Evaluation of Material Attractiveness to Non-State Actors of Various Nuclear Materials in Thorium Fuel Cycles

Evaluation of Material Attractiveness to Non-state Actors of Various Nuclear Materials in Thorium Fuel Cycles

By Eva Lisowski

Submitted to the Department of Nuclear Science and Engineering

In Partial Fulfillment of the Requirements for the Degree of Bachelor of Science in Nuclear Science and Engineering

At the Massachusetts Institute of Technology

May 2020

Signature of Author: ………………………………………………………………………….. Department of Nuclear Science and Engineering May 12, 2020

Certified by: …………………………………………………………………………………... Benoit Forget Associate Department Head and Professor of Nuclear Science and Engineering Thesis Supervisor

Certified by: …………………………………………………………………………………... Michael Short Associate Professor of Nuclear Science and Engineering Undergraduate Chair

2 Evaluation of Material Attractiveness to Non-state Actors of Various Nuclear Materials in Thorium Fuel Cycles

Eva Lisowski

Submitted to the Department of Nuclear Science and Engineering on May 12, 2020, in partial fulfillment of the requirements for the degree of Bachelor of Science in Nuclear Science and Engineering

Abstract

Thorium-based fuel cycles for advanced nuclear reactors have been explored to utilize thorium resources in nations where uranium is scarce, increase fissile material utilization, and enhance proliferation resistance. As a stepping stone, thorium-based fuels have been paired with pressure tube heavy water reactors because of their high neutron economy and online refueling capability. However, thorium fuel cycles have raised proliferation concerns regarding the presence of U-233 following the irradiation of fuel bundles. The presence of Pa-233, which decays into pure U-233, and the creation of Pu-239 due to the neutron capture of U-238 in mixed lightly-enriched uranium (LEU)/Thorium fuels, are also causes for proliferation concern. Based on a method developed in a previous study, the material attractiveness to non-state actors of fissile materials present in a 40%LEU/60%Th fuel lattice concept was evaluated for six metrics: bare critical mass (BCM), heat content, net weight, acquisition time, dose rate, and processing time & complexity. The lattice, composed of 35 fuel pins and a central ZrO2 displacer rod, was modeled and depleted in the OpenMC reactor physics software, over a range of burnups up to 40 MWd/kg followed by two years of cooling. It was found that the material attractiveness of uranium isotopes in the irradiated fuel bundle was Very Low due to the high fraction of U-238 present in the fuel and the assumed lack of enrichment capabilities among non-state actors. However, for a state with basic enrichment capabilities, this fuel may be attractive. The attractiveness of plutonium isotopes was also found, as expected in a thorium-cycle, to be Very Low. However, the low BCM and heat content of this mixture reveals that it could be attractive to states that can easily acquire the material and do not need to rely on the theft of many fuel bundles to acquire an IAEA Category I quantity of material. Further investigation of the material attractiveness to states is required. Material attractiveness evaluations are important to informing future decisions regarding which fuel bundles to select when designing advanced reactor facilities and developing methods to safeguard them.

Thesis Supervisor: Benoit Forget

Associate Department Head and Professor of Nuclear Science and Engineering

3 Acknowledgments

I would like to express special appreciation and gratitude to the people below who made the research and writing process successful and assisted me at every step towards my goal:

Professor Forget, my thesis supervisor, for his vital support and assistance, for spending time to help me develop and better understand my calculation methodology, and providing me with both essential and bonus resources to allow me to successfully complete my project, and expand my knowledge and confidence in the field of reactor physics, and especially for having faith in my ability to meet project deadlines despite the extenuating circumstances under COVID-19.

Professor Hiroshi Sagara, my lab supervisor during my fall 2019 semester study abroad at the Tokyo Institute of Technology, for welcoming me as a member of his lab, for introducing me to his research in nuclear non-proliferation and inspiring me to write a thesis based on his lab’s material attractiveness methodology, and for giving me the foundation for a successful research project.

The MIT Computational Reactor Physics Group, led by Professor Forget, for supporting me financially by providing me with the computational resources needed to perform complex reactor physics calculations, and Google, which donates free computing credits for use of its Google Cloud Platform Compute Engine.

Dr. Jiankai Yu, a postdoctoral fellow in the MIT Computational Reactor Physics Group, for his patience and sympathetic attitude at every point during my project, for training me in the use of OpenMC, helping me troubleshoot many software difficulties, and answering my many questions throughout the entire computation process.

Professor Mike Short, my academic advisor, for his support throughout my academic career at MIT, for helping me through transitioning my initial research concept to a fully developed thesis idea, and for keeping me on track by advising me throughout the research process.

Brandy Baker and Heather Barry, the NSE academic administrative staff members, for their tireless efforts in continuous support of NSE undergraduates, especially graduating seniors dealing with repercussions due to COVID-19, and for their reminders and motivation that encouraged me to meet project deadlines to the best of my ability despite the extenuating circumstances.

My partner and best friend, Daniel, for his love and patience throughout the school year, and for his support and perseverance through long-distance separations, time zone differences, and clashing sleep schedules.

Finally, I would like to thank everyone above for showing resilience, perseverance, and flexibility through remote communication and social isolation amidst the COVID-19 pandemic.

4 Table of Contents

Abstract 3 Acknowledgements 4 Table of Contents 5 List of Figures 6 List of Tables 7 1. Introduction 8 1.1 Motivation 8 1.2 Objectives 9 2. Background 10 2.1 232U/233U Ratio in Thorium Fuel Cycles 10

2.2 35-LEU/Th-ZrO2-rod Lattice Concept 14 2.3 Material Attractiveness 16 3. Methodology 19 3.1 Identification of Fuel Cycle & Materials 19 3.2 The OpenMC Monte Carlo Code 19 3.3 Model Thorium Fuel Bundle & Simulate Depletion 20 3.4 Material Attractiveness Evaluation of Direct-use Materials 23 4. Results and Discussion 26 4.1 Depletion Summary 26 4.2 Uranium NED: Material Attractiveness to Non-state Actors 28 4.3 Plutonium NED: Material Attractiveness to Non-state Actors 28 5. Conclusion 32 5.1 Future Options 32 Appendix A: Fuel Bundle Geometry and Depletion Code 34 Appendix B: Bare Critical Mass Calculation Code 40 References 42

5 List of Figures

Figure 1: Products of multiple-neutron captures on Th-232. 12

Figure 2: Neutron capture cross section data for Th-231 production and Th-233 production. 13

Figure 3: 35-LEU/Th-ZrO2-rod Lattice Concept modeled in the reference study. 14

Figure 4: k-infinity vs. Burnup for the 35-LEU/Th-ZrO2-rod bundle concept and Maximum burnup for LEU/Th lattices. 15

