# Fast broadband “cluster” spin-glass dynamics in Pb(Fe1/2Nb1/2)O3

C. Stock  School of Physics and Astronomy, University of Edinburgh, Edinburgh
EH9 3JZ, UK  B. Roessli  Laboratory for Neutron Scattering and Imaging (LNS),
Paul Scherrer Institut (PSI), 5232 Villigen PSI, Switzerland  P. M. Gehring
NIST Center for Neutron Research, National Institute of Standards and
Technology, Gaithersburg, Maryland 20899-6100, USA  J. A. Rodriguez-Rivera
NIST Center for Neutron Research, National Institute of Standards and
Technology, Gaithersburg, Maryland 20899-6100, USA  Department of Materials
Science, University of Maryland, College Park, MD 20742  N. Giles-Donovan
Centre for Medical and Industrial Ultrasonics, James Watt School of
Engineering, University of Glasgow, Glasgow G12 8QQ, United Kingdom  S.
Cochran  Centre for Medical and Industrial Ultrasonics, James Watt School of
Engineering, University of Glasgow, Glasgow G12 8QQ, United Kingdom  G. Xu
NIST Center for Neutron Research, National Institute of Standards and
Technology, Gaithersburg, Maryland 20899-6100, USA  P. Manuel  ISIS Neutron
and Muon Source, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX,
United Kingdom  M. J. Gutmann  ISIS Neutron and Muon Source, Rutherford
Appleton Laboratory, Chilton, Didcot OX11 0QX, United Kingdom  W. D. Ratcliff
NIST Center for Neutron Research, National Institute of Standards and
Technology, Gaithersburg, Maryland 20899-6100, USA  T. Fennell  Laboratory for
Neutron Scattering and Imaging (LNS), Paul Scherrer Institut (PSI), 5232
Villigen PSI, Switzerland  Institut Laue-Langevin, 71 avenue des Martyrs, CS
20156, 38042 Grenoble Cedex 9, France  Y. Su  Jülich Centre for Neutron
Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Forschungszentrum
Jülich, Lichtenbergstrasse 1, D-85747 Garching, Germany  X. Li  Shanghai
Institute of Ceramics, Chinese Academy of Sciences, Shanghai 201800, China  H.
Luo  Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai
201800, China

###### Abstract

PbFe1/2Nb1/2O3 (PFN) is a relaxor ferroelectric (Tc $\sim$ 400 K) consisting
of magnetic Fe3+ (S=${5\over 2}$, L$\approx$0) ions disordered throughout the
lattice and hosts a spin-glass phase at low temperatures built from spatially
isolated clusters of Fe3+ ions, termed a “cluster glass” (W. Kleemann et al.
Phys. Rev. Lett. ${\bf{105}}$, 257202 (2010)). We apply neutron scattering to
investigate the magnetism and dynamics of this phase in a large single crystal
which displays a low temperature spin glass transition ($T_{g}\sim$ 15 K,
found with magnetization), but no observable macroscopic and spatially long-
range antiferromagnetic order. The static response in the low temperature
cluster glass phase (sampled on the timescale set by our resolution) is found
to be characterized by an average magnetic spin direction that lacks any
preferred direction. The dynamics that drive this phase are defined by a
magnetic correlation length that gradually increases with decreasing
temperature. However, below $\sim$ 50 K the opposite is found with spatial
correlations gradually becoming more short range indicative of increasing
disorder on cooling, thereby unravelling magnetism, until the low temperature
glass phase sets in at Tg $\sim$ 15 K. Neutron spectroscopy is used to
characterize the spin fluctuations in the cluster glass phase and are found to
be defined by a broadband of frequencies on the scale of $\sim$ THz, termed
here “fast” fluctuations. The frequency bandwidth driving the magnetic
fluctuations mimics the correlation length and decreases until $\sim$ 50 K,
and then increases again until the glass transition. Through investigating the
low-energy acoustic phonons we find evidence of multiple distinct structural
regions which form the basis of the clusters, generating a significant amount
of local disorder. We suggest that random molecular fields originating from
conflicting interactions between clusters is important for the destruction of
magnetic order and the eventual formation of the cluster glass in PFN.

##  I Introduction

Spin glasses are an example of a non-equilibrium phenomena and occur in
magnets which have a large ground state spin degeneracy originating from
conflicting interactions. [1] Spin glasses have been heavily studied in
disordered metallic compounds with the magnetic susceptibility predominately
characterized by a range of slow frequencies. Relaxor ferroelectrics display
some analogous traits to that of spin glasses with polar correlations
displaying disorder and the dielectric properties being defined by a range of
frequencies. The magnetic relaxor PbFe1/2Nb1/2O3 (PFN) is a material were both
properties exist with relaxor ferroelectricity characterized by short-range
polar correlations and also a low temperature spin-glass phase based upon
clustering of magnetic Fe3+ (S=${5\over 2}$, L$\approx$0) ions. In this paper
we apply neutron scattering to study the static and dynamic magnetic
properties in PFN. We will show that the magnetism is driven by a broad
frequency band of fluctuations on the $\sim$ THz frequency scale. The results
demonstrate the dynamic magnetic response in an insulating spin-glass built
from clusters.

Relaxor ferroelectrics are materials that display exceptionally large
piezoelectric coefficients [2], particularly in single crystal form, making
them of interest for applications. [3, 4, 5, 6, 7] PbMg1/3Nb2/3O3 (PMN) and
PbZn1/3Nb2/3O3 (PZN) are prototypical relaxors [8] that display a temperature
broadened and frequency dependent dielectric response [9, 10, 11, 12]. This
behavior contrasts with conventional ferroelectrics, such as PbTiO3 [13, 14,
15, 16, 17] (PT) where the dielectric constant is measured to have a well
defined, and frequency independent peak, as a function of temperature
indicative of ferroelectric order. Nonmagnetic relaxors PMN and PZN display
two distinct temperature scales. [18] First, at a high temperature defined by
the Burns temperature [19], an energy broadened [20, 21] zone center
transverse optic mode softens to a minimum in energy. [22, 23, 24, 25, 26, 27,
28, 29, 30] This softening is concomitant with the development of spatially
short range [31, 32, 33, 34, 35, 36] and dynamic [37, 38, 39] structural
correlations characterized by momentum broadened diffuse scattering in the
neutron and x-ray cross sections [40, 41, 42]. This contrasts to expectations
for a bulk ferroelectric transition, defined by a resolution limited Bragg
peak as found in PbTiO3 [13]. A second lower temperature scale is defined when
long-range ferroelectric correlations can be stabilized under an applied
electric field [43, 44] and where local polar nanoregions become static.

These two temperature scales characterizing the dielectric properties of PMN
and PZN can be understood in terms of random fields from the quenched dipolar
disorder, originating from the valence difference on the $B$ site of the
A$B$X3 perovskite lattice. These disorder induced random dipolar fields [45,
46, 47] have been suggested to be a unique experimental realization in
condensed matter physics of a continuous phase transition in the presence of
random fields [48]. The case of discrete, or Ising, symmetries in a random
field have been extensively studied in magnetic systems where the application
of a magnetic field can induce random fields [49] in the magnetic Hamiltonian
through the “Aharony-trick”. [50] Given that applied magnetic fields also
break the rotational symmetry, such an experimental technique cannot be
applied to continuous phase transitions and hence has been confined to the
study of Ising magnets [51]. Random fields in continuous symmetries have been
investigated in soft matter systems, such as liquid crystals [52, 53], or
quantum liquids (such as liquid helium) [54] in aerogel or aerosil frameworks.

PbFe1/2Nb1/2O3 (PFN) is a relaxor ferroelectric with similar dielectric
properties to that discussed for PMN and PZN discussed above [55, 56]
undergoing a ferroelectric transition at $\sim$ 400 K and displaying spatially
short-range polar correlations similar to the polar nanoregions in PMN and PZN
[19]. Interest in PFN originates from efforts in multiferroics [57, 58, 59]
where strong coupling between ferromagnetism and ferroelectricity is sought
affording external control of either magnetic or ferroelectric domains. In
this context, PFN has shown large magnetoelectric coupling and domain
switching at room temperature [60, 61] illustrating that magnetic relaxors
maybe a promising avenue to study for coupling between spatial magnetic and
polar correlations. Evidence for coupling between dielectric and magnetic
properties has been further reported in Ref. 62, 63, 64, 65 based on
dielectric and magnetic susceptibility.

PFN belongs to a series of compounds which include PbFe1/2Ta1/2O3 (PFT) [66]
and PbFe2/3W1/3O3 (PFW) [67] that contain a magnetic Fe3+ (S=5/2, L$\approx$0)
ion [68]. Depending on the subtle change in the iron concentration in PFN (see
for example Ref. 69), it can display an antiferromagnetic transition at TN
$\sim$ 150 K or a complete paramagnetic response at all temperatures. Studies
of the bulk ordered magnetic moment in samples where a magnetic Bragg peak
have been observed have reported heavily reduced magnetic moments from the
predicted $gS=5$ $\mu_{B}$ for Fe3+ ions with an example of only 2.55
$\mu_{B}$ reported in Ref. 70. This supports the notion that PFN is based on
spatially separated clusters. At lower temperatures defined by Tg $\sim$ 15 K,
PFN undergoes a glass transition characterized by a differing zero/field
cooled susceptibilities. This spin-glass phase has been found to be
characterized by clusters of Fe3+ rich regions and hence termed a “cluster
glass”. [71]

In this paper, we investigate the magnetic and structural properties in single
crystalline PFN that define the cluster glass phase. This manuscript is
divided into four parts including this introduction. We first outline the
various neutron instruments used to investigate the structural and magnetic
response. We then discuss the structural properties by studying the nuclear
Bragg peaks, momentum broadened diffuse scattering, and long wavelength
acoustic phonons. This is followed by an investigation of the static and
dynamic magnetic properties using both polarized and unpolarized
diffractometers and triple-axis spectrometers. We show in this section a
crossover to where the magnetism is increasingly disordered as the temperature
is decreased. We then finish the paper with a discussion of the crossover from
paramagnetism to glassy dynamics in PFN on cooling and its origin.

##  II Experiment

Neutron scattering experiments were performed on different instruments to
compare the structural and magnetic responses in PFN. This section outlines
details on sample characterization, different neutron experiment setups, and
then the formalism used to describe the data through the remainder of the
paper.

###  II.1 PFN sample and characterization

The single crystal was grown using the modified Bridgeman technique (outlined
in Ref. 72) following the procedure in Ref. 73. The structural properties of
our single crystal have previously been studied with neutron spectroscopy [74]
which observed a zone center transverse optic phonon softening at $\sim$ 400 K
consistent with the formation of ferroelectric order. Raman scattering work
[75] on a piece of our large single crystal previously reported that this
corresponded to two ferroelectric transitions from a cubic to tetragonal
($P4mm$) then rhombohedral ($R3m$) unit cells (on cooling) which is consistent
with the structural properties reported using x-ray diffraction [76]. This
transition has further been confirmed by us using high resolution neutron
powder diffraction (HRPD,ISIS) on a piece of our single crystal confirming the
structural transitions reported in Refs. 76, 75. We note that due to the
comparatively poorer momentum resolution afforded by neutrons and also the
formation of domains which preserve the bulk high temperature structural
symmetry, we index the reciprocal lattice using an average cubic unit cell at
all temperatures in this paper.

