# Neutron Depolarization due to Ferromagnetism and Spin Freezing in CePd1-xRhx

M. Seifert  Physik-Department, Technical University Munich, D-85747 Garching,
Germany  P. Schmakat  Physik-Department, Technical University Munich, D-85747
Garching, Germany  Heinz Maier-Leibnitz Zentrum (MLZ), Technical University
Munich, D-85748 Garching, Germany  M. Schulz  Heinz Maier-Leibnitz Zentrum
(MLZ), Technical University Munich, D-85748 Garching, Germany  P. Jorba
Physik-Department, Technical University Munich, D-85747 Garching, Germany  V.
Hutanu  Heinz Maier-Leibnitz Zentrum (MLZ), Technical University Munich,
D-85748 Garching, Germany  C. Geibel  Max-Planck-Institute for Chemical
Physics of Solids, D-01187 Dresden, Germany  M. Deppe  Max-Planck-Institute
for Chemical Physics of Solids, D-01187 Dresden, Germany  C. Pfleiderer
Physik-Department, Technical University Munich, D-85747 Garching, Germany
Centre for QuantumEngineering (ZQE), Technical University of Munich, D-85748
Garching, Germany  Munich Center for Quantum Science and Technology (MCQST),
Technical University Munich, D-85748 Garching, Germany

(August 27, 2024)

###### Abstract

We report neutron depolarization measurements of the suppression of long-range
ferromagnetism and the emergence of magnetic irreversibilities and spin
freezing in CePd1-xRhx around $x^{*}\approx 0.6$. Tracking the temperature
versus field history of the neutron depolarization, we find clear signatures
of long-range Ising ferromagnetism below a Curie temperature $T_{\rm C}$ for
$x=0.4$ and a spin freezing of tiny ferromagnetic clusters below a freezing
temperature $T_{\rm F1}$ for $x>x^{*}$. Under zero-field-cooling/field-heating
and for $x>x^{*}$ a reentrant temperature dependence of the neutron
depolarization between $T_{\rm F2}<T_{\rm F1}$ and $T_{\rm F1}$ is
microscopically consistent with a thermally activated growth of the cluster
size. The evolution of the depolarization as well as the reentrant temperature
dependence as a function of Rh content are consistent with the formation of a
Kondo-cluster glass below $T_{\rm F1}$ adjacent to a ferromagnetic quantum
phase transition at $x^{*}$.

##  I Motivation and Outline

Quantum phase transitions (QPTs), defined as zero temperature phase
transitions, represent a well-established roadmap in the search for new
properties of correlated electron systemsLöhneysen  _et al._ (2007); Senthil
_et al._ (2004); Si and Steglich (2010); Vojta (2003, 2018). The perhaps
simplest example of a magnetic QPT is associated with the suppression or the
emergence of long-range ferromagnetic orderBrando  _et al._ (2016) as a
function of a non-thermal control parameter such as hydrostatic
pressurePfleiderer  _et al._ (2001); Grosche  _et al._ (1995); Uhlarz  _et
al._ (2004); Saxena  _et al._ (2000), an applied magnetic fieldThessieu  _et
al._ (1997); Perry  _et al._ (2001); Paglione  _et al._ (2003); Gegenwart  _et
al._ (2002), or chemical compositionFuchs  _et al._ (2014); van der Marel  _et
al._ (2003). In clean materials a variety of escape routes to ferromagnetic
quantum criticality have been identifiedBrando  _et al._ (2016). For instance,
the coupling of the magnetization to electronic soft modes may generically
lead to a first-order QPTs Belitz  _et al._ (2005); Brando  _et al._ (2016);
Pfleiderer (2005), or new forms of order masking the QPT such as spin density
wave orderBrando  _et al._ (2008); Abdul-Jabbar  _et al._ (2015); Friedemann
_et al._ (2018) or superconductivitySaxena  _et al._ (2000); Aoki  _et al._
(2001); Huy  _et al._ (2007); Aoki  _et al._ (2019); Pfleiderer (2009);
Haslbeck  _et al._ (2019). In systems featuring defects and disorder, a
ferromagnetic QPT may drive the appearance of intermediate phases such as
frustrated magnetism or the formation of magnetic clustersMiranda and
Dobrosavljević (2005); Vojta (2010), as well as electronic phase
segregationPfleiderer  _et al._ (2010); Uemura  _et al._ (2007); Schmakat  _et
al._ (2015); Benka  _et al._ (2022).

An increasing number of studies suggest an intricate interplay of microscopic
(atomic) and mesoscopic scales at QPTs. For instance, the temperature
dependence of the electrical resistivity at the pressure-tuned quantum phase
transition of pure itinerant-electron ferromagnets such as Ni3Al or ZrZn2, for
unexplained reasons, is characteristic of disordered ferromagnets Niklowitz
_et al._ (2005); Grosche  _et al._ (1995). Related high-pressure studies in
the itinerant helimagnet MnSi, connect the anomalous temperature dependence of
the resistivity empirically with the presence of strongly fluctuating
topological spin textures Ritz  _et al._ (2013a, b). Moreover, microscopic
studies of the ferromagnetic QPT in LaCrGe3 suggest the formation of short-
range order as an escape route of ferromagnetic quantum criticality Gati  _et
al._ (2021); Rana  _et al._ (2021). Perhaps most specific so far, recent
studies of the transverse field Ising transition in LiHoF4 under small tilted
magnetic fields revealed the presence of a line of mesoscale quantum
criticality, i.e., purely due to magnetic domainsWendl  _et al._ (2022). The
additional presence of defects and disorder at QPT, due to the strongly
enhanced (or even singular) response functions may readily generate the
formation of clusters in the sub-micrometer region or strongly fluctuating
ferromagnetically correlated patches akin superparamagnetism. In turn, the
interplay of disorder and defects with QPT has attracted considerable
theoretical interest as the cause of novel forms of quantum correlations such
as quantum Griffiths phases Vojta (2010).

While it may be intuitive that mesocale textures may be important in the
surroundings of QPTs, it is experimentally difficult to prove their existence
and to determine their character. Namely, in bulk materials processes on
mesoscopic length scales imply correlation lengths that require ultra-small
angle scattering. Also, the size of such textures implies large characteristic
time-scales that are difficult to determine experimentally in bulk systems.
Taken together, this raises the question for experimental methods capable to
provide such information. In the study reported here we explore the potential
of neutron depolarization measurements to provide such information for the
ferromagnetic to intermediate valent transition in the compositional series
CePd1-xRhx.

Since the seminal work of Halpern and Holstein in the early 1940s Halpern and
Holstein (1941), it has been known that the polarization of a neutron beam
decreases rapidly when traversing a bulk material with ferromagnetic domains.
Numerous studies have demonstrated a great sensitivity of the polarization to
the existence of ferromagnetic domains and superconducting flux lines Rauch
and Weber (1968); Drabkin  _et al._ (1969); Maleev and Ruban (1970); Weber
(1974); Rekveldt and van Schaik (1979); van der Valk and Rekveldt (1982);
Mitsuda and Endoh (1985); Mirebeau  _et al._ (1986); Rosman and Rekveldt
(1990); Mirebeau  _et al._ (1992); Rekveldt (1999); Yusuf  _et al._ (2000);
Sato  _et al._ (2004); De Teresa  _et al._ (2006); Rekveldt  _et al._ (2006);
Pfleiderer  _et al._ (2010); Treimer  _et al._ (2012a, b, 2013); Seifert  _et
al._ (2017); Deepak  _et al._ (2021). Likewise, it is also long known that
slow ferromagnetic fluctuations in the paramagnetic state can cause a
depolarization of a neutron beam as wellBakker  _et al._ (1968); Drabkin  _et
al._ (1968); Takahashi  _et al._ (1995). Moreover, combining neutron
depolarization measurements with neutron imaging, spatially resolved
information may be obtained Schulz  _et al._ (2010); Schulz (2010); Schulz
_et al._ (2016); Jorba  _et al._ (2019). This technique known as neutron
depolarization imaging (NDI) has been used, e.g., to map out magnetic stray
fields Kardjilov  _et al._ (2008). Developing a three-dimensional
reconstruction of NDI, neutron depolarization tomography has also been used to
determine ferromagnetically polarized regimes Schulz  _et al._ (2010). It is
thereby important to note that the spatial resolution of the imaging and
tomography is currently limited to a few hundred mikro-meters at best due to
beam divergence and detector resolution. Thus, neutron depolarization
measurements offer a probe that allows to distinguish the existence of
ferromagnetically correlated regimes on microscopic scales (neutron
depolarization) from metallurgical inhomogeneities on macroscopic scales
(NDI).

The depolarization of a neutron beam depends on the spatial extent and the
average magnetization of the magnetic domains generating the Larmor precession
along the trajectory of the neutron beam. Given the wavelength (and hence
velocity) of the neutrons this implicitly yields a characteristic time-scale,
notably the time required for traversing individual domains, at which the
polarization of the neutron beam is affected. In other words, a depolarization
is expected when magnetized patches are (i) sufficiently large, and (ii)
sufficiently long-lived, and (iii) the uniform magnetization is sufficiently
large. As compared to conventional bulk and transport properties, neutron
depolarization measurements provide microscopic information as inferred from
the threshold of the depolarization process. This has long been appreciated,
though material-specific and experimental set-up-specific details have neither
been reported nor exploited.

