Sakura Pascarelli European Synchrotron Radiation Facility, Grenoble, France

Transcription

1

2 X-RAY ABSORPTION SPECTROSCOPY: FUNDAMENTALS AND SIMPLE MODEL OF EXAFS Sakura Pascarelli European Synchrotron Radiation Facility, Grenoble, France Part I: Fundamentals o X-ray Absorption Fine Structure: basic principles X-ray Absorption X-ray Absorption Fine Structure Simple Theoretical Description Derivation o EXAFS Equation Part II: Fundamentals o X-ray Absorption Fine Structure: data analysis EXAFS Analysis: near neighbor R, N, and atomic species XANES Analysis: ormal valence and coordination chemistry Part III: Examples o Applications Maor historical EXAFS breakthroughs Selection o recent results at the ESRF Page 1 l S. Pascarelli l HERCULES l 16

3 Part I Fundamentals o X-ray Absorption Fine Structure: basic principles Sakura Pascarelli European Synchrotron Radiation Facility, Grenoble, France Page l S. Pascarelli l HERCULES l 16

4 Basic Principles: X-ray Absorption X-ray Absorption Fine Structure Simple Theoretical Description Derivation o EXAFS Equation Page 3 l S. Pascarelli l HERCULES l 16

5 X-ray Absorption Page 4 l S. Pascarelli l HERCULES l 16

6 Main X-ray based techniques Two undamental X-ray-matter interactions: photoelectric absorption scattering (elastic, inelastic) Two amilies o experimental techniques: spectroscopy exchange o energy (electronic structure, local structure o matter) absorption (XAS, EXAFS, XANES,..) emission (XES, HERFD,..) inelastic scattering (IXS, RIXS, X-ray Raman, etc..) elastic diusion no exchange o energy (microscopic geometric structure) diraction or crystalline solids (XRD, GIXRD,.) scattering or amorphous solids, liquids (XRS, WAXS, SAXS, ) Page 5 l S. Pascarelli l HERCULES l 16

7 The Absorption Coeicient m I I t I = I exp[-mt] linear absorption coeicient mt = ln [ I / I ] Page 6 l S. Pascarelli l HERCULES l 16

8 The Absorption Coeicient m synchrotron source monochromator incident lux monitor transmitted lux monitor t polychromatic X-rays monochromatic X-rays I sample I 1. Measure I and I as a unction o E X. Calculate: m t = ln [I /I ] Page 7 l S. Pascarelli l HERCULES l 16

9 m/r [barns/atom] FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES The Absorption Coeicient m m depends strongly on X-ray energy E and atomic number Z, the density r and atomic mass A m r Z 4 A E 3 m has sudden umps (absorption edges) which occur at energies characteristic o the element. Page 8 l S. Pascarelli l HERCULES l 16

10 m [cm -1 ] FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES Germanium total absorption coeicient 1 1 photoelectric absorption E(eV) elastic scattering inelastic scattering Photoelectric absorption dominates the absorption coeicient in this energy range Page 9 l S. Pascarelli l HERCULES l 16

11 Photoelectric Absorption X-rays (light with wavelength.6 l 1 Å or energy 1 E kev) are absorbed by all matter through the photoelectric eect An X-ray is absorbed by an atom when the energy o the X-ray is transerred to a core-level electron (K, L, or M shell) which is eected rom the atom. The atom is let in an excited state with an empty electronic level (a core hole). Any excess energy rom the X-ray is given to the eected photoelectron. Page 1 l S. Pascarelli l HERCULES l 16

12 De-excitation: Fluorescence and Auger Eect When X-rays are absorbed by the photoelectric eect, the excited core-hole will relax back to a ground state o the atom. A higher level core electron drops into the core hole, and a luorescent X-ray or Auger electron is emitted. X-ray Fluorescence: An X-ray with energy = the dierence o the core-levels is emitted. Auger Eect: An electron is promoted to the continuum rom another core-level. X-ray luorescence and Auger emission occur at discrete energies characteristic o the absorbing atom, and can be used to identiy the absorbing atom. Page 11 l S. Pascarelli l HERCULES l 16

