# Neutron Balance – Continuity Equation

The mathematical formulation of neutron diffusion theory is based on the balance of neutrons in a differential volume element. Since neutrons do not disappear (β decay is neglected) the following neutron balance must be valid in an arbitrary volume V.

rate of change of neutron density = production rate – absorption rate – leakage rate

where

Substituting for the different terms in the balance equation and by dropping the integral over (because the volume V is arbitrary) we obtain:

where

• n is the density of neutrons,
• s is the rate at which neutrons are emitted from sources per cm3 (either from external sources (S) or from fission (ν.Σf)),
• J is the neutron current density vector
• Ф is the scalar neutron flux
• Σa is the macroscopic absorption cross-section

In steady state, when n is not a function of time:

## The Diffusion Equation

In previous chapters we introduced two bases for the derivation of the diffusion equation:

Fick’s law:

which states that neutrons diffuses from high concentration (high flux) to low concentration.

Continuity equation:

which states, that rate of change of neutron density = production rate – absorption rate – leakage rate.

We return now to the neutron balance equation and substitute the neutron current density vector by J = -D∇Ф. Assuming that ∇.∇ = ∇2 = Δ  (therefore div J = -D div (∇Ф) = -DΔФ) we obtain the diffusion equation.

References:
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.