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A Reduced Moment Magnetic Ordering in a Kondo Lattice Compound: Ce8pd24ga

The magnetic ground state of the antiferromagnet Kondo lattice compound Ce8Pd24Ga has been investigated using neutron powder diffraction, inelastic neutron scattering and zero-field muon spin relaxation measurements. The neutron diffraction analysis, below TN (3.6(0.2)K), reveals a commensurate type-C antiferromagnetic structure with the ordered state magnetic moment of \~0.36 mB/Ce-atom along the cubic<111>direction. The analysis of the inelastic neutron scattering (INS) data based on the crystal field (CF) model reveals a doublet ground state with a ground state moment of 1.29 mB/Ce-atom. The observed magnetic moment from neutron diffraction, which is small compared to the expected value from CF-analysis, is attributed to screening of the local Ce moment by the Kondo effect. This is supported by the observed Kondo-type resistivity and a small change in the entropy of Ce8Pd24Ga at TN. The zero-field muon spin relaxation rate exhibits a sharp increase below TN indicating ordering of Ce moments, in agreement with the neutron diffraction data. The present studies reveal that the physical properties of Ce8Pd24Ga are governed by the onsite Kondo compensation, the moment stabilizing intersite RKKY interaction and the crystal field effect.


I. INTRODUCTION
Studies of Ce-based heavy fermion compounds are of fundamental importance for understanding their complex magnetic and transport properties. A number of recent discoveries have led to the resurgence of interest in the properties of the binary compounds which crystallise in the well known cubic Cu 3 Au-type structure. [1][2][3][4] Among these, CeIn 3 and CePd 3 exhibit interesting properties.
CeIn 3 shows antiferromagnetic ordering with T N = 10 K at ambient pressure, and becomes superconducting with T C =0.175 K at a pressure of 2.55 GPa. 1 On the other hand CePd 3 is a canonical intermediate valence compound which has been extensively investigated using many experimental techniques. [5][6][7][8] Despite of a low carrier density, 0.3 electrons per formula unit, the low temperature properties of CePd 3 are consistent with a simple Fermi-liquid ground state. 9 The commonality between CeIn 3 and CePd 3 is the presence of a broad peak in the resistivity, at 70 K and 150 K respectively, indicating the onset of a coherent Kondo lattice ground state.
In heavy fermion compounds the stability of the magnetic ordered state depends on a delicate balance between the long-range intermoment coupling driven by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and the on-site Kondo hybridization between the 4f electrons and the conduction electrons. 10,11 The strength of both the RKKY and Kondo interactions depends on the coupling parameter ( J), T RKKY ~ ( J) 2 and T K ~ exp(-1/ J ), where J is the exchange parameter and is the density of states of the conduction electrons at the Fermi level. In the strong coupling limit, the so-called intermediate valence state as found in CePd 3 , the ground state is non-magnetic.
In this case both charge and spin fluctuations play an important role in the physical properties of the materials. For the medium to weak coupling limit, known as the heavy fermion regime, there can be a variety of interesting ground states, including reduced moment modulated antiferromagnetism, a zero temperature quantum critical point (QCP), unconventional superconductivity, and anomalous ferromagnetism. In this case the physical properties are governed by the spin fluctuations only. The coupling strength is generally related to the number of holes in the s-p or d-like outer shells of a given ligand and to its distance with respect to the Ce ions. 12,13 By decreasing the number of these holes it is possible to dehybridize the 4f states leading to a dramatic change in the magnetic and transport properties of the material. Such a dehybridization effect in CePd 3 has been observed when the compound was alloyed with an appropriate ligand or by introduction of small atoms such as B, Si or Ge at the interstitial site in the Cu 3 Au unit cell. 14-18 The interstitial impurities cause a dramatic cross-over from archetypal intermediate valence behaviour in CePd 3 to local moment character as seen through resistivity, magnetic susceptibility and inelastic neutron scattering studies. 14-18 The crystal structure, magnetic and transport properties of Ce 8 Pd 24 Ga have been investigated by various groups. 3,[19][20][21][22] This compound is closely related to CePd 3 as the title compound may also be written CePd 3 Ga 0.125 . Our recent high resolution neutron powder diffraction study at 300 K on Ce 8 Pd 24 Ga revealed that the crystal structure is composed of distorted perovskite and Cu 3 Au subcells arranged with the perovskite like units centred on the corners of the cube. 21 In the present work we have investigated the magnetic ground state of Ce 8 Pd 24 Ga using neutron powder diffraction and muon spin relaxation ( SR) measurements below and above T N . The results of our inelastic neutron scattering data 21 are discussed in the context of the studies presented here.

