Magnetic and electrical properties of (Pu,Lu)Pd3
M. D. Le, K. A. McEwen, E. Colineau, J.-C. Griveau, R. Eloirdi
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Magnetic and electrical properties of (Pu,Lu)Pd M. D. Le
Helmholtz-Zentrum Berlin für Materialen und Energie,Hahn-Meitner-Platz 1, D-14109 Berlin, Germany ∗ K. A. McEwen
Department of Physics and Astronomy, and London Centre for Nanotechnology,University College London, WC1E 6BT, UK
E. Colineau, J.-C. Griveau, and R. Eloirdi
European Commission, Joint Research Centre,Institute for Transuranium Elements, Postfach 2340, 76125 Karlsruhe, Germany (Dated: August 7, 2018)We present measurements of the magnetic susceptibility, heat capacity and electrical re-sistivity of Pu − x Lu x Pd , with x =0, 0.1, 0.2, 0.5, 0.8 and 1. PuPd is an antiferromagneticheavy fermion compound with T N =
24 K. With increasing Lu doping, both the Kondo andRKKY interaction strengths fall, as judged by the Sommerfeld coefficient g and Néel tem-perature T N . Fits to a crystal field model of the resistivity also support these conclusions. Theparamagnetic effective moment µ eff increases with Lu dilution, indicating a decrease in theKondo screening. In the highly dilute limit, µ eff approaches the value predicted by interme-diate coupling calculations. In conjunction with an observed Schottky peak at ∼
60 K in themagnetic heat capacity, corresponding to a crystal field splitting of ∼
12 meV, a mean-fieldintermediate coupling model with nearest neighbour interactions has been developed.
I. INTRODUCTION
The AnPd series of compounds, withAn=U, Np, or Pu, are rare examples of actinideintermetallic compounds in which the 5 f elec-trons are well localised around the ionic sites.UPd is a very interesting compound which ex-hibits four quadrupolar ordered phases below8K , whilst there are indications that NpPd may also show quadrupolar order at low tem-peratures . These two compounds crystallisein the double-hexagonal close-packed (dhcp)structure, in contrast to PuPd which adoptsthe AuCu structure, with lattice parameter a = f electrons as shown by recentphotoelectron spectroscopy measurements .Early measurements of the bulk propertiesand neutron diffraction studies show it to be antiferromagnetic, with a transition temperature ∼
24 K, and a G-type structure, where near-est neighbour moments are aligned antiparallel.The same study found that the high temperatureresistivity shows a Kondo-like behaviour, in-creasing with decreasing temperature. This be-haviour, together with a high Sommerfeld coef-ficient deduced from recent heat capacity mea-surements , led us to make a further investiga-tion of the properties of PuPd .Our aim has been to study how the compe-tition between the Kondo effect and the RKKYexchange interaction affects the physical prop-erties of this compound. This may be accom-plished by doping with a non-magnetic ion,which increases the distance between localised f -electrons, and hence decreases the RKKY in-teraction. Finally, the single-ion properties ofthe Pu ion may be investigated in the highly di-IG. 1: (Color online) Lattice parameters andtransition temperatures for Pu − x Lu x Pd .lute limit.In this work, we present in section II the ex-perimental details and in section III the mea-surements of the magnetic susceptibility, andheat capacity of Pu − x Lu x Pd . These mea-surements are then analysed using a localisedmoment mean field model in section IV. Fi-nally, section V presents measurements of theelectrical resistivity and Hall coefficient ofPu − x Lu x Pd , which are assessed in terms of asimple crystal field model. II. EXPERIMENTAL DETAILS
Polycrystalline samples of PuPd , LuPd andPu − x Lu x Pd , with x = .
1, 0.2, 0.5, and 0.8,were produced at ITU by arc melting appropri-ate amounts of the constituent elements under ahigh purity argon atmosphere on a water-cooledcopper hearth, using a Zr getter. The AuCu structure was confirmed by x-ray diffractionfor each sample, and the lattice parameters areshown in figure 1. The data show a linear de-pendence of the lattice parameter with increas-ing Lu dilution, in accordance with Vegard’sLaw, which also confirms the stoichiometry ofthe samples. In addition, there also appears tobe a linear dependence of the transition tem-perature, T N , with doping. These temperatures were determined from magnetic susceptibilityand heat capacity measurements described inthe next section. At T N there is a maximumin the susceptibility, c , and hence this temper-ature was determined by numerically differen-tiating the data to find d c dT =
0. In the heat ca-pacity, C p , there is a lambda step at T N , whichwas determined by differentiating the data tofind the minima of dC p dT . The values of T N de-duced from these two measurements are in closeagreement, whereas the inflexion points of theresistivity data do not correlate with T N as de-termined from c or C p . Nevertheless, the resis-tivity inflexion points show the same decreasingtrend with Lu-doping as T N . Errors in T N quotedin table I were determined by the width in tem-perature of the lambda step for C p or the step in d c dT .The magnetisation and susceptibility weremeasured using a SQUID magnetometer (Quan-tum Design MPMS-7), whilst the heat capac-ity was determined by the hybrid adiabatic re-laxation method in a Quantum Design PPMS-9 for PuPd and LuPd , and in a PPMS-14for Pu − x Lu x Pd . Small samples with massless than 5 mg were used for the heat capacitymeasurements so that the decay heat does notsignificantly affect the measurements. The X-ray diffraction, magnetisation and heat capacitymeasurements were made immediately after thepreparation of samples in order to minimise theeffects of radiation damage.Finally, thin parallel-sided samples of eachcomposition were extracted, polished andmounted for electrical transport measurements.As these measurements were made some threemonths after synthesis, we observed significantradiation damage which manifested in a highresidual resistivity at low temperatures. Thisprompted us to anneal the samples at 800 o C for12 h, and to remeasure the electrical transportproperties. The remeasured data is presented insection V.2IG. 2: (Color online) The magnetisation at2 K of Pu − x Lu x Pd . Lines are guide to the eye. III. MAGNETISATION AND HEATCAPACITY MEASUREMENTSA. Magnetisation
Figure 2 shows the magnetisation at 2 K,which shows that the Pu-rich compositions arenot saturated at 7 T. This is not surprising be-cause we expect a J = / &
100 T, when the splitting between thelowest two CF levels is ≫ . Lu . Pd shows some evidence ofsaturation, however. The magnetic susceptibil-ity is shown in figure 3, and the inverse suscep-tibility in figure 4. The data in the paramagneticphase above the Néel temperature are well fittedby a modified Curie-Weiss Law MH = Nµ µ B k B T − q CW + c ( H ) (1)where N is the number of Pu atoms in the com-pound, and q CW is the paramagnetic Curie tem-perature. We recall that in the Weiss mean fieldtheory, (- q CW ) q CW corresponds to the (anti-)ferromagnetic transition temperature. However,this theory does not take into account single ioneffects such as the crystal field which are ex-pected to be significant in PuPd .There is a field dependent residual sus-ceptibility c ( H ) which mainly arises from impurities, the encapsulation and the sam-ple holder. In addition, the Pauli sus-ceptibility of the conduction electrons mayalso contribute to c and can be estimatedfrom the electronic Sommerfeld coefficientof LuPd , g + LuPd =3.2(1) mJmol − K − , whichyields c Pauli ≈ . ( ) × − µ B T − f.u. − . Thisis significantly lower than the observed valuesof the residual susceptibility, which are of theorder of 10 − µ B T − f.u. − , indicating that theconduction electron susceptibility contributionis negligible. The fitted parameters to the Curie-Weiss relation for each sample are given in ta-ble I. The quoted error is deduced from calcu-lating the covariance matrix of the parametersfrom the covariance matrix of the data assum-ing that this is diagonal and proportional to e − where e is the standard error in the measuredmoment or