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Teaching Flux in the Age of Desktop Monte Carlo

1. Introduction

Asked to define the scalar flux, too many students reach for an explanation involving surfaces and areas. Something along the lines of “scalar flux is the rate at which particles cross the surface into a sphere . . .” This inadequate response is not surprising. Students are often introduced to flux in various special monodirectional cases in which the rate at which particles cross a surface can be evaluated using even scalar flux (along with the implicit knowledge of the directional dependance of the field). That the units of flux involve something per unit area further compounds students’ belief that flux has something to do with particles crossing a surface. The misappropriation in radiation transport theory of the word “flux” for what is truly a volumetric concept does not help. My assertion is that no understanding of scalar flux in terms of surface crossing is adequate for our students. They must come to understand scalar flux in terms of path-length rate density.

2. Competing interpretations of scalar flux

Scalar flux, or fluence rate, is a central concept for nuclear engineering analysis. It is defined in several related ways, one of which is under the control of an international standards body. The International Commission on Radiation Units defines1 fluence as dN number of particles incident on a sphere Φ= = (1) da cross sectional area of sphere with the (unstated) understanding that the cross sectional area da is very small compared to macro- scopic spatial variations in the particle population and that the period of observation is finite. The fluence rate, called scalar flux in most nuclear engineering literature, is then simply the rate of change of the fluence. This definition centers on the rate at which particles enter a sphere, and it is often asserted that the definition is founded on some idealized measurement with a spherical detector. This is peculiar, because detectors do not count particles that enter them; all detectors measure reaction rates within a detector volume.

The definition of scalar flux as the rate at which particles enter a sphere, divided by the cross sectional area of the sphere, is useless as a concept for computation of the flux, and it makes no obvious connection to the reaction rate density. In nuclear reactor analysis, in shielding analysis, indeed in all applications of radiation, the scalar flux is required as a means to describe the particle population in a manner suited to the computation of reaction rate densities. But reaction rate density is a volumetric concept, while the ICRU definition of flux emphasizes surfaces (of a sphere) and areas (cross section of a sphere). In the development of radiation transport theory, and its approximate cousins like diffusion theory, the flux is introduced in yet another way. Most careful developments move from the particle number density—a concept easily understood—to the flux as particle speed times the number density. This approach is followed because this product naturally arises in computing reaction rate density.

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Holloway, J. (2007, June), Teaching Flux In The Age Of Desktop Monte Carlo Paper presented at 2007 Annual Conference & Exposition, Honolulu, Hawaii. 10.18260/1-2--3042