Nuclear Fuel Cycle

The general solution above simplifies further depending on whether the half-life is short or long compared to the irradiation time.

SHORT-LIVED APPROXIMATION: $T_{1/2} \ll$ irradiation time

If the fission product half-life is much shorter than the time the reactor has been operating (e.g. $^{131}$I with $T_{1/2}$ = 8 days in a reactor running for months), then the exponential term $e^{-\lambda t}$ becomes negligibly small (effectively zero), and:

$\boxed{A_{\text{eq}} = R = F \cdot Y}$

The activity reaches a constant equilibrium value. Production and decay are balanced --- for every atom created by fission, one atom decays. The activity depends only on the fission rate and the yield, not on how long the reactor has been running.

LONG-LIVED APPROXIMATION: $T_{1/2} \gg$ irradiation time

If the fission product half-life is much longer than the irradiation time (e.g. $^{137}$Cs with $T_{1/2}$ = 30.17 years in a reactor running for 1 year), then we can use the approximation $e^{-\lambda t} \approx 1 - \lambda t$ (first-order Taylor expansion), giving:

$A(t) = R\left(1 - e^{-\lambda t}\right) \approx R \cdot \lambda t$

$\boxed{A(t) = F \cdot Y \cdot \lambda \cdot t = F \cdot Y \cdot \frac{0.693 \cdot t}{T_{1/2}}}$

The activity builds up linearly with time. It has not yet begun to approach equilibrium. Longer irradiation gives proportionally more activity.

Summary of Approximations

• Equilibrium (short-lived) --- When $T_{1/2} \ll t_{\text{irradiation}}$: Use $A = F \cdot Y$. Activity is constant; production equals decay.
• Linear (long-lived) --- When $T_{1/2} \gg t_{\text{irradiation}}$: Use $A = F \cdot Y \cdot \lambda \cdot t$. Activity grows linearly; negligible decay during operation.
• Full solution (general case) --- When $T_{1/2}$ is comparable to $t_{\text{irradiation}}$: Use $A = R(1 - e^{-\lambda t})$.