Electron-phonon coupling in heavy fermion systems

274

Journal of Magnetism and Magnetic Materials 63 & 64 (1987) 274 280 North-Holland, Amsterdam

INVITED

PAPER

ELECTRON-PHONON

C O U P L I N G IN H E A V Y F E R M I O N SYSTEMS*

B LUTHI and M YOSHIZAWA

Phystkahsches InstttuI, Unwers~tatFrankfurt, Post[ach 111932 D-6000 Frankfurt, Fed Rep Germany We give a rewew on electron-phonon effects Thermal expansion, elastic constant and ultrasomc attenuation give ewdence of a strong electromc Grunelsen parameter leading to a strong strain dependence of the width of the heavy fermaon band One can account for the observed effects m terms of a two band model Ultrasonic attenuatton phenomena m the superconducting phase are discussed

1. Introduction

2 1 Couphng mechamsm

Since the last r e p o r t [1] on e l e c t r o n - p h o n o n p h e n o m e n a m s y s t e m s with u n s t a b l e m a g n e t i c ions, s o m e i m p o r t a n t p r o g r e s s has b e e n m a d e , e s p e c m l l y in the field of h e a v y ferm~ons H e r e we w o u l d like to r e v i e w this m a t t e r , c o n c e n t r a t e on h e a v y f e r m t o n systems a n d l e a v e aside the c a s e of m i x e d v a l e n c e f l u c t u a t i o n s w h i c h has b e e n c o v e r e d In the r e v i e w m e n t i o n e d a b o v e W e will discuss n o r m a l state a n d s u p e r c o n d u c t i n g state p r o p e r t i e s In the f o r m e r we r e v i e w e s p e c i a l l y new e w d e n c e for the e l e c t r o n p h o n o n c o u p l i n g m the h e a v y f e r m l o n b a n d a n d c h a r a c teristic f e a t u r e s for s o u n d p r o p a g a t i o n m th~s low t e m p e r a t u r e state In t h e s u p e r c o n d u c t i n g state we m e n t i o n new e x p e r i m e n t s on s o u n d a t t e n u a tion and discuss their s i g n i f i c a n c e A c c o r d i n g l y we dw~de th~s r e v i e w into two s e c t i o n s N o r m a l state p r o p e r t i e s a n d s u p e r c o n d u c t i n g state p r o pert~es

Two important electron-phonon coupling m e c h a n i s m s h a v e b e e n d e d u c e d a n d a n a l y s e d so far M a g n e t o e l a s t l c i n t e r a c t i o n of the strain with l o c a l i s e d e l e c t r o n s for T > T* a n d a G r u n e l s e n p a r a m e t e r c o u p l i n g for l o n g i t u d i n a l s o u n d w a v e s for T < T*, w h e r e T* d e n o t e s the f l u c t u a t i o n temperature The Grunelsen parameter coupling is a special case of the g e n e r a l d e f o r m a t i o n potential coupling M a g n e t o e l a s t i c I n t e r a c t i o n has b e e n r e v i e w e d b e f o r e for stable 4 f - i o n s [2] In the c o n t e x t of h e a v y f e r m l o n systems it has b e e n d i s c u s s e d in last y e a r s c o n f e r e n c e [1] H e r e we e m p h a s i z e a g a i n that o n e o b s e r v e s for C e - c o m p o u n d s c l e a r c u t m a g n e t o e l a s t l c effects, t e a c o u p l i n g of the v a r i o u s strain m o d e s El to the q u a d r o p u l a r o p e r a t o r s Ov of the Ce~+-ions for T > T * The magnetoelastic Hamiltonian reads 9~= - ~ l ,gl ~1 O1 , T y p i c a l c o m p o u n d s e x h i b i t i n g such effects are (table 1) e g CeAI2, CeAI~, CeB6, C e C u 6 , CePb~ F o r T < T* the V a n V l e c k p a r t of the strain s u s c e p t i b i l i t y (arising f r o m o i l - d i a g o n a l q u a d r u p o l a r m a t r i x e l e m e n t s ) can still be o p e r a t i v e b e c a u s e only the g r o u n d s t a t e of the c r y s t a l field split J = 5/2 Is s t r o n g l y m o d i f i e d by K o n d o i n t e r a c t i o n s For heavy fermIon U r a n i u m c o m p o u n d s no e w d e n c e of m a g n e t o elastic i n t e r a c t i o n has b e e n o b s e r v e d so far Of g r e a t t m p o r t a n c e ~s the e l e c t r o n - p h o n o n

