Why effective and environmental half-lives may be unbounded - a Gabriel's Horn analogy

This article proposes a new framework for interpreting the relationships between physical, effective, and environmental half-lives of radionuclides, grounded in one-compartment, single first-order model. The novel, central component of this approach is a hyperbolic decay function that describes the relationship between the effective decay constant (λ_eff) and the environmental half-life (T_1/2env) of a given radionuclide. This relationship reveals five clearly distinct scenarios for the environmental fate of radionuclides - ranging from classical decay, through seemingly stable concentrations, to accumulation. The function resembles the mathematical concept known as Gabriel's Horn, characterized by infinite length within a finite volume, aptly reflecting the complex and non-intuitive dynamics of radionuclide activity concentrations in biotic and abiotic compartments. Based on literature data (mainly concerning long-lived radionuclides such as ¹³⁷Cs and ⁹⁰Sr), it is demonstrated that some cases described as difficult to interpret-or even omitted altogether-find clear mathematical justification within proposed approach. This new perspective organizes and systematizes current knowledge on environmental half-life variability and may significantly enhance the interpretation of radiological data and risk assessments related to long-term radiation exposure.