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4.7.3    Attenuation of fast neutrons: neutron moderation and diffusion

In materials containing atoms of low atomic mass, neutrons of all energies can lose a significant fraction of their energy in a single elastic collision and such materials are referred to as moderators. In heavy nuclei appreciable energy loss in a collision is only possible at high energies where inelastic scattering can occur. The neutron dose rate from a point source of fast neutrons falls off with distance r approximately as exp(−Σ_remr)/4πr², where Σ_rem depends on the medium where Σ has been defined earlier. This macroscopic cross-section is called the removal cross-section and since all interactions tend to remove energy from the beam its value is not too different from the total macroscopic cross-section (Nσ_nT) of the material, but is slightly lower. This exponential fall off is only approximate and holds less well for media in which hydrogen is the principal fast neutron attenuator. In the table overleaf the removal cross-section refers to a fission neutron source.

In the slowing-down region the average number of collisions, , to slow a neutron from energy E₁ to energy E₂ is equal to ln(E₁/E₂)/ξ, where ξ is the average change per collision in the logarithm of the energy. At energies below that at which scattering becomes entirely elastic, ξ is independent of energy and is approximately equal to 2/(A + ). The spatial distribution of neutrons of energy E₂ which have slowed down from a point source of energy E₁ is of the form exp(−r²/4τ)² where τ is referred to as the Fermi Age and is the mean square distance a neutron migrates in slowing down from E₁ to E₂. It is given by:

where D is the diffusion coefficient and equal to (3Σ_nT − 3bΣ_nn)^{− 1} and b is the average value of cos Ψ where Ψ is the angle of scatter of a neutron in a collision. The table refers to the age of neutrons from a fission source slowing down to an energy of 1.46 eV. This value, which is just above the thermal region, is appropriate to the age determined from the measured spatial distribution of the resonance neutrons detected by indium foils.

The root mean square distance a neutron travels from the position where it is thermalized to the point where it is absorbed is the thermal diffusion length, L, and is equal to

(4/π)^1/4(D_th/Σ_nA)^1/2

where D_th is the value of the diffusion coefficient averaged over the thermal neutron spectrum and Σ_nA is assumed to have a l/υ dependence and is evaluated at an energy kT_n where T_n is the temperature of the medium.

Properties of moderators and shielding materials†
(At 20 °C unless stated otherwise)

† The data in this table are obtained from old sources as they are not needed in modern calculations though they remain valuable in compilations like this: a, experimental value; b, derived from components; c, UK Nuclear Data File (see for example Report AEEW-M 1208); d, taken from ANL 5800; e, ENDF/B data (see BNL 50274). Values not marked are obtained from approximate formulae.
‡ Composition in 10³ kg m⁻³: H, 0.023; O, 1.22; C, 0.0023; Mg, 0.005; Al, 0.078; Si, 0.775; K, 0.03; Ca, 0.1; Fe, 0.03; Na, 0.037.

§ Because of its large absorption resonance integral and its small value of ξ, almost no neutrons slowing down in uranium reach thermal energies.

The slowing down time, t_s, is the time taken for neutrons to slow down from an energy E to the thermal energy E₀ and is independent of E when E » E₀. The mean thermal neutron lifetime, t_th, refers to thermal diffusion before the neutron is absorbed in the medium. In the table values of t_s and t_th have been obtained from the simple formulae and  where the averages are taken over the slowing-down and thermal regions respectively.

M.G. Sowerby / R.A. Forrest

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