We note that our sample does not show a sharp peak in the magnetic
susceptibility [74] characteristic of spatially long-range magnetic order
which is highly sensitive to Fe3+ ordering and relative Fe3+ and Nb5+
concentrations. While single crystal neutron diffraction on our sample finds
equal Nb:Fe concentrations within a few percent, small changes in this
relative concentration have been shown to tune the presence of a spatially
long-range antiferromagnetic component in samples [77], with even the complete
absence of long-range magnetic order reported in some samples [78]. We
emphasize that single crystal samples that have displayed this peak in the
susceptibility, and that have been analyzed with neutron diffraction, have
shown both spatially short-range and long-range components. As far as we are
aware however, all reported samples show ferroelectric transitions at
$\sim$400 K and a glass transition below $\sim$ 15 K, both of which are
displayed in our sample studied here. The presence of a spatially long-ranged
magnetic component is not required for the formation of the low temperature
spin-glass phase which we study in this paper.

###  II.2 Instruments

Structural diffuse scattering with unpolarized neutrons (TASP and SPINS) : The
use of cold neutrons has been shown to be versatile in studying structural
properties, and in particular the momentum broadened diffuse scattering cross
section, in conventional nonmagnetic relaxors like PMN. [79] Motivated by this
previous work, we investigated the temperature dependent structural response,
both spatially long- and short- range, using cold triple-axis spectrometers
TASP (PSI) and SPINS (NIST). On SPINS, the collimation sequences was set to
$open$-80′-$S$-80′-$open$ with the incident and final energies fixed to Ei=5.0
meV using a PG(002) vertically focussed monochromator and flat PG(002)
analyzer crystals. Cooled Beryllium filters were placed before and after the
sample to remove higher order contamination of the neutron beam. On TASP, the
same collimation sequence was used as on SPINS with Ei=4.7 meV and a Beryllium
filter was used on the scattered side.

Magnetic diffuse scattering with polarized (DNS and D7) and unpolarized (TASP
and PRISMA) neutrons: The magnetic diffuse scattering in PFN, indicative of
spatially short-range correlations, was investigated using several
instruments. Unpolarized neutron scattering was performed using TASP (with the
same configuration stated above) and the time of flight instrument PRISMA
(ISIS, UK). PRISMA was used in diffraction mode, integrating over energy
transfers and using time-of-flight for the kinematics as described in Ref. 80.

To confirm the magnetic nature of the scattering, characteristic of the
underlying average magnetic structure of the Fe3+ ions, and also to separate
nuclear and magnetic contributions to the neutron cross section, we initially
used the DNS [81] diffractometer (MLZ, Garching, Germany). The incident energy
was fixed at Ei=3.7 meV with no energy analysis on the scattered side (two-
axis mode) therefore approximately measuring the energy integrated structure
factor $S(\vec{Q})\equiv\int dES(\vec{Q},E)$. The incident beam was
monochromated using a horizontally and vertically focused PG002 monochromator
and polarized with $m=3$ Schärpf supermirrors. The direction of polarization
at the sample was defined using an $XYZ$ geometry described in the Appendix.
The flipping ratio was 20 $\pm$ 1\. Experiments on D7 [82] (ILL) involved a
similar setup to that of DNS (MLZ) but used an incident energy of Ei=3.6 meV.

Magnetic and structural dynamics measured with a cold triple-axis (TASP and
SPINS): Cold triple-axis measurements were done on SPINS at NIST
(Gaithersburg, USA) and TASP (PSI, Switzerland) to investigate low-energy
magnetic dynamics and acoustic phonons. On SPINS (NIST), the collimation
sequence was set to $open$-80′-$S$-80′-$open$ with a cooled Beryllium filter
on the scattered side to remove higher order contamination of the neutron
beam. The incident beam energy was selected with a vertically focused PG(002)
graphite monochromator and the final energy was fixed at Ef=5.0 meV using a
flat PG(002) analyzer crystal. On TASP (PSI), the collimation was set to
$open$-$open$-$S$-$open$-$open$ with Ef=4.7 meV with a curved analyzer. A
cooled Beryllium filter was used on the scattered side.

Structural dynamics measured with a thermal triple-axis (BT4): To probe the
structural dynamics and in particular the zone center transverse optic mode,
we used the BT4 thermal triple-axis measurements at NIST (Gaithersburg, USA).
The collimation was set to $open$-80′-$S$-80′-$open$ with a graphite filter
placed on the scattered side of the sample to remove higher order
contamination of the neutron beam. The incident energy was selected with a
vertically focused PG(002) graphite monochromator and the final energy was
fixed to Ef=14.7 meV using a flat PG(002) analyzer crystal.

###  II.3 Neutron Cross Section

The measured neutron scattering intensity at a particular energy transfer
defined $\hbar\omega\equiv E\equiv E_{i}-E_{f}$ on a reactor based instrument,
with a monitor detector before the sample, is directly proportional to the
structure factor $S(\vec{Q},E)$. This in turn is related to the imaginary part
of the susceptibility $\chi^{\prime\prime}(Q,\omega)$ by,

$I(\vec{Q},E)\propto
S(\vec{Q},E)\equiv{1\over\pi}[n(E)+1]\chi^{\prime\prime}(\vec{Q},E),$ (1)

where $[n(E)+1]$ is the Bose thermal population factor. For the purposes of
this paper, we divide the susceptibility up into two terms depending on
momentum and energy transfer as follows,

$\chi^{\prime\prime}(\vec{Q},E)=\chi(\vec{Q})F(E),$ (2)

with $\int_{-\infty}^{\infty}dE\ F(E)=1$. We describe the particular form for
both the momentum dependent $\chi(\vec{Q})$ and energy dependent $F(E)$ in the
text below.

##  III Results

###  III.1 Structural properties:

Figure 1:  Temperature dependence of the structural properties of our 1 cm3
crystal. $(a)$ illustrates the lattice constant as a function of temperature
and $(b)$ the full width of the $\vec{Q}$=(0,0,1) Bragg peak as a function of
temperature. $(c)$ compares the temperature dependent structural diffuse
scattering near $\vec{Q}$=(0, 0, 1) with the structural intensity at
$\vec{Q}$=(1/2, 1/2, 1/2) measured with polarized neutrons on DNS. $(d)$
display representative scans through the $\vec{Q}$=(0,0,1) Bragg peak measured
on SPINS (note the errorbars are approximately the size of the data points).
Figure 2:  The structural diffuse scattering near $\vec{Q}$=(0, 0, 1) at $(a)$
T=300 K and $(b)$ T=3 K measured on SPINS. The temperature dependence of a
line cut through the diffuse scattering at $\vec{Q}$=(H, 1.07, 0) (r.l.u.) is
shown in $(c)$.

The magnetic and structural energy scales are different in PFN. PFN has a
magnetic Curie-Weiss temperature of $\Theta_{CW}$ $\sim$ \- 250 K (Figure 1 of
Ref. 74) while the zone center soft transverse optic mode reaches an energy
minimum at $\sim$ 400 K [74, 83] defining the ferroelectric transition. We
first present the structural properties below the ferroelectric transition
temperature, but at temperatures relevant to the magnetic response which we
discuss later in this paper.

####  III.1.1 Static structural properties:

We first experimentally investigate the structural properties of our PFN
crystal with neutron scattering and are summarized in Fig. 1 applying the cold
neutron spectrometer SPINS to investigate the elastic scattering near the
$\vec{Q}$=(0,0,1) Bragg peak. Fig. 1 $(a-b)$ plot the measured lattice
constant $a$ and also the longitudinal full-width at half maximum $\delta
2\theta$ as a function of temperature below 300 K (note this is below the
ferroelectric transition at $\sim$ 400 K). Representative $\theta-2\theta$
scans are illustrated in Fig. 1 $(d)$. The lattice constant in panel Fig. 1
$(a)$ shows a monotonic decrease as the temperature is lowered with no
observable discontinuity that would indicate a bulk structural transition and
a similar featureless trend is seen in the longitudinal linewidth Fig. 1 $(b)$
which only shows a very small (on the scale of the resolution) and monotonic
change with temperature.

In Fig. 1 $(c)$, we plot the temperature dependence of the momentum broadened
structural diffuse scattering near $\vec{Q}$=(0,0,1) which displays an
increase in intensity on cooling. This scattering is outside the momentum
resolution of the spectrometer and indicative of spatially short range
correlations. The data at $\vec{Q}$=(${1\over 2}$,${1\over 2}$,${1\over 2}$)
are taken at a position sensitive to the magnetic scattering discussed below,
however applying polarized neutrons to extract the nuclear component as
discussed in the Appendix. There is no observable change in the nuclear
component of the scattering at $\vec{Q}$=(${1\over 2}$,${1\over 2}$,${1\over
2}$). In terms of the nuclear Brillouin zone this is the $R$ point zone
boundary (taking the space group as $Pm\overline{3}m$) and temperature
dependent structural diffuse scattering has been reported at this point in
PMN. [84] Such zone boundary scattering is not clearly evident in our PFN
crystal and is suggestive that the Fe3+ and Nb5+ ions, though in approximately
equal 50 % stochiometric amounts, are disordered in our single crystal,
consistent with previous powder diffraction [85] and dielectric [86] studies.
Temperature dependent nuclear diffuse scattering in PFN exists near the
nuclear Bragg peaks, indicative of spatially short range correlations, but not
at the zone boundaries.

Mesh scans near $\vec{Q}$=(0,1,0) (Fig. 2 $(a,b)$) illustrate the presence of
both resolution limited Bragg peaks (discussed above) and momentum broadened
ridges of scattering. The diffuse ridges of scattering extend along the
[1$\overline{1}$0] directions and are qualitatively consistent with the
diffuse scattering reported in non magnetic relaxors PMN [32, 34, 35] and PZN
[87, 47]. The temperature dependence is plotted in Fig. 1 $(c)$ (discussed
above) and is further illustrated in Fig. 2 $(c)$ through a contour plot of
the elastic momentum line scans along the (H, 1.07, 0) direction. The cut
extends through the two ridges of the “butterfly” nuclear diffuse scattering
shown in panel Fig. 2 $(a,b)$ and further shows a smooth increase in intensity
as the temperature is lowered. The presence of diffuse scattering which grows
with decreasing temperature is indicative of localized polar correlations
analogous to prototypical relaxors PMN and PZN. There presence indicates
localized structural disorder as characterized extensively in prototypical
relaxors.

####  III.1.2 Low-energy lattice vibrations and phonons:

Figure 3:  The temperature dependence of the optic and acoustic phonons near
$\vec{Q}$=(2, 0, 0). $(a-c)$ illustrates constant momentum cuts at
$\vec{Q}$=(2, -0.15, 0). The temperature dependence of the linewidth $2\Gamma$
of the transverse $(d)$ acoustic and $(e)$ optic phonons.

