J. MATER. CHEM., 1991, 1(6), 1079-1080 MATERIALS CHEMISTRY COMMUNICATIONS Defect Calculations in Solids beyond the Dilute Limit Robert A. Jackson," James E. Huntingdon" and Richard G. J. Ball" a Department of Chemistry, University of Keele, Staffordshire ST5 5BG, UK Materials and Chemistry Division, AEA Technology, Harwell Laboratory, Oxfordshire OX11 ORA, UK Preliminary results are presented on the calculations of defect parameters beyond the dilute limit. The method employs lattice-energy minimisation with a defect supercell. Defect parameters for UO, are calculated as a function of concentration. Keywords: Defect; Lattice-energy minimisation Recent years have seen considerable progress in the calculation of defect properties of inorganic solids. A wide range of materials have been investigated, and quantitative agreement has been established in many cases.A number of general reviews are available (see for example Catlow and Mackrodt,' Catlow2) and specific applications include nuclear fuels3 and transition metal oxide~.~ In general, these calculations have employed the Mott-Littleton method' which embeds the defect in a perfect lattice, and they are therefore restricted to concentrations of defects in the dilute limit. In this communication we present the initial results of the application of a method which enables higher defect concen- trations to be studied. The method is based on a standard lattice-energy minimisation calculation, but employs a super- cell composed of a number of basic unit cells.Because this supercell is repeated throughout all space, a concentration of defects beyond the dilute limit can be established. The defect concentration can be controlled by varying the size of the supercell and/or the number of defects within it. A related approach has been used by Cormack6 to investigate shear plane ordering in transition metal oxides; the emphasis of the present work is in the calculation of defect energies and structures. Additionally, in contrast to the Mott-Littleton approach, calculations of lattice properties (elastic constants, relative permittivities etc.) are possible for the defective solid. As an example of the application of the method, U02 is chosen; this has been the subject of a series of Mott-Littleton calculations, both of its basic defects3-' and of the behaviour of fission gas within the However, during the operation of a nuclear reactor, high concentrations of defects and fission products may be generated within the U02 fuel, so there is clearly a need for calculations beyond the dilute limit.The method involves (i) generation of the supercell, (ii) incorporation of defects and (iii) lattice-energy minimisation. Step (i) is a straightforward process involving repetition of the basic unit cell by lattice translation. Step (ii) involves addition, removal or substitution of atoms within the supercell to create the appropriate defect. The method necessarily involves an ordering of the defect distribution over the super- cells, It is, however, possible to investigate a range of ordering schemes by varying the supercell size and shape and the distribution of defects within the supercell itself.The calcu- lations reported here relate to one particular ordering scheme, that of a central distribution where defects are placed at the centre of the supercell, which has a 2 x 2 x 2 index consisting of 96 atoms. The defect distribution and the variation of shape of the supercell at this relatively small size have minimal effect on the results obtained; the effect of alternative distributions on larger supercells will be the subject of a future paper. Step (iii), lattice-energy minimisation, employs the THBREL code." This program has been used extensively in the calcu- lation of structural and energetic properties of a wide range of materials.In carrying out this step for a defective supercell care must be taken to avoid minimisation problems. Diffi- culties can be overcome by performing the minimisation in stages, i.e. by selectively 'freezing' parts of the cell while others are allowed to relax. In the final calculation, however, the whole structure is allowed to relax to a minimum-energy configuration. As with any atomistic simulation study, the specification of interatomic potentials is of paramount importance. In this study, potentials are taken from the earlier Mott-Littleton study of Jackson et d3It should be noted that for higher defect concentrations, cation-cation potentials could become important.In the plutonium substitution calculations described below, it was found that the inclusion of U4+-Pu4+ and Pu4+-Pu4+ potentials could improve the speed of -0.3484 22 -0.3486 .-CBI + P2 -0.3488 L aQ $ -0.3490 a a -0.3492 .-c 3 c..-c;-0.3494 v) -0.3496 IIII111111 102030405060108090100 pu4+ ions (Yo) Fig. 1 U0,-PO, substitution energy for a supercell of index 2 x 2 x 2 (96 atoms). 0,Central distribution; *side-ordered distribution; 0, random distribution Table 1 Comparison calculations for basic defect energies defect U, supercell/eV U, Mott-Littleton/eV anion vacancy anion interstitial 17.33 -12.20 17.16 -12.29 cation vacancy cation interstitial 80.11 -61.22 80.27 -61.43 cation Frenkel 18.88 18.71 anion Frenkel 5.12 4.87 Schottky trio (unbound) 11.27 10.04 convergence of the minimisation.These potentials were obtained by electron-gas methods.I2 Results of supercell calculations on U02 are presented below. Table 1 is a comparison of basic defect formation energies calculated by the present method at low concen-trations with those calculated using the Mott-Littleton meth-od~logy.~The variation of substitutional defect formation energies with concentration is shown in Fig. 1, illustrating the minimal effect of various distribution schemes on the results obtained. Fig. 2 shows the variation of lattice parameter with 5.446 0 5.444 0 $ 5.442 .I-0 a g 5.438 .-.I- s 5.436 0 1.434 1 5.4305-432 L----2 10 20 30 40 50 60 70 80 90 100 Pu4+ ions (%) Fig. 2 Lattice parameter us.Pu4+ concentration for a 2 x 2 x 2 (96 ion) supercell J. MATER. CHEM., 1991, VOL. 1 increasing Pu4+ concentration,which is approximately linear, in agreement with the predictions of Vegard's Law. In this short communication we have presented preliminary results of a programme of work to investigate the consequences of defect concentrations beyond the dilute limit. The validity of the method is borne out by the close agreement between the low concentration and dilute limit results. This method has many potential applications in solid-state chemistry for the study of properties associated with non-stoichiometry, fission product incorporation etc. that have hitherto not been poss-ible using Mott-Littleton methods.Further calculations are in progress in these areas. The authors are grateful for the provision of computer time from the SERC and ULCC. Some of this work was funded as part of the corporate research programme of AEA Tech-nology. References 1 C. R. A. Catlow and W. C. Mackrodt, Computer Simulation of Solids, Lecture Notes in Physics no 166, Springer-Verlag, Berlin, 1982. 2 C. R. A. Catlow in Defects in Solids: Modern Techniques, ed. A. V. Chadwick and M. Terenzi, NATO AS1 Series B: Physics vol. 147, Plenum, Oxford, 1987. 3 R. A. Jackson, A. D. Murray, J. H. Harding and C. R. A. Catlow, Philos. Mag., 1986, A53, 27. 4 S. M. Tomlinson, C. R. A. Catlow, and J. H. Harding, J. Phys. Chem. Solids, 1990, 51, 477. 5 N. F. Mott and M. J. Littleton, Trans. Faraday Soc., 1938, 34, 485. 6 A. N. Cormack, Solid State lonics, 1983, 8, 187. 7 C. R. A. Catlow, Proc. R. SOC. London, Ser. A, 1977,353, 533. 8 R. A. Jackson and C. R. A. Catlow, J. Nucl. Muter., 1985, 127, 161. 9 R. A. Jackson and C. R. A. Catlow, J. Nucl. Muter., 1985, 127, 167. 10 R. G. J. Ball and R. W. Grimes, J. Chem. SOC.,Faraday Trans., 1990, 86, 1257. 11 R. A. Jackson and C. R. A. Catlow, Molecular Simulation, 1988, 1, 207. 12 J. H. Harding and A. H. Harker, UKAEA Harwell Report, AERE R-10425. 1982. Communication 1/0452I K; Received 29th August, 1991