Matter distribution and spin-orbit force in spherical nuclei

We investigate the possibility that some nuclei show density distributions with a depletion in the center, a semibubble structure, by using a Hartree-Fock plus Bardeen-Cooper-Schrieffer approach. We separately study the proton, neutron, and matter distributions in 37 spherical nuclei mainly in the $s\text{−}d$ shell region. We found a relation between the semibubble structure and the energy splitting of spin-orbit partner single particle levels. The presence of semibubble structure reduces this splitting, and we study its consequences on the excitation spectrum of the nuclei under investigation by using a quasiparticle random-phase-approximation approach. The excitation energies of the low-lying $4^{+}$ states can be related to the presence of semibubble structure in nuclei.

Figure 1

Values of $\mathcal{P}\left(E_{\mathrm{HF}+\mathrm{BCS},\mathrm{HF}}\right)$, defined in Eq. (3), obtained with the four interactions considered, for all the nuclei analyzed.

Figure 2

Values of the depletion fraction $\mathcal{F}$, defined in Eq. (4), for proton (red squares) and neutron (blue circles) density distributions. Open (solid) symbols indicate the results obtained with the D1M (D1S) interaction. Lines are drawn to guide the eyes.

Figure 3

Same as in Fig. 2 for the flatness index $\mathcal{S}$ defined in Eq. (5).

Figure 4

Proton (red dashed lines), neutron (blue dotted lines), and matter (black solid lines) density distributions, normalized to unity, for ${}^{28}\mathrm{Si}$, ${}^{30}\mathrm{Si}$, ${}^{30}\mathrm{S}$, and ${}^{32}\mathrm{S}$. The calculations have been carried out with the D1M interaction.

Figure 5

Same as described in the caption of Fig. 4 but for the ${}^{48}\mathrm{Ca}$ and ${}^{52}\mathrm{Ca}$ nuclei.

Figure 6

Charge distributions for some of the nuclei studied, calculated with the D1M (red dashed curves) and D1S (blue dotted curves) interactions, compared with the empirical ones (black full curves) taken from the compilation of Ref. [38].

Figure 7

Elastic electron scattering cross sections on ${}^{30}\mathrm{Si}$ target calculated for an incident electron energy of $300\mathrm{MeV}$, within the distorted wave Born approximation by using the charge densities shown in Fig. 6. The results obtained with the D1M (red dashed curve) and D1S (blue dotted curve) interactions are compared to those corresponding to the empirical density (black solid curve). The two vertical lines indicate the range probed by the experiment as indicated in Ref. [38].

Figure 8

Proton distribution for the ${}^{40}\mathrm{Ca}$ nucleus obtained by using the mean-field potential of Eq. (8). The full line has been obtained without the gaussian term, $B=0$, in Eq. (8). The dashed line is the result obtained by inserting the Gaussian term with the parameters indicated in the text.

Figure 9

HF+BCS spin orbit splitting $s_{nl}$, defined in Eq. (10), for neutron $1p$, $1d$, and $2p$ s.p. states. The results obtained for the Si, S, and Ca isotope chains are shown as a function of $A$. Green squares, blue triangles, red circles, and black diamonds indicate the values calculated with the D1M, D1S, D1MTd, and D1ST2a interactions, respectively. The vertical dotted lines indicate the nuclei with $N=14$ and 16.

Figure 10

HF+BCS spin orbit splitting $s_{nl}$, as defined in Eq. (10), for proton $1p$, $1d$, and $2p$ s.p. states. The results obtained for the $N=16$, 18, and 22 isotone chains are shown as a function of $A$. Green squares, blue triangles, red circles, and black diamonds indicate the values calculated with the D1M, D1S, D1MTd, and D1ST2a interactions, respectively. The vertical dotted lines indicate the nuclei with $Z=14$ and $Z=16$.

Figure 11

Density distributions of protons, ${}^{34}\mathrm{Si}$ and ${}^{36}\mathrm{S}$, and of neutrons, ${}^{34}\mathrm{Ca}$ and ${}^{36}\mathrm{Ca}$, of the four nuclei considered in Table 7. The four lines indicate the results obtained by using the four interactions.