# A reassessment of the Burns temperature and its relationship to the diffuse
scattering, lattice dynamics, and thermal expansion in the relaxor
Pb(Mg1/3Nb2/3)O3

P. M. Gehring,1 H. Hiraka,2,3 C. Stock,4 S.-H. Lee,1 W. Chen,5 Z.-G. Ye,5 S.
B. Vakhrushev,6 and Z. Chowdhuri1 1NIST Center for Neutron Research, National
Institute of Standards and Technology, Gaithersburg, Maryland 20899-6100, USA
2Department of Physics, Brookhaven National Laboratory, Upton, New York
11973-5000, USA  3Institute for Materials Research, Tohoku University, Sendai,
980-8577, Japan  4ISIS Facility, Rutherford Appleton Laboratory, Chilton,
Didcot, OX11 0QX, UK  5Department of Chemistry, Simon Fraser University,
Burnaby, BC, V5A 1S6, Canada  6Ioffe Physical-Technical Institute, 26
Politekhnicheskaya, 194021 St. Petersburg, Russia

###### Abstract

We have used neutron scattering techniques that probe time scales from
$10^{-12}$ s to $10^{-9}$ s to characterize the diffuse scattering and low-
energy lattice dynamics in single crystals of the relaxor PbMg1/3Nb2/3O3 from
10 K to 900 K. Our study extends far below $T_{c}=213$ K, where long-range
ferroelectric correlations have been reported under field-cooled conditions,
and well above the nominal Burns temperature $T_{d}\approx 620$ K, where
optical measurements suggest the development of short-range polar correlations
known as “polar nanoregions” (PNR). We observed two distinct types of diffuse
scattering. The first is weak, relatively temperature independent, persists to
at least 900 K, and forms bow-tie-shaped patterns in reciprocal space centered
on $(h00)$ Bragg peaks. We associate this primarily with chemical short-range
order. The second is strong, temperature dependent, and forms butterfly-shaped
patterns centered on $(h00)$ Bragg peaks. This diffuse scattering has been
attributed to the PNR because it responds to an electric field and vanishes
near $T_{d}\approx 620$ K when measured with thermal neutrons. Surprisingly,
it vanishes at 420 K when measured with cold neutrons, which provide $\sim 4$
times superior energy resolution. That this onset temperature depends so
strongly on the instrumental energy resolution indicates that the diffuse
scattering has a quasielastic character and demands a reassessment of the
Burns temperature $T_{d}$. Neutron backscattering measurements made with 300
times better energy resolution confirm the onset temperature of $420\pm 20$ K.
The energy width of the diffuse scattering is resolution limited, indicating
that the PNR are static on timescales of at least 2 ns between 420 K and 10 K.
Transverse acoustic (TA) phonon lifetimes, which are known to decrease
dramatically for wave vectors $q\approx 0.2$ Å-1 and $T_{c}<T<T_{d}$, are
temperature independent up to 900 K for $q$ close to the zone center. This
motivates a physical picture in which sufficiently long-wavelength TA phonons
average over the PNR; only those TA phonons having wavelengths comparable to
the size of the PNR are affected. Finally, the PMN lattice constant changes by
less than 0.001 Å below 300 K, but expands rapidly at a rate of $2.5\times
10^{-5}$ 1/K at high temperature. These disparate regimes of low and high
thermal expansion bracket the revised value of $T_{d}$, which suggests the
anomalous thermal expansion results from the condensation of static PNR.

###### pacs:

77.84.Dy, 77.65.-j, 77.80.Bh

##  I Introduction

The concept of nanometer-scale regions of polarization, randomly embedded
within a non-polar cubic matrix, has become central to attempts to explain the
remarkable physical properties of relaxors such as PbMg1/3Nb2/3O3 (PMN) and
PbZn1/3Nb2/3O3 (PZN). Ye_review ; Park ; Bokov_rev ; Vug06:73  The existence
of these so-called “polar nanoregions” (PNR) was first inferred from the optic
index of refraction studies of Burns and Dacol on PMN, PZN, and other related
systems, Burns  and later confirmed using many different experimental
techniques including x-ray and neutron diffraction, Bonneau89:24 ; Mathan91:3
; Zhao ; Hirota06:75  207Pb NMR, Blinc  and piezoresponse force microscopy.
Shvartsman04:69  Early small-angle x-ray scattering and neutron pair
distribution function (PDF) measurements on PMN by Egami et al. cast doubt on
the nano-domain model of relaxors. Egami  However, the recent PDF analysis of
Jeong et al., which shows the formation of polar ionic shifts in PMN below
$\approx 650$ K, and which occupy only one third of the total sample volume at
low temperatures, provides convincing support for the existence of PNR.
Jeong05:94  Neutron inelastic scattering data published by Naberezhnov et al.
on PMN offered the first dynamical evidence of PNR in the form of a prominent
broadening of the transverse acoustic (TA) mode that coincides with the onset
of strong diffuse scattering at 650 K, Naberezhnov ; Koo  roughly the same
temperature ($\approx 620$ K) as that reported by Burns and Dacol in their
optical study of PMN. This temperature, commonly known as the Burns
temperature, and denoted by $T_{d}$ in the original paper, is widely viewed as
that at which static PNR first appear. Likely condensing from a soft TO mode,
distinctive butterfly-shaped and ellipsoidal diffuse scattering intensity
contours centered on $(h00)$ and $(hh0)$ Bragg peaks, respectively, are seen
below $T_{d}$ in both PMN Vakhrushev_JPCS ; You ; Xu_TOF  and PZN. Xu06:5
Similar diffuse scattering patterns were subsequently observed in solid
solutions of PMN and PZN with PbTiO3 (PMN-$x$PT and PZN-$x$PT); however these
patterns appear only on the Ti-poor (relaxor) side of the well-known
morphotropic phase boundary (MPB). Mat06:74 ; Xu04:70  The polar nature of the
strong diffuse scattering, and thus its association with the formation of PNR,
was unambiguously established by several electric field studies of PMN
Vakhrushev_efield ; Stock:unpub  and PZN-8%PT, Gehring_efield ; Xu_memory ;
Xu_redistribution ; Xu06:5  all of which showed dramatic changes in the shape
and intensity of the diffuse scattering as a function of field strength and
field orientation. The Burns temperature $T_{d}$ thus represents what is
arguably the most important temperature scale in relaxors and is several
hundred degrees Kelvin higher than the critical temperature $T_{c}$ (which for
PMN is $\approx 210$ K, but defined only in non-zero electric field).

Figure 1:  Top - temperature dependence of the TA phonon energy width
$\Gamma_{TA}$ measured with thermal neutrons by Wakimoto et al. Waki_sm
Middle - diffuse scattering intensity contours measured at (110) below (300 K)
and just above (650 K) the Burns temperature $T_{d}$ using cold neutrons by
Hiraka et al. Hiraka  Bottom - diffuse scattering temperature dependence
measured with cold neutrons (open circles) and thermal neutrons (dashed line)
by Hiraka et al. Hiraka

