# Kondo behavior, ferromagnetic correlations, and crystal fields in the heavy
Fermion compounds Ce3X ( X=In, Sn)

C. H. Wang1,2, J. M. Lawrence1, A. D. Christianson3, E. A. Goremychkin4, V. R.
Fanelli2, K. Gofryk2, E. D. Bauer2, F. Ronning2, J. D. Thompson2, N. R. de
Souza4,5, A. I. Kolesnikov3, K. C. Littrell3 1University of California,
Irvine, California 92697, USA
2Los Alamos National Laboratory, Los Alamos, NM 87545, USA
3Neutron Scattering Sciences Division, Oak Ridge National Laboratory, Oak
Ridge, TN, 37831, USA
4Argonne National Laboratory, Argonne, IL 60439, USA
5Australian Nuclear Science and Technology Organisation, Lucas Heights, NSW
2234, Australia

###### Abstract

We report measurements of inelastic neutron scattering, magnetic
susceptibility, magnetization, and the magnetic field dependence of the
specific heat for the heavy Fermion compounds Ce3In and Ce3Sn. The neutron
scattering results show that the excited crystal field levels have energies
$E_{1}$ = 13.2 meV, $E_{2}$ = 44.8 meV for Ce3In and $E_{1}$ = 18.5 meV,
$E_{2}$ = 36.1 meV for Ce3Sn. The Kondo temperature deduced from the
quasielastic linewidth is 17 K for Ce3In and 40 K for Ce3Sn. The low
temperature behavior of the specific heat, magnetization, and susceptibility
can not be well-described by J=1/2 Kondo physics alone, but require
calculations that include contributions from the Kondo effect, broadened
crystal fields, and ferromagnetic correlations, all of which are known to be
important in these compounds. We find that in Ce3In the ferromagnetic
fluctuation makes a 10-15 $\%$ contribution to the ground state doublet
entropy and magnetization. The large specific heat coefficient $\gamma$ in
this heavy fermion system thus arises more from the ferromagnetic correlations
than from the Kondo behavior.

###### pacs:

71.27.+a, 71.70.Ch, 75.20.Hr

##  I Introduction

In heavy fermion (HF) compounds, it is very common to establish the Kondo
energy scale $T_{K}$ from the linear coefficient of specific heat $\gamma$
through Rajan’s formula $T_{K}$= $\pi JR/3\gamma_{0}$ derived for the
degenerate ($2J+1\geq$ 2) Kondo modelRajan  where $J$ is the total angular
momentum. In previous studies of the specific heat of the HF compounds Ce3X
(X=In, Sn)YYChen ; Lawrence  which crystallize in the Cu3Au cubic structure,
this formula was used to determine the Kondo temperature, which was found to
be 4.8 K for Ce3In and 16.7 K for Ce3Sn. The crystal electric field (CEF)
excitation energy was estimated to be $T_{CEF}$=65 K.

Most HF compounds reside close to a quantum critical point (QCP)where
antiferromagnetic (AFM) or ferromagnetic (FM) correlations are present. This
makes the previous analysis inappropriate in so far as it assumes that the
magnetic correlations do not contribute to $\gamma$. Indeed, the Wilson ratios
($\pi^{2}R\chi_{0}/3C_{J}\gamma_{0}$) which were determined previously for
Ce3In and Ce3Sn are 11.5 and 7.0 respectivelyYYChen ; Lawrence , indicating
that ferromagnetic correlations dominate the susceptibility.

Inelastic neutron scattering(INS) experiments on single crystals of compounds
that are close to a QCP, such as CeRu2Si2CeRu2Si2  or CeNi2Ge2CeNi2Ge2
exhibit two classes of excitations. At most $Q$ in the Brillouin zone, the
scattering has the characteristic Kondo energy dependence and is
$Q$-independent or only weakly $Q$-dependent. Similar behavior is observed in
intermediate valence compounds for which it is clear that the behavior of the
low temperature susceptibility, specific heat and INS spectra are close to the
Kondo impurity prediction, as though the onset of lattice coherence has only a
minor effect on these measurementsYbInCu4 ; slowcrossover . Near the QCP,
however, large $Q$-dependent scattering is observed with maximum intensity at
the critical wavevector $Q_{c}$ ($Q_{c}$ = 0 for FM and $Q_{c}$ = $Q_{N}$ for
AFM) where ordering occurs in the nearby magnetic state. This scattering
represents the short range order. It is dynamic and critically slows down, or
softens, as the QCP is approached by lowering the temperature or changing a
control parameter. These fluctuations affect the specific heat and can result
in non-Fermi liquid behavior.

