PHYSICAL REVIK% B VOLUME 18, NUMBER 10 15 NOVEMBER, 1978

Role of phonons and Coulomb interactions in fluctuating-valence europium compounds

O. L. T. de Menezes, A. Troper, and A. A. GomesCentro Brasileiro de Pesquisas Ffsicas, Av. Wenceslau Braz, 71, Rio de Janeiro, Brasil

(Received 18 April 1978)

An extended Anderson model including Coulomb interactions and electron-phonon coupling is discussed

within the Hartree-Pock approximation. This model is used to describe valence fluctuations in europiumintermetallic compounds. The total 4f occupation number and the 4f level width are shown to be in

qualitative agreement with experimental data. A connection to charge screening invoked in a previous work is

also made, and the conclusions tend to support the phenomenological approach developed in that work.

I. INTRODUCTION

A phenomenological model to discuss qualita-tively the valence of europium atoms in interme-tallic compounds was recently developed. ' Em-phasis was given on the role of the charge screen-ing in the valence-Quctuation phenomena. Thesystematic trend of the europium valence, in-cluding homogeneous mixed valence states in alarge variety of intermetallic compounds couldbe then qualitatively understood.

Although successful in this qualitative under-standing, drawbacks of this picture should bepointed out. The first one concerns the origin andvalue of the effective 4f level width b„, togetherwith the general approach of Ref. 1 in consideringseparately 4f ions and d-band electrons coupledonly through a phenomenological interaction y~.Clearly the values of 6 play a fundamental role tounderstand experimental results, e.g. , isomershifts, ' and x-ray photoemission' (XPS) measure-ments in Quctuating systems. Although it is pos-sible, within the framework of this picture, suita-bly extended to include temperature effects, 4 toextract from experimental data, excitation ener-gies between two europium configuration, and theeffective level widths, the problem of the originof these widths remains. In particular, one mustunderstand how 6 changes when one crosses froma Eu" configuration to a Eu" by changing of thex concentration in compounds like Eu(A, -„B„)„where A, and 8 are transition metals.

Another limitation of the model developed in Ref.1 is to understand the existence of magnetic orderassociated with the 4f ' configuration and a non-magnetic state for the 4f' configuration. In thatwork' only limiting cases x =0 and x =1 could bedescribed using Hund's rule, but in the x = 1 casethe origin of the long-range magnetic order wasnot discussed. Both problems involve among otherinteractions, the hybridization between the d bandand localized f states, which is completely absentin Ref. 1.

In this work one intends to improve the previousone' in several respects. Although conserving theone-center picture, one introduces one-electronhybridization between the local and conductionstates. Then one chooses an Anderson-type ap-proach for the f electrons in a metal. The usualAnderson model, incorporating intraorbital Cou-lomb interaction will be supplemented here byintroducing (i) local Coulomb interaction betweenthe local f electrons and the d band, (ii) the elec-tron-phonon intt. raction between d-band states andlattice vibrations. So, one expects to discuss amechanism which may account for the featuresobserved in Eu intermetaQic compounds.

The motivation of this extension goes as follows:hybridization introduces naturally a source of f-level width; intra-f-orbital Coulomb interaction isusually invoked to describe local moment forma-tion and magnetic ordering. The coupling between

f and d electrons introduces in a natural way cor-relations between f level an-d d-band occupationat the impurity site. Finally and perhaps moreimportantly, electron-phonon interactions are ex-pected to be present in the valence-Quctuatingsystems as suggested by lattice deformations ob-served in these systems. ' However it should bestressed that one incorporates the coupling be-tween d electrons and the lattice and not betweenthe f occupation number and the lattice. " -The

reason for this choice follows from the role playedby the d electrons in the cohesion of these sys-tems. If an f electron hops to the d band, thechange in the d electron density is expected to al-ter the cohesion forces and so lattice distortionfollows.

In the literature several extensions of the An-derson model have been discussed in connectionwith valence-fluctuation systems. The Coulombinteraction between d and f electrons has beenintroduced in the Anderson model by Khomskiiand Kocharjan, ' within the Hartree-Fock approxi-mation in order to discuss the y —e —y' transitionsin Ce and metal-insulator transition in SmS.

18 5131 1978 The American Physical Society

5182 O. L. T. DE MENEZES, A. TROPER, AND A. A. GOMES 18

These authors assumed a suitable ground state ofexcitonic nature and an extra f-level width ap-pears due to this excitonic pairing in addition tothe usual level width associated with hybridization.

