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4.7.2 Neutron cross-sections

If n(E) dE is the number of neutrons per unit volume of a material having an energy between E and E + dE, then the neutron flux, Φ, is defined as Φ(E) = n(E)υ, where υ is the neutron velocity. The number of neutrons giving a particular type of reaction per unit volume per unit time in a material containing N nuclei per unit volume is ∫ NΦ(E)σ_R(E) dE, where σ_R(E) is the nuclear cross-section for the particular type of interaction R. The product Nσ_R(E) is called the macroscopic cross-section, Σ_R, and its reciprocal can be shown to be the mean free path.

After a neutron interacts with a nucleus, charged particles (e.g. a proton or α particle), γ-rays or one or more neutrons may be emitted. In heavy materials fission may take place. These reactions are known as (n, p), (n, α), (n, γ), (n, Xn) [where X has a value typically of 1 to 3] and (n, f) reactions. The (n, γ) reaction is referred to as radiative capture and the corresponding cross-section written as σ_nγ. The (n, n) interaction indicates neutron scattering though it does not mean that the same neutron is emitted as produced the interaction. If after the interaction the nucleus remains in the ground state the event is referred to as elastic scattering and the cross-section is written σ_nn. If it is left in an excited state it is referred to as inelastic scattering and is denoted by (n, n') with the cross-section written as σ_nn′. The fission cross-section is written σ_nf and the total cross-section for all events is written σ_nT. The absorption cross-section, σ_nA, is defined as σ_nA = σ_nT − σ_nn − σ_nn'. The transport cross-section (σ_tr) is defined as σ_nT − bσ_nn where b is the average value of cos ψ and ψ is the angle of neutron scatter in a collision.

The energy range of interest in most applications of nuclear energy (i.e. for fission and fusion reactors) is 10⁻⁵ eV to 20 MeV. Evaluated cross-section values are available in a number of nuclear data files that have been prepared both locally and internationally—see below. In Figures A, B, C, D, E and F a selection of cross-sections is shown. Certain features may be distinguished.

At low energies absorption cross-sections tend to be proportional to 1/υ (i.e. to E^−1/2) where υ is the neutron velocity and for ¹⁰B and ⁶Li this law is good to ±5% up to about 100 keV (see Figures A and C). For light nuclides, scattering is dominated by potential scattering which can be considered to be the deflection of a neutron by the nuclear field without the formation of a compound nucleus, and the cross-section tends to be independent of energy. Departures, however, arise at low energies in crystalline materials (e.g. C and Be) when the neutron wavelength is comparable to the lattice spacing, and in molecular compounds (e.g. H₂ and H₂O) when the collision energy is comparable to the vibrational quanta of the molecular oscillations (see Figure B).

At intermediate energies the cross-sections show sharp maxima at neutron energies at which the reacting particles have the same energy (including binding energy) as one of resonance levels in the compound nucleus. Light nuclei have larger spacings between these resonances than heavier nuclei (cf. Figures C and D). It is thus more probable to find resonances at low energies in heavy nuclei and such resonances can be seen in Figures A and B for U and Pu.

For heavy nuclei having an odd number of neutrons (e.g. ²³⁵U) fission is possible at all neutron energies and such materials are called fissile (see Figures A, D and E) and are used as fuel in fission reactors. For heavy nuclei having an even number of neutrons (e.g. ²³⁸U) fission is only significant above a threshold energy (see Figure F). Since neutron capture in such materials produces a fissile material they are called fertile. See Section 4.7.1 for a further discussion on fission.

All (n, n′) and (n, 2n) reactions have threshold energies. In some light nuclides (n, p) and (n, α) reactions have appreciable cross-sections at all energies but in the heavier nuclides these cross-sections always show an effective threshold (due to the coulomb potential barrier) at an MeV or so, even when the reaction is energetically possible with low energy neutrons (see Figure F).

Because of the strong dependence of cross-section with energy it is usually necessary in practical calculations to make use of group cross-sections which are values averaged over chosen energy intervals and weighted according to the neutron energy spectrum expected in the system, i.e.

Figure D. Cross-section of a heavy nuclide in the slowing down region.(Above 20 eV the resonances are too numerous to be shown correctly on this scale.)

For a general comparison of the cross-section values for the interactions of neutrons in various materials it is often appropriate to consider four energy regions. The highest energy region around 14 MeV which is appropriate to fusion reactors (and for some activation analyses) is considered in section 4.7.4. The other three regions are applicable to fission reactor applications.

At high energies the cross-sections can be averaged over a fission neutron spectrum which can be expressed as Φ(E) = (4/πT³)^1/2E^−1/2 exp(−E/T) where T is typically 1.35 MeV. In the tables the average values have been taken over the energy range 1 keV to 20 MeV, although most of the contribution comes from a much narrower range around 2 MeV.

At intermediate energies the spectrum in a well moderated system is nearly proportional to 1/E and the average cross-section is equal to

Figure E. Non-threshold cross-sections in fast region. (Note different scales on left and right.)

The numerator is referred to as the resonance integral (R.I.) since it gives the cumulative absorption in the various resonances as a neutron slows down from energy E₂ to E₁. For cross-sections it is customary to quote the resonance integrals rather than the average values, and the tables show values where E₁ = 0.5 eV and E₂ = 100 keV though the values are insensitive to the upper limit.