Hence INS in single crystals can separate the Kondo behavior from the
contributions due to magnetic correlations. Since the spectral weight in the
magnetic correlations is typically small, INS in polycrystals will be
dominated by the $Q$-independent Kondo scattering. INS can also be used to
directly determine the CEF excitations. Under these circumstance, INS provides
a better way to determine $T_{K}$ and $E_{CEF}$ than through analysis of the
specific heat. In this paper, we employ INS to determine both $T_{K}$ and
$E_{CEF}$. We have re-measured the magnetic susceptibility, and have extended
the specific heat measurement, which in the previous report was measured down
to 1.8 K in zero applied magnetic field, to $T$ = 400 mK and $B=$ 9T. We have
also measured the low temperature magnetization to 13 T.

In the Ce3X compounds, the Ce ions sit on the face centers of the cubic
lattice and are subject to a crystalline electric field (CEF) of tetragonal
symmetry. In this case, the Hamiltonian is described as:

$H_{CF}=B_{2}^{0}O_{2}^{0}+B_{4}^{0}O_{4}^{0}+B_{4}^{4}O_{4}^{4}$,

where $B_{l}^{m}$ and $O_{l}^{m}$ are the crystal field parameters and Steven
operators, respectively. The sixfold degenerate 4$f^{1}$, J=5/2 state splits
into three doublets. Diagonalizing the Hamiltonian, the atomic wave functions
are given by:Aviani ; Fischer

$\Gamma_{7}^{(1)}=\eta|\pm 5/2>+\sqrt{1-\eta^{2}}|\mp 3/2>$
$\Gamma_{7}^{(2)}=\sqrt{1-\eta^{2}}|\pm 5/2>-\eta|\mp 3/2>$
$\Gamma_{6}=|\pm 1/2>$

Depending on the admixture of the $J$=5/2 and 3/2 states, the inelastic
neutron scattering spectra will exhibit one or two inelastic excitations. Low
energy transfer quasielastic scattering will also be observed if the
instrumental resolution is adequate. From the INS spectra, the crystal field
energies and wavefunctions can be determined from the ampitudes and energies
of the excitations. The Kondo effect, which arises from the hybridization of
the 4 $f$-electron with the conduction electrons, broadens the peak line-
widths proportional to $k_{B}T_{K}$. The quasielastic scattering peak width
$\Gamma_{QE}$ can be equated to the Kondo energy $k_{B}T_{K}$ of the ground
state doublet.

In what follows, we will use the CEF parameters and the Kondo energies derived
from the neutron scattering to calculate the Kondo contribution to the
specific heat, susceptibility and magnetization. All the Kondo calculations
utilizedRajan ; Hewson ; Sacramento  employ the same Bethe-Ansatz calculation,
making intercomparison possible.

##  II experiment

All samples were prepared by arc melting in an ultra-high-purity argon
atmosphere. After melting the samples were sealed under vacuum and annealed at
5000C for 2 weeks and cooled slowly to room temperature. The magnetization was
measured in a 14 T Quantum Design Vibrating Sample Magnetometer at the
National High Magnetic Field Laboratory (NHMFL) at Los Alamos National
Laboratory. The specific heat was measured in a Quantum Design PPMS system.
The magnetic susceptibility measurements were performed in a commercial
superconducting quantum interference device (SQUID) magnetometer.

We performed inelastic neutron scattering on a 29 gram sample of Ce3In and a
37 gram sample of Ce3Sn using the high energy transfer chopper spectrometer
(HET) at ISIS (at the Rutherford Appleton Laboratory) and the low resolution
medium energy chopper spectrometer (LRMECS) at IPNS (at Argonne National
Laboratory). For Ce3Sn, the quasi-elastic neutron spectrometer (QENS, at IPNS)
was also used to measure the low energy scattering. To increase the dynamic
range of the INS spectrum, a variety of incident energies ($E_{i}=$ 15 meV, 35
meV, 60 meV, 100 meV for HET and 35 meV for LRMECS) and temperatures ( 4.5 K,
100 K, 150 K, 200 K and 250 K for HET; 10K, 100K, 150K for LRMECS) were
employed. The HET data have been normalized to vanadium to establish the
absolute value. All the data have been corrected for absorption ( which is
very obvious for Ce3In case), total scattering cross section, and sample mass.