II. EXPERIMENTAL DETAILS
A polycrystalline sample (mass ~ 20 g) of Ce 8 Pd 24 Ga was prepared by arc melting the stoichiometric amounts of the constituent elements with purity 99.9% on a water-cooled copper hearth under high purity argon atmosphere. The ingot was turned over and remelted several times to ensure homogeneity. The neutron diffraction, inelastic neutron scattering and muon measurements were carried out at ISIS the pulsed neutron and muon source at Rutherford Appleton Laboratory, UK. Temperature dependent lattice parameters were measured between 4 K and 300 K using the high resolution neutron powder diffractometer HRPD. The magnetic structure was investigated as low as 1.9 K using the medium resolution neutron diffractometer ROTAX that views a cold methane moderator making the instrument suitable for magnetic structure studies. For both the ROTAX and HRPD diffraction experiments polycrystalline sample material was filled into a cylindrical vanadium sample container of 11 mm diameter (a flat plate type Al-sample holder was used for HRPD) and mounted in a standard 'orange-type' He-cryostat. Both HRPD and ROTAX are equipped with 3 detector banks, each of which provides neutron diffraction data in a distinct d-spacing and resolution range. Inelastic neutron scattering measurements were carried out using the time-of-flight spectrometer HET using an incident neutron energy of 40 meV. The Ce 8 Pd 24 Ga sample was cooled down to the lowest possible temperature of about 17 K in a closedcycle refrigerator. Zero field + SR measurements were carried on the MuSR spectrometer between 1.5 K and 6 K in a standard He-cryostat. The sample was mounted on an Al-sample holder with a silver mask. The neutron and SR measurements were performed on the same sample of Ce 8 Pd 24 Ga.

A. Neutron diffraction
The neutron powder diffraction studies show that down to the lowest temperature of 1.9 K Ce 8 Pd 24 Ga crystallises in the cubic structure with space group Pm-3m (no. 221), in agreement with our previous room temperature structurual analysis. 21 It is to be noted that the unit-cell of Ce 8 Pd 24 Ga is obtained by doubling the unit-cell of CePd 3 in all three cubic directions. The refinement of the site occupancies shows that the Ce and Pd sites are fully occupied, while the Ga site is only 78(4)% occupied. A detailed description of the crystal structure refinement and the crystallographic parameters at 300 K was given earlier. 21 Figure 1 shows the temperature dependence of the cubic lattice parameter obtained from Rietveld analysis. The lattice parameter decreases linearly with temperature from 300 K and becomes nearly temperature independent at low temperatures (below 15K), a typical behaviour expected for a system with thermally excited phonons. The temperature dependence of the lattice parameter was analysed using an Einstein expression of the form a(T)=a 0 + /[exp( E /T)-1], where a 0 is the lattice parameter at 0 K, is the Einstein constant, and E is an effective Einstein temperature. The solid line in Figure 1 shows the resultant fit. The fitted parameters are: a 0 =8.3669(1) Å, =0.0170(2) Å and E =157(5) K. The Einstein model yields a reasonable fit to the data over the entire temperature range suggesting that the electronic contribution to the temperature dependence of the lattice parameter is small. We have estimated the Grüneisen parameter, ~ 1.73 for Ce 8 Pd 24 Ga, using the above values of and E , and taking a typical value of the bulk modulus ~1Mbar.  Figure 2). These additional magnetic Bragg intensities are assumed to arise from the antiferromagnetic ordering of Ce 3+ moments at 1.9 K, in agreement with the heat capacity and resistivity measurements. 20,21 The observed magnetic reflections can be indexed on the basis of the crystallographic unit cell of Ce 8 Pd 24 Ga, i.e. with the wave vector q=(0, 0, 0). It is important to note that the same magnetic peaks can be indexed with respect to a smaller cubic CePd 3 -type unit cell with a propagation vector [½, ½, 0]. This type of order was labelled type-C antiferromagnetic structure of a primitive cubic lattice earlier. 23 In the Ce 8 Pd 24 Ga setting the Ce atoms occupy site 8g, site symmetry 3m, that leads to the admissible magnetic site symmetry 3m' restricting the Ce moments to directions along the 3-fold axis along <111>. 24 The corresponding magnetic space group is Pm3m' 25 corresponding to a triple-q structure model in terms of the [½, ½, 0] propagation vector, thus preserving the cubic symmetry. This non-collinear model, which has magnetic moments oriented along symmetry-equivalent <111> directions, represents the highest possible symmetry for the magnetic structure for fitting the magnetic intensities with one structure parameter only, i.e. the magnetic moment magnitude of the Ce 3+ ions. This magnetic structure cannot be discriminated by the diffraction experiments from other models of lower symmetry, e.g. a single-q structure. The restriction of the number of variable parameters was a main consideration for the model setup in view of the very weak observed magnetic powder diffraction intensities. The proposed structure takes into account the averaging of magnetic intensities over magnetic domains.
