J. MATER. CHEM., 1991,1(6), 1035-1039 Atomistic Lattice Simulations of the Ternary Fluorides AMF, (A=Li, Na, K, Rb, Cs; M=Mg, Ca, Sr, Ba) Neil L. Allan,*' Mark J. Dayer,b Daniel T. Kulpband William C. Mackrodtb a School of Chemistry, University of Bristol, Cantocks Close, Bristol BS8 ITS, UK ICI Chemicals and Polymers Ltd., P.O. Box 8, The Heath, Runcorn, Cheshire WA7 400, UK Atomistic lattice calculations are reported of the structure and stability of the series of ternary inorganic fluorides AMF, (A=Li+, Na+, K+, Rb+, Cs+; M=Mg2+, Ca2+, Sr2+, Ba2+). The predicted low-temperature structures agree with the available experimental data, with good agreement between the experimental and theoretical lattice constants. When A+ and M2+ are comparable in size, low temperature phases with the ferroelectric lithium niobate structure are predicted.As yet no phases with this structure appear to have been observed. Keywords: Atomistic lattice simulation ; Ternary Fluoride Solid inorganic fluorides appear to be playing an increasingly important role in areas such as novel glasses for low-loss optical fibres, solid-state lubricants, thin-film solid electrolytes and lasers,' in addition to their established use as fluorination catalysts, principally in the synthesis of fluorocarbons. In common with other classes of ionic insulators, detailed infor- mation of the lattice and defect properties, many of which control the applications currently being exploited, is sparse, so that the design of new materials from fundamental consider- ations is limited.However, recent theoretical studies, mainly of oxides,2 have shown that many important solid-state properties related to phase stability, stoichiometry, matter and charge transport, impurity segregation and the like can be calculated to within experimental accuracy using atomistic lattice simulations. Furthermore, the structures of inorganic fluorides are commonly close packed and hence generally less complex than those of the corresponding oxides and chlorides, owing presumably to the greater ionicity and reduced size and polarisability of the fluoride ion, which makes their study even more suited to these techniques. However, apart from work, mainly by Catlow and c~-workers,~*~ on the binary alkali-metal and alkaline-earth-metal fluorides, there would seem to be relatively few reports on more complex fluorides, which is surprising in view of the range of complex oxides that have been ~tudied.~Accordingly, this paper reports atomistic lattice calculations of the structure and stability of a range of ternary fluorides of the type AMF,, where A =Li, Na, K, Rb, Cs; M =Mg, Ca, Sr, Ba.Theoretical Methods The theoretical methods used in this study are similar to those used previously for a wide range of ceramic oxides.2 The calculations are all formulated within the framework of an ionic model, which is generally more applicable to fluorides than it is to oxides, with integral charges assigned to the constituent ions, i.e.+I for Li-Cs, +2 for Mg-Ba and -1 for F. Two-body interatomic potentials have been used throughout and are based on a modified form of the Kim- Gordon electron-gas approach.6 As ionic polar- izability was included by means of the Dick-Overhauser shell model.7 Following the approach for complex oxides adopted in recent studies of high-T, ternary cuprates,8 the potentials for A+-F-and M2+-F- were exactly those derived for the binary systems, AF and MF2,9t all of which were based on a single F--F-potential, which has been retained in the present study. The shell parameters for A+, M2+ and F-are retained from the parent binary fluorides. The associated shell-model polarizability of F-is 0.99 A3. The lattice simulations reported are all static simulations of perfect lattices, which give the crystal structure and lattice energy of the low-temperature phase.At 0 K, the lattice structure is determined by the condition that it is in mechan- ical equilibrium i.e. aqaxi=0 in which E is the internal