1.2 Radioactive Decay Kinetics & Half-Life Calculations

The Fundamental Law of Radioactive Decay

Radioactive decay is a purely stochastic (probabilistic) nuclear process. For any single unstable nucleus, it is physically impossible to predict the exact instant of disintegration. However, for a macroscopic collection of $N$ identical radioactive atoms, the rate of nuclear disintegration ($-dN/dt$), termed the activity ($A$), is strictly proportional to the number of radioactive atoms present at that moment:

$$A = -\frac{dN}{dt} = \lambda N$$

where $\lambda$ is the characteristic decay constant of the specific radioisotope (expressed in units of $\text{time}^{-1}$, such as $\text{s}^{-1}$, $\text{hr}^{-1}$, or $\text{day}^{-1}$). The decay constant represents the constant fractional probability per unit time that a given nucleus will decay.

Exponential Decay Equations

Integrating the differential decay equation yields the classical exponential decay law:

$$N(t) = N_0 e^{-\lambda t}$$

Because activity $A$ is directly proportional to $N$ ($A = \lambda N$):

$$A(t) = A_0 e^{-\lambda t}$$

where:

• $A(t)$ = Activity remaining after elapsed time $t$,
• $A_0$ = Initial activity at time $t = 0$,
• $\lambda$ = Decay constant,
• $t$ = Elapsed time (in the same units as $\lambda$).

---

Half-Life ($T_{1/2}$) and the Half-Life Formula

The half-life ($T_{1/2}$) is defined as the time required for one-half (50%) of the radioactive atoms in a sample to disintegrate, or for the activity of the sample to decrease to exactly half its initial value ($A = A_0 / 2$).

Setting $A(t) = A_0 / 2$ and $t = T_{1/2}$ in the exponential decay law:

$$\frac{A_0}{2} = A_0 e^{-\lambda T_{1/2}} \implies \frac{1}{2} = e^{-\lambda T_{1/2}}$$

Taking the natural logarithm of both sides:

$$\ln\left(\frac{1}{2}\right) = -\lambda T_{1/2} \implies -0.69315 = -\lambda T_{1/2}$$ $$\lambda = \frac{\ln(2)}{T_{1/2}} \approx \frac{0.693}{T_{1/2}}$$

Substituting $\lambda$ back into the exponential equation yields the highly practical half-life decay equation utilized across all industrial radiography field operations:

$$A(t) = A_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} = A_0 \cdot (0.5)^n$$

where $n = t / T_{1/2}$ represents the number of elapsed half-lives (which does not need to be an integer).

Step-by-Step Decay Factor Progression

Worked Field Calculation Example

Problem: A radiographer receives a newly loaded Iridium-192 source certified at 100 Curies on September 1. What will be the source activity after 148 days? (Half-life of Ir-192 = 74 days).

1. Calculate the number of elapsed half-lives ($n$): $$n = \frac{t}{T_{1/2}} = \frac{148\text{ days}}{74\text{ days}} = 2.0$$
2. Calculate remaining activity using the half-life formula: $$A = A_0 \cdot (0.5)^n = 100\text{ Ci} \times (0.5)^2 = 100\text{ Ci} \times 0.25 = 25\text{ Ci}$$

Answer: Exactly 25 Curies remain.

Now consider a fractional half-life problem: What is the activity after 45 days?

1. $n = 45 / 73.83 = 0.6095$
2. $A = 100 \times (0.5)^{0.6095} = 100 \times 0.6554 = 65.54\text{ Ci}$ Radiographers use printed manufacturer decay charts or daily calculation tables derived from this exact mathematical formula to set radiography exposure times.

---

Specific Activity and Source Pellet Geometry

Specific activity is defined as the radioactivity per unit mass of the isotope, expressed in Curies per gram (Ci/g) or Terabecquerels per gram (TBq/g):

$$\text{Specific Activity} = \frac{A}{m} = \frac{\lambda N_A}{M}$$

where $N_A$ is Avogadro's number ($6.022 \times 10^{23}\text{ atoms/mol}$) and $M$ is the molar mass.

Why Specific Activity Dictates Radiographic Quality

In non-destructive examination, the sharpness of the radiographic image depends on minimizing geometric unsharpness ($U_g$):

$$U_g = \frac{F \cdot d}{D}$$

where $F$ is the physical dimension of the radiation focal spot (the source pellet diameter), $d$ is the object-to-film distance, and $D$ is the source-to-object distance. A smaller focal spot ($F$) produces sharper, higher-resolution radiographs that reveal minute weld cracks and lack of fusion.

Because Iridium-192 can be activated to exceptionally high specific activity (often 200 to 450 Ci/g), a 100-Curie Ir-192 source pellet measures only 2.5 mm to 3.0 mm in diameter. Conversely, Cobalt-60 has a lower specific activity (typically 50 to 100 Ci/g), meaning a 100-Curie Co-60 source requires a larger physical pellet (4 to 6 mm diameter), resulting in greater geometric unsharpness.

---

Artificial Radioisotope Production in Nuclear Reactors

None of the primary radioisotopes used in modern industrial radiography exist in nature; they are artificially created by exposing stable target elements to high thermal neutron fluxes inside nuclear research reactors:

• Iridium-192 Production: Stable Iridium-191 ($37.3\%$ natural abundance) captures a thermal neutron via the $(n, \gamma)$ reaction: $${}^{191}_{77}\text{Ir} + {}^1_0n \longrightarrow {}^{192}_{77}\text{Ir} + \gamma$$
• Cobalt-60 Production: Stable natural Cobalt-59 ($100\%$ abundance) undergoes neutron capture: $${}^{59}_{27}\text{Co} + {}^1_0n \longrightarrow {}^{60}_{27}\text{Co} + \gamma$$
• Selenium-75 Production: Enriched Selenium-74 targets capture a neutron: $${}^{74}_{34}\text{Se} + {}^1_0n \longrightarrow {}^{75}_{34}\text{Se} + \gamma$$