PHYSICAL REVIEW B VOLUME 18, NUMBER 7 1 OCTOBER 1978

Configuration fluctuations and the dilute-magnetic-alloy problem

Samuel P. BowenDepartment of Physics, Virginia Polytechnic Institute and~State University, Blacksburg Virginia 24061

(Received 5 January 1977)

A new theoretical treatment of the Anderson model of transition-metal or rare-earth impurities in a simplemetal is presented. This study treats the impurity Coulomb correlation exactly and the host impurity

coupling to lowest order in the sense of a self-consistent degenerate perturbation theory. The central result ofthis work is that if an impurity interconfigurational excitation energy (ICEE) is close to the Fermi energy inthe atomic limit, a temperature-dependent shift of the ICEE is found for the interacting system. Thistemperature-dependent shift is shown to give a good description of the experimental observations on dilute

magnetic alloys.

I. INTRODUCTION

For many years the problem of transition-metalor rare-earth impurities in noble metal hostshas been studied theoretically be examining thesimpler S-shell Anderson model. ' This modelHamiltonian has g noninteracting part K, repre-senting the single impurity atom states and theconduction electrons of the host. In the usual no-tation' Ko is written as

1Ho= Q f~ ng + 2 UBg ngg+QCf nf,

Within this Anderson model, the impurity andhost conduction electron systems are coupled to-gether by a one-electron hopping term V whichrepresents a conduction electron hopping on andoff the impurity site

&=+V~,(c~~,c~+ cf c„,) .Rs

In many applications of this model the couplingmatrix elements V are regarded as small. Thoughthis is not essential to much of what follows, themajor thrust of this work will assume a smallVIe.

The major thrust of this study, besides assuminga small V~, is to treat the impurity correlationand many-electron configurations exactly, and toexamine under what circumstances such a systemcould reproduce the experimental properties ofdilute magnetic alloy systems. The consequencesof this study form a new theory of the dilute alloywhich is a significant departure from previoustheoretical descriptions. Because the results areso different it is well to briefly summarize themain points, even though heuristic and detailedpresentations are also given below.

First, the method used here is rooted in theionic picture of Schrieffer' and Hirsi (SH). Pro-ducts of many-electron impurity states and host

conduction-band states are taken as initial statesfor the alloy system. The equations of motion ofthe stepping operators4 between many-electronstates which differ by one electron are studiedin a formalism which combines elements of Feen-berg perturbation theory, ' evaluation of truncatedsecular equations as in degenerate perturbationtheory, and a self-consistent determination ofthermodynamic correlation functions. '

Second, the strongest effect of the impurity-hostcoupling is found in what the "Kondo" picture'would describe as a strong-coupling regime, andwhich in this picture is best described as the neardegeneracy of the unperturbed states of the alloysystem. To be more explicit the strongest effectof the V~„coupling is found when the energies ofan n electron impurity configuration and the N electron host Fermi sea are approximately degeneratewith n + 1 electron- impur ity configuration and theN+ 1 electron-Fermi sea of the host. That a smallmatrix element should show its greatest effect whenthe unperturbed states are degenerate is wellknown. However, in this context the conditionsmanifesting a many-electron degeneracy have notbeen considered before. As is shown below in anheuristic discussion the alloy many-electron de-generacy implies that the impurity configurationenergies E„and E„„arenot degenerate but differby an energy equal to the conduction-electronchemical potential p,, that is,

E„„-E„=p, or E„-E„,= p.

The physical relevance of this condition restson two consequences of this study. First, theunperturbed energy E„„-E„—p. occurs as oneof the atomic limit interconfigurational exci-tation energies (ICEE's) of impurity Green'sfunction when the host and impurity are in equil-ibrium. Second, our calculation for the coupledsystem derives a termperative-dependent energyshift 5E„ for these poles of the impurity Green's

18 3400

18 CON FIGURATION F LUCTUATION S AN D THE. . . $401

function in thermodynamic equilibrium. Thetemperature dependence of this shift in the inter-configurational citation energy (ICEE), E„„—E„—p, —5E„ is shown in this paper to give a fairlygood description of the experimental dilute alloyeffects.

The departure from the traditional perspectiveof the Friedel-Anderson' or Kondo' approaches tothis problem is apparent when one identifies theinterconfiguratiogal energies of the Andersonmodel a„,—p. or &„,+ U- p, . The considerationsdescribed above lead us to examine the conse-quences of one of these energy differences beingclose to zero. In the traditional approaches tothis problem, both of these energies are con-sidered as being large in absolute value and ofopposite sign. In this work a "ma.gnetic" responseis obtained for c~, —p. &0 and z„,—p, + U&0 as forprevious studies, but the parameter values forthe model are quite different. The essential dif-ference between our perspective and previous ap-plication using the Anderson-model Hamiltonianis that we regard the model pa, rameters as arisingfrom an ionic context, while previous treatmentshave used Hartree-Fock (HF) calculations todetermine z~, and U. As discussed in Appendix D,these HF estimates tend to emphasize the oneelectron energies far from the Fermi energy. Itis an interesting commentary on the tendency ofHF estimates to be too large that the experimen-tally estimated lif teime broadening' of the d or-bitals

6= mN(0)~ V~ ~

~, (1.3)

where N(0) is the density of conduction-electronstates at the Fermi energy, has been consistentlysmaller than the initial HF calculations predicted. "The most desirable procedure for the presentwork mould be to evaluate the model Hamiltonianparameters using a many-electron calculationprocess of the type suggested by Hirst. ".

The a priori evaluation of model parametersin the ionic context for even a simple Andersonmodel or for any more realistic model representingan iron group or rare-earth impurity is quite beyondthe scope of the present effort. The aim here isto explore the consequences of this new perspec-tive, examine what model parameter values areneeded to reproduce experiment, and raise theobservation that a previously unexplored constel-lation of hypotheses seems to explain many facetsof the dilute alloy problem. The fact that mymethod begins by treating the many-electron ef-fects of the impurity and represents a new and,possibly, powerful theoretical tool for such prob-lems lends merit to the possibility that this des-cription of the dilute alloy problem may be phys-

ically relevant.The work done in this paper differs from pre-

vious theoretical treatments in that the Anderson"HF criterion for the presence of a magnetic mo-ment is definitely not used. While Anderson'streatment has been criticized by many' on thegrounds that it lacks rotational invariance anddoes not hold in the b, -0 limit, it still retainsa strong following, especially in the experimentalliterature. The criterion for magnetic behaviorto be developed in this paper is simply a questionof whether the impurity is weakly enough coupled tothe Fermi sea to be able to independently orient itsspin or whether it is strongly correlated with therest of the conduction electrons.

The results of this paper also differ markedlyfrom the large body of literature" which is basedon the "Kondo" themes of the impurity inducing amany-body correlated state in the host conduction-electron gas. In particular, the widely held ideaof the Kondo effect being caused by a narrow res-onance at the Fermi energy which is not an im-purity interconfigurational excitation energy isrejected. Bather, the perspective af this workis that the ICEE peaks are not necesarily wide inenergy, that estimates of the widths of the z~ and

c„+U peaks in the 8-shell model have been derivedthrougn interpretations of the model in a HF senseand could be too large. The perspective of thispaper attempts to explore the proposition that thedilute magnetic alloy behavior is caused by many-electron effects localized on the impurity site andnot spread out through a cloud of surrounding con-duction electrons. This picture of the magneticbehavior of the ion being localized is quite con-sistent with the experimental results of neutronsca,ttering and other microscopic probes. "

Because of the recent application of Wilson's"Kondo calculations to this Hamiltonian, "a briefcomment of comparison is in order. The theo-retical methods used here are complimentary tothose used by Wilson et al. , though a direct com-parison may be difficult because of the necessaryapproximations required in this work. Wherethe Wilson eI, a/. method constructs explicit many-electron states for truncated many-electron Hamil-tonians, the method of this paper examines step-ping operators which step between many-electronstates whose electron counts differ by one. Wherethe Wilson method starts with small numbers ofelectrons in the truncated system and builds iter-atively to larger number of electrons, the ap-proach of this paper attempts to evaluate thethermodynamic expectation values of steppingoperators in the thermodynamically most probablemany-electron states. Of necessity this evaluationcan only be determined approximately and the ap-

3402 SAMUEL P. BOWEN 18

proximations must be carefully examined. How-ever, both calculations for the susceptiblity agreequite closely in the high-T and low-T regimes. Inthe intermediate temperature ranges, the twomethods are not in agreement. The root of thisdisagreement is examined elsewhere, but onepoint bears discussion here. There is only onepart of the Wilson procedure that is not justifiedin a rigorous fashion. The process of calculatingwith a sequence of truncated Hamiltonians hasbeen placed on sound footing in another contextby the studies of Masson. " The numerical.stability of the convergence for many-electronstates with -70 electrons is well demonstratedby the careful numerical studies of both Wilsonand Krishnamurti. Wilson's a priori anzatz whichreplaces the uniformly dense unperturbed con-duction-electron energies c-„=(lkl —Ikz I)vz withthe discretized c„=+I/A" sequence where & = 1.5,2, 3 underlies the whole Wilson method and hasnot been placed on the same solid foundation asthe other elements of their calculations. TheWilson discretization anzatz replaces host conduc-tion-electron energies near the Fermi energy,which in most models of pure system display analmost constant density of states near ez, with ahost density of states that is logarithmically sing-ular at z„. While such a discretization enablesthe application of the powerful renormalizationgroup methods to this problem it must still beregarded as an hypothesis motivated by the wealthof perturbative studies on this problem.

A test of the Wilson discretization anzatz is wellbeyond the scope of this study. It may well bethat such a test is not possible in the context ofthe current approach of this study. However, acomparison of the two methods will be made else-where. This discussion has been made only toindicate that in principle the theoretical approachof this study is not in contradiction with the gen-eral schema represented by the Wilson et al. cal-culations and that at least one element of Wilson'smethod is open to question. The differences be-tween this paper and previous studies lie in ap-proximations and hypotheses the nature of whichleave open the question of the microscopic des-cription of the dilute alloy problem. It is in thiscontext that the following study is presented.Further analysis and possibly new experimenta-tion may finally resolve the differences betweenthis approach and previous studies. In the mean-time, the consequences of the ionic picture ofSchrieffer and Hirst (SH) and the many-electrondegeneracy of the alloy will be presented below.

To gain a more heuristic understanding for thephysical picture used here, we need to adopt ageneral perspective. The problem we examine

involves the coupling of states of a "free" Hamil-tonian H, by a relatively small perturbation V.Let Ig„& and Igs& represent two states of H, whichwill be specified more precisely below, The gen-eral consideration of Hamiltonian secular matricessuggests that a small perturbation can quite signif-icantly alter the spectrum of H, if the unperturbedstates are close to being degenerate in energy. "Let us examine under what conditions for thisAnderson model the small coupling V could beespecially important in this sense. In other words,we seek to examine the states coupled by V tosee under what circumstances they could be closein energy On. e of the two types of states which arecoupled by the potential V is

I& &= lt.(N )&l&.(N)& (1.4)

where Ig~(N„)& is an N~-electron state of the hostconduction electrons, and

I P~(N)& is an N-electronimpurity state. In a realistic model If'(N)& mightrepresent a d" of f~ configuration of the d or fshell of the impurity. The second state is

ly.&=I 0'(N. I)&l—q.(N+ 1)& . (1.5)

Note first that both of these composite states cor-respond to having N~+N electrons. The energiesof these two states are

E„=U~(N~)+ E„,E =U'(N —1)+E „.

%e thus seek to explore the consequences of thecondition that E„=E~, i.e., that

EN,1-EN- ~UF(NE) - UE(NE-I)j=o . (1.7)

There are actually quite a large number of dif-ferent states

I gz(Nz)& and correspondingly manydifferent types of composite states lg„& and If'&for the same impurity configurations. This makesthe evaluation of (1.7) somewhat difficult, unlesswe adopt a thermodynamic perspective whichsubstitutes the thermodynamically most probablestate energies U~(Nz) and U~(N~ —1) into this ex-pression. In this case, the difference

U~(N~) —U~~(N~ —1)= p, =g~, (1.8)

This result says that the energies of states intwo adjacent configurations of the impurity shellshould have a difference which is on the order ofthe chemical potential of the host conduction elec-

where p, is the chemical potential of the host con-duction electrons. This reduces the condition forthe approximate degeneracy of the composite statesto the statement that the impurity interconfigura-tional excitation energy (ICEE) should be close tozero,

CONFIGURATION F I. UCTUATIONS AND THE. . . 3403

trons. This could mean, for example, that thetwo impurity configurations might differ in energyby something on the order of 5-8 eV. A Prioriverification of whether any atoms satisfy thiscondition is difficult because the configurationenergies are those appropriate to the impurityinside the host metal and not to the free-atomenergy differences. An impurity atom whichcould reasonably be a candidate for this pres-cription is Mn, "based on its d"s'- d" 's' coreto valence transitions.

It is important at this point to emphasize thateven though the preceding paragraphs have beenwritten in the language of the conduction electronsinteracting with the d or f shell, we would arguein agreement with Flynn ' that the impurity cell isalmost always netural and that the interconfigura-tional excitation energies (ICEE) which we dis-cuss are similar in magnitude to the core-valencetransitions discussed by Flynn and correspondin the neutral atom to d""'s'-cPs' transitions andothers.

A second important consequence of (1.9) be-comes immediately clear when it is real. ized thatthis small quantity E~„E„—p, -is exactly one ofthe ICEE's at which the atomic limit impurityGreen's function has its poles." Thus, the neardegeneracy of composite states in the host+ im-purity system can give rise to ICEE resonancesin the t matrix which are near the Fermi energy.It is worth noting that this degeneracy of com-posite states in the alloy requires the close match-ing of two physically distinct quantities: the ener-gy difference between impurity configurations andthe host Fermi energy. The mismatch of thesequantities will be seen below to give a significantdepression of the effects. of the coupling V on thecomposite system.

The rest of this paper will pursue the logicalconsequences of the heuristic picture presentedabove. Our analysis employs impurity parametervalues which in the usual Kondo picture would beregarded as strong J coupling, i.e., large~(O) Z.„=(~/v)[~„—(s„,+ U)-'].

There are at least two distinct ways in whichthis parameter regime can be realized. The firstof these is by taking the large V-„~ limit, in whichthe d-lifetime broadening 6 dominates all otherparameters in the problem. In this case the tem-perature behavior of thermodynamic quantitieslike the susceptibility is determined by comparingthe temperature T with T~ =4/mt', where ks isthe Boltzmann constant. For temperature belowT~ we have a constant Pauli-like susceptibility

This is not the regime that we will study here,for as is well-known in such a regime it makeslittle sense to begin with an ionic or atomic pic-ture of the impurity.

