Modal disorder and phase transition in Rb0.91Nb0.96W1.04O5.98. Interpretation of X-ray diffuse scattering using the group theory approach

Journal of Solid State Chemistry 230 (2015) 325–336

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Modal disorder and phase transition in Rb0.91Nb0.96W1.04O5.98. Interpretation of X-ray diffuse scattering using the group theory approach Dorota Komornicka a, Marek Wołcyrz a,n, Adam Pietraszko a, Wiesława Sikora b, Andrzej Majchrowski c a b c

Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 1410, 50-950 Wrocław 2, Poland Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Kraków, Poland Institute of Applied Physics, Military University of Technology, 2 Kaliskiego Street, 00-908 Warszawa, Poland

art ic l e i nf o

a b s t r a c t

Article history: Received 19 March 2015 Received in revised form 8 July 2015 Accepted 13 July 2015 Available online 17 July 2015

A complex scheme of atomic displacements (modes) that break the Fd 3¯ m symmetry of the high-temperature (Tc 4 395 K) cubic phase of Rb0.91Nb0.96W1.04O5.98 and ultimately lead to a phase transition to the tetragonal phase was determined using the group theory approach. The resulting set of modes was used to construct a model of the disordered crystal that provides diffuse scattering (particularly characteristic extinctions) that is highly consistent with the experimental results. The resulting solution reveals a disordered structure of cubic Rb0.91Nb0.96W1.04O5.98, which is a system of intersecting {111}-type planes in which Nb/W atoms (statistically occupying the centers of oxygen octahedra) are shifted along three symmetry-equivalent 〈110〉 directions parallel to these planes. Oxygen atoms also move in a characteristic manner, but their shifts are considerably smaller and do not substantially affect the diffuse scattering pattern. The movements of Rb atoms are large but uncorrelated. The obtained picture of the local structure of cubic Rb0.91Nb0.96W1.04O5.98 makes it necessary to change the interpretation of existing physical measurements, particularly dielectric measurements. Furthermore, the determined structure of the low-temperature tetragonal phase that exists below 395 K was found to be non-polar (space group I 4¯ 2d). Group theory analysis provides a coherent picture of the phase transition from the disordered cubic phase to the ordered tetragonal phase. At Tc, in a multimodal crystal of the high-temperature phase, mode symmetry breaking occurs, and each of the four displacive modes is decomposed: only 1/4 of the atoms of every mode of k¼ (x,x,x) retain their 〈110〉-type in-plane displacements; the displacements of the remaining atoms undergo reorientation to fulfill the conditions imposed by the k¼ (0,0,0) mode. The former group of displacements defines the direction of the appearing tetragonal axis. & 2015 Elsevier Inc. All rights reserved.

Keywords: Rubidium niobotungstate Local structure Disordered structure Short range order X-ray diffuse scattering Phase transition Mode crystallography

1. Introduction RbNbWO6 belongs to the group of defective pyrochlores, the so-called beta-pyrochlores, AB2X6 (where A ¼monovalent cation, X ¼oxygen or fluorine, and B ¼cation octahedrally coordinated by X). The beta-pyrochlores maintain the same B2X6 network as the alpha-pyrochlores, A2B2X7 (often written as A2B2X6X′), but with half of the A-type cations and the X′ anion being lost, thus destroying the A2X′ network. However, the remaining A-type cations gain a freedom of movement that can lead to interesting physical properties, such as ion exchange, fast ion conduction and even superconduction. n

Corresponding author. E-mail address: [email protected] (M. Wołcyrz).

http://dx.doi.org/10.1016/j.jssc.2015.07.021 0022-4596/& 2015 Elsevier Inc. All rights reserved.

Crystals of RbNbWO6 were first synthesized by Babel et al. [1], and their structure was determined (Fd3¯ m space group, a¼ 10.35 Å) (Fig. 1). Subsequent studies by Sleight et al. [2] and Pannetier [3] showed that at room temperature, RbNbWO6 crystals are tetragonal with lattice parameters for a face-centered cell: a¼ 10.360 Å, c¼ 10.379 Å or a ¼10.345 Å, c ¼10.368 Å. Observing the loss of a second harmonic signal, Sleight et al. [2] determined the temperature of the phase transition to a centrosymmetric space group to be 393 K. Pannetier [3] reported a similar value (398 K) and observed a notable increase in the background of X-ray powder diffraction patterns measured at temperatures above the transition temperature to the cubic phase. Although the structure of the low-temperature tetragonal phase of RbNbWO6 remained unsolved prior to this paper, the structure of the cubic phase was refined by Bydanov et al. [4] and

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Fig. 1. View of the average crystal structure of RbNbWO6 in the cubic phase.