heat capacity as determined by the MultiVu software supplied by Quantum Design.Figure 4 shows the inverse susceptibility of thedifferent compositions with the residual suscep-tibility c subtracted.The magnitudes of the effective momentsare all significantly higher than the LS -couplingvalue, 0.85 µ B . Any crystal field interaction willonly decrease this effective moment because asthe crystal field split levels become further sepa-rated and hence thermally de-occupied, their an-gular momentum will cease to contribute to themoment. The effective moment with zero crys-tal field splitting in intermediate coupling on theother hand is approximately 1.4 µ B , as calcu-lated using the theory outlined in section IV.This value may be decreased slightly by a largecrystal field, and suggests that we should use in-termediate coupling to calculate the single-ionproperties of Pu + .An alternative reason for the higher than ex-pected effective moment may be due to somehigh moment paramagnetic impurity. How-ever, analysis of the x-ray diffraction patternsshowed that the only detectable impurity isLu O which is non-magnetic. There may alsobe some trace amounts of oxides of Pu which isnot observed in the diffraction pattern. PuO isa Van Vleck paramagnet which may contribute3IG. 3: (Color online) The magneticsusceptibility of Pu − x Lu x Pd .FIG. 4: (Color online) The inverse magneticsusceptibility of Pu − x Lu x Pd . Thin solid linesare fits to the modified Curie-Weiss Law foreach compound. The mean field intermediatecoupling calculation is shown as a thick solidline, whilst the dashed line shows the single ionsusceptibility calculated from the crystal fieldin LS -coupling.to the impurity term in equation 1, whilst Pu O is an antiferromagnet with an effective momentof 2.1 µ B . However, one would need approx-imately 6 mol % Pu O in PuPd order for itbe responsible for the increased effective mo-ment compared to the LS -coupling expectation,at which concentration it should be detectablein the X-ray diffraction pattern, which is not the case. Furthermore, the enhanced value of µ eff for Pu O which also has a free ion Pu + con-figuration suggests that intermediate coupling isappropriate in these cases. B. Heat Capacity
The heat capacity at zero field is shown infigure 5, whilst details of the results in appliedfields up to 14 T are in figure 6. We note thatat high temperatures, C p tends to the classi-cal Dulong-Petit limit, 3 NR = . − K − .For Pu . Lu . Pd , the derivative in the heat ca-pacity shows two minima, which stem from thestep-like nature of the transition. The highertemperature inflexion point at ∼
22 K corre-sponds well with the peak in the inverse suscep-tibility, but the lower temperature peak at ∼
21 Kdoes not match any feature in the magnetic sus-ceptibility. Nevertheless, these two anomaliesraise the possibility that there may indeed betwo transitions in this compound. Moreover,as can be seen in Figure 6, the heat capacityof Pu . Lu . Pd also shows indications of twotransitions.An estimate of the electronic specific heat C el = g T and Debye temperature q D was ob-tained using the approximation C ∼ g T + p N A k B (cid:18) T q D (cid:19) (2)which is valid at low temperatures ( T ≪ q D ),from a plot of C p / T vs T shown as an inset infigure 5. However, the magnetic heat capacitycomplicates the determination of g because theNéel temperature is very low in some of the Lu-rich compounds. This increases the low tem-perature C p and hence the estimate of g fromthe straight line intercept. For this reason, forthe Lu-rich compositions, we show the resultsof fitting the data in the region above T N ( g + ) inaddition to that below 8 K ( g − ) in table I. ForPu-rich compositions, the data above T N willbe affected by the Schottky anomaly at approx-imately 17 K, and will be unreliable. Thus itappears from these estimates that the electronic4 omposition T N (K) g ± (mJ mol − K − ) q D (K) µ eff ( µ B Pu − ) q CW (K) C p c T > T N T < T > T N PuPd . Lu . Pd . Lu . Pd . Lu . Pd . Lu . Pd TABLE I: Transition temperatures and other parameters derived from the magnetic susceptibilityand heat capacity of Pu − x Lu x Pd .FIG. 5: (Color online) Heat capacity ofPu − x Lu x Pd at zero applied magnetic field.The inset shows C p / T vs T and solid linesthere show fits to C / T ∼ g + AT . LuPd datais shown in orange in the insetheat capacity first increases with increasing x until x ≈ .
