research papers

J. Appl. Cryst. (2012). 45, 1309–1313 doi:10.1107/S0021889812039015 1309

Journal of

AppliedCrystallography

ISSN 0021-8898

Received 6 July 2012

Accepted 12 September 2012

# 2012 International Union of Crystallography

Printed in Singapore – all rights reserved

Glass-like structure of a lead-based relaxorferroelectric

Alexei Bosak,a* Dmitry Chernyshovb and Sergey Vakhrushevc,d

aEuropean Synchrotron Radiation Facility, 6 rue Jules Horowitz, Grenoble, 38043, France, bSwiss–

Norwegian Beam Lines at the European Synchrotron Radiation Facility, 6 rue Jules Horowitz,

Grenoble, 38043, France, cIoffe Physico-Technical Institute, St Petersburg, 194021, Russian

Federation, and dSt Petersburg State Polytechnical University, Politekhnicheskaya ulica 29, St

Petersburg, 195251, Russian Federation. Correspondence e-mail: bossak@esrf.fr

It has been shown [Bosak, Chernyshov, Vakhrushev & Krisch (2012). Acta Cryst.

A68, 117–123] that detailed experimental three-dimensional maps of diffuse

scattering in lead-based relaxors do not agree with those expected from the

concept of polar nanoregions and/or polar nanodomains. Instead, the observed

diffuse scattering can be successfully parameterized in terms of a pseudo-

dynamical matrix, having a shape very similar to that of the thermal diffuse

scattering. Here the parameterization is extended and used to generate the

disordered atomic configurations. The analysis of the resulting displacement

patterns retrieved for lead ions shows that a static snapshot of the relaxor

structure corresponds to the specific dipole glassy state that is characterized by

local polarization and its projection onto the selected direction. The recovered

structural model agrees with the observed behaviour of dielectric susceptibility

as well as the existence of a wide-range hierarchy in the relaxation times in these

materials.

1. IntroductionRelaxor ferroelectrics have been known for more than 50

years (Smolenskii & Agranovskaya, 1958) and have been

arousing significant interest because of their numerous

unusual properties, such as the appearance of a broad peak in

the real part of the dielectric susceptibility as a function of

temperature. No structural change has been found to accom-

pany this peak, and one of the proposed scenarios is based on

a diffuse phase transition where local chemical disorder

provokes the formation of polar nanoregions (PNRs) or

nanosized domains (PNDs) (Cowley et al., 2011). On the other

hand, the peak of dielectric susceptibility decreases in

magnitude and shifts to a higher temperature with increasing

probe frequency over a very large frequency domain (Vieh-

land et al., 1990). There are also numerous observations of

history effects, a divergence of response for field-cooled and

zero-field-cooled samples, and specific relaxation behaviour;

all these properties can be reproduced within a dipole glass

theory (Timonin, 2009, 2010). Although the problem of

relaxors is still not completely understood, it is generally

agreed that the core of relaxor response lies in correlated local

polar deviations from the average nonpolar structure. Here we

do not attempt to review the large volume of experimental and

theoretical literature in this field; our goal is to show what kind

of local polar structure can be recovered from an accurate

X-ray diffuse scattering experiment and to propose this

information as the discriminating input for theoretical spec-

ulations.

The ultimate aim of a diffraction experiment is the retrieval

of a real disordered structure, which can be represented as an

average structure plus local deviations from the average.

While Bragg scattering carries information on the average

structure, the deviations from the average are encoded in the

diffuse scattering component. There is no unique recipe for

how to recover a disordered structure from the diffraction

data; in most cases Monte Carlo (MC) algorithms are used

(Welberry, 2010). The direct MC algorithm is typically based

on phenomenologically introduced interactions between

crystal building blocks (atoms or molecules) and may be

augmented by an Ising-like Hamiltonian to model the occu-

pational disorder. The small number of parameters that define

the model (see below) are varied in order to provide the best

possible agreement between observed and calculated diffuse

scattering intensities. Although this approach proved to be

very successful for many molecular crystals, for relaxor

compounds all published applications, using either direct or

reverse MC, were based on limited and distorted data sets and

therefore failed to predict real diffuse scattering maps (Bosak

et al., 2012). The collection of distorted data sets has recently

been augmented (Pasciak et al., 2012) by a study in which the

FlatCone multianalyser (IN20, Institut Laue–Langevin)

created artefacts related to the unreported but excessive

reciprocal-space layer thickness.