To explore the potential of neutron depolarization measurements in studies of
ferromagnetic QPTs we decided to study CePd1-xRhx Kappler  _et al._ (1988);
Kappler (1991); Sereni  _et al._ (1993, 2005); Deppe  _et al._ (2006); Pikul
_et al._ (2006); Sereni  _et al._ (2006, 2007); Westerkamp  _et al._ (2009);
Brando  _et al._ (2010); Schmakat (2015). The rare earth compound CePd1-xRhx
undergoes a ferromagnetic QPT as a function of Rh content $x$, where
ferromagnetism is continuously suppressed above $x=$0.6$$ Deppe  _et al._
(2006); Pikul  _et al._ (2006); Sereni  _et al._ (2005, 2006, 2007). In the
vicinity of the QPT an exponentially decreasing tail of the onset of
ferromagnetic correlations has been observed suggestive of disorder-induced
smearing. At high values of $x$ and high temperatures the magnetic properties
are characteristic of strong fluctuations that gradually freeze with
decreasing temperaturesWesterkamp  _et al._ (2009). To determine the inherent
length and time-scales of this freezing process and to assess the
metallurgical homogeneity of CePd1-xRhx we performed neutron depolarization
measurements in several compositions of intermediate Rh content. Furthermore,
we investigated the effect of the magneto-crystalline anisotropy on the
neutron depolarization of ferromagnetic CePd1-xRhx ($x=0.4$), where we find
well-behaved properties of an easy ferromagnetic axis.

Summarized in Fig. 1 are our main results. For all compositions studied, the
onset of neutron depolarzation we observe as a function of temperature is in
good agreement with data inferred from the magnetization, ac susceptibility,
and specific heat in the same samples as reported previouslyWesterkamp (2008).
For a Rh content up to $x\approx 0.6$ all data consistently exhibit the
characteristics of long-range ferromagnetic order. This is shown in blue
shading in Fig. 1, where the Curie temperature is denoted as $T_{\rm C}$. In
comparison, for CePd1-xRhx with $x>0.6$ all properties consistently exhibit
the onset of a spin frozen state below a temperature denoted $T_{\rm F1}$, as
shown in green shading in Fig. 1. The absence of depolarization above $T_{\rm
F1}$ is characteristic of a freezing of small, rapidly fluctuating clusters.
The temperature and magnetic field history of the neutron depolarization
reveals the absence of depolarization below a temperature $T_{\rm F2}$ under
zero-field-cooling/field-heating, where $T_{\rm F2}$ roughly tracks $T_{\rm
F1}$ for increasing $x$. The pronounced reentrance of the depolarization
between $T_{\rm F2}$ and $T_{\rm F1}$ under zero-field-cooling/field-heating
reveals a thermally activated increase of the cluster size in the spin-frozen
state. Taking into account the rapidly increasing Kondo screening for
increasing $x$, our depolarization data of the spin-frozen state reveal
microscopic signatures consistent with a Kondo cluster glass (KCG) as proposed
before based on the bulk propertiesWesterkamp (2008); Westerkamp  _et al._
(2009).

Figure 1: Magnetic phase diagram of CePd1-xRhx. $T_{\rm C}$, $T_{\rm F1}$, and
$T_{\rm F2}$ denote the Curie temperature, the spin freezing temperature, and
the reentrance temperature, respectively. Shown are values of $T_{\rm C}$,
$T_{\rm F1}$, and $T_{\rm F2}$ as inferred from the neutron polarization
$P(T)$ recorded in our study, and the ac susceptibility, $\chi^{\prime}_{\rm
ac}(T)$, magnetization, $M(T)$, specific heat, $C(T)$, and thermal volume
expansion, $\beta(T)$ of the same samples reported in Refs. Westerkamp, 2008;
Sereni  _et al._ , 2007. With increasing $x$ ferromagnetism (blue shading) is
suppressed and spin-frozen behaviour characteristic of Kondo Cluster Glass
(green shading) emerges.

The presentation of our paper is organized as follows. We begin with a
pedagogical introduction to neutron depolarization measurements in Sec. III.
Starting with the theory of neutron depolarization in Sec. III.1, we report
estimates of the sensitivity of neutron depolarization on the length and time
scales at which ferromagnetic regions begin to generate a noticeable
depolarization in Sec. III.2. This is followed by key methodological aspects
in Sec. III.3, where we report an experimental set up that permits to record
the neutron depolarization under bipolar field sweeps needed for measurements
of different temperature and field histories. In addition, we used spherical
neutron polarimetry to discriminate a precessional rotation of the neutron
polarization from a generic depolarization in ferromagnetic samples.

The presentation of our experimental results in Sec. IV proceeds in two
subsections. We begin with the dependence of the neutron depolarization on the
temperature and field history in Sec. IV.1. This is followed by the evidence
for a precessional rotation of the polarization in the ferromagnetic state of
a single crystal sample in Sec. IV.2, where we consider the role of the
combination of magnetic anisotropy, temperature and small applied magnetic
fields. In Sec. V we discuss our observations in the context of previous
studies of the bulk properties and present our arguments in support of a Kondo
cluster glass. Our paper closes with a summary of the main conclusions in Sec.
VI.

##  II Introduction to $\mathrm{CePd}_{1-x}\mathrm{Rh}_{x}$

CePd and CeRh represent isotructural siblings that crystallize in the
orthorhombic CrB structure. Whereas pure CePd orders ferromagnetically at
$T_{\rm C}=$6.5\text{\,}\mathrm{K}$$Thornton  _et al._ (1998), CeRh shows the
key characteristics of a non-magnetic intermediate valent system down to the
lowest temperatures studied. It has long been recognized that the
substitutional series CePd1-xRhx permits to explore the evolution from
ferromagnetic order to intermediate valent behavior under isostructural
conditions Kappler  _et al._ (1988); Kappler (1991); Sereni  _et al._ (1993)
in the additional presence of strong disorder Sereni  _et al._ (2005); Deppe
_et al._ (2006); Pikul  _et al._ (2006); Sereni  _et al._ (2007). Information
on the spontaneous magnetic moment as extrapolated to zero temperature is,
however, incomplete. In polycrystalline samples ordered moments of 0.87, 0.8
and 0.47 $\mu_{\rm B}\,{\rm f.u.^{-1}}$ for $x=0$, 0.25 and 0.5, respectively,
have been reportedNieva  _et al._ (1988); Sereni  _et al._ (1993). This
compares with easy axis moments along the c-axis of 1.54 and 0.5 $\mu_{\rm
B}\,{\rm f.u.^{-1}}$ in single crystals for $x=0.4$ and 0.6Deppe  _et al._
(2006); Schmakat (2015), respectively, where the bulk properties are
characteristic of an unchanged easy-axis anisotropy up to large $x$ Deppe  _et
al._ (2006); Brando  _et al._ (2010).

Seminal studies have established that the evolution of the properties of
CePd1-xRhx as a function of $x$ share two unusual characteristics with respect
to a composition $x^{*}\approx 0.6$ Kappler  _et al._ (1988); Kappler (1991);
Sereni  _et al._ (1993, 2005); Deppe  _et al._ (2006); Pikul  _et al._ (2006);
Sereni  _et al._ (2006, 2007); Westerkamp  _et al._ (2009); Brando  _et al._
(2010); Schmakat (2015). On the one hand, with increasing $x$ the lattice
constant decreases linearly up to $x^{*}$, consistent with Vegard’s law,
followed by a fast superlinear decrease for $x>0.6$. On the other hand, the
Weiss temperature, $\Theta_{\rm P}$, as inferred from the inverse
susceptibility above $\sim 100\,{\rm K}$ is negative for all $x$. It decreases
weakly up to $x^{*}$ followed by a steep decrease between $x^{*}$ and $x=1$.

In the light of the intermediate valent properties of CeRh, it has been argued
that the Weiss temperature reflects a mean-value of the Kondo temperature
$T_{\rm K}\approx|\Theta_{\rm P}|$ Koelling  _et al._ (1985). Thus, above
$x^{*}$ the Kondo temperature appears to increase rapidly such that it exceeds
the temperature scales on which ferromagnetism is observed. This suggests that
the magnetic properties for $x>x^{*}$ originate from a combination of
intermediate valent fluctuations, Kondo screening and disorder.

As a function of increasing $x$ the suppression of ferromagnetism in
CePd1-xRhx, notably the transition temperature $T_{\rm C}$ as inferred from
the bulk properties, crosses the increase of the average value of $T_{\rm K}$
in the regime of $x^{*}$ Sereni  _et al._ (2007); Westerkamp  _et al._ (2009).
A more detailed inspection reveals an unusual concentration dependence of the
signatures of ferromagnetism for intermediate values of $x$ Pikul  _et al._
(2006); Westerkamp  _et al._ (2009). Namely, with increasing $x$ the curvature
of $T_{\rm C}$ is initially negative such that $T_{\rm C}$ may be extrapolated
to zero approximately around $x^{*}$. However, as the composition approaches
$x^{*}$ a change of curvature is observed at the onset of ferromagnetic
correlations as inferred from pronounced maxima in $\chi^{\prime}_{\rm
ac}(T)$. These maxima in $\chi^{\prime}_{\rm ac}(T)$ may be discerned up to
$x=0.87$. The absence of a maximum in $\chi^{\prime}_{\rm ac}(T)$ down to 20
mK for $x=0.9$ suggests the suppression of ferromagnetic correlations for a
critical composition close to $x_{c}=0.87$.

Associated with the change of curvature at the onset of ferromagnetic
correlations is a change of the character of the processes underlying the
maximum in the susceptibility around $x^{*}$. For $x<x^{*}$ the bulk
properties are consistent with a transition to long-range ferromagnetism. In
contrast, for $x\geq 0.6$ the maxima in the susceptibility exhibit a
pronounced frequency dependence characteristic of a freezing process similar
to that observed in spin glasses Westerkamp  _et al._ (2009). We therefore
denote the onset of ferromagnetism and spin freezing with $T_{\rm C}$ and
$T_{\rm F1}$, respectively. Interestingly, in the regime of the freezing
process at $T_{\rm F1}$ the relative temperature shift between 3 % an 10 % per
decade of the excitation frequency overlaps with the behaviour of canonical
metallic spin glasses and superparamagnets, where shifts between $\sim 1\,\%$
and $\sim 30\,\%$ are observed. In addition to the freezing seen in the
susceptibility, the magnetization exhibits a small hysteresis already above
$T_{\rm F1}$, where the hysteresis is seen between zero-field-cooled/field-
heated and field-cooled/field-heated temperature sweeps. In turn, the
hysteresis has been attributed to the formation of ferromagnetic clusters,
which freeze at $T_{\rm F1}$.