For the HET data, the low Q data were obtained from averaging the low angle
detectors with angles ranging from 11.5 degrees to 26.5 degrees. The high Q
data were obtained from the high angle detector bank at an angle 136 degrees.
For the LRMECS experiment, the low Q data were obtained by averaging over the
low angle detectors with average angle equal to 13 degrees; and the high Q
data were obtained from high angle detectors where the average angle was 87
degrees. The QENS data were collected at 7 K. This inverse geometry
spectrometer has 19 detector banks with $Q$ from $Q=0.36{\AA}$ to
$Q=2.52{\AA}$, each with a slightly different final energy ($E_{f}$ from 2.82
meV to 3.36 meV). For every fixed $Q$, we removed the Ce 4$f$ form factor to
obtain a spectrum representing the $Q=0$ scattering and then summed all 19
spectra together to obtain a total $S(Q=0,\Delta E)$ spectrum. In order to
compare the QENS spectrum with the spectra from the direct geometry
spectrometers HET and LRMECS, we multiplied the QENS spectrum $S(Q=0,\Delta
E)$ by the 4$f$ form factor apppropriate for HET at $E_{i}$ = 35 meV.

To subtract the nonmagnetic (background, single phonon, and multiple phonon)
contributions, we measured the non-magnetic counterpart compounds La3In and
La3Sn. For the specific heat, we obtained the magnetic contribution by direct
subtraction, i.e. $C_{mag}=C(Ce)-C(La)$. For the INS data we used La3In and
La3Sn to determine the scaling of the nonmagnetic scattering between low $Q$
and high $Q$ as $h(\Delta E)=S(La,LQ)/S(La,HQ)$. Using this factor we scaled
the high $Q$ data (where the nonmagnetic scattering dominates) to the low $Q$
data (where the magnetic scattering dominates) in Ce compounds to determine
the nonmagnetic contributionMurani ; Eugene ; Jon .

##  III results and discussion

Fig. 1(a) and (c) directly compare the low $Q$ INS spectra of Ce3In and La3In;
the data were collected on HET with incident energy $E_{i}=$ 15 meV (a) and 60
meV (c) at 4.5 K. Fig. 1(b) compares the low $Q$ and high $Q$ data for Ce3In
collected on LRMECS with an incident energy $E_{i}$= 35 meV at 10 K. The low
$Q$ data for Ce3Sn and La3Sn, which were collected on HET, are compared in
Fig. (d), (e) and (f) where the incident energies are $E_{i}$=15 meV (d), 35
meV (e) and 60 meV (f) at 4.5 K. In these spectra, two excited energy levels,
corresponding to crystal field excitations, are observed for both Ce3In and
Ce3Sn. The spectra (a) and (d), which compare the Ce3In(Sn) and La3In(Sn)
scattering at low energy transfer ($\Delta E<$ 9 meV), exhibit obvious
quasielastic scattering which as mentioned above arises from Kondo scattering.

Figure 1:  Inelastic neutron scattering spectra for Ce3In and Ce3Sn together
with that of their nonmagnetic counterpart compounds obtained from HET and
LRMECS. The data collected on HET are at 4.5 K and on LRMECS are at 10 K. (a)
$E_{i}$=15 meV, (b) $E_{i}$=35 meV, and (c) $E_{i}$=60 meV spectra of Ce3In
and La3In. (d) $E_{i}$=15 meV, (e) $E_{i}$=35 meV and (f) $E_{i}$=60 meV
spectra for Ce3Sn and La3Sn. All spectra are for low $Q$ except in (b) where a
high-$Q$ spectrum is included for comparison.

The magnetic contribution $S_{mag}$ to the scattering of Ce3In, obtained using
the method described above, is shown in Fig. 2. The solid lines represent a
fit to the CEF model. Since the inelastic peaks are relatively broad, the line
widths $\Gamma_{i}$ are taken to be finite. In this case, the magnetic
scattering is described as:

$S_{mag}=\frac{2N}{\pi\mu_{B}^{2}}f^{2}(Q)(1-e^{-\Delta
E/{k_{B}T}})\chi^{\prime\prime}(Q,\Delta E)$

$\chi^{\prime\prime}(Q,\Delta E)=\Sigma\chi_{i}(T)\Delta
E(\frac{\Gamma_{i}}{2\pi})/[(\Delta E-E_{i})^{2}+\Gamma^{2}_{i}]$

Here $i$=0,1,2, $E_{0}$ = 0 corresponds to the quasielastic scattering, and
$f^{2}(Q)$ is the Ce 4$f$ form factor. The CEF model fitting was performed
simultaneously on six data sets at three different incident energies
($E_{i}$=15 meV, 35 meV and 60 meV) and at two different temperatures (4.5 K
and 150 K). Fig. 2(a)-(d) are the data collected on HET. In Fig. 2(f) the
LRMECS data are displayed for comparison. The resulting CEF fitting parameters
are shown in Table I. The ground state is the $\Gamma_{7}^{(1)}$ doublet, the
first excited state is the $\Gamma_{7}^{(2)}$ doubletendnote1  at the energy
13.2 meV, and the second excited state is the $\Gamma_{6}$ doublet at the
energy 44.8 meV. The quasielastic line width $\Gamma_{QE}$ = $\Gamma_{0}$ =
1.49 meV, implies that $T_{K}$ = $\Gamma_{QE}/k_{B}$ = 17 K.