At lower energies, when the neutrons are in thermal equilibrium with their surroundings, Φ_th = E exp(−E/kT_n)/(kT_n)², where T_n is the temperature of the surroundings. If the cross-section varies as 1/ν then the average cross-section in a thermal flux can be shown to be equal to σ₀(πT₀/4T_n)^1/2,

Figure F. Threshold reactions in fast region. (Curves refer to (n, 2n) reactions unless stated otherwise.)
where σ₀ is the cross-section corresponding to an energy kT₀. It is customary to refer cross-sections to the value σ₀ at a standard neutron velocity of 2200 ms⁻¹, which corresponds to a neutron energy kT₀ of 0.025 30 eV and a temperature of 293.59 K. Values of σ₀ have been given in the Table of Nuclides (section 4.6.1) and values averaged over a thermal spectrum with T_n= T₀ are given in the following tables. The departure of these average values from the value of σ₀(π/4)^1/2 is indicative of the departure of the particular cross-section from the 1/ν characteristic. The integrations were taken over the range 0.0001 eV to 1.0 eV although most of the contribution comes from a narrower range around 0.025 eV.

The table of neutron cross-sections gives properties of nuclides which have been grouped according to their function in nuclear reactors. Cross-sections are given in barns (10⁻²⁸ m²). For fissionable materials the number of neutrons produced per fission and denoted by ν are given as average values over the three spectral regions.

In systems which are not well moderated, such as fast nuclear reactors, the above division into three energy regions is not appropriate. There are virtually no thermal neutrons and no region at intermediate energies where the flux is proportional to E⁻¹. Detailed calculations require the neutron energy region to be split into a large number of groups with appropriate equations and corresponding cross-sections for each group and each nuclide present. For rough estimates and comparisons the average cross-sections table can be used for systems typical of a plutonium–uranium oxide fuelled, liquid metal cooled fast reactor with a fertile blanket.

Neutrons can be detected by both active and passive devices. Active detectors include fission chambers, BF₃ ion chambers, proton recoil counters and their response to neutrons is determined by the appropriate cross-section for the particular interaction employed in the device. The table of neutron cross-sections for reactor materials includes materials used in these detectors. The passive detectors contain materials which become radioactive when irradiated in a neutron flux and on removal from the flux the radioactivity can be measured. The activity A(t) (particles s⁻¹) at a time t seconds after the removal of a thin foil from a neutron flux having an energy spectrum Φ(E) in which the foil has been irradiated for T seconds is given by

where V, ρ and A are the volume, density and atomic weight of the foil material respectively, N_A is Avogadro's constant, λ (s⁻¹) is the decay constant of the radioactive species produced by the irradiation and σ_act(E) is the activation cross-section of the material. Values of σ_act averaged over the thermal and fission energy regions are given in the table together with the activation resonance integral. The materials listed either have a predominant resonance or a threshold and may be used to detect neutrons over a narrow energy region. Additional decay properties of the isotopes listed may be found in the Table of Nuclides (section 4.6.1).

Unless otherwise specified the values in the tables are mainly obtained from the JEF-2 library [which is mainly for fission applications] (Nordborg et al, 1991) and the average values were calculated by C. Nordborg and M. Konieczny of the NEA Data Bank. Other libraries that are available include EFF-2 [fusion applications] (Nordborg et al., 1991), EAF-3 [Fusion and other activation applications] (Kopecky et al. (1991)), ENDF/B-VI [fission and fusion applications] (Dunford, 1991) and JENDL-3 [fission and fusion applications] (Kikuchi et al., 1991).

References

Neutron cross-sections
       S. F. Mughabghab, M. Divadeenam and N. E. Holden (1981) Neutron Cross-sections, Volume 1, Neutron
           Resonance Parameters and Thermal Cross-sections, Part A: Z = 1–60, Academic Press, New York.
       S. F. Mughabghab (1984) Neutron Cross-sections, Volume 1, Neutron Resonance Parameters and Thermal
           Cross-sections, Part B: Z = 61–100, Academic Press, New York.
       V. McLane, C. L. Dunford and P. F. Rose (1988) Neutron Cross-sections, Volume 2, Neutron Cross-section
           Curves, Academic Press, New York.
       J. E. Lynn (1968) Theory of Neutron Resonance Reactions, Clarendon Press, Oxford.
       C. Nordborg, H. Gruppelaar and M. Salvatores (1991) ‘Status of JEF and EFF Projects’, Nuclear Data for
           Science and Technology, Springer-Verlag, Berlin, p. 782.
       C. L. Dunford (1991) ‘Evaluated Nuclear Data File ENDF/B-VI’, Ibid., p. 788.
       Y. Kikuchi and members of the JNDC (1991) ‘Japanese Evaluated Nuclear Data Library—JENDL-3’, 
            Ibid., p. 793.
       J. Kopecky, H. Gruppelaar and R. A. Forrest (1991) ‘European Activation File for Fusion’, Ibid., p. 828.
Data centres
   For USA and Canada
       National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY 11973, USA.
   For other OECD countries (e.g. Western Europe and Japan)
       NEA Data Bank, Le Seine Saint-Germain, 12, boulevard des Iles, F-92130 Issy-les-Moulineaux, France.
    For all other countries
       IAEA Nuclear Data Section, P.O. Box 100, A-1400 Vienna, Austria.
In addition there are also some National Data Centres.

Activation detectors for neutron foil detectors

^* Cross-section values from JEF-2.2 corrected where necessary for isomer ratio
^a Cross-section values from EAF-4.0.
^# Cross-section values include isomer contribution.

Neutron cross-sections

 is the average number of neutrons per fission.

Average neutron cross-sections in a fast reactor core and blanket

The values in the table are obtained from the UK Nuclear Data Library.
* (n, α) cross-section.
m indicates cross-section in millibarns.

M.G. Sowerby/R.A. Forrest

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