Structures of α-K3MoO3F3 and α-Rb3MoO3F3: Ferroelectricity from

Oct 29, 2013 - Special Issue. Published as part of the Crystal Growth & Design virtual special issue on Anion-Controlled New Inorganic Materials. Syno...
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Structures of α‑K3MoO3F3 and α‑Rb3MoO3F3: Ferroelectricity from Anion Ordering and Noncooperative Octahedral Tilting Published as part of the Crystal Growth & Design virtual special issue on Anion-Controlled New Inorganic Materials Allyson M. Fry and Patrick M. Woodward* Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210-1185, United States S Supporting Information *

ABSTRACT: The room temperature crystal structures of α-K3MoO3F3 and α-Rb3MoO3F3 have been solved via combined Rietveld refinements of synchrotron and neutron powder diffraction data. These two compounds are part of a broader family of A2BMO3F3 compounds that have been studied for their dielectric properties, but until now the complex crystal structures of the ferroelectric phases of these compounds were not known. At room temperature and below, these two isostructural compounds are tetragonal with I41 space group symmetry and unit cell parameters of a = 19.38613(3) Å, c = 34.86739(8) Å for αK3MoO3F3 and a = 20.0748(4) Å, c = 36.1694(1) Å for α-Rb3MoO3F3. Their structures are related to the cubic double perovskite structure but are considerably more complicated due to noncooperative octahedral tilting and long-range orientational ordering of the polar MoO3F33− units. The pattern of octahedral tilting is equivalent to that seen in the α-K3AlF6 structure, which has I41/a symmetry, but orientational ordering of MoO3F33− units lowers the symmetry to I41. The polar space group symmetry is consistent with earlier reports of ferroelectricity in these compounds. Hence orientational ordering of the MoO3F33− units is directly responsible for the ferroelectric behavior. high degree of twinning,10 and (2) the pseudocubic symmetry of these phases leads to considerable peak overlap that makes powder diffraction studies challenging.7 Recent work by Abakumov et al. proposed that the structure of the room temperature α-phase of K3MoO3F3 should be related to the complex structure of α-K3AlF6 due to similarities in their electron diffraction patterns.10,11,13,14 α-K3AlF6 is a member of a relatively small group of compounds that undergo phase transitions that have been labeled noncooperative octahedral tilting (NCOT). Although tilts of the octahedra are ubiquitous in perovskites, normally the tilting follows a pattern that maintains the corner-sharing connectivity of the octahedral network.15 In ordered A2BB′X6 double perovskites where there is a large difference in the charge and radius of the B and B′ cations, NCOT phase transitions can occur. Large amplitude (∼45°) rotations of rigid B′X6 octahedra (where B is an alkali or alkaline earth cation, and B′ is a smaller, more highly charged cation) break the corner-sharing connectivity of the BX6 and B′X6 octahedra. These rotations lower the coordination number of many of the A cations while at the same time increasing the coordination number of many B cations. As a result, the coordination environments of the A and B cations end up much more similar than they are in the parent perovskite structure, where the coordination numbers are 12

1. INTRODUCTION Ferroelectricity, pyroelectricity, piezoelectricity, and second harmonic generation are properties that can only arise in materials that crystallize with a non-centrosymmetric structure. Ferroelectrics and pyroelectrics have an additional requirement that the structure be polar.1 An obvious strategy for designing polar materials is to incorporate polyatomic cations or anions that are polar. Unfortunately this does not guarantee that crystals containing such ions will also be polar. Despite the appeal of using polar building units (PBU) as a means to create polar materials from the bottom-up, this approach has only yielded relatively few examples of polar materials.2−4 For almost a century, there has been interest in harvesting the dipole moments of the polar polyatomic anions, MO3F33− (M = Mo, W) discovered by Pauling, to create functional materials.5 These materials are attractive because the MoO3F33− anion has a dipole moment of nearly ∼6 D.2 There are a number of A2BMoO3F3 or A2BWO3F3 (A, B = Na, K, Rb, Cs) compounds that are known to have properties consistent with a polar material,6−8 but the complex structures of these compounds have for the most part remained elusive.9 The useful properties these compounds, especially K3MoO3F3, have made them the objects of many structural studies.6,7,10−12 Most of these compounds are known to go through one or two phase transitions above room temperature with their highest phase being a cubic double perovskite phase and the lower temperature phase(s) being ferroelectric.7 The difficulty in determining the structures of the ferroelectric phases is 2-fold: (1) single crystals of perovskite type compounds often have a © 2013 American Chemical Society

