Shell-model Energies Research Articles

Study of Nuclear Structure of 42-43 Ti isotopes by using F7MBZ & F742 interactions

In this work, The energy levels and lowered electric quadruple transition probability B(E2) were calculated with the use of the OXBASH code in the F7shell and the F7MBZ& F742 effective interaction for the 42-43Ti isotopes. All nucleon combinations in the) 0 7/2, 1 3/2, 0 5/2, and 1 1/2 ( orbits are included in the model space. Additionally, the shell model does a good in describing the invading states' energies. Computations have been done using effective interactions F7MBZ and F742 in full f7 space, As well as the shell model, Lowlying states, and binding energies computed, and it was found that the calculated rates of electric quadrupole transitions between the isotopes 42-43Ti were consistent with the existing experimental published data.

Weighted variation after projection method for low-lying nonyrast states* *Supported by the National Natural Science Foundation of China (11975314), the Key Laboratory of Nuclear Data foundation (JCKY2022201C158), and the Continuous Basic Scientific Research Project (WDJC-2019-13, BJ20002501)

We propose a simple algorithm to further improve the previous variation after projection (VAP) wave functions for low-lying nonyrast states. We attach a weight factor to each calculated energy; then, the sum of these weighted energies is minimized. It turns out that a low-lying nonyrast VAP wave function can be further optimized when the weight factor for the corresponding energy is far larger than the other ones. Based on the improved WVAP wave functions, the energy-variance extrapolation method is applied to estimate the exact shell model energies. The calculated results for nuclei in the and model spaces clearly show that the extrapolated energies for all the calculated states are very close to the exact shell model ones within keV.

Asymptotic equivalence of the shell-model and local-density descriptions of Coulombic systems confined by radially symmetric potentials in two and three dimensions

Asymptotic equivalence of the shell-model and local-density (LDA) descriptions of Coulombic systems confined by radially symmetric potentials in two and three dimensions is demonstrated. Tight upper bounds to the numerical constants that enter the LDA expressions for the Madelung energy are derived and found to differ by less than 0.5% from the previously known approximate values. Thanks to the variational nature of the shell-model approximate energies, asymptotic expressions for other properties, such as mean radial positions of the particles and number densities, are also obtained. A conjecture that generalizes the present results to confining potentials with arbitrary symmetries is formulated.

New variational Monte Carlo method with energy variance extrapolation for large-scale shell-model calculations

We propose a new variational Monte Carlo (VMC) method with an energy variance extrapolation for large-scale shell-model calculations. This variational Monte Carlo is a stochastic optimization method with a projected correlated condensed pair state as a trial wave function, and is formulated with the M-scheme representation of projection operators, the Pfaffian and the Markov-chain Monte Carlo (MCMC). Using this method, we can stochastically calculate approximated yrast energies and electro-magnetic transition strengths. Furthermore, by combining this VMC method with energy variance extrapolation, we can estimate exact shell-model energies.

Self-consistent calculations of isospin admixtures in the ground states of the N= Z nuclei in the mass region of 50–100

In the present study, based on spherical shell-model wave functions and energies with the pairing and the residual Fermi interaction treated in pn QRPA, the isospin admixture probabilities in the ground state of N=Z even-even nuclei from Z=28 to Z=50 are investigated. In these calculations, considering the restoration of the isotopic invariance for the nuclear part of the Hamiltonian, the residual interaction strength parameter in the quasiparticle space has been obtained in such a way that it is self-consistent with the Woods-Saxon form of the shell model potential. As a result of calculations, it has been found that the calculated isospin admixture probability is proportional to a higher power of Z than the results of other theoretical studies.

The continuum shell-model neutron states of209Pb

The neutron strength distributions of the three high-spin 1k 17/2, 2h 11/2 and 1j13/2 states of209Pb have been obtained within the formalism of the core-polarisation effect where the effect of interaction of the neutron shell-model states of209Pb with the collective vibrational states (originating also from the giant resonances) have been taken into consideration. The theoretical results have been discussed in the light of works on 1k 17/2, 2h111/2 and 1j13/2 neutron orbitals of209Pb. The shell-model energies of the neutron states have been obtained by Skyrme-Hartree-Fock method

Precise estimation of shell model energy by second-order extrapolation method

A second order extrapolation method is presented for shell model calculations, where shell model energies of truncated spaces are well described as a function of energy variance by quadratic curves and exact shell model energies can be obtained by the extrapolation. This new extrapolation can give more precise energy than those of first order extrapolation method. It is also clarified that first order extrapolation gives a lower limit of shell model energy. In addition to the energy, we derive the second order extrapolation formula for expectation values of other observables.

Random phase approximation vs exact shell-model correlation energies

The random phase approximation (RPA) builds in correlations left out by mean-field theory. In full 0-hbar-omega shell-model spaces we calculate the Hartree-Fock + RPA binding energy, and compare it to exact diagonalization. We find that in general HF+RPA gives a very good approximation to the ``exact'' ground state energy. In those cases where RPA is less satisfactory, however, there is no obvious correlation with properties of the HF state, such as deformation or overlap with the exact ground state wavefunction.

