# Recovery of harmonic–like behaviour of the polar mode in BaTiO3 at high
pressures

A. Herlihy  Department of Chemistry, University of Warwick, Gibbet Hill,
Coventry, CV4 7AL, UK  ISIS Neutron and Muon Facility, Rutherford Appleton
Laboratory, Didcot, OX11 0QX, UK  T. A. Bird  Department of Chemistry,
University of Warwick, Gibbet Hill, Coventry, CV4 7AL, UK  C. J. Ridley  ISIS
Neutron and Muon Facility, Rutherford Appleton Laboratory, Didcot, OX11 0QX,
UK  C. L. Bull  ISIS Neutron and Muon Facility, Rutherford Appleton
Laboratory, Didcot, OX11 0QX, UK  School of Chemistry, University of
Edinburgh, David Brewster Road, Edinburgh, EH9 3FJ, UK  N. P. Funnell
<EMAIL_ADDRESS>ISIS Neutron and Muon Facility, Rutherford Appleton
Laboratory, Didcot, OX11 0QX, UK  M. S. Senn<EMAIL_ADDRESS>Department
of Chemistry, University of Warwick, Gibbet Hill, Coventry, CV4 7AL, UK

(August 27, 2024)

###### Abstract

The local structure of high pressure BaTiO3 has been interrogated by neutron
total scattering methods, up to pressures of 4.18 GPa at ambient temperature.
Competitive refinements of cubic, tetragonal and rhombohedral distortion modes
against pair distribution functions indicate contrasting local structure
behaviour of temperature- and pressure-induced cubic BaTiO3. Suppression of
the mode amplitude, isotropy of the order parameter direction and loss of
sensitivity to correlated Ti displacements at high pressure all suggest a
high-pressure local structure that is more consistent with the harmonic
approximation, rather than an order-disorder model which better describes
high-temperature cubic BaTiO3 in the vicinity of the tetragonal phase
transition.

##  I Introduction

BaTiO3 is often given as a classic example of a proper ferroelectric where,
due to the second-order Jahn-Teller effect, an off-centering of the Ti4+
cation from its TiO6 octahedron results in a net polarisationBersuker (1966).
The resulting ferroelectric properties and high dielectric constant make
BaTiO3 a very attractive material for use in devices such as capacitorsAcosta
_et al._ (2017), and the perovskite-structured material (shown in Figure 1a)
has become the prototypical ferroelectric; intensively studied to understand
the link between ferroelectricity and crystal structure. Despite many decades
of investigation, there remains an ongoing debate about the nature of the
ferroelectric phase transition. Above its Curie temperature
(T${}_{\textup{C}}$), BaTiO3 adopts a cubic structure. Below
T${}_{\textup{C}}$, the structure is reduced to a tetragonal symmetry and on
decreasing temperature further, BaTiO3 transforms to an orthorhombic and,
finally, rhombohedral structureHippel  _et al._ (1946); Megaw (1947); Cochran
(1960); Hayward and Salje (2002).

A popular theory, suggested by Cochran et al.Cochran (1960) describes a
displacive model whereby Ti4+ cations are displaced microscopically along
⟨100⟩, ⟨110⟩, and ⟨111⟩ directions for the tetragonal, orthorhombic and
rhombohedral phases respectively. This model however fails to address key
observations such as the strong diffuse X-ray scattering in all but the
rhombohedral phaseComés  _et al._ (1968); Ravy  _et al._ (2007); Paściak  _et
al._ (2018) and the presence of first-order Raman excitations in the cubic
phaseQuittet and Lambert (1973). In 1968, Comés et al.Comés  _et al._ (1968)
proposed an order-disorder (OD) model, also commonly referred to as the
‘eight-site’ model, where the crystallographically-rich phase diagram of
BaTiO3 is rationalised due to correlations of local Ti displacements along the
eight ⟨111⟩ directions. Correlated displacements of the Ti atom in successive
⟨100⟩ directions give rise to the observed average symmetry, and it is this
underlying disorder that appears to simultaneously reconcile the perceived
average symmetry with the anomalous experimental results, discussed above.

