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 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.

中文翻译：

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Geiger-Nuttall定律在重团衰变中的局限性

Geiger-Nuttall定律是放射性衰变中最简单的关系，涉及半衰期和衰变能量。最初仅限于$\alpha$随后实现了单个同位素的衰变，统一了不同同位素的归纳和在一组恒定系数下的衰变模式。这促使人们研究这种概括的可能性。我们还研究了是否存在通用的盖革-纳塔尔定律，该定律可以同时描述所有衰变模式和包括重团簇在内的原子核。我们表明，盖革-努特尔定律及其推广的有效性取决于以下假设：半衰期可以线性近似地表示为衰减能量与库仑势垒高度之比的平方根的函数。系统地计算12个衰变模式下整个核图的比率，发现该比率在整个范围（0到1）之间变化。线性逼近无法在单一系数集下统一所有核和衰变模式，因此与以前的权利要求相比，不可能有通用的Geiger-Nuttall定律。在聚类衰减中，比率在0.6-1范围内变化，其中非线性变得很明显，因此不可能对重聚类进行广义的Geiger-Nuttall描述。在正在进行的统一尝试中，可能有必要超越盖革-纳特尔定律，并加入与衰减能量和/或其平方根成比例的附加项。