theoretical approaches to x-ray absorption fine structure

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Theoretical approaches to x-ray absorption fine structure

J. J. Rehr

Department of Physics, University of Washington, Seattle, Washington 98195

R. C. Albers

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

Dramatic advances in the understanding of x-ray absorption fine structure (XAFS) have been madeover the past few decades, which have led ultimately to a highly quantitative theory. This reviewcovers these developments from a unified multiple-scattering viewpoint. The authors focus onextended x-ray absorption fine structure (EXAFS) well above an x-ray edge, and, to a lesser extent,on x-ray absorption near-edge structure (XANES) closer to an edge. The discussion includes bothformal considerations, derived from a many-electron formulation, and practical computationalmethods based on independent-electron models, with many-body effects lumped into various inelasticlosses and energy shifts. The main conceptual issues in XAFS theory are identified and their relativeimportance is assessed; these include the convergence of the multiple-scattering expansion,curved-wave effects, the scattering potential, inelastic losses, self-energy shifts, and vibrations andstructural disorder. The advantages and limitations of current computational approaches areaddressed, with particular regard to quantitative experimental comparisons.

CONTENTS

I. Introduction 621A. X-ray absorption 622B. Early history of XAFS 623C. Structural information in XAFS 625D. Multiple scattering 628E. Other improvements 629F. Outline 629

II. Key Approximations 630A. One-electron golden-rule approximation 630B. Scattering potential 630

1. Muffin-tin approximation 631a. Charge density and potential 632b. Atomic configurations 632

2. Interstitial potential 633a. Muffin-tin corrections: warping 633b. Excited-state effects 633c. Energy reference 634

3. Relativistic effects 634C. Many-body effects 635

1. Inelastic losses 6352. Extrinsic losses 635

a. Self-energy 635b. Mean free path 636c. Energy shifts 636

3. Intrinsic losses and the initial- and final-state rules 636

a. Amplitude-reduction factorS 02 636

b. Excitations 637c. Phasor sum 637

III. Curved-Wave Multiple-Scattering Theory 638A. Multiple-scattering expansion 638B. Full multiple-scattering methods 639

1. Band-structure and exact diagonalizationmethods 640

2. Full multiple-scattering cluster methods 641C. Path-by-path methods 642

1. Exact path methods 6422. Small-atom approximations 6423. Separable approximation 642

D. Alternative methods and extensions 643E. Example of multiple-scattering approaches 644

IV. Debye-Waller Factors and Disorder 644A. Introduction 644B. Formal properties 645C. Phenomenological models 647

1. Correlated Debye model 6472. Correlated Einstein model 648

D. Radial distribution function 648E. Cumulant expansion 649

V. Conclusions 649Acknowledgments 650Appendix: Muffin-Tin Radii 651References 652

I. INTRODUCTION

X-ray absorption fine structure (XAFS) refers to theoscillatory structure in the x-ray absorption coefficientjust above an x-ray absorption edge. This turns out to bea unique signature of a given material; it also dependson the detailed atomic structure and electronic and vi-brational properties of the material. For this reason,XAFS is a very important probe of materials, sinceknowledge of local atomic structure, i.e., the species ofatoms present and their locations, is essential to progressin many scientific fields, whether for biology, chemistry,electronics, geophysics, metallurgy, or materials science.However, extracting this information with precision inthe often complicated, aperiodic materials of importance

in modern science and technology is not easy, even withthe subtle and refined experimental techniques currentlyavailable.

Over the past three decades the technique of XAFShas made great strides toward the goal of providing suchinformation. The existence of intense new synchrotronx-ray sources alone was not enough to achieve this goal,even though such facilities spurred considerableprogress. In addition, the full success of the XAFS tech-nique must be attributed in large part to advances intheory, which have led ultimately to a highly quantita-tive understanding of the phenomena. The purpose of

621Reviews of Modern Physics, Vol. 72, No. 3, July 2000 0034-6861/2000/72(3)/621(34)/$21.80 2000 The American Physical Society


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this article is to review these advances, many of whichhave taken place only within the last decade. We focusprimarily on extended XAFS (EXAFS), the fine struc-ture in the absorption well above an x-ray edge, which isnow largely understood in terms of a high-ordermultiple-scattering theory. To a lesser extent, we alsodiscuss progress in understanding the fine structure closeto an edge, i.e., the x-ray absorption near-edge structure(XANES), from the same unified one-particle, multiple-scattering viewpoint. However, the theory of XANES isnot as yet fully quantitative and requires different physi-cal considerations. Although this field is the subject of

many current investigations, a complete understandingremains elusive. Therefore this review is generally re-stricted to the regime where the multiple-scattering ex-pansion can be assumed to be convergent and the theoryis well established. This region typically begins severaleV or more above an absorption edge. Earlier reviewsof the field are given, for example, by Lee et al.(1981),and in Koningsberger and Prins (1988). We begin with asummary of the fundamental physics of x-ray absorptionand a brief historical review of the theory in this field.We then summarize the key ideas and outline the con-tent of the remainder of this review.

A. X-ray absorption

X-ray absorption spectroscopy measures the absorp-tion of x rays as a function of x-ray energy E. Morespecifically, the x-ray absorption coefficient (E)dlnI/dx is determined from the decay in the x-raybeam intensityIwith distancex (Fig. 1). If the absorp-tion coefficient is plotted as a function ofE (Fig. 2), theexperimental data show three general features: (1) anoverall decrease in x-ray absorption with increasing en-ergy; (2) the presence of a sharp rise at certain energiescalled edges, which roughly resembles step-function in-

creases in the absorption; and (3) above the edges, aseries of wiggles or oscillatory structure that modulatethe absorption, typically by a few percent of the overallabsorption cross section.

The first feature is illustrative of the well-understood

quantum-mechanical phenomenon of x-ray absorptionby atoms, as described, for example, by Fermis goldenrule in standard texts (e.g., Messiah, 1966). The energyposition of the second feature is unique to a given ab-sorption atom and reflects the excitation energy ofinner-shell electrons. The third feature is the XAFS thatis of primary interest in this review. When interpretedcorrectly, this feature contains detailed structural infor-mation, such as interatomic distances and coordinationnumbers. The XAFS spectrum is defined phenomeno-logically as the normalized, oscillatory part of the x-rayabsorption above a given absorption edge, i.e.,

E E 0E /0 , (1)

where 0(E) is the smoothly varying atomic-like back-ground absorption (including contributions, if any, fromother edges), and 0 is a normalization factor thatarises from the net increase in the total atomic back-ground absorption at the edge in question. In practice,this normalization factor is often approximated by themagnitude of the jump in absorption at the edge. Theerror in this approximation can be accounted for by asmall (typically about 10%) correction to the XAFSDebye-Waller factor, which is called the McMaster cor-rection (see Sayers and Bunker, 1988).

Each absorption edge is related to a specific atompresent in the material and, more specifically, to a

quantum-mechanical transition that excites a particularatomic core-orbital electron to the free or unoccupiedcontinuum levels (ionization of the core orbital). Thenomenclature for x-ray absorption reflects this origin inthe core orbital (Fig. 3). For example,Kedges refer totransitions that excite the innermost 1s electron. Thetransition is always to unoccupied states, i.e., to stateswith a photoelectron above the Fermi energy, whichleaves behind a core hole. The resulting excited electronis often referred to as a photoelectron and in a solidgenerally has enough kinetic energy to move freelythrough the material. This occurs even in insulators,

FIG. 1. Schematic view of x-ray absorption.I0is the intensityof the incoming x-ray beam, which has a cross-sectional dimen-sion (width)a . Inside the slab of absorbing material (of totaldepthx), the intensity isi , and there is a loss of intensitydIineach infinitesimal slab of the materiald x . After the x ray hastraversed a distance x into the slab, the intensity has beenreduced toII0e

x, where is the definition of the absorp-tion coefficient. This figure was redrawn; it is based on Fig. I.1of Muller (1980).

FIG. 2. Schematic view of x-ray absorption coefficient as afunction of incident photon energy. Four x-ray edges areshown:K, L1 , L2 , andL 3 . Note that the overall decrease inabsorption as a function of energy is punctuated by four sharp,step-function-like increases at each edge. Above each edge arethe oscillatory wiggles known as the EXAFS. This figure wasredrawn; it is based on Fig. I.2 of Muller (1980).

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since the excited states are almost always extendedstates (quasifree states in molecules and conduction-band states in solids). The energies of the edges (or ion-ization energies) are unique to the type of atom thatabsorbs the x ray, and hence themselves are signatures

of the atomic species present in a material.The generally weak oscillatory wiggles (Fig. 4) beyondabout 30 eV above the absorption edge were eventuallytermed EXAFS (extended x-ray absorption fine struc-ture) by Prins and Lytle (Lytle, 1965); see Lytle (1999)for a discussion of the history of this nomenclature. Asnoted above, this fine structure contains precise infor-mation about the local atomic structure around the atomthat absorbed the x ray. In contrast, the region closer toan edge is often dominated by strong scattering pro-cesses as well as local atomic resonances in the x-rayabsorption and is generally not as readily interpreted asEXAFS. This region of strong scattering is referred to asthe x-ray absorption near-edge structure, or XANES,and typically lies within the first 30 eV of the edge posi-tion. The more general term XAFS was introduced byRehr et al. (1986) to refer to the fine structure in bothXANES and EXAFS, following the recognition thatthey both have a common origin, namely, the scatteringof a photoelectron by its environment.