5, whereupon it falls as x rises fur-ther. The spread in the fitted parameters whendata from different ranges of temperatures in theregion 25 < T <
40 K for g + , and 2 < T < g − was taken as an estimate of the errors inthese parameters.The Debye temperature q D was determinedfrom fitting the high temperature data, and ap-pears to be independent of Lu-doping. This sug-gests that the phonon contribution to the heat ca- FIG. 6: (Color online) Heat capacity ofPu − x Lu x Pd in applied magnetic field. Thearrows indicate the direction of increasing field.pacity is constant through the series. A good es-timate of this contribution is given by the heatcapacity of the non-magnetic isostructural com-pound LuPd , which also has a negligible elec-5IG. 7: (Color online) Deduced magnetic andelectronic contribution to the heat capacity C mag p = C p (Pu − x Lu x Pd ) - ( C p (LuPd )- g + Lu T ).The solid line is a crystal field calculation. . tronic heat capacity, g Lu = . ( ) mJmol − K − .We have thus extracted the additional electronic and magnetic heat capacity of Pu − x Lu x Pd bysubtracting that of LuPd , as C mag p = C p ( Pu − x Lu x Pd ) − ( C p ( LuPd ) − g + Lu T ) (3)This extracted quantity, scaled by the Pu con-centration, is shown in figure 7.The magnetic heat capacity for all the com-pounds shows a peak at ∼
60 K, which we at-tribute to a Schottky anomaly from the crys-tal field (CF) splitting. The cubic CF onthe Pu + ions splits the six-fold ground mul-tiplet ( J = in LS -coupling) into a doubletand quartet. The energy gap, D CF , betweenthese two levels determines the temperature ofthe Schottky peak, such that D CF ∼
12 meVcorresponds to a peak at 60 K. The magni-tude of this peak, however, is determined bywhether the doublet ( C p =6.3 Jmol − K − ) orquartet ( C p =2 Jmol − K − ) is the ground state.The data in figure 7 thus suggest a doubletground state.This is supported by the magnetic entropy,shown in figure 8, deduced by numerically in-tegrating the magnetic heat capacity, S ( T ) = FIG. 8: (Color online) Deduced magneticcontribution to the entropy of Pu − x Lu x Pd ,calculated by numerically integrating C mag p / T . . R T C mag p T dT . From the very low heat capacity ofPuPd at low temperatures, we believe the mag-netic entropy from 0 to 2 K is negligble, andhave not included this range in the integration.The value of the entropy at the Néel tempera-ture is approximately R ln ( ) for PuPd , whichis above the value, R ln ( ) , expected for a dou-blet ground state. If the electronic heat capac-ity g − Pu =
76 mJmol − K − is subtracted fromthe integral, then we obtain S ( T N ) ≈ R ln ( . ) .The remaining discrepancy may be due to (i) anincomplete subtraction of the phonon contribu-tion, as the heat capacity of LuPd may not beexactly analogous to the phonon heat capacityof PuPd , and (ii) a larger value of g (see thediscussion in section IV).Finally, the inset to figure 5 shows what ap-pears to be a hump at ∼
17 K in the heat capacityof PuPd . This feature was initially attributed tothe Schottky anomaly from the CF splitting inreference . However it is more likely due to aSchottky anomaly from the splitting of the dou-blet ground state in the ordered phase, and in-deed such a feature is observed in the mean fieldcalculations in the next section. A splitting of D MF = . ∼
17 K, whichis reasonable.6IG. 9: (Color online) Calculated dependenceof the CF parameter L and the nearestneighbour exchange parameter J on L subjectto constraints described in the text. The verticalline indicates the optimal parameters. IV. MEAN FIELD CALCULATIONS
As noted in section III A, the measured ef-fective moment for both PuPd and the dopedcompounds is approximately 1 µ B /Pu, in con-trast to the expected LS -coupling value of g J p J ( J + ) = . µ B from a Hund’s rule H / ground state for a Pu + ion. LS -coupling is a good approximation when boththe Coulomb ( H C ) and spin-orbit ( H s o ) interac-tions are large but H C ≫ H so . In contrast, when H C ≪ H so , j j -coupling is a better approxima-tion, whereupon we obtain µ eff = 2.86 µ B . In be-tween these limits, for the case of intermediatecoupling, the effective moment is a function of H C and H so , and the full Hamiltonian, includingboth