Muon spin rotation and neutron scattering study of the non-centrosymmetric tetragonal compound CeAuAl3

We have investigated the non-centrosymmetric tetragonal heavy-fermion compound CeAuAl3 using muon spin rotation (muSR), neutron diffraction (ND) and inelastic neutron scattering (INS) measurements. We have also revisited the magnetic, transport and thermal properties. The magnetic susceptibility reveals an antiferromagnetic transition at 1.1 K with a possibility of another magnetic transition near 0.18 K. The heat capacity shows a sharp lambda-type anomaly at 1.1 K in zero-filed, which broadens and moves to higher temperature in applied magnetic field. Our zero-field muSR and ND measurements confirm the existence of a long-range magnetic ground state below 1.2 K. Further the ND study reveals an incommensurate magnetic ordering with a magnetic propagation vector k = (0, 0, 0.52) and a spiral structure of Ce moments coupled ferromagnetically within the ab-plane. Our INS study reveals the presence of two well-defined crystal electric field (CEF) excitations at 5.1 meV and 24.6 meV in the paramagnetic phase of CeAuAl3 which can be explained on the basis of the CEF theory. Furthermore, low energy quasi-elastic excitations show a Gaussian line shape below 30 K compared to a Lorentzian line shape above 30 K, indicating a slowdown of spin fluctuation below 30 K. We have estimated a Kondo temperature of TK=3.5 K from the quasi-elastic linewidth, which is in good agreement with that estimated from the heat capacity. This study also indicates the absence of any CEF-phonon coupling unlike that observed in isostructural CeCuAl3. The CEF parameters, energy level scheme and their wave functions obtained from the analysis of INS data explain satisfactorily the single crystal susceptibility in the presence of two-ion anisotropic exchange interaction in CeAuAl3.


I. INTRODUCTION
The antiferromagnetic (AFM) s-f exchange coupling, J sf , between conduction and localized spins in heavy fermion (HF) rare-earth systems is responsible for two competing effects: the screening of the on-site localized moments due to the Kondo effect and the intra-site RKKY interaction among the magnetic impurities which may induce a long-range magnetic ordering. The Doniach phase diagram describes this competition [1]. Firstly, the Néel temperature, T N , rises on increasing the absolute value of the exchange interaction constant J sf or hybridization strength, V sf , between conduction and localised electron states. Then, T N is passing through a maximum with further increases in J sf (or V sf ), and finally it tends to decrease to zero at the "quantum critical point" (QCP). Such a decrease of T N down to the QCP has been observed in many Cebased HF compounds [2]. At present, various theoretical scenarios exist to explain the observed behaviour of the systems close to QCP and they are classified into two major categories: 1) local QCP, where T K →0 at QCP [2][3][4] and 2) spin density wave scenario where T K remains finite at QCP [5,6]. Above the QCP, a very strong HF character will eventually reduce the T K and these systems exhibit non-Fermi-liquid (NFL) properties [7][8][9].
Cerium-based HF intermetallic compounds with the general formula CeTX 3 (1-1-3 stoichiometry, with T=transition metals and X=Si, Ge and Al), have recently attracted considerable experimental and theoretical interest [10][11][12][13][14][15]. The reason is due to the discovery of many novel ground state properties in the tetragonal non-centrosymmetric crystal structure, such as unconventional superconductivity in CeTSi 3 , T = Ir and Rh, and CeCoGe 3 , at around QCP under pressure [16][17][18][19]. The tetragonal CeAuAl 3 belongs to the above class of compounds and could have a similar strength of Kondo and RKKY interactions. The thermal and transport properties of CeAuAl 3 at low temperatures suggest the presence of strongly correlated electrons in a "magnetically ordered" phase [20][21][22]. Furthermore, CeAuAl 3 shows a large electronic coefficient (γ elec. ) at zero-field, ≈ 227 mJ/mol-K 2 , and a large coefficient of the quadratic term in the magnetoresistivity, ≈ 4.84 µΩcm/K 2 . CeAuAl 3 has been reported to order antiferromagnetically at ~1.3 K [20,21].
