Neutron Flux Spectra | Definition & Types

Neutron flux spectrum represents the distribution of neutron energies within some medium. From the physics point of view, the main differences among reactor types arise from differences in their neutron energy spectra.

The basic classification of nuclear reactors is based upon the average energy of the neutrons, which cause the bulk of the fissions in the reactor core. From this point of view, nuclear reactors are divided into two categories:

Thermal Reactors. Almost all of the current reactors built to date use thermal neutrons to sustain the chain reaction. These reactors contain neutron moderator that slows neutrons from fission until their kinetic energy is more or less in thermal equilibrium with the atoms (E < 1 eV) in the system.
Fast Neutron Reactors. Fast reactors contain no neutron moderator and use less-moderating primary coolants because they use fast neutrons (E > 1 keV) to cause fission in their fuel.

thermal vs. fast reactor neutron spectrum

Comparison of neutron spectra in a typical LWR and a sodium-cooled fast breeder reactor. Note that, the fast reactor spectrum is highly affected by the elastic scattering cross-section of used coolant.

The main differences between these two types are, of course, in neutron cross-sections, that exhibit significant energy dependency. It can be characterized by capture-to-fission ratio, which is lower in fast reactors. There is also a difference in the number of neutrons produced per one fission, which is higher in fast reactors than in thermal reactors. These very important differences are caused primarily by differences in neutron fluxes. Therefore it is very important to know detailed neutron energy distribution in a reactor core.

Classification of neutrons according to their kinetic energies

Cold Neutrons (0 eV; 0.025 eV). Neutrons in thermal equilibrium with very cold surroundings such as liquid deuterium. This spectrum is used for neutron scattering experiments.
Thermal Neutrons. Neutrons in thermal equilibrium with a surrounding medium. Most probable energy at 20°C (68°F) for Maxwellian distribution is 0.025 eV (~2 km/s). This part of the neutron’s energy spectrum constitutes the most important spectrum in thermal reactors.
Epithermal Neutrons (0.025 eV; 0.4 eV). Neutrons of kinetic energy are greater than thermal. Some reactor designs operate with an epithermal neutron spectrum. This design allows reaching a higher fuel breeding ratio than in thermal reactors.
Neutrons of kinetic energy below the cadmium cut-off energy (~0.5 eV) are strongly absorbed by 113-Cd.
Source: JANIS (Java-based nuclear information software) www.oecd-nea.org/janis/
Cadmium Neutrons (0.4 eV; 0.5 eV). Neutrons of kinetic energy below the cadmium cut-off energy. One cadmium isotope, ¹¹³Cd, absorbs neutrons strongly only if they are below ~0.5 eV (cadmium cut-off energy).
Epicadmium Neutrons (0.5 eV; 1 eV). Neutrons of kinetic energy above the cadmium cut-off energy. These neutrons are not absorbed by cadmium.
Slow Neutrons (1 eV; 10 eV).
Resonance Neutrons (10 eV; 300 eV). The resonance neutrons are called resonance for their special behavior. At resonance energies, the cross-sections can reach peaks more than 100x higher than the base value of the cross-section. At these energies, the neutron capture significantly exceeds the probability of fission. Therefore it is very important (for thermal reactors) to quickly overcome this range of energy and operate the reactor with thermal neutrons, increasing the probability of fission.
Intermediate Neutrons (300 eV; 1 MeV).
Fast Neutrons (1 MeV; 20 MeV). Neutrons of kinetic energy greater than 1 MeV (~15 000 km/s) are usually named fission neutrons. These neutrons are produced by nuclear processes such as nuclear fission or (ɑ,n) reactions. The fission neutrons have a Maxwell-Boltzmann energy distribution with mean energy (for ²³⁵U fission) of 2 MeV. Inside a nuclear reactor, the fast neutrons are slowed down to thermal energies via neutron moderation.
Relativistic Neutrons (20 MeV; ->)

The reactor physics does not need this fine division of neutron energies. The neutrons can be roughly (for purposes of reactor physics) divided into three energy ranges:

Thermal neutrons (0.025 eV – 1 eV).
Resonance neutrons (1 eV – 1 keV).
Fast neutrons (1 keV – 10 MeV).

Even most reactor computing codes use only two neutron energy groups:

Slow neutrons group (0.025 eV – 1 keV).
Fast neutrons group (1 keV – 10 MeV).

Region of Fast Neutrons

The first part of the neutron flux spectrum in thermal reactors is the region of fast neutrons. All neutrons produced by fission are born as fast neutrons with high kinetic energy.

At first, we have to distinguish between fast neutrons and prompt neutrons. The prompt neutrons can sometimes be incorrectly confused with the fast neutrons. But there is an essential difference between them. Fast neutrons are categorized according to kinetic energy, while prompt neutrons are categorized according to the time of their release.