The other strong J-coupling situation corres-ponds to the degeneracy of two composite statesof the alloy and in the S-shell model to eithere„or e~, + U being small [cf. Eq. (1.6)]. For thismodel z~, is the energy difference between %=1and N=O configurations, while z„,+ U is the en-ergy difference between E= 2 and %=1 configura-tions. Both of these quantities are measured withrespect to the Fermi energy and so do notexplicitly display the chemical potential. 24

The degeneracy of composite states of the sys-tem and the resulting placement of an ICEE nearthe Fermi energy implies that we must encounterstatistical fluctutations in the grand canonicalensemble between the impurity configurationsconnected by the ICEE. The work reported belowdisplays the connection between the configurationfluctuation phenomena" and the dilute magneticalloy phenomena.

This paper consists of five sections. SectionII will briefly review the ionic approach of Hirstand Schrieffer (SH). Section III will lay the theo-retical frame work for doing degenerate pertur-bation theory in the grand canonical ensemble.In particular, we examine here the atomic limit,display the relationship to the Hartree-Fock ap-proximation (HFA), and examine the effects ofthe coupling on the impurity states. In Sec. IV,we describe the main theoretical calculations un-derlying this report and display their applicationto a few experimental properties. Many of thetheoretical details are relegated to appendicesA. , B, C, and D, hopefully to enhance the read-ability of our results. Finally, in Sec. V, theextension of these ideas to realistic many-elec-tron atoms and to other problems is examined.

II. THE IONIC APPROACH AND THE ICEE'S

Hirst' in his important work outlined a des-cription of transition. or rare earth ions in metalswhich incorporated the earlier ionic-based per-spective of Schrieffer. " Hirst observed thatwhen the impurity potential is strong enough togive a fairly narrow virtual bound state (VBS) atthe impurity site, the traditional one-electronpicture is no longer valid and a correlated many-electron basis becomes important. This occursbecause the strong impurity potential and the nar-rowness of the VBS implies that the electronsspend more time in the impurity, thus enhancingthe importance of the local-local Coulomb inter-actions and the correlation that they imply. It is

SAMUEL P. BOWEN 18

this sort of argument which led Hirst to beginthe description of the dilute alloy with the impurityatomic states in the zeroth approximation.

The principle accomplishment of Hirst's ap-proach was to show that by starting with the es-sentially atomic point of view it is possible toderive from first principles a Hamiltonian withthree parts,

where II„,represents in zeroth order the ionHamiltonian which contains Coulomb interactionsbetween the electrons, spin-orbit interactions,and interactions with the impurity nucleus, theclosed-shell electrons, the lattice ions and theconduction electrons in their lowest states. Thisessentially atomic Hamiltonian has correlatedmany-electron states 4","'(n) which contain n elec-trons in the d (or f) shell and have term energiesE„(n). Here e represents the irreducible repre-senation to which the many-electron state 4'"'(o.)belongs. Where necessary, the rows of the irre-ducible representation will be labeled by the index

In a second quantized notation the creation op-erators for the correlated many-electron stateswill be written as bt(o. ). These operators are nthorder polynomials of the d (or f) shell creationoperators ct,. The form of the polynomials canbe determined by standard methods. "

The conduction-electron states and other closed-shell electron states are determined by a con-strained restricted Hartree-Fock (CHHF) pro-cedure described in detail by Hirst. " Physically,this CBHF procedure determines one-electronorbitals for those electrons which are orthogona, lto the localized shell states and which minimizethe ground- state energy. The conduction-electronstates are thus not only separated from the im-purity states, but also contribute to the impurityscreening. Far away from the impurity, theseconduction-electron states are little changed fromthe host states. Near the impurity the conduction-electron states are more strongly altered.

In the CRHF calculation the conduction electronsare treated as independent quasiparticles and thevariational CRHF calculation determines an ef-fective interaction between the distorted conductionelectrons and the d (orf ) shell impurity electrons.If we represent the conduction-electron annihiationoperators as c~ and the d-shell orbitals by c„„then the equation of motion for c-„,which is derivedfrom the CBHF variation can be written as'

[cf~,$C] =&I, cf + Q Vfgc~ (2 I)

In this Eq. (2.1) both the z~, and the V~ arise asvariational parameters of the CBHF procedure.This equation exhibits the conduction-electron

band energies c~ and the effective interaction ma-trix element V«between the conduction electronsand the d (or f) shell orbitals. This already spec-ifies the form of Xl which arises between the d-shell orbitals and the conduction electrons.

While Hirst did not explicitly emphasize themagnitude of V«which arises in his CRHF method,it must be quite small for his analysis to be valid.In particular, Hirst limited his discussion to first-order perturbation theory, neglecting the lifetimebroadening- of the atomic states. Hirst argued thatthis was permissible if the minimum intercon-figurational excitation energy E,„„

E,„,= min ~E~(n) —E~,(P) ~,was relatively large. Hirst estimated that E,„,should be roughly on the order of the Slater Il

integrals and thus might be expected to be as largeas 25 V. This estimate neglects one of the impor-tant characteristics of the CRHF calculation,namely, that it includes the screening of the dstates by the conduction electrons in zeroth order.While it is impossible to evaluate E,„,unequivocal-ly, short of doing a CRHF calculation itself, wecan gain insight into the allowed range of valuesby reviewing the appropriate estimates by Her-ring ' for E„,. It may be recalled that Herringestimated from comparison with atomic spectra,and other estimates of theories of conduction-;electron screening that the "effective" Slater in-tegral for the 3d shell could be in the range 3-6eV. This range of values was. obtained essentiallyby examining energy differences between 3d"4s'and 3d" '4s' configurations instead of energy. dif-ferences within a pure d shell alone.

Our interest in this paper is to look at Hirst'smodel in the grand canonical ensemble where thecorresponding version of E,„,= min~E„(o. )—E„„(P)+ p,

~is small and to do degenerate per-

turbation theory using essentially secular equa-tions or Feenberg techniques. "

At this stage it is appropriate to make again abrief comparison between the Schrieffex-Hirstionic approach and the traditional Friedel-Ander-son (FA) perspective. In one sense these twopictures simply represent alternative bases fromwhich to expand the states of the impurity andhost. The FA approach tends to emphasize theitinerant aspect, but typically fails to treat care-fully the inherent correlation effects because of areliance on Hartree-Fock approximations for thelocal states. As Wohlleben and Coles" have notedsuch an approach has led generally to theoreticalresults of questionable applicability to existingsystems. P resumably, this itinerate approachcould be made to properly describe these sys-tems, but it seems to require very complicated

CON FIGURATION FLUCTUATIONS AND THE. . .

linear combinations of states to diagnolize theHamiltonian.

The SH approach, on the other hand, beginsin a representation which already recovers theproper atomic limitas V„~-0. As notedby many re-searchers the FA approach fails to recover thislimit at all. As will be shown belom the SH ap-proach must be studied in a fashion that includesthe lifetime broadening of the impurity states,especially if the impurity ICEE is degenerate witha quasicontinuum of host conduction-electronstates.

The first task is to consider what kinds of en-ergies and operators associated with the correlatedstates are actually involved in the interaction(2.1) For an understanding of the resistivity, forexample, it is necessary to determine the t-matrixfor scattering from the impurity. As is well-known, " this is proportional to the one-electronimpurity Green's function. This means that theg-matrix energies will be the same as those whichoccur as poles in the impurity one-electronGreen's function.

While the exact nature of the EMCEE poles of theone-electron Qreen's function for many-electronsystems has been known for several years, manyworkers continue to compare conduction-electronsingle-particle energies with many-electron atom-ic-configuration energies as mell as with the HFKoopman's energies. A s emphasized by Hubbard"the excitation energies of the one-electron Green'sfunction correspond to the energy at which elec-trons bearing certain single-particle quantumnumbers can be added to the system. In particu-lar, then the "one-electron" energies should cor-respond to all possible ICEE differences

(2.2)

between configurations of the impurity systemwhich differ by one electron. The irreduciblerepresentations of the many-electron states Fz"'and I'"""must be related to each other in sucha way that the direct product of the many-electronrepresentations, 1 z"' x I'" ", must contain theone-electron representation.

In general, one must expect that for given one-electron quantum numbers (n, l, m „m,}, there willbe many resonances of differipg weights appearingin the Qreen's function. A particular one-electronorbital wave function could be part of severaldifferent many-electron states In what follows,we examine explicitly those cases in which atleast two impurity configurations will be com-peting for population in the grand canonical en-semble.

For purposes of calculation, it will be conveni-ent to have operators that correspond to the en-

ergy differences (2.2). Physically, we needpt(n, P) operators which annihilate the (n 1—)-electron states and replace them with the e-elec-tron states. If we consider only the impurityHamiltonian it is straightforward to show that theoperator corresponding to an interconf igurationalexcitation (2.2) is

(2.3)

It is the perspective of our ionic based approachthat if me calculate the properties of the systemusing as basis vectors the conduction electronsc@, and the interconfiguration excitation operators(2.3), then we have an almost diagonal "Hamil-tonian" for small coupling. The success in carry-ing out this program relies on a method whichutilizes the above operators as "states" and re-duces calculations of thermodynamic Green'sfunctions to considerations of secular equations,resolvents, and approximation procedures likethose of Feenberg. " It is to a brief discussionof this method that the next two sections aredevoted.

III. PROPERTIES OF THE GREEN'S FUNCTION

AND THE ATOMIC LIMIT

In this section, we examine properties of theone-electron impurity Green's function withspecial emphasis on the atomic limit. Also, thebasic properties of the theoretical techniquewhich we use in this paper will be developed inthis section. The essential criteria for the de-velopment of the scheme are that it should startwith a proper treatment of atomic correlation,that exact properties of the Green's function bereproduced, and that any scheme reproduce theatomic limit as the coupling V-„~ goes zero.

First, we examine the exact properties of theone-electron impurity Green's function. We ex-amine here the retarded Green's function for theimpurity one-electron orbital e„, which may bewritten as

(3 1)

Here (~ ~ ~ ) reflects the thermal average. (, J isan anticommutator, and e(t) is a unit-step func-tion."

Hubbard" has analyzed this Qreen's functionin the atomic limit and has detailed its exact prop-erties. The most important of these character-izes the spectrum of the Fourier transform G„,(&u).

Following Zubarev, 40 If ~p) and~ q) represent

exact eigenstates of the Hamiltonian H —p,N, whichhave different numbers of electrons, then theGreen's function G~, (&o) may be written

3406 SAMUEL P. 18

G (&u)=-g ' " ' (e & —e ) (32)l& Ic l &l'

8 ~, s) —(Ep-E )frere E~, E, are the eigenvalues of the states

~ p)and ~q), respectively, P is (SENT)

' and I is thegrand canonical partition function for the system.

Hubbard, Zubarev, and many others have an-alyzed the calculation of thermodynamic Green'sfunctions from many points of view. These haveincluded straightforward evaluation if the eigen-states are known, truncati:on and approximatesolution of the hierarchy of equations of motion,and the standard perturbation expansions of quan-tum field theory.

The theoretical methods of this paper rely on anextension and generalization~' of the ideas ofI onke~' and Feenberg. 4' This generalization placesthe calculation of one-electron Green's functionsinto the context of standard matrix theory, buton a space whose vectors are second-quantizedoperators and whose scalar product is the grandcanonical ensemble average of the anticommuta-tor of second-quantized operators.

The essential ideas are the following: insteadof working with the Hamiltonian H —p.N and directlyseeking its eigenvectors and eigenvalues, it provesto be slightly more convenient to work with theI. operator which is defined as the commutatorwith H- p.N, ~

La = [n, H pN j, - (3.3)

where n is an operator. The time dependenceof an operator 8 is given by (h = 1)

(3.4)

The Fourier transformed Green's function G~,(~)may then be written as

G~~(Q)) —(((47+15 —L) Cg8& Cgsj) y (3.5)

where 5 =0' and & is real. This expression maybe written in a more transparent form if it isrecognized that a scalar product between opera-tors A and 8 can be defined-for a Hilbert spaceof second-quantized Fermion operators, 4'

(3.6)

(3.V)

If we were dealing with finite dimensional ordiagonal representations for L, the determina-'tion of the various Gr'een's functions which areinvolved in the theory would require knowledgeof the secular matrix

(3.8)

This scalar product shows the one-electron Green'sfunction to be a diagonal matrix element of aresolvent (w —L) ' of a symmetric operator L:

for a convenient orthonormal set of basis vectors.This sort of calculation is familar from standardquantum- mechanics calculations and band-struc-ture studies in solids. The only essential dif-ference here is that the vectors are certainsecond-quantized operators, and the matrix ele-ments of L wiIl be temperature dependent. Thistemperature dependence simply reflects thethermal equilibrium of our system and will bethe source of an ICEE shift with temperaturewhich is one of the main results of this paper.The evaluation of the temperature dependence ofthe matrix elements would be trivial if we hadall of the information required to accomplish theevaluation in representation (3.2). This is ofcourse not the case, and the evaluation of thematrix elements will be obtained through thesolution of thermodynamic consistency equationswhich express the property of detailed balancein thermal equilibrium.