Received: September 6, 2013 Revised: October 26, 2013 Published: October 29, 2013 5404

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Figure 1. Superstructure reflections in α-K3MoO3F3 synchrotron data used to illustrate the choice of I41 symmetry.

within this temperature range; however, lowering the temperature to 140 K sharpened the peaks causing the supercell peaks to become more prominent. Therefore, all refinements of the laboratory XRPD data were done on the 140 K scans. Laboratory XRPD of α-A3MoO3F3 (A = K, Rb) were initially fit with a structural model that assumed they were isostructural with α-K3AlF6 in space group I41/a,14 as expected from similarities in their electron diffraction patterns.10,13 Like αK3AlF6, both α-K3MoO3F3 and α-Rb3MoO3F3 have large unit cells, a ≈ √5adp and c ≈ 4adp (adp is the typical double perovskite lattice parameter of ∼8 Å). Because of the size of the unit cell and the large number of crystallographically independent sites, 52, it was not possible to refine atomic coordinates using the laboratory XRPD data. To obtain a more accurate description of the structure, synchrotron and neutron powder diffraction data were collected on α-K3MoO3F3 and αRb3MoO3F3. We should note that the similar scattering factors of oxygen and fluorine to both X-rays and neutrons make it nearly impossible to differentiate the two in the refinements. However the Mo−O and Mo−F bond lengths are significantly different, ∼1.75 Å and ∼2.00 Å respectively, which allows for a relatively straightforward, albeit indirect, method of differentiating the oxygen and fluorine sites.19 The α-K3AlF6 structure possesses I41/a space group symmetry. While the α-A3MoO3F3 (A = K, Rb) phases appear to be isostructural with α-K3AlF6 this space group is centrosymmetric which is inconsistent with reports of ferroelectricity in these compounds.6,7 This inconsistency led us to take a very careful look at the systematic absences in the diffraction patterns. In the diffraction patterns of K3MoO3F3 and RbMoO3F3, the vast majority of reflections could be fit with the I41/a unit cell of α-K3AlF6, but as discussed below there were a few weak reflections that could not be accounted for with this space group and unit cell. Neither the synchrotron nor the neutron diffraction patterns exhibited peak splitting that would indicate a distortion of the tetragonal unit cell. Hence our search for a space group consistent with the ferroelectricity of these phases was limited to lower symmetry tetragonal subgroups of I41/a. Removal of the center of inversion from the space group I41/ a (88) lowers the space group to the non-centrosymmetric space group, I41 (80). In I41/a the hk0 reflections must satisfy the h, k = 2n reflection condition, whereas this condition is relaxed in I41 due to the loss of the a-glide plane (although the body centering dictates that hkl reflections must satisfy h + k + l = 2n for both space groups). Therefore, the presence of reflections such as (350) (at 7.13° 2θ) and (570) (at 10.54°

and 6, respectively. In addition to K3AlF6, NCOT has been observed in Sr3WO6,15 Rb2KCrF6, and Rb2KGaF6.12 In this study we report the room temperature structures of the polar compounds α-K3MoO3F3 and α-Rb3MoO3F3, which show NCOT of the MoO3F33− anions. The structures were determined from combined refinements of synchrotron X-ray and time-of-flight neutron powder diffraction data. Not only do the crystal structures determined here shed light on the physical properties of this family of ferroelectric phases, they reveal a rare case of orientational ordering of the highly polar MoO3F33− anion.

