Figure 7 Generating an Angle and a Solid Angle. A solid angle is two-dimensional by virtue of having no radial position, i.e., the solid angle remains fixed independent of radius. This provides the key to the mathematical formulation for a solid angle. Again, we proceed by scaling up in dimensionality. Contrariwise, line is generated by sweeping , through angle about the origin .
Emission occurs over the hemisphere defined from to and to .
While angles are measured in radians , solid angles are measured in steradians . The probability of emission of a particle at angle should properly be weighted by . As particles originate inside the solid, their probability of escaping from the solid and being emitted is inversely proportional to the path length they must travel through the material to escape. In Figure 8, we depict this conceptually for particles originating a depth beneath the surface. Thus, emission at angle is proportional to or .