research papers
Monoclinic distortion, polarization rotation and _{0.5}Bi_{0.5}TiO_{3}
in the ferroelectric Na^{a}Department of Physics, University of Siegen, WalterFlex Strasse 3, Siegen 57072, Germany, ^{b}Institute of Electromechanical Design, Technische Universität Darmstadt, Darmstadt, Germany, ^{c}Electronic Materials Research Laboratory, Key Laboratory of the Ministry of Education and International Center for Dielectric Research, Xian Jaotong University, Xian, People's Republic of China, ^{d}Physics Department, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, England, ^{e}Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, England, and ^{f}Materials Science and Engineering, Tel Aviv University, Wolfson Building for Mechanical Engineering, Tel Aviv 6997801, Israel
^{*}Correspondence email: gorfman@tauex.tau.ac.il
The relationship between _{0.5}Bi_{0.5}TiO_{3} (NBT) has been of interest for the last two decades. Originally, the average structure was held to be of rhombohedral (R3c) symmetry with a fixed polarization direction. This has undergone a series of revisions, however, based on highresolution Xray diffraction, total neutron scattering, and optical and The recent experimental findings suggest that the true average symmetry is monoclinic (space group Cc), which allows for a rotatable spontaneous polarization. Neither polarization rotation nor its potentially important real role in enhanced is well understood. The present work describes an in situ investigation of the average monoclinic distortion in NBT by timeresolved singlecrystal Xray diffraction under external electric fields. The study presents a highresolution inspection of the characteristic diffraction features of the monoclinic distortion – splitting of specific Bragg reflections – and their changes under a cyclic electric field. The results favour a model in which there is direct coupling between the shear monoclinic strain and the polarization rotation. This suggests that the angle of polarization rotation under a subcoercive electric field could be 30° or more.
and physical properties in the ferroelectric Na1. Introduction
Na_{0.5}Bi_{0.5}TiO_{3} (NBT), a perovskitebased ferroelectric, has been the focus of attention for over two decades (Vakhrushev et al., 1985; Roleder et al., 2002; McQuade & Dolgos, 2016). NBT is of interest because of its potentially important role as an end member of many leadfree substitutes to replace the commercially dominant PbZr_{1−x}Ti_{x}O_{3} piezoelectric (Takenaka et al., 1991, 2008; Shrout & Zhang, 2007; Panda, 2009; Rödel et al., 2009). NBT is also an interesting model system in the crystallography of distorted perovskites. Phase transitions in NBT are realized through symmetrylowering shifts of the A/B cations and tilting of the TiO_{6} octahedra. This symmetrylowering results in the formation of domains that are spontaneously polarized, electromechanically active and switchable by an external electric field. Although all these phenomena are ubiquitous in many other perovskitebased materials (Mitchell, 2003), NBT is one of the most complex and a number of unresolved controversies remain. The structure–property relationships of NBT continue to be the subject of debate and prompt continued research work on this unusual material.
The commonly accepted crystallographic reference for NBT comes from the neutron powder diffraction work of Jones & Thomas (2002). This work reported a transformation from an average rhombohedral (R3c) to an average tetragonal (P4bm) phase at ∼580 K, following the reorientation of the spontaneous polarization from the bodydiagonal to the celledge direction (the pseudocubic cell setting with the lattice parameter a ≃ 3.9 Å is used for lattice directions and reflection indices throughout this paper.). Most surprisingly, the TiO_{6} octahedra undergo a change in their tilt system from a^{} a^{} a^{} to a^{0} a^{0} c^{+} (according to Glazer's notation system; Glazer, 1972). The absence of a displacive path for such a is expressed by the lack of any group–subgroup relationship between the R3c and P4bm space groups. In some respects, this thermally driven R–T phase transformation resembles the compositionally driven one in PbZr_{1−x}Ti_{x}O_{3} (PZT) at the morphotropic phase boundary (MPB, x ≃ 0.48; Jaffe et al., 1954, 1971). In PZT, the Zrrich side has been recognized as monoclinic rather than rhombohedral close to the MPB, although recent work has shown that the structure is extremely complex, with mixing of disordered and ordered monoclinic regions (Noheda et al., 1999; Yokota et al., 2009; Zhang et al., 2011, 2014, 2018; Gorfman et al., 2011).
