Ab initio translationally invariant nonlocal one-body densities from no-core shell-model theory

Background: It is well known that effective nuclear interactions are in general nonlocal. Thus if nuclear densities obtained from ab initio no-core shell-model (NCSM) calculations are to be used in reaction calculations, translationally invariant nonlocal densities must be available.

Purpose: Though it is standard to extract translationally invariant one-body local densities from NCSM calculations to calculate local nuclear observables like radii and transition amplitudes, the corresponding nonlocal one-body densities have not been considered so far. A major reason for this is that the procedure for removing the center-of-mass component from NCSM wave functions up to now has only been developed for local densities.

Results: A formulation for removing center-of-mass contributions from nonlocal one-body densities obtained from NCSM and symmetry-adapted NCSM (SA-NCSM) calculations is derived, and applied to the ground state densities of ${}^4\mathrm{He}$, ${}^6\mathrm{Li}$, ${}^{12}\mathrm{C}$, and ${}^{16}\mathrm{O}$. The nonlocality is studied as a function of angular momentum components in momentum as well as coordinate space.

Conclusions: We find that the nonlocality for the ground state densities of the nuclei under consideration increases as a function of the angular momentum. The relative magnitude of those contributions decreases with increasing angular momentum. In general, the nonlocal structure of the one-body density matrices we studied is given by the shell structure of the nucleus, and cannot be described with simple functional forms.

Figure 1

The translationally invariant local one-body density obtained from a NCSM calculation ($N_{max}=10,\hslash \omega =20$ MeV) based on the JISP16 $\mathit{NN}$ interaction for the proton distribution of ${}^{12}\mathrm{C}$ as function of the momentum transfer $q$. The solid line (red) shows the direct construction in momentum space, while the solid triangles (black) give the local density obtained by integrating the nonlocal density over the momentum $\mathcal{K}$. As comparison a local density obtained from a HFB mean field calculation based on the Gogny interaction [33] is shown by the filled solid (blue) circles.

Figure 2

The $K=0$ component of the translationally invariant nonlocal one-body density obtained from a NCSM calculation ($N_{max}=10,\hslash \omega =20$ MeV) based on the JISP16 $\mathit{NN}$ interaction for the proton distribution of ${}^{12}\mathrm{C}$ as function of the momenta $q$ and $\mathcal{K}$. Panel (a) depicts the contribution of $l_q=0$, (b) of $l_q=2$, (c) of $l_q=4$, and (d) of $l_q=6$.

Figure 3

The $K=0$ component of the translationally invariant nonlocal one-body density obtained from a NCSM calculation ($N_{max}=8,\hslash \omega =20$ MeV) based on the JISP16 $\mathit{NN}$ interaction for the proton distribution of ${}^{16}\mathrm{O}$ as function of the momenta $q$ and $\mathcal{K}$. Panel (a) depicts the contribution of $l_q=0$, (b) of $l_q=2$, (c) of $l_q=4$, and (d) of $l_q=6$.

Figure 4

The $K=0$ component of the translationally invariant nonlocal one-body density obtained from a NCSM calculation ($N_{max}=14,\hslash \omega =20$ MeV) based on the JISP16 $\mathit{NN}$ interaction for the proton distribution of ${}^4\mathrm{He}$ as function of the momenta $q$ and $\mathcal{K}$. Panel (a) depicts the contribution of $l_q=0$, (b) of $l_q=2$, (c) of $l_q=4$, and (d) of $l_q=6$.

Figure 5

The $K=0$ component of the translationally invariant nonlocal one-body density obtained from a NCSM calculation ($N_{max}=10,\hslash \omega =20$ MeV) based on the JISP16 $\mathit{NN}$ interaction for the proton distribution of ${}^{12}\mathrm{C}$ as function of the local coordinate $\zeta$ and the nonlocal coordinate $Z$. Panel (a) depicts the contribution of $l_{\zeta }=0$, (b) of $l_{\zeta }=2$, (c) of $l_{\zeta }=4$, and (d) of $l_{\zeta }=6$.