Figure 5: Discrete phases in nuclear explosive device (NED) development. 16

Figure 6: Generic steps to make a uranium-based NED. 18

Figure 7: 35-LEU/Th-ZrO2-rod Lattice Concept modeled in OpenMC. 20

Figure 8: k-effective vs. Burnup for the 35-LEU/Th-ZrO2-rod bundle concept with 40% LEU modeled in OpenMC. 26

Figure 9: Number of atoms vs. Time [d] of uranium isotopes in the 35-LEU/Th-ZrO2-rod fuel bundle. 27

Figure 10: Number of atoms vs. Time [d] of plutonium isotopes in the 35-LEU/Th-ZrO2-rod fuel bundle. 27

6 List of Tables

Table 1: Comparison of some nuclear characteristics of U-233, U-235, and Pu-239. 11

Table 2: Scales of categorization for the material attractiveness for non-state actors. 16

Table 3: Specifications for Thorium-based fuels in PT-HWRs. 21

Table 4: Definition of the Attractiveness Level Bins. 25

Table 5: Masses and isotopic fractions of plutonium isotopes in the 35-Th/LEU-ZrO2-rod Fuel Bundle after zero to 40 MWd/kg of depletion and two years of cooling. 29

Table 6: Estimation of the Heat Content of a bare critical sphere containing Pu-238, Pu-239, Pu- 240, Pu-241, and Pu-242. 29

Table 7: Overall material attractiveness evaluation of Plutonium isotope mixture. 31

1. Introduction

As nations aim to develop clean energy sources and ensure energy security, new advanced nuclear reactor (ANR) concepts are becoming increasingly more attractive, along with innovative fuel cycles to complement them. In particular, thorium-based fuel cycles are becoming attractive in nations lacking an abundance of uranium resources, such as India, which contains over 800,000 tons, the largest reasonably assured resource of naturally-occurring thorium in the world [1], and has been working on designs for a thorium-based Advanced Heavy Water Reactor (AHWR) [2]. However, with new technologies come new weapons-proliferation vulnerabilities, and safeguard methods must evolve to address the possibility of theft and diversion of fissile material by non- state actors, such as terrorists. One method to evaluate the theft and diversion attractiveness level of nuclear materials was developed in a joint US-Japan study and published in 2013, and has been used primarily to examine materials in traditional Uranium-Plutonium fuel cycles [3]. This thesis aims to expand use of this method to begin to include thorium fuel cycles.

1.1 Motivation

Prior to the development of this material attractiveness concept, studies explored how doping plutonium with different isotopic ratios could increase decay heat [4], one characteristic of fissile material that can make it difficult to steal, due to health risks. Recently, however, studies have expanded proliferation resistance evaluation to include the additional factors of bare critical mass (BCM), heat content, processing time and complexity, net weight, acquisition time, and radiation dose rate. For example, one study compared the material attractiveness of various inert matrix fuels of high temperature gas-cooled reactors (IMF-HTGR) with fuels in mixed oxide light water reactor

8 fuel cycles (MOX-LWR) [5]. Concerning thorium fuel cycles, prior studies have investigated the U-232/U-233 isotopic ratio in the spent fuel of various pressurized water reactors (PWRs) and liquid-metal cooled fast breeder reactors (FBRs), fueled with natural or lightly-enriched uranium (LEU)-thorium fuel, and how these isotopic ratios and their decay heat affect proliferation resistance [6]. More recently, evaluations of the proliferation resistance of India’s theoretical AHWR have also been conducted using the metrics of BCM, heat content, mass of the source object, and dose rate [2]. However, as there are many thorium fuel bundle concepts that are not yet fully understood, studies on the proliferation resistance of thorium fuel cycles have been limited. The goal of this thesis is to contribute to addressing this knowledge gap by evaluating each of the six metrics used in the 2013 joint US-Japan method [3] for nuclear materials present in an additional thorium fuel cycle concept. In doing so, this project aims to contribute to better understanding the proliferation resistance of new thorium fuel cycles that may be deployed in the coming decades, and inform future decision-making regarding which thorium fuel bundles to incorporate into new reactor designs.

1.2 Objectives

In order to acquire a more comprehensive evaluation of attractiveness of thorium-based fuel cycle concepts to nuclear weapons proliferation by non-state actors and contribute to the development of effective safeguards, this thesis research project will (1) simulate one pre-designed thorium fuel bundle in a Canada deuterium uranium (CANDU) reactor core at various burnups, and (2) categorize various nuclear isotopes after irradiation based on material attractiveness. The evaluation of lattice concepts containing fuels of varying LEU/Th ratios will inform a better understanding of the effect LEU content has on the proliferation resistance of thorium-based fuels. Although this thesis will only look at one LEU/Th ratio, its goal is to contribute to this knowledge base and develop a methodology that can easily be extended for further analysis.

2. Background

2.1 232U/233U Ratio in Thorium Fuel Cycles

One of the primary concerns regarding the proliferation resistance of thorium fuel cycles is, similar to Plutonium-239 in uranium fuel cycles, the creation of Uranium-233 via neutron capture of Thorium-232. Although the fissile isotopes Uranium-235 and Plutonium-239 have historically been the target desirables for nuclear weapons fabrication, Uranium-233 has been seen, in some ways, as favorable due to its lower spontaneous fission rate, among other characteristics. Table 1 summarizes a comparison between these three fissile isotopes.

Table 1: Comparison of some nuclear characteristics of U-233, U-235, and Pu-239. [6]

Like Plutonium-239, Uranium-233 does not occur naturally, and can only be produced by the single-neutron capture process, depicted by Equations 1 and 2, during the operation of nuclear reactors partially-fueled by fertile Thorium-232 [6].

(1) (2)

While Plutonium-239 and Uranium-233 have relatively low decay heats, another significant feature of this decay chain is the existence of sister-products that have relatively high decay heats: Plutonium-238, and Uranium-232, respectively. The production of U-232 during reactor burnup as depicted in Equations 3 and 4 [6] contributes to the proliferation resistance of thorium fuel by greatly increasing the emitted dose rate of the uranium isotope mixture.

11 (3) (4)

Essentially, a higher fraction of U-232 in the resultant used fuel correlates to a higher decay heat, and therefore greater proliferation resistance. The ratio between U-233 and U-232 in thorium- based fuel has been observed to vary depending on reactor type and burnup levels, and one study concluded that the proliferation resistance of thorium fuel cycles greatly depends on the method of implementation [6]. Thus, in order to acquire U-233 to build a Nuclear Explosive Device (NED), the goal would be to take the fuel out of the reactor core at a burnup that optimizes U-232/U-233 ratio with a high mass of U-233. (Or, for a non-state actor, to steal fuel that has been taken out at such a burnup, possibly using insider information.) Another consideration is the decay of Protactinium-233 into pure U-233, as shown in Equation 1. Instead of waiting until U-233 itself is produced in the reactor core, it is possible to simply remove the fuel from the reactor at a burnup when there is little U-233, but a high mass of Pa-233 in the fuel, subsequently allowing the Pa-233 to decay into U-233 outside of the reactor core, and avoiding the production of the undesirable U-232. Figure 1 shows the products of multiple-neutron captures on Th-232, illustrating how this can be possible.