Gorfman & Thomas (2010) have reported a highresolution Xray diffraction study of multidomain NBT `single' crystals. They demonstrated that the angular separation between the different components of the {hkl}^{1} Bragg reflections (each component is diffracted from a similarly oriented set of ferroelastic domains) violates rhombohedral R3c symmetry and favours monoclinic Cc average symmetry instead. This average monoclinic symmetry of roomtemperature NBT has since been confirmed by many other authors (Aksel et al., 2011; Ma et al., 2013; Gorfman et al., 2012; Kitanaka et al., 2014; Levin & Reaney, 2012). The most important implication of monoclinic symmetry is that it allows the polarization to rotate (Vanderbilt & Cohen, 2001) and thus leads to greater susceptibility to an external electric perturbation (Fu & Cohen, 2000). The direct effect of the polarization rotation on the lattice parameters might be the reason for the strong electromechanical coupling in both NBT and PZT. For the case of NBT, the true nature of the monoclinic symmetry remains controversial, especially as the average structure is reported to be different from the local one (Aksel et al., 2013). Neutron scattering studies of pairdistribution function in NBT by Keeble et al. (2013) support the local monoclinic symmetry of the Bi positions, where Bi atoms may displace in two different `monoclinic' directions. At the same time, extended Xray absorption fine structural studies by Rao et al. (2016) show that the of the Bi sites is rhombohedral (consistent with R3c), while the apparent monoclinic symmetry averages out the combination of different orientation variants of unit cells of rhombohedral symmetry. Regardless of whether the apparent monoclinic distortion has a true localscale origin or results from the averaging, attempts to test the susceptibility of the monoclinic phase `in action' (dynamically under an electric field) are still rare and mainly limited to the use of static electric fields (Ogino et al., 2014; Kitanaka et al., 2014).
The aim of this work is to investigate the apparent monoclinic distortion in NBT under an alternating subcoercive (<14 kV cm^{−1}) external electric field, test the polarization rotation and clarify if this rotation can give rise to high We have implemented a stroboscopic dataacquisition system and highresolution Xray diffractometer (beamline P08 at the PETRA III storage ring) to collect reciprocalspace maps (RSMs) around the family of most representative Bragg reflections (Gorfman & Thomas, 2010). We have observed that the monoclinic splitting is indeed strongly sensitive to the external electric field: electricfieldinduced shifts of the peaks amount to a piezoelectric effect of as much as 124 pC N^{−1}. The positions of the Bragg peaks in are consistent with the existence of 12 monoclinic domains, in which the polarization vector rotates in one of the 12 monoclinic {110} mirror planes. Most importantly, we report that the average shear lattice strain is nonlinear with electric field and this nonlinearity can be well accounted for by the polarization rotation, with the maximum angle of polarization rotation reaching 35°.
2. Experimental details
The NBT single crystal was grown at the Shanghai Institute of Ceramics by the topseeded solutiongrowth method (as described by Ge et al., 2008) and doped with Mn. The crystal was cut to a 0.5 mm thick plate with the surface parallel to (001) and the edges along the [110] and [100] crystallographic directions. Thin (∼100 nm) gold electrodes were sputtered onto the faces to apply the electric field along [001]. We designed a sample stage, which serially connects the electrodes with a highvoltage supply via a 1 kΩ activeprobe monitor of the capacitive current. The current and polarization hysteresis loops were monitored continuously during the measurement.
