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where N 0 is the initial quantity of atoms at time t = 0. Half-life T 1/2 is defined as the length of time for half of a given quantity of radioactive atoms to undergo radioactive decay: = /. Taking the natural logarithm of both sides, the half-life is given by
Instead, the half-life is defined in terms of probability: "Half-life is the time required for exactly half of the entities to decay on average". 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.
Thus, the amount of material left is 2 −1 = 1/2 raised to the (whole or fractional) number of half-lives that have passed. Thus, after 3 half-lives there will be 1/2 3 = 1/8 of the original material left. Therefore, the mean lifetime is equal to the half-life divided by the natural log of 2, or:
The time constant τ is the e −1 -life, the time until only 1/e remains, about 36.8%, rather than the 50% in the half-life of a radionuclide. Thus, τ is longer than t 1/2. The following equation can be shown to be valid: = / = / /.
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.
The partial half-life is merely an alternate way to specify the partial decay constant λ, the two being related through: / = . For example, for decays of 132 Cs, 98.13% are ε (electron capture) or β + decays, and 1.87% are β − decays. The half-life of this isotope is 6.480 days, [2] which corresponds to a total decay constant of 0.1070 ...
This is a list of radioactive nuclides (sometimes also called isotopes), ordered by half-life from shortest to longest, in seconds, minutes, hours, days and years. Current methods make it difficult to measure half-lives between approximately 10 −19 and 10 −10 seconds.
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.