2. Normal state properties H e r e we discuss p h e n o m e n a such as c o u p l i n g m e c h a n i s m , s o u n d p r o p a g a t i o n in the h e a v y f e r m i o n state a n d m a n i f e s t a t i o n of a n a r r o w b a n d in this state

* Supported by the SFB 65

0 3 0 4 - 8 8 5 3 / 8 7 / $ 0 3 50 © E l s e v i e r S c i e n c e P u b l i s h e r s B V ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )

B Lutht, M Yoshtzawa / Electron-phononcouphng m HF systems couphng m the heavy fermlon regmn It was first noted [3] that the volume thermal expansion/3 of heavy fermlon compounds is very large and has a hnear temperature dependence typtcal for ttlnerant electrons, in contrast to thermal expanston from locahsed electrons, which give rise to Schottky type effects [4] An electromc Grunelsen parameter ~ = flca/C gwes a measure of the electron lattice couphng wtth ca denoting the bulk modulus and C the spectfic heat 1) was found to be very large m the heavy ferm,on region [3,5], typically of the order of 100 compared to 1 in ordmary metals and of order of 10 for valence fluctuatton compounds (table l) The Grunetsen parameter couphng has ~mportant consequences for longitudinal sound propagation, ultrasomc attenuation, thermal expansion and superconducttvtty, as will be discussed below [6-8] A somewhat analogous couphng has been used to describe the phase dmgram of cerium and was called K o n d o volume collapse [9] We write for the electron strata couphng generally

Ek

=

E ° - ~ , dkr~,, [

where dkr ts a deformation potential tensor c o m p o n e n t For electron-volume strain coupling

dkr, = ~ E ~ , I e Ek = E°(1 - llEv)

or

11 = -- 01n Ek/O~

(1)

275

tf the energy of the electron is measured from the Fermi energy This form of deformatmn potential exhlbtts clearly breathing character The couphng constant ts of order d,, ~ [~EF ~ f~T* One can compare the phenomenologicaily introduced Grunelsen parameter couphng to elect r o n - p h o n o n coupling mechanisms in microscoptc theories The most transparent formulatton is the quaslpartlcle Hamlltonlan used m the theory of K o n d o lattices [7,10]

Y( = ~

(k2 ) ~

- tx C~,,Cko~+ ~ E t ( k ) f ~ , f ~ ko

+70~.(fLCk~+C~[k~),

(2)

k~

where c~, f~ are creation operators for the hght and heavy bands The heavy fermlon energies have a dispersion of the form E t ( k ) = eF--tz + al T* + a2 T * ( k - kF)/kF and the hybrtdlsatlon parameter ts 3'o -~ T'fT%F~F The e l e c t r o n - p h o n o n interaction related to Grunelsen parameter couphng can be mtroduced by f~ = - 0 I n T*/O~v which is identical to the ortgmal formulation [3] Introducing phonon operators bq one obtains for the e l e c t r o n - p h o n o n Hamlltonlan [10]

Table 1 Physical parameters for heavy fermlon materials

Material CeCu2S12 CeAI2 CeAI~ CeCu6 CeB6 CePbs UPts UBels* Hes**

Fluctuation temp T* (K)

Specific heat 3,(To) (mJ/mol K2)

5 6 3 4 7

1100 135 1200 1450 260

26 8 1

422 1100 39700

Grunelsen parameter [l(To)

Ref t e m p To (K)

Transition temp TN (K)

6O

Supercond trans T~ (K) 07

38 -200 71

01 2 2 1 ( T o = 3 3)

11

* Ref [36]), ** (334 bar)

60 94 -2 2

1 1 0 02

0 54 0 97 0 0026

276

B Lutht, M

Yoshlzawa / Electron-phonon couphng m H F systems

Grunelsen parameter

1

" = flcB/C = - (Oln T/Oev)S + f;+,.~Ck..)(bq + b+-,) " Y . Iql 1 2 ~ ./2 Mt%N x (El(k) + E~(k + q)) [~+,[k(bq + b+-q)

(3)

