Heavy-fermion formation in USn3: Static and dynamical properties

S. Kambe,1 H. Sakai,1 Y. Tokunaga,1 T. D. Matsuda,1 Y. Haga,1 H. Chudo,1 and R. E. Walstedt21Advanced Science Research Center, Japan Atomic Energy Agency, Tokai-mura, Ibaraki 319-1195, Japan

2Physics Department, University of Michigan, Ann Arbor, Michigan 48109, USA�Received 13 December 2007; revised manuscript received 14 February 2008; published 7 April 2008�

USn3 is a heavy-fermion system with an electronic specific heat coefficient ��170 mJ /K2 mol. In order tofurther characterize the heavy-fermion phenomena for USn3, the Knight shift and spin-lattice relaxation timeT1 of 119Sn NMR have been measured. The static �specific heat� and dynamical �T1� properties in the heavy-fermion state can be described in a quantitatively consistent way in terms of a spin-fluctuation model with twoconstant energy scales. However, it is necessary to introduce a T-dependent “effective” Ruderman–Kittel–Kasuya–Yosida interaction JQ �JQ�a+b�T� in order to describe the crossover from an incoherent, localizedstate to a coherent, heavy-fermion state. In addition, a universal scaling behavior is proposed for the crossoverregime. The parameters obtained are used to predict the T dependence of the thermal expansion coefficient.

DOI: 10.1103/PhysRevB.77.134418 PACS number�s�: 76.60.�k, 75.30.Mb

I. INTRODUCTION

In f-electron itinerant systems, heavy-fermion �HF� stateshave been observed at low temperatures in certaincompounds.1 In such HF systems, the f-electron momentsare localized above the effective Fermi temperature T*. AtT�T*, a crossover from the incoherent localized state to thecoherent HF state takes place, driven by the Kondo interac-tion between the localized f moments and the conductionelectrons, which usually predominates over the Ruderman–Kittel–Kasuya–Yosida �RKKY� interaction between thef-electron moments. In the HF state, the static magnetic sus-ceptibility ��0,0� and the �Sommerfeld� electronic specificheat coefficient �=Cel /T become quite large compared withordinary metals.

The T dependence of the dynamical susceptibilityIm ��q ,�� also clearly reflects the formation of a HF state.For example, in CeRu2Si2,2 the spin-lattice relaxation rate1 /T1T� Im ��q ,�� increases with decreasing T and becomesconstant below T�T*. In ordinary metals, 1 /T1T� constantbehavior is observed for a wide temperature range �Korringabehavior�. In contrast, in HF systems, such behavior is ob-served when an HF state �i.e., Fermi liquid� is formed below�T*. At temperatures well above T*, T1 becomes constant,which is characteristic behavior for systems with localizedmoments.

Up to now, the static �uniform susceptibility, specific heat,etc.� and dynamical �spin-lattice relaxation time, inelasticneutron scattering, etc.� properties have been discussed sepa-rately in HF systems. In this study, we have tried to repro-duce the T dependence of the specific heat and spin-latticerelaxation time of the HF system USn3 in a quantitativelyconsistent way using the spin-fluctuation model for itinerantmagnets by Moriya.3 This approach gives a good account ofboth quantities for temperatures below �T*. However, T de-pendence of the “effective” RKKY interaction has to be in-troduced in order to describe the crossover regime.

In this paper, data for the T dependences of the Knightshift and spin-lattice relaxation time at the Sn site in USn3are presented. Among UX3 compounds with the same AuCu3�fcc� structure, where X is a IVB element �X :Si ,Ge,Sn,Pb�,USn3 shows the largest4,5 ��170 mJ /K2 mol and ��0,0�=9.0�10−3 emu /mol at T�0 K. These parameters give a

Wilson ratio of �2, indicating that USn3 is a typical heavy-fermion compound.

II. EXPERIMENT

A powder USn3 sample was prepared for NMR measure-ments by crushing a well-characterized single crystal.6 Theresistivity of the sample showed good metallic behavior witha residual resistivity of �1.7 �� cm. 119Sn �I=1 /2� NMRmeasurements were performed using a conventional pulsedspectrometer with a 12 T superconducting magnet. The /2− pulse sequence has been used to excite nuclear spin-echo signals. Field-sweep NMR spectra were taken at a fre-quency of 119 MHz using digital averaging of the nuclearspin-echo signals. Spin-lattice relaxation time �T1� data werealso obtained with this method.