13 synchrotron source XAS measurements monochromator I F I I sample XAS measures the energy dependence o the X-ray absorption coeicient μ(e) at and above the absorption edge o a selected element. μ(e) can be measured two ways: Transmission: The absorption is measured directly by measuring what is transmitted through the sample: μ (E)t I = I e μ(e) t = ln (I/I ) Fluorescence: The re-illing the deep core hole is detected. Typically the luorescent X-ray is measured: μ(e) ~ I F / I Page 1 l S. Pascarelli l HERCULES l 16

14 X-ray Absorption Fine Structure Page 13 l S. Pascarelli l HERCULES l 16

15 Absorption coeicient m Absorption FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES What is XAFS? X-ray Absorption Fine Structure: oscillatory variation o the X-ray absorption as a unction o photon energy beyond an absorption edge As K-edge in InAsP E (ev) E(eV) Proximity o neighboring atoms strongly modulates the absorption coeicient Page 14 l S. Pascarelli l HERCULES l 16

16 Absorption coeicient m Absorption Mean Free Path (A) FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES EXAFS and XANES XAFS is also reerred to as X-ray Absorption Spectroscopy (XAS) and is broken into regimes: XANES EXAFS X-ray Absorption Near-Edge Spectroscopy Extended X-ray Absorption Fine-Structure which contain related, but slightly dierent inormation about an element s local coordination and chemical state. XANES : transitions to unilled bound states, nearly bound states, continuum - low energy photoelectrons EXAFS: ~ 5 1 ev rom edge, transitions to continuum high energy photoelectrons Energy o photoelectron (ev) Page 16 l S. Pascarelli l HERCULES l E E(eV)

17 Absorption coeicient m Absorption Mean Free Path (A) FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES EXAFS and XANES 1. Approximations can be used to interpret EXAFS, that are not valid or XANES. Cannot be analized in the same way 3. Dierent inormation content. XANES: local site symmetry, charge state, orbital occupancy EXAFS: local structure (bond distance, number, type o neighbors.) Energy o photoelectron (ev) Page 16 l S. Pascarelli l HERCULES l E E(eV)

18 EXAFS qualitatively Absorption coeicient = probability o photon absorption ~ probability o electron presence at origin = amount o wave at origin isolated atom condensed matter E kin e - e - E l R E The kinetic energy o the eected photoelectron E kin is: l = p/k p k E kin E E m m Page 18 l S. Pascarelli l HERCULES l 16

19 Where do the oscillations come rom? Due to a quantum eect, the autointererence o photoelectron wave modiies the absorption coeicient value: 1. As E is scanned above E, E kin is varied, and consequently k and l. p k E kin E E m m. The outgoing and backscattered parts o the wave interere either constructively or destructively, depending on the ratio between l and R. 3. It is the intererence between outgoing and incoming waves that gives rise to the sinusoidal variation o m(e) l R e - requency ~ distance rom neighbors amplitude ~ number and type o neighbors Page 19 l S. Pascarelli l HERCULES l 16

20 l = R/3 l = R/3.5 l = R/4 photoelectron energy increases wavelength l decreases The probability o absorption oscillates due to constructive and destructive intererence Page l S. Pascarelli l HERCULES l 16

21 m (E) Absorption FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES The EXAFS signal c We re interested in the energy dependent oscillations in μ(e), as these will tell us something about the neighboring atoms, so we deine the EXAFS as: c E E m Dm m E E We subtract o the smooth bare atom background μ (E), and divide by the edge step Dμ (E ), to give the oscillations normalized to 1 absorption event m (E) Dm E (ev) E(eV) Page 1 l S. Pascarelli l HERCULES l 16

22 EXAFS: c (k) XAFS is an intererence eect, and depends on the wave-nature o the photoelectron. It s convenient to think o XAFS in terms o photoelectron wavenumber, k, rather than X-ray energy m k c (k) is oten shown weighted by k or k 3 to ampliy the oscillations at high-k: EE Page l S. Pascarelli l HERCULES l 16