The suppression of ferromagnetic order as well as the suppression of the
underlying spontaneous magnetic moment as a function of increasing $x$ in
CePd1-xRhx reflect the hybridization of the Ce 4f electrons with the valence
electrons of the surrounding ligands. Consistent with the evolution of $T_{\rm
K}$ inferred from $\Theta_{\rm P}$, Rh ligands in Ce compounds are known to
result in much larger Kondo temperatures than observed for Pd ligands.Koelling
_et al._ (1985) This is accompanied by strong local variations of $T_{\rm K}$
due to the random distribution of Rh and Pd atoms. Both aspects are consistent
with the entropy and the slope of $\chi^{\prime}_{\rm ac}(T)$ at 2 K, which
are suggestive of unscreened magnetic moments even when the average of $T_{\rm
K}$ exceeds several ten K.

Taken together, CePd1-xRhx differs distinctly from other strongly disordered
Ce-based systems such as CeNi1-xCux, in which the Ce valence remains nearly
trivalent and in which a percolative cluster scenario was proposedMarcano  _et
al._ (2007). In recognition of the broad distribution of Kondo temperatures on
local scales, which are believed to be responsible for the cluster formation,
the low-temperature state in CePd1-xRhx has been denoted a Kondo-cluster glass
(KCG). However, this interpretation was so far empirical, without evidence of
specific microscopic signatures expected of a Kondo screening.

Last but not least, the possible role of quantum correlations in CePd1-xRhx
has been addressed in several studies. Namely, putative evidence of a power-
law dependence in the specific heat, $C(T)/T\sim T^{\lambda-1}$, with
$\lambda=0.6$ and 0.67 for $x=0.87$ and 0.9, respectively Sereni  _et al._
(2005); Pikul  _et al._ (2006); Sereni  _et al._ (2007), as well as the ac
susceptibility, raise the question whether $T_{\rm C}$ near $x=0.87$ is
somehow connected with an additional form of ferromagnetic quantum
criticality. This is contrasted by the Grüneisen ratio $\Gamma$, which
displays a log-divergence, $\Gamma\propto\ln T$, as opposed to a power-law
divergence expected for quantum criticality ($\Gamma\propto\beta/C$ where
$\beta$ represents the volume thermal expansion coefficient and $C$ the
specific heat). Moreover, a power-law form of the magnetic-field dependence of
the magnetization at 50 mK was reported for very large $x$, also consistent
with the absence of quantum criticality. Rather, it has been pointed out that
these properties are consistent with the scenario of a quantum-Griffiths-
phase. However, whereas theory predicts that the values for $\lambda$ inferred
from the specific heat and susceptibility should be the same Vojta (2009),
they are found to be different experimentally, posing an unresolved
inconsistency.

The properties reported so far for CePd1-xRhx consistently point at an unusual
interplay of strong correlations with disorder on multiple scales Kappler  _et
al._ (1988); Kappler (1991); Sereni  _et al._ (1993, 2005); Deppe  _et al._
(2006); Pikul  _et al._ (2006); Sereni  _et al._ (2006, 2007); Westerkamp  _et
al._ (2009); Brando  _et al._ (2010); Schmakat (2015). Namely, the bulk
properties and limited microscopic information suggest the formation of
ferromagnetic clusters that undergo a complex freezing process subject to a
distribution of Kondo screening and thermal activation. The properties of
CePd1-xRhx reported so far raise questions for the actual size of the
underlying microscopic length and time scales as well as the magnetic
character of the clusters. In turn this raises the question of the role of the
disorder and the distribution of Kondo scales and, last but not least, the
relevance of the Kondo screening in the freezing process of the clusters.

##  III Neutron depolarization

Neutron depolarization measurements are based on the principle of neutron
radiography by means of a polarized neutron beam. Since the neutron interacts
with magnetic fields via its spin, neutron depolarization measurements allow
to identify and determine, within limits, spatially resolved ferromagnetic
correlations in bulk materials, i.e., ordered regions such as domains or spin
clusters Schlenker and Shull (1973); Schulz  _et al._ (2016). Due to the large
penetration depth of neutrons, neutron depolarization measurements allow to
use complex sample environment such as cryostats, electro magnets, and
pressure cells.

Examples of neutron depolarization measurements include the three-dimensional
imaging of ferromagnetic domains in bulk samples Schulz  _et al._ (2010), as
well as the detection of inhomogeneous field distributions Kardjilov  _et al._
(2008); Piegsa  _et al._ (2009); Sales  _et al._ (2018) as generated, e.g., by
screening currents in the vicinity of superconductors or due to the flux lines
penetrating superconductors Weber (1974); Treimer  _et al._ (2012a, b, 2013);
Treimer (2014). By means of neutron depolarization measurements the
metallurgical homogeneity of ferromagnetic materials can be characterized when
the magnetic properties vary sensitively with chemical composition and
internal stress/strain Jorba  _et al._ (2019). Moreover, measurements as a
function of temperature allow to map out the distribution of the Curie
temperature spatially and hence to infer compositional inhomogeneities across
larger sample volumesPfleiderer  _et al._ (2010); Schmakat (2015); Jorba  _et
al._ (2019).

In the following the basic principles of neutron depolarization are reviewed
in Sec. III.1, followed by material-specific estimates of the spatial and
temporal threshold for depolarization to occur in Sec. III.2. In Sec. III.3
the experimental setups used in our study are described.

###  III.1 Neutron depolarization in a ferromagnet

We begin with a summary of the formal description of neutron depolarization
measurements of a ferromagnet focussing on a few limiting cases which are
required for the interpretation of our results. For the discussion presented
in the following we consider a polarized neutron beam, the polarization $P$ of
which is given as

$P=\frac{I_{+}-I_{-}}{I_{+}+I_{-}}$ (1)

where $I_{+}$, $I_{-}$ represent the intensities with respect to the
polarization axis. In the following we denote the polarization of the incident
neutron beam by $P_{0}$. The neutron beam is transmitted through a
ferromagnetic sample, in which it traverses a series of magnetic domains
labelled $i$ with intrinsic fields $\textbf{B}_{i}$. We assume further that
these intrinsic fields are oriented randomly. The classical equation of motion
for the neutron spin $\textbf{s}_{i}$ that couples to the ferromagnetic domain
$i$ is given by the Larmor equation

$\frac{d}{dt}\textbf{s}_{i}(t)=\gamma\textbf{s}_{i}(t)\times\textbf{B}_{i}$
(2)

where $\gamma$ is the gyromagnetic ratio of the neutron. For constant magnetic
field Eq. 2 describes a precession of the spin with respect to the direction
of the field at the Larmor frequency $\bm{\omega}_{L}=-\gamma\textbf{B}_{i}$.

Assuming an average field $B_{0}=\langle B_{i}\rangle$ per domain, an average
domain length $\delta$ in the direction of flight and infinitesimally thin
domain walls the polarization in the $y$-direction may be written as Halpern
and Holstein (1941); Mitsuda  _et al._ (1992)

$\frac{P}{P_{0}}=\left[\left\langle\frac{B_{\parallel}^{2}}{B_{0}^{2}}\right\rangle_{\\!B_{0}}+\left\langle\frac{B_{\perp}^{2}}{B_{0}^{2}}\right\rangle_{\\!B_{0}}\left\langle\cos\left(\gamma
B_{0}\frac{\delta}{v}\right)\right\rangle_{\\!\delta}\right]^{N}$ (3)

where $y$ is perpendicular to the beam direction and parallel to the guide
field (representing the quantization axis). Further, $B_{\parallel}$ and
$B_{\perp}$ represent the components of the magnetic field $B_{0}$ in each
domain parallel and perpendicular to the direction of travel. $N=d/\delta$
represents the average number of magnetic domains on a neutron path through
the sample and $v$ is the neutron velocity. Here the brackets
$\langle...\rangle$ denote the average over the magnetic field or the average
domain size as denoted by the subscript. The argument of the cosine function
corresponds to the Larmor phase collected in each domain
$\varphi_{L}=\omega_{L}\tau=\gamma B_{0}\delta/v$.

Eq. 3 may be expressed analytically for some limiting cases. First, as
proposed by Halpern and HolsteinHalpern and Holstein (1941), averaging over
the ensemble of randomly oriented domains for small spin rotations per domain
$\omega_{L}\tau\ll 2\pi$ results in

$\frac{P}{P_{0}}=\exp\left(-\frac{1}{3}\gamma^{2}B_{0}^{2}(T)\frac{d\delta}{v^{2}}\right)$
(4)

where $d$ is the sample thickness in the direction of flight of the neutron
and the magnetic flux $B_{0}$ is assumed to be temperature-dependent. The
second case, where $\omega_{L}\tau\geq 2\pi$, represents a large spin rotation
per domain. Eq. 3 then yields for the polarization

$\frac{P}{P_{0}}\approx\exp\left(-N\right).$ (5)

Assuming that the average domain size is constant, the depolarization is then
expected to be constant below the Curie temperature $T_{\rm C}$.

Based on the solutions described by Eqs. 4 and 5 it may be concluded that for
temperatures below $T_{\rm C}$ the domain configuration leads to a
depolarization of the neutron beam, while in the paramagnetic state the
polarization is not affected. For the evaluation of our data we assumed that
the magnetic field in a single domain may be described using a temperature
dependence as follows

$B_{0}^{2}=\mu_{0}^{2}M^{2}_{0}\left[\frac{T-T_{\rm C}}{T_{\rm
C}}\right]^{\beta}\left[1-\Theta(T-T_{\rm C})\right],$ (6)

where $M_{0}$ represents the spontaneous magnetization in each domain, $\beta$
is system specific exponent, $\mu_{0}$ is the vacuum permeability, and
$\Theta(x)$ is a Heaviside function that serves to account for an idealized
spontaneous symmetry breaking below $T_{\rm C}$. For $\beta=1/2$ Eq. 6
corresponds to the mean field approximation for ferromagnets. In real systems
a smoothed version of the step function such as a Gaussian error function may
be used to represent the transition and the emergence of a finite
magnetization.

