arXiv:1003.3484v1 [cond-mat.str-el] 17 Mar 2010
Neutron scattering and scaling behavior in URu2Zn20and YbFe2Zn20
C. H. Wang1,2, A. D. Christianson3, J. M. Lawrence1, E. D. Bauer2, E. A. Goremychkin4,
A. I. Kolesnikov3, F. Trouw2, F. Ronning2, J. D. Thompson2, M.D. Lumsden3,
N. Ni5, E. D. Mun5, S. Jia5, P. C. Canfield5, Y. Qiu6,7and J. R. D. Copley6
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
5Ames Laboratory, Iowa State University, Ames, IA, 50011
6National Institute of Standards and Technology, Gaithersburg, Maryland 20899-6102, USA
7Department of Materials Science and Engineering,
University of Maryland, College Park, Maryland 20742, USA
(Dated: March 19, 2010)
The dynamic susceptibility χ′′(∆E), measured by inelastic neutron scattering measurements,
shows a broad peak centered at Emax = 16.5 meV for the cubic actinide compound URu2Zn20 and 7
meV at the (1/2, 1/2, 1/2) zone boundary for the rare earth counterpart compound YbFe2Zn20. For
URu2Zn20, the low temperature susceptibility and magnetic specific heat coefficient γ = Cmag/T
take the values χ = 0.011 emu/mole and γ = 190 mJ/mole-K2at T = 2 K. These values are roughly
three times smaller, and Emaxis three times larger, than recently reported for the related compound
UCo2Zn20, so that χ and γ scale inversely with the characteristic energy for spin fluctuations,
Tsf = Emax/kB. While χ(T), Cmag(T), and Emax of the 4f compound YbFe2Zn20 are very well
described by the Kondo impurity model, we show that the model works poorly for URu2Zn20
and UCo2Zn20, suggesting that the scaling behavior of the actinide compounds arises from spin
fluctuations of itinerant 5f electrons.
An important property of heavy fermion (HF) materi-
als is a scaling law whereby the low temperature magnetic
susceptibility χ and specific heat coefficient γ = C/T
vary as 1/Tsf. Here kBTsf is the spin fluctuation en-
ergy scale which can be directly observed as the maxi-
mum Emaxin the dynamic susceptibility χ′′(∆E), mea-
sured through inelastic neutron scattering. Such scal-
ing receives theoretical justification1–4from the Anderson
impurity model (AIM), where the spin fluctuation tem-
perature Tsf is identified as the Kondo temperature TK.
This model assumes that fluctuations in local moments
dominate the low temperature ground state properties
of HF materials. For 4f electron rare earth HF com-
pounds, the AIM appears to give an excellent description
of much of the experimental behavior, including the tem-
perature dependence of the magnetic contribution to the
specific heat Cmag, the susceptibility χ, and the 4f oc-
cupation number nf, as well as the energy dependence of
the inelastic neutron scattering (INS) spectra χ′′(∆E) of
polycrystalline samples5. The theoretical calculations1–4
show that these properties are highly dependent on the
orbital degeneracy NJ(= 2J +1 for rare earths). In par-
ticular, for large degeneracy (NJ> 2) both the calculated
γ(T) and χ(T) exhibit maxima at a temperature αTK
where α is a constant that depends on NJ. This kind
of behavior is observed in rare earth compounds such as
YbAgCu45, CeIn3−xSnx6, YbCuAl7, and YbFe2Zn208.
It is reasonable to apply the AIM, which assumes lo-
cal moments, to rare earth compounds where the 4f or-
bitals are highly localized and hybridize only weakly with
the conduction electrons. On the other hand, in ura-
nium compounds, the 5f orbitals are spatially extended
and form dispersive bands through strong hybridization
with the neighboring s, p, and d orbitals. Photoemission
spectroscopy in 4f electron systems shows clear signals
from local moment states at energies well below the Fermi
level; the weak hybridization between the f electron and
the conduction electron leads to emission near the Fermi
energy ǫF that can be described in the context of the
Anderson impurity model as a Kondo resonance9.
