Finite valence nucleon number and rotation-vibration interactions

Background: A characteristic observable of nuclear collective motion is the relative $B\left(E2\right)$ values from the $\gamma$ band to the ground band in even-even deformed nuclei. The Alaga rules provide an idealized set of benchmarks for these observables. However, deviations from the Alaga rules are universally observed and have been traditionally and successfully interpreted in terms of parameterized $\gamma$-band–ground-band bandmixing. An alternate approach, partial dynamical symmetries, has no bandmixing whatsoever and is parameter free, yet mimics closely the effects of bandmixing, due solely to the effects of finite valence nucleon number.

Purpose: To investigate the relation between these two seemingly contradictory approaches to understand how they can produce such similar results.

Method: To derive approximate relations between the two formalisms.

Results: A consistent relationship is found linking bandmixing to finite valence nucleon number effects on interband $\gamma$ to ground-band $B\left(E2\right)$ values.

Conclusions: Two disparate approaches to one of the iconic characteristics of deformed nuclei are shown to be intimately related. Moreover, a systematic difference in their predictions also emerges naturally from the derivation. The qualitative linkage of valence nucleon number and the separation of vibrational and rotational degrees of freedom has long been assumed but never before explicitly demonstrated through complementary models.

Figure 1

$Z_{\gamma }$ values as a function of half the number of valence nucleons extracted for rare-earth transitional and deformed nuclei. Figure taken from Ref. [13].

Figure 2

PDS calculations for $B\left(E2\right)$ ratios from the $2^{+}$ and $6^{+}$ states of the $\gamma$ band to the ground band calculated with the exact PDS formulas and with the approximations derived in this paper.