Fig. 1 shows the experimental equipment on the P08 highresolution fourcircle diffractometer at the PETRA III storage ring. The arbitrary function generator (HMF2550, Hameg) and highvoltage amplifier (AMT3B20, Matsusada) produce a triangularshaped AC highvoltage signal/electric field with an amplitude of 14 kV cm^{−1}. This field is significantly smaller than the coercive field of ∼45 kV cm^{−1} reported for an Mndoped NBT single crystal (Ge et al., 2010). We used an (APD) singlephoton counting detector and Si(111) analyser crystal and measured the scattering intensity as a function of ω (rocking) and 2θ (scattering) angles around the [004]* position of The output of the APD detector was introduced directly into a custombuilt stroboscopic dataacquisition system (Gorfman et al., 2010, 2013; Gorfman, 2014; Choe et al., 2015, 2017). The system implements the working principle of a multichannel analyser: it assigns detector counts to one of 10 000 time channels, where each channel has a fixed time delay to the beginning of latest highvoltage cycle. Each point in the RSM was collected for 10 s = 1000 electricfield cycles. The frequency of the applied electric field was 100 Hz and the time resolution (channel width) was 1 µs. The Xray energy was set to 15.1 keV (λ = 0.827 Å), just below the `Bi' L_{2} giving an average for the measured reflection of 〈t〉 = sinθ/2μ = 5.3 µm. This is ∼2.5 times deeper than the which was previously used in the experiment of Gorfman & Thomas (2010).
3. Results
Fig. 2(a) reproduces an ω versus 2θ RSM of one of the {002} reflections from our previous studies (Gorfman & Thomas, 2010) (measured using a homelaboratory highresolution PANalytical MRD diffractometer). This RSM contains two Bragg peaks, separated along the 2θ axis and diffracted from two families of ferroelastic domains. Different 2θ angles mean different lengths of the corresponding reciprocallattice vectors [H = (2sinθ)/λ]. Accordingly, such splitting violates the rhombohedral symmetry of the domains, which would constrain the average pseudocubic lattice parameters to be a = b = c and α = β = γ. The work of Gorfman & Thomas (2010) reports on the detailed analysis of such splitting in many other families of reflections, which includes 41 different RSMs. The results of this work clearly suggested that the above constraint must be lifted to a = b c and α = β γ, corresponding to an average monoclinic lattice with a mirror plane  to ^{2}. The structure of roomtemperature NBT must therefore be described by the monoclinic Cc which is a of rhombohedral R3c.
In the present work, we selected the most representative set of Bragg reflections to measure the field dependence of the monoclinic distortion. Figs. 2(c)–2(e) display stroboscopically collected RSMs of the {004} reflection (measured on the P08 beamline at PETRA III), corresponding to three different time channels or electric field states, marked by the circles in Fig. 2(b). Each RSM contained 5928 intensity values on the mesh of 78 × 76 points along the ω and 2θ directions, respectively. An animated set of 250 RSMs (after binning of every 40 channels to improve the counting statistics) is available in the supporting information. The varying separation of peaks on the 2θ axis suggests that the monoclinic distortion is field dependent. Fig. 3(a) shows three RSMs of an {004} reflection in Cartesian coordinates, with the horizontal axis X  [110]* and the vertical axis Y  [001]*, where X and Y are the coordinates of the scattering vector. Both the X and Y axes on these maps lie in the diffraction plane and the Y axis is parallel to the scattering vector. This Y axis corresponds to the Δ(2θ) = 2(Δω) dashed line in Fig. 2(d). The open slit of the detector is perpendicular to the diffraction plane, thus giving rise to automatic intensity integration along the Z direction.
We have fitted these RSMs using the superposition of two Moffat twodimensional distribution functions, each of which has adjustable parameters: the positions of the peaks (x_{0}, y_{0}), full widths at half maxima (σ_{x}, σ_{y}), integrated intensities I and shape parameter β. This means we used 12 model parameters to describe the intensity distribution over 5928 points on each RSM. The details are given in the supporting information. Fig. 3(b) shows three RSMs, calculated using the bestfit values of the parameters. The graphs in Figs. 3(c) and 3(d) then cut along the X and Y directions, clearly showing the Y separation of the peak components. An animated version of this figure (in the supporting information) shows that the split peaks have significantly different time and electricfield dependencies.
