The set of stable gases created in PWR is of the order of 0.31 at/fission. As can be seen from Table 1.1, the gases thus formed, especially Xenon, rep-resent significant amounts. These fission gases are mostly insoluble in bulk UO2 and can lead to macroscopic phenomena like swelling and gas release. These phenomena directly aﬀect the thermo-mechanics of the fuel rods and may shorten their lifetime. Due to their characteristics, the fission gas be-haviour needs to be understood properly as it concerns the safety of the nu-clear reactors. The fission gas behaviour will be discussed further in Section 1.4.
Defects in nuclear fuel
No material existing in nature is a perfect crystal. Atom arrangements in real materials do not completely follow perfect crystalline patterns. The imperfec-tions in the materials are called defects. These defects are best described in terms of their dimensions. The 0-dimensional defects aﬀect isolated sites in the crystal structure, and are hence called « point defects ». The defects other than the point defects are extended in space and are referred to as « extended defects ». The 1-dimensional defects are lines along which the crystal pattern is broken and are called dislocations. The 2-dimensional defects are surfaces, such as the external surface and the grain boundaries along which the diﬀer-ent crystallites are joined. The 3-dimensional defects are those which change the crystal pattern over a volume. These defects are discussed below.
Point defects are defects that occur only at or around a single lattice point. They are not extended in space in any dimension. During elastic collisions, if the energy transferred by the projectile is greater than the binding energy of the atoms in their lattice sites, the lattice atoms are displaced from their origi-nal positions. The displaced atoms, also known as Primary Knock-On atoms (PKA), can cause an avalanche of other atomic collisions if they have enough kinetic energy resulting in a succession of cascade displacement to form point defects. Point defects typically involve at most a few extra or missing atoms. Slightly larger defects in an ordered structure are usually dislocation loops (discussed later). The diﬀerent point defects are shown in Fig.1.6 and are described below.
Vacancy defects are lattice sites which would be occupied in a perfect crystal, but are vacant. If a neighboring atom moves to occupy the va-cant site, the vacancy moves in the opposite direction to the site which used to be occupied by the moving atom. The stability of the surrounding crystal structure guarantees that the neighboring atoms will not simply collapse around the vacancy.
Interstitial defects are atoms that occupy a site in the crystal structure at which there is usually not an atom. They are generally high energy configurations, but small atoms in some crystals can occupy interstices without high energy. They may be the same type of atom as the others (self interstitial) or an impurity atom.
A Frenkel pair defect arises when a lattice atom is pushed to an intersti-tial site as a result of ballistic collision with an energetic particle or when the crystal is heated at suﬃciently high temperature, provided that the energy absorbed by the atom is higher than its displacement energy. According to Olander , Frenkel pair defects are most commonly ob-served in UO2 under irradiation.
Schottky Defect and other defects including a U vacancy in UO2 x The Schottky defect is characterized by one uranium vacancy and two oxygen vacancies. For stoichiometric UO2, the Schottky defect is the most probable defect including a U vacancy .
As the name suggests, these defects are extended in space. These can be categorized as one-dimensional line defects, two-dimensional planar defects and three-dimensional bulk defects.
One-dimensional defects refer mainly to dislocation linear defects. The atoms of the crystal lattice around a dislocation are misaligned. Dislo-cation networks are likely to be formed in nuclear fuels due to the slow-ing down of fission fragments in high temperature conditions during in-reactor operations. There are two basic types of dislocations, the edge dislocation and the screw dislocation. « Mixed » dislocations, combining aspects of both types, are also common. Fig.1.7 depicts a schematic representation of the diﬀerent types of dislocations. Edge dislocations are caused by the termination of a plane of atoms in the middle of a crystal. Screw dislocations result from shear distortion such that the atoms over the cut surface are shifted in a direction parallel to the dis-location line. For the mixed dislocations the shift is neither parallel nor perpendicular to the dislocation line.
Two-dimensional defects include surfaces, grain boundaries and stalk-ing faults. These defects act as the internal interfaces that separate neighboring regions within the same crystal structure but with diﬀerent orientations. Solids are made of a number of small crystallites which are also known as « grains ». All the grains are separated by boundaries which are called « grain boundaries » and the atoms in this region are not in perfect arrangement. Grain boundaries occur where the crystal-lographic direction of the lattice changes abruptly. This usually occurs when two crystals begin growing separately and then meet. Fig.1.8(a) represents a grain boundary.
Stacking faults occur most commonly in close-packed structures. They are formed by a local deviation of the stacking sequence of layers in a crystal. An example would be the ABCABCBCABCA stacking sequence as shown in Fig.1.8(b).
