Defect processes in Be12X (X = Ti, Mo, V, W)

The stability of intrinsic point defects in Be12X intermetallics (where X  =  Ti, V, Mo or W) are predicted using density functional theory simulations and discussed with respect to fusion energy applications. Schottky disorder is found to be the lowest energy complete disorder process, closely matched by Be Frenkel disorder in the cases of Be12V and Be12Ti. Antitisite and X Frenkel disorder are of significantly higher energy. Small clusters of point defects including Be divacancies, Be di-interstitials and accommodation of the X species on two Be sites were considered. Some di-interstitial, divacancy and X2Be combinations exhibit negative binding enthalpy (i.e. clustering is favourable), although this is orientationally dependent. None of the Be12X intermetallics are predicted to exhibit significant non-stoichiometry, ruling out non-stoichiometry as a mechanism for accommodating Be depletion due to neutron transmutation.


Introduction
The use of beryllium (Be) in structural applications has been mostly limited to that of an alloying element, owing to the extreme toxicity of Be dust produced in machining. It is, nonetheless, employed due to its low atomic mass and unique neutronic properties; most notably as a window for x-rays and a neutron reflector and moderator in fission reactors [1].
Recently, Be has been used as a first wall material in experimental nuclear fusion reactors [2,3], since it causes relatively small radiative losses in the event it contaminates the plasma during a transient event. In future reactors, Be is proposed as a neutron multiplying material for tritium ( 3 T) breeding [4][5][6], since 9 Be exhibits a low threshold for the (n, 2n) reaction when bombarded with fast neutrons. In these applications, Be will be subject to temperatures of 600 °C [5,7] and a high flux of neutrons resulting in the generation of 3 T and helium (He) through transmutation reactions. Several studies have indicated that this will cause an unacceptable degradation of its mechanical and thermal properties [8][9][10], along with retention of a high 3 T inventory, producing a radiological hazard [11,12].
Be-rich intermetallics have been proposed as an alternative to elemental Be for nuclear applications, since they maintain similar neutronic properties to pure Be but perhaps offer a significant advantage in terms of 3 T retention and radiation tolerance [13,14]. In particular, the Be 12 X series, where X is a transition metal, have proved promising, with studies showing that Be 12 Ti and Be 12 V have adequate neutronic properties for use as a multiplier [15]. In comparison to Be, however, the irradiation response of Be intermetallics has not yet been adequately characterised, although several studies have investigated the response of Be 12 Ti, showing it to compare favourably to pure Be in terms of embrittlement, swelling and tritium retention [14].
Further work must be carried out to identify the fundamental processes occurring during radiation damage in Be intermetallics. In particular, a greater understanding is required of how the point defects generated in damage cascades interact and lead to macroscopic changes in the microstructure. Recent work by Allouche et al [16] on the isomorphic Be 12 W structure used density functional theory to simulate the interactions of vacancies and hydrogen, finding that the intermetallic exhibits a significantly greater Be vacancy formation enthalpy in comparison to pure Be.
The study reported here contributes to our understanding of the fundamental processes occurring during radiation damage of Be intermetallics by predicting the formation and migration of intrinsic defects. It builds upon previous work focused on impurity behaviour in pure Be [17,18].
After a brief description of the computational methodology, we review the crystal structure of the Be 12 X series, identifying all stable interstitial sites for intrinsic defects. The formation energies of all intrinsic point defects are evaluated, identifying the mechanisms relevant to stoichiometric, nonstoichiometric and radiation damage conditions.

Computational methodology
Density functional theory (DFT) simulations were mostly carried out using the Perdew, Burke and Ernzerhof (PBE) scheme of the generalised gradient approximation for the exchangecorrelation functional [19]. Ultra-soft pseudo potentials with a consistent cut-off of 480 eV (converged to 10 −3 eV atom −1 ) were used throughout. All simulations were performed using the CASTEP code [20].
Defect calculations were performed in supercells constructed from 2 × 2 × 2 full Be 12 X unit cells containing 208 atoms. A high density of k-points, with spacing of approximately 0.3 nm −1 was used for the integration of the Brillion Zone, following the Monkhost-Packing scheme [21]. This corresponds to k-point grids of 2 × 2 × 4 for defect calculations.
As these materials are metallic, density mixing and Methfessel-Paxton [22] cold smearing of bands were employed with a width of 0.1 eV. Calculations were not spin-polarised, and during defect and elastic calculations no symmetry constraints were applied. All parameters, including the k-point spacing were converged to at least 10 −3 eV atom −1 .
For atomic relaxation in defective cells, the energy conv ergence criterion for self-consistent calculations was set to 10 −7 eV and that for the forces on atoms to less than 0.01 eV Å −1 . The cell was relaxed the stress component less than 0.05 GPa.