Chernaya et al. [5] using single-crystal neutron diffraction. Both studies confirmed the high-temperature structure previously determined by Babel et al. [1] and reported some anomalies in the atomic displacement factors: extremely high values for Rb, high values for Nb/W and rather typical values for O. Both studies interpreted the phase transition to the lower symmetry phase as evidently being due to the ordering of rubidium cations as a result of collective dipole–dipole interactions. Moreover, Chernaya et al. [5] indicated the possibility of the Nb/W atoms1 displacing during the phase transition from 16c position along three directions mutually at 120°, which are equivalent and perpendicular to threefold axes. Measurements of the physical properties of RbNbWO6 began with optical measurements performed by Sleight et al. [2]. Their crystals appeared to be inhomogeneous and nonstoichiometric: some were completely birefringent, but others contained regions that were apparently cubic. Their attempts to generate a ferroelectric loop at room temperature were unsuccessful. Dielectric dispersion measurements performed by Beleckas et al. [6] showed that at lower frequencies, non-Debye relaxational dispersion exists. This observation was attributed to screened hopping of Rb cations. At higher frequencies, the Debye-type dielectric dispersion was caused by a relaxational soft mode. According to Beleckas et al. [6], at Tc ¼290 K, RbNbWO6 undergoes a ferroelectric phase transition caused by dynamic Rb ordering and shifting of Nb/W atoms. Astafiev et al. [7] performed dielectric, optical and ionic conductivity measurements on RbNbWO6 crystals containing 13 at% H2O. These crystals exhibited a tetragonal-to-cubic phase transition at 370 K. Other degrees of hydration resulted in a shift of Tc (from 350 K to 390 K). The dielectric measurements revealed untypical behavior of dielectric permittivity vs. T. A series of papers concerning Brillouin scattering, heat capacity and Raman scattering in RbNbWO6 crystals was published by Mączka et al. [8–10]. These papers suggested the occurrence of two consecutive phase transitions at 356.5 and 361.6 K, with a tetragonal phase below the first and a cubic phase above the second temperature point. Temperature-dependent Raman scattering studies [10] revealed the presence of additional Raman bands not allowed for the Fd3¯ m symmetry and associated with the substitutional disorder of Nb and W along with displacive disorder of the Rb atoms. Another additional band was attributed to the off-center displacement of Nb/W atoms due to the structural phase transition to the tetragonal phase at 360 K. However, this band was active to considerably higher temperatures (approximately 650 K), which was attributed to the presence of small domains of local tetragonal distortion embedded in the paraelectric matrix of cubic symmetry. 1 For simplicity, we will use term atoms although RbNbWO6 has ionic character and consists of Rb1 þ , Nb5 þ , W6 þ and O2  ions.

Our X-ray diffraction studies of RbNbWO6 single crystals – aimed at solving a structure of an unknown low-temperature tetragonal phase and confirming the high-temperature phase – revealed the existence of correlated structural disorder in the cubic phase manifested as the existence of streaks of diffuse scattering (DS) that disappeared when the crystals underwent a phase transition to the tetragonal phase. The characteristic shape of DS – 1-dimensional streaks running along 〈111〉n directions in the reciprocal lattice – indicates the presence of short-range order resulting from correlations occurring in {111} crystal planes.2 Such strong additional short-range periodicity combined with specific structural disorder can strongly influence all essential physical properties and change their interpretation. The typical approach for solving a problem that involves explaining the origin of structured DS is to find a proper model of short-range order, or in other words, to propose an appropriate model for correlations occurring in the crystal. Several examples of this approach were described by Withers [11]. In this work, however, it was not possible to find a simple ‘model of disorder’ that consists of uniform displacements or substitutions of atoms forming the structure. Although the diffuse scattering pattern itself appeared to be simple (relatively sharp and uniform streaks), diffuse scattering extinction rules that are complex and difficult to explain hinder the construction of a model. Therefore, we decided to focus our attention on a group theory approach, so-called ‘mode crystallography’ (see, e.g., [12,13]), which could suggest the possible directions of atomic displacements leading to the cubictetragonal phase transition and producing the required diffraction effects. Therefore, the aims of the present paper are as follows: (i) to structurally describe the cubic-tetragonal phase transition in RbNbWO6 and to determine the crystal structure of the unknown tetragonal phase and (ii) to explain, by modeling the X-ray diffuse scattering, the nature of short-range order in the high-temperature cubic phase leading to a specific local structure. A realistic model of cubic RbNbWO6 could be helpful in explaining some controversial or unclear points in interpretations of physical properties found in the literature.

2. Experimental RbNbWO6 single crystals were grown by means of the top seeded solution growth (TSSG) method, similarly to KNbW2O9 2 Note that 〈uvw〉 stands for a set of all symmetrically equivalent lattice directions [uvw] both in direct and in reciprocal space (then with the addition of n sign); similarly, {hkl} stands for a set of all symmetrically equivalent net planes (hkl).

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crystallization described in details by Macalik et al. [14]. Crystals used as seeds were obtained by means of spontaneous crystallization from 20 mol% solution of RbNbW2O9 in Rb2W2O7 at 1220 K under conditions of low-temperature gradients in a platinum crucible. The RbNbWO6 single crystals obtained by TSSG method were confined with crystallographic faces of approximately 1 cm in diameter and transparent when pulled out from the melt at elevated temperatures. During cooling to room temperature as-grown crystals became opaque due to multiple cracking caused by phase transitions. Several samples (0.002–0.050 mm3 for diffraction experiments and EPMA WDS measurements, 1–20 mm3 for optical and electric measurements), transparent and confined with crystallographic faces, were extracted from the bulk as-grown RbNbWO6 crystal and used in our studies. Atomic composition of the samples was determined by means of electron probe micro analysis (EPMA) with CAMECA SX100 device equipped with wavelength dispersive spectrometers (WDS). Analyses were performed at 15 keV/20 nA with standards for Rb (rubidium glass), Nb (99.5% niobium) and W (CaWO4). Second harmonic generation was observed qualitatively using 1064 nm pulsed YAG:Nd laser (LASER SYSTEM LS-2137/2 M) with t (pulse) 10 ns, E  30 mJ and repetition rate 10 Hz. The spontaneous polarization was measured between 300 and 420 K by a charge integration technique using a Keithley 6517B Programmable Electrometer. The temperature was stabilized by an Instec STC 200 temperature controller. The complex dielectric permittivity, εn ¼ε′–iε″, was measured between 290 and 410 K by the Agilent 4284A Precision LCR Meter in the frequency range between 135 Hz and 2 MHz. The overall error was less than 5%. The single crystal samples had dimensions ca. 5  3  1 mm3. Silver electrodes were stuck on the opposite faces. The sample before measurements was kept at 310 K and blown dry with dry nitrogen for 2 h. The dielectric measurements were carried out in a controlled atmosphere (N2). Powder X-ray diffraction measurements were performed on a PANalytical X'Pert diffractometer equipped with a PIXcel solidstate linear detector. X-ray diffraction patterns were obtained using CuKα radiation (λ¼ 1.5418 Å), generated at 45 kV and 30 mA, in transmission mode, with a focusing mirror forming the convergent X-ray beam irradiating samples placed in rotated capillaries or in reflection mode with flat sample Bragg–Brentano geometry. An Anton Paar HTK 1200 N high-temperature chamber was used to heat the capillary samples from room temperature to above the phase transition temperature (423 K) and to cool the samples. The temperature stability was better than 70.1 °C. Data reduction was conducted, and crystallographic calculations were performed using PANalytical X'Pert HighScore Plus software. Crystals of suitable quality and with dimensions of 0.15  0.13  0.12 mm3 (for structure determination) and 0.25  0.30  0.23 mm3 (for DS registration) were chosen for singlecrystal X-ray diffraction experiments. Data collection for both conventional crystal structure analysis and for X-ray DS modeling was performed using an Oxford Diffraction X'Calibur four-circle single-crystal diffractometer equipped with a CCD camera using graphite-monochromated MoKα radiation (λ¼0.71073 Å) generated at 50 kV and 25 mA. For high-temperature measurements up to 423 70.5 K, an Oxford Diffraction high-temperature blowing system that provided hot air was applied. For the crystal structure determination, three-dimensional sets of X-ray diffraction data were collected over the 2θ range of 3–59°, with Δω¼1.2° and the recording time of a single frame equal to 40 s. Respective values for DS registration were: 2θ¼3–92°, Δω¼0.8° and t¼ 55 s. The lattice parameters were calculated from the positions of all measured reflections. The intensities of the Bragg reflections for structure determination were corrected for

