arXiv subjects
Publications and source records attributed to G. F. Bertsch.
Mass differences are an often used as signature and measure for shell closure. Using the angular-momentum projected Generator Coordinate Method and the Skyrme interaction SLy4, we analyze the modification of mass differences due to static deformation and dynamic fluctuations around the mean-field ground state.
We analyze the accuracy of BCS-based approximations for calculating correlation energies and odd-even energy differences in 2-component fermionic systems with a small number of pairs. The analysis is focused on comparing BCS and projected BCS treatments with the exact solution of the pairing Hamiltonian, considering parameter ranges appropriate for nuclear pairing energies. We find that the projected BCS is quite accurate over the entire range of coupling strengths in spaces of up to about 20 doubly degenerate orbitals. It is also quite accurate for two cases we considered with a more realistic Hamiltonian, representing the nuclei around 117Sn and 207Pb. However, the projected BCS significantly underestimates the energies for much larger spaces when the pairing is weak.
We use the Quasiparticle Random Phase Approximation (QRPA) and the Skyrme interactions SLy4 and SkM* to systematically calculate energies and transition strengths for the lowest 2+ state in spherical even-even nuclei.The SkM* functional, applied to 178 spherical nuclei between Z=10 and 90, produces excitation energies that are on average 11% higher than experimental values, with residuals that fluctuate about the average by -35%+55%. The predictions of SkM* and SLy4 have significant differences, in part because of differences in the calculated ground state deformations; SkM* performs better in both the average and dispersion of energies. Comparing the QRPA results with those of generator-coordinate-method (GCM) calculations, we find that the QRPA reproduces trends near closed shells better than the GCM, and overpredicts the energies less severely in general. We attribute part of the difference to a deficiency in the way the GCM is implemented.
A new theoretical approach is presented that combines the Hartree-Fock variational scheme with the exact solution of the pairing problem in the finite orbital space. Using this formulation in the sd-space as an example, we show that the exact pairing significantly improves the results for the ground state energy
We describe a procedure for mapping a self-consistent mean-field theory (also known as density functional theory) into a shell model Hamiltonian that includes quadrupole-quadrupole and monopole pairing interactions in a truncated space. We test our method in the deformed N=Z sd-shell nuclei Ne-20, Mg-24 and Ar-36, starting from the Hartree-Fock plus BCS approximation of the USD shell model interaction. A similar procedure is then followed using the SLy4 Skyrme energy density functional in the particle-hole channel plus a zero-range density-dependent force in the pairing channel. Using the ground-state solution of this density functional theory at the Hartree-Fock plus BCS level, an effective shell model Hamiltonian is constructed. We use this mapped Hamiltonian to extract quadrupolar and pairing correlation energies beyond the mean field approximation. The rescaling of the mass quadrupole operator in the truncated shell model space is found to be almost independent of the coupling strength used in the pairing channel of the underlying mean-field theory.
We apply the configuration-interaction method to calculate the spectra of two-component Fermi systems in a harmonic trap, studying the convergence of the method at the unitary interaction limit. We find that for a fixed regularization of the two-body interaction the convergence is exponential or better in the truncation parameter of the many-body space. However, the conventional regularization is found to have poor convergence in the regularization parameter, with an error that scales as a low negative power of this parameter. We propose a new regularization of the two-body interaction that produces exponential convergence for systems of three and four particles. From the systematics, we estimate the ground-state energy of the four-particle system to be (5.05 +- 0.024)hbar omega.
We examine the global performance of an extended self-consistent mean field theory of 2+ excitations in even-even nuclei. 518 nuclei are included in our survey, comprising all but 39 of the known nuclei tabulated by Raman et al. The theory is based on a quantal collective Hamiltonian in five dimensions in which the potential energy and the tensor of inertia are obtained from constrained triaxial Hartree-Fock-Bogoliubov calculations. The only parameters in theory are those of the finite-range, density-dependent Gogny D1S interaction. The following properties of the lowest 2+ excitations are calculated: excitation energy, reduced transition probability, and spectroscopic quadrupole moment. We find that the theory is very reliable to classify the nuclei by shape. Quantitatively the performance of the theory in deformed nuclei is excellent: average excitation energies and transition quadrupole moments are within 5% of the experimental values, and the dispersion about the averages are 20% and 10%, respectively. The performance is not as good on spherical and soft nuclei. Including all nuclei in the performance evaluation, the average transition quadrupole moment is 10% too high and the dispersion about the mean is 20%. For the energies, the average is 13% too high and the dispersion is 40%.
We present an {\it ab initio} calculation of small numbers of trapped, strongly interacting fermions using the Green's Function Monte Carlo method (GFMC). The ground state energy, density profile and pairing gap are calculated for particle numbers $N = 2 \sim 22$ using the parameter-free "unitary" interaction. Trial wave functions are taken of the form of correlated pairs in a harmonic oscillator basis. We find that the lowest energies are obtained with a minimum explicit pair correlation beyond that needed to exploit the degeneracy of oscillator states. We find that energies can be well fitted by the expression $a_{TF} E_{TF} + Δ{\rm mod}(N,2)$ where $E_{TF}$ is the Thomas-Fermi energy of a noninteracting gas in the trap and $Δ$ is a pairing gap. There is no evidence of a shell correction energy in the systematics, but the density distributions show pronounced shell effects. We find the value $Δ= 0.7\pm 0.2ω$ for the pairing gap. This is smaller than the value found for the uniform gas at a density corresponding to the central density of the trapped gas.
We reexamine neutron-proton pairing as a phenomenon that should be explanable in a microscopic theory of nuclear binding energies. Empirically, there is an increased separation energy when both neutron and proton numbers are even or if they are both odd. The enhancement is present at some level in nearly all nuclei: the separation energy difference has the opposite sign in less than 1% of the cases in which sufficient data exist. We discuss the possible origin of the effect in the context of density functional theory (DFT) and its extensions. Neutron-proton pairing from the Hartree-Fock-Bogoliubov theory does not seem promising to explain the effect. We demonstrate that much of the increased binding in the odd-odd system might be understood as a recoupling energy. This suggests that the DFT should be extended by angular momentum projection to describe the effect.
We present a Fortran program to compute the distribution of dipole moments of free particles for use in analyzing molecular beams experiments that measure moments by deflection in an inhomogeneous field. The theory is the same for magnetic and electric dipole moments, and is based on a thermal ensemble of classical particles that are free to rotate and that have moment vectors aligned along a principal axis of rotation. The theory has two parameters, the ratio of the magnetic (or electric) dipole energy to the thermal energy, and the ratio of moments of inertia of the rotor.
We discuss the systematics of the 2+ excitation energy and the transition probability from this 2+ to the ground state for most of the even-even nuclei, from O16 up to the actinides, for which data are available. To that aim we calculate their correlated J=0 ground state and J=2 first excited state by means of the angular-momentum and particle-number projected generator coordinate method, using the axial mass quadrupole moment as the generator coordinate and self-consistent mean-field states only restricted by axial, parity, and time-reversal symmetries. The calculation, which is an extension of a previous systematic calculation of correlations in the ground state, is performed within the framework of a non-relativistic self-consistent mean-field model using the same Skyrme interaction SLy4 and a density-dependent pairing force to generate the mean-field configurations and mix them.
The density functional theory of nuclear structure provides a many-particle wave function that is useful for static properties, but an extension of the theory is necessary to describe correlation effects or other dynamic properties. Here we propose a procedure to extend the theory by mapping the properties of the self-consistent mean-field Hamiltonian onto an effective shell-model Hamiltonian with two-body interactions. In this initial study, we consider the sd-shell nuclei Ne-20, Mg-24, Si-28, and Ar-36. Our first application is in the framework of the USD shell-model Hamiltonian, using its mean-field approximation to construct an effective Hamiltonian and partially recover correlation effects. We find that more than half of the correlation energy is due to the quadrupole interaction. We then follow a similar procedure but using the SLy4 Skyrme energy functional as our starting point and truncating the space to the spherical $sd$ shell. The constructed shell-model Hamiltonian is found to satisfy minimal consistency requirements to reproduce the properties of the mean-field solution. The quadrupolar correlation energies computed with the mapped Hamiltonian are reasonable compared with those computed by other methods.
