A

Example Calculation Using
PFC

Radiation
transport is fundamental to the analysis of the radiation environments produced
by nuclear materials and processes. It
is used, inter alia, for calculating the
effectiveness of biological shields, tissue dose profiles associated with
radiation oncology treatments, radiation environments in the transport of
radioactive materials in both normal and accident conditions, critical mass and
power distributions in fissile assemblies, self-shielding of fertile materials
undergoing activation for the production of medical and industrial isotopes,
and scattering environments produced in experiments with particle
accelerators. The *A Monte
Carlo Primer, A Practical Approach to Radiation Transport* (the *Primer*), and its companion *A Monte Carlo Primer Volume 2, Solutions to
Exercises* (the *Solutions*),
provide a basic description of the Monte Carlo method as applied to radiation
transport plus an in-depth explanation of techniques commonly used to exploit
the method in realistic situations.

The
Probabilistic Framework Code (PFC), a *Primer* and
the *Solutions*. The following example illustrates the power
of the

Consider a
point, isotropic source of neutral, monoenergetic particles (which we can
consider for the moment to be neutrons) located at the center of a sphere of
pure scattering material that is ten mean free paths thick to the source
particles. Assume that the sphere is
surrounded by vacuum. Assume further
that inside the sphere the particles scatter isotropically
in the laboratory coordinate system and that their mean free path is
constant. We wish to determine the
angular dependence of the current of particles escaping from the sphere. For a discussion of the mean free path of a
particle, isotropic scattering in the laboratory system, the vacuum boundary
condition, and particle current see Chapter 3 of the *Primer* or see other references on radiation transport.

Let us assume
the source is located at the origin of a Cartesian coordinate system and that
the mean free path of the source particles is 1 cm. Because the sphere is a pure scattering
material we know that all of the source particles will escape. Furthermore, because the sphere is relatively
thick to the source radiation, and because the particles scatter isotropically within the sphere, the particle flux at
points far from both the source and the outer boundary should be nearly
isotropic. We expect the vacuum boundary
condition to perturb this isotropy, however, and thus we expect the escaping
current of particles to be anisotropic.
To understand the following problem description we recommend the reader
refer to Chapter 5 of the *Primer*. As is often the case with *Primer*.

In order to
use PFC to score the current of particles leaking from the sphere of scattering
material we will modify subroutine ‘Bdrx’ to score
the leaking particles as a function of the cosine of the angle between the
particle trajectory, given by the unit vector , and the outer normal to the sphere at the point of escape,
given by the unit vector **n**. This geometry is described on page 199 of the
*Primer*. The dot product **p**· is given by eqn 3.48
of that volume, where **p** = r**n** and r is the
radius of the sphere in the present calculation. Thus to normalize the result shown in eqn
3.48 for this calculation the dot product must be divided by the radius of the
sphere. We define ten equal intervals of
the cosine over the range [0,1] and score the leaking particles in these
bins. We must add common ‘Stats’ and the
associated double precision declarations (see subroutine ‘Stats’) to subroutine
‘Bdrx’ as given in Table 5.19 of the *Primer.*
Then, following statement number 15 of ‘Bdrx’
we add the following (or equivalent) lines of Fortran:

i=INT((x*u+y*v+z*w)+1)

bscore(i) = bscore(i)+1.

bsumsq(i) = bsumsq(i)+1.

In addition to these changes in subroutine ‘Bdrx’ we need to make a few changes in subroutine ‘Stats.’ In statements 22 and 24 in the PFC library version of ‘Stats’ given in Table 5.9 of the Primer we replace the subscript ‘1’ with an ‘i.’ Prior to statement 22 we add

DO i = 1,10

We then delete statements 25 through 27 and replace them with

END DO

Using the
geometry and cross section definitions shown in Tables 5.23 and 5.24 of the *Primer* we are now ready to solve the
present problem. Running
start particles with a
start random number of one gives the results shown in the table
below. It is clear from these results
that the leakage is a maximum at a cosine of one and a minimum at a cosine of
zero; i.e., that the maximum in the leakage current per unit cosine interval
occurs in a direction normal to the surface of the sphere. This is clearly shown in the normalized polar
plot of similar results obtained using a large number angular intervals that is
presented in the figure below. In this
plot the radius r of the curve, , represents the normalized probability of a particle leaking
from the sphere of scattering material at the angle , where is the angle between
the particle trajectory and the outer normal to the sphere at the point of
escape. The radial values are plotted at
the center of the associated cosine intervals.
The leakage probability is normalized so that the maximum probability is
one. Because the uncertainties are
relatively small in these results such uncertainties are not indicated in the
figure, although some statistical fluctuations are apparent. We see that, not only is the leakage
probability a maximum for directions normal to the sphere, but the probability
decreases rapidly as the angle to the normal increases.

