Each of the above material properties is the probability of the particular
interaction.
Figure 16
depicts the material properties as functions of the
incident cone angle, 
.
At any particular value of 
, the incident particle has the
following probabilities:
,
,
,
and
for specular and diffuse transmission, specular and diffuse reflection, and
absorption, respectively.
Each of these curves must be input (or generated) by point value as a function
of cone angle.
Figure 16: Material Properties vs. Incident Cone Angle.
Material properties of much greater complexity could easily be implemented in
Monte Carlo analysis (indeed, this is one of its great strengths).
For example, at each incident angle, we could have specified outgoing
directional distributions for transmission and reflection, in which case we
would need to specify a surface of outgoing directions for each point
on the abscissa of
Figure 16
(obviously, we would have to discretize
the abscissa intelligently).
However, in most cases, we lack sufficient data to make this level of effort
worthwhile.
So, we restrict our attention to two specific outgoing directional
distributions---specular and diffuse, and represent properties as shown in
Figure 16.
Finally, we note that we have found it sufficient to provide data every
in 
and to perform linear interpolation between
points.
For efficiency, we store these data together with their differences in tabular
form.
Thus, each material requires a storage space of 
words for particle/material interactions.
Our procedure for determining particle/surface interactions is then as follows.
After the intersection is obtained, we determine the incident angle.
At this incident angle, we look up (from the material property table) the
probabilities of interactions.
We choose a random number from a uniform distribution (between 0 and 1) of
random numbers to determine the specific interaction for this particle.
It is necessary to sum the probabilities for this, that is, if the random number
is less than 
, the particle is made to undergo a specular
transmission; if the random number is greater than 
, but less
than 
, then the particle is diffusely
transmitted; etc.
For efficiency, it is wise to sum all properties during the input phase (i.e.,
instead of storing 
, 
, etc., we store
, 
, etc.) at each angle, so the
sums need not be calculated during the solution phase.