Since the first proposal of these two contending models, a multitude of
experimental and computational studies have favoured either one of these two
possible scenarios. Local probes tend to support an OD modelRavel  _et al._
(1998); Zalar  _et al._ (2003); Laulhé  _et al._ (2009), for example, our
symmetry-motivated analyses of pair distribution functions (PDFs) of BaTiO3
have shown that Ti displacements are rhombohedral-like across all known
phasesSenn  _et al._ (2016). However, the observation of heavily-damped
modesLuspin  _et al._ (1980); Harada  _et al._ (1971); Yamada and Shirane
(1969) appears at odds with an OD model, and supports the soft-mode
explanation. Furthermore, there is not yet consensus—within the OD
interpretation—on the exact nature of the disordered local arrangements of Ti
cations, where some reports (via solid state NMRZalar  _et al._ (2003))
suggest a local tetragonal distortion and others support a rhombohedralRavel
_et al._ (1998); Stern (2004) distortion.

More recently, additional work has come out in support of the soft mode model
Paściak  _et al._ (2018), where diffuse scattering is attributed to the
overdamped anharmonic soft phonon branch. This results in a local probability
distribution for the Ti atoms that has a minimum coinciding with the average
crystallographic position and a maximum along ⟨111⟩ directions with an average
magnitude of ca. 0.15 Å. It seems that a wealth of experimental and
computational observations can either be explained by invoking an OD scenario
or considering highly over-damped, anharmonic, soft phonon modes that imply
the Ti atoms spend a substantial amount of time off-centre. Regardless of the
perspective adopted, it is clear that the local symmetry deviates
substantially from the average crystallographic symmetry over short length
scales and long time periods, indicating a significant departure from the
harmonic soft mode/displacive picture. Consideration of the long range
ordering of dynamic ⟨111⟩ Ti displacements projected onto the ⟨100⟩ directions
appears to reconcile these two modelsSenn  _et al._ (2016).

Clearly, the investigation of the temperature-induced phase transitions of
BaTiO3 has been extensive, and a wide range of techniques have been utilised
to investigate the average and local structure of the perovskite material
Culbertson  _et al._ (2020). However, challenges associated with in situ high
pressure measurements have perhaps limited investigation of the local
structure of BaTiO3 in other regions of the phase diagram.

It is predicted that modest hydrostatic pressure will initially act to
suppress ferroelectric distortions in ABO3 perovskites due to the increasing
influence of short-range electronic repulsions over long-range Coulomb ionic
interactions which favour polar distortionsKornev  _et al._ (2005). This is
born out by the well-established average structure phase diagram of BaTiO3
that indicates that at ambient temperature, there is a tetragonal-to-cubic
phase transition at ca. 2 GPa Hayward and Salje (2002); Ishidate and Abe
(1997); Bull  _et al._ (2021). However, high-pressure Raman studies show
evidence for persistent disorder within the cubic phase, with the suggestion
that this disorder results from off-centre Ti atoms and grain
boundary/intergrain stress Venkateswaran and Naik (1998). X-ray absorption
spectroscopy (XAS) of the Ti K edge also suggests that Ti remains displaced
until 10 GPa, above which the Ti is centred, and local and average symmetries
are reconciledItié  _et al._ (2006). Together, these results might imply that
the high temperature and high pressure behaviour mimic each other from both an
average (crystallographic) and local structure perspective. However, neither
of these studies appear to have allowed for robust refinement of models with
competing symmetries against the local probe data.

On the other hand, PDFs generated from total scattering experiments and their
sensitivity to short-range atom–atom correlations are well suited to this kind
of modelling that interrogates the precise local symmetry breaking behaviour
in BaTiO3. Whilst X-ray PDF work has been carried outEhm  _et al._ (2011), the
insensitivity of X-rays to the lighter oxygen atoms often fail to resolve the
level of detail available to neutron measurements. The lack of high-pressure
neutron PDF studies of BaTiO3, and indeed of crystalline materials more
generally can be attributed to the often opposing requirements of high
pressure and PDF experiments. It is only relatively recently that high-
pressure neutron PDF measurements have been achieved for crystalline
materialsPlayford  _et al._ (2017); Herlihy  _et al._ (2021).

