shape whose depth decreases exponentially with increasing energy and whose radius and diffuseness are independent of energy. The dispersive component is determined from the imaginary part of the optical-model potential by making use of the dispersion relation which connects these two quantities. The imaginary part is written as the sum of a volume and a surface-peaked contribution. The dispersion relation then implies that the real dispersive contribution is also the sum of volume and surface-peaked components. The parameters of the complex mean field are determined by fitting the available differential and polarization cross sections in the energy domain [4, 40 MeV] and the total cross sections in the domain [1,120 MeV]; these data are contained in previous published or unpublished reports, but new measurements of the total cross sections are presented from 1 to 25 MeV. Good fits to these cross sections, and also to unpublished total cross sections for energies up to 165 MeV, are obtained despite the fact that the number of adjusted parameters is quite small because of our use of the constraint implied by the dispersion relation.The real part of the mean field is well approximated by a Woods-Saxon shape whose radius decreases with increasing energy between 5 and 25 MeV; its depth is approximately constant from 5 to 15 MeV and then decreases with increasing energy; these findings are in keeping with recent empirical evidence. When the neutron energy decreases below 2.5 MeV, the potential radius decreases; it increases again when the neutron energy decreases below -14.5 MeV. In the domain -20 MeV<E<0 the deduced potential accurately reproduces the experimental single-particle energies as well as the asymptotic values of the single-particle wave functions as measured from sub-Coulomb pickup reactions; it also yields excellent agreement with the spreading width of the deeply bound 1${h}_{11/2}$ hole state. The rms radii, absolute spectroscopic factors, and occupation numbers are calculated for the valence particle and hole states. At the Fermi energy (-6 MeV), the mean field can be identified with the Hartree-Fock potential, for which the present analysis yields a depth of 46.4 MeV, a radius of 1.24${A}^{1/3}$ fm, and a diffuseness of 0.68 fm. In the energy domain 4\ensuremath{\le}E\ensuremath{\le}10 MeV, the already good agreement between the predicted and measured cross sections is further improved if the imaginary part of the mean field is allowed to have its strength depend upon the neutron orbital angular momentum, and its surface diffuseness is allowed to be energy dependent.

Extrapolation from positive to negative energy of the Woods-Saxon parametrization of the n- 208Pb mean field

The real part V( r); E) of the nucleon-nucleus mean field is assumed to have a Woods-Saxon shape, and accordingly to be fully specified by three quantities: the potential depth U v ( E), radius R V ( E) and diffuseness a v ( E). At a given nucleon energy E these parameters can be determined from three different radial moments [ r q ] v = (4 π/ A) ∝ V( r; E) r q d r. This is useful because a dispersion relation approach has recently been developed for extrapolating [ r q ] V ( E) from positive to negative energy, using as inputs the radial moments of the real and imaginary parts of empirical optical-model potentials V( r; E) + iW( r; E). In the present work, the values of U v ( E), R v ( E) and a v ( E) are calculated in the case of neutrons in 208Pb in the energy domain −20 < E < 40 MeV from the values of [ r q ] V ( E) for q = 0.8, 2 and 4. It is found that both U V ( E) and R v ( E) have a characteristic energy dependence. The energy dependence of the diffuseness a a ( E) is less reliably predicted by the method. The radius R V ( E) increases when E decreases from 40 to 5 MeV. This behaviour is in agreement with empirical evidence. In the energy domain −10 MeV < E < 0, R V ( E) is predicted to decrease with decreasing energy. The energy dependence of the root mean square radius is similar to that of R V ( E). The potential depth U v slightly increases when E decreases from 40 to 15 MeV and slightly decreases between 10 and 5 MeV; it is consequently approximately constant in the energy domain 5 < E < 20 MeV, in keeping with empirical evidence. The depth U v increases linearly with decreasing E in the domain −10 MeV < E < 0. These features are shown to persist when one modifies the detailed input of the calculation, namely the empirical values of [ r q ] v ( E) for E > 0 and the parametrization [ r q ] w ( E) of the energy dependence of the radial moments of the imaginary part of the empirical optical-model potentials. In the energy domain −10 MeV < E < 0, the calculated V( r; E) yields good agreement with the experimental single-particle energies; the model thus accurately predicts the shell-model potential ( E < 0) from the extrapolation of the optical-model potential ( E > 0). In the dispersion relation approach, the real part V( r; E) is the sum of a Hartree-Fock type contribution V HF( r; E) and of a dispersive contribution ΔV( r; E). The latter is due to the excitation of the 208Pb core. The dispersion relation approach enables the calculation of the radial moment [ r q ] ΔV ( E) from the parametrization [ r q ] w ( E): several schematic models are considered which yield algebraic expressions for [ r q ] Δ V ( E). The radial moments [ r q ] HF( E) are approximated by linear functions of E. When in addition, it is assumed that V HF( r; E) has a Woods-Saxon radial shape, the energy dependence of its potential parameters ( U HF, R HF, a HF) can be calculated. Furthermore, the values of ΔV( r; E) can then be derived. It turns out that ΔV( r; E) is peaked at the nuclear surface near the Fermi energy and acquires a Woods-Saxon type shape when the energy increases, in keeping with previous qualitative estimates. It is responsible for the peculiar energy dependence of R V ( E) in the vicinity of the Fermi energy.

Equilibrium deformations for the Ra-Th region of nuclei

New parameters of the single-particle Nilsson potential for the Ra-Th region have been found. They are: κ p = 0.0561, μ p = 0.702, κ n = 0.0588, μ n = 0.442. The fit is made to the experimental single-particle energy levels in the new quadrupole and hexadecapole equilibrium deformation points, obtained by the shell-correction method with a simple consistency condition. The equilibrium deformations calculated by this method are in good agreement with the experimental data. Experimental values of the quadrupole moments of 222Rn, 222Ra and 224Ra are reproduced for the first time within the Nilsson-plus-Strutinsky method.

Optical oscillator strengths for isoelectric ions of silicon

An analytic atomic independent particle model potential adjusted to experimental single-particle energy levels is used to generate wave functions for the valence and excited states of the ions P +1, S +2, Cl +3, Ar +4 and K +5 of the silicon isoelectronic sequence. Using these wave functions in conjuction with the Born approximation and the LS-coupling scheme, we calculate optical oscillator strengths for excitations from the 3p 2( 3P 0) ground state. Good agreement is obtained with experimental and theoretical data.