Figure 6

The $K=0$ component of the translationally invariant nonlocal one-body density obtained from a NCSM calculation ($N_{max}=14,\hslash \omega =20$ MeV) based on the JISP16 $\mathit{NN}$ interaction for the proton distribution of ${}^6\mathrm{Li}$ as function of the local coordinate $\zeta$ and the nonlocal coordinate $Z$. Panel (a) depicts the contribution of $l_{\zeta }=0$, (b) of $l_{\zeta }=2$, (c) of $l_{\zeta }=4$, and (d) of $l_{\zeta }=6$.

Figure 7

The $K=0$ component of the translationally invariant nonlocal one-body density ${\rho }_{l_q{l}_{\mathcal{K}}}(q,\mathcal{K})$ obtained from a NCSM calculation ($\hslash \omega =20$ MeV) based on the JISP16 $\mathit{NN}$ interaction for the proton distribution of ${}^6\mathrm{Li}$ as function of the momenta $q$ and $\mathcal{K}$ and the size of the model space. The first column contains angular momentum slices obtained with $N_{max}=6$, the second with $N_{max}=10$, the third with $N_{max}=12$, and the fourth with $N_{max}=14$. The rows represent different angular momentum slices $l_q=l_{\mathcal{K}}$ from 0 to 6.

Figure 8

The $K=0$ component of the translationally invariant nonlocal one-body density ${\rho }_{l_q{l}_{\mathcal{K}}}(q,\mathcal{K})$ obtained from a NCSM calculation ($N_{max}=12$) based on the JISP16 $\mathit{NN}$ interaction for the proton distribution of ${}^6\mathrm{Li}$ as function of the momenta $q$ and $\mathcal{K}$ and the oscillator parameter $\hslash \omega$. The first column contains angular momentum slices obtained with $\hslash \omega =15$ MeV, the second with $\hslash \omega =20$ MeV, and the third with $\hslash \omega =25$ MeV. The rows represent different angular momentum slices $l_q=l_{\mathcal{K}}$ from 0 to 6.

Figure 9

The $K=0$ component of the translationally invariant nonlocal one-body density ${\rho }_{l_{\zeta }{l}_Z}(\zeta ,Z)$ obtained from a NCSM calculation ($N_{max}=12$) based on the JISP16 $\mathit{NN}$ interaction for the proton distribution of ${}^6\mathrm{Li}$ as function of the local coordinate $\zeta$, the nonlocal coordinate $Z$, and the oscillator parameter $\hslash \omega$. The first column contains angular momentum slices obtained with $\hslash \omega =15$ MeV, the second with $\hslash \omega =20$ MeV, and the third with $\hslash \omega =25$ MeV. The rows represent different angular momentum slices $l_{\zeta }=l_Z$ from 0 to 6.

Figure 10

The $K=0,l_q=0$ component of the translationally invariant nonlocal one-body density obtained from a NCSM calculation based on the JISP16 $\mathit{NN}$ interaction for the proton distribution of ${}^{12}\mathrm{C}$ [panel (a)] and ${}^4\mathrm{He}$ (panel (b)) as a function of the nonlocal momentum $\mathcal{K}$ at fixed momenta $q$ as indicated in the legend. The distributions are normalized by the factors indicated in the legend.

Figure 11

The $K=0$ component of the translationally invariant one-body density obtained from NCSM calculations based on the JISP16 $\mathit{NN}$ interaction for the proton distributions of ${}^4\mathrm{He}$, ${}^6\mathrm{Li}$, ${}^{12}\mathrm{C}$, and ${}^{16}\mathrm{O}$ as a function of the nonlocal momentum $\mathcal{K}$ when integrated of the local momentum $q$ [panel (a)]. Panel (b) depicts the local densities for the same nuclei as function of the momentum transfer $q$ when integrated over the nonlocal momentum $\mathcal{K}$.