Markov chain models of nuclear transmutation: Part I - Theory

This paper presents mathematical models of the nuclide transmutation and decay chains based on discrete-time and continuous-time Markov chains. The models describe the transmutation chains of individual atoms as stochastic processes, and can provide details about the underlying processes in minor actinide burning and fissile material breeding. In terms of expected values, the continuous-time Markov chain model is consistent with the Bateman equations, and it can be used to derive time-dependent transmutation trajectory probabilities in the nuclide chains, including decay chains which end in a stable nuclide and actinide transmutation chains which end with fission. It is shown that transmutation trajectory probabilities in fact constitute the general solution of the Bateman equations. A method was developed to count labeled transitions in the nuclide chains, and the models of the actinide transmutation chains were used to obtain closed formulas for the calculation of finite-time integrated and asymptotic fuel cycle performance parameters, such as time-dependent fission probabilities and the distribution of fissioned daughter nuclides, the average time until fission, static and dynamic D-factors and the average neutron balance of different nuclides. As a demonstration of application, a detailed analysis of transmutation and breeding in Generation IV fast reactors will be presented in Part II of the article.