Recent studies of the neutron diffuse scattering in PMN by Stock et al., Stock
and especially those by Hlinka et al. Hlinka_110  and Gvasaliya et al.,
Gvasaliya ; Gvasaliya2  have proven to be extremely important because they
demonstrated the utility of cold neutron spectroscopy to the study of
relaxors. Generally speaking cold neutrons are ill-suited to lattice dynamical
studies because the longer wavelengths necessarily limit their use to the
study of comparatively fewer (lower-$Q$) Brillouin zones. On the other hand
cold neutron spectrometers provide significantly better energy ($\hbar\omega$)
and wave vector ($q$) resolution than do their thermal neutron counterparts.
In addition, both the (110) and the (100) Brillouin zones of PMN are
accessible using cold neutrons with wavelengths $\lambda\approx 4.26$ Å. The
combined cold and thermal neutron study of PMN by Hiraka et al. exploited this
fact and uncovered several major new findings, two of which are summarized in
Fig. 1. Hiraka  The first finding is that the temperature at which the strong
diffuse scattering vanishes depends on whether the measurement is made with
cold or thermal neutrons, i. e. the value of $T_{d}$ depends on the energy
resolution of the spectrometer. This fact, which is illustrated in the bottom
panel of Fig. 1, indicates that the diffuse scattering in PMN, and most likely
other relaxors, contains a substantial quasielastic component. However, a
consensus on this issue is lacking; while the finding of Hiraka et al. is
consistent with the study of Gvasaliya et al., Gvasaliya ; Gvasaliya2  it
contradicts that of Hlinka et al. Hlinka_110 ; eres  A second major finding is
displayed in the middle panel of Fig. 1 where intensity contours of the
diffuse scattering measured with cold neutrons around (110) are shown to
exhibit markedly different reciprocal space geometries at 300 K and 650 K.
These data unambiguously demonstrate the presence of two distinct types of
diffuse scattering in PMN. As the PNR are absent at 650 K $\geq T_{d}$, the
physical origin of the bow-tie-shaped diffuse scattering cross section is
believed to come primarily from the underlying chemical short-range order. We
shall refer to this as CSRO, although another commonly-used term for this is
compositionally/chemically ordered regions (COR). Burton

An intriguing perspective on the Burns temperature in PMN is provided in the
top panel of Fig. 1 where the TA phonon energy width $\Gamma_{TA}$, which is
inversely proportional to the phonon lifetime, is plotted as a function of
temperature for PMN at the scattering vector $\vec{Q}=(2,0.2,0)$. These data
were measured with thermal neutrons by Wakimoto et al. Waki_sm  and are
consistent with those of Naberezhnov et al. in that the TA mode begins to
broaden at $T_{d}\approx 620$ K, the same temperature where the strong diffuse
scattering first appears. These data also show that the TA broadening achieves
a maximum value (minimum lifetime) near 420 K, which coincides with the value
of $T_{d}$ measured with cold neutrons. These data raise the question of how
to interpret the Burns temperature $T_{d}$ properly in that they paint a
picture, markedly different from that currently held, in which the PNR are
dynamic below $\sim 650$ K and condense into static entities only at a much
lower temperature of 420 K.

The goal of our study then is to determine the intrinsic value of the Burns
temperature $T_{d}$ and to clarify its relationship to the diffuse scattering,
lattice dynamics, and structure in PMN. To this end we have carried out
extensive measurements of the neutron diffuse scattering cross section from 10
K to 900 K, far below $T_{c}$ and well above the nominal value of $T_{d}$,
that probe timescales from $10^{-12}$ s to $10^{-9}$ s. We find that a
300-fold improvement in energy resolution over that used by Hiraka et al.,
Hiraka  obtained using neutron backscattering techniques reproduces the same
onset temperature for the diffuse scattering; hence the intrinsic Burns
temperature $T_{d}$ for PMN is 420 K. At the same time an enormous change in
the thermal expansion is observed near 420 K, which is indistinguishable from
zero at low temperature. Given the revised value of $T_{d}$, this result
implies the existence of a direct influence of the PNR on the intrinsic
structural properties of PMN.

We also present new data on the effects of the PNR on the lattice dynamics
through measurements of the temperature and wave vector dependence of the
long-wavelength TA phonon energy width $\Gamma_{TA}$ measured near (110) from
25 K to 900 K. We find that TA modes with reduced wave vectors $q\ll 0.2$ Å-1
exhibit the same energy width at all temperature whereas those with $q\approx
0.2$ Å-1 exhibit a strongly temperature-dependent broadening similar to that
shown in the top panel of Fig.1. This behavior contrasts with that observed in
thermal neutron studies of the TO mode, which exhibits a broadening for all
$q\leq 0.2$ Å-1. Previous neutron scattering work on PMN-60%PT by Stock et
al., Stock06:73  a material in which there is no strong, temperature dependent
diffuse scattering, and thus no polar nanoregions, found no evidence of any TA
phonon broadening. In this context, our data lend extremely strong support to
the PNR model: the lifetimes of TA modes with wavelengths comparable in size
to the PNR are strongly diminished by the PNR, whereas long-wavelength (low
$q$) TA phonons simply average over the PNR and are unaffected.

##  II Experimental Details

The neutron scattering data presented here were obtained using the BT9 thermal
neutron triple-axis spectrometer, the SPINS cold neutron triple-axis
spectrometer, and the cold neutron High Flux Backscattering Spectrometer
(HFBS), all of which are located at the NIST Center for Neutron Research
(NCNR). On BT9, measurements of the phonons and diffuse scattering were made
at a fixed final (thermal) neutron energy $E_{f}=$14.7 meV ($\lambda=2.36$ Å)
using the (002) Bragg reflection of highly-oriented pyrolytic graphite (HOPG)
crystals to monochromate and analyze the incident and scattered neutron beams,
respectively. Horizontal beam collimations were 40′-47′-S-40′-80′ (S =
sample). A special, non-standard high $q$-resolution configuration was
employed to measure the thermal expansion in which the (004) Bragg reflection
from a perfect Ge crystal was used as analyzer and horizontal beam
collimations were tightened to 15′-47′-S-20′-40′. The choice of Ge was
motivated by the close matching between the PMN (022) (1.431 Å) and Ge (004)
(1.414 Å) $d$-spacings, which provides a significant improvement in the
instrumental $q$-resolution. Xu_Acta  On SPINS, which sits on the cold neutron
guide NG5, the phonon and diffuse scattering measurements were made at a fixed
final neutron energy $E_{f}=4.5$ meV ($\lambda=4.264$ Å) also using the (002)
Bragg reflection of HOPG crystals as monochromator and analyzer. A liquid-
nitrogen cooled Be filter was located after the sample to remove higher order
neutron wavelengths from the scattered beam, and horizontal beam collimations
were set to guide-80′-S-80′-80′. The resultant elastic ($\hbar\omega=0$)
energy resolution for the SPINS measurements was $\delta E=0.12$ meV half-
width at half-maximum (HWHM).

The High-Flux Backscattering Spectrometer was used to look for dynamics that
might be associated with the strong diffuse scattering below $T_{d}$. This
instrument uses a mechanically-driven Si(111) monochromator to Doppler shift
the energies of incident neutrons over a narrow range centered about 2.08 meV.
Neutrons are backscattered from the monochromator and proceed towards the
sample where they are scattered into a 12 m2 array of Si (111) crystals that
serve as analyzer. These neutrons are then backscattered a second time by the
analyzer, which selects the final neutron energy $E_{f}=2.08$ meV, into a
series of detectors positioned about the sample. The effective angular
acceptance of each detector is $\approx 15^{\circ}$. The HFBS instrument is
described in further detail elsewhere. Meyer03:74  The elastic energy
resolution for the HFBS measurements described here was $\delta E=0.4$ $\mu$eV
(HWHM).

Two high-quality single crystals of PMN, labeled PMN #4 and PMN #5, were used
in this study; both were grown using a top-seeded solution growth technique.
Ye  The crystal growth conditions were determined from the pseudo-binary phase
diagram established for PMN and PbO. The PMN #4 and #5 crystals weigh 2.7 g
(0.33 cm3) and 4.8 g (0.59 cm3), respectively. At 300 K the mosaic of each
crystal measured at (220) is less than 0.04∘ full-width at half-maximum
(FWHM). Loss of PbO, the formation of a pyrochlore phase, and the reduction of
Nb5+ are known to occur in PMN single crystals when subjected to high
temperatures under vacuum for extended periods of time. This process results
in a dramatic blackening of the crystal, which is normally of a transparent
gold/amber color. While dielectric measurements on such darkened crystals
reportedly show little difference from those on unheated samples,
Vakhrushev_private  our measurements reveal a diminishment of the diffuse
scattering intensity after sustained and repeated heating. Therefore
experiments on the larger PMN crystal #5 were limited to 600 K or less, while
PMN crystal #4 was used to obtain data above 600 K.

