\section*{Problem 8}

\subsection*{Part A}

The one delayed-group model assumes all six precursor groups can be lumped into
a single effective group with average parameters. This introduces errors:

\begin{itemize}
  \item Each group has different decay constants (\(\lambda_i\)) ranging from
  0.012 to 3.01 s\(^{-1}\), corresponding to half-lives from 0.2 to 80 seconds

  \item The one-group model cannot capture the multi-timescale behavior - early
  time dynamics are dominated by fast-decaying groups, late time by slow groups

  \item Effective parameters (\(\beta_{eff}\), \(\lambda_{eff}\)) are only
  approximate averages that work reasonably for long-term behavior but miss
  short-term details
\end{itemize}

This weakness matters most for short transients where individual group dynamics
are important, less so for long-term steady-state calculations.

\subsection*{Part B}

For six delayed groups with prompt jump approximation:

Power is given by:

\[N(t) = \frac{\Lambda}{\beta - \rho} \sum_{i=1}^{6} \lambda_i C_i(t)\]

Precursor concentrations evolve as:

\[\frac{dC_i(t)}{dt} = \frac{\beta_i}{\Lambda}N(t) - \lambda_i C_i(t), \quad i = 1,\ldots,6\]

In matrix form, define the state vector:

\[\mathbf{C}(t) = \begin{bmatrix} C_1(t) \\ C_2(t) \\ C_3(t) \\ C_4(t) \\ C_5(t) \\ C_6(t) \end{bmatrix}\]

The precursor equation becomes:

\[\frac{d\mathbf{C}}{dt} = \mathbf{A} \mathbf{C}(t)\]

where the matrix \(\mathbf{A}\) is:

{\tiny
\[\mathbf{A} = \frac{1}{\beta - \rho} \begin{bmatrix}
\beta_1 \lambda_1 - \lambda_1(\beta - \rho) & \beta_1 \lambda_2 & \beta_1 \lambda_3 & \beta_1 \lambda_4 & \beta_1 \lambda_5 & \beta_1 \lambda_6 \\
\beta_2 \lambda_1 & \beta_2 \lambda_2 - \lambda_2(\beta - \rho) & \beta_2 \lambda_3 & \beta_2 \lambda_4 & \beta_2 \lambda_5 & \beta_2 \lambda_6 \\
\beta_3 \lambda_1 & \beta_3 \lambda_2 & \beta_3 \lambda_3 - \lambda_3(\beta - \rho) & \beta_3 \lambda_4 & \beta_3 \lambda_5 & \beta_3 \lambda_6 \\
\beta_4 \lambda_1 & \beta_4 \lambda_2 & \beta_4 \lambda_3 & \beta_4 \lambda_4 - \lambda_4(\beta - \rho) & \beta_4 \lambda_5 & \beta_4 \lambda_6 \\
\beta_5 \lambda_1 & \beta_5 \lambda_2 & \beta_5 \lambda_3 & \beta_5 \lambda_4 & \beta_5 \lambda_5 - \lambda_5(\beta - \rho) & \beta_5 \lambda_6 \\
\beta_6 \lambda_1 & \beta_6 \lambda_2 & \beta_6 \lambda_3 & \beta_6 \lambda_4 & \beta_6 \lambda_5 & \beta_6 \lambda_6 - \lambda_6(\beta - \rho)
\end{bmatrix}\]
}

\subsection*{Part C}

The prompt jump approximation error is likely smaller than the one-group error.

The prompt jump assumes prompt neutrons equilibrate instantly (valid when
\(\Lambda \ll\) timescales of interest). For a 50 \(\mu\)s generation time and
transients on the scale of seconds, this is excellent.

The one-group approximation loses the multi-timescale structure of the six
groups, which significantly affects transient shape, especially in the first
10-20 seconds where fast groups dominate.

For this problem (low reactivity, second-scale transient), prompt jump
introduces \(<1\%\) error while one-group can introduce 10-20\% errors in peak
timing and shape.
