Diffusion Coefficient | Definition & Calculation

From diffusion theory, the diffusion coefficient is expressed in terms of the macroscopic cross-section and mean free path as:

where Σ_s is the macroscopic scattering cross-section and λ_s 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:

where:

λ_tr is the transport mean free path
Σ_a is the macroscopic absorption cross-section.
Σ_tr is the macroscopic transport cross-section
μ₀ is the average value of the cosine of the angle in the lab system

In a weakly absorbing medium where Σ_a << Σ_s the diffusion coefficient can be approximately calculated as:

The transport means free path (λ_tr) is an average distance a neutron will move in its original direction after an infinite number of scattering collisions.

is the 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):

Physical Interpretation

where

Consider neutrons passing through the plane at x=0 from left to right due to 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 a 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, Σ_s, we will study the 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 (Σ_s=σ_s.N_H2O) to a decrease in the macroscopic scattering cross-section, Σ_s=σ_s.N_H2O, 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.

Solution:
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.

Density:

M_C = 12

N_C = ρ . N_a / M_C

= (2.2 g/cm³)x(6.022×10²³ nuclei/mol)/ (12 g/mol)

= 1.1×10²³ nuclei / cm³

the microscopic cross-section

σ_s^12C = 4.8 b

the macroscopic cross-section

Σ_s^12C = 4.8×10^-24 x 1.1×10²³ = 0.528 cm^-1

the diffusion coefficient is then:

D = 1 / (3 x 0.528 x 0.9445) = 0.668 cm

the transport mean free path

λ_tr = 3 x D = 2.005 cm

Diffusion Coefficient and the Fick’s Law

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:

where:

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.

The generalized Fick’s law (in three dimensions) is:

where J denotes the diffusion flux vector. Note that the gradient operator turns the neutron flux, a scalar quantity, into the neutron current, a vector quantity.

Diffusion Coefficient and the Diffusion Equation

The diffusion equation:

Diffusion Coefficient and Diffusion Length

During the diffusion equation solution, we often meet with a very important parameter that describes the 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 is called the diffusion length).

References:

Nuclear and Reactor Physics:

J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.
E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See above:

Neutron Diffusion