Finally, Fig. 4 shows the field dependence of the key model parameters for both contributing Bragg peaks. These key parameters are the peak positions (Figs. 4a and 4b) and (Figs. 4c and 4d) along Y and X, respectively. In the following we will introduce this model, which will help us to calculate the monoclinic distortion parameters as a function of electric field.
4. Modelling of the field dependence of RSMs
4.1. Monoclinic distortion, polarization rotation and monoclinic domains
We now discuss whether the measured changes in the peak positions and widths may be explained by a model of electricfielddependent monoclinic distortion and polarization rotation. Fig. 5 shows how the pseudocubic is distorted after a transition from a cubic, (a = b = c and α = β = γ = 90°), to a monoclinic, Cc (a = b c and α = β γ), structure. The figure shows the orientation of the pseudocubic basis vectors a_{1}, a_{2}, a_{3} in a monoclinic domain relative to the Cartesian reference frame e_{1}, e_{2}, e_{3}, aligned with the edges of the cubic The monoclinic distortion can be modelled using four free parameters, c, a, ψ and ξ. Here, c and a are the unitcell lengths in and out of the monoclinic mirror plane, respectively, and ψ and ξ are the shearing angles of the as shown in Fig. 5(a). We also assume that all these free parameters can be expressed as a function of the polarization rotation angle ρ, the angle between the monoclinic P_{M} and rhombohedral P_{R} polarization directions. The positive and negative polarization rotation angles correspond to the monoclinic M_{A} (ρ 0) and M_{B} (ρ 0) phases, respectively (Vanderbilt & Cohen, 2001; Zhang et al., 2014). The loss of the threefold rotational symmetry results in the formation of monoclinic domains, in which the polarization vector rotates in a plane between the unitcell bodydiagonal directions towards one of the three edges (Fig. 5b). Therefore, a maximum of 24 monoclinic domains can be created in which the polarization rotation angle, ρ, is measured from one of the eight 〈111〉 bodydiagonal directions. Because the crystal response is strongly asymmetric with respect to the electric field direction, we must assume that e.g. the previous poling history of the sample kept only four rhombohedral domains with the polarization close to the `rhombohedral' [111], , and directions, resulting in the formation of only 12 monoclinic domains. We mark these domains as M_{mn}, where m = 1, 2, 3, 4 correspond to P_{R}  [111], P_{R}  , P_{R}  and P_{R}  , respectively, n = 1 corresponds to the domains where the electric field lies in the polarization rotation plane (see Fig. 5b) and n = 2, 3 correspond to the domains where the electric field is directed out of the polarization rotation plane.
4.2. Modelling of the positions of the diffraction peaks
We use the model and definitions above to simulate the positions in the {004} family of reflections, each diffracted from one of the 12 monoclinic domains. To do this we introduce the orientation matrix of a monoclinic domain (Fig. 5a),
The columns of this matrix are the coordinates of the vectors a_{i} in the Cartesian coordinate system e_{i}. The reciprocal orientation matrix [U_{B}] (the columns of which are the coordinates of the reciprocal basis vectors , such that ) can be obtained as
and therefore
where
are also the functions of the polarization rotation angle, A(ρ), B(ρ), C(ρ) and D(ρ). Transforming the coordinates of the vectors into the laboratory coordinate system X, Y, Z of Fig. 3 (here X  e_{1} + e_{2}, Y  e_{3} and Z  e_{1} − e_{2}) is done using the matrix equation [U_{B}]_{XYZ} = [XYZ] · [U_{B}] with
The functional form of the orientation matrices of all other monoclinic domains can be calculated using
where the columns of the T^{(mn)}] (see Table 1) are the coordinates of the Cartesian cubic axes of the domains M_{mn} in the e_{i} coordinate system. Finally, the positions of the Bragg peak diffracted from domain mn are described by the first two components (X and Y) of the third column of . Table 1 summarizes the matrices for all 12 monoclinic domains and the form of the reciprocal orientation matrices. The arrows indicate the polarization rotation direction induced by an [001]oriented electric field.
