# Point-Kernel Code Development for Gamma-Ray Shielding Applications
## 5 The PK Program Verification with MAVRIC
### Isotropic Gamma Disc with a Slab Shield  
A series of calculations were performed for an isotropic gamma disc of radius 50 cm and intensity 1 phot/s, emitting 1.27 \(\times\) 10\({}^{-4}\) phot/cm\({}^{2}\)/s. The slab shield thickness varied from 1 cm to 50 cm, and two typical shielding materials were considered (iron and concrete). The center of the slab shield was always positioned symmetrically between the source and the point detector at a fixed distance of 100 cm from the source. Two gamma lines of 0.5 MeV and 10 MeV were used independently for source photons, thus enveloping available data on \(B\)-factors. The obtained PK solution, with sensitivity study on material, slab shield thickness and gamma-ray energy, was then verified using the MAVRIC shielding sequence of the SCALE6.1.3 code package [19]. The Monaco MC module was used with following options: \"v7-27n19g\" cross section library, gamma source of 1 phot/s, point detector 100 cm from the gamma disc, and 500 batches with 1000 photons per batch. The CADIS variance reduction (VR) methodology [19; 33] was used only for thick shields, especially for iron, while analog MC was used for all others. The average MC relative error on the point detector for all simulation cases was well below 0.1% and CPU time was about 5 min, equal to the MC and VR parts. The selected results of 0.5 MeV gamma ray and iron slab shield are presented next. Figure 10 shows a good agreement of PK and MC buildup flux, with PK-to-MC relative errors shown in Figure 11 (with uncollided and buildup parts). Since bulk metal components like RPV will have thickness up to circa 25 cm, these results confirm PK applicability in practical shielding studies, even in high attenuating iron-based shields. Figure 11 also shows different \"conservatism\" trends in curves with increasing shield thickness, i.e., the uncollided PK flux is relatively higher than the MC part, but final MC buildup flux proved to be higher. For example, PK buildup flux is 50% lower than MC for 30 cm iron slab, but since the flux value is on the order of 1 \(\times\) 10\({}^{-12}\), this is an acceptable deviation. The inevitable differences have an origin in the MC process,  
Figure 9: The main window of PK GUI program for gamma line and slab shield. capturing many physical interactions of 0.5 MeV photons that are neglected in a simple PK integration technique. The comparison of \(B\)-factors in PK calculations is shown in Figure 12. A good agreement and similar trends in numeric and analytic \(B\)-factors were obtained, with \(B\)-factor values of over 200 for an iron slab of 50 cm. The high-valued \(B\)-factor is implying intensive gamma-ray scattering inside the iron shield, so relative ratio of uncollided to buildup flux is expected to be small (Figure 13). One can notice similar uncollided-to-total flux ratio trends in PK and MC solutions, ranging over 50% in 1 cm-thin shield to values well below 10% starting with 10 cm-thick shield. Figure 11: PK to MC relative errors of buildup flux for 0.5 MeV gamma rays. Figure 10: Comparison of PK and MC buildup flux. Finally, the mesh tally plot of the MC total flux with relative errors is shown in Figure 14 for 0.5 MeV photons and a 50 cm iron shield. The optimization of gamma-ray transport in the slab shield using the CADIS method is clear, preferentially splitting photons in forward-peaked direction towards the detector. This is a simple case of a deep penetration shielding problem with significant gamma flux attenuation for which effective VR technique is necessary. The CADIS methodology is based on the adjoint \(S_{N}\) solver Denovo [34; 35], and will optimize the final MC flux only at the point detector location, leaving other spaces weakly populated or with high statistical errors (see Figure 14b). In addition, a point detector has a specific MC estimator form [19] for which ray tracing is used at every photon interaction site, generally increasing the computational time as more details (surfaces) are added in the simulation model.