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Randomized form factor calculation

 

The randomized approach is based on the recognition that the formula defining the form factors can be taken to represent the probability of a quite simple event if the underlying probability distributions are defined properly.

An appropriate such event would be a surface j being hit by a particle leaving surface i. Let us denote the event that a particle leaves surface tex2html_wrap_inline1812 by tex2html_wrap_inline1888 . Expressing the probability of the ``hit'' of surface j by the total probability theorem we get:

equation237

The hitting of surface j can be broken down into the separate events of hitting the various differential elements tex2html_wrap_inline1814 composing tex2html_wrap_inline1824 . Since hitting of tex2html_wrap_inline1900 and hitting of tex2html_wrap_inline1902 are exclusive events if tex2html_wrap_inline1904 :

  equation242

Now the probability distributions involved in the equations are defined:

  1. Assume the origin of the particle to be selected randomly by uniform distribution:

    equation249

  2. Let the direction in which the particle leaves the surface be selected by a distribution proportional to the cosine of the angle between the direction and the surface normal:

    equation252

The denominator tex2html_wrap_inline1906 guarantees that the integration of the probability over the whole hemisphere yields 1, hence it deserves the name of probability density function. Since the solid angle of tex2html_wrap_inline1814 from tex2html_wrap_inline1812 is tex2html_wrap_inline1914 where r is the distance between tex2html_wrap_inline1812 and tex2html_wrap_inline1814 , and tex2html_wrap_inline1922 is the angle of the surface normal of tex2html_wrap_inline1814 and the direction of tex2html_wrap_inline1812 , the probability of equation 1.15 is:

displaymath1928

displaymath1930

equation260

where tex2html_wrap_inline1818 is the indicator function of the event `` tex2html_wrap_inline1814 is visible from tex2html_wrap_inline1812 ''.

Substituting these into the original probability formula:

equation265

This is exactly the same as the formula for form factor tex2html_wrap_inline1862 . This probability, however, can be estimated by random simulation. Let us generate n particles randomly using uniform distribution on the surface i to select the origin, and a cosine density function to determine the direction. The origin and the direction define a ray which may intersect other surfaces. That surface will be hit whose intersection point is the closest to the surface from which the particle comes. If shooting n rays randomly surface j has been hit tex2html_wrap_inline1948 times, then the probability or the form factor can be estimated by the relative frequency:

equation275

Two problems have been left unsolved:

Addressing the problem of the determination of the necessary number of attempts, we can use the laws of large numbers.

The inequality of Bernstein and Chebyshev [Rén81] states that if the absolute value of the difference of the event frequency and the probability is expected not to exceed tex2html_wrap_inline1952 with probability tex2html_wrap_inline1954 , then the minimum number of attempts (n) is:

equation282

The generation of random distributions can rely on random numbers of uniform distribution in [0..1] produced by the pseudo-random algorithm of programming language libraries. Let the probability distribution function of the desired distribution be P(x). A random variable x which has P(x) probability distribution can be generated by transforming the random variable r that is uniformly distributed in [0..1] applying the following transformation:

equation286


next up previous
Next: Analytic and geometric methods Up: Form factor calculation Previous: Form factor calculation

Szirmay-Kalos Laszlo
Mon Oct 21 14:07:41 METDST 1996