The d-f Coulomb interaction was invoked alsoby Haldane, ' together with a two-Channel descrip-tion of the coupled Anderson model: one channel(fast) associated with the d electrons is used toensure overall charge neutrality during the rela-tively slow f-electron-number fluctuation. With-in the framework of the Tomonaga approximationthis problem is shown' to be equivalent to a fer-mionlike problem (the Anderson Hamiltonian)coupled to a boson field (the screening i'ield). A

formally identical picture is provided by Rise-borough's work, ' where the fermion operators n&

are coupled, via an electron-phonon coupling, tothe phonons. In both points of vj.ew, due to theformal similarity one would say, following Hal-dane, ' that valence fluctuations are purely elec-tronic phenomena. The inclusion of the electron-yhonon coupling to the. d states enables one tobridge between cohesive energy arguments andvalence fluctuations. Also, as will be discussedin the approximations of this model, electron-phonon interaction will play a fundamental role inthe screening of the charge and so besides pro-viding a microscopic view of the phenomenologicalscreening potential, ' it connects also the electron-ic nature of valence fluctuations to lattice distor-tions.

This work is divided in four sections. SectionII is devoted to describe the model and approxi-mations to deal with the many-body terms. InSec. III the formalism is developed. Section IVcontains the numerical results and some discus-sions to generalize the one-electron problem toeuropium atoms. Finally, Sec. V includes gener-al. discussions and final comments.

II. MODEL AND APPROXIMATIONS

The conduction states are assumed to be of dcharacter, and will be discussed within the tight-binding approximation. In second-quantizationform the Hamiltonian for conduction states reads

If(d) — Q T(&)dt di,f, o

where T,f is the hopping integral leading to a dis-persion relation e(i+ and de ((f„) are creation(destruction) operators for a d electron with spincr at the site i. For simplicity, one neglects theCoulomb interaction tJ«among d states and so oneassumes that E(l. (1) contains all the relevant in-formation for the unperturbed d band.

Concerning the f states, one adopts the one-

center picture, "' as a zeroth-order approxima-tion, for valence-Quctuating systems. The one-center Hamiltonian for f electrons reads then

If .( = Zr6fsp +Uffmotsqk p no ofoafoa ' (2)(I)- ~ f f f f

In E(I. (2) fqt (fq ) creates (destroys) an electronwith spin 0 and energy e& at the origin. For themoment a nondegenerate level will be adopted; anapproximate description of the many-electron na-ture of Eu atoms will be made later on.

The interactions between the d conduction statesand the localized states are of two kinds.

(i) A one-electron hybridization term betweenthe local f states and the conduction d states

(lllIX)=Vuy~~oafqa+VyuZ fqo&qo ~

This mixing term contains the usual phenomeno-logical aspects, namely, it acts only at the ori-gin, and the matrix element

~ V~& ~' is to bethought of as a parameter of the theory. Alterna-tive formulations would involve' the different pari-ty of d and f electrons and a next-neighbor over-lay. This implies k-dependent matrix elementsand for simplicity one adopts the phenomenologi-cal approach.

(ii) A Coulomb interaction (intrasite) between the4f electrons and the d band; the complete termreads

H =Z Ugf no~ noa ~

(~f) —~ 4 f0 ~ 0

The Hamiltonians E(I. (1)-(4) are the terms ofthe usual extended Anderson Hamiltonian with theCoulomb interaction among d and f electrons. Ifthe one-center picture is applied to valence-fluc-tuating systems another term is to be included toaccount for peculiar effects associated to thesesystems. One refers to lattice deformations in-duced by valence changes. As mentioned in theIntroduction such deformations are understandablesince the transfer of a 4f electron to the conduc-tion band is expected to change the elastic forcesbetween neighboring atoms. If the cohesive energyin these metals is to be ascribed to the itinerantd states, it seems physically reasonable to expectthat the electron-phonon interaction between delectrons and lattice vibrations should play an es-sential role in the formulation of the valence-fluctuation problem. In view of these remarks oneconsiders the following term:

B"" = rApaaq

+ C ( iq. Ri &~&-, i)&&q q~ q lay

4, q, a

ROLE OF PHONONS AND COULOMB INTERACTIONS IN. . .

where a-„(a~) stand for creation (annihilation)operators for a phonon of wave vector q and ener-gy ~q Cq is the electron-phonon coupling. In-stead of adding Eq. (5) to the model Hamiltonian,one follows Bari' and adds the effective electron-electron interaction among d electrons mediatedvia phonon excitation, namely,