In order to clarify these ideas, let us considerthe evaluation of the electron Green's function inthe atomic limit. %'e first consider only the con-figurations for a shell of equivalent d (orf) elec-trons. As indicated in Sec. II, the atomic eigen-operators satisfy

Hb~ (a) =E„(e)b'„(o.) . (3.9)

f, = b„('S)b'„('S), (3.11)where '$ deriotes the filled shell; also, all the b„are written with respect to some canonicalordering of the single-particle orbitals c„,. Judd 'has discussed the utilization of this sort of in-variant I~ in second-quantized treatments of /"

atomic shells. The ICEE vectors P„(c,P) can henormalized by using the thermal inner product

ilg„(n, P)ll'=(( P„(n, P), (7j„(n,P))) . (3.12)

We write the normalized ICEE vectors without thetilde, P„(o., P). These vectors are precisely theorthonormal eigenvectors of L (in the atomiclimit), and they connect impurity configurationswith a single-electron difference: using (2.2),

(3.13)

This orthonormal set of eigenvectors of L is com-plete in this case so we can expand the c~,

' s inthe atomic limit impurity Green's function using

By utilizing the properties of the Lie group" as-sociated with the electron states in an atomicshell, an orthonormal set of interconfigurationexcitation operators can be constructed by forming

(3.10)

where the operator I~ is the invariant of the filledshell. ~' For a d shell this may be written as

18 CON FIGURATION F LUCTUAT ION S AND THE. . .

the standard eigenfunction expansion. The resultfor this case is

((c~„(u&—L) 'c„))

Q",,'=Q, ('S S) =n„-c„/((n„-&) '~' .It can be explicity verified that

LQ' = (E,+ U)@~;

(3.18)

(3.1 f)

Ie = c~)cg)cg)c« (3.15)

The operator acting between the one- and two-electron configurations is

(+)4s 4 2( St S) —egg(cggcg~cg~cg~) cggc qa

which may be simplified as n„;c„,. A simple cal-culation shows that the normalized version of thisoperator is

~((cg, 4.(o, P)))1'(3 14)

u —~„(o., P)

For the atomic limit (3.14) is nothing more thana rewrite of (3.2), since it is possible in thiscase to compute directly the grand canonicalpartition function as implied in (3.2). The realutility of (3.14) is that the inner product can bewritten out in terms of equal time expectationvalues of related matrix elements of the inverseof (3.8) without the partition function. The evalua-tion of these equal time expectation values by thefluctuation dissipation theorem4' yields self-con-sistency equations which reproduce the partitionfunction and all the thermal averages. In thelarge interacting system, where no other tractablemeans exists for calculating the thermal averages,this formalism is potentially quite useful.

To ground these ideas in specifics, let-us nowspecialize to the Anderson model. - We first discussthe atomic limit (Vg, =0). Because of the simplicityof this model we can enumerate the quantum statesof &p p¹The impurity states are fou r in total:a no-electron state with energy zero, two one-electron states with energy &„„and a single two-electron state with energy e«+&«+ U. The spec-trum of the one-electron Qreen's function mustreflect energy differences between those many-electron states which differ by one electron. Theonly such differences are four in number:&,&, e«+ U, and C«+ U. For a given spin values, the one-electron Qreen'8 function will alwaysexhibit two poles whenever Uc 0. This exactproperty is to be contrasted with the characteristicsof the Hartree-Fock treatment which attempts toreplace these two resonances for each spin direc-tion with a single pole at e~, + U(n~;&, where (n~;&is the average occupation number. These two reso-nances at &~, and &„,+ U are infinitely sharp in theatomic limit (V-„~=0); we should expect theirbroadened form to be present in the interactingcase for small V-„„.

Let us write out the operators (3.10) and (3.11)for this simple model. First,

The operator c„changes the number of electronsfrom one to zero and is the only such operator,but it is not orthogonal to P~', . ',

((c y(+))) (n &1 /2 (3.18)

c =~(Nds ~ ds Cs (3.20)

giving the one-electron Green's function

(3.21)

The thermodynamic self-consistency conditionswhose solutions generate the underlying grandcanonical ensemble partition function are ob-tained from (3.20) as the two equations (s = + 1)

(n, &= (1 —(n; & )f ', ' + (n;&f,", (3.22)

where f,'"' = (.P~t,'"'&f,',"'& The f,'"' ar.e calculcatedusing the standard Green's function techniques.For the atomic limit, the f,'"' are simply Fermifunctions

f (r) (egE + 1)-1 (3.23)

and the solution of the linear system of equationsis immediate. It is then a simple matter to de-

The Schmidt orthogonalization procedure will gen-erate the required vector from a linear combina-tion of c„, and P„' . The normalized form of thisvector is

(3.19)

which may be verified to have eigenvalue &„,. The(+) notation is less cumbersome for this modelthan (3.10) and is the same as Schrieffer's" wherethe vectors d,',"have eigenvalues Z,"'=e„+U/2a U/2. To simplify further the notation in thefollowing'we define the operator N',",' to equal n„,when r =+1 and to give 1 —n„, when ~=-1.

The reader familiar with Hubbard's analysis ofhis atomic Hamiltonian in the narrow band-metalcontext will recall that the operators N,',"-'c„, arosenaturally in his study. The use of the inner pro-duct (3.6) and Judd's Lie group analysis suggestmore profound properties of these two operatorsthan may have been apparent in Hubbard's earlierwork. The operators (t),'", ' are an orthonormalbasis of eigenfunctions of L containing an oddnumber of Fermion operators. The collection ofsuch operators is dense in a Hilbert space" andthe one-electron operator c„, can be expandedin this orthonormal basis as

SAMUEL P. BOWEN 18

termine all of the thermal properties of the im-purity from the solutions for (n«) and (n~().

The zero-field susceptibility x~ is obtained bycalculating the magnetic moment to linear orderin the field, in terms of the f,"functions, and itis given by

2~2 sf (-)) I f (+)XT ] f~+f (+) sE(-) I (] +f (-) f (+))

( sf (+)i f (-)+ I&a«.»l(l, f(-) f(~))

(3.24)

8C„= — (FI - i(N))„eT (3.26)

will exhibit a Schottky anomaly with a heat-capaci-ty peak occurring at a temperature T~ which isdefined as

a,T,= ,' min((c„(, (v-.„+U~'I. (3.27)

For this simple isolated-atom model, if the im-purity is magnetic the entropy change from low tohigh. temperatures will be k~ ln2, while for thenonmagnetic impurity the change is k~ ln4. In thepresence of an interaction between impurity andhost the scattering of the conduction electrons isrelated to the t matrix which can be written

t"„-„,(()=QdVI~VP~G~, (())) . (3.28)

An approximation for this can be obtained in what

where we assume g=2 and p.~ is the Bohr mag-neton.

If E,'-' & 0 and E,'+' &0, which corresponds to theone-electron states being the ground state of theisolated atom, the impurity is magnetic and showsa Curie susceptibility

g r p, ))/kgT (3.25)

for any temperature k)) T low compared to ~E,"~

or ~E,( ) ~. If E,"is negative or if E,' ' is positive,the susceptibility goes to zero as T-O, in thefirst case because the ground state is the two-electron singlet state, and in the second case,because the no-electron state is lowest in energy(U-0) and the impurity is unoccupied at lowtemperatures. One of these cases is apparentlyrealized for dilute ThCe alloys. "

In terms of atomic values of these parameters&, and U, atomic hydrogen values yield the mag-netic type of impurity z, &a~+&~+U, while atomichelium values for a and U yield the nonmagneticsinglet ground-state atom.

For any of these cases, the heat capacity at con-stant volume for the impurity in equilibrium with aparticle bath at chemical potential p. ,

might be termed the atomic limit if we put theatomic limit G~, ((o) into the expression for the fmatrix. For all U4 0, the t matrix will have res-onances at &=E,"and will never have a singleresonance.

Let us now examine corrections to this atomiclimit t matrix ~n an heuristic fashion with asimple application of perturbation theory. If thereis no lifetime broadening of the impurity ICEE's,then the scattering I, matrix has an imaginary part,

lmf(~) = g V-„,V-„.,[(I ( „-))2~5(~ c„)

+ Q„-)27(5(&u —e„—U)1,

(3.28)

consisting of two 6 functions. In this approxima-tion, unless one of the ICEE's is at the Fermi en-ergy there would be no resistivity from this im-purity at low temperatures.

%'e next examine the effect of coupling the im-purity states to the conduction electrons in twosteps: first, we consider the effect of lifetimebroadening,

d =7(N(0) V.„~, (s.so)

and then we examine in second-order perturbationtheory the change of the term energies of each con-figuration of the impurity. If the impurity ICEE'sare lifetime broadened, the normalized resistivity

R/RO=7(N(0) Imt(0)

is given by

~ - (I —Q„-))CP Q„-)~'

In the usual choice of impurity parameters,U& t~„~E("-)~&a, and the 4' in the denominatorscan be neglected, and we can write the lowest-order resistivity in terms of an effective exchangeparameter using (1.6):

2a c +U—[wN(0)Z ) (1+0

(3.32)

where we have assumed icsT &~ E~,

"'~

and thus Q~,)If one of the ICEE energies is small, say

0&&„,+ U «-a'„„ then the resistivity is dominatedby the (+) resonance which is closest to the Fermienergy. In this case the resistivity becomes

Q„-)cPR, (E,'+')'+ cP (s.ss)

If E,"is very small the resistivity would be closeto the unitarity limit for this channel, while if~E,")

~is comparable to or larger than 6 the re-

sistivity could be quite small.One of the major results of this work, which we

18 CON FIGU RATION F LUCTUATION S AN D THE. . .

derive below in- second-order perturbation theoryand verify in our many-body approximation schemelater, is the fact that the coupling of the impurityconfigurations to the conduction electrons givesrise to a temperature dependent ICEE shift 6E,'"'.In the case we are studying the ICEE becomes

(s.s4)

It will be shownlater that, if a„,+ U&0 and smalland g~, & 0 and large, then 6E,"' is positive and islogarithmically increasing at T-0. In Sec. IV,it is demonstrated that this circumstance can re-cover a Kondo-like resistivity with a quite differ-ent effective coupling constant than is usually de-rived.

As will be shown later,

4J

LU

CO

O

N09

OI

0 KIM AND MAPLE La, „Ce„xs.03

THIS WORK

(~) + 1 s 13 1 'Eg I

8 (s.ss)

in a bertain temperature range where 6E,"issmall. Expanding the resistivity to lowest orderin the shift yields

PRESSURE (kbar)

FIG. 1. This figure shows the pressure dependence ofthe resistivity derivative 8 p/8 lnT. The derivativeis normalized to 1 at zero pressure. The open circlesare experimental points and the solid line i.s thesimple theory. See text for full explanation.

Q, -,&A'—= Q (d'

)2 2 1+%(0)Z'„,

1.13 l a„lk~T

(3.36)

where

(0), 2 (eds + U)A

(&, +U)'+A' ' (3.37)

Bp cg+US lnT [(ed + U)'+ A']' (3.38)

Using the assumed linear pressure variation ofthe ICEE we may write this as

This is a Kondo-like result, but with a very dif-ferent and possibly quite small J ff .

An experimental verification of the functionalform in (3.36) and (3.37) is provided by pressurestudies of the dependence of the derivative ofthe resistivity with respect to lnT. ConventionalKondo explanations". of this effect require ex-tremely complex variations of T~ with pressure.This work results in a very simple predictionwhich is well reproduced by experiment. If weassume that pressure changes the ICEE in afashion that varies linearly with pressure, thenthe theory predicts that

Figure 1 of this paper shows a fit of this result toexperimental data" for A =8.16, X0=1.6, and

P~ = 14.66, and negligible additive constant. Sucha close fit from such a simple assumption con-cerning the pressure change of the ICEE at thevery least lends credibility to the considerationsof this work.

In Sec. IV and the appendices we will discuss thetemperature range in which this result obtains,but first it will be demonstrated that such a shiftarises naturally in second-order perturbationtheory of the impurity configuration terms.

We seek to apply the perturbation V (1.3) tostates of H, (1.2). In particular, we seek to deter-mine the changes in the energies of

ly, &= IIF,&,

ly,.&=c,', Iq,&,

i(s& —Cd) Cd) I PE& y

where if'& is any conduction-electron state of thehost. One almost immediately observes thatfirst-order perturbation theory produces nochange, and we must evaluate the second-ordercontributions. For the impurity vacuum stateiIItz& there is only one intermediate state cdt, cp if'&and the change in the energy of impurity vacuumls

Bp AxslnT (] +X2)2

where

X =Xo -P/Po.

(3.3S)

(3.40)

AE (yJ — g ITdfks (s.41)~s ks

In arriving at this result we have followed thestandard method of doing the perturbation calcula-tion for each definite many-electron state in

3410 SAMUEL P. BOWEN

which f» =0 or 1 an'd using f„,=f„, A. fter obtainingthe expression for any particular lP~) we can thentake the thermal average of both sides of the equa-tion and get the temperature-dependent shift.Upon averaging the fk, becomes the conduction-electron Fermi function. For the lg~), there aretwo types of intermediate states,

curly ) and c;;cu

For the two-particle state, the intermediate stateis cz, ct; lg~) and the energy shift is

(y )—+Q ku( fks

eus +U —e»

The result for the ICEE is

(3.43)

2 2

&(r) g(r) P kuf ks g kuf ks~E' —+U —6 — ~E ——E )4S kS k ds ks )

(3.44)

plus terms like the fk, terms but with fk, replacedby 1. These latter terms we neglect as being in-corporated into new values of e,",. In Sec. IV, weshall recover this same result in our degenerateFeenberg perturbation theory and there will writethe shift as a function A', where

Vku fk71' & —+U-ih - q—ks

2

ds ks& ——& —-zS ' (3.4S)

and in anticipation of results in Sec. V I have in-cluded lifetime broadening in the intermediatestates. As will be discussed more fully in thatsection, A' displays a log(T) behavior for hightemperature and a 1 —T' behavior at low tempera-tures.

Before closing this section, let us make somebrief comments about the impurity susceptibilityin the lifetime broadened atomic limit. Insertinga constant lifetime broadening into each ICEEresonance in the impurity Green's function changesthe

f(r) —(yt(r)y(r)) (3.46)

from the usual Fermi functions into somethingcharacteristic of a system coupled to a systemwith a constant density of states near the Fermienergy, i.e.,

These give a perturbed energy of2 2

~ (~ ) P ~kufks— ,gV u(1-fks)

kgs +U —6'k s ~ Cps ks

k

(3.42)

f ")=———imp —+—(I'+i g(r))1 1 1 P2 r 2 2r (3.47)

where g is the digamma function. If I' is muchlarger than ld")l, a straightforward evaluation of(3.24) using (3.47) gives exactly the strong coup-ling result(1. 10). At this point it is worthwhileobserving that in Appendix D we use the thermo-dynamic Hilbert space formalism introduced else-where to illustrate why HFA poorly indicates thepresence of an ICEE near the Fermi energy.