The heat capacity, magnetic susceptibility and resistivity measurements existing in the literature clearly show the influence of the CEF at around 10-50 K. The nuclear magnetic resonance (NMR) study of 27 Al in CeAuAl 3 shows that the Ce magnetic moments are ordered and their magnitude reduced by ~ 25% at 0.50 K, most likely due to Kondo screening [22].
Furthermore, those systems with strongly correlated electrons can show spin, charge, and lattice degrees of freedom that sustain low energy magnetic or crystal electric field (CEF) excitations very similar in energy scales of lattice vibrations (or phonons). In most of these systems, these excitations remain decoupled and therefore they can be studied independently. Particularly interesting are those systems where strong CEF-phonon (or spin-phonon) coupling exists.
Recently, we have investigated the non-centrosymmetric tetragonal CeCuAl 3 compound using inelastic neutron scattering and found the presence of three excitations in the paramagnetic phase at 1.25, 10 and 20 meV [23]. Based on Kramer's theorem, we cannot expect more than two CEF excitations for Ce 3+ (4f 1 ) in the paramagnetic phase. The observed three CEF excitations in CeCuAl 3 have been explained based on the CEF-phonon coupling model (called magneto-elastic coupling) [23]. In order to investigate whether or not the CEF-phonon coupling is also present in other members of the non-centrosymmetric compounds with the general formula CeTX 3 , we are currently investigating many compounds of this family using inelastic neutron scattering [10][11][12]23]. In the present work, we have investigated the tetragonal CeAuAl 3 compound using various bulk characterization techniques, muon spin rotation (µSR) as well as using neutron scattering (both elastic and inelastic). Our study reveals the presence of two CEF excitations in the paramagnetic phase indicating the absence of CEF-phonon coupling in CeAuAl 3 . Further, neutron diffraction (ND) reveals an incommensurate magnetic structure with a magnetic propagation vector k = (0, 0, 0.52) and a spiral structure of Ce moments coupled ferromagnetically within the ab-plane.

II. EXPERIMENTAL DETAILS
Polycrystalline samples of CeAuAl 3 and LaAuAl 3 were prepared by the standard arc melting method starting with a stoichiometric mixture of the high purity elements (Ce, La: 99.9%, Cu: 99.99%, Al: 99.999%). The as-cast samples were annealed for a week at 850 o C under high vacuum to improve the phase formation. The phase purity of LaAuAl 3 was checked using X-ray diffraction (XRD) at room temperature, and of CeAuAl 3 using ND at 9 K. The XRD study was carried out at room temperature by using a RIGAKU / MAX system model ROTAFLEX Ru-300 with a graphite monochromator.
The heat capacity (C p ), magnetization and electrical resistivity (ρ) have been measured using a commercial physical properties measurements system (PPMS) of Quantum Design. The C p was measured by using an adiabatic heat pulse type calorimeter between 0.350 K to 300 K. The magnetization and DC susceptibility measurements were carried out using a SQUID magnetometer and a vibrating sample magnetometer, both from Quantum Design from 1.8 K to 50 K. Finally, the electrical resistivity was measured by means of a standard 4-probe technique with the leads attached to the sample using silver epoxy paint from 0.350 K to 300 K. All these techniques were mounted on a superconducting coil with an applied magnetic field up to 9 T. We have used a homemade mutual inductance thermally anchored to the mixing chamber of a 3 He- 4 He dilution refrigerator, which enables measurements to be performed from 0.09 K up to 3.5 K in the frequency range between 333 Hz and 13.3 kHz and (µ 0 H) max. ≈ 10 -3 T. The sample was fixed to a sample holder centred inside the secondary coil of the susceptometer with apiezon Ngrease. As-measured values were calibrated by using DC susceptibility values because the out-ofphase signal obtained from the lock-in amplifier was below sensitivity limits.
