Introduction cation is smaller than the minimum size for a given coordination it is no longer fully coordinated, but has room with which to rattle about [5]. Figure 1.1: The calculation of the radius ratio for octahedral and cubic coordinations. The cation - anion radius ratio can therefore be used to determine the most stable coordinated polyhedra. Table 1.1 lists the critical ratio values, where the term ligancy has the same meaning as coordination number (i.e. the number of atoms bonded to a central atom [2]). In this approach, when the ratio becomes less than any of the critical values given in Table 1.1, the next lower structure becomes preferred. Polyhedron Ligancy Minimum radius ratio Example Compound Cubo-octahedron 12 1.000 none exist Cube 8 0.732 CaF 2 Octahedron 6 0.414 NaCl Tetrahedron 4 0.225 ZnS Table 1.1: The minimum ionic radius ratios for coordinated polyhedra stability, after [2]. Using ThO 2 as an example of a compound crystallizing in the fluorite structure, the ionic radius of 8 fold coordinate Th 4+ = 1.05Å, while the 4 fold coordinate of 18
Introduction O 2 = 1.38 Å [6]. Thus, the minimum radius ratio = 0.761, which is considerably above the minimum for a cubic coordination, but still well below the critical value of 1.000 required to form the cubo-octahedron structure. Therefore, Th atoms will be cubically coordinated by O atoms. Th atoms are FCC packed, thus forming a tetrahedron about each O atom and the resulting space group is Fm3m [7, 8], see Figure 1.2. Indeed, this description of the fluorite lattice pertains to many other compounds, such as halides (SrF 2 ), oxides (CeO 2 ) and intermetallics (NiMgBi). In this work, only oxides are discussed, predominantly UO 2. After examining Figure 1.2, it is clear that there are many unoccupied interstices in the fluorite structure. UO 2 is the most common material used for nuclear fuel [9], in part because of its ability to accommodate fission products in these voids, thereby reducing problematic fuel swelling. Figure 1.2: Unit cell of fluorite. The yellow atoms represent 4+ cations and the red atoms represent 2- anions. It should be noted that there are exceptions to Pauling s first rule, as it is solely based on simple geometry. Firstly, if the anions are smaller than the cations (as is the 19
Introduction case for the anti-fluorite structures Li 2 O, Na 2 O and K 2 O), then the anion - cation radius ratio must be used (as opposed to the cation - anion). Other exceptions, including UO 2, occur because this rule assumes ions to be rigid spheres, which of course they are not. Partial covalency also contributes to further exceptions. Although Pauling s first rule is useful when making simple predictions, it is clear that more rigorous techniques, such as atomistic simulation, are required in order to generate more accurate predictions. 1.1.2 A 2 B 2 O 7 Pyrochlore Oxides Pyrochlore oxides are named after the mineral pyrochlore, (NaCa)(NbTa)O 6 F/(OH), with which they share a similar structure [10]. A 2 B 2 O 7 pyrochlores are ternary metallic oxides whose crystal chemistry is complex enough to make them favourable for a wide range of applications. In this thesis, A and B cations are only considered as having charges of 3+ and 4+ respectively. However, there is another entire series of pyrochlore compounds consisting of 2+ and 5+ cations. Figure 1.3: Unit cell of pyrochlore. Blue spheres represent A 3+ cations, yellow B 4+ and red O 2. 20
Introduction Figure 1.4: The cationic sublattice of pyrochlore. 1.2 Point Defects All of the topics addressed in this thesis are concerned with deviations from the perfect periodic lattice. Though there are many ways in which crystal imperfections are manifest, such as dislocations, surfaces and pores, all of these are essentially conglomerations of zero dimensional point defects. Point defects consist of vacant lattice sites, atoms in non-regular lattice positions (so called interstitials, a term first coined by Wagner [13]) as well as impurity atoms. Many properties (e.g. conductivity, luminescence and diffusion) are influenced by the existence of these defects, which is particularly true for inorganic solids. Furthermore, it is not possible to produce a single crystal free of defects. Finally, as temperature increases, defects become even more important. In this section, the different types of point defect and their means of creation are discussed. Kröger - Vink notation is used throughout [14]. 22