Table 1:  CEF model fitting parameters for Ce3In and Ce3Sn. | Ce3In | Ce3Sn
---|---|---
$B_{2}^{0}$(meV) | -2.203$\pm$0.015 | -1.660$\pm$0.017
$B_{4}^{0}$(meV) | 0.066$\pm$0.001 | 0.038$\pm$0.0009
$B_{4}^{4}$(meV) | -0.154$\pm$0.004 | -0.263$\pm$0.003
$\eta$ | 0.94 | 0.89
$E_{1}$ (meV) | 13.2 | 18.5
$E_{2}$ (meV) | 44.8 | 36.1
$\Gamma_{QE}$ (meV) | 1.49$\pm$0.07 | 3.52$\pm$0.16
$\Gamma_{1}$ (meV) | 5.98$\pm$0.07 | 9.37$\pm$0.038
$\Gamma_{2}$ (meV) | 2.06$\pm$0.37 | 6.28$\pm$0.38
$\chi^{2}$ | 2.4057 | 2.1528
$\lambda$ (mole-Ce/emu) | 62 | 85
Figure 2:  Magnetic contribution $S_{mag}$ to the inelastic neutron scattering
spectra of Ce3In for data taken on HET at T=4.5 K and 150 K with different
incident energies $E_{i}$=15 meV, 35 meV, and 60 meV and taken on LRMECS at
T=100 K with $E_{i}$=35 meV. The solid lines represent the quasielastic and
crystal field contributions obtained from least squares fitting as described
in the text.

In Fig. 3(a)-(e) we display the magnetic contribution to the Ce3Sn scattering
collected from HET at 4.5 and 100 K and at three incident energies (15, 35,
and 60 meV). Data from QENS at 7 K (Fig. 3(f)) are included for comparison.
The CEF fits are also included (solid lines); as for the Ce3In case, the fits
were performed simultaneously on six different spectra at different incident
energies and temperatures. The intensity and form factor of the QENS data have
been adjusted to that of the HET spectra at $E_{i}$=35 meV (spectra (b)) to
make a direct comparison. The fitting parameters yield a similar crystal field
scheme as for Ce3In: the $\Gamma_{7}^{(1)}$ doublet is the ground state,
$\Gamma_{7}^{(2)}$ is the first excited state with energy 18.5 meV, and the
second excited state is the $\Gamma_{6}$ doublet at the energy 36.1 meV. The
Kondo temperature $T_{K}=$ 40 K is higher than for Ce3In, and the excited
state linewidths are broader, reflecting stronger 4$f$-conduction
hybridization.

Figure 3:  (a)-(f) Magnetic contribution $S_{mag}$ to the inelastic neutron
scattering spectra for Ce3Sn. The temperatures and incident energies are given
in the plot. The solid lines represent the CEF model. (f): the magnetic
contribution to the INS spectra collected from QENS at 7 K. The solid line in
(f) is the CEF model fit for the $E_{i}$ = 35 meV spectra.

Due to the large CEF excitation energies, the low temperature behavior of the
magnetic specific heat should be dominated by the $\Gamma_{7}^{(1)}$ doublet
ground state. This is confirmed by the fact that the magnetic entropy (Fig.
4(a) inset) reaches Rln2 near 20 K but only reaches Rln4 near 70 K. For a
doublet ground state, the Kondo model predicts $\gamma_{0}=\pi R/6T_{K}$ for
the linear coefficient of specific heat Rajan . In previous results for Ce3In
a value $T_{K}=$ 4.8 K was deduced using this formulaYYChen . In addition, the
specific heat coefficient $C/T$ showed a peak near 2 K whose existence was
somewhat uncertain since the lowest measured temperature was only 1.8 K. We
have extended the specific heat measurement down to 400 mK. In Fig. 4(a) we
plot $C_{mag}/T$ and find a peak at $T$ = 2.6 K. Comparison of the data to the
prediction $\gamma^{K}(T)$ of the Kondo model which is calculated using the
value $T_{K}=\Gamma_{QE}/k_{B}=17K$ deduced from our neutron data, shows that
the Kondo prediction is much smaller than the experimental value; indeed,
$\gamma^{K}_{0}$ is only half of $\gamma^{exp}_{0.4K}$ (Table II). Given the
large Wilson ratio reported earlierYYChen , the obvious explanation is that
ferromagnetic (FM) fluctuations dominate the low temperature specific heat,
increasing the specific heat above the Kondo value and giving rise to the peak
at 2.6 K representing the onset of short range FM order.