The muon spin rotation (μSR) experiments were performed on the MuSR spectrometer in the longitudinal geometry configuration at the ISIS Facility, UK. At ISIS, pulses of spin polarized muons are implanted into the sample at 50 Hz and with a full width at half maximum (FWHM) of ~70 ns. These implanted muons decay with a half-life of 2.2 µs into positrons, which are emitted preferentially in the direction of the muon spin axis. Each positron is time stamped and therefore the muon polarisation can be followed as a function of time. The μSR spectrometer comprises 64 detectors. The detectors before the sample (F) are summed together as well as the detectors after the sample (B). The muon polarisation then can be determined by, where N F and N B are the counts in the forward and backward detectors respectively and α is a calibration constant. The sample was mounted on an Ag-plate and covered with a thin Ag-foil with GE-varnish. The sample mount was then inserted into an Oxford Instruments dilution fridge with a temperature range of 0.05 K to 300 K. Any Ag exposed to the muon beam would give a time independent background.
Neutron diffraction measurements were carried out using the time-of-flight (TOF) diffractometer GEM at the ISIS facility. The powdered sample of CeAuAl 3 was inserted into a copper-can with a diameter of 6 mm and placed inside a standard Oxford He-3 system with a base temperature of 0.3 K. The measuring time was 6 hours at each temperature and data were collected at 0.3, 0.75 and 2 K. Measurements at 9 K were also performed with the sample filled into a vanadium-can and mounted inside a He-4 cryostat to characterize the sample quality. Each of the six detector banks of GEM provides a diffraction pattern for each measurement. The data from the six arrays are used in a multi-pattern Rietveld analysis. In order to correct variations of the detector efficiency across the detector banks, the neutron counts from the standard vanadium sample normalized the data. Further, the MARI data were presented in absolute units, mbarn/meV/sr/f.u., by using the absolute normalization obtained from the standard vanadium sample measured in identical conditions.

A. Structural characterization
First, we discuss the structural characterization of CeAuAl 3 using ND at 9 K and of LaAuAl 3 using XRD at 300 K. The analysis of ND and XRD data reveals that both samples were single phase and crystallize in the BaNiSn 3 -type tetragonal structure [24]. Fig. 1 [20][21][22]24] and are quite similar from a structural point of view. Figure 2 shows the C p data of CeAuAl 3 obtained under an applied magnetic field up to 7 T. Our results of C p exhibited the λ-type anomaly at T N = 1.1(1) K at zero field (ZF), close to the values previously reported in the literature [20,21]. This anomaly shifts to high temperatures on increasing the applied magnetic field (at 7 T it shifts to 5 K), and it becomes round and broad.

B. Heat capacity
There is no Schottky anomaly up to 10 K in the ZF data, which indicates that the CEF levels are higher than 10 K. This is in agreement with our INS study discussed in Section G. Further, the C p measurement of LaAuAl 3 at zero-field is shown in Fig. 2 exhibiting very low values at low temperature, as it can be expected. The linear T-contribution due to the conduction electrons, γ elec , is 3.24 mJ/mol-K 2 and the T 3 -phonon-lattice contribution is ~0.166 mJ/mol-K 4 . The calculated Debye temperature [25] is 227 K. These results seem to be in good agreement with those values existing in the literature [20,21]. Fig. 3 shows the electronic contribution to C p for CeAuAl 3 at different applied magnetic fields. We can see how the magnetic field destroys the enhancement of γ elec. , especially above 1 T. This breaking of the Kondo effect could be due to the reduction of density of states (DOS) at Fermi level (E F ) induced by the magnetic field [26]. On the other hand, C p results allow us to estimate the Kondo temperature, T K , in CeAuAl 3 . The inset of Fig. 2 shows the magnetic entropy in our system, and it is obtained from the experimental C p as, ( ) = ∫ � − � ⁄ , where C L is the C p of LaAuAl 3 . Assuming that our system behaves as a simple two level model with an energy splitting of k B T K [27] we can evaluate a T K of about 3.7 K, close to a previous estimation of T K = 4.5 K [21]. Now we analyse the field dependent of γ elec presented in Fig. 3 based on a theoretical model, which was proposed to explain field dependent of effective mass (m * ) of quasi-particles observed from the de Haas-van Alphen effect (dHvA) study for heavy fermion systems by Wasserman