matrices [

Symmetry dictates that the polarization rotation angles are the same for all monoclinic domains from groups M_{m1} (ρ = ρ_{1}) and M_{m2}/M_{m3} (ρ = ρ_{2}). The following notation is introduced in Table 1:
Fig. 6 represents the righthand column of Table 1 in the form of a schematic drawing of the positions of the Bragg reflection, diffracted from all 12 monoclinic domains. Note that the separation of the Bragg peaks along the X axis might be significantly smaller than the peak width (arising from e.g. crystal mosaicity, later defined as W_{004}^{(0)} and W_{400}^{(0)}), and therefore cannot be seen in Fig. 3 directly. Instead, this separation can be extracted from the fielddependent peak broadening, displayed in Fig. 4(b). Following equations (7) and (8) and the scheme in Fig. 6, the broadening can be simulated as
We further assume that the monoclinic distortion angles ψ and ξ are so small that we can replace all the trigonometric functions above by the corresponding firstorder Taylor expansions. This brings us to the following expressions,
Here, the Δ sign stands for the difference between e.g. a zerostate value and a nonzerofield value. The equations allow for the direct evaluation of the monoclinic distortion parameters Δc_{1} = Δc(Δρ_{1}) and Δξ_{1} = Δξ(Δρ_{1}) (domains M_{1n}), and Δa_{2} = Δa(Δρ_{2}) and Δψ_{2} = Δψ(Δρ_{2}) (domains M_{2n} or M_{3n}).
4.3. Nonlinear electric field response and the model of polarization rotation
According to equations (10) and (11), the monoclinic distortion parameters Δξ_{1}, Δψ_{2} and Δa_{2} have the same field dependencies as the peakshape parameters W_{004}, W_{400} and P_{400}, respectively. Therefore (Fig. 4), all the derived monoclinic distortions (except for Δc_{1}) are essentially nonlinear with respect to the magnitude of the electric field.
In the following we address the question of whether this nonlinearity can be accounted for by polarization rotation. More specifically, we will test if the monoclinic strains can be described as linear functions of the polarization rotation angle ρ, rather than of the magnitude of the electric field E, so that Δξ_{1} = F_{ξ}Δρ_{1}, Δψ_{2} = F_{ψ}Δρ_{2} and Δa_{2} = F_{a}Δρ_{2}. To derive the ρ_{1}(E) and ρ_{2}(E) dependence, we shall assume that the free energy ΔG (Devonshire, 1954) has a quadratic dependence on ρ with its minimum at ρ = 0, so that the total free energy (including the term describing the interaction of electric field and spontaneous polarization) is
where G_{0} is the energy expansion coefficient, P is the length of the polarization vector and χ is the angle between the polarization and electric field directions, as marked in Fig. 5(b):
where ρ_{R} = arccos(1/3^{1/2}) ≃ 54.57° is the angle between the cube edges and the body diagonal. Substituting equations (13) and (14) into (12) and locating the position of the global minimum by equating gives the polarization rotation angles Δρ in the domains M_{n1} and M_{n2}/M_{n3}:
with W = G_{0}/2P. Using our assumption that the change in the monoclinic distortion parameters is linear with respect to polarization rotation, we rewrite equations (15) and (16) as
Equations (17)–(19) can be solved numerically for any given electric field magnitude E, so that the unknown model parameters W, F_{ξ}, F_{ψ} and F_{a} can be found from the best fit to the experimental values. The solutions are shown in Figs. 7(a)–7(c), where both observed [from equations (10) and (11)] and simulated [according to equations (17)–(19)] monoclinic distortion parameters Δξ_{1}, Δψ_{2} and Δa_{2} are displayed. These figures show that our simplified model above is highly effective in accounting for the nonlinear dependence of all three nonlinear monoclinic distortion parameters. This good match between observed and simulated monoclinic distortion parameters points strongly to a close connection between electricfieldinduced polarization rotation and lattice strain, clearly suggesting that the corresponding piezoelectric effects are principally intrinsic rather than extrinsic in origin. Fig. 7(d), however, represents the Δρ_{1} = F_{ξ}Δξ_{1} and Δρ_{2} = F_{ψ}Δψ_{2} polarization rotation angles, showing that this nonlinearity must assume rotation of the polarization vector by an angle larger than 35°.