III. EXPERIMENTAL RESULTS

A. Static susceptibility

Figure 1 shows the cubic �AuCu3-type� crystal structureof USn3. The U site has cubic local symmetry, whereas forthe Sn site, it is tetragonal. The local principal axis for the Sn

�

��

n

FIG. 1. �Color online� Crystal �fcc� structure of USn3. n� indi-cates the local symmetry axis of the Sn site.

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site �indicated as n� in Fig. 1� is important for analyzing theNMR results.

Figure 2 shows the T dependence of the static susceptibil-ity ��0,0� of the NMR sample. At high temperatures, ��0,0�shows Curie–Weiss �CW� behavior �see inset�. Below 30 K,��0,0� starts to saturate and finally becomes independent ofT below 6 K. The latter behavior signals the formation of aHF state. Thus, we consider that T*�30 K in USn3. Thedata for ��0,0� have been least-squares fitted to the CW

function ��0,0�=�const�0,0�+�ef f

2

3kB�T−� above 50 K, yielding�const�0,0�=3.0�10−5 emu /mol, an effective moment �ef f=2.44�B, and a Weiss temperature =−58 K. The constantterm �const�0,0� usually represents the Van Vleck orbital anddiamagnetic susceptibilities. The value obtained for�const�0,0� is quite small compared with ��0,0�, indicatingthat the dominant term in USn3 is the spin susceptibility�spin�0,0�. In the inset to Fig. 2, the T dependence of1 / ���0,0�−�const�0,0�� is presented. The straight line ob-tained above 50 K confirms an ideal CW behavior at hightemperatures. The effective moment �eff=2.44�B is rathersmaller than the U3+ ionic value 3.87�B, indicating that the5f electrons have an itinerant nature below 300 K.

In band calculations, the noninteractive Pauli spin suscep-tibility is estimated as 6.6�10−4 emu /mol from the densityof states at the Fermi level,7 yielding a 14� enhancement of��0,0� in the HF state.

B. NMR spectrum and Knight shift

Figure 3 shows a field sweep spectrum at an NMR fre-quency of 119 MHz for the 119Sn at 1.6 K. This is a typicalpowder-pattern spectrum for I=1 /2 in a site of tetragonal�uniaxial� symmetry. The Knight shifts K� for applied mag-netic field H �n� and K� for H� to the principal axis n� of theSn site are determined based on fitting procedures usingan axially symmetric powder pattern with Gaussianbroadening.8 From K� and K�, the isotropic and anisotropiccomponents, Kiso�K� +2K�� /3 and Kani�K� −K�� /3,have been estimated.

Figure 4 shows the T dependence of Kiso and Kani. SinceKiso�Kani, the overall shift is basically isotropic in this com-pound. Kiso and Kani are plotted vs ��0,0� �the so-called K-�plots� in Fig. 5, showing good linearity in both cases. Hyper-fine coupling constants A�0�iso=68 kOe /�B for Kiso andA�0�ani=8.8 kOe /�B for Kani are determined from the slopesof linear fits to these data �Table I�. Since hyperfine cou-plings of this magnitude cannot be explained by classicaldipolar-dipolar interactions, transferred hyperfine fields dueto hybridization between U 5f and Sn 5s, p orbitals are con-sidered to be the underlying mechanism. Since Kiso is mainlydriven by spin polarization transferred to the Sn 5s orbital,the point on the plot where Kiso=0 corresponds to where thestatic spin susceptibility �spin�0,0�=0. As shown in Fig. 5,the extrapolation of the Kiso-��0,0� plot very nearly inter-

12x10-3

10

8

6

4

2

0

χ(0,

0)

3002001000T (K)

USn3

500

400

300

200

100

01/(χ

(0,0

)−χ c

onst(0

,0))

3002001000T (K)

FIG. 2. �Color online� T dependence of the static susceptibility��0,0�. In the inset, a Curie–Weiss plot of 1 / ���0,0�−�const�0,0��vs T is presented.