Dependence of the shell-model single-particle energies on different components of the nucleon-nucleon interaction

The spin-orbit splittings in the spectra of nuclei with mass numbers 5, 15, and 17 are studied within the framework of shell-model configuration mixing calculations including $2\ensuremath{\Elzxh}\ensuremath{\omega}$ excitations. The contributions of the two-body spin-orbit and tensor components of the nucleon-nucleon interaction are studied in various model spaces. It is found that the effects of the two-body spin-orbit interaction are dominant and quite sensitive to the size of the model space considered. The effects of the tensor interaction are weaker. The correlations effects which are included in the larger (0+2) $\ensuremath{\Elzxh}\ensuremath{\omega}$ shell-model space reduce the spin-orbit splitting in the case of $A=5$ by 20%, and enhance it for $A=15$ by about the same 20%. However, it is found that the correlations have a very small effect on the ${d}_{3/2}\ensuremath{-}{d}_{5/2}$ splitting in $A=17.$

Shell-model calculations for the properties of nuclei with A = 86–100 near the proton drip line

Extensive shell-model calculations in the mass range A = 86–100 on the neutron-deficient side of the nuclear chart are performed. The complete spectra of all isotopes are computed and compared to experimental excitation energies where available. Beta decay half lives are evaluated with both free and effective Gamow-Teller operators. The binding energies are obtained using a five parameter fit in addition to the shell-model energies. From the binding energies we deduce the proton drip line.

Fragmentation of the high spin discrete neutron and proton states of209Pb and209Bi

The distribution of the discrete 1h 9/2, 2f 7/2, 1i 13/2 and 2f 5/2 proton states of209Bi and 2g 9/2, 1i 11/2, 1j 15/2 and 2g7/2 neutron states of209Pb have been obtained within the particle-vibration coupling model calculation and compared with the experimental datas baising on the most recent high resolution stripping reaction on208Pb using 480 MeV12C projectile. The optimised shell model energies arising from the core-polarisation effect have profound influence for both the study of the structures of the high spin continuum shell model states of208Pb and stability of superheavy nuclei.

Gamma decay of high spin states in 25Mg above 6.1 MeV.

New high spin states in $^{25}\mathrm{Mg}$ (up to 13.3 MeV in excitation) have been located using the $^{12}\mathrm{N}$,p\ensuremath{\gamma}${)}^{25}$Mg reaction at a laboratory beam energy of 23.5 MeV. Gamma decay from 28 neutron unbound levels has been observed (24 of which are reported here for the first time) up to 7 MeV above the neutron drip line. Tentative identification of the yrast line up to ${(17/2}^{+}$ has been made, along with an extension of the ground-state rotational band up to ${(15/2}^{+}$ and the location of several other states with ${J}^{\ensuremath{\pi}}$${=(11/2}^{+}$ and ${(15/2}^{+}$. Extensive comparison with the s-d shell model and the cranked Nilsson-Strutinsky model has been made for the states in $^{25}\mathrm{Mg}$. Very good agreement is observed between the shell model and experimental level energies and \ensuremath{\gamma}-decay branches. Potential energy surfaces for ${J}^{\ensuremath{\pi}}$${=(11/2}^{+}$, ${(13/2}^{+}$, and ${(17/2}^{+}$ in the cranked Nilsson-Strutinsky model show a noncollective oblate local minimum at (${\ensuremath{\epsilon}}_{2}$,\ensuremath{\gamma})=(0.34,60\ifmmode^\circ\else\textdegree\fi{}) for ${(11/2}^{+}$ and prolate configurations at (0.2,-120\ifmmode^\circ\else\textdegree\fi{}) for ${(13/2}^{+}$ and ${(17/2}^{+}$. These states compete with the collective ground-state band members at \ensuremath{\gamma}\ensuremath{\approxeq}0\ifmmode^\circ\else\textdegree\fi{} and become yrast for ${(17/2}^{+}$. The stabilization of these prolate aligned states relative to more deformed, collective states of the same spin results from shell effects at high spin in the cranking model, and provides a description of the configuration in the yrast ${8}^{+}$; ${(11/2}^{+}$, ${(13/2}^{+}$, and ${(17/2}^{+}$; and ${4}^{+}$ states in $^{24}\mathrm{Mg}$, $^{25}\mathrm{Mg}$, and $^{26}\mathrm{Mg}$, respectively.

An isospin invariant form of the interacting boson model for odd nuclei (IBFM-3)

Abstract The interacting boson-fermion model IBFM for odd nuclei is extended to incorporate isospin symmetry in nuclei where valence neutrons and protons occupy the same orbits. A mapping is established with shell-model states. A boson-fermion hamiltonian is derived for the special case of charge-independent pairing forces and is found to reproduce the shell-model energies exactly. A comparison of the extension (IBFM-3) with the earlier version IBFM-2 shows that the latter omits some low-lying shell-model states.