With this in mind, we undertake the first analysis of neutron total scattering
measurements of BaTiO3 at pressures up to 4.2 GPa, in order to directly
investigate the nature of the pressure-induced tetragonal-to-cubic phase
transition of BaTiO3 over a range of length scales. Building on our recently
developed symmetry adapted PDF analysis (SAPA) Bird  _et al._ (2021)
technique, whereby distortion modes grouped by irreducible representation are
refined against local structure measurements, we analyse the high-pressure PDF
data, revealing pressure-induced suppression of the local Ti off-centerings.
We apply the same modelling approach to previously-published variable
temperature PDFsSenn  _et al._ (2016) in order to determine how the departure
of local from average symmetry compares for pressure vs temperature. Our
analysis of ambient temperature, variable pressure PDFs points toward a
gradual pressure-induced suppression of the anharmonic potential implicit in
describing the OD behaviour of BaTiO3, towards a more harmonic-like potential,
more consistent with a soft-mode picture.

##  II Experimental Details and Data Analysis

Polycrystalline BaTiO3 (also used for the variable-temperature study described
in reference 14) was measuredDAT  on the high-pressure instrument PEARLBull
_et al._ (2016), at the ISIS Neutron and Muon Facility. The powder sample was
loaded into a null-scattering Ti–Zr single-toroidal gasket, with a gas
loaderKlotz  _et al._ (2013), used to fill the remaining gasket volume with an
argon gas pressure transmitting media (PTM). A Paris–Edinburgh (PE) press,
equipped with zirconia-toughened alumina (ZTA) anvils was used to apply loads
of 3, 25, 40 and 50 tonnes to the sample. The lattice parameters of BaTiO3
were determined from Rietveld refinement against the Bragg data and the known
equation of stateBull  _et al._ (2021) used to calculate sample pressures of
0.24(2), 1.19(2), 2.55(6) and 4.18(8) GPa. Neutron powder diffraction patterns
were collected for a minimum of 11 hours each to ensure sufficient signal-to-
noise ratio at high $Q$ (where $Q=(4\pi\sin\theta)/\lambda$). Stacked vanadium
discs were measured in the same way with an argon PTM, and analogous data
collections were performed at loads of 8, 20, 30 and 45 tonnes, corresponding
to pressures roughly equivalent to those of the measured BaTiO3 data. Total
scattering data were collected and treated using the same procedure described
in references 26 and 27, without the added complication of needing to model
the PTM, since argon gas is a relatively weak neutron scatterer. That being
said, scattering due to the PTM was observed, with the presence of the (111)
Bragg reflection in the diffraction pattern at 4.18 GPa, suggesting the PTM
had crystallised. However, the absence of any significant sample peak
broadening indicated that hydrostatic conditions remained and there was no
evidence of an argon contribution to the PDF (i.e. no misfits in regions where
an Ar–Ar peak would be expected).

Data were reduced using the MANTID software package Arnold  _et al._ (2014) to
correct for the effects of attenuation by the ZTA anvils and normalised by a
vanadium standard to account for flux profile and detector efficiencies.
Scattering from the gasket and anvils were accounted for by subtracting data
from the vanadium measurements, and total scattering patterns (S(Q)s) were
produced by applying a scaling and offset value such that S(Q)→1 at Qmax. PDFs
(shown in Figure 1c) were obtained via Fourier transform of the S(Q) function
using the program StoG, distributed with the RMCProfile package Tucker  _et
al._ (2007).

