Thermal, Epithermal and Fast Neutron Spectra

Neutron capture cross section for ¹¹³Cd for a range of neutron energies. The 'Cadmium cut-off' energy is defiend to be at 0.0253 eV.

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Properties of Key Fissile and Breeder Nuclei

Key nuclear data for the nuclides ²³²Th, ²³³U, ²³⁵U, ²³⁸U, ²³⁹Pu and ²⁴¹Pu is provided in the table below. To complement the table, a brief description of practice for defining and quantifying thermal and epithermal neutron fluxes is first given.

In the literature, the ‘cadmium cut-off’ energy defines the boundary between terming a neutron to be in the thermal or epithermal energy regime. The definition arises from ¹¹³Cd, which has a particularly large neutron absorption coefficient below neutron energies of 0.55 eV, above this energy the probability of neutron absorption rapidly reduces, see the Figure to the right. The probability of capturing thermal neutrons is sometimes quoted at the velocity 2200 ms^-1. This corresponds to the mode neutron velocity for a Maxwellian energy distribution at 20 °C ($E = 0.0253$ eV).

The (n,γ) reaction rate can be described as,$r = \phi_{th} \sigma_0 + \phi_{epi} I_0(\alpha)$ for thermal and epithermal neutrons combined. Where the rate of neutron absorption is, $r$ (s^-1); $\phi_{th}$ and $\phi_{epi}$ are the conventional thermal and epithermal neutron fluxes (cm^-2), respectively; $\sigma_0$ is the neutron capture cross-section at 2200 ms^-1 (barns); $I_0$ is the infinite dilution resonance integral (cm²); and $\alpha$ is the epithermal flux distribution parameter [Verhijke 2000].

Whereas the thermal energy spectrum can be described simply by the neutron flux and the corresponding cross-sections for absorption, describing the absorption of epithermal neutrons is more complex as a result of the presence of resonances. The conventional epithermal neutron flux term, $\phi_{epi}$, is specifically defined as the integrated flux of all neutrons whose energies lie in the epithermal range. The equation that determines the infinite dilution resonance equation (which is equivalent to the thermal neutron absorption cross-section) includes a term that makes the magnitude of its cross-section inversely proportional to the energy of a neutron (this corrects for the non-uniform neutron flux distribution for different energies, not accounted for by the $\phi_{epi}$ term). In real reactors the neutron flux is not only non-uniform, but also not an ideal distribution, the energy dependence correction to $\phi_{epi}$ is therefore also modified, this time by the epithermal flux distribution term, $\alpha$ [Yucel and Karadag 2004, Karadag and Yucel 2004].

The notation in the table belowis such that $\sigma_a$ is the cross-section for thermal $(t)$ or fast $(f)$ neutron absorption, of which $\sigma_f$ fissions and $\sigma_c$ captures; epithermal cross-sections use the same notation style, replacing $\sigma$ with $RI$; $\bar{\nu}$ is the mean number of neutrons emitted by a fission event; $\eta$ is the net number of fission-neutrons yielded per neutron absorbed; $\beta$ is the fraction of those yielded neutrons that are emitted only after the β^-decay of a fission fragment or one of its daughters, they are hence the delayed neutron fraction. In addition to the data presented in the table below a series of figures are given in the section on selected nuclear data (below), these show evaluated nuclear data for a range of neutron energies regarding capture and fission cross-sections and fission neutron yields. The data is taken from the National Nuclear Data Center.

Neutronic properties of fissile ²³³U, ²³⁵U and ²³⁹Pu and fertile ²³²Th and ²³⁸U for thermal (average energy (0.05 eV) in a Maxwellian distribution at 300°C), epithermal and fast neutrons. The use of the terms, $\alpha$ should not be confused with the epithermal flux distribution term, also denoted $\alpha$ [Kazimi et al. 1999, Rubbia et al. 1995, NNDC 2010].

Nuclear Data	232Th	233U	235U	238U	239Pu	241Pu
Cross-section (barns)
Thermal
Absorption $\sigma_a(t)$	4.62	364	405	1.73	1045	1121
Fission $\sigma_f(t)$	0	332	346	0	695	842
$\alpha(t) = \sigma_c(t)/ \sigma_f(t)$		0.096	0.171		0.504	0.331
$\eta_{th}$		2.26	2.08		1.91	2.23
Epithermal
Infinite dilution epithermal resonance integral ($I_0$)	0	764	275	0	301
$I_{0a}$	85.6	882	405	278	474	740
$I_{0f}$		746	272		293	571
$\alpha = I_{0c}/I_{0f}$		0.182	0.489		0.618	0.296
$\eta_{epi}$		2.10	1.63		0.618	0.296
Fast (averaged)
Absorption $\sigma_a(f)$	0.317	2.948	2.321	0.345	2.213	2.948
Fission $\sigma_f(f)$	0.0086	2.684	1.814	0.037	1.751	2.426
$\alpha(f) = \sigma_c(f)/ \sigma_f(f)$	36.04	0.0096	0.280	8.414	0.264	0.215
Approx. yield for any neutron energy
Total neutron yield $\bar{\nu}$	2.16	2.48	2.43	2.54	2.87	2.97
Delayed neutron fraction $\beta$	0.026	0.0031	0.0069	0.017	0.0026	0.0050
Fission energy yield (MeV) (excluding neutrino)	188.5	191.0	193.5	198.0	198.8	202.0