&n n &n n &fI + ~n tf (4.1)

as notation for the matrix elements of a Hamilton-ian II or L operator and examine formulas fordiagonal and off-diagonal matrix elements of theresolvent, (&u -H)„'„,. For conceptual definitenessin this application the indices n, n' (and below l, l',etc.) may be thought of as labeling quantum num-bers of many-electron correlated states as wellas one-electron states. The essential point ofFeenberg's method of perturbation theory hasbecome the modus oPexandi of band-structurecalculations. This is the property of truncatingthe secular equation in some fashion and solvingthe remaining determinant exactly. This meansthat the self-energy expressions in the Feenbergpicture always include the summation restrictionswhich arise in any determinant expression. In thediscussion that follows, our approximations willrely on two special properties of this problem.First we study a weak coupling limit for the V's,and thus include only Vk~ terms in the self-energy.Actually, this is not precisely the manner inwhich to describe the calculation scheme. Be-cause of the infinite number of conduction-elec-tron states our secular equation must be infinitedimensional. This causes no real difficulty be-cause the conduction-electron part of the secularequation is block diagonal with few simple matrixelements to the impurity operators. The off-diagonal matrix elements connecting vectors con-taining conduction electrons and those few vectorswhich are wholly impurity states are found onlyin a few rows or columns. As we begin with thediagonal secular equation for the atomic sub-

IV. INTERACTING IMPURITY AND CONDUCTION

ELECTRONS

In order to evaluate the impurity Green's func-tion when interactions are present, it becomesnecessary to deal with nondiagonal secular ma-trices of L. The most useful technique for study-ing this case has been the Feenberg point of view.To illustrate the approximations to be used belowand review% the essential aspects of the methodlet us adopt

COÃ FIGURATION FLUCTUATIONS AND THE. . .

spaces, the resulting matrix is essentially dia-gonal blocks connected by a few rows and columns,all other entries being zero. This situation mightbe described" as a generalized bordered deter-minant. When the matrix elements in these bor-ders are already small, the matrix elements ofthe resolvent can be approximated by

tron model using the d or f shell p„(o., p) couldgive rise to significant lifetime narrowing forhalf -filled shells.

Since L is a symmetric operator we must expectthe pd(",) rows of the secular equations to have sym-metric matrix elements with each ck, . Operatingwith L on P~",) yields

((o -a)„„'=[(0—e„-z(n,m)] ',where the self-energy Z(n, (d) is written

(4 2) Llg" =E~ yd" + gvkdNds Ck, /&Ndk

~ ~nS ~'z(n (o) =Z "' +"~

,(~) (d —eg (n, (o)

and the reduced self-energy is

(4.8)

()Lck. = &k.ck. + Q Vk. &N —.&' '4'"'. (4.6)

As the vectors in this equation are orthonormaltheir coefficients represent the matrix elementsof L which will be used in constructing the self-energy expressions.

Note, in particular, that the matrix elementwhich will contribute to the lifetime broadeningof the resonance associated with p(ds) is modulatedby a population factor. While in the 8-shell modelbeing studied here these population factors pro-duce no significant lifetime narrowing of the re-sonances, similar factors arising in a many elec-

e, (n, )a)=e, +~ '1 )

+ . (4.4)& ~uzi'

z(&nl) + gl sly Go

The further reduced self energies e, (nl, . . .l„„(d)have similar expressions, but more restrictions.As we-seek a quasiparticle approximation for eachresonance of the final Green's function, the formof the higher-order reduced self-energies willnot be important. If the summation restrictionsinvolve the conduction electrons, they can be ig-nored, "but the atomic summation restrictionsmust be obeyed. Within this approximation schemethe off-diagonal matrix elements of the resolventwill be written as

((d-H), ,'=[(d-e, ((d)] 'V„.[(o eg. (l, a))]-'.(4.8)

To determine the self-energies and the inter-mediate states which must go into the Feenbergexpressions for the self-energy, it is necessaryto construct the secular equation for L with res-pect to an orthonormal basis. We adopt as theinitial vectors in this basis the conduction-elec-tron operators ck, and the impurity eigenvectors

We derive the various rows of the secularequation for the Anderson model L from the stan-dard equations of motion. The first set of rowsis given by

r+ &N(r)u/s ZVkd(cdscks —

ckscds )cds ' (4 7)da d'

This expression does not manifest the desiredsymmetry and, in fact, does not directly involve

ck, alone as a vector on the right-hand side. Ap-plication of the Schmidt orthogonalization proce-dure to the second term yields the desired results:

Ly( ) -g ( )y( ) +&N(")&&/s QV„C„k

(-&)+&&Nd; & Vkdgdks

ky'

+&N( )u/S ~Vkd Cds ks Cds

dSd'k

Vkd cksc«cds,us~

where the new vector pd~ is defined as

ydk, = (n„- -&n;,&)ck,/En„-& (1 - &n„-&)]' '.

(4.8)

(4.9)

Now the first three terms of this equation form anorthonormal basis set, but the last two terms arenot orthogonal to the vector pd(",). This meansthat we must first determine the overlap betweenp's and the vectors cd;ck cd„and-c„,c„cd„and--J.

then construct appropriate linear combinations ofthe vectors to yield an orthonormal basis. Theessential outlines of this procedure are given inAppendix A. The effect on the above equation isto add to the diagonal matrix element E," a temp-erature-dependent energy shift aE,"~ which arisesfrom the orthogonalization of the different three-operator vectors.

This temperature-dependent energy shift of theICEE can be thought of as arising from the coup-ling of the conduction electrons to the impurity.Jf there were no c(mpling of the impurity to thehost in this energy range, the energy of a transi-tion from ~d" '; n& to (d"; P& would only be e„(o., P).However, with the conduction electrons mixed intothe impurity states the transition is not possiblewithout corresponding transitions of conduction-electron states. The temperature dependence ofthe shift merely reflects the thermal distributionof the intermediate conduction-electron stateswhich are mixed into the impurity configurations.

3412 SA, MUEL P. BOWEN 18

If lifetime broadening of the intermediate statesis neglected the shift at T =0 is just the formalenergy shift from second-order perturbationtheory (3.41).

In Appendix A it is shown that the energy shiftcan be written as

(„) n, &nsg (,) (I -&nag) ( )&~

where the parameters A.," are positive definitequantities defined by

(4.10)

--&X("»")(")=+V„-,&c„=,y,(")

&.

This latter definition is related to the equal timeexpectation value of an off-diagonal matrix ele-ment of the resolvent,

G„s; ((d) =((c),-, , ((o —L) 'y„-))., (r&

(4.11)

(4.12)

Since this off-diagonal matrix element of the re-solvent will depend on the same self-energy as the

Gs, (&o), the temperature dependence of the shiftmust be determined self-consistently. This isdone in Appendix B. If we define as e(" ((0) theenergy part of the denominator of the impurityGreen's function,

(4.12)

then the off-diagonal Green's function can be writ-ten in the weak coupling limit as

(4.14(~ —~)-„)[(d —e(")(a )]

'

In terms of the resonance self-energy Zs(") (ar) andthe shift of the energy, e," is defined as

e(r)(&) g(r) +g~(r)+g(r) (&) (4.i5)

If one were to approximate the impurity reson-ance self-energy Z„", by neglecting its real partand giving it a constant imaginary part, then theimpurity response would be characterized by thepresence of two broadened resonances at E," +DE~,"~

whose energies change with temperature. Such asimple approximation ean already semiquantita-tively account for some of the features of the di-lute magnetic alloy problem. As will be discussedbelow and in more detail in Appendix B, the temp-erature dependence of the A~" is of the form —lnTat high temperatures and 1 —T' at low tempera-tures. For certain values of the physical para-meters one of the resonances e" can approach theFermi energy as T-0. This can give rise to theincrease and saturation of the resistivity, thethermopower peaks, and Schottky-like anomaliesin the heat capacity as well as other observedeffects.

Actually there is no justification for neglectingthe real part of the self-energy, even in the weak-eoupling limit since it gives a contribution whichis of the same size as the shift hE~"~. Accordinglywe must study the real part of the self-energy.As detailed in Appendix A, the derivation of theimpurity resonance energy e(" yields in the weakcoupling limit

() 1;((o))( &x(")& ' '~(") '&x('» '

ds

where y(w) is the usual one-electron self-energyfamiliar from previous studies of this model,

2

y( )=Z

(4.i6)

(4.i V)

The coefficient A, is

~(+) 1 &%ts& )„{-)'s s&~

sd

(4.i8)

The remaining contribution Y, ((d) is derived inAppendix A and is

I,((d) =(1-&n,g)f' )y((d)+&n„gf(-;)) ((d-2es —0')

+—[(&n„&—&n„-,&s))((o, n,')

—(1 -&ns, &—&ns,-&)8D ((u —2es —U, a')] .

(4.19)Here f," =(est("' ps",)& and the function 8D (E, I') de-pends explicitly on the cutoff of the conduction-band density of states,

(4.20)

and f,(e) is here the Fermi function. The prop-erties of this function 8D are discussed in Appen-dix C. As noted there, at high temperatures 8~has a lnT dependence, but in al1. eases dependslogarithmically on the cutoff i.e., 8~- lnD forlarge D. For purposes of continuity, let us notethat if we write

(4.2i)

then the shift parameters X~" can also be expressedin terms of Hn to order 1/D in the weak-V couplinglimit, that is, as shown in Appendix 8,

X',")=8 (e~', e'„') +O(1/D) . (4.22)

We now consider what tractable approximationscan be made on these expressions which will yieldsimple behavior. The essential idea that we useis to seek a Lorentzian approximation for eachinterconfigurational energy peak of the one-elec-tron Green's function. Because the imaginaryparts of the self-energy expressions are onlyweakly dependent on + for energies near the cen-

18 CONFIGURATION FLUCTUATiONS AND THE. . . 3413

(4.24)

This is an extremely satisfying result from aphysical point of view because it is independentof the cutoff D when D is large, and depends onthe impurity para, meters only. Thus, the magneticimpurity problem is properly an impurity effectin that the temperature of the shift depends onlyon host density of states and impurity energies andnot on the band width D. Writing

y&+) y(-)a s s (4.25)

one finds that the temperature behavior of the im-purity resonances is determined by two tempera-tures T'"' which are defined as

Z'«& [(@«)+ 5E«&)2+ (l «))2]1/&1s + s + s (4.26)

The weak temperature dependence of the T'"' dueto the 5E,"' is not important for the cases we con-sider. The analytic behavior of A, is quite simplein three ranges. First, consider T' '&T"; thiscan correspond to a„,«0 and e„,+U& 0. Thismeans in the S-shell model that the no-electronconfiguration is very high in energy, the two one-electron configurations are lowest in energy, butthat the two-electron configuration is close to theone-electron states in energy. If the temperatureT is greater than either T" or T' ', Appendix Ashows that A,' is zero to order 1/D and there isno shift of the ICEE with temperature. If the tem-perature is lower than T', but still higher thanT", the A.

' ' is almost constant while the X" stillvaries logarithmically. For this intermediatetemperature range, T"& T & T' ', the ICEE shiftwill be logarithmic in T:

3.56T&-)(4.27)

At low temperatures, that is, T lower than bothT" and T' ', both A.'"' are almost constant and a

simple Pauli-Sommerfeld behavior is found forA '

(4.28)

Consideration of the obverse case T"&T' ' willyield similar results, but with an overall change

ter of the conduction band, their ~ dependence isnot too important. We simply write as constants

e&r) @(r)+gg.&r) &(r) I (r) (4.23)

and set the frequencies in y, (a&) equal to e(,",~, anddetermine the expression for 5E,". After a cer-tain amount of algebra ver& pleasant cancellationsoccur and we find that to lowest order in 6

Qf sign for A'.At any temperature, the S-shell impurity Green's

function and thus also the t matrix will show tworesonances for each spin direction:

(4.29)

The interplay between the two effects of the couplingof conduction electrons to the impurity, namelythe lifetime broadening and the temperature-depen-dent shift of the ICEE's, can reproduce many ofthe experimental details associated with the dilutemagnetic alloy problem.

Before we discuss the experimental effects forthis simple S-shell model, it should be pointedout that for even more complex atoms we will ex-pect that only two types of ICEE resonances willhave energies near the Fermi energy and thatseveral features of this simple model will carryover to more realistic models. The main importof our results is that the essential physics of themagnetic impurity is taking place primarily onthe impurity site and that an understanding of theproblem will require a determination of the im-purity paramete rs: interconf igurational excita-tion energies (ICEE's) and the coupling parametersV„-„ for the many-electron states. In this paper,it can only be suggested a posteriori what valuesthe parameters might have in order to reproducemagnetic impurity effects. The calculation of theparameters must await a procedure like Hirst'sCRHF or something more sophisticated usingmany-electron stepping operators and many-elec-tron state matrix elements.

The parameters which seem to give the bestdescription of the various magnetic impurity ef-fects are very simple. We need an effective & tobe rather small, one of e("' (measured relativeto the Fermi energy) also small so that as T- 0the shift 5E,'" tends to pull that ICEE resonancealmost to the Fermi energy. When this happensthe resistivity almost saturates, the heat capacitydisplays a. very low-T Schottky-like peak and thethermopower will show appropriate behavior.

A natural question which needs answering con-cerns the proposed pulling toward the Fermi ener-gy of an ICEE: Why doesn't the: resonance prdceedon through zero, thus reversing the relative place-ment of the two relevant configuration energies,E» Np and E„„—(N-+ 1)p, , measured with respectto the grand canonical ensemble? The answeris probably contained in the energetics of pro-cesses left out of the model. In particular, latticeenergy associated with the different "size" of dif-

SAMUEL P. 80%EN 18

ferent configurations can be of enough importanceto force an impurity to retain the relative energyranking of its configurations from high tempera-tures. This circumstance couM possibly be re-versed in the discussion of intermetallic com-pounds, where a change in the energy ordering ofthe whole lattice could be energetically favorable.The extension of this model to the resistivity" ofCeA1, compounds appears to bear out these ideas.This will be discussed elsewhere.

Let us now proceed to examine the consequencesof having one of the E~"~ close enough to the Fermienergy that as T 0 the 5E,'"~ shift pulls that ICEEdown almost to the Fermi energy. In most of thefollowing discussion I will consider E~'~ ~ 0 andsmall, while E," «0, and comment only brieflyon the other case.

A. Resistivity

The temperature dependence of the impurityresistivity in our model reflects the movementthrough the Fermi energy of the leading half of anICEE Lorentzian. Because the only change fromthis model which will transpire as we considertransition-metal atoms is the number of such re-sonance peaks at the Fermi energy, a direct com-parison with the actual temperature dependenceshould be possible if the theory is properly scaled.