In case the data are recorded with a two-dimensional detector, spatially
resolved information may be obtained using Eqs. 4 and 5. This provides
spatially resolved information on the magnetic ordering temperature $T_{\rm
C}$, referred to in the following as a $T_{\rm C}$ map.

Another limit assumes a mono-domain ferromagnetic state, i.e., the sample
supports a uniform magnetization without domainsMitsuda  _et al._ (1992). It
is important to note that such a state does not cause a depolarization of a
monochromatic beam (a polychromatic beam featuring a distribution of
wavelengths may depolarize somewhat). Denoting the angle between the
polarization and the magnetization by $\alpha$, Eq. 3 yields

$P=P_{0}\left[\cos^{2}(\alpha)+\sin^{2}(\alpha)\cos\left(\gamma
B_{0}\frac{d}{v}\right)\right].$ (7)

If the magnetization is neither parallel nor antiparallel to the polarization
($\alpha\neq n\pi$ for $n=0,1,2,\dots$) the polarization vector precesses with
respect to the direction of the internal field $B_{0}$. A seeming decrease of
the polarization would then be due to a rotation of the polarization away from
the direction for which the polarization is analyzed. Moreover, a change of
the magnetization caused, for example, by a variation of the temperature, $T$,
or an applied magnetic field, $B=\mu_{0}H$, may result in an oscillation of
the polarization due to the cosine term. This has been observed in our study
of the ferromagnetic single-crystal CePd1-xRhx ($x=0.4$) as described in Sec.
IV.2.

###  III.2 Sensitivity of neutron depolarization

In this section we present an estimate of the length- and time-scales of
multi-domain ferromagnetic order at which the setup we used for our neutron
depolarization study described in Sec. III.3 was able to detect a
depolarization. Several publications have addressed this questionRekveldt and
van Schaik (1979); van der Valk and Rekveldt (1982); Mirebeau  _et al._
(1986); Rosman and Rekveldt (1990); Mirebeau  _et al._ (1992); Mitsuda  _et
al._ (1992); Takahashi  _et al._ (1995); Rekveldt (1999); Sato  _et al._
(2004); Rekveldt  _et al._ (2006). However, to the best of our knowledge
quantitative values have not been reported before.

We consider multi-domain ferromagnetic order with an average domain size
$\delta$. For decreasing $\delta$ the Larmor phase $\varphi_{L}$ collected in
each domain eventually will no longer be sufficient to depolarize the neutron
beam at a level that exceeds the resolution of the setup. This case is
described by Eq. 4. Solving this equation for $\delta$ and replacing
$P/P_{0}=1-\Delta P$ yields

$\delta=\frac{-3\log(1-\Delta P)v^{2}}{\gamma^{2}B_{0}^{2}d}.$ (8)

where $\Delta P$ is the minimal change of polarization that may be resolved.
In our setup the resolution corresponded typically to
$\sim$1\text{\,}\mathrm{\char 37\relax}$$ as estimated from the scatter of the
data points.

Shown in Fig. 2 is an evaluation of Eq. 8 for a typical sample thickness of
$d=$1\text{\,}\mathrm{mm}$$ and different neutron wavelengths. The minimum
average domain length $\delta$ required for a depolarization of
$1\text{\,}\mathrm{\char 37\relax}$ is shown as a function of the average
field $B_{0}$ in each domain. The abscissa at the top of both panels of Fig. 2
displays $B_{0}$ in units of the magnetic moment $\mu$ for CePd1-xRhx using a
unit cell volume of approximately ${187}\,{\rm\AA^{3}}$.Sereni  _et al._
(2007) For order of magnitude estimates, $B_{0}$ may be inferred from the
magnetization at sufficiently large magnetic fields.

Depicted in green shading is the regime, where a notable depolarization is
expected. The dashed lines depict the 1 % threshold for wavelengths of
$\lambda=$3\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ and
$\lambda=$7\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$, corresponding to the
wavelengths at the beam-line ANTARES available for monochromatic
depolarization measurements. In comparison, the solid black line represents a
calculation for a polychromatic spectrum ranging from
$\lambda=$4\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ to
$8.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ as available at the beam-line
ANTARES and used in the setup shown in Fig. 3 (a).

With increasing $B_{0}$ the average size of the domains, $\delta$, that may be
detected decreases as shown in Fig. 2 (a). The ordinate on the right hand side
of Fig. 2 (a) represents the wave vector $q=2\pi/\delta$ associated with
$\delta$. Remarkably, for strong ferromagnets the neutron depolarization is
sensitive to ferromagnetic domains down to the sub-nm scale. For instance, the
unscreened magnetic moment of CePd1-xRhx of $2\mu_{B}$ Pikul  _et al._ (2006)
as marked in Fig.2 (a) by a vertical gray line, implies a spatial sensitivity
of $2\text{\,}\mathrm{nm}$ for the experimental setup shown in Fig. 3 (a). In
turn, if the average size of ferromagnetic spin clusters exceeds
$\sim$2\text{\,}\mathrm{nm}$$ a depolarization larger than
$1\text{\,}\mathrm{\char 37\relax}$ is expected.

Figure 2: Characteristic length- and time-scales beyond which neutron
depolarization is expected in a multi-domain ferromagnet as a function of
magnetic flux $B_{0}$ per domain. Shown in green shading is the parameter
range of a noticeable depolarization exceeding $\Delta
P=$1\text{\,}\mathrm{\char 37\relax}$$ for a sample thickness of
$d=$1\text{\,}\mathrm{mm}$$, where the threshold corresponds to monochromatic
wavelengths of $3.0\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, and
$7.0\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, as well as a polychromatic
spectrum from $4.0\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ to
$8.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. The abscissa shown at the top of
each panel represent the magnetic flux per domain as expressed in units of the
magnetic moment $\mu$ in CePd1-xRhx. Marked in gray is the unscreened moment
of CePd1-xRhx. (a) Estimated length scale $\delta$ as a function $B_{0}$. The
ordinate on the right hand side displays a corresponding wave vector
$q=2\pi/\delta$. (b) Estimated flight time $\tau$ across a domain as a
function $B_{0}$. The ordinate shown on the right hand side displays a
corresponding frequency, $f=1/\tau$.

It is important to emphasize that the magnetic domains do not need to be
static for a depolarization to occur. Rather, the lifetime of the domains must
exceed the time, $\tau$, needed by the neutron to travel across the domains
given by $\tau=\delta/v$, where $v$ is the neutron velocity. As shown in Fig.
2 (b) typical time scales are of the order of ns, which decreases rapidly with
increasing $B_{0}$. The ordinate on the right hand side of Fig. 2 (b)
represents an associated frequency $f=1/\tau$ in the spirit of a fluctuation
rate, say, in a superparamagnet. For large $B_{0}$ the neutron depolarization
measurements are sensitive to ferromagnetically correlated regimes that
fluctuate at a time-scale less than pico-seconds. Interestingly, in real
materials this corresponds to the time scale of the movement of domain walls
at distances that are typical of domain sizes. In turn, this implies that the
effective field seen by the neutron does not need to be static on time scales
exceeding $\tau$.

Taken together, these considerations highlight the potential of neutron
depolarization measurements as a probe of ferromagnetic textures that
fluctuate on a nm scale in the sub-ns regime. The temporal resolution
associated with the unscreened moment of CePd1-xRhx of $2\mu_{B}$ Pikul  _et
al._ (2006) for the experimental setup shown in Fig. 3 (a) as marked by a gray
line in Fig.2 (b) corresponds to $\sim$4\text{\,}\mathrm{ps}$$.

###  III.3 Experimental setups

For our studies it proved to be essential to record data under different
temperature versus field histories. Before turning to an account of the
terminology used in our paper, we note that the expression ”zero-field”
implies the nominal presence of a guide field of roughly 0.5 mT required to
maintain the polarization of the neutron beam. Keeping this in mind, we
distinguish the following temperature versus field histories, noting that all
data were collected while heating the sample at the same constant rate in
order to avoid systematic errors of the sample temperature recorded. First,
data recorded after cooling in zero-field (zfc) while heating in zero-field
are denoted zfc-zfh (zero-field-cooled - zero-field-heated). Second, data
recorded after cooling in zero-field to base temperature and the application
of a finite field while heating under this applied magnetic field are denoted
zfc-fh (zero-field-cooled - field-heated). Third, after cooling the sample in
the presence of a magnetic field applied at sufficiently high temperatures of
several K, data were recorded while heating the sample in the same unchanged
applied magnetic field. This is denoted fc-fh (field-cooled - field-heated).

Figure 3: Schematic depictions of the three setups used for the neutron
depolarization measurements reported in this paper. (a) The neutron beam
passed a polarizer (P) and a spin flipper (SF). The polarization was
maintained by guide fields (G1 and G2) in the $y$-direction and hence
perpendicular to the neutron path, which was parallel to the $z$-direction
across the sample. The sample was placed in a cryostat which was located in a
Helmholtz pair of coils generating a magnetic field at the position of the
sample. The polarization was analyzed in the $y$-direction using an analyzer
(A) and detected by a CCD camera (D) in combination with a LiF/ZnS converter
and scintillator film. (b) Magnetic fields in positive and negative
$y$-direction could be applied by adding two horizontal guide fields (G2 and
G3) pointing in $x$-direction. (c) Schematic depiction of the setup used for
spherical polarization analysis with CryoPAD. Rotatable coupling coils (CC)
and two precession coils (PC) were located between two superconducting sheets
(SC) that permitted to adjust and to analyze the polarization in arbitrary
directions.

