Relations Involving Static Quadrupole Moments of $2^+$ states and B(E2)'s

We define the ``quadrupole ratio'' $r_Q = \dfrac{Q_0(S)}{Q_0(B)}$ where $Q_0(S)$ is the intrinsic quadrupole moment obtained from the static quadrupole moment of the $2_1^+$ state of an even-even nucleus and $Q_0(B)$ the intrinsic quadrupole moment obtained from $B(E2)_{0 \to 2}$. In both cases we assume a simple rotational formula connecting the rotating frame to the laboratory frame. The quantity $r_Q$ would be one if the rotational model were perfect and the energy ratio $E(4)/E(2)$ would be 10/3. In the simple vibrational model, $r_Q$ would be zero and $E(4)/E(2)$ would be two. There are some regions where the rotational limit is almost met and fewer where the vibrational limit is also almost met. For most cases, however, it is between these two limits, i.e. $0 < r_Q < 1$. There are a few cases where $r_Q$ is bigger than one, especially for light nuclei.