B. Early history of XAFS

Roughly 70 years have elapsed since the phenomenonof EXAFS was first observed, and it has taken nearly

that long to realize that accurate, detailed, structural in-formation could be extracted from it and to develop aquantitative theory. Early progress was impeded by ex-perimental limitations of the then available x-raysources. For recent historical reviews, see, for example,Lytle (1999) and Stumm von Bordwehr (1989). More-over, the theoretical interpretation was also not clear cutat that time. This is not surprising, since a full treatmentturns out to depend on many complicated details ofatomic and molecular structure, high-energy electron-atom scattering, many-electron processes, vibrationalstructure, and disorder. Indeed, much of this physics hasonly been unraveled within the past 30 years, and a fullyquantitative theory was not developed until the presentdecade. Remarkably, many of the developments neededto explain EXAFS are complementary to advances inour understanding of ground-state electronic structure,e.g., the development of the density-functional formal-ism (Kohn and Sham, 1965) for band-structure calcula-tions or low-energy excited states. Unfortunately, how-ever, many of these modern electronic-structure codes,such as, for example, the linear muffin-tin orbital code ofAndersen (1975; see also Skriver, 1984) or the full-

potential linear augmented plane-wave codes (e.g.,Blahaet al., 1997), are not completely applicable at thehigh energies typically encountered in XAFS (i.e., theyrequire additional modifications, such as hugely ex-panded basis sets and an energy-dependent exchange-correlation potential; see Sec. III.B.1).

In the early years, the precise origin of EXAFS wascontroversial (Parratt, 1959; Azaroff, 1963; Lytle, 1999).From general principles of quantum-mechanical transi-tion rates, it was expected that x-ray absorption shouldbe governed by the Fermi golden rule in terms of asquared transition matrix element times a density of

FIG. 3. The relationship between the x-ray absorption edgesand the corresponding excitation of core electrons. Shown arethe excitations corresponding to the K, L, and M x-ray ab-sorption edges. The arrows show the threshold energy differ-ence of each edge. Any transitions higher in energy (to unoc-cupied states above the Fermi energy EF) are also allowed.This figure was redrawn; it is based on Fig. 1 of Grunes (1983).

FIG. 4. The relationship between the x-ray absorption coeffi-cient (E), the smooth atomiclike background 0(E), and(E) for a CuKedge. Usually is plotted as a function ofk .Here the x axis for the plot has been converted to energyunits to make it consistent with the other plots. The values for

have been multiplied by a factor of 3 relative to the (E)plot to make it show up better. Notice that oscillates aroundzero.

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states for available energy levels. Kronig (1931) origi-nally interpreted the XAFS oscillations in crystals as adensity-of-states effect, due to the strong diffraction ofelectrons by the crystal lattice, the so-called long-range-order theory. However, the very fine structure present inthe density of states due to long-range Bragg scattering,e.g., the sharp van Hovetype peaks observed in band-structure calculations, is generally much too detailed toexplain the observed EXAFS, and attempts to matchEXAFS peaks with predictions of long-range-ordertheories proved unsatisfactory. Shortly thereafter, Kro-nig presented an alternative theory for small molecules,in which the oscillatory structure in EXAFS was attrib-uted to the influence of neighboring atoms on the tran-sition matrix element in the golden rule, i.e., a short-range-order effect (Kronig, 1932). This type of theory ismost often used today to explain the origin of XAFS,although, as we shall see later, both long- and short-range-order theories can be reconciled when appropri-ate broadening is introduced.

The short-range-order theory reflects the quantum-mechanical wavelike nature of the final, excited, photo-electron state. That is, the dominant wiggles in the

XAFS spectrum are interpreted as a quantum-interference phenomenon. The outgoing photoelectroncan be viewed as a quantum wave that spreads out overthe solid, much as a rock thrown in a pond creates anexpanding spherical wave in water. In the same way thatwater waves reflect off of any obstacles in the pond and,in turn, reflect other waves back toward the originalpoint of the splash, so will other atoms reflect the elec-tron wave back towards the original atom (see Fig. 5).The amplitude of all the reflected electron waves at theabsorbing atom add either constructively or destruc-tively to the outgoing photoelectron wave and hence

modulate the matrix element between the initial and fi-nal states that controls the strength of the transition.Because this interference pattern changes with the en-ergy of the photoelectron (note that the de Brogliewavelength of the electron wave varies inversely withthe wave vector k or momentum of the electron), thematrix element, and consequently the absorption, willexhibit similar oscillations.

The modern resolution to the controversy betweenthe short- and long-range-order approaches lies in theenergy-dependent competition between scatteringstrength and inelastic losses. A crucial element is therecognition that a high-energy, excited photoelectronstate is not infinitely long lived, but must decay as afunction of time and distance and hence cannot probelong-range effects. This decay is due primarily to inelas-tic losses (i.e., extrinsic losses) as it traverses the ma-terial, either by interacting with and exciting other elec-trons in the solid, or by creating collective excitations(e.g., losing energy to plasmon production). In addition,the intrinsic lifetime of the core-hole state (i.e., intrin-

sic losses) must be considered. In phenomenologicalterms, the original outgoing wave of the excited photo-electron dies away as it moves further away from theabsorbing atom. Ultimately it becomes too weak to sig-nificantly reflect any waves off of distant atoms. The re-turning reflected waves also suffer this same type of ex-tinction. The net effect is that XAFS can only measurethe local atomic structure over a range limited by the netlifetime (or effective mean free path) of the excited pho-toelectron. This range is typically on the order of tens ofangstroms or an inverse lifetime of a few eV and roughlyfollows a universal dependence (see, for example, Seahand Dench, 1979). The short-range-order theory focuseson this short-range interference between several impor-tant scattering paths.

The long-range-order theory, on the other hand, em-phasizes the density of states of the extended, excited-state, energy levels of the entire material, which can beimportant at low energies or for strong scattering. In-deed, such an approach is often needed for descriptionsof ground-state electronic structure, for example, byband-structure codes. However, as we shall discuss lateron, properly broadening the results of this type of ap-proach with the inelastic losses and lifetime consider-ations in effect washes out the fine structure in the spec-tra due to the long-range nature of the energy states,thereby recovering the results of short-range-order

theory at sufficiently high energies. As noted above, thecore hole that is left behind by the excited electron (seeFig. 3) also has a finite lifetime, since higher-lying atomicelectrons can make transitions to fill this core hole (ei-ther directly, by falling into the orbital and emitting pho-tons as radiative transitions, or indirectly in two-electronnonradiative Auger transitions). Through the uncer-tainty principle, any state with a finite lifetime does nothave a sharp (or delta-function-like) energy level, but isbetter thought of as having a finite width, e.g., a Lorent-zian line shape. In practice, these intrinsic losses lead toan additional broadening of the spectrum and dominate

FIG. 5. Pictorial view of the multiple scattering of an outgoingwave off neighboring atoms. The topmost atom is the originalsource of the wave, which diffracts first off the atom at thelower left and finally off the atom at the lower right. Eachsuccessive outgoing spherical wave is weaker, which is re-flected in the thickness of the spherical wave fronts. This typeof path is called a triangular path.




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the effective mean free path at threshold. Core widthscan be as large as several eV for the deeper core statesof heavy atoms, but are relatively small (of the order of0.1 eV) for light atoms and shallow transitions, com-pared to the effective widths due to extrinsic losses(typically about 5 eV). Indeed, residual long-range ef-fects often persist, especially in materials with long life-times and strong scattering.

C. Structural information in XAFS

It was only in the early 1970s, following the pioneer-ing work of Sayers, Stern, and Lytle (1971), that it waswidely recognized that these interference patterns orwiggles in the XAFS spectra could be used to obtainquantitative information about the local structure nearan absorbing atom from a short-range-order theory.This wiggly structure has embedded within it informa-tion about near-neighbor distances, coordination num-bers, and fluctuations in bond distances. The main ob-stacle to extracting this information is the necessity of an

accurate theoretical model or an experimental referencecompound to calibrate the measurements. Theoreticalmodels are actually preferable, since experimental refer-ences are not generally useful beyond the first coordina-tion shell and often are not readily available. Fortu-nately, such theoretical models for XAFS are nowcomparable in accuracy to experimental reference mate-rials, and they are available throughout the periodictable; their development was due to many theoreticaladvances (for reviews, see, for example, Leeet al., 1981and Koningsberger and Prins, 1988) that have been fos-tered and tested by precision experiments over the pastthree decades using modern high-intensity synchrotron-radiation sources. In particular, an accurate, general,theoretical treatment has emerged during the past sev-eral years. Thus it now appears that the phenomenon ofXAFS in the extended regime (EXAFS), beyond about2030 eV from an edge, is generally well understood(Rehret al., 1992; Zabinskyet al., 1995; see also Binstedet al., 1987, and Filipponi et al., 1995), albeit within ac-ceptable tolerances. This degree of knowledge is suffi-cient to make the XAFS technique reliable for experi-mental structure determinations accurate to about 0.02 or better.