The neutron depolarization measurements were carried out at the beamline
ANTARES at FRM IICalzada  _et al._ (2009); Schulz and Schillinger (2015). A
schematic depiction of the setup used initially at ANTARES is shown in Fig. 3
(a). Polarized 3He spin filter cells were used for polarizing and analyzing
the neutron beam in the $y$-direction which is perpendicular to the neutron
beam denoted as $z$-direction. A spin flipper located directly after the
polarizer allowed to change the polarization direction from $+y$ to $-y$. This
was required for the polarization analysis. Tiny guide fields between the
components prevented the loss of polarization, otherwise expected in low-field
field regions, where parasitic external magnetic fields dominate. Coarse
neutron wavelength selection was achieved by means of a Beryllium filter,
resulting in a rather broad wavelength band from
$\sim$4\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ to
$\sim$8.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$. The neutron detector used
was based on a LiF/ZnS scintillator, which converted the neutrons into visible
light that was detected by a high resolution CCD camera. A large part of the
data reported in Sec. IV were measured using this setup.

Following our first measurements we modified this setup to permit studies of
different temperature versus field histories. This required a setup that
permitted measurements under arbitrary positive or negative magnetic field
strengths. As explained above, true zero-field conditions are very difficult
to achive and even small stray fields will cause a severe depolarization of
the neutron beam. Therefore, we used small guide fields around the sample
position in order to stabilize the polarisation axis.

To satisfy these conditions we installed two additional guide fields pointing
in the $x$-direction. As indicated in Fig. 3 (b) these guide fields were
placed immediately before and after the Helmholtz coils. This way an adiabatic
rotation of the polarization into the horizontal plane was realized that
allowed to apply magnetic fields along the positive and the negative
$y$-direction by means of the Helmholtz pair without an undefined zero-field
transition along the neutron path that would cause a severe depolarization of
the neutron beam. Moreover, a neutron velocity selector was installed and the
3He polarizers were replaced by polarizing V-cavities. In these experiments we
used a neutron wavelength of
$\lambda=$4.13\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ with a wavelength
spread of $\Delta\lambda/\lambda=10\%$ given by the velocity selector. Further
details of this setup may be found elsewhereSeifert  _et al._ (2017); Jorba
_et al._ (2019); Schmakat (2015).

For all of our neutron depolarization measurements the samples were cooled to
temperatures as low as $\sim 0.07\,{\rm K}$ by means of bespoke 3He/4He
dilution insert as combined with a pulse tube cooler. Magnetic fields were
generated with a pair of Helmholtz coils operated at room temperature.

Figure 4: Coordinate systems used in the neutron polarization measurements
with CryoPAD, where the crystallographic $a$\- and $c$-axis of the CePd1-xRhx
($x=$0.40$$) single crystal were determined by Laue neutron diffraction. (a)
Depiction of the sample shape and sample orientation in the $xy$-plane. (b)
Definition of the angle $\chi$ in the $yz$-plane. The crystallographic $a$\-
and $c$-axis of the sample resided in the $xy$-plane perpendicular to the
neutron beam. The $b$-axis was parallel to the $z$-direction. CryoPAD allowed
to adjust and analyze the polarization in any arbitrary direction defined by
the angles $\Theta$ in the $xy$-plane and $\chi$ in the $yz$-plane.

In addition, spherical neutron polarimetry was carried out using CryoPAD at
the beam-line POLI at FRM II Hutanu  _et al._ (2016), where a schematic
depiction of the setup is shown in Fig. 3 (c). The implementation of coupling
coils, precession coils, and magnetic shielding of the sample position
permitted a complete determination of the three-dimensional polarization
matrix. However, it was not possible to obtain spatially resolved information
across the sample because a 3He tube had to be used as a neutron detector. A
detailed description of the setup may be found elsewhereHutanu  _et al._
(2016).

Shown schematically in Fig. 4 is the coordinate system as well as the outline
and orientation of the CePd1-xRhx single crystal as investigated at POLI. In
the following the direction of the neutron polarization is denoted by the
polar angles $\Theta$ and $\chi$, where the angle $\Theta$ was measured clock-
wise in the $xy$-plane and $\Theta=0$ corresponded to the $y$-direction. The
angle $\chi$ was measured in the $yz$-plane starting at $\chi=0$ in the
$y$-direction.

In our studies we adjusted and analyzed the polarization always in the same
direction to be able to distinguish a generic depolarization from a spherical
rotation of the direction of the polarization. Therefore, both nutator angles
$\Theta$ and both angles $\chi$ as determined by the precession coils were
always kept the same. Finally, the single crystal sample was oriented such
that the crystallographic $ac$-plane corresponded to the plane perpendicular
to the neutron beam and thus the $xy$-plane in the coordinate system used to
account for the polarization. The magnetic easy axis of the system, which
corresponded to the crystallographic $c$-axis, hence resided in this plane.

The resistivity, ac susceptibility, magnetization, and specific heat of the
CePd1-xRhx samples we investigated in our neutron depolarization measurements
were examined rather comprehensively prior to our study as reported elsewhere
Deppe  _et al._ (2006); Westerkamp (2008); Sereni  _et al._ (2007). Details of
the sample preparation may be found in these papers. All samples were poly-
crystals with the exception of the sample with $x=$0.40$$ which was a single
crystal.

##  IV Experimental results

The presentation of the experimental data is organized in two parts. It begins
with the dependence of the neutron depolarization on the temperature and field
history in Sec. IV.1 for a wide range of compositions. This is followed by the
variation of the neutron depolarization due to the magnetic anisotropy in
single-crystal CePd1-xRhx for the ferromagnetic composition $x=0.4$ in Sec.
IV.2.

###  IV.1 Dependence of the neutron depolarization on temperature and field
history

Figure 5: Temperature dependence of NDI in CePd1-xRhx. (a1), (b1) and (c1)
$T_{\rm C}$ maps of CePd1-xRhx across the sample shape for $x=$0.40$$,
$x=$0.60$$, and $x=$0.65$$ as inferred from temperature scans at zero field
(see text for details). The color bar denotes the range of the transition
temperatures $T_{\rm C}$. (a2), (b2) and (c2) Distribution of transition
temperatures inferred from the $T_{\rm C}$ maps shown in panels (a1), (b1),
and (c1), where $\Delta T_{\rm C}$ corresponds to the FWHM of the Gaussian fit
shown.

Data reported in the following were recorded at the beam-line ANTARES using
the setups shown in Fig. 3 (a) and 3 (b). Shown in Figs. 5 (a1), (b1) and (c1)
are $T_{\rm C}$ maps across the sample cross-section of the samples with
$x=$0.40$$, $x=$0.60$$, and $x=$0.65$$ as inferred from the temperature
dependence of neutron depolarization imaging. As reported above, the
expression zero-field refers to a very small field
$B=$0.5\text{\,}\mathrm{mT}$$ required as a guide field to maintain the
neutron polarization. Color bars indicate the transition temperatures while
the thin black lines denote the outline of the sample shape.

Shown in Figs. 5 (a2), (b2) and (c2) are the corresponding histograms of the
distribution of ordering temperatures across the $T_{\rm C}$ map. The
distribution of transition temperatures was fitted with a Gaussian where the
values shown in the histograms represented the average value of $T_{\rm C}$,
and the asscoiated FWHM, $\Delta T_{\rm C}$. Namely, we found $T_{\rm
C}(x=$0.40$)=5.24\pm 0.06\,{\rm K}$, $T_{\rm C}(x=$0.60$)=2.74\pm 0.23\,{\rm
K}$, and $T_{\rm C}(x=$0.65$)=1.46\pm 0.26\,{\rm K}$.

With increasing $x$, values of $T_{\rm C}$ and $T_{\rm F1}$ decrease in
excellent agreement with the properties inferred from the bulk properties
reported in the literature as shown in Fig. 1. As $T_{\rm C}$ and $T_{\rm F1}$
decrease with increasing $x$ the FWHM $\Delta T_{\rm C}$ increases. This trend
may be explained with an increase of the effects of disorder. Moreover, the
sample with $x=$0.40$$ was a single-crystal as compared to the poly-
crystalline nature of the samples with $x=$0.60$$ and $x=$0.65$$. It might
also reflect the vicinity to the QPT and the concomitant increase of the
susceptibility to form ferromagnetic clusters.

Shown in Fig. 6 is the polarization as a function of temperature for
$x=$0.40$$, $x=$0.60$$, and $x=$0.65$$ as observed in different temperature
versus field histories. The emphasis is here on the effects of the field
strength, where the polarization represents an average of a $32\times 32$
pixel region in the center of each sample. Data under an applied field were
recorded for $B=$7.5\text{\,}\mathrm{mT}$$, $B=$15.0\text{\,}\mathrm{mT}$$,
and $B=$22.5\text{\,}\mathrm{mT}$$ (zfc-fh and fc-fh). For comparison also
shown are data recorded for zfc-zfh.

Figure 6: Polarization of CePd1-xRhx for $x=$0.40$$, $x=$0.60$$, and
$x=$0.65$$ as a function of temperature under zfc-zfh and fc-fh at various
magnetic fields. Data represent an average over a region of $32\times 32$
pixels at the center of each sample. Three data sets were recorded using
different external magnetic fields $B=$7.5\text{\,}\mathrm{mT}$$,
$B=$15.0\text{\,}\mathrm{mT}$$ and $B=$22.5\text{\,}\mathrm{mT}$$ under field-
cooling, respectively. The data recorded under zero-field are shown for better
comparision. Below $T_{\rm C}$ the polarization decreases with increasing
external field $B$ while the transition broadens as a function of temperature.
The arrows indicate the position of $T_{\rm C}$ as determined in zfc-zfh.

