The Solid Angle for a Spherical Surface Source and a Detector with an Arbitrary Aperture: An Elegant Proof

It has previously been proven that the solid angle (or geometry factor, also called the geometrical efficiency) for a spherically symmetric outward-directed surface source with an arbitrary radius and polar angle distribution and an arbitrary detector aperture is equal to the solid angle for an isotropic point source located at the center of the spherical surface source and the same detector aperture. The solid angle for a source-detector system is the fraction of source particles that enter the detector aperture and so it is not purely geometric-the angular distribution of the source is a factor. The previous proof relied on an analytic transport-theory (streaming only) calculation of the solid angle Ω for a spherically symmetric surface source and a point detector. In this paper, a new, more elegant proof is given. Since the net current across any portion of a plane that intersects the center of a spherical surface source is zero, the solid angle for a spherical surface source with any polar angle distribution and a uniform azimuthal angle distribution and any detector aperture is equal to the solid angle for an isotropic point source and the same detector. This insight obviates the calculus and trigonometry involved in the previous proof. (author)