4.3.1 The General Differential Equation

The number density $N_i$ of any fission product nuclide $i$ in the reactor is governed by the following differential equation:

$\frac{dN_i}{dt} = \underbrace{\sum_p \lambda_{pi} N_p}_{\text{(1) Decay of precursors}} + \underbrace{\sum_j \sigma_{ji} \phi N_j}_{\text{(2) Neutron capture}} + \underbrace{\sum_k y_{ki} \sigma_{fk} \phi N_k}_{\text{(3) Fission production}} - \underbrace{\lambda_i N_i}_{\text{(4) Radioactive decay}} - \underbrace{\sigma_i \phi N_i}_{\text{(5) Burn-up}}$

Let us explain each term carefully:

Term	Symbol	Meaning (in plain English)
(1) Decay in	$\sum_p \lambda_{pi} N_p$	Nuclide $i$ is produced when its precursor nuclide(s) $p$ decay into it. $\lambda_{pi}$ is the decay constant for that specific decay path, and $N_p$ is the number of precursor atoms present.
(2) Capture in	$\sum_j \sigma_{ji} \phi N_j$	Nuclide $i$ is produced when another nuclide $j$ absorbs a neutron and is transformed into $i$ . $\sigma_{ji}$ is the capture cross-section for that reaction, and $\phi$ is the neutron flux.
(3) Fission production	$\sum_k y_{ki} \sigma_{fk} \phi N_k$	Nuclide $i$ is produced directly from fission. $y_{ki}$ is the fission yield of nuclide $i$ from the fissioning of nuclide $k$ (e.g. $^{235}$ U), $\sigma_{fk}$ is the fission cross-section of nuclide $k$ , and $\phi$ is the neutron flux.
(4) Decay out	$\lambda_i N_i$	Nuclide $i$ is removed by its own radioactive decay. $\lambda_i$ is its decay constant.
(5) Burn-up	$\sigma_i \phi N_i$	Nuclide $i$ is removed when it absorbs a neutron (burn-up). $\sigma_i$ is its neutron absorption cross-section.

In practice, these coupled equations are solved numerically using computer codes such as FISPIN (BNFL) or ORIGEN (Oak Ridge National Laboratory). These codes handle hundreds of isotopes, time-varying flux, and multi-group cross-sections.

After Shutdown

When the reactor shuts down, the neutron flux $\phi$ drops to zero. All terms containing $\phi$ vanish, and the equation simplifies to:

$\frac{dN_i}{dt} = \sum_p \lambda_{pi} N_p - \lambda_i N_i$

This is simply the decay chain equation --- nuclides are produced only from the decay of their precursors and are removed only by their own decay.

After Shutdown

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