# Diffuse Scattering from Lead-Containing Ferroelectric Perovskite Oxides

Authors: D. J. Goossens
Journal: ISRN Materials Science
 (2013)

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## Abstract

Ferroelectric materials rely on some type of non-centrosymmetric displacement correlations to give rise to a macroscopic polarisation. These displacements can show short-range order (SRO) that is reflective of the local chemistry, and so studying it reveals important information about how the structure gives rise to the technologically useful properties. A key means of exploring this SRO is diffuse scattering. Conventional structural studies use Bragg peak intensitiesto determine the average structure. In a single crystal diffuse scattering (SCDS) experiment, the coherent scattered intensity is measured at non-integer Miller indices, and can be used to examine the population of local configurations. This is because the diffuse scattering is sensitive to two-body averages, whereas the Bragg intensity gives single-body averages. This review outlines key results of SCDS studies on several materials and explores the similarities and differences in their diffuse scattering. Random strains are considered, as are models based on a phonon-like picture or a more local-chemistry oriented picture. Limitations of the technique are discussed.


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## Body

## 1. Introduction

Single crystal diffuse scattering (SCDS) has been the subject of study since the earliest days of crystallography [1] and is seen in many patterns collected using film (e.g., [2]), film being an early variant of “area detector” and therefore very good for surveying large regions of reciprocal space—far better than an electronic point counter. The different forms of disorder present in crystalline materials, and how they are manifested in diffraction patterns have been well explored, both in monographs and papers (these include [3–14]).Diffuse scattering can be defined for the purposes of this paper as the coherently scattered intensity that isnot localised on the reciprocal lattice. It is the result of the two-body correlations in a crystalline material (see e.g., [15, 16]). These correlations may exist between atoms, molecules, and, in the case of neutron diffraction, magnetic moments [17–20].It has long been known that diffuse scattering, an example is shown in Figure1, contains information about the correlations in atomic and molecular thermal motions [21] as well as static short-range order. Modelling diffuse scattering is not simple, because locally the short-range order, SRO, need not obey the space group symmetry of the crystal. The global symmetry must be regained on averaging across the crystal but may not be present on the local scale of a few nanometres, or if one prefers, on the scale of a few unit cells. This renders many of the tools of conventional crystallography of limited use.Figure 1
A single exposure from a stationary single crystal of wüstite,Fe1-xO (false colour map, white = high intensity). High-energy X-rays (65.35 keV in this case) give a small wavelength, λ, and an almost flat Ewald sphere. Hence these data closely approximate the hk0 section. Every Bragg reflection (and also the origin) is surrounded by a motif of incommensurate scattering. Second-order incommensurate spots have been arrowed and some Miller indices noted for scale [22].In particular, the idea of a single, repeating unit cell is not an adequate “solution” of the local structure—indeed, the very idea of what is meant by “structure solution” is thrown into question. Conventional crystallography solves (a better word may be “determines”) very important aspects of the structure—those describing its global average. Analysis of SCDS reveals another important aspect—the pair correlations, and thus some deeper information on the population of local environments, which will be closely related to the crystal chemistry. But even SCDS is limited to two-body correlations. An experiment that could measure the phase shift of the scattered radiation (X-ray or neutron, in most cases) would be sensitive to higher-order correlations, and so on. At what level of detail one could consider the structure to be “solved” is almost a philosophical question. In practice, the level of detail depends on what is needed to give insight into the properties of the material. The structure is “solved” when the questions being asked can be answered.Though not the subject of this review, it can be noted that diffuse scattering data collection is difficult.The weak diffuse scattering must be measured in the presence of strong Bragg reflections, and so the experiments require low noise, high dynamic range, and detectors with good resistance to saturation [32, 33]. Data may require considerable correction and reconstruction [34–37] before they reach a useful format. Dedicated instruments for diffuse scattering are relatively rare, for example DNS at Jülich [20], D7 at the Institut Laue Langevin [19], Corelli at the Spallation Neutron Source [38], a custom-made laboratory X-ray machine [39, 40], the E2 flat cone instrument at the Helmholtz-Zentrum Berlin für Materialien und Energie [41], and so on. However, many other instruments are capable of collecting high quality data, including the powder diffractometer at the Australian Synchrotron [42], SXD at the ISIS neutron source [27, 43–45], Wombat at the OPAL reactor [35, 46], the SNBL beamline at ESRF [47], 11-ID-B at the Advanced Photon Source [48, 49], and others [50–52].The correlation structure is likely to be anisotropic, such that a detailed model of the SRO will often require a three-dimensional (3D) dataset [23, 53–56], in turn implying the need to measure the distribution of weak and delocalised features throughout significant volumes of reciprocal space. This requires long experiments, very carefully characterised instruments, and large crystals—especially for neutron diffuse scattering. Even at a synchrotron a 3D dataset will take hours to collect, inhibiting studies as functions of temperature, pressure, and so on. If the crucial section of reciprocal space can be observed with a single crystal setting, such studies are practicable. This is perhaps made easier by using high X-ray energies that flatten the Ewald sphere such that a single exposure is almost flat in reciprocal space and can be made through sample orientation to coincide with some important symmetry directions. Hence a particular reciprocal space plane can, as long as it contains the origin, be captured in a single exposure if the crystal is appropriately aligned (Figure 1) [57, 58].The importance of diffuse scattering in examining structures in ferroelectrics has long been recognised [59, 60], and highly structured diffuse scattering has been observed in many ferroelectric materials, including those of commercial importance: KDP (KH2PO4) [61], TGS ((NH2CH2COOH)3H2SO4) [62], benzil (C14H10O2) [2, 43, 63, 64], BaTiO3 [59, 65], NBT (Na0.5Bi0.5TiO3) [66], and so on.Ferroelectric phenomena are inherently related to correlations amongst structural features, resulting as they do from electric dipoles due to ionic displacements [67]. As noted, it is well established that the diffraction images of BaTiO3, a much-studied ferroelectric material, contain diffuse scattering (DS) [59, 65]. The observation of the evolution of the diffuse patterns through the ferroelectric phase transition [68] leads to ongoing debate over the character of the phase transition—for example, order-disorder as against displacive—in this material [69]. DS planes were observed and considered to be the result of displacements correlated in chains, with the correlations being either static or dynamic.In the modern relaxor ferroelectric materials [70], DS has often been interpreted using the idea of domain-like polar nanoregions (PNRs) embedded in a paraelectric matrix, which are then linked to the outstanding electromechanical properties [71]. While the PNRs offer a convenient and somewhat intuitive model for the behaviour, the actual need for the concept is, according to some recent work, not thoroughly established [72]. It is also subtly different from the idea of ferroelectric nanodomains (FND) essentially “butting up” against each other—the main difference being the latter requires no paraelectric matrix.Many of the most significant ferroelectric materials rely on lead, Pb, as a key ingredient in their crystal chemistry: PZT(PbZr1-xTixO3) ([73–76] and others), PZN (PbZn1/3Nb2/3O3) and PZN-PT (PbZn1/3Nb2/3O3-PbTiO3) ([76–78] and so on), and PMN (PbMg1/3Nb2/3O3) and PMN-PT (PbMg1/3Nb2/3O3-PbTiO3) ([78–81] and others). The search for replacement Pb-free materials has often proceeded by analogy, with the obvious replacement for Pb2+ being Bi3+ ([82, 83] and many many more) and some exploration of Sn2+ [84–86], despite its difficult chemistry.There are commonalities in the SCDS observed in many of the lead-based relaxors, particularly in their cubic phase [30, 87, 88]. A range of microscopic models have been developed to explain the systematics of the distribution of DS. Often these explanations draw on the idea of the PNR, whose geometry relates to the observed diffuse—atomic displacements correlated in two-dimensional sheets when the diffuse scattering is rod-like, one dimensional chains when the DS is sheet-like, and so on [27, 31, 44, 89–92].A recent report by Bosak et al. [23] indicates that the diffuse scattering from these compounds can be modelled in a thermal-like way; in particular the often-referred-to diffuse “butterfly” (see the motif around the 300 peak in Figure 2) can be modelled using a purely thermal diffuse scattering-like approach that does not imply the existence of PNRs and has more in common with a phonon-based approach. This model uses parameters which are analogous to elastic constants, but are not elastic constants as such. It could be argued that of the two reciprocal space slices modelled in Figure 4 of [23] (reproduced in Figure 2) the second (hkk) is less than convincing; it seems to give streaks that are rather too strong and motifs of diffuse scattering which are too strong in every second horizontal row of features along the vertical axis (an example is arrowed by the current author on Figure 2(b)). Having said that, the model is clearly very powerful with very few parameters and shows that a displacement-wave picture rather than an order/disorder picture has much to recommend it. How the B-site randomness folds in to this is an interesting question, and it would be interesting to apply this approach to, for example, the PZT materials modelled in [25], in which the Pb-Pb ferroelectric domains were found to be of very different shape, being one-dimensional chains, and in which the static strains due to chemical disorder and B-site cation size mismatch are weaker and should therefore make a thermal-like model more applicable.The quality of agreement between experimental data “exp” and calculation “model” is excellent in (a), perhaps less so in (b). (From Figure 4 of [23], used by permission of the IUCr, http://dx.doi.org/10.1107/S0108767311040281.)
(a)
(b)