et al [28]. Further Rasul et al [29] have shown that the mass enhancement occurring in the dHvA amplitude is the same as that found in the heat capacity and the results are in agreement with experiments on CeB 6 [28]. Following Wasserman et al [28] the expression for the field dependent γ elec (∝ m * ) can be written as follow: Here γ 0 is free-electron linear term of heat capacity, which is proportional to band mass (m b ), 2D is the conduction electron band width, n f is the mean occupancy of 4f-electron (for Ce 3+ state n f~1 ), N is the effective spin degeneracy of the conduction electrons and local f-electrons (the magnetic field lifts only the spin degeneracy of these electrons), T K is Kondo temperature, g is the electron g-factor, µ B is the Bohr magneton, J is angular momentum of f-electrons (which is related to the angular momentum m of the conduction electrons by m=-J. H is applied magnetic field. As assumed in the analysis of the field dependent effective mass of CeB 6 [28], we have used N=2, J=5/2 and further we used g=6/7 for Ce 3+ state. Hence we left with three variables, γ 0 , 2Dn f and T K . Keeping 2Dn f =0.5 eV, we varied γ 0 and T K and the good fit to the data was obtained for γ 0 =5.3x10 -4 (J/mol-K 2 ) and T K =4 K (quality of the fit can be seen in Fig.3 shown by the solid line). Further the validity of our analysis is also supported through a very similar value of T K estimated from our inelastic neutron scattering study discussed in section G.

C. Magnetic susceptibility of CeAuAl 3
Figure 4(a) shows the AC-susceptibility for 633 Hz between 90 mK to 3.5 K (red colour), and zero-field cooled DC-susceptibility at 10 -3 T (black colour) between 3.5 K to 300 K. The ACsusceptibility values are calibrated by using the low temperatures values of the DC-susceptibility between 1.8 K and 3.5 K, as commented in Section II. Fig. 4(a) shows one clear magnetic transitions at T N = 1.1 K in agreement with published work [20,21] and possibility of another transition near 0.18 K, which need further investigation. Fig. 4(a) also shows the temperature dependence of the T×χ for which the magnetic transitions temperatures are much better observed.
Both transitions did not reveal any systematic shift with frequency and respond within a normal linear regime on increasing the amplitude of the oscillating magnetic field. The thermal dependence of the reciprocal susceptibility, 1/χ, is represented in Fig. 4

(b). It shows a typical
Curie-Weiss (CW) law (T-linear scale) with a negative CW temperature, θ p =-9.8 K, and an effective magnetic moment, µ eff = 2.50 µ B , relatively close to 2.53 µ B of Ce +3 . The estimated temperature independent Pauli contribution, χ P ≅ 9.0×10 -4 emu/mol. The deviation from a CW behaviour at low temperature (below 50 K) reveals the existence of CEF effects which are well documented in the literature [20,21]. However, here we provide direct confirmation of the CEF in CeAuAl 3 by using the INS measurements that will be presented later.

D. Magnetization and Electrical Resistivity
Figure 5(a) shows the field dependence of high-field magnetization isotherms up to 9 T between 1.8 K and 50 K for CeAuAl 3 . The magnetization isotherms show a different field dependence at around 10 K within the paramagnetic phase. The magnetization isotherms tend to saturate for cooling down to 1.8 K (see Fig. 5(a)), but they are linear above 10 K. The saturation type behaviour could be due to the CEF effect or a presence of short range magnetic interactions above T N . No magnetic remanence is observed. The magnetic moment at 1.8 K, ≅ 1.3µ B /f.u., can be calculated from magnetization at 9 T. Fig. 5a shows a metamagnetic-type transition around 0.5 T at 1.8 K, close to T N . The overall low temperature behaviour of the magnetization can be explained based on CEF effects along with magnetic exchange. Figure 5b displays the thermal dependence from 0.35 K to 300 K of the electrical resistivity for CeAuAl 3 up to 7 T applied field, and for LaAuAl 3 at 0 T. The resistivity for CeAuAl 3 shows a linear decrease from 300 K to ≈ 100 K, and a small plateau between 8 K and 4 K. Anomalies at around 10-50 K are considered as coming from the influence of CEF. Both compounds show an average ratio, ρ(300 K)/ρ(0.35 K) ≈ 3.8 which could indicate a slight structural disorder, as the structural analysis has shown in Section II (see Table I). The inset of Fig. 5b shows the low temperature region on an expanded scale, however, there is no sharp transition seen near T N , but small change in the slope has been observed that is in agreement with the published results [20].