Let us suppose one could view this sphere from a distance with an imaging sensor that indicates the magnitude of the flux of particles striking each pixel in the sensor. Because the sphere is in a vacuum, the sensor would measure the number of particles leaking through the surface of the sphere as a function of position on the two-dimensional projection of the sphere perpendicular to the line of sight between the sphere and the sensor. This two-dimensional projection is a disk. The fluence measured at or near the center of the disk consists of particles that escape from the sphere in a direction almost normal to the sphere, while that measured at the outer edge of the disk consists of particles that escape from the sphere in a direction nearly tangent to the sphere. Therefore, based on the present results the apparent brightness of the disk would be greatest at the center and would decrease with distance from the center of the disk to a minimum at the edge. Although the analogy is not exact, when the disk is the sun, the escaping particles are visible photons, and the sensor is either a photographic plate or the human eye, this phenomenon is known as limb darkening.

Upper Cosine Boundary |
Leakage Probability |
Standard Deviation |

0.1 |
0.003021 |
5.5E-5 |

0.2 |
0.01385 |
0.00011 |

0.3 |
0.02812 |
0.00017 |

0.4 |
0.04553 |
0.00021 |

0.5 |
0.06751 |
0.00025 |

0.6 |
0.09350 |
0.00029 |

0.7 |
0.12442 |
0.00033 |

0.8 |
0.16074 |
0.00037 |

0.9 |
0.20436 |
0.00040 |

1.0 |
0.25891 |
0.00044 |

This problem
is similar to some of those discussed in the *Primer* and the *Solutions*. As is done in those volumes, let us consider
some of the ramifications of the present calculation. In the scoring done here, any particles
escaping from the sphere without experiencing a collision are combined with the
scattered particles in the most forward-directed angular interval. However, the uncollided particles are exactly
radially directed in this calculation and thus could
be expressed as a delta function at in the polar coordinates of the figure. The curve of the angular dependence should
thus properly have a narrow peak normal to the sphere. The area under this peak will depend on the
fraction of escaping particles that are uncollided. In the present calculation, for a sphere ten
mean free paths in radius, this fraction will be small.

If we vary the radius of the sphere we expect the shape of the angular leakage to change. As the sphere becomes smaller, the fraction of escaping particles that are uncollided will increase. Thus the narrow peak in the angular dependence of the leaking particles at will increase relative to the scattered contribution. In addition, the average number of collisions experienced by an escaping particle will decrease and thus the anisotropy of the scattered contribution will become increasingly forward peaked. In the opposite limit, as the radius of the sphere becomes much greater than the mean free path of the particles, the problem can be approximated by an infinite half-space of scattering material. This approximation is commonly made for calculating the angular dependence and polarization of photons escaping from stars.

In a physically realistic configuration a source of particles can never be a point. Instead, the source will have a non-zero radius and the uncollided contribution to the particle leakage in a configuration such as that modeled here cannot be a delta function in angle. For the present calculation the angular dependence of the escaping particles will vary not only with the radius of the sphere but also with the radius of the source. In practice the source may vary in strength per unit volume with distance from the center of the sphere.

All of these
variations can be easily incorporated into the Monte Carlo model used in this
example calculation, and the characteristics of the angular leakage can be
examined as a function of the problem parameters. However, because all Monte Carlo results
contain statistical uncertainties (over and above any systematic uncertainties)
comparisons between the results obtained from similar Monte Carlo calculations
can be misleading. In many cases it is
useful to make use of the technique of correlated sampling to isolate and
quantify the differences between such results.
Correlated sampling is discussed in Chapter 9 of the *Primer*.