## 2. Survey of Diffuse Scattering in Pb-Based Ferroelectrics

### 2.1.PbZr1-xTixO3

PZT,PbZr1-xTixO3, ceramics are of great commercial importance and find uses as ultrasonic transducers in medical imaging, in accelerometers, and elsewhere [77, 93–96]. The structural phase diagram is summed up in Figure 3. The physical properties are optimal close to the morphotropic phase boundary, MPB, where the structure is prone to critical fluctuations and therefore responds more strongly to, for example, an external poling field. On the figure are noted (T,x) coordinates of the measurements discussed below.Figure 3
Schematic phase diagram for PZT. The morphotropic phase boundary is noted, along with key structural phases. Temperatures andx values for the various datasets discussed are noted on the figure—results at 577°C (850 K) are taken from [24], while letters (a), (b), and (c) correspond to diffraction patterns shown in Figure 4.PT,PbTiO3, appears to be a well-ordered perovskite [24, 97] with little diffuse scattering related to the PNR structures discussed here. Antiferroelectric PZ, PbZrO3, on the other hand, does show diffuse scattering. The average structure at high temperatures shows 〈110〉p Pb2+ displacements [24]. Antiferroelectrically correlated displacements along one of the [110]p directions had been suggested in early diffraction work [98], though ordered—but with unrealistic degrees of octahedral distortion. This was largely remedied in terms of the average structure by introducing two substructures related by a shear of c/2 and allowing some disorder [99].PDF work has since suggested that the Pb2+ are displaced at all temperatures, and the correlations amongst these displacements determine the observed average structure [100], including development of correlated c-axis displacements (compared to the antiferroelectric displacements in the bc plane) as expected to be associated with the weak room temperature piezoelectric effect in PZ [101]. Given the lack of B-site randomness, PNRs are unlikely to form if random strain fields (see below) play a role, and this is observed in PZ.The average structure of PZT has been the subject of numerous studies (just a few include [24, 102–105]), and the work continues. It has been shown that at room temperature powder diffraction data are best modelled by a multiple-phase approach, in which the sample consists of a rhombohedral R3c component and a monoclinic Cm component, with the proportion of Cm phase increasing with x (Ti content) as the MPB is approached [102]. Whether this is definitive or is an indication of the limits of powder diffraction in studying disordered materials is almost a question of ideology. It does appear that at high temperatures, in the cubic phase, the Pb atoms show disorder [24] that is not present in the Rietveld modelling [102], despite tests that were undertaken. Further, the presence of highly structured diffuse scattering in PZT at room temperature [25, 106–108] unambiguously requires an average crystal structure that permits local ordering to occur. Given that the Rietveld fits in [102] are of extremely high quality and yet do not yield models that permit SRO in Pb displacements to occur, it must be concluded that the powder diffraction is simply not sensitive to crucial aspects of the structure of PZT, despite the excellent fit statistics and the all-but perfect visual quality of the fits. This is a salutary lesson.In the high-temperature cubic phase, it has been noted that the Pb atomsdo show disorder. For x=0 (PbZrO3) they are distributed across 12 〈110〉c-type directions (c implies indexing on the parent cubic structure); as Ti is doped in this, it switches to the 8 〈111〉c-type directions, and until for x>0.5 it switches back to the 〈110〉c-type directions, until at x=1 (PbTiO3) the Pb shows a single-ordered site [24]. Such a model accords extremely well with the modelling of electron diffuse scattering, which appears to show that the observed diffuse scattering can be explained by one-dimensional chains of Pb displacements that result in planes of diffuse scattering perpendicular to 〈111〉c directions [107], and also accords well with neutron diffraction results [109]. Such a model was implemented using Monte Carlo simulation and was indeed found to reproduce many details of the observed electron diffuse scattering [25, 106]. As for other Pb2+ -based materials, the chronic underbonding of Pb2+ in the environment determined by the B-site cations (whose bonding is in a sense stiffer than the Pb-O bonds [110, 111]) is central to the mechanism of the off-centring.Figure5 shows a calculated diffraction pattern corresponding to in Figure 4(a), alongside a representation of the domain structure from the MC model. In this model, the Pb2+ are displaced along the eight 〈111〉c shifts, but not in equal proportions. For the domain structure shown in Figure 5(a), ~51% of the Pb2+ are displaced along [111]c, and this can be seen in the preponderance of red arrows (pointing up and to the right) in the figure. This imbalance is posited to drive the system away from cubic, and it may be conjectured that the average structure observed at high temperatures [24] is showing a state in which thermal energy is both evening out the distribution of Pb2+ displacements across the eight 〈111〉c shifts and overwhelming the correlations between displacements, giving a real space picture of the structural phase transition as one in which the types of individual Pb2+ displacements do not change—though the proportions along the 〈111〉c directions do—but the correlations amongst them break down. Since the crystal chemistry around a given Pb2+ is not changing with T, this seems plausible.Observed electron diffuse scattering from PZT. (a)PbZr1-xTixO3 with x=0.3 (rhombohedral phase) viewed down the [531]c zone axis (c means relative to the cubic parent structure). (b) x=0.46 (monoclinic) viewed down [731]c. (c) x=0.7 (tetragonal) viewed down [731]c. Some reflections and lines of scattering have been marked.
(a)
(b)
(c)Calculated electron diffuse scattering from PZT. (a) The model domain structure and (b) the resulting diffraction pattern, to be compared to (a) in Figure4 [25].
(a)
(b)Contrary to this atomistic model, a model in which inhomogeneous strains are caused by inclusions of a tetragonal phase within a rhombohedral or monoclinic matrix has been proposed [108]. On the basis of synchrotron diffuse and inelastic scattering the model reproduces some of the structure seen around selected Bragg peaks via these strain-inducing inclusions through a Huang-like formalism. This is qualitatively different from the relaxors like PMN-PT and PZN-PT, in which the Huang formalism cannot reproduce the observed scattering.It is clear that the diffuse scattering in PZT varies strongly with composition [25, 106], as would be expected if random strain fields induced by B-site doping play a role in inhibiting long-range order in Pb2+ displacements and promoting nanodomain growth.Interestingly, a Huang component has been observed in PMN above the Burns temperature,Td [112], which is therefore not associated with domains but with “simple, isotropic defects” which are considered to be fluctuations in Nb5+ :  Mg2+ concentrations on the nanoscale. This component of the diffuse scattering in PMN and related materials has been noted elsewhere [28, 113, 114]. It is relatively independent of temperature, being related to the chemical ordering on the B-site.Earlier work using pair distribution function analysis showed that local structure in PZT and closely related La-doped PLZT ([(Pb1-yLay)1-α□α][(Zr1-xTix)1-β□β]O3 or (Pb1-3x/2Lax)(ZryTi1-y)O3) was “more complex than that implied by the crystallographic structure” [115], with the PDF showing a wider range of Pb-O distances than the average structure suggests. The existence of short-range-ordered polar clusters has been suggested to explain Raman and other results in PLZT [116]. An electron diffraction study of (Pb1-3x/2Lax)(ZryTi1-y)O3 [117] draws related conclusions, noting that the “fundamental dipolar units in these materials correspond to highly anisotropic 〈111〉 chain dipoles formed from off-centre Pb/La and coupled Ti/Zr displacements.”Hence the overall picture is one in which the Pb2+ environments drive atomic Pb2+ displacements which interact via other atoms (in fact, via the BO6 octahedra) and affect those other atoms. This is a common theme across the Pb2+ ferroic materials.


