Surface incompressibility of semi-infinite nuclear matter via constrained Hartree-Fock calculations

With a view to calculating the incompressibility \(\ddot \sigma \) of the nuclear surface, we develop a constrained Hartree-Fock treatment of semi-infinite nuclear matter. Our approach leads first to a proof of the “ \(\dot \sigma = 0\) ” theorem of Myers and Swiatecki, and then to a corroboration of a simple expression for \(\ddot \sigma \) , previously obtained in an intuitive way by Stocker. This expression is used to calculate \(\ddot \sigma \) for theS3, SkM and Gogny forces. The results are significantly different from those of a scaled HF calculation, but in very good agreement with the values of \(\ddot \sigma \) that are implied by RPA calculations of the breathing mode in various finite nuclei.     

Microscopic nuclear magnetic moment calculations including Hartree-Fock graphs

All magnetic moment graphs which may be non-zero to second order for closed shell ±1nuclei are tabulated. Their contributions to the magnetic moments of ¹¹B, ¹³C, ¹³N, ¹⁵O, ¹⁵N, ¹⁷O, ¹⁷F, ²⁷Al, ²⁹Si, ³¹P and ³³S are calculated. For L-S ± 1 nuclei Hartree-Fock graphs do not contribute to second order for central residual interactions (and are small otherwise), while for the j-j ± 1 nuclei considered they are roughly as important as the non-Hartree-Fock graphs and of the same sign as the first-order corrections. Calculations were performed with the Kallio-Kolltveit and Sussex interactions. The procedures used to check the calculations are discussed.     

Variational time-dependent Hartree-Fock calculations

A pseudopotential framework is used to develop a simplified version of time-dependent Hartree-Fock theory. Variational solutions of the equations are obtained for the alkali metal hydrides LiH, NaH and KH, and excitation energies, oscillator strengths and other properties are calculated. The results are in good agreement with more elaborate theoretical calculations.     

Hartree-Fock theory with constrained deformation

We study the variational Hartree-Fock problem with constrained deformation. If the deformation is defined by prescribed expectation values of the conventional multipole moments, we show that a necessary condition for the problem to have physically meaningful solutions is that the multipole operators are negligible outside the nucleus, so that they must be state dependent. In this way, a further non-linearity is introduced into the problem. An alternative formulation is to impose a given shape to the nuclear surface by means of two inequality constraints.     

Hartree-Fock calculations of super-heavy nuclei

Hartree-Fock calculations are performed in the region of super heavy nuclei using zero range (Skyrme) forces and the finite range force G-0; the latter predict shell closure for the (α-unstable) nucleus N = 218, Z = 126 while the former do not. Calculations performed using the density matrix expansion (DME or DMEX) approximations give results similar to the Skyrme force. The problem is traced to an inadequacy of the DME, incorrectly placing the nodeless n = 1 high ? states.     

Hartree-Fock calculations of bubble nuclei

Hartree-Fock calculations of ³⁶Ar and of ²⁰⁰Hg in spherical configurations exhibit densities which are very much depleted in the interior region. The depletion agrees well with previous theoretical predictions of bubble nuclei.     

Exponential convergence and acceleration of Hartree-Fock calculations

We show that we can expect an exponential behaviour for the convergence of the Hartree-Fock solution during the HF iteration procedure. We use this property to extrapolate some collective degrees of freedom, in this case the shape, in order to speed up the self-consistent calculation. For axially deformed nuclei we apply the method to the quadrupole moment which corresponds to a simple scaling transformation on the single particle wave functions. Results are shown for the deformed nuclei ²⁰Ne and ²⁸Si with a Skyrme interaction.     