The needed theoretical developments did not all comeat once, but through a series of continual refinementsthat were incorporated one by one. To give an overview

of these developments it is useful to review the heuristictreatment of XAFS, starting with the more precise,semi-phenomenological treatment of Sternet al.(1975),which led to the now standard EXAFS formula.

First, we summarize the main physical quantities ofinterest. The x-ray absorption coefficient(E), which isthe attenuation of the x-ray beam per unit distance, is astrong function of x-ray energy E and exhibitsabrupt jumps at absorption edges, i.e., the thresholds Ec for photoexcitation of electrons from deepatomic core levelsE c. If the x-ray photon has an energyhigher than the threshold energyE c, the extra energy is

taken up by the excited photoelectron as a higher kineticenergy. In order to separate the structural informationfrom the energy dependence of the absorption cross sec-tions, the normalized XAFS spectrum (see Fig. 4) isusually defined as the normalized oscillatory part of,as in Eq. (1). Conventionally, is often defined withrespect to the photoelectron momentum index kEc, as measured from threshold, i.e., one de-

fines (k) rather than (E) on an absolute energy scale.Note too that this definition ofk differs from the physi-cal momentump in a solid, which reduces to the Fermimomentum kF at threshold. Often the normalizationconstant 0is taken to be the jump in absorption 0atthe edge (edge-jump normalization), but more preciselyit should be considered as a function of energy in orderto make contact with theory, as discussed below. Thequantity 0 is the relatively smooth, atomic-like, back-ground absorption of an embedded atom, in the ab-sence of neighboring scatterers (see Fig. 4). This is theabsorption that would, in principle, be measured from amodified atom, as it exists in a condensed material, if

one could turn off all the interference by the reflectedwaves from the neighboring atoms; it thus differs fromthat of a free atom.

According to the Fermi golden rule, the x-ray absorp-tion coefficient is proportional to a transition matrixelement squared. For highly localized core electrons thetransition matrix element is proportional to the prob-ability k(r)

2 that the photoelectron is found at theatom where the photon is absorbed. (Typically, the coreorbitals are of size 1/Z in Bohr units, where Z is theatomic number. Throughout this paper we shall gener-ally use atomic units em1 in theoretical formu-

las; however, experimental distances will generally begiven in and energies in eV.) The total wave k canbe expressed in terms of an outgoing wave emitted atthe absorbing atom plus incoming waves that are scat-tered back by neighboring atoms (see Fig. 5). For a wavereflected straight back by a neighboring shell of atoms,the phase difference between these components is ap-proximately 2kR , whereR is the distance to the shell ofatoms andk1/ is the photoelectron wave number, being the de Broglie wavelength. Thus decreases withincreasing energy, and the modulation in arises fromthe alternating constructive and destructive interferencebetween these components as the photon energy is var-

ied. The amount of interference also depends on thestrength of the reflection from the neighboring atoms(the backscattering amplitude) and the number of scat-terers.

The validity of the Kronig short-range-order theorywas largely substantiated by Sayers et al. (1971), whodeveloped a quantitative parametrization of EXAFSthat has become the standard for much current work:

k R

S02NR

fkkR2

sin2kR2ce2R/(k)e2

2k2. (2)




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Here the structural parameters are the interatomic dis-tances R , the coordination number (or number ofequivalent scatterers) NR, and the temperature-dependent rms fluctuation in bond length , whichshould also include effects due to structural disorder. Inaddition,f(k) f(k)e i(k) is the backscattering ampli-tude, c is central-atom partial-wave phase shift of thefinal state, and (k) is the energy-dependent XAFSmean free path (not to be confused with the de Brogliewavelength). Although the overall amplitude factor S02

did not appear in the original formula, we have added ithere for completeness, since the resulting equation canbe obtained from a more detailed many-body theory dis-cussed below (Sec. II.C), provided the terms are appro-priately renormalized. Moreover, although the originalEXAFS formula referred only to single-scattering con-tributions from neighboring shells of atoms, the sameformula can be generalized (Rehr and Albers, 1990) torepresent the contribution fromNRequivalent multiple-scattering contributions of path length 2R as outlinedbelow. Thus we shall refer to Eq. (2) as the standardXAFS formula.

In a very transparent and simple form, this formulacontains all of the key elements that a correct theorymust have and also provides a convenient parametriza-tion for fitting the local atomic structure around the ab-sorbing atom to the experimental EXAFS data. Most ofthe eventual improvements of the theory can be viewedas successive refinements of these key elements. The de-pendence of the oscillatory structure on interatomic dis-tance and energy is clearly reflected by the sin(2kR)term. The decay of the wave due to the mean free pathor finite lifetime (including core-hole lifetime) of thephotoelectron is captured by the exponential terme2R/. This factor is largely responsible for the rela-

tively short range (generally a few tens of ) in a mate-rial probed by EXAFS experiment. The strength of thereflected interfering waves depends on the type andnumber of neighboring atoms through the backscatter-ing amplitude f(k) [Fig. 6(a)], and hence is primarilyresponsible for the magnitude of the EXAFS signal.Other factors, namely, the spherical-wave factors 1/kR2

and mean-free-path terms, are secondary but importantfor a quantitative behavior of the EXAFS amplitude.The phase factor argf(k) [Fig. 6(b)] reflects thequantum-mechanical wavelike nature of the backscatter-ing. A somewhat larger contribution to the overall phaseis given by the phase shift c at the absorbing atom,

since the photoelectron sees the potential created by thisatom twice. These phase shifts account for the differencebetween the measured and geometrical interatomic dis-tances, which is typically a few tenths of an and mustbe corrected by either a theoretical or an experimentalreference standard (see Fig. 7).

Another effect that we have not yet discussed is theDebye-Waller factor, which is given to a good approxi-mation bye2

2k2. This factor is due partly to thermaleffects, which cause all of the atoms to jiggle aroundtheir equilibrium atomic positions. These slight move-ments smear the sharp interference pattern of the rap-

idly varying sin(2kR) term that would be seen if the at-oms were unmoving (Fig. 8). Effects of structuraldisorder are similar, and they give an additive contribu-

tion to 2. This Debye-Waller effect becomes more pro-nounced the shorter the wavelength of the photoelec-tron, and hence it cuts off the EXAFS at sufficientlylarge energy beyond about k1/, which is typically oforder 10 1. Thus the Debye-Waller factor is essentialin EXAFS, but is often negligible in XANES, when

2k21. Moreover, since 2 generally increases withtemperatureT(the vibrations become larger at highertemperatures), the EXAFS tends to melt at high tem-peratures, being confined to successively lower regionsof energy. Finally, the overall amplitude factor S0

2 is amany-body effect due to the relaxation of the system in

FIG. 6. Comparison of thekdependence of the effective back-scattering amplitudefeff(,R)evaluated atRnn2.55 (solid

line) and for comparison the result for the plane-wave approxi-mation (long dashes), which is equivalent to setting R. Thedotted line, which is thek dependence of the amplitude evalu-ated at 2Rnn, illustrates the approximate linear dependence offeffon 1/R. The angle indicates that the amplitude is forscattering the wave directly backwards (at 180 degrees). This isthe relevant amplitude for conventional single-scattering EX-AFS. Part (a) gives the magnitude offeffand part (b) its phase.This figure was redrawn; it is based on Fig. 1 of Rehr et al.(1986).




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response to the creation of the core hole. Although S 02 is

weakly energy dependent, it is usually approximated bya constant. A fully quantitative theory has yet to be de-veloped.

The inclusion of inelastic losses and Debye-Wallerfactors was crucial to the success of the modern short-

range-order theory of XAFS. Moreover, using the rep-resentation of EXAFS given by Eq. (2), Sayers et al.(1971) showed that a Fourier transform of the XAFSwith respect tok corresponds to an effective radial dis-tribution function, with peaks near the first few near-neighbor distances. This important observation pavedthe way for using EXAFS as a general probe of micro-scopic structural information in molecules and solids.

The technique is especially important in noncrystallinematerials, where lattice methods like x-ray diffractioncan be inapplicable. Conversely, the validity of the long-range-order theory depends on the absence or weaknessof these decay factors.

A key development has been the development of anaccurate treatment of scattering by taking curved-waveeffects into account. A simplification made in early workis the approximate treatment of the shape of the elec-tron waves. At large distances from the center of a wave,the curvature of a spherical wave front lessens, and itbecomes more and more plane-wave-like. The math-ematics of the scattering of plane waves is considerably

simpler than that of curved waves and can be treated bya plane-wave scattering amplitude f(k). This approxi-mation, which is sometimes called the plane-wave ap-proximation, was originally used to derive the simpleresult of Eq. (2). A more precise treatment of the plane-wave approximation was later given by Lee and Pendry(1975), which identified two contributions. One is thesmall-atom or point-scattering approximation,namely, the use of plane-wave scattering amplitudes,and the second is the small-wavelength approxima-tion, which approximates the Hankel-function behav-ior of the outgoing spherical wave as simple exponen-tials. At low energies or short distances, where the

FIG. 7. Filtered total XAFS phase (2c) for the first co-ordination shell of Cu, Pt, and GeCl4from FEFF(solid curves)and from similarly filtered experimental data (dashed curves).This figure was redrawn; it is based on Fig. 3 of Rehr et al.(1991).

