Electron-scattering form factors for ${}^6\mathrm{Li}$ in the ab initio symmetry-guided framework

We present an ab initio symmetry-adapted no-core shell-model description for ${}^6\mathrm{Li}$. We study the structure of the ground state of ${}^6\mathrm{Li}$ and the impact of the symmetry-guided space selection on the charge density components for this state in momentum space, including the effect of higher shells. We accomplish this by investigating the electron scattering charge form factor for momentum transfers up to $q\sim 4{\mathrm{fm}}^{-1}$. We demonstrate that this symmetry-adapted framework can achieve significantly reduced dimensions for equivalent large shell-model spaces while retaining the accuracy of the form factor for any momentum transfer. These new results confirm the previous outcomes for selected spectroscopy observables in light nuclei, such as binding energies, excitation energies, electromagnetic moments, $E2$ and $M1$ reduced transition probabilities, as well as point-nucleon matter rms radii.

Figure 1

$N_{max}=12$ model space (all circles, filled and unfilled) specified by the proton, neutron, and total intrinsic $\left(S_p,S_n,S\right)$ spin values (vertical axis) across the Pauli-allowed deformation-related $\left(\lambda \mu \right)$ values (horizontal axis) for the $1^{+}$ ground state of ${}^6\mathrm{Li}$. The selected $\langle 6\rangle 12$ SA-NCSM model space is shown by filled circles and it includes the complete space up through the $6\hslash \Omega$ subspace, while the $8\hslash \Omega ,10\hslash \Omega$, and $12\hslash \Omega$ subspaces are selected based on symmetry considerations favoring large deformation. The SA-NCSM results in a model space, for which all circles are filled up through the $12\hslash \Omega$ subspace, coincide with the $N_{max}=12$ NCSM results.

Figure 2

Longitudinal $C0$ electron scattering form factors $F_L^2$ [the $L_0=0$ term in Eq. (2) with a procedure discussed in the text to produce a translationally invariant form factor] for the SA-NCSM $1^{+}$ ground state of ${}^6\mathrm{Li}$ calculated in the complete $N_{max}=12$ space (darker colors) and the $\mathrm{SU}\left(3\right)$-selected spaces, $\langle 2\rangle 12,\langle 4\rangle 12,\langle 6\rangle 12,\langle 8\rangle 12$, and $\langle 10\rangle 12$ (lighter colors), for $\hslash \Omega =15$ MeV or $b=1.66$ fm (blue), $\hslash \Omega =20$ MeV or $b=1.44$ fm (red), and $\hslash \Omega =25$ MeV or $b=1.29$ fm (black) for (a) the bare JISP16 interaction, as well as for (b) the bare ${\mathrm{NNLO}}_{\mathrm{opt}}$ interaction. Experimental data are taken from Ref. [36].

Figure 3

Longitudinal $C0$ electron scattering form factors $F_L^2$ (with a procedure discussed in the text to produce a translationally invariant form factor) for the SA-NCSM $1^{+}$ ground state of ${}^6\mathrm{Li}$ calculated for $\hslash \Omega =20$ MeV or $b=1.44$ fm and with the bare JISP16 interaction. The outcome for the $\mathrm{SU}\left(3\right)$-selected spaces, $\langle 6\rangle 8$ (red dots) and $\langle 6\rangle 12$ (blue dots), accurately reproduces the corresponding results for the complete $N_{max}=8$ space (solid, red) and $N_{max}=12$ space (solid, blue), with larger-space $N_{max}=12$ results lying slightly closer to experiment [36].

Figure 4

Most dominant OBDMEs (with an absolute value $>0.02$) labeled by $n_1,n_2\left(\lambda \mu \right)$ for a particle-hole ${\left(n_1\right)}^1{n}_2^{-1}$ configuration, for the SA-NCSM $1^{+}$ ground state of ${}^6\mathrm{Li}$ calculated in laboratory coordinates (with the Lawson term employed in the SA-NCSM calculations ensuring a $0s$ CM wave function component, i.e., spurious CM excitations eliminated) and in the $N_{max}=12$ complete space with the JISP16 bare interaction for $\hslash \Omega =15$ MeV (blue, left bars), 20 MeV (red, middle bars), and 25 MeV (green, right bars). Error bars are defined by the range from the lowest value to the largest value of each OBDME over the set of $\mathrm{SU}\left(3\right)$-selected spaces.

Figure 5

Most dominant $\mathrm{SU}\left(3\right)$ contributions to the translationally invariant $F_L$, denoted as $F_L^{\left(\lambda \mu \right)}$, for the longitudinal $C0$ form factor, labeled by $n_1,n_2\left(\lambda \mu \right)$ for a particle-hole ${\left(n_1\right)}^1{n}_2^{-1}$ configuration. The $N_{max}=12$ SA-NCSM $1^{+}$ ground state of ${}^6\mathrm{Li}$ is calculated with the JISP16 bare interaction and for $\hslash \Omega =15$ MeV (left), 20 MeV (center), and 25 MeV (right). Note that the vertical axis scale is reduced by an order of magnitude from the top to the bottom panels. Results are very similar to the ones obtained with the ${\mathrm{NNLO}}_{\mathrm{opt}}$ bare interaction. Components $6,2\left(40\right)$ and $7,3\left(40\right)$, marked by $*$, lie almost on the top of the $4,4\left(22\right)$ and $5,5\left(22\right)$ components, respectively, for all $q$.

Figure 6

Same as Fig. 5, but for the ${\mathrm{NNLO}}_{\mathrm{opt}}$ bare interaction and for $\hslash \Omega =20$ MeV, with (solid) and without (dashed) removing the CM contribution (due to the $0s$ CM component of the wave functions used to calculate the OBDMEs). See Fig. 5 for curve labeling.

Figure 7

Longitudinal $C0$ translationally invariant form factors, constructed from a subset of selected $\left(\lambda \mu \right)$ OBDMEs, for the SA-NCSM $1^{+}$ gs of ${}^6\mathrm{Li}$ calculated with the bare ${\mathrm{NNLO}}_{\mathrm{opt}}$ interaction (similarly for JISP16) for (a) $\hslash \Omega =15$ MeV, (b) $\hslash \Omega =20$ MeV, and (c) $\hslash \Omega =25$ MeV, and compared to experiment (“$+$”) [36]. The $\left(\lambda \mu \right)$ components [and $\left(\mu \lambda \right)]$ included are as follows: all (grey solid), $\left(00\right)$ (red dashed), $\left(00\right)+\left(20\right)$ (red solid), $\left(00\right)+\left(20\right)+\left(40\right)$(green dot-dashed), $\left(00\right)+\left(20\right)+\left(40\right)+\left(60\right)$ (blue dotted), and $\left(00\right)+\left(20\right)+\left(40\right)+\left(60\right)+\left(80\right)$ (black long-dashed). Deviations due to the $\mathrm{SU}\left(3\right)$-based space selection are indicated by the curve thickness.