these terms and the crystal field ( H cf ) andZeeman interactions ( H Z ) must be calculated.The strength of H C and H so , parameterisedby the Slater ( F k ) and spin-orbit ( x ) integrals,is fixed by the atomic environment of the un-filled shell electrons. Thus in practice, inter-mediate coupling refers to the case where thevalue of F k and x are determined either fromab-initio (Hartree-Fock) calculations, or fromexperimental measurements using optical spec- FIG. 10: (Color online) Measured andcalculated magnetic heat capacity of PuPd .The estimated electronic heat capacity, D C mag p / T = C mag p − C ic p T is given in the bottompanel and has a mean value of122 mJmol − K − .troscopy. Using parameters determined exper-imentally by Carnall from the spectra of dilutePu + in LaCl , the effective moment is 1.44 µ B .It is conceivable that in a metallic system likePu − x Lu x Pd there may be small changes to F k and x compared to the insulating salts on whichthe measurements of were made . Neverthe-less, a 10 % change in F k and x to make the sys-tem more LS -like only yields µ eff = µ B . Inorder to obtain µ eff ∼ µ B , we must double themagnitude of F k and x compared to their mea-sured values, which is probably unphysical.Given that the crystal field interaction issmall, as judged by the ∼
12 meV split betweenthe doublet ground state and first excited quartetdeduced from the heat capacity measurements, H cf has little effect on µ eff . Thus we believethat the lower than expected effective moment ismost likely due to Kondo screening. Nonethe-less, a mean field calculation which can modelthe antiferromagnetic order and the single ionintermediate coupling behaviour is still valuableto interpret the heat capacity and magnetisationdata. Such a calculation, carried out using theMcPhase package , is detailed below.We have assumed a nearest neighbour only7xchange interaction between 5 f electrons,which is reasonable given the G-type antiferro-magnetic structure where nearest neighbour mo-ments align in antiparallel. Thus, there are threefree parameters in the calculations: two crystalfield (CF) parameters, L and L , and one ex-change parameter J . There are two other non-zero CF parameters, but they are fixed by thecubic point symmetry of the Pu + ions such that L = p / L and L = − p / L . It shouldbe noted that the parameters used here corre-spond to the Wybourne normalisation , ratherthan the usual Stevens normalisation (usuallydenoted B ml ). This is because the Stevens opera-tor equivalents O ml are valid only within a singlemultiplet of given J , whereas we now requireoperators that can span all the allowed J values.The two CF parameters are fixed by the re-quirement that they result in a doublet groundstate with a quartet at ∼
12 meV. This fixes arelation between L and L as shown in fig-ure 9. The Néel temperature T N then fixes arelation between J and the crystal field parame-ters, and finally the magnetisation below T N wasused to fixed all three values, yielding L = L = J = − .
204 meV.The magnetisation is calculated by includingin the Hamiltonian a Zeeman term, − µ B ( L + S ) · H ; numerically diagonalising the energymatrix and calculating the expectation value ofthe moment operator L + S . The calculatedinverse susceptibility is shown as a solid blackline in figure 4 for comparison with the mea-sured data. Unfortunately, better agreementwith the data within the constraints of the mean-field intermediate coupling model is only pos-sible by increasing the Coulomb or spin-orbitintegrals to unphysical values. A more likelyexplanation is the suppression of the effectivemoment by Kondo screening, which is not con-sidered in the current model.The heat capacity is calculated by numeri-cally differentiating the internal energy, h U i = (cid:229) n E n exp ( − E n / k B T ) / Z , by the temperature,and the entropy by subsequently numericallyintegrating this. The calculated heat capacity,shown in figure 10, shows a shoulder around ∼