Let us first demonstrate that at high tempera-tures the leading temperature dependence of theresistivity is logarithmic as found by Kondo. "This is easily done on realizing the dc resistivityis proportional to

(~(r))1 (r) ~&N(0) Imt Q ( („) --

~, /- )~—--(~(—„)),--. (4.30)

sr s e + s

At high temperatures, A', is small and proportion-al to In(T/~t~ ~). Expanding the I orentzian toleading order in small As' yields

(¹())I'(r)g — ~ @(r)ds s S

(&("')' (I'"')' & (@'"')'+(I'"')'

x ln ~ ~," (4.31)RENT )

for the leading logarithmic dependence of the re-sistivity. Clearly, as the temperature decreases,such an expansion fails to hold and the logarithmicbehavior is folded into the shape of the Lorentzianresonance. These considerations would suggestthat it would be quite rare that an impurity wouldexhibit a strictly linear behavior in -lnT for verymany decades of temperature. - This is clearly inaccord with the best experimental observations. '

It is worthwhile noting in passing that (4.31)recovers the behavior of the resistivity used inthe pressure study of Sec. III.

1.2—CL

1.1—IL

NORMALIZED RESISTIVITY0.9——Theory0.8 .—

M 0 7 ExperimentsV)

+ Cu —0.01 at % Fe

O 0.5—M

0 q o CutaAuep 0 08 a'I % Fe

~ Au —0.0025 at % Fe0

I I l ) I I I I I I I

10 10 10

~ Cu95Au5 —0.01 at % Fe

10

I i s I » I

10 10 10

TEMPERATURE IN UNITS OF RESONANCE LIFETIME BROADENING

FIG. 2. Normalized resistivity of Loram et al. (Ref.62) shown with the solid curve which is the theory ofthis paper.

A numerical solution of the thermodynamic self-consistency equation for this model is used togenerate the resistivity for parameters scaled togive a Kondo effect as we have been discussing.This calculation yields two characteristic tempera-tures of two types: the high-temperature T~ ~

=~e~, ~/& which signals the onset (as T decreases)

of the high-temperature logarithmic behavior,and the low-temperature T"= I',"/& which givesthe transition from ~' logarithmic to the Sommer-feld low-T behavior. For most alloys we may ex-pect that T' ' is large enough to be buried by thehost phonon contributions to the high-temperatureresistivity. Thus for most alloys T~ ~ is basicallyunobservable. This leaves one parameter, T",which plays a role like that of the Kondo tempera-ture. Low- temperature experiments should thengive a universal curve (when scaled to some tem-perature T" as a function of T/T". Figure 2

shows the comparison of the theoretical resistivitywith the scaled resistivities of Loram et al."forCuFe systems. The most important feature ofthe agreement is not the high-temperature regime,but the excellent comparison in the intermediaterange where A' is changing and where the peak ofthe Lorentzian is approaching the Fermi energy.This is the first time a microscopic model whichcan be made compatible with the high-temperatureHund's rule spin values has shown such agreementwith experiment in this range. " Comparison ofT' of Fig. 2 and the Loram definition of T„an~,suggests that „„~,= 10

A short comment is perhaps appropriate hereon the magnetoresistivity. Examination of (4.31)shows that the resistivity is caused by both a spin-up and a spin-down resonance at the Fermi energy.This may at first seem strange since the atomicconfiguration involved is a singlet ('8) state. Itmust be remembered that the resonance energies

18 CONFIGURATION FLUCTUATIONS AND THE. . .

which appear in the t matrix are the interconfigura-tional excitation energies, and the Zeeman energyof the level e(„' arises from the ('S) energy in thedifference

E'& =E,('S, 0) —E,('S, m, =s—'). (4.32)

The ICEE's can thus be split in a magnetic fieldand, as T 0, the resistivity measurements wouldobserve the lower Zeeman-split energy resonanceto pass through the Fermi energy; the resistivitythen would have a maximum and saturate at a de-creased value at T =0. This splitting out of theICEE resonances at the Fermi surface well ex-plains the negative magnetoresistivity associatedwith these impurities84 and well rationalizes thelarge-field behavior of several systems. If theICEE is very broad, it maybe difficult to resolvea peak in the resistivity, but for very narrow re-sonances (Mn) this effect has been observed. "

Another effect of a magnetic field is to changethe effective Kondo temperatures T ' at which the~' begins to approach a constant. The major con-sequence of this would be that the resonance ener-gy e" does not reach zero because the saturationvalue of A' [E(ls. (4.30) and (4.28)] is less than»(s~, + U)/&. In the situation where the magnetic-field induced change of the effective temperatureT" dominates the Zeeman splitting of the reson-ance, a decrease in the resistivity with increasingfield will be observed, and the "Kondo" tempera-ture will behave like

T = 10T(' = [(i&, H}' (I'(+&)']~'.10

Because this expression reflects quite stronglythe character of the S-shell model, comparisonwith experiments will not be made here. We justnote that the temperature dependence is in accordwith experiment for several systems. '

The Daybell and Steyert" data on CuCr showed thatthe temperature of the "knee" of the resistivityshifted up with magnetic field in a fashion con-sistent with this expression.

In the low-temperature range the resistivity dis-plays a Sommerfeld T' dependence, which arisesnot so much from the movement of the ICEE, butfrom the thermal population factors. This reflectsthe change in the population of the impurity levelsas the coupled Fermi system approaches completedegeneracy. The low-temperature expression whichwhich follows from K(I. (4.30) for this Andersonmodel is

B. Thermoelectric power

The impurity thermopower S~„ is most simplycalculated using the simple expression given byMacDonald"

(4.33)

where p(v) is the resistivity at energy (d from theFermi energy. In contrast to the resistivity, thethermopower is partially dependent on the polesin the t matrix which are not close to the Fermienergy; it is proportional to

T (A](r&) g&(r&&(r&

@Jim]&] P [(&(r))&+(e(v&)2]2 ' (4.34}

(4.36)

The impurity parameter values which give rise toa "Kondo" resistivity can concomitantly show apeak in the low-temperature thermopower, be-cause as T-0 the second term in (4.36) goes al-most to zero and the low-temperature S will thenhave a negative linear slope in T. If T & T" soA' varies like log(T), the temperature of the peakis given by

(4.37)

where the resonance parameters have been writtenin the notation of (4.23). For the low temperatures,the e',' are approaching zero, in the example beingconsidered, while the e,', ' have not changed muchfrom a~. The sign of the low-temperature thermo-power will depend on the dominant contributionsfrom the resonances which are near, but notclosest to the Fermi energy.

For the sake of definiteness, let us examinethe situation where E~'~ is positive and small,but E ' is negative and quite large. Then the S-shell impurity contribution to the thermopower is

((a,'T (1 —(n„-))A'E,(-&

3lel 1Imtl z ](E,' '&'+6*]' )(n„-)A'(E( &- AA /(()[A2+ (E(+& AAt/&)2] 2

where we have neglected AA'/&( compared to E,'-'.If T"& T & T'-', the shift A' increases logarith-mically as T decreases and to lowest order inE,"-&A'/» we may write, assuming ~E,'-'

~

&a,

TS~ — (1 —(n„-)) ( &

S

3416 SAMUEL P. BOVE N

Thus, if T"& T„,~, a peak can occur ig. S. Onthe other hand if T" is too large the low-tem-perature thermopower may appear to have nopeak and may just be linear in the temperature.Because of the fact that the thermopower isdominated by the strongest scattering mecha-nism, phonon contributions may preclude ex-perimentally seeing a peak if T&" is too large.

If we had considered the case E'-' small andE"large, we would have found similar behaviorbut with a change of sign. Remember that A' hasan overall sign change in. this case and also, thethermopower will reflect another change of signdue to the (+) resonances far from the Fermienergy contributing a positive term to $. Thesign of the low-temperature dS/dT in this theoryis dependent upon impurity configuration energyspacing. If E" is small then dS/dT&0 as T-D.Or if E' ' is small, then dS/dT&0 as T-D.

The sign of 8 at low temperatures is dependentupon the nature of the interconfigurational ex-citation energies of the impurity in the host. Aswill be discussed briefly below, if this picture iscorrect then the predominance of negative S for theiron-group atoms" suggests the importance ofd"- 's'-d"s' core to valence transitions in theseimpurities as opposed to actual ionization of theimpurity site." This line of discussion suggeststhat, like many other properties of the magneticimpurity problem in this theory, many-electronproperties of the impurity dominate the system'sreponse to external probes. (A calculation of Sis shown in Fig. 3.)

C. Zero-field susceptibility

The expression for the zero-field magneticsusceptibility of the impurity is given even in theinteracting case by the same expression (3.24) as

0.20

-0.2Oo —0.4O

-0.6-0.8

t2J'—I 0O

Qa —I.2X0UJXI-

I

10

I

10

TEMPERATURE IN UNITS OF RESONANCE LIFETIME BROADENING

FIG. 3. This figure shows the impurity contributionto the thermopower for the same theoretical para-meters which gave a good fit to the resistivity in Fig. 1.

for the free atom, except that the expression forf&r) (y& &r)y&r)) (4.38)

must reflect the coupled system. By examiningthe expressions for these thermal averages wefind from Appendices A and C that

1 1 il'1 Pe "' 'p

2 m™

&2 2m 2m(4.39)

where $ is the complex digamma function, P= I/ksT and e,',"' and e,',"' are the real and imaginarypart of the Green's function pole corresponding tothe rth ICEE.

The temperature dependence of the impuritysusceptibility will then be determined by the be-havior of the f,'"' For .an impurity coupled to aconduction-electron gas with a constant densityof states at the Fer.mi surface, the temperaturedependence is determined by the degeneracy tem-perature T"' associated with the xth ICEE. IfT&T'"', then

sf &"' 0.204 1() y T ~Tebs a

(4.40)

so at high temperatures the susceptibility can beCurie-like. However, if T& T'"', the temperaturedependence for the f &"' becomes characteristic ofa degenerate Fermi system. As is shown in Ap-pendix C for T & T'"',

ey(r) ] e(r)J 2 — 2s O(T 2)sc&r)&& (e&r))2+(e&r))2

1s Xs 3S(4.41)

The temperature behavior of the impurity sus-ceptibility at lower temperatures is dominated bythe smallest T'"'. For temperatures larger thanthis T'"' the susceptibility will display a Curie-Weiss behavior, but at temperatures comparableto or lower than T'"' the susceptibility will appearto be more Pauli-like.

The physical basis of this behavior shows somesimilarities to the fluctuation picture presented byWohlleben and Coles" and others. " In fact, theaverage fluctuation frequency given by

SQ)y = kgTf

in this picture plays exactly the role of thesmallest temperature T'"' here: if the tempera-tures are high compared to the energy k~T'"', theoccupation densities of the magnetic sublevels ofthe impurity can be significantly altered by theexternal magnetic field. In this temperature rangethere is sufficient thermal energy so that themagnetic degrees of freedom of the impurity aremore or less uncoupled from the conduction-elec-tron system. Thus the moment fluctuations re-semble an almost free impurity with its charac-teristic Curie susceptibility. If the temperature

18 CON FIGURATION F LUCTUAT ION & AND THE. . .

is low or comparable to T'"', the degeneracy ofthe conduction-electron system begins to dominatethe impurity system's magnetic response. Nowthe coupled composite many-particle states com-prised of both impurity and host electrons becomedominated by the density of states of the coupledsystem. Effectively, for T& T'"', the impuritycannot change its spin independently of the manyconduction electrons to which it is coupled. Thespin fluctuations are dominated by the density ofstates of the coupled states and appear to be Pauli-like in their behavior. Thus, just as in the free-electron" case, the new ground state in the pres-ence of a magnetic field, and the states just abovethe ground state in energy, all have momentsparallel to the field. The antiparallel momentstates are much higher in energy compared toAT. Such a distribution of many-electron statesgives rise to a susceptibility that reflects the mo-ments of the ground state and its nearby statesand is then effectively a Pauli susceptibility.

I et us now examine briefly the analytic sus-septibility results of the S-shell Anderson modelto see if there are any consequences of generalapplication The first such result occur's for thehigh-temperature susceptibility. It has been ob-served experimentally" that the high-tempera-ture Curie-Weiss susceptibility of iron-groupimpurities gives a somewhat reduced effectivemoment when compared to the atomic moment.It is of immediate interest to note that preciselythat condition which gives the characteristic re-sistivity of such magnetic impurities, namely,that one of the ICEE resonances is close to theFermi energy and i.s pulled toward the Fermienergy by the coupling, contributes to a reducedmoment. In the S-shell model the reduction in themoment comes about because of the change in theweighting of the different impurity conf igurationsdue to the conduction-electron coupling.

Second, the S-shell model spin susceptibilityat high temperatures is determined from the hightemperature expansion of the digamma function:

O.739, which is very close to the CuMn ratio ofO.V5 +0.05.'4 Further comparisons yield reason-able agreement with atoms across the iron group.The "gneiss" temperature 8„ is positive unlessE,"' becomes too large. However, estimates ofthe values of E,"which wi11 give a Kondo resis-tivity and bring the ICEE resonance near theFermi energy as T-O, that is

@(+) 1 -(E(-))2+ (T(-))2ln S

(E (+ ))2i (P (+))2

S

(4.43)

suggests that 0„ is always positive, though pos-sibly less than characteristic temperatures mea-sured in other effects.

The low-temperature analysis for the suscep-tibility, which gives the susceptibility in the de-generate Pauli region, yields

8 $" )) kT'@

2 3 P(+) 3 P(+)s 8

where g" is a thermal weighting factor

(4 44}

(-)(+) fo

]+4f ( )(1 f ( )) (4.45)

0.8—

0.6—

which approaches 1 as f,' '-1. This form of thesusceptibility is calculated assuming the e")resonance approaches the Fermi energy and thatksT &I'('/((. This form is very close to the spinfluctuation results for low temperatures. "

The susceptibili, ty has also been calculated nu-merically for the whole temperature range in anormalized form and Figs 4 and 5 show the results.

At this point a brief comparison of these resultswith the standard perturbative result on the sus-ceptibility by Scalapino" may be employed toshow a possible relationship between this work

where

0.5403 P.' I9

k T T (4.42) 0.5—

0.4—

0.5—

e„=0.6V5r&",

where I',"' is the linewidth of the ICEE and E,"is the high-temperature energy of the (+}ICEE.Note first that for this model in the atomic limitthe susceptibility is y/ps2= I/ks T, so that the num-erator in (4.42) is the effective moment squaredin Bohr magnetons. This yields a ratio for theeffective moment to the Hund's rule moment of

CI

N 0.2

0.1—KO

010 10 10 10 1 10 10 10

TEMPERATURE IN UNITS OF RESONANCE LIFETIME BROADENING

FIG. 4. This figure shows the normalized impuritysusceptibility plotted against temperature which isnormalized to the ICEE lifetime broadening, for thesame parameters as in Fig. 1.