The magnetic moment refinement of Ce 8 Pd 24 Ga was performed with the program GSAS, 26 using diffraction patterns from the three banks in a Rietveld refinement. The magnetic form factor of Ce 3+ was calculated in dipole approximation using analytical approximations as tabulated in the International Tables for Crystallography. 27 The resultant Rietveld fit of the forward scattering bank data is shown in Figure 2. The refined moment value is 0.36(2) B /Ce-atom. A schematic representation of the magnetic structure of Ce 8 Pd 24 Ga is displayed in Figure 3. The magnetic moment magnitude as a function of temperature is plotted in Figure 4. The magnetic moment of the Ce ion is saturated at low temperatures and almost constant at about 0.35 B /Ce-atom between 1.9 K and 3 K, rapidly dropping off to zero as the transition temperature is approached. Moment values of 0.2 B , corresponding to moment components of 0.1 B along the cubic axes, are close to the detection limit of the powder diffraction experiment. Figure 4 also compares the temperature variation of the magnetic moment with a phenomenological power-law behaviour 28  =0.208 (4). The solid line in Figure 5a represents the fitted response with this set of CF parameters, which agrees well with the observed magnetic response. Figure 5b represents the crystal-electric-field energy level scheme of the Ce 3+ ions in Ce 8 Pd 24 Ga with the above set of CF parameters. The ground state is 3/2 , which makes Ce 8 Pd 24 Ga an Ising system. The calculated value of the ground state magnetic moment, < 1 g J J z 1 > (where g J is the Landé gfactor), is 1.29 B . By applying the CEF theory in the lowest order approximation, the magnetocrystalline anisotropy energy E a can be expressed in term of the second-order CEF parameter B 2 0 as follows E a =-3/2 B 2 0 <3J z 2 -J(J+1)> sin 2 (2) where is the angle between the easy magnetization direction and the z-axis (i.e. the crystal electric field axis, which is <111> in the present case). The observed positive sign of B 2 0 indicates that the magnetic moment is along the <111> direction in Ce 8 Pd 24 Ga, which is in agreement with the neutron diffraction analysis.

C. Zero field muon spin relaxation
The zero field muon spin relaxation is a unique technique and has played an important role to investigate dynamical fluctuations and static inhomogeneities in reduced moment heavy fermion systems. 33 In order to shed more light on the nature of the magnetic ground state in Ce 8 Pd 24 Ga, zero field SR measurements have been carried out on a polycrystalline sample below and above T N . Figure. 6 shows zero field SR spectra from Ce 8 Pd 24 Ga measured at various temperatures. The zero field SR spectra show a very small depolarization rate in the paramagnetic state, while below 3.8 K an onset of strong muon depolarization is observed, which is in agreement with the antiferromagnetic ordering observed by neutron diffraction. Surprisingly, even well below the magnetic ordering temperature there is no clear sign of a coherent oscillation in the zero field SR spectra that would give evidence of a muon spin precession around a static internal magnetic field.