Here we propose a new approach to generating model

disordered structures for relaxor perovskites, which consists of

two steps. In the first step we create the parameterization with

a minimal number of parameters, based on the similarity

between thermal diffuse scattering and the relaxor-specific

component (Bosak et al., 2012). At the second stage we make

further use of the phonon-like parameterization – we generate

the disordered structure as a superposition of polar displace-

ments induced by phonon-like wave modulation. The disor-

dered displacement pattern naturally combines zero-average

polarization with nonzero local polarization and inherits the

correlation properties of the parameterization.

The generated disordered structure can be compared with

the microscopic models used to reproduce the macroscopic

physical behaviour of relaxors. While the ‘mainstream’

concept of polar nanoregions is incompatible with the

experimental findings, an alternative well developed dipole

glass approach (Timonin, 2009), rarely inspected against

diffuse scattering data, is shown here to give favourable

results.

2. Experimental procedure and parameterization

We have undertaken a synchrotron X-ray diffuse scattering

study of lead magnesium niobate–titanate, PbMg1/3Mb2/3O3–

PbTiO3 (PMN–PT); the diffuse scattering data set has been

collected at the Swiss–Norwegian Beam Lines at ESRF with a

mar345 image-plate detector. The complete description of the

experiment and data processing has been published elsewhere

(Bosak et al., 2012). Full three-dimensional maps of the

selected reflections can be found there as well. Here we recall

two-dimensional sections for more detailed analysis (Fig. 1).

In order to parameterize the experimental data we used an

approach exploiting a similarity between diffuse scattering in

relaxors and thermal diffuse scattering (TDS). For the one-

phonon TDS in the limiting case E << kBT (where E denotes

energy, kB the Boltzmann constant and T temperature), we

represent the intensity as

I / ~ff 2PbðQÞ �Q

T� RðQÞ�1

�Q; ð1Þ

neglecting all the ions but lead. Here Q is the scattering vector,

and the atomic scattering factor fPb, Debye–Waller factor WPb

and Pb ion delocalization over a shell of radius r0 (Zhukov et

al., 1995) are merged to give ~ffPb ¼ fPbðQÞ exp½�WPbðQÞ� �

sinð2�r0QÞ=Q. The symmetric pseudo-dynamic matrix �ðQÞ is

defined as

���ðQÞ ¼ ð1� �Þ�pc��ðQÞ þ ��fcc

��ðQÞ; ð2Þ

�pc��ðQÞ ¼ 2�11½1� cosð2�Q�Þ�

þ 2�44½2� cosð2�Q�Þ � cosð2�Q�Þ�; ð3Þ

�fcc��ðQÞ ¼�11f2� cosð2�Q�Þ½cosð2�Q�Þ þ cosð2�Q�Þ�g

þ ð2�44 ��12Þ½2� cosð2�Q�Þ cosð2�Q�Þ�; ð4Þ

���ðQÞ ¼ ð�44 þ�12Þ sinð2�Q�Þ sinð2�Q�Þ; ð5Þ

in the approximation of two-shell interactions (� 6¼ � 6¼ �, 0 �

� � 1 and �ij follow the Voigt notations, as for the elastic

tensor). For � = 1 equation (2) coincides with the expression

for a face-centred cubic lattice with nearest-neighbour inter-

actions only (in the actual primitive cubic lattice this corre-

sponds to next-nearest neighbours only). We have to recall

that our parameterization is focused on the lead sublattice

only and is targeted to reproduce the X-ray diffraction

patterns, where the contribution from lead dominates.

Besides the scaling factor, the model presented here

depends on three parameters only, i.e. �11/�12, �11/�44 and �.

Within the given formalism any values of � give the correct

shape of the diffuse clouds in the near proximity of Bragg

nodes, but the agreement at intermediate q is better for �close to 1. In the case of � = 1 extra peaks at R points [q =

( 12

12

12 )] are produced. The diffuse spots with non-integer Miller

indices (R points) visible in Fig. 1(b) may point either to a

correlated Mg/Nb disorder (nearest neighbours tend to be of a

different type, thus locally resembling the NaCl structure) or

to a significant contribution of Pb-correlated disorder, which

can be mimicked when � approaches 1 (see Fig. 1). The former

interpretation is supported to some extent by the electron

diffraction/electron microscopy observations, where ordered

domains of a few nanometres in size were showed in PMN

(Husson et al., 1988; Hilton et al., 1990).

