Empirical evidence for magic numbers of superdeformed shapes

We present for the first time many types of empirical evidence that point to the existence of preferred neutron/proton numbers for superdeformed (SD) shapes. We use a simple premise based on the pairing correlations to obtain the proposed empirical evidence. In particular, plots of γ-ray energy ratio such as $R$($I$) $=$ $E_{\gamma }$($I$ → $I$-2)/ $E_{\gamma }$($I$-2 → $I$-4) vs $N$ and $Z$, $R$($I$) vs $I$ plots, nuclear softness parameter values, and the number of SD bands for a given $N$ and $Z$ are used to pinpoint the $N$, $Z$ numbers that are most favored as the deformed magic numbers. The proton and neutron magic numbers so obtained not only confirm the earlier theoretical predictions made for the chain of particle numbers corresponding to the SD shapes but also verify the increase in deformation with the particle number within each chain. The analysis also leads to several new predictions for the occurrence of the SD bands.

Figure 1

(a)–(d) γ-ray energy ratio $R$($I$) vs neutron number $N$ for deformation 1.5:1 group of SD bands for the $A$ $=$ 57–137 mass region. The line in the graph shows the rigid rotor value for the ratio. The nuclides approaching the rigid rotor line have been identified and are pointed by dark circles.

Figure 2

(a)–(d) γ-ray energy ratio $R$($I$) vs proton number ($Z$) for deformation (1.5:1) group of SD bands for the $A$ $=$ 57–137 mass region. The line in the graph shows the rigid rotor value for the ratio. The nuclides approaching or lying on the rigid rotor line have been identified and pointed by dark circles.

Figure 3

(a)–(d) Same as in Fig. 1, but for the deformation 1.7:1 group of SD bands for the $A$ $=$ 189–198 mass region.

Figure 4

(a)–(d) Same as in Fig. 2, but for the deformation 1.7:1 group of SD bands for the $A$ $=$ 189–198 mass region.

Figure 5

(a),(b) Same as in Fig. 1, but for the deformation 2:1 group of SD bands for the $A$ $=$ 148–154 mass region. The top panel shows the graph for the neutron number ($N$), while the bottom panel is for the proton number ($Z$).

Figure 6

(a)–(f) Plot of $R$($I$) vs the angular momentum $I$. The solid black line is the rigid rotor behavior for the particular shape of the SD band.

Figure 7

(a)–(d) Softness parameter vs the proton number ($Z$) and the neutron number ($N$) for the SD bands (a) in the $A$ $=$ 80 mass region, (b) in the $A$ $=$ 130 mass region, (c) in the $A$ $=$ 150 mass region, (d) in the $A$ $=$ 190 mass region. The arrows point to the neutron numbers which have the smallest value for the softness parameter.

Figure 8

(a),(b) Number of SD bands vs (a) the neutron number $N$ and (b) the proton number $Z$.

Figure 9

Plot of the measured deformation vs mass number for the SD bands corresponding to the magic numbers of protons/neutrons from $A$ $=$ 80 to $A$ $=$ 190.