20 K in accordance with the data which arisesfrom a Schottky peak due to the splitting ofthe ground state doublet in the ordered phase.We can also estimate the electronic heat capac-ity by subtracting this calculated C ic p from themeasured electronic and magnetic heat capac-ity, C mag p , the result of which is shown in thebottom panel of figure 10. The spike near T N is due to the differences in the sharpness ofthe calculated and measured transitions in thistemperature range. Overall however, the meanvalue, ¯ g =
122 mJmol − K − , over the full tem-perature range is in fair agreement with that de-rived from the low temperature heat capacity, g − Pu =76 mJmol − K − .Similar mean-field heat capacity calculationsfor the other compositions, where the exchangecoupling J was reduced to reflect the lower T N ,did not yield the broad transitions seen in fig-ure 7, but rather the sharp lambda anomalies ex-pected of an antiferromagnetic transition. Thusa subtraction to deduce their electronic specificheat becomes increasingly untenable. The broadtransitions observed in the data are probably dueto disorder in the system as a result of the Ludoping.Finally, the calculated internal fields in themodel are 226 T (180 T) at 1 K (20 K), whichagree well with the molecular field of 217 Tdetermined from fitting the measured resistiv-ity using a simple CF model, as described in thenext section. V. ELECTRICAL TRANSPORTMEASUREMENTS
Electrical transport measurements were car-ried out using thin parallel-sided samples ex-tracted after crushing the polycrystalline buttonsproduced by arc-melting. The first transportmeasurements were completed some months af-ter the production of the samples, so there weresignificant aging effects in the Pu samples. Thisprompted us to anneal the PuPd sample, andre-measure its resistivity, resulting in a large de-crease in the residual resistivity r from 225 µ W cm to 11 µ W cm. Subsequently the resistivity8f the other compositions was also re-measuredafter annealing. The values of r show a rapidincrease with Lu doping up to x = . x , as summarised in ta-ble II. This is due to the increasing number ofdefects caused by Lu substitution. Finally, mea-surements of the Hall coefficient and longitu-dinal resistivity of PuPd and the x =0.1,0.2,0.5compositions in field were also carried out.The resistivity of LuPd is well fitted by theBloch-Grüneisen relation r = C T q D Z x x ( e x − )( − e − x ) dx (4)where x = T / q D , with C = ( ) µ W cmand q D = ( ) K. It was taken to be rep-resentative of the non-magnetic contributionto the resistivity of Pu − x Lu x Pd , and usedto estimate the magnetic resistivity as Dr = r (Pu − x Lu x Pd ) − r (LuPd ). This quantity isplotted in the case of zero applied magnetic fieldin figure 11. The in-field measurements showedlittle change from the zero field data, and thedata for PuPd agree well with previous mea-surements , albeit with a slightly lower resid-ual resistivity.Qualitatively, the behaviour of the resistivitymay be divided into a high temperature Kondo-like regime, where the resistivity increases withdecreasing temperature until ∼
50 K, followedby the onset of coherence, from where it fallssharply with temperature, and shows no clearanomaly at T N . At low temperatures, the re-sistivity follows an exponential temperature be-haviour, in contrast the power law behaviour ex-pected in metals from the Bloch-Grüneisen re-lation. Electrons scattering from antiferromag-netic magnons will give rise to an exponentialtemperature dependence, as will spin-disorderscattering from the localised 5 f moments them-selves. A fit to the electron-magnon resistiv-ity assuming an isotropic magnon dispersion w = √ D + Dk , such that r e − m = r + B D r k B T D e − D / k B T (5) × " + D k B T + (cid:18) D k B T (cid:19) yields, however, a spin-gap D = T N , if the shoulder at17 K corresponds to a Schottky peak whicharises from a gap of approximately 3.2 meV. Incontrast, this very splitting between the doubletground states in the ordered phase is predictedby the spin-disorder resistivity model describedbelow.Above ∼
70 K, the resistivity is well fit-ted by a r − r log ( T ) term , where r isthe residual resistivity, and r is proportionalto the interaction between the conduction elec-trons and Kondo impurities. The fit is shownin figure 11,with parameters r = . ( ) µ W cm and r = ( ) µ W cm for PuPd . Thevalue of r initially decreases with Lu dopingto 15(2) µ W cm for x = . µ W cm for x = . µ W cm for x = .
