Unexpected phase transition sequence in the ferroelectric Bi₄Ti₃O₁₂

The high-temperature phase behaviour of the ferroelectric layered perovskite Bi₄Ti₃O₁₂ has been re-examined by high-resolution powder neutron diffraction. Previous studies, both experimental and theoretical, had suggested conflicting structural models and phase transition sequences, exacerbated by the complex interplay of several competing structural instabilities. This study confirms that Bi₄Ti₃O₁₂ undergoes two separate structural transitions from the aristotype tetragonal phase (space group I4/mmm) to the ambient-temperature ferroelectric phase (confirmed as monoclinic, B1a1). An unusual, and previously unconsidered, intermediate paraelectric phase is suggested to exist above T_C with tetragonal symmetry, space group P4/mbm. This phase is peculiar in displaying a unique type of octahedral tilting, in which the triple perovskite blocks of the layered structure alternate between tilted and untilted. This is rationalized in terms of the bonding requirements of the Bi³⁺ cations within the perovskite blocks.

1. Introduction

Bi₄Ti₃O₁₂ is an n = 3 member of the Aurivillius family of layered perovskite ferroelectrics (Aurivillius, 1950; Subbarao, 1962), (Bi₂O₂)A_n−1B_nO_3n+1. The symmetry of the ideal, paraelectric aristotype phase is tetragonal, space group I4/mmm (Fig. 1). A complex combination of octahedral tilting and atomic displacements leads to the room-temperature ferroelectric phase adopting the monoclinic space group B1a1 (standard setting Pc) (Rae et al., 1990). Early variable-temperature crystallographic studies of Bi₄Ti₃O₁₂ using electron microscopy (Nistor et al., 1996), powder X-ray diffraction (PXRD) (Hirata & Yokokawa, 1997) and high-resolution powder neutron diffraction (PND) (Hervoches & Lightfoot, 1999) concluded that Bi₄Ti₃O₁₂ did indeed adopt the aristotype I4/mmm structure at temperatures well above the ferroelectric Curie temperature, T_C ≃ 675°C. However, these analyses did not cover the nature of the evolution of the ambient-temperature monoclinic phase towards this aristotype tetragonal phase; for example, the present author's previous study (Hervoches & Lightfoot, 1999) only reported data at two intermediate temperatures (500 and 650°C). Since that study, several authors have re-examined the nature of this complex structural evolution and differing structural models for various intermediate phases have been proposed. Kennedy and co-workers (Zhou et al., 2003), using synchrotron PXRD, suggested an intermediate orthorhombic phase, space group Cmce, over a narrow temperature range above T_C. In contrast, the optical microscopy measurements of Iwata et al. (2013) suggested a tetragonal phase above T_C, but one of lower symmetry than the aristotype (either P4/nnc or P4₂/nmc). We shall return to these studies later. Meanwhile, insights from theory, facilitated specifically by the combination of improvements to first-principles calculations and by the development of techniques for symmetry mode analysis, have helped to guide experimentalists towards possible crystallographic models that may not have been previously considered. Thus, Perez-Mato et al. (2008) illustrated the true complexity of the symmetry lowering from I4/mmm to B1a1 as consisting of the interplay of six distinct normal modes, three of which are necessary for the observed symmetry breaking. The presence of more than one primary symmetry-breaking mode at a single transition event can, in general, be considered as an exception to the Landau theory, and in this particular case the DFT calculations (Perez-Mato et al., 2008), showed that the couplings among the relevant modes did not favour the simultaneous condensation of three primary order parameters. This gives rise to a scenario with several different possible pathways of symmetry descent from the aristotype phase to the experimentally observed ferroelectric phase at ambient temperature.

Figure 1 Crystal structures of (a) the aristotype (tetragonal, I4/mmm) and (b) the ambient-temperature (monoclinic, B1a1) phases.

It is clear that considerable ambiguities still remain over the exact details of the thermal evolution of the crystallographic nature of Bi₄Ti₃O₁₂. In order to shed further light on this, we have now carried out a more detailed high-resolution PND study, which supersedes our earlier work and provides significant new insights into this unusual phase transition behaviour. In particular, we make use of the methods of symmetry mode analysis (Perez-Mato et al., 2010; Campbell et al., 2006) to explore systematically the possible phase transition pathways on cooling from the aristotype I4/mmm phase. This study leads to a rather unexpected result.

2. Experimental

2.1. Synthesis

Bulk polycrystalline Bi₄Ti₃O₁₂ was synthesized using a conventional mixed-oxide solid-state route. Stoichiometric amounts of Bi₂O₃ and TiO₂ (99.9%, Alfa Aesar) were dried at 100°C for 48 h, ground and pressed into pellets of 13 mm diameter and approximately 2 mm thickness. Samples were subsequently heated at 700°C for 24 h, followed by 850°C for 24 h and cooled at a rate of 10°C min⁻¹. Pellets were then re-ground to produce final powders suitable for characterization. Phase purity was confirmed by PXRD using Cu Kα₁ radiation with a wavelength of 1.5406 Å on a Panalytical EMPYREAN diffractometer (2θ step size of 0.017°, 1 h total scan time).

2.2. Powder neutron diffraction

Time-of-flight neutron powder diffraction experiments were conducted using the high-resolution powder diffractometer at the ISIS neutron spallation source at the Rutherford–Appleton Laboratories. The polycrystalline sample (∼3 g) was sealed in a thin-walled quartz tube mounted in a cylindrical vanadium can. Data were collected at a series of temperatures commencing at 20°C, and subsequently at selected intervals for the temperature regime 150 ≤ T ≤ 1000°C. Each scan was counted for a 15 µAh incident proton beam (ca 20 min), except for longer scans (∼3 h, 120 µAh) at temperatures of 450, 615, 635, 655, 670, 685, 705, 850 and 1000°C.

2.3. Diffraction data analysis

All PND data were analysed by Rietveld refinement using the GSAS software package with the EXPGUI interface (Toby, 2001). Refinement strategies were kept as uniform as possible across all datasets. For each dataset, two diffraction histograms were used [detector banks centred at 2θ = 168° (bank 1) and 90° (bank 2)]. The same set of profile parameters were refined in each case: twelve background coefficients and three peak-shape parameters for each histogram, three diffractometer constants and two histogram scale factors in total. In addition to these common parameters, appropriate lattice parameters, atomic positional coordinates and all isotropic atomic displacement parameters (ADPs) were refined; specific details are outlined for each crystallographic model in the relevant sections, as necessary. Symmetry mode analysis was carried out using the ISODISTORT suite (Campbell et al., 2006). Further details of each of the crystallographic models, together with example Rietveld plots, are provided in the supporting information.

3. Results and discussion

3.1. Ambient-temperature phase

The aristotype structure (Fig. 1), space group I4/mmm, has the approximate unit cell metrics a_T ≃ 3.9 Å, c_T ≃ 33 Å. Both experimental (single-crystal X-ray, Rae et al., 1990) and theoretical (Perez-Mato et al., 2008) work have identified the monoclinic B1a1 model (Fig. 1) as the stable ground state of the ferroelectric phase (note: B1a1 is a non-standard setting of space group Pc, No. 7). This model has a unit cell of approximately double the volume of the aristotype (a_M ≃ b_M ≃ a_T; c_M ≃ c_T; β ≃ 90.0°). In PL's earlier work (Hervoches & Lightfoot, 1999) this was approximated to an orthorhombic model, space group B2eb, with a_O ≃ b_O ≃ a_T; c_O ≃ c_T. The key difference between the (correct) monoclinic and (approximate) orthorhombic models lies in the presence of an additional `octahedral tilt' mode in the former. Octahedral tilt modes are prevalent in the structural chemistry of perovskites, and it has been recognized that simultaneous condensation of two or more tilt modes can give rise to spontaneous polarization in layered perovskites, hence precise identification of these modes is critical to the understanding of ferroelectricity in layered perovskites (Benedek et al., 2015). Although the additional tilt mode was not incorporated into the model used for the ambient-temperature phase in our earlier work (Hervoches & Lightfoot, 1999), with hindsight it can be seen that there is some evidence for its presence in the derived B2eb models presented in that work (e.g. anomalously large ADP of atom O1 in Tables 2 and 3). Hence, we first of all confirm that the B1a1 model is a more valid description of the ambient-temperature structure than the approximate B2eb model.