As it is observed in Fig. 2 for C p , the transition in the field dependent ρ is slightly shifted to higher temperatures with applied magnetic field. The LaAuAl 3 resistivity was used to calculate: 1) the phonon contribution from other impurity contributions (≈54.8 µΩcm), a Debye temperature of ≈170 K (by using a Bloch-Grüneisen-Mott law) [20] and, 2) the magnetic contribution, ∆ρ, to the electrical resistivity, ρ. Fig. 3 (left y-axis) shows the coefficient of T 2 contribution (∆ρ ∼ AT 2 ), A, of the magnetic resistivity as a function of applied magnetic field. It shows that the field dependence of A is very close to the field dependence of γ elec and hence similar theoretical model can be applied to understand the field dependent of A as applied for γ elec in section B.

E. Muon Spin Relaxation
To shed light on the two-phase transitions seen in the AC-susceptibility, we have investigated the temperature dependence of the muon spin relaxation in zero-field (ZF). Fig. 6 shows the ZF asymmetry μSR spectra of CeAuAl 3 at selected temperatures between 0.05 K and 3 K. At 3 K, the μSR spectra exhibit a typical behaviour expected from the static nuclear moment. The ZF μSR spectra were fitted using a static Gaussian Kubo-Toyabe (GKT) function [30] multiplied by an exponential decay under a constant ground A gnd , where A 0 is the initial zero-field asymmetry parameter, σ is the nuclear contribution, and λ is the electronic relaxation rate mainly arising from the local 4f moment of the Ce ion. The static GKT function results from a Gaussian distribution of local magnetic fields at the muon site which arise from the nuclear spins [30]. The exponential decay, exp(-(λt) β ), is the magnetic contribution which results from the dynamic magnetic fields which arise from the fluctuating electronic spins.
The multiplicative nature of the nuclear and magnetic contributions is only valid if these processes are independent, as it was assumed in our case. We had estimated the value of A gnd and β~0.5 from the fit of 3 K data and these values were kept fixed to reduce the number of fit parameters. Fig. 7 (a-c) shows the temperature dependence of A 0 , σ, λ parameters obtained after fitting the ZF-μSR spectra. It is clear that at 1.1 K A 0 and σ exhibit a sharp drop, while λ exhibits a peak. A 0 drops nearly 2/3 to its high temperature values (Fig. 7a) indicating the bulk nature of the long range magnetic ordering. In the ordered state the internal fields arising from the electronic moment ordering are high compared to those from the nuclear moment. Hence the muons mainly sense the electronic magnetic field below T N and, as a result, σ cannot be measured, which is seen in Fig. 7b. The absence of any frequencies oscillations in the μSR spectra at 0.055 K (i.e. below T N ) indicates that internal fields at the muon sites are high and outside the time windows of the µSR spectrometer. This limitation is due to the broad pulse width (~80 ns) of the muon pulses at ISIS. Further, the divergence of λ (see Fig. 7c) above T N also confirms that the transition is magnetic and magnetic moment fluctuations start slowing down well above T N . It is to be noted that the transition temperature estimated by µSR is in agreement with that estimated by C p and susceptibility measurement. It is also of interest to note that the temperature dependence of λ shows a clear Arrhenius-like behaviour (see the inset of Fig. 7c), i.e.
