Basis Representation for Nuclear Densities from Principal Component Analysis

TL;DR

The paper introduces a PCA-based approach to represent nuclear densities by deriving an orthogonal basis from a diverse set of proton densities calculated with RCHB theory. The first five principal components capture over $99.999 ext{ extpercent}$ of the variance, enabling highly accurate density reconstructions with only a handful of parameters and outperforming traditional Fourier-Bessel and Sum-of-Gaussians methods in both accuracy and convergence. This density-centric basis provides a practical tool for experimental density interpretation and for theories where densities are central, such as orbital-free DFT and reaction models relying on density folding. The method demonstrates universal applicability across the nuclear chart and offers a scalable, robust representation for density-centric frameworks.

Abstract

We develop an efficient method to represent nuclear densities using basis functions extracted via Principal Component Analysis (PCA). Applying PCA to densities of 75 nuclei calculated with the relativistic continuum Hartree-Bogoliubov (RCHB) theory yields an orthogonal set of components that efficiently capture the dominant features of nuclear density distributions, which can be used as basis functions for nuclear density representation. The first five basis functions account for more than 99.999\% of the total variance, demonstrating the efficiency of these PCA basis functions. The PCA basis achieves significantly higher accuracy and faster convergence than the Fourier-Bessel and Sum-of-Gaussians methods for reconstructing both theoretical and experimental densities. This approach provides an efficient and robust representation of nuclear densities, offering a practical tool for experimental density representation and for theories where densities play a central role, such as the orbital-free density functional theory, or the double folding model for nuclear reactions.

Figures (6)

• Figure 1: Cumulative explained variance ratio curves for PCs derived from proton density distributions of 75 nuclei with PCA. The specific values up to PC4, PC5, and PC6 are indicated in the figure.
• Figure 2: Radial spatial distributions of principal components PC1-PC6 for the nuclear proton density distribution, with explained variance ratio indicated for each component.
• Figure 3: Comparison of the representation performances for proton density distributions of $^{88}$Sr, $^{208}$Pb, $^{94}$Zr, and $^{154}$Sm from RCHB calculations by the PCA, SOG, and FB methods with the number of fitting parameters fixed at 5.
• Figure 4: Comparison of the representation performances for proton density distributions of $^{88}$Sr, $^{208}$Pb, $^{94}$Zr, and $^{154}$Sm from RCHB calculations by the PCA, SOG, and FB methods with the number of fitting terms fixed at 5. The numbers in the bracket of the caption represent the number of fitting parameters included in each method.
• Figure 5: The absolute differences between the proton root-mean-square radii from the represented densities of the PCA, SOG, and FB methods and the target values from the RCHB calculations of 75 adopted nuclei. The differences under the cases with different numbers of parameters adopted in these three methods are presented.