Macroscopic Cross Section and Atom Density — From N = ρN_A/M to Σ = Nσ and λ = 1/Σ

Principle and mechanism

Number density: single element, compound, mixture

Enrichment is usually given in w/o (mass fraction). For 3 w/o UO₂ the mean atomic weight of uranium follows from 1/M_U = 0.03/235.04 + 0.97/238.05, giving M_U = 237.96 and M_UO₂ = 269.96. The uranium mass fraction is 0.8815 and the U-235 atom fraction 0.0304. At a density of 10 g/cm³, N_UO₂ = 2.231×10²² and N₂₅ = 6.78×10²⁰ cm⁻³.

For water, N = 1.0×6.022×10²³/18.015 = 3.343×10²² molecules/cm³, and the number of hydrogen atoms is twice that. At 2200 m/s, σ_a = 0.664 b per molecule, so Σ_a = 0.0222 cm⁻¹, and σ_s = 103 b per molecule, so Σ_s = 3.44 cm⁻¹ (Lamarsh Table II.3). The scattering mean free path λ_s is 0.29 cm and the absorption mean free path λ_a is 45 cm: at 2200 m/s values, about 155 scatterings (Σ_s/Σ_a) per absorption. The 103 b is larger than the sum for free atoms (about 44 b) because of chemical binding in the thermal range, which is why the molecular cross section is used.

From beam attenuation to the mean free path

The fraction that reacts in a thin layer dx is Σ_t·dx. So dI = −Σ_tI·dx, and the uncollided beam decreases as I(x) = I₀e^{−Σ_t x}. The probability that a neutron reaches x without reacting is e^{−Σ_t x}. The probability that it then reacts for the first time within dx is p(x)dx = Σ_t e^{−Σ_t x}dx, whose integral from 0 to ∞ is 1. The mean distance travelled is λ = ∫x·p(x)dx = 1/Σ_t. After one λ the uncollided beam falls to e⁻¹ = 36.8 %, and after 3λ to 5.0 %.

When scattering is forward-peaked, the actual diffusion distance is longer than λ_s. This is captured by the transport mean free path λ_tr = 1/Σ_tr, with Σ_tr = Σ_t − μ̄₀Σ_s, μ̄₀ = 2/(3A), and the diffusion coefficient D = λ_tr/3. For hydrogen μ̄₀ ≈ 0.67, so the difference is large.

Reaction rate and 1/v and non-1/v corrections

The reaction rate is R = Σφ [reactions/cm³·s], and power is P = E_R·Σ_f·φ·V. Most thermal absorption cross sections are proportional to 1/v. The absorption rate is then F_a = Σ_a(E₀)·nv₀ = Σ_a(E₀)·φ₀, independent of the spectrum, where E₀ = 0.0253 eV, v₀ = 2,200 m/s, and φ₀ is the "2200 m/s flux".

For nuclides that depart from 1/v, the Westcott non-1/v factor g is applied: F_a = g_a(T)·Σ_a(E₀)·φ₀.

(Lamarsh Table 3.2, Westcott AECL-1101)

One-group calculations use the thermal flux φ_T integrated over the whole Maxwellian. In that case φ₀ = (√π/2)(T₀/T)^½·φ_T (T₀ = 293.61 K), so the average cross section is:

Σ̄_a = (√π/2)·g_a(T)·(T₀/T)^½·Σ_a(E₀)

The factor is 0.887 at 20 °C, 0.634 at 300 °C and 0.585 at 400 °C. Multiplying water's Σ_a of 0.0222 by 0.887 gives 0.0197 cm⁻¹, matching the thermal average in Lamarsh Table 5.2. When the reaction rate is written with φ₀, however, only g is applied, without this factor; mixing the two conventions applies the 0.886 factor twice. Also, T is the neutron temperature, not the moderator temperature: absorption hardens the spectrum, so the neutron temperature is higher than the moderator's.

What it means in a plant

• Core design: core analysis codes multiply the nuclide densities N_i (which change with burnup) by group cross sections to build Σ, then perform diffusion or transport calculations. In the four-factor formula, η = νΣ_f^F/Σ_a^F and f = Σ_a^F/Σ_a are both ratios of Σ.
• Moderator: the slowing-down power ξΣ_s and the moderating ratio ξΣ_s/Σ_a are also defined with Σ. Water has a large Σ_s, so its slowing-down length is short.
• Temperature and density feedback: when moderator temperature rises, density falls and N and Σ fall with it, while spectrum hardening also changes the average cross sections. This is where the MTC starts.
• Linear power and burnup: Σ_fφ is the fission rate.