Figure 1:  a) The average structure unit cell of cubic BaTiO3, with an arrow
indicating the shortest atom–atom correlation within the structure (Ti–O). b)
The longer-range structure and circles with radii of 4 and 30 Å indicating the
minimum and maximum range of atom–atom correlations probed by our variable
range PDF refinements. c) Variable-pressure PDFs measured on PEARL (offset in
the y-direction with increasing pressure for clarity). The yellow arrow
indicates the features arising due to the Ti–O correlations and the horizontal
arrows correspond to the probe distances shown in Figure 1b.

PDF modelling and Rietveld refinements were carried out using TOPAS Academic
software v6Coelho (2018). We performed small-box variable range PDF
refinementsSmith  _et al._ (2008); Culbertson  _et al._ (2020), with the
minimum of the fitting range ($r_{\textup{min}}$) kept constant at 1.2 Å, and
the maximum ($r_{\textup{max}}$) varied from from 4 to 30 Å in steps of 1 Å.
Therefore, the overall fitting range was varied between 2.8 and 28.8 Å, such
that increasingly large length-scale atom–atom correlations were probed with
increasingly large $r_{\textup{max}}$ values, as depicted in Figure 1b. This
is in contrast to so-called ‘box-car’ refinementsUsher  _et al._ (2016); Hou
_et al._ (2018) where the fitting range is held constant and shifted along the
PDF, resulting in the progressively reduced influence of the immediate local
structure on the refined small-box model. We used a $P1$ unit cell, refining
only the polar distortion modes associated with Ti and O which transform as
the Γ${}_{4}^{-}$ irreducible representation (irrep.), and fixing Ba modes to
zero to avoid a floating origin of the unit cell. The most general order
parameter direction (OPD) associated with this irrep is three dimensional
(a,b,c). The Ti($T_{1u}$), O($A_{2u}$) and O($E_{u}$) modes, that form a basis
of this irrep, thus have three branches each, where particular constraints on
the branched mode amplitudes correspond to higher symmetry OPDs.

Rather than allowing distortion modes to refine freely, we constrained the OPD
to be consistent with cubic (0,0,0), tetragonal (a,0,0) and rhombohedral
(a,a,a) symmetries in order to test these three specific local distortion
behaviours. We did not consider other order parameters such as (a,a,0),
(a,b,0) or (a,a,b) as the aim of this work was to resolve the OD behaviour of
BaTiO3 at the tetragonal to cubic phase transition. We found that
unconstrained refinements of the Ti($T_{1u}$), O($A_{2u}$) and O($E_{u}$)
modes resulted in non-physical coupling, particularly for refinements of the
PDFs measured at 2.55 and 4.18 GPa, where for $r_{\textup{max}}$ values of
greater than 10 Å, Ti and O atoms refined to displace in the same, rather than
opposite, directions (see Supplementary Information (SI)SI ). In order to
maintain the correct relative displacements associated with the modes, a ratio
of 1:$-$1.6:$-$1.3 for Ti($T_{1u}$):O($A_{2u}$):O($E_{u}$) displacements,
respectively, was applied. These values were calculated by averaging ratios
determined by fitting rhombohedral (a,a,a) models against high quality
diffraction data measured at 15 and 293 K on GEMSenn  _et al._ (2016). Mode
amplitude values reported herein refer to the A${{}_{\textup{P}}}$ values
defined in ISODISTORTCampbell  _et al._ (2006), as the parent-cell-normalized
amplitude, and are assigned the mode-specific notation,
$|$Q(Γ${}_{4}^{-}$)$|$. We found that although the refined
$|$Q(Γ${}_{4}^{-}$)$|$ values differ slightly depending on the precise ratio
used, the relative values and fitting statistics of each refinement remain
essentially constant. Lattice parameters determined from Rietveld refinements
of the diffraction patterns (see SISI ) were fixed for all small-box PDF
refinements, constraining the metric symmetries to those known from the
average structures. The $beq\\_r\\_r2$ function (discussed further in
reference 28) was used to describe the correlated thermal motion that leads to
r-dependent broadening, with isotropic displacement parameters fixed to the
lower limits found for the three models (see SISI  for further details). The
sensitivity of our modelling approach to the limited Qmax (20.32 Å-1)
available on PEARL was thoroughly investigated and reported within Appendix A.