Introduction Figure 1.5: The left figure is Frenkel disorder in an ionic crystal, where the red square denotes a vacancy left by a cation which has moved to an interstitial site. The right figure is Schottky disorder. Several compounds readily exhibit Frenkel disorder. For example, UO 2, CaF 2 and CeO 2, all demonstrate the Frenkel anion type of disorder (i.e. anion interstitials and anion vacancies) while AgCl, AgBr and Fe 3 O 4 all exhibit cation Frenkel disorder. In this defect process, disorder, structural energy as well as entropy are increased. It is important to understand the factors which determine the concentrations of these types of defects. Recalling Equation 1.1, the configurational entropy can be expressed as: S c = k ln Ω (1.3) where Ω is the number of ways in which to arrange the defects in the crystal and k is Boltzmann s constant, leaving only a value for Ω, which can be expressed as: Ω v = N! (N n v )! n v! (1.4) 24
Introduction on 1 + 92 U 235 92 U 236 57 La 139 + 42 Mo 95 + 2 0 n 1 (1.15) The total energy released by a fission event can be expressed as [35]: E = (M o ΣM i )c 2 (1.16) where M o is the mass of the original nucleus (U 235 in the example case) and M i is the mass of the resulting nuclei (2 M neutron + La 138.955 + Mo 94.945 ). An energy of 200 MeV results from the mass change of 0.215 amu. Figure 1.6: Fission product yield for a PWR fuel rod after 2.9% burnup, reproduced from [36] and after 1% burnup after data given in [37]. The fragments of the fission process quickly emit neutrons after the fission event because they are proton deficient. The fission products are left as a charge deficient pair and continue toward a stable state through β decay or via excited states. Though rare, uranium is also known to split into three and four fission products. The left hand plot in Figure 1.6 displays the number of each fission product resulting from the fission of 100 uranium atoms, while the right hand plot is the concentration of fission products in weight-ppm for a fuel pin with a different history. Although both sets of data originate from U-235 fuel, differing only in burnup percentage, 30
Introduction 1.3 Surfaces A useful way of considering a surface, at least from the perspective of the ions directly beneath the surface, is as a giant defect [42] and the positions of these ions are relaxed as if they were responding to a point defect. The ultimate effect of this relaxation can be significant. In fact, many properties and phenomena of ionic crystals are governed by surfaces, such as mechanical strength, catalysis and crystal morphology. A convenient way of envisioning a surface is as a cleaved crystal consisting of stacked planes. According to Tasker [43], there are then three types of surface, as shown in Figure 1.7. Figure 1.7: The types of ionic surface according to Tasker [43]. The horizontal arrows indicate planes along which the crystal can be cut without forming a dipole. A Type 1 surface is a series of neutral planes, each plane consisting of a stoichiometric ratio of anions and cations. A Type 2 surface consists of charged planes, but arranged symmetrically such that there is no dipole moment perpendicular to the surface, so long as the surface is cut between neutral blocks. A Type 3 surface is a series of alternately charged planes, but however the crystal is oriented, it is 33
Introduction Figure 1.8: A bar with a concentration gradient in the x direction, after [58]. Assume two planes, E and E, are separated by a distance r. If there are n 1 atoms diffusing per unit area in plane E and n 2 atoms diffusing per unit area in plane E, and if the average jump frequency is Γ, it follows that the net flux is: J = 1 6 (n 1 n 2 )Γ = (number of atoms) (area)(time) (1.23) Concentrations c 1 and c 2 can be written as: n 1 r = c 1 and n 2 r = c 2 (1.24) This can then be substituted into Equation 1.23: J = 1 6 (c 1 c 2 )r Γ (1.25) Generally, concentration changes are slow enough that they can be expressed as [52]: c 1 c 2 = r ( ) c x Which is essentially Fick s first law, if the diffusion constant is expressed as: D = 1 6 Γ r2 (1.26) (1.27) Equation 1.27 is the definition of the diffusion coefficient in solid state systems. It consists of a geometric factor (= 1 6 in this case because of the six directions in space 37