Having discussed the continuous static response at low temperatures where
magnetism is relevant, we now discuss the phonons. Fig. 3 $(a-c)$ displays a
series of constant momentum scans at $\vec{Q}$=(2, -0.15, 0) finding two peaks
characterizing the lower energy transverse acoustic and a higher energy
transverse optic modes. The data displays phonons with the solid curve a fit
to the sum of two simple harmonic oscillators with each oscillator being
defined by antisymmetric lorentzians,

$F(E)\propto\left({1\over{1+\left({{\omega-\Omega_{0}}\over\Gamma}\right)^{2}}}-{1\over{1+\left({{\omega+\Omega_{0}}\over\Gamma}\right)^{2}}}\right)$
(3)

with $\hbar\Gamma$ the energy linewidth, inversely proportional to the
lifetime $\tau$, and with an energy position of $\hbar\Omega_{0}$. The
lorentzian at $+\hbar\Omega$ in Eqn. 3 accounts for the cross section for
neutron energy gain (energy transfer negative $E<0$). This is required for the
cross section to follow detailed balance which implies the imaginary part of
the susceptibility $\chi^{\prime\prime}(\vec{Q},E)$ is an odd function in
energy.

Figure 4:  A comparison of the low-energy T2 acoustic phonons near
$\vec{Q}$=(1, 1, 0) measured on the non magnetic relaxor $(a)$ PMN and the
magnetic relaxor $(b)$ PFN with identical spectrometer configurations on
SPINS.  Figure 5:  The temperature dependence of the T2 acoustic phonons in
PFN measured on SPINS. $(a-c)$ show constant momentum cuts $\vec{Q}$=(1.035,
0.965, 0) and $(d-e)$ illustrate scans taken at $\vec{Q}$=(1.065, 0.935, 0).

Both acoustic and optic modes at $\vec{Q}$=(2,-0.15,0) are underdamped and
well defined in energy and this is confirmed in Fig. 3 $(d,e)$ which do not
show any large anomaly over the temperature range studied. This response is in
agreement with non magnetic relaxors PMN and PZN measured at similar positions
in momentum away from the nuclear zone center where the waterfall effect [23,
20, 88] is present. However, this is in contrast to the phonon response near
$\vec{Q}$=(1,1,0). Figure 4 illustrates a comparison between constant momentum
cuts in non magnetic PMN (taken from Ref. 89) compared to PFN. Both scans are
performed at the same position in momentum using the same instrumental setup
on SPINS. These scans are taken along the transverse direction near the
$\vec{Q}$=(1,1,0) Bragg position and are therefore sensitive to the $T_{2}$
acoustic phonons which propagate along [1$\overline{1}$0] and are polarized
along [110]. The PMN data displays a resolution limited acoustic phonon
described by the lineshape above, however the PFN data is indicative of a
broader energy lineshape. Given the lack of any other model, we have chosen to
fit the data to two damped harmonic oscillators as illustrated by the curves
in Fig. 4 to provide a description of this multi component lineshape.

As noted in Refs. 90, 89, 79, the dynamic lineshape near $\vec{Q}$=(1,1,0) is
complicated over measurements near $\vec{Q}$=(2,0,0) by the presence of a
strong elastic diffuse scattering cross section which contributes to a low-
energy quasielastic-like [91, 92, 93] response. This has been interpreted as
either a coupling between harmonic phonons with relaxational dynamics [90, 89]
or in terms of quasielastic scattering indicative of non harmonic relaxation.
However, as displayed in Fig. 5, the lineshape near $\vec{Q}$=(1,1,0) is much
more complex than reported previously in PMN. Fig. 5 $(a)$ shows that at T=500
K the linshape is dominated by a single underdamped phonon mode. At T=200 K
(Fig. 5 $(b)$) the lineshape becomes more intricate and is well described by
two temporally damped harmonic oscillators. At low temperature of T=50 K (Fig.
5 $(c)$), the lineshape is described by two distinct components with a sharp
underdamped phonon mode and a much broader in energy overdamped mode. The
development of a multi component lineshape is also found at larger momentum
transfers as illustrated in Fig. 5 $(d,e)$ which display constant momentum
scans at T=500 K and 250 K. The multiple components are particularly clear at
250 K in panel $(e)$.

Figure 6:  The magnetic diffuse scattering in PFN. $(a)$ The temperature
dependence of the magnetic intensity at $\vec{Q}$=(1/2, 1/2, 1/2) measured on
energy integrating instruments DNS (MLZ) and PRISMA (ISIS). These are compared
against measurements on TASP (PSI) with an energy resolution of $2\delta$E=0.1
meV. Note that all measurements have been normalized to the intensity at the
base temperature. $(b-c)$ illustrate the purely magnetic diffuse scattering
taken at T=3 and 300 K, respectively, measured using polarized neutrons at DNS
(MLZ). We note that the intense scattering at the Bragg position of
$\vec{Q}$=(-1,-1,0) is due to incomplete polarization of the neutron beam
discussed in the appendix. This is compared to the extracted nuclear cross
section in panels $(d-e)$.

While non magnetic PMN displays a broadening of the acoustic phonons near
$\vec{Q}$=(1,1,0) [89, 90, 79], the lineshapes shown above in PFN are more
complicated and require at least two components to parameterize. We will
discuss this below in the context of the magnetic properties, however these
indicate a strong amount of localized structural disorder in PFN not observed
in prototypical relaxors PMN and PZN. We note that this breakdown of well
defined phonons is not due to any distinct low temperature distortion given
the continuous response observed in the statics discussed in the previous
section.

###  III.2 Magnetism in PFN:

We now discuss the magnetic properties originating from the Fe3+ ($S$=5/2, $L$
$\approx$ 0) moments. First, we show energy integrated data which is an
approximate measure of $S(\vec{Q})$ that defines the static magnetic response.
Second, we show triple-axis measurements which are energy resolved to discuss
the static (on the timescale of our resolution) and dynamic magnetism. We note
that, owing to resolution, neutrons are sensitive to dynamics on $\sim$ THz
frequency scale in comparison to the slower dynamics $\sim$ MHz probed with,
for example, muons.

####  III.2.1 Energy integrated magnetic diffuse scattering:

Figure 7:  The magnetic diffuse scattering measured with energy integrating
neutron instruments. Polarized neutrons are used from D7 (ILL) to compare
magnetic and nuclear cross sections at T=3 K and 300 K in $(a-d)$. Unpolarized
measurements from PRISMA (ISIS) are showin in $(e-g)$.

Figure 6 illustrates a summary of the elastic magnetic properties as a
function of temperature using both polarized (DNS) and unpolarized neutrons
(PRISMA and TASP). Fig. 6 $(a)$ plots the temperature dependence of the
momentum integrated magnetic scattering at $\vec{Q}$=(${1\over 2}$,${1\over
2}$,${1\over 2}$) corresponding to antiferromagnetically correlated spins
along the three crystallographic directions. Three different datasets from
PRISMA, DNS, and TASP are shown normalized to base temperature. Differences
are seen in the temperature dependence of TASP data in comparison with PRISMA
and DNS. While the PRISMA and DNS data illustrate a consistent temperature
dependence with a gradual and continuous increase of intensity with decreasing
temperature, TASP displays more of a well defined intensity increase at low
temperatures. We note that none of these data sets is representative of a
sharp increase of intensity, indicative of a clear phase transition defined by
a critical temperature where a spatially long-range magnetic order parameter
appears that subsequently saturates at low temperatures.

The difference between the temperature dependences from the three different
data sets can be understood by the differing energy integration of the
corresponding instruments resulting from the differing energy resolutions.
[94] Both DNS and PRISMA are two-axis diffractometers which integrate the
dynamics owing to the lack of any energy analysis. TASP is a triple-axis with
an analyzer PG(002) crystal that selects a particular final energy. The energy
resolution on TASP for the data discussed here is $2\delta E$ =0.1 meV where
$\delta E$ is the half-width at half maximum in energy. We therefore conclude
the presence of a significant amount of dynamic spectral weight which
gradually slows and enters the elastic resolution with decreasing temperature
which causes a gradual increase in intensity for diffractometers that
integrate over the dynamics. It also results in the lower onset temperature on
TASP as the dynamics which slow will enter the time window on TASP at a lower
temperature.

Figs. 6 $(b-e)$ illustrate the polarization analysis described in the Appendix
separating nuclear and magnetic contributions. Figs. 6 $(b-c)$ illustrate the
purely magnetic contribution at 3 K and 300 K showing the presence of magnetic
correlations which develop at $\vec{Q}$=(${1\over 2}$,${1\over 2}$,${1\over
2}$). We note that this is not a Bragg peak as the magnetic correlations are
extended in momentum indicative of spatially short-range correlations. Figs. 6
$(d,e)$ display the nuclear cross section which shows the absence of
observable nuclear scattering at the $\vec{Q}$=(${1\over 2}$,${1\over
2}$,${1\over 2}$) antiferromagnetic position, however does illustrate the
diffuse nuclear scattering near $\vec{Q}$=(0,0,$\pm$1) discussed above.

A more detailed momentum dependence is displayed in Fig. 7 taken on the D7
(ILL) polarized diffractometer in panels $(a-d)$ and unpolarized data from
PRISMA in panels $(e-g)$. The polarized data in panels $(a-d)$ are separated
into magnetic and nuclear components for datasets taken at T=3 K and T=300 K.
While the low-temperature polarized data show symmetric momentum broadened
peaks at $\vec{Q}$=(${1\over 2}$,${1\over 2}$,${1\over 2}$), the high
temperature T=300 K maps shown in panels Figs. 7 $(c)$ and Fig. 6 $(c)$
illustrate an unusual lineshape in momentum. In particular, the T=300 K
magnetic cross section (Fig. 7 $(c)$) displays a “wheel-like” response which
is unphysical given the average cubic nature of the crystal structure. Such
lineshapes have been reported in the study of energy integrated diffuse
scattering and have been referred to as “Catherine Wheels”. [95, 96, 97, 98]
The extended and curved lineshape is further confirmed using PRISMA in panels
Fig. 7 $(e-g)$ and is arguably more pronounced. We note that PRISMA integrates
over a broader energy range than DNS, owing to the fact it is on a spallation
source, and therefore this is suggestive that the lack of energy analysis is
the origin of this unphysical lineshape. We discuss this in the Appendix and
show that it is an instrument artifact resulting from the energy integration
on D7, DNS, and PRISMA combined with the presence of thermally excited
magnetic dynamics sampled through the energy gain neutron cross section.

####  III.2.2 Static Magnetism

Figure 8:  Correlated elastic magnetic scattering measured on TASP at $(a)$
T=2 K and $(b)$ 70 K. The solid red lines are fits to the lineshape discussed
in the text. $(c)$ The inverse correlation length as a function of
temperature. Note the local maximum at $\sim$ 15 K highlighted by the inset.

