%ALTON, RIVES, AND KHALID

~Experimental work performed at Oak Ridge National Laboratory,Oak Ridge, Tenn.

tResearch supported in part by a grant from the NationalResearch Council of Canada.

~Present address: Department of Physics, North Carolina Uni-versity at Raleigh, Raleigh N. C.

'R. L. Douglass, Phys. Rev. 129, 1132 (1963).~S. A. Friedberg and E. D. Harris, Proceedings of the Eighth

International Conference on Low Temperature Physics, London,1962 (Butterworth, London, 1962), p. 302.

3B. Luthi, J. Phys. Chem. Solids 23, 35 (1961).'Peter Carruthers, Rev. Mod. Phys. 33, 92 (1961).'Joseph Callaway, Phys. Rev. 132, 2003 (1963).Joseph Callaway and Robert Boyd, Phys. Rev. 134, A1655(1964).

C. Kittel, Phys. Rev. 110, 836 (1958).C. Kittel and E. Abrahams, Rev. Mod. Phys. 25, 233 (1953).D. Walton, Proceedings of International Conference on PhononScattering in Solids, edited by H. J. Albany (CEN, Saclay,France, 1972).

' L. Neel, R. Pauthenet, and B. Dreyfus, Prog. Low Temp. Phys.4, 344 (1964).

"R. Wyckoff, Crystal Structures (Interscience, New York, 1953),Vol. III, Chap. 12.

' A. Brooks Harris and Horst Meyer, Phys. Rev. 127, 101 (1962).' A. James Henderson, Jr. , David G. Onn, and Horst Meyer,

Phys. Rev. 185, 1218 (1969)."Roland Gonano, Earle Hunt, and Horst Meyer, Phys. Rev.

156, 521 (1967)."G. P. Rodrigue, H. Meyer, and R. V. Jones, J. Appl. Phys.

31, 376S (1960).' J. E. Kunzler, L. R. Walker, and J. R. Gait, Phys. Rev.

119, 1609 (1960)."D. Edmonds and R. Peterson, Phys. Rev. Lett. 2, 499 (1959)."S.S. Shinozaki, Phys. Rev. 122, 388 (1961).' R. Pauthenet, Ann. Phys. (N.Y.) 3, 424 (1958); J. Phys. Radium

20, 388 (1959).' R. Aleonard, J. Phys. Chem. Solids 15, 167 (1960)."A. Brooks Harris, Phys. Rev. 132, 2398 (1963)."D. Walton, Rev. Sci. Instrum. 37, 734 (1966)."R. Pohl, Phys. Rev. 118, 1499 (1960).' R. I. Joseph and E. Schlomann, J. Appl. Phys. 31, 3765 (1960).' W. G. Nilsen, R. L. Comstock, and L. C. Walker, Phys. Rev.

139, A472 (1965).

PHYSICAL REVIEW B VOLUME 8, NUMBER 3 1 AUGUST 1973

Effect of Pressure on the Kondo Temperature of Cu:Fe=Existence of a Universal ResistivityCurve

James S. Schilling~ and W. B. HolzapfelPhysik Department der Technischen Universitat Mu'nchen, 8046 Garching, Germany

and Max-Planck-Institut fur Festkorperforschung, 700 Stuttgart, Germany(Received 22 January 1973)

The electrical resistivity of Cu-110-ppm Fe has been measured for temperatures from 1.3-40 K atpressures to 82 kbar. The Kondo temperature T~ is observed to increase with pressure, having doubled invalue by 82 kbar. When plotted versus T/1~, the spin-resistivity curves are found to accurately overlap forall measured pressures, thus confirming tfie existence of a universal resistivity curve p = p(T/T~) forCu:Fe. Within the experimental uncertainty of 1.7%, the saturation value of the resistivity p(T = 0 K) doesnot change with pressure. This indicates that to this accuracy the spin S and the potential scattering at themagnetic impurity remain constant. From the pressure dependence of T~ one obtains the volumedependence of the effective exchange parameter J,tr. Approximately the same volume dependence is foundfor a series of CuAu:Fe alloys using their known average atomic volume. The Cu:Fe high pressure and theCuAu:Fe-alloy data are discussed within the context of a simple Fermi-gas model based on theSchrieffer-Wolff transformation. The pressure dependence of the resistivity of pure copper, p„h,„,„,has also beenstudied and can be understood using the Bloch-Griineisen formula with known values of the compressibilityand the Griineisen parameter. A method for experimentally determining deviations 5 from Matthiessen'srule in Kondo alloys is also presented. Such deviations can be very large; in fact, for T & 30 K, we findthat 6 = 1.3pp„,„,„.

I. INTRODUCTION

Because of its importance to the general problemof magnetism in metals, the problem of localizedmagnetic states in a metallic environment has formany years been an object of considerable study.Investigations on dilute magnetic alloys have beenpursued with particular intensity following Kondo'ssuccess in 1964 in explaining the resistance mini-mum anomaly. Kondo postulated an s-d exchangeinteraction X, „=—J,«s ~ S between conduction elec-

trons with spin s and magnetic impurities with spinS and calculated the scattering to third order inperturbation theory. For antiferromagnetic (nega-tive) values of the effective exchange parameterJ',«, the resultant resistivity is found to increaselogarithmically with decreasing temperature. Thisis in approximate agreement with measurementson a large number of dilute magnetic alloys. ~'3

Such a logarithmic divergence is, of course, notpossible for arbitrarily low temperatures, as theresistivity at T= 0 K cannot exceed the unite, ry lim-

EFFECT OF PRESSURE ON THE KONDO TEMPERATURE OF. . . 1217

it. This saturation effect at low temperaturesis clearly visible, for example, in the resistivityof Cu:Fe and Y:Ce. That such deviations from alogarithmic dependence are to be expected is seenfrom the fact that for temperatures below a char-acteristic temperature TE, the Kondo tempera-ture, the perturbation series expansion for theresistivity diverges [T» = T» exp[i/J, «n(E»)], whereT» is the Fermi temperature and n(E„) is the den-sity of states yer atom at the Fermi surface}. Attemperatures below TE, strong spin correlationsmediated by the impurity spin are set up betweenconduction electrons near the impurity and it be-comes necessary to treat the above problem usingmore sophisticated theoretical techniques. Fortu-nately, a considerable mathematical apparatus wasavailable from many-body theory which facilitatedfurther work on the Kondo problem. ' Whereassome of the ensuing theories were only applicablefor T& T~ or T& T&, Hamann and Suhl and%ongmere able to derive approximate expxessions forthe resistivity valid at all temperatures. Theyfind that the resistivity decreases from its unitarylimit value at T = 0 K, proceeding with a logarith-mic dependence through T~, to a high-temperatureplateau for T» T». The measured resistivity isindeed found to have this general behavior. How-ever, a careful comparison between experimentand theory over a wide temperature range revealsonly qualitative agreement. Despite enormousefforts, it has to date not been possible, startingwith the deceptively innocuous looking s-d Hamil-tonian, to derive an exact expression for the tem-perature dependence of any experimentally acces-sible quantity. All available expressions involveapproximations whose exact implications areunclear. In short, although qualitative agreementbetween experiment and theory does exist in theKondo problem, it has not as yet been possible toobtain the kind of agreement which would enableone to reliably extract the sort of detailed infor-mation on dilute alloy systems which is needed.

