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Introduction to Inelastic x-ray scattering

Introduction to Inelastic x-ray scattering. Michael Krisch European Synchrotron Radiation Facility Grenoble, France krisch@esrf.fr. Outline of lecture. Introduction short overview of IXS and related techniques IXS from phonons why X-rays? complementarity X-rays <-> neutrons

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Introduction to Inelastic x-ray scattering

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  1. Introduction to Inelastic x-ray scattering Michael Krisch European Synchrotron Radiation Facility Grenoble, France krisch@esrf.fr

  2. Outline of lecture Introduction short overview of IXS and related techniques IXS from phonons why X-rays? complementarity X-rays <-> neutrons instrumental concepts & ID28 at the ESRF study of single crystal materials study of polycrystalline materials revival of thermal diffuse scattering Example I: plutonium Example II: supercritical fluids Other applications Conclusions

  3. W d E f , r k n f o t o h p r k , E q 2 i i photon r E Q , • Energy transfer: Ef - Ei = DE =1 meV – several keV • Momentum transfer: = 1 – 180 nm-1 Introduction I – scattering kinematics

  4. quasielastic phonon, magnons, orbitons Compton profile plasmon valence electron excitations core-electron excitation Introduction II - schematic IXS spectrum S. Galombosi, PhD thesis, Helsinki 2007

  5. Magnons Phonons Introduction III – overview 1 Lattice dynamics - elasticity - thermodynamics - phase stability - e--ph coupling Lecture today! Spin dynamics - magnon dispersions - exchange interactions Lecture on Friday by Marco Moretti Sala!

  6. 3/2¯  Ee  = 4.85 neV  = 141 ns ±3/2¯ 1/2¯ 1/2¯ 0 nuclear level scheme 57Fe delayed scattering prompt scattering Introduction IV – overview 2 Nuclear resonance Lecture by Sasha Chumakov on Tuesday!

  7. Detector Sample RRowland p Kin Kout Q Spherical crystal Introduction V – IXS instrumentation Energy analysis of scattered X-rays - DE/E = 10-4 – 10-8 - some solid angle Rowland circle crystal spectrometer p = Rcrystal·sinqB Rcrys = 2·RRowl

  8. Introduction VI – IXS at the ESRF ID28: Phonons ID32: soft X-ray IXS ID20: Electronic and magnetic excitations ID18: Nuclear resonance

  9. Relevance of phonon studies Phase stability Thermal Conductivity Sound velocities and elasticity Superconductivity

  10. Vibrational spectroscopy – a short history Infrared absorption - 1881 W. Abney and E. Festing, R. Phil. Trans. Roy. Soc. 172, 887 (1881) Brillouin light scattering - 1922 L. Brillouin, Ann. Phys. (Paris) 17, 88 (1922) Raman scattering – 1928 C. V. Raman and K. S. Krishnan, Nature 121, 501 (1928) TDS: Phonon dispersion in Al – 1948 P. Olmer, Acta Cryst. 1 (1948) 57 INS: Phonon dispersion in Al – 1955 B.N. Brockhouse and A.T. Stewart, Phys. Rev. 100, 756 (1955) IXS: Phonon dispersion in Be – 1987 B. Dorner, E. Burkel, Th. Illini and J. Peisl, Z. Phys. B – Cond. Matt. 69, 179 (1987) NIS: Phonon DOS in Fe – 1995 M. Seto, Y. Yoda, S. Kikuta, X.W. Zhang and M. Ando, Phys. Rev. Lett. 74, 3828 (1995)

  11. X-rays and phonons? “ When a crystal is irradiated with X-rays, the processes of photoelectric absorption and fluorescence are no doubt accompanied by absorption and emission of phonons. The energy changes involved are however so small compared with photon energies that information about the phonon spectrum of the crystal cannot be obtained in this way.” W. Cochran in Dynamics of atoms in crystals, (1973) “…In general the resolution of such minute photon frequency is so difficult that one can only measure the total scattered radiation of all frequencies, … As a result of these considerations x-ray scattering is a far less powerful probe of the phonon spectrum than neutron scattering. ” Ashcroft and Mermin in Solid State Physics, (1975) b – tin, J. Bouman et al., Physica 12, 353 (1946)

  12. Nobel Prize in Physics 1994: B. N. Brockhouse and C. G. Shull Press release by the Royal Swedish Academy of Sciences: “Neutrons are small magnets…… (that) can be used to study the relative orientations of the small atomic magnets. ….. the X-ray method has been powerless and in this field of application neutron diffraction has since assumed an entirely dominant position. It is hard to imagine modern research into magnetism without this aid.” X-rays and magnons?