A starting model for the B1a1 phase was derived in two independent ways: the first was adapted from that reported by Rae et al. (1990) (we note that the model in ref. 3 uses an origin shift of Δy = 0.25 relative to the setting used here). The second was derived from our own B2eb model (Hervoches & Lightfoot, 1999) by appropriate manual symmetry lowering. Careful Rietveld refinement of both models led to identical derived parameters and quality of fit, giving confidence in the reliability and robustness of this model. On close inspection of these models, using the ISODISTORT software, it was found that the most significant distortion modes, relative to the I4/mmm parent phase, were those that can be attributed to the B2eb distortion plus one additional mode (the previously mentioned additional octahedral tilt). There are many other distortion modes allowed in the B1a1 model, (the B1a1 model allows 55 variable atomic coordinates, whereas the B2eb model allows only 27), but since these were all of relatively minor magnitude, we have chosen to simplify this refinement model to allow easier comparison of the significant changes in the structure on proceeding to higher temperatures. Specifically, positional constraints were introduced such that all atoms conformed to the B2eb model, except O(1) and O(1)', which are the `in-plane' oxygen atoms of the middle layer of the perovskite blocks. This approximation does not significantly affect the derived parameters discussed below. Details of Rietveld refinement outcomes for all models discussed are given in Table 1. The most significant distortion modes, which consist of three different octahedral tilt modes, in addition to the polar atomic displacement mode, are shown in Fig. 2. In detail, there are three tilt modes, designated by the irreducible representation (irrep) notations X₁⁻, X₃⁺ and X₂⁺. Further details of these modes are given in the supporting information. In addition to these three tilt modes, the dominant polar mode is designated Γ₅⁻. This represents polarization along the a axis of the B1a1 setting of the unit cell. Although a polarization (Γ₃⁻) is also allowed along the c axis, this was found to be relatively weak in preliminary refinements (most noticeably in minor polar displacements of the O atoms in the octahedra), and subsequently constrained to zero.

Table 1
Details of the crystallographic models and Rietveld refinements for each of the three observed phases

N_xyz is the number of refined atomic coordinates (in the case of the B1a1 model, constraints have been added – see text. In all other cases, all atomic coordinates were refined freely). N_tot is the total number of refined parameters (see text).

Temp (°C)	Space group	Significant modes	Unit cell (Å, °)	Nxyz	Ntot	χ^2/Rwp
20	B1a1	X1^−, X3^+, X2^+, Γ5^−	a = 5.4447 (1), b = 5.4094 (1), c = 32.8504 (7), β = 90.047 (3)	30	78	11.86,† 2.35
				55	101	9.83, 2.15
685	P4/mbm	X2^+, M1^+	a = 5.4563 (1), c = 33.2475 (4)	18	72	6.30, 2.39
				18	68	6.38,‡ 2.41
				18	64	6.59, 2.45
	Cmce	X2^+	a = 33.2475 (5), b = 5.4561 (4), c = 5.4566 (4)	11	58	7.01, 2.53
	P42/ncm	X3^+	a = 5.4562 (1), c = 33.2479 (5)	15	65	8.50, 2.78
	P4/nbm	X1^−	a = 5.4563 (1), c = 33.2476 (5)	16	70	7.95, 2.69
1000	I4/mmm	None	a = 3.8771 (1), c = 33.4026 (4)	6	52	8.50, 3.06

†The two refinements shown are those with `partially constrained' and `fully independent' coordinates, described in the text.
‡In the case of the P4/mbm model, the refinement was carried out (i) with all U_iso values independent (χ² = 6.30), (ii) with the cation U_isos treated as `paired' according to parent atom type (χ<sup>2</sup> = 6.38), (iii) with all the U<sub>iso</sub>s treated as `paired' according to parent atom type (χ² = 6.59); the second model is reported in Tables 2 and 3.

Figure 2 Details of the three individual tilt modes that contribute to the B1a1 model. Simulated mode amplitudes of 0.5 Å (ISODISTORT) have been used for illustration. Each plot shows an isolated triple-octahedral block view perpendicular and parallel to the unit cell axis c in the B1a1 setting. (a) The X3^+ mode, showing out-of-plane (ab) tilting. (b) The X1^− mode, showing anti-phase rotation of the two outer octahedral layers only around c. (c) The X2^+ mode, showing rotation of the inner octahedral layer only around c.

3.2. Thermal evolution of the structure; ambient temperature to T_C

The partially constrained model described above for the ambient-temperature refinement was applied sequentially to all datasets proceeding upwards in temperature. Refinements proceeded smoothly and led to stable refinements, with comparable qualities of fit up to 670°C. Lattice metrics show a smooth convergence of the a and b unit cell parameters on increasing T, with an apparent coalescence at T_C (Fig. 3). Although the β angle remains very close to 90° throughout this temperature range (Fig. S1), and cannot be used in isolation as an unambiguous measure of crystal symmetry, the continued presence of key distortion modes (see below) confirms that monoclinic symmetry is retained all the way to T_C.

Figure 3 Full-range (top) and expanded (bottom) thermal evolution of the lattice metrics. (a) a and b parameters, (b) c parameter and (c) unit cell volume.

ISODISTORT was used to derive mode amplitudes from each refinement: plots of the thermal evolution of the four most significant modes are given in Fig. 4. From these data, it can be immediately seen that the tilt modes X₁⁻ and X₃⁺, together with the polar mode Γ₅⁻ show a gradual trend towards zero at T_C but, importantly, just below T_C a `plateauing' is seen. In contrast, the X₂⁺ mode shows a smaller variation throughout most of the temperature regime, followed by a more abrupt transition to a larger value at T_C and above.

Figure 4 Thermal evolution of the four most significant modes: (a) X1^− and X3^+ tilt modes and Γ5^− polar mode. Note that in the expanded plot on the right, all three modes feature a plateau towards TC. (b) X2^+ tilt mode. The mode amplitudes are normalized relative to the common parent unit cell (parameter Ap in ISODISTORT).

The `plateauing' behaviour of the X₁⁻, X₃⁺ and Γ₅⁻ modes is suggestive of a first-order, rather than a continuous evolution of the structure through T_C.

3.3. Nature of the phase immediately above T_C

Fig. 4 shows that the X₁⁻, X₃⁺ and Γ₅⁻ modes display a decreasing and then `plateauing' trend on heating towards T<sub>C</sub>, whereas the X<sub>2</sub><sup>+</sup> mode increases, and indeed becomes the strongest mode approaching T<sub>C</sub>. From this it might be inferred that, of the four largest ambient-temperature modes, the X<sub>2</sub><sup>+</sup> mode persists uniquely above T<sub>C</sub>. This is a curious and unexpected result, and a detailed comparison of selected models was undertaken to confirm it and establish a rationalization of the behaviour. Weak superlattice peaks at the X-point (1/2,1/2,0) of the parent I-centred tetragonal Brillouin zone can be clearly seen in the raw data (Fig. 5). The next step was to determine the possible space groups for the phase directly above T<sub>C</sub>, taking into account the continued presence of the X<sub>2</sub><sup>+</sup> mode, but with the absence of the X<sub>1</sub><sup>−</sup>, X<sub>3</sub><sup>+</sup> and Γ<sub>5</sub><sup>−</sup> modes. Again, ISODISTORT was used to derive the simplest crystallographic models based on the requirement for the X<sub>2</sub><sup>+</sup> mode as the primary-order parameter. Three models are derived (see Table 1 and the supporting information for further details). Each of these retains a `doubled' unit cell volume compared with the parent I4/mmm phase; i.e. (unit cell axes for some models are switched in order to conform to standard space-group settings). As a comparison, models derived using only X₁⁻ or X₃⁺ modes were also refined for the 685°C dataset (Table 1). As a final check, two models were considered which permit X₂⁺/X₁⁻ or X₂⁺/X₃⁺ combinations. Neither of these produced significantly improved fits compared with the model chosen below (see supporting information).

