Solutions of the Diffusion Equation – Non-multiplying Systems
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 non-multiplying system, i.e., in system where fissile isotopes are missing and therefore the fission cross-section is zero. The neutrons are emitted by external neutron source. We will assume that the system is uniform outside the source, i.e. D and Σa are constants.
Solution for the Point Source
Let assume the neutron source (with strength S0) as an isotropic point source situated in spherical geometry. This point source is placed at the origin of coordinates. In order to solve the diffusion equation, we have to replace the Laplacian by its spherical form:
We can replace the 3D Laplacian by its one-dimensional spherical form, because there is no dependence on angle (whether polar or azimuthal). The source is assumed to be an isotropic source (there is the spherical symmetry). The flux is then a function of radius – r only, and therefore the diffusion equation (outside the source) can be written as (everywhere except r = 0):
If we make the substitution Φ(r) = 1/r ψ(r), the equation simplifies to:
For r > 0, this differential equation has two possible solutions exp(r/L) and exp(-r/L), which give a general solution:
Note that, B is not usually used as a constant, because B is reserved for a buckling parameter. To determine the coefficients A and C we must apply boundary conditions.
To find constants A and C we use the identical procedure as in the case of infinite planar source. From finite flux condition (0≤ Φ(r) < ∞), that required only reasonable values for the flux, it can be derived, that A must be equal to zero. The term exp(r/L)/r goes to ∞ as r ➝∞ and therefore cannot be part of a physically acceptable solution for r > 0. The physically acceptable solution for r > 0 must then be:
Φ(r) = Ce-r/L/r
where C is a constant that can be determined from source condition at x ➝0.
If S0 is the source strength, then the number of neutrons crossing a sphere outwards in the positive r-direction must tend to S0 as r ➝0.
So that the solution may be written: