Theory and Methods
Point-Kernel Method
ZapMeNot calculates the uncollided photon flux using the point-kernel equation:
\(\phi(r) = \frac{S}{4\pi r^{2}}e^{-\mu r}\)
where
\(\phi(r)\) = space dependent uncollided photon flux
\(S\) = point isotropic source of mono-energetic photons
\(r\) = distance from source location to flux location
\(\mu\) = total attenuation coefficient for medium
When multiple materials exist in the model, sum \(\mu_{i} r_{i}\) over all of the material regions between the source and the flux location.
Uncollided Photon Exposure Rate
The exposure rate from uncollided photons can be calculated as:
\(D_{u}(r) = \phi(r) \Re\)
and
\(\Re = 1.835\cdot 10^{-8} E\left ( \frac{\mu _{en}\left ( E \right )}{\rho } \right )_{air}\)
where \(\Re\) has units of R cm2,
\(E\) is the photon energy in MeV, and
\(\mu _{en}/\rho\) is the mass energy deposition coefficient of air in cm2/g.
Buildup Factors
The buildup factor \(B(r)\) is defined as the ratio of the dose from all photons (collided and uncollided) to the dose from uncollided photons. The buildup factor will vary with distance from the source location and the material traversed. As such the total exposure from all photons can be calculated as:
\(D_{T}(r) = D_{u}(r) B(\mu r)\)
where
\(D_{T}(r)\) = the space dependent dose from collided and uncollided photons
\(B(\mu r)\) = the space dependent buildup factor
Buildup factors are generally derived from analytical results and are approximated by mathematical functions. The form of approximation used in ZapMeNot is the geometric progression (GP) form. The GP coefficients for calculating the buildup factor as provided in ANSI/ANS-6.4.3-1991 “Gamma-Ray Attenuation coefficients and Buildup Factors for Engineering Materials.”
Note that the buildup factors in GP form as provided in ANSI/ANS-6.4.3 are only valid for distances up to 40 mean free paths. This is generally not a limitation, as the uncollided flux at some energy E traversing 40 mean free paths has been reduced by a factor of at least \(10^{-13}\). However, extrapolation of buildup factors out to 80 mean free paths has been implemented in ZapMeNot based on methods described in “Development of New Gamma-Ray Buildup Factor and Application to Shielding Calculations” by Harima, et al., 1991.
Quadrature
The point kernel method can be applied directly for point sources. However, distributed sources required an additional level of modeling. The point kernel method is extended to distributed sources using the method of numerical quadrature. Put simply, the source volume or area is subdivided into a spatial quadrature and the source within a spatial mesh is represented as a point source. Summing the dose responses resulting from the point sources returns the dose response resulting from the distributed source. The spatial mesh is not required to be uniform. The quadrature weights are normally computed as follows:
\(W_{k}=\frac{V_{k}}{\sum_{k=1}^{N}V_{k}}\)
where
\(W_{k}\) = the source mesh weight
\(V_{k}\) = the volume or area of the kth mesh
Multiple Photon Energies
A typical photon source will emit photons with a number of discrete photon energies, each with a unique intensity. These photon sources are modeled by performing a numerical quadrature analysis for each photon energy with an associated energy-dependent buildup factor. Photon energies are limited to a range of 15 keV to 15 MeV to ensure applicability of the attenuation coefficients and buildup factors presented in ANSI/ANS-6.4.3-1991 “Gamma-Ray Attenuation coefficients and Buildup Factors for Engineering Materials.”
Frequently the number of discrete photon energies can be in the 100’s. However, many of these photons energies are low enough that the photon contribution to dose is small. Many of the photon energies may also have a very low intensity due to a small decay branching fraction. The evaluation is therefore simplified by limiting the number of discrete photon energies to 40. A 40 photon energy-group structure is used in the event that a source has more than 40 discrete photon energies. The energy-group structure is linearly distributed between the highest and lowest photon energies in the source. The photon intensity of each energy group is determined such that the energy flux of photons within the group is preserved.