Three-dimensional defects, also referred as volume or bulk defects, in-clude pores (voids), cracks and bubbles. Voids are small regions where there are no atoms, and which can be thought of as clusters of vacan-cies. In the nuclear fuels, voids are produced during the fuel fabrication process. The porosity of sintered UO2 can be classified either as open pores associated to the pellet surface or closed pores isolated from the surface. Fig.1.9(a) indicates the pores presented in as-fabricated UO2 disk sintered at 1400 C during 4 hours.
During nuclear operations, a large amount of insoluble gases (mainly Xe and Kr) precipitate as bubbles and start to agglomerate in the fuel matrix. The behaviour of these gaseous bubbles needs to be considered as they can significantly alter the physical and mechanical properties of the fuel. As an illustration in Fig.1.9(b), Michel et al.  presented the formation of a multimodal bubble population during the implantation of UO2 thin film with Xe ions which was then followed by an annealing at 1500 C.
Fission gas behaviour
Under irradiation, the crystallographic network of UO2 undergoes changes that have an eﬀect on its physio-chemical properties. Fission gases play a significant part in this evolution and their behaviour needs to be understood. About 15% of the generated fission products consist of the noble gases, xenon and krypton. The solubility of these noble gases in the UO2 matrix is extremely
low  and they either tend to be released from the fuel or precipitate in the form of small nanometer size clusters leading to macroscopic phenomena like Fission Gas Release (FGR) and swelling in the fuel. Both swelling and FGR are detrimental to fuel safety.
If the gas is released from the fuel, the pressure within the fuel rod is correspondingly increased, subjecting the cladding to additional stress. The thermal conductivity of the gap is lowered, since xenon has a lower conductiv-ity than the cover gas, helium. This further increases the temperature which accelerates gas release and, eventually, the cladding is subjected to further stresses that can ultimately result in failure.
On the other hand, if the fission gases are retained in the fuel, they sig-nificantly precipitate as bubbles. As the density of the gas in such bubbles may be considerably lower than that of the gas in the solid fuel (especially for inter-granular bubbles, or at high temperature), gas atoms residing in bubbles occupy more volume than either the fissile atoms they replaced or fission-product atoms that segregate as solid phases. The precipitation of fission gases thus leads to swelling of the fuel to a larger degree than the volume ex-pansion that would occur if the xenon and krypton had remained dispersed on an atomic scale in the fuel matrix. Swelling contributes to the pellet-cladding interaction, again exposing the cladding to higher stress and ultimately limit-ing the life time of the fuel rod. Pellet-cladding interaction due to swelling of fuel in the fuel rod is depicted in Fig.1.10.
To better understand the evolution and release of gaseous fission prod-ucts, numerous studies have been conducted for several decades on the dif-fusion of Xe and Kr atoms in UO2, as well as nucleation and growth of bub-bles/cavities. The mechanism of FGR out of the fuel can be understood as a two step process. The first and basic step is the transfer of gas atoms from within the grain to the grain boundaries (intra-granular fission gas release). The second step is the coalescence and inter-connection of grain-boundary bubbles to give pathways to gas atoms at the grain boundaries to go outside the fuel (inter-granular fission gas release). The latter is clearly observed in high temperature conditions (ramp tests or annealing tests). We will be discussing the diﬀerent mechanisms associated with these two steps in the following sections.
Mechanisms of FGR
Recoil and Knock-out
At temperatures below 1000 C, where the thermally activated processes do not dominate, the fission products formed at the external surface of the UO2 pellets can escape by the direct recoil or knock-out mechanisms. The fission fragments having large kinetic energies (60 to 100 MeV) can dissipate their energy to the fuel lattice, primarily by interaction with the electrons of the ma-terial. However, a fission fragment close enough to a free surface (< 6 to 7 mm), can escape from the fuel due to its high kinetic energy. This mechanism of release is known as (direct) recoil release.
Similarly, when the fission fragments make elastic collisions with the nu-clei of atoms of the lattice, these atoms can also become energetic particles, called primary knock-ons with mean energy of 100 keV. These can then be released, or transfer part of their energy to neighbouring atoms, thereby, creating higher order knock-ons with mean energy of 200 eV . The in-teraction of a fission fragment, a collision cascade or a fission spike with a stationary gas atom near the surface can cause the latter to leave the fuel. This mechanism of release is known as knock-out release.
These release mechanisms in fuel operating at low powers were studied by Olander , Wise , and Lewis . Recoil release can contribute significantly in the release of short-lived radioactive nuclides in failed rods and in experiments  on low temperature fission product release in which high density gas surrounds the fuel [17, 18]. The release by knock-out is negligible compared with that of recoil for short-lived isotopes. These release mechanisms, active at low burn-ups, contribute less than 1% to the release of generated gas [17, 18].