It is evident that the second therm is the one which resembles most the phenomenologlcal one of eq (1) This IS made more transparent by ehmlnatlng the C k via a transformation [7,10] eq (3) was used for an estimate of the superconducting transition temperature [7] All other interactions of the deformation potential type are less important and describe the weaker coupling Its couphng constant dr which IS of order E~ (Fermi energy of the c o n d u c n o n electrons) is changed by the same transformation to drT*/E~ --~ T* ~ T * (Grunelsen parameter coupling) This weaker couphng shows up m small elastic constant anomalies, especmlly for transverse waves (UPt3) [11], CeCu6 [12,13] and m ultrasonic attenuation phenomena to be discussed below An ~mportant point ~s the near conservation of heavy quasi-particles in the narrow resonance near the Fermi energy Microscopically th~s is implemented (at least for Ce-compounds) by the pruning of the K o n d o resonance to the Ferm~ energy Experimentally this particle conservation lmphes a normal bulk modulus ca and a posmve Polsson ratio v This point has been stressed before [1] by comparing heavy fermlon systems (ca, ~,> 0) and valence fluctuation compounds (ca-->0, v < 0 ) where one has a single particle f-band and conduction band hybridization

(4)

which means that a strata wave is accompanied by a temperature wave. which for high llmaterials could be detected in principle [6] b) Another consequence is that the difference between longitudinal adiabatic and mothermal elastic constants is noticeable for the heavy fermlon state, since C~, -- Cl = [ 3 2 T c 2 / c ~ = . 2 y c ~

(5)

The relatwe &fference (Cs--CT)/CT amounts typically to 1% for f~ = 100 mstead of 1 0 - 4 for normal metals c) In the heavy fermlon state one is measuring c. For q l ~ 1 (q wave number of soundwave, l electron mean free path) a&abatic sound propagatlon means [6] to~-.~ (3/2"rr)(vs/vv) 2 which IS fulfilled for heavy fermlon metals with electromc relaxation times ~"~ 10-12s and sound velocity v. of the same order as the Ferm~ velocity vF d) The &fference between c. and CT leads to an anomalously large Landau-Placzek ratio R which determines the ratio of the quaslelastlc hne L) and the Brdloum hnes 21118] R = (cJcv)--1 This central peak p h e n o m e n o n m heavy fermlon compounds has not been observed so far e) Still another important consequence ss a possible ultrasonic attenuation mechanism, similar to what is found in flmds [8.14]

a. .

20v ~

Here K is the thermal conductivity and C.. C.~ the specific heats at constant strain E and stress ~r. respectively Since ( C ~ - C.) = T[32ca we get

--¢'02"2T/(- 092"2 (I)~F)2 T C T

(6)

2 2 Sound propagaUon effects

a.s

The heavy fermlon state with its high density of electronic states and large Grunelsen parameters has a profound influence on sound propagation Here we mention some typical effects a) Using Maxwell's relation (dT/de)s = -(aT/aS').(OS/Oe)T we can write for the

Here we use for K the gaskinettc expression K=~ Cv~'r Because of the large Grunelsen parameter ~ and the large specific heat C, this energy dissipation mechanism ~s as important as the one due to friction Because of the specml structure of C(T) and ~-(T) an ultrasonic attenuation peak is expected and has been found [15]

or2

~rpv 2 \ v,)

B Luthz, M Yoshtzawa / Electron-phonon couphng m HF systems

We shall comment more about this transport coefficient and some difficulties in the next section Ultrasonic attenuation calculations with Gruneisen parameter coupling have been given by different authors [6,8,15,16] In the AkhelzerBoltzmann equation theory [16] employed by some authors the energy dependent electron phonon couphng (eq (1)) does not lead to the usual ultrasonic attenuation formula (see eq (9)) discussed for ordinary deformation potential coupling [1,17] but rather to eq (6) with VF =

In addition to the speofic heat [18] other thermodynamic quantities such as thermal expansion and elast.1c constants show low temperature anomalies which are direct mamfestatlons of a narrow band with a large density of states [1,6] Relevant experimental reformation ex.1st for CeAI3 (thermal expansion [5], elastic constants [19]), UPt3 (thermal expansion [20], elastic constants [6,11]), CeCu6 (elastic constants [12,13] UBe13 [21] In fig 1 we show experimental data of elastic constants for various heavy fermton materials These low temperature elastic constants exhibit typical Grunelsen parameter effects which can be described us.1ng a two band model for heavy fermlon quaslpartlcles [6] A one band model ~s not sufficient to explain the different temperature dependences of CeCu6, UBe13 on the one hand and UPt3, CeAI~ on the other hand Likewise for ultrasonic attenuaUon w~th Grunelsen parameter couphng, a single band is not sufficient for dlss.1patlon [16] In a two band model the formulas for specific heat, ther-

/)s

2 3 Mam[estaaon of a narrow heavy fermton band The high specific heat y-values found in heavy fermlon systems imply a narrow band of heavy quast particles with a width typtcally of the order of the fluctuation temperature T* The low T* .1mphes also that the number of f-electrons is very close to one

32 2

~a

32

o

uPt3

°%~o~ °° °°

29 3: 29

o oo

2

~

.