where E a is an activation energy and k B is the Boltzmann constant. This shows that the spin dynamics within CeAuAl 3 are based on a thermally activated process with a barrier energy of E a = 0.0037±0.001 K and λ₀ = 0.077±0.008 μs⁻¹. This type of activation behaviour has been observed for CeInPt 4 with E a = 0.009 K, which remains paramagnetic down to 0.040 K [31]. In order to decouple the nuclear contribution from the electronic contribution we also measured the temperature dependent µSR spectra in applied longitudinal field of 5×10 -3 T. The data were fitted with Eq. 3 but without the KT term (i.e. σ =0). The temperature dependent A 0 and λ (not shown here) are also in agreement with those values given in Fig. 7. In order to obtain an estimate of the internal field at the muon sites, we also measured the field dependence of µSR spectra for applied fields between 0 and 0.25 T at 0.06 K. The initial asymmetry increases with the field and reaches 0.20 at a field of 0.25 T compared to a value of 0.26 at 3 K in zero-field, which indicates that the internal fields on the muon sites are larger than 0.25 T. As it was not possible to get information about the magnetic structure of CeAuAl 3 from our µSR study, we therefore carried out a ND study, and the results will be presented in the next section. intensities as a function of Q-(stronger at smaller-Q and falling towards higher Q) indicate that these are due to the long range magnetic ordering of the Ce-moment. For the estimation of the magnetic propagation vector, an automatic indexing procedure using a grid search in Fullprof program was used [32]. The neutron diffraction data allow for a direct observation of the propagation vector compared to indirect estimation from NMR [22]. In principle, the propagation vector can be refined in the Rietveld fitting process, but uncertainties of the zero-shift of GEM detector bank-1 at a d-spacing of 20 and the small number of weak magnetic reflections in other detector banks hampered variation in our case. Therefore, the propagation vector was manually adjusted until the observed extra peaks were successfully indexed using k=(0, 0, 0.52), which is close to (0, 0, 0.55) proposed by the NMR study [22].
A symmetry analysis using the SARAh program [33] for an incommensurate structure with k=(0, 0, 0.52) for Ce atoms at (0, 0, 0) indicates that there are four one-dimensional representations, labelled Γ 1 to Γ 4 , and one two-dimensional representation Γ 5 in the little group.
Only Γ 2 and Γ 5 enter the decomposition of Γ mag = Γ 2 + Γ 5 . Γ 2 and Γ 5 correspond, respectively, to an ordering of the Ce sites along the c-axis (one component, imposing a sinusoidal structure) and in the ab-plane (two basis vectors with real and imaginary components along a and b, each of both enabling spiral arrangements rotating in opposite directions, or if linked together, a helicoidally structure with an elliptical envelop controlled by the linear combination of the two vectors). A good fit to the data (magnetic Bragg factor for bank-1 R B =6 %) was obtained using FullProf [32] using a single basis vector of the representation Γ 5 (see Fig. 9). The fit using Γ 2 was not able to explain the intensities of the observed magnetic peaks, as expected from the NMR results [22]; in particular, Γ 2 does not contribute to the strongest magnetic peak at 20 Å. The magnetic structure of CeAuAl 3 is hence a simple helicoidal structure (Fig.10), for which Ce moments are ferromagnetically aligned in the ab-plane and for which moments rotate by an angle in radians given by ϕ=2π×K×t where t is a translation along the c-direction. For magnetic moments in neighbouring planes containing Ce-atoms at (0, 0, 0) and at the centering translation (½, ½, ½), respectively, the rotation angle is ϕ=93.6°, in agreement with the model proposed using NMR results. The magnetic moment is 1.05(09) µ B in the a-b plane. The relatively large error of the Ce-moment is due to the magnetic structure analysis being dominated by the strong magnetic Bragg reflection at 20 Å, a d-spacing region which on the GEM diffractometer is affected by a systematic error of typically 10% due to low neutron count rates and uncertainties of the wavelength-dependent neutron flux determination.. The direction and absolute value of the magnetic moment is compared to the estimated moment value from the CEF analysis in the next section.
