From diffusion theory
, the diffusion coefficient
is expressed in terms of the macroscopic cross-section
and mean free path
where Σs is the macroscopic scattering cross-section and λs is the scattering mean free path.
However, we can express the diffusion coefficient from the more advanced transport theory in terms of transport and absorption cross-sections:
- λtr is the transport mean free path
- Σa is the macroscopic absorption cross-section
- Σtr is the macroscopic transport cross-section
- μ0 is average value of the cosine of the angle in the lab system
In a weakly absorbing medium where Σa << Σs the diffusion coefficient can be approximately calculated as:
The transport mean free path (λtr) is an average distance a neutron will move in its original direction after infinite number of scattering collisions.
is average value of the cosine of the angle in the lab system at which neutrons are scattered in the medium. It can be calculated for most of the neutron energies as (A is the mass number of target nucleus):
Consider neutrons passing through the plane at x=0 from left to right as the result of collisions to the left of the plane. 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. The diffusion coefficient determines the rate of diffusion.
Operational changes that affect the diffusion length
The diffusion coefficient is very important parameter in thermal reactors and its magnitude can be changed during reactor operation. Since the diffusion coefficient is dependent on the macroscopic scattering cross-section, Σs, we will study impacts of operational changes on this parameter.
Change in the moderator temperature
The diffusion coefficient, D, is sensitive especially on the change in the moderator temperature.
In short, as the moderator temperature increases, the diffusion coefficient also slightly increases.
This increase in the diffusion coefficient is especially due (Σs=σs.NH2O) to a decrease in the macroscopic scattering cross-section, Σs=σs.NH2O, caused by the thermal expansion of water (a decrease in the atomic number density).
Calculation of Diffusion Coefficient
The scattering cross-section of carbon at 1 eV is 4.8 b (4.8×10-24 cm2). Calculate the diffusion coefficient and the transport mean free path.
We will calculate the diffusion according to the advanced formula:
First, we have to determine the atomic number density of carbon and then the scattering macroscopic cross-section.
MC = 12
NC = ρ . Na / MC
= (2.2 g/cm3)x(6.022×1023 nuclei/mol)/ (12 g/mol)
= 1.1×1023 nuclei / cm3
the microscopic cross-section
σs12C = 4.8 b
the macroscopic cross-section
Σs12C = 4.8×10-24 x 1.1×1023 = 0.528 cm-1
the diffusion coefficient is then:
D = 1 / (3 x 0.528 x 0.445) = 1.42 cm
the transport mean free path
λtr = 3 x D = 4.26 cm
Diffusion Coefficient and the Fick’s Law
The use of this law in nuclear reactor theory
leads to the diffusion approximation
The 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 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 unit of cm and it is given by:
- φ is the neutron flux, which is the number of neutrons crossing through some arbitrary cross-sectional unit area in all directions per unit time.
The generalized Fick’s law (in three dimension) is:
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.
Diffusion Coefficient and the Diffusion Equation
The diffusion equation:
Diffusion parameters for thermal neutrons of 0.025 eV in some materials
Diffusion Coefficient and Diffusion Length
During solution of the diffusion equation we often meet with very important parameter that describes behavior of neutrons in a medium.
The solution 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 called the diffusion length).
Solution of diffusion equation in two different non-multiplying diffusion media