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Published in Splinter Robert, Illustrated Encyclopedia of Applied and Engineering Physics, 2017
[nuclear] Equilibrium that is reached through a mediator stage. In nuclear science, the transient state is accomplished through a parent daughter transition where the half-life of decay for the daughter is shorter than the parent but not intrinsically negligible. The transient equilibrium in radioactive decay is represented by the Bateman equation, linking the parent decay half-life time (τp) to the daughter decay half-life time (τd) by the definition of the activity of parent (Ap = λρΝρ; dNpļdt = −λρΝρ, λρ the decay constant of the parent) and offspring (Ad = λdNd; dNd/dt = −XdNd) respectively, captured as Ad/Ap = τp/(τp — τd) BR, where BR is the branching ratio (ratio of decaying particles under a specific process of decay to the total number of decaying particles). The Bateman equation defines the activity of the daughter as Ad = BR {Ap (0)[λd/(λd − λp)][e−λpt − e−λdt]} + Ad (0)e−λdt. The time of maximum activity (tmax) for the daughter activity in transient equilibrium has an inherent maximum defined as tmax = 1.44 [τdτp 1 (τp—τd)] ln (τp/τd)which may exceed the parent activity under certain conditions, in which case there will be noequilibrium state.
Radioactive Materials and Radioactive Decay
Published in Robert E. Masterson, Nuclear Engineering Fundamentals, 2017
It is easily possible to generalize what we have just done to examine the concentrations of the isotopes in a radioactive decay chain that consists of n nuclides where each nuclide has its own value of λn. The resulting equations for the time-dependent nuclide concentrations are called the Bateman equations in honor of Englishman Harold Bateman (1882–1946), who first provided a general solution to them in 1910 (see Figure 6.31).*
An Inline Burnup Algorithm
Published in Nuclear Science and Engineering, 2023
P. Cosgrove, E. Shwageraus, J. Leppänen
Evaluating this matrix exponential is difficult,32 although for burnup matrices this can be accurately done using the Chebyshev rational approximation method33,34 (CRAM). Other accurate methods of solving the Bateman equation include using more familiar Ordinary Differential Equation (ODE) solvers.29,35 The Bateman equation is valid in a homogeneous region, whereas in reactor problems, one is generally interested in how the many different regions of the problem evolve. Thus burnup problems tend to discretize the reactor geometry into many separate, homogeneous burnable regions in which reaction rates are obtained and where the Bateman equation is solved separately. For large, finely discretized reactor geometries, this may impose a substantial computer memory burden unless care is taken.29 Although relatively unexplored, a spatially continuous approach to depletion has also been proposed and preliminarily investigated.36
Evaluating Quantities of Interest Other Than Nuclide Densities in the Bateman Equations
Published in Nuclear Science and Engineering, 2023
Olin W. Calvin, Micah D. Gale, Sebastian Schunert
The Bateman equations mathematically describe the nuclide number densities (NNDs) in a given system as radioactive decay events occur, resulting in the production of the decay products of the nuclides originally present in the system.1 The Bateman equations originally accounted for only radioactive decay since they were developed before the discovery of the neutron. However, it is simple to extend the Bateman equations to account for neutron transmutation, as shown in Eq. (1) adapted from Ref. 2:
Introduction of the Adding and Doubling Method for Solving Bateman Equations for Nuclear Fuel Depletion
Published in Nuclear Science and Engineering, 2023
Olin W. Calvin, Barry D. Ganapol, R. A. Borrelli
The Bateman equations, named after mathematician Harry Bateman, form the mathematical model describing the abundances of various nuclides in a system as time passes and radioactive decay occurs, transforming radioactive nuclides into their decay products.1 While the Bateman equations originally considered only radioactive decay, they can be modified to account for the neutron transmutation that occurs in nuclear systems. This is shown in Eq. (1) adapted from Ref. 2: