XANES spectroscopy. Andrej Mihelič Mentor: prof. dr. Alojz Kodre. December 2002

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1 XANES spectroscopy Andrej Mhelč Mentor: prof. dr. Alojz Kodre December 2002 Abstract The physcs of x-ray absorpton near-edge structure (XANES) spectra s descrbed brefly usng the multple-scatterng (MS) approach. Ths method s used for calculatons of the scatterng processes n small clusters of atoms. Further, the dfferences between XANES and x-ray absorpton fne structure (EXAFS) regmes are ponted out. A few llustratve examples of Zn XANES spectra are shown and the reasons for dfferences among them are dscussed. At the end, a comparson s made between the calculated and measured spectra of ZnO, and a comment s made on the falures of the muffn-tn approxmaton. 1 Conventons Rydberg atomc unts ( = e 0 = m e = 1, c = 2/α = ) are used throughout the followng two sectons: dstances are expressed as multples of Bohr s radus a 0 and energes are measured n rydbergs, a 0 = 4πɛ 0 2 e 2 0 m e = m, 1 ry = 2 2m e a 2 0 = ev. The knetc energy operator thus obtans the form H 0 = 2 and a free electron s wave number k s n smple relaton wth ts (knetc) energy E: k = E. Furthermore, we wll use L to denote a par L = (l, m), where l and m are the orbtal and magnetc quantum numbers, respectvely. Fnally, vectors and matrces (or, more generally, tensors) wll be wrtten n bold, for example: r, G 0, etc. 2 Introducton Especally n the last decades, x-ray absorpton spectrometry has ganed much of ts reputaton of beng a powerful analytcal and research tool, mostly due to the use of synchrotron radaton sources. In ths knd of spectrometry, nteractons of photons wth matter are studed by measurng the photoabsorpton cross-secton. For ncdent lght wth polarsaton ˆɛ and frequency ω the cross secton s gven by the Ferm s golden rule [1, 2] σ = 4π2 ω f ˆɛ p e k r 2 δ(e E f + ω) f f f D 2 δ(e E f + ω), (1) where, f, E, and E f denote the ntal and fnal states and ther energes; k stands for the wave vector of the ncdent beam. Only the dpole contrbuton to the total cross-secton has been 1

2 consdered n Eq. (1). By exctng an atom usng an x-ray source, the electron confguraton of the atom s changed; one (usually a core-shell) electron or more electrons populate unoccuped bound or contnuum states. The success of ths spectrometry les n the fact that the photoelectrons act as very senstve probes that can feel the charge dstrbuton and the arrangement of the neghbourng atoms around the absorbng atom, or, wth other words, they can feel the arrangement of the neghbourng atoms (chemcal envronment). When the photon energy s large enough, the photoeffect n one of the core shells can occur. Ths results n the steplke shape of the absorpton spectrum: the ncreased photoabsorpton cross-secton due to the knockng-out of an electron s called absorpton edge (Fg. 1). We shall see n the next secton that one way of understandng ths exctaton process n a bound atom s descrbng t by means of multple scatterng: the photoelectron s wave s scattered on atoms surroundng the absorbng atom (Fg. 2). The cross-secton of a bound atom therefore depends on the postons and types of the neghbours and s dfferent from that of an solated atom XANES EXAFS µd E [ev] Fgure 1: The K-shell absorpton edge of cobalt. The XANES and EXAFS regons are marked wth arrows. Fgure 2: The wave functon of the photoelectron s scattered on the neghbourng atoms. When the photon energy ω s low, the photoelectrons can populate ether unoccuped bound states or low-lyng contnuum states. Ths part of the spectrum (Fg. 1) s called the XANES regon; the acronym stands for X-ray Absorpton Near-Edge Structure. On ts hgh-energy end, XANES extends up to Extended X-ray Absorpton Fne Structure [3, 4, 5, 6, 7], EXAFS n short, where the photoelectron knetc energy s hgh. The lmtng energy that dvdes XANES from EXAFS s by no means exactly defned snce the transton from the one regme to the other s smooth: as ω s ncreased, the transton starts happenng gradually. As a rule of thumb, near-edge structure ends approxmately where the electron wavelength equals the dstance from the absorbng atom to ts nearest neghbours [8], whch usually means about ev above the edge. In the XANES regme, the electron s knetc energy s small and the scatterng on the neghbourng atoms tends to be strong for ths reason, whle the effect of the scatterers becomes smaller at hgher energes; n EXAFS regon, the photoelectrons are only weakly scattered. The underlyng theory s the same n both cases, but complcatons arse at low energes because the usual Born-lke scatterng expanson s dvergent. 2

3 3 The theory of XANES 3.1 Multple scatterng formalsm In ths secton we dscuss scatterng from potentals v of several atoms or ons, each of whch makes a non-zero contrbuton only wthn a sphercally non-overlappng scatterng regon of fnte radus (Fg. 3), V = v. (2) Outsde the sphercal regons n the ntersttal regon the potental s set to zero. Ths Fgure 3: The muffn-tn potental conssts of non-overlappng sphercal regons. approxmaton s often used when descrbng condensed materals and s known as the muffntn approxmaton. The scatterng parameters of each of the scatterers,. e., the scatterng ampltudes and phase shfts, are determned separately for each scatterer and are therefore pure atomc quanttes. Propagaton of a photoelectron n such muffn-tn potental V s descrbed by the Hamltonan H = H 0 + V, (3) where H 0 s the knetc-energy operator. One s seekng the statonary soluton ψ wth the energy E, H ψ = E ψ, (4) whch can be twsted around a lttle bt to yeld (E H 0 ) ψ = V ψ. (5) Wrtng Eq. (3) n ths way suggests treatng the term V ψ as a source and solvng the nhomogeneous Eq. (5) by means of the Green s functons: we pretend that the rght-hand sde of Eq. (5) represents the nhomogenety. To be more specfc, f r φ s the soluton of the homogeneous part of Eq. (5) (E H 0 ) r φ = 0 (6) and the free-electron Green s functon G 0 s defned wth relaton (E H 0 )G 0 (r, r ; E) ( 2 + k 2 )G 0 (r, r ; E) = δ(r r ), (7) then the general soluton of Eq. (5) s gven as a sum of the soluton of the homogeneous equaton and the partcular soluton [9, 10] r ψ = r φ + d 3 r G 0 (r, r ; E) r V ψ. (8) 3

4 If the photoelectron s only weakly scattered by the potental V, the soluton r ψ s close to the free-electron soluton r φ. Furthermore, when V s dentcally zero everywhere, the exact equalty r ψ = r φ holds, as expected. It s apparent from Eqs. (4) and (6) that we are only lookng for solutons of the total Hamltonan ψ that have the same energy as φ whch means we are only consderng elastc scatterng. The formal soluton of the operator equaton (5) s gven by the Lppman-Schwnger equaton [9, 10, 11], 1 ψ = φ + V ψ. (9) E H 0 ± η Because of ts sngularty, the operator 1/(E H 0 ) has been modfed by an magnary term n the denomnator, ±η, where η s nfntesmally small and postve, η 0 +. The form of Eq. (9) remnds us mmedately of Eq. (8). In fact, the comparson of both equatons reveals that G ± 0 (r, 1 r ; E) = r E H 0 ± η r and G ± 0 = 1 E H 0 ± η, (10) whch becomes clear after nsertng a complete system d 3 r r r nto Eq. (9). So, what s the meanng of the superscrpt ±? It can be shown (see ref. [10]) that r G ± 0 r = G ± 0 (r, r ; E) = 1 4π exp(±k r r ) r r, (11) where k = E. Obvously, G + 0 and G 0 descrbe how outgong and ncomng sphercal waves propagate n free space, respectvely. Through the rest of the paper we shall use only the + form of the propagators so the superscrpt wll be omtted. The Lppman-Schwnger equaton can be solved formally f the transton operator T s ntroduced [9, 10, 11], If we defne the propagator G of the whole system as V ψ = T φ. (12) G(E) = (E H 0 V + η) 1 = ([G 0 ] 1 V ) 1, (13) the followng mportant relatonshps can be deduced from Eq. (12) [9, 10]: G = G 0 + G 0 T G 0 = G 0 + G 0 V G (14) and T = V + V G 0 T = V + V GV. (15) Rememberng that V = v and that the potentals are non-overlappng, we can wrte the transton operators of ndvdual atoms as t = v + v G 0 t. (16) The physcal meanng of the operators T and t s not obvous yet, but we wll try to explan t by means of the weak-scatterng expansons. When the scatterng s weak,. e., the photoelectron s path s not strongly affected by the potentals v, equatons (14) (16) can be wrtten 4