The zfc-zfh data recorded in the sample with $x=$0.40$$ shows a sharp drop of
the polarization at $T_{\rm C}$ consistent with spontaneous ferromagnetic
order forming large domains in zero field or in the presence of small applied
magnetic fields as described by Eq. 5. The spontaneous depolarization under
zfc-zfh sets in at a well-defined transition temperature and saturates rapidly
below $T_{\rm C}$. For increasing applied magnetic field a small broadening is
observed at $T_{\rm C}$ while the strength of the depolarization increases
slightly.

In comparison to the sample with $x=$0.40$$ the temperature dependence under
zfc-zfh for $x=$0.60$$ displays only a weak and gradual decrease just below
$T_{\rm F1}$ consistent with small domains and/or weak internal fields as
described by Eq. 4. Here the size of the depolarization increases remarkably
under fc-fh in a small field of $7.5\text{\,}\mathrm{mT}$. When further
increasing the magnetic field the temperature dependence qualitatively and
quantitatively changes only slightly.

For $x=$0.65$$ the depolarization below $T_{\rm F1}$ almost vanishes under
zfc-zfh. This suggests that spontaneous correlations are virtually suppressed
for this Rh concentration on the scales sensitive to neutron depolarization,
i.e., the magnetic properties must be featuring very small domains or clusters
which are spatially separated from each other. Alternatively, the zfc-zfh data
for $x=0.65$ may reflect a strongly fluctuating state. Yet, in the presence of
a small applied magnetic field a noteable depolarization is observed
consistent with a weak form of ferromagnetism. This suggests that a small
applied field stabilizes a ferromagnetic character on length and time scales
sensitive to neutron depolarization. We will return to this point in Sec. V.

Figure 7: Polarization of CePd1-xRhx for $x=$0.40$$, $x=$0.60$$, $x=$0.65$$,
and $x=$0.70$$ as a function of temperature under zfc-zfh, zfc-fh, and fc-fh.
In the zfc-fh and fc-fh measurements a field of $7.5\text{\,}\mathrm{mT}$ was
applied. The ferromagnetic transition temperature $T_{\rm C}$ is denoted by
arrows. The increase of the polarization well below $T_{\rm C}$ under zfc-fh
denoted $T_{\rm F2}$ underscores to the formation of a cluster glass below
$T_{\rm F1}$ for $x\geq 0.6$. The evidence for spontaneous ferromagnetic
correlations that are sufficient to depolarize the neutron beam vanishes for
$x>$0.60$$. However, in this regime a small applied field enhances the
depolarizing effects.

The pronounced depolarization below a characteristic temperature $T_{\rm F1}$
observed under zfc-zfh and fc-fh is contrasted by a pronounced reentrance of
the polarization observed under zfc-fh as illustrated in Fig. 7 for
$7.5\text{\,}\mathrm{mT}$. For ease of comparison also shown in Fig. 7 are the
data recorded under zfc-zfh and fc-fh at $7.5\text{\,}\mathrm{mT}$ shown in
Fig. 6. The key signature observed under zfc-fh at $7.5\text{\,}\mathrm{mT}$
with decreasing temperature concerns a recovery of the polarization at a
temperature $T_{\rm F2}$ well below $T_{\rm F1}$. For the compositions
exhibiting a well defined initial decrease of the polarization under fc-fh,
the reentrant behaviour is only observed for $x=$0.60$$ and $x=$0.65$$. For
$x=$0.40$$ the reentrance is almost absent with a tiny recovery of
polarization below $T_{\rm F2}\ll T_{\rm C}$.

Finally, above a critical concentration around $x=$0.65$$ no significant
spontaneous depolarization is observed under zfc-zfh as a function of
temperature. Nonetheless sizeable reentrant behaviour under zfc-fh is still
observed, as shown in Fig. 7 for $x=$0.70$$. Thus the reentrant behaviour
under zfc-fh prevails as a key signature of the magnetic properties up to high
Rh concentrations. Moreover, when taken together with the data recorded for
$x=$0.60$$ and $x=$0.65$$ we observe a decrease of $T_{\rm F2}$ with
increasing $x$ that roughly tracks the decreases of $T_{\rm F1}$.

###  IV.2 Variation of the neutron polarization due to magnetic anisotropy

An important facet of the interpretation of the neutron polarization concerns
the difference between a generic depolarization and a possible spherical
rotation of the direction of the polarization. Such a rotation may be caused
by the magnetic anisotropy of the material. Previous studies of the
magnetization of CePd1-xRhx are consistent with an easy magnetic $c$-axis
Deppe  _et al._ (2006). To follow up on the role of the magnetic anisotropies
at zero magnetic field, we tracked the anisotropy of the depolarization as
determined at the instrument POLI at FRM II using the 3D polarization analysis
device CryoPAD.

Figure 8: Anisotropy of the depolarization as a function of crystallographic
orientation for a single crystal of CePd1-xRhx with Rh concentration
$x=$0.40$$. For the data shown in (a) and (b) the angles $\Theta$ and $\chi$
were scanned separately, while the polarization was always set and analyzed in
the same direction. (a) Polarization dependence as a function of $\Theta$ in
the crystallographic ac-plane. The ac-plane was almost perpendicular to the
neutron beam. The phase shift in the oscillation suggests that the easy axis
of the magnetization is rotated roughly by an angle of $55\text{\,}\deg$ with
respect to the x-axis. (b) Polarization as a function of $\chi$, defining the
angle of the polarization in the yz-plane where $\chi\approx$90\text{\,}\deg$$
corresponds to the beam direction and roughly to the crystallographic b-axis
of the sample.

Our measurements were performed on the single-crystalline sample with
$x=$0.40$$. This represents a composition for which the bulk properties
provide unambiguous evidence of long-range ferromagnetism consistent with the
neutron depolarization reported above. The orientation of the sample with
respect to the coordinate system of the instrument is illustrated in Fig. 4.
The ac-plane of the sample was oriented in the xy-plane of the coordinate
system of the instrument, with an angle of approximately $55\text{\,}\deg$
between the crystallographic c-axis and the y-axis of the set-up. The
polarization as a function $\Theta$ and $\chi$ are shown in Fig. 8 (a) and 8
(b), respectively. Both angular dependences confirm that the polarization was
equal to one and constant above $T_{\rm C}$ as expected for the paramagnetic
state. Below the transition temperature a sinusoidal variation was observed as
a function of $\Theta$ and $\chi$. The variation in $\chi$ was much weaker
than the variation in $\Theta$ which underlines the fact that the easy axis of
the magnetization was almost perpendicular to this crystallographic plane.

In the angular scan of $\chi$ there was also a small additional oscillation
which suggests that the surface at which the sample was attached to the sample
holder, i.e., the crystallographic ac-plane, was not oriented perfectly
perpendicular to the neutron beam. Two well-defined minima and maxima were
observed for a complete angular scan of the polarization vector. The maxima
correspond to the case where the easy axis of the magnetization (in this case
the crystallographic c-axis) was almost parallel or anti-parallel to the
polarization vector. The minima correspond to a maximum in depolarization
where the polarization vector was almost perpendicular to the preferred
magnetization axis.

The small amount of depolarization for the case when the polarization was
parallel or antiparallel to the anisotropy axis suggests that the magnetic
domains were dominantly aligned parallel or antiparallel to the c-axis. For
such a regular pattern of domains a depolarizing effect may still be expected
due to a finite beam divergence resulting in slightly different neutron path
lengths across the sample.

It is in particular instructive to explore the interplay of an applied
magnetic field with the effects of magnetic anisotropy, domain populations,
neutron depolarisation and spherical precession of the polarization. Namely,
the application of a magnetic field results in an effective internal magnetic
field $B_{\textrm{int}}$ that changes the ratio of the domain populations.
Moreover, for a small applied magnetic field the internal field will point
along the easy axis, even if the easy axis is not parallel to the applied
field.

As the Larmor phase collected by the neutrons depends on their wavelength, a
polychromatic beam such as that used in the setup shown in Fig. 3 (a)
naturally depolarizes when it precesses in a magnetic field. In contrast, for
a monochromatic beam featuring a narrow wavelength band the polarization
decreases by a few percent only even after several Larmor precessions. If, in
addition, the strength of $B_{\textrm{int}}$ changes due to changes of the
magnetization, the total Larmor phase will change also. This finally causes
oscillations in the polarization when the magnetization varies monotonically
as a function of temperature or the applied magnetic field. In summary, we
expect oscillations in the polarization when three conditions are fulfilled:
(i) The magnetization of the sample changes, (ii) a monochromatic neutron beam
is used, and (iii) the easy axis is not parallel to the polarization.

The polarization observed at POLI in a single-crystalline sample suggests that
the magnetic domains in CePd1-xRhx ($x=0.4$) dominantly support a
magnetization along the easy magnetic c-axis of the material characteristic of
a 3d Ising ferromagnet and consistent with the magnetization Deppe  _et al._
(2006). At zero magnetic field the populations of up and down domains are
equal such that the integrated internal field vanishes,
$B_{\mathrm{int}}=$0$$. In the presence of an applied magnetic field this
ratio changes according to the internal field. The precession of the
polarization with respect to the internal field may then be described by Eq. 7
where $B_{0}$ is replaced by $B_{\mathrm{int}}$. If $B_{\mathrm{int}}$
changes, e.g., due to changes of the applied magnetic field or changes of the
magnetization as a function of temperature, and if the angle between the
polarization and the internal field is finite, the cosine part of Eq. 7 causes
oscillations in the polarization. These oscillations were observed both in
temperature and field scans slightly above the Curie temperature $T_{\rm
C}=$5.26\text{\,}\mathrm{K}$$. At lower temperatures the strong depolarization
prevents the appearance of oscillations.

