A mathematical model to describe the alpha dose rate from a UO₂ surface

Highlights

• Bragg peak shift due to attenuation through fuel can be approximated as linear.

• Bragg curve maintains functional form with peak shift.

• Single function can be manipulated to describe all particle paths.

• Clarification over error made in literature regarding the probability of alpha escape.

• Results comparable to Monte Carlo attempts while improving computational efficiency.

Abstract

A model to determine the dose rate of a planar alpha-emitting surface has been developed. The approach presented is a computationally efficient mathematical model using stopping range data from the Stopping Ranges of Ions in Matter (SRIM) software. The alpha dose rates as a function of distance from irradiated UO₂ spent fuel surfaces were produced for benchmarking with previous modelling attempts. This method is able to replicate a Monte Carlo (MCNPX) study of an irradiated UO₂ fuel surface within 0.6% of the resulting total dose rate and displays a similar dose profile.

Introduction

The role of nuclear power in its potential to combat climate change is well-established (NIRAB, 2020). Despite this, due to high cost and concerns over safety, its future as a major energy resource is uncertain (Verbruggen, 2008). A drawback of nuclear power is the complex waste forms that it produces, and the potential decommissioning challenges associated with radioactive materials (Seier and Zimmermann, 2014). To develop a comprehensive plan of how to deal with this waste there must be an understanding of what could happen when moving, treating and storing such waste. In order to do this safely, predictive tools are required to highlight the potential risks and how to mitigate them.

An important component of nuclear power production is the management of spent fuel. Whether the fuel is to be reprocessed or placed in a geological disposal facility, the maintenance and assessment of fuel integrity during storage is crucially important. Upon exposure to water, dissolution of the fuel matrix and a release of highly radioactive fission products can occur (Johnson and Smith, 2000; Poinssot et al., 2005; Shoesmith, 2007). In many storage practices this exposure is possible. In the case of a geological repository, it is even expected due to the large time scales associated with the fuel being in one location. A detailed understanding of these degradation mechanisms and the conditions that drive them could improve the effectiveness of any control measures, influence facility design and ultimately, reduce the cost. Developing modelling tools to predict the rate of radioactive dose and dose profile through the fuel-water interface is a critical part of a wider effort to provide accurate predictions of fuel dissolution rates in the event of a containment breach.

Spent nuclear fuel consists of predominantly, UO₂ (≈95%); the remaining material is comprised of fission products and other actinides (Bobrowski et al., 2017). The reactivity of UO₂ in water is so low it is almost considered inert (Shoesmith et al., 1996), however, if oxidised the uranium valence state converts from U(IV) to the much more soluble U(VI); hence, in the presence of oxidising species, UO₂ will corrode more rapidly leading to a faster release of the radioactive isotopes held within the fuel matrix (Shoesmith, 2007; Bright et al., 2019; Springell et al., 2015).

G-value is the yield of a particular species resulting from ionisation (Bobrowski et al., 2017; Buxton, 2008). It can be used for relating instantaneous yields, normally of radical species, or equilibrium yield of molecular species. The G-value used commonly in disposal environments is the yield of molecular species, used to convert the energy lost by ionising radiation in water, to the number of molecules of a given species produced. This is the method whereby dosimetry results can be used to determine dissolution kinetics in chemical reaction and diffusion models (Liu et al., 2019; Nielsen et al., 2008a; Nielsen et al., 2008b; Eriksen et al., 2008).

Alpha particles can generate energetic species which are able to react with each other and their surrounding environment (Draganić et al., 1971; Elliot and Bartels, 2009). This process can produce oxidising conditions near the solid-water interface, which, due to the short penetration depth of alpha particles in water, varies rapidly with distance from the surface. In many senses beta and gamma radiation is more ubiquitous than alpha across the fuel cycle because they are more penetrating. However, the more rapid decay of beta and gamma emitting nuclides results in alpha radiation being dominant at the fuel interface when considering the timescales relevant to disposal environments (>1000 yrs) (Liu et al., 2019). These considerations indicate the importance of developing an accurate model of alpha radiation across the fuel-water interface.

Most radiolysis models utilise the linear energy transfer (LET) curve for calculating the dose received by a medium per decay. This is because the rate at which a corpuscle is stopped is equivalent to the rate the energy is transferred to the medium; hence, a linear energy transfer between the two. The functional form of the LET is described by the following relation.

$$LET=-\frac{dE}{dx}$$

where E is the energy lost by the ion and x is the length over which it is lost. High LET radiation refers to slower heavier ions and low refers to fast moving electrons. To model radiolysis from high LET radiation the chemical events occur so frequently along its path we assume a uniform cylindrical region, known as the penumbra (Elliot and Bartels, 2009). In order to obtain the LET function of a single alpha particle, the stopping powers of each material it is traversing through is required. The stopping power of a material can be derived from Bethe-Bloch theory.

Bethe-Bloch theory describes the average energy lost by a charged particle due to Coulomb interactions between the particle and the electrons of atoms within the medium (Grimes et al., 2017). At the basis of all stopping power models, lies the Bethe-Bloch equation (Bloch, 1933). The equation, with additional corrections, does well to predict the stopping ranges of high velocity ions through a variety of media (Lindhard et al., 1963). The Stopping Ranges of Ions in Matter (SRIM), a program created by Ziegler and Biersack, contains a comprehensive database of experimental values to use alongside a corrected Bethe-Bloch model (Ziegler et al., 2008). SRIM generates the stopping corrections required from compounds containing common elements. This process is known as the core and bond (CAB) approach. It uses the interaction between the traversing ion in the atomic centres and adding the stopping from the materials bonding electrons (Ziegler et al., 2008). The accuracy of this SRIM software had been tested through many compounds (Sørensen and Andersen, 1973; Andersen and SøSrensen, 1972; Ishiwari et al., 1984; Montanari and Dimitriou, 2017) and found to predict the stopping of H and He ions within 2% at the Bragg peak (Ziegler et al., 2008). The LET curve can be extracted from the ionisation output.

The modelling approaches used for determining alpha dose rates can be split into two categories: analytical derivations; utilising stopping power ratios, tables and geometries (Sunder, 1998; Nielsen and Jonsson, 2006; Poulesquen et al., 2006; Hansson et al., 2020), or Monte Carlo methods utilising nuclear Monte Carlo simulators such as GEANT4 or MCNP (Kumazaki et al., 2007; Miller, 2006; Mougnaud et al., 2015; Tribet et al., 2017). The Monte Carlo simulators are often computationally demanding whereas the analytical approach often oversimplifies stopping power and refrains from treating the energy distribution of particles separately. The model described in this study utilises the SRIM software, alongside geometrical considerations, in an attempt to produce a fast and accurate method for determining dose rates from planar alpha emitting surfaces, particularly UO₂. This study also highlights issues in dimensional analysis when simulating this geometry and contradicts a theoretical analysis by Hansson et al. (2020). The following model was built in Python with the use of the math, random and Numpy libraries.