Diffusion Coefficient

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

diffusion coefficient - basic

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:
diffusion coefficient - transport

where:

  • 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:

diffusion coefficient - transport formula

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

diffusion coefficient - angle

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):

diffusion coefficient - angle2

Physical Interpretation

Ficks Law - physical interpretation

where

Ficks Law - equation

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.

transport mean free path - graphically

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 (Σss.NH2O) to a decrease in the macroscopic scattering cross-section, Σss.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.

Solution:
We will calculate the diffusion according to the advanced formula:

diffusion coefficient - example 2

First, we have to determine the atomic number density of carbon and then the scattering macroscopic cross-section.

Density:

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 0.9445) = 0.668 cm

the transport mean free path

λtr = 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.

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:

Ficks Law - equation

where:

  • 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:diffusion coefficient - equation
  • 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:
Ficks Law - 3D

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 equation - general

Table of diffusion parameters

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:

Diffusion Length - equation

The term L2 is called the diffusion area (and L called the diffusion length).

diffusion equation - two media

Solution of diffusion equation in two different non-multiplying diffusion media

Nuclear and Reactor Physics:

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

Advanced Reactor Physics:

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