Figure 1: Products of multiple-neutron captures on Th-232. [6] Pa-233, the desired isotope, and U-232, the undesirable isotope, are highlighted in yellow.

Both U-233 and U-232 stem from the neutron capture of Th-232, and subsequent beta decay of Pa-233 and Pa-232, respectively. However, the key difference is in the length of the half-lives in each part of the decay chain. Pa-233 can be acquired after only neutron capture of Th-232 to Th-233, which has a relatively short half-life of 22.12 minutes, lending to a quick buildup of Pa- 233 in the reactor core. The half-life of Pa-233, by contrast, is 27 days, so the creation of U-233

12 will be delayed for a time. In order for U-232 to build up in the reactor core, however, there is an additional neutron capture required. In addition, the half-life of Pa-232 is relatively long, 1.31 days, when compared to Th-233, so U-232 should build up more slowly than Pa-233. Furthermore, as illustrated in Figure 2, Th-231 production is only viable at fast neutron energies, whereas the cross section for Th-233 production is relatively high for thermal neutrons, making the path to Pa-233 production more likely than U-232 production over a short amount of time. (Once U-233 has been produced, there is also the conversion between U-233 and U-232 occurring via neutron capture, as depicted in Equation 4.)

Figure 2: Neutron capture cross section data for Th-231 production (top) and Th-233 production (bottom). [7]

13 In conclusion, when evaluating the attractiveness of thorium fuel to produce U-233 for use in NEDs, there are two pathways to be considered: the production of U-233/U-232 mixtures directly in the reactor core, and the production of Pa-233 in the reactor core, then subsequent decay of Pa- 233 into pure U-233 outside of the core.

2.2 35-LEU/Th-ZrO2-rod Lattice Concept

A literature review of studies on various thorium fuel bundle designs was conducted, and a single lattice concept was selected from a study that examined ten lattice concepts with thorium- based fuels for their potential use in a CANDU-like PT-HWR. Since the study concluded that “the

35-element bundle with a central ZrO2 displacer rod appears to offer the best compromise between maximizing burnup, fissile utilization, and bundle power, while reducing coolant void reactivity” [8], this fuel bundle was selected for material attractiveness analysis. The original depiction is reproduced in Figure 3.

Figure 3: 35-LEU/Th-ZrO2-rod Lattice Concept modeled in the reference study. The center rod is a Zr-4 tube filled with ZrO2. The original study examined LEU wt% ranging from 10% to 70%. [8] For this thesis, the fuel pins consist of a 40/60 weight percent LEU/Th oxide mixture.

14 Furthermore, although fractions of LEU ranging from 10 wt% to 70 wt% were tested in this configuration, it was determined that greater than 35% LEU was required to achieve a desirable burnup (>20MWd/kg), as depicted in the bottom graph in Figure 4. However, flatter k-infinity vs. burnup curves, shown in the top graph of Figure 4, are more favorable for producing U-233 in blanket fuel [8].

Figure 4: k-infinity versus burnup for the 35-LEU/Th-ZrO2-rod bundle concept (top) and Maximum burnup for LEU/Th lattices (bottom). [8] For this thesis, 40 wt% LEU was selected to optimize between Max. Burnup and flatter k-infinity vs. Burnup.

15 Since lower LEU fractions achieve flatter k-infinity vs. burnup curves, 40% LEU was chosen for this thesis project to optimize between burnup and blanket fuel favorability.

2.3 Material Attractiveness

Defined as “the relative utility of nuclear material for an adversary to assemble a NED,” [3] material attractiveness evaluates just one of three types of factors important to evaluating the overall security of a nuclear material. These three factors – material factors, technical factors, and institutional factors – encompass evaluations intrinsic to the material, the nuclear facility itself, and the nature of domestic and international regulations. All of these factors contribute to the nuclear security and attractiveness of a nuclear material [3]. This thesis, however, maintains focus on the material factors, regardless of the facility or regulatory characteristics involved. Six material factors are identified and used to characterize the attractiveness of a nuclear material. They are divided into three phases based on the chronological order in which they occur during an incident of theft and weapons production: Acquisition, Processing, and Utilization [3]. Figure 5 illustrates the meaning of each of these phases and Table 2 summarizes the metrics and criteria that are used to evaluate a material in each phase.

Figure 5: Discrete phases in nuclear explosive device (NED) development. [5]

Table 2: Scales of categorization for the material attractiveness for non-state actors. [5]

16 2.3.1 Acquisition Phase Net Weight The net weight is defined as the product of the mass of the item containing the attractive material and the number of this item that must be stolen in order to acquire an IAEA Category I quantity of the material [3]. For example, after irradiation, a quantity of U-233 will exist in the nuclear reactor spent fuel assembly. Therefore, the appropriate net weight will be the weight of the fuel assembly multiplied by the number of fuel assemblies that cumulatively contain a Category I quantity of U- 233 [9]. Acquisition Time Defined in the 2013 joint study, the acquisition time is “the product of the number of thefts needed to secure a NED-sufficient quantity of nuclear materials and the time required for each theft.” For one theft event, the acquisition time begins when an adversary begins removing the material from the facility until the adversary successfully crosses the site boundary with the material [3]. It is important to note that this estimation does not take any technical or institutional factors into account, and assumes an unguarded, unalarmed nuclear facility. This situation can best be imagined as a scenario in which the non-state actors had insider information, or an ally with access to the facility. “Any delay in acquisition of a Category I quantity results from an intrinsic characteristic of the nuclear material,” [3] not any facility response system, or physical barrier. Radiation Dose Rate The radiation dose rate is calculated based on the item and quantity with the highest dose rate that must be stolen during one theft attempt, and is measured in Gy/h at one meter from the item [3].

2.3.2 Processing Phase Processing Time and Complexity Once the material has been successfully removed from the facility, it will require processing in order to separate the fissile material from other isotopes or fission products. Various methods exist that can be accomplished at a garage-scale facility, such as solvent extraction and multi-step chromatography processes. Thus, the metric for estimating processing time and complexity can only be evaluated qualitatively. Figure 6 depicts the generic steps required to make a uranium-

17 based NED, and the qualitative processing time and complexity evaluation based on at which point in the processes the material has been stolen [3].