Crystallography
It has been established that the crystal structure of this family of materials exhibits tetragonal symmetry and spacegroup I 4 /mmm [23]. Several studies also report Be 12 Ti as being hexagonal with spacegroup P 6 /mmm [24,25]. In fact, Gillam et al [23]. suggested that the large disordered hexagonal cell (with dimensions a = 29.44 and c = 7.33 Å) originally proposed by Raeuchle [25] is a derivative of the smaller tetragonal structure. This was confirmed in our recent work [26].
Within the I 4 /mmm structure, the transition metal occupies the 2a lattice site (0,0,0) with 20 fold coordination by Be (see figure 1). Be occupies three symmetrically distinct sites within the structure, here named Be1, Be2 and Be3. The Be1 site (¼,¼,¼) has 8f symmetry, coordinated by 10 Be sites and two transition metal sites. The Be2 site (x,0,0) has 8i symmetry and is coordinated by 9 Be sites and one transition metal. Finally, the Be3 site (x,½,0) has 8j symmetry and is coordinated with 10 Be sites and two transition metal sites. Thus, the Be2 and Be3 sites exhibit special positions, so that different compounds with the same crystal structures have different position values. To part validate the choice of the model, and in par ticular the choice of the exchange-correlation functional, the calculations of perfect crystals structures were performed using the local density approximation (LDA) and PBE functional, the results of which are compared with experimental data in table 1. It is clear that results generated using PBE are in closer agreement with experiment than LDA. (Further comparison between PBE and LDA results are discussed later: in figure 3 for phase stability and for the formation energies of vacancy and anti-site defects in Be 12 Ti in the appendixtable A2.) The simulated lattice volumes are ~1% below their experimental volumes, typical for DFT predictions on metallic and intermetallic systems when using the PBE exchange correlation terms [19], and significantly closer to the experimental value than those predicted using the LDA functional.
While the perfect structure of this family of materials has been well characterised, interstitial sites within the I 4 /mmm structure are identified here for the first time. This is achieved using the brute force approach described by Murphy [32], by seeding a Be 12 Ti unit cell with a dense grid of Be and Ti interstitials with 0.03 nm spacing and performing a single point calculation to find the energy of these (unrelaxed) sites. The 20 lowest energy and symmetrically distinct defects were reproduced in a 2 × 2 × 2 supercells and geometry optimised to find the final position of the interstitial site. Four interstitial sites were identified as present within the I 4 /mmm structure in figure 2. Three interstitial sites can accommodate both Be and Ti, a 2b site labelled i1, a 4b site labelled i2, and an 8h site labelled i3. The Wyckoff notation uniquely describes their location. A further site was found to be stable for the X species at (0,½,½) (4c symmetry) and is labelled i4. Be interstitials placed on this  site move to a neighbouring Be3 site, displacing the Be atom to the i1 site when geometry optimised. In all cases, these present as typical interstitials in a complex structure, remaining on high symmetry sites and perturbing the surrounding lattice in a roughly symmetrical manner rather than forming a dumbbell as is common in simple metallic structures [33,34].

Point defects
The formation enthalpies (E f ) of a Be vacancy and interstitial, denoted in Kröger-Vink notation [35] V Be and Be i , are presented relative to their elemental reference states in table 2, following for example: For these materials, the V Be2 site systematically exhibits the lowest E f . V Be1 has the next highest E f , except for Be 12 Ti where the order of the Be2 and Be3 sites is reversed. For all cases there is only a small relative difference in E f between all three sites, reaching a maximum of 0.32 eV for V Be2 and V Be3 in Be 12 Mo. For Be i the i2 site systematically exhibits the lowest E f , with the i1 and i3 sites significantly higher. All Be i species have considerably higher E f than V Be species.
The formation energies of V X and X i are presented in table 3. Both V X and X i have significantly higher E f than V Be and Be i . For all materials V X exhibit lower E f than the lowest enthalpy interstitial site, X i4 .
Another important defect process resulting from radiation damage is antisite disorder, the E f of which are presented in table 4. In all materials the accommodation of Be on a X site has a large energy penalty, likely due to the size mismatch between Be and the X species. The lowest E f defect is X Be2 , likely as this site is only coordinated with one X site rather  [16]. b Middleburgh and Grimes (DFT data) [17].  than two as for the Be1 and Be3 sites. In the case of Be 12 W the energy penalty for accommodating X on a Be site is noticeably larger than for other materials.