2. EXPERIMENTAL SECTION Synthesis. A3MoO3F3 (A = K, Rb) were prepared via traditional solid state synthesis in an argon filled glovebox, due to the hydroscopic nature of the fluoride salts. Polycrystalline 1 g samples were prepared by combining stoichiometric amounts of starting materials MoO3 (Alfa Aesar, ≥85%) and either KF (Acros, ≥95%) or RbF (Alfa Aesar, 99.7%) were ground in the glovebox, placed in a crimped Ag tube (outer diameter 5.5 mm, wall thickness 0.25 mm, tube length ∼7 cm) to maintain the dry Ar atmosphere, and sealed with a H2/O2 torch outside of the glovebox. During sealing, the end of the tube containing the sample was immersed in ice water to ensure that the reactants did not react. The tubes were heated in a box furnace to 600 °C for 72 h. Diffraction. XRPD data were collected on a Bruker D8 powder diffractometer (40 kV, 50 mA, sealed Cu X-ray tube) equipped with an incident beam Ge 111 monochromator and Lynx Eye position sensitive detector. The diffractometer is equipped with an Anton Paar TTK 450 Camera to access low temperature data. Low temperature data was collected on these samples to damp thermal displacements and therefore aid in the observation of very small supercell peaks. Neutron and synchrotron data were collected for A3MoO3F3 (A = K, Rb). Time-of-flight neutron powder diffraction (NPD) data at 300 K and 150 K were collected at the Spallation Neutron Source, Oak Ridge National Laboratory on POWGEN powder diffractometer. Synchrotron X-ray diffraction (λ = 0.413961 Å) data at 300 K and 140 K were collected on beamline 11-BM at the Advanced Photon Source, APS, at Argonne National Laboratory. Structural refinements of laboratory XRPD data were carried out using TOPAS Academic software package using the Rietveld method.16 The 300 K and 140 K diffraction patterns were identical except for an expected shift of all of the peaks corresponding to a contraction of the lattice parameters. Therefore, we conclude that no phase transitions occur between these two temperatures. All refinements were performed on the 300 K data. Combined Rietveld refinement of synchrotron and NPD was performed with the GSAS software package.17,18

3. RESULTS AND DISCUSSION Laboratory XRPD was collected on A3MoO3F3 (A = K, Rb) at both 140 K and 300 K. No phase transitions are observed 5405

