The distribution of cations in FeSbO 4 A computer modelling study

1. The distribution of cations in FeSbO4 A computer modelling study Ricardo Grau-Crespo*, Nora H. de Leeuw and C. Richard A. Catlow School of Crystallography of Birkbeck College and Department of Chemistry of University College London *Email: <r.grau-crespo@mail.cryst.bbk.ac.uk> FeSbO4 is a material that is commercially used to catalyse the synthesis of acrylonitrile, an intermediate in the production of acrylic plastics and fibres. Experimental studies using diffraction techniques have shown that FeSbO4 has a rutile-like structure, in which the Fe and Sb cations are distributed in the octahedral positions, with no apparent long-range preferential order. The existence of any order in the distribution of cations at a shorter range or in particular directions within the crystal are much more difficult to investigate using experimental methods. In the present work we employ quantum-mechanical computer modelling techniques to study the cation distribution in the material. By comparing the stabilities of all the different cationic configurations in FeSbO4 supercells, we conclude that Fe and Sb cations strongly prefer to alternate along the c direction of the crystal, but there is no definite order in their distribution in the other two directions. We show that the distribution of cations in this manner can explain some of the magnetic properties observed in this material, including two-dimensional antiferromagnetism and spin-glass behaviour. Statistics of configurationsWe estimate the probability of occurrence of a given configuration using Boltzmann statisticswhereis the free energy corresponding to the configurational entropy(T = 1273 K is a typical synthesis temperature and Ωm is the multiplicity of the independent configuration m). Methodology We employ supercells of different sizes and shapes, for which all possible cationic configuration are calculated. A computer program was developed to find the independent configurations according to the symmetry of the crystal. The VASP code was used to relax and obtain the energy of each cationic configuration. Rutile-like structure of FeSbO4 - Density Functional Theory - Basis Set of Plane Waves - Ultrasoft Pseudopotentials - Generalized Gradient Approximation (GGA) - Perdew 91 exchange – correlation functional - Structure relaxation via conjugated gradients Supercell 1x1x2: two different configurations Supercell 1x1x3: three different configurations Domains of ordered distributions of cations as in configuration 3? Configuration 2 has a lower energy and also a larger multiplicity (and hence a larger configurational entropy). The free energy of configuration 1 is more than half an eV higher than the free energy of configuration 2, and therefore the Boltzmann probability of configuration 1 is very close to zero, which means that the alternation of different cations along the c-axis is more favourable than having vertical chains of identical cations. Also note that the γ angle in configuration 1 deviates from the experimental angle of 90º, which does not occur for configuration 2. Structure suggested from electron diffraction experiments (Berry et al, 1987) Supercell 1x1x6: -FeSbFeSb- vs. -FeFeSbFeFe- : The cationic arrangement with complete Fe-Sb alternation along the c-axis was found to be considerably more stable than the most stable configuration in the 1x1x3 supercell. conf 1 conf 2 Ω 2 4 TS / eV 0.08 0.15 E / eV -79.00 -79.49 Frel / eV 0.56 0 P 0.01 0.99 a, b / Å 4.54 4.56 c / Å 6.20 6.12 α, β / deg 90 90 γ / deg 89.07 90 conf 1 conf 2 conf 3 Ω 2 6 12 TS / eV 0.08 0.20 0.27 E / eV -118.40 -118.53 -119.06 Frel / eV 0.86 0.61 0 P ~ 0 0.004 0.996 a, b / Å 4.530 4.585 4.562 c / Å 9.318 9.228 9.206 α, β / deg 90.00 90.00 90.00 γ / deg 89.11 89.64 89.62 E(conf3) < E(conf2) < E(conf1). The lowest – energy configuration 3 also has the largest multiplicity and hence it has the minimum configurational free energy, well below the values for configurations 1 and 2. E E + 0.71 eV Distribution of probabilities in the 2x2x2 supercell Supercell 2x2x2 There is not just one configuration with appreciable probability, but all 11 distributions can occur, since the energy differences between them are small. Therefore, even at short range, a large degree of disorder should be expected in the a and b directions, contrasting with the rather well-ordered alternation of cations along the c-axis. 180 inequivalent configurations! The configuration energies increases with the number Nv of vertical Fe – Fe (or Sb - Sb) pairs. A gap of around 1 eV exists between the energies of the group with Nv = 0 (strict vertical Fe – Sb alternation) and the rest of the configurations, which is large enough to reduce to zero the probabilities of configurations with Nv ≠ 0. Probabilities vs. free energies for the 11 most stable configurations (those with Nv=0). The curve represents the Boltzmann exponential function. Conclusions There is a clear preference for the alternation of Fe and Sb cations along the lines of edge-sharing octahedra in the 001 direction of the crystal These chains of alternating Fe and Sb are connected to each other through octahedra corners with no apparent preferential order The experimentally observed 2D character of the magnetic correlations in the material is caused by the low probability of vertical Fe-Fe pairs The similar strength of the magnetic coupling for diagonal and vertical pairs can lead to magnetic frustration, which is an essential ingredient of the spin glass behaviour. Acknowledgments Dr. J. Cockroft (Birkbeck), Dr. G. Sankar(RI), Dr. S. Carling (RI) and Prof. S. T. Bramwell (UCL) for useful comments / EPSRC for the financial support of this work (grant GR/S01986/01) / ORS scheme for a studentship award. 2D AFM correlations The low probability of vertical Fe-Fe pairs prevents the extension of magnetic correlations along the c-axis. This effect has been observed experimentally. Magnetic frustration The similar values for Jv and Jd give rise to frustration in triangular configurations, which can produce the spin- glass behaviour observed in experiment. Magnetic Configurations in the 1x1x2 supercell 1 FM 1 AFM 2 FM 2 AFM E / eV -80.41 -80.60 -80.35 -80.74 N(α) – N(β) 9.66 0.00 9.76 0.00 Jv  Jd Frustration H = – 2Jv ∑{i,j}SiSj – 2Jd ∑{k,l}SkSl (S=±5/2) Jv = [E1(AFM) – E1(FM)]/8S2 ≈ - 43 K and Jd = [E2(AFM) – E2(FM)]/16S2 ≈ - 45 K Negative magnetic coupling (antiferromagnetic correlations) in agreement with experiments. Spin Glass References R.Grau-Crespo et al, Chem. Mater.16, 1954 – 1960 (2004) R. Grau-Crespo et al, J. Mater. Chem.13, 2848 – 2850 (2003)