To isolate the static magnetic cross section from the dynamics, we use the
energy analysis on the TASP triple-axis spectrometer. Representative scans
through the correlated magnetic scattering are shown in Fig. 8 $(a,b)$ at T=2
K and 70 K. To model the elastic lineshape we have followed Ref. 74 and fit a
lorentzian squared with one direction integrated to account for the coarse
vertical integration of the spectrometer, hence resulting in a power of
${3\over 2}$,

Figure 9:  $(a,b)$ Contour plots of the magnetic scattering as a function of
temperature measured on SPINS (Ef=5.0 meV, $2\delta E$=0.25 meV) and TASP
(Ef=2.9 meV, $2\delta E$=0.1 meV). $(c)$ The elastic magnetic intensity and
$(d)$ dynamic component measured on TASP with Ef=2.9 meV.

$\chi(\vec{Q})\propto{1\over{\left[1+\left({|\vec{Q}-\vec{Q}_{0}|\xi}\right)^{2}\right]^{3\over
2}}}.$ (4)

The inverse correlation length $1/\xi$ is plotted in Fig. 8 $(c)$. On cooling
from room temperature, the spatial correlations increase gradually until
$\sim$ 50 K. On further cooling these shorten until $\sim$ 15 K where spatial
correlations increase once again. This lower temperature coincides with
anomalies in the temperature dependent magnetization and attributed to the
formation of a glass phase [99, 100] and the temperature dependence of the
correlation length is similar to that reported in ferromagnetic spin-glasses
[101]. The lengthscale studied here never gets particularly large and only
reaches a maximum of $\sim$ 4 unit cells. This behavior is different from
typical critical scattering where the correlation length scale becomes large
and diverges on cooling and neither inflection points in the correlation
length can be considered as a phase transition in PFN. As discussed in the
Appendix in the context of our polarized neutron results, the magnetic
structure that we measure with neutrons is isotropic on average. However,
given the issue of domains, it is not possible from our bulk experiments to
determine if the structure is uniaxial as proposed in Ref. 102. However,
unlike Ref. 102, we do not observe a magnetic Bragg peak indicative of a
spatially long-range component.

####  III.2.3 Magnetic dynamics:

In Fig. 9 $(a)$ we show a contour plot of constant momentum scans at
$\vec{Q}$=(${1\over 2}$,${1\over 2}$,${1\over 2}$) taken on SPINS (NIST)
showing the temperature dependence of the low-energy magnetic dynamics. The
data indicate a gradual slowing of the dynamic magnetic response as the
temperature is decreased and the fluctuations collapse to the elastic line
which defines static magnetism on the timescale of the instrument resolution.
The same dynamics are studied with finer resolution using TASP (PSI) with
Ef=2.9 meV in Fig. 9 $(b)$ and indicates a freezing of the dynamics on the
timescale of TASP at a lower temperature of $\sim$ 20 K. Similar to the
comparison above (Fig. 6), the measured temperature where static magnetism is
observable is determined by instrumental resolution with finer energy
resolution illustrating lower temperature scales.

In Fig. 9 $(c,d)$ we plot the parameters of a fit to the TASP data to the
following lineshape, convolved with the spectrometer resolution, indicative of
a static (on the timescale of the spectrometer resolution) and dynamic
magnetism defined by a single energy scale,

$F(E)=\chi_{0}[n(E)+1]{{\omega\Gamma}\over{\omega^{2}+\Gamma^{2}}}+B\delta(E),$
(5)

where $\chi_{0}$ is the dynamic intensity and is related to the real part of
the susceptibility by the Kramers Kronig relation and $\Gamma$ is the single
dominant energy scale inversely proportional to the fluctuation timescale. The
amplitude $B$ is the component which is resolution limited and residing in the
elastic line (denoted as a $\delta$-function in the lineshape above). Fig. 9
$(c)$ plots the elastic component defined by the parameter $B$ as a function
of temperature and shows a gradual increase in intensity with decreasing
temperature starting $\sim$ 50 K. As illustrated above in our discussion of
energy integrated diffuse scattering, this temperature dependence is tied to
the spectrometer resolution and not indicative of a phase transition to static
magnetic order. We note that muon spectroscopy studies magnetism on the $\sim$
MHz scale which is within the experimental resolution here characterized by
the elastic $\delta$-function in Eqn. 5. In Fig. 9 $(d)$ we plot $1/\chi_{0}$
as a function of temperature which displays a peak below $\sim$ 15 K at a
similar temperature where we see an anomaly in the correlation length above
(Fig. 8) and magnetization [99, 100]. Based on the freezing of the dynamics,
characterized by an elastic contribution to the magnetic neutron cross
section, and also the peak in the susceptibility, we relate this low
temperature to a glass transition in analogy to prototypical spin-glasses
[103, 104].

We investigate the magnetic dynamics over a broader energy range in Fig. 10
using the TASP triple-axis which highlights a problem as the data cannot be
well described by this single energy scale lineshape (Eqn. 5). The magnetic
scattering displayed by PFN at low temperatures shows a continuous intensity
distribution over all measured energy transfers. This contrasts with typical
critical scattering of a magnet where the cross section is peaked at a finite
energy defining the characteristic timescale $\tau$ of the fluctuations which
become static at the ordering temperature (examples shown in Refs. 105, 106).
While the single energy scale is an effective approximation, describing data
over a small energy range (like in Fig. 9) or with coarser energy resolution
(as in Ref. 74), the finer energy resolved experiments presented here
illustrate the neutron cross section is not consistent with this analysis.

Figure 10:  Fits to the bandwidth model described in the text at $(a)$ 2 K and
$(b)$ 160 K. $(c)$ illustrates the bandwidth as a function of temperature. The
data has been corrected for the background by measuring intensity at a
reciprocal space position away from $\vec{Q}=({1\over 2},{1\over 2},{1\over
2})$ as a function of energy transfer.

Given that Fig. 10 $(a)$ displays a featureless cross section at all
observable energy transfers with no observable peak in energy, we consider a
model that describes a bandwidth of energy scales. Instead of the approach
applied above where the susceptibility is

$\chi(E)\propto{1\over{\Gamma-iE}}$ (6)

we integrate the susceptibility over a range of frequencies,

$\chi(E)\propto\int_{\Gamma_{1}}^{\Gamma_{2}}d\gamma{1\over{\gamma-iE}}.$ (7)

This analysis is motivated by work on non-Fermi liquids [107] and has recently
been applied to understand the unusual quasielastic lineshape in hybrid
perovskites [108]. Performing the integral and taking the imaginary part,
which is proportional to the neutron cross section, we find

$F(E)\propto{1\over{\Gamma_{2}-\Gamma_{1}}}\left[\arctan\left({E\over\Gamma_{1}}\right)-\arctan\left({E\over\Gamma_{2}}\right)\right].$
(8)

A fit of this lineshape to the magnetic scattering in PFN is shown in Fig. 10
$(a-b)$ at T=2 K and 160 K and provides a good description of the continuous
range of scattering over the entire dynamics range. The fits have also
included an elastic contribution at E=0 defined by the energy resolution which
parameterizes scattering from localized magnetic ions that are static on the
timescale of the resolution of TASP. In comparison to the frequency range
sampled by muons, we note that the frequency scale was on the order of $\sim$
MHz, which is within the elastic line measured here with neutrons.

The bandwidth obtained from this fitting procedure, defined by
$\Delta\Gamma\equiv|\Gamma_{1}-\Gamma_{2}|$, is plotted in Fig. 10 $(c)$. It
is worth noting that the energy range of the scans essentially fixes the
maximum bandwith that can be derived. In this experiment applying cold neutron
triple-axis spectrometers, this energy range is confined to below
$\Delta\Gamma_{max}\sim$ 10 meV and therefore the errorbars on the
measurements at high temperatures are large when it is expected the bandwidth
exceeds the dynamic range of the spectrometer.

Fig. 10 $(c)$ shows a decrease in $\Delta\Gamma$ with decreasing temperature
from 200 K until below $\sim$ 40 K the bandwidth is seen to increase. This is
a similar temperature scale to where the elastic, or static, component of the
low-energy neutron cross section was found to increase in Fig. 9 above. The
range of frequencies contributing to the imaginary part of the susceptibility
$\chi^{\prime\prime}(\omega)$ makes this response analogous to a glass
response. [109, 110]

Figure 11:  $(a-c)$ Representative constant energy cuts at T=2 K with the
horizontal bar a measure of the calculated momentum resolution. Plots of the
momentum half-widths of the correlated magnetic scattering as a function of
temperature at $(d)$ 0.5 meV, $(e)$ 1.0 meV, and $(f)$ 2.0 meV.

In Fig. 11 we investigate the momentum dependence of the dynamic correlated
magnetic scattering with temperature. Example constant energy scans are
illustrated in Figs. 11 $(a-c)$ and the peak half widths in momentum, at each
energy transfer, is shown in Figs. 11 $(d-f)$. The lineshape used was defined
as follows to fit the momentum dependence of the dynamic component of the
scattering cross section,

$\chi(\vec{Q})\propto{1\over{1+\left({{|\vec{Q}-\vec{Q}_{0}|}\over{\Gamma_{q}}}\right)^{2}}}.$
(9)

The plots of the $\Gamma_{q}$ in Fig. 11 illustrates a qualitatively similar
temperature dependence to the bandwidth $\Delta\Gamma$ discussed above. On
decreasing temperature from 300 K, the linewidth initially decreases,
indicative of larger spatial dynamic correlations. However, below some “cross
over” temperature (indicated by the arrow in Fig. 11) the linewidth increases
indicative of spatially shorter dynamic correlations. It is interesting to
note that this cross over temperature occurs at progressively lower
temperatures, when sampled at lower energies. The measured unraveling of
magnetism in PFN depends on the energy scale experimentally sampled.

Based on this study of the dynamics, we characterize the fast dynamics that
drive the cluster glass which forms at low temperatures. We find there is a
temperature range $\sim$ 30-40 K where the frequency range over which the
spins fluctuate reaches a minimum. Below this temperature, PFN becomes more
disordered spatially and temporally. This temperature range also defines where
the magnetic fluctuations become static (albeit spatially short-ranged) on the
resolution of cold neutron spectroscopy. However, below this a second
temperature scale $\sim$ 15 K develops where the correlation lengths increase
which is coincident with a peak in $1/\chi_{0}$. This coincides with previous
reports of the formation of a cluster spin-glass phase.

##  IV Discussion and conclusions

The magnetic static and dynamics discussed above show a spin response
characterized by a range of frequencies on the scale $\sim$ THz. On decreasing
temperature, we observe a development of correlations characterized by an
increase in the magnetic fluctuation timescale and increasing correlation
lengths. However, over a particular crossover temperature range this trend
towards spatially long range order is interrupted and the static and dynamic
correlation lengths decrease and the frequency response is characterized by a
broadband of fluctuations. We note this magnetic response occurs on an
underlying lattice with considerable disorder evidenced by our acoustic phonon
investigation.