Figure 5 (a) Portion of the raw PND at 685°C (above), highlighting continued presence of the X-point peaks (arrowed) above TC, together with the corresponding region at 1000°C. (b) Corresponding Rietveld fit using the P4/mbm model. (c) Corresponding Rietveld fit at 1000°C (I4/mmm model): the peak at d ≃ 2.40 Å is a Bi2Ti2O7 impurity due to partial decomposition under these conditions.

These refinements firstly confirm the validity of the X₂⁺ mode, (and not the X₁⁻ or X₃⁺ modes) in accounting for the presence of the weak X-point superlattice peaks above T_C. Second, amongst the options displaying the X₂⁺ mode, they clearly demonstrate a preference for the tetragonal (P4/mbm) model rather than the orthorhombic (Cmce) model; even the third P4/mbm model shown in Table 1, with constrained U_iso parameters, shows a significant improvement in fit due to the few additional structural variables.

The difference between the two models, P4/mbm and Cmce, is worthy of further description and analysis (see also supporting information). In both cases, the X₂⁺ mode corresponds to a `rigid' rotation of the central octahedral layer of the perovskite block around the `long' unit cell axis (i.e. the c axis in the B1a1 and P4/mbm models). In the case of the Cmce model, this rotation occurs in all the perovskite blocks, acting in the same sense around c in each block. However, in the P4/mbm model the rotation occurs in only alternate perovskite blocks along c, with every other block having zero rotation. This mode therefore breaks the lattice centering, but maintains the tetragonal symmetry of the parent. This can be regarded as a `2k' mode [k1 = (1/2,1/2,0) and k2 = (1/2,1/2,1)], whereas the mode of the same symmetry leading to the Cmce model has only a single k contribution (k = 1/2,1/2,1). Hence, although the crystal system may be regarded as `higher symmetry' for the P4/mbm case, the Cmce model has the smaller (primitive) asymmetric unit cell and a smaller number of refineable atomic coordinates (Table 1). The ground-state model B1a1 resembles the Cmce model in this sense: i.e. its X₂⁺ mode condenses at a single k-point. This means that there is no group–subgroup relationship between the P4/mbm and B1a1 models whereas there is such a relationship between the Cmce and B1a1 models. So, why do we propose that the P4/mbm model is preferred? First, there is no evidence from refined lattice parameter metrics for any deviation from tetragonal symmetry, although this in itself is insufficient evidence. More importantly, the tetragonal model has effectively `lower' symmetry and allows additional distortion modes which are not permitted in the Cmce model. Specifically, there is a mode of M₁⁺ symmetry which is active in the P4/mbm model. This mode affects primarily the Bi sites within the triple-perovskite blocks. As shown in Fig. 6, the M₁⁺ mode allows displacement of these Bi atoms along the c axis, such that in blocks where the X₂⁺ mode requires the octahedral rotation to be present, the Bi atoms are displaced towards the central octahedral layer, whereas in blocks with zero octahedral rotation, the Bi atoms are displaced away from the central layer. This subtle feature is presumably not coincidental, but allows the overall energetics of the system to be optimized by cooperation with the unusual tilt system. Further support for this correlation comes from bond-valence sum analysis, as described in the Discussion.

Figure 6 M1^+ mode (present in the P4/mbm model but not in the Cmce model). The arrows describe a displacive degree of freedom of the Bi sites within the perovskite blocks: note that the Bi displacement alternates `towards' or `away from' the central octahedral site in alternate layers (see text).

The fact that this unusual type of transition (i.e. where a 2k mode leads to a situation of alternating `tilted' and `untilted' perovskite blocks) has apparently not been seen in other families of layered perovskites is perhaps due to the peculiar bonding preferences of the non-spherical Bi³⁺ cation; further work may be merited to explore this phenomenon. The final crystallographic model for the P4/mbm phase is presented in Table 2, and selected bond lengths in Table 3.

Table 2
Refined crystallographic model for the intermediate P4/mbm phase at 685°C, lattice parameters: a = 5.4563 (1), c = 33.2475 (4) Å

Atom	x	y	z	Uiso (Å^2)
Bi1(1)†	0.0000	0.5000	0.5650 (1)	0.0452 (5)
Bi1(2)	0.0000	0.0000	0.0715 (1)	0.0452 (5)
Bi2(1)	0.0000	0.5000	0.7106 (2)	0.0356 (5)
Bi2(2)	0.0000	0.0000	0.2113 (2)	0.0356 (5)
Ti1(1)	0.0000	0.5000	0.0000	0.019 (1)
Ti1(2)	0.0000	0.0000	0.5000	0.019 (1)
Ti2(1)	0.0000	0.5000	0.8705 (3)	0.0161 (7)
Ti2(2)	0.0000	0.0000	0.3717 (3)	0.0161 (7)
O1(1)	0.7531 (9)	0.2531 (9)	0.0000	0.052 (2)
O1(2)	0.3135 (7)	0.8135 (7)	0.5000	0.023 (1)
O2(1)	0.7497 (5)	0.2497 (5)	0.2516 (2)	0.0271 (7)
O3(1)	0.0000	0.5000	−0.0592 (3)	0.073 (3)
O3(2)	0.0000	0.0000	0.4412 (3)	0.056 (2)
O4(1)	0.0000	0.5000	0.8175 (2)	0.052 (2)
O4(2)	0.0000	0.0000	0.3174 (2)	0.042 (2)
O5(1)	0.7473 (6)	0.2473 (6)	0.6164 (2)	0.052 (2)
O5(2)	0.2517 (6)	0.7517 (5)	0.8822 (1)	0.029 (1)

†Cation U_iso values were constrained `pairwise' according to the parent phase symmetry.

Table 3
Key bond lengths for the intermediate P4/mbm phase at 685°C

Bond	Length (Å)	Bond	Length (Å)
Bi1(1)—O1(2) × 2	2.596 (5)	Ti1(1)—O1(1) × 2	1.905 (7)
Bi1(1)—O1(3) × 2	2.736 (1)	Ti1(1)—O1(1) × 2	1.953 (7)
Bi1(1)—O5(1) × 2	2.593 (6)	Ti1(1)—O3(1) × 2	1.969 (11)
Bi1(1)—O5(1) × 2	2.562 (6)	Ti1(2)—O1(2) × 4	1.990 (1)
Bi1(2)—O3(1) × 4	2.759 (2)	Ti1(2)—O3(2) × 2	1.955 (10)
Bi1(2)—O5(2) × 4	2.468 (4)	Ti2(1)—O3(1)	2.337 (14)
Bi2(1)—O2(1) × 2	2.305 (6)	Ti2(1)—O4(1)	1.761 (13)
Bi2(1)—O2(1) × 2	2.301 (6)	Ti2(1)—O5(2) × 2	1.980 (5)
Bi2(1)—O4(2) × 3	2.882 (3)	Ti2(1)—O5(2) × 2	1.955 (5)
Bi2(2)—O2(1) × 4	2.349 (5)	Ti2(2)—O3(2)	2.310 (13)
Bi2(2)—O4(1) × 4	2.891 (3)	Ti2(2)—O4(2)	1.805 (13)
		Ti2(2)—O5(1) × 4	1.969 (2)

3.4. The parent phase

The X-point reflections noted above are seen to persist to a temperature of at least 850°C, but disappear by 1000°C. We can therefore conclude that in the region 850–1000°C there is a final transition from P4/mbm to the parent structure I4/mmm. Only weak reflections violating this model (which are due to partial sample decomposition to the pyrochlore Bi₂Ti₂O₇) are seen at 1000°C [Fig. 5(c)]. Unfortunately, we were not able to collect further datasets in this temperature region in order to define a more precise transition temperature, or to probe further the nature of this transition. The I4/mmm model refines straightforwardly and is in agreement with that previously presented in the work by Hervoches & Lightfoot (1999).