Introduction which an atom can jump), an elementary jump distance squared (= r 2 which is proportional to the lattice parameter) and an effective jump frequency (= Γ). 2 (A more detailed discussion of random walk and the diffusion coefficient becomes rather complicated and is not warranted by this thesis and can be found elsewhere [52,61].) There are several possible mechanisms for the diffusion of ions and atoms, those of which are predominant in the crystalline solids are now described. The interstitial mechanism is the diffusion of an atom via interstices, see Figure 1.9. Figure 1.9: The interstitial and vacancy diffusion mechanisms. An obvious prerequisite of this type of diffusion is that there are defects residing in interstitial positions. This mechanism is most likely to occur with small solutes, as there is a local deformation of the lattice. In Figure 1.9, where atoms labelled 1 and 2 must move in order for the interstitial atom to move into its new position. Another type of diffusion mechanism is known as the vacancy mechanism. In this case, adjacent atoms are able to jump into these unoccupied sites (denoted by the square in Figure 1.9). There is much less distortion to the lattice when atoms diffuse via this mechanism compared to diffusion via the interstitial mechanism. 2 An interesting aside is that Perrin used Einstein s kinetic theory of random walk to determine Avogadro s number [59, 60]. 38
Introduction Due to the severe lattice distortions required, larger atoms occupying interstitial sites are unable to diffuse via an interstitial mechanism. However, an alternative is afforded by the interstitialcy mechanism, in which an interstitial atom forces an adjacent atom on a regular lattice site into an interstitial site, see Figure 1.10. Figure 1.10: The interstitialcy and crowdion diffusion mechanisms. This type of diffusion can be collinear (if the displaced atom moves in the direction of the red arrow in Figure 1.10) or non-collinear (if it moves in other directions than the red arrow). A related diffusion mechanism occurs when an interstitial occurs in a close packed direction, and is known as the crowdion mechanism. The extra atom in the row displaces several atoms from their regular sites, see Figure 1.10. Although this mechanism is not likely for ionic materials because of the strong Coulombic forces involved, it may be important after irradiation, especially in metals. It has already been stated that it is not unlikely for several types of defects to be present simultaneously. Each of the diffusion mechanisms discussed is dependent on point defects. Therefore, several types of diffusion mechanism can co-exist, with one generally predominating. It follows that diffusion is directly related to the concentration of defects and their mobility. 39
Atomistic Simulation: Bulk and Surface Methods two parts, is that a parameter η can be optimized to determine the width of the Gaussian peaks, such that both parts converge quickly and independently. Catlow and Norgett [84] determined an optimum value for this width, which is given by: η = 6 Nπ 3 V 2 (2.3) where η is the width parameter, N is the total number of species and V is the unit cell volume. The charge distribution components of Ψ 1 and Ψ 2 cause the Gaussian distributions to completely drop out of Ψ, thus leaving the overall potential, Ψ, completely independent of the width parameter, η. However, the speed of convergence is dictated by this parameter. The definition of the Madelung constant dictates that the charge distribution on the reference point is not considered to contribute. In other words, ions do not feel their own electrostatic potential. Therefore, Ψ 1 can be expressed as the difference: Ψ 1 = Ψ a Ψ b (2.4) where Ψ a is the potential of a continuous series of Gaussian distributions of sign the same as the actual lattice and Ψ b is the potential of a single Gaussian charge distribution on the reference ion, see Figure 2.1. (a) Ψ a (b) Ψ b (c) Ψ 1 = Ψ a Ψ b Figure 2.1: The development of Ψ 1 from the difference of a lattice of Gaussian distributions and a Gaussian distribution at a reference point. 45
Atomistic Simulation: Bulk and Surface Methods The remaining part of the overall potential is Ψ 2. It is evaluated at the reference point and has three contributions from each lattice point: Ψ 2 = q i q i erfc( η r ij ) (2.11) 4πɛ o r ij j The three contributions are therefore: the point charge associated with the ion j, the Gaussian distribution contained in the sphere of radius r ij at the j lattice point and the Gaussian distribution occurring outside of the same sphere, see Figure 2.2. Figure 2.2: Graphical representation of Ψ 2 of the Ewald summation. With equations for Ψ 1 and Ψ 2 determined, an expression for Equation 2.2 can be formulated, making use of Equations 2.10 and 2.11: [ 4π ( 1 Ψ = q i q j V C j k G 2 e + q i q i erfc( η r ij ) 4πɛ o r ij j G 2 4η e i(g r)) ] 2q2 i ɛ o η π (2.12) 47
