Defect processes in Be$_{12}$X Beryllides

The stability of intrinsic point defects in Be$_{12}$X 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 Be$_{12}$V and Be$_{12}$Ti. 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 X$_{2Be}$ combinations exhibit negative binding enthalpy (i.e. clustering is favourable), although this is orientationally dependent. None of the Be$_{12}$X 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 Be exhibits a low threshold for the (n,2n) reaction when bombarded with fast neutrons a . In these applications, Be will be subject to temperatures of 600 o 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 a note: here by Be we mean 9 Be, while acknowledging 10 Be is present as a trace impurity. in terms of 3 T retention and radiation tolerance 13,14 . In particular, the Be12X series, where X is a transition metal, have proved promising, with studies showing that Be12Ti and Be12V 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 Be12Ti, 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 Be12W 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 Be12X 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 Computational methodology Computational methodology Computational methodology
Density functional theory (DFT) simulations were carried out using the Perdew, Burke and Ernzerhof scheme of the generalised gradient approximation (GGA) for the exchange-correlation 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 x 2 x 2 full Be12X 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 x 2 x 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 convergence criterion for self-consistent calculations was set to 10 -7 eV and that for the forces on atoms to less than 0.01 eVÅ -1 . The cell was relaxed the stress component less than 0.05 GPa.

Crystallography Crystallography Crystallography Crystallography
It has been established that the crystal structure of this family of materials exhibits tetragonal symmetry and spacegroup I4/mmm 23 . Several studies also report Be12Ti as being hexagonal with spacegroup P6/mmm 24,25 . In fact, Gilliam et al. 23    While the perfect structure of this family of materials has been well characterised, interstitial sites within the I4/mmm structure are identified here for the first time. This is achieved using the brute force approach described by Murphy 32 , by seeding a Be12Ti 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 x 2 x 2 supercells and geometry optimised to find the final position of the interstitial site. Four sites can accommodate both Be and Ti, a 2b site labelled i1, a 4b site labelled i2, and an 8h site labelled i3. 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 (Ef) of a Be vacancy and interstitial, denoted in Kroሷ ger-Vink notation 35 VBe and Bei, are presented relative to their elemental reference states in table 2, following for example:

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 EB of nearest neighbour vacancies and interstitials were calculated with respect to the lowest enthalpy isolated defect of each type, that is, two VBe2 or two Bei2 (see table 5). Also, the cluster X2Be was considered, formed from XBe and VBe (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 VBe1VBe3 cluster from two (lowest energy) isolated VBe2 defects (which is positive and thus the cluster is not stable).  and only moderately repulsive in Be12Mo and Be12W. 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. Pc/mmm phase, which is closely related to the Be12Ti I4/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 Ef values for vacancy, interstitial and anti-site 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 Be12Ti, 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 Xi species from a damaged lattice. In the case of Be12V, antisite disorder will also be significant, being within 0.3 eV/defect of the Schottky process.

Nonstochiometry
Due to the defect disorder processes examined previously, deviation from stoichiometry may occur.
This has been assessed by calculating the energy to dissolve a formula unit of the nearest 0 K reference state into Be12X, creating defects in the Be12X lattice. For instance in the case of Be12Ti, where Be17Ti2 phase incorporation results in VBe formation: where BeBe are Be atoms on Be sites in the host Be12Ti lattice. A full list of these equations and reference states is presented 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 fig. 3.  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 Be12Ti, 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 Be17V2, Be17Ti2, Be2W and Be2Mo, to accommodate excess X.
From this point of view, Be12V and Be12Ti 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, Be22W and Be22Mo should be considered as possible candidates over Be12W and Be12Mo.

Conclusions
Density functional theory simulations have been carried out in order to predict defect properties of Be12X materials. Calculated perfect lattice parameters were in good agreement with experiment.
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 I4/mmm structure: the i2 site (4b) is most stable for Be self-interstitials but for Xi 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 VBeVBe, BeiBei and X2Be were investigated, with some combinations exhibiting favourable binding enthalpy, particularly for X2Be, 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.
Be12Mo, Be12V and Be12W can only exhibit modest non-stoichiometry at elevated temperatures, while Be12Ti is essentially a true line compound due the relatively high energy required to form a defect relative to its nearest reference state. The ability to accommodate the removal of Be atoms (X excess) without severely affecting the compounds stability is a noteworthy property for neutron multiplier purposes, where Be atoms are continually consumed to maintain the necessary neutron flux. Consideration of which secondary phases might form is therefore important.