A very similar situation has been observed for the antiferromagnet YbCu 3 Al 2 (T N =2 K) and ferromagnetic CePdSb (T C =17.5 K). 34,35 This might suggest that the average dipolar field, experienced by the muon spins at the muon stopping site, is equal to zero in Ce 8 Pd 24 Ga. In this scenario the sharp increase of the muon depolarization rate observed below T N in Ce 8 Pd 24 Ga could be understood as a result of the distribution of internal fields with zero mean at the muon site.
In the paramagnetic state the zero field SR spectra are well described using a product of a temperature-independent Kubo-Toyabe (KT) function, arising from the Ga and Pd nuclear moments, and an exponential relaxing function (ERF), exp(-1 t), representing the additional muon depolarization due to fluctuating electronic 4f-magnetic moments. The dynamic relaxation rate, 1 , is related to the interaction amplitude and to the rate of the spin fluctuations. In our analysis, we have also added a small time and temperature independent background contribution (a bg =0.026, estimated from the spectra at 1.6 K) arising from muon stopping in the silver mask. Fits to data collected at temperatures above T N yield good agreement between observed and calculated spectra as shown in Figure 6. The estimated KT relaxation rate is KT =0.0306 s -1 . The comparison of this value for KT  where G KT ( KT , t) is the well-known Kubo-Toyabe function 33 and the asymmetry parameter a 2 preceding the additonal second term. The first and third terms are also used for fitting above T N whereas the second term contributes below T N only. The functional form of the second term has been suggested for a muon stopping site of high symmetry in an antiferromagnetic matrix. 36 At such a site the mean field is zero. From the neutron powder diffraction experiments it follows that 22(4)% of the Ga site is vacant in Ce 8 Pd 24 Ga, hence one expect a different value of moment for the Ce atoms around this vacant site, which justifies to use the second term and, hence, the two components model in our analysis. In the analysis, the values of total asymmetry, a=a1+a2, a bg and KT were kept fixed. The temperature dependent relaxation rates, 1 and 2, and asymmetries, a 1 and a 2 , obtained from the fits are displayed in Figure 7a and 7b. Below T N 2 increases sharply on cooling, while 1 shows a very weak temperature dependence (Figure 7a). On the other hand, the asymmetry a 2 decreases sharply when the temperature is raised towards T N , accompanied by a compensation rise of a 1 (Figure 7b). These types of two components in the muon depolarization rate have been also observed in antiferromagnetic CeAl 2 and the heavy fermion YbBiPt. 33 The presence of two components has been attributed to two types of domains in these materials. In The temperature dependence of 2 has been modelled using the same functional power-law term that was plotted with the ordered magnetic moment in Figure 4, namely 2 (T) ~ (1 -d ) (Figure   7a), yielding T N =3.51(1) K, =0.429(1) and d=2.05 (2). The value of T N obtained from the muon data is in agreement with that obtained from the neutron diffraction study. The phenomenological parameter of d=2.05, reflecting spin-wave excitations, 28 is very close to 2 predicted for a cubic antiferromagnetic system. 37 We do not clearly understand at present the disagreement with the much higher value of d=10 employed to obtain a saturated ordered magnetic moment below 3 K ( Figure 4). One of the reasons could be a few data points with a large errors near T N in the diffraction data.

IV. CONCLUSIONS
In the present study, we have investigated the antiferromagnetic Kondo lattice compound Ce 8 Pd 24 Ga using neutron diffraction, inelastic neutron scattering and zero-field muon spin relaxation measurements. The chemical formula of the title compound can also be written as (T c =0.5 K and T N =5 K), with an estimated ordered state moment of 0.02 B /U-atom. 38 24 Ga shows an onset of the coherence in the resistivity, i.e. Kondo lattice behaviour, implying the importance of interactions between the magnetic ions of the lattice. In this scenario, while gaining the energy through the singlet formation, the system is also losing the magnetic interaction energy between the sites. This implies that the singlet formation should not be fully complete when the intersite interaction is sufficiently strong, and hence the system orders magnetically with a reduced magnetic moment. This is because the fully Kondo compensation of the 4f-moments would lead to an overall loss of energy instead of an energy gain. Thus, the Kondo compensation model explains the reduced moment magnetic ordering in Ce 8 Pd 24 Ga.