23 Amplitude o FT FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES Qualitative picture o local coordination in R space The requencies contained in the EXAFS signal depend on the distance between the absorbing atom and the neighboring atoms (i.e. the length o the scattering path). A Fourier Transorm o the EXAFS signal provides a photoelectron scattering proile as a unction o the radial distance rom the absorber. R 1 R R 3 R(Å) Page 3 l S. Pascarelli l HERCULES l 16

24 Amplitude o FT FUNDAMENTALS OF X-RAY ABSORPTION FINE STRUCTURE: BASIC PRINCIPLES Quantitative structural determination Structural determinations depend on the easibility o resolving the data into individual waves corresponding to the dierent types o neighbors (SS) and bonding conigurations (MS) around the absorbing atom. absorber As atom As In As As As P InAs x P 1-x As Page 4 R(A) l S. Pascarelli l HERCULES l 16 In As As In As P As In In In

25 c: sum o damped waves c (k) is the sum o contributions c (k) rom backscattered wavelets: c c k c k Each c (k) can be approximated by a damped sine wave o the type: k A ksin k Path expansion implies photoel w/short mean ree path true only or EXAFS The larger the number o neighbors, the larger the signal N k e k k R k The stronger the scattering amplitude, the larger the signal Damping o the amplitude at large k, due to static and thermal disorder Each shell contributes a sinusoidal signal which oscillates more rapidly the larger the distance Page 5 l S. Pascarelli l HERCULES l 16

26 Frequencies: Single and Multiple Scattering paths The sum over paths in the EXAFS equation includes many shells o atoms (1 st neighbor, nd neighbor, 3 rd neighbor,... ), but can also include multiple scattering paths, in which the photoelectron scatters rom more than one atom beore returning to the central atom. SS g (r) = R 1 MS g (r) = 4 R 1 R1 RR 3 3 MS g 3 (r) = R 1 + R + R 3 R 1 R 1 R MS g 3 (r) = R 1 + R 3 EXAFS can give inormation on the n-body distribution unctions g n (r). Page 6 l S. Pascarelli l HERCULES l 16

27 Amplitudes A k ~ N shape o the envelope o each wave indicative o nature o backscatterer atom: k e k absorber As atom As In..1 As P InAs x P 1-x As As.4. As.4. In Page l S. Pascarelli l HERCULES l k(a -1 ) k(a -1 )

28 Scattering Amplitude and Phase-Shit: (k) and (K) The scattering amplitude (k) and phase-shit (k) depend on atomic number. The scattering amplitude (k) peaks at dierent k values and extends to higher-k or heavier elements. For very heavy elements, there is structure in (k). The phase shit (k) shows sharp changes or very heavy elements. These scattering unctions can be accurately calculated (i.e. with the programs FEFF, GNXAS, etc.), and used in the EXAFS modeling. Z can usually be determined to within 5 or so. Fe and O can be distinguished, but Fe and Mn cannot be. Page 8 l S. Pascarelli l HERCULES l 16

29 Calculating (k) and (k) These days, we can calculate (k) and (k) easily using dierent sotware codes. These programs take as input: 1. a list o atomic x,y,z coordinates or a physical structure. a selected central atom The result is a set o iles containing the (k), and (k) or a particular scattering shell or scattering path or that cluster o atoms. Many analysis programs use these iles directly to model EXAFS data. A structure that is close to the expected structure can be used to generate a model, and used in the analysis programs to reine distances and coordination numbers. Page 9 l S. Pascarelli l HERCULES l 16

30 XAFS vs Diraction Methods Diraction Methods (X-rays, Neutrons) Crystalline materials with long-range ordering -> 3D picture o atomic coordinates Materials with only short-range order (amorphous solid, liquid, or solution) -> 1D radial distribution unction containing interatomic distances due to all atomic pairs in the sample. XAFS 1D radial distribution unction (centered at the absorber) Element selectivity Higher sensitivity to local distortions (i.e. within the unit cell) Charge state sensitivity (XANES) Structural inormation on the environment o each type o atom: distance, number, kind, static and thermal disorder 3-body correlations Investigation o matter in the solid (crystalline or amorphous), liquid, solution or gaseous state with same degree o accuracy Page 3 l S. Pascarelli l HERCULES l 16