4. Discussion

In our previous paper (Hervoches & Lightfoot, 1999) it was suggested that the driving force for any structural transition from the aristotype I4/mmm phase was most likely the result of significant underbonding at the A site of the perovskite block [Bi(1) site, Fig. 1]. Our current study supports that assertion, but leads to an unexpected means of relieving this underbonding: via activation of the 2k X₂⁺ tilt mode (Fig. 4), together with the M₁⁺ Bi displacement mode (Fig. 6). Bond-valence sum analysis (Brese & Keeffe, 1991) provides a semi-quantitative means of rationalizing this (we do not attempt a fully quantitative justification as the bond-valence method is semi-empirical and uses parameters derived from ambient-temperature crystal data). In Table 4, we compare bond lengths around the Bi(1) site for four idealized high-temperature models: (i) the parent I4/mmm phase; (ii) the Cmce phase (which allows a single k X₂⁺ mode, but does not allow the M₁⁺ Bi displacement mode); (iii) the P4/mbm phase, with only the 2k X₂⁺ mode activated; and (iv) the P4/mbm phase, with both the 2k X₂⁺ mode and the M₁⁺ mode activated. Each of the lower-symmetry models are derived from the same I4/mmm parent, with fixed mode amplitudes of −0.5 Å for X₂⁺ and 0.5 Å for M₁⁺, derived from ISODISTORT.

Table 4
Comparison of the bonding environments of Bi(1) for selected idealized models

For the P4/mbm models two different Bi(1) sites are generated in alternate perovskite blocks; these are shown in separate rows. Σv is the bond valence sum for Bi(1), taking into account the 12 Bi—O bonds. For each, bond, the bond valence is v = exp((R₀ − d)/b), where d is the individual bond length, R₀ is a constant for a particular bond type (e.g. 2.09 Å for Bi—O) and b = 0.37 Å, a universal constant.

Bond (Å)	I4/mmm (i)	Cmce (ii)	P4/mbm (iii)	P4/mbm (iv)
Bi1—O5	2.55 × 4	2.55 × 4	2.55 × 4	2.67 × 4
			2.55 × 4	2.44 × 4
Bi1—O3	2.75 × 4	2.75 × 4	2.75 × 4	2.73 × 4
			2.75 × 4	2.77 × 4
Bi1—O1	2.96 × 4	2.81 × 2; 3.13 × 2	2.81 × 2; 3.13 × 2	2.67 × 2; 3.01 × 2
			2.96 × 4	3.10 × 4
Σv (Bi1)	2.21	2.23	2.23 2.21	2.13 2.45

What can be seen here is that the Bi(1) site is dramatically underbonded (bond valence sum, Σv = 2.21 valence units) in the parent phase (the bond valence sum simply represents the nominal oxidation state of the bonded atom in this model, hence a Σv of 3.0 would be expected for optimally bonded Bi³⁺). On permitting a small distortion due to the X₂⁺ tilt mode within the Cmce model, it can be seen that eight of the twelve Bi—O bonds remain unaffected [those to O(3) and O(5); see Fig. 1], whereas the set of four bonds (at 2.96 Å) to the O(1) atom in the central octahedral layer split pairwise, `2 long + 2 short'. The net bonding environment around Bi(1) is improved by the introduction of the two shorter bonds (at 2.81 Å), which outweighs the weaker bonding to the two longer bonds (3.13 Å), resulting in a slight increase in the Σv (2.23 v.u.). Incorporating the X<sub>2</sub><sup>+</sup> mode only into the P4/mbm model does not significantly change this (Σv = 2.23 v.u.), but allowing the additional degree of freedom from inclusion of the M<sub>1</sub><sup>+</sup> mode makes a significant difference, the Bi(1) site within the `tilted' layer benefiting from an increase in Σv to 2.45 v.u. [and with only a minor disadvantage in bonding to the other Bi(1) site].

The above argument is only intended to illustrate the general features of the differing distortion modes and their effect on the bonding around the Bi(1) site. The observed magnitudes of the X₂⁺ and M₁⁺ modes in our final refinement of the P4/mbm model (Tables 2 and 3) are of the order of −1.0 and 0.3 Å, respectively, leading to actual Σv values of 2.38 and 2.39 v.u. In comparison, our experimentally determined Σv for the Bi(1) site in the Cmce model is only 2.33, confirming the validity of the above generic argument.

The present study demonstrates that Bi₄Ti₃O₁₂ undergoes a highly unusual structural evolution versus temperature, displaying two distinct phase transitions, incorporating several competing active modes. Previous experimental studies have failed to recognize the details of these modes, and hence have proposed incorrect or incomplete models for the intermediate phase (the P4/mbm phase in this study). Our own earlier study (Hervoches & Lightfoot, 1999) did not see the intermediate phase, in part due to the larger temperature intervals used in that experiment. Two more recent studies are also worthy of comment. Kennedy and co-workers (Zhou et al., 2003) proposed, from synchrotron PXRD, an intermediate orthorhombic model just above T_C (in the temperature interval 675 < T < 695°C). It is significant to note, however, that despite the same apparent space group (Cmce), this is not the same model as that considered in the present work. Kennedy's suggestion (a full model was not proposed) arises from the I4/mmm parent via condensation of the X₃⁺ mode, not the X₂⁺ mode, and was based on an implicit assumption of a single k mode. Hence, the unit cell requires a different setting of the space group Cmce to the one considered here: the two alternatives can be seen in the group–subgroup graph suggested in Perez-Mato et al. (2008) and are also shown in Table S1 of the supporting information. Iwata's study, using optical microscopy on single crystalline samples, identified the tetragonal (not orthorhombic) nature of the intermediate phase at 800°C, and indeed this offers support to our own proposed model having tetragonal symmetry. However, they failed to identify the superlattice due to the X-point distortion. Instead, that study proposed an M-point distortion from the I4/mmm parent, leading to a unit cell of the same size as the parent (a_T ≃ 3.9 Å, c_T ≃ 33 Å), but with symmetry lowered from body-centred to primitive. Unfortunately, a full structural model was again not proposed, nor details of the specific mode(s) underlying the proposed distortion. However, it is clear that in a model with only Γ and M-points active, there can be no octahedral tilting, only atomic displacements along the c axis.

It is perhaps not surprising, with hindsight, that the subtleties of the P4/mbm option were missed in all of the previous studies; its identification relies on careful observation and modelling of the weak X-point peaks above T_C, for which neutron diffraction is so well suited.

Two structural transitions are observed in Bi₄Ti₃O₁₂ in this study; I4/mmm to P4/mbm in the region 850–1000°C and P4/mbm to B1a1 at T_C ≃ 675°C. In the case of the I4/mmm to P4/mbm transition, the scenario is the usual Landau-type transition with a single irrep for the order parameter or distortion, which is responsible for the symmetry break. This primary mode corresponds to the X₂⁺ irrep and involves its two distinct wavevectors. But, as an induced effect, a secondary mode with symmetry M₁⁺ is also present. The X₂⁺ primary distortion mode is essentially the same as that present in the lower-temperature phase (reducing its symmetry from orthorhombic to monoclinic), but with the difference that in this intermediate phase the modes corresponding to the two wavevectors, k1 and k2, of the irrep are concomitantly condensed, and thus maintain a tetragonal symmetry, whereas in the monoclinic phase at lower temperatures only one of the modes associated with one of the two wavevectors is involved. Only the simultaneous condensation of the two waves allows the presence of the mentioned secondary M₁⁺ mode, and this could be the fundamental reason favouring this phase. Phenomenologically, the condensation of the M₁⁺ distortion can be regarded as a consequence of a coupling in a trilinear fashion (Benedek et al., 2015; Extebarria et al., 2010), i.e. the free energy is lowered by a term of the form Q_k1Q_k2Q_M1⁺, where Q represents the magnitude of each mode.