Hartree-Fock and Hartree-Bogolyubov calculations inp shell nuclei

Hartree-Fock (HF) and Hartree-Fock-Bogolyubov (HFB) calculations have been performed for the 1p shell nuclei. Nuclear deformations are assumed to be at most axially symmetric. The HFB transformation is restricted to allow forp-p andn-n pairing only.Volkov's force, a soft-core, two-body interaction of semi-realistic nature, is used which does not produce any single-particle spin-orbit splittings. Coulomb force and the usual correction for centre-of-mass motion are taken into account. The calculations are carried out in a single-particle basis including all states up to principle oscillator quantum numberN=3 (in some cases,N=4). Binding energies, rms radii, density distributions, and quadrupole moments are calculated and found to be in reasonable agreement with experiment. Large Hartree-Fock energy gaps are obtained. They prevent the pairing correlations considered from becoming effective in an HFB approach and from changing the HF ground-state properties appreciably. In non-selfconjugate nuclei, the Pauli principle, rather than the Coulomb interaction, yields large differences between the charge and mass distributions. A theorem on selfconsistent symmetries is proved. The coefficients of the HFB transformation turn out to be real, if time-reversal and angular momentum projection flip are selfconsistent symmetries.     

Hartree-Fock and shell model calculations for giant nuclei

The properties of giant nuclear systems like Uranium/Uranium etc. (A?476) have been studied in statical, spherical Hartree-Fock approximation. Various effective forces of Skyrme type were used to calculate binding energies, density distributions and single particle spectra. A shell model of Woods-Saxon type is shown to be more or less equivalent with respect to the shell structure derived from Skyrme models and easy generizable to slightly deformed giant nuclei. Strong magic shells could be found even in this region of the periodic table, but it seems that they cannot contribute significantly to a stabilization of these systems. The total binding energy is in good agreement with liquid drop model extrapolations. Only very briefly we discuss possibilities and problems for calculating deformation energies.     

Exact shell-model and projected Hartree-Fock calculations for Mg

The results of a projected Hartree-Fock calculation are compared with those of an exact shell-model calculation for states of J ? 6 in the nucleus ²⁴Mg. The realistic Kuo interaction is used, and gives reasonable agreement with experiment.     

Relativistic models for nuclear structure calculations: Comparative study of mean-field and Hartree-Fock approximation for superheavy nuclei

The relevance of exchange effects for the stability of superheavy nuclei is examined within a linear QHD-II model by comparing Hartree-Fock with meanfield results. To allow a scan of the complete superheavy regime the recently developed local density approximation (LDA) for the exchange potential is employed for the Hartree-Fock level calculations. It turns out that, while many nuclear properties obtained with the LDA approach differ significantly from the corresponding mean-field results, the predictions of the two methods for shell closures are very similar. Furthermore, a comparison with a nonlinear variant of QHD-II shows that many nuclear properties obtained with the LDA in the framework of linear QHD-II are somewhere in-between the corresponding linear and nonlinear mean-field results. This indicates that the LDA exchange partially includes nonlinear contributions, which supports the interpretation of the meson self-coupling as a parametrization of many-body effects.     

A Hartree-Fock nuclear mass formula

We recall the main features of the recently published mass formula, HFBCS-1, based on the Hartree-Fock-BCS method, and compare its extrapolations out to the neutron drip line with those given by the fine-range droplet model. A new Hartree-Fock-Bogolyubov mass formula, HFB-1, is described: the rms error of the fit to 1888 masses is 0.766 MeV, compared with 0 .738 MeV for HFBCS-1, but there are no substantial changes in the predictions relevant to the r-process. After a critical examination of various questions relating to the effective nucleon mass and to the requirements of the relativistic mean-field theory, we conclude that the greatest remaining ambiguity concerns the nature of the pairing force. Received: 21 March 2002 / Accepted: 16 May 2002 / Published online: 31 October 2002 RID="a" ID="a"e-mail: pearson@lps.umontreal.ca     

A simplified Hartree-Fock method for opacity calculations

Complex atoms can be treated by using few parameters; thereby close approximation to accurate Hartree-Fock results will be obtained. This is achieved by optimization of effective potentials in the Hartree-Fock equations. The method is suitable for large-scale computations of atomic and ionic wave functions. Parameters are presented for ground states of neutral atoms up to Z = 36.     