5f electron systems, no local states are seen, but rather
broad 5f band emission is observed near ǫF. The Ander-
son lattice model is sometimes employed to understand
the f-derived band in actinide systems10while in some
systems itinerant-electron band models are employed11.
Hence, despite the common occurrence of scaling, we
might expect differences between the uranium and the
rare-earth based heavy fermion materials in the details
of the thermodynamics and the spin fluctuations. Never-
theless, we have recently shown12that the actinide com-
pound UCo2Zn20exhibits a maximum in the susceptibil-
ity and a specific heat coefficient that are strikingly simi-
lar to those seen in the rare earth compound YbFe2Zn20.
It is thus of interest to test whether a local moment
AIM/Kondo description, which has been shown to give
excellent agreement with the data for the Yb compound
(see Ref. 8 and also Fig. 3 of this paper), may also be
valid for 5f HF compounds.
To accomplish this, we present herein the results of INS
experiments on polycrystalline URu2Zn20together with
results for the magnetic susceptibility and specific heat of
single crystalline samples. We also present the INS data
on single crystal YbFe2Zn20. Both compounds belong
to a new family of intermetallic compounds RX2Zn20(R
= lanthanide, Th, U; X = transition metal)8,13–16which
crystallize in the cubic CeCr2Al20type structure (Fd3m
space group)14,17. In this structure, every f-atom is sur-
rounded by 16 zinc atoms in a nearly spherical array of
cubic site symmetry, which leads to small crystal field
splittings. Because the R-atom content is less than 5%
of the total number of atoms, and the shortest f/f spac-
ing is ∼ 6 Å, these compounds are possible candidates
for studying the Anderson impurity model in periodic f
The crystals were grown in zinc flux8,12. The mag-
netic susceptibility measurements were performed in a
commercial superconducting quantum interference de-
vice (SQUID) magnetometer. The specific heat experi-
ments were performed in a commercial measurement sys-
tem that utilizes a relaxational (time constant) method.
For URu2Zn20, we performed inelastic neutron scatter-
ing on a 40 gram powder sample on the low resolu-
tion medium energy chopper spectrometer (LRMECS)
at IPNS, Argonne National Laboratory, on the High-
Resolution Chopper Spectrometer (Pharos) at the Lujan
center, LANSCE, at Los Alamos National Laboratory,
and on the time-of flight Disk Chopper Spectrometer
(DCS) at the NIST Center for Neutron Research. For
YbFe2Zn20 the INS spectrum was obtained for two co-
aligned crystals of total mass 8.5 grams, using the HB-3
triple-axis spectrometer at the High Flux Isotope Reactor
(HFIR) at Oak Ridge National Laboratory (ORNL); the
final energy was fixed at Ef= 14.7 meV, and the scatter-
ing plane was (H,H,L). The data have been corrected
for scattering from the empty holder but have not been
normalized for absolute cross section. For the Pharos
and LRMECS measurements of URu2Zn20, we used the
non-magnetic counterpart compound ThCo2Zn20to de-
termine the scaling of the nonmagnetic scattering be-
tween low Q and high Q; for YbFe2Zn20, we measured
at Q = (1.5,1.5,1.5) and (4.5,4.5,4.5) and assumed that
the phonon scattering scales as Q2dependence18. As-
suming that the magnetic scattering scales with the Q-
dependence of the 4f or 5f form factor, we subtracted
the nonmagnetic component to obtain the magnetic scat-
tering function Smag(∆E)5,19,20.
III. RESULTS AND DISCUSSION
The magnetic susceptibility χ(T) and the specific heat
C/T of URu2Zn20are displayed in Fig. 1 and compared
to the data for UCo2Zn20. Fits of the data to a Curie-
Weiss law (Fig. 1(a)) at high temperature give the ef-
fective moments µeff= 3.61 µBfor URu2Zn20and 3.44
µB for UCo2Zn20. The Curie-Weiss temperatures are θ
The lines are Curie-Weiss fits.
T of URu2Zn20. Insets: Susceptibility and specific heat of
UCo2Zn20; the data are from Bauer et al12.
(a) Magnetic susceptibility χ(T) for URu2Zn20.
