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Nuclear surface energy solving Hartree-Fock equations with Gogny interactions using Lagrange mesh

Dany Davesne, Alessandro Pastore, Jesus Navarro

TL;DR

The paper develops a fast, accurate Hartree–Fock framework for finite-range Gogny interactions in semi-infinite nuclear matter using a Lagrange-mesh discretization. By enabling larger box sizes and exact derivatives, the method reduces Friedel oscillation artifacts and yields surface-energy values $a_s$ with keV-level precision, while systematically exploring the impact of the spin–orbit term. Across a range of Gogny parametrizations, the study finds $a_s$ clustering around ~18 MeV, with spin–orbit coupling lowering $a_s$ by approximately $1.2$–$1.9$ MeV, and notes some parametrizations yield $a_s>19$ MeV without spin–orbit, suggesting suitability considerations for fission studies. The work also establishes a linear correlation between the spin–orbit strength $W_0$ and $a_s$ and lays groundwork for future extensions to include tensor terms and pairing via HFB.

Abstract

Hartree Fock equations for finite range interactions in a slab of nuclear matter are presented and solved using an algorithm based on the Lagrange mesh method. This approach is faster and more efficient than the Numerov algorithm commonly used in the literature. Thanks to the improved numerical accuracy, we were able to perform calculations with sufficiently large boxes to minimize the impact of Friedel oscillations on the final results, achieving a precision on the surface energy within a few dozens of keV. Results are presented for several Gogny interactions that have not been previously discussed. In addition, the inclusion of the spin orbit term is examined, showing a net reduction of 1.2-1.9 MeV in the surface energy.

Nuclear surface energy solving Hartree-Fock equations with Gogny interactions using Lagrange mesh

TL;DR

The paper develops a fast, accurate Hartree–Fock framework for finite-range Gogny interactions in semi-infinite nuclear matter using a Lagrange-mesh discretization. By enabling larger box sizes and exact derivatives, the method reduces Friedel oscillation artifacts and yields surface-energy values with keV-level precision, while systematically exploring the impact of the spin–orbit term. Across a range of Gogny parametrizations, the study finds clustering around ~18 MeV, with spin–orbit coupling lowering by approximately – MeV, and notes some parametrizations yield MeV without spin–orbit, suggesting suitability considerations for fission studies. The work also establishes a linear correlation between the spin–orbit strength and and lays groundwork for future extensions to include tensor terms and pairing via HFB.

Abstract

Hartree Fock equations for finite range interactions in a slab of nuclear matter are presented and solved using an algorithm based on the Lagrange mesh method. This approach is faster and more efficient than the Numerov algorithm commonly used in the literature. Thanks to the improved numerical accuracy, we were able to perform calculations with sufficiently large boxes to minimize the impact of Friedel oscillations on the final results, achieving a precision on the surface energy within a few dozens of keV. Results are presented for several Gogny interactions that have not been previously discussed. In addition, the inclusion of the spin orbit term is examined, showing a net reduction of 1.2-1.9 MeV in the surface energy.

Paper Structure

This paper contains 7 sections, 23 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Gogny interactions
  3. Semi infinite nuclear matter
  4. Lagrange mesh
  5. Results
  6. Conclusions
  7. Acknowledgments

Figures (3)

  • Figure 1: Neutron density obtained in a full HF calculation using D1 interaction (without spin-orbit term). The inset shows the presence of Friedel oscillations.
  • Figure 2: Evolution of the surface energy coefficient $a_s$ for the Gogny D1 interaction without (panel a) and with spin-orbit term (panel b) as a function of half of the size of the slab ($L$) and for various number of integration points in the $k_t$ direction. The dotted lines represent the zero and are there just to guide the eye. See text for details.
  • Figure 3: Evolution of the surface energy coefficient $\bar{a}_s$ in function of the spin-orbit parameter $W_0$ for two selected Gogny interactions. The star symbols indicate the nominal value of $W_0$ for each interaction. See text for details.