energy, and the (Xi}are the variables that define the structure, namely the three lattice vectors, the atomic positions in the unit cell, and, in the case of the shell model, the shell displacements. The last of these represent the electronic polarization of those ions which are not at a centre of inversion symmetry in the lattice and, for polarizable ions such as Cs+ or F-, can make an appreciable contribution to the internal energy. AMF3 Compounds We have based our study of AMF3 compounds on the most common structures adopted by AMO, oxides," since the oxide and fluoride ions have similar radii under normal conditions of temperature and pressure.These structures fall into two classes. (i) The first arises when A is large enough for the formation of close-packed layers AX3 which can be stacked in various ways. The simplest such structure is the cubic perovskite (Fig. 1) in which the AX3 layers are cubic close packed. Orthorhombic variants, in which the M-X-M bridges linking the MX6 octahedra are not linear, are also common (Fig. 2). Known fluorides with the cubic perovskite structure, which are relevant to the present work, include KMgF,, RbMgF,, RbCaF,, CsCaF, and BaLiF,. In all except the last of these structures it is the larger univalent ion which is 12-co-ordinate with a nearest-neighbour separation of 2-'I2a,, where a, is the lattice constant.The divalent ions are six-co-ordinate with a smaller nearest-neighbour separation of a,/2. In contrast BaLiF, has an 'inverse perovskite' structure in that the large barium ion is 12-co-ordinate and the smaller lithium ion six- co-ordinate. Known orthorhombic perovskites include t Full details of the shell-model parameters and two-body potentials are available from the author*. OF eM (-J*......:. ..I.. Fig. 1 'Ideal' cubic perovskite structure Fig. 2 Tilting of MX, octahedra in orthorhombically distorted AMF, perovskites (after ref. 30) NaMgF,, KCaF, and RbCaF,. Other stacking sequences of the AX, layers are known.In oxides" there is a large number Of possible structures, whereas there appear to be only three types of hexagonal fluoride struct~re.'~-~~ These are the RbNiF,, CsCoF,, and CsNiF, structures (Fig. 3) in which the CsCoF, 0 Co,Ni0Cs,Rb B A €3 J. MATER. CHEM., 1991, VOL. 1 AF3 layers are packed in the order ABCACBA..., ABABCBCACA... and ABA ..., respectively. Oxides with these structures are BaTiO,, BaRuO, and BaNiO,." Of the fluor- ides to be considered here RbMgF, has the RbNiF, struc-ture," while at high pressure CsMgF, exhibits both the RbNiF, and CsCoF, structures.I6 (ii) In oxides the second class of structures occurs when A and M are approximately the same size and the size is suitable for octahedral co-ordination.These adopt structures which are either a random or ordered arrangement of A and M ions both of which are six-co-ordinate. Examples considered in this work are the lithium niobate and ilmenite structures" (Fig. 4), both of which contain hexagonally packed anion layers and differ solely in the distribution of the cations between the octahedral holes. The cation stacking sequence along the c-axis in the LiNbO, structure is LiNbLiNb ... and in the ilmenite structure, LiNbNbLi ... We have been unable to find reports of any ternary fluoride with either type of structure. Indeed, Babel l2 has commented that fluoride struc- tures of the ilmenite type 'do not seem to exist'. Previous attempts to rationalise the structures adopted by 0 Nb 0 Fig.4 Lithium niobate crystal structure. The ilmenite structure is very similar; the stacking sequence of cations A and B along the c axis is ABBA ... in place of ABAB ... CsNiF, RbNiF, A B C A C B A B C A Fig. 3 RbNiF,, CsCoF, and CsNiF, crystal structures (after ref. 13) J. MATER. CHEM., 1991, VOL. 1 ternary systems such as these have been largely in terms of the radii of the constituent ions.12-14 By defining a tolerance factor, t, t =(rA+rX)/2'12(rM+rx) where rA, rM and rx are the ionic radii of ions A, M and X, then the