Both samples were mounted with an [001] axis oriented vertically, giving
access to reflections of the form $(hk0)$. For the high temperature
experiments, PMN crystal #4 was wrapped in quartz wool, mounted in a niobium
holder secured by tungsten wire, and then loaded into a water-cooled furnace
capable of reaching temperatures from 300 K to 1800 K. PMN crystal #5 was
mounted onto an aluminum sample holder assembly placed inside an aluminum
sample can, and then loaded inside the vacuum space of a closed-cycle 4He
refrigerator that provides access to temperatures from 10 K to 700 K. Each
sample has a cubic lattice spacing of $a=4.05$ Å at 300 K; thus 1 rlu
(reciprocal lattice unit) equals $2\pi/a=1.55$ Å-1.

##  III Origins of the Diffuse Scattering: PNR versus CSRO

Figure 2:  Diffuse scattering intensity contours in PMN measured on SPINS near
(100) are shown using a linear gray scale at (a) 300 K, (b) 650 K, and (c) 900
K. Diffuse scattering intensity contours measured on BT9 at 900 K near (300)
using thermal neutrons are shown in (d). The data shown in panels (a) and (b)
are from Hiraka et al. Hiraka  Schematic diagrams of the diffuse scattering
intensity contours below (Low-$T$) and at $T_{d}$ (High-$T$) are shown in the
upper right. Figure 3:  Contour plot of the difference in diffuse scattering
intensity near (110) measured at 550 K and 400 K. The result is the
characteristic figure “8” pattern, observed previously by Vakhrushev et al.
Vakhrushev

The diffuse scattering in PMN crystal #5 was studied by Hiraka et al. Hiraka
at and below the nominal Burns temperature $T_{d}$ with good energy resolution
($\delta E\approx 0.12$ meV HWHM) using cold neutrons. Data were measured in
the (100) and (110) Brillouin zones and are represented schematically in the
upper right portion of Fig. 2. The horizontal and vertical axes in this figure
correspond to the components of the scattering vector $\vec{Q}=(hk0)$, which
are measured in reciprocal lattice units (rlu). The “Low-$T$” ($T<T_{d}$)
regime is dominated by contributions from the PNR, where the diffuse
scattering intensity contours near (100) resemble a butterfly and those near
(110) resemble an ellipse for which the long axis is oriented perpendicular to
$\vec{Q}$. This is shown in Fig. 2 (a) where diffuse scattering data near
(100) at 300 K reveal an intense butterfly-shaped pattern. Hiraka  We note
here that we see no evidence of any “transverse component” to the diffuse
scattering in any of our PMN crystals like that reported previously by
Vakhrushev et al. Sergey_NSE  We assume that this component was due to the
crystal imperfections mentioned by these authors in their PMN sample, which
gave rise to a powder ring in their data at (100).

The same butterfly/ellipsoidal diffuse scattering geometry was shown to
persist in single crystals of PMN-$x$PT and PZN-$x$PT in studies by,
respectively, Matsuura et al. and Xu et al. for compositions spanning the Ti-
poor (relaxor) side of the morphotropic phase boundary (MPB). Mat06:74 ;
Xu04:70  These results also completely refute those of La-Orauttapong et al.
who reported that the orientation of the strong diffuse scattering varies with
Ti content in PZN-$x$PT and concluded that the PNR orientation changes with
doping. Gop ; comment  For Ti-rich (tetragonal) PMN-$x$PT compositions just
beyond the MPB, Matsuura et al. found that the strong, temperature dependent
diffuse scattering vanishes and is replaced by critical scattering. Mat06:74
Matsuura et al. also found that the $q$-integrated diffuse scattering
intensity increases with Ti content on the Ti-poor side of the MPB, peaks near
the MPB, then drops dramatically on crossing the MPB. This finding is
significant because it suggests that an intriguing and direct correlation
exists between the PNR and the piezoelectric coefficient $d_{33}$, which
exhibits the same dependence on Ti content. Park  A model based on pancake-
shaped, ferroelectric domains has been used successfully to fit the three-
dimensional diffuse scattering distributions measured in PZN-$x$PT with high-
energy x-rays. Xu04:70 ; Welberry74:06 ; Welberry38:05  A similar type of
real-space structure has been proposed to explain the diffuse scattering in
the relaxor KLixTa1-xO3. Waki74:06  On the other hand, alternative models
explaining the same diffuse scattering distributions have also been proposed.
Pasciak

In the “High-$T$” regime ($T\geq T_{d}$) there are no PNR, and the associated
butterfly-shaped diffuse scattering is no longer present. The weak diffuse
scattering that remains is thus argued to originate primarily from the
underlying chemical short range order (CSRO), which reflects weak correlations
between the Mg2+ and Nb5+ cations on the $B$-site of the perovskite $AB$O3
structure. In this regime the shapes of the diffuse scattering contours are
radically different, resembling a bow-tie in both $\vec{Q}=(h00)$ and
$\vec{Q}=(hh0)$ Brillouin zones in which the diffuse scattering extends mainly
parallel to $\vec{Q}$. The only difference between the contours near (100) and
(110) appears to be in the orientation of the triangular regions of diffuse
scattering, which point in towards (100), but away from (110). Data taken near
(100) at 650 K are displayed in Fig. 2 (b). Hiraka  At this temperature the
diffuse scattering intensities, shown using a linear gray scale, are much
weaker than those of the butterfly pattern at 300 K. We further note that the
intensities increase (become darker) from left to right in panels (b) and (c),
which corresponds to increasing $Q$. Given the $(\vec{Q}\cdot\vec{u})^{2}$
dependence of the neutron diffuse scattering cross section, this intensity
signature implies the presence of some short-range, correlated displacements
$\vec{u}$ since otherwise, if $u=0$, there would be no $Q$-dependence. Thus
the weak diffuse scattering is not solely due to CSRO.

In Fig. 3 we plot the difference between the diffuse scattering intensities
measured near $\vec{Q}=(110)$ at 400 K and 550 K. The resulting contours
approximately reproduce the “figure 8” pattern observed previously by
Vakhrushev et al., Vakhrushev , and indicates that the ellipsoidal diffuse
scattering has a strong temperature dependence. By contrast, the bow-tie-
shaped diffuse scattering is effectively subtracted out in this analysis,
which confirms that it has little to no temperature dependence. Thus the high-
temperature diffuse scattering is not associated with the Burns temperature
$T_{d}$ or with the formation of long-range polar correlations below $T_{c}$.
We note that such a strictly incoherent treatment of the high-temperature
(CSRO) and low-temperature (PNR) components of the total diffuse scattering,
as described here, ignores the inevitable cross terms that must exist between
them. On the other hand, if the high-temperature scattering does arise
primarily from CSRO, then it should be largely independent of the ionic
displacements that give rise to the butterfly-shaped diffuse scattering (PNR)
below $T_{d}$. The relative weakness of the high-temperature diffuse
scattering compared to that at low temperatures also suggests such cross terms
should be weak, and this appears to be supported by the simple subtraction
analysis presented in Fig. 3 in that one effectively recovers the ellipsoidal
(not bow-tie) intensity contours. For this reason we believe it is a
reasonable first approximation to treat the two diffuse scattering components
as being nearly independent.

Figure 4:  Diffuse scattering intensity measured on BT9 at 300 K as a function
of reduced momentum transfer relative to (100) (open circles) and (200) (solid
circles). The dotted lines isolate the diffuse scattering component from the
Bragg peak. The inset shows the orientation of the scan relative to the
familiar butterfly pattern measured near (200) at 100 K. Figure 5:  Diffuse
scattering contours measured on BT9 near (200) are shown at (a) 100 K, (b) 300
K, and (c) 600 K.