(b) Specific heat C/T vs
= -145 K and -65 K for the Ru and Co cases, respec-
tively. For URu2Zn20, the magnetic susceptibility χ(T)
increases monotonically as the temperature decreases to
the value χ(2K) ≃ 0.0111 emu/mole. At 2 K, the suscep-
tibility of UCo2Zn20is about 0.0372 emu/mole, which is
3.3 times larger than for the Ru case. The specific heat
is plotted as C/T vs T in Fig. 1 (b). For URu2Zn20
C/T has the magnitude γ ≃ 190 mJ/mole-K2at 2 K. At
low temperature C/T follows the T2behavior expected
for a phonon contribution, which permits the extrapo-
lation of the Sommerfeld coefficient to the value γ ≃
188 mJ/mole-K2. From the inset to Fig. 1(b), it can
be seen that for UCo2Zn20, γ(2K) is approximately 500
mJ/mole-K2, while at Tmax= 4.1 K, γ = 558 mJ/mole-
K2; these values are 2.6 and 2.9 times larger than for
As mentioned above, the characteristic energy for spin
fluctuations can be determined from the inelastic neu-
tron scattering experiments. In Fig. 2 we plot the Q-
averaged dynamic susceptibility χ′′(∆E) of URu2Zn20
as a function of energy transfer ∆E.
mined from the scattering function through the formula
Smag= A(n(∆E)+1)f2(Q)χ′′(∆E), where (n(∆E)+1)
is the Bose factor and f2(Q) is the U 5f form fac-
tor.Both the Pharos data and the LRMECS data
for χ′′(∆E) for URu2Zn20 exhibit broad peaks with
peak position Emax at an energy transfer ∆E ≃ of or-
This is deter-
FIG. 2: Low temperature dynamic susceptibility χ′′vs ∆E of
URu2Zn20. Error bars in all figures represent ±σ.(a) Pharos
data at T=7 K (Ei = 35 meV). (b) LRMECS data at T=10
K (Ei = 60 meV). The lines represent Lorentzian fits with
E0=13.5 meV± 1.9 meV and Γ= 9.5 meV ± 0.6 meV. In-
set: low temperature dynamic susceptibility of UCo2Zn20; the
data are from Bauer et al12. The line is a fit to a Lorentzian
with E0=3 meV ± 1.2 meV and Γ= 5 meV ± 0.4 meV. The
arrows indicate the peak positions predicted by the AIM for
NJ = 10 (See Table I).
der 16 meV. The dynamic susceptibility χ′′(∆E) can
be fit by a Lorentzian power function as χ′′(∆E)=
χ′(0)∆E(Γ/π)/[(∆E − E0)2+ Γ2] with the parameters
E0 = 13.5 meV and Γ= 9.5 meV, giving Emax = 16.5
meV. As shown in the inset to Fig. 2, for UCo2Zn20,
χ′′(∆E) shows a peak centered near Emax= 6 meV. Fits
of this data to an inelastic Lorentzian give E0= 3 meV
with Γ = 5 meV, for which Emax = 5.8 meV. We note
that these values of Emaxare nearly equal to the values of
kBθ derived from the high temperature susceptibility; i.e.
the temperature scale for the suppression of the moment
is identical to the energy scale of the spin fluctuation.
Given that γ(2K)Co/γ(2K)Ru = 2.6 (alternatively
γ(Tmax)Co/γ(2K)Ru= 2.9), that χ(2K)Co/χ(2K)Ru=
3.3, and that Emax(Ru)/Emax(Co) = 2.8, we see that
at low temperature these compounds exhibit a factor-of-
three scaling of χ, γ, and Emaxto an accuracy of about
We next examine whether such scaling arises due to
FIG. 3: (a) Specific heat Cmag(T) (from Torikachvili et al8)
and (b) magnetic susceptibility χmag(T) for YbFe2Zn20. (c)
The dynamic susceptibility χ′′(∆E)/χ′′(Emax) determined at
the (3/2, 3/2, 3/2) zone boundary point. The lines are fits,
for the J = 7/2 case, to Rajan’s predictions for Cmag and
χmag and to Cox’s predictions for χ′′(∆E)/χ′′(Emax). In all
three cases, there is only one common adjustable parameter
T0, set to the value 69 K to give the best agreement with
the applicability of the AIM to these actinide compounds.
