In the previous section, we dealt with the multiplication system and defined the infinite and finite multiplication factors. This section was about conditions for a stable, self-sustained fission chain reaction and maintaining such conditions. This problem contains no information about the spatial distribution of neutrons because it is a point geometry problem. We have characterized the effects of the global distribution of neutrons simply by a non-leakage probability (thermal or fast), which, as stated earlier, increases toward a value of one as the reactor core becomes larger.
To design a nuclear reactor properly, predicting how the neutrons will be distributed throughout the system is highly important. This is a very difficult problem because the neutrons interact differently with different environments (moderator, fuel, etc.) in a reactor core. Neutrons undergo various interactions when they migrate through the multiplying system. To a first approximation, the overall effect of these interactions is that the neutrons undergo a kind of diffusion in the reactor core, much like the diffusion of one gas in another. This approximation is usually known as the diffusion approximation, based on the neutron diffusion theory. This approximation allows solving such problems using the diffusion equation.
In this chapter, we will introduce the neutron diffusion theory. We will examine the spatial migration of neutrons to understand the relationships between reactor size, shape, and criticality and determine the spatial flux distributions within power reactors. The diffusion theory provides a theoretical basis for neutron-physical computing of nuclear cores. It must be added many neutron-physical codes are based on this theory.
First, we will analyze the spatial distributions of neutrons, and we will consider a one-group diffusion theory (mono-energetic neutrons) for a uniform non-multiplying medium. That means that the neutron flux and cross-sections have already been averaged over energy. Such a relatively simple model has the great advantage of illustrating many important features of the spatial distribution of neutrons without the complexity introduced by the treatment of effects associated with the neutron energy spectrum.
See also: Neutron Flux Spectra.
Moreover, mathematical methods used to analyze a one-group diffusion equation are the same as those applied in more sophisticated and accurate methods such as multi-group diffusion theory. Subsequently, the one-group diffusion theory will be applied in simple geometries on a uniform multiplying medium (a homogeneous “nuclear reactor”). Finally, the multi-group diffusion theory will be applied in simple geometries on a non-uniform multiplying medium (a heterogenous “nuclear reactor”).
Derivation of One-group Diffusion Equation
The derivation of the diffusion equation depends on Fick’s law, which states that solute diffuses from high concentration to low. But first, we have to define a neutron flux and neutron current density. The neutron flux is used to characterize the neutron distribution in the reactor, and it is the main output of solutions of diffusion equations. The neutron flux, φ, does not characterize the flow of neutrons. There may be no flow of neutrons, yet many interactions may occur (I = Σ.φ). The neutrons move in random directions and hence may not flow. Therefore the neutron flux φ is more closely related to densities. Neutrons will exhibit a net flow when there are spatial differences in their density. Hence we can have a flux of neutron flux! This flux of neutron flux is called the neutron current density.
In chemistry, Fick’s law states that:
Suppose the concentration of a solute in one region is greater than in another of a solution. In that case, the solute diffuses from the region of higher concentration to the region of lower concentration, with a magnitude that is proportional to the concentration gradient.
In one (spatial) dimension, the law is:
- J is the diffusion flux,
- D is the diffusion coefficient,
- φ (for ideal mixtures) is the concentration.
The use of this law in nuclear reactor theory leads to the diffusion approximation.
Fick’s law in reactor theory stated that:
The current density vector J is proportional to the negative of the gradient of the neutron flux. The proportionality constant is called the diffusion coefficient and is denoted by the symbol D.
In one (spatial) dimension, the law is:
- J is the neutron current density (neutrons.cm-2.s-1) along the x-direction, the net flow of neutrons that pass per unit of time through a unit area perpendicular to the x-direction.
- D is the diffusion coefficient, it has the unit of cm, and it is given by:
- φ is the neutron flux, the number of neutrons crossing through some arbitrary cross-sectional unit area in all directions per unit time.
where J denotes the diffusion flux vector. Note that the gradient operator turns the neutron flux, which is a scalar quantity into the neutron current, which is a vector quantity.
The physical interpretation is similar to the fluxes of gases. The neutrons exhibit a net flow in the direction of least density. This is a natural consequence of greater collision densities at positions of greater neutron densities.
