Solutions of the Diffusion Equation – Multiplying Systems
in various geometries that satisfy the boundary conditions. In this equation ν is number of neutrons emitted in fission and Σf is macroscopic cross-section of fission reaction. Ф denotes a reaction rate. For example a 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 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 provides an illustrative insights, how can be the neutron flux distributed in a reactor core.
Solution for the Infinite Reactor
Let assume a uniform infinite reactor, i.e. uniform infinite multiplying system without an external neutron source. This system is in a Cartesian coordinate system and under these assumptions (no neutron leakage, no changes in diffusion parameters) the neutron flux must be inherently constant throughout space.
Since the neutron current is equal to zero (J = -D∇Ф, where Ф is constant), the diffusion equation in the infinite uniform multiplying system must be:
The only solution of this equation is a trivial solution, i.e., Ф = 0, unless Σa = νΣf. This equation (Σa = νΣf) is known as the criticality condition for an infinite reactor and expresses the perfect balance (critical state) between neutron absorption and neutron production. This balance must be continuously maintained in order to have steady state neutron flux.
Infinite Multiplication Factor
As can be seen we can rewrite the diffusion equation in following way and we can define a new factor – k∞ = νΣf / Σa:
A non-trivial solution of this equation is guaranteed when k∞ = νΣf / Σa = 1. On the other hand we have no information about the neutron flux in such critical reactor. In fact the neutron flux can have any value and the critical uniform infinite reactor can operate at any power level. It should be noted this theory can be used for a reactor at low power levels, hence “zero power criticality”.
In power reactor core, the power level does not influence the criticality of a reactor unless thermal reactivity feedbacks act (operation of a power reactor without reactivity feedbacks is between 10E-8% – 1% of rated power).
See also: Reactor Criticality