Evolution of the $N=50$ gap from $Z=30$ to $Z=38$ and extrapolation toward ${}^{78}$Ni

The evolution of the $N=50$ gap is analyzed as a function of the occupation of the proton $\pi f_{5/2}$ and $\pi p_{3/2}$ orbits. It is based on experimental atomic masses, using three different methods of one- or two-neutron separation energies of ground or isomeric states. We show that the effect of correlations, which is maximized at $Z=32$ could be misleading with respect to the determination of the size of the shell gap, especially when using the method with two-neutron separation energies. From the methods that are the least perturbed by correlations, we estimate the $N=50$ spherical shell gap in ${}_{28}^{78}$Ni${}_{50}^{}$. Whether ${}^{78}$Ni would be a rigid spherical or deformed nucleus is discussed in comparison with other nuclei in which similar nucleon-nucleon forces are at play.

Figure 1

Energy of the two first states of the $N=50$ isotones having an odd-$Z$ value [8].

Figure 2

(a) Experimental binding energies of the states 9/2${}^{+}$ (5/2${}^{+}$) located below (above) the $N=50$ magic number. The experimental values of the atomic masses are from Refs. [6, 7, 9]. The uncertainties are smaller than the symbols. (b) Difference in binding energies of these two states surrounding the gap at $N=50$. The dashed lines are the linear fits using the data at $Z=30,34,36$ (see text).

Figure 3

(a) Experimental binding energies of the two last neutrons in the $N=52$ isotones and the $N=50$ isotones. Experimental atomic masses are taken from Refs. [6, 7, 9]. The uncertainties are smaller than the symbols. (b) Difference of the two-neutron binding energies between the $N=52$ and $N=50$ isotones.

Figure 4

Energy of the first quadrupole excitation built on the ground state of the nuclei of interest: (a) The $2^{+}\to 0_{\mathrm{g}.\mathrm{s}.}^{+}$ transitions of the even-even nuclei. (b) The $13/2^{+}\to 9/2_{\mathrm{g}.\mathrm{s}.}^{+}$ transitions of the $N=49$ isotones and the $9/2^{+}\to 5/2_{\mathrm{g}.\mathrm{s}.}^{+}$ transitions of the $N=51$ isotones. The $E\left(4_1^{+}\right)/E\left(2_1^{+}\right)$ ratios, $R_{4/2}$, for the $N=52$ and $N=48$ isotones are shown in the inset. The experimental information is from Refs. [8, 11, 12, 13, 14].

Figure 5

Pictorial illustration of the total binding energies, BE, and various neutron binding energies, $B{E}_{1n}$, $2B{E}_{1n}$, $B{E}_{2n}$, and $B{E}_{2n}^{*}$ for the Se isotopes, showing the correlation energy of the 0${}_{\mathrm{g}.\mathrm{s}.}^{+}$ state, V(0${}^{+}$), and the residual interaction of the 8${}^{+}$ spherical state, V(8${}^{+}$), in ${}^{82}$Se${}_{48}$.

Figure 6

Experimental binding energies of the two last neutrons in $N=50$ isotones: Values of $2B{E}_{1n}$(50) (red triangles, cf. Fig. 2) and of $B{E}_{2n}^{*}$(50) [green squares, cf. Eq. (1)]. The difference between the two lines is, by definition, the interaction energy between the two neutrons located in the $\nu g_{9/2}$ orbit, which couple to $J=8^{+}$. The uncertainties are smaller than the symbols.

Figure 7

Distance in energy between the 0${}_{\mathrm{g}.\mathrm{s}.}^{+}$ and the 8${}_1^{+}$ states for the $N=48$ isotones as a function of the proton number. The energies of the 8${}_1^{+}$ state are fixed to 0.

Figure 8

Difference of the binding energies of the two states surrounding the gap at $N=50$ [cf. Fig. 2b]. When using the averaged DME value of ${}^{48}$Ca (see text), the $N=50$ gap would amount to 4.2 MeV for $Z=28$. The solid line gives the evolution of the $N=50$ gap from $Z=28$ to $Z=38$.