Atomistic Simulation: Bulk and Surface Methods 2.1.2 Short Range Potential The Ewald sum accounts for the long range, attractive Coulomb interaction, but is unable to describe what occurs when two charged atoms are brought near one another. Equation 2.1 accounts for this short range interaction with the term Φ s r. It is important to understand how this term originates and what different forms can be adopted to describe it, as the success of this study is largely a function of the quality and moreso the accuracy with which these potentials describe this interaction. The charge distributions of two adjacent atoms are able to overlap if they are brought near enough to one another. This causes two repulsive interactions, which if the distance between these atoms becomes sufficiently small causes the overall force between them to become repulsive, even if the ions are oppositely charged. The two terms are (a.) the Pauli term, which is a result of the Pauli exclusion principle [85,86] and (b.) the nuclear - nuclear repulsion. The generalized statement of the Pauli exclusion principle is that no two fermions can occupy the same quantum state. When electron clouds overlap (shown as charge distributions in Figure 2.3), for the Pauli exclusion principle to be satisfied, the ground state charge distribution of an electron is forced to occupy a higher energy state, thus creating an increase in electronic energy. This increase in energy gives rise to the repulsion. Figure 2.3: The electronic charge distribution as atoms near one another, where the orange circles denote nuclei (reproduced from [25]). 48
Atomistic Simulation: Bulk and Surface Methods potential is best suited to calculations concerning liquids and gases. In this work, only solids are considered, thus necessitating an alternative to the Lennard-Jones potential. If the short range repulsive term from Equation 2.15 is combined with the van der Waals attractive term, one arrives at the so called Buckingham potential (though Born and Mayer certainly deserve at least some recognition for their effort) [100]: Φ ij = Ae r ρ C r 6 (2.17) where A, ρ and C are the adjustable parameters whose description of the short range interaction largely determine the success of the calculations described later in this thesis. Figure 2.4 depicts the influence of the short range potential on the overall ionic interaction. The derivation of these parameters is discussed in the following section. Figure 2.4: The overall potential for Sr 2+ - O 2, and its relation to the long range Coulomb interaction and short range Buckingham potential. The short range parameters can be thought of as having loose physical meaning. 51
Atomistic Simulation: Bulk and Surface Methods Figure 2.6: The shell model, where the orange atom is the core and the blue atom represents the charge of the massless shell. The red arrows represent polarization, black arrows represent Coulombic interactions and blue arrow represents short range interaction. It is common for the parameters Y and k to be fit to dielectric and elastic constants. This fitting is primarily concerned with the high frequency dielectric constant, ɛ, as it arises solely from the electronic polarizability, rather than having ionic polarizability contributions, as does the static dielectric constant, ɛ s. Although the shell model is phenomenological in nature, its use is warranted by its success in previous studies, (see for example Catlow et al. [110]). The strength of this model (versus other models, such as the rigid ion model or the point polarizable ion model) is that any force acting on an ion is assumed to do so via the shell, thus coupling short range interactions to the polarizability, see Figure 2.6. Clearly, this provides a framework by which it is possible to model more of the interactions occurring between species than if the shell model was not used (i.e. the rigid ion model alone). There are however limitations to this model. First and foremost is that the shell 57
