# Point-Kernel Code Development for Gamma-Ray Shielding Applications
## 2 The Point-Kernel Approach in Radiation Shielding  
The PK method has a long history in radiation shielding problems and is mostly used for gamma-ray simulations, since complexities of the neutron cross sections demand alternative computational approaches such as MC or deterministic method of discrete ordinates (\(S_{N}\)) [20]. The gamma source of arbitrary shape (linear, planar or volumetric) can be represented by an equivalent number of point sources, so total point detector response can be found as a summation, i.e., numerical integration of radiation received from those point sources. The exponential attenuation kernel is used for such point source, and depends on source-detector distance \(R\), gamma-ray energy \(E_{\gamma}\) and shielding material properties (mass attenuation coefficient \(\mu/\rho\)) [1,2]. For an isotropic point gamma emitter of \(S_{p}\) intensity (phot/s) located in a uniform medium with linear attenuation coefficient \(\mu\) (cm\({}^{-1}\)), the uncollided gamma flux at a distance \(R\) from the source is given as  
\[\phi(R)=\frac{S_{p}}{4\pi R^{2}}\exp(-\mu R). \tag{1}\]  
The shield thickness is given as \(\mu R\), which is equal to the number of gamma-ray mean free paths inside the shield. The PK attenuation kernel is defined as the point detector response from a point source emitting one particle per second, so it has the following form:  
\[G(R)=\frac{\exp(-\mu R)}{4\pi R^{2}}. \tag{2}\]  
However, every practical shield is thick enough to produce multiple scatterings of gamma rays, increasing their probability of reaching the detector. The flux correction is introduced by a multiplicative buildup factor \(B>1\), so the corresponding kernel is given as  
\[G(R)=\frac{\exp(-\mu R)B(\mu R,E_{\gamma})}{4\pi R^{2}}. \tag{3}\]  
This corrective \(B\)-factor is crucial in shielding calculations, and tables and curves of \(B\)-factors are readily available as functions of gamma-ray energy and shield thickness. To permit an analytical integration of general, distributed gamma source, \(B\)-factors are typically expressed in parametric form with tables of fitting coefficients that depend on shielding material and gamma-ray energy. In our work, we used a well-known Taylor form given as  
\[B(\mu R,E_{\gamma})=A\exp(-\alpha_{1}\cdot\mu R)+(1-A)\exp(-\alpha_{2}\cdot\mu R), \tag{4}\]  
where \(A\), \(\alpha_{1}\) and \(\alpha_{2}\) are readily available from the classic literature on the gamma radiation shielding [1,2]. The total or buildup flux would be calculated by merely multiplying Equation (1) using the Taylor or alternatively Berger form of the \(B\)-factor given as [1,2]  
\[B(\mu R)=1+\mu R(C\cdot e^{D\cdot\mu R}), \tag{5}\]  
where tabulated parameters \(C\) and \(D\) again depend on the gamma-ray energy and shielding material. The first term represents the uncollided, and the second term the scattering contribution. These forms balance accuracy with simplicity to permit analytical integration, while more complicated forms, such as the modified geometric progression of Y. Harima [21], will be fully implemented in future program versions. A comprehensive review on the current status of \(B\)-factors calculations and applications can be found in [22].