We extended the diffuse scattering measurements to temperatures well above
$T_{d}$ using PMN crystal #4, which we reserved for very high temperature
experiments. Previous data taken on PMN and PZN crystals heated to 1000 K
revealed significant evidence of sample decomposition, so we limited our
measurements to 900 K. Diffuse scattering intensity contours at 900 K for the
(100) and the (300) Brillouin zones are presented in Fig. 2 (c) and 2 (d),
respectively. Although the diffuse scattering near (100) is quite weak, it is
consistent with the bow-tie geometry observed in Fig. 2 (b) at 650 K. We
exploited the $Q^{2}$ dependence of the neutron diffuse scattering cross
section to obtain higher intensity by using thermal neutrons to access (300).
The (300) diffuse scattering intensity contours are shown in Fig. 2 (d), where
the bow-tie pattern observed at 650 K is still present at 900 K. The contours
are truncated for $h>3.02$ rlu because of a mechanical limit on the maximum
scattering angle available on BT9, but the triangular region on the low-$Q$
side of (300) is clearly evident. That the bow-tie-shaped diffuse scattering
persists to such high temperature provides what is perhaps the most convincing
evidence that it arises mainly from CSRO.

During the course of our measurements we noticed that the diffuse scattering
in PMN crystal #4 had diminished and was noticeably weaker than that in PMN
crystal #5, which we had never exposed to temperatures above 650 K. We also
observed a broad ring of scattering passing directly through $\vec{Q}=(300)$,
which is shown in Fig. 2 (d). This feature was never observed prior to heating
this crystal to 900 K. Given the length of time spent at high temperatures,
this feature most likely corresponds to a powder ring arising from partially
decomposed regions of PMN crystal #4, which had turned entirely black after
exposure to high temperatures. These regions do not affect any other data
presented in this paper because the 900 K measurements on PMN crystal #4 were
the last ones performed on this sample. Therefore the powder ring appears only
in Fig. 2 (d).

It is instructive to compare these results with those on PMN-60%PT, a
composition that lies well beyond the morphotropic phase boundary and
undergoes a first-order ferroelectric transition from a cubic to a tetragonal
phase near 540 K. An extensive study of this material using neutron and high-
energy x-ray scattering methods found no sign of the strong, butterfly-shaped
diffuse scattering at low temperatures. Stock06:73  This result lends further
support to our association of the strong, temperature dependent diffuse
scattering with the PNR, which are absent in PMN-60%PT. Neutron measurements
on PMN-60%PT do, however, reveal the presence of bow-tie shaped diffuse
scattering intensity contours at all temperatures studied, which supports the
identification of such diffuse scattering with chemical short-range order
between cations on the $B$ site of the PMN perovskite $AB$O3 structure. This
picture is supported by theoretical work Burton99:60  as well as 93Nb NMR,
Laguta03:67  electron microscopy, Boul94:108  and polarized Raman scattering
Svit03:68  measurements. All of these studies suggest that there is no
temperature dependence to the bow-tie shaped diffuse scattering below $\approx
1000$ K, which is consistent with our results on PMN over the extended
temperature range.

During our study of PMN we discovered that the diffuse scattering near (200)
is not as weak as previously believed. Vakhrushev ; You ; Hirota ; Gop  To
confirm this finding, we made detailed measurements of the diffuse scattering
intensity near (200) at 300 K and 100 K along a trajectory in reciprocal space
that follows one wing of the butterfly intensity contour; this is shown by the
dashed line in the inset to Fig. 4. The results of the 300 K scan are compared
to an identical scan measured in the (100) zone, both of which are shown in
Fig. 4. These data demonstrate that the (100) diffuse scattering cross
section, represented by the dotted lines passing through the open circles, is
substantially larger than that at (200), designated by the solid circles. This
result supports the model of Hirota et al. in which the (unexpectedly) weak
(200) diffuse scattering cross section observed in PMN and other relaxors can
be explained by the presence of a uniform shift or displacement of the PNR
relative to the non-polar cubic matrix along the direction of the local PNR
polarization. Hirota  Indirect evidence for the existence of this shift has
been obtained from neutron scattering measurements of the anisotropic response
of the diffuse scattering in PZN-8%PT to an electric field applied along the
[001] direction. Gehring_efield

Fig. 5 shows diffuse scattering intensity contours measured on BT9 at 100 K,
300 K, and 600 K near(200); these data illustrate that the (200) diffuse
scattering intensity follows the same temperature dependence as that measured
in other Brillouin zones, where the diffuse scattering is much stronger. As
the temperature is raised the diffuse scattering intensity decreases in the
same manner as that previously measured and observed in the (100), (110), and
(300) Brillouin zones. This proves that the diffuse scattering measured at
(200) has the same origin as that in other zones, i. e. that it is associated
with the formation of PNR. At 600 K in panel (c) one can already see the
emergence of the bow-tie-shaped diffuse scattering that is otherwise obscured
by the stronger PNR-related diffuse scattering at lower temperatures. These
data are important because they support the mode-coupling analysis of Stock et
al. Stock , which assumes that the diffuse scattering in PMN in the (200) and
(220) Brillouin zones is much weaker than that in the (110) zone. Thus we
emphasize that while the neutron diffuse scattering cross-section near (200)
is not zero, it is small and consistent with previous structure factor
calculations.

##  IV Diffuse Scattering Dynamics: A Reassessment of the Burns Temperature

Figure 6:  Diffuse scattering intensity contours in PMN measured on the NCNR
Disk Chopper Spectrometer (DCS) near (100) are plotted on a logarithmic gray
scale at 300 K. Open circles represent the reciprocal space locations of the
detectors of the High Flux Backscattering Spectrometer relative to the
butterfly-shaped intensity contours. Each open circle is a separate detector,
and the detector numbers are indicated. These DCS data are taken from Xu et
al. Xu_TOF  Figure 7: (Color online) (a) Contour plot of the diffuse
scattering intensity measured on the HFBS as a function of energy transfer and
temperature. Contours are shown on a linear intensity scale; dashed lines
indicate the full-width at half-maximum (FWHM) of the peak linewidth at each
temperature. Data were summed over detectors 10-16 as illustrated in Fig. 6.
Panels (b), (c), and (d) show inelastic scans at 700 K, 300 K, and 100 K. The
horizontal bar in panel (c) represents the instrumental FWHM elastic energy
resolution ($2\delta E$).

The reciprocal space geometry of the strong diffuse scattering in PMN was
first characterized using x-ray diffraction and is consistent with the neutron
scattering data we have presented here. Vakhrushev_JPCS ; You  The energy
resolution provided by x-ray diffraction ($\delta E\approx 1000$ meV) is
typically much broader than that of thermal neutrons ($\delta E\approx 1$
meV); thus it was assumed that the strong diffuse x-ray scattering originated
from low-energy, soft, transverse optic (TO) phonons that were captured by the
large energy resolution. You ; Comes70:26  However the cold neutron data of
Hiraka al. provide a much narrower elastic energy resolution of $\approx 0.12$
meV HWHM and show, unambiguously, that the diffuse scattering cross section
contains a component that is static on timescales of at least $\sim 6$ ps
below 420 K as illustrated in Fig. 1. This result was subsequently confirmed
on a separate PMN crystal by the neutron study of Gvasaliya et al. which
employed comparable energy resolution. Gvasaliya_JPhysC05  Hence the observed
strong diffuse scattering cannot simply be the result of a soft, low-lying TO
phonon. The TO mode must condense and/or broaden sufficiently to produce the
elastic diffuse scattering cross section observed by Hiraka et al. Such a
scenario is in fact suggested by the corresponding thermal neutron data taken
on BT9 using a somewhat coarser energy resolution of $\approx 0.50$ meV HWHM.
As shown in Fig. 1, an apparent elastic diffuse scattering cross section is
observed up to temperatures as high as 650 K.

Figure 8:  (a) Temperature dependence of the diffuse scattering measured on
the HFBS; data are integrated in energy from $\pm$ 5 $\mu eV$ and in $Q$ over
detectors 10-16 as illustrated in Fig. 6. (b) Temperature dependence of the
diffuse scattering intensity measured on SPINS at (1.05,0.95,0) by Hiraka et
al. Hiraka  The SPINS energy resolution of $\delta E$=120 $\mu$eV HWHM
provided the energy integration.