This magnetic response differs from disordered or frustrated magnets.
Frustrated magnets [111] generally show a gradual slowing of dynamics with
decreasing temperature and similar results are found in disordered two
dimensional cuprates [106, 105]. Qualitatively similar results are observed in
disordered metallic compounds [112] which enter into a spin-glass phase at low
temperatures. In such systems, there are two lengthscales that are important,
first the cluster [71] size consisting of a high concentration of magnetic
ions, and second the lengthscale being the distance between the magnetic
clusters. The spin-glass phase is driven by frustrating interactions that
become relevant once the correlation length becomes large enough. Unlike
reports in conventional spin-glasses [113] where the dynamics appear to be
characterized by a single energy scale, the dynamics in the cluster glass
discussed here are governed by a band of frequencies. The effects of localized
clusters has a strong impact on the spin fluctuations. The presence of
spatially localized regions, which presumably form the basis of the clusters,
with differing chemical environments is corroborated by our phonon results
illustrative of multiple components and hence regions in the crystal.

However, such a model does not explain the change in spin correlations in PFN
at low temperatures where the spatial correlation length decreases on
decreasing temperature. Such a scenario based on two length scales with
interactions becoming frustrated would be suggestive of a saturation of static
and dynamic correlations. It might be tempting to associate this with the
onset of local polar correlations characterized by the temperature dependence
of the diffuse scattering near the nuclear Bragg peaks described above. Indeed
such a scenario has been proposed in lightly doped SrTiO3 where spin and polar
freezing are coupled. [114] However, the temperature scale of the inception of
static polar correlations in PFN is much higher than where we observe an
unfolding of magnetism when measured with comparable resolution. The
temperature scale where we observe a decrease in magnetic spatial correlation
lengths, characterizing increased disorder on cooling, is near where static
magnetism is observed through neutron scattering intensity at $E=0$ and
$\vec{Q}$=(${1\over 2}$,${1\over 2}$,${1\over 2}$).

We therefore propose that the origin of the disordering of the magnetism in
PFN on cooling is the result of random fields induced by the local molecular
field from static magnetism. Such a scenerio has been found to drive similar
temperature responses in dilute ferromagnets. [115] We speculate and mention
two different mechanisms for this based on competing interactions. First,
while PFN is not a ferromagnet, there has been the suggestion for the presence
of competing antiferromagnetic and ferromagnetic exchange interactions. [116]
The exchange pathway in PFN is likely to be dominated by superexchange through
the oxygen ion and this could be ferromagnetic if the pathway is 90∘ or
antiferromagnetic if the pathway is 180∘ [117] resulting in differing
localized molecular fields, based on the local environment which is randomized
owing to the disorder on the Fe3+/Nb5+ site occupancy. A second possibility is
for the existence of iron rich spatially localized regions which are coupled
in different manners depending on the environment. Both mechanisms, result
from conflicting interactions resulting in a local random field which
inherently changes depending on the strength of the localized static
magnetism. With the presence of disorder and clustering of magnetic Fe3+
spins, it is inevitable for the presence of localized random molecular fields.
We note that this induces a new random molecular field as static magnetic
order gradually develops with decreasing temperature. In turn, this would
result in unravelled magnetism evidenced through shortened spatial correlation
lengths. We note that this mechanism depends on the correlation length
increasing on decreasing temperature and this does not depend on the small
changes of Fe3+ concentration or ordering that may induce a magnetic Bragg
peak. Ultimately, the presence of a glass transition at lower temperatures
where magnetism is unravelled is not dependent on the subtle details of Fe3+
concentrations and may explain why this lower temperature anomaly is present
in all reported samples.

The presence of significant disorder in the sample and also the correlation
lengths of only a few unit cells amplifies the role that this field would
impose on spins. The presence of a random field contribution to the magnetic
Hamiltonian would result in the lineshape describing the magnetic
$S(\vec{Q},E)$ to have two components. [118] The static component measured at
the elastic line would be characterized by a lorentzian squared momentum
dependence and the dynamics a lorentzian. This is in agreement with the data
analysis presented above. While we observe two different temperature scales
for the onset of static spatially short-range magnetism and also polar
correlations, the onset of static diffuse scattering at higher temperatures
likely enhances the disorder and random fields at lower temperatures.

We note, however, that other samples of PFN have reported a magnetic Bragg
peak indicative of spatial long-range order set by the resolution of the
diffractometer [119] and supported by NMR [120]. These have been suggested to
be spatially distinct, hence the term “cluster” in Ref. [71]. We note Bragg
peaks in the neutron response defining antiferromagnetic order have been
observed to coexist with intense magnetic diffuse scattering. It is not clear
from the available published data how the integrated spectral weights,
reflecting the relative volume fractions, compare. Also, magnetic order has
been reported to be sensitive to relative Fe/Nb concentrations [121, 122]. We
note that similar issues have been discussed in the context of the unit cell
shape in relaxors PMN and PZN are discussed elsewhere. [123, 87, 124, 125,
126, 127, 128, 129] In some respects, this is analogous to skin regions
reported in random field Ising magnets. [130]

Having discussed the disordering of the magnetic correlations in PFN, we now
mention the low-temperature anomaly in Fig. 8 $(c)$ (see inset) where an
increase in spatial ordering at $\sim$ 15 K occurs evidenced by a subtle
increase in the correlation length on cooling. This low temperature scale is
not reflected in the dynamics probed with triple-axis spectrometers discussed
above, indicative that the fluctuations associated with this temperature are
within the resolution of the neutron instruments applied here. Higher
resolution measurements such as with either spin-echo or muon spectroscopy are
required to probe the dynamics in this temperature range.

In summary, we have reported a neutron study of the structural and magnetic
correlations in PFN. Based on the low-energy acoustic phonons, we find
evidence that the sample consists of multiple chemically different regions
presenting significant disorder in our single crystal. The magnetic dynamics
on cooling from room temperature initially show increased spatial correlation
lengths yet at a crossover temperature the static and dynamic correlation
lengths decrease. This is followed by a transition that we suggest represents
a glass transition comparing our neutron and susceptibility results. The
magnetic dynamics at low temperatures are well described by a broadband of
contributing frequencies analogous to glasses. We propose a model where the
disordering of magnetic correlations at low temperatures is driven by random
fields resulting from the local molecular field originating from the spatially
short range Fe3+ magnetic correlations. These random fields eventually drive
PFN into a magnetic low temperature glass “cluster” phase with the dynamics
described by a broadband of frequencies.

## Acknowledgements

We are grateful to the EPSRC and the STFC for funding. N.G.-D. supported by
EPSRC/Thales UK iCASE Award No. EP/P510506.

##  V Appendix A - Neutron polarization analysis

Figure 12:  A visualization of the sampled region in momentum space on the DNS
diffractometer by scanning the angle at the sample position. Each point
represents a detector and illustrates the detector density used to generate
the plots for the DNS experiment. Figure 13:  Colormaps illustrating the
polarized neutron cross section at T=3 K for the different cross sections
measured on DNS (MLZ). The $X$ channel gives the total magnetic cross section
in the spin-flip channel and the residual correlated scattering near
$\vec{Q}$=(1/2, 1/2, 1/2) is the result of incomplete polarization of the
neutron beam. Figure 14:  Colormaps illustrating the polarized neutron cross
section at T=200 K for the different polarized cross sections measured on DNS
(MLZ).

In this appendix, we outline the polarization analysis used to separate the
purely magnetic and nuclear cross sections discussed above. For the discussion
here, we focus our analysis using the DNS (MLZ) and the D7 (ILL) neutron
diffractometers which both consist of arrays of neutron detectors confined to
a horizontal scattering plane with the detectors equally spaced in angle. For
example, the scattered beam at DNS was measured with 21 detectors equally
spaced 5∘ apart in the horizontal scattering plane. These have been converted
to momentum transfer in the main text of the paper and the conversion is
illustrated in Fig. 12. Both DNS and D7 operate with an $XYZ$ polarization
analysis where the neutron spin direction can be tuned to be along three
orthogonal axes. The $Z$ axis defined as being vertical to the horizontal
scattering plane and $X$ is defined as being parallel to the average momentum
transfer $\vec{Q}$ of the scattered detector bank. The $Y$ axis is defined as
being perpendicular to the $X$ and $Z$ axes.

The magnetic scattering cross section and how it is measured in neutron
scattering has been outlined in several books [131, 132] and review articles
[133, 134]. For completeness we summarize the important points here.
Unpolarized neutron scattering is sensitive to the component of the magnetic
moment, perpendicular to the wavevector transfer defined by $\vec{Q}$. This
component of the magnetic moment is defined by $\vec{S}_{\perp}$. For the
uniaxial polarization experiments used here, the components of
$\vec{S}_{\perp}$ perpendicular to the neutron polarization will appear in the
spin flip (SF) channel and the components of $\vec{S}_{\perp}$ parallel to the
neutron polarization will be sampled in the non spin flip (NSF) channel.
Nuclear scattering will appear in the non spin flip channels. These
polarization dependent rules were used in this experiment to separate out
purely magnetic and nuclear scattering cross sections.

An example plot of all six channels for an uniaxial polarization experiment is
illustrated in Fig. 13 for PFN aligned in the (HHL) scattering plane using the
DNS diffractometer at T=3 K. All of the magnetic scattering occurs in the SF
channel with the polarization aligned along $X$. The NSF channel for the
neutron beam polarized along $X$ is a measure of the incomplete polarization
of the neutron beam. This originates predominately from several experimental
aspects associated with polarized neutrons. First, the flipping ratio, defined
as the ratio of the intensities in the NSF and SF channels. On both DNS and D7
this was measured to be 20. Second, we note that the $X$ direction is defined
as parallel to the average momentum transfer $\vec{Q}$ of the scattered
detector bank. This results in some feed through of intensity between the NSF
and SF channels for the SF measurements in the $X$ channel as $\vec{Q}$ is not
exactly along with the $X$ direction for all detectors. This net feedthrough
is evident in Fig. 13 which shows some correlated scattering in the NSF
channel with the neutron polarization along $X$. The extra scattering which
occurs at large momentum transfers in the NSF channels originates from nuclear
scattering not sampled in the SF channel.

Figure 15:  Simulated magnetic diffuse scattering in PFN applying the model
discussed in the main text. $(a,b)$ Simulated magnetic diffuse scattering
illustrating the effects of different integration ranges on the resulting
pattern at T=300 K. $(c)$ Further simulation of the magnetic diffuse
scattering at T=3 K. Figure 16:  The kinematics for diffuse scattering
measurements without energy analysis. The peaks (or maximum intensity in the
energy integrated neutron cross section $S(\vec{Q})$) is illustrated assuming
a ridge of dynamical scattering centered around a single momentum position
like that measured using triple-axis spectroscopy in PFN.

When comparing the SF and NSF channels for the $Y$ and $Z$ polarizations (when
the polarization is aligned perpendicular to the average momentum transfer) it
can be seen that the correlated scattering around $\vec{Q}$ =$({1\over
2},{1\over 2},{1\over 2})$ in Fig. 13 is isotropic and consistent with having
the same intensity in all four of these channels. The intensity in these four
channels is also, within experimental error, ${1\over 2}$ of the intensity
measured for the SF channel with the neutron polarization aligned along $X$
which samples the entire magnetic cross section. This scattering resembles
magnetic scattering with no preferred magnetic moment direction as is the case
for isotropic magnetic scattering. This isotropic scattering among these four
channels is not indicative of spatially long-range magnetic ordering and is
consistent with the short magnetic correlation lengths reported in this
compound and the lack of any sharp, or well defined magnetic ordering in the
data presented in the main text. The magnetic scattering in PFN is isotropic
with no preferred ordering direction.