Under these circumstances it mould be desirableto establish a set of fixed points or theorems whichthe exact solution to the 8-d Hamiltonian must sat-isfy. This mould be useful not only in helping todecide which approximations should be made insolving the Kondo problem, but also it would serveas a cheek on the validity of existing theories. Onesuch fixed point was supplied by Mattisa whoproved formally that for antiferromagnetic s-d ex-change, the ground state (T = 0K) of the magneticimpurity problem for S= —,

' is a singlet. As waspointed out by Heeger, 3 this theorem implies thatthe integral of the impurity-conduction-electron-spin correlation is fdsr(f s(r)) = ——,'. Several so-called ground-state theories, valid only for T & Tz,are found to satisfy this theorem. On the other

hand, Zittartz and Muller-Hartmann 0 have dem-onstxated that this theorem is violated by the the-ories of Hamanne and Suhl and Kong, ' amongothers. As is confirmed experimentally, theselatter theories are thus not expected to be reliableat temperatures much below T~.

Another theorem which seems to be satisfied byall theoretical expressions derived3' from X,~is that T~ merely plays the role of a sealing tem-perature; i. e. , if f is the property in question,then f=f(T/T»). The possibility of the existence ofsuch "universal" curves seems to have been firstpointed out by Heeger. 3 Work by Anderson andYuval'2 seems to imply that f=f (T/T») followsrather generally from X,„~. In fa,ct, the existenceof universal curves appears to also follom fromspin-fluctuation theories 3' ~ based on the moregeneral Wonf or Anderson Hamiltonians which in-clude X, ~ as a special case. Experimentally, onecould test for the existence of a universal depen-dence f=f(T/T») by measuring the temperature de-pendence of f for different values of T». Resistivitymeasurements by Loram et af. on a series ofCuAu:Fe alloys suggest the existence of a univer-sal dependence of p on T/T». However, it has beenshown~5 that in alloys the local atomic environ-ment of a magnetic ion can play a decisive role indetexmining its magnetic state; Stax has alsopointed out~6 that it is quite possible that CuAu isnot a homogeneous host alloy to Fe. One would ex-pect, on the other hand, that applying uniform pres-sure to a sample should vary its properties in arelatively well-defined may. A preliminary reporton high-pressure resistivity measurements onCu:Fe was given by Schilling et ul. and Summerset n/. ' High-pressure experiments on I.a:Ceand V:Ce'9'~' indicate a rapid increase of T& withpressure in these systems, but are not able to testfor a universal dependence.

A further result common to calculations basedon both R 6'~ and the Anderson '~ and %olffHamiltonians is that at T = 0 K the exchange scat-tering resistivity has reached its unitary limitvalue p„„=28po, where pa is the resistivity fors-wave resonant impurity scattering. ~ If the or-dinary non-spin-flip potential scattering (phaseshift hv) at each impurity is also included, the uni-tary limit value is found+ to be reduced by thefactor cos 5~. This would imply that if T~ couldbe shifted, without changing po, S, or 5~, the re-sistivity at T = 0 K would remain constant.

Dilute Cu: Fe would seem to be a suitable alloyon which to study the questions of the existence ofa universal resistivity curve and the change of S or5„with pressure. The resistivity of Cu: Fe hasbeen extensively studied over a wide concentrationrange ~' to very lom temperatures. In addition,its spin-scattering xesistivity exhibits a transition

1218 J. S. SCHILLING AND W. B. HOLZAPFEL

from an approximate logarithmic temperature de-pendence above 10 K to a T ~ dependence belowabout 2 K, thus allowing an accurate estimate ofthe value of the resistivity at T = 0 K.

II. EXPERIMENTAL

The alloys used in these experiments were pre-pared from 99.999+%-pure Cu and 99.998%-pureFe. The concentration of magnetic impuritiesin pure Cu as estimated by the manufacturer isquoted as being typically Fe & 0. 7 ppm and Cr & 1

ppm. The resistivity ratio of pure Cu samplesprepared in the same manner as the alloy sampleswas found to be psoo „/p«—- 950, which is compat-ible with a total impurity content less than 5 ppm.A very weak Kondo effect was also observed onthis sample, indicating an Fe or Cr impurity con-tent of less than 0.2 ppm.

The alloys were prepared by induction meltingthe components on a water-cooled silver boat inhigh vacuum. The sample was held in the moltenstate for a total period of 15 min, being rotatedseveral times and jostled continually. Initially a0. 5-at. % Fe alloy was prepared and then furtherdiluted to obtain a nominal Fe concentration of 114ppm. Chemical analysis of this sample indicated(110+5)-ppm Fe. After being cold rolled into 50-p-thick foils, the samples were cut to size and tem-pered for 15 min by Joule-heating the sample to700'C in a water-cooled chamber under high vac-uum. The homogeneity of these samples was con-firmed to 1 p. by electron-microprobe examination.More detailed information on the homogeneity ofCu: Fe alloys is provided by a careful study oftheir magnetization by Hirschkoff gt a). ' Theyconclude '.hat a 110-ppm alloy might be expected tocontain 1 ppm of Fe-Fe pairs. Such effects can bevery important at extremely low temperatures(- 10 mK), where the main temperature dependenceof the resistivity is due to scattering from atompairs, the single-ion scattering having saturatedout. In the temperature range of our experiments(above 1.3 K), the temperature dependence of theresistivity from single-ion scattering is large andone would expect the effect of Fe pairs on our mea-surements to be very small indeed. This expecta-tion is confirmed by the low-temperature resistiv-ity measurements of Star. To test for possibleclustering in our samples we also carried out resis-tivity measurements on 55- and 27-ppm Cu: Fe al-loys which revealed a linear dependence of theKondo effect on Fe concentration. In light of theabove, it seems reasonable to assume that the re-sistivity measured in these experiments is due toscattering from isolated Fe ions.

High pressures are applied to the samples usingthe quasihydrostatic Bridgeman technique of op-posing anvils. The high-pressure cell and anvil

v, P/ Zg vz

FIG. 1. Arrangement of samples and electrical leadson epoxy disk (4-mm diam) mounted in pressure cell.Cu: Fe and Cu are the Kondo and pure copper samples,respectively; Pb serves as internal manometer; Au-0.5% V detects pressure-cell deformation. I and V labelcurrent and potential leads, respectively.

assembly is mounted inside a massive Cu-Beclamp which retains the applied force after remov-al from a hydraulic press. The applied pressureis changed at room temperature after each low-temperature resistivity measurement. The B4Canvils used are semiconductors and thus serve asinsulators at He temperatures, thereby simplifyingelectrical insulation problems. The samples aremounted between two epoxy disks each 4 mm indiam and 0. 3 mm thick. These disks in turn aresurrounded by an annular Pyrophyllite ring 10 mmin diameter and 0. 6 mm thick, which is itself sup-ported by a belt made of a special hardened steel.