5 as seres n v and t (Dyson s expanson [9]): G = G 0 + G 0 v G 0 + G 0 v 1 G 0 v G (17), 1 = G 0 + G 0 t G 0 + G 0 t 1 G 0 t G T = t + t 1 G 0 t + t 2 G 0 t 1 G 0 t +... (18) t = v + v G 0 v + v G 0 v G 0 v +... (19) Let us frst try to understand those last equatons. Propagaton of a partcle as descrbed by v t t 2 1 t G 0 t 1 v v = Σ G 0 t v 1 T Σ Σ (a) (b) 1 Σ 1 Σ Σ 2 1 G 0 t Fgure 4: (a) Sngle scatterng (left), double scatterng from dfferent atoms (mddle), and double scatterng from the same atom (rght). (b) Graphcal representaton of the T -operator. G n Eq. (17) conssts of a seres of terms whch represent the order of the scatterng process. The frst term descrbes how an unscattered wave propagates, whle the second term descrbes free propagaton G 0 to the th atom where the wave s scattered once by ts potental v and then heads forward agan (Fg. 4a, left). In the same way, the thrd term scatters the ncdent wave two tmes (Fg. 4a, mddle and rght), etc. Smlarly, the meanng of the T -operator can be nferred from Eq. (18) and wth the help of Fg. 4b: the propagatng wave can be scattered on the th atom, on the th atom and on the 1 th atom, etc. Note also that accordng to Eq. (18) successve scatterng from the same centre cannot occur. Ths s because t already accounts for scatterng from v to arbtrary order (Fg. 5). The operator t s made up of terms descrbng v v t = v + G v G 0 G0 v v Fgure 5: Graphcal representaton of t. scatterng on the potental v alone: whle the electron-wave propagates n the potental of the th atom, t can, n prncple, be scattered nfntely many tmes. Though n the case of strong scatterng the expansons (17) (19) do not converge when appled to state vectors descrbng the photoelectron wave, the physcal nterpretaton remans the same. In ths case, however because of the large wavelength of the photoelectron wth respect to the nearest neghbour dstance the whole scatterng process cannot be decomposed to separate scatterng events from dfferent atoms or groups of atoms. Despte descrptve and elegant formulaton wth the operator algebra, one cannot avod computng the matrx elements belongng to dfferent operators when performng real calculatons. Nevertheless, the operators t and G can be cast nto matrx form qute ncely usng the 5