##  III Results and Discussion

Neutron diffraction patterns indicate that the measured average structure of
BaTiO3 at variable pressure is consistent with previous literaturePruzan  _et
al._ (2002); Hayward and Salje (2002); Bull  _et al._ (2021). The neutron
diffraction patterns (see SISI ) at 0.24 and 1.19 GPa exhibited clear peak
splitting (particularly the (200)/(002) reflection), indicative of a
tetragonal symmetry, and Rietveld refinements confirmed a $P4mm$ average
crystal structure. Above 2 GPa, BaTiO3 goes through a well documented phase
transition to an average cubic symmetry ($Pm\bar{3}m$), confirmed again by
Rietveld refinement at 2.55 and 4.18 GPa.

Figure 2:  $R_{w}$ and $|$Q(Γ${}_{4}^{-}$)$|$ values for variable range
refinements for cubic (0,0,0), tetragonal (a,0,0) and rhombohedral (a,a,a)
OPDs against variable pressure (left) and temperature (right) PDFs.
$|$Q(Γ${}_{4}^{-}$)$|$ values for the cubic model were fixed at zero and are
plotted as such. $R_{w}$ values for tetragonal and rhombohedral modes in the
high pressure cubic data are almost exactly coincident, and cannot be visually
distinguished.

Refining small-box models over an increasing range of r (Å) of the PDF
provides information on the correlation length scale. This is particularly
relevant for materials with OD behaviour such as BaTiO3 where a local
rhombohedral distortion may be observed over a short length scale, for example
one unit cell, however, longer length scales will increasingly resemble the
average structure. Comparisons of fitting statistics ($R_{w}$) and
$|$Q(Γ${}_{4}^{-}$)$|$ values (shown in Figure 2) for cubic, tetragonal and
rhombohedral models provide insight into the evolution of local displacements
of Ti and O atoms as a function of pressure. We demonstrate that even for
limited Qmax values available on PEARL, our data is sensitive to subtle
changes in the local structure (see Appendix A).

We compare our findings for the local structure of BaTiO3 at high pressure
with analogous results for the thermally-induced phase transition. The same
modelling approach has been applied to PDFs measured at 293, 350, 410 and 500
K using the GEM instrument at ISIS (and processed with a Qmax of 20 Å-1 for a
fairer comparison), and previously published in support of persistent OD
behaviour at high temperature Senn  _et al._ (2016). The average structure of
BaTiO3 is tetragonal at 293 and 350 K and cubic at 410 and 500 K, inviting a
comparison of the local structure of BaTiO3 at high pressure and high
temperature.

At 0.24 and 1.19 GPa consistent improvements in $R_{w}$ over all
$r_{\textup{max}}$ (shown in Figure 2) indicate that the local and medium-
range structure of BaTiO3 is best described by a rhombohedral displacement of
the Ti atom. The refined $|$Q(Γ${}_{4}^{-}$)$|$ values for an (a,0,0) OPD are
approximately $\sqrt{3}/2$ smaller than those of an (a,a,a) OPD suggesting
that we are essentially resolving a projection of the [111] type displacement
onto the [100] direction.

The results for the local structure of BaTiO3 at 1.19 GPa are essentially very
similar to those found for the structure at 0.24 GPa and a decrease in
$|$Q(Γ${}_{4}^{-}$)$|$ of ca. 15 % points towards a small pressure-induced
hardening of the local potential describing the off-centre displacements.
These results are comparable with those of the variable temperature PDFs,
measured at 293 and 350 K. $R_{w}$ values at 293 and 350 K again favour the
rhombohedral-type displacements up to $r_{\textup{max}}$ = 10 Å, after which,
fitting statistics favour the tetragonal model, indicating sensitivity of the
PDF to the average, long-range structure. $|$Q(Γ${}_{4}^{-}$)$|$ values are in
good agreement with the variable pressure results. Again, relative
$|$Q(Γ${}_{4}^{-}$)$|$ values for (a,0,0) compared to (a,a,a) OPDs suggest the
resolution of the [111] type displacement onto the [100] direction.

