Radioactive Half-Life

The radioactive half-life for a given radioisotope is a measure of the tendency of the nucleus to "decay" or "disintegrate" and as such is based purely upon that probability. The tiny nuclear size compared to the atom and the enormity of the forces which act within it make it almost totally impervious to the outside world. The half-life is independent of the physical state (solid, liquid, gas), temperature, pressure, the chemical compound in which the nucleus finds itself, and essentially any other outside influence. It is independent of the chemistry of the atomic surface, and independent of the ordinary physical factors of the outside world. The only thing which can alter the half-life is direct nuclear interaction with a particle from outside, e.g., a high energy collision in an accelerator.

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The predictions of decay can be stated in terms of the half-life , the decay constant, or the average lifetime. The relationship between these quantities is as follows.

Graph of decayCalculation
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Nuclear Decay Probability

Radioactive decay is a statistical process which depends upon the instability of the particular radioisotope, but which for any given nucleus in a sample is completely unpredictable. The decay process and the observed half-life dependence of radioactivity can be predicted by assuming that individual nuclear decays are purely random events. If there are N radioactive nuclei at some time t, then the number DN which would decay in any given time interval Dt would be proportional to N:

where l is a constant of proportionality (decay constant).

Without any further assumptions, this leads to the exponential radioactive decay result:

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and also implies that the decay rate and amount of emitted radiation also follow the same type of relationship:

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Developing the Decay Expression

Although radioactive decay involves discrete events of nuclear disintegration, the number of events is so large that it can be treated like a continuum and the methods of calculus employed to predict the behavior. The result from the decay probability can be put in the differential form:

This can be integrated directly to giveln N = -lt + C where C is a constant of integration.

Taking the exponent of both sides gives
so the standard form of the decay equation is:



which also applies to all other quantities which are proportional to N.

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Radioactive Decay Constant

The rate of radioactive decay is typically expressed in terms of either the radioactive half-life, or the radioactive decay constant. They are related as follows:

The decay constant is also sometimed called the disintegration constant.The half-life and the decay constant give the same information, so either may be used to characterize decay. Another useful concept in radioactive decay is the average lifetime. The average lifetime is the reciprocal of the decay constant as defined here.

For example, free neutrons decay with a halflife of about 10.3 minutes. This corresponds to a decay constant of .067/s and an average lifetime of 14.8 minutes or 890 seconds.

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