FIG. 3. �Color online� 119Sn field sweep NMR spectrum at119 MHz in a powder sample of USn3 �T=1.6 K�. The edge posi-tions �indicated by arrows� correspond to shift parameters K� �H �n��and K� �H�n��. Solid line is a fitted curve based on the axiallysymmetric powder pattern with Gaussian broadening.

FIG. 4. �Color online� T dependence of the isotropic Kiso

and anisotropic Kani Knight shifts �Kiso�K� +2K�� /3 and Kani

�K� −K�� /3�.

χ(0,0)

FIG. 5. �Color online� Knight shift versus static susceptibility��0,0� plot �K-� plot�. The solid lines were obtained with least-squares fits. The slopes of the lines correspond to the hyperfinecoupling constants A�0�iso and A�0�ani.

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sects the origin, indicating that ��0,0��spin�0,0� in USn3.This is consistent with the small nonspin susceptibility �constwhich was estimated from the static susceptibility measure-ments. The origin of Kani is apparently hybridization with theSn 5p orbital, which gives an anisotropic, dipolar hyperfinefield. The large Kiso compared with Kani indicates that thehybridization between the U 5f and Sn 5s orbitals is compa-rable with that between the U 5f and Sn 5p orbitals since the5s hyperfine field is expected to be much larger than that ofthe 5p electrons.

Since the K-� plot is linear down to 1.6 K in the HF state,the hyperfine coupling constant at the Sn site is not modifiedby the HF effects in this compound. In some HF compoundsthe K-� plot becomes nonlinear due to T-varying hyperfineinteraction contributions.9 In contrast, in USn3 there is nosuch complication from T-dependent couplings; thus, a quan-titative analysis is possible in detail as described below.

C. Spin-lattice relaxation time T1

Figure 6 shows the T dependence of �1 /T1�� at the 119Snsite for H�n� measured at the K� position in Fig. 3.

At high temperatures, the T dependence of T1 becomesweak. This regime corresponds to a crossover from the HF toa localized momentlike state. In the localized moment state,T1 is independent of T and is expressed as the exchange-narrowing limit case,10

1/T1ex = �2�A�0�/��2J�J + 1�3n�ex

. �1�

Here, the effective total angular momentum J and ex-change frequency �ex are estimated to be 0.82 and3.1�1012 s−1, respectively, from the CW parameters��ef f =2.44�B and =−58 K�, and n is the number of neigh-boring magnetic sites around Sn, i.e., n=4 in the presentcase. These parameters lead to 1 /T1ex�1.9�105 s−1, whichis much larger than the observed value at 240 K. Even at

high temperatures, the 5f electrons still have some itinerantcharacter in USn3.

At low temperatures, �1 /T1�� is proportional to T below�10 K, i.e., Korringa behavior appears in the HF state. Foritinerant systems with magnetic exchange enhancements,the usual Korringa relation between T1, K and S��e /�N�2�h /4kB�: T1TK2=S is modified with K� � term,leading to an extended Korringa relation for ligand sites,11

T1TK2 = nSK� �−1. �2�

The estimated K� ��0.6 is smaller than 1 in USn3, suggest-ing that ferromagnetic exchange enhancement seems to bepresent in this compound. However, an alternative explana-tion is the cancellation of antiferromagnetic fluctuations atthe 119Sn, as discussed below. The present case indicates thatwe should be careful in applying the extended Korringa re-lation to ligand-site NMR results when the hyperfine formfactor is a critical element. In fact, in the paramagnetic stateof the isostructural antiferromagnet UIn3 �TN=88 K�, K� � isfound to be less than 1 owing to a similar cancellation.12

In order to estimate the anisotropy of hyperfine fluctua-tions, �1 /T1�� for H �n� has also been measured at 20 and120 K �not shown�. Generally, �1 /T1��,� at tetragonal sites isexpressed as

�1/T1�� =2�N

2 kBT

�B2 �

q

A�q��2 Im ��q,�n��

�n

2�N

2 kB

�B2 �

q

f�q�2A�0��2 Im ��q,�n��

�n,

�1/T1�� =�N

2 kBT

�B2 �

q�A�q��

2 Im ��q,�n��

�n

+ A�q��2 Im ��q,�n��

�n

�N

2 kB

�B2 �

q� f�q�2A�0��

2 Im ��q,�n��

�n

+ f�q�2A�0��2 Im ��q,�n��

�n , �3�

where �N is the nuclear gyromagnetic ratio, A�q� is the hy-perfine coupling constant, f�q� is the hyperfine form factorwhich reflects the local symmetry of Sn site, Im ��q ,�n� isthe dynamical susceptibility, and �n=119 MHz is the NMRfrequency. The off-diagonal hyperfine coupling13 is ignoredhere since the hyperfine coupling is very nearly isotropic atthe Sn site. Thus, the hyperfine form factor f�q� is the samefor H �n� and H�n� .

