  Geiger-Nuttall Law is the simplest relation in radioactive decay relating the
half-life and the decay energy. Initially restricted to $\alpha$ 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 non-linearity 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.