3418 SAMUEL P. B0%EN 18

and other more traditional approaches. %e willshow that it is possible to reproduce Scalapino'sresults in an asymptotic expansion to logarithmicaccuracy, but a more careful examination willshow that this expansion is not physical forScalapino's choice of parameter values. Also,the same 8,'«as was derived for the resistivitywill be recovered.

Scalapino used the standard thermodynamicperturbation expansion for the partition function,calculated an expansion for the free energy, andevaluated the impurity susceptibility exactly tosecond order in the coupling V, and to logarithmicaccuracy in fourth order. The result of this eval-uation was

1+N&0)J 1+N& )J0&n +IB B

(4.46)

To recover this expansion here it is necessaryto evaluate our expression for X' ~ (3.24) usingan approximation for f,'"'. To do this we seekan expansion in the lifetime broadening of theICEE resonance, keeping the shift of the real partof the energy in the Fermi function. The appropri-ate expansion is

(5E(+)) (& ) D

BH ')&(n~,-) ~' BE s&+& ns&+&

—(1 —{~,.&),g

—2(1 —());,) —{n„))

(4.50)@( ~) 6g (4') )Is s

For the parameter range under study the lastterm vanishes, and

BS,(E, r) E+D EBE (E+D)+r -E+r '"""'"

Neglecting the 1 compared to the i and z+ D wefind for the susceptibility

limy y T & ~ gE(-)le

1

D+ cd + U 5E"

ID+ g, -5E,' '

(4.52)

() i I ar rf r + 8 y 8

s + ~ s s s

4g a T't'r,'"'E(r) ) E(r)

s s(4.47) P(0)j= V f&&'(0) (4.53)

The zeroth-order expansion of this second termis Scalapino's first term

e'"'= c + —U(1+x) —6E&r& (4.48)

it is straightforward to show that, in the sameregime (s~, &0, c~, + U&0, T &T'"') that Scalapinostudied,

&a 8 (+)X),=~ T 1+

B~ (5E.' )B

(4.49)

where in deriving (4.49) the fact has been used that,to lowest order in the magnetic field, 5E,'"'= 5E,' '.Using the properties of the dD functions and 5E,'"'from the Appendices, we can show that

where E,'"' includes the shift (4.24) and I","' is thelifetime broadening of the resonance. This ex-pression is valid if I","'« ~E&"&

~and kT «E~("' .

To recover Scalapino's result we must keep onlythe Fermi function term and neglect the otherterms. For accurate numerical approximationsthis is not a consistent scheme since the secondterm in the expansion dominates the first for someenergy and temperature ranges. This is particu-larly true if T & T'"'. Remembering that

The first-order expansion of the denominators5E;& yields for the next term a term

&~(0)~)*~~ i,"")~2e

U(4.54),

if we insist on writing expressions in terms of theeffective J of (4.53). If one of the resonances dom-inates the shift as in the resistivity, the sameeffective 7 as found in Sec. IG is recovered here.Now if one seeks to recover only the log(T) partof A' without regard for the fact that A' dependson I', and only displays this behavior for T "& T & T", then one achieves a log(T) term in theexpansion for the susceptibility. However, it iswell known that perturbation expansions are oftenasymptotic and, when only a particular functionalform of an expansion [log(T)] is sought, thereis little control of the region of convergence orapplicability. As illustrated above there is aregime within which an asymptotic expansion ofScalapino's form can be found for g&~. However,there are at least two grounds on which it can beargued that this expansion will not be seen ex-perimentally. First, the expansion for f&"&is dom-

18 CONFIC URATION FLUCTUATIONS AND THE. . . 3419

D. Heat capacity

The heat capacity of the impurity-host systemwhich shows the "Kondo" resistivity should alsoshow what is effectively a Schottky anomaly atlow temperatures. This interpretation cannot betaken too literally since the energies of the im-purity resonances are themselves temperaturedependent. However, the heat capacity will showa peak at low temperatures.

The heat capacity is calculated using

,Catomi c+ C tntv v V (4.55)

where Cv atomic can be expressed in terms off,"' functions, giving a formula like the atomiclimit, but. calculated with the fully interactingf~"i. The second term is

inated by the Pauli terms (I'/Z) instead of theFermi function first term. Second, the logarith-mic behavior of A' will not appear in the samerange of temperatures as does the range in whichScalapino's first term is found. The actual tem-perature dependence of the susceptibility is con-trolled rather more strongly by the degeneracytemperatures T", one of which is dominated bythe ICEE broadening if E," is small. The sus-ceptibility displays the transition from a Curiebehavior to a degenerate Pauli susceptibility whichdepends more on the lifetime broadening than onthe shift of the ICEE. It is possible for impurityparameters of the sort which give a "Kondo" effectand ICEE Quctuations that lnT behavior at hightemperatures can be seen. The major point isthat coefficients of the expansion are differentfrom those which the usual theoretical treatmentswould predict.

16,15

14

12

11C3

10V)

4Jcn

LLI 72LLI

0N

0 2

00

I I

1 2I I I

4 5I I I I I

6 7 8 9 10

TEMPERATURE IN UNITS OF RESONANCE LIFETIME 8ROADENING

FIG. 5. This figure shows the normalized inversesusceptibility for the same parameters as in Fig. 1.

would correspond to a 0.08-at. /p error in theexperimental concentration of impurities in thealloy. %hile this possibility has not been exploredin detail for this system, such an error is notunreasonable for heat capacity measurements onimpur stses.

It is of interest to note that the best theoreticalfit to the low-T data suggests that the second peakin the heat capaci. ty due to the other ICEE is cloSeenough to the lowest one that it may have already beenseen experimentally (see Fig. 6). Further study ofthis possibility should be made in the future.

At this point we give a brief comparison of thelow-temperature ratio Cv/Txcalculated in thistheory and in Wilson's calculation of the spin-&Kondo model. Using the expressions (4.56) and(3.24) for Cv and X, it can be shown that in thestrong coupling limit (b. dominating all otherquantities)

int e x&-'C~„"=- g (1-(n„-)), ' +(n„-) P gCv 41r

Tx 3 (4.57)

, s(n;, )S (4.56)

When the interacting system f," are evaluated thelow-temperature behavior of Cv is dominated byCv" and shows a peak around the smallest T . Ifthe T" is small enough to give the "Kondo" re-sistivity, this same condition gives rise to a lowT peak in the heat capacity. In Fig. 6 is a plotof the heat capacity calculated for a set of param-eters which closely approximates the experimentalheat capacity of La, „Ce„alloys as measured byBader and Phillips. " For this calculation, the

' theoretical parameters have been fitted to givethe experimental temperature scale, and themagnitude of the heat capacity was multiplied bya factor of 0.88 to bring the magnitude of thetheory into accord with experiment. This factor

and

&v 8& 4& y 5m yksT 96 9A (1+y')' 276, (1+y )'

4@~ y'3mb, 1+y') '

(4.58)

(4.59)

where y= 6/e and is very small. Neglecting they terms yields the desired result.

in agreement with Wilson. " On the other hand, ifthe medel parameters are fixed so that the (+)resonance approaches the Fermi energy as T -0,which condition gives the "Kondo" effect in thispaper, the ratio gives exactly the same valuefound by Wilson, "

~s for the Kondo Hamiltonian.This result obtains from the following low-tem-perature expressions

3420 SAMUEL P. BOWEN 18

I I I'I I I III I I I I I I III I I' I I I I II I I I I I II

0 BADER et al, 0.64 &&'~ Ce, H=O

1.80 — —THIS WORK

l.25—

C3

~ I.OO—O

.75—O

&l

FIG. 6. This figureshows the fit of the theor-etical heat capacity with atemperature sc ale chosento fit the experimentaltemperatures. See textfor full explanation.

.25—

0.OI

I I I I I I IIII.O

I 'I I I I I IIIO.O

I I I I I II50.0

In another paper, we examine a comparison ofthe calculated values of Tx for the Krishnamurty,Wilkins, Wilson'0 (KWW) method and the calculationsof this paper. In general the high-temperatureand low-temperature results agree, but a sub-stantial difference occurs in the temperature rangein which the susceptibility changes from Curie-Weiss to Pauli-like. The KWW result gives thechhrige over at consistently lower temperaturesfor the same parameters than do the calculationsof this work. While the two methods are notdirectly comparable, the major difference seemsto be that the KWW density of states for the con-duction electrons has a logarithmic peak at theFermi energy. The calculations of this workemploy a constant conduction electron density ofstates near the Fermi energy. The differencebetween these two methods will be discussed inmore detail elsewhere. "

guage this equation is

LCkq 6kqMks+Q Vkmsc~ ~ (5.1)

where m= -l, . . . , +l and s=+& label the orbitalsof the impurity shell. There are two commentswhich are relevant here. First, the Vk, ariseas variational parameters in the CRHF-like pro-cedure. It is especially important not to attemptto evaluate these coupling parameters using resultsof calculations which do not fully consider themany-electron configurations of the impurity.

Second, it is probably necessary to redo Hirst'sformulation of this problem so that instead of(5.1) an equation of motion involving the Juddstepping operators Q„(n, P) is determined by somevariational principle analogous to Hirst's CRHF.This would probably yield instead of (5.1), anequation of the form

V. TRANSITION-METAL IMPURITIES IN METALS (5.2)

In this section, the many-electron extensionof the results of the simple 8-shell calculationand other implications for the impurity with apartially filled d or f shell will be briefly andqualitatively discussed. Most of the emphasis willbe placed on the iron group, though the ex-planations are quite general.

The first element of unfinished business is thequestion of the lifetime broadening of the ICEE's.Perhaps the easiest way to approach this is tobegin with Hirst's equation of motion of the con-duction electrons. Written in our operator lan-

In the comparisons with experiment the smallsize of the ICEE widths found suggests that many-electron matrix elements like V&k B„should be usedin place of the CRHF one-electron variationalparameters Vi, , The estimation of these many-electron matrix elements requires a more so-phisticated reexamination of the (n —1,n, n+ 1)electron impurity Hamiltonian in a variationalscheme which more carefully treats the many-electron states.

Short of pursuing such a fundamental approachat this time, a relatively crude estimate of the

18 CONFIGURATION FLUCTUATIONS AND THE. . . 3421

effective line width variation across a lfamily likethe Sd family will be attempted. The main ideais that the effective matrix element in the Green. 's

function is

(5 3)

The variation of the linewidth across a group likethe 3d's will reflect the variation of

(5.4)

as n varies from n=1 to n= 10. The simplestestimate of

(5.5)

can be obtained by assuming that both the n and Pstates are single determinants. Then the form

$„,(a, P) would be

(ct, P) c . . .c Igcg . . ~ cgno 1 +0- 1 Bno

where the n, and P, label the m, s, in each deter-minant. Recalling from Sec. III that Ie is a pro-duct of all cc~ for all states of the shell, a cal-culation of la 1' yields, for a d shell, the expec-tation of n, —1 factors of n„and 9 -n, factors of1-n . The most naive estimate of this is to re-place each n„by the fractional occupation 0 &x& l of each orbital. Then, we estimate lal' to beproportional to

(~)"p- (l ~) "p

(5.6)

This dependence gives relatively narrower ICEElinewidths for the resonances near the Fermi sur-face as one passes through the iron-group family.Thus, this estimate would make the n, = 1, x = 0.1and the n, = 5, x=0.5 linewidths compare as theratio of about 1:10 3.

The preceeding estimate can be evaluated muchmore precisely by using the true correlated com-binations of products of creation operators whichcreate the terms of the shell. These linear com-binations can be obtained from standard tablessuch as Slater. " The coefficients a can be ex-panded in terms of Wigner 3-j symbols, Racah'scoefficients of fractional parentage" and thermalexpectation values of polynomials on one-electronnumber operators. These expectation values mustbe in principle evaluated in the grand canonicalensemble or equivalently as solutions of thermo-dynamic self-consistency equations in a Green'sfunction calculation.

Even with the exact expressions for the expansionparameters, it is difficult at this time to improve.much on the estimates above. The lifetime broad-ening of an ICEE could be much smaller than anyone electron estimate, and we should expect thatimpurities with half-filled or close to half-filledshells will have much narrower ICEE's than atoms

on either end of the series.The major implication of our study to this point

is that the presence of impurity ICEE's near thehost Fermi energy is what most strongly in-fluences the alloy properties. The equation ofmotion of each Q„(n, P) may be manipulated exactlyas in the simple S-shell model. There will bethe diagonal terms and the mixing terms. Thethermal shift of the ICEE energies will arise inthe same fashion as before, though now therewill be many more sets of terms. The essentialeffects, however, will arise from those ICEE'swhich are nearest to the Fermi surface. Onemight expect only two types of ICEE's to be im-portant for a many-electron atom. This suggeststhat the major difference between a realistic im-purity and the S-shell model is a matter of how

many of each type of ICEE there are. Many of thesimplest properties of this model should beapplicable.

Thus, for every alloy there will be some smallesteffective degeneracy temperature

(@2+l 2) I /212m

which will determine the changeover from a lo-garithmic ICEE shift and a Curie-like susceptibilityto a Sommerfeld 1-T' shift and a Pauli-like sus-ceptibility.

To illustrate one application of these ideas, thevariation of the resistivity through the iron groupwill be briefly examined. As is well know. n, "atlow temperatures this quantity exhibits a singlepeak at about Cr or Mn, which peak in the tradi-tional Hartree-Fock" picture is represented by asin'( —,', mN) functional form. At an elevated tem-perature the resistivity of those e'lements in thecenter of the family (Cr, Mn, Fe) is less than the"outer" members of the family. In many earlytreatments" of this problem this behavior wasregarded as experimental verification ofAnderson's HF treatment of this model. In whatfollows, we will examine the explanation of theseresults from the perspective of this study, usingthe result that an ICEE is near to and pulledtoward the Fermi energy by the conduction-electron coupling and that the ICEE linewidth issmallest at the center of the family.

Let us first examine the low-temperature re-sistivity. If every impurity at low temperaturesfits into the picture I have been presenting here,then the resistivity should be simply a measureof the number of interconfigurational channelsfrom n electrons to m+1 electrons which aresaturated for that specie of impurity. If oneassumes a Hund's rule ground state for the im-purities, it is easy to show that N„„„, the num-

SAM UK I. P. BOW EN 18

ber of channels between d"-d"", is simply pro-portional to Schrieffer's result which has theresidual resistivity proportional to 2S„, whereS„ is the Hund's rule spin of the d" ground state.However, as noted by Schrieffer" and others theexperimental data on residual resistivities isreally not good enough to distinguish between theSchreiffer result and the HF result. Furthermore,as will be discussed below some of the im-purities at the ends of the family may not havereached the unitarity limit at low T and this coulddecrease their residual resistivities below thestraight lines of our Schrieffer-like prediction.This last effect would appear to be approximatedby the HF sin'(, 0)7n) result and thus lessen thevalue of residual resistivities as a distinguishingexper iment.