### 2.2. PbZn1/3Nb2/3O3

AboveTC PbZn1/3Nb2/3O3 (PZN) is cubic (Pm3-m) and average structure analysis that shows disordered off-centre displacements of Pb atoms [118]. According to early studies by Kuwata et al. [119, 120], on cooling PZN undergoes a phase transition to the rhombohedral phase [121, 122]. The existence of the R phase in bulk was questioned [91] after measurements of splitting of the (111) reflection, and an unusual bulk structure was posited in which a sample/grain showed a rhombohedral outer shell and a cubic inner core. This has been shown to be unnecessary [121, 122]. That PZN from PbMg1/3Nb2/3O3 (PMN) which stays cubic down to 5 K [123]. It has been suggested that the phonon behaviour and diffuse scattering are very similar in the two materials [124].You [125] recorded two-dimensional X-ray DS for PMN, showing “butterfly” and ellipsoidal-shaped DS, and related it to transverse optic soft modes producing correlations along the cubic 〈110〉 directions (later works have shown that a coupling of optic and acoustic-like displacements is needed to explain the intensity variation of DS in PMN [126]).Some three-dimensional DS studies of PZN [27, 35, 44, 46] have applied m3-m symmetry to the diffuse features even in the rhombohedral phase, as these appear to obey this symmetry within experimental resolution—the rhombohedral angle ~89.95°.The diffuse scattering from PZN has been studied extensively. It was suggested that locally polarised “regions (probably of a size of the order of several unit cells)” could explain measurements of the index of refraction of PLZT ((Pb1-3x/2Lax)(ZryTi1-y)O3) and also the discovery of a second important temperature, the Burns temperature (Td) well above TC, above which it was posited that the domains ceased to exist [127]. PZN shows similar behaviour, with diffraction experiments showing that a signature for the posited domains could indeed be observed in the scattering from PZN and closely related PMN [126, 128, 129].Using single high-energy X-ray exposures and tilting the sample, rather in the manner of a transmission electron microscope diffraction experiment, Xu and coworkers were able to show that the rods of scattering observed in PZN were indeed〈110〉-type rods and not the intersections of, for example, the hk0 sections with out-of-plane rods in directions like 〈111〉 [54].Figure6 shows X-ray diffuse scattering in the hk0, hk0.5, and 111 sections of reciprocal space, measured on a single crystal of PZN at 300 K at the advanced photon source. These data clearly show the key diffuse features: the rods of scattering along 〈110〉-type directions (green arrow in Figure 6(a)), and the “butterfly” shapes where the rods intersect (boxed in Figure 6(a)), the ((1/2)(1/2)(1/2))-type spots that indicate B-site ordering (box “A” in Figure 6(b)) and that arise from intersection of the streaks with the layer (box “B” in Figure 6(b)). Also apparent is the variation of intensity along the length of the rods.X-ray diffuse scattering from PZN,PbZn1/3Nb2/3O3, showing the (a) hk0, (b) hk0.5, and (c) 111 sections of reciprocal space. Select features are highlighted. To indicate scale, note that the 2-00 is boxed in (a) [26].
(a)
(b)
(c)Figure7 shows three slices of diffuse scattering, measured using neutron diffraction at the ISIS-pulsed neutron source. There is no energy analysis undertaken, and so the scattering is treated as wholly elastic. That this induces little or no loss of symmetry—as it does when the scattering is the result of phonons [43]—is indicative that the scattering is not of predominantly thermal origin, or that the thermal fluctuations are slow on the energy scale of thermal neutrons. Again, the key features are visible. More apparent than in Figure 6 is the size-effect scattering around, for example, the 004 Bragg peak and which is also apparent in the intensity of the 〈110〉 rods on the inside of a row of Bragg peaks compared to the outside (note green arrows showing direction of intensity transfer).Neutron diffuse scattering from PZN,PbZn1/3Nb2/3O3, showing the (a) hk0, (b) hk1, and (c) hk0.5 sections of reciprocal space. Select features are highlighted [27].
(a)
(b)
(c)The idea of a relatively static PNR lends itself to a local, atomistic model driven by the crystal chemistry—a real-space chemical picture rather than a reciprocal space phonon-based picture, as it was. Such a model is also reasonable given that local inhomogeneity appears to play a role in relaxor behaviour—relaxor behaviour is often induced by doping when the “pure” compound is a nonrelaxor ferroelectric. This is even the case in multiferroics, where the doping of BiFeO3 with BaTiO3 induces a change to relaxor-like behaviour [130, 131].Hence, in modelling neutron diffuse scattering, Welberry and coworkers developed models for the diffuse scattering from PZN based on what could be termed chemical principles [27]. For example, the displacing of the Pb2+ ions is considered to be driven by the fact that Pb2+ is heavily underbonded in the average unit cell of PZN and also possesses an electron lone pair. The suggestion is that an inherently polar atom in a large environment (as parameterised by bond valence sums [132]) will satisfy its bonding requirements by moving off-centre rather than isotropically contracting its environment, and this is especially true of a “soft” atom like Pb2+ [133] which can be considered as sitting within a “hard” framework established by the oxygen atoms and the B-site cations.The Pb atoms are considered to be displaced along〈110〉-type directions in the unit cell, and these displacements are correlated within two-dimensional domains (see Figure 8), such that the displacement direction is parallel to a 〈110〉-type direction that lies within the plane of the domain. In essence, this is quite similar to the structure posted earlier based on diffuse X-ray scattering [54, 131]—except that there is no paraelectric matrix, but rather the volume is virtually entirely occupied by ferroelectric nanodomains (FND) of a range of sizes and orientations; this has led to the introduction of the term FND in distinction to PNR [72].Figure 8
Schematic diagram of posited PNRs (domains) in PZN in the plane of one of the domains, axes as noted (section is normal to[11-0]). Grey arrows lie in the plane and point left (L), black point right (R). Red (pink) dots point into (out of) the page. Other directions are intermediate.This model and its ramifications, including the response of the O ions to the Pb2+ framework and the alternating ordering of the Nb/Zn (frustrated by the 2 : 1 ratio of Nb to Zn), give rise to a qualitatively good agreement between observed and calculated data, including subtle details like the fact that every second diffuse rod is weak in the hk1 layer (Figure 7(b)). Further, the model gives an empirical chemical basis for the observed behaviour from which one can attempt to generalise in the search for Pb-free compounds. However, no underlying cause for the Pb2+ ions to form into the posited {110} domains is apparent.More significantly, it has been shown that the simple heuristic interpretation of the diffuse scattering that leads to the displacement directions being of〈110〉-type directions is misleading [89]. Investigation of these ideas shows that, at least in atomistic models, displacements indeed need not be along 〈110〉 directions, but that the domains themselves do need to be a single atomic layer thick, and with the established 〈110〉-type interfaces [134]. Having said that, high-temperature average structure determinations suggest that when the Pb site disorders, it is split across the 12 〈110〉 directions [118]. Thus in some sense the initial model remains the best supported.The question arises as to the power of the diffuse scattering to reveal the underlying structures. It has been shown that the diffuse scattering can be well modelled — and quantitatively—by a model that dispenses with static planar domains entirely [26]. This is supported by work discussed above [23] that also dispenses with PNRs and does a good job of modelling the scattering with just a handful of free parameters. Having said that, there is evidence that PNRs have been observed in relaxors using imaging techniques [135, 136] (with appropriate caveats about how representative of the bulk the surface may be), and so perhaps the ability to model the diffuse scattering with and without PNRs says more about the limits of the technique and the need to draw on multiple techniques than it says about PZN.