5. Lowfield piezoelectric coefficients of a single monoclinic domain
The intrinsic lowfield piezoelectric coefficients of a single monoclinic domain can be calculated from the experimental results as d_{ijk} = (E = 0), where x_{jk} and E_{i} are the components of the strain tensor and electric field vector, respectively, in the domainrelated Cartesian coordinate system . The strain tensor for the monoclinic distortion (Fig. 5a) is:
The piezoelectric coefficients d_{3jk} describe the strain in response to the electric field applied along (parallel to the polarization rotation plane). Such orientations are realized in the M_{m1} domains, so that the field dependence of the monoclinic distortion parameters Δc_{1} and Δξ_{1} can be used to calculate the piezoelectric coefficients d_{333} and d_{323}. Similarly, the first and second rows in the tensor representation of the piezoelectric coefficients d_{1jk} = d_{2jk} describe the strain in response to the electric field being parallel to the and axes (out of the polarization rotation plane). Such orientations of the electric field are realized in the monoclinic domains M_{m2} and M_{m3}, respectively. Therefore, the field dependence of the monoclinic distortion parameters Δψ_{2} and Δa_{2} gives the piezoelectric coefficients d_{112} and d_{111}, respectively:
The numerical values of the corresponding piezoelectric constants are d_{333} = 124.1 pC N^{−1}, d_{323} = 20.36 pC N^{−1}, d_{112} = −43.93 pC N^{−1} and d_{111} = 4.29 pC N^{−1}. The field dependence of the strain suggests that the lowfield piezoelectric coefficients, d_{323}, d_{112} and d_{111} are associated with the polarization rotation.
6. Discussion
The macroscopic piezoelectric coefficients of NBT materials (ceramics and single crystals) range between 20 and 100 pC N^{−1} (Ge et al., 2010; Foronda et al., 2014; Hiruma et al., 2010, 2009). These have the same order of magnitude as the values in the previous paragraph. Therefore, our results suggest that electromechanical coupling in NBT is predominantly intrinsic. The intrinsic character of the electromechanical coupling is seen in the shifts of the angular positions in the {004} family of twin reflections. We have also argued that some components of the strain can be explained straightforwardly by polarization rotation. This suggestion follows from the nonlinear dependence of the monoclinic parameters (Δξ_{1}, Δψ_{2} and Δa_{2}) on the electric field. We must stress, however, that using a polarization rotation argument to explain this dependence produces an unexpectedly large amplitude for the polarization rotation angle: 25° between [111] and [001] (M_{A} phase) and up to 35° between [111] and [110] (M_{B} phase). This would mean that the polarization rotation represents such a `soft mode of structural changes' that even a subcoercive electric loading of 14 kV cm^{−1} can change it greatly. Can a polarization vector really rotate so much in a ferroelectric crystal? Our preliminary analysis favours a positive answer to this question and might lead to the suggestion of different origins for such extreme `structural softness'.
Firstly, the large polarization rotation may originate from the intricate structural flexibility of perovskites, where a `soft' displacement of A/B cations from the centres of the corresponding oxygen AO_{12}/BO_{6} cages occurs. This explanation assumes that the above displacements are longrange ordered and that the polarization rotates through coherent changes of atomic position in every of the crystal. Such coherent structural changes would appear as a change in the `Bragg intensities' and could be analysed using the standard formalism (see e.g. Gorfman et al., 2016, 2013, 2006; Tsirelson et al., 2003; Schmidt et al., 2009). Indeed, we observe changes in the integrated intensity for both 400 and 004 Bragg peaks by ∼10% (see Fig. S1 in the supporting information) but we have not been able to measure enough Bragg reflection intensities to model the structural changes within the Cc space group.