The arrangement of the samples on the epoxydisk is shown in Fig. 1. The three samples (Cu: Feor Cu; Pb and Au:V) are arranged in series. Es-sential for an accurate pressure determination atlow temperatures is the presence of an internalmanometer in the high-pressure cell itself. ThePb sample serves this function well because thepressure dependence of its superconducting tran-sition temperature T, is known to 10% accuracy.In addition, the width of the superconducting tran-sition b, T, gives a measure of the pressure inhomo-geneity in the pressure cell. Typical values of thepressure inhomogeneity over the area of the Cu: Fesample as estimated from b, T, lie between 2% and7% of the measured pressure.

Applying or removing pressure from the pres-sure cell was found to not only change the amountof defect scattering in the sample but also to alterthe sample's dimensions, the magnitude of thischange depending on the pressures involved and

EFFECT OF PRESSURE ON THE KONDO TEMPERATURE OF. . .

the past history of the pressure cell. Due to thenature of these investigations, it mas essential tobe able to estimate any changes in the length/areafactor n of the Cu: Fe sample. Onemaytoestimatechanges in a is to include within the pressure cellan additional sample whose resistivity has a verylom intrinsic pressure dependence. This is thecase for the alloy Au+0. 5-at. % V where at roomtemperature we estimate that (hp/-pnP) & 0. 04%/kbar. In order to minimize both changes in n andvariation in defect scattering in measurements atdifferent pressures, the pressure cell mas initiallypreloaded to 90 kbar and then unloaded to zeropressure before commencing with high-pressuremeasurements. The success of this procedurewill be shomn in the Sec. III. Complete detailsof the high-pressure techniques employed in theseexperiments mill be described elsemhere. ~

The cryostat used in the high-pressure measure-ments was of standard design. Temperatures be-tween 1.3 and 60 K mere determined by a calibratedGermanium resistor supplemented by a copper-constantan thermocouple for higher temperatures.Both temperature sensors are located directly nextto the high-pressure anvils. The accuracy of thetemperature determination is always better than0. 5%.

Resistivity measurements were carried out usingthe standard four-terminal technique. The currentsource is constant to better than 10 during the 6hr needed to make a temperature run. The samplepotentials are first amplified by a galvanometricphotocell amplifier and then displayed on an inte-grating 5~ digit digitalvoltmeter. The relative ac-curacy of the measurement is better than 10

The resistivity measurements at zero pressuremere carried out in a separatecryostatonaCu: Fesample with known dimensions. The relative ac-curacy of the measurement is better than 10, theabsolute accuracy being limited to about 1% by un-certainties in the length/area determination. Thesplinters of Cu: Fe alloy used in the high-pressuremeasurements are taken from this sample. Acomparison of the zero-pressure and high-pressuredata allows a determination of the absolute resis-tivity scale for all data. The experimental detailsof the zero-pressure measurement will be de-scribed in a future publication.

lnT/T»[ln T/T +» S(S+1)]'i') '

mhere the Kondo temperature including potentialscattering is nom given by

T» —Tjp exp [I/tE(Ey ) elegy cos 5r]

It is noted that this expression is of the formp = p(T/T») expected for a universal resistivitycurve, assuming 8, 5~, and po are constants. Theresistivity given by Eg. (1) is shown schematicallyin Fig. 2 3 for several values of 1g. Although itis well known that this expression agrees well withexperiment only for T & T&, its qualitative featuresare probably correct at all temperatures. FromFig. 2 it is seen that as T» increases (8, 5„, and

po being held constant), the spin-scattering curve isshifted bodily to higher temperatures. A change ineither 8, 5~, or po as Tz increases mould have theeffect of changing the shape of the spin-scatteringcurve as it is shifted.

From Eq. (1) it is seen that the saturation valueof the resistivity is given by

p»„(T = 0 K) = 2Spo cos»5„.

This saturation value is thus a sensitive function ofSy Po and 5y.

IV. RESULTS OF EXPERIMENT

A. Temperature Dependence of Resistivity

In Figs. 3(a) and 3(b) are shown the results ofresistivity measurements on Cu-110-ppm Fe atpressures to 82 kbar and temperatures from 1.3to 40 K. From the data it is immediately seenthatincreasing the pressure shifts the spin-scatteringcurve to su'cessively higher temperatures. Ap-plying pressure to Cu: Fe therefore increases theKondo temperature T~. The rapidly rising resis-

2spp

2 5 ppcps 5y --2

3-

C7O

III. RESULTS OF THEORY

Before presenting the results of the high-pres-sure resistivity measurements on Cu: Fe, let usconsider the following approximate expression forthe resistivity derived from R, ~ by Hamanne withthe unitary limit value given by Schxieffer andgeneralized by Fischer~ to include potential scat-tering:

2Sp sin2@

I

lOOLOQl 10 SOQT(K)

FIG. 2. Spin-resistivity curves from Hamann's ex-pression [Eq. 0.)] for increasing values of the Kondotemperature Tz. Because Eq. O.) is of the form p= p(T/Tz), increasing Tz shifts the spin-resistivity curvebodily along the temperature scale.

14.8-

14.6-

«00

& 14.4-E

14.2-V

1QI-

tivity of the Cu host has been subtracted out} areplotted as a function of the reduced temperatureT/T» and are seen to overlap within the experimen-tal accuracy over the entire range T& T(p „). Theoverlap for the othex' pressures is similarly accu-rate. Thus, over the pressure and temperaturerange considered here, the present experimentsconfirm the existence of a universal resistivitycurve p= p(T/Tz) for Cu: Fe.

It should be pointed out that the fact that the re-sistivity data shown in Figs. 4(a) and 4(b) still haveminima, even though the host resistivity has been

13.6-

13l-

EP

II

CL

14.6.

—,14.4-

E

C 14.2-

07 1

I & I I I 1 I I i I I

2 4 6 8 10 20 30 40

temperature T {K)

114-

112-

8 no-

5~~1G8-

CL'3 1GS-

1G4-

u2- (a)

P(kbar)

44

"; 82,4

p4

X

45 "

00 0

O, O'CI

I I I I I I I I

0.5'

1.0 l5

13.6-

(b) 11.4

0,4-07 1

I I I I I I I I I t

2 4 6 8 10 20 30 40temperature T(K)

FIG. 3. Total measured resistivity of Cu: Fe versustemperature fox different pressures. Dots are our data,crosses are from Loram eg aE. (Ref. 11). The solidlines connecting dots are to distinguish data at differentpressures. c is Fe concentration (110 ppm). (a) Rela-tive accuracy is 10 . Pressure cell was not preloaded.(b) Relative accuracy is 10 . Pressure cell was pre-loaded.

tivity above 20 K is due to phonon scattering. It isalso seen that with increasing pressure the resis-tivity minimum fills up and that the temperature ofthis minimum, T(p &,), increases slightly. Itwillbeshown that the former effect is due to the shiftingof the spin-scattering curve to highex temperatureswhereas the latter follows from the reduction of thephonon scattering with increasing pressure.