Worked examples

Example 1 — PWR core quantities (PWR: 1,000 MW electric, efficiency 33 %, UO₂ 100,000 kg, ρ 10 g/cm³, 3 w/o, 150 assemblies × 200 rods, core height 3.5 m, 200 MeV per fission, 3×10⁻¹¹ J deposited in the fuel, σ_f 400 b)

The loading is taken as the UO₂ mass (3,000 kg if it is the U mass). The fission rate in (5) is 3.03×10⁹ W ÷ 3.204×10⁻¹¹ J; dividing the full thermal power instead gives 28.9 kW/m.

Example 2 — three-group balance (Na fast reactor, three groups). Group 3 data: σ_γ 0.001 b, φ₃ 3×10¹⁵. Scattering transfers: σ_1→3 0.06 b (φ₁ 6×10¹⁴), σ_2→3 0.18 b (φ₂ 1×10¹⁵).

• N = 0.97×6.0×10²³/29 = 2.01×10²² cm⁻³. The actual atomic weight of Na is 22.99, which would give N = 2.54×10²², but the example uses the stated value.
• F_a = Nσ_γφ₃ = 6.02×10¹⁰; F_out = 0 (lowest group); F_in = N(σ_1→3φ₁ + σ_2→3φ₂) = 4.33×10¹² /cm³·s
• With s₃ = 0, F_diff = −D₃∇²φ₃ = F_in − F_a − F_out = 4.27×10¹² /cm³·s (net leakage)

Example 3 — scattering in natural UO₂ (natural UO₂, ρ 10.6, σ_s: U-238 8.3, U-235 10, O 4.2 b): N_UO₂ = 10.6×6.022×10²³/270.03 = 2.364×10²². The scattering cross section per molecule is 0.0072×10 + 0.9928×8.3 + 2×4.2 = 16.71 b, so Σ_s = 0.395 cm⁻¹ and λ_s = 2.53 cm. The oxygen contribution (0.199) is about the same as that of U-238 (0.195).

Regulatory and design criteria

The items below are examples from US NRC practice and US standards.

• Cross sections themselves are not regulatory values; what is regulated is the validation of nuclear design methods and nuclear data.
• NUREG-0800 SRP 4.3 (Nuclear Design): reviews the validation and uncertainties of calculation methods and cross-section libraries.
• 10 CFR 50 Appendix A, GDC 10, 11, 26 and 28: requirements on the results of nuclear design.
• ANSI/ANS-19.6.1: reload startup physics tests (critical boron concentration, rod worth, temperature coefficient) confirm the predictions.
• Nuclear data versions: the physical definitions are unchanged. Evaluated nuclear data have been updated from ENDF/B-VI → VII.1 (2011) → VIII.0 (2018) → VIII.1 (2024), and thermal cross sections such as the U-235 σ_f differ by a few barns between versions (about 582–587 b).

Easily confused pairs

Key points at a glance

1. Atom density: N = ρN_A/M [cm⁻³]; for a compound N_i = n_i·ρN_A/M_mol.
2. Macroscopic cross section: Σ = Nσ [cm⁻¹], for a mixture Σ = ΣN_iσ_i — the reaction probability per unit length.
3. Beam attenuation: dI = −ΣI·dx → I(x) = I₀e^−Σx.
4. First-collision probability: p(x)dx = Σe^−Σxdx.
5. Mean free path: λ = ∫x·p(x)dx = 1/Σ — 36.8 % remain uncollided after 1λ.
6. Transport mean free path: λ_tr = 1/(Σ_t − μ̄₀Σ_s), μ̄₀ = 2/3A, D = λ_tr/3.
7. Reaction rate: R = Σφ; power P = E_R·Σ_fφV.
8. Thermal corrections: 1/v absorbers use the φ₀ = nv₀ basis as is; non-1/v absorbers multiply by the Westcott g; the φ_T-based average multiplies by (√π/2)(T₀/T)^½.
9. Example: H₂O Σ_s 3.44 cm⁻¹ → λ_s 0.29 cm; Σ_a 0.0222 → λ_a 45 cm.
10. Bookkeeping: count both oxygen atoms in UO₂, keep w/o and a/o apart, convert UO₂ mass to U mass (×0.8815) when needed, and use 1 b = 10⁻²⁴ cm²; if a stated value is non-standard (such as Na at 29 g/mol), note the discrepancy.