Limitations of the Geiger-Nuttall law in heavy cluster decay

Abstract

Geiger-Nuttall law is the simplest relation in radioactive decay relating the half-life and the decay energy. Initially restricted to $\ensuremath{\alpha}$ decay of individual isotopes, generalizations unifying different isotopes and decay modes under a single set of constant coefficients were subsequently achieved. This motivates investigating to what extent such generalizations are possible. We also examine whether there exists a universal Geiger-Nuttall law that can simultaneously describe all decay modes and nuclei including heavy clusters. We show that the validity of Geiger-Nuttall law and its generalizations hinges on the assumption that half-life can be approximated linearly as a function of the square root of the ratio of the decay energy to the Coulomb barrier height. Systematic calculation of the ratio across the nuclear chart for 12 decay modes reveals that it varies over its whole range between 0 and 1. Consequently, no linear approximation can unify all the nuclei and decay modes under a single set of coefficients, and thus no universal Geiger-Nuttall law is possible in contrast to previous claims. In cluster decay, the ratio varies within 0.6--1 where nonlinearity becomes significant such that no generalized Geiger-Nuttall description of heavy clusters is possible. In the ongoing attempts of unification, it might be necessary to go beyond the Geiger-Nuttall law and incorporate additional terms proportional to the decay energy and/or its square root.