In Figs. 4(a) and 4(b) some of the data (the resis-

'11.0-

E

gl0.8-

CL

10.6-

1G4-

1G2-

Cu-110ppm Fe

(b)

401

4

P(kbar)

0ba &.53 ~

o

0"39 ~0 ~ g~w 0

a 28O~

G05 G1 0.5 1.0 15T/T»

FIG. 4. Reduced resistivity of Cu: Fe versus reducedtempexature fox three px'essures. ~= p~- p~~~. p&

and p~~ are the resistivities of Cu: Fe and Cu, respec-tively. The data shown in (a) and (b) are taken fromFigs. 3(a) and 3(b). The values of T~ used for the givenpressures are listed in Table I.

E FF EC T OF PRE SSURE ON THE KONDO TEMPERAT URE OF. . . 1221

TABLE I. Shown for Cu: Fe as a function of pressureP are the Kondo temperature T~, the volume change hV/V, the density of states at the Fermi surface n(E~), andthe effective exchange parameter J@&. Data taken on thesame sample are arranged in groups.

P(kbar) T~ (K) —AV//V( 0/) n(Ep) (states/eV atom) J«(eV)

27~ 145+ 3.555+ 482+ 5.7

38+ 1

28 R 0.839+ 153 + 0.8

55+ 3.972+ 6.5

24. 0

31.637.740. 350.6

35.2

31.234, 639.0

41.047. 5

1.752.823.394.79

2.42

1.802.483.27

3.384.30

0, 294

0.2920.2900.2900.288

0.291

0.2920 ~ 2910 ~ 290

0.2900 ~ 289

—0.418

—0.435—0.448—0.451—0.470

—0.443

—0.435—0.443—0.451

—0 ~ 454—0.465

'See Ref. 35. 'See Ref. 30.

subtracted out, is due to large deviations fromMatthiessen's rule, as will be shown shortly. Forthis reason it is not possible for the curves to over-lap for T& T(p „).

In Table I are listed for each pressure the valuesof TE which give the best overlap. Since our zero-pressure data agree well with the resistivity data(0. 3-300 K) of Loram et al. in our temperaturerange (1.3-40 K), we use their value of Tz = 24 Kfor Cu: Fe at zero pressure determined by theslope of a T~ plot at the lowest temperatures. TheKondo temperature at high pressures is determinedby the temperature shift necessary for the curvesto overlap. Using Eq. (2) and the known pressuredependence of n(Ez), 30 one can calculate the pres-sure dependence of the effective exchange param-eter J,«. From Table I we see that a pressure of82 kbar is sufficient to approximately double T~,implying that J,« increases in magnitude by about12 p.

B. Unitary Limit

In Fig. 3(a) the data at 27, 45, 55, and 82 kbarwere taken, in the given order, on the same samplewithout having preloaded the pressure cell. Thesaturation value of the resistance, R(T =0 K), canbe determined for each pressure by using the known

T dependence of the resistivity at low tempera-tures. ' 6 For the data shown in Fig. 3 the satu-ration value of the resistance (not resistivity) isfound to increase by 14% between 27 and 82 kbar.The most likely explanation for this shift is thatthe initial application of pressure causes irrevers-ible deformation of the pressure cell (and the sam-ple) leading to an increased amount of temperature-independent defect scattering. The data shown inFig. 3(a) a,re thus shifted vertically to have thesame value of R(T= 0 K) for all pressures Changes.in the length/area factor of the Cu: Fe sample

would also cause R(T=0 K) to shift. This possi-bility will be discussed below.

In an effort to minimize changes in defect scat-tering for different pressures, the pressure cellfor the data shown in Fig. 3(b) was initially pre-loaded to 90 kbar and then unloaded completely be-fore proceeding with measurements at successivelyhigher pressures. The data at 28, 39, and 53 kbarwere taken on a different sample than the data at55 and 72 kbar. The success of the preloading pro-cedure is demonstrated by the fact that between 28and 53 kbar, R(T = 0 K) increases by only 0. 14%and between 55 and 72 kbar by 0. 063%.

A Au: V sample is included in the pressure cellto supply information on changes in the length/areafactor o. of the Cu: Fe sample, as was discussed inSec. II. These measurements indicate that for theAu: V sample between 28 and 53 kbar o. decreasesby 1.4% and between 55 and 72 kbar increases byonly 0. 2%. From the compressibility alone onewould expect an increase in n of 0. 5% and 0. 3%,respectively. The measured values of o give arough indication of the deformation of the pressurecell as the pressure is increased and they there-fore represent the accuracy to which one can es-timate changes in the length/area factor for theCu: Fe sample. The actual changes in R(T = 0 K)with pressure for Cu: Fe are much less than thechanges in n. Therefore, to the accuracy of ourmeasurement (1.4% between 28 and 53 kbar and0. 2% between 55 and 72 kbar), the saturation valueof the resistivity is independent of pressure.

As was pointed out in Sec. 0, a zero-pressuremeasurement on a Cu: Fe sample withknowndimen-sions is used to determine the absolute resistivityscale of the high-pressure data. For pressuresbelow about 20 kbar, rather large irreversible de-formations of the sample take place whether thepressure is being increased or decreased. It isthus not possible to carry out a measurement nearP= 0 on a sample with the same dimensions andamount of defect scattering as the high-pressuredata. The P= 0 measurement was actually carriedout on a tempered Cu: Fe sample whose length/areafactor was known to about 1'Pp. In order to comparethis measurement to the high-pressure data, dif-ferent amounts of temperature-independent defectscattering p~ are added to the P = 0 data until thesedata, when shifted in temperature, overlap bestwith the P= 28-kbar data. The amount of defectscattering added, p~=35 p,Qcm, agrees within 40%with that measured on a pure Cu sample before andafter pressure application. The confirmation ofthe existence of a universal resistivity curve forCu: Fe at pressures below 28 kbar is thus not ascomplete as for the other pressures. However,the accuracy of the overlap of the P= 0 and 28-kbardata and the fact that the necessary temperature

J. S. SCHILLING AND %. B. HOLZAPFEL

shift (Tr increase) between these two pressures isproportionately the same as for the other pressures(see Table I), supports the existence of a universalresistivity curve for pressures below 28 kbar .

The saturation value of the resistivity is deter-mined from the P=O data to be p„„(T=OK)=11.6+ 0. 6 IfQ cm/at. %, where the accuracy is limitedby uncertainties in the Fe-concentration determi-nation. This value of the saturation resistivity isin reasonable agreement with estimates from othersources. ~ '~5 Using Efi. (3) it is found that for aCu host p„„(T= 0 K) = 3. 8(2S) cos~b» pQ cm/at. /g.

