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The following apply for the nuclear reaction: a + b ↔ R → c in the centre of mass frame , where a and b are the initial species about to collide, c is the final species, and R is the resonant state .
In nuclear physics, the Bateman equation is a mathematical model describing abundances and activities in a decay chain as a function of time, based on the decay rates and initial abundances. The model was formulated by Ernest Rutherford in 1905 [ 1 ] and the analytical solution was provided by Harry Bateman in 1910.
A Markov process is called a reversible Markov process or reversible Markov chain if there exists a positive stationary distribution π that satisfies the detailed balance equations [13] =, where P ij is the Markov transition probability from state i to state j, i.e. P ij = P(X t = j | X t − 1 = i), and π i and π j are the equilibrium probabilities of being in states i and j, respectively ...
Nuclear reactions may be shown in a form similar to chemical equations, for which invariant mass must balance for each side of the equation, and in which transformations of particles must follow certain conservation laws, such as conservation of charge and baryon number (total atomic mass number). An example of this notation follows:
The "Six-factor formula" is the neutron life-cycle balance equation, which includes six separate factors, the product of which is equal to the ratio of the number of neutrons in any generation to that of the previous one; this parameter is called the effective multiplication factor k, also denoted by K eff, where k = Є L f ρ L th f η, where ...
where λ A and λ B are the decay constants of radionuclide A and B, related to their half-lives t 1/2 by = / /, and N A and N B are the number of atoms of A and B at a given time. Secular equilibrium occurs when d N B / d t = 0 {\displaystyle dN_{B}/dt=0} , or
It has the standard resonance form of the Lorentz, or Cauchy distribution, but involves relativistic variables s = p 2 , here = E 2 . The distribution is the solution of the differential equation for the amplitude squared w.r.t. the energy energy (frequency), in such a classical forced oscillator,
In physics and chemistry, specifically in nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI), and electron spin resonance (ESR), the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization M = (M x, M y, M z) as a function of time when relaxation times T 1 and T 2 are present.