°ooo C33 %0° 0

o

CB

-

0

°° °%

U

"'"

29

I 50

~O ~ 12 86 | C

I 100

~ 460"----

_

..""

• •

12 82

1274

,'~"

_

455

,=s ~

12700

i

Temperature

5'0

0

". • c,1

..... ~ / "

•

J

I

05

10

15

2 d

">

" /

J 100

oo"

200

CeCu 6

Fig

•

U

I 150

•

.o

°o

0

u

CeA[ 3

E

°°

291

b

u 465 --..

o

~'~

E

47o

. . . . . o o. . . . . . .

o

31 8

277

i i I I 150 200 250 T(K)

"

UBe'3 i

-1

098 6Hz

<~ 0.1

i

I 10

,

J 10

J

100

T(K)

d e p e n d e n c e o[ e]astlc constants f o r h e a v y f e r m l o n matermls (a) UPt3, ref [11], (b) CeAI~, rcf [19], (c)

CeCu6, ref [13], (d) bBe13, ref [21]

B Luthl M Yoshtzawa / Electron-phonon couphng m HF systems

278

mal expansion and elastic constants are [6] (for special case of scaling factor A = e -n. and cubic systems) specific heat C = CI + C2,

thermal expansion ~- ( ~ 1 C I

~- ~2C2)/CB,

(8)

elastic constant CS = ~ 2 U, -[- ~ 2 02 _ ( T / C ) ( ~ 1 - a 2 ) 2 C 1 C 2

Eq (8) describes correctly the temperature dependence of the elastic constants in the heavy fermlon region for the materials shown in fig 1 For the simple case where the specific heat varies hnearly with temperature, TT, Cs varies hke T 2 with a positive or negative prefactor, depending on the values of ~ , tie, ~/~, 72 Detailed comparison between experiment and fits of eq (8) for thermal expansion and Cs have been given elsewhere [6,13] Recently an ultrasonic attenuation peak was found In UPt~ around 12 K [15] It can be interpreted with an expression like eq (6) [6,8,15,16] It was noted [6] however that the ~-deduced from the experiment, using eq (6) or the equivalent expression a . . = to2~'(c~- co)/2pv ~, (c~ background elastic constant, c0 low frequency elastic constant} is a factor of 50 larger than the electron colhslon time %on determined from electrical resistivity or thermal conductivity This is not explained yet In addition to this ultrasonic attenuation anomaly one observes an increase in attenuation with decreasing temperature for both longitudinal and transverse waves m UPt3 [15,17] This attenuation p h e n o m e n o n can be explained with deformation potential couphng The corresponding formula for ql ~ l,

au ~ = d~ Nq21/ pVsVF

(9)

was discussed before [ 1,17] and it was noted that d~-N/VF~-(dl m * / m ) 2 ~ - ( T * m * / m ) 2 gives a stronger enhancement than in the case of elastic constant Because of the factor v 3 in the denominator the transverse attenuation can be

even more pronounced than the longitudinal one, as observed [15] Recent experiments on the pressure dependence of the specific heat for CeAI~ [22] and UPt~ [23] give additional interesting results Whereas the pressure dependence of 3' for UPt~ gives the same Grunelsen parameter fl as the one used for thermal expansion and elastic constants, for CeAI3 the same experiment gives just the opposite sign (ll = 183) This means that for UPt~ one can hope to understand the value of tl from density of states arguments, whereas this is not the case for CeAl~

3. Superconducting properties There have been a number of experiments performed on ultrasonic propagation in heavy fermlon superconductors Here we address two points for which experimental reformation exists

3 1 Observation of a peak a n o m a l y in U P h and UBe 1 Contrary to what IS observed m normal type BCS superconductors an ultrasomc attenuation peak was observed for longitudinal waves of the superconducting transition m U B e ~ [21,24] and UPt~ [25] In analogy to superfluld 3He it was suggested that this attenuation peak arises from couphng of sound to low lying collective modes of the superconductor [21,24] Contrary to the case of superfluld ~He however one is not in a regime ql > 1 but ql ~ 1 Therefore such a process is highly unlikely A more promising explanation can be sought again with Grunelsen parameter coupling [6,8] Using the correlation function approach [6] one obtains a . ~ -- C2~o° or a . , -- C2~o2(wlth ~-- C) Experiments exhibit such behavlour very closely [24.25] C 2 exhibits a sharp structure for T T . which shifts with magnetic field The frequency dependence has not been checked so far 3 2 Ultrasomc phase

attenuation,

in superconductwe

Especially for UPt3, temperature variations of ultrasomc attenuation have been precisely

Yoshzzawa / Electron-phonon couphng m HF systems

B Luth,, M

19

o8 E

_ UPt

3

~

a

_i ~,

17

~92 MHz,long t ~ ' % ~ " - . . ~

o75

"-. U

~

°

15

-.