Atomistic Simulation: Bulk and Surface Methods Figure 2.7: The two region approach to calculating defect energies, where the inner black sphere represents a defect, the orange sphere represents the boundary of Region I, the grey sphere Region IIa, while Region IIb tends to infinity. Region I is a sphere of ions including the defect. The total energy of these ions is calculated explicitly as defined by the Coulombic interaction and short range potential discussed previously. The size of Region I is a very important consideration in these types of calculations. Therefore, Region I must be chosen large enough such that the defect energy converges appropriately, but computational efficiency must also be taken in to account. Figure 2.8 demonstrates the effect of Region I size on the defect energy. The Region I sizes in this thesis have been chosen to err on the side of defect energy convergence, and are found in Table 2.2. 63
Atomistic Simulation: Bulk and Surface Methods Figure 2.9: A surface simulation cell, where the z axis is normal to the surface. A major difference in calculating surface energies and bulk defect energies is how long range electrostatic interactions between ions are computed. In the case of bulk defects, as mentioned in Section 2.1.1, the Coloumbic term of the total energy, Equation 2.1, is determined via the 3D Ewald summation. When considering surfaces, it is useful to consider this type of summation in 2D. A 2D treatment converges less quickly than the 3D surface created through slabs and gaps but the construction of 3D surfaces often results in an infinite array of dipoles which will not converge. Unfortunately, the 2D Ewald sum has a different form than the 3D. For a full description of the 2D sum, first published by Parry [123] (dutifully corrected in [124]) and Heyes [125], the reader is directed elsewhere [126]. 67
Solution of Fission Products in UO 2 Figure 3.1: The chemical state of fission products, where orange denotes volatile fission products, grey denotes metallic precipitates, blue denotes oxide precipitates and green indicates products in solid solution. Elements labelled with more than one colour denote the possibility of an alternate chemical state, with the preferential chemical state denoted by the top colour. The noble gases Kr and Xe are insoluble in UO 2 and migrate to grain boundaries [141], dislocations or pre-existing pores [142, 143] where they aggregate into bubbles which lead to fuel swelling. Fuel swelling is a performance limiting factor and therefore, understanding the behaviour of Kr and Xe is imperative to improving fuel performance. Several studies have focused on the determination of the position of Kr and Xe within the UO 2 lattice and this will be discussed further in Section 3.3. The other volatile fission products, the halogens Br and I, are not considered in this work (and in fact have a much lower yield than Kr or Xe). However, other studies have concentrated on the determination of their behaviour in UO 2. Iodine, for example, has been found to diffuse two orders of magnitude faster than Xe [144], and therefore tends to be released in the fuel clad gap. Unlike the noble gases kryp- 74
Segregation of Fission Products to UO 2 Surfaces individual planes though there is no dipole perpendicular to the surface (so long as the block is cut at the appropriate layer, see Section 1.3) and is therefore denoted as Type 2. The (100) surface is of the unstable Type 3 distinction. In order to create a physically stable (100) surface, a series of vacancies must be formed on the surface. In the case of cation terminated UO 2, half of an oxygen layer is moved from the bottom of the simulated bulk, to the surface which serves to neutralize the dipole. If a surface consisting of only one unit cell is initially considered, there are two ways to configure the neutralizing oxygen ions: diagonally opposing or next to each other (see Figure 4.1 A and B). However, if a larger surface repeat unit is considered, there will be many more possible configurations of the oxygen ions than for a single unit cell. In fact, if a surface region of 2x2 unit cells is considered, to arrange the 8 oxygens in the 16 available positions, there will be =16!/8!8! (= 12,870) ways to arrange the surface oxygens. However, this number is reduced by the symmetry of the repeat block to 153 unequivalent configurations of the oxygen ions [44]. Of these 153 configurations, Abramowski [44] found those denoted as A and B to be the lowest energy surfaces, and in addition, an AB hybrid, see Figure 4.1. Figure 4.1: The three anion terminations of the (100) UO 2 surface considered in this work, calculated to be low energy by Abramowski [44]. As mentioned above, the A and B configurations can be formed in a single unit cell repeat. However, the AB hybrid requires the larger 2x2 repeat unit. The AB 90