6 angular-momentum representaton [2, 12, 13] G jj LL = L, j; k G 0 L, j ; k (1 δ jj ) (20) t jj LL = L, j; k t L, j ; k δ j δ j. (21) Indces j, j and are used to denote the atoms and L and L are the angular momenta (l, m) and (l, m ), respectvely; k = E s the wave number of the ncdent wave. 1 For example, G jj LL descrbes how the partal wave wth angular momentum L transforms to L when t propagates from j th to jth atom. The factor (1 δ jj ) has been added n Eq. (20) to exclude consecutve scatterngs from the same scatterer. Wth defntons (20) and (21), the propagator matrx G can be wrtten as G = G 0 + G 0 TG 0 + G 0 TG 0 TG (22) where T = t and the multplcatons used are smply matrx multplcatons. expresson s a geometrc seres for 3.2 Even more about photoabsorpton cross-secton Ths last G = (1 G 0 T) 1 G 0. (23) Armed wth the multple scatterng (MS) formalsm, we are ready to proceed to the next problem. Let H be the total Hamltonan H = H 0 + V and G(E) = 1/(E H + η) the Green s functon that descrbes the propagaton of the whole system. By nsertng an arbtrary complete set of egenstates, we obtan G(E) = f f E E f + η. (24) f The expresson for photoabsorpton cross-secton [Eq. (1)] can be smplfed further f we use Eq. (24) and the dentty (P s the Cauchy s prncpal value) lm ɛ 0 the cross-secton formula can be rewrtten as 1 ω ± ɛ = P 1 πδ(ω); (25) ω σ = f = f f D 2 δ(e E f + ω) D δ(e H + ω) f f δ(e H + ω)d, (26) whch gves us σ = 1 π D Im G(E + ω)d θ(e + ω µ). (27) Snce we are nterested n the absorpton of lght, the step functon θ(e +ω µ) has been added to assert that only exctatons to free states above the chemcal potental µ are consdered. 1 The r-representaton of the state vector L, ; k s r L, ; k = l j l (k r R ) Y l,m (r R ), where R s the poston of the th scatterer. Y l,m s the sphercal harmonc of angular momentum l wth projecton m and j l the sphercal Bessel functon of order l. 6

7 In some cases, the wave functons r and r f can be determned from electronc-structure calculatons and then used drectly to obtan σ through Eq. (1). For example, n band structure calculatons n crystals, the egenstates and egenvalues of the complete Hamltonan H can be obtaned usng ether Bloch and Wanner functons or by means of the tght-bndng (LCAO) methods [8, 14]. In that case, the state vectors and f represent the statonary solutons of electrons movng n a perodc potental. Conversely, the problem can be solved usng the MS theory. At frst we note that the complete set of egenstates f can be chosen to correspond to the atomc egenstates. 2 We magne that f other atoms are brought to the vcnty of the excted atom, the local electron densty around the absorbng atom s affected by the scatterng of the photoelectron s wave functon on the neghbourng potentals. The backscattered electron wave changes the partcle densty around the central atom. 3 To be precse: n almost exactly the same way as n Eq. (27) t can be shown that the fnal densty of states s equal to n(e) = f f δ(e H) f = 1 Im G(E). (28) π Snce, as can be seen from Eq. (14), the total Green s functon G conssts of the free propagator G 0 and the term descrbng the propagaton of the scattered wave, the total densty of states can also be dvded nto the atomc part, n 0, and the change due to the scatterng, n scat, n(e) = n 0 (E) + n scat (E). (29) It s now clear that the densty of fnal states s changed because of the presence of the scatterers. Thus, we are able to renterpret the behavour of our physcal system: the solutons can be understood ether as statonary solutons of the complete Hamltonan or as solutons that are changed by the multple scatterng from surroundng atoms. For nstance, n an N 2 molecule, the photoelectrons can get trapped to a vrtual bound state of the molecule ( shape-resonance ), as seen through the eyes of molecular physcs. Also, ths can be understood as a strong scatterng between two ntrogen atoms. So far, we have sad almost nothng about the strong scatterng whch occurs when the knetc energy of the knocked-out electron s small. The expanson (17), needed n the crosssecton formula [Eq. (27)], converges f the matrx norm GT s less than one [15], whch s usually true for the energes n the EXAFS regon. In the XANES regme, however, ths s not the case and the seres dverges. Despte that, the soluton can stll be found for a small cluster of atoms f Eq. (23) s used; matrx nverse must be calculated n that case. The formal procedure that led us to Eq. (23) conssted of dsassemblng an expresson to a seres and then assemblng t back to a compact form. Eq. (23) therefore holds for the case of strong scatterng as well [2]. The dmensons of the G and t matrces depend on the number of scatterers ncluded n the calculaton. The cluster sze must be relatvely small (a few hundred atoms at most [2]) f the sze of the matrces s to be manageable. But s ths restrcton to a small cluster really reasonable and does t gve any useful results? The answer s yes, fortunately. Solvng the problem for a small cluster of atoms can be used even for a system wth a macroscopc number of partcles: contrbutons from successve scatterng weakens because of the 1/r dependence of the scattered wave s ampltude. And most mportantly, the mean-free-path of the electron, λ(e), s fnte, so 2 In x-ray absorpton, the vector represents a core-level state whch s almost unaffected by the surroundng potental and s to be taken as an atomc one too. Moreover, the sngle-electron approxmaton s normally used, so that the kets and f represent the ntal and fnal one-electron states of the photoelectron. 3 For the detaled nterpretaton of the local densty of states see [9]. 7

8 that the electron can only probe nsde a regon of dameter of approxmately 2λ around the absorbng atom. 4 In the present-day calculatons, Matthess prescrpton s used to construct scatterng potentals [2, 8]: the potentals are sphercally averaged (t jj LL δ LL ) and the muffn-tn rad are determned self-consstently. At the end of ths secton let us emphasze that the mportance of MS approach was realzed qute early: n sold-state physcs, ths formalsm s known as the Korrnga-Kohn-Rostoker method [16]. 4 Example: Zn In the XANES regme, the photoelectrons populate unoccuped states above the chemcal potental µ. Snce because of the scattered wave the densty of the fnal states changes, and along wth t the charge densty, the shape of the absorpton edge of the same element s dfferent n dfferent chemcal envronments. Also, the electron bndng (. e., onsaton) energes change σ/σ 0 3 Zn vapour 2 Zn(C 2 H 5 ) 2 Zn metal 1 ZnO E [ev] Fgure 6: K-edge near-edge regons of Zn oxde, Zn metal, Zn-dethyl, and Zn vapour. spectra are shfted along the ordnate for clarty. The for the same reason, whch results n shftng of the absorpton edge towards lower or hgher energes. As an example, n Fg. 6, the K-edge absorpton spectra of znc oxde, znc metal, 4 Formally, the fnte electron mean-free-path can be accounted for by replacng η n Eq. (10) wth -Σ(E), where Σ(E) s the complex self-energy. See [8]. 8

