The method of invariant embedding has been applied to calculations of gamma-ray buildup factors for point isotropic sources in infinite homogeneous media up to depths of 100 mean free paths (mfp) without bremsstrahlung. A comprehensive survey of buildup factors was performed to estimate errors due to energy, angle and space meshes adopted in the transport calculations by the present method. It is confirmed numerically that the exposure buildup factors can be calculated up to depths of 100 mfp with an error less than 10% by using the present method. The exposure buildup factors for water, iron and lead for typical source energies of 10 MeV, 1.0 MeV and 0.1 MeV are provided up to depths of 100 mfp. Those buildup factors are found to agree well with other data including the moments method calculations and the Monte Calro calculations by EGS4. These results indicate that the method of invariant embedding is able to provide gamma-ray buildup factor up to depths of 100 mfp with an accuracy enough to be used in shielding calculations.

https://www.tandfonline.com/doi/pdf/10.1080/18811248.2002.9715225

Calculation of Gamma-Ray Buildup Factors up to Depths of 100 mfp by the Method of Invariant Embedding, (I) : Analysis of Accuracy and Comparison with Other Data

Point kernel codes that simulate gamma-ray transport often use build-up factors to take scattered photons into account. This study introduces a new method, for computing multi-layer shield build-up factors. This method, based on an empirical formula for calculating double-layer shield build-up factors, is iterative. For an N-layer shield, each iteration of the method treats the first and the second layer of the shield. It replaces these layers by a single equivalent layer composed of an appropriate material and, hence, it turns the N-layer shield into an (N - 1)-layer shield. In order to determine the equivalent layer of an appropriate material, a neural network approach is developed: some neural networks trained on a large set of various configurations provide the equivalent material for any double-layer configuration. The method is implemented into MERCURE-6.3 straight-line attenuation code and is validated by comparison between MERCURE-6.3 results and reference data for one-dimensional geometries. Reference data obtained from transport calculations performed using the Sn transport code TWODANT. The comparisons prove the accuracy and sturdiness of the method.

An iterative method for calculating gamma-ray build-up factors in multi-layer shields.

An improved method to calculate the gamma-ray buildup factors including bremsstrahlung has been developed. The exposure buildup factors with bremsstrahlung were computed by the present method for lead, iron and water at the source energy of 10.0 MeV up to depths of 100 mfp. The accuracy of the present method was checked by comparison with the calculations by use of EGS4. Excellent agreement was obtained between the calculations by both methods about the exposure buildup factors per energy (energy spectrum of transmitted photons) for lead up to depths of 10 mfp and the ratio of the exposure buildup factor with bremsstrahlung to that without bremsstrahlung for lead, iron and water up to depths of 40 mfp. It is confirmed that the present method has an accuracy sufficient to be used to the generation of an improved set of gamma-ray buildup factors including bremsstrahlung.

https://www.tandfonline.com/doi/pdf/10.1080/18811248.2003.9715349

Calculation of Gamma-Ray Buildup Factors up to Depths of 100 mfp by the Method of Invariant Embedding, (II) : Improved Treatment of Bremsstrahlung

A New Monte Carlo code (EBUF) is developed to calculate improved point isotropic photon exposure buildup factors in media. Variance reduction techniques are used to perform calculations up to 60 mean free paths. EBUF accounts for coherent scattering and bound-electron Compton scattering. Bremsstrahlung photons and annihilation gamma rays as well as K and L X-rays are considered. The most recent cross-section data are used. The EBUF exposure buildup factors compare very well with those from the ANS-6.4.3 Working Group (ANS-6.4.3) when the same initial conditions are assumed: no coherent scattering, free-electron Compton scattering, and only K X-ray fluorescence. Next, a detailed physics treatment is used to calculate a representative set of exposure buildup factors in aluminum, iron, lead, water, air, and concrete over a large energy range (20 keV to 10 MeV). The effects of L X-rays are shown for lead at low energy. The EBUF factors are in good agreement with the SN1D code results for low-Z media. ...

New Photon Exposure Buildup Factors

The DANTSYS discrete ordinates computer code system is applied to quantitative estimation of water kerma rate distributions in the vicinity of discrete photon sources with energies in the 20- to 80...

Two-Dimensional Discrete Ordinates Photon Transport Calculations for Brachytherapy Dosimetry Applications

Improvements of gamma-ray transport calculations in Sn codes aim at taking into account the bound-electron effect of Compton scattering (incoherent), coherent scattering (Rayleigh), and secondary s...

Improvement of Gamma-Ray Sn Transport Calculations Including Coherent and Incoherent Scatterings and Secondary Sources of Bremsstrahlung and Fluorescence: Determination of Gamma-Ray Buildup Factors

Exposure and energy absorption buildup factors for aluminum, iron, lead, and water are calculated by the SN1D discrete ordinates code for an isotropic point source in a homogeneous medium. The calc...

Determination of Point Isotropic Buildup Factors of Gamma Rays Including Incoherent and Coherent Scattering for Aluminum, Iron, Lead, and Water by the Discrete Ordinates Method

This study proposes a new approximating formula for calculating gamma-ray buildup factors in multilayer shields. The formula combines the buildup factors of single-layer shields with products and quotients. The feasibility of the formula for reproducing the buildup factors was tested by using point isotropic buildup factors calculated with the SNID discrete ordinates code as reference data. The dose buildup factors of single-, double-, and multilayer shields composed of water, aluminum, iron, and lead were calculated for a spherical geometry in the energy range between 10 MeV and 40 keV and for total thicknesses of up to 30 mean free paths. The calculation of the buildup factors takes into account the bound electron effect of Compton scattering (incoherent scattering ), the coherent scattering, the pair production, and the secondary sources of bremsstrahlung and fluorescence. The tests have shown that the approximating formula reproduces the reference data of double-layer shields very well for most cases. With the same parameters and with a new physical consideration that takes into account in a global way the degradation of the gamma-ray energy spectrum, the buildup factors of three- and five-layer shields were also very well reproduced.

P. Ridoux

Papers

Improvement of Gamma-Ray Sn Transport Calculations Including Coherent and Incoherent Scatterings and Secondary Sources of Bremsstrahlung and Fluorescence: Determination of Gamma-Ray Buildup Factors

Determination of Point Isotropic Buildup Factors of Gamma Rays Including Incoherent and Coherent Scattering for Aluminum, Iron, Lead, and Water by the Discrete Ordinates Method

A New Approximating Formula for Calculating Gamma-Ray Buildup Factors in Multilayer Shields