The present parameterization not only gives a very detailed

match to the observed diffuse scattering but also automatically

provides the q�2 decay of diffuse intensity observed experi-

mentally (Gvasaliya et al., 2005). However, the para-

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1310 Alexei Bosak et al. � Glass-like structure of a lead-based relaxor ferroelectric J. Appl. Cryst. (2012). 45, 1309–1313

Figure 1Experimental (left) and model (right) diffuse scattering (�11/�12 = 1.25,�11/�44 = 2.5, � = 0.95) intensity distribution of PMN–PT in the (a) hk0and (b) hhl planes. Selected symmetry elements of the crystal wereapplied for the reconstruction of the experimental patterns.

meterization alone does not warrant that the corresponding

disordered structure can be found. In order to relate that

parameterization to some kind of realistic structure, in the

next chapter we propose a procedure that maps local dis-

placements of the lead sublattice.

It might be useful to note that the characteristic ‘butterfly’

shape of h00 spots with the local intensity minimum in the

equatorial plane [see the detailed description of three-

dimensional shape given by Bosak et al. (2012)] appears when

�12>�11ð1��44=�11Þ1=2��44, i.e. for �11/�44 = 2.5 the

limiting case for such a shape to be observed is �11/�12’ 2.67.

3. Modelling of real structure

The class of solutions for the correlated displacements

producing the pattern of equation (1) coincides with the

population of waves with Bose statistics at E << kBT. Formally,

any E/kBT range can be explored after subsequent modifica-

tion of equation (1) (Bosak et al., 2012), but for the region of

interest no substantial difference is apparent. 3N sine dis-

placement waves with randomly chosen phases were gener-

ated on a 64� 64� 64 unit-cell real-space cluster; polarization

vectors and the squared amplitudes were obtained by the

diagonalization of the corresponding pseudo-dynamic matrix

at each q point. As generated, the displacement pattern would

correspond to a snapshot of the phonon population with

E(q, j) << kBT. To satisfy the condition of fixed (by modulus)

Pb displacement from the average position, all the final

displacement vectors are normalized to the value of 0.07

lattice units as determined in the diffraction experiment

(Zhukov et al., 1995). Scattering intensity was calculated

directly and averaged over the Laue symmetry of the crystal in

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J. Appl. Cryst. (2012). 45, 1309–1313 Alexei Bosak et al. � Glass-like structure of a lead-based relaxor ferroelectric 1311

Figure 2Parameterization of scattering in the (a) hk0 plane and (b) hhl plane (leftpart of panels) compared with the calculated scattering from real-spacepatterns (right part of panels). Eight clusters are generated as describedin the text; Laue symmetry is applied to the scattering intensity.

Figure 3Dependence of diffuse intensity on the model cluster traced for aselection of directions in the proximity of the (200) Bragg node. Data arerepresented on a double logarithmic scale; solid lines correspond to a q�2

fit for the first eight points in each series.

Figure 4Arbitrarily chosen XY cuts of real-space clusters generated as describedin the text for (a) � = 0, (b) � = 0.95, (c) � = 1 and (d) randomdistribution. Arrows correspond to the scaled XY projection of the fixed-length displacement vector of Pb. Note the visual similarity of patterns(b) and (c) to the random distribution (d), while the statistical propertiesare different.

order to increase the signal-to-noise ratio; up to eight clusters

were generated for the same purpose.

While the procedure of displacement renormalization is

generally illegitimate, in this particular case it does not alter

significantly the generated scattering pattern. It practically

coincides with the analytical parameterization (see Fig. 2) and,

within the limitations of parameterization, with the experi-

mental data. As expected, it satisfies automatically the

condition of q�2 decay (see Fig. 3), in contrast to all PNR-

related models where the decay is at least q�4.