From theoretical DFT calculations Perez-Mato et al. (2008) concluded that a direct transition from the parent phase I4/mmm to the monoclinic phase with symmetry B1a1, as generally assumed, was very unlikely, as the energy landscape did not favour the necessary simultaneous condensation of the three primary distortion modes associated with different irreps that are present in the monoclinic phase. Such `avalanche' first-order transitions are known for two order parameters, but not for three, and in the present case there is no indication in the mode couplings that would promote such an exceptional scenario. The existence of an intermediate phase where one of the order parameters previously condenses solves this puzzle, and the second transition between the P4/mbm and B1a1 symmetries is now reduced to the simultaneous activation of two additional primary distortion modes, together with a change of direction (change from a 2k to a 1k distortion) of the order parameter associated with the intermediate phase. This second phase transition is then analogous to the one in Aurivillius compounds with perovskite blocks with only two layers (Perez-Mato et al., 2004), the difference being the additional previous condensation of the X₂⁺ mode, which involves only the central octahedral layers and it is therefore specific to materials with three layers in the perovskite blocks. The instability of this additional distortion was shown (Perez-Mato et al., 2008) to be very strong, comparable with the polar strongest one, and therefore its condensation at a higher temperature than the rest is not unreasonable, but the unexpected feature is the involvement of the two wavevector branches of this distortion, which was overlooked in previous theoretical calculations as a competitive configuration of the distortion.

From the above analysis the transition at T_C takes place between two phases that are not group–subgroup related, and therefore it is necessarily a first-order or discontinuous transition. This is supported by the `plateau' behaviour of the tilts and polar mode just below T_C (Fig. 4). The condensation of the additional modes is concomitant with the disappearance of the M₁⁺ mode and the reduction to zero of the amplitude for one of the two wavevectors involved in the X₂⁺ distortion. The X₂⁺ distortion remains in the ground state of the system, but with its amplitude significantly reduced due to its unfavourable coupling (Perez-Mato et al., 2008), with the other spontaneous distortions. Even if the intermediate phase is the result of a 1k X₂⁺ distortion, resulting in a Cmce symmetry (instead of the P4/mbm space group proposed here) then, although there would be a group–subgroup relation between the symmetries of the intermediate and the monoclinic phase, the transition would also be first order, since this symmetry break implies an avalanche transition, with two irreps being activated simultaneously. We note that there is no rule forbidding an intermediate phase with a less dense set of lattice translations (and a larger primitive unit cell) than the lower-temperature phase, and this always happens in cases where the high rotational symmetry is maintained by combining several distortion modes with symmetry-related k-vectors (multi-k case).

5. Conclusions

In conclusion, we have shown that Bi₄Ti₃O₁₂ undergoes a highly unusual phase evolution versus temperature. The aristotype tetragonal phase (space group I4/mmm) undergoes a two-step transition to the monoclinic ground state (space group B1a1) on cooling. At temperatures still well above T_C (∼675°C), an octahedral tilt mode described by irrep X₂⁺ acts to reduce the symmetry to an intermediate centrosymmetric phase. The activation of this mode above T_C has not been considered in previous studies. Of the two simplest models considered, a tetragonal phase with symmetry P4/mbm is determined to be the most likely. Significantly, the X₂⁺ mode present in this model affects octahedral tilts in alternating perovskite-like blocks along the c axis, a phenomenon which has not been seen in other families of layered perovskites as far as we are aware. This can be rationalized in terms of the most favourable way to optimize bonding interactions around the Bi atoms within the perovskite blocks. On lowering the temperature through T_C, two further octahedral tilt modes are activated simultaneously, coupled with the polar displacive mode, leading directly to the monoclinic (B1a1) ground state. Despite the fact that three significant modes appear together below T_C, the polar Γ₅⁻ mode appears dominant, and Bi₄Ti₃O₁₂ may hence be described as a proper ferroelectric.

The research data supporting this publication can be accessed at https://doi.org/10.17630/b7c11904-b4b0-4acf-a02a-b1e53aec3672.

Supporting information

Link https://doi.org/10.17630/b7c11904-b4b0-4acf-a02a-b1e53aec3672
Unexpected phase transition sequence in the ferroelectric Bi4Ti3O12 (dataset)

Computing details

(BI4TI3O12-655_phase_1)

Crystal data

Bi4O12Ti3	c = 33.2011 (5) Å
Mr = 1171.61	β = 89.9609 (17)°
Monoclinic, B1c1	V = 988.32 (4) Å^3
a = 5.46098 (9) Å	Z = 4
b = 5.45098 (9) Å

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
Bi1	0.0203 (11)	0.7539 (7)	0.56776 (5)	0.0435 (5)*
Bi2	0.01892	0.7453 (7)	0.711235 (34	0.0304 (5)*
Ti1	0.021 (2)	0.25	0.5	0.0111 (6)*
Ti2	0.0508 (13)	0.2560 (14)	0.62901 (6)	0.0110 (6)*
O1	0.267 (2)	0.522 (2)	0.50354 (15)	0.0449 (12)*
O2	0.3032 (10)	0.4954 (11)	0.24836 (12)	0.0209 (7)*
O3	0.0705 (10)	0.2151 (10)	0.55901 (7)	0.0363 (10)*
O4	0.0367 (15)	0.2875 (11)	0.68201 (6)	0.0386 (11)*
O5	0.2814 (15)	0.5139 (13)	0.61298 (10)	0.0277 (10)*
O6	0.3131 (14)	0.0238 (11)	0.62271 (10)	0.0334 (12)*
Bi1a	0.0203 (11)	0.7461 (7)	0.43224 (5)	0.0435 (5)*
Bi2a	0.01892	0.7547 (7)	0.288760 (34	0.0304 (5)*
Ti2a	0.0508 (13)	0.2440 (14)	0.37099 (6)	0.0110 (6)*
O1a	0.337 (2)	0.0488 (18)	0.49646 (15)	0.0449 (12)*
O2a	0.3032 (10)	0.0047 (11)	0.75165 (12)	0.0209 (7)*
O3a	0.0705 (10)	0.2849 (10)	0.44099 (7)	0.0363 (10)*
O4a	0.0367 (15)	0.2125 (11)	0.31799 (6)	0.0386 (11)*
O5a	0.2814 (15)	−0.0139 (13)	0.38702 (10)	0.0277 (10)*
O6a	0.3131 (14)	0.4762 (11)	0.37829 (10)	0.0334 (12)*

(BI4TI3O12-20_phase_1)

Crystal data

Bi4O12Ti3	c = 32.8504 (7) Å
Mr = 1171.61	β = 90.048 (3)°
Monoclinic, B1c1	V = 967.53 (6) Å^3
a = 5.44472 (11) Å	Z = 4
b = 5.40938 (11) Å

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
Bi1	0.00544	0.7548 (3)	0.566878 (35	0.0066 (3)*
Bi2	0.0056 (4)	0.7326 (4)	0.711318 (29	0.0091 (3)*
Ti1	0.0454 (9)	0.25	0.5	−0.0004 (4)*
Ti2	0.0585 (6)	0.2523 (9)	0.62818 (6)	−0.0004 (4)*
O1	0.3072 (11)	0.5178 (10)	0.50768 (8)	0.0130 (9)*
O2	0.2787 (5)	0.4921 (5)	0.24990 (8)	0.0062 (5)*
O3	0.0945 (4)	0.1900 (4)	0.55923 (7)	0.0114 (6)*
O4	0.0625 (5)	0.2983 (5)	0.68159 (5)	0.0059 (6)*
O5	0.2999 (5)	0.5243 (5)	0.61185 (7)	0.0033 (6)*
O6	0.3594 (5)	0.0408 (5)	0.62386 (7)	0.0069 (7)*
Bi1a	0.00544	0.7452 (3)	0.433122 (35	0.0066 (3)*
Bi2a	0.0056 (4)	0.7674 (4)	0.288681 (29	0.0091 (3)*
Ti2a	0.0585 (6)	0.2477 (9)	0.37182 (6)	−0.0004 (4)*
O1a	0.3602 (11)	0.0362 (11)	0.49232 (8)	0.0130 (9)*
O2a	0.2787 (5)	0.0079 (5)	0.75010 (8)	0.0062 (5)*
O3a	0.0945 (4)	0.3100 (4)	0.44077 (7)	0.0114 (6)*
O4a	0.0625 (5)	0.2017 (5)	0.31841 (5)	0.0059 (6)*
O5a	0.2999 (5)	−0.0243 (5)	0.38815 (7)	0.0033 (6)*
O6a	0.3594 (5)	0.4592 (5)	0.37714 (7)	0.0069 (7)*

(BI4TI3O12-685-X2-CMC_phase_1)