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Lorentz and polarization factors. Analytical absorption correction was applied. Conventional X-ray crystal structure analysis was performed using the SHELX-97 software package [15]. A structure of tetragonal phase was refined taking into account three coexisting twins (SHELX instructions: TWIN and BASF). Eighteen reciprocal planes were extracted for the analysis of X-ray diffuse scattering data from the set of registered frames via CrysAlis software [16]. Final plane images were averaged over 7 adjacent slices, with the thickness covering 6% of the lattice constant, i.e., the 1kl plane was prepared as the average of 0.97kl to 1.03kl reciprocal planes. Neither absorption nor polarization corrections were applied to the data used for the diffuse scattering analysis. Test crystals for simulations were constructed using our own procedures or using routines included in the DISCUS program [17]. The size of the test crystals was 50  50  50 unit cells, and they contained 9 million atoms. Diffraction effects were calculated for the test crystals using the DISCUS program. The KUPLOT routine, which, together with DISCUS, is a part of the DIFFUSE package [17], is a tool for the graphical presentation of diffraction results. The DISCUS program calculates the Fourier transform according to the standard formula for kinematic scattering. The finite-size effect was avoided by applying periodic boundary conditions: Δh ¼1/Δx, where Δh is the grid size of the calculated reciprocal plane and Δx is the corresponding dimension of the modeled crystal.

3. Sample characterization, phase transition and crystal structure of the tetragonal phase From EPMA WDS measurements, performed on three crystals from the same batch as the samples used for diffraction measurements (total number of measurements ¼14), the following average atomic ratio has been obtained: Rb:Nb:W:O¼ 10.27 (6):10.76(10):11.73(9):67.17(4). In this formula numbers in parentheses stand for standard deviations calculated from the 14points series of measurements. The absolute experimental errors of atomic contributions (at%) determination are: 0.18, 0.27 and 0.20, for Rb, Nb and W, respectively, for experimental conditions applied. Recalculating the above values into atomic stoichiometry we obtain the following formula: Rb0.91(2)Nb0.96(2)W1.04(2)O5.98(2). It means that our samples crystalized with excess of W over Nb in octahedra centers (48%Nb, 52%W) and Rb deficiency (91%). Therefore, the above nonstoichiometric formula is used in all cases we mention our crystals. Single crystals of Rb0.91Nb0.96W1.04O5.98 were qualitatively checked for second harmonic generation and the pyroelectric effect. Optical measurements revealed second harmonic generation in the lowtemperature phase and the disappearance of second harmonic generation above Tc. Dielectric measurements revealed a lack of pyroelectric current, both for sample heating from room temperature to 423 K and cooling down. Both of these phenomena indicate a noncentrosymmetric and nonpolar space group for the low-temperature phase. Permittivity measurements showed anomaly at 395 K occurring for frequencies greater than 100 kHz. For lower frequencies the effect was masked by large contribution of conductivity. X-ray powder diffraction patterns for the powdered sample of Rb0.91Nb0.96W1.04O5.98 were measured both at room temperature (293 K) and as a function of temperature between 293 and 423 K. The room temperature phase was indexed to a tetragonal body-centered cell, and the lattice parameters were obtained by Pawley refinement. The final lattice parameters were as follows: a¼ 7.3201 Å, c ¼10.3819 Å (Rwp ¼3.44% for 252 refined peaks). The experimental powder diffraction pattern and the results of the full-pattern fitting are presented in Fig. 2.

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Fig. 2. Results of Pawley refinement for the tetragonal body-centered cell of the low-temperature Rb0.91Nb0.96W1.04O5.98 phase. Inset shows tetragonal splitting of two cubicphase reflections (denoted by subscript C) into four tetragonal ones (denoted by T).