Before doing so, we first check the validity of the AIM
for the rare earth 4f compound YbFe2Zn20. We apply
Rajan’s Coqblin-Schrieffer model3, which is essentially
the AIM in the Kondo limit (nf ≃ 1) for large orbital
degeneracy. In Fig. 3, we compare the data for Cmag(T)
and χmag(T) (where the data for LuFe2Zn20 has been
subtracted to determine the magnetic contribution) to
Rajan’s predictions for the J=7/2 case3. In these fits,
the only adjustable parameter is a scaling parameter T0;
we find that the value 69 K gives the best agreement with
To fit to the dynamic susceptibility χ′′(∆E) we use the
results of Cox et al4, obtained using the noncrossing ap-
proximation (NCA) to the AIM. This calculation, which
was performed for the J = 5/2 case, gives the peak po-
sition of the dynamic susceptibility at low temperature
as Emax= 1.36 kBTCox
(see Fig. 5 in Cox et al4). The
scaling temperature TCox
is related to Rajan’s scaling
temperature T0via TCox
= T0/ 1.15 (see the caption of
Fig. 2 in Cox et al4). Hence for the J = 5/2 case, we
have Emax= 1.18 kBT0. In the absence of comparable
theoretical results for other values of J, we will assume
that this relationship between T0and Emax is approxi-
mately true for the J = 7/2 and 9/2 cases; the error is
probably of order 20%. For YbFe2Zn20, we then expect
Emax= 7 meV. The lineshape for χ′′(∆E)/χ′′(Emax) was
determined from Fig. 4 in Cox et al4using this value for
Emax. Albeit we have only determined χ′′(∆E) at one
location in the zone, it is clear from these plots that the
NJ = 8 AIM in the Kondo limit does an excellent job
of fitting the susceptibility χ(T), magnetic specific heats
Cmag, and characteristic energy Emaxof this rare earth
Turning now to the actinide compounds, we note that
Rajan’s calculations3for a 2J+1 Kondo impurity give
the following zero-temperature limits for the specific
heat, and magnetic susceptibility:
χ0= (2J + 1)CJ/2πT0
where R is the gas constant and CJ is the Curie con-
stant. To test these scaling laws, we first note that ura-
nium has a possible 5f3state for which J = 9/2 and
µeff=3.62µB(CJ= 1.64 emu K/mole) or a possible 5f2
state for which J = 4 and µeff=3.58µB(CJ= 1.60 emu
K/mole). The high temperature Curie-Weiss fit of χ(T)
for URu2Zn20gives an experimental value for the Curie
constant close to these free ion values. In what follows,
we choose J = 9/2, but we note that the analysis is not
significantly different for the J = 4 case. We estimate
T0 from the low temperature value for γ, and then de-
termine χ0. To estimate Emaxwe use the above-stated
rule Emax= 1.18 T0, which as mentioned we expect to
be correct here to 20%. The calculated results are listed
in Table I, along with the similar results for J = 5/2 and
J = 1/2.