Secondly, the large polarization rotation may originate from variations in the local structure and shortrange order parameters. In this model, by contrast with the first, the displacement of the atoms varies from one et al., 2003; Thomas et al., 2010; Gorfman et al., 2015) or total neutron scattering (Keeble et al., 2013). A reverse Monte Carlo simulation of the atomic pairdistribution function in NBT (Keeble et al., 2013) demonstrated that the Bi atoms in the monoclinic {110} planes do indeed differ and involve two coexisting directions of bismuth displacements. From this starting point, it follows that one might suggest that the application of an external electric field in a particular direction switches a subpopulation of atoms, thus changing the distribution of Bi atoms over two metastable states. Such a redistribution would produce a change in the average direction of the polarization vector relatively easily.
to another, so that the polarization can only be defined on average. The strong structural disorder in NBT has been documented by the observation of diffuse Xray scattering (KreiselThirdly, the large polarization rotation in NBT is strongly reminiscent of that in the compositionally driven polarization rotation in PbZr_{1−x}Ti_{x}O_{3} at the morphotropic phase boundary. The work of Zhang et al. (2014) shows that even a minor change in the composition x near the morphotropic phase boundary leads to the rotation of the average direction of the Pb displacement vector by a large angle of well beyond 35°. This large polarization rotation is commonly considered as one of the origins of enhanced piezoactivity at this particular boundary in the phase diagram.
We finally note the ongoing discussion of the true structural origin of the monoclinic phase in NBT. One suggestion is that the monoclinic symmetry can be mimicked by an adaptive phase mechanism (Jin et al., 2003; Wang, 2007), which is a microstructural material state made of periodically arranged nanodomains. Here the effective polarization rotation can be driven by the dynamics in the hierarchical nanodomain pattern, where each domain would have rhombohedral symmetry. The polarization direction of such an assembly is given by the volume average of polarization in the individual sets of nanodomains. Provided that the domains are sufficiently small, the Bragg diffraction pattern of such an adaptive structure would be indistinguishable from the equivalent Bragg diffraction pattern from a truly longrange monoclinic phase.
Finally, the demonstrated timeresolved reciprocalspace mapping approach has the potential to uncover the origins of electromechanical coupling in other ferroelectric perovskites.
Supporting information
Animation 1. DOI: https://doi.org//10.1107/S2052252518006784/fc5023sup1.mp4
Animation 2. DOI: https://doi.org//10.1107/S2052252518006784/fc5023sup2.mp4
Animation 3. DOI: https://doi.org//10.1107/S2052252518006784/fc5023sup3.mp4
Animation 4. DOI: https://doi.org//10.1107/S2052252518006784/fc5023sup4.mp4
Additional figure. DOI: https://doi.org//10.1107/S2052252518006784/fc5023sup5.pdf
Footnotes
^{1}{hkl}_{PC} denotes all possible reflections, obtained by applying the cubic m3m group to the reflection hkl. For cubic symmetry, the lengths of such reciprocalspace vectors would be the same. However, when the symmetry is distorted the lengths of the vectors vary, giving characteristic splitting of reflections for every crystal system.
^{2}These lattice parameters will look conventionally monoclinic (a_{m} ≠ b_{m} ≠ c_{m}, α_{m} = γ_{m} = 90°, β_{m} ≠ 90°) when the unitcell vectors are transformed as a_{m} = a + b, b_{m} = a − b and c_{m} = c. In this case the mirror plane becomes parallel to (010)_{m}
Acknowledgements
We acknowledge Dr Genziana BussoneGrifone and Dr Ute Rutt for their support of the P08 beamline staff at the PETRA III storage ring. We acknowledge Prof Haosu Luo (Shanghai Institute of Ceramics) for providing the single crystal of NBT.
Funding information
The following funding is acknowledged: Bundesministerium für Forschung und Technologie (award No. 05K13PSA to H. Choe and S. Gorfman). We are grateful for financial support provided by the EPSRC (Materials World Network: Nanoscale Structure–Property Relationships in LeadFree Morphotropic Phase Boundary Piezoelectrics).
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