This behavior across polarization channels is further consistent with the
weaker magnetic response at T=200 K illustrated in Fig. 14. In this case the
overall magnetic cross section is reduced owing to the temperature, yet the
the same pattern found above for T=3 K is repeated with generally isotropic
scattering found in the SF and NSF channels for $Y$ and $Z$ polarizations.

The properties of uniaxial polarization analysis of isotropic magnetic
scattering was used to extract and disentangle the magnetic and nuclear cross
sections in the main text. On D7, the nuclear scattering was measured directly
through the SF and NSF channels when the neutron was aligned along $X$. For
DNS, where the temperature dependence was studied in detail, we measured the
SF and NSF with the neutron polarization along the $Z$ axis. The isotropic
scattering inherent to the underlying isotropic nature of the Fe3+ moments
implies that the magnetic scattering is the same in both channels. However,
the nuclear cross section is only present in the NSF channels. By subtraction,
this allowed us to extract the nuclear background and plot the purely magnetic
and nuclear components shown in Figs. 6 and 7 in the main text.

##  VI Appendix B - Diffuse scattering and “Catherine Wheels”

In this appendix, we simulate the neutron diffuse scattering cross section
measured in energy integrating two-axis mode in an attempt to understand the
unusual momentum dependence discussed above in the magnetic channel at high
temperatures. Using a conventional triple-axis spectrometer with both a
monochromator and analyzer crystal, the measured intensity provides a direct
measure of $S(\vec{Q},\omega)$ with the energy transfer defined as
$\hbar\omega=E_{i}-E_{f}$. The diffractometers DNS and D7 operate in two-axis
mode where the analyzer is removed which ideally provides a measure of
$S(\vec{Q})\equiv\int_{-\infty}^{\infty}d\omega S(\vec{Q},\omega)$. DNS and D7
operate in a fixed Ei mode with the incident energy defined by a
monochromator. PRISMA uses time of flight to assign an energy to every
detected neutron under the assumption of elastic scattering where
Ei$\equiv$Ef. Given the constraints of the instrument, these only provide an
approximate measure of the $S(\vec{Q})\equiv\int_{-\infty}^{\omega_{c}}d\omega
S(\vec{Q},\omega)$ with the cutoff energy $\hbar\omega_{c}$ being fixed by
kinematics of the instrument used.

To test whether the unusual lineshape measured on D7 (Fig. 7) originates from
dynamics, we have assumed, for simplicity, that the neutron scattering cross
section can be approximated by a single relaxational lineshape multiplied by a
lattice lorentzian accounting for exponentially decaying spatial correlations.
This is written with the measured neutron scattering cross section
$I(\vec{Q},\omega)$, proportional to the structure factor $S(\vec{Q},\omega)$,
which is in turn related to the imaginary part of the susceptibility
$\chi^{\prime\prime}(\vec{Q},\omega)$ written above,

$\displaystyle\chi^{\prime\prime}(\vec{Q},\omega)\propto{{\omega\Gamma}\over{\omega^{2}+\Gamma^{2}}}\times{{\sinh(a/\xi)}\over{\cosh(a/\xi)-\cos(\vec{Q}\cdot\vec{a})}}.$
(10)

$\Gamma$ is the single dominant relaxation rate proportional to the inverse of
the lifetime $\tau$, $\xi$ is the spatial correlation length, and $\vec{a}$ is
the lattice constant. This form for the susceptibility was found to provide a
good approximation of the scattering previously measured with a thermal
triple-axis. [74]

To calculate an estimate of the diffuse scattering resulting from dynamic
magnetism we have performed the following procedure. 1) We have taken the form
of $S(\vec{Q},\omega)$ given by Eqn. 10 at each energy transfer as a function
of scattering angles $A4=2\theta$ and sample rotation angle $\psi$. 2) We have
then calculated the momentum transfer values based on these two angles
assuming the scattering is entirely elastic ($\hbar\omega=0$). 3) We have then
integrated the intensities over energy and imposed an energy cutoff which we
assume is determined by the instrument configuration. The results of this
calculation are present in Fig. 15 integrating in energy from -10 meV to two
cutoff energy values of 1 meV (panel $a$) and 10 meV (panel $b$). At T=300 K,
we have taken the measured value of $\Gamma\sim$ 10 meV found in Ref. 74 and
this is consistent with expectations based on the size of the Fe-O-Fe magnetic
exchange [135]. The value of $\Gamma$ restricts the cross section and implies
that energies below -10 meV do not contribute significantly to the integral of
$S(\vec{Q}$).

We note that this calculation and the resulting diffuse scattering pattern in
an energy integrated experiment like done on DNS or D7 depends on the
kinematics of the experiment. This is schematically shown in Fig. 16 which
outlines the scattering triangle for a particular geometry of
$\vec{Q}=\vec{k}_{i}-\vec{k}_{f}$ and how the apparent “elastic” cross section
is influenced by the dynamics. Three scattering triangles are illustrated with
the solid line defining $\vec{Q}$ showing an elastic scattering process. The
dashed lines define inelastic (energy gain) scattering processes at
$\vec{Q}=({1\over 2},{1\over 2},{1\over 2})$, but illustrate how they appear
if assumed to be elastic with $|\vec{k}_{i}|\equiv|\vec{k}_{f}|$ (solid
points). It can be seen that such processes appear away from the
$\vec{Q}=({1\over 2},{1\over 2},{1\over 2})$ position. The position is defined
by the Ewald Sphere with a radius defined by $|\vec{k}_{i}|$ in the case of
DNS and D7. We note that the situation on PRISMA is more complex given the
range of incident and scattered wavelengths. In such a case all scattering is
defined as being elastic and fixed via geometry. [136]

The scattering cross section from our simulation of the diffuse magnetic
scattering in Figs. 15 $(a,b)$ are qualitatively in agreement with the unusual
magnetic scattering in Fig. 7 $(d)$ which mimics “Catherine Wheels”. Figs. 15
$(a,b)$ compare the effects on the scattering cross section with energy
integration illustrating that the upper cuttoff on the integration only
controls the extent of the scattering in momentum. The main result of the
scattering occurs due to the energy gain side of the integration. This is
further confirmed by the low temperature simulation at T=2 K in Fig. 15 $(c)$,
which should be compared to Fig. 15 $(a)$, where it can be seen that the
scattering cross section is described by momentum broadened points near
$\vec{Q}$=(${1\over 2},{1\over 2},{1\over 2}$). We therefore conclude that the
unusual diffuse scattering cross section experimentally measured in Fig. 7
$(c,f,g)$ originates from dynamics and is indicative of fluctuations of
localized Fe3+ moments and sampled through the thermally excited scattering on
the energy gain side (negative energy transfers).