5. This increase suggests that the Kondointeraction is strengthened at half doping.The magnetic resistivity of Pu . Lu . Pd does not show the Kondo behaviour of the othercompositions, but rather increases with increas-ing temperature with a plateau region around30-80 K. This behaviour and also the exponen-tial temperature dependence of the low temper-ature part of the resistivity is characteristic ofa simple crystal field spin-disorder resistivitymodel . This model is based on the scatter-ing of conduction electrons with spin s by a lo-calised moment J through an exchange interac-tion − G s · J , giving the resistivity in the firstBorn approximation as r s − f = p Nm ∗ ~ e E F G ( g − ) Z × (cid:229) m s , m ′ s , i , i ′ h m ′ s , y ′ i | s · J | m s , y i i p i f ii ′ (6)9IG. 11: (Color online) Zero-field magneticresistivity of (Pu,Lu)Pd on a logarithmic scale.Inset is an enlarged view of the Pu . Lu . Pd composition. Solid lines are fits to the relation r − r log ( T ) described in the text.where the occupation factor for the crystal fieldlevel at E i is p i = exp ( − E i k B T ) and the conduc-tion electron population factor is f ii ′ = [ + exp ( − E i − E i ′ k B T )] − . The wavefunctions | y i i andenergies E i are determined by diagonalising thecrystal field Hamiltonian.In the absence of a crystal field, the 2 J + r ( ) s − f = p Nm ∗ ~ e E F G ( g − ) J ( J + ) (7)In our case, with a J = multiplet in a cu-bic crystal field, and in the absence of a mag-netic field, equation (6) reduces to a sum ofexponential functions, because the wavefunc-tions | y i i are fixed, and the crystal field pa-rameter can only change the splitting D CF be-tween the quartet and doublet. There is thus auniversal behaviour, with the resistivity tendingto r ( ) s − f at T ≫ D CF , and then falling exponen-tially as the temperature falls below some levelsuch that the excited crystal field states are nolonger populated. This temperature is approxi-mately 0 . D CF , so the model suggests a splitting Composition r r ( ) s − f B mf B icmf ( µ W cm) ( µ W cm) (T) (T)PuPd . Lu . Pd . Lu . Pd . Lu . Pd . Lu . Pd TABLE II: Parameters for the spin-disorderresistivity model for Pu − x Lu x Pd described inthe text. The crystal field parameter B =0.041(1) meV was determined using thedata for PuPd , and thereafter fixed in thefitting of the other compositions. D CF ≈ . ∼
60 K but is similar to the splitting ofthe ground state doublet ( D MF ) in the orderedphase, as determined by the shoulder at ∼
17 Kin the heat capacity data.In order to accommodate this splitting, weintroduce a molecular field B mf term. The lowtemperature exponential increase is then gov-erned primarily by the split doublet, with a sec-ond exponential step at higher temperatures dueto the crystal field splitting. It turns out thatthis second step is not observed in the case ofPuPd because the two steps merge into eachother. Indeed a fit to the data below T N with allparameters in the spin-disorder resistivity modelvarying freely yields a CF splitting of 14 meV,in agreement with the heat capacity data. As B mf decreases in line with T N with increasingLu doping, whilst the CF splitting remains con-stant, the two steps become more pronounced inthe calculations. These two steps are observedin the case of Pu . Lu . Pd , but the second stepis masked by the Kondo screening in the othercompositions.Using this simple crystal field model we ob-tained the parameters shown in table II. Fig-ure 12 shows the resulting fit to the data with10IG. 12: (Color online) Zero-field magneticresistivity of (Pu,Lu)Pd with residualresistivity r subtracted. Solid lines are fitsusing the crystal field model described in thetext r subtracted. The CF splitting was fixed forall Lu doped compositions to the value deter-mined from fitting the PuPd data. The molec-ular field determined from mean field calcu-lations, B icmf , with exchange parameters J = -0.202, -0.199, -0.1925, and -0.174 meV, forcompositions x =0.1,0.2,0.5 and 0.8 respectively,is also shown in the table. Apart from the caseof Pu . Lu . Pd , the fitted B mf is consistentlylower than the calculated B icmf . This is due to theoverestimation of T N in the mean field approxi-mation, because we have estimated J from T N ,and B icmf is proportional to J . Thus, B icmf is alsooverestimated.Pu . Lu . Pd shows an upturn at low tem-peratures, which cannot be accounted for bythe current model. In addition, this upturn af-fects the fit by decreasing the ratio between themaximum and minimum resistivity, and hence r ( ) s − f . The exponential increase in the resis-tivity also occurs at a higher temperature andover a broader temperature range in this com-position than in the others, which explains theanomalously high B mf . These features may beartefacts of the sample, because whilst an agedPu . Lu . Pd showed the upturn at low tem-peratures, the annealed sample did not, whereas FIG. 13: (Color online) Temperaturedependence of the Hall coefficient R H = r xy / B and magnetoresistance at 9 T (inset). The solidblack line is a fit using the measured magneticsusceptibility as described in the text . x = R R -6.97(8) -2.31(3) -0.74(1) -0.028(1) TABLE III: (Color online) Fitted Hall constantsof Pu − x Lu x Pd . All values are in ( − m C − ) .both aged and annealed Pu . Lu . Pd samplesshowed the upturn. Furthermore the resistivityof the aged Pu . Lu . Pd sample is lower thanthat of the annealed sample. This suggests thatthe annealing had not fully repaired the radi-ation damage, and thus the resistivity may bestrongly affected by crystallographic defects.Nevertheless when the fits were repeated us-ing the calculated B icmf as fixed parameters, thefitted r ( ) s − f changed by less than 10%. The fitsthus showed that the f -conduction electron in-teraction decreases with Lu doping, with a slightincrease for the x = . x = . x = .
