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On the Excess Charge Problem of Atoms

Dirk Hundertmark, Nikolaos Pattakos, Marvin Raimund Schulz

TL;DR

The paper sharpens non-asymptotic bounds on the maximal number of electrons an atom of charge $Z$ can bind, proving $N_c(Z)<1.1185Z+O(Z^{1/3})$ for large $Z$ and offering explicit lower-order terms. By extending the Benguria–Lieb–Nam weight-argument to powers $|x|^s$ with $s\in(1,3]$, it develops mean-field quantities $\alpha_{N,s}$ and $\beta_s$, proves radial symmetry of minimizing measures, and establishes quantitative links between finite-$N$ and mean-field bounds via IMS localization and Hardy-type estimates. The authors derive concrete bounds for $s=2$ and $s=3$, including $N_c(Z) < 1.12 Z + 4 Z^{1/3}$ for $Z\ge4$ and a leading coefficient $b(3)\approx1.1185$ for the cubic weight, highlighting a substantial qualitative and quantitative difference between fermionic and bosonic atoms at finite $Z$. These results advance the understanding of ionization limits and demonstrate tight, explicit control over excess charge in interacting fermionic systems with finite nuclear charge.

Abstract

This paper establishes new bounds on the maximum number of electrons $ N_c(Z) $ that an atom with nuclear charge $Z$ can bind. Specifically, we show that \begin{equation*} N_c(Z) < 1.1185Z + O(Z^{1/3}) \end{equation*} with an explicit bound on the lower order term $O(Z^{1/3})$. This result improves long--standing bounds by Lieb and Nam obtained in 1984, respectively 2012. Our bounds show the fundamental difference between fermionic and bosonic atoms for finite $Z$ since for bosonic atoms it is known that $\lim N_c(Z)/Z = t_c \approx 1.21$ in the limit of large nuclear charges $Z$.

On the Excess Charge Problem of Atoms

TL;DR

The paper sharpens non-asymptotic bounds on the maximal number of electrons an atom of charge can bind, proving for large and offering explicit lower-order terms. By extending the Benguria–Lieb–Nam weight-argument to powers with , it develops mean-field quantities and , proves radial symmetry of minimizing measures, and establishes quantitative links between finite- and mean-field bounds via IMS localization and Hardy-type estimates. The authors derive concrete bounds for and , including for and a leading coefficient for the cubic weight, highlighting a substantial qualitative and quantitative difference between fermionic and bosonic atoms at finite . These results advance the understanding of ionization limits and demonstrate tight, explicit control over excess charge in interacting fermionic systems with finite nuclear charge.

Abstract

This paper establishes new bounds on the maximum number of electrons that an atom with nuclear charge can bind. Specifically, we show that \begin{equation*} N_c(Z) < 1.1185Z + O(Z^{1/3}) \end{equation*} with an explicit bound on the lower order term . This result improves long--standing bounds by Lieb and Nam obtained in 1984, respectively 2012. Our bounds show the fundamental difference between fermionic and bosonic atoms for finite since for bosonic atoms it is known that in the limit of large nuclear charges .

Paper Structure

This paper contains 16 sections, 24 theorems, 338 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic notation and main result
  3. The Benguria--Lieb--Nam Argument: Playing with weights
  4. Symmetry of minimizing sequences
  5. Mean--Field type bounds
  6. Upper bounds on the weighted kinetic energy
  7. Bounds on maximal excess charge
  8. Proof of the Main Theorem
  9. The Case of a Quadratic Weight
  10. The Case of a Cubic Weight
  11. Technical Details
  12. Density Argument for $\alpha_{N,s}$
  13. Funk-Hecke formula and Multipol Moments
  14. Explicit constant in an inequality due to Lieb
  15. Bound on the Groundstate Energie of the Bohr Atom
  16. ...and 1 more sections

Key Result

Theorem 2.2

Let $s\in (1,3]$ then there exists $c(s)>0$ such that where

Figures (2)

  • Figure 1: Numerical approximation of the values of $\alpha_{N,s}$ for various $s$ and $N$. Starting from an initial sample set of vectors the values of $\alpha_{N,s}$ have been obtained using a Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm implemented in Python. For $s=2.5$ and $s=3.0$ the values of $\alpha_{N,s}$ seem to be almost identical for $N\leq 20$. In the plot $\alpha_{16,1}>\alpha_{18,1}$ which is contrary to Lemma \ref{['ch5:3:lem:monotonicity in N']} and due to the fat that numerical approximation is difficult for small $s\geq 1$.
  • Figure 2: Values of $b(s)$ according to Proposition \ref{['ch5:4:lower_bnd_on_beta']}, where $t_0$ was computed numerically. The lower bounds $b_{num}(s)$ on $\beta_s^{-1}$ have been found by choosing explicit measures in \ref{['ch5:4:eq:radial lower bound']} and numerical optimization. The exact value of $\beta_s^{-1}$ has to be between both lines. The values $b(3)$ and $b_{num}(3)$ differ by approximately $3 \, \%$.

Theorems & Definitions (63)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 53 more