Results for the local structure of high pressure cubic BaTiO3 point toward a
departure from the local structure behaviour of the high pressure tetragonal
structure, and perhaps more interestingly, from the local structure of the
high temperature cubic structure. At 2.55 and 4.18 GPa, $|$Q(Γ${}_{4}^{-}$)$|$
at $r_{\textup{max}}$ = 4 Å becomes suppressed by ca. 1/2 (cf. 0.24 GPa) and
the magnitudes of $|$Q(Γ${}_{4}^{-}$)$|$ with OPD (a,0,0) and (a,a,a) are
approximately equal. Over all $r_{\textup{max}}$ there is negligible
difference between the $R_{w}$ values for models of tetragonal and
rhombohedral Ti displacements. At 2.55 GPa the difference between cubic models
and models with off centre displacements decreases approximately linearly
until $r_{\textup{max}}$ = 20 Å, after which the difference in $R_{w}$ drops
below significance, whereas at 4.18 GPa this occurs at $r_{\textup{max}}$ = 10
Å. At 4.18 GPa, by 16 Å, $|$Q(Γ${}_{4}^{-}$)$|$ refines to zero, suggesting
that the correlation length of the Ti displacements is below four unit cell
lengths. The suppression of $|$Q(Γ${}_{4}^{-}$)$|$, isotropy of the
displacement with respect to the different OPD, and the reduction in
correlation lengths are all consistent with the ferroelectric instability in
BaTiO3 being well-described by the harmonic approximation at elevated
pressures.

On the other hand, our results against previously-published high temperature
PDF data clearly favour an (a,a,a) OPD, consistent with the model of chains of
rhombohedrally displaced off-centre Ti atoms, which retain substantial
correlations along ⟨100⟩ directions. At 410 and 500 K, refined
$|$Q(Γ${}_{4}^{-}$)$|$ values over $r_{\textup{max}}$ = 4–10 Å) are similar to
those observed at lower temperatures (at $r_{\textup{max}}$ = 4 Å,
$|$Q(Γ${}_{4}^{-}$)$|$ at 293 K = 0.094 Å, 350 K = 0.074 Å, 410 K = 0.071 Å,
500 K = 0.071 Å), but drop to values that are ca. 2/3 of those observed over
longer $r_{\textup{max}}$. The persisting sensitivity to off-center
displacements in the high temperature cubic regime is consistent with the
model of correlated chains of [111] displacements projected along the [100]
axis and lends further support to the OD model for the temperature-induced
phase transition.

We find that our observed high pressure trends of the local structure agree
with the work of Ravy et al.Ravy  _et al._ (2007) who report diminishing
diffuse scattering planes at high pressure and broadening of the diffuse
features indicative of a decrease in correlation length of Ti chains, which
they discuss in the wider context of pressure-induced Ti-centering.
Correlation lengths of ca. six unit cell lengths (ca. 24 Å) implied by
broadened diffuse features at ca. 4 GPa are also in good agreement with our
results, where diffuse scattering is sensitive to chain correlations and the
PDF method will average chain and non-chain interactions. While the reported
diffuse scattering is sensitive to chain correlations, it is less sensitive to
the precise nature of the local symmetry breaking. On the other hand, the
method we report here for analysing our high-pressure PDFs has a higher degree
of sensitivity to the local symmetry breaking at low $r_{\textup{max}}$ ,but
will average over chain and non-chain interactions at high $r_{\textup{max}}$,
and thus the two approaches should be viewed as providing complimentary
information.

We stress that although XAS measurements suggest continual off-centre Ti
displacements up to 10 GPaItié  _et al._ (2006), the sensitivity of the
technique is limited to the immediate local environment of the probe atom,
extending as far as the next-nearest neighbour only. This makes it difficult
to judge how these results differ from those expected from the root mean
square displacement of a harmonic oscillator—estimated to be 0.05 Å at 4.18
GPa from our $r_{\textup{max}}$ = 4 Å refinements (see Figure 2).