### 2.3. PbMg1/3Nb2/3O3

Any discussion of PbMg1/3Nb2/3O3-PbTiO3 and PbMg1/3Nb2/3O3 must relate to PZN, as the two materials are highly analogous.PbMg1/3Nb2/3O3 (PMN) is a heavily studied relaxor. Some work has used a dipole glass model to interpret diffuse scattering from PMN [137]. Earlier again it was suggested based on powder diffraction data that heavily displaced polar clusters of ions, particularly Pb2+ and O2-, were required to model the average structure [123, 138]. Butterfly-shaped and ellipsoidal diffuse scattering was seen in X-ray diffraction experiments and explained in terms of transverse optic soft modes along the cubic 〈110〉 directions [125]. This has been elaborated more recently [126]. As with PZN, the polar nanoregion has been invoked, with the T dependence of the DS being attributed to the growth of PNRs with cooling [139]. The relation of the lattice dynamics in PMN and PNRs (or local polar correlations) has been explored in some detail using a range of experimental techniques [140–144] and also ab initio calculations [145, 146].As noted in the discussion of PZN, the diffuse scattering from PbMg1/3Nb2/3O3-PbTiO3 appears to be able to be parameterised using a thermal-like model that dispenses with PNRs [23, 147, 148] and also to be calculated from an ab initio approach [149] using recently established potentials [150]. Further molecular dynamics results suggest that Pb atoms show 〈111〉p displacements, as expected [151] and that while the PNR does seem to be a useful model for PMN at low temperatures [152], it is a sense-limiting case resulting from correlations that, while present at higher temperatures, do not necessarily produce domains as such. This is quite plausible; when there is substantial randomness in the system, components of the displacements can be correlated but a random component that increases with temperature may prevent them actually assembling into a physical domain in which all ions share a common displacement direction. Further, diffuse scattering is sensitive to pair correlations—without distinguishing between static and dynamic in the case of X-ray scattering of neutron energies well above the excitation energies—and so correlations amongst components of displacements on pairs of interacting atoms can show structure that is not apparent in the absolute displacements of the atoms. Importantly, these results depend on the local strains due to the B-site ordering of Nb and Mg as a mechanism for inhibiting large domains from forming, providing a strong link between B-site doping [153] and the introduction of relaxor behaviour [152]. An interesting comparison of scattering from PMN and PMN-60PT [28] shows that increasing PT doping greatly reduces the structure in the DS, leaving features that are weakly elongated in the radial direction (Figure 9). These have been interpreted as short-range chemical order scattering due to weak B-site ordering. As the B-site becomes heavily occupied by Ti4+, the strains will grow weaker, removing the barrier to PNR growth and preventing the formation of the characteristic rods of 〈110〉p scattering.Diffuse scattering in thehk0 layer of (a) PMN-60PT at 100 K and (b) PMN at 15 K. The radical change in the form of the scattering with PT fraction (x) and with T is apparent. Reprinted with permission from Figure 4 of [28] (http://link.aps.org/doi/10.1103/PhysRevB.73.064107). Copyright 2006 by the American Physical Society. (Readers may view, browse, and/or download material for temporary copying purposes only, provided that these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.)
(a)
(b)Relatively straight-forward crystal chemical considerations and structural work [154, 155] show that the Pb2+ displacements are very large—typically 0.5 Å—which is why the treatment in [23] is “thermal-like” but not truly thermal. One approach to this has been to suggest that atomic shifts possess two components—one from a soft mode and the other a uniform displacement of the PNR [126].On the assumption that the PNRs are real, diffuse scattering has been used to explore their response to applied field [156], and it is found that for a field applied along [001] the diffuse scattering in the (hk0) plane is affected, indicating a possible redistribution of nanodomains of different polarisations, at least those perpendicular to the field, as some become more energetically favourable than others. Similar effects have been modelled for PZN in the nanodomain picture [44] with good qualitative agreement with observations [92].Further results, obtained by measuring diffuse scattering with different neutron energies to probe different time scales, suggest that the PNRs in PMN are (relatively) static [143] up to 420 K, and that the Burns temperature, as measured using diffuse scattering, may depend on the energy of the probe used—as the “static” PNRs become more dynamic above 420 K, they fluctuate on the timescale of a slow probe, but not a faster probe, causing different measurements of Td and implying a switch from a slowly to more quickly fluctuating PNR state. This is similar to but different from molecular dynamics results which, when used to interpret diffuse scattering over the range 10–700 K, suggest that strongly cooperative displacement behaviour sets in at 400 K on cooling [152]. This could also be interpreted as a value for Td, as the PNRs/FNDs are the result of cooperative displacements.Direct observation of PNRs through powder diffuse scattering (PDF analysis) has been claimed [151] though what actually manifests in the scattering are displacements and correlations. The work suggests that PNRs are “dispersed in a pseudocubic host lattice” [157] and that the PNRs begin to overlap as the sample cools to close to 200 K and increase in size and volume fraction of the sample but are affected by random fields due to the Mg/Nb disorder and are limited from growing further [158]. Other work also suggests 200 K as an important temperature—where possibly the domains truly manifest [152] and, most crucially, where the Curie temperature, TC  ~213 K [123, 138], is found. Hence, the overall picture is one in which on cooling correlations set in at Td but do not necessarily induce an actual domain structure, at least not one in which the displacements do not possess a significant random component, with both correlated and uncorrelated components fluctuating in time and space; close to 200 K, the domains lock in, possibly as they start interacting with each other more strongly and stabilising their existence, and the Curie temperature is obtained.


### 2.4. PbSc0.5Nb0.5O3 and PbSc0.5Ta0.5O3

There are a range of other Pb-containing ferroic materials. Two significant materials are PbSc0.5Nb0.5O3 and PbSc0.5Ta0.5O3 [159, 160].Diffuse scattering has been observed in PbSc0.5Ta0.5O3 and derivatives [29, 161, 162] and is of similar form to that seen in PMN and PZN—rods of scattering perpendicular to 〈110〉 directions such that “the atomic ferroic shifts correlate within {110} planes of the real space” [29]. Studies as a function of pressure and Ba A-site doping “chemical pressure” suggest that pressure suppresses the diffuse scattering and decouples B-site/Pb displacements and is interpreted in terms of PNRs [161, 162]. The B-site stoichiometry of PST allows full anticorrelated ordering, which is seen in a cell doubling along all cubic directions. As a result, at room temperature, the diffuse scattering appears very similar to that from PZN; the diffuse spots which appear at (1/2)(1/2)(1/2)-type positions in PZN are sharp peaks occurring at 111-type positions for the doubled cell (Figure 10). That the increase in pressure suppresses the diffuse scattering accords with the bond valence picture of the driver for Pb2+ displacement, in that the Pb2+ environment will be reduced in size, lessening the need for displacement off-centre to increase the Pb2+-bonding valence. This would also serve to decrease the B-site/Pb displacement correlations, as a Pb2+ which is “happier” in its bonding environment will distort the oxygen lattice to a smaller degree, reducing any transmitted effects.Figure 10
X-ray diffuse scattering fromPbSc0.5Ta0.5O3 as a function of temperature. The unit cell here is doubled relative to the parent perovskite cube. Note that the diffuse features at 300 K are very similar to those present in Figures 6 and 7, allowing that the indices here are doubled. Images taken with permission from [29] (http://hasyweb.desy.de/science/annual_reports/2007_report/part1/contrib/42/20848.pdf) and Figure 1 of [30] (http://link.aps.org/doi/10.1103/PhysRevB.79.224108). Copyright 2009 by the American Physical Society (see the following: Readers may view, browse, and/or download material for temporary copying purposes only, provided that these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society).At lowT, the diffuse scattering becomes less pronounced because the polar order is quite long ranged. The intersection of the 〈110〉 diffuse streaks with the hk1 becomes more apparent though it can be discerned at 300 K.Diffuse scattering has been observed in PbSc0.5Nb0.5O3 as well [30, 163–165] and again tends to show the characteristic 〈110〉p (p = parent cell) diffuse streaks. These weaken substantially on cooling, rather like PST, whereas in doped PSN (with Ba and Bi in this very thorough study [30]) the diffuse scattering is more strongly maintained on cooling, something also apparent in Ba-doped PST [161]. This is strong evidence that, as discussed in relation to PMN, it is randomness that prevents the PNR from continuing to grow on cooling. In PST and PSN when the B-site cations are in a 1 : 1 ratio and can be strong ordered, such that there is a cell doubling, there is not the randomness that exists in PMN and PZN where the B-site cations are in ratio 2 : 1 and cannot alternate. Hence, on cooling, the PNR growth is less inhibited in PST/PSN, and the diffuse scattering sharpens to the point of almost vanishing—the ferroelectric order becomes virtually long ranged. This can be “remedied” by doping into the A site, introducing greater disorder and restoring the randomness and preventing large domains from forming.Average structure determination, allowing the Pb to disorder across split sites, has shown that the Pb2+ in PSN are indeed displaced (by ~0.40 Å at 2 K), but the displacement directions are not well determined by these neutron powder diffraction data [165]. The observed diffuse scattering in the powder diffraction background suggests that on cooling the “correlation among Pb displacements along 〈111〉pc directions increases to form polar domains” (pc = parent cell). They conclude that in the paraelectric phase the Pb2+ is distributed on many split sites around the Wyckoff position and that the minimum energy configuration changes with temperature, appearing to be most likely 〈111〉pc at lower temperatures and 〈100〉pc at the highest temperature considered.