Segregation of Fission Products to UO 2 Surfaces Figure 4.2: A 2x2x6 repeat unit of the anion terminated (100)AB surface of UO 2, where the white atom represents the segregating species. It is also important to note that a concern here, and indeed with any atomistic calculation, is the size of the simulation cell. For example, Figure 4.2 depicts a simulation cell of 2x2x6 unit cells in the x, y and z directions respectively. Initially this work began using a simulation cell of 1x1x6 unit cells in size. Unfortunately, this was far too restrictive, giving rise to horizontal, unphysical defect-defect interactions. This defect-defect interaction is evident in in Figure 4.3(a). In this plot, there is a pronounced gap between the sets of data labelled a and d config., and the sets of data labelled b and c config. The meaning of each data label will be discussed in Section 4.4.1. 94
Segregation of Fission Products to UO 2 Surfaces Figure 4.4: The binding energy of the {(Me U ) :(V O ) } cluster plotted as a function of separation. An interesting side note about Figure 4.4 is that the binding energy at 7Å begins to follow the Coulombic trend line, as is to be expected. However, at a separation of 12Å, the binding energy deviates from the Coulombic trend. This drop in energy can be accounted for by a defect nearing the Region I - Region II boundary. If Region I was enlarged, it is expected that the binding energy would continue to follow the Coulombic trend. What is also interesting from Figure 4.4, is that even at a separation of 13.5Å, the binding energy is still 25% of the maximum binding energy at a separation of 2Å. This behaviour suggests that at realistic defect concentrations (i.e. above the dilute limit, in Figure 4.4 at 13.5Å this corresponds to just below 2%), defects can only be isolated to a certain extent. For the divalent substitutional cation and oxygen vacancy defect cluster mentioned, there are several equivalent nearest neighbour positions for those oxygen vacancies. However, when the cluster is deep within the bulk, it does not matter 97
Segregation of Fission Products to UO 2 Surfaces which of these nearest neighbour positions the oxygen vacancy occupies. As the cluster nears the surface, the equivalence of the oxygen vacancy sites with respect to the cation substitutional site at the nearest neighbour becomes broken. In particular, the distance of the oxygen vacancy to the surface is different in different nearest neighbour configurations. It is therefore expected that the energies of the different nearest neighbour configurations will be different, due to the extent of relaxation of the planes of atoms being a function of the distance from the surface. For the {(Me U ) :(V O ) } cluster, there are four unique nearest neighbour configurations with respect to the (111) surface; see Figure 4.5. A calculation was carried out for each of these configurations and subsequently labelled according the position of the oxygen vacancy. The middle plot in Figure 4.4 shows that configuration a has a slightly higher energy than the other three cluster configurations at the Region I - Region II boundary. This can be accounted for by the position of the oxygen vacancy in the a configuration being closer to the boundary than the other three. Figure 4.5: The four unique {(Me U ) :(V O ) } cluster configurations with respect to the (111) surface (i.e. surface normal), where the arrow indicates the direction of that surface. The notation of the cluster configurations is maintained throughout the text. Similarly, the cluster configurations with respect to the (110) surface must be considered. These are shown in Figure 4.6. In this case, there are only three unique configurations of the oxygen vacancy with respect to this surface. 98
Segregation of Fission Products to UO 2 Surfaces Figure 4.6: The four unique {(Me U ) :(V O ) } cluster configurations with respect to the (110) surface, where the blue plane indicates this surface. The configuration of the {(Me U ) :(V O ) } cluster is more complicated with respect to the (100) surface. As mentioned in the in Section 4.3, the (100) surface of fluorite is a polar surface and therefore inherently unstable. It was also mentioned that defects must be introduced to this surface in order to neutralize the dipole normal to the surface, and that there are three anion terminations considered; see Figure 4.1. However, instead of considering the cluster in each case, it is possible to rely only on two sets of configurations. Since only half of the anion sites on the surface are filled, the oxygen vacancies associated with the divalent fission product can therefore reside on one of two places: either on a site that is directly under a surface oxygen atom or directly below a void on the surface. Revisiting Figure 4.1, this behaviour is evident, as is the similarity of the termination denoted as AB to the A and B configurations. As mentioned, the AB surface termination can be considered a hybridization of the two surfaces. Therefore, the number of unique cluster configurations required for the three (100) surface terminations can be reduced, see 99
