Method of False Rates for Radioactive Decay Chain Calculation

Michael Erle Lord

doi:10.11648/j.ijssam.20200503.11

Abstract

Many engineering applications involving radioactive materials requires the time history of the radioactivity of nuclides within a decay chain. The system of differential equations with initial conditions or initial value problem describing radioactive decay of a parent and daughter nuclides was posited by Ernest Rutherford who was awarded a Noble Prize in 1910 for this work. Harry Bateman (1910) provided an analytic solution to the radioactive decay chain problem with the constraint that initial inventory of all daughter elements is zero. Required data for the decay chain calculation consists of the parent and all daughters’ radioactive half-life. The half-life for essentially all radionuclides has been established and is available from multiple sources. Solutions other than Bateman’s (non-zero initial conditions) can be computed analytically but become unwieldy for longer decay chains. For this reason, many applications use a numerical solution. However, a numerical solution can require constraints on the time step size. The proposed method of false rates provides a unique algorithm for the decay chain activities. The method treats the decay chain with arbitrary initial conditions and the calculation is analytic or exact. The method is unexpectedly simplistic. An example decay chain calculation compares the solutions by the method of false rates with a numerical method. The comparison is a verification of the method of false rates calculation. The method of false rates is easily coded as a stand-alone application or as a sub-module of a more general code such as a contaminant transport model.

Highlights

Radioactive materials have applications in many disciplines
Most radioactive decay chain calculators are specific to a stated application
The method of false rates is not limited by the initial concentration constraint imposed by Bateman’s solution

Summary

Introduction

Radioactive materials have applications in many disciplines. A short list includes a) Nuclear medicine b) Reactor fuel for power generation c) Inspection of manufacture metal materials and integrity of welds d) Radiocarbon dating e) Production of nuclear weapon materials f) Environmental remediation of nuclear waste Computer models for radioactive processes are found in the private sector [1,2,3]. The radioactive decay chain calculation of the private sector codes is only a minor portion of the possible algorithmic solutions of numerous mathematical problems. For the private sector models the source code is not provided and specific numerical algorithms are not disclosed. Bateman’s solution requires zero initial mass for all daughters in Eq 2. The method of false rates is not limited by the initial concentration constraint imposed by Bateman’s solution. For nuclear waste management regulatory limits are expressed in terms of nuclide activity, ) = , which represents a mass rate of change. In certain applications, such as the remediation of nuclear waste sites, the initial mass of daughter elements can be greater than zero.

Decay of Parent and Single Daughter

Method of False Rates for Multiple Daughters

Example of Decay Chain Calculation

Other Decay Modes

Conclusion