Essential Ingredient for Radial-Composition Correlations in Two-Component Many-Body Systems: Short-Range Attractive Central Force

Abstract

The linear correlation between RMS radius difference and composition asymmetry in two-component many-body systems is a robust feature observed across nuclear experiments, diverse theoretical models, and metallic nano-alloy cluster calculations. By employing random-interaction ensembles within a Hartree-Fock framework, we demonstrate that this correlation is not a trivial consequence of many-body symmetries. Instead, we identify the short-range, attractive central potential as the essential ingredient for its emergence, a mechanism underpinned by the Moshinsky transformation and the virial theorem within a harmonic-oscillator approximation of such a potential.

Figures (5)

• Figure 1: Experimental $\Delta R_{\rm np}$zenihiro2018directdeterminationneutronskindoi:10.1142/S0218301304002168PhysRevC.46.1825PhysRevLett.129.042501PhysRevLett.131.202302 versus isospin asymmetry $I$. A linear fit indicates a strong correlation with a Pearson coefficient $\rho \approx 0.85$pearson1895.
• Figure 2: (Color online) Distributions of Pearson's $\rho$ for the $\Delta R_{\rm np}-I$ correlation across different random ensembles: random quasi-particle (RQE), central, tensor, spin-orbit, and short-range (Short). Radial matrix elements for random central, tensor, and spin-orbit ensembles are randomized via Eq. (\ref{['eq:rand-v-ele']}), while the random short-range ensemble employs the radial potential in Eq. (\ref{['eq:v_short']}) with $V_0=1$. All distributions are based on 1000 samples from each ensemble with a bin size of 0.1. The shaded region highlights the peak near $\rho=0.95$ for the random central-force ensemble. Note the broken $y$-axis; the random short-range ensemble exhibits an exceptionally high rate ($P(\rho>0.9) \approx 78.7\%$) of strong $\Delta R_{\rm np}-I$ correlation.
• Figure 3: (Color online) Average of radial matrix elements ($\overline{V^S_{n^\prime nl}}$) for samples with $\rho > 0.85$ from the random central-force ensemble, plotted against corresponding matrix elements of a 3D harmonic oscillator (HO) potential ($\langle n^\prime l|V(r)=r^2|nl\rangle$). Symbols distinguish different $n'-n$ selections. The $\overline{V^S_{n^\prime nl}}$ values exhibit a rough linear scaling with $\langle n'l|r^2|nl\rangle$ for $n'-n = 0, 1$, while suppress toward 0 for $|n'-n|>1$, mimicking the HO selection rules.
• Figure 4: (Color online) Hartree-Fock (HF) mean-field single-particle properties of samples with the $\Delta R_{\rm np}-I$ correlation ($\rho > 0.85$) in the random central-force ensemble. (a) Proton and neutron level schemes for a representative sample ($\rho = 0.94$), revealing clear gaps at HO magic numbers (2, 8, 28). (b) Distributions of the gap supremacy $\langle g\rangle/\langle G\rangle$ and (c) the gap equality $\Delta G/\langle G\rangle$. Compared to the stochastic background of randomly distributed levels, samples with $\rho>0.85$ cluster at lower ratios, signifying characteristics of an HO-like mean field, i.e., the near-degeneracy of intra-shell levels and equal spacing of major gaps.
• Figure 5: (Color online) Correlations between the RMS radial difference $\Delta R_{\rm Au-Ag}$ and atom-number asymmetry $I_{\rm Au-Ag}$ in the alloy cores of Au$_x$Ag$_{309-x}$ clusters based on many-body calculations from Ref. PhysRevLett.120.256101. Distinct piecewise linearities ($\rho \geq 0.97$) correspond to different shell-filling stages, while the transient region ($x=145-166$) reflects irregular shell occupation in the 2nd shell, as detailed in Ref. PhysRevLett.120.256101.