67 MHz,transv

~I

AC 5u5c 065

i 05

0 ~-"

I 10

E ~IT~

11

¢1

09 15

T/Tc

b u~

4

E

2

'~>

.'ill6

0

03

06

09

T (K) O3

UB¢13

~

GHz

Oo

170

02

• H=0

+-"

eu

C

~c I I,

o H :2r

~

4

"

.%~~2d~

01

279

Therefore the usefulness of power laws is somewhat limited T h e r e are other examples The T 3 law for the specific heat in UBei3 [3 l] indicative for a p-wave axial like superconductor has been found also for a strong coupling d-electron superconductor HfV2 [32] In addition power laws in ultrasonic attenuation have been found theoretically for gapless superconductors [33] The ultrasonic attenuation in the superconducting phase is caused mainly via the coupling to the order parameter Of special importance for the investigation of sound wave-order parameter coupling is the polarization dependence of ultrasonic attenuation and dispersion A distract polarizat,on dependence of ultrasonic attenuation in UPt3 has been observed [27] The question is whether this dependence arises from the amsotropy of the gap state and/or from amsotroplc coupling between sound and order parameter Up to now the ultrasomc experiments on heavy fermlon superconductors cannot decide on the central question of s-type or p-type superconductlwty More experimental and theoretical mformat]on is needed [34]

o

4. Conclusion 0

Q2

OZ. 06 T (I<)

08

10

Fig 2 Ultrasomc attenuation in the superconducting state of UPq and U B e ~ (a) UPh, ref [25], (b) UPh, ref [27], (c) UBeI~, ref [21]

measured by several authors [25,26,27] (see fig 2) In the superconducting phase, ultrasonic attenuation follows a power law a - - t o 2 T " T h e r e are, however, some discrepancies between the exponents [25,26] ( T 2 o r T 3) Because there exist predictions for such power laws for p-wave triplet type superconductors, it IS Important to check such laws For UPt~, specific heat and thermal conductwity, show a T 2 temperature dependence [28] These power laws, together w~th the T 2 dependence of ultrasonic attenuat]on, stand for the p-wave (polar like) superconductwity [29] On the other hand, group theory excludes the possIbdIty of polar p-wave superconductor In solids due to strong spin-orbit coupling, contrary to the case of 3He [30]

The coupling mechanisms for the electronphonon interaction shows a rich variety in heavy fermlon materials Although magnetoelastic interactions and rigid band type deformation potential coupling effects can be observed in certain cases, the most important interaction is the Grunelsen parameter coupling, a strain modulation of the heavy fermion band width Some consequences of this type of coupling have been discussed The various anomalies in thermal expansion, elastic constants and ultrasonic attenuation can be quant,tatively interpreted Ultrasonic attenuation in the superconducting state gives a variety of interesting effects Up to now these experiments cannot conclusively determine the ground state of the heavy fermlon superconductors It would be interesting to apply the Gruneisen parameter coupling to these attenuation results High magnetic field effects on elastic constants and ultrasonic attenuation in UPt3 reveal

