Moderating Power and Moderating Ratio — From the Derivation of ξ to the Purity Sensitivity of Heavy Water, by Calculation

Principle and mechanism

Step 1: energy distribution after one collision, and α

Assume elastic scattering on a nucleus of mass number A at rest, isotropic in the centre-of-mass frame. The energy after scattering, E′, is then uniformly distributed between αE and E.

• α = [(A−1)/(A+1)]²: the minimum energy ratio, reached in a head-on collision (E′_min = αE)
• Mean energy after scattering ⟨E′⟩ = (1+α)E/2; mean loss ⟨ΔE⟩ = (1−α)E/2

Hydrogen has α = 0, so a neutron can lose all its energy in one collision and loses half on average. Carbon has α = 0.716, so even the maximum loss is 28 % and the average only 14 %.

Step 2: deriving ξ

ξ is the average decrease in the logarithm of energy per collision.

ξ = ⟨ln(E/E′)⟩ = ∫ ln(E/E′) · dE′/[(1−α)E] (from αE to E) = 1 + α·lnα/(1−α)

In the model of a free nucleus at rest with isotropic scattering, E drops out and ξ depends on A only. Lethargy therefore increases by the same average amount in every collision, and the number of collisions is estimated as n ≈ ln(E₀/E)/ξ. From 2 MeV to 0.025 eV, ln(8×10⁷) = 18.2, so n = 18.2/ξ. For large A, ξ ≈ 2/(A + 2/3) (error below 1 % for A ≥ 10; not used for hydrogen).

"Mean fractional loss" and ξ are different quantities. For hydrogen the fractional loss is 0.5 while ξ is 1: the logarithmic average gives more weight to collisions with large losses.

Step 3: ξ of a compound and the moderating power

For a molecular moderator, ξ is the average of the elemental ξ values weighted by scattering cross section.

ξ̄ = Σ ξ_iΣ_s,i / Σ Σ_s,i

For H₂O, with two σ_s(H) = 20.4 b and σ_s(O) = 3.76 b: (40.8×1 + 3.76×0.120)/44.56 = 0.926 (textbook table 0.920). Taking a simple average of the elemental values, (1+1+0.12)/3 = 0.71, is wrong. For D₂O: (6.78×0.725 + 3.76×0.120)/10.54 = 0.509.

The moderating power multiplies this by Σ_s. With molecular number densities of 3.34×10²² (H₂O) and 3.32×10²² cm⁻³ (D₂O), ξΣ_s comes out at 1.38 and 0.178 cm⁻¹ respectively, consistent with the textbook values of 1.35 and 0.176.

Step 4: why the moderating ratio means "absorption versus moderation"

In a 1/E slowing-down spectrum the slowing-down density is q = ξΣ_s·φ(E)·E, and the absorption rate per unit lethargy is Σ_a·φ(E)·E. Their ratio is

(absorption per unit lethargy) / (slowing down through unit lethargy) = Σ_a / ξΣ_s = 1 / moderating ratio

A moderating ratio of 5,670 means one absorption for every 5,670 neutrons slowing down through one unit of lethargy. The resonance escape probability uses the same denominator: in the homogeneous NR approximation p = exp[−N_F·I_eff/(ξΣ_s)], so the larger the moderating power, the larger p.

Step 5: the scorecard (values from one textbook table)

Calculated directly from 2200 m/s absorption cross sections (H 0.332 b, D 0.000519 b, O 0.00019 b), the moderating ratio is 62 for light water and about 4,400 for pure heavy water. The difference appears to come from different cross-section data sets and averaging methods (2200 m/s values versus Maxwellian averages, etc.), so compare values only within one table.