9 dethyl znc, and znc monatomc vapour are shown. Zn vapour s an exceptonal case among the shown spectra. Whle ts shape cannot be explaned by means of the MS theory, t can be decomposed to contrbutons of separate exctaton channels [17, 18], the shape of whch s well known. It has therefore been added for completeness. At frst we note that the apparent postons of the edges are shfted one to the other. Moreover, the pre-edge resonance that s attrbuted to the 1s 4p transton n the case of the atomc spectrum [19] becomes broader and lower quasatomc resonance (an altered bound state) n Zn et 2, whereas n Zn metal and Zn oxde ths spectral feature cannot be dscerned. Also further on, the dfferences between ndvdual speces are apparent. An electron of low knetc energy s scattered strongly and possbly many tmes by the neghbourng atoms [MS to nfnte order s used to descrbe ths process see Eq. (23)]. In contrast wth sngle scatterng whch prevals n the hgh-energy regon, multple scatterng depends strongly on angles between scatterers and s thus very senstve to local envronment of the absorbng atom [8]: Fg. 6 demonstrates strong dependence of the shapes of the spectra on the atoms among whch Zn resdes. Wth the muffn-tn approxmaton and wth use of the Matthess prescrpton we mentoned n the prevous secton, a serous approxmaton s made. For example, f the potental n the ntersttal regon oscllates consderably around ts mean value, the error made by replacng the real potental wth a constant one could be enormous. The goal s therefore to have the ntersttal regon as small as possble [8] snce t s here where the crude approxmaton s made. In Fg. 7, the dfference between the measured and calculated ZnO spectra are shown. The 2.0 measured calculated 1.5 σ/σ E[eV] Fgure 7: The measured (dots) and calculated (lne) XANES spectra of znc oxde. The calculated spectrum was shfted to lower energes for approxmately 7 ev to match the measurement. general agreement of the two spectra s qute good but for the detaled quanttatve analyss, the muffn-tn approxmaton has to be mproved (or bypassed). A few successful solutons have been proposed, e. g., the use of the fnte-dfference method n the ntersttal regon [20]. Nevertheless, the shape of the XANES spectra seems to depend prmarly on the structure of matter and less on the detaled propertes of the potentals [8]. Furthermore, precse calculatons requre substantal computng power. For example, even wth the sphercally-symmetrzed muffn-tn 9

10 potentals (obtaned by Matthess prescrpton), the calculaton of the ZnO XANES spectrum from Fg. 7 for a cluster of about 180 atoms took more than 10 hours on a 350 MHz PC pentum II usng the FEFF 8.0 code [12]. 5 Concluson We have seen that the nformaton bured n the near-edge regon of absorpton spectra s understood qute well already: the XANES analyss can gve detals about the electronc structure and the local arrangement of atoms around the absorbng atom. XANES studes already seem to be a powerful and promsng tool for the determnaton of the local structure of varous chemcal samples. A very mportant part of ths semnar was certanly to llumnate the approach to the nearedge processes by the MS technque. Such local treatment of the scatterng nsde a small cluster of atoms can be used to explan the local chemcal envronment and local densty of states and s thus sutable for studyng amorphous materals, such as glasses or gels, as well as for crystallne matter, where other technques have also been developed. Further attempts are beng made to brng XANES analyss to a more quanttatve level. The frst steps n ths drecton are relaxng the muffn-tn shackles and the next ones may be even more sophstcated (e. g., takng care of correlaton effects between electrons to go beyond one-electron approxmaton). References [1] J. J. Sakura, Advanced Quantum Mechancs, Addson Wesley (1967) [2] A. L. Ankoudnov, Relatvstc Spn-Dependent X-ray Absorpton Theory, dssertaton, Unversty of Washngton (1996) [3] P. A. Lee, J. B. Pendry, Phys. Rev. B 11, 2795 (1975) [4] C. A. Ashley, S. Donach, Phys. Rev. B 11, 1279 (1975) [5] W. L. Schach, Phys. Rev. B 8, 4028 (1973) [6] J. J. Boland, S. E. Crane, J. D. Baldeschweler, J. Chem. Phys. 17, 142 (1982) [7] X-Ray Absorpton Prncples, Applcatons, Technques of EXAFS, SEXAFS and XANES, ed. by D. C. Konngsberger and R. Prns, John Wlley (1988) [8] P. J. Durham n ref. [7], chap. 2 and the references theren [9] P. Lloyd, P. V. Smth, Adv. Phys. 21, 69 (1975) [10] J. J. Sakura, Modern Quantum Mechancs, Rev. Ed., Addson Wesley (1994) [11] E. Merzbacher: Quantum Mechancs, John Wlley (1970) [12] A. L. Ankudnov, B. Ravel, J. J. Rehr, S. D. Conradson, Phys. Rev. B 58, 7565 (1998) [13] J. J. Rehr, R. C. Albers, Phys. Rev. B 41, 8139 (1990) [14] N. W. Ashcroft, N. D. Mermn, Sold state physcs, Saunders College (1976) 10

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