Mathematically this effect may be described by a multiplication of Eq. 5,
recognizing that the sample is strongly ferromagnetic, and Eq. 7. To account
for the finite distribution of Curie temperatures across the sample, the
Heaviside function is replaced by an error function centred at $T_{\text{C}}$

$\begin{split}\operatorname{erf}_{\text{depol}}(T,T_{\rm C},\Delta T_{\rm
C},N)=&\frac{1}{2}\left[\operatorname{erf}\left(m\cdot(T-T_{\text{C}})\right)+1\right]\\\
&\cdot\left(1-P_{\text{offset}}\right)+P_{\text{offset}}\end{split}$ (9)

which varies between $P_{\text{offset}}=3^{-N}$ and 1. The parameter $m$
represents the slope at $T=T_{\text{C}}$ such that

$\Delta T_{\text{C}}=\frac{2\sqrt{\ln 2}}{m}$ (10)

represents the half-width, $\Delta T_{\text{C}}$, of the distribution of
$T_{\text{C}}$. The polarization in the presence of a magnetic anisotropy,
i.e., an angle between the easy axis and the crystallographic c-axis, is then
given by

$\begin{split}P=P_{0}&\cdot\left[\cos^{2}(\alpha)+\sin^{2}(\alpha)\cos\left(\gamma\mu_{0}M(B,T)\frac{d}{v}\right)\right]\\\
&\cdot\operatorname{erf}_{\textrm{depol}}\left(T,T_{\rm C},\Delta T_{\rm
C},N\right).\end{split}$ (11)

Experimental evidence for this behavior may be observed in the single-crystal
with $x=$0.40$$ using the setup illustrated in Fig. 3 (b). Data were recorded
at the beam-line ANTARES after a major instrument upgradeCalzada  _et al._
(2009); Schulz and Schillinger (2015) in which an additional neutron velocity
selector was installed to monochromatize the beam. For the measurements
reported here the crystal was oriented with the crystallographic c-axis under
an angle of approximately $\alpha=$45\text{\,}\deg$$ with respect to the
polarization of the incident beam.

Figure 9: Comparison of several fc-fh temperature scans for different magnetic
fields of a CePd1-xRhx, $x=$0.40$$ single crystal. Applying small fields leads
to a decrease of the polarization well above the ordering temperature $T_{\rm
C}=$5.26\text{\,}\mathrm{K}$$ determined from the zfc-zfh scan. Additionally,
oscillations close to $T_{\rm C}$ appear in the polarization signal if small
fields are applied. The width of the transition as determined from a fit of
Eq. 5 to the zf data set is $\Delta T_{\rm C}=$57\text{\,}\mathrm{mK}$$.

Shown in Fig. 9 is the polarization as a function of temperature across the
phase transition at $T_{\rm C}=$5.26\text{\,}\mathrm{K}$$. Under zfc-zfh,
shown in Fig. 9 (a), a sharp drop of the polarization was observed at $T_{\rm
C}$ in agreement with the data shown in Fig. 6. Also, the width of the
transition of $\Delta T_{\rm C}=$57\text{\,}\mathrm{mK}$$ compares well with
the $T_{\rm C}$ maps shown in Fig. 5.

In comparison, the temperature dependence of the polarization became more
complex under fc-fh as recored in various applied magnetic fields between $0$
and $30\text{\,}\mathrm{mT}$, shown in Figs. 9 (b) through 9 (f). In a small
applied field of $2.5\text{\,}\mathrm{mT}$ the onset of the decrease of the
polarization shifted to higher temperatures and exhibited considerable
broadening. Similar behavior was observed in several neutron depolarization
measurements reported in the literature Bakker  _et al._ (1968); Drabkin  _et
al._ (1968); Rauch and Weber (1968); Drabkin  _et al._ (1969); Maleev and
Ruban (1970); Mitsuda and Endoh (1985); Endoh and Ishikawa (1986), which were
attributed to a change of the temperature dependence to the slowing down of
ferromagnetic fluctuations close to $T_{\rm C}$, such that they satisfy the
conditions for depolarizing the neutron beam.

Further, when increasing the applied magnetic field oscillations in the
polarization as a function of temperature emerged close to $T_{\rm C}$. These
oscillations may be explained by a precession of the polarization with respect
to the effective internal field comprising the interplay of the magnetization
under the applied magnetic field and the easy magnetic axis of the sample.
This interpretation is corroborated by the observation of a polarization that
is nominally negative for $B=$10\text{\,}\mathrm{mT}$$ which cannot be
accounted for by a depolarization alone.

Similar oscillations of the polarization were also observed in magnetic field
sweeps. As explained above, the setup shown in Fig. 3 (b) allowed to measure
the polarization in bipolar field cycles. Shown in Fig. 10 is the polarization
as a function of applied field between $+$50\text{\,}\mathrm{mT}$$ and
$-$50\text{\,}\mathrm{mT}$$ at various temperatures below and above $T_{\rm
C}=$5.26\text{\,}\mathrm{K}$$ for the single-crystal sample with $x=$0.40$$.
Rough fits of the data illustrating the frequency and amplitude of the
oscillations were extracted using a gaussian-damped cosine function.

Already at $7.2\text{\,}\mathrm{K}$, shown in Fig. 10 (a), a pronounced
oscillation may be discerned, i.e., well above $T_{\rm C}$. The frequency of
the oscillation increases with decreasing temperature, consistent with an
increase of the internal field. The amplitude of the oscillation is damped for
increasing magnitude of the applied field due to the finite wavelength spread
of the neutron beam. This effect compares with similar behaviour seen, e.g.,
in neutron spin-echo measurements Franz  _et al._ (2019). The oscillation is
smeared out below $T_{\rm C}$ where a small applied magnetic field of
$20\text{\,}\mathrm{mT}$ is already sufficient to completely depolarize the
neutron beam, i.e., $P=$0$$, as shown in Fig. 10 (e).

Figure 10: Polarization as a function of magnetic field of single-crystal
CePd1-xRhx with $x=$0.40$$ for various temperatures. An oscillation in the
polarization is already present at temperatures well above $T_{\rm
C}=$5.26\text{\,}\mathrm{K}$$. To illustrate key characteristics of the
oscillations a fit using a gaussian-damped cosine function is shown.

A direct comparison of Eq. 11 with the temperature dependence observed
experimentally is not satisfactory due to the large number of parameters and
the lack of information of the precise temperature dependence of the
magnetization $M(T)=B_{\textrm{int}}(T)/\mu_{0}$. In contrast, an evaluation
of the magnetic field dependence is possible as the depolarization term stays
constant and the field-dependence of the magnetization $M(B)$ is roughly
linear in small applied magnetic fields. In turn, this permits to infer the
angle $\alpha$ between $B_{\textrm{int}}$ and the polarization as a function
of temperature as shown in Fig. 11.

With increasing temperature above $T_{\rm C}=5.26\,{\rm K}$ the angle $\alpha$
decreases. As the magnetic anisotropy inferred from the bulk properties is
unchanged up to $100\text{\,}\mathrm{K}$Westerkamp (2008) the temperature
dependence may be attributed to the decrease of the easy-axis susceptibility
and the associated decrease of the life-time of fluctuations in the
paramagnetic state vis a vis with the time needed of a neutron to traverse the
regime of a fluctuation. As depolarizing effects decrease with increasing
temperature above $T_{\rm C}$, the combination of these time-scales may be
effectively viewed in terms of a magnetic field causing dominantly a rotation
of the polarization direction.

Figure 11: Angle $\alpha$ between the c-axis of a CePd1-xRhx single-crystal
with $x=$0.40$$ and the direction of the neutron polarization $P$ above the
ferromagnetic transition temperature $T_{\rm C}=5.26\,{\rm K}$. Data points
were inferred from the field sweeps shown in Fig. 10 using Eq. 11. A
continuous decrease of $\alpha$ is observed with increasing temperature. The
black line is a guide to the eye.

##  V Discussion

For the interpretation of our experimental results it is helpful to recall at
first the notions and terminology used in conventional spin freezing processes
starting from a paramagnetic state. In so-called canonical spin glasses the
separation and interactions between the spins are sufficiently small, such
that the frozen state is characterized by an ensemble of essentially randomly
oriented, uncorrelated microscopic spins. In systems with a larger density of
spins and larger interactions, clusters of correlated spins may form under
decreasing temperature. The associated frozen state is commonly dubbed a
cluster glass. Finally, in the limit of strongly interacting densely packed
spins, correlated regimes may form that behave essentially like very large
macroscopic spins and the behaviour is referred to as superparamagnetism. For
completeness we note that spin-frozen states which emerge from long-range
ordered states are referred to as a reentrant spin glass – a misleading
expression as the long-range ordered state actually exhibits the reentrant
temperature dependence.

Regardless of the precise character of the spin-frozen state, neutron
depolarization is expected if the same threshold conditions are satisfied as
in a multi-domain ferromagnet. To distinguish long-range ferromagnetism from a
spin-frozen state is, in turn, not straight-forward and requires consideration
of further information such as the bulk properties. In particular, apart from
quantitative differences of the size of the depolarization, the onset of the
depolarization will be insensitive to the precise magnetic field and
temperature history. The former depends on the precise alignment of the
domains, whereas the latter depends on the interactions and the size of the
correlated regimes.

Further, it is also important to distinguish a generic depolarization from the
rotation of the polarization axis. Both effects may be present simultaneously
as reported in the literature, e.g., for Ni and observed in our study of
CePd1-xRhx for $x=0.4$. This compares with Fe1-xCrx Benka  _et al._ (2022),
where a pronounced depolarization was recently reported deep in the
paramagnetic state far above the freezing temperature observed in the bulk
properties. In this context we also wish to note that reentrant spin glasses
may exhibit a depolarization for all temperatures below the onset of
sufficiently strong ferromagnetic correlations. If the long range ordered
state is ferromagnetic the depolarization may start below the Curie
temperature and prevail unchanged into the spin glass regimeBenka  _et al._
(2022). If, in contrast, the long-range ordered state is antiferromagnetic, a
depolarization may only be expected below the spin-glass temperature.

