Decay Constant Calculator

The decay constant lambda is the probability per unit time that any single unstable nucleus will disintegrate. Because that probability does not depend on the age of the nucleus or on how many neighbours it has, radioactive decay obeys first-order kinetics: dN/dt = -lambda N, whose solution is N(t) = N0 times e raised to minus lambda t. Setting N/N0 equal to one half and solving gives lambda = ln(2) / t-half, which is why the constant 0.693147 appears in the formula. Lambda carries units of reciprocal time, so a value quoted per second cannot be compared with one quoted per year without conversion. Two derived quantities follow immediately: the mean lifetime tau = 1/lambda, which is 1.4427 times the half-life, and the activity A = lambda N, measured in becquerel (one disintegration per second) or curie (3.7 times 10 to the 10th Bq). Real decay constants span more than twenty orders of magnitude, from roughly 4,200 per second for polonium-214 (half-life 164 microseconds) to about 4.9 times 10 to the minus 11 per year for thorium-232 (half-life 1.4 times 10 to the 10th years). The main caveats are statistical and structural. The exponential law is a large-number average; with only a few hundred atoms remaining, actual counts fluctuate according to Poisson statistics and a single measurement can deviate noticeably from the predicted curve. Many nuclides also decay by more than one route, so the measured lambda is the sum of partial decay constants: potassium-40, for example, decays by beta emission about 89.3 percent of the time and by electron capture about 10.7 percent, and only the total governs the observed half-life. Decay constants are essentially immune to temperature, pressure, and chemical bonding, though electron-capture nuclides such as beryllium-7 show shifts near one percent between chemical environments, and fully stripped ions in storage rings can display radically altered lifetimes. When solving for lambda from measured counts, correct for detector background, dead time, and any ingrowth of radioactive daughters.

Henri Becquerel found in 1896 that uranium salts fogged photographic plates without prior illumination, and Marie and Pierre Curie isolated polonium and radium in 1898, giving the phenomenon its name. The quantitative decay law came from Ernest Rutherford and Frederick Soddy at McGill University in Montreal. Working in 1902 and 1903 on the short-lived thorium emanation now known as radon-220, they observed that its activity fell off geometrically with time and published their transformation theory in the Philosophical Magazine, treating radioactivity as the spontaneous change of one element into another. Rutherford introduced the practical descriptor half-value period, later shortened to half-life, and the exponential constant in their fitted curves became the decay constant. Egon von Schweidler showed in 1905 that the exponential law follows purely from assuming a fixed decay probability per nucleus per unit time, which recast lambda as a statistical quantity rather than an empirical fitting parameter and predicted the fluctuations later confirmed in counting experiments. Hans Geiger and John Nuttall reported in 1911 an empirical relation between the decay constant of alpha emitters and the energy of the emitted particle, and George Gamow explained that relation in 1928 using quantum tunnelling through the nuclear Coulomb barrier, the first successful theory of a decay rate from first principles. Enrico Fermi supplied the corresponding theory for beta decay in 1934. Evaluated modern values are maintained in the Evaluated Nuclear Structure Data File and published through the IAEA and Brookhaven National Laboratory.