Motivated by these results, we looked for evidence of a dynamic component of
the strong diffuse scattering using the NCNR High-Flux Backscattering
Spectrometer (HFBS), which provides an elastic energy resolution $\delta
E=0.4$ $\mu$eV HWHM. We oriented PMN crystal #4 in the $(hk0)$ scattering
plane, which is the same geometry used for the triple-axis studies discussed
in the previous sections. Fig. 6 displays the resulting locations of each of
the 16 HFBS detectors relative to the butterfly-shaped diffuse scattering
pattern at (100) measured previously using the NCNR Disk Chopper Spectrometer
(DCS). Xu_TOF  From this figure it can be seen that detectors 10 through 16
sample different parts of the wings of the strong diffuse scattering. In
particular, detectors 13 and 14 lie close to the (100) Bragg peak. Because the
instrumental $Q$-resolution of the HFBS is relatively poor, we checked for the
presence of Bragg contamination in the integration analysis by removing
contributions from detectors 13 and 14. This did not change any of the
results. Moreover the PMN study by Wakimoto et al. showed the (202) Bragg peak
intensity changes by less than 10% between 50 K and 300 K. Waki_coupling
Therefore, in the following analysis we have integrated the intensity from
detectors 10 through 16.

The energy dependence of the diffuse scattering as a function of temperature
is illustrated in Fig. 7. Panel (a) shows a color contour plot of the peak
linewidth in energy as a function of temperature after subtraction of a high
temperature background. This background was taken to be the average of the
intensity measured at 600 K and 700 K because no strong diffuse scattering is
observed with either cold neutron triple-axis or neutron backscattering
methods at these temperatures. The distance between the dashed lines
represents the measured energy width (full-width at half-maximum) as a
function of temperature. Typical energy scans are displayed in panels (b),
(c), and (d) at 700 K, 300 K, and 100 K, respectively. We see that the peak
centered at $\hbar\omega=0$ is equal to the instrumental energy resolution
$\delta E=0.4$ $\mu$eV HWHM at all temperatures. Based on this analysis, we
conclude that the strong diffuse scattering is elastic on timescales of at
least $\tau\approx\hbar/\delta E\sim 2$ ns. Our results are thus consistent
with the neutron-spin echo study on PMN of Vakhrushev et al. Sergey_NSE

Fig. 8 (a) shows the temperature dependence of the diffuse scattering
intensity integrated over $\pm 5$ $\mu$eV. These data are compared to those
measured on PMN crystal #5 using the cold neutron spectrometer SPINS, which
are plotted in panel (b). That the temperature dependences from the
backscattering and SPINS measurements agree within error in spite of a
300-fold (120 $\mu$eV/0.4 $\mu$eV) difference in energy resolution proves that
static, short-range polar order first appears at much lower temperatures than
has been understood from previous data measured with much coarser energy
resolution. These data thus demand a revised value for the Burns temperature:
$T_{d}=420\pm 20$ K. We mention, however, that if the diffuse scattering
energy width obeys an Arrhenius or some power law form, which has been
suggested for spin glasses in Ref. Som82:25, , then a more detailed analysis
based on data taken closer to the onset of the diffuse scattering at 420 K
will be required to confirm (or reject) the existence of such alternative
dynamic contributions to the diffuse scattering energy width.

To gain a better understanding of the apparent quasielastic nature of the
diffuse scattering, we examined the temperature dependence of the low-energy
transverse acoustic (TA) modes in greater detail. In particular we focused on
measurements of the temperature dependent broadening of the transverse
acoustic phonon over a range of reduced wave vectors $q$ that approach the
zone center ($q=0$). These data are discussed in the following section.

##  V TA and TO Phonons: Effects of the PNR

An extensive series of inelastic measurements were made on both PMN crystals
#4 and #5 in the (110) Brillouin zone using the SPINS spectrometer in an
effort to map out the temperature dependence of the TA mode for reduced wave
vectors $q\ll q_{wf}$ and $q\approx q_{wf}$, where $q_{wf}=0.14$ rlu $\approx
0.2$ Å-1 is the wave vector associated with the so-called “waterfall” effect,
and below which the long-wavelength, soft TO modes in PMN, PZN, and PZN-8%PT
are observed to broaden markedly at temperatures below $T_{d}$. Gehring_pzn8pt
; Gehring_Aspen ; Waki_sm ; Gehring_sm ; Gehring_pzn  This zone was chosen
because the TA phonon dynamical structure factor for (110) is much larger than
that for (100). The TA phonon energy lineshapes were studied at two very
different values of the reduced wave vector $q$, measured relative to the
(110) zone center, for temperatures between 100 K and 900 K. These data are
presented in Fig. 9. The data shown on the left-hand side of this figure
correspond to $\vec{Q}=(1.035,0.965,0)$ or $q=\sqrt{2}\cdot 0.35$ rlu = 0.05
rlu = 0.077 Å-1, which is less than half of $q_{wf}$. These data show that the
TA lineshape at this small wave vector remains sharp and well-defined at all
temperatures and has an intrinsic energy width that is larger than the
instrumental resolution (shown by the small, solid horizontal bar at 500 K).
By contrast, the data on the right-hand side of Fig.9 correspond to
$\vec{Q}=(1.1,0.9,0)$ or $q=\sqrt{2}\cdot 0.10$ rlu = 0.14 rlu = 0.22 Å-1,
which is nearly equal to $q_{wf}$. In this case it is quite evident that the
TA lineshape broadens dramatically, becoming increasingly ill-defined below
$T_{d}$, especially at 300 K, but then sharpens at lower temperature, e. g. at
100 K. This behavior is the same as that observed for the soft, zone-center TO
mode in PMN by Wakimoto et al. measured at (200). Waki_sm

Figure 9:  TA phonon lineshapes in PMN measured on SPINS with cold neutrons at
$\vec{Q}=(1.035,0.965,0)$ and (1.1,0.9,0) from 100 K to 900 K.

In addition to these cold neutron measurements of the TA mode using SPINS,
other measurements were made using thermal neutrons on BT9 to characterize the
TO mode lineshapes for reduced wave vectors $q\approx q_{wf}$ from 300 K to
900 K. In these experiments data were taken using a water-cooled furnace with
PMN crystal #4 in both the (300) and (100) Brillouin zones for $q=q_{wf}=0.14$
rlu and are shown in Fig.10. As can be seen on the left-hand side of this
figure, both the TA and TO modes are well-defined at 900 K. The scattering
intensity below 1-2 meV increases sharply because of the comparatively coarser
energy resolution intrinsic to BT9 ($\approx 0.5$ meV HWHM), which uses
thermal neutrons, compared to that on SPINS. Even so, the TA mode is easily
seen at both 900 K and 700 K. However at 500 K the TA mode is less well-
defined. This occurs in part because the mode has broadened, but also because
the low-energy scattering has increased, which results from the onset of the
strong diffuse scattering due to the PNR. At 300 K the TA mode is nearly
indistinguishable from the sloping background, and the TO mode has become
significantly damped. One also sees a gradual softening of the TO mode from
900 K down to 300 K that is of order 1-2 meV. The same data were taken in the
(100) zone, which appear in the right-hand portion of Fig. 10. Essentially the
same trends are observed in this zone, with the TA mode again becoming almost
indistinguishable from background at 300 K. One difference is that the TA mode
is better separated at all temperatures from the steep increase in scattering
seen at low energies. These data are consistent with the conjecture of Stock
et al. Stock  that the low-energy TA phonon is coupled to the diffuse
scattering centered around the elastic position. The inset on the top panel on
the right-hand side displays the full TA-TO phonon spectrum at 900 K out to 20
meV.

Figure 10:  TA and TO phonon lineshapes in PMN measured on BT9 with thermal
neutrons near the waterfall wave vector at $\vec{Q}=(3,-0.14,0)$ and
(1,-0.14,0) from 300 K to 900 K.  Figure 11:  The square of the phonon energy
is plotted versus temperature for the zone-center TO modes measured at (200)
(open diamonds) and (220) (solid diamonds), taken from Wakimoto et al. and
Stock et al. Waki_sm ; Stock , as well as for three TA modes measured at
(1.035,0.965,0) (open triangles), (1.1,0.9,0) (open circles), and (3,-0.14,0)
(solid circles). Linear behavior is consistent with that expected for a
conventional ferroelectric soft mode.