The results of heating the sample from 353 K to 423 K and subsequently cooling to 353 K are presented in Fig. 3. A fully reversible phase transition with the disappearance of several Bragg peaks is observed in the diagram. The Tc was estimated for 395 K. The well-known cubic phase with Fd3¯ m symmetry occurs above Tc. The conventional crystal structure analysis performed on single crystal of Rb0.91Nb0.96W1.04O5.98 at room temperature allowed us to determine structure of the low-temperature tetragonal phase (space group I4¯ 2d). Essential details regarding the crystal structure and its refinement are summarized in Table 1 (complete data for the crystal structure determinations are presented in Supplementary Data [18] (Table S1 and attached CIF file). Despite a relatively large contribution of diffuse scattering to the intensities of the Bragg reflections and taking into consideration three possible twining entities, the refinements results are quite good (cf. R-factors). A drawing of the tetragonal structure is presented in Fig. 4.

The unit cell of the tetragonal phase is half of the cubic one with aT ¼(aC þbC)/2, bT ¼(–aC þbC)/2 and cT ¼cC. Atomic shifts from ideal positions in the cubic phase are relatively large in the case of Nb/W atoms (0.16 Å) and rather small (0.05 Å) for O atoms.

4. Diffuse scattering and short-range order in the cubic phase 4.1. Shape and extinctions of diffuse scattering The cubic phase of Rb0.91Nb0.96W1.04O5.98 is characterized by the existence of well-defined narrow streaks of X-ray DS running through the Bragg reflections along the symmetry-equivalent 〈111〉n directions in reciprocal space. Formally, the observed structured DS can be described as intensities located at H7 k points of reciprocal space, where H is a reciprocal lattice vector and k¼ε〈111〉n is a continuous set of modulation wave vectors, where ε is continuous. When the crystal undergoes the phase

Fig. 3. High-temperature powder diffraction patterns showing a reversible phase transition between tetragonal (below 395 K) and cubic (above 395 K) phases of Rb0.91Nb0.96W1.04O5.98. Black frames mark characteristic reflections that disappear during transformation to the cubic phase. Inset magnifies the 2θ range 37–39.5°.

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Table 1 Essential crystallographic data for two polymorphic phases of RbNbWO6. Formula

Rb0.91Nb0.96W1.04O5.98

RbNbWO6

Crystal system Temperature Space group

tetragonala 293.0(5) K I 4¯ 2d (122)

cubicb 423 K Fd 3¯ m (227)

Cell parameters Cell volume Z

a¼ 7.3108(10) Å c ¼ 10.364(2) Å 553.93(16) Å3 4

a ¼10.383(2) Å 1119.4(6) Å3 8

Atomic positions

Nb/W 8d Rb 4a O1 16e O2 8c Nb 0.5220(8); W

Occupation parameters

0.52281(5) 0 0.8097(5) 0 0.4780(8); Rb 1

0.25 0 0.1849(5) 0 (fixed)

0.125 0 0.4988(5) 0.3134(5)

origin choice 1c Nb/W 16c Rb 8b O 48f

0.125 0.5 0.188(1)

0.125 0.5 0

0.125 0.5 0

origin choice 2d 0 0 0.375 0.375 0.313(1) 0.125

0 0.375 0.125

a

Results from this paper; complete data for crystal structure determination are presented in CIF file attached as Supplementary Data. Results from [4]; c Setting used by MODY program; d Setting used in all crystal structure determination papers. b

transition from the tetragonal phase to the cubic phase and back, DS appears above Tc and disappears below Tc, and the transition is fully reversible. DS streaks are best viewed in the reciprocal lattice sections perpendicular to the set of symmetry-equivalent 〈110〉n directions. A method for constructing sections is shown in Fig. 5 for the [110]n direction. Pictures of other symmetry-equivalent directions, i.e., [1 1¯ 0]n, [101]n, [10 1¯ ], [011]n and [01 1¯ ]n, are exactly the same. Two representative sections, perpendicular to the [110]n direction, are presented in Fig. 5, which have 110 (Fig. 5a) and 220 (Fig. 5b) reciprocal lattice points as their origins. We will call such sections the 110 and 220 sections, respectively. Characteristic extinctions of DS are observed. The extinction rules can be formulated as follows: (i) there is no DS in any of the six symmetry-equivalent 000 sections; (ii) in 110, 330, 550…, i.e., in odd nn0 sections (see Fig. 5a), streaks exist that run along all rows of non-extinct Bragg reflections if it does not interfere with condition (i); and (iii) in 220, 440, 660…, i.e., in even nn0 sections (see Fig. 5b), every second streak is extinct. Note that the coincidence of conditions (i) and (ii) leads to the appearance of two pairs of extinct streaks in every odd section. They arise from lines being geometrical sections of a current plane with four symmetryequivalent 000 sections perpendicular to the following directions in the reciprocal lattice: [011]n (see extinct streaks A in Fig. 5),

[1¯ 01]n (streak B), [01¯ 1]n (streak C) and [101¯ ]n (streak D). In the case of even nn0 planes, extinctions of this type are not observed because they overlap with already extinct streaks according to condition (iii). A set of experimental sections of the reciprocal space obtained for cubic Rb0.91Nb0.96W1.04O5.98 is presented in Fig. 6. 4.2. Application of group theory approach to the determination of atomic displacements The characteristic shape of streaks extending along 〈111〉n indicates that correlation effects occur in the {111} planes of the crystal and their lack between planes. A natural explanation for this phenomenon is the occurrence of correlated and most often uniform movements of certain atoms, as has been proposed in many papers [see, e.g., [19–23]]]. However, in the case of Rb0.91Nb0.96W1.04O5.98, it was practically impossible to guess a simple model that fulfills all three extinction conditions described in Section 4.1. Therefore, we decided to focus our attention on the group theory approach, which could suggest possible directions of atomic displacements that produce the required diffraction effects. The applied symmetry analysis method [24] is based on a decomposition of the respective full representation of the crystal space group, calculated for a given wave vector k and a given set of equivalent positions, into its irreducible representations (IRs).