  13. Burkel, Dorner and Peisl (1987) Thermal neutrons: Ei = 25 meV ki = 38.5 nm-1 DE/E = 0.01 – 0.1 Hard X-rays: Ei = 18 keV ki = 91.2 nm-1 DE/E  1x10-7 IXS versus INS Brockhouse (1955)

  14. Inelastic x-ray scattering from phonons HASYLAB DE = 55 meV 0.083 Hz B. Dorner, E. Burkel, Th. Illini, and J. Peisl; Z. Phys. B 69, 179 (1987)

  15. W d E f , r k n f o t o h p r k , E 2q i i photon r E Q , IXS scattering kinematics = - E E E i f momentum transfer is defined only by scattering angle r r = Q 2 k sin( ) q i

  16. IXS v = 7000 m/s v = 500 m/s INS DE IXS from phonons – the low Q regime No kinematic limitations: DE independent of Q Q = 4p/lsin(q) DE = Ei - Ef Disordered systems: Explore new Q-DE range • Interplay between structure and dynamics on  nm length scale • Relaxations on the picosecond time scale • Excess of the VDOS (Boson peak) • Nature of sound propagation and attenuation

  17. bcc Mo single crystal ruby helium Ø 45 m t=20 m Diamond anvil cell IXS from phonons – very small samples Small sample volumes: 10-4 – 10-5 mm3 • (New) materials in very small quantities • Very high pressures > 1Mbar • Study of surface phenomena

  18. kin E, Q kout IXS – dynamical structure factor Scattering function: Thermal factor: Dynamical structure factor:

  19. Comparison IXS - INS IXS • no correlation between momentum- and energy transfer • DE/E = 10-7 to 10-8 • Cross section ~ Z2 (for small Q) • Cross section is dominated by photoelectric absorption (~ l3Z4) • no incoherent scattering • small beams: 100 mm or smaller INS • strong correlation between momentum- and energy transfer • DE/E = 10-1 to 10-2 • Cross section ~ b2 • Weak absorption => multiple scattering • incoherent scattering contributions • large beams: several cm

  20. Efficiency of the IXS technique L = sample length/thickness, m = photoelectric absorption, Z = atomic number QD = Debye temperature, M = atomic mass

  21. IXS resolution function today • DE and Q-independent • Lorentzian shape • Visibility of modes. •Contrast between modes.

  22. APS IXS resolution function tomorrow Sub-meV IXS with sharp resolution DE = 0.89 (0.6) meV at Petra-III DE = 0.62 meV at APS Dedicated instrument at NSLS-II E = 9.1 keV DE = 0.1 – 1 meV Y.V. Shvydk’o et al, PRL 97, 235502 (2006), PRA 84, 053823 (2011)

  23. Monochromator : q Si(n,n,n ), = 89.98 º B q n=7 - 13 l tunable 1 DT DT DE DE 1/K at room temperature 1/K at room temperature Instrumentation for IXS IXS set-up on ID28 at ESRF Analyser : q Si(n,n,n ), = 89.98 º B n=7 - 13 l constant 2

  24. 9- analyser crystal spectrometer KB optics or Multilayer Mirror Beamline ID28 @ ESRF Spot size on sample: 270 x 60 mm2 -> 14 x 8 mm2 (H x V, FWHM)

  25. An untypical IXS scan Stokes peak: phonon creation energy loss Diamond; Q=(1.04,1.04,1.04) Anti-Stokes peak: phonon annihilation energy gain dscan monot 0.66 –0.66 132 80

  26. kin E, Q kout Phonon dispersion scheme Diamond Diamond (INS + theory): P. Pavone, PRB 1993