Crystal data

Bi4O12Ti3	b = 5.4561 (4) Å
Mr = 1171.61	c = 5.4566 (4) Å
Orthorhombic, Cmca	V = 989.82 (4) Å^3
a = 33.2475 (5) Å	Z = 4

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
Bi1_1	0.56839 (5)	0.0	0.0	0.0476 (5)*
Bi2_1	0.71086 (4)	0.0	0.0	0.0345 (5)*
Ti1_1	0.0	0.0	0.0	0.0143 (11)*
Ti2_1	0.87141 (7)	0.0	0.0	0.0171 (7)*
O1_1	0.0	0.2081 (8)	0.2955 (8)	0.0370 (9)*
O2_1	0.25	0.2495 (7)	0.25	0.0317 (7)*
O3_1	−0.05914 (8)	0.0	0.0	0.0671 (8)*
O4_1	0.81737 (6)	0.0	0.0	0.0456 (7)*
O5_1	0.61720 (5)	0.2480 (6)	0.2494 (6)	0.0386 (5)*

(1000-TWOPHASE_phase_1)

Crystal data

Bi4O12Ti3	c = 33.4028 (4) Å
Mr = 1171.61	V = 502.11 (2) Å^3
Tetragonal, I4/mmm	Z = 2
a = 3.87760 (4) Å

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
Bi1	0.5	0.5	0.06860 (8)	0.0619 (7)*
Bi2	0.5	0.5	0.21108 (6)	0.0564 (9)*
Ti1	0.5	0.5	0.5	0.042 (2)*
Ti2	0.5	0.5	0.37126 (10)	0.0239 (11)*
O1	0.5	0.0	0.0	0.0834 (15)*
O2	0.5	0.0	0.25	0.0432 (11)*
O3	0.5	0.5	0.44101 (12)	0.0789 (12)*
O4	0.5	0.5	0.31832 (9)	0.0552 (10)*
O5	0.5	0.0	0.11726 (7)	0.0461 (8)*

(1000-TWOPHASE_phase_2)

Crystal data

Bi1.02O3.50Ti	a = 10.4476 (9) Å
Mr = 317.07	V = 1139.0 (3) Å^3
Cubic, Fd3m	Z = 16

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq	Occ. (<1)
bi	0.044 (9)	0.044 (9)	0.968 (12)	0.10 (8)*	0.17
o	0.457 (3)	0.125	0.125	−0.009 (5)*
o'	0.181 (8)	0.125	0.125	0.00 (4)*	0.167
ti	0.5	0.5	0.5	0.23 (14)*

(BI4TI3O12_P4MBM-3_phase_1)

Crystal data

Bi4O12Ti3	c = 33.2475 (5) Å
Mr = 1171.61	V = 989.82 (3) Å^3
Tetragonal, P4/mbm	Z = 4
a = 5.45630 (6) Å

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
Bi1_1	0.0	0.5	0.56500 (13)	0.0452 (5)*
Bi1_2	0.0	0.0	0.07155 (12)	0.0452 (5)*
Bi2_1	0.0	0.5	0.71056 (16)	0.0356 (5)*
Bi2_2	0.0	0.0	0.21129 (16)	0.0356 (5)*
Ti1_1	0.0	0.5	0.0	0.0195 (12)*
Ti1_2	0.0	0.0	0.5	0.0195 (12)*
Ti2_1	0.0	0.5	0.8705 (3)	0.0161 (7)*
Ti2_2	0.0	0.0	0.3717 (3)	0.0161 (7)*
O1_1	0.7531 (9)	0.2531 (9)	0.0	0.052 (2)*
O1_2	0.3135 (7)	0.8135 (7)	0.5	0.0230 (14)*
O2_1	0.7497 (5)	0.2497 (5)	0.25160 (19)	0.0271 (7)*
O3_1	0.0	0.5	−0.0592 (3)	0.073 (3)*
O3_2	0.0	0.0	0.4412 (3)	0.056 (2)*
O4_1	0.0	0.5	0.8175 (2)	0.052 (3)*
O4_2	0.0	0.0	0.3174 (2)	0.042 (2)*
O5_1	0.7473 (6)	0.2473 (6)	0.61643 (17)	0.0516 (18)*
O5_2	0.2517 (6)	0.7517 (6)	0.88217 (14)	0.0295 (13)*

Geometric parameters (Å, º)