Defect clustering
Since larger defects such as voids might form through coalescence of smaller clusters, how defects interact, particularly whether there is a driving force for association, is a key indicator of how the microstructure of a material will evolve during irradiation. As an initial step towards cluster formation, the binding enthalpies E B of nearest neighbour vacancies and interstitials were calculated with respect to the lowest enthalpy isolated defect of each type, that is, two V Be2 or two Be i2 (see table 5). Also, the cluster X 2Be was considered, formed from X Be and V Be (which provides more volume for the larger X atom than a single vacancy). In all cases the propensity to form a cluster is indicated as a negative binding energy, for example, which is the lowest energy orientation cluster incorporating two lowest energy isolated vacancies (where out of plane indicates the two vacancies are orientated out of the basal plane). Alternately, will yield the binding energy to form the V Be1 V Be3 cluster from two (lowest energy) isolated V Be2 defects (which is positive and thus the cluster is not stable). Table 5 shows that for all materials the strongest cluster association by far is between V Be2 and V X , while two V X defects are strongly repelled in all materials. That some orientations of the V Be V Be and V Be V X divacancy are favourable suggests such clusters may act as nucleation sites for void formation, however, given that this cluster energy is strongly orientation dependent, further simulations of larger clusters are required to validate such a hypothesis.
Values of E B for two Be i on nearest neighbour sites are presented in table 6. Be i Be i clusters include those on the i4 site, despite this site not being stable for individual Be i defects. For most materials and interstitial combinations, E B (Be i Be i ) is repulsive, thus there is usually no driving force to form such clusters. One exception to this is E B (Be i4 Be i4 ) which is slightly  negative in both Be 12 Ti and Be 12 V and only moderately repulsive in Be 12 Mo and Be 12 W. Thus, we predict no driving force for the formation of larger interstitial defect clusters, through a model where isolated interstitial defects associate initially into pairs, other than via two interstitials reorienting onto i4 sites. The binding energies for V Be and X Be to form X 2Be are presented in table 7. E B ranges from strongly negative (−4.55 eV) to strongly positive (4.26 eV). The lowest energy sites are split between Be2-Be3 for Be 12 V and Be 12 Mo, and Be2-Be2 for Be 12 Ti and Be 12 W. The latter case might have been anticipated, as this position corresponds roughly to the arrangement of atoms in the Be 17 Ti 2 Pc/mmm phase, which is closely related to the Be 12 Ti I 4 /mmm phase [23].

Defect disorder processes
Defects can be generated in a material through several different processes (disorder reactions) that can occur thermally, or be driven by radiation damage cascades. Table 8 shows the energies associated with Frenkel, Schottky and Antisite processes (normalised by the number of defects for each process).
The range of E f values for vacancy, interstitial and antisite defects leads to a range of values for each disorder process, which can vary by as much as 2 eV or as little as 0.2 eV. While the minimum value could be the most significant in equilibrium processes, radiation damage is not an equilibrium process and thus higher enthalpy configurations may also be important.
For all of these materials, Schottky disorder is the lowest enthalpy disorder process, while Be Frenkel disorder exhibits a similar albeit slightly higher energy. Indeed for Be 12 Ti, Schottky and Frenkel disorder energies are essentially identical. Conversely, X Frenkel disorder is a much higher energy process in all materials. Thus, we predict a strong thermodynamic driving force for the removal of X i species from a damaged lattice. In the case of Be 12 V, antisite disorder will also be significant, being within 0.3 eV/defect of the Schottky process.