The square of the energies of both the TA and TO modes presented in the two
previous figures are plotted versus temperature in Fig. 11. This is done to
draw attention to the similarity between the low-energy lattice dynamics of
PMN and the behavior expected for a conventional ferroelectric, for which the
soft mode frequency squared $\omega^{2}$ varies linearly with $(T-T_{c})$.
Data for the zone-center soft mode measured at (200) and (220) have been taken
from Wakimoto et al. and Stock et al. and added to the top of Fig. 11 for ease
of comparison. Stock ; Waki_sm  The data taken by Wakimoto et al. are based on
energy scans measured at constant-$Q$ at the zone center, whereas Stock et al.
determined the zone-center TO energies by extrapolating from data obtained at
non-zero $q$. The extrapolation technique permits phonon energies to be
extracted in the temperature range where the zone-center TO mode is heavily
damped. These values for the zone-center TO phonon energy have been confirmed
by infrared spectroscopy. Bovtun04:298  What this figure immediately reveals,
then, is a surprising ferroelectric character of the zone center TO mode above
$T_{d}$ and below $T_{c}$, and a corresponding change in the behavior of the
TA mode energy that is bracketed by the same two temperature scales, which are
denoted by the two vertical dash-dotted lines. We note that no such change in
the acoustic phonons was observed in PMN-60%PT where PNR and the associated
strong diffuse scattering are absent; therefore the softening of the TA mode
is directly related to the development of PNR. Stock06:73  A further
interesting feature is the minimum that is present in all of these data near
420 K, which is the same temperature at which the strong diffuse scattering
first appears when measured with high energy resolution, namely the revised
value of the Burns temperature. Therefore, the onset of the diffuse scattering
is directly associated with the softening of the TO mode; this is yet further
evidence that it is associated with the formation of static, short-range polar
correlations.

It is very important to note that the square of the TA phonon energy measured
at $\vec{Q}$=(1.1,0.9,0), which corresponds to $q\sim 0.14$ rlu, shows a much
more pronounced minimum at 420 K than does that measured at
$\vec{Q}$=(1.035,0.965,0), which corresponds to $q\sim 0.05$ rlu. This shows
that long-wavelength TA phonons exhibit a much weaker response to the
formation of static short-range, polar correlations. This can be understood in
terms of a simple physical picture in which those phonons with wavelengths
comparable to the size of the PNR are strongly affected (damped) by the PNR
whereas longer wavelength phonons simply average over the PNR and are thus not
affected by the presence of static, short-range polar correlations. This idea
was previously proposed to explain the anomalous damping of the TO mode for
wave vectors $q\leq q_{wf}$ near the zone center. Gehring_pzn8pt  However, no
strong diffuse scattering is seen in PMN-60%PT and thus no PNR are present,
even though the anomalous TO broadening is still observed; hence this TO
broadening, which gives rise to the waterfall effect, cannot be associated
with the presence of static, short-range polar correlations. On the other
hand, the idea that the acoustic phonons are affected by PNR is confirmed by
the absence of any such acoustic phonon broadening in PMN-60%PT. Thus PNR have
a significant effect on the low-energy acoustic phonons over a limited range
of reduced wave vectors that may be related to the size of the PNR. In light
of the diffuse and inelastic scattering data that have been analyzed so far,
we now turn to the detailed measurement of the thermal expansion in PMN.

##  VI Thermal Expansion: Invar-like Effect Below $T_{d}$

A variety of experimental techniques have been used to measure the thermal
expansion in both single crystal and powder samples of PMN, which includes
x-ray diffraction, laser dilatometry, and neutron diffraction. Shebanov ;
Arndt ; Dkhil  Here we present extremely high $q$-resolution neutron
measurements ($\delta q\approx 0.0005$ rlu = 0.0008 Å-1 HWHM) of the cubic
unit cell lattice parameter $a$ of PMN crystal #5 using the special triple-
axis configuration described in the experimental section. This configuration,
which employs a perfect single crystal of Ge as analyzer, also provides an
exceptional energy resolution $\delta E\approx 5-10$ $\mu$eV HWHM. The
resulting data are plotted in Fig. 12 in terms of the strain $a/a_{0}-1$,
where $a_{0}$ is the lattice parameter at 200 K. Measurements were made on
heating from 30 K to 580 K using a closed-cycle 4He refrigerator after having
first cooled to 30 K. X-ray data measured by Dkhil et al. Dkhil  on a single
crystal of PMN, shown as open squares, are provided for comparison.

Figure 12:  Lattice strain of single crystal PMN derived from the (220) Bragg
reflection measured from 30 K to 580 K on BT9. Data were obtained on heating
using a perfect Ge(004) analyzer, an incident neutron energy $E_{i}=14.7$ meV,
and horizontal beam collimations of 15′-47′-S-20′-40′ to obtain extremely high
$q$-resolution. X-ray data from Dkhil et al. are indicated by the open squares
for comparison. Dkhil .

Several features are interesting to note. First, at low temperature the system
exhibits an invar-like effect in which the cubic lattice parameter changes by
less than 0.001 Å; indeed, the data below 320 K are consistent with a thermal
expansion of zero. At higher temperatures, however, the average thermal
expansion is $2.5\times 10^{-5}$ 1/K. That these disparate regions of null and
high rates of thermal expansion bracket the revised value for the Burns
temperature suggests that a direct connection exists between the onset of
static, short-range polar correlations and the structural properties of PMN.
This behavior seems to be consistent with that of PZN, which also exhibits an
increase in the thermal expansion at temperatures above that where the diffuse
scattering first appears. Agrawal87:34

There is ample evidence of similar behavior reported by other groups in
samples of PMN and PMN-$x$PT. A low-temperature invar-like effect was observed
in single crystal PMN-10%PT, where a transition to a high rate of thermal
expansion was found at 400 K; the thermal expansion for this sample at high
temperature is $1\times 10^{-5}$ 1/K, which is very close to that measured
here. Gehring_pmn10pt  The x-ray study of ceramic samples of PMN by King et
al. also show a transition between low and high rates of thermal expansion,
but larger values for both. King  X-ray and neutron work conducted by Bonneau
et al. on PMN powders yielded onset temperatures and values for the thermal
expansion consistent with our single crystal measurements. Bonneau91:91
Finally, an invar effect was also observed in the laser dilatometry study by
Arndt and Schmidt on a ceramic sample of PMN, which covered a range from 300 K
to 800 K. Arndt

Even though there is a general trend towards a larger thermal expansion for
temperatures above that where the diffuse scattering is onset, there is some
sample dependence and some differences between powders and single crystals. As
noted by Ye et al., Ye10pt  powder measurements of PMN yield a different slope
for the thermal expansion measurements than do those of Dkhil et al. Dkhil
Also, studies using neutron strain scanning techniques found different thermal
expansion coefficients as a function of depth in single crystal samples.
Conlon04:70 ; Xu05:74  Therefore, part of the discrepancy observed between
different samples may be associated with surface effects. In this regard, we
also note the presence of a change in the slope of the strain curve between
400 K and 500 K; since this change is not observed in other PMN studies, we
believe this feature to be sample-dependent and thus extrinsic.