FIG. 8. Temperature depen-dence of EXAFS,k 2(k) data,for Ag (K edge) at 300, 500,and 800 K. This figure was pro-vided by M. Newville.




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curvature is more pronounced, it is especially importantto remove the second approximation. This is fairly easyto do at the single-scattering level. Indeed, some earlywork (Sayers et al., 1971) was based on the point-scattering approximation, although that alone is insuf-ficient to correct the theory. Subsequently, fast and ac-curate single-scattering, curved-wave theories weredeveloped (Muller and Schaich, 1983), but at the ex-pense of a somewhat more complex mathematical treat-ment. Later it was shown (Rehr et al., 1986; Rehr andAlbers, 1990) that the entire theory, including multiple-scattering terms, can be recast exactly as in the standardEXAFS equation, but with an effective scattering ampli-tudefeff(k,R) replacing the plane-wave scattering ampli-tude (Fig. 6). A similar result is implicit in the formula-tion of Lee and Pendry (1975) in terms of a scatteringfactor Z. Moreover, a simplified but accurate,asymptotic, spherical wave approximation for feff wasdeveloped (Rehret al., 1986), which explains the physi-cal origin of curved-wave effects in terms of an addi-tional phase shift in each partial wave of expl (l1)/2kR2 . Tables of curved-wave phases and ampli-

tudes were also calculated and made available (McKaleet al., 1986; Rehr et al., 1991), thus improving on theplane-wave phases and amplitudes (Teo and Lee, 1979).

D. Multiple scattering

Another key development in the theory is that of anaccurate treatment of multiple-scattering effects. Oneobvious weakness of the phenomenological EXAFStheory is that it takes into account only the simplest, butusually the most dominant, form of scattering, i.e., re-flections by neighboring atoms directly back to the ab-sorbing atom (this type of scattering is often calledbackscattering or single scattering). More gener-ally, atoms can reflect the electron wave onto other at-oms that then reflect off of still other atoms (Fig. 5).Such multiple scattering is now known to be essential foraccurate calculations of the absorption coefficient inmost materials. In particular, multiple scattering by at-oms along a linear path, the so-called focusing or shad-owing effect, can exceed the backscattering contribu-tions in magnitude (Lee and Pendry, 1975). The beautyof the formulation of Sayers et al.(1971) is that a Fou-rier transform made it possible to filter the contributionsto the absorption by a path-length criterion. For ex-ample, if the scattering observed in the transform in-

volved the shortest path length (twice the nearest-neighbor distance), backscattering was the only possibletype of scattering that contributed significantly, andhence Eq. (2) was essentially correct in this respect, ex-cept for small corrections due to the leakage of longer-distance contributions into the first-shell signal. How-ever, in order to reliably extract more general atomic-structure information beyond nearest-neighbor bonddistances, it has proved crucial to include multiple-scattering effects, as we discuss below.

The formal multiple-scattering theory (Beeby, 1964)of XAFS has been derived based on both the Greens-

function method (Schaich, 1973; Ashley and Doniach,1975) and the wave-function approach (Lee and Pendry,1975). The pioneering EXAFS theory paper by Lee andPendry (1975) showed that the two approaches wereequivalent and gave the first quantitative treatment ofboth backscattering and multiple-scattering effects. Itwas also found that the much simpler plane-wave ap-proximation for the backscattering terms, together with

an ad hoc inner potential shift, could be a fairly goodapproximation at high energies (Leeet al., 1981). How-ever, Rehret al.(1986) showed that curved-wave correc-tions are actually important at all energies and intro-duced an asymptotic spherical wave approximation,which largely accounted for the main curved-wave ef-fects in backscattering. The plane-wave approximationalso simplifies multiple-scattering calculations, as shownby Lee and Pendry (1975), but the results generally con-tain significant errors both in phase and in amplitude.While somewhat better, the spherical wave approxima-tion also proved to be unsatisfactory in a fully quantita-tive theory. Unfortunately, since Clebsch-Gordan coef-

ficients proliferate for higher-order scattering, an exacttreatment of curved-wave contributions based on angu-lar momentum algebra (Messiah, 1966) is computation-ally demanding and was a major bottleneck in the devel-opment of precise theoretical calculations. Despite thesedifficulties, analytical curved-wave theories were subse-quently developed for up to third-order multiple scatter-ing (i.e., four-leg paths; Gurman et al., 1986). Iterativemethods for going to even higher order were developedby Natoli and collaborators (Brouderet al., 1989). How-ever, these methods are all generally limited to smallclusters or low energies due to the large angular mo-menta and consequent complexity of the calculations.Moreover, the inclusion of thermal vibrations and disor-der in such exact methods is nontrivial without ap-proximations such as an average only over the exponen-tial dependence e ik R for a given path length R . Forthese reasons, neither the plane-wave approximation,the spherical wave approximation, nor exact methodsprovide practicable schemes for accurate multiple-scattering calculations of EXAFS much beyond the firstcoordination shell.

A strategy that overcomes all of these computationalproblems was developed by the present authors (Rehrand Albers, 1990). This method is based on a rapidlyconvergent separable representation of the electronpropagator, which permits fast, accurate calculations of

any multiple-scattering path. However, the calculationswere then limited by a second problem, the exponentialproliferation of multiple-scattering paths of increasinglength. Fortunately, the overwhelming majority of suchpaths turn out to have negligible amplitude and can beeliminated with efficient multiple-scattering path filtersand a fast path-generation and sorting algorithm devel-oped by Zabinsky et al. (1995). The implementation ofthese algorithms into theab initioEXAFS codes knownas FEFFnamed after the effective scattering amplitude

feff in the theorynow makes accurate, high-order,multiple-scattering calculations of XAFS in general ma-




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terials routine. Due to the success of this prescription,the multiple-scattering formulation of EXAFS can nowbe regarded as a well-understood problem, although thetreatment of inelastic losses and disorder should be im-proved. This and other approaches will be reviewed ingreater depth later in the paper.

E. Other improvements

Yet another key improvement to early EXAFS theo-ries came with the replacement of the phenomenologicalmean-free-path term byab initiotheoretical calculationsof inelastic losses (Lee and Beni, 1977). This treatmentwas based on the complex, energy-dependent, electron-gas self-energy of Hedin and Lundqvist (1971) in thelocal-density approximation. This development also re-placed the ground-state exchange models by an energy-dependent exchange potential more appropriate for ex-cited states. The use of a complex, energy-dependentself-energy provides a more accurate, system-dependentdescription of the damping of the electronic waves in a

material. However, it also complicates the theory, sincethe one-electron Hamiltonian is then non-Hermitianand requires complex phase shifts. Although the Hedin-Lundqvist model usually leads to excessive loss in theXANES regime (Tyson et al., 1992; Zabinsky et al.,1995; Roy and Gurman, 1999), it is generally quite reli-able for EXAFS. Tables of atomic EXAFS phase shiftsand plane-wave scattering amplitudes based on theHedin-Lundqvist model of inelastic losses (Teo and Lee,1979) served as the original theoretical XAFS standardsfor many years. The Hedin-Lundqvist model continuesto be the most widely used self-energy for XAFS calcu-lations.

Perhaps the most pervasive and important influenceof all of the improvements to the phenomenologicalshort-range-order theory was the development of fast abinitiocomputer codes for the calculation of EXAFS, to-gether with the ready availability of faster computers.Suchab initiocodes replace the need for tables and givea more accurate treatment of the local environment thanfree-atom approximations on which the tables werebased. Most of these codes now include multiple-scattering effects, which are important for treating theEXAFS beyond the first coordination shell. For ex-ample, the earliest such EXAFS code, known as EX-CURV(Binstedet al., 1987, 1991), incorporated fast algo-rithms for precise calculations of curved-wave single,double, and triple scattering effects and also includedEXAFS analysis tools. A significantly improved set oftheoretical standards is based on an automated, ab initiocode FEFF3 developed by the present authors (Rehret al., 1991; Mustre de Leon et al., 1991).1 This imple-mentation of the theory includes both inelastic losses(which are neglected by McKaleet al., 1986) and curved-wave effects (which are neglected in the Teo-Lee

tables), together with an ab initiotreatment of the mo-lecular potential based on overlapped atomic densities.With all of these ingredients, the theoretical XAFS cal-culations give quantitative agreement with experimentfor a wide range of materials. This formulation typicallyyields distance determinations to better than 0.02 andcoordination numbers to within 1 (Rehr et al., 1991).As a result, many of the discrepancies between EXAFSand other techniques have been eliminated (Dagget al.,1993). But neglecting one or the other, as in variousearlier approaches, usually leads to significant errors.The accurate parametrization (Mustre de Leon et al.,1991) of the Hedin-Lundqvist self-energy made calcula-tions of inelastic losses much easier to implement andspeeded up ab initio calculations considerably. Yet an-other ab initio multiple-scattering EXAFS code isGNXAS (Filipponi and DiCicco, 1995; Filipponi et al.,1995), which is based on an efficient grouping ofmultiple-scattering paths for structural analysis.2

