# Point-Kernel Code Development for Gamma-Ray Shielding Applications
## 5 The PK Program Verification with MAVRIC
### Volumetric Gamma Slab with a Slab Shield  
These calculations were performed for several combinations of thicknesses and slab shield materials, as presented in Figure 4, so the presented case is for the volume source emitting 1 phot/s of energy 1 MeV in the water layer of 25 cm followed by 25 cm of ordinary concrete. The height and width of slabs were 200 cm, giving in turn \(1\times 10^{-6}\) phot/cm\({}^{3}\)/s in water. The gamma flux was calculated on the slab shield external surface, considering self-absorption in the water. The MAVRIC results of the adjoint gamma flux and weight windows of group 12 (1.0-0.8 MeV) in the \"v7_27n19g\" library show the \"recipe\" for MC biasing, i.e., photon splitting is more intense in the vicinity of the detector (Figure 19). It is clear how 1 MeV photons are attenuated by penetrating the shield, but contribution from other groups is necessary for total photon flux. For the PK approach, this is a composite shield 50 cm in thickness, with an uncollided flux of 2.72 \(\times\) 10\({}^{-8}\) phot/cm\({}^{2}\)/s. The obtained buildup factors were 15.9, 14.2 and 10.0 for the Goldstein, Blizard, and Broder methods, respectively. Taking the first one as the most conservative, the buildup flux was 4.32 \(\times\) 10\({}^{-7}\) phot/cm\({}^{2}\)/s; this is comparable to the uncollided MC solution of 3.39 \(\times\) 10\({}^{-8}\)\(\pm\) 0.04% and total MC flux of 5.51 \(\times\) 10\({}^{-7}\)\(\pm\) 0.9%. This PK-to-MC flux differ  
Figure 19: MAVRIC results (group 12, 1.0–0.8 MeV) for the slab source with self-absorption: (**a**) adjoint \(S_{\rm N}\) gamma flux; (**b**) weight windows; (**c**) MC photon flux; (**d**) MC photon flux relative error. The rightmost layer is part of the global unit filled with dry air. Figure 18: Total MC flux distribution of 1 MeV photons leaving spherical surface using CADIS technique: (**a**) mesh tally total gamma flux; (**b**) relative errors of mesh tally flux. ence is only \(-21.6\%\), but PK total CPU time was a fraction of second (\(>\)1 ms) compared to approximately 12 min for MAVRIC simulation, giving significant speedups for a little loss in accuracy. Of course, increasing the number and/or thickness of shielding layers can generally worsen PK agreement with MC. This behavior is shown next for the concrete layer extending from the initial 25 cm to 100 cm using steps of 25 cm. Figure 20 shows a comparison of PK and MC total gamma flux at external composite shield surface while relative errors of flux components are shown in Figure 21.  
One can notice higher values of MC flux components relative to PK and correspondingly increasing relative errors with increasing concrete layer thickness. However, this PK flux is still an acceptable approximation taking that absolute flux value is order of \(1\times 10^{-13}\) phot/cm\({}^{2}\)/s at a composite shield surface. Actually, this flux difference is a consequence of various composite \(B\)-factor models, which are compared and shown in Figure 22. The Goldstein model of shield homogenization proved to be the most conservative one, so it was used for all PK buildup flux calculations. For an increasingly thicker concrete shield, these composite \(B\)-factor models converge to the same value. Figure 21: PK to MC relative error for 1 MeV gamma ray in concrete shield (volume source). Figure 20: Total gamma flux (PK and MC) from 1 MeV gamma-ray volume source. Lastly, the influence of layer ordering on composite \(B\)-factor and MC simulation was examined. The specific volume sources were 5 cm lead (\(S_{A}\) = 5 \(\times\) 10\({}^{-6}\) phot/cm\({}^{2}\)/s) and 50 cm water (\(S_{A}\) = 5 \(\times\) 10\({}^{-7}\) phot/cm\({}^{3}\)/s). All other control and numerical MAVRIC parameters remained the same. MAVRIC results of total gamma flux with relative errors for lead-water are shown in Figure 23a,b while water-lead results are given in Figure 23c,d.