Figure 6: Generic steps to make a uranium-based NED. [3]

2.3.3 Utilization Phase Bare Critical Mass (BCM) The bare critical mass of a fissile material is simply the smallest mass of the material required (without reflectors, a moderator, or an external neutron source) in order for the material to sustain a fission chain reaction, leading to an explosion when uncontrolled. Heat Content The heat content of a material is calculated based on the decay heat or heat generation of the material multiplied by the bare critical mass, and is measured in Watts [3]. In order for a material to be attractive, the heat content of a usable quantity must not be so high that the adversaries are incapacitated before they are able to construct and detonate the NED.

3. Methodology

3.1 Identification of Fuel Cycle & Materials

Extensive literature review was conducted in order to select an appropriate thorium fuel model. Due to the limited timescale and scope of this project, the chosen model was simulated as a single assembly, as opposed to an entire reactor core, and has been modeled in a prior study for comparison of results. To neglect the proliferation considerations associated with fuel reprocessing, the fuel bundle chosen was suitable for a once-through-thorium (OTT) cycle. In addition, because a fuel cycle with reprocessing can assume clean fissile material separation regardless of reactor core burnup, the protactinium decay pathway depicted in Figure 1 is more significant in an open fuel cycle. The reactor type, power, and other specifications were also determined in the literature review of this phase.

3.2 The OpenMC Monte Carlo Code

OpenMC is an open-source Monte Carlo code initially developed by the MIT Computational Reactor Physics Group (CRPG) [10]. The code was initially developed for full core reactor simulations on large scale computing platform but eventually added many useful capabilities for many other types of analysis. OpenMC supports both neutron and photon transport, and can perform calculations in eigenvalue or fixed source mode. OpenMC recently added depletion capabilities that facilitated this study. In addition to being open source, which facilitates access to all, OpenMC has an extensive Python interface which provides a modern approach to reactor simulations and facilitates parametric studies and data post-processing [11]. OpenMC also distributes pre-processed nuclear data libraries in its native HDF5 format and depletion chains.

19 In this study, the ENDF/B-VII.1 library was chosen and the simplified CASL PWR depletion chain with 255 nuclides [12] was selected since it is quite accurate for thermal spectrum reactors.

3.3 Model Thorium Fuel Bundle & Simulate Depletion

The selected thorium fuel bundle depicted in Figure 3 was modeled and simulated in OpenMC [10] [11], and the input script is reproduced in Appendix A. Calculations were conducted on the Ubuntu 18.04 system, locally when building the geometry, then in Google Cloud Console when running the longer, more complex depletion calculations. The geometric model in OpenMC of a

35-LEU/Th-ZrO2-rod lattice is shown in Figure 7.

Figure 7: 35-LEU/Th-ZrO2-rod Lattice Concept modeled in OpenMC. The central ZrO2- rod is depicted in grey, and the rings of identical fuel elements are depicted in various different colors in the geometric model, because OpenMC tallies the depletion of each fuel element individually. Heavy water (D2O) as moderator and coolant is depicted in blue, surrounding the individual fuel elements as well as the entire fuel bundle, based on the geometry of a standard CANDU reactor calendria and fuel channels (lattice pitch of 28.575cm).

The depletion simulation placed this lattice in a 700 MWe Canada Deuterium-Uranium (CANDU) reactor with 380 fuel channels and twelve bundles per channel, mirroring the

20 simulations performed in the reference paper [8], with standard CANDU dimensions and some relevant specifications given in Table 3, reproduced from the reference study.

Table 3: Specifications for Thorium-Based fuels in PT-HWRs. [8]

Because the clad thickness and distances from bundle center to each ring of fuel elements were not given in the reference paper, these values were based off standard CANDU dimensions, acquired from the “Modeling a CANDU Bundle” example in the OpenMC documentation [11]. The standard distance from bundle center to fuel element rings was slightly altered to prevent overlap with the central ZrO2 rod (see code in Appendix A for exact values) and make the water

21 channels somewhat even. To shorten the calculation time, the bundle geometry was modeled and depleted in 2D. This is a valid approximation because leakage is minimal is very large heavy water reactors and the isotopic ratios will be very similar to a much costlier simulation performed in 3D. Assuming the widely accepted thermodynamic efficiency value of 33%, the depletion was performed with a power of 9390 W/cm for one bundle. Time steps from zero to forty MWd/kg were taken to mirror the simulation performed in the reference paper, as shown in the top graph of Figure 4, with step sizes of 0.25, 1, and 2 MWd/kg, that were determined using the maximum step size guidance in the OpenMC documentation of 2 MWd/kg. Then, fuel bundle cooling was simulated by taking identical step sizes of one month over a cooling period of two years. Over the course of depletion, 50,000 neutrons were tracked per batch, and results were calculated for 150 total batches, with 50 inactive batches to ensure the source neutron distribution converged before accumulating tallies [11]. The “constant extrapolation on predictor and constant midpoint on corrector” (CE/CM) integration method was used, defined mathematically as in Equation 5 [11].

(5)

where y’, the change in the nuclide density vector y at time t, is calculated by numerically solving a differential equation. At first, a predicted matrix Ap is constructed using cross-section and reaction rates for the current time step n. This predicted value is used to calculate an estimated nuclide density vector ym, which is said to be at the midpoint of yn and yn+1, and based on the time step size h/2. Then, a new OpenMC calculation is performed to evaluate corrected cross-sections and reaction rates to form the corrected matrix Ac. Finally, the new nuclide density vector yn+1 is calculated at tn+h, using the full step size. When calculating new estimated y values in both cases, the expm function is used to integrate y’ numerically. In the case of OpenMC, a high order Chebyshev Rational Approximation Method (CRAM) is used to approximate the exponential matrix evaluation.

22 The desired depletion results included k-infinity versus burnup in order to verify that the 35-

LEU/Th-ZrO2-rod lattice reaches criticality and is viable in a power reactor, and the isotopic composition of the irradiated fuel as a function of time and burnup. The final goal of this phase was to acquire the fraction of each significant isotope (U-232, U-233, U-235, Pu-239, Th-233, and Pa-233) in the bundle at each desired burnup.

3.4 Material Attractiveness Evaluation of Direct-use Materials

These six metrics of material attractiveness were calculated or evaluated for each direct-use material in the irradiated fuel bundle after depletion and two years of cooling: Bare Critical Mass (BCM), Heat Content, Net Weight, Acquisition Time, Dose Rate, Processing Time and Complexity [3]. Direct-use materials are fissile isotopes that can be used directly to build a nuclear explosive device (NED): U-235, Pu-239, U-233, and mixtures of these isotopes with their non- fissile counterparts: U-238, Pu-238, and U-232, respectively. The order of calculation is different than the chronological order shown in Figure 5.