Based on Eq. �3�,

TABLE I. Transferred hyperfine coupling constants A�0� inkOe /�B for the 119Sn in USn3 obtained from K-� plots.

A�0�� A�0�� A�0�iso A�0�ani

86�2 59�2 68�2 8.8�0.5

FIG. 6. �Color online� T dependence of 1 /T1 at the 119Sn site forH�n� .

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2�1/T1�� − �1/T1��

�1/T1��

A�0��2�

q

f�q�2 Im ��q,�n��

A�0��2 �

q

f�q�2 Im ��q,�n��

. �4�

The experimental value of2�1/T1��−�1/T1��

�1/T1���2�0.2 at 20 and

120 K then arises from the anisotropy of the transferred hy-perfine coupling constant A�0��

2 /A�0��2 �2.1 �see Table I�.

From Eq. �4�, this fact indicates that the dynamicalsusceptibility Im ��q ,�� is isotropic i.e., Im��q ,�n��

=Im ��q ,�n��, in agreement with the local cubic symmetryof the U site, which makes the main contribution to the mag-netic susceptibilities i.e., ��0,0� and Im ��q ,��.

IV. ANALYSIS AND DISCUSSION BASED ON THE SCRMODEL OF DYNAMIC MAGNETISM

A. Analysis based on the usual self-consistentrenormalization model

In this section, the electronic specific heat4 and the presentspin-lattice relaxation results are interpreted in a quantita-tively consistent way based on the framework of the self-consistent renormalization �SCR� model. Originally, the SCRmodel was developed by Moriya in order to interpret weakmagnetism in itinerant systems.3 More recently, this modelwas adapted to describe the HF state.14

In the SCR model, the dynamical susceptibility is charac-terized by two energy scales, T0 and TA, which correspond tomagnetic fluctuation energy in � and q spaces, respectively.The q dependence of the RKKY interaction JQ is expressedas JQ−JQ+q=2TA��q� / �qB � �2 around the antiferromagneticwave vector Q, where qB is the zone-boundary vector. Thus,the dispersion of the RKKY interaction can be defined as�JQJQ−J0=2TA. Since this JQ includes all q-dependenteffects such as s− f mixing in addition to the original RKKYinteraction,14 JQ represents effective RKKY interaction. Usu-ally, T0 has a magnitude of T* and is connected with the localmagnetic susceptibility. The framework of the SCR modelfor application to HF systems has been described in previousworks.15,16

Since no magnetic phase transition takes place in USn3,the nature of the magnetic correlations must be decided insome other way. In the present analysis, correlations havebeen presumed to be antiferromagnetic �see below�.

For the case of antiferromagnetic correlations, the generalsusceptibility ��Q+q ,�� has been formulated to be14

1

��Q + q,��=

1

��Q,0�+ B�q�2 −

i�

�L�L

= 2TA�y�T� + � �q��qB�

2

−i�

2T0� ,

y�T� 1

2TA��Q,0�,

TA B�qB�2

2,

T0 TA�L�L

=

TA�q�q

�Q � q � 0� ,

JQ =�L − �Q

�L�L, �5�

where B is the dispersion constant, �L and �q are the localand q-dependent susceptibilities, and �L and �q are the localand q-dependent relaxation rates, respectively. The T depen-dence of y�T� is determined based on the fluctuation-dissipation theorem in order to guarantee internal consis-tency on condition that the total spin fluctuation �S2� isconstant.3

In addition, two dimensionless parameters y0 and y1 areintroduced to characterize the state of the system,

y0 y�T = 0 K�, y1 2JQ

2TA=

4JQ

2�JQ. �6�

Here, y0 is a measure of the deviation from the quantumcritical point, since y0=0 corresponds to a quantum phasetransition at T=0 K. y1 reflects the strength of dispersion ofthe effective RKKY exchange interaction JQ.

