Magnetic dipole moments, electric quadrupole moments, and electron scattering form factors of neutron-rich $sd\text{−}pf$ cross-shell nuclei

Magnetic dipole and electric quadrupole moments are calculated for neutron-rich sd-pf cross-shell nuclei. These nuclei include open shell isotopes with the number of protons less than 20 and neutrons greater than 20, for which experimental data are available. Shell model calculations are performed with full $sd$ shell model space for $Z$−8 valence protons and full $pf$ shell model space for $N$−20 valence neutrons, where the remaining 20 neutrons are frozen in $s$, $p$ and $sd$-shells. Also, magnetic and Coulomb electron scattering form factors are calculated for some of these nuclei. Excitation out of major shell space are taken into account through a microscopic theory which allows particle-hole excitation from the core and model space orbits to all higher orbits with $2\hslash \omega$ excitation. Effective charges are obtained for each isotope. Core polarization (CP) is essential for obtaining a reasonable description of the electric quadrupole moments and enhance the Coulomb form factors but has no effect on the dipole magnetic moments but squeezes the magnetic form factors. The magnetic static and dynamic properties can be described by free $g$ factors for the model space nucleons without introducing CP effect, on the contrary to the electric static and dynamic properties, which cannot be described properly by the model space nucleons without taking into account CP effects.

Figure 1

Comparison between the experimental and theoretical magnetic moments $\mu$, using free $g$ factors (a) and effective $g$ factors (b). The experimental values are taken from Ref. [30].

Figure 2

Quadrupole moments for Ar, K, and S isotopes calculated with CP (a) and with BM effective charges (b) in comparison with the experimental values of Ref. [30].

Figure 3

Total elastic magnetic form factors for ${}^{27}\mathrm{Al}$ (a). The contribution of each multipole is shown in (b). The solid curves are with CP and the dashed curve with MS only. The data are taken from Ryan et al. [38]. The right panel (c) shows the multipole contributions at different ranges of $q$.

Figure 4

Elastic magnetic form factors for ${}^{27}\mathrm{Al}$. The CP contribution of different multipoles are denoted by the cross symbols. Model space magnetic form factor and that including CP are denoted by the dashed and solid curves, respectively.

Figure 5

Longitudinal C2 form factor for the $1/2^{+}\left(0.844\mathrm{MeV}\right)$ state in ${}^{27}\mathrm{Al}$, calculated with MS (dashed curve) in comparison with that including CP effect (solid curve) (a). The right panel (b) shows a comparison of the calculated C2 form factors using CP effect (solid curve) with that of the effective charge model (dashed curve). The experimental data are taken from Ryan et al. [38].

Figure 6

Longitudinal $C2$ form factors for ${}^{27}\mathrm{Al}$. The CP contribution is denoted by the cross symbols. Model space and total $C2$ form factors are denoted by the dashed and solid curves, respectively. Elastic magnetic and longitudinal $C2$ form factors are calculated including CP effects for one isotope of each neutron-rich nucleus considered in this work. The results are shown in Figs. 7–12, for ${}^{34}\mathrm{Al}$, ${}^{35}\mathrm{Si}$, ${}^{43}\mathrm{S}$, ${}^{38}\mathrm{Cl}$, ${}^{39}\mathrm{Ar}$, and ${}^{40}\mathrm{K}$. Small suppression of the magnetic form factors is noticed with CP effects. Enhancement of the form factor is noticed with the inclusion of CP effects for the longitudinal C2 form factors for the first lope and a suppression occurs in the form factors for the second lope, for $q\ge 2.5\mathrm{f}{\mathrm{m}}^{-1}$.

Figure 7

Elastic magnetic form factors (a) with the contribution of the different multipolarities (b) and $C2$ longitudinal form factor (c) for $4^{-}$ ground state of ${}^{34}\mathrm{Al}$. Dashed curves represent the MS form factors and the solid curves represent the $\mathrm{MS}+\mathrm{CP}$ form factors.

Figure 8

Elastic magnetic form factors (a) with the contribution of the different multipolarities (b) and $C2$ longitudinal form factor (c) for $7/2^{-}$ ground state of ${}^{35}\mathrm{Si}$. Dashed curves represent the MS form factors and the solid curves represent the $\mathrm{MS}+\mathrm{CP}$ form factors.

Figure 9

Elastic magnetic form factors (a) with the contribution of the different multipolarities (b) and $C2$ longitudinal form factor (c) for $7/2^{-}$ ground state of ${}^{43}\mathrm{S}$. Dashed curves represent the MS form factors and the solid curves represent the $\mathrm{MS}+\mathrm{CP}$ form factors.

Figure 10

Elastic magnetic form factors (a) with the contribution of the different multipolarities (b) and $C2$ longitudinal form factor (c) for $2^{-}$ ground state of ${}^{38}\mathrm{Cl}$. Dashed curves represent the MS form factors and the solid curves represent the $\mathrm{MS}+\mathrm{CP}$ form factors.

Figure 11

Elastic magnetic form factors (a) with the contribution of the different multipolarities (b) and $C2$ longitudinal form factor (c) for $7/2^{-}$ ground state of ${}^{39}\mathrm{Ar}$. Dashed curves represent the MS form factors and the solid curves represent the $\mathrm{MS}+\mathrm{CP}$ form factors.

Figure 12

Elastic magnetic form factors (a) with the contribution of the different multipolarities (b) and $C2$ longitudinal form factor (c) for $4^{-}$ ground state of ${}^{40}\mathrm{K}$. Dashed curves represent the MS form factors and the solid curves represent the $\mathrm{MS}+\mathrm{CP}$ form factors.