F. Outline

In the remainder of this paper, we review in moredetail the modern theoretical approaches to XAFS cal-culations. We focus primarily on the conceptual ideasneeded for an accurate theory of the XAFS spectra .For example, because of the large kinetic energy of thephotoelectron state compared to chemical energies, thephysical considerations needed in XAFS theory arerather different from those in modern ground-stateelectronic-structure calculations or in XANES. We alsofocus especially on deep-core-level absorption, whichgreatly simplifies the treatment of the initial state. Shal-low core absorption and optical absorption often requirequite different theories (Zangwill and Soven, 1980; Che-likowsky and Louie, 1996), such as the time-dependentlocal-density approximation. This review is not intendedto address fully the case of XANES, in which, for ex-ample, the multiple-scattering expansion fails. Nor dowe discuss the analysis of XAFS experiment (Binstedet al., 1987; Koningsberger and Prins, 1988; Filipponiand Di Cicco, 1995; Filipponiet al., 1995; Newvilleet al.,1995; Westreet al., 1995); the development of which hasfollowed advances in the theory. Because the fine struc-ture in both EXAFS and XANES involves fundamen-tally similar quantum-interference effects, they will bebe discussed from a unified viewpoint, hence the short-ened acronym XAFS. However, this synthesis is, as yet,not totally successful. For many reasons, especially dueto atomic, chemical, and many-body corrections, thequantitative treatment of XANES remains a challengingand generally unsolved problem. For example, variousatomic effects can become important in XANES andrelated spectroscopies (see, for example, de Groot, 1994;Kotani, 1997). For these reasons, we shall generally limit

1Details on obtaining the FEFF codes can be obtained athttp://leonardo.phys.washington.edu/feff/

2Details on obtaining the GNXAScodes are availlable at http://www.aquila.infn.it/gnxas/




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this review to a treatment of the XAFS , rather thanthe full absorption spectrum .

The outline for the rest of the paper is as follows: inSec. II we discuss the physical processes involved inx-ray absorption and how they are included in moderntheories of XAFS. In Sec. III we go into great detail toshow how the multiple-scattering theory can be solved.This section has been emphasized, since it is the key toextracting the structural information from XAFS. In thelast two sections (IV and V), we introduce additionalcomplications due to thermal disorder, show how wellthe theory compares to experiment, and summarize thestate of XAFS theory. Finally, we mention some connec-tions between XAFS and related spectroscopies.

II. KEY APPROXIMATIONS

A. One-electron golden-rule approximation

The dominant mechanism for the absorption of hard(10 keV) x rays by matter is the photoexcitation of

electrons. Compton scattering is negligible for thinsamples and pair production in the energy range of in-terest is forbidden. Moreover, the weakness of the elec-tromagnetic field (even for current third-generation syn-chrotron x-ray sources) is such that only processes thatare first order in the field are important. In consequence,the x-ray absorption coefficient is proportional to thetransition rate as given by Fermis golden rule. Mostpractical calculations of x-ray absorption are based onthe one-electron approximation of the golden rule,

f

fpAri2EfE i. (3)

Unless otherwise specified, for clarity, we shall useatomic units (em1) in the formalism presentedin this paper; when results are calculated for comparisonwith experiment, energies will be given in eV and dis-tances in . The reduction to an effective one-electronapproximation is discussed in detail in Sec. II.C.2. Thewave functions i and fin this equation refer, respec-tively, to the initial and final eigenstates of the effectiveone-electron HamiltoniansHfor the initial state andHfor the final state, with energies E iand Ef, respectively.That is, the statesiand fare generally calculated withdifferent self-consistent potentials in a SCF approxi-mation (Bagus, 1965), where SCF is an acronym for

self-consistent field. The use of the final-state one-particle Hamiltonian to calculate fis equivalent to thefinal-state rule. The quantity pis the momentum op-erator and A(r) is the vector potential of the incidentelectromagnetic field, which may be taken to be a clas-sical wave of polarization k, i.e., A(r,t) A0e

ikr.For deep-core excitations, the spatial dependence of

the electromagnetic field (vector potential A) can usu-ally be neglected (dipole approximation), i.e., e ikr1.Quadrupole corrections are of order (Z)2, whereZisthe nuclear charge seen by a core electron and 1/137 is the fine-structure constant. Likewise, local-

field effects are usually negligible for deep-core excita-tions (Zangwill and Soven, 1980). Several formallyequivalent representations of the dipole operator exist.For example, using the commutator H,rp/m , themomentum formof the dipole operator can be replacedby theposition form,

m r. (4)

HereHis the effective one-particle Hamiltonian and the transition frequency. Similarly, one can derive anacceleration form. Partly owing to its simplicity andpartly for numerical and physical considerations, the po-sition form seems preferable for XAFS calculations anddeep-core absorption.

B. Scattering potential

As noted in the Introduction, the oscillatory structurein the XAFS spectra results from the interference be-tween contributions to the photoelectron wave functionfrom various paths the excited photoelectron can take asit scatters off nearby atoms. In an effective one-electrontheory this excited electron, as it traverses the solid ormolecule, behaves as a quasiparticle that moves in aneffective complex-valued optical potential. Such alossy potential is needed to calculate the phase shifts,the scattering, and also the damping of the electron. Inthis section we discuss how such a potential is con-structed for practical calculations, as well as the currentadvantages and limitations involved.

The nature of the effective one-particle scattering po-tentialVhas been considered by several authors (see,for example, Sham and Kohn, 1966; Hedin and Lun-

dqvist, 1969; Lee and Beni, 1977; and Fujikawa and He-din, 1989). Because of the excited-state nature of thephotoelectron, its potential differs significantly from thatused in ground-state calculations based on the local-density approximation familiar from band-structuretheory. The optical potential V appears in the non-Hermitian but otherwise Schrodinger-like, one-particleDyson equation for the photoelectron (quasiparticle)states,

12 2VE E (5)(in atomic units), or its relativistic generalization in

terms of the Dirac equation. Here the operatorVE VcoulE (6)

consists of the net Coulomb potential Vcoulfelt by thephotoelectron and a self-energy (E), which isanalogous to the exchange-correlation potential Vxc inground-state calculations (Kohn and Sham, 1965). Thefunctional form of(E) depends on the particular ap-proximation used (Sec. II.C.2.a), but a local approxima-tion appears to be an excellent approximation for high-energy excited states, partly due to the decrease of thede Broglie wavelength with increasing energy. It is also




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crucial that the energy dependence of(E) be takeninto account. Indeed, the variation in (E) is typicallyabout 10 eV over the experimental energy range of EX-AFS. This variation corresponds physically to turningoff the exchange interaction with increasing energy. Theoptical potential is also dependent on the particularN-particle final state being considered and hence de-pends on both the core-hole state and the atomic/molecular configuration of the system (Natoli et al.,1990). Finally, for practical considerations a sphericalmuffin-tin geometry, without corrections, is usually ad-equate for modern EXAFS calculations. In the final two

subsections we discuss the nature of the potential used,both within the muffin tin itself and in the interstitialregion between the muffin tins. While muffin-tin poten-tials may be of questionable validity near an x-ray edgeor for total-energy ground-state calculations, nonspheri-cal corrections generally diminish rapidly with increas-ing energy and can generally be neglected in the EXAFSregime. However, muffin-tin jumps can lead to oscilla-tory behavior of the phase shifts (Loeffen and Pettifer,1996) and can also show up in the calculated backgroundabsorption0 (Hollandet al., 1978; Rehret al., 1994).

1. Muffin-tin approximation

Almost all practical XAFS calculations currently relyon a muffin-tin geometry (cf. Figs. 9, 10, and 11), whichconsists of a spherical scattering potential centered oneach atom and a constant value in the interstitial regionbetween atoms. A tremendous amount of mathematical

machinery has been built up over the last century thatrelies on spherical symmetry. The spherical muffin-tingeometry makes it possible to apply this machinery in avery efficient way. In the multiple-scattering approachthat most directly connects to the underlying atomicstructure of a material (see Sec. III), the muffin tins pro-vide atomic scattering centers that are described com-pletely by atomic phase shifts, which are calculated from

spherically symmetric atomiclike potentials inside themuffin tins. Greens-function propagators are employedto connect these scattering centers. Accurate calcula-tions depend on good potentials, since they determinethe strength of the scattering at each site.

The actual potential in a material (solid or cluster ofatoms) is, of course, more complicated. Near the centerof each atom, the charge density of the atomic core willbe large and dominate the potential. Hence the potentialis approximately atomic-like very close to the nucleus,where the spherical approximation is highly accurate. Inthe outer regions of the atom and between the atoms,the bonding properties of a material determine the dis-tribution of charge, and the potential is generally aniso-tropic. The degree of anisotropy depends on the type ofbonding present (ionic, covalent, or metallic) and thetypes of atoms that are bonding. In close-packed metals,the electronic wave functions overlap so strongly thatthe bonding is much more flat and isotropic than inother, more open systems, where there can be strongspatial variations.

In EXAFS spectroscopy the kinetic energy of the ex-cited electron is large, and the electron is less sensitiveto the details of the potential at the outer edges of theatom and in the regions between the atoms. The elec-tron is mainly scattered by the inner parts of the atomicpotential and moves more or less freely in the average

potential within the flat interstitial region. It is for thisreason that a spherical muffin-tin potential works sowell. In the near-edge region (XANES) and for ground-state total-energy calculations, the details of the shape ofthe interatomic potential are much more important. InSec. II.B.2.a we shall return to a further discussion ofnon-muffin-tin effects.