In the usual SCR model, T0, TA, and JQ are assumed to beT independent. This can be justified below �T*, and thus,interpretation of the electronic specific heat and spin-latticerelaxation rate at temperatures below 30 K are consideredfirst.

Figure 7 shows the T-dependence of the electronic spe-cific heat Cel /T as estimated by Norman et al.4 It should benoted that specific heat results for our sample are in goodagreement with this previous result. Since Cel is obtained bysubtraction of the large phonon contribution from the totalspecific heat,4 there is a degree of uncertainty about the es-timated Cel above 10 K. The electronic specific heat Cel canbe expressed as a function of y�T� and T0.14 We have ob-tained y0�0.22, y1�1, T0�33 K �Table II�, and the T de-pendence of y�T� by fitting Cel /T �TA cannot be determinedfrom the specific heat14�. The deviation of Cel /T at high tem-peratures may be partly due to an overestimation of the pho-non contribution to the specific heat. Figure 8 shows the Tdependence of y�T�=1 / �2TA��Q ,0��. At high temperatures,y�T� exhibits CW behavior, i.e., ��Q ,0��1 /T. In contrast,

FIG. 7. �Color online� Closed circles are the T dependence ofthe electronic specific heat Cel /T from Ref. 4, which has been esti-mated by subtraction of the phonon part from the total specific heat.The solid line is Cel /T calculated using the spin fluctuation param-eters in Table II and the equation for Cel /T �Ref. 14�.

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y�T� is nearly constant below 10 K �inset of Fig. 8�.The values obtained for T0, y0, and y1 are similar to those

for other HF compounds with the same magnitude of ��i.e., for CeRu2Si2 �Ref. 15�: ��350 mJ /K2 mol, T0=14 K,TA=16 K, y0=0.31, and y1=1.6�. In the SCR model,14 therelaxation rate at q=Q is estimated as �Q�T��2T0y�T�.Actually, �Q�T� agrees with the T dependence of half widthof magnetic quasiparticle excitations spectra in USn3; ��T�determined in inelastic neutron scattering measurements17

�Fig. 9�. On the other hand, the local relaxation rate is esti-mated as �L�T��0.53T0y1+�Q�T��5.1�102 K+�Q�T��5.6�102 K at 0 K, which is considerably larger than�Q�0 K��46 K. Although such a high energy magnetic ex-citation has not been detected in the neutron scatteringmeasurements,17 this may be due to a weak and broad spec-trum for �L�T�.

It should be noted that T1 is found to be independent ofthe applied magnetic field H between 3 and 7 T. Althoughthe Zeeman splitting energy 2�BH is comparable with thecharacteristic energy of the spin fluctuations T0, the spin-lattice relaxation is insensitive to H in the present measure-ments.

Using the relation Sel�T�=�0TCel /TdT, the T dependence of

the electronic entropy Sel�T� can be estimated. For the HFstate of systems with the degeneracy of quasiparticlesN�2J+1=2 �i.e., J=1 /2 case�, Sel�T� becomes �R ln 2 atT�T*�T0. Actually, Sel�T� seems to reach R ln 2 around40 K�T0 in USn3 �not shown�. Recently, the Kadowaki–Woods plot for highly degenerate HF systems �N�2� wasfound to be different from the usual behavior of N�2systems.18 However, the point for USn3 is found on theKadowaki–Woods plot for the N�2 case.18 These facts in-dicate that the effect of degeneracy is not pronounced inUSn3 as well as other U-based HF compounds,18 althoughorbital degeneracy can affect physical properties because ofthe cubic symmetry. Since a N�2 case is assumed in thepresent SCR model,14 the consistency of the present analysisis confirmed.

From Eq. �5�, the dynamical susceptibilityIm ��Q+q ,�� can be expressed as

Im ��Q + q,�� =1

2TA

� �

2T0

�y�T� + � �q��qB�

2�2

+ � �

2T0 2

.

