Question

A small container of Ra–Be is embedded in the middle of a sphere of paraffin wax of a few cm radius so as to form a source of (predominantly) thermal neutrons. This source is placed at the centre of a very large block of graphite. Derive an expression for the density of thermal neutrons at a large distance r from the source in terms of the source strength Q, the diffusion coefficient D and diffusion length L.

A small BF3 counter is placed in the graphite at a distance of 3 m from the above source contains 1020 atoms of 10B. The cross-section of 10B for the thermal neutron capture follows a 1/v law and has a magnitude of 3,000 barns for a neutron velocity v = 2, 200 m/. If the counting rate is 250/min, calculate the value of Q. Given L = 50 cm; D = 5 × 105 cm2/.

Answer 1

The diffusion equation for the steady state in the absence of sources at the point of interest is

Writing the Laplacian for spherical geometry (1) becomes

Equation (2) is easily solved, the solution being

As K is positive, the first term on the right hand side tends to ∞ as r → ∞. Therefore, C1 = 0 if the flux is required to be finite everywhere including at ∞.

We can calculate the constant C2 by considering the current J through a small sphere of radius r with its centre at the source. The net current

where we have used (4) The net number of neutrons leaving the sphere per second is

But as r → 0, the total number of neutrons leaving the sphere per second must be equal to the source strength Q. Thus from (6)