A Student's Guide to Numerical Methods by Ian H. Hutchinson

Author:Ian H. Hutchinson [Hutchinson, Ian H.]
Language: eng
Format: epub, mobi
Tags: Computers, General, Computer Science, Mathematics, Matrices, Measurement, Numerical Analysis, Science, Physics, Mathematical & Computational
ISBN: 9781107095670
Google: KDwPCAAAQBAJ
Amazon: 1107095670
Publisher: Cambridge University Press
Published: 2015-04-30T13:32:39.910343+00:00

Figure 8.7 Isotropic scattering (an idealized approximation) gives particles emerging equally in all directions Ω. With heavy targets, υ is not changed in magnitude, only in direction. Eq. (8.20) is the result.

Here d2Ω = sin θdθdχ is the element of solid angle, and the integral is over the angular position (θ, χ) on the surface of the sphere in velocity-space at constant total velocity υ. In other words, the second term is the average of the distribution function over all directions, at υ. This type of collision scatters the velocity direction, thus tending to remove any anisotropy (variation with angles θ or χ.)

It should be noticed that in these examples where self-collisions can be ignored the collision term generally consists of two parts. The first is negative, the removal or “sink” rate of particles that collide with whatever targets happen to be present (−n2σ υf in eq. (8.20)). The second is positive, the “source” rate of particles from all mechanisms. When non-reactive gases are being treated, the source is only the re-emergence of particles from collisions. But in other situations, such as neutron transport in a reactor, generation of new particles from reactions or spontaneous emission from the target medium may be equally important.

For multiple target species j, the sink term is the sum of collisions with all target types. And this is often written in shorthand as −∑t × (υf), with

(8.21)

and referred to in reactor-physics literature as the “macroscopic cross-section.” This terminology is unfortunate because the quantity ∑t has units m−1 not m2, and is an inverse attenuation-length, not a cross-section. When the targets are stationary, ∑t is isotropic: the rate of collisions is independent of the direction of particle velocity. The source term, by contrast, is not usually isotropic because it includes the emergence of particles from pure scattering events. Scattering, even from stationary targets, usually partially retains any anisotropy in the distribution function itself. (The conditions of eq. (8.20) are a non-typical idealization.)

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