We turn now to the evolution of the nature of ferromagnetic correlations in
CePd1-xRhx as a function of increasing Rh content $x$, which is the result of
several microscopic interactions. Notably, magnetic moments develop with
decreasing temperature that interact by virtue of an exchange coupling.
Crystal electric fields and spin-orbit coupling partly quench the magnetic
moments and introduce magnetic anisotropies. The Kondo effect results in an
additional screening of these moments and changes of the concomitant
interactions. As a function of increasing Rh content the spontaneous moment at
zero temperature decreases from an almost unscreened large value for $x=0$.
Both the change of the lattice constant and the concomitant decrease of the
density of states at the Fermi level, as well as the increase of the Kondo
screening under increasing Rh content control the suppression of
ferromagnetism.

For all CePd1-xRhx samples studied we observe neutron depolarization. Up to
$x=0.65$ even a spontaneous depolarization under zfc-zfh conditions is
observed. This provides unambiguous microscopic evidence of the ferromagnetic
character of the spin correlations up to $x=$0.70$$, the largest value
studied. In addition, the $T_{\rm C}$ maps depicted in Fig. 5 show that the
distribution of the ordering temperature varies only slightly over the sample
cross-section with well-defined ordering and/or freezing temperatures. The
samples are hence metallurgically homogeneous on macroscopic scales. This
confirms that bulk properties like the magnetization reflect intrinsic
behavior.

The onset of the neutron depolarization at $T_{\rm C}$/$T_{\rm F1}$ is in
excellent agreement with the ordering/freezing temperature observed in the
bulk properties. The behaviour at $T_{\rm C}$/$T_{\rm F1}$ does not depend on
the temperature and field history, apart from a small broadening at $T_{\rm
C}$ for $x=0.4$. Indeed, for $x=0.4$ the strong easy-axis ferromagnetism
causes a Larmor rotation of the polarization even well above the Curie
temperature, whereas a pronounced depolarization is observed below $T_{\rm
C}$. The properties of the ferromagnetically ordered compositions are, hence,
perfectly consistent with an Ising ferromagnet without noticeable evidence of
disorder, e.g., such as depolarization above $T_{\rm C}$. This provides an
important point of reference for the emergence of the reentrant and glassy
behaviour near quantum criticality.

For $x>0.6$ the agreement of $T_{\rm F1}$ observed in the depolarization and
the bulk properties contrasts the observation of a small but finite hysteresis
in the magnetization for $T>T_{\rm F1}$ that has been attributed to the
formation of clustersWesterkamp  _et al._ (2009). The absence of
depolarization above $T_{\rm F1}$ thus shows that the size and the life-time
of the clusters inferred from the magnetization must be tiny and below the
threshold of depolarization, consistent with a decrease of the ordered moment
and the interactions under increasing Rh content. The behaviour we observe in
CePd1-xRhx contrasts that observed in the superparamagnetic regime of Fe1-xCrx
where a sizeable depolarization is observed well above the freezing
temperature Benka  _et al._ (2022). It underscores the formation of a cluster
glass at $T_{\rm F1}$ in CePd1-xRhx, however, consisting of tiny clusters.

A highly unconventional property emerges, finally, under zero-field-
cooling/field-heating. All samples with $x>0.6$ exhibit a pronounced
reentrance of the depolarization between $T_{\rm F1}$ and $T_{\rm F2}$. In
fact, even the ferromagnetic single-crystal with $x=0.4$ displayed a
reentrance at $T_{\rm F2}$, though barely noticeable. Here it may be helpful
to note that the reentrant temperature dependence of the depolarzation under
zfc-fh cannot be the signature of a reentrant spin-glass. Rather, the lack of
depolarization up to $T_{\rm F2}$ under zero-field-cooling/field-heating
implies that the clusters which undergo a freezing at $T_{\rm F1}$ must be
tiny in the absence of an applied field. This is consistent with the absence
of a depolarization above $T_{\rm F1}$ despite the presence of hysteresis in
the magnetization.

Yet, the applied magnetic field of 7.5 mT under which reentrance is observed
is small, and the energy scale associated with the applied field corresponds
to a temperature of several hundred milli-Kelvin. The Zeeman energy of the
applied magnetic field is hence roughly consistent with the values of $T_{\rm
F2}$. Thus, when heating the sample in a small applied magnetic field after
zero-field-cooling, a thermally activated formation of clusters may take place
at $T_{\rm F2}$, where the resulting clusters are sufficiently large to
generate a sizeable depolarization.

Considering the combination of energy scales in CePd1-xRhx mentioned above, it
is instructive to discuss the possible origin of the small size of the
clusters at zero magnetic field that undergo the spin freezing at $T_{\rm
F1}$. Experimentally we find that the freezing temperature $T_{\rm F1}$ and
the reentrance temperature $T_{\rm F2}$ decrease with increasing $x$ and
roughly track each other. As a function of increasing Rh content, this is
consistent with the reduction of the spontaneous magnetic moment and the
strength of the interactions, as well as the steep increase of the Kondo
screening and the distribution of Kondo temperatures for $x>x^{*}\approx 0.6$.
Namely, as the moment decreases the freezing temperatures decrease,
empirically suggesting that the additional Kondo screening above $x^{*}$
controls the small size of the clusters.

The setup shown in Fig.3 (a) yields detection thresholds of
$\delta>$2\text{\,}\mathrm{nm}$$ and $\tau>$4\text{\,}\mathrm{ps}$$ for an
unscreened moment of $2\,\mu_{\rm B}{\rm f.u.^{-1}}$ in CePd1-xRhx as
discussed in Sec. III.2. However, ferromagnetic fluctuations in this parameter
regime and on this time scale are not plausible based on the bulk properties
and the value of $T_{F1}$. Therefore, we attribute the change in polarization
at $T_{F2}$ to an increase of the average size of the clusters.

Figure 12: (a) Product $B_{0}^{2}\delta$ of average flux density per domain
$B_{0}$ and average domain length $\delta$ in transmission direction as a
function of external magnetic field $B=\mu_{0}H$. The behavior is monotonic,
however with increasing Rh content $x$ the initial value at zero applied field
vanishes for $x>$0.65$$. (b) The product $B_{0}^{2}\delta$ shown as function
of Rh concentration $x$ derived from zero-field measurements and under an
applied field $B=$7.5\text{\,}\mathrm{mT}$$ for each concentration. The
signature after zero-field-cooling vanishes at $x=$0.70$$ which implies that
$B_{0}^{2}\delta\rightarrow 0$.

Fitting the depolarization shown in Fig. 6 with Eq. 4, typical values of
$B_{0}^{2}\delta$ may be inferred, i.e., the average field per domain $B_{0}$
squared times the average domain size $\delta$. Shown in Figs. 12 (a) and 12
(b) is the extrapolated zero-temperature limit of $B_{0}^{2}\delta$ as a
function of the applied field $B=\mu_{0}H$ and the Rh concentration $x$,
respectively. Under small applied magnetic fields $B_{0}^{2}\delta$ increases
significantly. The absolute values of $T_{\rm F2}$ as compared to the applied
magnetic field suggest that this may be attributed to a thermally activated
increase of the average cluster size. The broad distribution of Kondo
temperatures at large values of $x$ support this suggestion. Likewise,
$B_{0}^{2}\delta$ decreases as a function of increasing Rh content, as
expected when approaching the intermediate valent properties of CeRh.

The average size $\delta$ of the clusters may, finally, be estimated when
taking into account the magnetic moment of CePd1-xRhxNieva  _et al._ (1988);
Sereni  _et al._ (1993); Schmakat (2015); Deppe  _et al._ (2006). In the
field-cooled state the cluster size decreases continuously from a value
exceeding $6\text{\,}\mathrm{nm}$ to $4\text{\,}\mathrm{nm}$ when the Rh
concentration increases from $x=0.4$ to 0.7. Under zero-field-cooling $\delta$
decreases from $5\text{\,}\mathrm{nm}$ to a value below the detection
threshold. This is consistent with basic estimates of the cluster size of
approximately 5 spins in the tail region of the phase diagramWesterkamp
(2008).

##  VI Conclusions

In conclusion, we have carried out neutron depolarization measurements of
ferromagnetism and spin freezing in CePd1-xRhx. We find clear signatures of
ferromagnetic correlations up to a Rh concentration of $x=$0.70$$, where the
3D polarization analysis of a single crystal with $x=0.4$ underscores well
behaved long-range ferromagnetic order. The ordering and freezing temperatures
are in good agreement with the bulk properties. A reentrant temperature
dependence of the depolarization under zero-field-cooling/field-heating of the
Rh compositions featuring spin freezing reveals thermally activated cluster
growth in the spin-frozen state. The sensitivity of our setup and the
estimated size of the ferromagnetic correlations provide microscopic
information consistent with the formation of a Kondo cluster glass, initially
proposed on the basis of the bulk properties.Westerkamp  _et al._ (2009);
Westerkamp (2008) The Kondo cluster glass emerges adjacent to a ferromagnetic
QPT. Taken together, our observations in CePd1-xRhx underscore the potential
of neutron depolarization as a microscopic probe of ferromagnetic quantum
phase transitions and concomitant escape routes.

###### Acknowledgements.

We wish to thank Peter Böni for fruitful discussions and the team at the
beamline ANTARES for support with our experiments. This work has been funded
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
FOR960 (Quantum Phase Transitions), TRR80 (From Electronic Correlations to
Functionality, Project No. 107745057) and the excellence cluster MCQST under
Germany’s Excellence Strategy EXC-2111 (Project No. 390814868). Financial
support by the European Research Council (ERC) through Advanced Grants No.
291079 (TOPFIT) and No. 788031 (ExQuiSid) is gratefully acknowledged. M.S.,
P.S., and P.J. acknowledge financial support through the TUM Graduate School.

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