We next consider the high temperature susceptibility, comparing the measured
value to the value calculated from the crystal field parameters of Table I in
the inset of Fig. 4(b). A molecular-field $\lambda$ = 62 mole-Ce/emu has been
added to compensate the reduction of the susceptibility at high temperature
due to the Kondo effect ($1/\chi^{HT}=1/\chi^{CEF}+\lambda$). At high
temperatures, when the crystal field states are excited, the effective Kondo
temperature $T^{HT}_{K}$ is larger than the Kondo temperature of the ground
state doublet. The molecular field constant is related to the effective Kondo
temperature via $\lambda=T^{HT}_{K}/C_{5/2}$ where $C_{5/2}$ is the free ion
Curie constant for cerium. This relation gives $T^{HT}_{K}=$ 77 K, which value
is essentially equal to the width $\Gamma_{1}$ of the first excited level seen
in the neutron scattering (Fig. 2 and table I).

At low temperatures, there should be three contributions to $\chi(T)$, as well
as to $M(H)$ and $C_{mag}$: one from the Kondo single ion impurity physics of
the ground state doublet, one from the FM fluctuations, and one from the
excitation of higher lying crystal field states. To carry out such an
analysis, we note first that in the Cu3Au crystal structure, the tetragonal
crystal field axis (i.e. the z-axis for the doublet wave functions) points
perpendicular to the face containing any given face-centered cerium atom;
hence there are three orthogonal tetragonal axes in the unit cell. When
applying a magnetic field in a polycrystalline sample, the field will point
along the tetragonal axis for $\frac{1}{3}$ of the cerium atoms but orthogonal
to the tetragonal axis (in the $x-y$ plane) for $\frac{2}{3}$ of the atoms.
The effective low temperature Curie constant is then
$C_{eff}=\frac{1}{3}C^{z}_{eff}+\frac{2}{3}C^{x}_{eff}$, where
$C^{z(x)}_{eff}=N(g^{z(x)}_{eff}\mu_{B})^{2}\frac{1}{2}(\frac{1}{2}+1)/3k_{B}$.
This is the form for a pseudo spin $\frac{1}{2}$ doublet where the CEF physics
is absorbed into the effective $g$-factor. Here
$g^{z(x)}_{eff}=\frac{12}{7}<J_{z(x)}>$ where $<J_{z(x)}>$ is the matrix
element of the angular moment component along the $z(x)$ axis. From the CEF
mixing parameter $\eta$, we determine $C_{eff}$ to be 0.48 emu-K/mole-Ce for
Ce3In and 0.41 emu-K/mole-Ce for Ce3Sn. (Table II).

Table 2:  Kondo single ion model calculation for Ce3In and Ce3Sn. | $<J_{z}>$ | $<J_{x}>$ | $M^{sat}_{CEF}$ | $C^{LT}_{eff}$ | $\chi^{K}_{0}(\frac{emu}{mole-Ce})$ | $\gamma^{K}_{0}(\frac{J}{moleCeK^{2}})$ | $\chi^{exp}_{0.4K}(\frac{emu}{mole-Ce})$ | $\gamma^{exp}_{0.4K}(\frac{J}{moleCeK^{2}})$
---|---|---|---|---|---|---|---|---
Ce3In | 2.0344 | 0.7171 | 0.991 | 0.4765 | 0.0076 | 0.256 | 0.064 | 0.467
Ce3Sn | 1.6684 | 0.9074 | 0.994 | 0.4086 | 0.0033 | 0.109 | 0.018 | 0.221
Figure 4:  (a) The magnetic contribution to the specific heat $C_{mag}/T$
versus $T$ for Ce3In. The solid line is the Kondo prediction $\gamma^{K}(T)$
calculated using $T_{K}=\Gamma_{QE}/k_{B}=$ 17 K. The inset is the magnetic
entropy of Ce3In. (b) The magnetic susceptibility $\chi(T)$ for Ce3In. The
solid line is the Kondo prediction $\chi^{K}(T)$ calculated using $T_{K}=$ 17
K and the low temperature Curie constant determined as described in the text.
The dashed line gives the sum of the Kondo and crystal field contributions
$\chi^{CEF+K}=\chi^{K}+(\chi^{CEF}-\chi^{LT}_{Curie})$. The inset is the
inverse susceptibility together with the calculated susceptibility (solid
line) $1/\chi^{HT}=1/\chi^{CEF}+\lambda$ obtained using the CEF fitting
parameters in table I. (c) The magnetization for Ce3In. The solid line
$M^{K}(B)$ is the Kondo calculation calculation using $T_{K}=$ 17 K. The
dashed line is the contribution from the ferromagnetic fluctuations.