B,t p~=-QCi; (Qnf ) (6)

Due to the many-body nature of the total Hamil-tonian one needs some approximate method to dealwith correlations. The extended version of theAnderson model has been treated using severalmethods. In particular for C« = 0, Haldane' usingTomonaga' s method reduces the problem to acoupled fermion-boson one, which is treated with-in the mean-field approximation. On the otherhand, a functional-integral technique, togetherwith the saddle-point method was used by Rise-borough' to treat the Anderson model coupledwith phonons through the f electron occupation.Due to the complexity of the Hamiltonian, Eq. (I),we decided to work within the simplest approxima-tion, namely, the Hartree-Fock approximation,as in other similar works"" dealing with valenceQuc tuations.

One starts by linearizing the electron-electroninteraction terms. The intra-f-orbital Coulombterm is linearized in the usual way, but the d-finteraction terms needs some discussion. Asmentioned in Ref. 7 an important question ariseswhen one considers the full Hartree-Fock lineari-zation of Eq. (4).

+&dt.f..&f.'. d..).The last two terms of Eq. (8) are those invoked

In Bari's work, ' this term arises through theclassical canonical transformation eliminatingphonon variables from the Hamiltonian. One hasmade in Eq. (6), the approximation of keeping onlyintrasite terms, where C«characterizes thestrength of the electron-phonon interaction be-tween d states and lattice vibrations. It should bementioned that the validity of (6) to describe pho-non-mediated coupling between d, electrons is animportant assumption of this work.

The total Hamiltonian is given by

./ -/(~) +/(f) +/(1m~) +II(&f) +/(~&)

by Khomskii and Kocharjan' to describe the extraf-level width due to excitonic pairing, which per-sist in the absence of one-electron hybridization.The existence of self-consistent solutions exhibi-ting nonzero values of the "order parameter"&do,fo, ) assume a ground state of different naturefrom the usual one assumed for the Andersonmodel. .We argue however that the behavior for thef-level width can be explained without this extraassumption, provided the first two terms of Eq.(8) are properly included. Moreover, one expectsthat the 4f width should go to zero in the absenceof one-electron hybridization", except for extrabroadening effects which are not included in thispicture. Hence, we neglect the second term ofEq. (8). The other terms of the total Hamiltonianare linearized in the usual way and one gets theeffective Hamiltonian

II —II(&)+ II(f)+ ~(lllh)

where

H = Q TIg~+ U„~Q n&~~0)6;,6;,

(10)

The mixing term remains unaltered. Concerpingthe nonhybridized f states, Eq. (11), the Hartree-Fock linearization introduces apart from the usualCoulomb shift U&&&n~ o), an extra shift due to therepulsion energy among d and f electrons. FromEq. (10) one observes that the U„& coupling intro-duces a localized perturbation, with strength pro-portional to the f-orbital occupation number,which may strongly deform the unperturbed d-banddensity of states. Within the Hartree-Fock ap-proximation one sees that the electron-phononterm acts twofoldly. If one rewrites this term as

c, Q&n',.!&» ~;..-gc...P«n", !,)~!.

two contributions appear. The first one, associa-ted with the translationally invariant occupationmembers, just shifts the center of the band, andso is unimportant. The second one however, in-troduces a new scattering source for d electrons.This term is nonlocalized in nature, due to usualFriedel-like oscillations, and the solution of such

O. L. T. DE Mg NEZES, A. TROPER, AND A. A. GOMES

T&')a'a +Vn'i j ia ja ia~i,a, &

(12)

where the local scattering potential is defined by

a potential scattering problem is mathematicallyand numerically complex. Cl,early, one could as-sume a perturbation localized at the central siteand next neighbors, treating the perturbation as-sociated with the latter within the Born approxi-mation. If one assumes the potential to be re-stricted to the central site, this term togetherwith the U«one, will provide a Slater-Koster im-purity problem" which is exactly treatable. Post-poning a better discussion for Sec. IV, for themoment one restricts to the central cell and ap-proximates the Eq. (10) by

where Ed(oi) has real and imaginary parts, i.e.,

Ed((a+i e) = Esd(od) iw—p d((o), e 0'

Ed (&u) being the Hilbert transform defined by

E", ((o) =(P ",d(o'."Pd(oi')

g (d-CO

E, and E, are the bottom and the top of the d band,respectively. Similarly 5'd o(u&) is the Hilberttransform associated with the perturbed 4 bandP d (o()o= —(I/m)lmg dot((o) .

