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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. [2]
Radioactive decay is a random process at the level of single atoms. According to quantum theory, it is impossible to predict when a particular atom will decay, regardless of how long the atom has existed. [2] [3] [4] However, for a significant number of identical atoms, the overall decay rate can be expressed as a decay constant or as a half-life.
In other words, the probability of a radioactive atom decaying within its half-life is 50%. [2] For example, the accompanying image is a simulation of many identical atoms undergoing radioactive decay. Note that after one half-life there are not exactly one-half of the atoms remaining, only approximately, because of the random variation in the ...
Time taken for half the number of atoms present to decay + / / s [T] Number of half-lives n (no standard symbol) = / / dimensionless dimensionless Radioisotope time constant, mean lifetime of an atom before decay
The surface emission rate (SER) is used as a measure of the rate of particles emitted from a radioactive source which is being used as a calibration standard. When the source is of plate or planar construction and the radiation of interest is emitting from one face, it is known as " 2 π {\displaystyle 2\pi } emission".
and are the half-lives (inverses of reaction rates in the above equation modulo ln(2)) of the parent and daughter, respectively, and BR is the branching ratio. In transient equilibrium, the Bateman equation cannot be simplified by assuming the daughter's half-life is negligible compared to the parent's half-life.
The activity of a sample decreases with time because of decay. The rules of radioactive decay may be used to convert activity to an actual number of atoms. They state that 1 Ci of radioactive atoms would follow the expression N (atoms) × λ (s −1) = 1 Ci = 3.7 × 10 10 Bq, and so N = 3.7 × 10 10 Bq / λ, where λ is the decay constant in s ...
Alternatively, since the radioactive decay contributes to the "physical (i.e. radioactive)" half-life, while the metabolic elimination processes determines the "biological" half-life of the radionuclide, the two act as parallel paths for elimination of the radioactivity, the effective half-life could also be represented by the formula: [1] [2]