�7�

Noting the isotropic nature of Im ��Q+q ,��, �1 /T1T��

based on Eq. �3� then becomes

�1/T1T�� �N

2 kB

�B2 �A�0��

2 + A�0��2 �

��q

f�Q + q�2 Im ��Q + q,�n��n

. �8�

Figure 10 shows the q dependence of the hyperfine formfactor at the Sn site for an isotropic hyperfine field. Thehyperfine form factor is a function of the two �x ,y� axessince the Sn site is located in the U plane �Fig. 1�. As shownin Fig. 10, f�Q+q�2 for Q= � ,� has a minimum at q= �qx ,qy�= �0,0�, indicating that antiferromagnetic fluctua-tions are canceled at the Sn site.

In the present analysis, an antiferromagnetic correlationQ= � ,� is assumed in order to interpret the specific heatand T1 consistently. Figure 11 illustrates the q dependence ofIm ��Q+q ,�n� around Q= � ,� calculated from the givenparameter values at several temperatures. As T decreases,

χ

FIG. 8. �Color online� T dependence of calculated y�T� fory0�0.22, y1�1, and T0�33 K. The inset shows the same plot ona semilogarithmic scale.

Γ

ΓΓ

FIG. 9. �Color online� T dependence of the relaxation rate�Q�T��2T0y�T� estimated from y�T� in Fig. 8. For comparison,the relaxation rate ��T� determined in neutron inelastic scatteringmeasurements �Ref. 17� is presented.

q=(-π,π)

q=(π,π)

q=(-π,-π)

f(Q+q)2

q qx y

q=(q ,q )x y

FIG. 10. �Color online� q dependence of the hyperfine formfactor f�Q+q�2=sin�qx /2�2+sin�qy /2�2 at Sn sites in USn3 for thecase of Q= � ,�. The origin of the plot �q= �0,0�� corresponds tof�Q�2.

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enhancement of Im ��Q+q ,�n� develops around q= �0,0�,i.e., antiferromagnetic fluctuations develop at low tempera-tures. Although f�Q+q�2 has a minimum at q= �0,0�, thespin-lattice relaxation at the Sn site probes the T dependenceof Im ��Q+q ,�n� because there is some overlap betweenf�Q+q�2 and Im ��Q+q ,�n� �see Figs. 10 and 11�.

Based on Eq. �8� and the T dependence obtained forIm ��Q+q ,�n� from the specific heat result, the T depen-

dence of 1 /T1T has been calculated by q integration. SinceA�0� has already been obtained, TA is the only adjustableparameter for T1, which is estimated as �44 K. Fromthe SCR model,15 TA has a magnitude of1 / �2�1+y0���0,0�T=0 K��70 K, which is roughly consistentwith the present estimate.

If a ferromagnetic correlation �Q= �0,0�� were assumed,the estimated value of �1 /T1T�� would become much largerthan the observed value due to the hyperfine form factor, orTA would become unnaturally large in order to reproduce theobserved T1. This indicates that antiferromagnetic correla-tions �Q= � ,�� are the most likely internally consistentexplanation for the specific heat and spin-lattice relaxationrate in USn3. In fact, antiferromagnetic correlations are alsosuggested by inelastic neutron scattering measurements.17 Itshould be noted that only type II antiferromagnetic�Q= � , ,�� ordering, i.e., no ferromagnetic ordering, hasbeen found in UX3 compounds up to now.19

Figure 12 shows the measured T dependence of �1 /T1T��

compared with calculated values. In the HF state below10 K, �1 /T1T�� has a weak T dependence. In the crossoverregime above 10 K, �1 /T1T�� decreases more rapidly withincreasing T. Here, the T dependence of �1 /T1T�� is wellreproduced by the calculations below 30 K, i.e., �T0. Above30 K, the observed �1 /T1T�� decreases more rapidly thanthat predicted by the SCR model. This is not unexpectedsince the present SCR model cannot treat the crossover fromthe HF to the incoherent state. The T1 data show that withincreasing T, low energy fluctuations disappear abruptly, ow-ing to the collapse of the HF state.

B. Analysis of the crossover regime with T-dependentJQ and TA

We now argue that, in the crossover regime, the charac-teristic parameters for the random phase approximation

TABLE II. Obtained characteristic parameters for USn3 based on the SCR model.