While the diffuse scattering pattern in the proximity of

Bragg nodes is weakly sensitive to the parameter �, the

influence on the real-space pattern is substantial, as follows

from inspection of Fig. 4. For � = 0 a tendency to the parallel

alignment of displacement vectors is evident, and a tendency

of antiparallel alignment may be suspected for � = 1. Even a

small deviation of � from 1 results in the weakening of the

tendency for antiparallel alignment.

As can be seen, no regions of any regular or irregular shape

with any preferential polarization can be identified. Actually,

for � > 0 it is hard to distinguish the displacement pattern

from the random one (Figs. 3b and 3c, to be compared with

Fig. 3d). This fact is in agreement with the inability of PNR

and/or PND models to reproduce the experimentally observed

diffuse scattering; on the other hand, a glass-like structure is

obvious.

4. Statistical properties of real-space structure

Once the real-space representations are available, statistically

weighted properties can be easily calculated. The local

polarization can be qualitatively related to the electric

susceptibility. In the very first attempt we calculate the local

polarization distribution using a moving sampling box of 4 �

4 � 4. As shown in Fig. 5, both the absolute value of local

polarization and its projection on a selected direction are

sensitive to the � value, which in turn corresponds to the

sensitivity of the model to high-q components of diffuse

scattering.

The shift of the centroid of local polarization (by modulus)

towards higher values (Fig. 5a) is characteristic of the

tendency to parallel alignment; the same effect manifests itself

in the broadening of the distribution of projected local

polarization (Fig. 5b). The bell-shaped distribution of the

projected polarization fits with the fact that, using two-

dimensional 93Nb NMR, the probability distribution of local

polarization in PMN is found to be Gaussian (Blinc et al.,

1999).

5. Conclusions

We have shown that parameterization of diffuse scattering in

lead-based ferroelectrics in terms of a pseudo-dynamical

matrix not only allows us to reproduce the observed diffuse

scattering, but also shows how to generate real-space config-

urations of local displacements of lead ions, compatible with

the available diffuse scattering data. The similarity between

the thermal diffuse scattering and the relaxor-specific scat-

tering component suggests a simple recipe to generate disor-

dered configurations as a sum of static waves of local

displacements. Future development of the proposed concept

might involve other sublattices; a more complex pseudo-

dynamical matrix could be evaluated from inelastic scattering

experiments or ab initio calculations.

Even such a simple model neglecting the contribution of

other sublattices points out that relaxor ferroelectrics belong

to a specific class of glass-like systems rather than being a

composite of nanoregions with any given shape, size and

polarization. This conclusion agrees with many observed

macroscopic properties of relaxors, such as glass-like relaxa-

tion and divergence between field-cooled and zero-field

susceptibilities (Timonin et al., 1995). Moreover, adding even a

single phonon-like wave would switch the system to a new

realization; notably the scattering pattern would remain

unaltered. A continuum of such structures, similar in the sense

of scattering but different microscopically, may serve as an

illustration of numerous metastable and nearly degenerate

states proposed within the dipole glass paradigm (Timonin,

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1312 Alexei Bosak et al. � Glass-like structure of a lead-based relaxor ferroelectric J. Appl. Cryst. (2012). 45, 1309–1313

Figure 5Absolute value of local polarization (a) and its projection onto the Xdirection (b) for the 4� 4� 4 sampling box and different � values; cyclicboundary conditions are imposed.

2009). We have to stress the fact that the proposed glass-like

real structure provides a q dependence of the diffuse scat-

tering intensity, including the realistic decay law q�2, that is

compatible with reliable experimental data, contrary to any

PNR-based models used to reproduce diffuse scattering so far.

The experimentally observed Gaussian-like shape of the local

polarization distribution is reproduced here as well; to our

knowledge such a comparison for real-space models has not

been reported before.

Further detailed studies of diffuse scattering evolution as a

function of temperature, pressure and electric field, coupled

with the microscopic level modelling proposed here, may help

to establish new links to the macroscopic properties.

We are grateful to Efim Kats (Institut Laue–Langevin,

Grenoble, France) and to Michael Krisch (ESRF, Grenoble,

France) for numerous fruitful discussions and encouragement.

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J. Appl. Cryst. (2012). 45, 1309–1313 Alexei Bosak et al. � Glass-like structure of a lead-based relaxor ferroelectric 1313