## References

  * Binder and Young [1986] K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986).
  * Park and Shrout [1997] S. E. Park and T. R. Shrout, J. Appl. Phys. 82, 1804 (1997).
  * Ye [1998] Z.-G. Ye, Key Eng. Mater. 155-156, 81 (1998).
  * Ye [2009] Z.-G. Ye, MRS Bull. 34, 277 (2009).
  * Bokov and Ye [2006] A. A. Bokov and Z. G. Ye, J. Mater. Sci. 41, 31 (2006).
  * Ye and Bokov [2004] Z. G. Ye and A. A. Bokov, Ferroelectrics 302, 473 (2004).
  * Ye [1996] Z. G. Ye, Ferroelectrics 184, 193 (1996).
  * Cowley  _et al._ [2011] R. A. Cowley, S. N. Gvasaliya, S. G. Lushnikov, B. Roessli, and G. M. Rotaru, Adv. Phys. 60, 229 (2011).
  * Viehland  _et al._ [1991] D. Viehland, J. F. Li, S. J. Jang, L. E. Cross, and M. Wuttig, Phys. Rev. B 43, 8316 (1991).
  * Viehland  _et al._ [1992] D. Viehland, J. F. Li, S. J. Jang, L. E. Cross, and M. Wuttig, Phys. Rev. B 46, 8013 (1992).
  * Pirc and Blinc [2007] R. Pirc and R. Blinc, Phys. Rev. B 76, 020101(R) (2007).
  * Pirc  _et al._ [2001] R. Pirc, R. Blinc, and V. Bobnar, Phys. Rev. B 63, 054203 (2001).
  * Shirane and Hoshino [1951] G. Shirane and S. Hoshino, J. Phys. Soc. 6, 265 (1951).
  * Shirane  _et al._ [1970] G. Shirane, J. D. Axe, J. Harada, and J. P. Remeika, Phys. Rev. B 2, 155 (1970).
  * Kempa  _et al._ [2006] M. Kempa, J. Hlinka, J. Kulda, P. Bourges, A. Kania, and J. Petzelt, Phase Transitions 79, 351 (2006).
  * Tomeno  _et al._ [2006] I. Tomeno, Y. Ishii, Y. Tsunoda, and K. Oka, Phys. Rev. B 73, 064116 (2006).
  * Hlinka  _et al._ [2006] J. Hlinka, M. Kempa, J. Kulda, P. Bourges, A. Kania, and J. Petzelt, Phys. Rev. B 73, 140101(R) (2006).
  * Gehring  _et al._ [2009] P. M. Gehring, H. Hiraka, C. Stock, S.-H. Lee, W. Chen, Z.-G. Ye, S. B. Vakhrushev, and Z. Chowdhuri, Phys. Rev. B 79, 224109 (2009).
  * Burns and Dacol [1983] G. Burns and F. H. Dacol, Solid State Commun. 48, 853 (1983).
  * Gehring  _et al._ [2001a] P. M. Gehring, S.-E. Park, and G. Shirane, Phys. Rev. B 63, 224109 (2001a).
  * Hlinka  _et al._ [2003] J. Hlinka, S. Kamba, J. Petzelt, J. Kulda, C. A. Randall, and S. J. Zhang, Phys. Rev. Lett. 91, 107602 (2003).
  * Naberezhnov  _et al._ [1999] A. Naberezhnov, S. Vakhrushev, B. Dorner, D. Strauch, and H. Moudden, Eur. Phys. J. B 11, 13 (1999).
  * Gehring  _et al._ [2000] P. M. Gehring, S.-E. Park, and G. Shirane, Phys. Rev. Lett. 84, 5216 (2000).
  * Gehring  _et al._ [2001b] P. M. Gehring, S. Wakimoto, Z. G. Ye, and G. Shirane, Phys. Rev. Lett. 87, 277601 (2001b).
  * Wakimoto  _et al._ [2002] S. Wakimoto, C. Stock, R. J. Birgeneau, Z.-G. Ye, W. Chen, W. J. L. Buyers, P. M. Gehring, and G. Shirane, Phys. Rev. B 65, 172105 (2002).
  * Cao  _et al._ [2008] H. Cao, C. Stock, G. Xu, P. M. Gehring, J. Li, and D. Viehland, Phys. Rev. B 78, 104103 (2008).
  * Vakhrushev and Shapiro [2002] S. B. Vakhrushev and S. M. Shapiro, Phys. Rev. B 66, 214101 (2002).
  * Kamba  _et al._ [2005] S. Kamba, M. Kempa, V. Bovtun, J. Petzelt, K. Brinkman, and N. Setter, J. Phys. Condens. Matter 17, 3965 (2005).
  * Nuzhnyy  _et al._ [2017] D. Nuzhnyy, J. Petzelt, V. Bovtun, M. Kempa, S. Kamba, J. Hlinka, and B. Hehlen, Phys. Rev. B 96, 174113 (2017).
  * Bishop  _et al._ [2010] A. R. Bishop, A. Bussmann-Holder, S. Kamba, and M. Maglione, Phys. Rev. B 81, 064106 (2010).
  * Vakhrushev  _et al._ [1989] S. B. Vakhrushev, A. A. Naberezhnov, N. M. Okuneva, and B. P. Toperverg, Ferroelectrics 90, 173 (1989).
  * You and Zhang [1997] H. You and Q. M. Zhang, Phys. Rev. Lett. 79, 3950 (1997).
  * Dkhil  _et al._ [2001] B. Dkhil, J. M. Kiat, G. Calvarin, G. Baldinozzi, S. B. Vakhrushev, and E. Suard, Phys. Rev. B 65, 024104 (2001).
  * Hirota  _et al._ [2002] K. Hirota, Z.-G. Ye, S. Wakimoto, P. M. Gehring, and G. Shirane, Phys. Rev. B 65, 104105 (2002).
  * Xu  _et al._ [2004a] G. Xu, G. Shirane, J. R. D. Copley, and P. M. Gehring, Phys. Rev. B 69, 064112 (2004a).
  * Matsuura  _et al._ [2006] M. Matsuura, K. Hirota, P. M. Gehring, Z.-G. Ye, W. Chen, and G. Shirane, Phys. Rev. B 74, 144107 (2006).
  * Stock  _et al._ [2010a] C. Stock, L. Van Eijck, P. Fouquet, M. Maccarini, P. M. Gehring, G. Xu, H. Luo, X. Zhao, J.-F. Li, and D. Viehland, Phys. Rev. B 81, 144127 (2010a).
  * Xu  _et al._ [2012] Z. Xu, J. Wen, E. Mamontov, C. Stock, P. M. Gehring, and G. Xu, Phys. Rev. B 86, 144106 (2012).
  * Vakhrushev  _et al._ [2005] S. Vakhrushev, A. Ivanov, and J. Kulda, PCCP 7, 2340 (2005).
  * Vakhrushev  _et al._ [1998] S. Vakhrushev, A. Naberezhnov, N. M. Okuneva, and B. N. Savenko, Phys. Solid State 40, 1728 (1998).
  * Vakhrushev  _et al._ [1995] S. Vakhrushev, A. Naberezhnov, N. M. Okuneva, and B. N. Savenko, Fizika Tverdogo Tela 37, 3621 (1995).
  * Gvasaliya  _et al._ [2003] S. N. Gvasaliya, B. Roessli, and S. G. Lushnikov, Europhys. Lett. 63, 303 (2003).
  * Ye and Schmid [1993] Z.-G. Ye and H. Schmid, Ferroelectrics 145, 83 (1993).
  * Stock  _et al._ [2007] C. Stock, G. Xu, P. M. Gehring, H. Luo, X. Zhao, H. Cao, J. F. Li, D. Viehland, and G. Shirane, Phys. Rev. B 76, 064122 (2007).
  * Westphal  _et al._ [1992] V. Westphal, W. Kleemann, and M. D. Glinchuk, Phys. Rev. Lett. 68, 847 (1992).
  * Fisch [2003] R. Fisch, Phys. Rev. B 67, 094110 (2003).
  * Stock  _et al._ [2004] C. Stock, R. J. Birgeneau, S. Wakimoto, J. S. Gardner, W. Chen, Z.-G. Ye, and G. Shirane, Phys. Rev. B 69, 094104 (2004).
  * Cowley  _et al._ [2009] R. A. Cowley, S. N. Gvasaliya, and B. Roessli, Ferroelectrics 378, 53 (2009).
  * Imry and Ma [1975] Y. Imry and S.-k. Ma, Phys. Rev. Lett. 35, 1399 (1975).
  * Fishman and Aharony [1979] R. Fishman and A. Aharony, J. Phys. C: Solid State Phys. 12, L729 (1979).
  * Birgeneau  _et al._ [1983] R. J. Birgeneau, H. Yoshizawa, R. A. Cowley, G. Shirane, and H. Ikeda, Phys. Rev. B 28, 1438 (1983).
  * Bellini  _et al._ [2001] T. Bellini, L. Radzihovsky, J. Toner, and N. A. Clark, Science 294, 1074 (2001).
  * Park  _et al._ [2002] S. Park, R. L. Leheny, R. J. Birgeneau, J.-L. Gallani, C. W. Garland, and G. S. Iannacchione, Phys. Rev. E 65, 050703(R) (2002).
  * Reppy [1992] J. D. Reppy, J. Low Temp. Phys. 87, 205 (1992).
  * Gridnev and Kamynin [2012] S. A. Gridnev and A. A. Kamynin, Phys. Solid State 54, 1018 (2012).
  * Brzezinska  _et al._ [2018] D. Brzezinska, R. Skulski, D. Bochenek, P. Niemiec, A. Chroback, L. Fajfrowski, and S. Matyjasik, J. Alloys Compd. 727, 299 (2018).
  * Cheong and Mostovoy [2007] S. W. Cheong and M. Mostovoy, Nat. Mater. 6, 13 (2007).
  * Kimura  _et al._ [2003] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Nature 426, 55 (2003).
  * dos Santos  _et al._ [2002] A. M. dos Santos, S. Parashar, A. R. Raju, Y. S. Zhao, A. K. Cheetham, and C. N. R. Rao, Solid State Commun. 122, 49 (2002).
  * Laguta  _et al._ [2016] V. V. Laguta, A. N. Morozovska, E. A. Eliseev, I. P. Raevski, S. I. Raevskaya, E. I. Sitalo, S. A. Prosandeev, and L. Bellaiche, J. Mater. Sci. 51, 5330 (2016).
  * Laguta  _et al._ [2017] V. V. Laguta, V. A. Stephanovich, I. P. Raevski, S. I. Raevskaya, V. V. Titov, V. G. Smotrakov, and V. V. Eremkin, Phys. Rev. B 95, 014207 (2017).
  * Yang  _et al._ [2004] Y. Yang, J.-M. Liu, H. B. Huang, W. Q. Zou, P. Bao, and Z. G. Liu, Phys. Rev. B 70, 132101 (2004).
  * Correa  _et al._ [2011] M. Correa, A. Kumar, S. Priya, R. S. Katiyar, and J. F. Scott, Phys. Rev. B 83, 014302 (2011).
  * Garcia-Flores  _et al._ [2011] A. F. Garcia-Flores, D. A. Tenne, Y. J. Choi, W. J. Ren, X. X. Xi, and S. W. Cheong, J. Phys.: Condens. Matter 23, 015401 (2011).
  * Turik  _et al._ [2012] A. V. Turik, A. V. Pavlenko, K. P. Andryushin, S. I. Shevtsova, L. A. Reznichenko, and A. I. Chernobabov, Phys. Solid State 54, 947 (2012).
  * Chillal  _et al._ [2014] S. Chillal, S. N. Gvasaliya, A. Zheludev, D. Schroeter, M. Kraken, F. J. Litterst, T. A. Shaplygina, and S. G. Lushnikov, Phys. Rev. B 89, 174418(R) (2014).
  * Chen  _et al._ [2016] L. Chen, A. A. Bokov, W. Zhu, J. Zhuang, N. Zhang, H. N. Tailor, W. Ren, and Z. G. Ye, Sci. Rep. 6, 22327 (2016).
  * Pavlenko  _et al._ [2012] A. V. Pavlenko, A. T. Kozakov, S. P. Kubrin, A. A. Pavelko, K. A. Guglev, L. A. Shilkina, I. A. Verbenko, D. A. Sarichev, and L. A. Reznichenko, Ceram. Int. 38, 6157 (2012).
  * Raevski  _et al._ [2019] I. P. Raevski, S. P. Kubrin, A. V. Pushkarev, N. M. Olekhnovich, Y. V. Radyush, G. R. Li, C. C. Chou, S. I. Raevskaya, V. V. Titov, and M. A. Malitskaya, Ferroelectrics 542, 36 (2019).
  * Pietrzak  _et al._ [1981] J. Pietrzak, A. Maryanowska, and J. Leciejewicz, Phys. Stat. Sol. 65, K79 (1981).
  * Kleemann  _et al._ [2010] W. Kleemann, V. V. Shvartsman, P. Borisov, and A. Kania, Phys. Rev. Lett. 105, 257202 (2010).