Let us now qualitatively examine the finite-tem-perature behavior of the resistivity. The impurityresistivity will be given roughly by

2+n.n+ l~n1'„+(c„-A„)2 ' (5.7)

where N„„„gives the number of channels I „ isthe effective width of the important ICEE, e„ is theICEE, and A„ the shift we have been studying.[We are assuming that at low temperatures theICEE value he, given by

(5 8)

is a small quantity. ] The low-temperature re-sistivity will be

(5.9)

If he, = 0 we achieve the unitarity limit. If .thetemperature increases to T &T(„'~, then we canwrite the resistivity as

1+ ——ln (+)

(5.10)

The amount by which the resistivity will havedecreased at this elevated temperature will depend .

on both A~, and T„'. We examine some relevantvalues of these quantities. First, if 6~0=0 andT„' is quite small, the impurity could have bothachieved the unitarity limit at T = 0 and exhibiteda significant decrease for T & T„'). From the per-spective of this study, this case would seem toapply to impurities near the center of the irongroup. A case which appears to be more applicableto impurities away from the center of the groupcorresponds to A~, W 0. In this case, two effectsare present which imply that the decrease of the

resistivity at the temperature T will be less thanthat in the just preceeding case. First, sincehe, w 0 the value of T„' is larger. This means that1n(T/T„') is smaller than in the preceeding caseat the same T. Second, the Ae, term can decreasethe contribution to the denominator and thus furthersuppresses a high-T decrease of L&.

There is yet a, third case which may be applicableto the atoms on either. end of the family. This isthe case in which b.e, is so large that the cor-responding T„' is comparable to or larger thanthe temperature T at which we are comparing theresistivities. In this ca,se the shift A will not havechanged much (T & T„' ) and would not vary likelog(T), and the resistivity will have undergonelittle or no change.

This qualitative discussion thus indicates that thebehavior of the resistivity across the iron groupat both low and high temperatures can be ration-ialized by the effects presented in this paper.

Further examina, tion of other consequences of thetheoretical ideas presented here are being pursued,and wi11 be discussed elsewhere.

VI. SUMMARY

The logical c.onsequences of considering theionic approach of Schrieffer and Hirst in a thermo-dynamic quantum-mechanical formalism whichincludes impurity lifetime broadening has beenpresented in some detail. Up to a question ofoverall temperature scale, or equivalently, thequestion of the lifetime broadening of the relevantmany electron ICEE impurity resonances, theconsequences of this ionic perspective seem todescribe impurity systems remarkably like thephysical alloys. The different types of temperaturedependences of bulk properties and related effectsseem very much like the experimental behavior ofmany alloy systems. Furthermore, our calcula-tions show a possible intimate link between Kondo-like behavior of transport properties, Pauli-likebehavior of the paramagnetic susceptibility, andvalence Quctuations of the transition metal oractinide series atoms.

ACKNOWLEDGMENTS

The author gratefully acknowledges the helpfulconversations of a large number of colleaguesduring the long study of this problem'. R. Silsbee,A. L. Ritter, J. Gardner, C. D. Williams, J. Long,and R. Orbach, to name a few. Their good physicalintuition and sound physical understanding has beenincorporated where ever possible. Any errors arethe sole responsibility of the author.

18 CON F IGU g. ATION F LU CTUAT ION S AND THE. . . 3428

APPENDIX A: DETAILS OF THE SECULAR MATRIX

6E,' =r ~ N(g ((Qd, (cd-, c%; —cX;cd-, )c„)).ds

(A2)

The evaluation of the commutators in (A2) yields

(N,"-,&~8"' =P V-„, ((N(„;))"(yt(') c-„;&k

(A3)

or, a.s written in Eq. (4.10} in terms of the d")

parameter,

@„) a (n~& (,) I-&n~& ( I))

v(N"'& ' (ng ') ' (A4)

To carry out the Schmidt orthogonalization wemust add in and subtract out the parts of the vec-tors cd—,cg, cd, an-d c,tt c dc -dwhich are along (I&d, .()The resulting row of the secular equation becomes

+r(N &1/2 V»(r)

k

r(N(r)&&/2 ~ Vkdllkdkds Il(tdkds

k

-(~) (y)—&N»&, /. g vkdllykd~llykd~

k

Here the vectors p kand Qkdd, are the unitrg (r)

vectors for the vectors

fdkds Cds Cks CdKskdsf'ds

(y)

Q k dds= C ks C ds C ds

—Ek ds 4 ds

(A5)

(A6)

(A7)

As indicated in Eq. (4.7), the first applicationof the Gram-Schmidt orthogonalization procedureyields a symmetrized form of the secular equation,but one which contains vectors c„-,cz-, c„, and

ck,-cd-, cd, ,which are not orthogonal to (t&d(",&. To carryout the orthogonalization it is necessary to find theOverlap between these vectors. This is the sam& ascalculating the matrix element

((@(r) I y(r) )) g(r) + / g(r) (Al)

where

of the corresponding vectors.At this stage, we start to utilize the assumed

weakness of the coupling between the conductionelectrons and the impurity resonances. In theevaluation of the self-energy of the impurity re-sonances, the weak-coupling condition will beinvoked, and the only term for the self-energywhich will be evaluated is (written in notation ofSec. II}

(A9)

ks& ~dk» ( dkdsl ~kdds

The matrix elements required are

((c I y(r&)) V (N(r)&1/2

((y «(r))) rv (N(-r»&/s

((@d~s «d's'))=rVkdll@d~sll/'&Nd'rs'&"'

((0 kdd „I0d'",)))= ~vkd II @~„ll/&Nd",)&)/'.

(A10)

(A11)

(A12)

(A13)

(A14)

We next evaluate the matrix elements to lowest,relevant order in V~. For (A13) and (A14) thismeans

The argument justifying this expression is thatthe third- and higher-order terms will be smallerand will be neglected. In what follows we makethe further approximation of determining each ma-trix element V,„ to only second order in V@, orequivalently to first order in & = vN(0)v~a.

In most circumstances, a determination to low-est order in 4 would also imply that our determin-ation of e, (n, (d) should be only to zeroth order in

For a low-density system, this would be cor-rect, but with a degenerate Fermi system the sumover intermediate states can lead in the thermo-dynamic limit to essential singularities of theself-energy for some values of the coupling con-stant. This is a general feature of secular equa-tions where the eigenvalues are not necessarilyentire functions of the matrix elements. As weproceed the reduced self-energies and all otherenergies will be evaluated to whatever lowest or-der in 4 which is required to yield nonsingulh, r re-sults.

As a first step toward evaluating the self-ener-gy, we must list the states "/" which have to beincluded. As can be seen in the relevant rom ofthe secular equation these vectors include

where

r +) y//p f(+)k~ -

(N(g&. /s (&N~ &

-&N' '&'™(c-k;(kd-,&) . (A8)

(y) -(r)The quantities [[Qdkds l[ and llQkdd, l[ are the norms

and

lid„„-,ll = llc,'~f~„li+o(v„-}

II y~d, ll = lick'~d~d, ll+ o(v„-„).It is straightforward to verify that

llcd~f-, c„ll'= (1-&ndg)f( '+((n„—nds)n~g

(A15)

(A16)

(A17)

SAMUEL P. BO%EN

and

)(c~~„;c„,l('=(n~g f,"+(ng-, ( l-ns, -n~-, )),where, as before,

f(t) ((t t(s)y'(r )) (A18)

SrVQ Vs

(d —e(vasss (0) ( (0 —f( s

(A21)

Similarly, for those terms which. involve the sameclass of integrals, w'e write the same approxima-tion. This yields for 1', the following result

Now the norms involve a knowledge of expecta-tions of the sort (nf„n„,g. Such (luantities can beevaluated from two-particle correlation functions,but at this point we adopt the form for such coef-ficients which holds to zeroth order in V~. Thisapproximation does not represent the neglect ofany singular terms and so should be accurate inthe weak coupling limit. Thus,

(n„,n„-„)=(n...)(n;„)+O(V„-) .These approximations have the obvious advantagethat our self-consistency equations are closed, inthat all thermal expectations can be computed bya limited number of correlation functions. Thedifferent contributions to the self-energy can nowbe written out:

p2g(y(r). &) (N(r))ds & dspT l7s

(„) Vfs 1+(N@:")

( )+( („&) v ((0),

(A19)

where

&,(~)=Z (V('; [(1 —(n g)f,' '+((n„gk

-(n„g)(nfl]/[(o —e(y - (o)]].

(n„-g = [exp(Pe„-,-)+ 1]-' . (A28)

Having done this it is now necessary to be carefulabout how the denominators e(P;(0) are approxi-mated. In particular, it is not admissable to ne-glect the imaginary part of the reduced self-en-ergy, because one of the contributions to theseintegrals is of the form 4ln4, where 4= w(v@)N(0), and this does not have a small-6power series expansion about 4= 0. This meansthat the lowest-order contribution from these Fer-mi function integrals must include a lifetimebroadening as well as any shifts. Thus the evalua-tion of these terms requires a specification ofboth the real and imaginary parts, e, and e„ofe. In order to determine these it is straightfor-ward to observe that

and

~cs~s, gscss 'ej~ss~sÃscss+ 0(vh() (A24}

V,((0)= (1 -(n~g)f, y(-v) +(n„gf~"y((d —2zs —p)2.(&,g -&n,g}Z ~ —e

VI n"-+(1-(n g -(n Q) Z "' . . (A22}97 —e h(ss' M

Now for the dilute alloy, it is entirely approp-riate to approximate the (nfl factor by the Fermifunction

+ g(Va, [(n gf,"+(1-(n gl7

-& 3)&;3]/[ — (0- .' )]].

I cfmcs~sss = ( ~s+ ~ rs) i™is~ass+ (Vrs() .(A25)

Note that these energies involve just the energy ofa conduction electron or hole along with the two-electron or non-electron impurity state.

To proceed in greater detail with the reducedself-energy of each of these intermediate statesis to introduce quite a bit more detail into thisproblem than is useful. Therefore we set

(A20)

To carry out the approximation scheme morecarefully we must now determine good approxima-tions for the various reduced energy denominat-ors e(Q; e). These energies will contain a diag-onal energy and a reduced self-energy arisingfrom the off-diagorial matrix elements. In allcases the diagonal energy must be determined,but one might hope to approximate the reducedself-energy because of our weak-coupling limita-tion. Care must be taken in these approximationsto examine the effect 6f sums over intermediatestates. In the case where the matrix elements areindependent of n, we may sum over intermediatestates in a convergent power series in the weakcoupling. 'Thus, for example, we approximate

e (Q u&)—b, ' —f'6

v~&nr) (, '), (A26)

and can rewrite V,(u)) as follows:

where f' is a dimensionless parameter (for themost part t'= 1). Using the results of AppendixC, we have immediately,

C 0 N F I G U R A T I 0 N FLUCTUATIONS A N 9 THE

+ —((s g -(s g) (f (&, &')

+ —(1 -(n, g -(n,g)

((y& & ((0 —g) P(r)))= (& —E(r) —5E(r) —I ( )) &

(A29)

It is shown in Appendix B that if Eq. (A29) givesthe form of the resonance, then to order 1/D thevalue of X " is given by

f()(e )df~ ~ (A27 )

)&(r) sz (E(r) + 5Eft& pie &) (A30)

The integral s defining 4~, and in Appendix Bthe integral s defining -X,'"', are quite simp le to or-der 1/D if the impurity resonances can be regard-ed as Lorene ians with constant lifetime broaden-ing and ene rgy -independent shift . In the followingwe seek two smal 1 parameters 5E,'"' and I","', soth at

e(r)(E(t&+ SE(t') } E(r) + 5E&r) + iI (r)S S S S S S

In terms of these parameters each resonance ofthe impurity will be a simple Lorentzian

where 4D is the real part of +~ as defined in Ap-pendix C . Note the peculiarity of the notation thatXs'" ' depends on the s resonance energies . Theimaginary part of &D is defined in (C13) in termsof 3(E,1"), and it can be easily shown that

f(x& ~f (E(r&+ 5E(t') I (t))' (A31 )

Al 1 that remains is to derive the functional formfor 5E',"' and I",'"'. The calculation for 6E',"' in-volves evaluating Y,(E',")+ 6E(")) and substitutingthis into (A25). After some algebra the expressionfor 6E,"including corrections to order 4 is easilyfound:

5E,"= -—(~(' -)&&-&) + —((n, g -(n,g), &.&(5E,"-5E-,")+, &.&

(~ - r&'&}

Similar equations are found for 5E,' ~ which differonly in the second and third lines . The correctioncoefficients in (A32) vanish exactly in the absenceof a magnetic field and if the impurity is on theaverage neutral . In any case, the correction termsare quite small and can be neglected. This givesthe result in Eq. (4.24):

APPEN MX B: DERIVATI ON OF THE 'A &'~

In this Appendix, the expression for X.,'"' givenby Eq. (4.22} is derived within the resonance ap-proximation of (A29). We want to evaluate

P V„-&c,'=,y~(, &}=~„PV„-((c„--„(~-L, )-)4~(;)))

5E(t') (y(+& y(-))S g S S (A33) (Bl)

Similar calculations yield the fol lowing lowest-order expressions:

where F „[ ] is the Hamann-Bloomfield func-t iona 1

and

r,(-& = ~(I+f '+f', -)) .

HI

z (+) n 1 g(+) ( &~ g(-& (A34)Cy„[((A, ((0 -1 ) 'E))]= —,f.( )

a 00

x Im((A, (&o -i6 - I,) '8)) .

(B2)

These last equations have also neglected corre c-tion terms like those in (A32), which terms are oforder 4' and will not be discussed here .