T0TA�L�L

TA

BqB2

2y0

1

2TA��Q ,0�0Ky1

4JQ

2�JQ−J0�

33�2 K 44�2 K 0.22�0.02 1�0.1

���

���

���

���

q=(π,π)

q=(-π,π)

q=(-π,-π)

��χ�� ω������

������

q=(q ,q )x y

qx y

q

FIG. 11. �Color online� Calculated T and q dependences ofIm ��Q+q ,�n� for the parameters given in Table II. Since Q= � ,�, the origin of the plot q= �0,0� corresponds to Im ��Q ,�n�.

FIG. 12. �Color online� T dependence of �1 /T1T�� at 119Sn sitefor H�n� . The solid line shows calculated values of �1 /T1T�� basedon the Eq. �8� and the parameters in Table II. The dashed line showsthe calculated values of �1 /T1T�� with T-dependent JQ shown inFig. 13.

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�RPA�-form magnetic susceptibility �Eq. �5�� will be modi-fied due to the formation of HF quasiparticles. However, T0is considered to be constant, while JQ and TA=2JQ / �2y1�become T dependent.20 In addition, y1 is assumed to be Tindependent in the present analysis since the RKKY interac-tions have no strong T dependence in paramagnetic HF sys-tems. In order to reproduce the experimental results withreasonable modifications, one may simply introduceT-dependent JQ and TA on the condition of constant total spinfluctuations �S2�. The constant TA determined in the previoussection is noted as TA�0 K� in this section.

In Fig. 12, the T dependence of �1 /T1T�� is shown, wherean optimized fit �dotted line� has been obtained with theT-dependent JQ plotted in Fig. 13. The other parameters y0,y1, and T0 have been fixed at the values presented in Table II.The experimental results for �1 /T1T�� can be accurately re-produced in this fashion. It should be noted that the elec-tronic specific heat Cel is unaffected by this modificationsince y�T� and T0 are unchanged.

The T dependence of JQ is obtained by fitting the data for�1 /T1T��. Figure 13 shows that the JQ−�T plot showsa very nearly straight line above 25 K, indicating thatJQ�T��a+b�T in the crossover regime. As a result of com-petition between the RKKY interaction and the Kondo effectin the crossover regime, the effective RKKY interaction de-creases with decreasing T.

Such a T dependence may be connected with a similarcharacteristic �T law for the line width of the quasielasticneutron scattering ���q� observed in USn3 �Ref. 17� andheavy-fermion compounds21–24 since JQ= ��L−�Q� / ��L�L�here.14 Such behavior may be interpreted theoretically interms of crystal field effects in Kondo systems,25 although noclear crystal field excitation is detected in USn3.17 In anycase, the present analysis suggests that theoretical develop-ments which include a T-dependent effective RKKY interac-tion are necessary to describe the formation of the HF state.

In the present analysis, TA�T�=2JQ�T� / �2y1� increaseswith increasing T, while T0 remains constant. From Eq. �5�,this indicates that �q�q=

T0

TA�T� decreases with increasing T inthe crossover regime. In order to characterize the crossoverregime to the HF state, it may be useful to define a hypo-thetical scaling function f�T /T0� by means of

�q�q =T0

TA�0 K�f�t�

t T/T0. �9�

Since �q�q is expressed approximately as an �-integrationof Im��q ,�� up to �q: �q�q��0

�qIm��q ,��d� using theKramers-Kronig relation, the scaling function f�t���q�q isrelated with the spectral weight of Im��q ,�� for a wideenergy range.

Figure 14 shows the estimated f�t�=TA�0 K� /TA�T� as afunction of the normalized temperature t=T /T0 in USn3. Forcomparison, f�t�=TA�0 K��q�q / �T0� is also presented us-ing �q�q data for q=Q determined by neutron scattering26,27

in CeRu2Si2 �T0=14 K and TA�0 K�=16 K�. Although a de-tailed formula of f�t� �e.g., an exponent T−0.5 for USn3� maydepend on other parameters such as y1 �see below�, the roughfeatures of f�t� may be universal for HF compounds. Theseeffects should be explained by a theory for the crossoverregime which goes beyond the SCR model. This kindof scaling function has been already proposed forCe1−xLaxRu2Si2 systems.26

In the present analysis, y1 is assumed to be T independent.If we assume that y1 is T dependent and TA is constant, wefind again JQ�T�=0.52TA�0�y1�T��c+d�T in the crossoverregime. This fact suggests that the �T behavior for JQ doesnot depend on T dependence of y1. In contrast, the T depen-dence of TA� f�t�−1 could be modified by a possible weak Tdependence of y1. Based on an improved model, the scalingfunction may be formulated as f�t ,y1� which can character-ize the crossover regime of HF system.

