High order decay equations, without any equilibrium assumptions and allowing any initial progeny presence, analyzed precisely with mathematical code. Visualizing High-Order Decay after Disequilibria Charles A. Wilson IV, 1 Katherine R. Hendrickson, 2 Amin M. Hamideh, 3 Kenneth L. Matthews II, 4 and Wei-Hsung Wang 5 Abstract: High-order decay equations are of- ten difficult to study without significant care taken with variables and assumptions. As par- ent and progeny activities evolve over time, the effects of uncertainties and approximations con- found the quality and interpretation of results. Of particular concern is the situation when decay equilibrium has been disturbed and progenies have arbitrary initial activities. To address this, code was created using Wolfram Mathematica to visualize the time-activity plots of the high or- der progenies of naturally occurring radioactive material after secular equilibrium is disturbed. The Bateman equation for an un-replenished parent was expanded to calculate activity vs. time for up to 13 progenies at different initial ac- tivities. The code uses the formula of Skrable et al., without parent production, expanded to the 13th progeny with arbitrary initial concen- tration. The code calculates and plots activity vs. time; it also reports the cumulative disinte- grations of each progeny over a user-specified time period for comparison to counting measure- ments. The code could also be modified to incor- porate additional production or branched decay schemes. We believe this code may be useful to health physicists and is intended to be accessible for anyone’s use. This paper presents the code with explanations and examples on how to use it. Health Phys. 115(6):791–796; 2018 Key words: operational topics; decay chain; naturally occurring radionuclides; spectrome- try, gamma INTRODUCTION High-order Bateman equations are commonly used in health physics to calculate parent and progeny activities. Typically, the Bateman equation can be solved relatively easily in closed form under condi- tions of equilibrium or that of a pure parent. Yet a complete solu- tion that allows for more compli- cated boundary conditions can better be used to facilitate the un- derstanding and visualization of the temporal behavior of progeny activities during more complicated high order decay. In situations where either it is known that progenies exist but are not in equilibrium with the parent or something disturbs equilibrium during processing the Bateman equation is more difficult to apply. This commonly occurs when one wants to compare calculations to experimental measurements. A simplifying assumption such as equilibrium may not be true; while equilibrium may exist in the en- vironment, sample collection or preparation may cause some prog- eny activity to be lost. An example of this issue is naturally occurring radioactive materials (NORM) that contain gaseous progenies (e.g., uranium and thorium decay se- ries). These gaseous progenies can escape if the sample is not sealed properly. The escape of radon gas during sample preparation is used as an example for this work. When disequilibrium occurs, an acceptable practice is to seal the sample and wait for equilib- rium to be restored. Unfortunately, waiting some time to reestablish equilibrium before making mea- surements is not time efficient, especially for a decay chain with long-lived progenies. Although some techniques are available to shorten the waiting time (Li et al. 2015), approximations may limit the accuracy of results. The full Bateman equation is necessary to correctly predict activity of the parent or progenies in a non- equilibrium situation. This project was motivated by a difficulty to find software or code that models radioactive transfor- mation when the original decay chain does not have a pure parent. This search focused on online published solutions for n th order Bateman equations, especially those that allow arbitrary initial progeny concentrations. While a variety of resources were discovered, few of these offer the ability to specify ar- bitrary initial progeny activities and often are limited to only a 1 J. Bennett Johnston Sr. Center for Advanced Micro- structures and Devices (CAMD), Louisiana State Uni- versity, 6980 Jefferson Highway, Baton Rouge, LA 70806; 2 University of Florida Department of Industrial and Systems Engineering, 1819 Lewis Turner Blvd, Ft Walton Beach, FL 32547; 3 Louisiana State University Radiation Safety Office, 112 Nuclear Science Building, Baton Rouge, LA 70803; 4 Louisiana State University Department of Physics and Astronomy, 202 Nicholson Hall, Baton Rouge, LA 70803; 5 Louisiana State Uni- versity Center for Energy Studies, 1067 Energy, Coast and Environment Building, Baton Rouge, LA 70803. The authors declare no conflicts of interest. Charles A. Wilson IV is the radiation safety officer at Louisiana State University’s Center for Advanced Microstructures and Devices (CAMD). He is currently a doctoral candidate studying environmental health physics in the Department of Environmental Sciences at LSU. He earned his master’s degree in medical physics and health physics from LSU in 2012. Charles was President of the Deep South Chapter of the HPS and a former Chair of the HPS Student Support Committee. He presently serves as Chair of the Society Support Committee and member on the IRPA task force. His email is cwils35@LSU.edu. Operational Topic Operational Radiation Safety www.health-physics.com 791 Copyright © 2018 Health Physics Society. Unauthorized reproduction of this article is prohibited.