It is worth comparing the magnetic structure and the direction of the magnetic moments of CeAuAl 3 to those of isostructural compounds, CeCuAl 3 and CeAgAl 3 . The compound CeCuAl 3 exhibits an AFM ordering at T N =2.5 K with apropagation vector k = (0.5, 0.5, 0) and moment along the c-axis [14], while CeAgAl 3 is a FM with T C =3 K [15] and easy magnetization axis in the ab-plane. It is interesting to note that even though the Cu, Ag and Au are isoelectronic the magnetic properties of these compounds change dramatically, which might indicate that magnetic exchanges, controlled through magnetostriction, magnetovolume pressure and chemical pressure, play an important role in determining the ground states of these compounds. ion (see Fig. 13). The observed small deviation from the F 2 (Q) behaviour could arise due to imperfect subtraction of phonon contribution, background coming from the closed-cycle refrigerator (CCR) and/or presence of short range magnetic correlations above the magnetic ordering temperature. Further, the intensity of the 24.7 meV peak also decreases on increasing Q up to 4 Å -1 , and then remains nearly constant. As the measured intensity of this peak is very small and also due to presence of phonon scattering at the same position, it was not possible to give any qualitative Q-dependent analysis for this excitation by using the Ce 3+ form factor F 2 (Q). Now, we present the analysis of the estimated magnetic scattering at 4.5 K, 50 K and 250 K (see Fig. 14(a-c)) based on the CEF theory and Kramer's theorem for the Ce 3+ ion (4f 1 ). By this way, we will achieve a full characterization of the CEF effects on the heat capacity and magnetic susceptibility commented at the end of this section.

G. Inelastic neutron scattering
Two magnetic excitations at around 5.1 and 24.7 meV have linewidths of 0.71(4) meV and 1.18 (11) meV, respectively, at 4.5 K suggesting that our sample is in a well crystallographically ordered state, which is in agreement with our diffraction analysis discussed above. Further, the smaller line width of the CEF excitations suggests that the hybridization between localized 4f 1 electronic states and conduction electrons must be smaller, which is in agreement with the reported smaller value of the Kondo temperature, T K =4.5 K [20,21]. We will develop this point further using low energy INS data. can be also determined using the high temperature expansion of the single crystal magnetic susceptibility [36] assuming isotropic exchange. Then, 2 0 can be written in terms of the Curie-Weiss temperatures, θ ab, when the applied magnetic field is in the ab-plane, and θ c when it is along the c-axis. For CeAuAl 3 , these values are θ ab =4.58 K and θ c =-194 K, and they were obtained from the single crystal susceptibility [37] which gives 2 0 =20.69 K (or 1.78 meV). This value of 2 0 is larger than that obtained from the INS data, which may indicate the presence of anisotropic exchange interactions in CeAuAl 3 .
The single crystal susceptibility of CeAuAl 3 [37] was analysed using the CEF parameters obtained from the INS analysis enhanced by a molecular field parameter that could describe the intensity of the anisotropy exchange coupling mentioned above. The form of the enhanced susceptibility is given by where ξ= {||a-axis, ||c-axis} and indicate the direction of the applied magnetic field when susceptibility is calculated, χ ξ CEF is the single ion susceptibility calculated by using H CEF , ξ λ is the molecular field parameter and 0 ξ χ is a constant temperature independent contribution. Fig. 15 shows two fits with Eq. 5. Continuous black lines give the first fit as it is obtained from the CEF at the Ce M 4,5 edges [38], which also gives |±1 2 ⁄ ⟩ as a ground state. It is to be noted that CEF analysis using the x-ray absorption spectroscopy is not an independent analysis and it does need information of CEF energy levels from other techniques such as INS study or heat capacity analysis.