Neutron strengths of the unbound 2h11/2, 1j13/2, and 1k17/2 shell-model states in 209Pb.

A broad based fragmented pattern of the three 2${h}_{11/2}$, 1${j}_{13/2}$, and 1${k}_{17/2}$ unbound shell-model neutron states in $^{209}\mathrm{Pb}$ has been obtained on the basis of the recent $^{208}\mathrm{Pb}$(\ensuremath{\alpha}${,}^{3}$He) reaction data. The present theoretical calculation shows that the shell-model energies of the three neutron states lie at 7.8 MeV excitation energy in $^{209}\mathrm{Pb}$. The dilution of the neutron strengths of these three high spin states also indicates that the coupling of the neutron particle states with the 11 vibrational states of $^{208}\mathrm{Pb}$ is strong indeed. The theoretical results on the damping of the three neutron states are in sharp contrast with the existing theoretical results on the quasiparticle phonon coupling model calculation.

Coulomb energies for multi-orbital shell-model calculations

A model is presented for calculating the Coulomb contribution to shell-model binding energies, for an arbitrary model space. The method is appropriate for both ground and excited states. The parameters of the model are determined by a fit to measured Coulomb displacement energies. As an example, Coulomb energies are computed for nuclei around mass 40.

Spectrum of the high spin neutron hole states of205Pb

The high resolution (3He,α) reaction on206Pb shows the distribution of the 2f 7/2, 1h 9/2 and 1i 1 3/2 neutron states of205Pb within the 6 MeV excitation energy of205Pb. The spectrum of these three-hole states is obtained within the hole-core vibrational coupling scheme. The shell model energies of the neutron hole states arising from the core-polarization effect are compared with the Bansal-French energy weighted sum rule. The possible implication of the present neutron hole energies has been discussed in the light of the deduced shell model wave functions of the collective states of206Pb.

The damping of the single-particle proton strengths in 1i11/2and 1j15/2states of209Bi

The 15 proton states 1h9/2, 2f7/2, 1i13/2, 3p3/2, 2f5/2, 3p1/2, 1j15/2, 2g9/2, 1i11/2, 3d5/2, 2g7/2, 4s1/2, 3d3/2, 2h11/2 and 1k17/2 have been mixed with the low-lying vibrational 3-, 51-, 52-, 7-, 2+, 4+, 6+ and 8+ states as well as the vibrational 1-' and 2+' states arising from the giant resonances in 208Pb to search for the broad fragmentations of the 1j15/2 and 1i11/2 states as observed in the most recent ( alpha ,t) reaction on 208Pb. The distribution of the shell-model strengths of these two high-spin states can be explained on the basis of the core polarisation effect on these two proton states of 209Bi. Knowledge of the altered shell-model energies of the proton states of 208Pb is necessary for the search for magicity in the superheavy nuclei.

Shell-model interaction energies in a relativistic hamiltonian formulation (II)

The two-nucleon interaction energies of light and medium nuclei are discussed in a picture of mesonic degrees of freedom and a pure shell model for nucleons. The role of the different mesons in the interaction energies and the domain of the best values of the coupling constants are explored by calculating the mean square deviation between theoretical and experimental two-nucleon interaction energies. Because of the redundancies between different meson contributions to the interaction energies, the values of the coupling constants are not uniquely determined, and rather large domains of best values are obtained. These domains include sets of values of the coupling constants which are compatible with those from other sources. The experimental data can be reproduced with the same quality as in conventional phenomenological potential models by taking the meson fields, σ, π, ρ and ω, and using only vector coupling for the vector fields.

Shell-model interaction energies in a relativistic hamiltonian formulation (I)

The non-covariant hamiltonian formulation of relativistic field theory is presented and solved as a secular problem in a discrete representation space. The shell-model two-nucleon interaction in the one-boson exchange picture is used to test the method before applying it to many-body systems. The results of the diagonalization treatment are convergent as a function of the discretized meson space and are close to the usual OBEP when considering the exchange of a single type of meson, thus establishing the numerical feasibility of the method. However a significant deviation between the two approaches appears for the sigma meson or for a mixture of several types of mesons with values of the coupling constants in the “physical” domain. Also, in the limited momentum space of the nucleons in the lowest shell-model state, the cut-off in the nucleon vertex form factors plays an unimportant role.

Microscopic Structure of the Magnetic High-Spin States inPb208

Consideration of both the effect of the one-pion and $\ensuremath{\rho}$-meson exchange potentials in the particle-hole interaction and the inclusion of 2p-2h configurations explains why the magnetic high-spin states discovered recently in $^{208}\mathrm{Pb}$ are close to the shell-model energies although the cross sections are only 50% of the single-particle estimate.