Furthermore, using our value of the saturation re-sistivity and 8= —, from susceptibility measure-ments, 3 one obtains 5~ = 0. Negligible potentialscattering, however, seems to be inconsistent withthe fact that Cu: Fe exhibits a sizable peak in thethermoelectric power. 3'+ EPR investigations3~

imply 8= 2 which results in the more reasonablevalue 5~ = 29'.

C. Phonon Scattering

between 31 and 57 kbar rW/b = 14%, in goodagree-ment with our measurements. It is interesting tonote that at temperatures near 300 K, the 57-kbarcurve in Fig. 5 is displaced downward by only about5% from the 31-kbar curve. This agrees well withthe downward shift of 4/0 expected for the high-temperature limit (p~ T).

The value of 5 at all pressures is determinedabsolutely by extrapolating the high-pressure datato P = 0 and comparing this with the zero-pressure

100

10-

The temperature dependence of the resistance ofpure Cu was measured on a single sample for pres-sures 31+2.4 kbar and 5V +4 kbar at temperaturesbetween 4 and 300 K. In a separate experiment, azero-pressure measurement was carried out on apure Cu sample with known dimensions. The re-sults of these measurements are shown in Fig. 5(note that the residual defect scattering has beensubtracted out). It is seen that for temperaturesbelow approximately 40 K, a p~ T~ law fits thedata quite well. This result is in agreement withthe low-temperature (T«eff = 343 K for Cu) depen-dence predicted by the Bloch-Griineisen expres-sion~3 p,„„~= bTS, where b~ 1/(2 e~~), r, is theradius of the signer-Seitz sphere, and 6& is theDebye temperature. At temperatures above 40 K,the data are seen to bend over, heading toward theapproximate p~ T dependence expected for T»e~. 33

Inspection of the pressure dependence of the datareveals that for temperatures below 40 K, the re-sistance curve for 53 kbar is shifted downwardfrom that at 32 kbar by roughly 15%. This decreaseis due mainly to an increase in the Debye tempera-ture with pressure, as will be seen below. Usingthe above low-temperature expression for the re-sistivity, one finds

(4)

C)II

I

LKI

I

~ 0.1-

0.01

100

10E

CDII

I

lQI

0.1

o 57 kbar~ 31 kbar

pure Cu

P=O

where off = —(jeff/eff)/(&V/V) is the {Debye) Grunei-sen parameter. Typical values of the Gruneisenparameter for Cu determined by thermal-expansionand specific-heat measurements vary from about1.5 to 2. 0 at temperatures from 4 to 300 K. Inthe following we set @=1.V. For the compressi-bility of Cu we use the expression~s b V/V= —6. 9X 10 P (kbar)+ 1.3&&10 ~Pa (kbar). One finds that

20 2QQ 300 43010 40 60 80 100temperature T(K}

FIG. 5. Resistivity of pure Cu sample versus tem-perature for three pressures. The residual defect re-sishvity p(T= 0 Ig has been subtracted out. The high-pressure data can be normalized by comparing them tothe I'= 0 data which were measured absolutely.

EFFECT OF PRESSURE ON THE KONDO TEMPERATURE OF. . . 1223

V

II

CL4 &OaIIII 44

14.8-

14.6-

14.4-

4&

~I4

04

O44O

14.2-

5~148-

.?IA

g 13.B-

136-

Cu-110 ppm Fe

pt (53 kbar)

( t Qhonon(53 ar)8

0

p-.„-—— (28 Mm)

(temp. shifted )

00

III

13.4-

per 1

13.0-1

I

2 5 10 20 50 1temperature T(K)

FIG. 6. The total resistivity of Cu: Fe at P= 53 kbaris shown before and after subtraction of the Cu host re-sistivity p~b, . The universal resistivity curve for T&20 K is estimated by the temperature-shifted reducedresistivity at P=28 kbar and for higher temperatures bya straight line, thus allowing an estimate of the deviation4 from Matthiessen's rule.

data which were measured absolutely. Using Eq.(4) with y= 1. I gives the following expression forthe resistivity due to phonon scattering:

p» „=2. 60x10 ~0(1+n V/V) 'ST p.Qcm. (5)

In Figs. 3(a) and 3(b) it was seen that with in-creasing pressure the resistivity minimum fills upand that the temperature at this minimum T(p „)increases by approximately 2 K between 0 and 82kbar. It is now possible to understand the originof these two effects. The shift of T(p „)is due toa decrease in the phonon scattering with increasingpressure. Assuming that near the minimum p,yf,~ lnT and p»„„——bT, it can be shown that dT „/T „=4b/5b. The former assumption is seen[Figs. 3(a) and 3(b)] to hold even for P= 82 kbarand the latter was shown in Fig. 5 to be valid. Thusby using Eq. (4) we find that between 0 and 82 kbaraT „=1.8 K, in good agreement with experiment.The increase of p(T „)with increasing pressureis due to the shifting of the spin-scattering curveto higher temperatures. Near the resistivity min-imum, the spin scattering dominates the phononscattering (p,yI 15p,„„„)and the spin-scatteringresistivity is essentially linear in lnT. Since the

10-8-

6-

3-

O

Qr

ar

05-

15 20 30T(K)

40

minimum itself moves only slightly with pressure,it would be expected that as p„„is shifted to highertemperatures, p(T „)should increase by a pro-portional amount. Our measurements confirm thisexpectation.

In Fig. 6 the Cu: Fe data at 53 kbar are shownbefore and after p,„„is subtracted off. As dis-cussed by Loram et al. , the fact that the reducedresistivity np still has a prominent minimum isdue to large deviations from Matthiessen's rule.To estimate the magnitude of these deviations, wealso show in Fig. 6 the reduced resistivity hp forP = 28 kbar which has been shifted in temperatureuntil it overlaps the 53-kbar data. Due to the ex-istence of a universal resistivity curve for Cu: Fe,the spin-scattering resistivity is thus quite well de-termined out to about 30 K. A straight line ap-proximates the spin-scattering curve to highertemperatures allowing the deviation from Matthies-sen's rule to be estimated to about 40 K. Thesedeviations are seen to be approximately as largeas the phonon scattering itself. In Fig. 7 we plotthe estimated deviation L as a function of tempera-ture. The deviation at lower temperatures is seento have a T dependence and is therefore propor-tional to the phonon resistivity. This result is inagreement with the findings of Whall et al. 36 on aseries of gold alloys and with theoretical yredic-tions3~'3 for temperatures low enough that the pho-non scattering is small compared to the impurityscattering. The deviation of 4 from a T' depen-dence at temperatures above 30 K could be at leastpartially due to an incorrect estimate of the spin-scattering resistivity for these temperatures.However, it is known experimentally'~ and expect-

FIG. 7. Deviation from Matthiessen's rule versus tem-perature estimated by method shown in Fig. 6. For T&30 K, 4 has approximately the same temperature de-pendence as p&o~.

J. S. SC HILLING AND W. B. HQLZAPF EL

ed theoretically that 4 levels off and has a max-imum at about 60 K for ll0 ppm Fe in Cu. Exten-sion of the present measurements to higher tem-peratures a d pressures is necessary for a t or-ough investigation of these questions.