Using the Lorentizian form for the diagonal ma-trgr. elements of the secular equation as determin-ed in Appendix A, we have

715(40 —&gz)((L —E "5Ei&)-(( kg% ( ) Plf8 }) h(( cs ) (~ ~ }2+52 (~ E(F) 5E(t))2~ ('Pit'))2 ((0 E&r& 5E(t))a+ (I'(t'))2

3426 SAMUEL P. BOWEN 18

Evaluating the functional F„we have APPENDIX C: PROPERTIES OF & (E, I')

where

"d&o f,(&d)(&d -e)7& ((0 —e) + 5

r~&X

((g E«) 6E«))2/ (1'«))»

(B4)

s.&E, r)= J dc (cl)

where

For the parameter ranges of this work we want

S~ for large D, and we will only list expressionsfor that limit. The definition is

f (e)(e E&t') 6E()'))g(&)

( E()) 6E(|))2i(Z (r))2 '

The evaluation of the sum over k yields

(B6) f,(z) =(e"+1}'

This integral may be written as

(f, (E, r) =f, +f„ (c2)

~ ~- &,'--.&,'!'& = ~~ &N,'"-.'&"' Z [s,(;.).~,(.;.)]k

where

I =in(D+E -iF) —ln(E-il'}, (C8)

=-&N",'&' 'N(0) m ( .)

1+

N (0) (i)(mrs)and

I =— dc 2cp~'+ 1 z2+y»0

(c4)

y=r+is. (c6)

where N(0) is the density of states at the Fermisurface. Comparing the defiriitions of sD(E, 1 )(Appendix C) and )((") we see that

)((r) —)f I (E(r)+ 6E&r) Z'&r))+ 6)((r)a D 3 3 y s 8 (B8)

In the following we show that 5A.,'"' is of order1/D and thus can be neglected when the cutoff be-comes large. In order to estimate this integralit becomes necessary to use a more realistic den-sity of states for the conduction electrons, whichstill cuts off at energies of the order of D. Forthis purpose, we adopt the I orentzian density ofstates of Hamann and Bloomfield

(ez'+ ]) = (e(" ]) i 2(em&)' 1) (c6)

Substituting (C6} into (C4} and using the definitionof the digamma function, "

ln —— 2, C7

In the derivation of I, the integration range hasbeen extended from D to infinity. This will giveerrors only of order e ~ . The integral I, may bewritten in terms of digamma functions if thefollowing identity is used:

2N(0)Dw(e'+D') ' (B9)

yields I, in terms of the variable z = py/2z as

f, ( ) =in(4z)+y( ) —2y(2z), (c8)

Actually, all calculations of this model could havebeen carried out using the density of states (B9),but it then would have been somewhat more com-plicated. The integral to be done is

N(e) "d&0 f,(&o)((d —e) e,

N(0) „)( (&o —e)'+5' ((u —e,)'+e', '

where g(z) is the digamma function. Using theduplication formula for the digamma function"yields another simpler form

f, ( ) =»(z) —0(-'+z) . (c9)

This last form yields for SD(E, 1) the following ex-pression,

The evaluation of the required contour integra-tions is straightforward, but lengthy, and will notbe reproduced here. The essential point is thatintegrating first over & yields an integral over +of the form

"d&0 2D&0 f,(&d)e,„z ((P+D' (&o —e,)'+e,''

which is clearly of order 1/D as D ~ and can beneglected in comparison with the 4~ term. u r =—[(D+E)2+r2]'i21

(C11)

SD (E, I') = ln (D+E - iI') +ia—z - g (~ +z),2m

(C10)

which is accurate to order eThe temperature behavior of S~(E, 1"}is deter-

mined by two characteristic temperatures T~ and

Tr ~, where

CONFIGURATION FLUCTUATIONS AND TH E. . . 8427

ksTr s =-(S'+ V)"'.1

For use in other appendices, define the real andimaginary part of f~ in the following way:

', (S, r) =~,'(Z, r)+'vf (Z, r).Then it is easy to show that if T (~7~ » then

s'(E, I') =ln ~ —(ksT) 'v' —' ", (C14)

(C13)

(c13)

';(z, r)-i.(",'""),o(';*),0.6433PE 0.426PII'Ss, r =-,' +

2m m

(C16)

Analytic expressions for the intermediate rangecan also be determined, but it becomes simplerjust to evaluate g(~+@) on a computer and solvewith self-consistency at the same time. An algo-rithm for g(&+a) can be easily constructed froma code for the gamma function in the complexplane. '

I

APPENDIX D: COMPARISON OF THE ATOMIC LIMITGREEN'S FUNCTION IN THE EXACT AND HF FORMS

In this appendix, we show the relationship be-tween the one-electron Green's function in theatomic limit and the Hartree-Fock approximationthereto. We wish to show that in the grand canon-ical ensemble the Hartree-Fock energy is a parti-cularly poor estimator of the existence of an in-terconfigurational excitation energy (ICEE) nearthe Fermi energy (i.e., near zero).

The demonstration begins by writing the one-electron quantum numbers as n and expanding c

e &n~~n+i ~" "n "n+i

where 4 is the non-normalized Judd stepping op-erator [cf. Eqs. (3.10) and (3.11), for example]

e(y„,y„.,) =At I~A„ (o3)

between configurations y„and y„„which are

l

f(EI')= —,—ta"( )~+(I T)', ,), . (C15)

If the temperatures are high (T& Trs) then,

(o4)

where the energies of the n-electron atomic eigen-states yn have been written as &„.

&n

Now it is a property of the Hartree-Pock approx-imation that it preserves the first moment of theone electron Green's function

((c, (&o —L,) c ))=—+—,((c,Lc ))+ ~ ' ~ .=1 1(o6)

This can be verified by recalling that the substanceof the Hartree-Fock approximation is to find the

~ matrix elements of L completely within the one-electron subspace, or equivalently to determinethe effect of I only on the c . This yields

L,c =((c,L,c ))c (o6)

where the neglected terms j.nvolve more than oneco ~

To complete the demonstration we examine thefirst moment of the exact Green's function,

((c.,«.))= g la,„,,„„~'&,„+&,„.,)n' "n' "n+1

x(E E —p),and observe that if there is an ICEE which is verysmall, its contribution could easily be dominatedby other transitions from the ground state of theatom. Therefore, Hartree-Fock calculationsshould not always be trusted as an indicator ofmany electron ICEE's close to the Fermi energy.

(ov)

created from the vacuum by the operators &~ andA~, respectively. The a, z coefficients areclosely related in spherical symmetry to theClebsch-Gordan coefficients, but their values arenot of importance at this point. By straightforwardcalculation it can be shown that

((@(r„,r„„),@(r„,r„„)))=&„„+&„„„, (o3)

where P„ is the probability at temperature T and&n

chemical potential p, (in the grand cononical en-semble) that n-electron state y„ is occupied. Theatomic-limit impurity Green's function is thengiven by

la~ I' +P )((c ((g7 g) ~c ))= g &II~ ~Ii+1 l'll 1'n+1

P. W. Anderson, Phys. Rev. 124, 41 (1961).2J. R. Schrieffer, J. App]. . Phys. 38, 1143 (1967).3L. L. Hirst, Kondens. Mater. 11, 255 (1970);J. Phys.Chem. Solids 35, 1285 (1974); J. Appl. Phys. 39, 849(1968).

The operators used are elements of the Lie algebraassociated with the S shell of the impurity and the

conduction electrons. See B.R. Judd, Topics in Atomicand Nuclear Theory (University of Canterbury, NewZealand, 1970), p. 21 ~

E. Feenberg, Phys. Rev. 74, 206 (1948).6S. P. Bowen, J. Math. Phys. 16, 620 (1975).~J. Kondo, Prog. Theor. Phys. 32, 37 (1964).J. Friedel, Nuovo Cimento Suppl. 1, 287 (1958).

3428 SAM UE L 'P. 80%EX. 18

J. Kondo, Solid State Physics, edited by F. Seitz,D. Turnbull and H. Ehrenreich (Academic, New York,1960), Vol. 23, p. 183; C. Bizzuto, Rep. Prog. Phys.37, 147 (1974).G. Bruner, Adv. Phys. 23, 941 (1974).P. W. Anderson and W. L. McMillan, in Theory ofMagnetism in Transition Metals, Varenna, 1966,edited by W. Marshall (Academic, London, 1967);see also Table I in Ref. 10.See Ref. 3.

3See Ref. l.~4H. Suhl and D. R. Fredkin, Phys. Rev. 131, 1063

(1963); A. C. Hewson and M. J. Zuckermann, Phys.Lett. 20, 219 (1966).

5See Ref. 9.C. S. Stassis and C. G. Shull, Phys. Rev. B 5, 1040(1972); N. Kroo and Z. Szentirmay, Phys. Lett. A 40,173 (1972).

~K. G. Wilson, Nobel Symp. 24, 68 (1973).8H. R. Krishnamurti, K. G. Wilson, and J. W. Wilkins,Phys. Rev. Lett. 35, 1101 (1975).

' D. Masson, "Hilbert Space and the Pad& Approximant, "in The pgde Approximant in Theoretic/ Physics,edited by G. A. Baker, Jr. and John L. Gammel (Aca-demic, New York, 1970), p. 197ff.

2 Situations where degeneracy breaking le ads to largeeffects from small interactions are widespread, super-conductivity and charge-density wave's being two examples.Examination of ICEK estimates for Mn fromC. Moores atomic Energy Levels as Derived fromthe Analysis of Optical Spectra, Natl. Bur. Stds. Circ.No. 467 (U. S. GPO, Washington, D. C., 1949)] givesvalues in the range 5-7 eV.J. J. Peters 2nd C. P. Flynn, Phys. Rev. B 6, 3343(1972).

23See Ref. 3.2 For most of the remainder of the article the one-elec-

tron energies ez~, etc. , will be measured with res-pect to the Fermi energy.M. B. Maple and D. K. Wohlleben, AIP Conf. Proc.18, 447 (1973).

26See Ref. 3.~See Ref. 2.J. C. Slater, Quantum Theory of Atomic Structure

'

(MeGrew-Hill, New York, 1960), Vol. II; B. R. Judd,Operator Techniques in Atomic Spectroscopy (Mc-Graw-Hill, New York, 1967).See Bef. 3.

3 As noted in Ref. 24, the conduction-electron energieseg~ are measured with respect to the Fermi energy,i.e., the chemical potential has already been sub-tracted off.

3~C. Herring, Magnetism IV, edited by G. T. Radoand H. Suhl (Academic, New York, 1966).

32S. P. Bowen, J. Math. Phys. 16, 620 (1975).3 D. K. WohllebenandB. B.Coles, in Magnetism V, edited

by G. T. Rado and H. Suhl (Academic, New York,1973), p. 3.

34See Bef. 14.See Ref. l.J. Hubbard, Proc. R. Soc. A 277, 237 (1963).

3~E. Feenberg, Phys. Rev. 74, 206 {1948);S. P. Bowen,J. Math. Phys. 16, 620 (1975).The unit-step function is +1 for t & 0 and + 0 for t & 0.J. Hubbard, Proc. R. Soc. A 276, 238 (1963).

D. Zubarev, Usp. Fiz. Nauk 71, 71 (1960) fSov. Phys. —

Usp. 3, 320 (1960)].4 See Ref. 32.42A. Lonke, J. Math. Phys. 12, 2422 (1971).43See Ref. 37.

See Refs. 24 and 30.See Ref. 42.

46See (a) B.R. Judd, Topics in Atomic and NuclearTheory (University of Canterbury, New Zealand,1970), Chap. V; (b) Ref. 28 (Judd, Chap. 5).

4~See Ref. 46 (a) (p. 22);also F.A. Berezin, The Method

of Second Quantization (Academic, New York, 1966),p. 28ff.

48See Ref. 46 (a).4~L. P. Kadanoff and G. Baym, Quantum Statistical

Mechanics (Benjamin, New York, 1962).See Ref. 2.

~ See Ref. 42.~ L. F. Bates and M. M. Newmann, Proc. Phys. Soc.

London 72, 345 (1958).~3K. S. Kim and M. B. Maple, Phys. Bev. B 2, 4696

(1970); W. Gey and E. Umlauf, Z. Phys. 242, 241(1971).See first of Ref. 53.

+See Ref. ' 53.See Ref. 32.J. C. Slater, Quantum Theory of Atomic Structure.(McGraw-Hill, New York, 1960), Vol. I.

~ In the thermodynamic limit the contribution to theself-energy from a single conduction-electron state isof measure zero and so countably many of these termscan be omitted or included with no effect.

~~A. Percheron et al. , Solid State Commun. 12, 1289(1973); K. Andres and J. K. Graebner, Phys. Bev.Lett. 35, 1779 (1975); B. Cornut and B. Coqblin, Phys.Rev. B 5, 4541 (1972).

OSee Ref. 9.6~C. 'Rizzuto, Rep. Prog. Phys. 37, 147 (1974).62J. W. Loram, T. K. %hall, and P. J. Ford, Phys.

Rev. B 2, 857 (1970).63P. E. Bloomfield and D. B. Hamann, Phys. Rev. Lett.

164, 856 (1967).4See Ref. 61.

6~P. Monod, Phys. Bev. Lett. 19, 1113 (1967).6M. D. Daybell and W. A. Steyert, Phys. Rev. Lett. 20,195 (1968).

TD. K. C. McDonald, Thermoelectricity: An Introduc-tion to the Principles (Wiley, New York, 1962).See Ref. 61, p. 219.

6~See Ref. 22.70See Ref. 33.7~N. Rivier and M. J. Zuckermann, Phys. Rev. Lett. 21,904 (1968).2J. H. Van Vleck, The Theory of Electric and MagneticSusceptibilities, (Oxford University, London, 1932).

73See Ref. 61, p. 158.4K. C. Hirschkoff, O. G. Symko, and G. C. Wheatley,J. Low Temp. Phys. 5, 155 (1971).

~~See Ref. 10.D. J. Scalapino, Phys. Rev. Lett. 16, 937 {1966).

T~S. D. Bader, N. E. Phillips, M. B. Maple, and C. A.Luengo (unpublished).M. Gomez, S. P. Bowen, and J. A. Krumhansl, Phys.Rev. 153, 1009 (1967).See Ref. 17.

&8 CONFIGURATION FLUCTUATIONS AND THE. . . 3429

See Ref. 18.S. P. Bowen (unpublished).

~See Slater in Ref. 28.~3See Judd in Ref. 28.

See Ref. 61.5See Ref. 61.

~6See Ref. 10.See Ref. 2.I. S. Gradshteyn and I. M. Rhyzik, Table of Integzals,

Series and Products (Academic, New York, 1965),p. 328.

~Philip J. Davis, in Handbook of Mathematica/ Eunc-tions, edited by M. Abramowitz and I. A. Stegun(Dover, New York, 1965), p. 259.A. K. Hershey, Approximations of Eunctions by Setsof Poles, NWL Technical Report RT-2564 (U.S.Naval Weapons Laboratory, Dahlgren, Va. 22448,1971).