Moderator-to-Fuel Ratio Calculator

The moderator-to-fuel ratio R is the number of moderator atoms (or molecules) per fuel atom in the reactor lattice. It quantifies how much neutron-slowing material surrounds each fuel nucleus. For light water reactors the ratio is typically 1 to 5 H2O molecules per uranium atom; for CANDU heavy water reactors it reaches several hundred D2O molecules per uranium atom. R controls both the resonance escape probability p and the thermal utilization f, and therefore has a dominant effect on k∞.

As R increases, p rises because more moderator slows neutrons past the U-238 resonance region before they can be captured. However, f simultaneously falls because more thermal neutrons are absorbed in the moderator rather than in the fuel. The product p×f therefore rises at low R (where p is small) and falls at high R (where f is small), passing through a maximum at the optimal ratio R_opt. Maximizing p×f is a central goal in lattice design.

Setting d(p×f)/dR = 0 and solving gives R_opt = [A + sqrt(A² + 4A×sigma_a_F/sigma_a_M)] / 2, where A = I_eff / (xi_M × sigma_s_M), sigma_a_F is the thermal absorption cross-section of the fuel, and sigma_a_M is the absorption cross-section of the moderator. This closed-form result depends only on the fuel enrichment, effective resonance integral, and moderator nuclear constants.

An under-moderated reactor operates at R below R_opt: increasing the moderator density raises k, meaning less moderation leads to lower reactivity. An over-moderated reactor operates at R above R_opt: adding more moderator reduces k. Light water reactors are deliberately designed slightly under-moderated, so that if the coolant heats up and its density drops (less moderator), k also drops. This negative moderator temperature coefficient is a passive safety feature required for all commercial LWRs.

Heavy water absorbs roughly 660 times fewer thermal neutrons per atom than light water (sigma_a_D2O ≈ 0.001 barns vs sigma_a_H2O ≈ 0.664 barns). This very low absorption means f stays high even at the large M/F ratios needed to keep p high with natural uranium (which has only 0.72% fissile U-235). Light water absorbs far more thermal neutrons, so the fuel must be enriched to at least 3% U-235 to compensate and maintain sufficient k∞.

The infinite-dilution resonance integral of U-238 is approximately 277 barns, valid when every U-238 atom sees the full unshielded neutron flux. In a real fuel pin the outer layer of U-238 atoms resonance-absorbs incoming neutrons first, shielding the interior atoms from the resonance flux. This self-shielding effect reduces the effective RI to roughly 10-30 barns for typical LWR fuel pin geometries. Heterogeneous lattices therefore have higher p than a homogeneous mixture with the same M/F ratio, which is one key advantage of separating fuel and moderator.

Higher enrichment increases the thermal absorption cross-section of the fuel, sigma_a_fuel = e×678 + (1-e)×2.73 barns, because U-235 absorbs much more strongly than U-238. A larger sigma_a_fuel shifts R_opt to a higher value (the fuel can dominate absorption at a higher M/F ratio before f drops too much) and also raises the maximum value of f at R_opt. Highly enriched uranium therefore tolerates more moderator before becoming over-moderated.

The slowing-down power (SDP) is the product xi × Sigma_s, where xi is the mean logarithmic energy decrement per collision and Sigma_s is the macroscopic scattering cross-section. A high SDP means the moderator decelerates neutrons rapidly, reducing time spent in the U-238 resonance energy region and raising p. H2O has a high SDP (0.920 × 49.2 = 45.3 barns per molecule), graphite has a low SDP (0.158 × 4.74 = 0.749 barns per atom). This is why graphite reactors require very large moderator volumes.

Yes. Select Graphite as the moderator type. Because graphite has a much lower slowing-down power than water, the parameter A = I_eff / (xi × sigma_s) is much larger, pushing R_opt to several thousand C atoms per U atom. A typical graphite reactor uses R of 300-3000. Natural uranium graphite reactors (like the UK Magnox reactors) operate at large lattice pitches to achieve the high M/F ratio needed for acceptable p values.

The formulas p = exp(-A/R) and f = sigma_a_F / (sigma_a_F + R × sigma_a_M) assume a perfectly mixed (homogeneous) fuel and moderator. Real reactors are heterogeneous: fuel rods are physically separated from the moderator. The key difference is that heterogeneous designs achieve higher p at the same M/F ratio because fuel self-shielding reduces the effective resonance integral. The optimum M/F ratio is therefore lower in real lattices than the homogeneous model predicts, but the qualitative shape of the p×f curve is the same.

This calculator estimates k∞ = eta × epsilon × p × f using user-supplied values of the reproduction factor eta and fast fission factor epsilon. For a 3.5% enriched UO2 fuel in a light water lattice, typical values are eta = 2.07 and epsilon = 1.05. The p and f values come directly from the M/F ratio and nuclear constants. This gives k∞ approximately 1.3-1.7 for fresh LWR fuel at operating conditions, consistent with published lattice physics results.

The criticality status compares k∞ to unity. Supercritical (k∞ > 1.001) means the lattice can theoretically sustain a chain reaction. Subcritical (k∞ < 0.999) means it cannot. Critical (k∞ ≈ 1) is the boundary. Note that k∞ is for an infinite lattice with no neutron leakage. A real finite reactor needs k_eff = k∞ × P_NL where P_NL (non-leakage probability) is 0.90-0.99 for power reactors, so the actual operating k∞ must exceed 1 by a margin sufficient to compensate for leakage.