Shell structure from nuclear observables

The appearance and disappearance of shells and subshells are determined using a previously introduced method of structural analysis. This work extends the approach and applies it to protons, in addition to neutrons, in an attempt to provide a more complete understanding of shell structure in nuclei. Experimental observables including the mean-square charge radius, as well as other spectroscopic and mass related quantities are analyzed for extrema. This analysis also uses differential observables among adjacent even-even nuclei to serve as the derivatives for these quantities of interest. Local extrema in these quantities indicate shell structure and the lack of local extrema indicate missing shell closures. The shell structure of low-mass nuclei is inconsistent likely as a consequence of the single-particle structure. Additionally, multiple shell features occurring in midshell regions are determined by combining information from two or more observables. Our results near stability complement previous observations further out.

Figure 1

(a) The first $2^{+}$ energy state of even-even nuclei from Ref. [19] and (b) its derivative using Eq. (2) as a function of neutron number for five isotopes at $A\sim 130$.

Figure 2

(a) Two-neutron separation energy and (b) its derivative from Ref. [21] for $A\sim 90$. (c) Binding energy minus liquid drop using Eq. (3) and (d) its derivative.

Figure 3

Energies of the (a) $2_1^{+}$, (b) $4_1^{+}$, (c) $6_1^{+}$, and (d) $8_1^{+}$ states from Ref. [19] for $A\sim 140$.

Figure 4

(a) Derivative in the two-proton separation energy and (b) binding energy minus liquid drop from Ref. [21] for $A\sim 50$. Energies of the first excited (c) $2^{+}$ and (d) $4^{+}$ from Ref. [19]. (e) Mean-square charge radius from Ref. [48] and (f) $B\left(E2\right)$ values from Ref. [32].

Figure 5

(a) Derivative in the two-proton separation energy and (b) binding energy minus liquid drop from Ref. [21] for $A\sim 90$. Energies of the first excited (c) $2^{+}$ and (d) $4^{+}$ from Ref. [19]. (e) Mean squared charge radius from Ref. [48] and (f) $B\left(E2\right)$ values from Ref. [32].

Figure 6

(a) Derivative in the two-proton separation energy and (b) binding energy minus liquid drop from Ref. [21] for $A\sim 150$. Energies of the first excited (c) $2^{+}$ and (d) $4^{+}$ from Ref. [19]. (e) Mean-square charge radius from Ref. [48] and (f) $B\left(E2\right)$ values from Ref. [32].

Figure 7

(a) Neutron and (b) proton shell features from $S_{2n}$ or $S_{2p},B_{\mathrm{Exp}.}-B_{\mathrm{LD}},E\left(2_1^{+}\right),E\left(4_1^{+}\right),B\left(E2\right)$, and $\langle r^2\rangle$ values. Blue diamonds indicate unexpected shell features and black squares indicate expected shell features found in at least two of the observables. Red circles indicate two or more expected shell features that are missing, and orange stars indicate a combination of both found and missing shell features. Symbols with a hollow center represent additional determinations of shell structure for ${}^{22}\mathrm{O}$ from Ref. [52], ${}^{24}\mathrm{O}$ from Ref. [53], ${}^{54}\mathrm{Ca}$ from Ref. [5], ${}^{130}\mathrm{Cd}$ from Ref. [54], ${}^{12}\mathrm{O}$ from Ref. [55], ${}^{36}\mathrm{S}$ from Ref. [56], ${}^{186-188}\mathrm{Pb}$ from Ref. [57], and otherwise from Refs. [1, 2, 3, 4]. For reference, dark gray squares indicate stable nuclides with half-lives greater than ${10}^{24}$ yr based on data from Ref. [19] and the light gray squares indicate all nuclides included in the 2012 AME [21].

Figure 8

Odd-$A$ ground-state spin and parity indicated by color and symbol for (a) odd neutron and (b) odd proton nuclides with data from Ref. [19].