3.4.1 Design Basis Threat

In order to evaluate the theft attractiveness of a material, it is necessary to assume a design basis threat (DBT) [13] [14], or a specific adversary against whom the material should be protected. The qualitative material attractiveness concept developed in the 2013 joint US-Japan study assumes the adversary to be a non-state actor, such as a terrorist group, acting without the large-scale, high-tech resources characteristic of an organized state seeking to build nuclear weapons. These characteristics are assumed: [3] • Willing to sacrifice their lives to successfully detonate a nuclear explosive device (NED). • Accepting of any nuclear yield. • Has access to “garage-scale” processing capabilities. • Possesses design (or can produce design) and has technical capability to build a rudimentary NED. • Does not have the capability of irradiation or enrichment

23 3.4.2 Processing Phase: Processing Time & Complexity Under the assumption that the DBT has only garage-scale reprocessing capabilities, it can be understood that a non-state actor may have the ability to separate individual elements using methods such as lab-scale column chromatography or solvent extraction, but does not have the ability to separate isotopes (i.e. cannot perform uranium enrichment). Therefore, a direct-use material for non-state actors will be comprised of all the isotopes of one element that exist in the fuel bundle at the time of attempted theft. That is, an NED containing U-233 will also contain U- 235, U-238, U-232, and U-234, due to lack of enrichment capabilities. Also during this phase, literature reviews were conducted to gain a fundamental understanding of the reprocessing methods necessary for element separation. Finally, Processing Time & Complexity was evaluated based on Figure 6 and Table 2, where Preferred, Usable, Impractical, and Impossible categorizations correspond to Categories 1 through 4, respectively.

3.4.3 Utilization Phase: Bare Critical Mass & Heat Content

The BCM of each direct-use material was calculated in OpenMC by modeling a sphere with a vacuum boundary condition (approximation of an open-air situation). The calculation used a built-in OpenMC function that iterated over various radius values until a k-effective of 1 was reached within 577 pcm. The heat content of a material is calculated based on the decay heat or heat generation of the material multiplied by the bare critical mass, and is measured in Watts [3]. The development of a method to calculate decay heat precisely using OpenMC was initiated, but not completed due to strict project deadlines. The concept behind this method was that it would calculate decay heat alongside BCM for each material by tallying neutrons and photons produced during the simulation. Instead, the heat content was estimated using known values for the decay heat of each individual isotope. The calculation code associated with BCM is reproduced in Appendix B.

3.4.4 Acquisition Phase: Net Weight, Acquisition Time, & Dose Rate Since all metrics included in evaluating the Acquisition Phase are based on the output results of the entire fuel bundle, these calculations are performed while processing the outputs of the depletion simulation in OpenMC, and are specifically defined in Section 2.3.1. The dose rate

24 of the fuel bundle was calculated to determine the risk to human health during the acquisition phase, which could occur as a theft of the fuel assembly from a nuclear facility. Due to the high decay heat of fission products directly after irradiation, it is almost impossible for a theft to occur directly after a fuel bundle is removed from the reactor core. The fuel bundle is more likely to be targeted after a cooling down period. Therefore, to calculate the dose rate, the fuel bundle was first further depleted for two years at zero power, in time steps of one month. Development of a method to calculate dose rate was initiated, but not completed due to strict project deadlines. The methodology would simulate the decay of isotopes in the irradiated and cooled fuel bundle, then tally the flux of neutrons and photons based on energy at 1 meter in air from the fuel assembly to a cylinder of water – a simple approximation of the geometry of an adult human. Finally, the dose rate would be calculated from the product of flux and conversion coefficients in the ICRP Publication 116 for each neutron and photon energy [15].

3.4.5 Final Material Attractiveness Evaluation

After each metric was calculated and categorized based on Table 2, the final material attractiveness evaluation was made based on the categories defined in Table 4, where Category 1 corresponds to High attractiveness level and Category 4 corresponds to a Very Low attractiveness level. The overall material attractiveness bin for the material is chosen based on the lowest level of material attractiveness present among the six individual metrics. That is, even if BCM of the material is Category 1, if the dose rate is Category 4, the overall material attractiveness is determined to be Very Low.

Table 4: Definition of the Attractiveness Level Bins. [3]

4. Results and Discussion

4.1 Depletion Summary

To verify that the 35-LEU/Th-ZrO2-rod fuel bundle was modeled correctly, graphs of k- effective versus burnup were generated, shown in Figure 8. When compared with Figure 4, these plots show that the change in k-effective of the 35-LEU/Th-ZrO2-rod fuel bundle modeled in OpenMC takes roughly the same shape as the fuel bundle modeled in the reference study.

Figure 8: k-effective vs. Burnup for the 35-LEU/Th-ZrO2-rod bundle concept with 40% LEU modeled in OpenMC. This graph matches the corresponding data in green in the graph from the original study, reproduced in Figure 4. The left graph shows the data with the same y-axis as the reference study, and the right graph shows the data with more appropriate y-axis scale.

Isotopic composition of the irradiated fuel as a function of time was acquired to summarize the changes in quantities of direct-use materials. The changes in uranium isotopes are depicted in Figure 9, and the changes in plutonium isotopes are depicted in Figure 10. Isotopic composition

26 was plotted over both depletion and cooling periods. Both graphs are plotted from the beginning of depletion to the end of 2 years of cooling.

Figure 9: Number of atoms vs. Time [d] of uranium isotopes in the 35-LEU/Th-ZrO2-rod fuel bundle. The amount of U-238 greatly outnumbers other uranium isotopes due to its presence in the 40%LEU-60%Th fuel elements.

Figure 10: Number of atoms vs. Time [d] of plutonium isotopes in the 35-LEU/Th-ZrO2- rod fuel bundle. The shift from depletion to cooling can clearly be observed around 1060 days, where the sudden increase in Pu-239 concentration is due to the decay of Np-239.

27 As expected, the quantity of U-233 increased while the quantity of U-235, fissile material used to seed the fuel, decreased over the course of the depletion. Additionally, the creation of Pu- 239 can be observed over time due to neutron capture by U-238.

4.2 Uranium NED: Material Attractiveness to Non-state Actors

The material attractiveness to non-state actors of the U-233 and U-235 contained in this fuel bundle was determined to be very low. The percentage of U-238 present in the mixture of all uranium isotopes after depletion and two years of cooling was calculated to be around 97.6%, which means that, even if the uranium isotopes are completely separated from all other isotopes in the fuel assembly, any mass of this mixture can never be critical. This is consistent with the statement in the reference paper that “one additional advantage of using homogenously mixed (LEU, Th) is that the U-233 produced will be denatured by the presence of U-238, which enhances the proliferation resistance of the fuel” [8]. A targeted evaluation of proliferation resistance for the

35-LEU/Th-ZrO2-rod Lattice Concept (40% LEU) not only confirms prior knowledge, but also demonstrates exactly how denatured the U-233 will become.

