Variational calculation of the ground state of closed-shell nuclei up to $A=40$

Variational calculations of ground-state properties of ${}^4\mathrm{He},{}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$ are carried out employing realistic phenomenological two- and three-nucleon potentials. The trial wave function includes two- and three-body correlations acting on a product of single-particle determinants. Expectation values are evaluated with a cluster expansion for the spin-isospin dependent correlations considering up to five-body cluster terms. The optimal wave function is obtained by minimizing the energy expectation value over a set of up to 20 parameters by means of a nonlinear optimization library. We present results for the binding energy, charge radius, one- and two-body densities, single-nucleon momentum distribution, charge form factor, and Coulomb sum rule. We find that the employed three-nucleon interaction becomes repulsive for $A\ge 16$. In ${}^{16}\mathrm{O}$ the inclusion of such a force provides a better description of the properties of the nucleus. In ${}^{40}\mathrm{Ca}$ instead, the repulsive behavior of the three-body interaction fails to reproduce experimental data for the charge radius and the charge form factor. We find that the high-momentum region of the momentum distributions, determined by the short-range terms of nuclear correlations, exhibits a universal behavior independent of the particular nucleus. The comparison of the Coulomb sum rules for ${}^4\mathrm{He},{}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$ reported in this work will help elucidate in-medium modifications of the nucleon form factors.

Figure 1

Point proton densities in ${}^4\mathrm{He}$. The solid green line refers to the “experimental” result; see text for details. The dash-dotted brown line is the GFMC result for AV18+UIX [55].

Figure 2

Point proton densities in ${}^{16}\mathrm{O}$. The green line refers to the “experimental” result; see text for details.

Figure 3

Point proton densities in ${}^{40}\mathrm{Ca}$. The green line refers to the “experimental” result; see text for details.

Figure 4

Point proton densities for AV18+UIX. ${}^4\mathrm{He},{}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$ are the results of this work. ${}^8\mathrm{Be}$ and ${}^{12}\mathrm{C}$ are VMC results collected in Ref. [55].

Figure 5

Two-nucleon densities of ${}^4\mathrm{He}$.

Figure 6

Two-nucleon densities of ${}^{16}\mathrm{O}$.

Figure 7

Two-nucleon densities of ${}^{40}\mathrm{Ca}$.

Figure 8

Operator two-nucleon densities in ${}^4\mathrm{He}$. $\mathbb{1},\tau ,\sigma ,\sigma \tau ,t,t\tau$ correspond to operators $p=1,...,6$ in Eq. (4).

Figure 9

Operator two-nucleon densities in ${}^{16}\mathrm{O}$.

Figure 10

Operator two-nucleon densities in ${}^{40}\mathrm{Ca}$.

Figure 11

Proton momentum distributions in ${}^4\mathrm{He}$. Averages are calculated on ${\mathcal{N}}_c={10}^7$ configurations. The brown line is the VMC result for AV18+UIX [61].

Figure 12

Proton momentum distributions in ${}^{16}\mathrm{O}$. Averages are calculated on ${\mathcal{N}}_c={10}^7$ configurations for AV18, and on ${\mathcal{N}}_c=8\times {10}^6$ configurations for AV18+UIX.

Figure 13

Proton momentum distributions in ${}^{40}\mathrm{Ca}$. Averages are calculated on ${\mathcal{N}}_c={10}^7$ configurations for AV18, and on ${\mathcal{N}}_c=8\times {10}^6$ configurations for AV18+UIX.

Figure 14

Proton momentum distributions for AV18+UIX.

Figure 15

Integrated strengths for AV18+UIX. See text for details.

Figure 16

Longitudinal elastic form factors for ${}^4\mathrm{He}$. Shaded areas indicate propagated Monte Carlo statistical errors in the Fourier transforms. Experimental data are from an unpublished compilation by I. Sick, based on Refs. [68, 69, 70, 71, 72].

Figure 17

Longitudinal elastic form factors for ${}^4\mathrm{He}$. Results employing the AV18+UIX potential are reported for CVMC and GFMC, the latter with and without MECs. Experimental results are the same as in Fig. 16.

Figure 18

Longitudinal elastic form factors for ${}^{16}\mathrm{O}$. Shaded areas indicate propagated Monte Carlo statistical errors in the Fourier transforms. Experimental data are from an unpublished compilation by I. Sick, based on Refs. [75, 76, 77].

Figure 19

Longitudinal elastic form factors for ${}^{40}\mathrm{Ca}$. Shaded areas indicate propagated Monte Carlo statistical errors in the Fourier transforms. Experimental data are from an unpublished compilation by I. Sick, based on Refs. [77, 78, 79].

Figure 20

Coulomb sum rules for $A\le 40$. Symbols with statistical error bars show GFMC calculations employing the AV18+IL7 potential [67]. The curves show CVMC results for AV18+UIX. Shaded areas indicate propagated Monte Carlo statistical errors in the Fourier transforms.

Figure 21

Central correlation functions for AV18 and AV18+UIX for $A=4$ and $A>4$ (the same two-body correlations have been employed in ${}^{16}\mathrm{O}$ and ${}^{40}\mathrm{Ca}$; see text for details).

Figure 22

Radial correlation functions for $A=4$ and AV18. $\tau ,\sigma ,\sigma \tau ,t,t\tau ,ls,ls\tau$ correspond to operators $p=2,...,8$ in Eqs. (4) and (6).

Figure 23

Radial correlation functions for $A=4$ and AV18+UIX.

Figure 24

Radial correlation functions for $A>4$ and AV18.

Figure 25

Radial correlation functions for $A>4$ and AV18+UIX.