# Thermal, Epithermal and Fast Neutron Spectra

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## General Properties of Key Fissile and Breeder NucleiEdit

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.

Nuclear Data

232Th

233U

235U

238U

239Pu

241Pu

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].

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/E1 + 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 DataEdit

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

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