Fig. 4. View of the Rb0.91Nb0.96W1.04O5.98 crystal structure in the tetragonal phase and (Nb/W)O6 octahedron with values of atomic shifts from ideal cubic positions.

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Fig. 5. Scheme for constructing nn0 sections of the reciprocal space perpendicular to the [110]* direction with plots of the 110 (a) and 220 (b) sections. Diffuse scattering streaks are shown as full lines. Extinct streaks due to the geometrical section with non-parallel 000 sections (condition ii) are shown as dotted lines. See the text for the meanings of the A–D marks. The sizes of the Bragg reflections are exaggerated.

Such a decomposition occurs when the basis used for describing the original function space is transformed to the special symmetry-adapted basis. The decomposition is equivalent to splitting the original function space into sub-spaces attributed to individual IRs. New coordinates, called basis vectors (BVs), can be divided into subsets that are attributed to individual IRs and transform within the respective subspaces. The calculated BVs can be used for constructing the ordering mode in the crystal by taking a linear combination of these BVs attributed to one or more IRs. The calculations were performed using the computer program MODY [25]. In the case of space group G ¼Fd 3¯ m (no. 227) and the modulation vector k ¼(x, x, x), resulting from the 〈111〉n directions of diffuse scattering streaks, there are three IRs of k-vector group, G(k), for each of the occupied Wyckoff positions (16c for Nb/W, 48f for O and 8b for Rb): two, τ1 and τ2, 1-dimensional; the third, τ3, 2-dimensional (see Table 2 and Tables S2–S3 in Supplementary Data [18]; IR symbols are used after Kovalev [26]). The atomic displacements are calculated as linear combinations of the BVs obtained (for the current IR) for the pairs of modulation vectors ki and -ki (i¼ 1.4) forming opposite arms of a wave vector star. In this approach, successive modes (correlated displacements) are constructed, encompassing all atoms forming a common orbit of G(k). There are 2 orbits for the Nb/W 16c and Rb 8b positions and 4 orbits for the O 48f position.

In the case of the Nb/W 16c position (Table 2), two IRs are active for two orbits, but one, τ2, is inactive for orbit 1, which prevents atoms belonging to it from correlated shifts. A similar situation is observed in the case of the Rb 8b position (Table S3 in Supplementary Data [18]), where τ2 is inactive for both orbits. There are no limitations for displacement of oxygen atoms of the 48f position (Table S2 in Supplementary Data [18]). All three IRs are active for all 4 orbits. Note that the amplitudes of atomic displacements do not need to be constant but can change their quantities and sense as modulated functions along modulation vector k. However, they remain constant on the planes perpendicular to k. As shown in Table 2 and Tables S2–S3 (Supplementary Data [18]), while τ1 and τ3 allow for atomic displacements of Nb/W and O in directions different from 〈110〉, e.g., 〈UPQ〉, 〈RRR〉 or 〈UVV〉, τ2 limits atomic shifts to the set of 〈110〉 directions. It is also the case for Rb τ3. There is a rule concerning DS produced by displacement disorder: no DS is observed for a zero reciprocal space layer perpendicular to the direction of displacement. Because we observe total DS extinctions in all six 000 sections of reciprocal space perpendicular to all symmetry-equivalent 〈110〉 directions, it is clear that only specific combinations of displacements of (U, U, 0)-type can give the expected extinctions. Therefore, only τ2 should be taken into account as promising source of modes for Nb/W, O and Rb displacements. Additionally, τ3 can be considered for Rb atoms.

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Fig. 6. Experimental sections of the reciprocal space of cubic Rb0.91Nb0.96W1.04O5.98. Even n (left side) and odd n (right side) nn0 sections are presented. The geometry is the same as in Fig. 5. Note the total DS extinction in the 000 section and the characteristic DS extinctions of certain streaks (their directions are indicated by arrows and denoted by A–D marks; see text).

Fig. 7 shows the most important exemplary scheme of atomic displacements in (111) planes of types A and B (cf. Fig. 1) according to τ2/ k1, k5 modes for all atoms forming the structure. All shown shifts are parallel to the plane; Nb/W atoms from the B plane do not shift due to inactivity of τ2 for the orbit to which these atoms belong. Rb atoms (shown in the A plane) do not shift in a correlated way due to the inactivity of τ2 for both orbits, but they can move in a non-correlated manner.

4.3. Implementation of disorder When constructing a model of a disordered crystal, it is not sufficient to apply monomodal atomic shifts obtained from the symmetry analysis described in 4.2. To obtain diffuse scattering effects, it is necessary to implement correlated disorder into tested models of crystals. The procedure was as follows: one {111}-parallel packet consisting of A- and B-type planes (cf. Fig. 7) was randomly chosen; the second step was to shift atoms from the

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Table 2 Atomic amplitude components of displacements determined by MODY program for Nb/W at 16c position of Fd 3¯ m space group for three irreducible representations (IRs) of wave vector group G(k) (τ1 to τ3). Displacements are calculated as linear combinations of basis vectors obtained (for the current IR) for the pair of modulation vectors ki and  ki (i¼ 1..4) forming opposite arms of wave vector star. Displacements are correlated among atoms forming each of two orbits (1 and 2) of G(k). a, b, c, d denote definite atomic subsets of the same displacements (for details, see the note below the Table). Nb/W Fd 3¯ m 16c 0.125, 0.125, 0.125 Orbit

k1 ¼(x, x, x) and k5 ¼ –k1 ¼ (  x,  x,  x)

k2 ¼(x,  x,  x) and k6 ¼–k2 ¼ (-x, x, x)

k3 ¼(  x, x,  x) and k7 ¼ –k3 ¼ (x,  x, x)

k4 ¼ (  x, –x, x) and k8 ¼–k4 ¼(x, x,  x)