following rules have been pr~posed'~,'~ linking the value of t with the structure of the compound AMX,: 0.76 <t <0.88* orthorhombic perovskite 0.88 <t <1.003 cubic perovskite 1.OO <t <1.1 3 hexagonal perovskite Babel" comments that this very simple idea is much more successful at rationalising the structures of fluoride perovskites than those of the oxide perovskites for which it was originally developed, presumably because of the greater ionicity of the fluorides.In this scheme when t= 1, both cations are in contact with the anion in the 'ideal' cubic structure. For t<0.88, the A cation is too small to touch the anions in the cubic structure and the M-F-M links bend, tilting the MF6 octahedra to bring some anions into contact with the A cations. If t >1, hexagonal structures form in which there is some degree of face sharing of MF6 octahedra (Fig.3). When t is very large, as in CsNiF, (Fig. 3), there is no corner sharing of these octahedra at all. Table 1 lists the values of the tolerance factor t for the systems studied in this work. The ionic radii have been taken from Shannon and Pre~itt,'~ with an effective ionic radius of 1.33 8, for the fluoride ion. For consistency with the work of Babel14 ionic radii for the 12-co-ordinate cations have been assumed to be 6% greater than the radii for the same ions when octahedrally co-ordinated. The values for the tolerance factor t' for the 'inverse perovskite' structure, in which the co- ordinations of the A and M ions are reversed, are given by These are also listed in Table 1. It is clear from Table 1 that there are several combinations of cations of similar size (e.g.Li+ and Mg2+, Na+ and Ca2+, K+ and Ba2+), for which both t and t' are less than 0.76 in which case the rules given above do not apply; it is for these combinations that the corresponding ternary fluorides might be expected to adopt the lithium niobate or ilmenite structures.For each AMF, compound we have calculated the lattice energy and lattice constants of the possible structures: cubic, orthorhombic, hexagonal perovskite, lithium niobate and ilmenite. Triclinic distortions of the orthorhombic structure were also investigated. For each system the predicted low- temperature phase, i.e. that with the lowest lattice energy, is listed in Table 2. For the cubic and orthorhombic perovskites N denotes 'normal' 12-co-ordination of the univalent cation; R ('reversed') denotes octahedral co-ordination of this ion (the inverse perovskite structure).The corresponding lattice con- stants and lattice energies are listed in Tables 3 and 4. It is Table 1 Values of the tolerance factors t and t' for the AMF, systems studied ion MgZ+(0.72) Ca2+(1.00) Sr2+(1.16) Ba"(1.36) Li'(0.74) Na +(1.02)K+(1.38) 0.73/0.7I 0.83/0.63 0.96/0.55 0.64/0.82 0.73/0.72 0.85/0.62 0.60/0.87 0.69/0.77 0.79/0.67 0.56/0.95 0.63/0.83 0.73/0.72 Rb +(1.49) Cs'( 1.70) 1.00j0.52 1.08/0.49 0.88/0.60 0.95/0.56 0.83/0.64 0.89/0.60 0.76/0.69 0.82/0.65 t and t' are defined in the text. The ionic radius in A for the six- co-ordinate cation17 is given in parentheses after each ion.1037 Table 2 Calculated low-temperature phases of AMF, compounds ion Mg2+ Ca2+ Sr2+ Ba2+ Li + LiNbO, OR OR CR" Na+ ON" LiNb0, OR OR K+ CN" ON" LiNbO, LiNbO, Rb+ RbNiF," (H) CN" ON LiNbO, cs + CSN~F,~(H) CN" CN ON ~~~~ C= Cubic perovskite, 0=Orthorhombic perovskite, H =hexagonal perovskite, N =normal (see text: univalent ion is 12-co-ordinate), R = reversed (see text: divalent ion is 12-co-ordinate)." Experimental data are available. In each case the predicted structure is that observed. References to experimental structures are as follows: LiBaF,,26 NaMgF,,I4 KMgF3,27 KCaF,,I4 RbMgF,," RbCaF,,'* CSC~F,.~~ Only high-pressure phases with RbNiF, and CsCoF, structures are known.16 See the text.clear from Table 2 that, where experimental data are available, the predicted structures are those observed at low temperature, while lattice constants are predicted typically to within 1% of the measured values. Attempted synthesis of CaMgF,, under normal pressure conditions, leads only to materials with the composition Cs4Mg,Flo.'