The phase-shifted model of polar nanoregions proposed by Hirota et al.
provides a plausible starting point from which to understand the anomalous
invar-like behavior in PMN below the Burns temperature $T_{d}$. This model is
based on the observation that the Pb, Mg, Nb, and O ionic displacements
obtained from the diffuse scattering measurements of Vakhrushev et al.
Vakhrushev  on PMN can be decomposed into a center-of-mass conserving
component, consistent with the condensation of a soft, transverse optic mode,
and a uniform scalar component, corresponding to an acoustic phase shift. In
so doing Hirota et al. were able to reconcile the discrepancies between the
structure factors of the diffuse scattering and the soft TO modes and, in
particular, to explain the weakness of the diffuse scattering intensity
observed in the vicinity of the (200) Bragg peak. Hirota  The idea that the
PNR are uniformly displaced with respect to the underlying cubic lattice in a
direction parallel to the PNR polarization has already been used by Gehring et
al. to explain the persistence of the strong diffuse scattering in a single
crystal of PZN-8%PT against large electric fields applied along [001].
Gehring_efield  In that study it was shown that the diffuse scattering near
(003), which measures the projections of the ionic displacements along [001],
decreases under an external [001] field as expected. By contrast, the diffuse
scattering near (300), which measures the projections of the ionic
displacements along [100], i. e. perpendicular to the field direction, remains
unaffected even for field strengths up to 10 kV/cm. Thus a uniform polar state
is not achieved. This surprising behavior can be understood if one assumes
that the electric field applied along [001] removes or reduces the PNR shifts
along [001] while preserving those along [100]. The diffuse scattering should
then decrease anisotropically with field, as is observed. In the context of
the invar-like behavior of PMN below $T_{d}$, we speculate that such uniform
shifts of the PNR could effectively stabilize the lattice against subsequent
thermal expansion at lower temperature. Such a scenario is consistent with the
fact that both PMN-10%PT and PMN-20%PT retain cubic structures down to low
temperature when examined with high $q$-resolution neutron diffraction
methods. Gehring_pmn10pt ; Xu68:03

##  VII Discussion and Conclusions

We have presented a comprehensive neutron study of the diffuse scattering in
the relaxor PMN. We have greatly extended the timescales of previous neutron
measurements by taking data on three different spectrometers, BT9, SPINS, and
the HFBS, which provide elastic energy resolutions of 500 $\mu$eV, 120
$\mu$eV, and 0.4 $\mu$eV HWHM, respectively. While the backscattering data
represent a 300-fold improvement in energy resolution over those obtained by
Hiraka et al. Hiraka  on SPINS, both yield an onset temperature of $420\pm 20$
K for the diffuse scattering. This indicates that the PNR in PMN are static on
timescales of at least 2 ns below 420 K, but are apparently dynamic at higher
temperatures. Thus the true Burns temperature in PMN, which was originally
interpreted by Burns and Dacol as the condensation temperature of static
regions of local, randomly-oriented, nanometer-scale polarization, Burns  is
$420\pm 20$ K, not 620 K. Independent evidence of the existence of purely
dynamic, short-range, polar correlations has been reported in PMN-55%PT, a
composition with no strong diffuse scattering and thus no static PNR, by Ko et
al. who observed phenomena that are typically related to the relaxation of
dynamic PNR. Ko  These include significant Brillouin quasielastic scattering,
a softening of the longitudinal acoustic phonon mode, and a deviation of the
dielectric constant from Curie-Weiss behavior over an 80 K temperature
interval above $T_{c}$.

Previous measurements of the temperature dependence of the neutron diffuse
scattering have been extended in this study to 900 K, well above $T_{d}$. In
so doing we have unambiguously established the existence of two types of
diffuse scattering based on the observation of two markedly different
temperature dependencies, one of which vanishes at $T_{d}$ and one of which
does not. We associate the strong, temperature-dependent, diffuse scattering
with the formation of static, short-range, polar correlations (PNR) because of
its well documented response to an external electric field, and because it
first appears at the same temperature at which the soft (polar) TO mode
reaches a minimum in energy. We associate the weak, temperature-independent,
diffuse scattering, which shows no obvious change across either $T_{c}$ or
$T_{d}$, with chemical short-range order because it persists to extremely high
temperature. We further confirm the observations of Hiraka et al., Hiraka  who
first characterized the distinctly different reciprocal space geometries of
both types of diffuse scattering, and we show that the bow-tie shape of the
weak diffuse scattering persists up to 900 K $\gg T_{d}$.

Key effects of the strong, temperature-dependent, diffuse scattering, and thus
of the PNR, on the low-energy lattice dynamics of PMN are also highlighted in
this study. The neutron inelastic measurements on PMN-60%PT by Stock et al.
prove conclusively that PNR cannot be the origin of the anomalous broadening
of long-wavelength TO modes observed in PMN, PZN, and other compounds, also
known as the waterfall effect, because PMN-60%PT exhibits the same effect but
no strong diffuse scattering (no PNR). Stock06:73  By contrast, many studies
have shown that PNR do broaden the TA modes in PMN Naberezhnov ; Koo ; Stock
and in PZN-$x$PT, Gop  but that such effects are absent in compositions with
no PNR such as PMN-60%PT. Stock06:73  Our cold neutron data show that these
effects are $q$-dependent. Whereas long-wavelength TA modes with reduced wave
vectors $q\ll 0.2$ Å-1 remain well-defined and exhibit a nearly constant
energy width (lifetime) from 100 K to 900 K, shorter wavelength TA modes with
reduced wave vectors $q\approx 0.2$ Å-1 broaden substantially, with the
maximum broadening occurring at $T_{d}=420$ K. This result motivates a very
simple physical picture in which only those acoustic phonons having
wavelengths comparable to the size of the PNR are significantly scattered by
the PNR; acoustic modes with wavelengths much larger than the PNR are largely
unaffected because they simply average over the PNR. Models describing this
effect have been discussed elsewhere. Stock  In particular, very recent work
by Xu et al. has revealed the presence of a phase instability in PZN-4.5%PT
that is directly induced by such a PNR-TA phonon interaction. This is shown to
produce a pronounced softening and broadening of TA modes in those zones where
the diffuse scattering is strong, and provides a natural explanation of the
enormous piezoelectric coupling in relaxor materials. Xu_Nature

In addition, we have performed neutron measurements of the thermal expansion
with extremely high $q$\- and $\hbar\omega$-resolution over a broad
temperature range extending well below $T_{c}$ and far above $T_{d}\sim 420$
K. In agreement with many other studies, our single crystal samples of PMN
exhibit little or no thermal expansion below $T_{d}$, behavior that is
reminiscent of the invar effect, but an unusually large thermal expansion
coefficient of $2.5\times 10^{-5}$ 1/K above $T_{d}$, where the strong diffuse
scattering is absent. The crossover between null and large coefficients of
thermal expansion coincides closely with $T_{d}$, which suggests that the
appearance of static PNR strongly affects the thermal expansion in PMN and
thus provides a structural signature of the formation of short-range, polar
correlations in zero field. The model of uniformly displaced, or phase-
shifted, PNR proposed by Hirota et al., which successfully predicts the
anisotropic response of the strong diffuse scattering to an electric field,
offers a simplistic, yet plausible, framework in which to understand this
anomalous behavior.

Finally, it is satisfying to note that the revised value of $T_{d}=420\pm 20$
K is consistent with the dielectric susceptibility data of Viehland et al.,
from which a Curie-Weiss temperature $\Theta=398$ K was derived. Viehland
Such good agreement between $T_{d}$ and $\Theta$ solidifies our identification
of the strong diffuse scattering with the condensation of the soft TO mode,
which reaches a minimum frequency at $T_{d}=420$ K. However this begs the
question of how one should interpret the original Burns temperature of $\sim
620$ K. At present there are two broadly divergent opinions on this issue, one
of which considers $\sim 620$ K to be a meaningful temperature scale in PMN,
and one of which does not. As it turns out, this debate is closely tied to
another on how many temperature scales are needed to describe the physics of
relaxors. We offer no final resolution to this discussion. Instead, we close
our paper with a brief summary of the primary studies supporting these
contrasting points of view, which is by no means comprehensive, so that the
readers may draw their own conclusions.

