Generalization of the differential equation model for both $B\left(E2\right)\uparrow$ and the excitation energy $E(\mathrm{g}.\mathrm{s}.\to 2_1^{+})$ of even-even nuclei, and its application to the study of the $B\left(E2\right)$ problem in ${}^{46}\mathrm{Ar}$

We generalize the recently developed differential equation model for the reduced quadrupole transition probability $B\left(E2\right)\uparrow$ to include the corresponding excitation energy $E2$ for the transition from the ground state to the first $2^{+}$ state of a given even-even nucleus. Accordingly both these quantities of a given nucleus can satisfy a common differential equation, in which both of them individually can be expressed in terms of their derivatives with respect to the neutron and proton numbers. We use this equation to obtain two recursion relations in both of them connecting in each case three neighboring even-even nuclei from lower- to higher-mass numbers and vice versa. Then we demonstrate their numerical validity using the available $E2$ data throughout the nuclear chart and also explore their possible utility in predicting the unknown $E2$ values. Finally we apply the model to analyze ‘‘the $B\left(E2\right)$ problem in ${}^{46}\mathrm{Ar}$,” which refers to the recent experimental observation of a large $B\left(E2\right)$ value contradicting its prevailing value.

Figure 1

Known $E2$ values plotted as isolines for even-even nuclei. Panels (a)–(c) show these isolines for various isotopes from $Z=16$ onwards with varying neutron number $N$, while panels (d)–(f) show the same for various isotones from $N=36$ onwards with varying proton number $Z$. Other possible isolines are not shown here to avoid clumsiness of the graphs.

Figure 2

Plot of the percentage errors of the computed $E2$ values for all the anchor nuclei against their proton number $Z$. Panel (a) shown as $E2$-F refers to the results of the relation (5) while panel (b) marked as $E2$-B shows those of the relation (6). The vertical solid lines are drawn just to focus larger deviations of any at the magic and semimagic numbers.

Figure 3

Same as Fig. 2 plotted versus neutron number $N$.

Figure 4

Both calculated (DEM) and known (PRT) [4] $E2$ values presented for various isotopes of Cr, Fe, Ni, and Zn. Recent [4, 12] shell-model-calculated values (SM1 and SM2) are also given for the sake of comparison.

Figure 5

Similar to Fig. 4 but for $Z=40$, 48, 64, and 88. The experimental data points marked as EXP refer to those of Raman et al. [3].

Figure 6

The DEM predictions (DEM), the Raman data [3] (EXP), and the data of Mengoni et al. [10] (MNG) of $E2$ and $B\left(E2\right)\uparrow$ for Ar isotopes are presented in panels (a) and (b), respectively. The experimental uncertainties are indicated by vertical lines.