Hollandet al. (1978) proposed another effect of thepotential in the outer parts of an atom embedded in asolid. They suggest that it can give rise to additionalweak backscattering of the photoelectron and hence ex-tra oscillations in the absorption that resemble EXAFS

FIG. 9. Schematic drawing of a one-dimensional muffin-tin po-tential. The solid line between the atoms shows how the flatinterstitial potential truncates the true shape of the potential(cf. dashed line).

FIG. 10. Schematic drawing of a two-dimensional sphericalapproximation inside a Wigner-Seitz or Voronoi polyhedroncentered around each atom. This is an illustration of the effectof spherical muffin-tin-like approximations for real crystals.This figure was redrawn; it is loosely based on Fig. 50 of Ziman(1971).

FIG. 11. Figure illustrating why a muffin-tin potential is sonamed (two-dimensional drawing). This figure was redrawn; itwas based on Fig. 2.7 of Harrison (1970).




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structure, but with an effective distance R that is ap-proximatelyhalfthe near-neighbor distance. Additionalwork on this subject, which has been termed atomicXAFS, has been done by Rehr et al.(1995). In particu-lar, Rehret al.(1995) suggest that atomic XAFS is sen-sitive to the nature of the bonding potential. The effecthas been corroborated, for example, by Wende et al.(1997), Koningsberger et al. (1999), and Ramaker andOGrady (1999). However, a definitive theory of atomicXAFS will likely require corrections to the muffin-tinapproximation. The effects of multielectron excitationson the atomic background can also be important (Filip-poni and Di Cicco, 1996), but are typically smaller inmagnitude.

a. Charge density and potential

The potential within a muffin tin depends on thecharge density in this region, and it is essential to useapproximations that conserve overall charge neutralityfor an accurate determination of the energy reference.Because the muffin tin is centered on a given atom, most

of the charge density is that due to the atom in question,with only small contributions in the outer parts due tothe tails of neighboring atoms charge distributions.

Perhaps the best charge density to use would be thatproduced by a modern full-potential self-consistentelectronic-structure calculation (e.g., Blahaet al., 1997).Such calculations make no shape approximations (i.e.,they make no muffin-tin or other restrictions on theshape of the charge density or potential) and include allmodifications of the charge density due to bonding andother interactions with neighboring atoms. However, aswe shall discuss in Sec. II.B.1.b, such a calculationshould be modified for XAFS by including effects due tothe presence of the core hole. That is, to be consistentwith the final-state rule, the one-electron potential un-der consideration is usually taken to be the potential ofthe final state.

In practice, full-potential calculations are time con-suming and not yet readily available for many complexsystems studied by XAFS spectroscopies. Remarkably,they are not usually necessary (except, perhaps, close toan edge), since scattering off the atomic core is far stron-ger than scattering off the outer parts of the potential.Bigger errors come from the treatment of the self-energy. Thus simpler prescriptions for the potential haveproved to be accurate enough for a wide range of appli-cations for large excitation energies. The easiest and

most often used approach is based on overlapping, neu-tral, atomic-charge densities, which turns out to be avery good approximation to the total ground-statecharge distribution. This prescription, pioneered byMattheiss (1964) in the early days of electronic-structurecalculations, requires placing the charge density from aneutral-atom calculation on each atomic center and thenusing a direct superposition of these charges to obtainthe total charge distribution around any given atom. Amuffin tin mainly has the charge density due to the cen-tral atom. At the edges of the muffin tin, however, somecharge density from neighboring atoms spills in, thus re-

moving the spherical symmetry. A proper treatment ofthis nonspherical charge distribution involves solving aset of coupled differential equations for the radial wavefunctions of different l (angular momentum) instead ofthe conventional equation for each l. This is fairly easyto do and is computationally inexpensive. Suggestionsfor how to do this can be found in many places in theelectronic-structure literature, e.g., Evans and Keller(1971) and Siegel et al. (1976). These corrections areusually neglected in EXAFS calculations, and the chargedensity is spherically averaged, using, for example, theefficient Loucks (1967) algorithm. The Coulomb poten-tial is easily calculated for such spherically symmetriccharge densities (Loucks, 1967).

The neutral atomic-charge density may be calculatedin several ways. Both conventional (nonlocal exchange)Hartree-Fock and local-density-approximation (LDA)atomic codes are readily available. There are also tablesof charge densities available, such as those by Hermanand Skillman (1963) and Clementi and Roetti (1974).For heavy atoms, relativistic effects are important andatomic densities based on a Dirac equation are most

often used, e.g., using the semirelativistic code of Des-claux (1975) and a generalization of the Dirac-FockDesclaux code by Ankudinov and Rehr (1996). LDAversions of these approaches are usually used for com-putational efficiency and typically allow a number of dif-ferent ground-state exchange-correlation potentials tobe used. For example, the von Barth and Grossmann(1982) exchange-correlation potentials usually yield ac-curate atomic-charge densities.

In standard electronic-structure calculations a ground-state LDA exchange-correlation potential is typicallyused for crystalline solids, while a Hartree-Fock ex-change is often used for molecular calculations. As

noted above, for XAFS applications it is important totake into account the energy dependence of theexchange-correlation potential, which includes the qua-siparticle character of the excited electron and reflectsthe decreasing importance of the Pauli principle withincreasing energy. XAFS calculations based on ground-state calculations miss this additional energy depen-dence and therefore differ systematically from spectrameasured experimentally; see Sec. II.C.2.a.

b. Atomic configurations

In determining the muffin-tin scattering potential, animportant consideration is whether to use (1) (initial-

state rule) the ground-state charge density, or (2) (final-state rule) the charge density with the core hole present(which was left behind by the excited photoelectron). Inthe latter case, which is currently used most often bydefault, the question of the appropriate atomic configu-ration for the final state with the core hole is not unam-biguous. For the absorbing atom a reasonable approxi-mation is a neutralatomic configuration of a free atomof atomic number Zc1 with a missing electron in agiven core level, corresponding to the fully relaxed pri-mary channel (Rehret al., 1978). This choice is appro-priate for low-energy XAFS, and since the primary




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channel usually dominates the XAFS, it is also an ad-equate approximation at high energies as well. Alterna-tively one could choose the same configuration as in theinitial state with an appropriate hole and an extra elec-tron in the lowest unoccupied state. Ionic configurationsare appropriate only for highly ionized materials, but,due to charge-transfer effects, the net charge on an atomis usually much smaller than the chemical valence. An-other choice appropriate for high energies (see, for ex-ample, Lee and Beni, 1977) is a completely unrelaxedpotential. Usually we have found that XAFS calculatedwith a neutral, fully relaxed potential gives very goodagreement with experiment. However, self-consistentpotentials with a better treatment of many-body effectswould be desirable in the future to help avoid these am-biguities.

2. Interstitial potential

The interstitial region causes more complications forXAFS than for typical ground-state electronic-structurecalculations or for XANES. These complications are the

following: (1) the most efficient multiple-scatteringtreatments assume that the interstitial potential has nospatial variation (i.e., is constant), which is not generallytrue; (2) for excited-state calculations the interstitial re-gion is lossy and energy dependent; and (3) the criticalparameter for determining structural information (thewave number k of the excited photoelectron) dependson the choice made for the average value of the poten-tial in the interstitial region.

a. Muffin-tin corrections: warping

As discussed above, the electronic charge density inany real material is more complex than that implied bythe muffin-tin approximation. Inside the muffin tins thecharge density is not perfectly spherical, and in the in-terstitial region the charge density is not constant. In-stead, it is often common to find a buildup of chargebetween neighboring atoms, where there is significantbonding. This same effect can be intuitively pictured as aconsequence of overlapping atomic-charge densities,which also naturally leads to a buildup of charge be-tween neighboring atoms. In order to evaluate the valid-ity of the muffin-tin approximation, we review how non-muffin-tin corrections to the potential are treated inground-state electronic-structure calculations.

In the electronic-structure literature the spatial varia-

tion of the charge density in the interstitial region issometimes called warping of the potential or chargedensity. Methods that include this effect are called full-potential techniques (since the muffin-tin potentialcaptures only the spherical part of the potential andmisses the spatial dependence in the interstitial region).Such effects are often important for sensitive bondingproperties.

For EXAFS this effect is of less importance. In addi-tion to the simplification provided by the kinetic energyof the photoelectron being typically much larger thanthe size of the warping potential, in the muffin-tin ap-

proximation the spatial variations in the interstitial re-gion are also reduced by subtracting out the averageinterstitial potential. This greatly reduces the perturba-tion on the photoelectron. It is only in the XANES re-gime that warping begins to be a potentially seriousproblem, and then only within a few eV of the edge.

In band-structure approaches the warping has mainlybeen treated by taking advantage of the periodicity ofthe crystal. Any periodic function in a given crystal canbe expanded in reciprocal-lattice plane waves. Matrixelements between the basis set used to expand theground-state wave functions are typically reexpanded ina plane-wave basis, which then makes it mathematicallyrather easy to calculate the matrix elements that involvethe interstitial potential. For most XAFS calculationalmethods, this periodicity is not assumed (except possiblyin band-structure or exact diagonalization approaches;see Sec. III.B), and the simplicity introduced throughthe plane-wave expansions is not possible. Thus includ-ing warping within the XAFS formalism has been foundto be very difficult. Although a scattering formalism hasbeen developed by Natoli et al. (1986), it is difficult to

apply in practice. Alternative approaches based on dis-crete basis sets have also been implemented in XANES(Ellis and Goodman, 1984; Jolyet al., 1999), but appearto be computationally difficult to extend to EXAFS en-ergies.

b. Excited-state effects

The spectrum of photoelectron (quasiparticle) states,moving in the optical potential Vdefined by Eq. (5), isdetermined from the solution to an electron-atom scat-tering problem. The unperturbed system contains a pho-toelectron of energy E and angular momentum L

(l,m) moving in a uniform (lossy) optical potential,which is conventionally taken to be the average valueVin t(E)V(E) in the interstitial region. Note that ingeneral this uniform potential is energy dependent andcomplex valued, and hence the concept of a fixed innerpotential in XAFS is not well defined. With Vin t(E) asan energy reference, the inelastic loss in a system ismostly accounted for by the uniform mean-free-pathterm calculated at the interstitial density. The scatteringperturbation is then defined as the difference with re-spect to the uniform potential, which we denote by alower-case v. This potential is usually expressed as asum over a set of local potentials vj relative to atomic

sitesR

j, i.e.,

vVEVin tj

vjrRj. (7)

Some treatments, e.g., Teo and Lee (1979), use a realmuffin-tin zero. Moreover, since there is virtually no in-elastic loss in V(E) in the high-density core region ofatoms, the scattering potential v(r,E) is often amplify-ing in the core region and hence leads to increases in thebackscattering amplitudes and phases compared withthose defined with respect to the vacuum level (Ekardtand Thoai, 1981).




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c. Energy reference

The variation ofVin t(E) over the range of XAFS en-ergies roughly amounts to the magnitude of the ground-state exchange hole, i.e., VxckF/in atomic units,wherek Fis the Fermi momentum and is typically about10 eV. To circumvent this ambiguity in the comparisonof theoretical XAFS standards with experiment, it isuseful to define a fixed energy referenceE 0 , the photo-

absorption energy threshold. Experimentally, thisthreshold corresponds to the energy at the onset of anabsorption edge. We emphasize that this threshold en-ergy is not necessarily the midpoint of the main-edgestep in the x-ray absorption coefficient, but rather at theonset of absorption for a given edge. Self-consistent cal-culations are needed to determine E 0 precisely (Anku-dinovet al., 1998). However, values ofE 0 that are typi-cally valid to within a few eV may be estimated from thechemical potential of a homogeneous electron gas atthe average interstitial-charge density. The errors intro-duced by the electron-gas approximation and the aver-aging of the interstitial potential and charge density are

such that these estimates are typically a few eV higherthan those from self-consistent calculations (Mulleret al., 1982; Mustre de Leon et al., 1991; Tyson et al.,1992; Ankudinovet al., 1998).

3. Relativistic effects

Relativistic effectsper sehave generally played a sec-ondary role in XAFS theory. Similarly, spin-dependenteffects are usually negligible and are manifestly impor-tant only when the dominant spin-independent contribu-tions are forced to cancel, as in x-ray magnetic circulardichroism; see, for example, Brouderet al.(1996), Ebert(1996), and Ankudinov and Rehr (1997). To understandwhy this is the case, it is useful to review some basicaspects of electronic-structure theory.

In atomic theory it has long been recognized that rela-tivistic effects are only important for very heavy (large-Z) atoms. For such atoms the kinetic energy of the elec-trons becomes quite large near the nucleus of the atomwhere relativistic corrections are essential. Besidesmodifying the wave functions and energy eigenvalues ofthe electronic states, these corrections also change theself-consistent potential. In contrast, at large distancesfrom the atom the kinetic energy of the outermost elec-trons responsible for bonding properties and the photo-ejected electrons are usually well into the nonrelativistic

regime. Thus relativity can strongly affect the produc-tion of the photoelectron and hence the atomic back-ground 0 (Grant, 1970) through the dipole matrix ele-ments, but have a weak effect on the propagation of aphotoelectron (Loucks, 1967).

Multiple scattering in turn involves knowledge of boththe photoelectron propagators connecting different at-oms and the scatteringtmatrices, which depend on thephase shifts of the electrons at a muffin-tin radius.Clearly the electron propagators, which vary asexp(ikR), are essentially nonrelativistic (assuming para-magnetic materials with spin-independent potentials),

and thus most of the propagation involves knowledge ofthe electrons behavior at large distances from thenucleus where relativistic effects are usually negligible,even for heavy atoms at the highest EXAFS energies(about 1500 eV). Thus the only place where relativitycan play a substantial role is in the phase shifts them-selves. Relativistic effects enter in two ways for such cal-culations. Because relativistic effects shift wave func-

tions to distances closer to the nucleus, the self-consistent potential must include relativistic effects.Secondly, for heavy atoms the wave function near thenucleus will be modified and affect the integration of thewave function for calculating the phase shifts, even if thecorrect self-consistent potential is known; this willmodify the phase shift. To account for the first effect, itis standard to use a fully relativistic solution for the coreelectrons to get their correct charge density. Within thelocal-density approximation for the electronic potentialthis is very easy to do and requires almost no computa-tional overhead.

For calculating the phase shift of the photoelectron,

some approximations have to be made. The chief prob-lem is symmetry. For the spherically averaged potentialsused in XAFS, the nonrelativistic wave functions are la-beled by spin and orbital angular momentum (l). Rela-tivistic wave functions have total angular momentum la-bels j as well as an additional label, usually (Rose,1961), which specifies whether the l s or the l scomponent is meant (e.g., whether the d 3/2 ord 5/2 com-ponent is being used). Because this difference betweenrelativistic and nonrelativistic theories arises from thespin-orbit terms in the Dirac equation, a conventionalapproach in many electronic-structure methods has beento use semirelativistic solutions for the wave function(Andersen, 1975; Koelling and Harmon, 1977). Such asapproach solves the Dirac equation, but averages thespin-orbit term over the two j components for each l .This removes the symmetry label problem. In the outerpart of the atom where the electron is clearly in thenonrelativistic regime, it is easy to match the semirela-tivistic wave function onto its nonrelativistic form to findthe phase shifts appropriate for a nonrelativisticmultiple-scattering formalism. This approximation isvery accurate and has been used in electronic-structurecalculations since the mid 1970s; it also works very wellfor XAFS calculations (Mustre de Leon et al., 1991).

For very heavy atoms (e.g., 5ds, 4fs, and 5fs) the

spin-orbit splitting of the outermost valence electronscan be significant, and a better theory is to retain therelativistic labels and to use the corresponding relativis-tic phase shifts (Tyson, 1994). Because the muffin-tininterstitial regions are in the nonrelativistic regime, onecan form the correct linear superposition (usingClebsch-Gordan coefficients) of the nonrelativisticGreens-function propagators to form propagators ofthe correct relativistic symmetry (Ankudinov and Rehr,1997). The multiple-scattering theory thus can bestraightforwardly generalized from the nonrelativistictreatment (Gonis, 1992).




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C. Many-body effects

1. Inelastic losses

As we have stressed, the inclusion of damping due toinelastic losses is a key ingredient in the modern short-range-order theory of XAFS. These losses give the finalphotoelectron state a finite lifetime and hence lead ef-fectively to a broadened one-electron spectrum. It is tra-ditional to differentiate two types of inelastic losses inthe photoabsorption, namely,intrinsicand extrinsicpro-cesses.Extrinsiceffects refer to losses in propagation ofthe photoelectron and include excitations such as plas-mons, electron-hole pairs, and inelastic scattering inwhich the photoelectron loses energy. Many of theseprocesses are the same as for an electron beam as ittraverses a solid (Quinn, 1962; Sevier, 1972; Powell,1974; Seah and Dench, 1979; Wagner et al., 1980). Theselosses lead to a decay of the final state, which has beenaccounted for phenomenologically by an energy-dependent mean-free-path (k), i.e, a path-dependentdecay factor exp2R/(k) of EXAFS amplitudes.However, such extrinsic losses are more precisely de-scribed in terms of a complex, energy-dependent self-energy (E), which gives both a real energy shift and adecay (cf. Figs. 12 and 13). Intrinsiclosses, on the otherhand, refer to excitations in response to creation of thecore hole. These losses are traditionally accounted forphenomenologically by a constant many-body

amplitude-reduction factor S02. However, more pre-cisely, the amplitude reduction is energy dependent andmust be described by a complex, path-dependent con-stant given by a phasor sum. We now summarize thecommonly used approximations for these quantities.

2. Extrinsic losses

a. Self-energy

The extrinsic losses and associated energy shifts canbe described in terms of a self-energy operator (E).This operator is complex valued and energy dependent

and arises from the dynamically screened exchange in-

teraction between the photoelectron and the system (cf.Figs. 12 and 13). This interaction is the analog of thereal-valued exchange-correlation potential Vxc () ofground-state calculations. The real part of (E) ac-counts for the energy dependence of the exchange andcauses systematic shifts of location of the EXAFS peakscompared to the positions obtained for the ground state,while the imaginary part gives rise to the mean freepath. As in conventional electronic-structure theory, themany-body nature and interactions of the electrons canbe approximated within a local-density approximation(E ,). The energy dependence of the screened ex-change is of the order of several eV and decays slowlywith energy, typically over a few hundred eV, but this iscrucial for quantitative calculations of XAFS. Ulti-mately, the exchange interaction vanishes, and the pho-toelectron can be treated as a classical particle.