4.3 Plutonium NED: Material Attractiveness to Non-state Actors

The material attractiveness to non-state actors of Pu-239 present in this fuel bundle was determined to be very low. First, the Processing Time & Complexity of separating plutonium from other fissile isotopes and fission products was evaluated as Impractical, or Category 3, because the fuel bundle must first undergo dissolution before further steps can be taken to construct a NED, as shown in Figure 6. In the Utilization phase, BCM and heat content of the plutonium mixture were estimated. The masses and isotopic fractions of plutonium isotopes in the fuel bundle after two years of cooling are summarized in Table 5.

28 Mass in Bundle (g) Isotopic Fraction Pu-238 0.822 0.0208 Pu-239 18.2 0.462 Pu-240 12.2 0.310 Pu-241 3.64 0.0923 Pu-242 4.51 0.114 Total 39.4

Table 5: Masses and isotopic fractions of plutonium isotopes in the 35-Th/LEU-ZrO2-rod Fuel Bundle after zero to 40 MWd/kg of depletion and two years of cooling.

Using a solid density of 19,800 kg/m3 [16], the BCM of a plutonium sphere containing the isotopes listed in Figure 10 was calculated to be 16.06 kg, roughly twice that of pure Pu-239. Since the development of a decay heat calculation methodology in OpenMC was not completed, heat content of the BCM was not calculated directly. However, in order to include the high decay heats of Pu-238 (567 W/kg) and Pu-240 (7.06 W/kg) in the evaluation, values from a prior study for the decay heats in W/kg of each isotope [4] were used to approximate the heat content of this bare critical sphere. The heat content values of each isotope are listed in Table 6, and the total heat content of the sphere was estimated to be 244 W.

Isotopic Fraction Mass in Critical Sphere (kg) Heat Content (W) Pu-238 0.0208 0.334 189 Pu-239 0.462 7.42 14.3 Pu-240 0.310 4.98 35.1 Pu-241 0.0923 1.48 5.04 Pu-242 0.114 1.83 0.220 Total 16.06 244

Table 6: Estimation of the Heat Content of a bare critical sphere containing Pu-238, Pu-239, Pu-240, Pu- 241, and Pu-242.

The Net Weight was calculated to be 2612 kg. Since the BCMs of U-233, U-235, and Pu- 239 are about 8.4 kg, 21 kg, and 7.5 kg, as shown in Table 1, and the IAEA Category I (CI) quantities of these isotopes are 2 kg, 5 kg, and 2 kg, respectively [14] – for unirradiated material – the CI quantity of the plutonium mixture if unirradiated was estimated to be 4 kg. This quantity is significant because NEDs can be built without acquiring a critical mass, by using a Be reflector, for example. Since there is only 39.4 g of plutonium mixture in one irradiated fuel assembly, a non-state actor would have to steal 102 fuel assemblies to acquire a CI quantity. Finally, using commonly known material densities [17] [18] and densities given in the reference paper [8], the mass of one 35 LEU/Th-ZrO2-rod fuel assembly was estimated to be 25.6 kg. The net weight was acquired by multiplying the mass of one assembly with the total number of assembles required to acquire a CI quantity. The time to acquire one fuel bundle was estimated to be 20 minutes, and the total Acquisition Time was estimated to be 2040 minutes. Due to lack of time and strict project deadlines, this estimated time is identical to the estimation made in a similar study, which summarizes, “A 10 min. operation time is assumed for the lifting by crane of a quantity not portable by man. An additional 10 min. of operation time is assumed for removal of the target materials to the off-site” [5]. While the CANDU fuel bundle in this project is instead a man-portable 25 kg, it was estimated that at least the same time would be required to operate a crane in order to remove the fuel bundle from wet storage. This calculation was completed under the assumption that only one fuel bundle could be stolen at a time.

The overall material attractiveness evaluation for plutonium in the 35 LEU/Th-ZrO2-rod fuel bundle was determined to be Very Low, based on five out of six of the material attractiveness metrics. The evaluations of each metric are summarized in Table 7.

Value Category H M L VL

BCM 16.06 kg 1

Heat Content 244 W/BCM 1

Net Weight 2612 kg 3 Acquisition Time 2040 min. 4 Dose Rate NE*

Processing Impractical 3 Time/Complexity *Not Evaluated

Table 7: Overall material attractiveness evaluation of Plutonium isotope mixture.

Although the BCM and Heat Content of this plutonium mixture look very attractive, the amount of material in one fuel bundle is small enough that a non-state actor would have to steal many fuel bundles in order to have a significant quantity of the material. Furthermore, while the Acquisition Time is quite high, it is important to recognize that, due to the relatively small mass and volume of the fuel bundle, it may be possible for a non-state actor to steal multiple bundles in one facility visit, thereby decreasing the overall Acquisition Time. For example, if a non-state actor is able to steal 10 or 11 fuel bundles at once, it will require only 200 minutes to acquire a C1 quantity of plutonium, thereby putting the Acquisition Time metric into Category 3. In this scenario, the overall material attractiveness level will instead be rated as Low.

5. Conclusion

A thorium-based lattice concept was simulated in a CANDU reactor using OpenMC software, and the attractiveness to non-state actors of various fissile materials was evaluated in order to inform future decisions when selecting thorium-based fuels for advanced nuclear reactors. The pursuit of thorium-fueled reactors has become especially attractive to nations such as India, where uranium is scarce, but thorium is abundant. The lattice concept investigated in this thesis was based on one lattice evaluated in a reference study that simulated ten lattice concepts with varying LEU/Th ratios. It consisted of 35 fuel elements made up of 40% LEU and 60% Th, and a central ZrO2 displacer rod. This fuel bundle was depleted from 0 to 40 MWd/kg, then simulated for a cooling period of 2 years under a zero power setting. The material attractiveness of uranium isotopes was confirmed to be Very Low due to the high fraction of U-238 in the fuel, which cannot be separated by non-state actors with limited reprocessing capabilities. Despite its low BCM and Heat Content, the plutonium mixture in this lattice was determined to have a Very Low material attractiveness rating. This was due to the high number of fuel bundles required to achieve an IAEA Category I quantity of the plutonium mixture.

5.1 Future Options

One of the original goals of this project was to hypothesize a quantitative figure-of-merit (FOM) that incorporated all six material attractiveness metrics into one single numerical value. This would allow for definitive comparison between attractive materials. Future work may expand this qualitative evaluation of material attractiveness to include a quantitative FOM.