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2θ), shown in Figure 1, rules out I41/a as a possible space group. Starting with the I41/a structure of α-K3AlF6 the program ISODISTORT was used to lower the symmetry of the structure to the non-centrosymmetric space group I41.20 Combined Rietveld refinements of α-K3MoO3F3 and α-Rb3MoO3F3 were carried out on TOF neutron and synchrotron data. Because of the large number of crystallographically independent sites, 104, constraints were necessary to refine the structure of αK3 MoO 3 F 3 from powder diffraction data. Initially the MoO3F33− units were treated as rigid bodies, with an average Mo-anion bond length of 1.9203 Å (taken from the MoO3F33− units in Na1.5Ag1.5MoO3F3),19 and only rotations of rigid octahedra were permitted. The refinements at this stage were stable and gave respectable fits to the diffraction data, confirming that the rotations of the MoO3F33− octahedra were similar to the rotations of AlF63− octahedra in α-K3AlF6. From electron diffraction studies of α-K3MoO3F3, and comparison to the rare examples where the MoO3F33− units are ordered, we expect local anion ordering of MoO3F33− in a fac configuration.21 Therefore, the opposite faces of the MoO3F33− octahedra were allowed to refine as two rigid faces independent of one another. The face with the shorter Mo− anion bond length was determined to be the oxygen face, and the fluorine face was determined to be the face with longer Mo−anion bond length. The rigid bodies were then removed and soft bond length constraints were added: Mo−O 1.75 ± 0.09 Å, Mo−F 2.00 ± 0.09 Å, and anion−anion 2.60 ± 0.10 Å. The anion−anion constraint was assumed based on a right angle formed between O−Mo−F where the hypotenuse would be the O−F bond length. The MoO3F33− is distorted and thus does not form a perfect O−Mo−F right angle, and therefore a larger tolerance of 0.10 Å was allowed for this constraint. Additionally, the displacement parameters were refined as one value for each type of cation and one value for O/F. The refinement of the α-Rb3MoO3F3 was carried out in a similar manner starting from the α-K3MoO3F3 structure. The refinements confirmed that α-Rb3MoO3F3 and α-K3MoO3F3 are isostructural. Details obtained in the refinements of α-K3MoO3F3 and αRb3MoO3F3 are given in Table 1, and the final fits to the synchrotron and TOF neutron data are shown in Figures 2−5. The atomic positions for both compounds are listed in the Supporting Information. The final refinement of the data shows that α-K3MoO3F3 and α-Rb3MoO3F3 have the same pattern of octahedral rotations as seen in α-K3AlF6. However, the refinements also show that the oxygen and fluorine positions are ordered. The orientational ordering of MoO3F33− ions, which has previously only been observed in (Ag3MoO3F3)(Ag3MoO4)Cl,22 Na3MoO3F3,3 and Na1.5Ag1.5MoO3F3,19 is responsible for destroying the a-glide plane and lowering the symmetry from the nonpolar I41/a structure of α-K3AlF6 to the polar I41 structure of α-K3MoO3F3 and α-RbMoO3F3. If viewed perpendicular to the c-axis the structure contains eight layers of octahedra. In each layer there are 10 octahedra contained within the unit cell, as shown in Figure 6. Six of the 10 octahedra (those shown in blue) do not rotate relative to their orientations in the cubic perovskite structure. Two rotate by ∼45° around either the a- or b-axes of the cubic perovskite structure (those shown in red), which leads the quadrupling of the c-axis of the unit cell. The final two octahedra rotate by ∼45° around the c-axis of the cubic perovskite structure (those

Table 1. Structural Refinements Obtained from Synchrotron and TOF Neutron Data for α-A3MoO3F3 (A = K, Rb) α-K3MoO3F3 space group a (Å) c (Å) V (Å3) Z calculated density (g/cm3) data type λ (Å) range (2θ) number of reflections Rwp Rexp data type range (ms) number of reflections Rwp Rexp parameters refined total Rwp total Rexp total χ2

α-Rb3MoO3F3

I41(80) I41(80) 19.38613(3) 20.0748(4) 34.86739(8) 36.1694(1) 13103.93(4) 14576.23(6) 80 80 3.226 4.168 Synchrotron 0.413961 0.5−23 0.5−29 3479 7417 13.80 14.99 11.34 12.38 TOF neutron 23−85 25−86 3269 2860 6.86 5.51 9.38 7.70 333 316 12.90 12.63 11.31 12.34 7.464 3.545

Figure 2. Observed synchrotron data for α-K3MoO3F3 given in black with the calculated pattern given in red. The difference between observed and calculated patterns is given in gray. The blue tick marks denote the allowed peak positions. The inset shows the fit from 8−12° 2θ.

shown in cyan), which leads to the expansion of the unit cell in the ab plane. All layers have a similar pattern of tilts of the octahedra, but the locations of the (red) octahedra that rotate about the a- or b-axes differ from one layer to the next. In contrast, the positions of the (cyan) octahedra that rotate about the c-axis are confined to four regions in the ab plane. In every other layer they are located at x ≈ 1/4, y ≈ 1/4 and x ≈ 3/4, y ≈ 3/4, as shown in Figure 6, while in the alternate layers they are located at x ≈ 1/4, y ≈ 3/4 and x ≈ 3/4, y ≈ 1/4. We can picture these octahedra being confined to the channels marked by the cyan colored crosses in Figure 7, which will hereafter be referred to as the c-axis rotation channels. Down each of these channels there is an alternation between MoO3F33− ions rotated by 45° 5406

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Figure 5. Observed TOF neutron data for α-Rb3MoO3F3 given in black with the calculated pattern given in red. The difference between observed and calculated patterns is given in gray. The blue tick marks denote the allowed peak positions.

