Impact of the Center of Mass Fluctuations on the Ground State Properties of Nuclei

TL;DR

The paper investigates how center-of-mass fluctuations affect ground-state energies in nuclear density functional theory and identifies limitations of conventional CoM corrections that mix in excited-state components. It advocates the Peierls–Yoccoz translational-symmetry restoration as a principled, state-free way to obtain a translationally invariant ground-state energy, casting the correction as a generator-coordinate projection rather than a simple mean-field term. Using the SeaLL1 EDF, the authors quantify the CoM correction, finding a strong, mass-dependent energy shift $|E_{CoM}| \approx 17 A^{-1/6}$ MeV and a corresponding kinetic component $T_{CoM}$, with the restored energy $E_{gs}$ consistently lower than the unprojected mean-field energy by $E_{gs}-E_{MF} \le 0$. The work demonstrates that translational symmetry restoration is essential for accurate binding energies and has implications for nuclear surface properties and reaction rates, supporting broader adoption of PY-like projections in nuclear DFT analyses. $

Abstract

Ground state properties across the entire nuclear chart are described predominantly and rather accurately within the density functional theory (DFT). DFT however breaks many symmetries, among them the most important being the translational, rotational, and gauge symmetries. The translational symmetry breaking is special, since it is broken for all nuclei, unlike the rotational and gauge symmetries. The center-of-mass (CoM) correction most commonly used in the literature, see \textcite{Vautherin:1972} and \textcite{Bender:2003} leads to a gain of 15,...,19 MeV, which varies rather weakly for medium and heavy mass nuclei. A better approximation to the CoM correction was suggested by \textcite{Butler:1984} and its magnitude varies between 10 and 5 MeV from light to heavy nuclei, a correction which is also significantly larger than the RMS energy error in the Bethe-Weizsäcker mass formula, initially proposed by \textcite{Gamow:1930}, which is at most 3.5 MeV, and which for heavy nuclei corresponds to about 0.2\% of their mass. The CoM energy correction due Butler {\it et al.} is also significantly larger than the RMS energy deviation achieved in any DFT evaluations of the nuclear masses performed without any symmetry restoration or zero-point energy fluctuations, with an energy RMS typically between 2 and 3 MeV. Here we analyze the CoM projection method suggested by \textcite{Peierls:1957} (PY), which leads to a translationally invariant many-body wave function, in a procedure fully equivalent to those suggested for restoring rotational and gauge symmetries. This is the only approach for the evaluation of the CoM energy correction to the mean field binding energies, which is not contaminated by contributions from excited states.

Figures (2)

• Figure 1: The most accurate estimates of $|E_{CoM}|$ reported so far in literature, along with the results of this work, with a solid line the fit and with two dashed lines the upper and lower error bars of the fit. We used SeaLL1 EDF Bulgac:2018 for the estimates for $E_{CoM}^{HF}$, see Eq. \ref{['eq:ehf']}, to generate Butler:1984 estimates.
• Figure 2: The many-body wave functions overlap $\langle \Phi({\bf a})|\Phi({\bf 0})\rangle \approx \exp[ - {\bf a}^2/(2 \sigma_\mathcal{O}^2 ) ]$ solid lines and with dashed lines for the Hamiltonian overlap $\langle \Phi ( {\bf a} ) | H | \Phi( {\bf 0} )\rangle$ normalized to $\langle \Phi ( {\bf 0} ) | H | \Phi( {\bf 0} )\rangle$. ${\cal H}({\bf a},{\bf 0})$, see Eq. \ref{['eq:gm']}, normalized to the absolute value $|{\cal H}({\bf 0},{\bf 0})|$ is shown in the inset with lines of the same color for each nucleus.