What it means in a plant

Why heavy water has the best moderating ratio yet needs a large core

The ratio of moderating powers is 0.176/1.35 = 0.13: to gain the same lethargy, a neutron must travel about 7.7 times farther than in light water. This is because σ_s of D (3.4 b) is one sixth that of H (20.4 b) and ξ is also smaller, at 0.725. The Fermi age τ = ∫D/(ξΣ_s)du grows too, so the slowing-down length √τ is 5.2 cm in light water and 11.4 cm in heavy water. Representative moderator-to-fuel volume ratios are 1.95 for a PWR and 17 for a PHWR, with power densities of 102 versus 7.7 MW/m³ (Lewis, Table 4.1). That is why a CANDU has a large calandria and a large heavy-water inventory. In return, absorption is very small, so natural uranium can be used. A light-water reactor is the opposite: its large moderating power allows a tight lattice, but its high absorption requires enrichment, and it is designed on the under-moderated side.

When light water gets into heavy water: purity sensitivity

With the light-hydrogen fraction x = N_H/(N_H + N_D) (oxygen excluded; with D purity in at%, x = 100 − purity), the moderating ratio of the mixture is

MR(x)/MR(0) ≈ (1 + 6.68x)/(1 + 540x)

Per water molecule, the absorption contribution of H₂O is about 540 times that of D₂O, and its moderating contribution (Σξσ_s) about 7.7 times. At x = 1 at%, the moderating contribution rises slightly, by a factor of 1.067, but the absorption contribution becomes 6.4 times larger, so the moderating ratio drops to 0.167 of its pure value. Calculated values on a 2200 m/s basis are about 4,370 pure → 2,860 at 0.1 at% → 1,890 at 0.25 at% → 730 at 1 at% → 133 at 10 at%. The moderating ratio halves at about 0.19 at% H. A light-water content of 0.25 wt% corresponds to a mole fraction of 0.278 %.

CANDU moderator is therefore kept at reactor-grade purity (roughly 99.75 wt% D₂O or higher). In one lattice calculation, reactivity fell by about 35 mk (fresh fuel) per 1 at% drop in D purity. Loss of purity is negative reactivity, so it is conservative from a control standpoint, but it directly erodes fuel economy. In operation, isotopic purity is analysed periodically and managed with upgrading (re-enrichment) facilities and leak monitoring (heat exchangers, etc.).

Regulatory and design criteria

• For example, US 10 CFR 50 Appendix A, GDC 11, 26 and 28: require inherent reactivity feedback and reactivity control. The choice of moderator and the design of the moderator-to-fuel ratio (under-moderation) follow from these requirements.
• Nuclear Suppliers Group Guidelines, INFCIRC/254 Part 1: deuterium and heavy water with a D/H atomic ratio exceeding 1:5,000 are controlled as trigger-list non-nuclear materials for reactors, because heavy water with its high moderating ratio makes natural-uranium reactors possible.

Easily confused pairs

Key points at a glance

1. α = [(A−1)/(A+1)]²; E′ after scattering is uniform between αE and E → mean loss (1−α)E/2.
2. ξ = ⟨ln(E/E′)⟩ = 1 + α lnα/(1−α) ≈ 2/(A+2/3); H = 1, D 0.725, C 0.158.
3. Collisions n = ln(E₀/E)/ξ = 18.2/ξ (2 MeV → 0.025 eV): H₂O about 20, D₂O about 36, graphite about 115.
4. Moderating power ξΣ_s [cm⁻¹]: lethargy gain per unit distance → light water 1.35 > heavy water 0.176 > graphite 0.060.
5. Moderating ratio ξΣ_s/Σ_a: Σ_a/ξΣ_s = absorption per lethargy / slowing down → heavy water 5,670 > graphite 192 > light water 71.
6. ξ of a compound: scattering-cross-section-weighted average (H₂O 0.92).
7. Light water: largest moderating power → compact core; high absorption → enrichment needed.
8. Heavy water: largest moderating ratio → natural uranium possible; small moderating power → large core and heavy-water inventory.
9. Purity: 0.19 at% light hydrogen halves the moderating ratio → reactor-grade purity control.
10. Which index? "Best moderator" is ambiguous: light water leads on moderating power, heavy water on moderating ratio. Quote moderating-ratio numbers from a single table, since heavy-water values in the literature range from 5,670 to 21,000 depending on purity and the Σa basis.