1 2

a (R1, R1, R1) b (U1, V1, V1) c (V1, U1, V1) d (V1, V1, U1)

b (  R1, R1, R1) a (  U1, V1, V1) c (  V1, V1, U1) d (  V1, U1, V1)

c (R1,  R1, R1) a (V1,  U1, V1) b (V1,  V1, U1) d (U1,  V1, V1)

d (R1, R1, a (V1, V1, b (V1, U1, c (U1, V1,

1n 2

a (0, 0, 0) b (0, U2,  U2) c (  U2, 0, U2) d (U2,  U2, 0)

b (0, 0, 0) a (0, U2,  U2) c (  U2,  U2, 0) d (U2, 0, U2)

c (0, 0, 0) a (  U2, 0, U2) b (U2, U2, 0) d (0,  U2,  U2)

d (0, 0, 0) a (U2,  U2, 0) b (  U2, 0,  U2) c (0, U2, U2)

1 2

a (R3, –R3, 0) or (S3, S3,  2S3) b (U3, P3, Q3) c (Q3, U3, P3) d (  2P3,  2Q3,  2U3)

IR τ3 (2 dimensional) b (  R3,  R3, 0) or (  S3,  S3, –2S3) a (  U3, P3, Q3) c (2P3,  2Q3,  2U3) d (  Q3, U3, P3)

c (R3, R3, 0) or (S3, S3,  2S3) a (Q3,  U3, P3) b (–2P3, 2Q3, –2U3) d (U3,  P3, Q3)

d (R3,  R3, 0) or (S3,  S3, 2S3) a (  2P3,  2Q3, 2U3) b (Q3, U3,  P3) c (U3, P3,  Q3)

IR τ1 (1 dimensional)  R1)  U1)  V1)  V1)

IR τ2 (1 dimensional)

Positions of Nb/W atoms: 1: (0.125, 0.125, 0.125); 2: (0.125, 0.875, 0.875); 3: (0.875, 0.125, 0.875); 4: (0.875, 0.875, 0.125); 5: (0.125, 0.625, 0.625); 6: (0.125, 0.375, 0.375); 7: (0.875, 0.625, 0.375); 8: (0.875, 0.375, 0.625); 9: (0.625, 0.125, 0.625); 10: (0.625, 0.875, 0.375); 11: (0.375, 0.125, 0.375); 12: (0.375, 0.875, 0.625); 13: (0.625, 0.625, 0.125); 14: (0.625, 0.375, 0.875); 15: (0.375, 0.625, 0.875); 16: (0.375, 0.375, 0.125). Description of atomic subsets: a¼ {1, 5, 9, 13}; b ¼ {2, 6, 10, 14}; c ¼{3, 7, 11, 15}; d¼{4, 8, 12, 16}. U1, U2, U3, V1, V3, P3, Q3, R1, R3, S3 can be arbitrary. n

IR is inactive for this orbit.

selected A and B planes according to a corresponding mode using the displacement vectors collected in Table 2 and Tables S2–S3 (Supplementary data [18]). The following rule was applied: the mode used for atom shifts for a current plane has to belong to arms of a k-vector star perpendicular to the plane, e.g., when the ¯ ¯ ) plane was chosen, the mode determined for k2 ¼ (x,  x, x) (111 and k6 ¼(  x, x, x) was used for shifting atoms. It was also permissible to use definite ’antimodes’, i.e., modes with opposite displacement components. Then, the next plane was chosen, and the above procedure was repeated until the specified number (N) of planes had been drawn. N can be regarded as an indicator of the degree of order. According to our experience, when N exceeds 1000, it ensures a sufficient number of randomly chosen planes to

have each atom move according to modes associated with the chosen IR. Models with N equal to 2000 and more do not provide new information to the model and give equivalent diffraction patterns. The values of atomic displacements observed when comparing atomic positions in the ideal cubic cell with the tetragonal one (see Fig. 4) were taken for modeling, i.e., 0.16 Å for Nb/W, 0.05 Å for O and 0.2 Å for Rb. Both schemes of atomic displacements, (i) with constant values and (ii) with values changing according to modulation functions, were applied and provided practically the same diffuse scattering patterns. We excluded ordering of Nb and W atoms. First, we did not observe such effects in our classical crystal structure analysis.

Fig. 7. Scheme of atomic displacements according to τ2/k1, k5 modes in type A (left) and type B (right) planes (cf. Fig. 1). All shifts are parallel to the planes. The lengths of the shift vectors are exaggerated. The pictures have the same scale and origin and can be superimposed.

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Fig. 8. Two representative sections of the model crystal generated for N ¼2000 with shifts of Nb/W atoms (Rb and O atoms are omitted). (a) Chosen (111) A-type plane; (b) chosen (1¯ 11) A-type plane. Planes intersect along the blue line. (c) and (d) are magnifications of the circular 1 and 2 areas, respectively. All atomic shifts are along the 〈110〉 directions. Red arrows (points in a and b) denote shifts lying in the respective {111} plane; green arrows (points in a and b) are shifts directing out of plane.