To sort the low temperature susceptibility into Kondo, FM, and CEF
contributions, we note that since the first CEF excited level is at 152 K, at
sufficiently low temperatures the Zeeman splitting of the $\Gamma_{7}^{(1)}$
ground state doublet will obey a Curie law $\chi^{LT}_{Curie}=C^{LT}_{eff}/T$.
Due to the Kondo effect, this Curie behavior will be replaced by the Kondo
contribution $\chi^{K}$, which we calculate using the same Curie constant
$C^{LT}_{eff}$ and using $T_{K}=$ 17 K (solid line, Fig. 4(b)). The
susceptibility from the combination of the ground state Kondo and the excited
crystal fields will then be of the form
$\chi^{CEF+K}=\chi^{K}+(\chi^{CEF}-\chi^{LT}_{Curie})$ where we subtract
$\chi^{LT}_{Curie}$ to avoid double counting the ground state contribution. As
for the specific heat, the resulting $\chi^{CEF+K}$ (dashed line in Fig. 4(b))
is much smaller than the experimental value at $T<$ 20 K. The excess can be
viewed as the contribution from the ferromagnetic fluctuations. Taking the
latter to be equal to the difference $\chi^{exp}-\chi^{CEF+K}(T)$, the FM
contribution is seen to increase below 10 K in a manner characteristic of
ferromagnetic short range order.

In Fig. 4(c) we exhibit the magnetization as measured up to 13 Tesla at $T$=2
K. Based on Hewson’s calculation of Kondo modelHewson , we can estimate the
Kondo contribution to the magnetization. Since the effective $g$-factors
differ in the $z$ and $x-y$ directions, we calculate
$M^{K}=\frac{1}{3}M^{K}(z)+\frac{2}{3}M^{K}(xy)$. The result is plotted as a
solid line in Fig. 4(c). After subtracting the Kondo contribution, we obtain
the contribution from the FM correlations (dashed line). This saturates at a
relatively small field $B\sim 2.5$ tesla with $M^{sat}$ = 0.095 $\mu_{B}$,
which is 10 percent of the saturation value 1.0 $\mu_{B}$ expected based on
the effective $g$-factors.

Figure 5:  (a) the magnetic specific heat $C_{mag}/T$ versus $T$ at B=0 T
(open circle) and B=9 T (solid triangle) for Ce3Sn. The solid line is the
Kondo contribution $\gamma^{K}(T)$ for $T_{K}=$ 40 K. The inset is the
magnetic entropy. (b) Magnetic susceptibility $\chi(T)$ for Ce3Sn. The solid
line is the Kondo contribution $\chi^{K}(T)$ calculated with
$T_{K}=\Gamma_{QE}/k_{B}=$ 40 K and the dashed line is the sum of the Kondo
and CEF contributions $\chi^{CEF+K}=\chi^{K}+(\chi^{CEF}-\chi^{LT}_{Curie})$.
The inset is the inverse susceptibility together with the value
$1/\chi^{HT}=1/\chi^{CEF}+\lambda$ (solid line) calculated from CEF fitting
parameters in table I. (c) $C_{mag}$ at $B$=0 T (open circle) and B=9 T (solid
triangle) for Ce3Sn. The thin solid line is $C^{K}(T)$ calculated with
$T_{K}=$ 40 K.

In the same way as for Ce3In, we calculate $\chi^{K}(T)$, $\chi^{CEF}(T)$,
$\chi^{CEF+K}(T)$, $\gamma^{K}(T)$ and $C^{K}(T)$ for Ce3Sn, comparing to the
measured data in Fig. 5. The high temperature susceptibility (Fig. 5(b) inset)
can again be fit with the sum of the CEF contribution calculated using the
parameters of Table I and a molecular field contribution (solid line). The
value $\lambda=$ 85 mole-Ce/emu of molecular field constant implies an
effective Kondo temperature at high temperature $T^{HT}_{K}$ = 105 K, which
again is essentially equal to the linewidth 9.4 meV of the first excited state
seen in the neutron scattering (Table I).