32 Although the 40/60 ratio of LEU/Th in this configuration was determined to be unattractive to non-state actors, future studies may evaluate the material attractiveness of this and other thorium lattice concepts to both state and non-state actors. In the case of uranium isotopes, the ability to create a high fraction of U-233 with a low fraction of U-232 in this lattice may be attractive to states that have basic reprocessing capabilities to separate out U-238, but may not be able to separate U-233 from U-232, which requires more advanced laser methods. In the case of the plutonium isotopic mixture, its low BCM and Heat Content makes it a possible choice for states that may want to avoid building enrichment facilities, but have easy access to CANDU reactors and do not have to rely on theft. This eliminates the acquisition time and net weight concerns. Furthermore, part of the goal of this thesis was to develop methods to evaluate material attractiveness using a relatively new, open-source, reactor physics software – OpenMC. Due to time limitations, the dose rate methodology was not completed, and may be pursued in future work. This project also presented a limited evaluation of Processing Time & Complexity and Acquisition Time, and future material attractiveness evaluations of thorium-based fuels may benefit from deeper literature review on these topics. Finally, this project simulated only one lattice concept and LEU/Th ratio from the reference study. Future studies may evaluate other geometries and LEU/Th ratios to acquire a more comprehensive understanding of the relationship between reactor geometry, burnup, LEU/Th ratio, and material attractiveness.

33 Appendix A: Fuel Bundle Geometry and Depletion Code

#Modelling the 35-LEU/Th + ZrO2 central rod lattice concept #This code is modified from the CANDU Bundle example provided at #https://docs.openmc.org/en/latest/examples/candu.html #And depletion was coded by following the example at #https://docs.openmc.org/en/latest/examples/pincell- depletion.html from math import pi, sin, cos import numpy as np import openmc import openmc.deplete

fuel_materials = [] for i in range(0,35): fuel = openmc.Material(name='fuel') #The below ratios were calculated assuming 40%LEU-60%Th Oxide Fuel. fuel.add_nuclide('Th232', 0.60611,'ao') fuel.add_nuclide('U235',0.019695,'ao') #5% U-235 enrichment fuel.add_nuclide('U238',0.37420,'ao') fuel.add_element('O', 2.0) fuel.set_density('g/cm3', 9.7) #from Bromley paper fuel_materials.append(fuel)

#Center displacer rod of ZrO2 center = openmc.Material(name='rod') center.add_element('Zr',1.0) center.add_element('O',2.0) center.set_density('g/cm3',4.3) #from Bromley paper

#Zircaloy-4 is approximated as just Zr clad = openmc.Material(name='zircaloy') clad.add_element('Zr', 1.0) clad.set_density('g/cm3', 6.0) #default density from example CANDU Bundle heavy_water = openmc.Material(name='heavy water') heavy_water.add_nuclide('H2', 2.0) heavy_water.add_nuclide('O16', 1.0) heavy_water.add_s_alpha_beta('c_D_in_D2O') heavy_water.set_density('g/cm3', 1.1) #default density from example CANDU Bundle

34 # Outer radius of fuel and clad r_fuel_ex = 0.6122 #example fuel radius r_clad_ex = 0.6540 #example clad radius clad_thickness = r_clad_ex-r_fuel_ex #clad thickness NOT GIVEN in the Bromley Thorium Fuel paper, so the value is calculated based on the example CANDU bundle r_clad = 0.57 #outer radius of Thorium fuel element, given in Bromley paper r_fuel = r_clad-clad_thickness

#Outer Radius of the center ZrO2 rod r_center = 2.4

# Pressure tube and calendria radii - given in Bromley paper pressure_tube_ir = 5.2 pressure_tube_or = 5.6 calendria_ir = 6.4 calendria_or = 6.6

# Radius to center of each ring of fuel pins; assumed based on the CANDU bundle example. This leads to overlapping in the OpenMC model, so the numbers were altered slightly. #ring_radii = np.array([2.8755, 4.3305]) – original values in OpenMC CANDU Bundle example

#To avoid overlapping, I slightly changed the numbers below. ring_radii = np.array([3.0755, 4.3305])

#Center Rod location center_radius = np.array([0])

# These are the surfaces that will divide each of the rings radial_surf = [openmc.ZCylinder(r=r) for r in (ring_radii[:-1] + ring_radii[1:])/2] water_cells = [] for i in range(ring_radii.size): # Create annular region if i == 0: water_region = -radial_surf[i] elif i == ring_radii.size - 1: water_region = +radial_surf[i-1] else: water_region = +radial_surf[i-1] & -radial_surf[i]

water_cells.append(openmc.Cell(fill=heavy_water, region=water_region))

35 #Also create a water region for the center tube radial_surf_tube = openmc.ZCylinder(r=r_center+clad_thickness) water_cell_tube = [] water_region_tube = -radial_surf_tube water_cell_tube.append(openmc.Cell(fill=heavy_water, region=water_region_tube)) plot_args = {'width': (2*calendria_or, 2*calendria_or)} bundle_universe = openmc.Universe(cells=(water_cells)) #bundle_universe.add_cell(water_cell_tube) #bundle_universe.plot(**plot_args)

#pin_cell universe for each fuel pin, divided between inner and outer pins for ease of iteration when creating fuel pins pin_universes_inner = [] pin_universes_outer = [] fuel_cells = [] for i in range(0,14): surf_fuel = openmc.ZCylinder(r=r_fuel) fuel_cell = openmc.Cell(fill=fuel_materials[i], region=-surf_fuel) clad_cell = openmc.Cell(fill=clad, region=+surf_fuel)

pin_universes_inner.append(openmc.Universe(cells=(fuel_cell, clad_cell))) fuel_cells.append(fuel_cell) for i in range(14,35): surf_fuel = openmc.ZCylinder(r=r_fuel) fuel_cell = openmc.Cell(fill=fuel_materials[i], region=-surf_fuel) clad_cell = openmc.Cell(fill=clad, region=+surf_fuel)

pin_universes_outer.append(openmc.Universe(cells=(fuel_cell, clad_cell))) fuel_cells.append(fuel_cell)

#Center rod universe surf_center = openmc.ZCylinder(r=r_center+clad_thickness) center_cell = openmc.Cell(fill=center, region=-surf_center) center_tube_cell = openmc.Cell(fill=clad, region=+surf_center) center_universe = openmc.Universe(cells=(center_cell, center_tube_cell))

#pin_universe.plot(**plot_args)

#center_universe.plot(**plot_args)