Single gas atom transport
The transport of single gas atoms in the bulk UO2 is determined by the diﬀu-sion rate of fission gas atoms and by the interaction with the fuel microstruc-ture, in particular, the trapping and re-solution of gas atoms by and from intra-granular bubbles, respectively [10, 20, 21]. In this section, the transport refers to the unperturbed diﬀusion of gas atoms (mainly Xe) in the bulk lattice with-out the interaction with the fuel microstructure. The interactions with the fuel microstructure like trapping and re-solution are discussed in the next sections.
Several studies based on density functional theory (DFT) [22, 23, 24, 25, 26, 27] and molecular dynamics (MD) with emperical potentials [28, 29, 30, 31, 32] show that Xe gas atoms prefer uranium vacancy trap sites. There is a general agreement between DFT and empirical potential calculations for the most favourable sites for Xe atoms in the UO2 lattice, with regard to thermo-dynamics. These are the divacancies (one uranium vacancy and one oxy-gen vacancy) in stoichiometric (UO2) matrices; a bound Schottky defect (one uranium vacancy and two oxygen vacancies) for hypostoichiometric (UO2 x) matrices; and single Uranium vacancies for hyperstoichiometric (UO2+x) ma-trices . The computational study of Nicoll et al.  showed a shift of the most probable site for Xe atoms from di- to tri-vacancies when the fuel be-comes hypostoichiometric. As a concequence, the lattice diﬀusion coeﬃcient for inert gas atoms in UO2 is influenced not only by the temperature, but also by the stoichiometry.
Above 1000 K, the single gas atom diﬀusion coeﬃcient in UO2 can be expressed by the Arrhenius law suggested by Matzke  as:
where D0 is the pre-exponential factor (= 5*10 4 to 2.90*10 12 m2/s ), DH represents the activation enthalpy (= 2.87 to 3.95 eV/atom ), and k is the Boltzmann constant (= 8.6173 eV/K).
Depending on the stoichiometry of uranium oxide fuel, although there is general agreement that Xe-release from UO2+ x is faster than release from UO2, and, conversely that the Xe-release is inhibited in UO2 x, there is es-sential disagreement on the details [36, 35, 37, 38].
Several kinds of doping are added to the UO2 fuel in order to improve diﬀerent aspects of fuel performance [39, 40, 41, 42]. Doping of UO2 with foreign cations can simulate the non-stoichiometry, hence it aﬀects the defect structure as well.
During irradiation, Xe atom mobility is increased due to the fission that creates more vacancies (irradiation enhanced diﬀusion) or by directly moving the Xe atoms in the displacement cascades (athermal diﬀusion).
Trapping of gas atoms
The migration of gas atoms in nuclear fuels involves more than simple lat-tice diﬀusion. The solubility of noble gases in the UO2 matrix is extremely low (See Section 1.5.5). A direct consequence of the low solubility is the tendency of the moving gas atoms to be easily trapped in more stable config-urations in various clusters or sinks, such as closed pores, vacancy clusters, precipitation of gases into bubbles, as well as trapping on dislocations or at grain boundaries [43, 44]. Experiments suggest that for burn-ups reached in power reactors, gas atom trapping by fission produced defects should be more important than trapping at natural defects in the as-manufactured fuel . TEM observations have shown that the most possible trap is the popula-tion of fission gas bubbles . Experiments on single crystals and sinters of stoichiometric UO2 indicate that the apparent diﬀusion coeﬃcient of Xe is dra-matically decreased for fission doses exceeding 1015 fissions/cm3 [36, 46]. The strong tendency of the gas atoms to precipitate into bubbles has been demonstrated for samples implanted with Kr and Xe ions, even at tempera-tures as low as 300-350 C . Volatile fission products such as Cs, I, and Rb can also form bubble-like structures in the bulk depending on the temper-ature and the chemical nature of the impurity . The factors aﬀecting the trapping of migrating fission products by bubbles are the diﬀusion coeﬃcient of the fission product and the bubble number density and size.
Other significant trapping sites for fission gas atoms are the dislocations. Nerikar et al.  investigated Xe segregation to edge and screw dislocations in UO2 and the segregation trends were found to be significantly dependent on the dislocation characteristics. Xe prefers to segregate to screw disloca-tion rather than to edge. The Xe segregation to dislocations is found to be thermodynamically favorable. Furthermore, MD simulations using empirical potentials of Xe diﬀusion around edge dislocations in UO2 were performed by Murphy et al.  to analyze the scenario for FGR via this pathway. The results suggest that the activation energy for Xe diﬀusion is dramatically re-duced for free gas atoms in the vicinity of a dislocation as compared to the bulk. However, Xe atoms diﬀusing along the dislocation aggregate to form small bubbles, which incorporate all of the isolated mobile Xe atoms and in-hibits fast diﬀusion of Xe along the dislocation core.
