Radioactive Decay

To understand the behavior of fission products, the rates of conversion of fertile to fissile materials, and a number of other phenomena related to reactor physics we must quantify the behavior of radioactive materials. The law governing the decay of a nucleus states that the rate of decay is proportional to the number of nuclei present. Each radioisotope—that is, an isotope that undergoes radioactive decay— has a characteristic decay constant lamda. Thus if the number of nuclei present at time t is N(t), the rate at which they decay is




Dividing by N(t), we may integrate this equation from time zero to t, to obtain





where N(0) is the initial number of nuclei. Noting that dN=N ¼ d ln(N), Eq. (1.34) becomes



Figure 1.7 illustrates the exponential decay of a radioactive material. The half-life,
t1/2, is a more intuitive measure of the times over which unstable nuclei decay.
As defined earlier, t1/2 is the length of time required for one-half of the nuclei to decay.
Thus it may be obtained by substituting N(t1/2 ) =N(0)/2 into
Eq. (1.35) to yield ln(1/2)= -0.693 =lamda(t1/2), or simply




A second, less-used measure of decay time is the mean time to decay, defined by





Before proceeding, a word is in order concerning units.
Normally we specify the strength of a radioactive source in terms
of curies (Ci) where 1 Ci is defined as 3.7x10¹⁰ disintegrations per
second, which is the rate decay of one gram of radium-226; the
becquerel (Bq), defined as one disintegration per second, has also
come into use as a measure of radioactivity. To calculate the
number of nuclei present we first note that Avogadro’s number,
No= 0.6023 x10²⁴, is the number of atoms in one gram molecular
weight, and thus the total number of atoms is just mNo/A where
m is the mass in grams and A is the atomic mass of the isotope. The
concentration in atoms/cm³ is then


where is


the density
in grams/cm<sup>3


Saturation Activity

Radionuclides are produced at a constant rate in a number of situations.
For example, a reactor operating at constant power produces
radioactive fission fragments at a constant rate. In such situations,
we determine the time dependence of the inventory of an isotope
produced at a rate of Ao nuclei per unit time by adding a source term
Ao to Eq. (1.33):
















where N(t) is the activity measured in disintegrations per unit time.
Note that initially the activity increases linearly with time, since for


t<<1,>t)~1(-t). After several half-lives, however,
the exponential term becomes vanishingly small, and the rate
of decay is then equal to the rate of production or
N(infinity) = Ao. This is referred to as the saturation activity.
Figure 1.8 illustrates the buildup to saturation activity given by
Eq. (1.42). To illustrate the importance of saturation
To illustrate the importance of saturation activity, consider
iodine-131 and strontium-90, which are two of the more important
fission products stemming from the operation of power reactors.
Assume a power reactor produces them at rates of 0.85x</sup>10¹⁸
nuclei/s