8, in agreement with the fitsto the Kondo parameter r .We now turn to the electrical transport prop-11rties in an applied magnetic field. The temper-ature dependence of the Hall coefficient R H andthe magnetoresistivity r xx in 9T are shown infigure 13. r xx ( T ) follows the same behaviour asthe zero field resistivity shown above, with theexception that there is a small peak just belowthe Néel temperature as shown in the inset to thefigure. This was not observed previously and isreminiscent of the superzone scattering near T N in the heavy rare earths .The temperature dependence of the Hall ef-fect may be described phenomenologically by ascaling of the magnetisation, as in r xy ( T ) = R B + R µ M ( T ) (8)where R is the ordinary and R the extraor-dinary Hall constant. In figure 13, the solidlines show fits of the Hall coefficient to this re-lation using the measured magnetisation data.Both the ordinary and extraordinary Hall con-stants were found to decrease with Lu doping,as shown in table III. R is proportional tothe conduction- f electron exchange interactionstrength G discussed above, so the decrease inits magnitude further indicates that this interac-tion becomes weaker with Lu doping.Finally, we observed that the magnetoresis-tance, r ( B ) , is linear with applied field for allsamples, and showed a negative slope at hightemperatures and a positive slope at low tem-peratures, except for LuPd where the slope wasalways positive. We interpret the negative mag-netoresistance to be a sign of the Kondo effect athigh temperatures, with the normal metallic be-haviour giving a positive magnetoresistance atlow temperatures. VI. CONCLUSIONS
We have completed extensive bulk propertiesmeasurements on antiferromagnetic PuPd andthe pseudo-binary compounds Pu − x Lu x Pd .The transition temperature was found to de-crease linearly from T N = to7(2) K in Pu . Lu . Pd . Heat capacity measurements show a Schot-tky anomaly at ∼
60 K, which was interpretedas arising from a crystal field splitting between adoublet ground state and an excited state quartetat ∼
12 meV. The deduced Sommerfeld coeffi-cient, g was found to be significantly higher thanthat expected for the free electron model, with avalue of the order of 0.1 Jmol − K − determinedby fitting the data directly and by subtracting thecalculated magnetic and measured phonon con-tributions. Direct fits to the data suggest that g decreases with increasing Lu substitution. Themagnetic heat capacity was calculated using amean field model which showed that the shoul-der in the data corresponds to a splitting of thedoublet ground state in the ordered phase witha gap of ∼ µ eff increases with Luconcentration, approaching the value expectedin intermediate coupling. These observationsarise from the Kondo interaction which sup-presses the effective magnetic moment but en-hances the electronic effective mass, m ∗ . Asthe Kondo interaction decreases with Lu dop-ing, µ eff is screened less, and g (cid:181) m ∗ falls.Electrical transport measurements supportthis decrease in the Kondo interaction with in-creasing Lu concentration, x , as parameters pro-portional to the f -conduction electron couplingin a crystal field model of the resistivity and Halleffect were found to fall as x increases. Acknowledgements
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