Separable character of ab initio No-Core Shell Model one-body densities

Abstract

Motivated by recent findings on the separability of optical potentials that are derived from folding off-shell densities with off-shell nucleon-nucleon amplitudes, we study the off-shell character of one-body density matrices created within the No-Core Shell Model (NCSM). Concentrating on nuclei with a 0$^+$ ground state from $^4$He through $^{48}$Ca, we investigate the off-shell character of their one-body density matrices in momentum space when using the momentum transfer and the average momentum as variables. A singular value decomposition of the one-body density matrices reveals that they can be characterized by only very few terms, depending on the mass number of the nucleus. These findings are independent of the nucleon-nucleon interactions employed, as well as from computational specifics as grid spacing and size of the model space.

Figures (13)

• Figure 1: The neutron density of $^{16}$O calculated with the NNLO$_{\rm opt}$ chiral potential Ekstrom13 as function of the angle $\theta_{q{\mathcal{K}}}$ at fixed values of $q$ and $\mathcal{K}$ for a series of $N_{\rm max}$ values. All calculations use $\hbar\omega = 20$ MeV.
• Figure 2: The $K=0$ component of the translationally-invariant off-shell one-body density obtained from NCSM calculations based on the NNLO$_{\rm opt}$ chiral potential Ekstrom13 for the neutron distributions of $^4$He (left column), $^{16}$O (middle column), and $^{40}$Ca (right column) as function of the momenta $q$ and ${\mathcal{K}}$. Each row shows $l_q=l_{\mathcal{K}}$ components from 0 to 4. All calculations use $\hbar\omega = 20$ MeV. The values of $N_{\rm max}$ for the different calculations are given in each panel. Note the change of scales for the higher partial waves.
• Figure 3: The calculated values of $\eta (R)$ as function of $R$ for $^4$He ($N_{\rm{max}}=16$), $^{16}$O ($N_{\rm {max}}=10$), and $^{40}$Ca ($N_{\rm{max}}=6$) based on the NNLO$_{\rm opt}$ chiral potential Ekstrom13 (red open symbols) and the Daejeon16 potential Shirokov:2016ead (grey hashed symbols). The filled symbols indicate calculations using $N_{\rm max}=0$ and $\hbar \omega=20$ MeV. A detailed explanation of the symbols is given in the text. The dashed line indicates our tolerance threshold of $10^{-5}$ for $\eta (R)$.
• Figure 4: The calculated values of $\eta(R)$ as a function of $R$ for $^{20}$Ne, $^{28}$Si, $^{32}$S (all at $N_{\max}=4$), and $^{48}$Ca (at $N_{\max}=2$) based on the NNLO$_{\rm opt}$ (red open symbols) and the Daejeon16 (grey hashed symbols) potentials. The filled symbols indicate calculations using $N_{\rm max}$=0 at $\hbar\omega = 20$ MeV with NNLO$_{\rm opt}$. The dashed line indicates our tolerance threshold of $10^{-5}$.
• Figure 5: The calculated value of $\eta(R)$ as function of $R$ for $^{12}$C, $^{16}$O, $^{20}$Ne (all at $N_{\rm max}=6$), $^{28}$Si ($N_{\rm max}=2$), $^{40}$Ca ($N_{\rm max}=4$), and $^{48}$Ca ($N_{\rm max}=2$). The blue symbols, shifted to the left, represent the LENPIC SMS chiral N$^3$LO $NN$ potential, while the green symbols, shifted to the right, represent calculations including the corresponding $3N$ interactions Reinert:2017usi. Symbols filled with a solid color represent $N_{\rm max} = 0$. For comparison, the open symbols represent the NNLO$_{\rm opt}$ interaction using $N_{\rm max}=10$ for $^{12}$C and $^{16}$O, $N_{\rm max}=6$ for $^{20}$Ne and $^{40}$Ca, $N_{\rm max}=4$ for $^{28}$Si, and $N_{\rm max}=2$ for $^{48}$Ca. The dashed line indicates the tolerance threshold of $\eta(R)=10^{-5}$, and all calculations were performed at $\hbar\omega=20$ MeV.