### 2.5. PbMg1/3Ta2/3O3

PMT, PbMg1/3Ta2/3O3, is clearly related to PMN and PZN in terms of its chemistry, and it is perhaps not surprising that the diffuse scattering accords with this. PMT is a relaxor ferroelectric with a mean Curie temperature Tcm~170K [166].Initial neutron diffuse scattering experiments used limited regions of reciprocal space—essentially, linear cuts through the (110) Bragg position [167]. These showed diffuse scattering whose intensity increased and width fell as T was reduced from room temperature to 100 K. These quantities then remained constant down to 20 K. The results suggested that displacements were static at low temperature and gave a low temperature correlation length, ξ, of ξ ~ 39 Å below 100 K, smaller than that has been suggested for PZN [137]. Given that the size difference between Mg and Ta is the largest exhibited in these Pb2+ containing compounds (the Rij [132] for relevant species are: Mg2+ 1.693 Å, Zn2+: 1.704 Å, Nb5+: 1.911 Å, Ta5+: 1.92 Å, and Sc3+: 1.849 Å). Hence the B-site randomness coupled to the most pronounced size difference gives rise to the strongest random strain field and serves to inhibit large domain growth. In PSN and PST, where the B-site cations are present in a 1 : 1 ratio and a relatively more similar in size, the domains seem able to grow so large, as T is reduced that the diffuse scattering coalesces into sharp features (Figure 10).Data collected over wider regions of reciprocal space using synchrotron X-rays at 175 K shows similar structure to that observed in PMN and PZN [31] (Figure 11) including evidence for partial B-site ordering.X-ray diffuse scattering fromPbMg1/3Ta2/3O3 in the (a) hk0 and (b) hhl planes, showing similar structure to that observed in PMN and PZN. The inset shows a close view of the 220 Bragg peak and the surrounding diffuse scattering. (Figure from [31], with permission of the IUCr, http://dx.doi.org/10.1107/S0021889811012635.)
(a)
(b)These data are modelled with Pb2+ displacements along 〈111〉pc and anisotropic PNRs whose longest dimension is in 〈110〉pc. PNR dimensions were estimated as ~7apc×7apc×20apc (apc being parent perovskite cube dimension). This is not greatly larger than the other estimate [167], except for the long axis which is approximately doubled. It should be noted however that the agreement between Figures 2 and  3 in [31] is moderate. The model does not capture the 〈110〉pc rods of diffuse scattering that connect the motifs of scattering around the Bragg peaks. Figure 12 shows a line profile, integrated over a wide band of pixels, taken in the direction of the broad black outline arrow in Figure 11(a). It has been fitted with a Gaussian on a linear background and shows quite clearly that, though weak, the diffuse scattering extends right through the Brillouin zone and is not localised in the vicinity of the Bragg spots. This large extent in reciprocal space strongly suggests that the PNRs do indeed need to be more pancake-like in real space than the model presented in [31] suggests, only enhancing their similarity to those in PMN and PZN.Figure 12
A line profile through the diffuse scattering shown in Figure11(a), along the direction of the black arrow, demonstrating that the streaks of scattering do not just form “butterflies” and similar shapes around the Bragg spots, but also extend through reciprocal space much like those in PZN (Figure 6).


## 2.1.PbZr1-xTixO3

PZT,PbZr1-xTixO3, ceramics are of great commercial importance and find uses as ultrasonic transducers in medical imaging, in accelerometers, and elsewhere [77, 93–96]. The structural phase diagram is summed up in Figure 3. The physical properties are optimal close to the morphotropic phase boundary, MPB, where the structure is prone to critical fluctuations and therefore responds more strongly to, for example, an external poling field. On the figure are noted (T,x) coordinates of the measurements discussed below.Figure 3
Schematic phase diagram for PZT. The morphotropic phase boundary is noted, along with key structural phases. Temperatures andx values for the various datasets discussed are noted on the figure—results at 577°C (850 K) are taken from [24], while letters (a), (b), and (c) correspond to diffraction patterns shown in Figure 4.PT,PbTiO3, appears to be a well-ordered perovskite [24, 97] with little diffuse scattering related to the PNR structures discussed here. Antiferroelectric PZ, PbZrO3, on the other hand, does show diffuse scattering. The average structure at high temperatures shows 〈110〉p Pb2+ displacements [24]. Antiferroelectrically correlated displacements along one of the [110]p directions had been suggested in early diffraction work [98], though ordered—but with unrealistic degrees of octahedral distortion. This was largely remedied in terms of the average structure by introducing two substructures related by a shear of c/2 and allowing some disorder [99].PDF work has since suggested that the Pb2+ are displaced at all temperatures, and the correlations amongst these displacements determine the observed average structure [100], including development of correlated c-axis displacements (compared to the antiferroelectric displacements in the bc plane) as expected to be associated with the weak room temperature piezoelectric effect in PZ [101]. Given the lack of B-site randomness, PNRs are unlikely to form if random strain fields (see below) play a role, and this is observed in PZ.The average structure of PZT has been the subject of numerous studies (just a few include [24, 102–105]), and the work continues. It has been shown that at room temperature powder diffraction data are best modelled by a multiple-phase approach, in which the sample consists of a rhombohedral R3c component and a monoclinic Cm component, with the proportion of Cm phase increasing with x (Ti content) as the MPB is approached [102]. Whether this is definitive or is an indication of the limits of powder diffraction in studying disordered materials is almost a question of ideology. It does appear that at high temperatures, in the cubic phase, the Pb atoms show disorder [24] that is not present in the Rietveld modelling [102], despite tests that were undertaken. Further, the presence of highly structured diffuse scattering in PZT at room temperature [25, 106–108] unambiguously requires an average crystal structure that permits local ordering to occur. Given that the Rietveld fits in [102] are of extremely high quality and yet do not yield models that permit SRO in Pb displacements to occur, it must be concluded that the powder diffraction is simply not sensitive to crucial aspects of the structure of PZT, despite the excellent fit statistics and the all-but perfect visual quality of the fits. This is a salutary lesson.In the high-temperature cubic phase, it has been noted that the Pb atomsdo show disorder. For x=0 (PbZrO3) they are distributed across 12 〈110〉c-type directions (c implies indexing on the parent cubic structure); as Ti is doped in this, it switches to the 8 〈111〉c-type directions, and until for x>0.5 it switches back to the 〈110〉c-type directions, until at x=1 (PbTiO3) the Pb shows a single-ordered site [24]. Such a model accords extremely well with the modelling of electron diffuse scattering, which appears to show that the observed diffuse scattering can be explained by one-dimensional chains of Pb displacements that result in planes of diffuse scattering perpendicular to 〈111〉c directions [107], and also accords well with neutron diffraction results [109]. Such a model was implemented using Monte Carlo simulation and was indeed found to reproduce many details of the observed electron diffuse scattering [25, 106]. As for other Pb2+ -based materials, the chronic underbonding of Pb2+ in the environment determined by the B-site cations (whose bonding is in a sense stiffer than the Pb-O bonds [110, 111]) is central to the mechanism of the off-centring.Figure5 shows a calculated diffraction pattern corresponding to in Figure 4(a), alongside a representation of the domain structure from the MC model. In this model, the Pb2+ are displaced along the eight 〈111〉c shifts, but not in equal proportions. For the domain structure shown in Figure 5(a), ~51% of the Pb2+ are displaced along [111]c, and this can be seen in the preponderance of red arrows (pointing up and to the right) in the figure. This imbalance is posited to drive the system away from cubic, and it may be conjectured that the average structure observed at high temperatures [24] is showing a state in which thermal energy is both evening out the distribution of Pb2+ displacements across the eight 〈111〉c shifts and overwhelming the correlations between displacements, giving a real space picture of the structural phase transition as one in which the types of individual Pb2+ displacements do not change—though the proportions along the 〈111〉c directions do—but the correlations amongst them break down. Since the crystal chemistry around a given Pb2+ is not changing with T, this seems plausible.Observed electron diffuse scattering from PZT. (a)PbZr1-xTixO3 with x=0.3 (rhombohedral phase) viewed down the [531]c zone axis (c means relative to the cubic parent structure). (b) x=0.46 (monoclinic) viewed down [731]c. (c) x=0.7 (tetragonal) viewed down [731]c. Some reflections and lines of scattering have been marked.
(a)
(b)
(c)Calculated electron diffuse scattering from PZT. (a) The model domain structure and (b) the resulting diffraction pattern, to be compared to (a) in Figure4 [25].
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(b)Contrary to this atomistic model, a model in which inhomogeneous strains are caused by inclusions of a tetragonal phase within a rhombohedral or monoclinic matrix has been proposed [108]. On the basis of synchrotron diffuse and inelastic scattering the model reproduces some of the structure seen around selected Bragg peaks via these strain-inducing inclusions through a Huang-like formalism. This is qualitatively different from the relaxors like PMN-PT and PZN-PT, in which the Huang formalism cannot reproduce the observed scattering.It is clear that the diffuse scattering in PZT varies strongly with composition [25, 106], as would be expected if random strain fields induced by B-site doping play a role in inhibiting long-range order in Pb2+ displacements and promoting nanodomain growth.Interestingly, a Huang component has been observed in PMN above the Burns temperature,Td [112], which is therefore not associated with domains but with “simple, isotropic defects” which are considered to be fluctuations in Nb5+ :  Mg2+ concentrations on the nanoscale. This component of the diffuse scattering in PMN and related materials has been noted elsewhere [28, 113, 114]. It is relatively independent of temperature, being related to the chemical ordering on the B-site.Earlier work using pair distribution function analysis showed that local structure in PZT and closely related La-doped PLZT ([(Pb1-yLay)1-α□α][(Zr1-xTix)1-β□β]O3 or (Pb1-3x/2Lax)(ZryTi1-y)O3) was “more complex than that implied by the crystallographic structure” [115], with the PDF showing a wider range of Pb-O distances than the average structure suggests. The existence of short-range-ordered polar clusters has been suggested to explain Raman and other results in PLZT [116]. An electron diffraction study of (Pb1-3x/2Lax)(ZryTi1-y)O3 [117] draws related conclusions, noting that the “fundamental dipolar units in these materials correspond to highly anisotropic 〈111〉 chain dipoles formed from off-centre Pb/La and coupled Ti/Zr displacements.”Hence the overall picture is one in which the Pb2+ environments drive atomic Pb2+ displacements which interact via other atoms (in fact, via the BO6 octahedra) and affect those other atoms. This is a common theme across the Pb2+ ferroic materials.