The second step consists of solving a genera-lized Anderson-Moriya problem Th.e bare f-fHartree-Fock propagator reads [cf. Eq. (11)]

goo(4&) =5o5io~((o &y) ~

V = Ug j n~p~~ —C&np . (13) where

Equations (12) and (13) in the absence of the mix-ing term B™~~)define the above-mentioned Slater-Koster problem. If the mixing is used, Eqs. (11).,(3), (12), and (13) define an Anderson-Moriya"problem coupled to a Slater-Koster one. Thesame formalism has been used in other contexts,namely, the formation of actinide-impurity mag-netic m'oment in disordered transition hosts" andrare-earth impurities in s-P hosts. " The mathe-matical solution of this problem together with thesuitable sum rules for the electron occupationsnumbers is the subject of the Sec. III.

III. FORMALISM

The effective Hartree-Fock Hamiltonian, Eqs.(10), (3), and (ll), can be treated in two steps.Using the Green's function formalism one definesan "intermediate" Green's function g to describethe scattering of the extended d states. In such away one solves the extended Slater-Koster pro-blem defined in Eqs. (12) and (13) through theDyson-like equation

The solution of the coupled equations is

G~~ 1-'" [g" ()]-'-IV, I'-"'()The G",f ((o) propagator is given by

G, fa(&0) =g i«(&o)+g foa(&o)VdfG', ia((o), (18a)

G~,' ((o) = g~o~~((u)VqdGod';(&u) (18b)

and from Eqs. (15), (16), (18a), and (18b) one gets

ej ~f +ff (+O o) +Udy g(NOa'

The Anderson-Moriya problem, formulated interms of a Dyson-like equation, can be defined by

G =g+gV'G,

where the matrix elements g between d and fstates are noted as g

dd and g~~. The f fpropato-ris obtained from the two coupled equations

(Vo(&) giia(&) +giOa(&)i'ydGOi a((O) ~

G." .(~) =g!" (o)VdyGoi' (~)

. where g has the following matrix elements:

gdd((O) Q gi k( RRid /')( (dg" )

(14) Gff. (~) =gfi (od)+ g' '(~i)1 V dd. .gfo(~)~8 ppk)

go (&)1-Vgll(od) [g"...((o)] '-I Vdg I'l.'."((o)

c&K being the band dispersion relation.Equation (11) yields for the diagonal matrix ele-

ment

g 'f(~)1 —Vg' ((o)

The 4f occupation number . is obtained from

(19)

g";((o)=g';"(&o)+g'".(oi) 1 V, ,g."((d) (15)&-Vg«&~& '

One has at the origin

- dd goo((d) Ed((d) EI Vdd() I VE()do()x()

Q(n~() ) = ——Im Q Go~~~ ((o)dooa CJ

8 cfQp=- -impa &-~ (o &i —

I Vd-i I ~d, o((o)

18 ROLE OF PHONONS AND COULOMB INTERACTIONS IN. . . 5135

where Z~ is the Fermi level.A convenient way to take into account the possi-

bility of f-bound states below the d band is to di-vide the integral of Eq. (20) into two parts: (i)

respectively.By imposing

g(n', .)+ 6n'= 1, (26)

(21)including the contribution from the extended statesin energy and (ii) if bound states are present, theycontribute as

ni ~1' "' ~I-IV., I'[»l,.()/s ]l...,

(22)

where E~, is position of the pole corresponding fothe 4f, propagator. As expected the total 4f oc-cupation goes to 1 when «~&- -~.

The localized-state occupation number is givenby Eg. (21) if bound states are absent or Eg. (21)plus (22) when a bound state exists below'-the dband.

On the other hand the total change of d occupa-tion number is

1 gg6n'= —-Imp t [Gtf, ((o) gf';,-((u)jdcu.

W

Using Eq. (19) one gets.

10 v Vp, (Z~)6n = —arctani

I'~ —l vs. l'[»g..(~)/3~1

l veal E, (+)

(23) .