31 EXAFS: typical applications Element selectivity Local structure sensitivity Local structure in non-crystalline matter Local environment o an atomic impurity in a matrix o dierent atomic species Study o systems whose local properties dier rom the average properties Detection o very small distortions o local structure Page 31 l S. Pascarelli l HERCULES l 16

32 Simple Theoretical Description Page 3 l S. Pascarelli l HERCULES l 16

33 Kr gas Rh metal Page 33 l S. Pascarelli l HERCULES l 16

34 photoelectron core hole continuum 1s electron 1 ˆ in principle, all electrons are involved multi body process Absorption coeicient 3 4 continuum single electron 1 1 ˆ i N i N r 1 1 N N i S sudden ˆ i r S slide 53 slide 54 l S. Pascarelli l HERCULES l 16 Page 34 ˆ ) ( i H I m slides 48-5 I r A H ˆ r A A ˆ ˆ slide 51 dipole ˆ ˆ i r

35 Absorption coeicient photoelectron ˆ continuum s electron continuum core hole i 1s Ei p E E i m( ) S r ˆ ˆ i : photon polarization : electron position i > relatively easy ground state o atom; i.e. 1s e - waveunction Approximations dipole + single electron + sudden > very complicated inal state strongly inluenced by environment r Page 35 l S. Pascarelli l HERCULES l 16

36 m( ) ˆ r i matrix elements actor into spin, radial and angular parts By looking at the non-zero matrix elements we get the dipole selection rules For 1-electron transitions: where q is the X-ray angular momentum edge initial state i inal state K, L 1 s (l=) p (l=1) L, L 3 p (l=1) s (l=), d (l=) Page 36 l S. Pascarelli l HERCULES l 16

37 Isolated atom: atomic absorption coeicient e - photoelectron ree to travel away undisturbed h outgoing spherical wave originating rom the absorbing atom m r ˆ i m ˆ r i d r r r * overlap integral o initial and inal state waveunctions: monotonically decreases as unction o E Page 37 l S. Pascarelli l HERCULES l 16

38 sum o the outgoing and all the incoming waves, one per each neighboring atom. * ˆ i r m * ˆ r r r r r d i m * * * * ˆ ˆ ˆ Re ˆ r r r d r r r r r r r d r r r r d r i i i i m EXAFS e - h a ib a ib a Non-isolated atom Page 38 l S. Pascarelli l HERCULES l 16 m ()

39 c k Re mm 1 c c: ractional change in m introduced by the neighbors d r * * r ˆ r r r ˆ r r i i * d r r ˆ r r Origin o EXAFS i (1) Intererence between outgoing waveunction and backscattered wavelets Dominant contribution to integral comes rom spatial region close to absorber atom nucleus, where the core orbital waveunction i. The region where i represents simultaneously the source and the detector or the photoelectron that probes the local structure around the absorber atom Page 39 l S. Pascarelli l HERCULES l 16

40 The EXAFS equation To model the EXAFS, we use the EXAFS Equation c k ~ S k R / l k k e e sin [kr ] N k kr where (k) and (k) are photoelectron scattering properties o the neighboring atom. (the sum is over shells o similar neighboring atoms) I we know (k) and (k), we can determine: R distance to neighboring atom. N coordination number o neighboring atom. mean-square disorder o neighbor distance. The scattering amplitude (k) and phase-shit (k) depend on atomic number Z o the scattering atom, so we can also determine the species o the neighboring atom. Page 4 l S. Pascarelli l HERCULES l 16

41 Derivation o the EXAFS equation Page 41 l S. Pascarelli l HERCULES l 16

42 The EXAFS equation: simple description With spherical wave e ikr /kr or the propagating photoelectron, and a scattering atom at a distance r = R, we get: D ~ Region III: amplitude o backscattering on B e ikr kr eic (k) e is(k) e ikr kr eic I II III A R B Region I: amplitude o outgoing wave Region II: amplitude o wave arriving on B Region II: amplitude o ingoing wave, backscattered rom B where the neighboring atom gives the amplitude (k) and phase-shit s (k) to the scattered photoelectron. Substituting into equation (1) and ater some math we get (or 1 scattering atom): c k Page 4 l S. Pascarelli l HERCULES l 16 k S sin [kr k] kr ~ k s k k c

