From superdeformation to extreme deformation and clusterization in the $N\approx Z$ nuclei of the $A\approx 40$ mass region

A systematic search for extremely deformed structures in the $N\approx Z$ nuclei of the $A\approx 40$ mass region has been performed for the first time in the framework of covariant density functional theory. At spin zero such structures are located at high excitation energies, which prevents their experimental observation. The rotation acts as a tool to bring these exotic shapes to the yrast line or its vicinity so that their observation could become possible with future generation of $\gamma$-tracking (or similar) detectors such as GRETA and AGATA. The major physical observables of such structures (such as transition quadrupole moments, as well as kinematic and dynamic moments of inertia), the underlying single-particle structure and the spins at which they become yrast or near yrast, are defined. The search for the fingerprints of clusterization and molecular structures is performed and the configurations with such features are discussed. The best candidates for observation of extremely deformed structures are identified. For several nuclei in this study (such as ${}^{36}\mathrm{Ar}$), the addition of several spin units above the currently measured maximum spin of $16\hslash$ will inevitably trigger the transition to hyper- and megadeformed nuclear shapes.

Figure 1

Energies of the calculated configurations in ${}^{40}\mathrm{Ca}$ relative to a smooth liquid drop reference $AI(I+1)$, with the inertia parameter $A=0.05$. This way of the presentation of the results has clear advantages as compared with the energy versus spin plots; see Sec. 4.1 in Ref. [20] for details. Different types of configurations are shown by different types of lines. The SD, HD, and MD configurations, which are yrast in respective deformation minima, are shown by thick lines with symbols.

Figure 2

Neutron single-particle energies (Routhians) in the self-consistent rotating potential as a function of the rotational frequency ${\mathrm{\Omega }}_x$. They are given along the deformation path of the yrast SD configuration in ${}^{40}\mathrm{Ca}$. Long-dashed, solid, dot-dashed, and dotted lines indicate $(\pi =+,r=+i), (\pi =+,r=-i), (\pi =-,r=+i)$, and $(\pi =-,r=-i)$ orbitals, respectively. At ${\mathrm{\Omega }}_x=0.0$ MeV, the single-particle orbitals are labeled by the asymptotic quantum numbers $\left[N{n}_z\mathrm{\Lambda }\right]\mathrm{\Omega }$ (Nilsson quantum numbers) of the dominant component of the wave function. Solid (open) circles indicate the orbitals occupied (emptied). The arrows indicate the particle-hole excitations leading to excited SD configurations; for these configurations only the changes (as compared with yrast SD configuration) in the occupation of the orbitals are indicated in the figure.

Figure 3

The same as Fig. 2, but along the deformation path of the yrast HD configuration in ${}^{40}\mathrm{Ca}$. The arrows indicate the particle-hole excitations leading to excited HD configurations.

Figure 4

The same as Fig. 2, but along the deformation path of the yrast MD [42,42] configuration in ${}^{40}\mathrm{Ca}$.

Figure 5

Calculated transition quadrupole moments $Q_t$ and $\gamma$ deformations of the yrast and excited SD and HD configurations in ${}^{40}\mathrm{Ca}$. The colors of the lines for different types of configurations roughly correspond to those used in Fig. 1. Red and orange (black and dark brown) are used for the SD (HD) configurations.

Figure 6

The same as Fig. 5, but for the normal- and highly deformed triaxial configurations in ${}^{40}\mathrm{Ca}$.

Figure 7

The same as Fig. 5, but for the MD configurations in ${}^{40}\mathrm{Ca}$.

Figure 8

The self-consistent proton density ${\rho }_p(y,z)$ as a function of $y$ and $z$ coordinates for the indicated configurations in ${}^{40}\mathrm{Ca}$ at specified spin values. The equidensity lines are shown in steps of $0.01 {\mathrm{fm}}^{-3}$ starting from ${\rho }_p(y,z)=0.01{\mathrm{fm}}^{-3}$.

Figure 9

The comparison of the results of the calculations with and without pairing for the configurations of ${}^{40}\mathrm{Ca}$ that do not require blocking procedure in the CRHB calculations. The results of the calculations without pairing are shown by solid lines with symbols. The results for paired analogs of unpaired configurations are shown by dashed lines of the same color.

Figure 10

The same as Fig. 1, but for ${}^{42}\mathrm{Sc}$.

Figure 11

The same as Fig. 5, but for ${}^{42}\mathrm{Sc}$.

Figure 12

The same as Fig. 8, but for ${}^{42}\mathrm{Sc}$.

Figure 13

The same as Fig. 1, but for ${}^{42}\mathrm{Ca}$.

Figure 14

The same as Fig. 5, but for ${}^{42}\mathrm{Ca}$.

Figure 15

The same as Fig. 1, but for ${}^{44}\mathrm{Ca}$. Note that for some configurations several aligned (terminating) states can be formed; the reasons for their formation are discussed in Sec. 6.6 of Ref. [20].

Figure 16

The same as Fig. 1, but for ${}^{44}\mathrm{Ti}$. Note that it was not possible to trace the middle parts of the [2,2] and [3,3] configurations in the calculations. Thus, they are shown by dotted lines.

Figure 17

The same as Fig. 5, but for ${}^{44}\mathrm{Ti}$.

Figure 18

The same as Fig. 1, but for ${}^{46}\mathrm{Ti}$.

Figure 19

The same as Fig. 5, but for ${}^{46}\mathrm{Ti}$.

Figure 20

The same as Fig. 1, but for ${}^{48}\mathrm{Cr}$.

Figure 21

The same as Fig. 1, but for ${}^{50}\mathrm{Cr}$.

Figure 22

The same as Fig. 1, but for ${}^{36}\mathrm{Ar}$.

Figure 23

The same as Fig. 5, but for ${}^{36}\mathrm{Ar}$.

Figure 24

The same as Fig. 8, but for ${}^{36}\mathrm{Ar}$.

Figure 25

The same as Fig. 1, but for ${}^{38}\mathrm{Ar}$.

Figure 26

The same as Fig. 5, but for ${}^{38}\mathrm{Ar}$.

Figure 27

The same as Fig. 1, but for ${}^{32}\mathrm{S}$. Dotted lines show expected diabatic continuations of the [2,2] and [21,21] configurations beyond the spin range where the convergence has been obtained.

Figure 28

The same as Fig. 5, but for ${}^{32}\mathrm{S}$.

Figure 29

The same as Fig. 8, but for ${}^{32}\mathrm{S}$.

Figure 30

The same as Fig. 1, but for ${}^{34}\mathrm{S}$.

Figure 31

The same as Fig. 5, but for ${}^{34}\mathrm{S}$.

Figure 32

The same as Fig. 8, but for ${}^{34}\mathrm{S}$.

Figure 33

The impact of rotation on the proton density distribution in selected configurations.

Figure 34

Three-dimensional representation of the evolution or proton density distribution with spin in the [42,42] configuration of ${}^{40}\mathrm{Ca}$.

Figure 35

Kinematic $\left(J^{\left(1\right)}\right)$ and dynamic $\left(J^{\left(2\right)}\right)$ moments of inertia of typical SD, HD, and MD configurations in indicated nuclei. The calculated $J^{\left(1\right)}$ and $J^{\left(2\right)}$ values are shown by solid and open symbols, respectively. Red circles, black squares, and blue triangles are used for the SD, HD, and MD configurations, respectively.