Figure 3. Observed TOF neutron data for α-K3MoO3F3 given in black with the calculated pattern given in red. The difference between observed and calculated patterns is given in gray. The blue tick marks denote the allowed peak positions.

Figure 4. Observed synchrotron data for α-Rb3MoO3F3 given in black with the calculated pattern given in red. The difference between observed and calculated patterns is given in gray. The blue tick marks denote the allowed peak positions. The inset shows the fit from 8−12° 2θ. Figure 6. One layer of 10 MoO3F33− octahedra in α-A3MoO3F3 (A = K, Rb) viewed down the c-axis. The cubic perovskite lattice parameters are marked as adp and bdp. The six blue octahedra are not rotated relative to the cubic double perovskite structure. The two red octahedra are rotated by ∼45° about bdp (rotated about adp in other layers). The two cyan octahedra are rotated by ∼45° around cdp.

about the c-axis, and K+ ions that sit on octahedral B-sites of the double perovskite structure. The structure also contains columns of K+ ions, sitting on the A-sites of the double perovskite structure that are not adjacent to one of the c-axis rotation channels. These regions are marked with red squares in Figure 7. As is the case in α-K3AlF6 the octahedral rotations lead to large changes in the local coordination environments of the potassium ions that sit on both the A and the B sites. The rotations of the MoO3F33− octahedra increase the average coordination number of the K+ ions that sit on the B-sites from six (in the cubic double perovskite structure) to values ranging from 6 to 8, depending upon the number of neighboring octahedra that rotate by ∼45°. The mechanism for the increase in B coordination as a function of B′ rotation is discussed in detail by Abakumov et al.14 In a typical perovskite the A-site cation occupies a 12 coordinate cavity with a cuboctahedron arrangement of anions

surrounding it. Once NCOT occurs, the coordination number of the A-site cation ranges from 6 to 12. The changes are more pronounced for the A-site cations located adjacent to the c-axis rotation channels. The coordination numbers of K+ (Rb+) ions that sit on A-sites that border the c-axis rotation channels ranges from 6 to 10, with an average coordination number of ∼8. In contrast, the K+ (Rb+) ions that sit on A-sites that are isolated from the c-axis rotation channels (marked with red squares in Figure 7) have coordination numbers that range from 8 to 12, with an average coordination number of 10. 5407

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Figure 8. A view of the MoO3F33− units in the α-A3MoO3F3 (A = K, Rb) structure looking down the c-axis, that illustrates the orientational ordering of these units. Red spheres represent oxygen and green spheres represent fluorine. The octahedra that rotate about the c-axis, and the A ions are not shown for clarity.

Figure 7. One layer of α-K3MoO3F3 viewed down the c-axis. The same colors are used to represent MoO3F33− octahedra as in the previous figure; yellow and purple spheres represent K+ ions that sit on the Bsite and A-site positions, respectively, in the cubic double perovskite structure. The c-axis rotation channels are highlighted by cyan crosses. The columns of K+ isolated from those channels are marked with red squares.