10 vVp (E )6n'= —are tan ~„( )

I vu. l'[&...—(~)» ~1w-~. -lv„, l z, ,(~)

g lv. ,l'[s&". .( )» 1

~ 1 —lVej l*[»u,o(~)/6~) s, .' (24)

Summing Eels. (21), (22), and (24) one has

{n, &+en = —arctan10 v Vp~(E~)

Q PQB g

+ (1-u), (25)

where ~=0 or 1, if one has a bound state or not,

where the factor of 10 in the first term accountsfor d-band degeneracy. The integral in Eq. (23)can be treated similarly as done for Eq. (20). TheEq. (23) becomes

one gets the sum rule which governs the self-consistent procedure and contains the essentialphysical requirement of charge neutrality. Twolimiting cases are to be pointed out. IfZ, (n~a )=1, which corresponds to the Eu ' con-figuration, then 6n~=0. The othe limit;Z, (n~o, )=0, corresponding to the Eus' shouldsatisfy for 6n~= i. This means that the f electronis transferred to the d band and the screening isperformed entirely by the d states. A word ofcomparison with Ref. 1 is in order. In that casetwo ionic configurations were present and theelectron transferred to the 4 band was screenedby the localized potential. In the present ap-proach, the two-level nature of the problem (twoconfigurations of the 4f ions) is replaced by theaverage occupations Z, (n~o, ), but the screeningassociated with the d band is still present.

I

rV. NUMERICAL RESULTS AND DISCUSSIONS

The ideas discussed in Sects. II and III could beapplied to general pseudobinary compounds likeEu(A, „B,)„,4 and B standing for transition met-als and n = 1, 2, 3, . . . . The concentration x rangesfrom x=0 to x=1. In these limits, theeuropium valence is assumed to be 3' and 2',respectively. One is interested in thi.s work inexhibiting a smooth and continuous change withconcentration of the europium valence state. Toour knowledge such behavior has only been ex-perimentally observed in the Eu(Ir, „Pt„),sys-tem. ' However, Table I of Ref. 1 may suggestother experimental possibilities. Based on thesecomments, one keeps the notation Eu(A, ,B,),throughout the work. Since no detailed informa-tion is available concerning the density of statesof the intermetallics EuA. , and EuB, to justify acoherent-potential approximation (CPA) calcula-tion, we use the simplest approach, namely, thevirtual-crystal approximation. One adopts a para-bolic Moriya density of states defined, by

pg(~) = (3/4&g) (1 —~'/&g'),

the half bandwidth A~ being chosen equal to 1 inthe numerical calculations. One assumes alsothat for a EuB~ compound the Fermi level lies atthe top of the d band, whereas in the EuA. 2 com-pound the A atom has one less electron than B.The Fermi level is fixed for x =0 imposing thed-band occupation equal to 9. For intermediateconcentrations the Fermi level lies between these

5136 0. L. T. DE MENEZES, A. TROPER, AND A. A. GOMES

V =(U„+C)g(n', .) -C.Ot

(28)

For x = 0 the potential V(x =0) is equal to —C, andwith the adopted band structure it is determinedto be C=0.3351. Then the difficulty of estimatingelectron-phonon interactions and phonon spectra iscircumvented using this approximation.

In the x = 1 limit, p~(Ez) = 0 since e~ lies at thetop of the d band. This implies that the Slater-Koster phase shift vanishes. So, in the sum rule,Eq. (26), only the Anderson-Moriya contributionsremain, which should be equal to 1. One may askif a self-consistent solution can be achieved for aneffective f level, ez, lying within the d band. Sincep„(E„)vanishes and so does p„,(Ez), thus imply-ing that the phase shift associated with 0 or 4

spins is equal to m. This situation violates thesum rule and is dropped out. Another possibilityis to try a solution involving a filled and an emptyf bound state. As mentioned above the constantVdz assumption (~ V~&~'= 0.05) implies in takingthese levels one far below and the other far abovethe d band respectively. Since one takes (n~«)= 1and (n~„)=0, one sees that a quite large value ofUzf (-203,~) is necessary. Although the validity ofthe Hartree-Pock scheme could be questioned, oneargues that this is an artifact of the constant~ V„z~' para. meter. So, much smaller values of

limits. The self-consistent determination of thef level occupation is defined by the Slater-Kosterproblem coupled to the Anderson-Moriya one. Aswas shown in Eq. (25), the Slater-Koster potentialdepends on the 4f occupation number. For x=0one has a Eu" configuration corresponding in thepresent picture to Z, (n ~0)=0. This means f lev-els far above the top of the d band. In fact, if onetakes

~ V,z~'= const, there exists a critical dis-tance between the f level and the top of the d bandabove which the Anderson-Moriya occupation num-ber can be neglected. In this case the sum rule,Eq. (26), with @=1 imposes the Slater-Kosterpotential to satisfy