Nonstochiometry
In neutron multiplier materials 9 Be atoms are depleted to generate additional neutrons. Thus, it is important to understand the stability of Be-X phases at different Be contents. First we investigate a wide range of Be-X compositions through convex hull analyses, then we focused on small deviation of stoichiometry from the Be 12 X composition. Figure 3 shows the convex hulls obtained for all four Be-X series. For completeness, these calcul ations were repeated with LDA exchange-correlation functional.
Within this analysis, all phases that lie on the line of lowest-energy phases are predicted to be stable, while those that lie above the convex hull are predicted to be unstable (notwithstanding temperature effects). Formation energies obtained with the PBE functional are significantly lower than those obtained with LDA, consistent with the fact that LDA calculations are known to over-delocalise electrons, which may lead to an apparent increase in stability of the parent metals. Nevertheless, except for Be 22 X, the two methods are qualitatively in agreement with one-another. Be 22 X is consistently predicted unstable with LDA and stable with PBE, while experimentally it is known to be stable (at room temperature) for W and Mo but unstable for Ti and V [36][37][38][39].
The PBE convex hulls for Be-Mo and Be-W are in perfect agreement with experimental phase diagrams [37,38], reporting only three stable compositions: Be 22 X, Be 12 X and Be 2 X, with all other candidate phases lying well above the convex hull. A single study also reports the formation of BeMo 3 [40], but this has not been confirmed by subsequent experiments. Our calculations predict this phase to be unstable. The Be-Ti and Be-V phase diagrams are more complex: the simulation results are in agreement with the experimental literature regarding the stability of Be 12 Ti, Be 12 V, Be 17 Ti 2 , Be 17 V 2 , Be 2 Ti and Be 2 V; but in contrast to the experimental observation, Be 22 Ti, Be 22 V and Be 3 V are also predicted to be stable (with the PBE functional), while Be 3 Ti is predicted to be unstable by a small margin [36]. It is important to note that the formation energies of these phases lie within meVs of the boundary of stability. Thus, these discrepancies are likely attributed to temperature effects (both enthalpy and entropy), as was shown to be the case for the closely-related Be-Fe and Be-Fe-Al systems [18].
To assess the possible deviation from stoichiometry that Be 12 X phases may accommodate, one needs to consider the formation of defects that lead to non-stoichiometry against the formation of a phase with neighbouring composition. This is quantified by the energy to dissolve a formula unit of the nearest 0 K reference state into Be 12 X, creating defects in the Be 12 X lattice. For instance in the case of Be 12 Ti, where Be 17 Ti 2 phase incorporation results in V Be formation: where Be Be are Be atoms on Be sites in the host Be 12 Ti lattice. A full list of these equations and reference states (which are chosen using a convex hull analysis) is presented table A1 in the appendix. The minimum energy for the incorporation of reference states is shown in table 9. The number of defects formed can then be calculated at temperature using the Arrhenius approximation and total deviation from stoichiometry calculated, as presented in figure 4.
All compounds exhibit almost no deviation from stoichiometry (note that the full range of the ordinate axis corresponds to a compositional variation of 0.08 at%). This is especially true for Be 12 Ti, while Be 12 V may accommodate limited Be substoichiometry, Be 12 W may accommodate very minor levels of Be hyper-stochiometric, and Be 12 Mo may accommodate small deviation on both side of the stoichiometric composition. This difference in behaviour can be attributed to the relative energy of the three lowest energy defect reactions resulting in V Be X Be and X 2Be formation. In Be 12 Ti the lowest energy defect is Ti 2Be with energy 1.48 eV, considerably higher than for other materials thereby promoting little deviation from stoichiometry. In the case of Be 12 V, V 2Be has a lowest energy, 0.84 eV, so that any non-stoichiometry will be accommodated by an excess of the transition metal. A similar profile occurs in Be 12 Mo, where Mo 2Be has energy 0.88 eV, however Be Mo has formation energy 0.81 eV which also allows for some non-stochiometry in the Be rich region. Be 12 W becomes hyperstochiometric as Be W has energy 0.88 eV.
The extent of non-stochiometry predicted in these materials is such as they may be considered line compounds. Thus, upon depletion of Be due to the (n,2n) reaction, these materials are likely to form secondary phases with composition Be 17 V 2 , Be 17 Ti 2 , Be 2 W and Be 2 Mo, to accommodate excess X. From this point of view, Be 12 V and Be 12 Ti are likely to be the least affect by the formation of secondary phases, not because of their ability to accommodate deveiations from stoicbiometry, but because the secondary phases have structural similarities to the parent phase [23] and not too dissimilar composition. Following the same logic, Be 22 W and Be 22 Mo should be considered as possible candidates over Be 12 W and Be 12 Mo.

Conclusions
Density functional theory simulations have been carried out to predict defect properties of Be 12 X materials. The PBE functional yielded better agreement with experiment compared to the LDA functional, both in terms of phase stability and lattice parameters (within 1% for PBE). Formation enthalpies of all possible vacancies, interstitials and anti-site point defects were calculated for all four materials. Four stable sites for intrinsic interstitials are identified for the first time in the I 4 /mmm structure: the i2 site (4b) is most stable for Be selfinterstitials but for X i species, the i4 (4c) site offers a similar or lower energy. In all cases, the Be sub-lattice accommodates defects more readily than the X sub-lattice. Formation energies of point defects were combined to predict the energies of intrinsic disorder processes. Of these, Schottky disorder was identified as the lowest energy process, while Be Frenkel disorder exhibited only slightly higher energy for the V and Ti containing materials.
Small clusters including V Be V Be , Be i Be i and X 2Be were investigated, with some combinations exhibiting favourable binding enthalpy, particularly for X 2Be , which in some cases has a notably lower enthalpy than that of simple antisite disorder. This is likely due to the large size discrepancy between Be and the X species.