  * Luo  _et al._ [2000] H. Luo, G. Xu, H. Xu, P. Wang, and Z. Yin, Jpn. J. Appl. Phys. 39, 5581 (2000).
  * Kozlenko  _et al._ [2014] D. P. Kozlenko, S. E. Kichanov, E. V. Lukin, N. T. Dang, L. S. Dubrovinsky, H.-P. Liermann, W. Morgenroth, A. A. Kamynin, S. A. Gridnev, and B. N. Savenko, Phys. Rev. B 89, 174107 (2014).
  * Stock  _et al._ [2013] C. Stock, S. R. Dunsiger, R. A. Mole, X. Li, and H. Luo, Phys. Rev. B 88, 094105 (2013).
  * Wilfong  _et al._ [2016a] B. Wilfong, M. Ahart, S. A. Gramsch, C. Stock, Z. Li, H. Luo, and R. J. Hemley, J. Raman Spectrosc. 47, 227 (2016a).
  * Bonny  _et al._ [1997] V. Bonny, M. Bonin, P. Sciau, K. J. Schenk, and G. Chapuis, Solid State Commun. 102, 347 (1997).
  * Zagorodniy  _et al._ [2018] Y. O. Zagorodniy, R. O. Kuzian, I. V. Kondakova, M. Maryško, V. Chlan, H. Štěpánková, N. M. Olekhnovich, A. V. Pushkarev, Y. V. Radyush, I. P. Raevski, B. Zalar, V. V. Laguta, and V. A. Stephanovich, Phys. Rev. Materials 2, 014401 (2018).
  * Bochenek and Niemiec [2018] D. Bochenek and P. Niemiec, Materials 11, 2504 (2018).
  * Hiraka  _et al._ [2004] H. Hiraka, S.-H. Lee, P. M. Gehring, G. Xu, and G. Shirane, Phys. Rev. B 70, 184105 (2004).
  * Zinkin  _et al._ [1997] M. P. Zinkin, M. J. Harris, and T. Zeiske, Phys. Rev. B 56, 11786 (1997).
  * Schweika and Boni [2001] W. Schweika and P. Boni, Physica B 297, 155 (2001).
  * Stewart  _et al._ [2009] J. R. Stewart, P. P. Deen, K. H. Andersen, H. Schober, J. F. Barthelemy, J. M. Hillier, A. P. Murani, T. Hayes, and B. Lindenau, J. Appl. Cryst. 42, 69 (2009).
  * Wilfong  _et al._ [2016b] B. Wilfong, M. Ahart, S. A. Gramsch, C. Stock, X. Li, H. Luo, and R. J. Hemley, J. Raman Spectrosc. 47, 227 (2016b).
  * Swainson  _et al._ [2009] I. P. Swainson, C. Stock, P. M. Gehring, G. Xu, K. Hirota, Y. Qiu, H. Luo, X. Zhao, J.-F. Li, and D. Viehland, Phys. Rev. B 79, 224301 (2009).
  * Lampis  _et al._ [1999] N. Lampis, P. Sciau, and A. G. Lehmann, J. Phys.: Condens. Matter 11, 3489 (1999).
  * Majumder  _et al._ [2006] S. B. Majumder, S. Bhattacharyya, R. S. Katiyar, A. Mannivannan, P. Dutta, and M. S. Seehra, J. Appl. Phys. 99, 024108 (2006).
  * Xu  _et al._ [2004b] G. Xu, Z. Zhong, H. Hiraka, and G. Shirane, Phys. Rev. B 70, 174109 (2004b).
  * Stock  _et al._ [2018] C. Stock, P. M. Gehring, R. A. Ewings, G. Xu, J. Li, D. Viehland, and H. Luo, Phys. Rev. Materials 2, 024404 (2018).
  * Stock  _et al._ [2012] C. Stock, P. M. Gehring, H. Hiraka, I. Swainson, G. Xu, Z.-G. Ye, H. Luo, J.-F. Li, and D. Viehland, Phys. Rev. B 86, 104108 (2012).
  * Stock  _et al._ [2005] C. Stock, H. Luo, D. Viehland, J. F. Li, I. P. Swainson, R. J. Birgeneau, and G. Shirane, J. Phys. Soc. Jpn. 74, 3002 (2005).
  * Gvasaliay  _et al._ [2005] S. N. Gvasaliay, B. Roessli, R. A. Cowley, P. Huber, and S. G. Lusnikov, J. Phys. Condens. Matter 17, 4343 (2005).
  * Gvasaliya  _et al._ [2004a] S. N. Gvasaliya, S. G. Lushnikov, and B. Roessli, Phys. Rev. B 69, 092105 (2004a).
  * Gvasaliya  _et al._ [2004b] S. N. Gvasaliya, S. G. Lushnikov, and B. Roessli, Crystallogr. Rep. 49, 108 (2004b).
  * Murani and Heidemann [1978] A. P. Murani and A. Heidemann, Phys. Rev. Lett. 41, 1402 (1978).
  * Welberry  _et al._ [2003] T. R. Welberry, D. J. Goossens, W. I. F. David, M. J. Gutmann, M. J. Bull, and A. P. Heerdegen, J. Appl. Cryst. 36, 1440 (2003).
  * Hohlwein  _et al._ [2003] D. Hohlwein, J.-U. Hoffmann, and R. Schneider, Phys. Rev. B 68, 140408(R) (2003).
  * Zeiske  _et al._ [1997] T. Zeiske, D. Hohlwein, and W. Prandl, Physica B 241, 628 (1997).
  * Hohlwein  _et al._ [1997] D. Hohlwein, H. M. Mayer, and T. Zeiske, Physica B 234, 1109 (1997).
  * Falqui  _et al._ [2005] A. Falqui, N. Lampis, A. Geddo-Lehmann, and G. Pinna, J. Phys. Chem. B 109, 22967 (2005).
  * Kumar  _et al._ [2008a] A. Kumar, R. S. Katiyar, C. Rinaldi, S. G. Lushnikov, and T. A. Shaplygina, Appl. Phys. Lett. 93, 232902 (2008a).
  * Aeppli  _et al._ [1984a] G. Aeppli, S. M. Shapiro, H. Maletta, R. J. Birgeneau, and H. S. Chen, J. Appl. Phys. 55, 1628 (1984a).
  * Chillal  _et al._ [2013] S. Chillal, M. Thede, F. J. Litterst, S. N. Gvasaliya, T. A. Shaplygina, S. G. Lushnikov, and A. Zheludev, Phys. Rev. B 87, 220403(R) (2013).
  * Nagata  _et al._ [1979] S. Nagata, P. H. Keesom, and H. R. Harrison, Phys. Rev. B 19, 1633 (1979).
  * Sherington [1975] D. Sherington, J. Phys. C: Solid State Phys. 8, L208 (1975).
  * Yamani  _et al._ [2015] Z. Yamani, W. J. L. Buyers, F. Wang, Y.-J. Kim, J.-H. Chung, S. Chang, P. M. Gehring, G. Gasparovic, C. Stock, C. L. Broholm, J. C. Baglo, R. Liang, D. A. Bonn, and W. N. Hardy, Phys. Rev. B 91, 134427 (2015).
  * Stock  _et al._ [2008] C. Stock, W. J. L. Buyers, Z. Yamani, Z. Tun, R. J. Birgeneau, R. Liang, D. Bonn, and W. N. Hardy, Phys. Rev. B 77, 104513 (2008).
  * Bernhoeft [2001] N. Bernhoeft, J. Phys.: Condens. Matter 13, R771 (2001).
  * Stock  _et al._ [2020] C. Stock, M. Songvilay, P. M. Gehring, G. Xu, and B. Roessli, J. Phys. Condens. Matter 32, 374012 (2020).
  * Sherrington [2019] D. Sherrington, J. Phys. A: Math. Theor. 52, 264001 (2019).
  * Kumar  _et al._ [2008b] A. Kumar, R. Katiyar, C. Rinaldi, S. Lushnikov, and T. Shaplygina, Appl. Phys. Lett. 3, 232902 (2008b).
  * Stock  _et al._ [2010b] C. Stock, S. Jonas, C. Broholm, S. Nakatsuji, Y. Nambu, K. Onuma, Y. Maeno, and J.-H. Chung, Phys. Rev. Lett. 105, 037402 (2010b).
  * Aeppli  _et al._ [1983] G. Aeppli, S. M. Shapiro, R. J. Birgeneau, and H. S. Chen, Phys. Rev. B 28, 5160 (1983).
  * Aeppli  _et al._ [1984b] G. Aeppli, S. M. Shapiro, R. J. Birgeneau, and H. S. Chen, Phys. Rev. B 29, 2589 (1984b).
  * Shvartsman  _et al._ [2008] V. V. Shvartsman, S. Bedanta, P. Borisov, W. Kleemann, A. Tkach, and P. M. Vilarinho, Phys. Rev. Lett. 101, 165704 (2008).
  * Aeppli  _et al._ [1982] G. Aeppli, S. M. Shapiro, R. J. Birgeneau, and H. S. Chen, Phys. Rev. B 25, 4882 (1982).
  * Stephanovich and Laguta [2016] V. A. Stephanovich and V. V. Laguta, Phys. Chem. Chem. Phys. 18, 7229 (2016).
  * Kuzian  _et al._ [2014] R. O. Kuzian, I. V. Kondakova, A. M. Daré, and V. V. Laguta, Phys. Rev. B 89, 024402 (2014).
  * Birgeneau [1998] R. J. Birgeneau, J. Magn. Magn. Mater. 177-181, 1 (1998).
  * Rotaru  _et al._ [2009] G. M. Rotaru, B. Roessli, A. Amato, S. N. Gvasaliya, C. Mudry, S. G. Lushnikov, and T. A. Shaplygina, Phys. Rev. B 79, 184430 (2009).
  * Blinc  _et al._ [2008] R. Blinc, V. V. Laguta, B. Zalar, B. Zupancic, and M. Itoh, J. Appl. Phys. 104, 084105 (2008).
  * Bhat  _et al._ [2004] V. V. Bhat, K. V. Ramunujachary, S. E. Lofland, and A. M. Umarji, J. Magn. Magn. Mater. 280, 221 (2004).
  * Bhat  _et al._ [2005] V. V. Bhat, A. M. Umarji, V. B. Shenoy, and U. V. Waghmare, Phys. Rev. B 72, 014104 (2005).
  * Xu  _et al._ [2003] G. Xu, Z. Zhong, Y. Bing, Z.-G. Ye, C. Stock, and G. Shirane, Phys. Rev. B 67, 104102 (2003).
  * Conlon  _et al._ [2004] K. H. Conlon, H. Luo, D. Viehland, J. F. Li, T. Whan, J. H. Fox, C. Stock, and G. Shirane, Phys. Rev. B 70, 172204 (2004).
  * Kisi and Forrester [2005] E. H. Kisi and J. S. Forrester, J. Phys. Condens. Matter 17, L381 (2005).
  * Xu  _et al._ [2006] G. Xu, P. M. Gehring, C. Stock, and K. Conlon, Phase Transit. 79, 135 (2006).
  * Phelan  _et al._ [2015] D. Phelan, E. E. Rodriguez, J. Gao, Y. Bing, Z. G. Ye, Q. Huang, J. S. Wen, G. Xu, C. Stock, M. Matsuura, and P. M. Gehring, Phase Trans. 88, 283 (2015).
  * Gehring  _et al._ [2004] P. M. Gehring, W. Chen, Z. G. Ye, and G. Shirane, J. Phys. Condens. Matter 16, 7113 (2004).
  * Brown  _et al._ [2018] K. L. Brown, C. P. J. Stockdale, H. Luo, X. Zhao, J. F. Li, D. Viehland, G. Xu, P. M. Gehring, K. Isida, A. D. Hillier, and C. Stock, J. Phys. Condens. Matter 30, 125703 (2018).
  * Hill  _et al._ [1991] J. P. Hill, T. R. Thurston, R. W. Erwin, M. J. Ramstad, and R. J. Birgeneau, Phys. Rev. Lett. 66, 3281 (1991).
  * Shirane  _et al._ [2004] G. Shirane, S. M. Shapiro, and J. M. Tranquada,  _Neutron Scattering with a Triple-Axis Spectrometer_ (Cambridge University Press, Cambridge, 2004).
  * Bacon [1975] G. E. Bacon,  _Neutron Diffraction_ (Oxford University Press, London, 1975).
  * Moon  _et al._ [1969] R. M. Moon, T. Riste, and W. C. Koehler, Phys. Rev. 181, 920 (1969).
  * Stewart [2000] J. R. Stewart, J. Appl. Phys. 87, 5420 (2000).
  * Benkhaled  _et al._ [2020] N. Benkhaled, M. Djermouni, A. Zaoui, I. Kondakova, R. Kuzian, and R. Hayn, J. Mag. Mag. Mater. 515, 167309 (2020).
  * Liu  _et al._ [2021] Z. H. Liu, H. Lane, C. D. Frost, R. A. Ewings, J. P. Attfield, and C. Stock, J. Appl. Crystallogr. 54, 957 (2021).