Gamma Radiation Decay and First-Order Kinetics

The transcript describes gamma radiation entering the period; this signal decays over time.
It states that the decay follows what is called first-order kinetics.

Gamma radiation: high-energy photons emitted by unstable nuclei during radioactive decay.
Radioactive decay: spontaneous transformation of unstable nuclei to more stable configurations, often emitting gamma photons.
First-order kinetics: rate of decay is proportional to the number of undecayed nuclei; mathematical form below.
Decay constant k: probability per unit time that a given nucleus will decay; units s^{-1}.
Activity A(t): rate of decay, A(t) = -dN/dt.
Detected signal: if the measurement is proportional to activity, the observed signal S(t) decays accordingly.
Half-life t{1/2}: the time for N to fall to N0/2; t{1/2} = \frac{\ln 2}{k}.
Time constant tau: tau = 1/k.

Differential equation: $\frac{dN}{dt} = -k N$
Solution: $N(t) = N_0 e^{-k t}$
Activity: $A(t) = -\frac{dN}{dt} = k N<em>0 e^{-k t} = A</em>0 e^{-k t}$
Half-life: $t_{1/2} = \frac{\ln 2}{k}$
Time constant: $\tau = \frac{1}{k}$
Signal: $S(t) = S_0 e^{-k t}$
Example: If N0 = 1.0 \times 10^6 and k = 3.0 \times 10^{-3} s^{-1} then
- t_{1/2} = \frac{\ln 2}{3.0 \times 10^{-3}} ≈ 231 \text{ s}
- N(t) = 1.0 \times 10^6 e^{-0.003 t}
- A(t) = (3.0 \times 10^{-3}) \times 1.0 \times 10^6 e^{-0.003 t} = 3.0 \times 10^3 e^{-0.003 t} \text{ counts/s}
- S(t) similarly decays.

Real-world relevance: medical imaging, nuclear safety, dating techniques; safety considerations regarding exposure.
Distinctions: activity vs. counts vs. dose; detection efficiency.

Exponential decay appears as linear on a semi-log plot: log A vs t yields a straight line with slope -k.

If multiple isotopes with different k values present, the observed decay is a sum of exponentials.
Gamma emission is not always the same product; can accompany alpha/beta decay.