43 For N scattering atoms, distributed around average distance R with a thermal and static disorder o (mean square disorder in R*), we have: c k ~ k k S N e sin [kr k] kr A real system will have neighboring atoms at dierent distances and o dierent types. We add all these contributions to get a version o the EXAFS equation: c Development o the EXAFS equation k ~ k k sin [ S N e kr k] kr To obtain this ormula we used a spherical wave or the photoelectron: * EXAFS takes place on a time scale much shorter than that o atomic motion, so the measurement serves as an instantaneous snapshot o the atomic coniguration e ikr kr Page 43 l S. Pascarelli l HERCULES l 16

44 But the photoelectron can also scatter inelastically*, and may not be able to get back to the absorbing atom. Also: the core-hole has a inite lietime**, limiting how ar the photoelectron can go. Using a damped wave-unction: ikr e kr r / l( k ) (including core hole lietime), the EXAFS equation becomes: c k ~ The photoelectron mean ree path The mean ree path l depends on k. (or the EXAFS k range, l ~ 1- Å) S The l and R - terms make EXAFS a local atomic probe. e where l(k) is the photoelectron s mean ree path k R / l k k e e sin [kr ] N k kr * Electrons that have suered inelastic losses will not have the proper wave vector to contribute to the intererence process. ** the photoelectron and core hole exist simultaneously Page 44 l S. Pascarelli l HERCULES l 16

45 S : Amplitude Reduction Term The amplitude reduction term is due to the relaxation o all the other electrons in the absorbing atom to the hole in the core level (slides 53-54): S N 1 where N1 accounts or the relaxation o the other N-1 electrons relative to these electrons in the unexcited atom: N-1. Typically S is taken as a constant:.7 < S < 1. which is ound or a given central atom, and simply multiplies the XAFS c. Note that S is completely correlated with N. This, and other experimental and theoretical issues, make EXAFS amplitudes (and thereore N) less precise than EXAFS phases (and thereore R). Usually S is ound rom a standard (data rom a sample with well-known structure) and applied to a set o unknowns as a scale actor. N 1 i Page 45 l S. Pascarelli l HERCULES l 16

46 Page 46 l S. Pascarelli l HERCULES l 16

47 How to calculate m energy density u carried by X-ray beam is: linear absorption coeicient m measures the energy density reduction due to the interaction with the system o atoms: u o E o A 1 du m u dx m A du dx m m A n A d dx d dx n ph ph m A n W i n = atomic density W i = transition probability Page 47 l S. Pascarelli l HERCULES l 16

48 X-ray Absorption I. Lets consider the interaction between: Monochromatic X-ray beam ( = pn + monoatomic sample EM ield (classic) + atom (quantistic) (semi-classical description) II. m ~ m photoelectric absorption or 1 < E < 5 kev III. Qualitatively, interaction process is: continuum or ree energy level s electron 1 core hole i >, E i >, E = E i + Page 48 l S. Pascarelli l HERCULES l 16

49 Transition probability: Golden Rule m depends on: atomic density n transition probability W i o atom rom i > to > m n A W i (1) time-dependent perturbation theory (power series o EM ield - atom interaction potential) The interaction is in general WEAK: can limit series to 1st order: Golden Rule W i p Hˆ r E () I i Ĥ I EM ield - atom interaction hamiltonian operator Hˆ Matrix element o H I between initial and inal state I E i r Density o inal states, compatible with energy conservation: E = E i + Page 49 l S. Pascarelli l HERCULES l 16

50 the interaction hamiltonian or photoelectric absorption (see Appendix 1) is (to 1 st order): Hˆ e i m A r I (3) the transition probability or photoelectric absorption o a monochromatic, polarized and collimated photon beam is [(3) into ()]: W i pe ik r A e ˆ m i r E (4) Page 5 l S. Pascarelli l HERCULES l 16