Segregation of Fission Products to UO 2 Surfaces Figure 4.7. Figure 4.7: The {(Me U ) :(V O ) } cluster configurations with respect to the (100) surface, for the three anion terminations considered (see Figure 4.1), where the blue plane indicates the (100) surface. Figure 4.7 demonstrates that classifying the anion terminations can be simplified by considering the anions either residing directly next to one another or diagonally across from one another. These two classifications can then be applied to the three terminations considered. The A configuration will always have the anions diagonally across from one another and therefore, only the left hand picture in Figure 4.7 applies. The B configuration will always have termination anions situated in horizontal rows, which applies to the right hand picture of Figure 4.7. The hybrid AB configuration, alternates between having anions next to one another and diagonally across from one another and therefore warrants the use of both pictures in Figure 4.7. Again, the notation for each of these cluster configurations is consistently used throughout this chapter. 100
Segregation of Fission Products to UO 2 Surfaces The segregation of rare-gas atoms through the UO 2 lattice has also been investigated. As mentioned in Chapter 3, the low energy solution site of these atoms is a matter of dispute. Here, both Kr and Xe are assumed to reside in the tri-vacancy site, consistent with results from previous studies on stoichiometric UO 2 [154]. It is worth noting that Kr may equally likely reside in a divacancy site, which would result in a charged defect cluster. However, at this time only neutral defects are supported by the periodic repeat computational code. There are three possible unique geometric configurations of the inert gas defect cluster with oxygen vacancies in nearest neighbour positions, see Figure 4.8. The configuration denoted (i) is the only configuration considered in this segregation study as it was found have the lowest solution energy in the bulk. The present work calculated solution energies of 1.4eV, 1.9eV and 2.2eV respective to configurations (i), (ii)and (iii) in Figure 4.8, which is in agreement with previous studies [36]. Figure 4.8: The three unique, nearest neigbour configurations of the {(V O ) :(Kr U /Xe U ) :(V O ) } cluster, where the defect labelled (i) was found to be of the lowest energy and is used in this work. As was the case with the divalent fission product defect clusters, the orientation of the inert gas defect cluster with respect to surface must be considered. The lowest energy tri-vacancy configuration ((i) in Figure 4.8) can be oriented in three non-equivalent ways with respect to the (111) surface, see Figure 4.9. 101
Segregation of Fission Products to UO 2 Surfaces Figure 4.9: The three unique {(V O ) :(Kr U /Xe U ) :(V O ) } cluster configurations with respect to the (111) surface, which is indicated by the arrow. The notation above is used throughout the text. A calculation for each of the configurations of defect clusters is performed around each uranium layer in increasing depth from the surface. Therefore, the label of each cluster configuration corresponds to a specific cluster. The reader is therefore referred back to Figures 4.5 through 4.9 while examining the results section. To this point, only the (111) surface has been considered for rare gas Kr and Xe, due to the computational demand of these calculations. Rather, emphasis was put on determining segregation trends for the less computationally intensive defect types. 4.5 Results and Discussion 4.5.1 The (111) Surface The energy, E S, is defined to be the difference in energy between a fission product in the bulk and at the surface. It is therefore possible to compare relative energies which elucidate segregation trends. The results of Ce 4+ and Zr 4+ segregation to the (111) surface are shown in Figure 4.10. It is clear from this figure that there is a segregation barrier, duly indicated by positive segregation energies: 0.232eV and 0.261eV for Ce 4+ and Zr 4+. 102
