3.2.1 Local Spherical System-Directional Distribution

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### 3.2.1 Local Spherical System-Directional Distribution

A particle is emitted from a solid surface into the ``space'' or hemisphere above the surface. Locations on the hemisphere are characterized by solid angle. We introduce the concept by analogy to linear dimensions. Recall the definition of geometrical quantities as follows:

Table 4 Analogy of Angles to Lengths. View Table

Referring to the Table 4 and Figure 7, a line is generated by sweeping a point linearly through space, and an area is generated by sweeping a line through space. Similarly as shown in Figure 7, an angle is generated by sweeping a line through an arc. A solid angle is generated by sweeping two ``orthogonal'' angles ( and ) through arcs.

Figure 7 Generating an Angle and a Solid Angle. View Figure

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. Recall that a differential angle is defined as follows:

where is the differential arc length and is the radius. A differential solid angle is defined by scaling up the differential arc length to a differential area and dividing by to make the solid angle independent of radial position (distance from the origin) as follows (see Figure 7(f)):

Viz., referring to Figure 7(f), note that length is generated by sweeping line (not ), through angle - i.e. the revolution is of a line normal to the vertical axis , about the vertical axis . Contrariwise, line is generated by sweeping , through angle about the origin .

In Figure 7, is the ``cone'' angle and is the ``azimuthal'' angle. Emission occurs over the hemisphere defined from to and to . While angles are measured in radians, solid angles are measured in steradians. A hemisphere contains steradians, as follows:

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 . The emission from many real surfaces is well modelled in this fashion. Such surfaces are termed Lambertian or ``diffuse'' surfaces.

Figure 8 Emission from Within a Solid. View Figure

We now turn our attention to emission from a general surface. In a specific time interval, the rate of emission of particles, , emitted over the hemisphere from differential area may be related to the rate of particles emitted per unit solid angle per unit projected surface area, , as follows:

If is constant, i.e., independent of and (as it is for Lambertian emission), then

Therefore, we obtain exactly half as many particles by weighting the solid angle with the projected area () for emission (recall that there are steradians in a hemisphere). In practice, we define a ``perfect'' emitter, which emits the maximum rate of particles over the hemisphere at each steradian. The ratio of the rates of particle emission from an actual (real) surface to that from a ``perfect'' emitter is termed the emittance, , which typically depends upon .

The quantity is typically measured, although it may be accurately calculated in some situations. Note that, we have specified that may be a function of and not of . This is a common assumption, as only biases in surface finish (such as may be caused by specific machining practices) cause a dependence upon , and these are rarely significant. However, variations in molecular structure of emitting surfaces cause a dependence upon , and this must in general be accounted for.

Then, over the entire hemisphere,

Since has a maximum of 1, has a maximum of 1. We term the total hemispherical emittance, as characteristic of the emission into the entire hemisphere above the point. Henceforth, we shall neglect all angular dependences in this introductory treatment, and proceed by examining emission from ``perfect'' surfaces, where is 1.

Thus, the directional distribution for emission is what we actually seek. This is given by the argument of the integral in equation (15), i.e., particles are emitted from perfect surfaces according to , as shown in Figure 9. As we see, per unit degree of cone angle-, the greatest number of particles are emitted at an angle of from the surface normal (over all in ). As tends to (normal to the surface), the solid angle tends to 0, and as tends to (grazing), the projected area tends to zero. The balance between these two factors results in a maximum for emission per unit cone angle from a ``perfect'' surface at .

Figure 9 Probability of Emission in dq. View Figure

In Monte Carlo simulation, it is often required to emit particles according to a directional distribution such as shown in Figure 9. In effect, we emit many particles to establish a directional distribution such as shown in Figure 9. One way to do this is termed the accept/reject method.

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