V. DiSCUSSION

A. Universal Resistivity Curves

In Sec. IV we have shown that, over the pressureand temperature range considered, the present ex-periments confirm the existence of a universal re-sistivity curve p =p (T/Tz) for Cu: Fe. The exis-tence of universal curves f=f(T/Tr) in the Kondo

problem is a result common to all approximate ex-pressions derived from the s-d Hamiltonian. Infact, this result appears to follow quite generallyfrom X „. Anderson and Yuval' have demon-strated an "asymptotically exact equivalence" be-tween the Kondo problem based on X~~ and the sta-tistica1 mechanics of a certain one-dimensionalgas. It is found that the renormalized partitionfunction of the problem is a function of terms likeT/Tz and j/Tr. This'implies that in expressionsfor measurable quantities calculated from this par-tition function the temperature T is always scaledby Tz. It is conceivable that terms involving T»and J' could appear separately from T; however,from an inspection of the spectral function, it would

seem intuitively reasonable that such terms tendto cancel out. The present work implies that suchterms, if they do exist, do not enter the expres-sion for the resistivity in any important way.

It should also be pointed out that the existenceof universal curves is a prediction which is not re-stricted to expressions derived from the s-d Ham-iltonian but also applies to spin-fluctuation theo-ries 3' based on the more general %oUf or Ander-son Hamiltonians which include K, ~ as a specialcase.

In Sec. III it was shown using Eq. (I) (Fig. 2)that the shape of the spin-scattering curve remainsunchanged as T~ increases only if pz, 5&, and S re-main constant. The present work supports this re-sult. As is seen from Eg. (3), the constancy of

pz, 5~, and S in these experiments is indicatedeven more directly by the fact that the saturationvalue of the resistivity p„„(T= 0 K) does notchange as T& increases. These considerations areexpected to hold on general grounds since p0, 5„,and 8 must enter any expression for the spin-scat-tering resistivity.

From Eg. (I) it is clear that other Kondo alloysneed not exhibit the same "universal" resistivitycurve as Cu: Fe because different values of theparameters p+ 6~, and 8 might be appropriate.All curves, however, should have the same quali-tative dependence as shown in Fig. 2. To investi-

-G50-

-0.45-

-G40-

-G30-

Ef (eV)6.0

I7.0

I

I

I

tD

O

l o'o 0

&o

IOI

)I(l

II

/

/II

/

r

1.45

~—Au-Cu Cu

it I I I I I I

140 135 19) 1.05 1.00 Q95I I

125 QO 115V/ VCu

FIG. 8. Effective exchange parameter J&z as a func-tion of the average atomic volume of sample relative tothat of Cu at P= 0. Circles are from our high-pressuredata on Cu: Fe, crosses are from Cu-Au: Fe data byLoram et gl. (Ref. 11), and triangles are preliminaryresults of high-pressure data on Au: Fe by Crone et al.6Lef. 42). The dashed line is a fit using the SW trans-formation fEq. (6)]. Values of the Fermi energy Ez givenare those of a free-electron gas.

gate this question we are currently carrying outresistivity measurements on Au: V and other al-loys over a wide range of temperature and pressure.

On the basis of the above results, it would seemreasonable to expect that the universal resistivitycurve for Cu: Fe exists over a much wider rangeof pressure and temperature than was accessiblein the present experiments. If this is true for oth-er alloys as well, then for those alloys where TE

increases with pressure, the application of veryhigh pressures would in fact enable one to deter-mine a portion of the universal curve which forP= 0 would lie at lower temperatures. This meth-od of mapping out p„„(T/Tz) would be especiallyadvantageous for rare-earth alloys such as Y:Ceand La: Ce, where the increase in T& with pres-sure is extremely rapid.

B. Volume Dependence of Tz

In these experiments it is found that a pressureof 82 kbar is sufficient to approximately double theKondo temperature of Cu: Fe. In Fig. & we plotthe values of J,«shown in Table I as a function ofrelative volume. Also shown in this figure are theresults of the resistivity measurements of Loram

EFFECT OF PRESSURE ON THE KONDO TEMPERATURE OF. . .eg al. on a series of Cu-Au: Fe alloys. It is in-teresting to note that (with the exception of theAu: Fe alloy) both alloy and high-pressure datashow roughly the same volume dependence. Thiswould seem to support the point of view that inthese experiments changes in the value of J,«aremainly a function of the macroscopic volume andare not dependent on details of the band structureor the immediate surroundings of the magnetic im-purity. In the simplest model, therefore, we as-sume that changes in sample volume (whether lat-tice contraction under pressure or lattice expan-sion due to alloying Au in Cu) merely have the ef-fect of shifting the Fermi level relative to the vir-tual bound states of the magnetic impurity. Thevalue of the Fermi energy and its volume depen-dence are estimated from the free-electron model.The position of the virtual bound states of the mag-netic impurity are assumed fixed with respect tothe bottom of the conduction band.

An expression which connects J«with the posi-tions of the virtual levels relative to E& is given inthe magnetic limit by the Schrieffer-Wolff (SW)transformation~:

(6)

where J, is the ordinary positive exchange integral,y is the covalent admixture matrix element, 8 isthe impurity spin, E' is the energy required totransfer an electron from the Fermi surface to theFe impurity in its ground state, and E is the en-ergy required to remove an electron from the im-purity and place it at the Fermi level. The aboveexpression for 8, which was initially dex'ived onlyto lowest order, was shown by Hamann~ to retainvalidity in an asymptotically exact path-integraltheory of magnetic alloys. Since 4, is estimatedto be small 0 [8,= 0. 2 eV/(2l+ I)= 0. 04 eV fortransition-metal impurities] and essentially pres-sure independent, it will be neglected in the fol-lowing. For the impurity spin we use the EPRvalues S= 2 for Fe in Cu. For Mn in Ag it hasbeendetermined that U= E'+E = 5 eV; we use thisvalue here in the absence of good spectroscopicmeasurements for Fe in Cu. Going from Au to Cuthe value of the free-electron-gas Fermi energy in-creases from 5. 5 to 7. 0 eV. Inserting the abovevalues in Eq. (6) we obtain the dashed curve inFig. 8. The values of the parameters used in thefit are v=0. 8 eV, E'=1.0 eV, and E =4. 0 eV forCu: Fe. Going from Au to Cu, E' thus decreasesfrom 2. 5 to 1 eV.