51 l S. Pascarelli l HERCULES l 16 urther simpliication: transition probability in dipole approximation: alternative and equivalent expression (see Appendix ): inally one gets [(5) into (1)]: 1! 1 r k i r k r k i e i i E A m e W r p ˆ i i E r A e W r p ˆ (5) (6) 1 r k i n e p m i E r r ˆ Dipole approximation Page 51

52 i i > and > are known (i waveunctions and energies can be calculated): 1) calculate W i ) calculate m in practice, one is interested in inverse process: 1) measure m ) extract EXAFS p e n 3) obtain inormation on local structure through > but, to obtain structural ino, one still needs to calculate i > and > or at least be able to express their structural properties in parametric orm m ˆ r re i i > relatively easy ground state o atom > in general very complicated in principle, all electrons are involved (multi body process, inal state strongly inluenced by environment) Page 5 l S. Pascarelli l HERCULES l 16

53 Single electron approximation large part o m due to elastic transitions: only 1 electron out o N modiies its state: leaves its deep core level all other N-1 passive electrons relax their orbitals to adapt to the new potential created by presence o core hole remaining part o m due to inelastic transitions: primary excitation o core electron provokes successive excitations o other (external) electrons (shake up, shake o processes) excess energy distributed among all excited electrons N 1 i where m m m m el inel N 1 N 1 ˆ r r el i i Slater determinant o passive electrons waveunctions, r, Waveunction, position vector, inal energy o active electron Page 53 l S. Pascarelli l HERCULES l 16

54 Sudden approximation and overlap actor I photoelectron energy is suiciently high (E > ew 1 ev above edge) time to exit atom << relaxation time o passive electrons its state not inluenced by passive electrons relaxation m el S r r ˆ i (7) where N 1 N 1 S (S ~.7 1.) i Allows to reduce interpretation o EXAFS to the calculation o the inal state o ONLY the photoelectron Page 54 l S. Pascarelli l HERCULES l 16

55 m( ) ˆ i r dipole operator = x, y, z X-ray prop direction q = +1,, -1 polarization states (q photon angular momentum) electron position vector photon polarization vectors linear polarization circular polarization with k // z dipole operator in terms o spherical harmonics Page 55 l S. Pascarelli l HERCULES l 16 55

56 m( ) ˆ r i matrix elements actor into spin, radial and angular parts By looking at the non-zero matrix elements we get the dipole selection rules where q is the X-ray angular momentum Page 56 l S. Pascarelli l HERCULES l 16

57 APPENDIX 1 = Page 57 l S. Pascarelli l HERCULES l 16

58 APPENDIX 1 Page 58 l S. Pascarelli l HERCULES l 16

59 APPENDIX 1 Page 59 l S. Pascarelli l HERCULES l 16

60 APPENDIX 1 Page 6 l S. Pascarelli l HERCULES l 16

61 APPENDIX Page 61 l S. Pascarelli l HERCULES l 16

62 APPENDIX Page 6 l S. Pascarelli l HERCULES l 16

63 APPENDIX Page 63 l S. Pascarelli l HERCULES l 16

64 APPENDIX 3 The absorption coeicient m t di I x at/cm N dx t a cm /at I I t di I x N t a t dx x ln I(t) ln I( ) N a I(t) I( ) e e N a m t Page 64 l S. Pascarelli l HERCULES l 16

65 APPENDIX 3 I(t) I( ) e N a e m t m is related to the atomic cross section: m N Na cm gr 1 r cm a in general you ind tabulated the mass absorption coeicient m/r: m r or a generic sample P x Q y..: t a a A N a A cm at at at mole gr mole at mole gr mole cm cm gr 3 m r tot x m r P A P M y m r Q A Q M Page 65 l S. Pascarelli l HERCULES l 16

66 APPENDIX 3 Recipe or calculating t or transmission XAS I(t) I( ) e N a e m t 1. Total absorption above the edge must not be too high: m above edge t = 5 I / I ~.14.7 ideally m above edge t = -3. Contrast at edge must be as large as possible: [ m above edge - m below edge ] t >.1 ideally [ m above edge - m below edge ] t = 1 I absorber is very dilute, and matrix absorbs a lot, then this is not possible luorescence detection Page 66 l S. Pascarelli l HERCULES l 16