From Table I, we can see that the expected values for
χ0and Emaxare closer to the experimental values for the
J = 9/2 case than for either the J = 5/2 or 1/2 cases. In
Fig. 4 we compare the experimental data to the predic-
tions (solid lines) for the temperature dependence of χ(T)
and Cmag(where the data for the corresponding Th com-
pound have been subtracted to determine the magnetic
contribution22) in the J = 9/2 case. For the energy de-
pendence of χ′′(∆E)/χ′′(Emax) at low temperature, we
utilize the results of Cox et al4, as outlined above. Again,
there is only one adjustable parameter, T0, which is deter-
mined from the low temperature specific heat coefficient
as equal to 208 K for the Ru case and 70 K for the Co
case. The fitting is very poor in several respects. First,
the expected values of Tmaxfor both χ(T) and Cmag(T)
are much higher than observed in the experiment, and
indeed for URu2Zn20 there is even no maximum in the
experimental curve for χ(T). Even more significant is
the fact that the experimental entropy developed below
20 K is much smaller than expected. Indeed the exper-
imental entropy at 20 K is less than Rln2, which would
be expected for a two-fold degeneracy (J=1/2). How-
ever, if we attempt to fit the data assuming J=1/2, we
find that very small values of T0are required to reproduce
the specific heat coefficients, and hence the characteristic
energy Emax would disagree markedly with the experi-
mental value (see Table I). Hence there appears to be a
very serious discrepancy between the data and the Kondo
In our previous paper12, we attempted to compare the
data for UCo2Zn20 to the predictions of the AIM cal-
culated using the NCA. The calculation assumed mixed
valence between the J = 4 and 9/2 states, and assumed
that a large crystal field splitting (∼ 200 meV) resulted
in a six-fold degeneracy (effective J = 5/2 behavior) at
low temperature. To confirm whether such a crystal field
excitation is present in these compounds, we measured
URu2Zn20 and ThCo2Zn20 on Pharos using large inci-
dent energies. In Fig. 5(a) we show the INS spectra for
energy transfers up to ∆E = 550 meV. The results ex-
hibit no sign of crystal field excitations. We believe that
a similar result will be valid for UCo2Zn20. Furthermore,
it is clear from Table I that such an effective J = 5/2 ap-
proach will overestimate TC
badly overestimate the entropy so that the use of the
AIM to describe this compound is problematic.
Hence, while the J = 7/2 AIM works extremely well8
for the susceptibility and specific heat and also repro-
duces the characteristic energy Emaxof the neutron spec-
trum of YbFe2Zn20, for these actinide compounds, the J
= 9/2 (or J = 4) AIM works well only for the low temper-
ature scaling, but very poorly for the overall temperature
dependence of χ(T) and C(T); in particular the theory
badly overestimates the entropy. For calculations based
on smaller values of NJ, the characteristic energy Emax
is badly underestimated by the theory. These results sug-
gest that the physics responsible for the low temperature
heavy mass behavior in these actinide compounds is not
that of local moments subject to the Kondo effect, as
for the 4f electron compounds, but is that of itinerant
5f electrons subject to correlation enhancement. In sup-
port of this, we note that when uranium compounds such
as UPd3 exhibit local moments, then intermultiplet ex-
citations can be observed at energies near 400 mev; no
such excitation is seen for metallic compounds such as
UPt323. The lack of such excitations in the Pharos data
(Fig. 5(a)) for URu2Zn20gives further evidence that the
5f electrons are itinerant, not localized, in these com-
Since the peaks observed in Cmag(T) for both the Ru
and Co cases and in χ(T) for the Co case occur at a
much lower temperature than the characteristic temper-
ature Emax/kB, they are very probably associated with
low temperature magnetic correlations, which exist only
in the vicinity of some critical wavevector QN, and which
yield only a fraction of Rln2 for the entropy. In this re-
gard, the behavior is similar to that of UBe13 or UPt3,
where Q-dependent antiferromagnetic fluctuations occur
on a much smaller energy scale (∼ 1 meV for UBe13and
0.2 meV for UPt3) than the scale of the Kondo-like24fluc-
max, underestimate Emaxand
FIG. 4: (a) Magnetic susceptibility χ(T), (b) magnetic specific heat Cmag(T), and (c) entropy Smag(T) for URu2Zn20; the
insets show the same quantities for UCo2Zn2012. The lines are fits using Rajan’s predictions for J=9/2. The open symbols in
(b) and (c) represent data corrected for the energy of the Einstein mode at 8 (7) meV in the U (Th) compounds22. (d): The
dynamic susceptibility χ′′(∆E)/χ′′(Emax) of URu2Zn20; the inset shows the data for UCo2Zn20. The lines are obtained using
Cox’s results, as explained in the text.
TABLE I: Experimental and theoretical values of key quantities for URu2Zn20 and UCo2Zn20. The values for the scaling
temperature T0 are obtained using γ2K = 188 mJ/mol-K2for URu2Zn20 and γmax = 558 mJ/mol-K2for UCo2Zn20. For
J=9/2 and 5/2, the Curie constant used in the calculation is the 5f3free ion value while for J=1/2, CJ is obtained from the
Curie-Weiss fit to the low temperature magnetic susceptibility.
tuations (13 meV for UBe13 and 6 meV for UPt325,26).
