盖革-纳托尔定律在重团簇衰变中的局限性
Limitations of the Geiger-Nuttall Law in heavy cluster decay
摘要：
盖革-纳托尔定律是放射性衰变中关于半衰期和衰变能的最简单的关系式。最初仅限于个别同位素的$α$衰变，随后实现了在一组常数系数下统一不同同位素和衰变模式的推广。这促使人们调查这种普遍化在多大程度上是可能的。我们还研究是否存在一个通用的盖革-纳托尔定律，可以同时描述所有的衰变模式和核，包括重集群。我们表明，盖革-纳托尔定律及其推广的有效性取决于这样的假设，即半衰期可以近似线性的衰变能量的库仑势垒高度的比率的平方根的函数。对12种衰变模式的核图表中的比率进行系统计算，发现它在0和1之间的整个范围内变化。因此，没有任何线性近似可以将所有的原子核和衰变模统一在一组系数之下，因此与先前的主张相比，没有普适的盖革-纳托尔定律是可能的。在团簇衰变中，该比率在0.6-1范围内变化，其中非线性变得显著，使得不可能对重团簇进行广义盖革-纳托尔描述。在目前统一的尝试中，可能有必要超越盖革-纳托尔定律，并纳入与衰变能量和/或其平方根成比例的额外项。
英文摘要：
Geiger-Nuttall Law is the simplest relation in radioactive decay relating the half-life and the decay energy. Initially restricted to $α$ 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.
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