* At high pressure16 phases with the RbNiF, and CsCoF, structures are obtained and this is consistent with the well known observation that high pressure stabilizes cubic (relative to hexagonal) stacking of the close-packed layers in these systems. Consequently our prediction of the CsNiF, structure (at zero pressure), with purely hexagonal packing of the CsF, layers, is reasonable, bearing in mind the similar ionic radii of Mg2+ and NiZ+. A comparison of Tables 1 and 2 indicates that predictions of structure simply on the basis of the values oft and tf are remarkably successful.These can be used straightforwardly to predict whether a given perovskite will adopt a normal or inverse structure as well as orthorhombic distortions from the cubic structure. For the orthorhombic perovskites Table 3 also lists the calculated M-F-M bond angles. As a general rule the smaller the value of t (or t),the smaller the M-F-M angle and the larger the orthorhombic distortion. Our conclusions concerning the existence of low-temperature orthorhombic phases for RbCaF, and KCaF,, but not for CsCaF,, agree well with previous static simulation and molecular-dynamics studies.19-22 It is particularly interesting that five of the compounds, in which the univalent and divalent cations are of comparable size (LiMgF,, NaCaF,, KSrF,, KBaF, and RbBaF,), are predicted to adopt the lithium niobate structure. The related ilmenite structure is higher in energy in each case. For these compounds the values of both t and t' are less than 0.77 except for KSrF, where Table 1 suggests an orthorhombic structure. This is the only case where the phase predicted by the calculations differs from that suggested by the tolerance fact or. Although, as already mentioned, no ternary fluoride with the lithium niobate structure seems to have been reported, Edwardson et al. have predicted a low-temperature phase with this structure for NaCaF, on the basis of a molecular- dynamics Lithium niobate is a well known ferroelectric materia1.24,25 Above the Curie temperature the lithium ions move from sites in which they are approximately octahedrally co-ordinated into the nearest anion plane where they are three-fold co- ordinate, while the niobiums move into the centre of the Nb06 octahedra (Fig.4), midway between adjacent anion oxygen planes. For NaCaF,, Edwardson et aLZ3have pre- dicted such displacements of 0.70 and 0.18 A for sodium and J. MATER. CHEM., 1991, VOL. I Table 3 Calculated lattice parameters (in A) for the low-temperature phases listed in Table 2 ion Mgz+ CaZ+ Srz+ BaZ+ ~ ~__________ Li + a =5.028 C= 13.51 a =5.190 b =5.703 a=5.421 b =5.627 a =3.98 l(3.988) c =7.345 c =7.700 e = 146.4 0= 160.5 Na + a= 5.349(5.350) b =5.462(5.474) a =5.764 c = 15.20 a =5.695 b =6.256 a =5.864 b =6.510 c =7.638(7.652)e = 149.7 0 = 134.3 c =8.080 e = 146.4 c =8.359 K+ a =3.989(3.989) a =6.12 l(6.164) a =6.406 a =6.645 b =6.221(6.209) C= 16.30 C= 17.15 c =8.706(8.757) 0 = 152.8 Rbf a =5.838(5.828) a =6.262(6.273) a =6.479 a=6.819 C= 14.21(14.20) b =6.26q6.274) b =6.622 C= 17.32 c =8.881(8.867) c =9.136 e= 167.8 0 = 149.9 cs + a =6.652 a =4.606(4.526) a =4.803 a=7.143 c =5.334 b=7.138 c= 10.12 e= 167.1 ~~ Experimentally known parameters are in parentheses.References to experimental work are as given in Table 2. For the orthorhombic perovskites 0 denotes the calculated MFM bond angle (see text). Table 4 Calculated lattice energies (per formula unit) in eV (kJ mol-') for the low-temperature phase listed in Table 2 ~~~ +ion Mg2+ CaZ Sr2+ BaZ+ +Li -42.00 -38.57 -37.14 -35.77 (-4052) (-3721) (-3584) (-345 1) +Na -40.9 1 -37.19 -35.63 -34.17 (-3947) (-3588) (-3438) (-3297)K+ -39.94 -36.19 -34.48 -32.77 (-3854) (-3492) (-3327) (-3162) Rb+ -39.5 1 -35.93 -34.22 -32.43 (-3812) (-3467) (-3301) (-3 129) cs+ -38.61 -35.05 -33.57 -31.76 (-3725) (-3382) (-3239) (-3064) calcium