The minimax technique in relativistic Hartree-Fock calculations

The online version of the original article can be found at     

Microscopic calculations of fission barriers and critical angular momenta for excited heavy nuclear systems

An extended version of Strutinsky's macro-microscopic method is used to calculate effective potential energies for rotating, excited heavy compound nuclei undergoing fission. Nuclear deformation is parameterized in terms of Lawrence's family of shapes. A two-center single-particle potential corresponding to these shapes is employed, with BCS pairing added. Statistical excitation is introduced by temperature-dependent occupation of (quasi-) particle energy levels. We calculate shell corrections to the energy, the free energy and the entropy as functions of deformation and temperature. The associated average quantities are derived from a temperature-dependent liquid drop model. The resulting static deformation energy is augmented by the rotational energy to yield the isothermal effective potential energy as a function of deformation, temperature and angular momentum. Moments of inertia are obtained from the adiabatic cranking model with temperature-dependent pairing included.We have also calculated the effective potential for constant entropy rather than constant temperature. Although this isentropic process physically is more appropriate than the isothermal process, it has not been treated before. For the same amount of excitation energy in the spherical state of the compound nucleus, the isentropic barriers turn out higher than the isothermal ones. For both processes we have extracted the critical angular momentum (defined as the one for which the barrier approximately vanishes) as a function of excitation. Our model is applied to the super-heavy nuclei ²⁷⁰110, ²⁷⁸110, ²⁹⁸114, ²⁹²118 and ³²²128, which have been tried to form in krypton and argon induced heavy ion reactions.     

The minimax technique in relativistic Hartree-Fock calculations

Using the set of trial spinors and the Dirac-Coulomb Hamiltonian (H _DC) we discuss the role of the minimax theorem in relativistic Hartree-Fock calculations. In principle, the minimax theorem guarantees the occurrence of an upper bound. We also consider a scaling of the functionsu _i and discuss the condition to derive the relativistic hypervirial theorem; the variational procedure represented by the condition serves as an example of the minimax technique. Single zeta calculations onH ₂ ⁺ ,H ₂ and He are analysed. The effect of enlarging the basis is investigated for the He atom. The “upper bound” obtained by usingcoherent basis spinors differs from the result of the (random) linear variation using the kinetically balanced basis set by an amount which is at most of orderc ⁻⁴. Use of thecoherent basis set is advocated. An erratum to this article is available at .     

Hartree-Fock calculations of surface-symmetry properties with finite-range forces

Complete Hartree-Fock calculations on asymmetric semi-infinite nuclear matter have been performed with a family of finite-range Gogny-type forces. For one of these forces all the computed values of the droplet-model symmetry coefficients J, L, M and Q are in good agreement with recent mass-formula fits.     

Average effective-interaction calculations in a simulated Hartree-Fock basis

The averaged effective interaction for mass-18 nuclei is computed through fourth-order perturbation theory in both a pure harmonic-oscillator basis and a simulated Hartree-Fock basis. Going to a Hartree-Fock basis does not eliminate the large fourth-order averages found earlier by Goode and Koltun using an harmonic-oscillator basis.     

Taylor expansions in Hartree-Fock and isoscalar nuclear shifts

We consider changes in the bulk properties of a nucleus when a particle is added to an even system. The main thing we show is that one can determine these changes by doing a Hartree-Fock calculation on the even system only. We must, however, also examine what happens away from equilibrium, or at least get derivates at equilibrium, in the even system. We get expressions for E0 isoscalar effective charges which involve, but do not determine, the nuclear compressibility, likewise expressions for E2 effective charges which involve but do not determine the quadrupolarizability. Illustrative examples using simple interactions are presented. E0 effective charge corrections in a major shell are larger by roughly a factor of two in orbits with high radial quantum number and E2 effective charge corrections are near unity. Finally, we consider simultaneous changes in the radial and quadrupole shapes.