Our results not only show robustly that the OD behaviour of BaTiO3 is
suppressed at high pressure, but also adds to an emerging research direction
on neutron local structure measurements of crystalline materials under
hydrostatic pressurePlayford  _et al._ (2017); Herlihy  _et al._ (2021), where
local structure analysis approaches such as the symmetry motivated approach we
have used here can be applied. Such experiments would provide fundamental
insight into the pressure induced mode softening in negative thermal expansion
materials like ScF3Greve  _et al._ (2010); Bird  _et al._ (2020), for example,
or pressure-induced phase behaviour of framework materials such as Prussian
blue analoguesChapman  _et al._ (2006); Boström  _et al._ (2021).

##  IV Conclusion

Although it might be tempting to conclude from the average structures that the
high temperature and high pressure tetragonal and cubic phases behave in an
analogous way, in terms of the local structure, our detailed high pressure PDF
study shows that this is not the case. Our symmetry motivated approach of
interrogating the local structure of BaTiO3 reveals that at high pressure, the
OD model provides a less satisfactory description. By 2.55 GPa already,
significant suppression of the mode amplitude over short $r_{\textup{max}}$,
isotropy of the OPD and loss of sensitivity to correlated Ti displacements at
high pressure all point towards a more harmonic character of the polar mode,
which contrasts the high temperature behaviour.

## Acknowledgements

We thank Professor David Keen for supplying the total scattering data from
GEM. A. H. thanks the Science and Technology Facilities Council and the
University of Warwick for a studentship. M. S. S. acknowledges the Royal
Society for a University Research Fellowship (UF160265) and the EPSRC for
funding (EP/S027106/1). We are grateful to STFC for the provision of neutron
beam time at ISIS, supported under experiment number RB1910162DAT .

##  Appendix A Validation of PDF sensitivity

Figure 3:  Refined $|$Q(Γ${}_{4}^{-}$)$|$ values for rhombohedrally
constrained Ti and O distortions for a room temperature BaTiO3 PDF measured on
GEM and processed with Qmax values ranging from 10–40 Å-1 in steps of 5 Å-1
(green filled markers). The unfilled marker represents the
$|$Q(Γ${}_{4}^{-}$)$|$ value for the lowest pressure (0.24(2) GPa), ambient
temperature measurement of BaTiO3 on PEARL. Figure 4:  PDFs of BaTiO3 measured
on PEARL at 0.24 GPa, processed with a Qmax of 20.32 Å-1, and on GEM at
ambient pressure and temperature, processed with Qmax values of 20 and 40 Å-1
(offset in the y-direction).

In order to evaluate the sensitivity of our modelling approach to limited Qmax
values of the PEARL data, PDFs were processed from room temperature S(Q)s
measured on GEM, from reference 14 with artificially lowered Qmax values of
10–40 Å-1 in steps of 5 Å-1. A rhombohedral model was fit to this contiguous
series of PDFs, over the $r$-range 1.2–10 Å and the resulting
$|$Q(Γ${}_{4}^{-}$)$|$ values for Ti are shown in Figure 3. They show notable
consistency over the range 20–40 Å-1 and crucially, the values refined against
our 0.24 GPa PEARL data with Qmax = 20.32 Å-1 are in excellent agreement,
falling within error of each other. At Qmax = 15 Å-1 the values become
inconsistent with those found for higher resolution PDFs. This establishes the
lower limit in Qmax with respect to extracting physically meaningful
displacements in this reported study and validates the sensitivity of the high
pressure PDFs with respect to the local order parameters that we seek to
probe. A 293 K total scattering measurement of BaTiO3 on the GEM instrument
provides a good comparison to the lowest pressure (0.24(2) GPa), ambient
temperature PEARL measurement. Figure 4 shows that PDFs measured on GEM and
PEARL (i.e. in a vanadium can and PE press respectively) and processed with
the same Qmax very closely agree, again, validating the high-pressure data.

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