36 #Center tube creation num_tubes = [1] angles = [0] for i, (r, n, a) in enumerate(zip(center_radius, num_tubes, angles)): tube_boundary = openmc.ZCylinder(x0=0, y0=0, r=r_center+clad_thickness) water_cells[i].region &= +tube_boundary

tube = openmc.Cell(fill=center_universe, region=-tube_boundary) bundle_universe.add_cell(tube)

#bundle_universe.plot(**plot_args) num_pins = [14, 21] angles = [0, 0] for i, (r, n, a) in enumerate(zip(ring_radii, num_pins, angles)): for j in range(n): # Determine location of center of pin theta = (a + j/n*360.) * pi/180. x = r*cos(theta) y = r*sin(theta)

pin_boundary = openmc.ZCylinder(x0=x, y0=y, r=r_clad) water_cells[i].region &= +pin_boundary

# Create each fuel pin -- note that we explicitly assign an ID so # that we can identify the pin later when looking at tallies if n == 14: pin = openmc.Cell(fill=pin_universes_inner[j], region=- pin_boundary) pin.translation = (x, y, 0) pin.id = (i + 1)*100 + j fuel_materials[j].id = pin.id #ID the fuel materials to match the pin ID if n == 21: pin = openmc.Cell(fill=pin_universes_outer[j], region=- pin_boundary) pin.translation = (x, y, 0) pin.id = (i + 1)*100 + j fuel_materials[14+j].id = pin.id #ID the fuel materials to match the pin ID

bundle_universe.add_cell(pin)

#bundle_universe.plot(**plot_args)

37 pt_inner = openmc.ZCylinder(r=pressure_tube_ir) pt_outer = openmc.ZCylinder(r=pressure_tube_or) calendria_inner = openmc.ZCylinder(r=calendria_ir) calendria_outer = openmc.ZCylinder(r=calendria_or) bundle = openmc.Cell(fill=bundle_universe, region=-pt_inner) pressure_tube = openmc.Cell(fill=clad, region=+pt_inner & -pt_outer) v1 = openmc.Cell(region=+pt_outer & -calendria_inner) calendria = openmc.Cell(fill=clad, region=+calendria_inner & - calendria_outer) box = openmc.rectangular_prism(width=28.575, height=28.575, boundary_type='reflective') water_region = box & +calendria_outer moderator = openmc.Cell(fill=heavy_water, region=water_region) root_universe = openmc.Universe(cells=[bundle, pressure_tube, v1, calendria,moderator]) geom = openmc.Geometry(root_universe) geom.export_to_xml() mats = openmc.Materials(geom.get_all_materials().values()) mats.export_to_xml() p = openmc.Plot.from_geometry(geom) p.color_by = 'material' p.colors = { clad: 'silver', center: 'gray', heavy_water: 'blue' } p.to_ipython_image() settings = openmc.Settings() settings.particles = 50000 settings.inactive = 50 settings.batches = 150 settings.source = openmc.Source(space=openmc.stats.Point()) settings.export_to_xml()

# tally is not manually determined by user in depletion

38 #Set up depletion material volume #h_fuel = 49.5 for i in range(35): fuel_materials[i].volume = r_fuel ** 2 * pi #print(fuel_materials[0].volume)

# define time step over depletion time_steps = [0.25, 0.25, 0.25, 0.25, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] time_steps = [(0.25, 'MWd/kg'), (0.25, 'MWd/kg'), (0.25, 'MWd/kg'), (0.25, 'MWd/kg'), (1.0, 'MWd/kg'), (1.0, 'MWd/kg'), (1.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (2.0, 'MWd/kg'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d'), (30.0,'d')]

#define changing power over depletion and cooling power_bundle = [0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0.00939e6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] #W/cm

# set up depletion operator chain_file = './chain_casl_pwr.xml' # you can download it from https://openmc.org/depletion-chains/ chain = openmc.deplete.Chain.from_xml(chain_file) op = openmc.deplete.Operator(geom, settings, chain_file)

# choose Predictor-Corrector scheme to run the depletion integrator = openmc.deplete.CECMIntegrator(op, time_steps, power_bundle) integrator.integrate()

39 Appendix B: Bare Critical Mass Calculation Code

#Based on Criticality Search example in OpenMC documentation #https://docs.openmc.org/en/stable/examples/search.html

# Initialize third-party libraries and the OpenMC Python API import matplotlib.pyplot as plt import numpy as np from math import pi, sin, cos import openmc import openmc.model

# Create the model. `radius` will be the parametric variable.

#This example is for the Plutonium Sphere def build_model(BCM_radius): # Create the NED (nuclear explosive device) materials fuel = openmc.Material(name='Pu Sphere') fuel.set_density('g/cm3', 19.8) fuel.add_nuclide('Pu238', 0.020837272086869064,'ao') fuel.add_nuclide('Pu239', 0.46223605860628364, 'ao') fuel.add_nuclide('Pu240', 0.31030506927878254,'ao') fuel.add_nuclide('Pu241', 0.09227164971432611, 'ao') fuel.add_nuclide('Pu242', 0.11434995031373879, 'ao')

# Instantiate a Materials object materials = openmc.Materials([fuel])

# Create a sphere of the fuel fuel_radius = openmc.Sphere(x0=0.0, y0=0.0, z0=0.0, r=BCM_radius, boundary_type='vacuum')

# Create fuel Cell fuel_cell = openmc.Cell(name='Critical Sphere') fuel_cell.fill = fuel fuel_cell.region = -fuel_radius

# Create root Universe root_universe = openmc.Universe(name='root universe', universe_id=0) root_universe.add_cells([fuel_cell])

# Create Geometry and set root universe geometry = openmc.Geometry(root_universe)

# Finish with the settings file settings = openmc.Settings() settings.batches = 300 settings.inactive = 20

40 settings.particles = 1000 settings.run_mode = 'eigenvalue'

# Create an initial uniform spatial source distribution over fissionable zones #bounds = [-0.63, -0.63, -10, 0.63, 0.63, 10.] #uniform_dist = openmc.stats.Box(bounds[:3], bounds[3:], only_fissionable=True) #settings.source = openmc.source.Source(space=uniform_dist)

# We dont need a tallies file so dont waste the disk input/output time settings.output = {'tallies': False}

model = openmc.model.Model(geometry, materials, settings)

return model

# Perform the search crit_radius, guesses, keffs = openmc.search_for_keff(build_model, bracket=[1, 100], tol=1e-2, bracketed_method='bisect', print_iterations=False) print('Critical Pu Sphere Radius: {:4.0f} cm'.format(crit_radius)) print('Critical Pu Sphere Radius:',crit_radius) vol_sphere = 4/3 * pi * crit_radius**3 BCM = 19.8 * vol_sphere / 1000 #kg - Bare Critical Mass print('BCM of Pu:', BCM, 'kg')

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