Consider neutrons passing through the plane at x=0 from left to right due to collisions to the plane’s left. Since the concentration of neutrons and the flux is larger for negative values of x, there are more collisions per cubic centimeter on the left. Therefore more neutrons are scattered from left to right, then the other way around. Thus the neutrons naturally diffuse toward the right.
Validity of Fick’s Law
It must be emphasized that Fick’s law is an approximation and was derived under the following conditions:
- Infinite medium. This assumption is necessary to allow integration of overall space but flux contributions are negligible beyond a few mean free paths (about three mean free paths) from boundaries of the diffusive medium.
- Sources or sinks. Derivation of Fick’s law assumes that the contribution to the flux is mostly from elastic scattering reactions. Source neutrons contribute to the flux if they are more than a few mean free paths from a source.
- Uniform medium. Derivation of Fick’s law assumes that a uniform medium was used. There are different scattering properties at the boundary (interface) between the two media.
- Isotropic scattering. Isotropic scattering occurs at low energies but is not true in general. Anisotropic scattering can be corrected by modification of the diffusion coefficient (based on transport theory).
- Low absorbing medium. Fick’s law derivation assumes (an expansion in Taylor’s series) that the neutron flux, φ, is slowly varying. Large variations in φ occur when Σa (neutron absorption) is large (compared to Σs). Σa << Σs
- Time-independent flux. Derivation of Fick’s law assumes that the neutron flux is independent of time.
To some extent, these limitations are valid in every practical reactor. Nevertheless, Fick’s law gives a reasonable approximation. For more detailed calculations, higher-order methods are available.
Neutron Balance – Continuity Equation
The mathematical formulation of neutron diffusion theory is based on the balance of neutrons in a differential volume element. Since neutrons do not disappear (β decay is neglected), the following neutron balance must be valid in an arbitrary volume V.
rate of change of neutron density = production rate – absorption rate – leakage rate
Substituting for the different terms in the balanced equation and by dropping the integral over (because the volume V is arbitrary), we obtain:
- n is the density of neutrons,
- s is the rate at which neutrons are emitted from sources per cm3 (either from external sources (S) or from fission (ν.Σf.Ф)),
- J is the neutron current density vector
- Ф is the scalar neutron flux
- Σa is the macroscopic absorption cross-section
In steady-state, when n is not a function of time:
The Diffusion Equation
In previous chapters, we introduced two bases for the derivation of the diffusion equation:
which states that neutrons diffuse from high concentration (high flux) to low concentration.
which states that rate of change of neutron density = production rate – absorption rate – leakage rate.
We return now to the neutron balance equation and substitute the neutron current density vector by J = -D∇Ф. Assuming that ∇.∇ = ∇2 = Δ (therefore div J = -D div (∇Ф) = -DΔФ) we obtain the diffusion equation.
The derivation of the diffusion equation is based on Fick’s law which is derived under many assumptions. Therefore, the diffusion equation cannot be exact or valid at places with strongly differing diffusion coefficients or in strongly absorbing media. This implies that the diffusion theory may show deviations from a more accurate solution of the transport equation in the proximity of external neutron sinks, sources, and media interfaces.
To solve the diffusion equation, which is a second-order partial differential equation throughout the reactor volume, it is necessary to specify certain boundary conditions. It is very dependent on the complexity of a certain problem. One-dimensional problems solutions of diffusion equation contain two arbitrary constants. Therefore, we need two boundary conditions to determine these coefficients to solve a one-dimensional one-group diffusion equation. The most convenient boundary conditions are summarized in the following few points:
Diffusion Length of Neutron
During the diffusion equation solution, we often meet with a very important parameter that describes the behavior of neutrons in a medium.
The solution of diffusion equation (let assume the simplest diffusion equation) usually starts by division of entire equation by diffusion coefficient:
The term L2 is called the diffusion area (and L is called the diffusion length). For thermal neutrons with an energy of 0.025 eV, a few values of L are given in the table below.
Physical Meaning of the Diffusion Length
It is interesting to try to interpret the “physical” meaning of the diffusion length. The physical meaning of the diffusion length can be seen by calculating the mean square distance that a neutron travels in the one direction from the plane source to its absorption point.
It can be calculated that L2 is equal to one-half the square of the average distance (in one dimension) between the neutron’s birth point and its absorption.
If we consider a point source of neutrons, the physical meaning of the diffusion length can be seen again by calculating the mean square distance that a neutron travels from the source to its absorption point.
It can be calculated that:
L2 is equal to one-sixth of the square of the average distance (in all dimensions) between the neutron’s birth point (as a thermal neutron) and its absorption.
This distance must not be confused with the average distance traveled by the neutrons. The average distance traveled by the neutrons is equal to the mean free path for absorption λa = 1/Σa and is much larger than the distance measured in a straight line. This is because neutrons in the medium undergo many collisions, and they follow a very zig-zag path through the medium.
Applicability of Diffusion Theory
Nowadays, the diffusion theory is widely used in the core design of the current Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs). It provides a strictly valid mathematical description of the neutron flux. Still, it must be emphasized that the diffusion equation (in fact the Fick’s law) was derived under the following assumptions:
- Infinite medium
- No sources or sinks.
- Uniform medium.
- Isotropic scattering.
- Low absorbing medium.
- Time-independent flux.
To some extent, these limitations are valid in every practical reactor. Nevertheless, the diffusion theory gives a reasonable approximation and makes accurate predictions. Nowadays, reactor core analyses and designs are often performed using nodal two-group diffusion methods. These methods are based on pre-computed assembly homogenized cross-sections, diffusion coefficients, and assembly discontinuity factors (pin factors) obtained by single assembly calculation with reflective boundary conditions (infinite lattice). Highly absorbing control elements are represented by effective diffusion theory cross-sections, which reproduce transport theory absorption rates. These pre-computed data (discontinuity factors, homogenized cross-sections, etc.) are calculated by neutron transport codes based on a more accurate neutron transport theory. In short, neutron transport theory is used to make diffusion theory work.
Two methods exist for the calculation of the pre-computed assembly cross-sections and pin factors.
- Deterministic methods that solve the Boltzmann transport equation.
- Stochastic methods are known as Monte Carlo methods that model the problem almost exactly.
These methods are very efficient and accurate when applied to the current Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs).
Solutions of the Diffusion Equation – Non-multiplying Systems
As was previously discussed, the diffusion theory is widely used in the core design of the current Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs). This section is not about such calculations but provides illustrative insights, how can be the neutron flux distributed in any diffusion medium. In this section, we will solve diffusion equations:
in various geometries that satisfy the boundary conditions discussed in the previous section.
We will start with simple systems and increase complexity gradually. The most important assumption is that all neutrons are lumped into a single energy group. They are emitted and diffuse at thermal energy (0.025 eV).
In the first section, we will deal with neutron diffusion in a non-multiplying system, i.e., in a system where fissile isotopes are missing, the fission cross-section is zero. The neutrons are emitted by an external neutron source. We will assume that the system is uniform outside the source, i.e., D and Σa are constants.
Solutions of the Diffusion Equation – Multiplying Systems
In the previous section, it has been considered that the environment is non-multiplying. In a non-multiplying environment, neutrons are emitted by a neutron source situated in the center of the coordinate system and then freely diffuse through media. We are now prepared to consider neutron diffusion in the multiplying system containing fissionable nuclei (i.e., Σf ≠ 0). In this multiplying system, we will also study the spatial distribution of neutrons, but in contrast to the non-multiplying environment, these neutrons can trigger nuclear fission reactions.
In this section, we will solve the following diffusion equation
in various geometries that satisfy the boundary conditions. In this equation, ν is the number of neutrons emitted in fission, and Σf is the macroscopic cross-section of the fission reaction. Ф denotes a reaction rate. For example, the fission of 235U by thermal neutron yields 2.43 neutrons.
It must be noted that we will solve the diffusion equation without any external source. This is very important because such an equation is a linear homogeneous equation in the flux. Therefore if we find one solution of the equation, then any multiple is also a solution. This means that the absolute value of the neutron flux cannot possibly be deduced from the diffusion equation. This is totally different from problems with external sources, which determine the absolute value of the neutron flux.
We will start with simple systems (planar) and increase complexity gradually. The most important assumption is that all neutrons are lumped into a single energy group. They are emitted and diffuse at thermal energy (0.025 eV). Solutions of diffusion equations, in this case, provide illustrative insights, how can be the neutron flux distributed in a reactor core.