Neutron Moderators in Nuclear Reactors

The moderator, which is of importance in thermal reactors, is used to moderate, that is, to slow down, neutrons from fission to thermal energies. The probability that fission will occur depends on incident neutron energy. Physicists calculate with fission cross-section, which determines this probability.

Nuclei with low mass numbers are most effective for this purpose, so the moderator is always a low-mass-number material. In a fast reactor there is no moderator, only fuel and coolant. The moderation of neutrons is undesirable in fast reactors. Commonly used moderators include regular (light) water (roughly 75% of the world’s reactors), solid graphite (20% of reactors) and heavy water (5% of reactors). Beryllium and beryllium oxide (BeO) have been used occasionally, but they are very costly.

Why the moderator is needed?

The probability of the fission U-235 becomes very large at the thermal energies of slow neutrons. This fact implies increase of multiplication factor of the reactor (i.e. lower fuel enrichment is needed to sustain chain reaction)

Why fast reactors don’t need moderator?

Fast reactors use fast neutrons to split uranium or plutonium nuclei. They use higher fuel enrichment to sustain chain reaction. The moderation of neutrons is undesirable in fast reactors.

Neutron Moderator
How does the neutron moderator work?
Source: http://hyperphysics.phy-astr.gsu.edu/
Neutron Moderator
How does the neutron moderator work? Fuel pin
Source: http://hyperphysics.phy-astr.gsu.edu/

Elastic Scattering and Neutron Moderators

To be an effective moderator, the probability of elastic reaction between neutron and the nucleus must be high. In terms of cross-sections, the elastic scattering cross section of a moderator’s nucleus must be high. Therefore, a high elastic scattering cross-section is important, but does not describe comprehensively capabilities of moderators. In order to describe capabilities of a material to slow down neutrons, three new material variables must be defined:

Key properties of neutron moderators:

  • high cross-section for neutron scattering
  • high energy loss per collision
  • low cross-section for absorption
  • high melting and boiling point
  • high thermal conductivity
  • high specific heat capacity
  • low viscosity
  • low activity
  • low corrosive
  • cheap
During the scattering reaction, a fraction of the neutron’s kinetic energy is transferred to the nucleus. Using the laws of conservation of momentum and energy and the analogy of collisions of billiard balls for elastic scattering, it is possible to derive the following equation for the mass of target or moderator nucleus (M), energy of incident neutron (Ei) and the energy of scattered neutron (Es).

equation momentum energy

where A is the atomic mass number.

In case of the hydrogen (A = 1) as the target nucleus, the incident neutron can be completely stopped. But this works when the direction of the neutron is completely reversed (i.e. scattered at 180°). In reality, the direction of scattering ranges from 0 to 180 ° and the energy transferred also ranges from 0% to maximum. Therefore, the average energy of scattered neutron is taken as the average of energies with scattering angle 0 and 180°.

Moreover, it is useful to work with logarithmic quantities and therefore one defines the logarithmic energy decrement per collision (ξ) as a key material constant describing energy transfers during a neutron slowing down. ξ is not dependent on energy, only on A and is defined as follows:

logarithmic energy decrement - equation

For heavy target nuclei, ξ may be approximated by following formula:
the logarithmic energy decrement per collision

From these equations it is easy to determine the number of collisions required to slow down a neutron from, for example from 2 MeV to 1 eV.

Example:
Determine the number of collisions required for thermalization for the 2 MeV neutron in the carbon.
ξCARBON = 0.158
N(2MeV → 1eV) = ln 2⋅106/ξ =14.5/0.158 = 92

Table of average logarithmic energy decrement for some elements

Table of average logarithmic energy decrement for some elements.

For a mixture of isotopes:

the logarithmic energy decrement for mixtures

We have defined the probability of elastic scattering reaction, we have defined the average energy loss during the reaction. The product of these variables (the logarithmic energy decrement and the macroscopic cross section for scattering in the material) is the macroscopic slowing down power (MSDP).

MSDP = ξ . Σs

The MSDP describes the ability of a given material to slow down neutrons and indicates how rapidly a neutron will slow down in the material, but it does not fully reflect the effectiveness of the material as a moderator. In fact, the material with high MSDP can slow down neutrons with high efficiency, but it can be a poor moderator because of its high probability of absorbing neutrons. It is typical, for example, for boron, which has a high slowing down power but is absolutely inappropriate as a moderator.

The most complete measure of the effectiveness of a moderator is the Moderating Ratio (MR), where:

MR  = ξ . Σs/Σa

Table of macroscopic slowing down power MSDP for some materials.

Table of macroscopic slowing down power MSDP for some materials.

The moderating ratio or moderator quality is the most complete measure of the effectiveness of a moderator because it takes into account also the absorption effects. When absorption effects are high, most of the neutrons will be absorbed by moderator, leading to lower moderation or lower availability of thermal neutrons.

Therefore a higher ratio of MSDP to absorbtion cross sections ξ . Σs/Σa is desirable for effective moderation. This ratio is called the moderating ratio – MR and can be used as a criterion for comparison of different moderators.

Examples:

  • Light water has the highest ξ and σs among the moderators (resulting in the highest MSDP) shown in the table, but its moderating ratio is low due to its relatively higher absorption cross section.
  • On the other hand, heavy water has lower ξ and σs, but it has the highest moderating ratio owing to its lowest neutron absorption cross-section.
  • Graphite has much heavier nuclei than hydrogen in water, despite the fact graphite has much lower ξ and σs, it is better moderator than light water due to its lower absorption cross-section compared to that of light water.
Table of moderating ratios MR for some materials.

Table of moderating ratios for some materials.

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