ions, respectively; the corresponding values reported here are slightly larger at 0.83 and 0.24 A.It is somewhat surprising that there are so few experimental reports of the ternary fluorides studied here.Consequently we have calculated their heat of formation, AH, from the corresponding binary fluorides. These are collected in Table 5. With the exception of KCaF,, for which a small positive heat of formation has been found, the calculated values of AH for the known fluorides LiBaF,, NaMgF,, KMgF,, RbMgF,, RbCaF, and CsCaF, are all negative. Of those that are apparently unknown, the heats of formation of LiMgF, and CsSrF, are calculated to be negative, while LiCaF, has been Table5 Values of AH at OK for the reaction AF(s)+ MF,(s) -,AMF,(s) ion Mg2+ Ca2 SrZ Ba2++ + +Li -0.15 0.10 0.05 -0.17 (-14.5) (9.2) (4.5) (-16.1) Na + -0.2 1 0.32 0.41 0.28 (-19.8) (31.3) (39.7) (26.8)K+ -0.47 0.09 0.33 0.45 (-45.7) (8.5) (3 1.6) (43.0) Rb+ -0.43 -0.04 0.20 0.40 (-4 1.3) (-3.7) (1 9.5) (38.8) +cs -0.47 -0.10 -0.09 0.13 (-45.7) (-9.2) (-8.5) (1 2.5) Units are eV (kJ mol-I).found to have a positive heat of formation comparable to that of KCaF,. The remaining systems, NaCaF,, KSrF, and KBaF,, which are predicted to have the lithium niobate structure, all have large positive heats of formation. As in the case of CsMgF,, which has been referred to previously, attempted synthesis could lead to phases with compositions different from those considered here. Conclusions In this paper we have shown that atomistic simulation tech- niques can be used to model a wide range of ternary fluorides and have examined the structure and stability of the series of compounds AMF, (A=Li, Na, K, Rb, Cs; B=Mg, Ca, Sr, Ba).Where data exist the agreement between experimental and calculated lattice constants is good. The calculations reported here, strictly speaking, refer to 0 K and no estimates of the entropic contribution to the free energy of formation at higher temperature have been made. The fluoride-fluoride potential seems to be widely transferable, probably as a result of the highly ionic character of these systems, and it is intended to use it further in future work. Our results indicate that some AMF, systems, notably those in which the A and M cations are comparable in size, should have a low-temperature phase with the lithium niobate structure. Despite the putative stab- ility of LiMgF, with respect to the binary fluorides, no such fluoride appears to have been reported and it is hoped that an experimental investigation of these compounds might be stimulated by the present work.References Znorganic Solid Fluorides-Chemistry and Physics, ed. P. Hagen-muller, Academic Press, London, 1985. W. C. Mackrodt, Solid State Zonics, 1984, 12, 175. C. R. A. Catlow, ref. 1, ch. 5, and references therein. A. N. Cormack, C. R. A. Catlow and S. Ling, Phys. Rev. B, 1989,40, 3278. S. M. Tomlinson, C. Freeman, C. R. A. Catlow, H. Donnerberg and M. Leslie, J. Chem. SOC., Faraday Trans 2, 1989,85, 367. W. C. Mackrodt and R. F. Stewart, J. Phys. C, 1979, 12, 431. B. G. Dick and A. W. Overhauser, Phys. Rev., 1958, 112, 90. N. L. Allan and W. C. Mackrodt, J.Am. Ceram. 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Klein, J. Chem. Phys., 1989, 90,5005. 23 P. J. Edwardson, L. L. Boyer, R. L. Newman, D. H. Fox, J. R. Hardy, J. W. Flocken, R.A. Guenther and W. Mei, Phys. Rev. B, 1989, 13, 9738. 24 M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Oxford University Press, Oxford, 1977. 25 P. W. Haycock and P. W. Townsend, Appl. Phys. Lett., 1986, 48, 698. 26 W. L. W. Ludekens and A. J. E. Welch, Acta Crystallogr., 1952, 5, 841. 27 R. C. DeVries and R. Roy, J. Am. Chem. SOC., 1953,75, 2459. 28 A Bulou, C. Ridou, M. Rousseau, J. Novet and A. W. Hewat, J. Phys. (Paris), 1980, 41, 87. 29 M. Rousseau, J. Y. Gesland, B. Hennion, G. Heger and B. Renker, Solid State Commun., 1981, 38, 45. 30 J. Geller and E. A. Wood, Acta Crystallogr., 1956, 9, 563. Paper 1/02721B;Received 7th June, 1991