## 2.2. PbZn1/3Nb2/3O3

AboveTC PbZn1/3Nb2/3O3 (PZN) is cubic (Pm3-m) and average structure analysis that shows disordered off-centre displacements of Pb atoms [118]. According to early studies by Kuwata et al. [119, 120], on cooling PZN undergoes a phase transition to the rhombohedral phase [121, 122]. The existence of the R phase in bulk was questioned [91] after measurements of splitting of the (111) reflection, and an unusual bulk structure was posited in which a sample/grain showed a rhombohedral outer shell and a cubic inner core. This has been shown to be unnecessary [121, 122]. That PZN from PbMg1/3Nb2/3O3 (PMN) which stays cubic down to 5 K [123]. It has been suggested that the phonon behaviour and diffuse scattering are very similar in the two materials [124].You [125] recorded two-dimensional X-ray DS for PMN, showing “butterfly” and ellipsoidal-shaped DS, and related it to transverse optic soft modes producing correlations along the cubic 〈110〉 directions (later works have shown that a coupling of optic and acoustic-like displacements is needed to explain the intensity variation of DS in PMN [126]).Some three-dimensional DS studies of PZN [27, 35, 44, 46] have applied m3-m symmetry to the diffuse features even in the rhombohedral phase, as these appear to obey this symmetry within experimental resolution—the rhombohedral angle ~89.95°.The diffuse scattering from PZN has been studied extensively. It was suggested that locally polarised “regions (probably of a size of the order of several unit cells)” could explain measurements of the index of refraction of PLZT ((Pb1-3x/2Lax)(ZryTi1-y)O3) and also the discovery of a second important temperature, the Burns temperature (Td) well above TC, above which it was posited that the domains ceased to exist [127]. PZN shows similar behaviour, with diffraction experiments showing that a signature for the posited domains could indeed be observed in the scattering from PZN and closely related PMN [126, 128, 129].Using single high-energy X-ray exposures and tilting the sample, rather in the manner of a transmission electron microscope diffraction experiment, Xu and coworkers were able to show that the rods of scattering observed in PZN were indeed〈110〉-type rods and not the intersections of, for example, the hk0 sections with out-of-plane rods in directions like 〈111〉 [54].Figure6 shows X-ray diffuse scattering in the hk0, hk0.5, and 111 sections of reciprocal space, measured on a single crystal of PZN at 300 K at the advanced photon source. These data clearly show the key diffuse features: the rods of scattering along 〈110〉-type directions (green arrow in Figure 6(a)), and the “butterfly” shapes where the rods intersect (boxed in Figure 6(a)), the ((1/2)(1/2)(1/2))-type spots that indicate B-site ordering (box “A” in Figure 6(b)) and that arise from intersection of the streaks with the layer (box “B” in Figure 6(b)). Also apparent is the variation of intensity along the length of the rods.X-ray diffuse scattering from PZN,PbZn1/3Nb2/3O3, showing the (a) hk0, (b) hk0.5, and (c) 111 sections of reciprocal space. Select features are highlighted. To indicate scale, note that the 2-00 is boxed in (a) [26].
(a)
(b)
(c)Figure7 shows three slices of diffuse scattering, measured using neutron diffraction at the ISIS-pulsed neutron source. There is no energy analysis undertaken, and so the scattering is treated as wholly elastic. That this induces little or no loss of symmetry—as it does when the scattering is the result of phonons [43]—is indicative that the scattering is not of predominantly thermal origin, or that the thermal fluctuations are slow on the energy scale of thermal neutrons. Again, the key features are visible. More apparent than in Figure 6 is the size-effect scattering around, for example, the 004 Bragg peak and which is also apparent in the intensity of the 〈110〉 rods on the inside of a row of Bragg peaks compared to the outside (note green arrows showing direction of intensity transfer).Neutron diffuse scattering from PZN,PbZn1/3Nb2/3O3, showing the (a) hk0, (b) hk1, and (c) hk0.5 sections of reciprocal space. Select features are highlighted [27].
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(c)The idea of a relatively static PNR lends itself to a local, atomistic model driven by the crystal chemistry—a real-space chemical picture rather than a reciprocal space phonon-based picture, as it was. Such a model is also reasonable given that local inhomogeneity appears to play a role in relaxor behaviour—relaxor behaviour is often induced by doping when the “pure” compound is a nonrelaxor ferroelectric. This is even the case in multiferroics, where the doping of BiFeO3 with BaTiO3 induces a change to relaxor-like behaviour [130, 131].Hence, in modelling neutron diffuse scattering, Welberry and coworkers developed models for the diffuse scattering from PZN based on what could be termed chemical principles [27]. For example, the displacing of the Pb2+ ions is considered to be driven by the fact that Pb2+ is heavily underbonded in the average unit cell of PZN and also possesses an electron lone pair. The suggestion is that an inherently polar atom in a large environment (as parameterised by bond valence sums [132]) will satisfy its bonding requirements by moving off-centre rather than isotropically contracting its environment, and this is especially true of a “soft” atom like Pb2+ [133] which can be considered as sitting within a “hard” framework established by the oxygen atoms and the B-site cations.The Pb atoms are considered to be displaced along〈110〉-type directions in the unit cell, and these displacements are correlated within two-dimensional domains (see Figure 8), such that the displacement direction is parallel to a 〈110〉-type direction that lies within the plane of the domain. In essence, this is quite similar to the structure posted earlier based on diffuse X-ray scattering [54, 131]—except that there is no paraelectric matrix, but rather the volume is virtually entirely occupied by ferroelectric nanodomains (FND) of a range of sizes and orientations; this has led to the introduction of the term FND in distinction to PNR [72].Figure 8
Schematic diagram of posited PNRs (domains) in PZN in the plane of one of the domains, axes as noted (section is normal to[11-0]). Grey arrows lie in the plane and point left (L), black point right (R). Red (pink) dots point into (out of) the page. Other directions are intermediate.This model and its ramifications, including the response of the O ions to the Pb2+ framework and the alternating ordering of the Nb/Zn (frustrated by the 2 : 1 ratio of Nb to Zn), give rise to a qualitatively good agreement between observed and calculated data, including subtle details like the fact that every second diffuse rod is weak in the hk1 layer (Figure 7(b)). Further, the model gives an empirical chemical basis for the observed behaviour from which one can attempt to generalise in the search for Pb-free compounds. However, no underlying cause for the Pb2+ ions to form into the posited {110} domains is apparent.More significantly, it has been shown that the simple heuristic interpretation of the diffuse scattering that leads to the displacement directions being of〈110〉-type directions is misleading [89]. Investigation of these ideas shows that, at least in atomistic models, displacements indeed need not be along 〈110〉 directions, but that the domains themselves do need to be a single atomic layer thick, and with the established 〈110〉-type interfaces [134]. Having said that, high-temperature average structure determinations suggest that when the Pb site disorders, it is split across the 12 〈110〉 directions [118]. Thus in some sense the initial model remains the best supported.The question arises as to the power of the diffuse scattering to reveal the underlying structures. It has been shown that the diffuse scattering can be well modelled — and quantitatively—by a model that dispenses with static planar domains entirely [26]. This is supported by work discussed above [23] that also dispenses with PNRs and does a good job of modelling the scattering with just a handful of free parameters. Having said that, there is evidence that PNRs have been observed in relaxors using imaging techniques [135, 136] (with appropriate caveats about how representative of the bulk the surface may be), and so perhaps the ability to model the diffuse scattering with and without PNRs says more about the limits of the technique and the need to draw on multiple techniques than it says about PZN.