Second, if these effects (including ordering that results from specific modes, identical as for atomic displacements) occurred in the cubic phase, they would also remain in the tetragonal one. However, we did not observe such effects. Therefore, we claim that the Nb/W subsystem is randomly disordered, producing substantial diffuse scattering contributing to the background. Fig. 8 shows two representative sections of the model crystal generated for N ¼2000 with shifts of Nb/W atoms (Rb and O atoms are omitted in the pictures for clarity). It is observed that the simple method of disorder implementation produces quite uniform crystals (both intersecting planes are similar in the topological sense). As it will be shown in Section 4.4., crystals of this type produce very acceptable DS patterns despite their simplicity and a type of unavoidable ‘coarseness’. 4.4. Results of diffuse scattering modeling Modeling of DS in cubic Rb0.91Nb0.96W1.04O5.98 was performed in three preliminary steps for τ2 (see Table 2 and Tables S2-S3 in Supplementary Data [18]) for all atoms and for τ3 for Rb: (1) the only disordered subsystem consisted of Nb/W atoms, and correlated disorder was introduced into the 16c position; Rb and O subsystems were ordered; (2) the only disordered subsystem was connected with O atoms, and correlated disorder was introduced into the 48f position; the Nb/W and Rb subsystems were ordered; and (3) the only disordered subsystem was within Rb atoms, and correlated disorder was introduced into the 8b position; Nb/W and O subsystems were ordered. The results of the DS calculations in step 1 clearly show an agreement between the observed extinctions and streak distributions obtained from the model for τ2. For Nb/W displacements, in accordance with τ2, we observe fully extinct DS in the 000 sections of all directions, characteristic ‘every-second’ streak extinctions in even sections and the lack of streaks at geometrical

sections of a given plane with appropriate 000 planes for odd sections. The consideration of this type of modes only leads to the conclusion that displacive disorder of Nb/W atoms has to be the primary source of DS in Rb0.91Nb0.96W1.04O5.98. On the other hand, all three IRs, including τ2, do not provide good results in the case of the O subsystem and τ3 for the Rb one (steps 2 and 3). Both subsystems’ shifts according to τ2 or τ3 break, at least partially, the extinction rules (Fig. 9). In the case of the oxygen subsystem, it is true that in zero and odd sections, the extinction rules are fulfilled, but in even sections, very weak streaks are visible that should be extinct (Fig. 9b, left side, yellow arrows). In the case of the rubidium subsystem, most extinctions are broken in 000 and in even and odd sections (Fig. 9, right sides). However, their contributions to the total diffuse scattering are drastically smaller than that of the Nb/W subsystem, i.e., 2 to 3 orders of magnitude (cf. intensity levels given in squares in Fig. 9c and Fig. 10). The reason for this result is, in the case of O – extremely small atomic shifts, as observed in tetragonal phase. Rb atoms most likely do not displace according to τ3-defined modes; their movements are uncorrelated, which is manifested in high atomic displacement parameters and ionic conductivity. Moreover, both contributions are strongly masked by high background originating in randomly disordered Nb and W atoms together with additional common experimental background. As a final step, we constructed an aggregated model with Nb/W and O subsystems with short-range order implemented according to τ2 and with the Rb subsystem randomly disordered. The results of calculations for this model with N ¼2000 are presented in Fig. 10. 5. Mechanism of the cubic-tetragonal phase transition Group theory analysis, similar to that performed in 4.2., allowed the path of the phase transition from the cubic to the

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Fig. 9. Comparison of nn0 sections of reciprocal space generated for O atoms displaced according to τ2 and all remaining atoms in their ideal positions (left sides of pictures) and of Rb atoms displaced according to τ3 and all remaining atoms in their ideal positions (right sides of pictures): (a) 000 section, (b) 440 section, and (c) 550 section. Note that the intensity scale is different for all left-side pictures than for the right-side ones. The intensities of the most intense streaks in points framed by squares are described in percent of intensity marked in Fig. 10 in the 550 section. The streak description is the same as that in Fig. 6. Streaks holding the extinction rules are marked in white, and streaks breaking them are marked in yellow. Bragg reflections are excluded from the simulations.

tetragonal phase to be confirmed. The mechanism is as follows: when the temperature decreases to Tc, component x of the modulation vector k¼ (x, x, x) approaches zero. Calculations performed using the MODY program for the wave vector group of G ¼Fd 3¯ m and k ¼(0,0,0) result in 10 IRs. One of these IRs, the 3-dimensional IR τ8, provides atomic displacements leading from Fd 3¯ m to its subgroup I 4¯ 2d. Appropriate modes for Nb/W and O atoms are presented in Tables S4 and S5 in Supplementary Data [18]. It can be seen that in the case of Nb/W, atomic displacements appear along the [110], [1¯ 10], [11¯ 0] and [110] directions of the cubic lattice only; [001] becomes a direction of a tetragonal axis. Consequently, the Nb/W 16c position of Fd 3¯ m transforms into the 8d position of I 4¯ 2d, and the displacement parameter determined from the atomic positions presented in Table 1 is A ¼0.031. Some fraction of O atoms from the 48f position does not move and gives the 8c position of I4¯ 2d. The remainder transform into the 16e positions of I 4¯ 2d after shifts with B¼ 0.006 and C ¼0.002. Rb atoms cannot move according to the inactivity of τ8 for the 8b position in Fd 3¯ m, and the nonparametric Rb 8b position transforms into 4a without any atomic shifts. From the perspective of mode description, in Rb0.91Nb0.96W1.04O5.98 multimodal crystal of the high-temperature phase, a mode symmetry breaking occurs at the transition point. Each of the four displacive modes is decomposed into three groups of atoms. Only one group from each displacive mode (e.g., subset d in the case of τ2/k1 ¼ (x, x, x), k5 ¼(  x, x,  x) mode) keeps its 〈110〉-type displacements in-plane. The displacements of the remainder (subsets b and c) undergo reorientation to fulfill the conditions imposed by the τ8/k¼ (0, 0, 0) mode. The former group of displacements define the direction of the appearing tetragonal axis, cT: it is created as one of three cubic axes, and it is perpendicular to them (Fig. 11). Atoms that were not moved within mode τ2/k1 ¼(x, x, x), k5 ¼(  x,  x,  x), i.e., subset a, gain displacements in-plane, but opposite to those for atoms from subset d. Other modes are broken in similar way, leading to stronger or weaker tetragonal twinning.

