k-Eigenvalue Calculations

A k-eigenvalue (criticality) calculation determines the effective neutron multiplication factor \(k_{\text{eff}}\) and the fundamental-mode fission source distribution of a system. MC/DC implements this via the standard power iteration algorithm.

The k-Eigenvalue Equation

In steady state the Boltzmann equation with fission becomes an eigenvalue problem:

\[\hat{\Omega}\cdot\nabla\psi + \Sigma_t\,\psi = \int \Sigma_s\,\psi'\,dE'\,d\Omega' + \frac{1}{k}\,\frac{\chi}{4\pi}\int \nu\Sigma_f\,\phi'\,dE'\]

where \(k = k_{\text{eff}}\) is the eigenvalue. A system is

  • critical if \(k = 1\),

  • supercritical if \(k > 1\),

  • subcritical if \(k < 1\).

Power Iteration

MC/DC solves the eigenvalue problem using power (source) iteration:

  1. An initial guess for \(k^{(0)}\) is provided (default: 1.0).

  2. Source particles are sampled from the current fission source distribution.

  3. Particles are transported; fission sites are banked.

  4. The new eigenvalue estimate is updated:

    \[k^{(i+1)} = k^{(i)} \;\frac{W_{\text{fission}}^{(i+1)}}{W_{\text{source}}^{(i)}}\]

    where \(W\) denotes the total statistical weight.

  5. The fission bank becomes the source for the next cycle.

  6. Repeat until convergence.

Users configure eigenmode via:

mcdc.settings.set_eigenmode(N_inactive=50, N_active=200, k_init=1.0)

  • N_inactive — Cycles discarded for fission source convergence.

  • N_active — Cycles used for tally accumulation.

  • k_init — Initial \(k\) guess.

Inactive vs. Active Cycles

The first N_inactive cycles allow the fission source distribution to converge from the (often arbitrary) initial guess to the fundamental eigenmode. Tallies are not accumulated during inactive cycles to avoid bias.

During the N_active cycles, tally scores are accumulated and batch statistics (mean, standard deviation) are computed for both \(k_{\text{eff}}\) and spatial quantities.

Shannon Entropy

MC/DC can optionally track the Shannon entropy of the fission source distribution as a convergence diagnostic. The spatial domain is divided into \(M\) mesh cells, and the entropy at cycle \(i\) is:

\[H^{(i)} = -\sum_{m=1}^{M} p_m \log_2 p_m\]

where \(p_m\) is the fraction of fission source weight in cell \(m\). A plateau in \(H\) over successive cycles indicates that the source has converged.

Gyration Radius

The gyration radius measures the spatial spread of the fission source around its centre of mass:

\[R_g = \sqrt{\frac{\sum_j w_j\,|\mathbf{r}_j - \mathbf{r}_{\text{cm}}|^2} {\sum_j w_j}}\]

MC/DC reports gyration radius diagnostics in the output when running eigenvalue problems (see the C5G7 — k-eigenvalue example for an example).

Tips for k-Eigenvalue Simulations

  • Sufficient inactive cycles: Too few inactive cycles biases \(k_{\text{eff}}\) and spatial tallies. Check Shannon entropy convergence.

  • Particle count: Start with \(10^3\)–\(10^4\) particles per cycle for development, then increase to \(10^5\)–\(10^6\) for production.

  • Initial source distribution: A uniform distribution over the fissile region is a reasonable default.

References

  • T. M. Sutton and A. Morel. “Iteration Acceleration Techniques for Monte Carlo Eigenvalue Calculations.” Transactions of ANS (1996).

  • F. B. Brown. “On the Use of Shannon Entropy of the Fission Distribution for Assessing Convergence of Monte Carlo Criticality Calculations.” PHYSOR (2006).