In Fig. 8 it is seen that the 8% curve gives goodqualitative agreement with the Cu: Fe-high-pressureand Cu-Au: Fe-alloy data. One obvious method oftesting the above simple model would be to carryout high-pressure measurements on Au: Fe. Forthe 8% curve shown in Fig. 8 one would expect a

very weak pressure dependence of T~. More gen-erally, it can be inferred from Eq. (6) that if J,«is less for Au: Fe than for Cu: Fe, as is the case,then dZ,«/d V should also be less. This resultholds independent of the exact values assumed forthe parameters in Eq. (6). Preliminary high-pres-sure measurements on Au: Fe4~ shown in Fig. 8confirm this expectation, although the agreementwith the 8% curve is rather poor. Assuming thatthe pressure dependence of J,« for Au: Fe shownin Fig. 8 is correct, thexe could be a number ofreasons for this discrepancy. The values of T~ forAu: Fe and Cu: Fe were determined by fits to dif-ferent theoretical expressions and are thereforenot directly comparable. Vfhereas the value T&= 24 K obtained from resistivity measurements onCu: Fe seems reasonable, one can only estimatethat for Au: Fe, T~ & 0. 5 K or equivalently 1Z,« l

«0. 2V eV. In addition, as is seen from Eq. (2),different amounts of potential scattering for Fe inAu and Cu could also affect the determination ofJ,«. Shifting Tz for Au: Fe to Lower temperaturesand increasing U would improve the agreement sub-stantially. Unfortunately, there is at the momentno independent information to justify this procedure.

Besides possible large errors in the values ofthe parameters used, it could weLL be that the sim-ple model presented here is itself inadequate;there are a variety of modifications which mightbe necessary. Firstly, the virtual bound state ofFe is likely to be located energetically in or nearthe d bands of Cu or Au which lie only 2. 1 and 2. 5

eV, respectively, below the Fermi level. Appli-cation of pressure would tend to increase the over-lap of band and virtual 3d wave functions and bx'oad-en the virtual bound state, thereby increasing theinterband mixing. If this were the dominant effect,it would explain why in all known high-px'essure ex-periments on Kondo alloys (Cu: Fe, Au: Fe,Au: V, 39 Y:Ce, + and La: Ce ~) the Kondo temper-ature is found to increase with pressure (note thatin the simple model T~ could decrease with increas-ing pressure if E «E'). Secondly, in the simplemodel we have also neglected any influence of theimmediate local surroundings on T~. The ionic ra-dius of Au is about 40% larger than that of Cu andthus for a given lattice parameter the cage formedby the surrounding ions within which the Fe" ionxesides is smaller for Au than Cu. This effectcould have the tendency to shift T& for Au comparedwith Cu to relatively higher temperatures. Thixdly,it should also be noted that we have completely ig-nored numerous orbital effects such as orbit-orbitexchange, mixed spin and orbit exchange, andcrystal-field splitting. As pointed out by Hirst, 3

the orbital angular momentum of M impurities incubic metals is not necessarily quenched. In fact,EPR studies on Fe in Cu indicate that g4 2 where

1226 J. S. SC HIL LING AND W. B. HOLZAPFEL

g is the Lande g factor. The assignment to a mag-netic impurity of orbital as well as spin degreesof freedom can be carried out in the context of ageneralized s-d interaction model. ~' However, aspointed out by Hirst, 46 the generalized Kondo prob-lem arising from the generalized s-d interactionmodel is very similar to that arising from K = —Js ~ S.It would, therefore, not be expected that considera-tion of the generalized model would change the mainconclusions of this paper. Inclusion of orbital ef-fects and crystal-field spiittings could, however,complicate the interpretation of the pressure de-pendence of J,ff.

To investigate the importance of such effects,further experimental work is necessary. Theabove modifications to the simple Fermi gas modelwill be discussed in detail in a future publication. ~

C. Potential Scattering

In Sec. IV it was pointed out that within the ex-perimental accuracy of 1.7% the saturation valueof the resistivity of Cu: Fe, p„„(T=OK), remainsconstant between 28 and 72 kbar. This result isconsistent with the volume dependence of the scat-tering from charge impurities which would give achange in resistivity over this pressure range ofroughly 1.5%. Since p„„(T=0K)= 2Spocos 5», itis therefore not possible within the accuracy ofthese experiments to detect any change in 6~. Thisis a result which, as pointed out previously, is in-dicated by the existence of a universal resistivitycurve.

There are several types of potential scatteringwhich contribute to the phase shift 6~. The sim-plest kind arises from the charge difference be-tween the magnetic ion and host ions. In addition,it has been shown 7 that scattering from potentialscatterers (nonmagnetic impurities, vacancies, andother crystal defects) located near the magneticion can also interfere significantly withthe exchangescattering from the magnetic impurity. A furthertype of potential scattering Vd is due to resonancescattering as the virtual bound state of the mag-netic impurity nears the Fermi level. From the'Schrieffer-Wolff transformation one obtains Vd= —(v /2)(1/E' —1/E ), where the phase shift 5»for this type of potential scattering is given by

.5»„= —tan ~[—,'wn(E„) V,]. Using the value of the pa-rameters for the fit in Fig. 8, one finds that forCu: Fe, 6~ is very small, cos26~ =0.99. Usingd '2the above simple model, the change in cos 6z uponapplication of 82 kbar would be less than 19O andthus not detectable in this experiment. The phaseshift 6&„does not become significant until the vir-tual d level is very near E~. However, it can beargued that this source of potential scattering maynot enter the expression [Eq. (3)] for the saturationvalue of the resistivity p„„(T= 0 K). When the vir-

tual level is far away from E~, 6~ =0; however, atT = 0 K the spin-scattering resistivity has reachedits unitary limit, implying 6~= —,'z, where 6~ is thephase shift for spin scattering. On the other hand,when the virtual bound state is at the Fermi level,it is to be expected that S=O, 6~=0, and 6&„= &m.

If it still holds for the virtual level near E& that6&+6&„=-,'w, then one would expect that as the vir-tual level approaches E&, the spin scattering wouldbe wiped out from below, p„„(T= 0 K), however,remaining constant. This model is consistent withhigh-pressure resistivity measurements on Y:Ce. ao

It would thus appear that 5» in Eq. (3) does not in-clude 6~ from this resonant scattering. This would

d

imply that our measurements are only capable ofdetecting changes in the "nonresonant" types ofpotential scattering.

D. Deviations from Matthiessen's Rule

In Sec. IV we described an experimental proce-dure for determining deviations from Matthiessen'srule in Kondo alloys. This method relies on theexistence of a universal resistivity curve as hasbeen demonstrated for Cu: Fe. It is seen fromFigs. 6 and 7 that below 40 K these deviations areactually larger than the phonon scattering itself.This result emphasizes the necessity, at all butthe lowest temperatures, of accurately det"rminingsuch deviations before attempting to extract infor-mation about the spin-scattering curves. For tem-peratures below 30 K it is found that these devia-tions are proportional to the phonon scattering inpure Cu, in agreement with experimental and theo-retical results. ~ The departure from this ruleseen at temperatures above 30 K is probably dueto the presence of a maximum in 4 at about 60 K.Extension of the present measurements to highertemperatures and pressures is necessary for athorough investigation of such effects.