A number of experimental studies report evidence of either structural or
dynamic changes in PMN in the temperature range 600 K – 650 K, starting with
the optical index of refraction measurements of Burns and Dacol. Burns  Siny
and Smirnova were the first to observe strong, first-order Raman scattering in
PMN at high temperatures, which, being forbidden in centrosymmetric crystals,
implied the presence of some type of lattice distortion. Siny  In 2002
Vakhrushev and Okuneva calculated the probability density function for the Pb
ion in PMN within the framework of the spherical layer model using x-ray
powder diffraction data. Vakhrushev_2002  It was shown that this probability
density evolves from a single Gaussian function centered on the perovskite
$A$-site to a double Gaussian form between 635 K and 573 K, and that the
positions of maximum density for the lead ion follow a power law
$(T_{d}-T)^{0.31}$ with $T_{d}=635$ K. This picture was developed further by
Prosandeev et al. within a model that ascribed the changes in the lead
probability density function and subtle variations in the thermal expansion
near 620 K to a crossover from soft-mode to order-disorder dynamics.
Prosandeev  In 2004 Egami et al. first proposed that at very high temperatures
the vibrations of the oxygen octahedra are sufficiently faster than those of
the Pb ions that, on average, a high-symmetry, local environment is obtained,
whereas at temperatures below $\sim 680$ K the Pb and O ions become
dynamically correlated. Egami_2004  This led to the subsequent
reinterpretation of the Burns temperature ($\sim 600$ K) as being the point
below which dynamic PNR first form, which was based on the dynamic PDF
measurements of Dmowski et al. obtained with an elastic energy resolution of 4
meV at six temperatures (680 K, 590 K, 450 K, 300 K, 230 K, and 35 K). Dmowski
Interestingly, after integrating their data from -5 meV to +5 meV, Dmowski et
al. saw evidence of a third temperature scale in PMN of order 300 K, and thus
distinct from both $T_{c}\sim 210$ K and $T_{d}\sim 600$ K, which they
associated with the formation of static PNR. Similar ideas have very recently
been put forth by Dkhil et al. based on extremely interesting acoustic
emission and thermal expansion measurements, and also in a general summary
written by Toulouse. Dkhil_3T ; Toulouse

A different approach was taken by Stock et al. who proposed a unified
description of the lead-based relaxors on the basis of striking similarities
between PMN and the Zn-analogue PZN, both of which display strong,
temperature-dependent, diffuse scattering; identical soft mode dynamics; yet
no long-range, structural distortion at low temperature in zero field.
Stock_PZN ; Xu_PZN  Arguing by analogy with magnetic systems, Stock et al.
considered a three-dimensional Heisenberg model with cubic anisotropy in the
presence of random fields in which the Heisenberg spin corresponds to the
local ferroelectric polarization, the cubic anisotropy represents the
preferential orientation of the polarization along the ¡111¿ axes, and the
isotropic random magnetic field corresponds to the randomly-oriented local
electric fields that originate from the varying charge of the $B$-site cation.
Following a suggestion by Aharony, Stock et al. considered the case where the
Heisenberg term dominates over the random field term, and the cubic anisotropy
term is the weakest of the three. In this picture there would be just two
distinct, static, temperature scales. For $T>T_{d}$, the cubic anisotropy is
irrelevant and therefore the system should behave like a Heisenberg system in
a random field. In this case the excitation spectrum is characterized by
Goldstone modes and therefore no long-range order is expected in the presence
of random fields. Aharony  Instead the system forms polar nanoregions in a
paraelectric background. The second temperature scale $T_{c}$ appears at low
temperatures where the cubic anisotropy term becomes important, and in this
limit the system should resemble an Ising system in the presence of a random
field. This model thus explains the local ferroelectric distortion
characterized by the recovery of the soft-optic mode and, although a 3D Ising
system in equilibrium should display long-range order in the presence of small
random fields, as is observed in magnetic systems, nonequilibrium effects with
long time scales become dominant. The presence of such nonequilibrium effects
may explain the lack of long-range ferroelectric order in PMN and PZN as well
as the history dependence of physical properties such as the linear
birefringence. The phase-shifted nature of the polar nanoregions may also
create another energy barrier against the ordered phase at $T_{c}$. Stock et
al. are also able to explain the physics of compounds beyond the MPB, such as
PMN-60%PT, within this model, for which only one temperature scale ($T_{c}$)
exists. As the Ti concentration Ti increases, the relaxor phase diagram
crosses over to a ferroelectric regime, and this can be understood as an
increase in the strength of the cubic anisotropy term that is simultaneously
accompanied by a reduction of the random fields as must occur in the limit of
pure PbTiO3. Stock06:73  It should be mentioned that other studies have
invoked random fields to explain the static and dynamic properties of
relaxors. Westphal ; Pirc ; Fisch

Independent of the validity of either of the two pictures summarized above,
the seminal finding of our study of remains that the strong diffuse scattering
in PMN first appears at a temperature that depends sensitively on the energy
resolution of the measurement. This fact inevitably raises interesting
questions about the significance of the previous value of the Burns
temperature ($\sim 620$ K). If the strong diffuse scattering in PMN results
from the soft TO mode, then other techniques based on x-ray diffraction,
thermal neutron scattering, or neutron PDF, which provide comparatively much
coarser energy resolution, will be unable to discriminate between low-energy,
dynamic, short-range polar correlations and truly static ($\hbar\omega=0$) PNR
because any low-energy, polar correlations will be integrated into the elastic
channel by the broad energy resolution. Thus as the TO mode softens, the
associated low-energy, polar, spectral weight will fall within the energy
resolution at temperatures above $T_{d}=420$ K, and the net result will be an
artificially higher value of $T_{d}$ that increases with the size of the
energy resolution. This effect should be especially pronounced for phonon
modes that are broad in energy (i. e. that have short lifetimes), and this is
clearly the case for the soft TO mode in PMN, which exhibits a nearly
overdamped lineshape and a minimum frequency precisely at $T_{d}=420$ K. While
it is true that the structure factor of the strong diffuse scattering is
inconsistent with that of the soft TO mode, the phase-shifted model of Hirota
et al. provides a way to reconcile this discrepancy. In this respect, we
simply suggest that the previous value of $T_{d}\sim 620$ K might not
represent a physically meaningful temperature scale in PMN. Future studies
examining the values of $T_{d}$ in other relaxor compounds with improved
energy resolution are thus of interest.

Our reassessment of the Burns temperature $T_{d}$ immediately clarifies an
intimate relationship between the formation of static, short-range polar
correlations and the consequent effects on both the low-energy lattice
dynamics and structure of PMN. Cold neutron triple-axis and backscattering
spectroscopic methods conclusively show the existence of static, short-range
polar correlations, only below $T_{d}=420\pm 20$,K on timescales of at least 2
ns. Thermal neutron measurements of the lattice dynamics reflect the presence
of these static PNR through the presence of a distinct minimum in both the
soft TO and TA mode energies, both of which occur at 420 K. The effect of PNR
on the lattice dynamics is evident only for TA modes having wave vectors of
order 0.2 Å-1, a fact that could be exploited to determine the size of the
PNR. At the same time an enormous change in the coefficient of thermal
expansion is seen near $T_{d}$, below which the crystal lattice exhibits
invar-like behavior.

##  VIII Acknowledgments

We would like to thank A. Bokov, Y. Fujii, K. Hirota, D. Phelan, S. Shapiro,
S. Wakimoto, and G. Xu for stimulating discussions. This study was supported
in part by the U. S. - Japan Cooperative Neutron-Scattering Program, the
Natural Sciences and Engineering Research Council (NSERC) of Canada, the
National Research Council (NRC) of Canada, the Japanese Ministry of Monbu-
Kagaku-shou, RFBR grants 08-02-00908 and 06-02-90088NSF, the U. S. Dept. of
Energy under contract No. DE-AC02-98CH10886, the U. S. Office of Naval
Research under grant No. N00014-06-1-0166, and by the NSF under grant No.
DMR-9986442. We also acknowledge the U. S. Dept. of Commerce, NIST Center for
Neutron Research, for providing the neutron scattering facilities used in this
study, some of which are supported in part by the National Science Foundation
under Agreement No. DMR-0454672.

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