Most of the neutrons produced in fission are prompt neutrons. Usually, more than 99 percent of the fission neutrons are prompt neutrons. Still, the exact fraction is dependent on the nuclide to be fissioned and is also dependent on an incident neutron energy (usually increases with energy). For example, fission of ²³⁵U by thermal neutron yields 2.43 neutrons, of which 2.42 neutrons are the prompt neutrons, and 0.01585 neutrons (0.01585/2.43=0.0065=ß) are the delayed neutrons.

Characteristics of Prompt Neutrons

See also: Delayed Neutrons

The presence of delayed neutrons is perhaps the most important aspect of the fission process from reactor control.

Delayed neutrons are emitted by neutron-rich fission fragments that are called delayed neutron precursors.

These precursors usually undergo beta decay, but a small fraction of them are excited enough to undergo neutron emission.

The emission of neutrons happens orders of magnitude later compared to the emission of the prompt neutrons.

About 240 n-emitters are known between ⁸He and ²¹⁰Tl. About 75 of them are in the non-fission region.

It is suggested to group together the precursors based on their half-lives to simplify reactor kinetic calculations.

Therefore delayed six delayed neutron groups traditionally represent neutrons.

Neutrons can also be produced in (γ, n) reactions (especially in reactors with heavy water moderator), and therefore they are usually referred to as photoneutrons. Photoneutrons are usually treated no differently than regularly delayed neutrons in the kinetic calculations.

The total yield of delayed neutrons per fission, v_d, depends on:
- An isotope that is fissioned.
- The energy of a neutron induces fission.

Variation among individual group yields is much greater than variation among group periods.

In reactor kinetic calculations, it is convenient to use relative units as delayed neutron fraction (DNF).

At the steady-state condition of criticality, with k_eff = 1, the delayed neutron fraction is equal to the precursor yield fraction β.

In LWRs, the β decreases with fuel burnup. This is due to isotopic changes in the fuel.

Delayed neutrons have initial energy between 0.3 and 0.9 MeV with an average energy of 0.4 MeV.

Depending on the type of the reactor, and their spectrum, the delayed neutrons may be more (in thermal reactors) or less effective than prompt neutrons (in fast reactors). The effectively delayed neutron fraction – β_eff must be defined to include this effect into the reactor kinetic calculations.

The effectively delayed neutron fraction is the product of the average delayed neutron fraction and the importance factor β_eff = β . I.

The weighted delayed generation time is given by τ = ∑_iτ_i . β_i / β = 13.05 s, therefore the weighted decay constant λ = 1 / τ ≈ 0.08 s^-1.

The mean generation time with delayed neutrons is about ~0.1 s, rather than ~10^-5 as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.

Their presence completely changes the dynamic time response of a reactor to some reactivity change, making it controllable by control systems such as the control rods.

Delayed Neutrons Energy Spectra

The key properties of delayed neutrons, which are very important for the nuclear reactor design, also belong to delayed neutron energy spectra. The energy spectra of the delayed neutrons are the poorest known of all input data required because it is very difficult to measure them.

Depending on the type of the reactor, and their spectrum, the delayed neutrons may be more (in thermal reactors) or less effective than prompt neutrons (in fast reactors). To include this effect into the reactor kinetic calculations, the effectively delayed neutron fraction – β_eff must be defined.

Parameters of Delayed Neutrons — Delayed neutrons are traditionally represented by six delayed neutron groups, whose yields and decay constants (λ) are obtained from nonlinear least-squares fits to experimental measurements.

Prompt Neutron Energy Spectra - Dependence on fissioning nucleus. — Prompt Neutron Energy Spectra – Dependence on fissioning nucleus.
Source: Madland, David G., New Fission-Neutron-Spectrum Representation for ENDF, LA-9285-MS, April 1982.

The vast of the prompt neutrons and even the delayed neutrons are born as fast neutrons (i.e., kinetic energy higher than > 1 keV). But these two groups of fission neutrons have different energy spectra. Therefore they contribute to the fission spectrum differently. Since more than 99 percent of the fission neutrons are the prompt neutrons, it is obvious that they will dominate the entire spectrum.

Therefore the fast neutron spectrum can be described by the following points:

Almost all fission neutrons have energies between 0.1 MeV and 10 MeV.
The mean neutron energy is about 2 MeV.
The most probable neutron energy is about 0.7 MeV.

The fast neutron spectrum can be approximated by the following (normalized to one) distribution:

The neutrons released during fission with an average energy of 2 MeV in a reactor on average undergo many collisions (elastic or inelastic) before they are absorbed. As a result of these collisions, they lose energy so that the reactor spectrum is not identical to the fission spectrum. It is always ‘softer’ than the fission spectrum. The fact is that the fission spectrum is part of the reactor spectrum.