The major difference between the structures of α-K3AlF6 and α-K3MoO3F3 (α-Rb3MoO3F3) compounds is the ordering of the anions. The ordering of the anions causes the loss of inversion center, rendering the structure polar. Each of the sites in the I41/a structure splits into two sites upon lowering to the I41 structure which doubles the number of atomic positions. Interestingly the fluorine faces of the MoO3F33− octahedra point toward the c-axis rotation channels, while the oxygen faces point toward the columns of K+ ions that are isolated from those channels (Figure 8). The MoO3F33− octahedra that make up the c-axis rotation channels sit on the 41 screw axes, which results in a helical procession of the anions down the c-axis (Figure 9). The dipole moment of the MoO3F33− unit is aligned parallel to the Mo displacement, with the negative end of the dipole pointing toward the fluorine face.2 The c-axis of I41 is the polar axis and therefore the dipole moment contributions in the a and b directions will cancel by symmetry. For the octahedra that are not rotated relative to ap or bp, the dipole contribution in the c direction corresponds with the direction of the fluorine apex. The fluorine apexes of the unrotated octahedra are pointing down with respect to the c-axis, while for those octahedra that make up the c-axis rotation channels the fluorine apexes are all pointing up with respect to the c-axis. The difference in the number of PBUs that are not rotated, 48, versus the number that are rotated about the c-axis, 16, results in a net dipole moment pointing along the c-axis resulting in the ferroelectric response that has previously been reported.6,7 This study confirms that α-K3MoO3F3 and α-Rb3MoO3F3 possess the same pattern of octahedral tilting as seen in αK3AlF6, with the addition of anion ordering. From preliminary low temperature laboratory XRPD, it appears that αCs3MoO3F3 and α-A3WO3F3 (A = K, Rb) adopt the same

Figure 9. A view of the orientational ordering of the MoO3F33− units in α-A3MoO3F3 (A = K, Rb), looking perpendicular to the c-axis. The cyan octahedra that make up the right chain are rotated by ∼45° about the c-axis and define the c-axis rotation channel. The procession of fluorine face shows the chiral polarity associated with the 41 screw axis. The left chain shows the alignment of MoO3F33− units in the columns that either do not undergo rotations (blue octahedra) or rotate by ∼45° about either the adp or bdp axes (red octahedra). The dipole moment direction along c for each chain is marked by arrows.

structure. The rotations of octahedra around all three of the parent perovskite axes result in these compounds adopting extremely large unit cells, with volumes of 13103.94(4) Å3 for α-K3MoO3F3 and 14576.23(6) Å3 for α-Rb3MoO3F3. These 5408

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compounds are the first NCOT compounds in which the longrange anion ordering has been confirmed. The addition of anion ordering to the complex α-K3AlF6 structure further lowers the symmetry by removing the glide plane and the center of inversion, which doubles the number of crystallographically unique atoms with respect to the I41/a symmetry of the α-K3AlF6 structure. The structures of α-A3MoO3F3 (A = K, Rb) are part of a relatively small family of compounds where NCOT has been confirmed. The phase transition temperatures and available structural information for fluorides and oxyfluorides that are known to undergo NCOT phase transitions are shown in Figure 10.23,12,25,13,26,6,7 Using the nomenclature established for

Figure 11. A view of β-K3AlF6 looking down the c-axis. The AlF63− octahedra that rotate by ∼45° about the c-axis are represented by cyan octahedra, the AlF6 octahedra that do not rotate are represented by blue octahedra. In response to these rotations 80% of the K+ ions that sit on the B-sites of the cubic double end up with a bicapped pentagonal prismatic coordination (purple polyhedra), while 20% retain octahedral coordination (yellow polyhedra).

Figure 10. Phase transitions for fluoride and oxyfluoride compounds where noncooperative octahedral tilting (NCOT) has been observed. The polymorphs marked with a question mark are proposed structures that have not been experimentally verified. The space group symmetries given in parentheses correspond to oxyfluorides where the MO3F33− units are orientationally ordered.