10 vV(x=0)p, (Z )1 —V(x = 0)&~(E~)

where V(x =0) denotes the value of the Slater-Kos-ter potential corresponding to x=0 and it is givenfrom Eq. (13) by

V(x=0) = C&n—,'Recall that 5n," is the change in the d occupationnumber at the origin. If one approximates 5n," by5n, defined in Eq. (24), which amounts to assumea perfect local screening, one may use the sumrule, Eq. (26), to rewrite the potential for an ar-bitrary concentration x:

Uzz (e.g. , —2A,) can be used if hybridization isproperly taken into account. Then the boundarycondition, x =1, is only satisfied by the magneticsolution (n~«)= 1 and (n~»)= 0.

In principle the complete self-consistent pro-blem for intermediate concentrations could besolved numerically using Eq. (28) together withthe equations of Sec. III. One has preferred how-ever to restrict the parameter space of the pro-blem introducing another simplification concerningthe potential V. This is based on the following re-mark concerning the Slater-Koster phase shift.Suppose that the. potential V is constant. For anattractive potential (it is true for Eu" configura-tion) the phase shift starts from zero at the bot-tom of the d band, increases to attain a maximumand decreases to zero at the top of the band. Onefixes such potential to warrant a screening equalto 1 for x =0. This gives the same value of C=0.3351 obtained before. At the top of the bandone obtains also the same behavior as before. Forintermediate concentrations the U~f repulsiveterm in Eq. (28) does not destroy the tendency ofthe phase shift observed for a potential which isattractive and constant. Then the sum rule Eq.(26) which contains the more important physicalaspect of the problem is satisfied in the sameway, that means the charge difference I -Z, (n~„)is always attracted in the d band.

%ithin the above discussed approximations onesolves the self-consistent problem.

One uses the model density of states [Eq. (27)];the Fermi level as a function of concentrationbeing defined in terms of the band fillings at thelimiting concentrations. It is assumed that thed-band occupation number changes by 1 in goingfrom x =0 to x =1. Vhthin the proposed simplifi-cation of constant V= -C, the parameters of themodel are fixed by the boundary conditions at x = 0and x=1, namely, Eu" and Eu", respectively.For intermediate concentrations, the self-consis-tent procedure is defined by Eqs. (2), (23), andthe general sum rule, Eq. (26). The results areshown in Fig. 1, which can be compared qualita-tively with the experimental observations of thevalence state of the Eu(Ir, „Pt„),system. ' Forcomparison one has plotted also in Fig. 1 the re-sults obtained within the phenomenological modelin Ref. 1 (cf. Fig 3of Ref.. 1). In both cases, thescreening condition plays a central role. How-ever, in the present work, hybridization effectsgive a natural origin for the width introducedphenomenologically in Ref. 1. One believes thatpresent approach can be considered as a step fur-ler in avoiding the phenomenological aspects ofRef. 1. The 4f-level width, due to distortions ofthe usual Lorentizian form were estimated com-

ROLE OF PHONONS AND COULOMB INTERACTIONS .IN. . . 5137

Ov, mp

I g~z,~g~ ) = Q A(Mz J o&, m, )O]= Oyfftg= tà j

tftyOy N2O2'

tN6O6 mvO7

(29a)Ops ltle

0.0 0.2 0.4 0.6 0.8 1.0 X

FIG. 1. Change of the total 4f occupation number withthe x concentration in the pseudobinary intermetallicsystems Eu(A& „B„)2. The full line represents the para-magnetic solution and the dotted line the magnetic one.The dashed line shows for comparison the results ob-tained in Bef. 1 within the phenomenological model. Theexperimental data are obtained from interpolation oflattice-parameter measurements in the Eu(Ir~ „Pt„)~system (Bef. 2).

puting the second moment of the f density ofstates. The behavior of the level width as a func-tion of the concentration shows a rapid increasefrom zero to attain a plateau value. of the order of600 K and decreases, in the same way, at theother limit of the concentration range. Such be-havior suggests that the fast-valence-fluctuationregime persists along the concentration range ex-cept in the borders of the concentration limits.