The solid lines in Fig. 5 represent the Kondo ground state doublet
contributions. In Fig. 5(b), the dashed line is $\chi^{CEF+K}(T)$. The excess
due to the FM correlations has a much smaller magnitude ($\sim$ 0.005 emu/mol-
Ce) than for Ce3In where the FM contribution is of order 0.05 emu/mol-Ce. A
similar statement holds for the FM contribution to the specific heat
coefficient, which is of order 0.4 J/mol-Ce-K2 for Ce3In but only 0.1 J/mol-
Ce-K2 for Ce3Sn (Figs. 4(a) and 5(b)). Hence the FM enhancement is smaller in
Ce3Sn than in Ce3In, consistent with the larger value of $T_{K}$.

In order to better understand these compounds, we measured the specific heat
of Ce3In under different applied fields (B = 0 T, 1 T, 3 T, 6 T and 9 T). The
results for the magnetic contribution $C_{mag}$ are shown in Fig. 6. The low
temperature peak in $C_{mag}$ moves to higher temperature when the field is
increased. Since the peak in the Kondo contribution to $C_{mag}$ is expected
to increase with field, we plot in Fig. 6(a)-(e) the Kondo contribution
$C^{K}(B)$ calculated for different applied fields using the theoretical
results of Sacramento and SchlottmannSacramento . In calculating $C^{K}(B)$,
we again account for the different effective $g$-factors in the $z$ and $x-y$
directions. The results indicate that the Kondo contribution is not expected
to alter significantly in applied fields of order 9 T, essentially because
$g_{eff}\mu_{B}B<k_{B}T_{K}$ for these fields. This makes it clear that the
peak does not arise from the Kondo scattering but must be due to the FM
fluctuations.

To quantify the FM contribution, we again assume that the measured magnetic
specific heat is the sum of the ground state doublet Kondo contribution
$C^{K}(B)$, the FM contribution $C^{FM}$, and a contribution $C^{CEF}$ due to
the excitation of higher lying CEF states. Since the FM fluctuations appear to
only contribute to the susceptibility below 10 K (Fig. 4 b) we assume that the
excess $C_{mag}$ \- $C^{K}(B)$ observed for $T>$ 10 K is primarily due to CEF
excitations. Given the large linewidths of the CEF excitations seen in the
neutron scattering, and concomitant large effective $T^{HT}_{K}$ at high
temperature, this contribution to the specific heat is much broader as a
function of temperature than would be the case for a simple CEF Schottky
anomaly. For simplicity, we approximate this CEF contribution as linear in
temperature, with slope equal to that observed in the range 8-15 K, and we
assume that since the CEF excitation energy is large, this contribution will
be unaffected by fields of order 9 T. We approximate the FM contribution
$C^{FM}$ as a Gaussian, centered at a temperature that increases with field.
The three contributions, Kondo, CEF, and FM, are plotted at the different
fields in Figs. 6(a)-(e). The solid lines, which represent the sum of all
three contributions, fit the data very well at all fields.

We plot the Gaussian peak temperature in Fig. 6(f), where it is seen to grow
linearly with field. This suggests Zeeman splitting, where at zero field the
splitting arises from the internal field in the regions of FM short range
order, and where the applied field increases the splitting. To determine the
internal field $B_{int}$, we calculate the Schottky anomaly
$C^{Schotkky}(B_{int})$ expected due to Zeeman splitting of a doublet with the
same effective $g$-factors as we have obtained from the neutron fits; we then
adjust $B_{int}$ until the peak temperature of the Schottky anomaly is the
same as that of the Gaussian peak temperature for $B$ = 0. This gives
$B_{int}$ = 9.5 T. We then calculate $C^{Schottky}(B_{int}+B_{app})$ to
determine peak position of the Schottky anomaly in an applied field $B_{app}$.
As can be seen in Fig. 6 (f), the Gaussian peak temperatures track the
expected Zeeman splitting very closely. On the other hand, the temperature
dependence of the Schottky specific heat calculated in this manner is
considerably broader than the Gaussian contributions $C^{FM}$ that are plotted
in Fig. 5. This means that, while the contribution of the FM short range order
to the specific heat is not of Schottky form, the increase of the Gaussian
peak position is the same as the Zeeman splitting expected for a total field
$B_{int}+B_{app}$ given the effective $g$-factors.

