SUMMARY

This paper discusses the challenges of using Monte Carlo calculations for radiation shielding analysis. Complex shielding scenarios often yield unreliable outcomes due to inadequate particle sampling. To enhance the efficiency of analog Monte Carlo simulations, the paper discusses variance reduction techniques offered by the MCNP code, specifically the weight window method for optimizing shielding calculations. However, for intricate shielding cases, especially with deep penetration, the existing MCNP’s weight window generator is less effective. To address this, an automated code named TRAWEI was developed, utilizing recursive Monte Carlo methodology to generate optimal weight parameters in a single run, thus reducing computation time. This paper verifies TRAWEI-generated weights using a simple reactor model and compares MCNP results utilizing TRAWEI-generated weights against those using MCNP’s weight window generator.

KEYWORDS

Shielding calculations, MCNP code, Variance reduction techniques, Weight window, TRAWEI code

1.     INTRODUCTION

Radiation shielding is a critical aspect of nuclear engineering and plays an important role in ensuring the safety of people, the public, and the environment near nuclear facilities. Monte Carlo methods are commonly used to analyze and create efficient radiation shielding solutions. By simulating the transport of radiation particles through shielding materials, Monte Carlo calculations offer a detailed and precise means of assessing shielding effectiveness, optimizing design parameters, and minimizing probable risks. Monte Carlo N-Particle Transport (MCNP) code [1] developed by Los Alamos National Laboratory serves as the established standard code utilized for performing such calculations due to its adaptability and reliability. However, challenges arise, particularly in circumstances involving deep penetration, in which particles pass through thick shield materials to reach target areas. Because of the substantial absorption and scattering, a large number of particles do not reach far into the desired areas. This often leads to insufficient particle sampling in the desired target areas, causing significant statistical fluctuations in analog Monte Carlo results. To enhance the accuracy and efficiency of these simulations, the weight window (WW) variance reduction technique is frequently used. This method aims to improve particle sampling by controlling the population of particles entering specific region through splitting and Russian roulette techniques [2]. Typically, users use an automatic inbuilt MCNP weight window generator (MCNP-WWG) to establish weight window parameters. However, for intricate shielding scenarios such as deep penetration, the generator necessitates multiple iterations of Monte Carlo calculations to produce an optimal set of variance reduction parameters, leading to significant computational demands. In response, several researchers proposed alternative methods, such as deterministic methods [3,4,5], MAGIC method, reduce density method [6] etc., to evaluate the weight window values implicitly and explicitly. However, each of these methods has its own set of advantages and disadvantages. Hence, this study aims to address the issue of multiple iterations linked with the

standard MCNP-WWG by introducing an automated tool utilizing recursive Monte Carlo (RMC) methodology [7, 8, 9]. This tool aims to generate optimal weight values in a single iteration, thereby minimizing computational time. Incorporating this methodology into Monte Carlo codes offers a more efficient approach to handling complex shielding scenarios. This paper includes an overview of weight window techniques, the theoretical foundation of the RMC methodology, the verification of the developed algorithm using a simple reactor model, and future prospects.

2.     THEORY

2.1.   WEIGHT WINDOW TECHNIQUE

The weight window (WW) is a crucial method employed in Monte Carlo simulations to effectively manage variance and enhance simulation efficiency. This method involves space-energy (or time)-dependent splitting and Russian roulette techniques to regulate particle populations within designated regions [2]. To implement the WW technique, the user specifies the lower weight bound and the width of the weight window for each region or cell. If the weight of a particle exceeds the upper weight bound, it is split such that the splitting particles have a weight within the weight window. Conversely, if the weight of a particle reaches the lower weight bound, Russian roulette is played, and the weight of the particle is either increased to be within the weight window or the particle is eliminated. The determination of weight window parameters typically relies on the reciprocal of particle importance. These values can either be manually inputted or automatically generated through a weight window generator. Particle importances are characterized as the anticipated scores generated by a unit weight particle upon entering a specific region. The MCNP code incorporates a built-in weight window generator, which automatically computes the lower weight window bound. By default, the upper weight window bound is set to five times the generated lower weight window bound [10].

2.2.   RECURSIVE MONTE CARLO METHODOLOGY

The concept of recursive Monte Carlo (RMC) method, initially proposed by Goldstein and Greenspan [11], has been adapted and revised to address deep penetration challenges in this study [7, 8, 9]. In the RMC methodology, the importance of individual cells or regions is determined by solving the forward transport equation and sequentially defining the cells one by one as source cells. The calculation process begins with the lowest energy particles within the designated target area and progresses to adjacent regions. If a particle reaches the boundary of a previously assessed region, the computation halts, and the known importance is utilized for the preceding evaluation. Once the importance for all regions within the same energy level is assessed, the procedure repeats for particles with higher energy levels. However, while calculating higher-energy particles, there’s a chance of scattering (elastic and inelastic), causing the particle’s energy to drop to lower levels. In such cases, the previously determined importance of lower-energy particles can be applied to subsequent assessments. Once importance is determined for all regions across all energy levels, weights are computed by taking the reciprocal of the importance. This approach offers the benefit of speeding up weight calculation by making use of previously estimated importances from nearby regions or lower-energy groups. It is an automated method that does not require any iterative processes.

The RMC method has been included in the TRAWEI code, an in-house code developed at the Helmholtz-Zentrum Dresden-Rossendorf (HZDR). TRAWEI is specifically designed to generate weights for the TRAMO code, which is used for solving particle transport problems [12, 13]. In the context of this study, the aim is to utilize the weight values produced by TRAWEI for MCNP calculations in a single step. TRAWEI has the capability to replicate MCNP meshes, thus facilitating the creation of an MCNP mesh within TRAWEI for weight calculations. Once the weights are generated, they are subsequently used in MCNP for the actual calculation process.

3.     VERIFICATION OF THE METHODOLOGY

In order to assess the effectiveness of weight values produced by the TRAWEI (RMC) code, a simple 3D reactor model with 90º symmetry was prepared [9], closely resembling a PWR model (cf. Figure 1). The model consists of 13 axially aligned cylindrical bodies interleaved with different shield materials. Its dimensions include an outer radius of 330 cm and a height of 120 cm, with a uniform volumetric neutron source positioned within the radial range of 0-150 cm. Using this model, MCNP simulations with an F1

tally was used to calculate the net neutron current across the outer cylinder surface of the biological concrete shield (at 330 cm). The MCNP simulations were performed in three ways:

• Analog MCNP simulation without weight window,
• MCNP simulation (MCNP-WWs) using MCNP-generated weight parameters,
• MCNP simulation (TRAWEI-WWs) using weight parameters generated by the TRAWEI (RMC)

To perform the MCNP simulation with weight window, a set of weight parameters for two energy groups (thermal and high) was generated using MCNP-WWG and the TRAWEI code, respectively, specifically optimized for the outer surface of the biological shield. Furthermore, the analysis of the MCNP (TRAWEI- WWs) simulation involved a comparison of its computational efficiency with that of analog MCNP and MCNP (MCNP-WWs) simulations. Typically, computational efficiency is assessed using a Figure of Merit (FOM), which is expressed as [1]: