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X-ray Absorption Fine Structure in Embedded Atoms J. J. Rehr,(1) C. H. Booth,(2) F. Bridges,(2) S. I. Zabinsky(1) (1)Department of Physics, University of Washington, Seattle, WA 98195 (2)Department of Physics, University of California Santa Cruz, Santa Cruz, CA 95064 (February 6, 2008) Oscillatory structure is found in the atomic background absorption in x-ray-absorption fine structure(XAFS).Thisatomic-XAFS orAXAFSarises from scattering within anembeddedatom, and is analogous to the Ramsauer-Townsend effect. Calculations and measurements confirm the existenceofAXAFSandshowthatitcandominatecontributionssuchasmulti-electronexcitations. Thestructureissensitivetochemicaleffectsandthusprovidesanewprobeofbondingandexchange effects on thescattering potential. 4 PACS numbers: 61.10.Lx, 71.10.+x, 71.45.Gm, 78.70.Dm 9 9 1 n a J 3 1 v 6 0 0 1 0 4 9 / t a m - d n o c : v i X r a 1 The main features of X-ray absorption spectra µ(E) is summarized as follows: After removing the pre-edge are due to one-electrontransitions from deepcore levels. absorption, a smooth spline function is fit to the ab- Inmoleculesandsolids,oscillatoryfinestructureexistsin sorption data µ(E) (Fig. 1), which simulates to lowest µ(E)due toscatteringofthe photoelectronbyneighbor- order the free atomic absorption µ (E) without XAFS 0 ingatoms. Thewellknowntechniqueofx-ray-absorption or other features present. Then a trial XAFS function fine structure (XAFS), which includes both extended- χ(E) = [µ(E)/µ (E)]−1 is obtained. A Fourier trans- 0 XAFS (EXAFS) and x-ray-absorption near-edge struc- form of χ with respect to wave number k defined with ture(XANES),isbasedontheanalysisofthisfinestruc- respect to threshold energy E , yields peaks in r-space 0 ture. In XAFS the oscillatory part χ is defined relative correspondingto the distribution ofneighbors to the ab- to an assumed smooth “atomic background” absorption sorbing atom. Initially, the transforms often have a spu- µ (E),i.e.,χ=(µ−µ )/µ .Acomplicationisthatµ (E) rious, r-space peak near 1 ˚A that is inconsistent with 0 0 0 0 is not necessarily smooth. For example, the background their known structures. Next an approximate fit of the may exhibit such well known structures as white lines, first few peaks of the r-space transform is made using resonances and jumps due to multi-electron transitions, theoretical XAFS standards [6]. This fit is transferred even well above threshold. Less well known, however, is back to energy space and subtracted from the experi- the possible fine structure in µ (E) itself, in molecules mental data to remove most of the low frequency XAFS 0 andcondensedsystems,asdiscussedbyHollandet al.[1] oscillations. A high order spline is used to smooth the The purposeof this Letter is to showthat this atomic x- remaining data (Fig. 1). The positions of the knots in ray-absorptionfinestructure(AXAFS)canproducelarge this spline are varied to follow the larger features in the oscillations,hasanXAFSlikeinterpretation,andcanal- residue. This spline function then becomes a new µ (E) 0 ter XAFS analysis. In view of recent advances in XAFS and a new XAFS function is extracted. The process is theoryandanalysistechniques[2,4,5],inwhichtheback- iterated to convergence, typically in several iterations. ground plays a crucial role, this structure is now partic- With this procedure the background µ (E) contains all 0 ularly important. the atomic fine structure and the spurious r-space peak Thisextrafinestructureoriginatesfromresonantscat- near 1 ˚A is eliminated. tering “in the periphery of the absorbingatom” [1]. The This procedure was tested on a theoretical absorption effect is like an internal Ramsauer-Townsend (RT) res- spectrum from FEFF 5X for a model of PrBa Cu O 2 3 7 onance where the incident electron is a spherical wave (PBCO) which included many XAFS shells and a back- created at the center of the atom, rather than a wave ground with the above RT-like resonance. PBCO was scattered by an atom. As the photoelectron electron ap- chosenbecausethecontributionfromhighershellsissig- proaches a potential barrier - in this case the edge of nificant. This check therefore tests both the fit to the an embedded atom potential - the reflection coefficient XAFSandtothebackground. Theextractedbackground oscillates with energy, with a pronounced increase just fits the simulated background well (Fig. 1). above threshold, followed by a dip and subsequent oscil- This procedure was then applied to Ba, Ce, and Pr lations that conserve integrated oscillator strength. We K-edge data of BaO, CeO , and PBCO. The data were 2 find that AXAFS can be the dominant background fine collected at T ≃ 80 K, at the Stanford Synchrotron Ra- structure and has features in the same energy range as diation Laboratory (SSRL) using (400) monochromator multi-electron transitions, complicating detection of the crystals. Details are given in a separate paper [10]. The latter. Using a new background subtraction technique extracted backgrounds of all these high energy K-edges [4],experimentalbackgroundsforBa,CeandPrK-edges (Fig. 2) are similar in shape and energy scale, exhibit- are obtained which exhibit AXAFS as large as 60% of ing the near-edgepeak anddipstructure consistentwith the XAFS amplitude. Theoretical calculations based on thatexpectedforaRTresonance. Themagnitudeofthis an ab initio XAFS/XANES code FEFF 5X [2] confirm structure is comparable to EXAFS amplitudes and is a these observations. To our knowledge, the only previous factoroffourlargerthanthestep-likestructuresobserved attempt to identify AXAFS [1] was only partly success- above the edge for the rare gas Kr or for the Rb and Br ful. Notable discrepancies between theory and exper- K-edge data for RbBr. iment were found at low energies and the work did not WenowbrieflydiscussthetheoryofAXAFSandshow deriveitsoscillatorycharacter. Webelieve,however,that that it has an interpretation analogous to the curved- evidence for AXAFS exists in many previous studies, al- wave theory of XAFS [7]. “Embedded” atoms in solids thoughnotheretoforeidentifiedassuch. Inparticularwe may be defined in terms of their respective scattering suggest that AXAFS is largely responsible for the spuri- potentials. The final state potentialv at the absorption 0 ous peak at about half the first neighbor distance often site consists of a bare atomic potential v , plus extra- a observed in XAFS Fourier transforms [3,4]. atomiccontributionsv fromthetailsoftheelectrondis- e A number of improved background subtraction tech- tributions of neighboring atoms. In the muffin-tin ap- niques have recentlybeen developed[4,5]. The approach proximation, adoptedhere[4]isbasedonaniterativeprocedure,which 2 v (r)=v (r)+v (r), (r ≤R ), Combining these ingredients, one finds that µ(E) can 0 a e mt =v , (r ≥R ), (1) be factored as in conventional XAFS theory [11], i.e., mt mt µ =µ (1+χ ), where µ is givenby Eq. (2) calculated 0 a e a where Rmt is the muffin-tin radius. For simplicity we with free atomic states |fai and the AXAFS χe is consideraone-electroncalculationofphotoabsorptionby ∞ 1 an embedded atom using the Fermi Golden rule and the χe ≃−Imk Z dr[Ra+(kr)]2δv(r). (3) dipole approximation, i.e., 0 An analogy to the curved-wave XAFS formula [7] is ob- µ (E)=4π2αω |hc|ǫˆ·~r|fi|2δ(E−E ), (2) tained by recognizing that the perturbation arises from 0 f Xf the periphery of the atom where one may approximate R+ by its asymptotic form, R+ ≃ c (kr)exp(ikr+iδa). a a l l where α ≃ 1/137 is the fine structure constant, ω is Here c (kr) is the curved wave factor [7] in the spherical l the X-ray energy (we use Hartree atomic units e = Hankel function h(+)(kr)=c (kr)exp(ikr)/kr. For sim- l m = h¯ = 1), E = ω − E is the photoelectron en- c plicity we model the perturbation as δv(r) ≃ v /[1+ mt ergy, ǫˆ is the x-ray polarization vector, and the final exp(ζ(R −r))], where ζ characterizesthe decay of the mt states |fi = (1/r)R (r)Y (rˆ) are calculated at en- 0 lm atomic potential tails near R . The integral (3) can mt ergy E = (1/2)k2 in the embedded atom potential f then be expressed as v . The normalized radial wave functions R (r) are 0 0 1 obtained by matching the regular solution of the ra- χ =− |f |sin(2kR +2δa+Φ ). (4) e kR2 e mt l e dial l-waveSchrödingerequation to the asymptotic form mt R0(r)=kr[jl(kr)cosδl−nl(kr)sinδl], (r ≥Rmt), where wherefe =|fe|exp(iΦe)isaneffectivecurved-wavescat- jlandnlaresphericalBesselfunctions,δlisthel−thpar- tering amplitude. With the above model the AXAFS tial wave’s phase shift, and l is fixed by dipole selection is analogous to a damped harmonic oscillator, f ∼ e rules. This matching procedure is equivalent to a calcu- exp(−2πk/ζ)/k. For comparison to experiment, the Eq. lation of the Jost function Fl(E) which guarantees final (4) should have a few additional factors as in the usual statenormalization,asdiscussedbyHollandetal.[1]and XAFS formula, namely an amplitude reduction factor byNewton[8]. InparticularHollandetal. showthatthe S2, a Debye-Waller factor, exp[−2σ2(R /R)2k2], and 0 mt atomiccross-sectioncanbewrittenasµ˜0/|Fl|2,whereµ˜0 a mean-free path term, exp(−2Rmt/λ). is a reduced matrix element which varies smoothly with AXAFS comparisons between the theoretical calcula- energy. All of the calculations of AXAFS reported here tions and experimental results presented here are in rea- are based on an analogous matching procedure for the sonableagreementwitheachother(Fig.2),especiallyfor relativistic spinor wavefunctions used in FEFF, without thesimpleoxides. ThediscrepancyattheedgeforBaOis any of the simplifying approximations of the following not fully understood, but may point to errors in FEFF’s discussion. Additionaldetailswillbegivenelsewhere[9]. muffin-tinpotentialandenergyreference. Thelongrange The formal relations [8] satisfied by the Jost function, oscillatory structure in the calculations is likely due to are very general and do not explicitly show the oscilla- a small discontinuity in FEFF’s muffin-tin potential at tory behavior of AXAFS. Thus to illustrate its nature R . mt we present a highly simplified model based on first or- Tocheckwhethermulti-electronexcitationsmightalso der perturbation theory with respect to the free atom bepresent,weusedtheZ+1modeltoestimatewherethe potential. We will assume that the free atomic back- step for a two-electron excitation would begin. In this groundhasnegligibleoscillatorystructure;samplecalcu- model excitation energies correspond to the ionization lations with large muffin-tin radii support this assump- energiesof Z+1atoms,and are99 eV for Baand 113eV tion. Using the spectral representation of the embedded for Ce, as indicated by arrows in Fig. 2. Small features atom Green’s function, the final state sum in Eq. (2) in the background were previously attributed to multi- can be expressed as Σf|fiδ(E−Ef)hf| = (−1/π)ImG0 electron excitations based on this model [4,5]. However, where G0 = (E − H0 +i0+)−1 is the embedded atom itislikelythatpartoftheobservedstructurecanalsobe Green’s function (operator) and H0 the embedded atom attributed to AXAFS. Hamiltonian. To first order in the perturbation δv = Wepointoutthatourcalculationsofthe atomicback- v0(r)−va(r),G0isgivenbyG0 ≃Ga+GaδvGa,whereGa grounds shown in Fig. 2 were all done with groundstate isthefreeatomicGreen’sfunction. Fordeepcoreabsorp- exchange potentials. We found that the usual Hedin- tion, the core states are highly localized so we need only Lundqvist(HL)self-energymodelusedinFEFF[6]gives evaluate G0 in positionspace for verysmallarguments r toolargeanoscillatoryamplitudenearthreshold. Thisis ′ andr ,whereδvisnegligible. TheradialpartofGa[8],is an indication of the sensitivity of the AXAFS to the ex- given by Ga(r,r′)= (−1/k)Ra(r<)Ra+(r>), where r>(<) change interaction. Evidently improvements to FEFF’s is the greater(lesser) of r and r′, and Ra+ = Sa + iRa muffin-tin potentials are necessary, and AXAFS may be is the outgoing part of the radial Schrödinger equation. useful in assessing various improvements. 3 It is well known that simple, monotonic approxima- [1] B.W.Holland,J.B.Pendry,R.F.PettiferandJ.Bordas, tions to the atomic background are not sufficient to ob- J. Phys.C 11, 633 (1978). tainaccurateXAFSdata,emphasizingtheimportanceof [2] J. J. Rehr, Jpn.J. Appl.Phys. 32, 8 (1993). improved background removal methods [4,5]. However, [3] A. I. Frenkel, E. A. Stern, M. Qian, and M. Newville, Phys. Rev.B 48, (1993). the atomic background µ and the XAFS χ are tightly 0 [4] G. G. Li, F. Bridges and G. S. Brown, Phys. Rev. Lett. linked by the definition χ = (µ−µ )/µ , so the back- 0 0 68, 1609 (1992). grounds obtained for theoretical and experimental stan- [5] M. Newville, P. L¯ıvin¸ˇs, Y. Yacoby,J. J. Rehr,and E. A. dards may differ. Thus an understanding of AXAFS is Stern, Phys.Rev.B 47, 14126 (1993). essential to obtain experimental backgrounds. This dif- [6] J. J. Rehr, R. C. Albers and S. I. Zabinsky, Phys. Rev. ferencealsoaffectsXAFSanalysis;ifonetriestoisolatea Lett. 69, 3397 (1992); J. J. Rehr, J. Mustre de Leon, S. Ce-Ostandardwithouttakingitsoscillatorybackground I. Zabinsky and R. C. Albers, J. Am. Chem. Soc. 113, into account, one cannot obtain a good fit to the first 5135 (1991). Pr-O peak in PBCO. The inclusion of extra-atomic con- [7] J. J. Rehr and R. C. Albers, Phys. Rev. B 41, 8139 tributions in the atomic background may at first seem (1990). arbitrary. For example, the XAFS could be defined with [8] R.G.Newton,ScatteringTheoryofWavesandParticles, 2nd Edition (Springer, NY1982). respecttothebareatomicbackground,whichisindepen- [9] S. I. Zabinsky, A. Ankudinov, J. J. Rehr, and R. C. Al- dent of the environment. However such a definition is bers (unpublished). problematical and inconsistent with multiple-scattering [10] C. H. Booth, F. Bridges, J. B. Boyce, T. Claeson, Z. X. theory based on independent scattering sites; also be- Zhao,andP.Cervantes,tobepublishedinPhys.Rev.B. cause the exchange interaction is not additive, it is not [11] P.A.LeeandJ.B.Pendry,Phys.Rev.B11,2795(1975). possible to construct the scattering potential by super- posing free atomic potentials. For the materials discussed in this paper, the AXAFS FIG. 1. Top curves: Ce K-edge absorption µ(E) (dotted) is quite large, and is the dominant contribution to the and µ0(E) (solid) from CeO2 vs. energy E above the Ce background fine structure, exceeding multi-electron ef- K-edge (40441 eV). All spectra in this paper have their step fects in magnitude. Ab initio calculations ofthe AXAFS height normalized to unity, and shifted as displayed. The oscillatory structure above the edge (dotted) is the XAFS. agree reasonably well with these observations and with Thesolidcurveistheexperimentallyobtained“atomicback- the simplifiedmodelintroducedhere. The sizeandchar- ground” absorption µ (E) (see text). Note the sharp dip in 0 acter of these backgroundfeatures, particularly their in- thisbackgroundat≃115eV.Middlecurves: residuefunction terferencewiththe firstcoordinationshellpeak,indicate and fit for Ba K-edge data from BaO. The residue functions that accurate fits to XAFS data must take them into are the difference between the data and the fit in E-space, account. AXAFS is also interesting in its own right, be- i.e., µres(E) = µ(E)/(χfit(E)+1). Bottom curves: simu- cause it depends critically on the scattering potential in lated background (solid) and extracted background (dotted) the outer part of the absorbing atom. Thus, it provides as a test of our extraction method. a new and useful probe of chemical effects, the electron self-energy, core-hole effects, and other contributions to FIG.2. Experimental (dotted lines) and theoretical (solid the embedded atom potentials. lines)backgroundabsorptions, µ (E),fortheBa,Ce,andPr We thank J.B.Boyce,G.G.Li,P.L¯ıvin¸ˇs,M.Newville, 0 K-edges of BaO, CeO , and PBCO. ∆E is the energy above 2 B. Ravel, E.A. Stern, and Y. Yacoby for comments and threshold, i.e., 37,444, 40,441, and 41,991 eV for the Ba, Ce discussionsconcerningbackgroundremovalmethodsand and Pr edges, respectively. Both the experimental and the T.Claesonforassistanceincollectingthedata. Oneofus theoretical backgrounds have been adjusted to fit the Vic- (JJR)alsothankstheInstitutfu¨rExperimentalphysikof toreenformulawitha4th-orderpolynomial. Thecalculations the FreieUniversitätBerlinforhospitality,wherepartof are currently limited by discontinuities at Rmt which can ef- this work was completed. SSRL is operated by the U.S fect the AXAFS amplitude. The BaO calculation has an ad- Department of Energy, Division of Chemical Sciences, ditionalthresholdenergyshiftof+20eV.Arrowsindicatethe and by the NIH, Biomedical Resource Technology Pro- positions of Z+1 excitation thresholds. gram,DivisionofResearchResources. Thedatawascol- lected under University PRT time. This work was sup- ported in part by DOE Grant DE-FG06-ER45415 (JJR and SIZ), and by NSF Grant DMR-92-05204 (CHB and FB). 4 1.1 1.05 1 0.95 0.9 0 100 200 300 400 500 1 0.98 0.96 0.94 0.92 0.9 0.88 0 100 200 300 400 500