## 2.3. PbMg1/3Nb2/3O3

Any discussion of PbMg1/3Nb2/3O3-PbTiO3 and PbMg1/3Nb2/3O3 must relate to PZN, as the two materials are highly analogous.PbMg1/3Nb2/3O3 (PMN) is a heavily studied relaxor. Some work has used a dipole glass model to interpret diffuse scattering from PMN [137]. Earlier again it was suggested based on powder diffraction data that heavily displaced polar clusters of ions, particularly Pb2+ and O2-, were required to model the average structure [123, 138]. Butterfly-shaped and ellipsoidal diffuse scattering was seen in X-ray diffraction experiments and explained in terms of transverse optic soft modes along the cubic 〈110〉 directions [125]. This has been elaborated more recently [126]. As with PZN, the polar nanoregion has been invoked, with the T dependence of the DS being attributed to the growth of PNRs with cooling [139]. The relation of the lattice dynamics in PMN and PNRs (or local polar correlations) has been explored in some detail using a range of experimental techniques [140–144] and also ab initio calculations [145, 146].As noted in the discussion of PZN, the diffuse scattering from PbMg1/3Nb2/3O3-PbTiO3 appears to be able to be parameterised using a thermal-like model that dispenses with PNRs [23, 147, 148] and also to be calculated from an ab initio approach [149] using recently established potentials [150]. Further molecular dynamics results suggest that Pb atoms show 〈111〉p displacements, as expected [151] and that while the PNR does seem to be a useful model for PMN at low temperatures [152], it is a sense-limiting case resulting from correlations that, while present at higher temperatures, do not necessarily produce domains as such. This is quite plausible; when there is substantial randomness in the system, components of the displacements can be correlated but a random component that increases with temperature may prevent them actually assembling into a physical domain in which all ions share a common displacement direction. Further, diffuse scattering is sensitive to pair correlations—without distinguishing between static and dynamic in the case of X-ray scattering of neutron energies well above the excitation energies—and so correlations amongst components of displacements on pairs of interacting atoms can show structure that is not apparent in the absolute displacements of the atoms. Importantly, these results depend on the local strains due to the B-site ordering of Nb and Mg as a mechanism for inhibiting large domains from forming, providing a strong link between B-site doping [153] and the introduction of relaxor behaviour [152]. An interesting comparison of scattering from PMN and PMN-60PT [28] shows that increasing PT doping greatly reduces the structure in the DS, leaving features that are weakly elongated in the radial direction (Figure 9). These have been interpreted as short-range chemical order scattering due to weak B-site ordering. As the B-site becomes heavily occupied by Ti4+, the strains will grow weaker, removing the barrier to PNR growth and preventing the formation of the characteristic rods of 〈110〉p scattering.Diffuse scattering in thehk0 layer of (a) PMN-60PT at 100 K and (b) PMN at 15 K. The radical change in the form of the scattering with PT fraction (x) and with T is apparent. Reprinted with permission from Figure 4 of [28] (http://link.aps.org/doi/10.1103/PhysRevB.73.064107). Copyright 2006 by the American Physical Society. (Readers may view, browse, and/or download material for temporary copying purposes only, provided that these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.)
(a)
(b)Relatively straight-forward crystal chemical considerations and structural work [154, 155] show that the Pb2+ displacements are very large—typically 0.5 Å—which is why the treatment in [23] is “thermal-like” but not truly thermal. One approach to this has been to suggest that atomic shifts possess two components—one from a soft mode and the other a uniform displacement of the PNR [126].On the assumption that the PNRs are real, diffuse scattering has been used to explore their response to applied field [156], and it is found that for a field applied along [001] the diffuse scattering in the (hk0) plane is affected, indicating a possible redistribution of nanodomains of different polarisations, at least those perpendicular to the field, as some become more energetically favourable than others. Similar effects have been modelled for PZN in the nanodomain picture [44] with good qualitative agreement with observations [92].Further results, obtained by measuring diffuse scattering with different neutron energies to probe different time scales, suggest that the PNRs in PMN are (relatively) static [143] up to 420 K, and that the Burns temperature, as measured using diffuse scattering, may depend on the energy of the probe used—as the “static” PNRs become more dynamic above 420 K, they fluctuate on the timescale of a slow probe, but not a faster probe, causing different measurements of Td and implying a switch from a slowly to more quickly fluctuating PNR state. This is similar to but different from molecular dynamics results which, when used to interpret diffuse scattering over the range 10–700 K, suggest that strongly cooperative displacement behaviour sets in at 400 K on cooling [152]. This could also be interpreted as a value for Td, as the PNRs/FNDs are the result of cooperative displacements.Direct observation of PNRs through powder diffuse scattering (PDF analysis) has been claimed [151] though what actually manifests in the scattering are displacements and correlations. The work suggests that PNRs are “dispersed in a pseudocubic host lattice” [157] and that the PNRs begin to overlap as the sample cools to close to 200 K and increase in size and volume fraction of the sample but are affected by random fields due to the Mg/Nb disorder and are limited from growing further [158]. Other work also suggests 200 K as an important temperature—where possibly the domains truly manifest [152] and, most crucially, where the Curie temperature, TC  ~213 K [123, 138], is found. Hence, the overall picture is one in which on cooling correlations set in at Td but do not necessarily induce an actual domain structure, at least not one in which the displacements do not possess a significant random component, with both correlated and uncorrelated components fluctuating in time and space; close to 200 K, the domains lock in, possibly as they start interacting with each other more strongly and stabilising their existence, and the Curie temperature is obtained.