280

B Luth~, M Yoshlzawa / Electron-phonon couphng m HF systems

mterestmg p h e n o m e n a [35] It could be posstble that with such experiments we can learn a great deal about the heavy electrons and about transttlons to other states such as spin denstty or charge density ground states We thank Prof P Fulde for a crttlcal readmg of thts paper One of us (M Y ) ts supported by A v H u m b o l d t fellowshtp References [1] B Luthl, J Magn Magn Mat 52 (1985)70 [2] B Luthl, m Dynamical Propertws of Sohds, vol 3, eds G K Horton and A A Maradudm (North-Holland, Amsterdam, 1980) [3] R Takke, M N1ksch, W Abmus, B Luthl, R Pott, R Schefzyk and D K Wohlleben, Z Phys B44 (1981) 33 [4] H R Ott and B Luthl, Z Phys B28 (1977) 141, Sohd State Commun 33 (1980) 717 [5] J Flouquet, J L Lasjaunms, J Peyrard and M Rlbault, J Appl Phys 53 (1982)2127 [6] M Yoshlzawa, B Luthl and K D Schotte, Z Phys B 64 (1986) 169 [7] H Razafimandlmby, P Fulde and J Keller, Z Phys B 54 (1984) 111 [8] K W Becker and P Fulde, Europhys Lett 1 (1986) 669 [9] J W Allen and R M Martin, Phys Rev Lett 49 (1982) 1106 M Lavagna, C Lacrolx and M Cyrot, Phys Lett 90A (1982) 210 [10] N d'Ambrumenfl and P Fulde, J Magn Magn Mat 47&48 (1985) 1, and in Theory of Heavy Fermtons and Valence Fluctuatlon~ eds T Kasuya and T Saso, Springer Series m Sohd State Soences 62 (1985) [11] M Yoshlzawa, B Luthl, T Goto, T Suzuki, B Renker, A de Vlsser, P Frmgs and J J M Franse, J Magn Magn Mat 52 (1985) 413 [12] T Suzuki, T Goto, A Tamakl, T Fupmara, Y Onukl and T Komatsubara, J Phys Soc Japan 54 (1985) 2367 [13] D Weber et al, to be pubhshed [14] L D Landau and E M Lffshltz, Flmd Mechamcs (Pergamon Press, New York, 1959) chap VIII [15] V Muller, D Maurer, K de Groot, E Bucher and H E Bommel, Phys Rev Lett 56 (1986) 248

[16] K D Schotte, D Forster and U Schotte, Z Phys 13 64 (1986) 165 [17] B Batlogg, D J Bishop, B Goldmg, E Bucher, J Hufnagl, Z Flsk, J L Smith and H R Ott, Phys Rev B3:~ (1986) 5906 [18] G R Stewart Rev Mod Phys 56 (1984)755 [19] M Niksch, B Luthl and K Andres, Phys Rev B22 (1980) 5774 [20] A de Vlsser, J J M Franse and A Menovsky, J Phys F15 (1985) L53 [21] B Goldmg, B Batlogg, D J Bishop, W H Haemmerle, Z Flsk, J L Smith and H R Ott, Proc 2nd Intern Conf on Phonon Physics (World Scientific Singapore, 1985) p 4(16 [22] G E Brodale, R A Fisher, N E Phdhps and J Flouquet, Phys Rev Lett 56 (1986)390 [23] G E Brodale, R A Fisher, N E Phillips, G R Stewart and A L Giorgi, Phys Rev Lett 57 (1986) 234 [24] B Goldmg, D J Bishop, B Batlogg, W H Haemmerle, Z Fisk, J L Smith and H R Ott, Phys Rev Lett 55 (1985) 2479 [25] V Muller, D Maurer, E W Scheldt, Ch Roth, K Luders, E Bucher and H E Bommel, Sohd State Commun 57 (1986) 319 [26] D J Bishop, C M Varma, B Batlogg, E Bucher, Z FIsk and J C Smith, Phys Rev Lett 53 (1984)1009 [27] B S Shwaram, Y H Jeong, T F Rosenbaum and D G Hmks, Phys Rev Lett 56 (1986) 1078 [28] J J M Franse, A Menovsky, A de Vlsser, C D Bredl, U Gottwtek, W Lleke, H M Mayer, U Rauchschwalbe G Sparn and F Steghch, Z Phys B 59 (1985) 15 [29] C M Varma, J Appl Phys 57 (1985) 3064 [30] K Ueda and T M R~ce, m Theory of Heavy Fermlons and Valence Fluctuations, eds T Kasuya and T Saso, Springer Series m Sohd State Soences 62 (1985) [31] H R Ott, H Rudlgler, T M Rice, K Ueda, Z Fisk and J L Smith, Phys Rev Lett 52 (1984)1915 [32] B Luthl, M Herrmann, W Abmus, H Schmldt, H Rletschel, H Wuhl, U Gottwlck, G Sparn and F Steghch, Z Phys B 60 (1985) 387 [33] K Makl, m Superconductwlty, ed R D Parks (Marcel Dekker, New York, 1969) chap 18 [34] P A Lee, T M Rlce, J W Serene, J L Sham a n d J W Wflkms, Comments on Sohd State Physics [35] I Kouroudls et al, J Magn Magn Mat 63&64 (1987) 389, and to be pubhshed [36] H R Ott, private commumcatlon