Such antiferromagnetic fluctuations are large only in the
vicinity of the wavevector QN and contain only a small
fraction of the spectral weight compared to the Kondo-
like fluctuations. Hence, it is not surprising that the poly-
crystalline averaged INS spectra in Fig. 5(b) shows no
obvious excitation in the energy transfer range 0.1 meV
to 4 meV; careful measurements on single crystals are
required to reveal such low energy, low-spectral-weight
Q-dependent magnetic fluctuations.
Given these considerations, we believe the character-
istic INS energies of 5.8 and 16.5 meV that we have
observed in UCo2Zn20and URu2Zn20represent Kondo-
like24fluctuations as observed in many uranium com-
pounds such as UBe1325, UPt326and USn327. The small
magnetic entropy remains a difficulty, however, even for
this case.To see this, consider the scaling product
γEmax/kB, which represents how the T-linear entropy
is generated by the damped spin excitation centered at
Emax. For a Kondo ion, this product takes the value
πJR/3. A crude approximation would be γEmax/kB =
2Rln(2J + 1), which might be expected to be valid even
for spin fluctuations arising in an itinerant electron sys-
tem; this approximation gives a similar value (∼ 39) for
the J = 9/2 case. The measured values for UCo2Zn20
and URu2Zn20are in the range 33-37, very close to the
FIG. 5: (a) The INS spectra of URu2Zn20 and ThCo2Zn20
taken on Pharos with incident energies Ei = 400 meV and
700 meV. The diamond is the estimated magnetic scattering
χ′′, obtained as described in the text. (b) The INS spectra of
URu2Zn20 in the energy range 0.1-4 meV taken on DCS with
incident energies Ei = 2.3 meV and 7.1 meV. The near equal-
ity of the high-Q and low-Q scattering suggests that all the
scattering observed in this energy range is due to background.
expected J = 9/2 value. Fig. 4 indicates, however, that
the compounds generate entropy in a manner that sat-
isfies this formula only at the lowest temperatures, but
then saturate above 10K. The point is that if the scal-
ing product has the right value, then the Rln(2J + 1)
entropy should continue to be generated up to tempera-
tures of order Emax/kB, much larger than 10 K for these
compounds. We emphasize that this should be true even
for itinerant 5f electrons.
The static and dynamic magnetic susceptibility and
the specific heat of URu2Zn20and YbFe2Zn20compounds
have been presented. The results show that the AIM
works very well to describe the magnetic susceptibility,
specific heat and dynamic susceptibility well of the com-
pound YbFe2Zn20 where the 4f electrons are localized.
In the actinide compounds URu2Zn20(UCo2Zn20), how-
ever, the fits to the AIM temperature dependence are
very poor even though the low temperature scaling be-
havior expected for a J = 9/2 Kondo impurity was ob-
served. An associated problem is that the magnetic en-
tropy generated by 20 K is too small compared to the
expected value. These results suggest that the spin fluc-
tuations in these actinide compounds arise from itiner-
ant rather than localized 5f electrons. Antiferromagnetic
fluctuations may affect the specific heat. While our neu-
tron scattering results for a polycrystalline sample saw
no signs of these fluctuations in the 0.1 to 4 meV range,
they may be observable as a small spectral weight signal
in single crystal experiments.
Research at UC Irvine was supported by the U.S. De-
partment of Energy, Office of Basic Energy Sciences,
Division of Materials Sciences and Engineering under
Award DE-FG02-03ER46036. Work at ORNL was sup-
ported by the Scientific User Facilities Division Office
of Basic Energy Sciences (BES), DOE. Work at ANL
was supported by DOE-BES under contract DE-AC02-
06CH11357.Work at the Ames Laboratory was sup-
ported by the DOE-BES under Contract No. DE-AC02-
07CH11358. Work at Los Alamos, including work per-
formed at the Los Alamos Neutron Science Center, was
also supported by the DOE-BES. Work at NIST utilized
facilities supported in part by the National Science Foun-
dation under Agreement NO. DMR-0454672.
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