The Asymptotic Drift-Diffusion Limit of Thermal Neutrons

Abstract

It is well known that in an infinite, source-free, purely scattering medium with physically realizable cross sections, neutrons attain a Maxwell-Boltzmann distribution characterized by the material temperature. In this work we look at how small variations to these conditions change the behavior of the thermal-neutron distribution in space and energy. Specifically, our analysis examines the influence of small amounts of absorption and a small source as well as temperature variations in the material. We restrict our study to regions away from boundary and initial layers. The result of the asymptotic analysis is that the amplitude of the neutron scalar flux satisfies a drift-diffusion equation in which the diffusion coefficient and drift velocity depend on the first-order anisotropy of the scattering kernel as well as the gradient of the material temperature. Additionally, through first order in the asymptotic expansion the neutron energy distribution is Maxwell-Boltzmann at the local material temperature.

Keywords:

• asymptotic analysis
• neutron transport theory
• thermal neutrons

APPENDIX: INVERTIBILITY OF FOR L > 0

Consider an infinite medium that is purely scattering with σ_s(E) > 0. Assume that the scattering law depends only on the dot product of the incident and scattered directions, which means the medium itself is rotationally invariant. This is sometimes referred to as an isotropic medium. In such an instance the transport equation for the steady angular flux of neutrons is (Lewis and Miller, 1984) (37) where (38) (39) Here μ₀ is the dot product of the incident and scattered angle and P_l is the lth-order Legendre polynomial. If we multiply the transport equation by and integrate over we obtain, via orthonormality of the spherical harmonics functions, (40) We can write this equation in operator form as (41) where (42) In this appendix we show that for physically realizable scattering functions, the operator is invertible for l > 0. We will begin with remarks about the l = 0 case.

Remark 1.

The operator is singular for physically realizable scattering functions.

Argument. There is a nonzero solution of the equation (43) given by (44) where M(E, T) is the Maxwell-Boltzmann distribution (for the scalar flux, not density) at temperature T, the temperature of the scattering medium, and is a constant (Bell and Glasstone, 1970). According to well-known results from statistical mechanics, the solution is unique except for the multiplicative constant . That is, the Maxwell-Boltzmann distribution is the only possible solution to the given steady-state problem. This means the operator has exactly one zero eigenvalue, with a corresponding eigenfunction equal to the Maxwell-Boltzmann distribution. This in turn means that the operator has one eigenvalue equal to unity, with eigenfunction equal to the Maxwell-Boltzmann distribution.

The next remark is a mathematical consequence of the physical reality that scattering does not create particles, but only changes their directions and energies.

Remark 2.

The spectral radius of .

Argument. The spectral radius of satisfies (45) where the supremum is taken over all functions in the function space and ‖f‖ is a valid norm of f. For our purposes we define the norm (46) and we choose our function space to include only functions of E whose norms are bounded. We recognize that the integral of the differential scattering cross section is the scattering cross section: (47)

The norm of satisfies (48) from which it follows that the spectral radius of is ⩽ 1. Equality holds for any non-negative function ϕ(E), so the spectral radius equals 1. We saw previously that exactly one eigenvalue equals 1; now we see that there are no eigenvalues with larger magnitude.

The next remark will also be useful.

Remark 3.

If the spectral radius of is less than 1, then the operator is invertible.

Argument. 1 − λ is an eigenvalue of with eigenfunction f if and only if λ is an eigenvalue of with eigenfunction f. If all eigenvalues of have magnitude less than unity (i.e., if the spectral radius ), then the eigenvalues of lie within a circle of radius centered in the complex plane at 1 on the real axis. Zero is outside of this circle and is therefore not an eigenvalue of ; thus, is invertible.

Remark 4.

The spectral radius of is less than 1 for l > 0.

Argument. The spectral radius satisfies (49) where the supremum is taken over all functions in the function space and ‖f‖ is a valid norm of f. We continue to use the norm (50)

In what follows we shall use the following identities and definitions: (51) (52) (53) Here . The form of Equation (53) makes it clear that ∫dE f_l(E′ → E) is a proper weighted average of P_l(μ₀), with non-negative weight function f(E′ → E, μ₀), for particles scattering with initial energy E′. The average of the absolute value of P_l(μ₀) is a function of E′ and will be a useful quantity in what follows, so we define it here: (54)

With these definitions, the norm of satisfies the inequality (55) Here |P_l|_max is the supremum over all E′ of the averaged |P_l(μ₀)|. P_l(μ₀) is between − 1 and 1 for all values of l, and μ₀. If l = 0, then P₀(μ₀) = 1 for all μ₀ and of course |P₀|_max = 1. In this case, the equal sign holds throughout Equation. (55(55) ), which is in keeping with our previous result that the spectral radius of is unity. However, for l > 0, ⟨|P_l(μ₀)|⟩ = 1 only if μ₀ = ±1 for every particle that scatters with initial energy E′—that is, only if every particle of energy E′ suffers only scattering events that either do not change the particle’s direction or that exactly reverse its direction. This is not a physically realizable scattering function for neutrons. That is, for physically realizable scattering functions, the left-hand side of Equation (55(55) ) is strictly less than the right-hand side for l > 0. It follows that for physically realizable scattering functions, the spectral radius of is strictly < 1 for l > 0. For a scattering law where μ₀ = ±1 for every scattering event (non physically realizable for neutrons), our argument would not hold.

Now we can combine our results to demonstrate invertibility.

Main Result. The operator is invertible for l > 0, for physically plausible scattering laws.

Argument. Remark 4 shows that the conditions of Remark 3 are satisfied and the operator is invertible.

Notes

That is, we assume that the source-rate density is small relative to the collision-rate density.