An appropriate generalization of the local-density ap-proximation for excited states is again based on a uni-form electron-gas model, e.g., the Hedin-Lundqvist GW/plasmon-pole self-energy HL (E ,) (Hedin andLundqvist, 1969; Lundqvist, 1977). Here GW refers toan approximation for the electron self-energy which ne-glects vertex corrections, where G is the Greens-function operator andWis the screened coulomb inter-action. This model was introduced into XAFS theory byLee and Beni (1977). However, full calculations within

this model are computationally time consuming. Thus inpractice it is preferable to use accurate analytic repre-sentations in both energy and density (Lu et al., 1989;Mustre de Leonet al., 1991) ofHL. Although there issome evidence for positive corrections to the Hedin-Lundqvist self-energy (Horsch, von der Linden, and Lu-kas, 1989; Mustre de Leon et al., 1991), the model ap-pears to be accurate to within a few eV at XAFSenergies. To avoid a discontinuity at the Fermi energy, itis convenient to shift the self-energy by a constant, i.e.,(E ,)HL (E ,)HL (EF,)Vxc (). Theexcited-state potential is therefore given by V(E ,r)

FIG. 12. Real part of the self-energy (E) in eV. The quasi-particle energyERe is also shown. For comparison pur-poses, a dashed line linear inE is also included.

FIG. 13. Energy dependence of the mean free path (E) andthe corresponding imaginary part of the self-energy (E). Forconvenience, the same (y-axis) numerical scale was used forboth quantities (one is positive and one is negative). However,(E) is in units of and the Imis in eV.




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V(r)Vxc()HLE ,(r)HLEF,(r). The ad-vantage of this formulation over other approximations,such as the nonlocal Hartree-Fock or the analogousLDA Dirac-Hara self-energies (Chou et al., 1987), isthat extrinsic losses are represented fairly accurately interms of the imaginary part of (E ,). A disadvantageis that the plasmon-pole approximation introduces aweak, unphysical cusp singularity at the onset of plas-mon excitations in the near-edge region. Other more so-phisticated approaches to the self-energy have been sug-gested, but these are mostly formal and have yet to bewidely implemented (Fujikawa and Hedin, 1989;Fujikawaet al., 1995).

b. Mean free path

The effect of the imaginary part of the self-energy issimilar to Lorentzian broadening of the XAFS spectrumwith a half-width Im (Schaich, 1973) and accounts forextrinsic losses. As noted above, the Hedin-Lundqvist(1971) self-energyHL (E) yields a good approximation

for these losses (Mustre de Leon et al., 1991) for EX-AFS. Thus the model generally gives good agreement(Penn, 1987) with experimental values of the XAFSmean-free-pathk/Im(E)for a variety of systems.This relation follows by expanding the relation (inatomic units) (1/2)k2(1/2)(k i/)2E(E). Thequantityis also in accord with the universal mean-free-path curve (Lindau and Spicer, 1974; Seah andDench, 1979), apart from a factor of 2, since the uni-versal curve measures the shorter-range decay inelectron-beam intensities, while the XAFS mean freepath measures the decay in the wave amplitude. Despitethe good agreement for EXAFS, at low energies the

Hedin-Lundqvist model based on the plasmon-pole ap-proximation tends to give too much loss and moreoverexhibits a sharp, unphysical jump at the onset of theplasmon excitation energy. This excessive loss can becorrected, for example, by using an ad hoc position-averaged approximation, as discussed, for example, byPenn (1987) and by Roy and Gurman (1999). However,such an averaging destroys the local behavior of(r).Clearly a better treatment is desirable.

c. Energy shifts

A comparison of XAFS peak energies, as calculated

by ground-state band-structure theory or measured inexperiment, reveals a systematic shift that can be repre-sented at low energies by an approximately linear scal-ing with energy, i.e., EexpEth(1), where is typi-cally about 0.05 (Materliket al., 1983). See, for example,Figs. 12 and 14. Such a scaling is actually consistent withthe effect of a real energy-dependent self-energy or ex-change interaction, as shown by Mustre de Leon et al.(1991). Although this effect is ignored in phenomeno-logical XAFS theories (e.g., theories containing only amean free path or an imaginary optical potential), itcould have been anticipated, since the presence of a

mean-free-path term implies the existence of a real en-ergy shift due to the dispersion relations satisfied by theself-energy operator.

3. Intrinsic losses and the initial- and final-state rules

a. Amplitude-reduction factor S02

Intrinsic losses correspond to excitations that arisedue to the creation of a core hole in the photoabsorptionprocess. Although the theory of XAFS is often couchedin a one-electron language, where a single electronmakes a transition from a core orbital to an excitedstate, the process is truly many body, and other electronsare generally excited as well. Examples of intrinsic pro-cesses are the well-known shakeup and shakeoff excita-tions in which more than one electron is excited in atransition. This secondary emission results from a relax-ation of theN1 electron Fermi sea in response to the

creation of a core hole. Because both of these processesinvolve the same final states, their quantum-mechanicalamplitudes can in principle interfere, and hence one can-not simply add their transition rates. We review here thetreatment of such intrinsic effects given by Rehr et al.(1978). This treatment ignores interference effects thatcan be important near the edge, but appears to be ad-equate for current EXAFS calculations. In any case, afully satisfactory theory has yet to be developed.

To understand the origin of such intrinsic effects, onemust examine the full many-body expression for thex-ray absorption spectrum,

f fNj1

N

pjArj iN2EfE i. (8)

In the Hartree-Fock or density-functional-theory ap-proximations, for example, the initial and final states areSlater determinants of one-particle states, calculatedwithdifferentself-consistent one-electron potentials, i.e.,the self-consistant-field approximation, as discussed, forexample, by Bagus (1965). The potential for the initialstate is that of the ground state, while that for the finalstates includes a core hole and a photoelectron. The cal-culation can be simplified if one assumes that the finalstate can be factored, i.e.,f 0

N1f , which is a

FIG. 14. Comparison of the ground-state band-structuretheory of x-ray absorption with experiment (for a Gd L1edge), showing the gradually increasing discrepancy betweenthe peaks in each curve due to linear scaling. This figure wasredrawn; it is based on Fig. 2 of Materlik, Muller, and Wilkins(1983).




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reasonable approximation at high energies, when thesudden approximation is valid. Then the dominantcontribution to the many-body dipole matrix element isgiven by

Mfifrc0N10

N1 , (9)

where the prime refers to states calculated in the pres-ence of the core hole. Thus the absorption in the lowest

energy or primary channel is reduced in magnitudefrom the one-particle expression by a many-body over-lap integral,

S02 0

N10N12. (10)

In this approximation the one-electron matrix elementfor the primary channel is consistent with thefinal-staterule, i.e., that the core orbital should be calculated withthe initial ground-state Hamiltonian and the final statewith the presence of a fully relaxed core hole (von Barthand Grossmann, 1982).

b. Excitations

Because of these relaxation effects, the observed EX-AFS amplitudes are reduced from their one-electronvalues by a phenomenological factor, which is usuallytaken to be a constant. To see this, one must examinethe contribution from excitations (see Natoli et al.,1990). Indeed, ifS0

21, there must be contributions frommultielectron transitions in which the (N1)-electronion is left in an excited state n

N1 with excitationenergyE n . The net absorption is then given by a sumover excited states or channelsn ,

n

nn

n rc2Sn

2 , (11)

where the many-body overlap integral Snn

N10N1 ; also, the energy of the photoelectron

is reduced in accordance with energy conservation bythe excitation energy En. The quantity Sn is usuallysmall (for n0) and would vanish by orthogonality inthe absence of relaxation. Thus the total absorption inthis approximation is equivalent to a convolution, i.e., toa broadened one-particle calculation:

0

dA, (12)

where is the one-electron spectrum and A()nSn2(En) is the excitation spectrum. Moreover,adding the effect of extrinsic losses (which adds addi-tional broadening) to this result is equivalent to calcu-lating the photoelectron state fwith a complex opticalpotential. Similar convolution expressions have alsobeen derived with other formalisms, e.g., the generalizedGW approach of Hedin (1989) and the multichannelmultiple-scattering theory of Natoli (1990). Due to com-pleteness of the n

N1 states, the sum of the weightsS n2

is unity. They can be interpreted in the sudden ap-proximation as the probabilities that the ion, initially inthe ground state, finds itself in thenth final excited state,

n(N1) . According to the Manne-Aberg theorem, the

centroid of the excitation spectrum corresponds toKoopmans theorem for the transition energy (Manneand Aberg, 1970). The difference between the centroidand the onset of absorption is defined as the relaxationenergy. Since Koopmans theorem involves the ground-state wave functions, if the excitation spectrum were a-function spectrum, theinitial-state rulewould be valid.The initial-state rule states that the photoelectronshould be calculated with the ground-state charge den-sity (i.e.

theoretical approaches to x-ray absorption fine structure

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