Analysis of the anomalies of the constancy of Grodzins E × B(E2) product rule

The constancy of the Grodzins product (GP) $E\left(2_1^{+}\right)\times B\left(E2\right)\uparrow$, implies independence from the vibration–rotation status of the nucleus, after allowing for the Z and A dependence. However, in some recent studies, the correlation of the variation of GP, with some collective entities like the $N_p{N}_n$ product have been pointed out. This anomaly of the constancy of the GP versus the variation is reviewed here. Detailed analysis of these applications for the violation of the GPR pointed out in these studies is given. Comparison is also made with the linearity relation of $B\left(E2\right)\uparrow$ versus $1/E\left(2_1^{+}\right)$ plots for a few elements, which provide a simple view of the GP variation with N and Z. The basic physics underlying this rule is presented.