Now we discuss the low energy excitations, especially quasi-elastic linewidths, measured on IN6 with an incident energy E i =3.1 meV at various temperatures between 2 K and 260 K. Fig. 16 shows the quasi-elastic response from CeAuAl 3 at various temperatures. It is clear that at 2 K we have a clear sign of low energy scattering and with increasing temperature the linewidth of the quasi-elastic line increases with temperature and the quasi-elastic intensity decreases. The former one gives the estimation of Kondo temperature, while the latter follows the behaviour very similar to dc-susceptibility. To analyse quantitatively the linewidth and intensity as a function of temperature we first analysed the data using a Lorentzian lineshape function. Although the fits were very good for the data above 50 K, the data below 50 K and especially at 2 K were not fitted very well to a Lorentzian lineshape. We therefore analysed the low temperature data using a Gaussian lineshape function, which showed excellent agreement with the data. The estimated linewidth and the intensity of the quasi-elastic line are plotted as a function of temperature in Fig.   17. It is interesting to see that the intensity (or inverse intensity) follows Curie-Weise type behaviour very similar to the DC-susceptibility. Furthermore, the linewidth exhibits nearly linear behaviour above 50 K, while it shows nearly T 2 behaviour at low temperature (see the inset in Fig. 17b). The value of the linewidth at 2 K is ~0.3 meV, which gives a Kondo temperature of 3.5 K. This value of T K is in excellent agreement with that estimated from the heat capacity [20]. The observation of a Gaussian line shape below 50 K suggests that the spin fluctuations are mainly due to inter-site spin-spin correlations (there are strong paramagnetic correlations at least up to 30 K) rather than single-site spin relaxation observed in many heavy fermion systems. This type of a Gaussian line shape and the presence of paramagnetic correlations has been observed in the heavy fermion compound YbBiPt [39]. Further, it is to be noted that the quasi-elastic response of YbAuCu 4 and YbPdCu 4 also show the presence of two components, Lorentzian and Gaussian, below 10 K [40]. The observation of the Gaussian component in these compounds has been attributed to a precursor of the magnetic order taking place below 1 K [40].

IV. CONCLUSIONS
We have investigated the heavy fermion antiferromagnetic compound CeAuAl 3 using muon spin rotation (µSR) and neutron scattering measurements, in addition to magnetization, transport and heat capacity studies. CeAuAl 3 shows a magnetic phase transitions at 1. shows a well-defined quasi-elastic line, which gives T K =3.5 K, in good agreement with the T K =4 K estimated from the heat capacity, and further shows evidence of slowdown of spin fluctuations below 30 K, which is well above the magnetic ordering temperature.
Finally, we would like to mention that most of the known heavy fermion (HF) compounds scale quite well with the Kadowaki-Woods ratio (KWR, A/γ elec 2 ), i.e. a ratio between T 2 -term of resistivity (A) and the square of electronic contribution (γ elec 2 ) to C p . The KWR is considered universal and has the value of ~10 -5 µΩcm (mol-K/mJ) 2 [41][42][43][44][45][46][47]. KWR could be constant under an applied magnetic field whenever the system is far away from a QCP. According to the Fermi liquid theory, KWR is proportional to a constant coupling of quasiparticles under exchange interaction, α 0 , and proportional to a parameter that characterizes the shape of Fermi surface (SFS) [42]. Therefore, the product of these two factors support the universal character of KWR in HF [42,47]. In our case, KWR=9.4 10 -5 µΩcm (mol-K/mJ) 2 , which is slightly enhanced with respect to most of known AFM heavy fermions [42][43][44][45][46][47] and it is quite unaffected by the existence of applied magnetic fields, at least up to 7 T. Then, 1) CeAuAl 3 would not be so close to the QPC as initially expected at the beginning, and 2) the shape of Fermi surface is the most likely factor to explain the enhancement of KWR, as the quasi-particle interaction α 0 hardly changes in most of known HF systems. On the other hand, the ratio of Wilson (WR) [48][49][50] which is also used to characterize HF compounds can provide insight into the types of interactions present. WR depends on two important contributions: 1) the electronic contribution to heat capacity, γ elec. and 2) the static magnetic susceptibility, χ 0 . Both are proportional to the density of state (DOS) at the Fermi energy, and so they should have similar changes when a magnetic field is applied [47]. In our CeAuAl 3 system, WR ≅ 1.6 which is close to the theoretical proposed WR=1.5 and also close to 1.46 observed in CeRu 2 Si 2 [47]. For most of strongly correlated systems WR >1 where the spin fluctuations are enhanced while charge fluctuations are suppressed.  to C p (■ from 0 to 7 T, Δ from ref [20], and ᐁ from ref [21])  [37]. The solid black line shows the best fit based on the CEF model including a molecular field parameters with fixed CEF parameters from the INS analysis; the blue dotted line shows the fit with both CEF parameters and molecular field parameters as variables. The latter fit agrees better with the susceptibility data, but does not explain the INS data.