Up to now, a nondegenerate f level was as-sumed. The question one raises naturally is un-der which assumptions the above results aremeaningful for many-electron ions like europium.In particular one may ask how physical quantitiesrelevant to europium intexmetallics may be ob-tained from the crossover curve obtained underthe hypothesis of a nondegenerate f level. Onewants to propose the following picture: the bindingenergy of an Eu atom is such that during the fluc-tuation process, six electrons remain tightlybound to the ion core by the strong ionic potential.So, only one electron is active during the Quctua-tion process. The natural set of states to describethis situation could consist in a complete set ofatomiclike states, properly symmetrized to pre-serve atomic symmetries, describing a 4f' con-figuration and a 4f'5d' configuration. In the liter-ature, to our knowledge, only few works" "dis-cuss the many-electron nature of the f ions. Oneadopts here the formulation introduced in Ref. 1V,which although more appropriate to a perturbationscheme may suggest a way to formalize the pre-sent picture. These authors" construct a many-electron ionic function starting from one-electronBloch wave functions (or their Wannier trans-forms) for f and d states.

The St'ates corresponding to an Eu ion in theEu'+ and Eu" configurations read

I

(29b)

where J and M~ stands, respectively, for theangular momentum and its z component, respec-tively. In E&l. (29a) and (29b) suitable com-binations of atomiclike functions, with the appro-priate symmetry of states Eu" (J,M~) andEu" (O', M~~) are provided by the proper choice ofthe coefficients A(Mz, J) and A(Mz, J').

~ 0) isdefined as the extended vacuum consisting of allcore electrons and filled bands. These states areassumed to be eigenstates of the uncoupled f and-@-band Hamiltonians and, in principle, intraionicinteractions are contained from the beginning,giving rise to spectroscopic energies. Then d f-hybridization or whatever interaction among d and

f electrons will mix these states. Moreover oneexpects that E&ls. (29a) and (29b) form a suitablecomplete set for perturbation theory, that is, agood zeroth-order approximation for the coupledsystem.

Our approximation goes in the reverse sense;one wants to suggest modifications to be made inE&ls. (29a) and (29b) in order to include &lualita-tively the effects of the coupling without makingany further perturbation theory. One has devel-oped an one-electron theory along this work.From this one-electron picture one extracts occu-pations numbers which include hybridization ef-fects, the role of d-d interactions mediated byphonons and Coulomb interactions. We proposethen to use this information to construct manyelectron states like Eqs. (29a) and (29b) but keep-ing the idea of a fixed "shell" of six electronsduring the fluctuation. So, one conserves sixpurely atomic f-like functions denoted by

~ f ).and adds two modified (weighted) ones:

Starting from these new functions, one constructsmany-electron functions using the above-g. en-tioned scheme.

5138 O. L. T. DE MENEZES, A. TROPER, AND A. A. GOMES 18

Symmetry requirements are incorporated againthrough the quantities C(M~, J). The final resultis clearly equivalent to multiplying Eqs. (29a) and(29b) by the weights Z, i(n~o .) and I - Z, i(n~„),respectively. Such a simplified picture neglectsall intraionic relaxation processes which occurwhen a f electron hops into the d band. In view ofthis picture, the isomer shift of the fluctuatingEu ion can be described from the crossovercurves of Fig. 1. Moreover, one argues that be-hind the experimental determination of the valencestates through the Mossbauer experiments theseassumptions are implicitly included. From theknown values, respect to a given source, of theisomer shift associated with Eu'+ and Eu" con-figurations, one fits the observed shifts usingweight factors like those introduced above. Then,the present picture connects these data to someelectronic parameter s.

One has discussed in this section the one-elec-tron results, in particular the crossover curvesand the 4f-level width. In Fig. I one has plottedthe paramagnetic solution (full line) and the mag-netic one (dotted line). The level-width behaviorfor the magnetic solution shows essentially thesame behavior as in the paramagnetic case.

It is experimentally known that a 4f' configura-tion shows a 7/~ net moment in agreement withHund's rule. In some cases, e.g. , EuPt„ thismagnetic moment couples with neighboring onesand produces long-range order. The magneticorder in EuPt has been usually ascribed to Ru-dermann-Kittel-Kasuya- Yosida interactions medi-ated by the highly mobile s electrons of the com-pound.