We discuss the systematics of ground-state quadrupole correlations of binding energies and mean-square charge radii for all even-even nuclei, from O16 up to the superheavies, for which data are available. To that aim we calculate their correlated J=0 ground state by means of the angular-momentum and particle-number projected generator coordinate method, using the axial mass quadrupole moment as the generator coordinate and self-consistent mean-field states only restricted by axial, parity, and time-reversal symmetries. The calculation is performed within the framework of a non-relativistic self-consistent mean-field model using the same non-relativistic Skyrme interaction SLy4 and a density-dependent pairing force to generate the mean-field configurations and mix them. (See the paper for the rest of the abstract).
We introduce a simple model to calculate the nuclear moment of inertia at finite temperature. This moment of inertia describes the spin distribution of nuclear levels in the framework of the spin-cutoff model. Our model is based on a deformed single-particle Hamiltonian with pairing interaction and takes into account fluctuations in the pairing gap. We derive a formula for the moment of inertia at finite temperature that generalizes the Belyaev formula for zero temperature. We show that a number-parity projection explains the strong odd-even effects observed in shell model Monte Carlo studies of the nuclear moment of inertia in the iron region.
In developing theories of nuclear binding energy such as density-functional theory, the effort required to make a fit can be daunting due to the large number of parameters that may be in the theory and the large number of nuclei in the mass table. For theories based on the Skyrme interaction, the effort can be reduced considerably by using the singular value decomposition to reduce the size of the parameter space. We find that the sensitive parameters define a space of dimension four or so, and within this space a linear refit is adequate for a number of Skyrme parameters sets from the literature. We do not find marked differences in the quality of the fit between the SLy4, the Bky4 and SkP parameter sets. The r.m.s. residual error in even-even nuclei is about 1.5 MeV, half the value of the liquid drop model. We also discuss an alternative norm for evaluating mass fits, the Chebyshev norm. It focuses attention on the cases with the largest discrepancies between theory and experiment. We show how it works with the liquid drop model and make some applications to models based on Skyrme energy functionals. The Chebyshev norm seems to be more sensitive to new experimental data than the root-mean-square norm. The method also has the advantage that candidate improvements to the theories can be assessed with computations on smaller sets of nuclei.
We calculate correlation energies associated with the quadrupolar shape degrees of freedom with a view to improving the self-consistent mean-field theory of nuclear binding energies. The Generator Coordinate Method is employed using mean-field wave functions and the Skyrme SLy4 interaction. Systematic results are presented for 605 even-even nuclei of known binding energies, going from mass A=16 up to the heaviest known. The correlation energies range from 0.5 to 6.0 MeV in magnitude and are rather smooth except for large variations at magic numbers and in light nuclei. Inclusion of these correlation energies in the calculated binding energy is found to improve two deficiencies of the Skyrme mean field theory. The pure mean field theory has an exaggerated shell effect at neutron magic numbers and addition of the correlation energies reduce it. The correlations also explain the phenomenon of mutually enhanced magicity, an interaction between neutron and proton shell effects that is not explicable in mean field theory.
Formulas are derived for the coupled quadrupolar and monopolar oscillations of a fermion condensate trapped in a axially symmetric harmonic potential. We consider two-component condensates with a large particle-particle scattering length and in the limits of frequencies either small (adiabatic) or large (diabatic) compared to the pairing gap. Using the method of coordinate scaling, we reproduce Stringari's results for adiabatic excitations and present new results for the diabatic limit. Unlike the adiabatic oscillations, the diabatic ones show sensitivity to the interaction between particles.
To examine the applicability of the time-dependent density-functional theory (TDDFT) for treating the electron-nucleus coupling in excited states, we calculate the strength distribution associated with the pi-pi* transition in ethylene. The observed optical transition strength at 7-8.5 eV region shows a complex structure arising from coupling to C-C stretch motion, to torsional motion, and to Rydberg excitations. The mean energy of the observed peak is reproduced to about 0.2 eV accuracy by the TDDFT in the local density approximation (LDA). The reflection approximation is used to calculate the peak broadening. Roughly half of the broadening can be attributed to the fluctuation in the C-C coordinate. The asymmetry in the line shape is also qualitatively reproduced by the C-C coordinate fluctuation. We find, in agreement with other theoretical studies, that the torsional motion is responsible for the progression of weak transition strength extending from the peak down to about 6 eV. The LDA reproduces the strength in this region to about factor of 3. We conclude that the TDDFT is rather promising for calculating the electron nucleus coupling at short times.