6. Conclusions The results of this study reveal the following picture of the crystal structure and polymorphism of Rb0.91Nb0.96W1.04O5.98. At temperatures below Tc ¼395 K, a non-polar tetragonal phase (space group I 4¯ 2d) occurs. Above Tc, the crystal transforms into a phase in which the average structure is described by space group

Fd 3¯ m but with additional short-range order described by the assembly of modes appropriate for the τ2 irreducible representation of the wave vector k¼(x, x, x) group. The modes break the symmetry of the Fd3¯ m group causing large, specifically oriented (along 〈110〉 directions) and correlated (in the {111} planes) atomic shifts of Nb/W and small shifts of O. Rb atom movements are not correlated. Proper modes restrict their action to the packets of A and B planes (see Fig. 1) perpendicular to all four 〈111〉 directions to form a statistical mixture of smaller or larger monomodal domains. This type of crystal with ‘modal disorder’ presents a diffuse scattering pattern that is in good agreement with the experimental results; in particular, it maps very well the characteristic DS extinctions. The nature of the crystal structure of the disordered cubic phase is somewhat unexpected. It would appear that the main origin of DS should be correlated shifts of Rb atoms, as has been reported for a number of similar materials. For example, in the case of Bi-based pyrochlores [19,20], the origin of DS is the existence of correlated {111} planes of Bi displacements that are uncorrelated from one plane to the next. In the case of an analogue to our crystal – CsTi0.5W1.5O6 [27], the authors reported a displacive disorder of the Cs cations that are shifted along 〈111〉 with correlations along the 〈110〉 directions. Mączka et al. [10] share this view, claiming that the appearance of the Raman bands not allowed for space group Fd 3¯ m can most likely be explained by assuming local symmetry breaking due to substitutional disorder of Nb/W and displacive disorder of Rb ions. According to our modeling, we can exclude both of the phenomena described above as origins of the observed DS. Another candidate phenomenon for explaining the DS could be correlated rotations of oxygen octahedra; however, this is also not the case. First, shifts of oxygen atoms, as obtained from their atomic positions in the tetragonal phase, are relatively small (0.05 Å) compared with the shifts of Nb/W (0.16 Å) and do not provide a considerable contribution to the total DS. Second, their movements according to every IR (including τ2) destroy, but to a very limited extent, the characteristic extinctions of DS. We believe that this effect of DS extinction impairment is masked by the high background arising from random Nb/W atomic disorder and uncorrelated Rb atom movements. Group theory analysis also provides a coherent picture of the phase transition from the cubic to tetragonal phase. At Tc, in a multimodal crystal of the high-temperature phase, a mode symmetry breaking occurs, and each of the four displacive modes is decomposed. Only 1/4 of the atoms of every displacive mode retain their 〈110〉-type in-plane displacements. These displacements

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Fig. 10. Calculated sections of the reciprocal space of cubic Rb0.91Nb0.96W1.04O5.98 for the model with N ¼ 2000. Even n (left side) and odd n (right side) nn0 sections are presented. The order of pictures and all descriptions are the same as in Fig. 6. Bragg reflections are excluded from the simulations. The intensity marked by a square in the 550 section serves as a reference point for the intensity estimations in Fig. 9.

determine the direction of the appearing tetragonal axis: it is created as one of three cubic axes, and it is perpendicular to them. Other modes are broken in a similar way, leading to stronger or weaker tetragonal twinning. The results obtained in this study require a new, critical look at the electrical properties of RbNbWO6 and the interpretation of the results obtained in a few papers regarding dielectric susceptibility measurements. The nonpolar I 4¯ 2d space group excludes ferroelectricity in Rb0.91Nb0.96W1.04O5.98. This finding is consistent with the

statement of Sleight et al. [2] that attempts to generate a ferroelectric loop for RbNbWO6 at room temperature were unsuccessful. They claim, more generally, that the dielectric properties of all AB2O6 phases are complicated by the significant ionic conductivity. Although two other papers [6,7] reporting on dielectric measurements appear to accept the ferroelectric nature of the low-temperature phase of RbNbWO6, their results arouse certain suspicions: in [6], the susceptibility maximum in a quite typical plot is achieved at T¼ 300 K, and in [7], the ε(T) plot is clearly untypical.

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Fig. 11. Scheme of atomic displacements according to the τ8/k¼ (0,0,0) mode (cf. Table S4 and S5 in Supplementary Data [18]) in type A (left) and type B (right) planes leading to the tetragonal structure. Three subsets of Nb/W atoms in plane A and one in plane B are indicated by double lines defining three 〈110〉-type directions. Only subset d (marked by red lines in plane A) retains its displacements in-plane, according to mode τ2/k1 ¼ (x, x, x), k5 ¼(  x,  x,  x) (cf. Fig. 7). Atoms from subsets b and c are shifted out-of-plane. The tetragonal axis, cT, lies out-of-plane. All atoms in plane B (subset a) gain displacements in-plane but opposite to those for atoms from subset d.

It is not our objective to explain all doubts associated with previous dielectric studies of RbNbWO6, but we agree with the statement made by Mączka et al. [10] that the phenomena observed in cubic phase can be attributed to the presence of small regions (although not polar) with lower local symmetry embedded in the matrix of the Fd 3¯ m structure.

Acknowledgements The authors are grateful to Dr P. Dierżanowski (Faculty of Geology, University of Warsaw) for EPMA WDS analysis, Dr. Ł. Marciniak (Institute of Low Temperature and Structure Research, Polish Academy of Sciences, Wrocław) for second harmonic generation measurements, and Dr. A. Piecha-Bisiorek (Department of Chemistry, University of Wrocław) for electric measurements.

Appendix A. Supplementary material Supplementary material associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jssc.2015.07.021.

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