The data in the above tableshows that when exposed to thermal neutrons ²³⁹Pu and ²⁴¹Pu absorb them at approximately 2.5 times the rate that ²³³U and ²³⁵U do. The breeder nuclei, ²³²Th and ²³⁸U, absorb them at a rate 2-3 orders of magnitude less. ²³⁹Pu and ²⁴¹Pu emit the most neutrons (on average) during fission, however their neutron capture/fission ratio is poor compared to ²³³U and ²³⁵U. Among the presently considered nuclides, ²³³U emits the most neutrons during fission that will go on to cause another fission reaction. This continues to be the case for an epithermal neutron flux. The capture/fission ratio for all nuclei is less favourable in the epithermal energy regime compared to thermal, however the ratio reduces the least for ²³³U.

The cross-sections for neutron absorption reduces by orders of magnitude for the fissionable isotopes listed above. Small quantities of the breeder nuclei will fission. ²³³U maintains a near identical ratio for captures-fission, however for ²³⁵U, ²³⁹Pu and ²⁴¹Pu the ratio is less good. The trend for other isotopes of these elements and also for minor actinides tends to be that the capture-fission ratio improves in the fast spectrum.

The delayed neutron fraction is a significant property when gauging the controllability of a reactor. Increasing the delayed neutron fraction correspondingly reduces the rate of change of reactivity in the core, thus allowing for more time to intervene and return the core reactivity to the desired steady state. ²³⁵U has the largest delayed neutron fraction among the presently considered nuclides, the ²³³U delayed neutron fraction is less than half that of ²³⁵U.Per fission, 2.5 MeV less is released for ²³³U than for ²³⁵U.

Between the two breeder nuclides, ²³²Th has a smaller epithermal resonance cross-section. For nuclei in this energy regime, ²³²Th is therefore less sensitive than ²³⁸U to negative Doppler reactivity feedback.

Verheijke, M.L., 2000, “On the Relationship between the Effective Resonance Energy and the Infinite Dilution Resonance Integral for (n,γ) Reactions”, Journal of Radioanalytical and Nuclear Chemistry, 246, pp. 161-163

Yucel, H. and Karadag, M., 2004, “Experimental determination of the α-shape factor in the 1/E^{1 + α} epithermal-isotopic neutron source-spectrum by dual monitor method”, Annals of Nuclear Energy 31, pp. 681-695

Karadag, M. and Yucel, 2004, H. “Measurement of thermal neutron cross-section and resonance integral for ¹⁸⁶W(n,γ)¹⁸⁷W reaction by the activation method using a single monitor”, Annals of Nuclear Energy 31, pp. 1285-1297

Kazimi, M.S., Czerwinski, K. R., Driscoll, M.J., Hejzlar, P. and Meyer, J.E., 1999, “On the use of Thorium in Light Water Reactors” Massachusetts Institute of Technology, USA, MIT-NFC-TR-016

NNDC, Updated October 2009, “σigma Evaluated Nuclear Data Files (ENDF) Retrieval and Plotting” Version 3.1, National Nuclear Data Center, USA. http://www.nndc.bnl.gov/sigma/ [Accessed 9th March 2010]

Rubbia, C., Buono, S., Gonzalez, E., Kadi, Y., Rubio, J.A., 1995, “A realistic Plutonium Elimination Scheme with Fast Energy Amplifiers and Thorium‑Plutonium Fuel” European Organisation for Nuclear Research, CERN/AT/95‑53 (ET)

Selected Nuclear Data

Neutron capture cross-section data for a range of neutron energies.

The data for all of the following figures has been taken from the National Nuclear Data Center, Evaluated Nuclear Data Files [NNDC 2010].

Fission cross-section data for a range of neutron energies.

ENDF experimental data for prompt and delayed neutrons emitted per fission as a function of neutron energy.

NNDC, Updated October 2009, “&simga;igma Evaluated Nuclear Data Files (ENDF) Retrieval and Plotting” Version 3.1, National Nuclear Data Center, USA.