temperature phases of α-Rb2KMF6 (M = Cr, Ga).7 The symmetry of β-Rb2KCrF6 is I4/m, which is a centrosymmetric space group. However, if the orientational ordering of the octahedra in the c-axis rotation channels seen in α-A3MoO3F3 (A = K, Rb) (see Figure 9) is retained in the β-polymorph, both the a glide and the center of inversion would be lost lowering of the symmetry from I4/m (87) to the maximal non-isomorphic subgroup I4 (79). This space group, which is polar, would be consistent with previous dielectric and optical measurements indicating that the structure is still polar up to the 522 K (538 K) phase transition for K3MoO3F3 (Rb3MoO3F3).6 A structural study of the intermediate temperature phase is needed to verify this hypothesis. Our ability to predict when NCOT phase transitions will occur is still evolving. Abakumov et al. suggest that the tolerance factor and difference in the ionic radii of the B and B′ cations can be used as indicators of when to expect phase transitions driven by NCOT.14 These phase transitions are expected when the difference in the ionic radii of B and B′ is large, 0.76−0.85 Å for fluorides and 0.50−0.66 Å for oxides, and the tolerance factor is significantly smaller than 1.0.14,24 The oxyfluorides in this study have an even larger difference in ionic radii of B and B′ of 0.78−1.06 Å. We suggest that a small difference in the radii of the A and B cations will also favor NCOT, provided the above conditions are met. The difference in ionic radii of the A and B cations is less than 0.15 Å for each of the known NCOT compounds, K3AlF6, Rb2KMF6 (M = Cr, Ga), Sr3WO6 and the compounds in this study.14,12,24 The fact that the A and B cations have such similar (or identical) radii favors NCOT because the noncooperative tilts tend to make the coordination numbers for the A and B cations similar.

K3AlF6, we use the labels α, β, γ, and δ to denote the patterns of octahedral tilting seen in each polymorph. The δ-polymorph is the high temperature phase common to all five compounds. In this polymorph the rotations of the octahedra are dynamic and the average crystal structure is cubic. The γ-polymorph, which thus far has only been observed for K3AlF6, has a large orthorhombic unit cell where one-sixth of the octahedra rotate by ∼45° about either the a- or the b-axes. In the β-polymorph, one-fifth of the octahedra rotate by ∼45° about the c-axis, leading to an expansion of the unit cell in the ab-plane and tetragonal symmetry. As a result of these rotations, 80% of the alkali metal cations that occupy octahedral sites in the cubic double perovskite structure end up with a bicapped pentagonal prismatic coordination, as shown in Figure 11.26 In the αpolymorph, which is also tetragonal and represented by the structures of K3MoO3F3 and Rb3MoO3F3 discussed in this paper, the ∼45° rotations about the c-axis of the β-polymorph are retained and ∼45° rotations about the a- and b-axes are also present. K3MoO3F3 and Rb3MoO3F3 undergo two phase transitions on heating above room temperature. We hypothesize that the first transition, at 436 K (423 K) for K3MoO3F3 (Rb3MoO3F3), would correspond to loss of coherent rotations about the a- and b-axes, leading to the β-polymorph shown in Figure 11 and having the same pattern of octahedral tilting as the low 5409

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4. CONCLUSIONS The structures of α-K3MoO3F3 and α-Rb3MoO3F3 have been determined using synchrotron and TOF neutron powder diffraction. These isostructural compounds have the same pattern of noncooperative octahedral tilting as α-K3AlF6. The presence of long-range orientational ordering of the polar MoO3F33− units destroys the a-glide plane of the nonpolar αK3AlF6 structure lowering the symmetry to the polar space group I41. This structure is consistent with the ferroelectric behavior of these two compounds. We propose that in A2BB′X6 compounds a tolerance factor below unity, a relatively large difference in the ionic radii size of the B and B′ cations, and a small difference in the ionic radii of the A and B cations favor the occurrence of noncooperative octahedral tilting (NCOT).

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ASSOCIATED CONTENT

S Supporting Information *

Crystallographic data in the form of .cif files and atomic positions for α-A3MoO3F3 (A = K, Rb); tables of the coordination numbers and bond valence sums. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the National Science Foundation (Award Number DMR-0907356) is acknowledged. Use of the Advanced Photon Source at Argonne National Laboratory was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC0206CH11357. Use of the Spallation Neutron Sources (SNS) was supported by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy.

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REFERENCES

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dx.doi.org/10.1021/cg401342q | Cryst. Growth Des. 2013, 13, 5404−5410