Bi1_1—Ti1_2	3.480 (3)	Ti1_2—Bi1_1^xv	3.480 (3)
Bi1_1—Ti1_2^i	3.480 (3)	Ti1_2—O1_2^xxviii	1.9904 (13)
Bi1_1—Ti1_2^ii	3.480 (3)	Ti1_2—O1_2^viii	1.9904 (13)
Bi1_1—Ti1_2^iii	3.480 (3)	Ti1_2—O1_2^xxix	1.9904 (13)
Bi1_1—Ti2_2^iv	3.445 (7)	Ti1_2—O1_2^xxx	1.9904 (13)
Bi1_1—Ti2_2^v	3.445 (7)	Ti1_2—O3_2	1.955 (10)
Bi1_1—Ti2_2^vi	3.445 (7)	Ti1_2—O3_2^iv	1.955 (10)
Bi1_1—Ti2_2^vii	3.445 (7)	Ti2_1—Bi1_2^iv	3.340 (6)
Bi1_1—O1_2^viii	2.596 (5)	Ti2_1—Bi1_2^v	3.340 (6)
Bi1_1—O1_2^ix	2.596 (5)	Ti2_1—Bi1_2^vi	3.340 (6)
Bi1_1—O3_2^iv	2.7359 (8)	Ti2_1—Bi1_2^vii	3.340 (6)
Bi1_1—O3_2^v	2.7359 (8)	Ti2_1—O3_1^xxxi	2.337 (14)
Bi1_1—O3_2^vi	2.7359 (8)	Ti2_1—O4_1	1.761 (13)
Bi1_1—O3_2^vii	2.7359 (8)	Ti2_1—O5_2	1.980 (5)
Bi1_1—O5_1^x	2.593 (6)	Ti2_1—O5_2^viii	1.955 (5)
Bi1_1—O5_1^xi	2.562 (6)	Ti2_1—O5_2^xxix	1.980 (5)
Bi1_1—O5_1^xii	2.593 (6)	Ti2_1—O5_2^ix	1.955 (5)
Bi1_1—O5_1^xiii	2.562 (6)	Ti2_2—Bi1_1^iv	3.445 (7)
Bi1_2—Ti2_1^iv	3.340 (6)	Ti2_2—Bi1_1^v	3.445 (7)
Bi1_2—Ti2_1^v	3.340 (6)	Ti2_2—Bi1_1^xiv	3.445 (7)
Bi1_2—Ti2_1^xiv	3.340 (6)	Ti2_2—Bi1_1^xv	3.445 (7)
Bi1_2—Ti2_1^xv	3.340 (6)	Ti2_2—O3_2	2.310 (13)
Bi1_2—O3_1^xvi	2.7588 (16)	Ti2_2—O4_2	1.805 (13)
Bi1_2—O3_1^xvii	2.7588 (16)	Ti2_2—O5_1^xxxii	1.969 (3)
Bi1_2—O3_1^xviii	2.7588 (16)	Ti2_2—O5_1^xxiii	1.969 (3)
Bi1_2—O3_1^xix	2.7588 (16)	Ti2_2—O5_1^xxiv	1.969 (3)
Bi1_2—O5_2^v	2.468 (4)	Ti2_2—O5_1^xxxiii	1.969 (3)
Bi1_2—O5_2^xiv	2.468 (4)	O1_1—Ti1_1^xxxiv	1.905 (7)
Bi1_2—O5_2^xx	2.468 (4)	O1_1—Ti1_1^viii	1.953 (7)
Bi1_2—O5_2^xxi	2.468 (4)	O1_2—Bi1_1^xxxv	2.596 (5)
Bi2_1—Bi2_2^iv	3.7674 (16)	O1_2—Bi1_1^xxiii	2.596 (5)
Bi2_1—Bi2_2^v	3.7674 (16)	O1_2—Ti1_2^i	1.9904 (13)
Bi2_1—Bi2_2^vi	3.7674 (16)	O1_2—Ti1_2^iii	1.9904 (13)
Bi2_1—Bi2_2^vii	3.7674 (16)	O2_1—Bi2_1^xxii	2.305 (6)
Bi2_1—O2_1^xxii	2.305 (6)	O2_1—Bi2_1^xv	2.301 (6)
Bi2_1—O2_1^xxiii	2.301 (6)	O2_1—Bi2_2^xxxiv	2.349 (5)
Bi2_1—O2_1^xxiv	2.305 (6)	O2_1—Bi2_2^iii	2.349 (5)
Bi2_1—O2_1^xxv	2.301 (6)	O3_1—Bi1_2^xvi	2.7588 (16)
Bi2_1—O4_2^iv	2.883 (3)	O3_1—Bi1_2^xvii	2.7588 (16)
Bi2_1—O4_2^v	2.883 (3)	O3_1—Bi1_2^xxxvi	2.7588 (16)
Bi2_1—O4_2^vi	2.883 (3)	O3_1—Bi1_2^xxxvii	2.7588 (16)
Bi2_1—O4_2^vii	2.883 (3)	O3_1—Ti1_1	1.969 (10)
Bi2_2—Bi2_1^iv	3.7674 (16)	O3_1—Ti2_1^xxxviii	2.337 (14)
Bi2_2—Bi2_1^v	3.7674 (16)	O3_2—Bi1_1^iv	2.7359 (8)
Bi2_2—Bi2_1^xiv	3.7674 (16)	O3_2—Bi1_1^v	2.7359 (8)
Bi2_2—Bi2_1^xv	3.7674 (16)	O3_2—Bi1_1^xiv	2.7359 (8)
Bi2_2—O2_1^x	2.349 (5)	O3_2—Bi1_1^xv	2.7359 (8)
Bi2_2—O2_1^xxvi	2.349 (5)	O3_2—Ti1_2	1.955 (10)
Bi2_2—O2_1^xxvii	2.349 (5)	O3_2—Ti2_2	2.310 (13)
Bi2_2—O2_1^xiii	2.349 (5)	O4_1—Bi2_2^iv	2.891 (3)
Bi2_2—O4_1^iv	2.891 (3)	O4_1—Bi2_2^v	2.891 (3)
Bi2_2—O4_1^v	2.891 (3)	O4_1—Bi2_2^vi	2.891 (3)
Bi2_2—O4_1^xiv	2.891 (3)	O4_1—Bi2_2^vii	2.891 (3)
Bi2_2—O4_1^xv	2.891 (3)	O4_1—Ti2_1	1.761 (13)
Ti1_1—O1_1^x	1.905 (7)	O4_2—Bi2_1^iv	2.883 (3)
Ti1_1—O1_1^xi	1.953 (7)	O4_2—Bi2_1^v	2.883 (3)
Ti1_1—O1_1^xii	1.905 (7)	O4_2—Bi2_1^xiv	2.883 (3)
Ti1_1—O1_1^xiii	1.953 (7)	O4_2—Bi2_1^xv	2.883 (3)
Ti1_1—O3_1	1.969 (10)	O4_2—Ti2_2	1.805 (13)
Ti1_1—O3_1^xvii	1.969 (10)	O5_1—Bi1_1^xxxiv	2.593 (6)
Ti1_2—Bi1_1^xxviii	3.480 (3)	O5_1—Bi1_1^viii	2.562 (6)
Ti1_2—Bi1_1	3.480 (3)	O5_1—Ti2_2^xxxii	1.969 (3)
Ti1_2—Bi1_1^xi	3.480 (3)	O5_1—Ti2_2^vii	1.969 (3)
Ti1_2—Bi1_1^viii	3.480 (3)	O5_2—Bi1_2^v	2.468 (4)
Ti1_2—Bi1_1^iv	3.480 (3)	O5_2—Bi1_2^vii	2.468 (4)
Ti1_2—Bi1_1^v	3.480 (3)	O5_2—Ti2_1	1.980 (5)
Ti1_2—Bi1_1^xiv	3.480 (3)	O5_2—Ti2_1^xxxv	1.955 (5)
O5_1^xi—Bi1_1—O5_1^xiii	96.3 (3)	O3_1^xxxi—Ti2_1—O4_1	180.0
O5_2^v—Bi1_2—O5_2^xiv	67.12 (12)	O3_1^xxxi—Ti2_1—O5_2	78.7 (3)
O5_2^v—Bi1_2—O5_2^xx	102.9 (2)	O3_1^xxxi—Ti2_1—O5_2^viii	78.6 (3)
O5_2^v—Bi1_2—O5_2^xxi	67.12 (12)	O3_1^xxxi—Ti2_1—O5_2^xxix	78.7 (3)
O5_2^xiv—Bi1_2—O5_2^xx	67.12 (12)	O3_1^xxxi—Ti2_1—O5_2^ix	78.6 (3)
O5_2^xiv—Bi1_2—O5_2^xxi	102.9 (2)	O4_1—Ti2_1—O5_2	101.3 (3)
O5_2^xx—Bi1_2—O5_2^xxi	67.12 (12)	O4_1—Ti2_1—O5_2^viii	101.5 (3)
O2_1^xxii—Bi2_1—O2_1^xxiii	72.64 (19)	O4_1—Ti2_1—O5_2^xxix	101.3 (3)
O2_1^xxii—Bi2_1—O2_1^xxiv	113.8 (4)	O4_1—Ti2_1—O5_2^ix	101.5 (3)
O2_1^xxii—Bi2_1—O2_1^xxv	72.64 (19)	O5_2—Ti2_1—O5_2^viii	87.77 (12)
O2_1^xxiii—Bi2_1—O2_1^xxiv	72.64 (19)	O5_2—Ti2_1—O5_2^xxix	157.4 (6)
O2_1^xxiii—Bi2_1—O2_1^xxv	113.7 (4)	O5_2—Ti2_1—O5_2^ix	87.77 (12)
O2_1^xxiv—Bi2_1—O2_1^xxv	72.64 (19)	O5_2^viii—Ti2_1—O5_2^xxix	87.77 (12)
O2_1^x—Bi2_2—O2_1^xxvi	71.01 (19)	O5_2^viii—Ti2_1—O5_2^ix	157.1 (6)
O2_1^x—Bi2_2—O2_1^xxvii	110.4 (4)	O5_2^xxix—Ti2_1—O5_2^ix	87.77 (12)
O2_1^x—Bi2_2—O2_1^xiii	71.01 (19)	O3_2—Ti2_2—O4_2	180.0
O2_1^xxvi—Bi2_2—O2_1^xxvii	71.01 (19)	O3_2—Ti2_2—O5_1^xxxii	78.5 (4)
O2_1^xxvi—Bi2_2—O2_1^xiii	110.4 (4)	O3_2—Ti2_2—O5_1^xxiii	78.5 (4)
O2_1^xxvii—Bi2_2—O2_1^xiii	71.01 (19)	O3_2—Ti2_2—O5_1^xxiv	78.5 (4)
O1_1^x—Ti1_1—O1_1^xi	90.0	O3_2—Ti2_2—O5_1^xxxiii	78.5 (4)
O1_1^x—Ti1_1—O1_1^xii	180.0	O4_2—Ti2_2—O5_1^xxxii	101.5 (4)
O1_1^x—Ti1_1—O1_1^xiii	90.0	O4_2—Ti2_2—O5_1^xxiii	101.5 (4)
O1_1^x—Ti1_1—O3_1	90.0	O4_2—Ti2_2—O5_1^xxiv	101.5 (4)
O1_1^x—Ti1_1—O3_1^xvii	90.0	O4_2—Ti2_2—O5_1^xxxiii	101.5 (4)
O1_1^xi—Ti1_1—O1_1^xii	90.0	O5_1^xxxii—Ti2_2—O5_1^xxiii	87.71 (14)
O1_1^xi—Ti1_1—O1_1^xiii	180.0	O5_1^xxxii—Ti2_2—O5_1^xxiv	156.9 (7)
O1_1^xi—Ti1_1—O3_1	90.0	O5_1^xxxii—Ti2_2—O5_1^xxxiii	87.71 (14)
O1_1^xi—Ti1_1—O3_1^xvii	90.0	O5_1^xxiii—Ti2_2—O5_1^xxiv	87.71 (14)
O1_1^xii—Ti1_1—O1_1^xiii	90.0	O5_1^xxiii—Ti2_2—O5_1^xxxiii	156.9 (7)
O1_1^xii—Ti1_1—O3_1	90.0	O5_1^xxiv—Ti2_2—O5_1^xxxiii	87.71 (14)
O1_1^xii—Ti1_1—O3_1^xvii	90.0	Ti1_1^xxxiv—O1_1—Ti1_1^viii	180.0
O1_1^xiii—Ti1_1—O3_1	90.0	Ti1_2^i—O1_2—Ti1_2^iii	151.5 (3)
O1_1^xiii—Ti1_1—O3_1^xvii	90.0	Bi2_1^xxii—O2_1—Bi2_1^xv	113.8 (4)
O3_1—Ti1_1—O3_1^xvii	180.0	Bi2_1^xxii—O2_1—Bi2_2^xxxiv	108.09 (11)
O1_2^xxviii—Ti1_2—O1_2^viii	90.0	Bi2_1^xxii—O2_1—Bi2_2^iii	108.09 (11)
O1_2^xxviii—Ti1_2—O1_2^xxix	180.0	Bi2_1^xv—O2_1—Bi2_2^xxxiv	108.23 (11)
O1_2^xxviii—Ti1_2—O1_2^xxx	90.0	Bi2_1^xv—O2_1—Bi2_2^iii	108.23 (11)
O1_2^xxviii—Ti1_2—O3_2	90.0	Bi2_2^xxxiv—O2_1—Bi2_2^iii	110.4 (4)
O1_2^xxviii—Ti1_2—O3_2^iv	90.0	Ti1_1—O3_1—Ti2_1^xxxviii	180.0
O1_2^viii—Ti1_2—O1_2^xxix	90.0	Ti1_2—O3_2—Ti2_2	180.0
O1_2^viii—Ti1_2—O1_2^xxx	180.0	Bi1_1^viii—O5_1—Ti2_2^xxxii	98.1 (3)
O1_2^viii—Ti1_2—O3_2	90.0	Bi1_1^viii—O5_1—Ti2_2^xxxix	98.1 (3)
O1_2^viii—Ti1_2—O3_2^iv	90.0	Ti2_2^xxxii—O5_1—Ti2_2^xxxix	156.9 (7)
O1_2^xxix—Ti1_2—O1_2^xxx	90.0	Bi1_2^v—O5_2—Bi1_2^vii	102.8 (2)
O1_2^xxix—Ti1_2—O3_2	90.0	Bi1_2^v—O5_2—Ti2_1	96.7 (2)
O1_2^xxix—Ti1_2—O3_2^iv	90.0	Bi1_2^v—O5_2—Ti2_1^xxxv	97.4 (2)
O1_2^xxx—Ti1_2—O3_2	90.0	Bi1_2^xxxix—O5_2—Ti2_1	96.7 (2)
O1_2^xxx—Ti1_2—O3_2^iv	90.0	Bi1_2^xxxix—O5_2—Ti2_1^xxxv	97.4 (2)
O3_2—Ti1_2—O3_2^iv	180.0	Ti2_1—O5_2—Ti2_1^xxxv	157.2 (6)