ACKNOWLEDGMENTS

The authors would like to express special grati-tude to J. Crone both for permission to use resultsof his preliminary high-pressure data on Au: Feas well as for able experimental assistance in thecourse of this work. We are indebted to R. Brunoand L. Hirst both for helpful discussions as wellas for criticaily reading the manuscript and offer-ing valuable suggestions. Helpful discussions withK. Schotte, U. Schotte, W. Brenig, W. Daniels,and W. Gotze are also acknowledged. One of theauthors (J.S. S. ) expresses gratitude both to E.Luscher for introducing him to the Kondo problemand to the Deutsche Forschungsgemeinschaft forfinancial support. Special thanks are due M.Bohning, Technische Universitat, Munich, forletting us use the excellent facilities of his labora-

EFFECT OF PRESSURE ON THE KONDO TEMPERATURE OF. . . 1227

tory. We gratefully acknowledge the assistance ofH. Pink, Siemens Research Labs, Munich, withanalyzing the samples. We are also indebted to J.

Nickl, Institut fur Festkorperchemie, Munich, foradvice on metallurgical problems and an electron-microprobe analysis.

~Present address: Ruhr Universitat, Experimentalphysik IV, 463Bochum, Germany.

'J. Kondo, Prog. Theor. Phys. 32, 37 (1964).~For a review of the experimental situation up to 1964, see G. I.

Van den Berg, in Progress i@ Low Temperature Physics, edited

by G. J. Gorter (North-Holland, Amsterdam, 1964), Vol. 4, p.194.

'A review of experimental and theoretical work up to 1969 is to befound by A. J. Heeger, in Solid State Physics, edited by F. Seitz,D. Turnbull, and H. Ehrenreich (Academic, New York, 1969),Vol. 23.

'J. R. Schrieffer, J. Appl. Phys. 38, 1143 (1967).'For a thorough review of theoretical work on the Kondo problem

up to 1969, see J. Kondo, in Ref. 3.D. R. Hamann, Phys. Rev. 158, 570 (1967).

'H. Suhl and D. Wong, Physics (N.Y.) 3, 17 {1967);J. Appl.Phys. 39, 1294 (1968). Note error in graph on p. 1300. Theresistivity is plotted versus T and not log T. The curves fordifferent J thus appear to change shape.

SD. C. Mattis, Phys. Rev. Lett. 19, 1478 (1967).9See, for example, J. Applebaum and J. Kondo, Phys. Rev.

170, 524 (1968);P. W. Anderson, Phys. Rev. 164, 352 (1967).' J. Zittartz and E. Muller-Hartmann, Z. Phys. 212, 380 (1968);

J. Zittartz, Z. Phys. 217, 155 (1968)."J.W. Loram, T. E. Whall, and P. J. Ford, Phys. Rev. B 2, 857

(1970).' P. W. Anderson and G. Yuval, Phys. Rev. Lett. 23, 89 (1969);

P. W. Anderson, G, Yuval, and D. R. Hamann, Phys. Rev. B1, 4464 (1970);Solid State Commun. 8, 1033 (1070).

"M. J. Levine, T. V. Ramakrishnan, and R. A. Weiner, Phys.Rev. Lett. 20, 1370 (1968).

'"N. River and M. J. Zuckermann, Phys. Rev. Lett. 21, 904(1968).

"K.C. Brog and Wm. H. Jones, Jr., Phys. Rev. Lett. 24, 58(1970).

' W. M. Star, Physica 58, 585 (1972)."J.S. Schilling, W. B. Holzapfel, and E. Luscher, Phys. Lett. A

38, 129 (1972)."G. P. Summers, J. G. Lipham, and Louis D. Roberts, Bull. Am.

Phys. Soc. 16, 82 (1972); Bull. Am. Phys. Soc. 17, 195 (1972);Bull. Am. Phys. Soc. 17, 571 (1972).

' B. Coqblin, M. B. Maple, and G. Toulouse, Int. J. Magn. 1, 333(1971).K. S. Kim and M. B. Maple, Phys. Rev. B 2, 4696 (1970); W.Gey and E. Umlauf, Z. Phys. 242, 241 (1971).

"M. B. Maple and J. Wittig, Solid State Commun. 9, 1611 (1971);

M. Dietrich, W. Gey, and E. Umlauf (unpublished).'D. R. Hamann, Phys. Rev. B 2, 1373 (1970).' K. Fischer, Z. Phys. 225, 444 (1969).Cu was supplied by the American Smelting and Refining Co. andFe by Johnson Matthey and Co.

'E. C. Hirschkoff, M. R. Shanabarger, O. G. Symko, and J. C.Wheatley, J. Low Temp. Phys. 5, 545 (1971); Phys. Lett. A35, 449 (1971).A. Eichler and J. Wittig, Z. Angew. Phys. 25, 319 (1968).

"W. B. Holzapfel, U. Klein, and J. S. Schilling (unpublished).J. S. Schilling and J. Crone (unpublished).Actually, in the high-temperature limit T-+ T and

p ~ (1/2)Spom J n (EF)S(S+1),for 5„=0. In theseexperiments this term is very small and will be neglected.

' H. L. Davis, J. S. Faulkner, and H. W. Joy, Phys. Rev. 167, 601(1968); N. E. Phillips, Phys. Rev. 134, A385 {1964).

"C. M. Hurd, J. Phys. Chem. Solids 28, 1345 (1967).' L. L. Hirst, Z. Phys. 241, 9 (1971)."See, for example, J. M. Ziman, Principles of the Theory of Solids

(Cambridge U.P., Cambridge, England, 1964), p. 191~

P. K. Sharma and N. Singh, Phys. Rev. B 1, 4635 (1970).'J. C. Slater, Phys. Rev. 111,713 (1958); W. C. Overton, Jr. andJ. Gaffney, Phys. Rev. 98, 969 (1955).T. E. Whall, P. J.Ford, and J. W. Loram, Phys. Rev. B 6,3501 (1972).

"M. Kohler and H. A. Neckar, Z. Phys. 126, 495 (1949).'E. H. Sondheimer and A. H. Wilson, Proc. R. Soc. Lond.190, 435 (1947); M. J. Rice and O. Bunce, Proc. R. Soc. B2, 3833 (1970).' J. S. Schilling, J. Crone, and W. B. Holzapfel (unpublished).' A. Blandin, J. Appl. Phys. 39, 1285 (1968).

'H. P. Meyers, L. Wallden, and A. Karllson, Philos. Mag.18, 725 (1968);C. Norris and L. Wallden, Solid State Commun.

7, 99 (1969).4~J. Crone, W. B. Holzapfel, and J. S. Schilling (unpublished)."O. Laborde and P. Radhakrishna, Solid State Commun. 9, 701

(1971)."A. C. Gossard, A. J. Heeger, and J. H. Wernick, J. Appl. Phys.

38, 125 (1967).L. L. Hirst, Z. Phys. 244, 230 (1971).For an application to 4fimpurities, see B. Coqblin and J, R. Schrieffer, Phys. Rev.185, 847 (1969),' L. L. Hirst, Int. J. Magn. 2, 213 (1972); and privatecommunication.

'K. -P. Bohnen and K. Fischer, Z. Phys. 248, 220 (1971).