During irradiation, an irradiation-induced re-solution mechanism is also operational (see section 1.5.4), hence the trapping and the re-solution rates can be balanced. Accordingly, the migrating atoms are at equilibrium be-tween the lattice and the traps. Speight  has shown that saturation of the intra-granular bubbles occurs relatively fast and many FGR models define an eﬀective diﬀusion coeﬃcient, considering the trapping and re-solution of gas atoms. Practically, the free, untrapped gas atoms still diﬀuse with their in-trinsic diﬀusion coeﬃcient, but the fraction of them reduces. The re-solution of gas atoms from the bubble to the bulk UO2 is discussed in the upcoming sections.
Irradiation induced re-solution of gas atoms
The low solubility of rare gas atoms in UO2 provides a strong driving force for the precipitation of gas atoms in bubbles. The trapped gas atoms can, however, simultaneously be freed from the traps during irradiation. Early ex-periments [52, 53, 54, 55, 56, 57] have shown that the population of gas in the matrix significantly increases due to irradiation induced re-solution of the gas, which thus, remains capable of diﬀusing out of the grains. The trapping eﬀect can saturate either if there is no more gas in the solid or if there is re-solution of gas into the bulk. This saturation of the trapping eﬀect has been demon-strated in experiments , which is in accordance with the conclusions of Lawrence .
Among g-rays, neutrons and fission fragments, which can all cause the re-solution of fission gases from bubbles, only the fission fragments are eﬃcient as observed experimentally. The reason for this being that the fission frag-ments have a higher kinetic energy (50-100 MeV) than fast neutrons (2 MeV), and they are also charged, thus, having a higher cross-section for transferring energy to the lattice, or to the gas atoms. The plausible mechanisms of irra-diation induced re-solution have been reviewed by Turnbull  and Olander . Olander and Wongsawaeng  have reviewed the re-solution of fission gas bubbles by two radiation induced mechanisms: heterogeneous mech-anism, based on the work of Turnbull  and homogeneous mechanism, based on the work of Nelson .
Homogeneous mechanism involves the re-solution of individual fission gas atoms, one by one, from the bubble by direct elastic collisions with fission frag-ments or energetic primary knock-on atoms (PKA). Heterogeneous mecha-nism, on the other hand, involves instantaneous complete or partial re-solution of the fission gas contained in a bubble by a fission fragment passing through the bubble due to the energy lost by it through electronic excitation. These en-ergetic ions first heat the electrons, then part of the energy is transferred from the hot electrons to the lattice through electron–phonon coupling. This pro-duces a cylindrical thermal spike which can easily reach temperatures above the melting point of the material, and if the spike intersects a gas bubble, a purely thermally-driven re-solution may occur .
Table of contents :
1.1 Rate of creation of the main fission products in PWR for enriched UO2 fuel calculated using the CATACOMB module 
1.2 Literature on thermal re-solution
1.3 Literature on experiments of PIA tests for FGR
1.4 Fuel Performance Codes and FGR modules
1.5 Initial conditions for the intra-granular bubbles and the thermodynamic properties of defects used for benchmark problem 
1.6 Computational efforts made in the different models used in the benchmark demonstration 
1.7 Computational efforts made in the different models used in the benchmark demonstration [176, Table 6]
2.1 Physical parameters used in the model
2.2 Computation time and speed up for test case using MPI and communications between sub domains
2.3 Analytical vs Numerical calculation of RS
2.4 Quantities used in the model for the cases of coalescence and vanishing of bubbles at T = 1800C
2.5 Properties of bubbles for coalescence test
2.6 Properties of bubbles for vanishing test
2.7 Conditions used in the model
3.1 Properties of the five bubbles with respect to the big bubble at t = 0 and t = 60 min
3.2 Evolution of quantities from t = 0 to t = 60 min
3.3 Initial conditions used for the analyses
3.4 Evolution of various parameters with time for the directed movement case
3.5 Evolution of various parameters with time for the directed movement case continued until possible
3.6 FGR during uncertainty analysis of Ceq
3.7 Ceqv values at the rightmost and leftmost end of the domain
4.1 Values of surface diffusion coefficient, Ds from literature
4.2 Evolution of various quantities with time for the combined directed and random movement case
5.1 Physical parameters used for the analysis
5.2 Characteristics of the dislocations and the pinned bubbles at time t = 1122 s