Symmetry codes: (i) x, y+1, z; (ii) −x−1/2, y+1/2, z; (iii) −x+1/2, y+1/2, z; (iv) −x, −y, −z+1; (v) −x, −y+1, −z+1; (vi) x−1/2, −y+1/2, −z+1; (vii) x+1/2, −y+1/2, −z+1; (viii) −y+1, x, z; (ix) y−1, −x+1, z; (x) x−1, y, z; (xi) −y, x, z; (xii) −x+1, −y+1, z; (xiii) y, −x+1, z; (xiv) y−1, −x, −z+1; (xv) y, −x, −z+1; (xvi) −x, −y, −z; (xvii) −x, −y+1, −z; (xviii) y−1, −x, −z; (xix) y, −x, −z; (xx) x, y−1, −z+1; (xxi) −y+1, x, −z+1; (xxii) −x+1, −y+1, −z+1; (xxiii) y, −x+1, −z+1; (xxiv) x−1, y, −z+1; (xxv) −y, x, −z+1; (xxvi) −y, x−1, z; (xxvii) −x+1, −y, z; (xxviii) x, y−1, z; (xxix) −x, −y+1, z; (xxx) y−1, −x, z; (xxxi) x, y, z+1; (xxxii) −x+1, −y, −z+1; (xxxiii) −y, x−1, −z+1; (xxxiv) x+1, y, z; (xxxv) −y+1, x+1, z; (xxxvi) x−1/2, −y+1/2, −z; (xxxvii) x+1/2, −y+1/2, −z; (xxxviii) x, y, z−1; (xxxix) x+3/2, −y+3/2, −z+1.

(BI4TI3O12_P4NBM_phase_1)

Crystal data

Bi4O12Ti3	c = 33.2476 (5) Å
Mr = 1171.61	V = 989.81 (4) Å^3
Tetragonal, P4/nbm(originch	Z = 4
a = 5.45626 (7) Å

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
Bi1_1	0.25	0.25	0.57190 (14)	0.0612 (17)*
Bi1_2	0.75	0.25	0.06522 (13)	0.0339 (12)*
Bi2_1	0.25	0.25	0.71133 (14)	0.0488 (17)*
Bi2_2	0.75	0.25	0.21122 (13)	0.0318 (13)*
Ti1_1	0.25	0.25	0.0	0.000 (3)*
Ti1_2	0.75	0.25	0.5	0.106 (8)*
Ti2_1	0.25	0.25	0.8683 (3)	0.036 (3)*
Ti2_2	0.75	0.25	0.37207 (19)	−0.0109 (14)*
O1_1	0.0	0.0	0.0	0.118 (4)*
O1_2	0.0	0.0	0.5	0.058 (3)*
O2_1	0.0030 (6)	−0.0030 (6)	0.25145 (18)	0.0251 (7)*
O3_1	0.25	0.25	−0.0587 (3)	0.078 (3)*
O3_2	0.75	0.25	0.4417 (2)	0.049 (3)*
O4_1	0.25	0.25	0.8179 (2)	0.050 (3)*
O4_2	0.75	0.25	0.3178 (2)	0.049 (3)*
O5_1	−0.0022 (5)	0.0022 (5)	0.61525 (14)	0.0311 (13)*
O5_2	0.5321 (4)	0.4679 (4)	0.88022 (14)	0.0266 (13)*

(BI4TI3O12_P42NCM_phase_1)

Crystal data

Bi4O12Ti3	c = 33.2479 (5) Å
Mr = 1171.61	V = 989.81 (4) Å^3
Tetragonal, P42/ncm(originc	Z = 4
a = 5.45625 (7) Å

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
Bi1_1	−0.0041 (5)	−0.0040 (5)	0.56844 (6)	0.0531 (6)*
Bi2_1	0.0043 (5)	0.0043 (5)	0.71123 (4)	0.0408 (7)*
Ti1_1	0.0	0.0	0.0	0.0337 (16)*
Ti2_1	−0.0026 (9)	−0.0026 (9)	0.87062 (8)	0.0176 (9)*
O1_1	0.25	0.25	−0.0003 (4)	0.102 (5)*
O1_2	0.75	0.25	0.0	0.063 (3)*
O2_1	0.25	0.25	0.25441 (17)	0.0000 (14)*
O2_2	0.75	0.25	0.75	0.054 (3)*
O3_1	−0.0087 (9)	−0.0087 (9)	−0.05907 (10)	0.0686 (10)*
O4_1	0.0330 (5)	0.0330 (5)	0.81801 (7)	0.0286 (9)*
O5_1	0.25	0.25	0.6213 (3)	0.036 (3)*
O5_2	0.75	0.25	0.1172 (2)	0.0229 (15)*
O5_3	0.25	0.25	0.3862 (3)	0.058 (3)*