Segregation of Fission Products to UO 2 Surfaces Figure 4.11: The calculated relative energies of the {(Ba U /Sr U ) :(V O ) } defect cluster as a function of depth from the (111) surface, where E S is the segregation energy. 106
Segregation of Fission Products to UO 2 Surfaces Figure 4.14: The calculated relative energies of the {(Ba U /Sr U ) :(V O ) } defect cluster as a function of depth from the (110) surface, where E S is the segregation energy. 113
Non-Stoichiometry in A 2 B 2 O 7 Pyrochlores Figure 5.12: Contour map of the energy difference between the split 48f oxygen vacancy and the split vacancy, where the pyrochlore compounds occurring in the blue region were considered as split and the orange were considered as isolated [244]. 5.3.2 Solution at the Dilute Limit BO 2 Excess Figure 5.13 shows the results for solution of BO 2 assuming that each defect is isolated. For some compositions, it is evident that there are two competing defect mechanisms for initial BO 2 excess non-stoichiometry. Thus, at small A cation radius and large B cation radius compositions, both A cation vacancies (Reaction 5.1, Figure 5.13a) and oxygen interstitials (Reaction 5.3, Figure 5.13c) will be the compensating defects since both reactions have similar energies. However, when moving toward more stable pyrochlore compositions, the A cation vacancy mechanism becomes considerably more favourable than the oxygen interstitial mechanism. For all compositions, Reaction 5.2 (Figure 5.13b), which involves B cation vacancy compensation, was found to be unfavourable. 137
Non-Stoichiometry in A 2 B 2 O 7 Pyrochlores Figure 5.13: Normalized energies for the three mechanisms of BO 2 accommodation in pyrochlore oxides assuming an isolated defect model. Maps a., b. and c. refer to Equations 5.1, 5.2 and 5.3 respectively. 138
Non-Stoichiometry in A 2 B 2 O 7 Pyrochlores Figure 5.14: Normalized energies for the three mechanisms of A 2 O 3 accommodation in pyrochlore oxides. Maps a., b. and c. refer to Equations 5.4, 5.5 and 5.6 respectively. 141
Non-Stoichiometry in A 2 B 2 O 7 Pyrochlores Figure 5.15: Solution energies for the three low energy solution mechanisms calculated assuming clustered defects. Maps a., b. and c. refer to Equations 5.1, 5.3 and 5.6 respectively. 145
Prediction of Rare Earth A 2 Hf 2 O 7 Pyrochlore Phases Figure 6.2: Contour map showing formation energies for combined cation antisite - anion Frenkel local disorder, highlighting the location of the compound Dy 2 Hf 2 O 7. Each composition for which a calculation was carried out is represented by a symbol whose meaning is discussed in the text. Figure 6.2 is a contour map which displays the formation energies of this cluster for 54 pyrochlore compositions. Each point on the map refers to a specific composition for which a calculation was carried out. Open circles are stable pyrochlore formers, whereas filled circles have not been observed experimentally to form a pyrochlore structure. The filled triangle symbols are compositions that have been observed once to form pyrochlores but only when synthesized under high pressure (300MPa) [256]. Further information on these contour maps can be found in Section 5.2.1. A useful characteristic of contour maps is their ability to predict properties of compounds for which calculations were not explicitly carried out. For example, al- 156
Segregation of Yttrium Ions to the Surfaces of t-zro 2 and below in the next layer (this is because the (101) surface is type II). Thus, the four configurations in successive layers are related by this symmetry operation, see Figure A.1. Figure A.1: The <101> surface of t-zro 2, denoting the oxygen layers above and below the Zr 4+ ion. A.4 Results Let us define the energy E S as the difference between the defect cluster in the bulk crystal (defined as the zero of energy) and at the surface (which may be negative i.e. more stable or positive i.e. less stable). (Note: E S is not the same as the experimental enthalpy for segregation [189]). In addition, let us define E T as the difference between the energy of the defect cluster at the surface and in the lowest energy trap site. It is these values which will be used to describe segregation trends. It should be noted that 0Åcorresponds to the cleavage plane which is equidistant between the top atomic layer and the next layer in a bulk material. 171