As mentioned earlier (Section 2.1.1), the stability of a nucleus (as characterized by the binding energy) is determined by the ratio of protons to neutrons (see Eq. (2.10)). The binding energy of isobar nuclei as a function of proton—neutron ratio forms a parabola or parabolas (see Figure 3.1), where those nuclei close to the minimum are stable while those farther away are undergoing radioactive decay in order to reach the optimal proton/neutron ratio. The radioactive decay is a random process for the individual nucleus so as to describe the kinetics of radioactive decay, a statistical approach has to be applied.

4.1 Kinetics of Radioactive Decay

4.1.1 Statistics of Simple Radioactive Decay

Let us consider that the probability of the decomposition of a radioactive nuclide in a At time interval is p:

p = XAt (4.1)

where A is a factor of proportionality. The probability of the process that the nuclide does not decompose in At is:

1 — p = 1 — AAt (4.2)

The probability that the nuclide does not decompose in another second, third, or more At interval can also be defined by Eq. (4.2). The probability that the nuclide does not decompose in the 2 X At interval is:

(1 — p)2 = (1 — AAt)2 (4.3)

The probability that the nuclide does not decompose in n X At is:

(1 — p)n = (1 — AAt)n (4.4)

Let us divide the total time of the observation (t) into n intervals:

— = At (4.5)

n

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Substituting Eq. (4.5) into Eq. (4.4), we obtain:

/ t n

(1 — p)n = 1 — An (4.6)

At n! n:

t n

1 — A — = e-At (4.7)

n

When the initial number of the radioactive nuclides is N0, the number of nuclides that do not undergo radioactive decomposing during t time (N) is:

N = N0 e—A (4.8)

Equation (4.8) describes the kinetics of the simple radioactive decay, i. e., the radioactive decay law, where A is the decay constant. The value of the decay con­stant characterizes the radionuclide; thus, it is independent of physical and chemi­cal conditions (pressure, temperature, chemical environment, etc.).

As seen in Eq. (4.8), the radioactive decay has first-order kinetics, having all characteristics of first-order reactions. It has a well-defined half-life (t1/2), i. e., the time needed to reduce the number of the radioactive nuclides to half:

N0 = N0 e—At1/2 (4.9)

and from here,

A = ^ (4.10)

t1/2

Half-life performance 7 and 10 times over gives the interval when the number of radionuclides decreases below 1% and 0.1% of the initial number, respectively.

The reciprocal of the decay constant (A) is the average lifetime (t) of the radionuclides, which is the time when the number of the radionuclides decreases by a factor of e (i. e., Euler’s constant). This amount of time should be required for the decomposition of all radionuclides if the rate of decay remains constant.

(4.11)