The entropy of the Gaussian contribution is about 15 $\%$ of Rln2 for all
fields. This corresponds to the estimate obtained from the magnetization
$M(B)$ where the saturation value of the FM contribution is about 10$\%$ of
the value 1.0 $\mu_{B}$ expected for the $\Gamma_{7}^{(1)}$ ground state
doublet for the measured value of $\eta$. Hence, the enhancement of $\chi_{0}$
and $\gamma_{0}$ arises from magnetic fluctuations which involve 10-15 $\%$ of
the 4$f$ electron degrees of freedom.

In Fig. 5(c), we compare the magnetic specific heat $C_{mag}$ at zero field
and $B=$ 9 T for Ce3Sn. The solid line is the Kondo contribution $C^{K}$. The
specific heat does not change with field for $B<$ 9 T. The most likely
explanation of this is that, as discussed above, the FM correlations make a
smaller contribution than in Ce3In. The excess specific heat $C_{mag}-C^{K}$
seen for $T>$ 6 K is presumably due to the CEF contribution, which should be
even broader in temperature in Ce3Sn than in Ce3In due to the larger Kondo
temperature.

Figure 6:  (a), (b), (c), (d) and (e): $C_{mag}$ of Ce3In in different applied
magnetic fields. The solid line sums the three contributions (Kondo, CEF, and
FM fluctuations) shown in the plot. (f): The peak position of $C_{mag}$, of
the Gaussian contribution $C^{FM}$ due to the FM fluctuations, and the
expected peak position in the Schottky anomaly $C^{Schottky}(B_{int}+B_{app})$
due to Zeeman splitting in the presence of an internal field $B_{int}$.

We have demonstrated that the large $\gamma$ observed in Ce3In arises more
from ferromagnetic correlations than from the single ion Kondo physics. This
reflects the fact that the system is close to a ferromagnetic quantum critical
point. In a $Q$-resolved INS experiment, the ferromagnetic correlations should
show up in the vicinity of $Q=$0 riding on a background of $Q$ independent
Kondo scattering. We can estimate that these FM correlations will have
10-15$\%$ of the total spectral weight in $Q$-space. Since the large $\gamma$
and the proximity to the QCP occurs when the Kondo temperature $T_{K}=$ 17 K
is fairly large, it is also reasonable to believe that when an appropriate
control parameter (e.g. alloying parameter x in Ce3-xLaxIn) drives this system
to the QCP, the Kondo temperature $T_{K}$ will remain finite, as expected for
example for a spin density wave type QCP.

We have observed strong FM fluctuations in the related compound Pr3In, which
is an antiferromagnet below 12 KAndy . A possibility for this behavior is that
AFM interactions between rare earth atoms on the face centers of the Cu3Au
structure are frustrated. If, for example, the atoms at (1/2 1/2 0) and (1/2 0
1/2) are aligned antiferromagnetically, the atom at (0 1/2 1/2) will be free
to point to any direction. Ferromagnetic next-nearest-neighbor interactions
could then stabilize ferromagnetism on this sublatticeCristian . In any case,
the FM correlations appear to be generic to this crystal structure.

In conclusion, we have used inelastic neutron scattering to determine the
crystalline electric field (CEF) splitting and Kondo energy scale in Ce3In and
Ce3Sn. For both compounds the crystal field excitation energy is large. For
Ce3In we have separated the magnetization $M(B)$, susceptibility $\chi(T)$ and
specific heat $C_{mag}$ into contributions from the Kondo effect, from the
CEF, and from FM fluctuations. The simplified model calculation for Ce3In
shows that the FM correlations make a 15 $\%$ contribution to the doublet
ground state entropy and that the large $\gamma$ arises mostly from the FM
correlations. This suggests Ce3In is close to a quantum critical point (QCP).
The Kondo temperature $T_{K}$ is expected to remain finite at the QCP, as
occurs for a spin density wave type QCP. INS experiments in single crystals of
these compounds would be very interesting.

##  IV acknowledgements

We thank Vivien Zapf for her assistance in the measurement at NHMFL and
Cristian Batista for his insightful comments. Research at UC Irvine was
supported by the U.S. Department of Energy, Office of Basic Energy Sciences,
Division of Materials Sciences and Engineering under Award DE-FG02-03ER46036.
Work at ORNL was supported by the Scientific User Facilities Division Office
of Basic Energy Sciences, DOE and was managed by UT-Battelle, LLC, for DOE
under Contract DE-AC05-00OR22725. Work at Los Alamos National Laboratory was
performed under the auspices of the U.S. DOE/Office of Science. Work at NHMFL-
PFF, Los Alamos was performed under the auspices of the National Science
Foundation, the State of Florida, and U.S. DOE. Work at ANL was supported by
DOE-BES under contract DE-AC02-06CH11357.

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