Thermal vs. Fast Reactors

The spectrum of neutron energies produced by fission varies significantly with certain reactor designs. The previous figure illustrates the difference in neutron flux spectra between a thermal reactor and a fast breeder reactor. Note that the neutron spectra in fast reactors also vary significantly with a given reactor coolant. For example, gas-cooled reactors have significantly harder neutron spectra than neutron spectra in sodium-cooled reactors. The main differences in the curve shapes may be attributed to the neutron moderation or slowing down effects.

Intermediate Energy Region

In fast neutron reactors, there is an insignificant number of neutrons that can reach the thermal or intermediate energies. On the other hand, in thermal reactors, neutrons have to be moderated to profit from the larger cross-sections at lower energies. In these reactors, the neutrons are predominantly absorbed only when they are in kinetic equilibrium with the thermal movement of the surrounding atomic nuclei.

There is an intermediate energy region (1 eV to 0.1 MeV) between the fast and thermal regions. For this region, the 1/E dependency is typical. If the energy (E) is halved, the flux Ф(E) doubles. This 1/E dependence is caused by the nature of the slowing down process. In this region, Σ_s(E) varies only a little. The elastic scattering removes a constant fraction of the neutron energy per collision (see logarithmic energy decrement), independent of energy. Therefore the neutron loses larger amounts of energy per collision at higher energies than at lower energies. The neutrons lose a constant fraction of kinetic energy per collision causes the energy-dependent neutron flux to “pile up” at lower energies.

Thermal Region

In the thermal region, the neutrons achieve thermal equilibrium with the atoms of the moderator material (in
the idealized situation where no absorption is present). That is, the neutrons behave as a strongly diluted gas in thermal equilibrium. These neutrons do not all have the same energy. There is a distribution of energies, usually known as the Maxwell-Boltzmann distribution:

in which k is the Boltzmann constant (k = 8.52⋅10^-5 eV/K). For the thermal neutron flux density, it thus holds that:

in which n₀ is the total thermal neutron density.

The most probable energy (for which the spectrum is maximum) is E = kT. At room temperature, this is 0.025 eV. The velocity corresponding with this energy is 2200 m/s. This energy is particularly important since reference data, such as nuclear cross-sections, are tabulated for a neutron velocity of 2200 m/s.

At a reactor temperature of 320°C (593 K), a value characteristic for PWRs, the most probable velocity is 3100 m/s, and the corresponding energy is 0.051 eV.

But this distribution only holds for complete thermal equilibrium. Unfortunately, some absorption will always be present in a nuclear reactor, and this equilibrium will never be complete. As a result of 1/v behavior, low energy neutrons are absorbed preferentially, which leads to a shift of the spectrum to higher energies.

On the other hand, the neutron leakage has an opposite effect. With decreasing energy the diffusion coefficient D decreases as a result of the increasing cross-sections, therefore the neutron leakage preferentially removes neutrons with higher energies. This effect strongly depends on the size of the multiplying system, but in most cases it is much less important than the presence of absorption.

Thermal reactor neutron spectrum.

Reactors with Spectral Shift Control

From the physics point of view, the main differences among reactor types arise from differences in their neutron energy spectra. The basic classification of nuclear reactors is based upon the average energy of the neutrons, which cause the bulk of the fissions in the reactor core. The neutron energy spectrum also influences fuel breeding. As was written in LWRs, the fuel temperature also influences the rate of nuclear breeding (the breeding ratio).

Instead of increasing fuel temperature, a reactor can be designed with so-called “spectral shift control”. The main idea of the spectral shift is based on the neutron spectrum shifting from the resonance energy region (with lowest p – resonance escape probability) at the beginning of the cycle to the thermal region (with the highest p – resonance escape probability) at the end of the cycle. In pressurized water reactors, chemical shim (boric acid) and burnable absorbers are used to compensate for an excess of reactivity of reactor core along the fuel burnup (long-term reactivity control). From the neutronic utilization aspect, compensation by absorbing neutrons in poison is not ideal because these neutrons are lost. For better utilization of the neutrons, fertile isotopes can absorb these neutrons to produce fissile nuclei (in radiative capture). These fissile nuclei would contribute to obtaining more energy from the fuel.

The spectral shift method can be used to offset the initial excess of reactivity. There are many different ways of such regulation in the core. Spectral shift control can be performed by coolant density variation during the reactor cycle or by changing the moderator-to-fuel ratio with some mechanical equipment. Some of the current advanced reactor designs use for spectrum shift movable water displacers to change the moderator-to-fuel ratio. A decrease in reactivity caused by fuel burnup is simply compensated by the withdrawal of these movable water displacers while changing the moderator-to-fuel ratio. This makes it possible to exclude chemical shim from the operational modes completely. This method promises significant natural uranium savings (up to 50% of natural uranium).

See also: Teplov, P.; Chibiniaev, A.; Bobrov, E.; Alekseev, P. The main characteristics of the evolution project VVER-S with spectrum shift regulation. 2014.

References:

Nuclear and Reactor Physics:

J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
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U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
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E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

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See above:

Neutron Nuclear Reactions