An interesting question to be discussed concernsthe persistence or not of magnetic order for in-creasing Ir concentration in Eu(Ir„Pt, „),. Thesmall concentration region x&0.1 has not, to ourknowledge, been experimentally studied: such anexperimental study is in our view very illumina-ting. In fact, if fast charge fluctuation starts tooccur, no more Curie-Vfeiss behavior is expectedto occur. According to our predictions, the widthincreases steeply as a function of concentration toattain a plateau of the order of 600 'K. Thiswidth would correspond to a fluctuation time of theorder of 10 "sec. Clearly this predi. ction dependson the particular choice of the value of the ~V~&)'matrix element. If experimental data become a-vailabl, one can check the validity of our as-

sumption for this parameter. On the other hand,if experimental results indicate that magnetic or-der persists over a broader concentration region,our

~ V~& ~' assumption should be revised takinginto account a better description of the hybridiza-tion matrix element (cf. Ref. 8).

V. FINAL REMARKS

In this section one summarizes the principalresults and indicates possible extensions of thiswork. One has developed an extended AndersonHamiltonian which includes among other interac-tions the coupling of the d band to lattice vibra-tions. A general formulation within the Hartree-Fock approximation was developed and the numer-ical results obtained using some simplifications.The-general results support a previous view' ofthe role of d-band screening of the extra chargetransferred during the fluctuation process. Thisscreening is connected to a phonon-mediated in-teraction among d electrons. From the numericalresults, crossover curves could be compared withexperimental data. Also, the 4f level width whichis computed from the self-consistent solutionshows a plateau for a large concentration rangeand a rapid decrease to zero in the neighborhoodof the concentrations limits g =0 and x = I) of thepseudobinary intermetallic system Eu(A, ,B„),.These two facts show the consistency of our viewof fluctuating systems, in particular respect tothe role of the d-band screening via a general sumrule. One has developed also a qualitative view ofthe many-electron ions like Eu and connectedsome experimental information, extracted fromisomer shifts measurements, to the microscopicparameters of the model via the total f occupationnumbers derived from the one-electron problem.Finally one has pointed out the limitations of thesuggested picture with respect to magnetic pro-perties.

The above-mentioned ideas can be generalizedto treat extended versions of the one-center Harn-iltonian discussed here. More specifically the socalled alloy analogy treatment of strong correla-tions" can be used to describe a collection of flevels and a d band. The same couplings invokedhere can be treated within this approximation andin particular the phonon-mediated coupling amongd states may be of central importance. "

18 ROLE OF PHONONS AND COULOMB INTERACTIONS IN ~ ~ ~ 5139

O. L. T. de Menezes, A. Troper, P. Lederer, andA. A. Gomes, Phys. Rev. B 17, 1997 (1978).

2E. R. Bauminger, I. Feiner, D Levron, I. Nowik, - andS. Ofer, Solid State Commun. 18, 1073 (1976).

3I. Nowik, M. Campagna, and G. K. Wertheim, Phys.Rev. B 38, 43 (1977).

O. L. T. de Menzes, A. Troper, and A. A. Gomes,Phys. Rev. 8 17, 4117 (1918).

~F. D. M. Haldane, Phys. Rev. B 15, 289 (1977); 15,2477 (1977).

6P. S. Riseborough, in Valence Instabilities and Related@arround-Band Phenomena, Rochester, N. F., 1976,

. edited by R. D. Parks (Plenum, New York, 1977),p. 399.

D. I. Khomskii and A. N. Kocharjan, Solid State Com-mun, 18, 985 (1976).

R. Jullien and B. Coqblin, Phys. Rev. B 8, 5263 (1973).~A. R. Bari, Phys. Rev. B 3, 2662 (1971).DA. C. Hewson and P. S. Riseborough, Solid State

Commun. 22, 379 (1977).~'J. C. Slater and G. F. Koster, Phys. Rev. 95, 1167

(1954).~P. %. Anderson, Phys. Rev. 124, 41 (1961);T. Moriya,Prog. Theor. Phys. 34, 329 (1965).

~3J. R. Iglesias-Siccardi, A. A. Gomes, R. Jullien,B. Coqblin, and F. Ducastelle, J. Low Temp. Phys. 27,593 (1977).

'4A. Troper, O. L. T. de Menezes, P. Lederer, andA. A. Gomes, Phys. Rev. B 18, 3709 (1978).

~~B. Alascio, in Ref. 6, p. 247.' J. H. Jefferson and K. W. H. Stevens, J. Phys. C 9,

2151 (1976).A. Bringer and H. Lustfeld, Z. Phys. B 28, 213 (1977).F. Brouers, F. Ducastelle, G. Treglia, M. Brauwers,and J. Giner, J. Phys. C 4, 207 (1976).

~F. Brouers, K. H. Benhemann, O. L. T. de 1Cenezes,A. Troper, and A. A. Gomes (unpublished).