C. Prediction of thermal expansion coefficient

Finally, we have predicted the T dependence of the elec-tronic thermal expansion coefficient = 1

VdVdT based on the T

dependence obtained for y. In antiferromagnetic HF systems,the thermal expansion coefficient is one of the few quantitieswhich can detect the staggered magnetic susceptibility��Q ,0� owing to the large Grüneisen parameter of HF sys-tems. For antiferromagnetic systems, may be expressedas28

FIG. 13. �Color online� The T dependence of JQ estimated fromthe fitting of 1 /T1T �Fig. 12� above 20 K using Eqs. �7� and �8�.

FIG. 14. �Color online� Hypothetical scaling functionf�t���q�q as function of t=T /T0 in USn3 �T0=33 K�. For compari-son, f�t� in CeRu2Si2 �T0=14 K� obtained from neutron scatteringdata �Refs. 26 and 27� is presented.

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=32DQ�T0

JQ

dy

dT, �10�

where DQ and � are the magnetovolume constant at q=Qand the compressibility, respectively, which can be deter-mined experimentally. Figure 15 shows the T dependence of �� 1

JQ

dydT � calculated from the T dependence of y �Fig. 8� for

the constant and T-dependent JQ cases.Since reflects the T dependence of JQ, as shown in Fig.

15, it will be interesting to compare the calculated with anexperimental one in USn3, although no such data have beenreported up to now. For the related case of the heavy-fermionsystem Ce1−xLaxRu2Si2, the experimental value28 issmaller than the calculated one for the fixed JQ case above�T0, which can be explained by values of JQ which increasewith T.

V. CONCLUSION

In the heavy-fermion compound USn3, the hyperfine cou-pling constant at the Sn site is large and isotropic, indicating

that the transferred hyperfine coupling is mainly due to hy-bridization between U 5f and Sn 5s orbitals. The spin-latticerelaxation rate 1 /T1T at the Sn site increases with decreasingT and finally becomes constant in the heavy-fermion state.

The T dependence of 1 /T1T and the electronic specificheat Cel in the HF state can be described quantitatively interms of a spin-fluctuation model.14 However, in order todescribe the crossover from the heavy-fermion tolocalizedlike state above T*, a T-dependent effective RKKYinteraction �JQ�a+b�T� and its dispersion strength��JQ=JQ−J0�, which were not included in the foregoingmodel, should be considered. In order to develop a realisticmodel for the formation of the heavy-fermion state, experi-mental determination of the RKKY dispersion in the para-magnetic state and a theoretical interpretation for it beyondthe usual spin-fluctuation model are suggested to be impor-tant.

In the crossover regime, a universal behavior character-ized by a scaling function f�T /T0� is proposed, which shouldbe deduced naturally from a realistic model.

From the estimation of thermal expansion coefficientbased on the SCR model, a thermal expansion coefficientmeasurement is found to be a good test to confirm the pro-posed T dependence of JQ.

It is proposed that the present quantitative analysis basedon T1 and Cel is a useful method for determining the type ofRKKY interaction �antiferromagnetic in the present case�which is present in a compound without magnetic phase tran-sition.

ACKNOWLEDGMENTS

We thank K. Miyake and Y. Takahashi for useful discus-sions on the SCR model and S. Raymond for sending us dataof �q�q in Ce1−xLaxRu2Si2 and for many important sugges-tions. We are grateful to K. Kubo, T. Hotta, K. Kaneko, N.Metoki, H. Kadowaki, H. Yasuoka, and G. H. Lander forstimulating discussions.

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FIG. 15. �Color online� The calculated T dependence of theelectronic thermal expansion coefficient based on the T depen-dences of y �Fig. 8� and JQ �Fig. 13� for T0=33 K. The solid anddashed lines correspond to the constant JQ and T-dependent JQ

cases, respectively.

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