## 2.4. PbSc0.5Nb0.5O3 and PbSc0.5Ta0.5O3

There are a range of other Pb-containing ferroic materials. Two significant materials are PbSc0.5Nb0.5O3 and PbSc0.5Ta0.5O3 [159, 160].Diffuse scattering has been observed in PbSc0.5Ta0.5O3 and derivatives [29, 161, 162] and is of similar form to that seen in PMN and PZN—rods of scattering perpendicular to 〈110〉 directions such that “the atomic ferroic shifts correlate within {110} planes of the real space” [29]. Studies as a function of pressure and Ba A-site doping “chemical pressure” suggest that pressure suppresses the diffuse scattering and decouples B-site/Pb displacements and is interpreted in terms of PNRs [161, 162]. The B-site stoichiometry of PST allows full anticorrelated ordering, which is seen in a cell doubling along all cubic directions. As a result, at room temperature, the diffuse scattering appears very similar to that from PZN; the diffuse spots which appear at (1/2)(1/2)(1/2)-type positions in PZN are sharp peaks occurring at 111-type positions for the doubled cell (Figure 10). That the increase in pressure suppresses the diffuse scattering accords with the bond valence picture of the driver for Pb2+ displacement, in that the Pb2+ environment will be reduced in size, lessening the need for displacement off-centre to increase the Pb2+-bonding valence. This would also serve to decrease the B-site/Pb displacement correlations, as a Pb2+ which is “happier” in its bonding environment will distort the oxygen lattice to a smaller degree, reducing any transmitted effects.Figure 10
X-ray diffuse scattering fromPbSc0.5Ta0.5O3 as a function of temperature. The unit cell here is doubled relative to the parent perovskite cube. Note that the diffuse features at 300 K are very similar to those present in Figures 6 and 7, allowing that the indices here are doubled. Images taken with permission from [29] (http://hasyweb.desy.de/science/annual_reports/2007_report/part1/contrib/42/20848.pdf) and Figure 1 of [30] (http://link.aps.org/doi/10.1103/PhysRevB.79.224108). Copyright 2009 by the American Physical Society (see the following: Readers may view, browse, and/or download material for temporary copying purposes only, provided that these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society).At lowT, the diffuse scattering becomes less pronounced because the polar order is quite long ranged. The intersection of the 〈110〉 diffuse streaks with the hk1 becomes more apparent though it can be discerned at 300 K.Diffuse scattering has been observed in PbSc0.5Nb0.5O3 as well [30, 163–165] and again tends to show the characteristic 〈110〉p (p = parent cell) diffuse streaks. These weaken substantially on cooling, rather like PST, whereas in doped PSN (with Ba and Bi in this very thorough study [30]) the diffuse scattering is more strongly maintained on cooling, something also apparent in Ba-doped PST [161]. This is strong evidence that, as discussed in relation to PMN, it is randomness that prevents the PNR from continuing to grow on cooling. In PST and PSN when the B-site cations are in a 1 : 1 ratio and can be strong ordered, such that there is a cell doubling, there is not the randomness that exists in PMN and PZN where the B-site cations are in ratio 2 : 1 and cannot alternate. Hence, on cooling, the PNR growth is less inhibited in PST/PSN, and the diffuse scattering sharpens to the point of almost vanishing—the ferroelectric order becomes virtually long ranged. This can be “remedied” by doping into the A site, introducing greater disorder and restoring the randomness and preventing large domains from forming.Average structure determination, allowing the Pb to disorder across split sites, has shown that the Pb2+ in PSN are indeed displaced (by ~0.40 Å at 2 K), but the displacement directions are not well determined by these neutron powder diffraction data [165]. The observed diffuse scattering in the powder diffraction background suggests that on cooling the “correlation among Pb displacements along 〈111〉pc directions increases to form polar domains” (pc = parent cell). They conclude that in the paraelectric phase the Pb2+ is distributed on many split sites around the Wyckoff position and that the minimum energy configuration changes with temperature, appearing to be most likely 〈111〉pc at lower temperatures and 〈100〉pc at the highest temperature considered.


## 2.5. PbMg1/3Ta2/3O3

PMT, PbMg1/3Ta2/3O3, is clearly related to PMN and PZN in terms of its chemistry, and it is perhaps not surprising that the diffuse scattering accords with this. PMT is a relaxor ferroelectric with a mean Curie temperature Tcm~170K [166].Initial neutron diffuse scattering experiments used limited regions of reciprocal space—essentially, linear cuts through the (110) Bragg position [167]. These showed diffuse scattering whose intensity increased and width fell as T was reduced from room temperature to 100 K. These quantities then remained constant down to 20 K. The results suggested that displacements were static at low temperature and gave a low temperature correlation length, ξ, of ξ ~ 39 Å below 100 K, smaller than that has been suggested for PZN [137]. Given that the size difference between Mg and Ta is the largest exhibited in these Pb2+ containing compounds (the Rij [132] for relevant species are: Mg2+ 1.693 Å, Zn2+: 1.704 Å, Nb5+: 1.911 Å, Ta5+: 1.92 Å, and Sc3+: 1.849 Å). Hence the B-site randomness coupled to the most pronounced size difference gives rise to the strongest random strain field and serves to inhibit large domain growth. In PSN and PST, where the B-site cations are present in a 1 : 1 ratio and a relatively more similar in size, the domains seem able to grow so large, as T is reduced that the diffuse scattering coalesces into sharp features (Figure 10).Data collected over wider regions of reciprocal space using synchrotron X-rays at 175 K shows similar structure to that observed in PMN and PZN [31] (Figure 11) including evidence for partial B-site ordering.X-ray diffuse scattering fromPbMg1/3Ta2/3O3 in the (a) hk0 and (b) hhl planes, showing similar structure to that observed in PMN and PZN. The inset shows a close view of the 220 Bragg peak and the surrounding diffuse scattering. (Figure from [31], with permission of the IUCr, http://dx.doi.org/10.1107/S0021889811012635.)
(a)
(b)These data are modelled with Pb2+ displacements along 〈111〉pc and anisotropic PNRs whose longest dimension is in 〈110〉pc. PNR dimensions were estimated as ~7apc×7apc×20apc (apc being parent perovskite cube dimension). This is not greatly larger than the other estimate [167], except for the long axis which is approximately doubled. It should be noted however that the agreement between Figures 2 and  3 in [31] is moderate. The model does not capture the 〈110〉pc rods of diffuse scattering that connect the motifs of scattering around the Bragg peaks. Figure 12 shows a line profile, integrated over a wide band of pixels, taken in the direction of the broad black outline arrow in Figure 11(a). It has been fitted with a Gaussian on a linear background and shows quite clearly that, though weak, the diffuse scattering extends right through the Brillouin zone and is not localised in the vicinity of the Bragg spots. This large extent in reciprocal space strongly suggests that the PNRs do indeed need to be more pancake-like in real space than the model presented in [31] suggests, only enhancing their similarity to those in PMN and PZN.Figure 12
A line profile through the diffuse scattering shown in Figure11(a), along the direction of the black arrow, demonstrating that the streaks of scattering do not just form “butterflies” and similar shapes around the Bragg spots, but also extend through reciprocal space much like those in PZN (Figure 6).


## 3. Conclusions

It appears that diffuse scattering, both single crystal and powder, does indeed give deep insight into the structure of these materials, but it does not deliver “unique” solutions but constrains the possible solutions. Into the future, energy-analysed diffuse scattering is likely to play an important role, as it provides insights onto the dynamic or static nature of the local correlations, adding important new information to the picture. Further, it is clear that modelling large windows of reciprocal space and preferably three-dimensional volumes are important when the scattering is so anisotropic and extended. Some models work well close to Bragg positions but less well near reciprocal cell boundaries, showing that measuring comprehensive data is necessary if models are to be adequately tested.The Pb-based relaxors show great commonality of behaviour and similarities in their diffuse scattering. It appears that the idea of the PNR (or FND) may not be mathematically necessary to describe the locus of the diffuse scattering but nevertheless has a strong basis in crystal chemistry, electron imaging, and molecular dynamics modelling and still appears to be a reasonable model for the local order. The questions of whether the domains are static or dynamic and what aspects of their behaviour are crucial in determining the properties are yet to be fully resolved across the family. Is a domain truly an observable polarised volume of the crystal, like a small (often one- or two-dimensional) ferroic crystal embedded in the larger, or is it a relatively evanescent region in which some components of the displacements are correlated while others are random? As yet no definitive picture has emerged although measurements on both PZN and PMN suggest that the PNRs are fluctuating relatively slowly on the time scale of a thermal neutron experiment, which is not to say that random components of the displacements are not fluctuating much faster. It does appear that randomness, whetherB-site or A-site, plays a central role, with the size mismatch and the level of doping being, along with T relative to Td, key in determining the degree of PNR-related scattering.The overall picture is one in which the Pb2+ ion is too small to “sit” in the centre of the O environment determined by the stiffer B-site-to-oxygen bonds, and as an inherently polar atom due to its unpaired electrons resolves this by moving off-centre. In systems with chemical disorder, these shifts are prevented from becoming long-range ordered even at low T by the random strain fields caused to the size mismatch of the B-site cations (or A-site when applicable, as in Ba-doped PST). Hence the nature of the local order and that of the relaxor properties appear to be determined by both the average Pb2+ environment and by the distribution of local environments—for example, PZ shows Pb2+ displacements but not PNRs, while PT, where Ti4+ is substantially smaller than Zr4+ [168], appears to show an ordered Pb2+ site, as might be more likely given that the A-site environment is smaller and so the Pb2+ is less underbonded; and PZN and its derivatives and close relatives, with significant randomness, do show the PNR scattering.



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