  27. S(Q,w)  (Q·e)2 ˆ Single crystal selection rules well-defined momentum transfer for given scattering geometry

  28. S(Q,w)  (Q·e)2 ˆ Single crystal selection rules well-defined momentum transfer for given scattering geometry

  29. Phonon dispersion and G-point phonons Brillouin light scattering Raman scattering

  30. Phonon dispersion and density of states • single crystals - triple axis: (very) time consuming - time of flight: not available for X-rays • polycrystalline materials - reasonably time efficient - limited information content

  31. VL~E/q IXS from polycrystalline materials - I At low Q (1. BZ) At high Q (50–80 nm-1) (Generalised) phonon density-of-states Orientation averaged longitudinal sound velocity How to get the full lattice dynamics?

  32. IXS from polycrystalline materials - II New methodology Lattice dynamics model + Orientation averaging Polycrystalline IXS data Q = 2 – 80 nm-1 least-squares refinement or direct comparison Validated full lattice dynamics Single crystal dispersion Elastic properties Thermodynamic properties I. Fischer, A. Bosak, and M. Krisch; Phys. Rev. B 79, 134302 (2009)

  33. IXS from polycrystalline materials - III Stishovite (SiO2) rutile structureN = 618 phonon branches 27 IXS spectra A. Bosak et al; Geophysical Research Letters 36, L19309 (2009)

  34. IXS from polycrystalline materials - IV SiO2 stishovite: validation of ab initio calculation single scaling factor of 1.05 is introduced

  35. IXS from polycrystalline materials - V Single crystal phonon dispersion the same scaling factor of 1.05is applied F. Jiang et al.; Phys. Earth Planet. Inter. 172, 235 (2009)

  36. Revival of thermal diffuse scattering • = 0.7293 Å • Dl/l = 1x10-4 • Angular step 0.1° • ID29 ESRF Pilatus 6M hybrid silicon pixel detector

  37. TDS: theoretical formalism with eigenfrequencies , temperature and scattering factor with eigenvectors Debye Waller factor , atomic scattering factor and mass .

  38. Diffuse scattering in Fe3O4 A. Bosak et al.; Physical Review X (2014)

  39. Diffuse scattering in Fe3O4 Fe3O4 A. Bosak et al.; Physical Review X (2014)

  40. (h0l)-plane T=80K (1.3 TCDW) T=295 K (400) (300) (301) (401) ZrTe3: IXS and (thermal) diffuse scattering M. Hoesch et al.; Phys. Rev. Lett. 2009

  41. Example I: phonon dispersion of fcc d-Plutonium Pu is one of the most fascinating and exotic element known • Multitude of unusual properties • Central role of 5f electrons • Radioactive and highly toxic strain enhanced recrystallisation of fcc Pu-Ga (0.6 wt%) alloy typical grain size: 90 mm foil thickness: 10 mm J. Wong et al. Science 301, 1078 (2003); Phys. Rev. B 72, 064115 (2005)

  42. Plutonium: the IXS experiment ID28 at ESRF • Energy resolution: 1.8 meV at 21.747 keV • Beam size: 20 x 60 mm2 (FWHM) • On-line diffraction analysis

  43. Plutonium phonon dispersion soft-mode behaviour of T[111] branch proximity of structural phase transition (to monoclinic a’ phase at 163 K) • Born-von Karman force constant model fit - good convergence, if fourth nearest neighbours are included

  44. Plutonium: elasticity Proximity of G-point: E = Vq VL[100] = (C11/r)1/2 VT[100] = (C44/r)1/2 VL[110] = ([C11+C12+2C44]/r)1/2 VT1[110] = ([C11 - C12] /2r)1/2 VT2[110] = (C44/r)1/2 VL[111] = [C11+2C12+4C44]/3r)1/2 VT[111] = ([C11-C12+C44]/3r)1/2 C11 = 35.31.4 GPa C12 = 25.51.5 GPa C44 = 30.51.1 GPa highest elastic anisotropy of all known fcc metals

  45. Plutonium: density of states Specific heat g(E) qD(T0) = 115K qD(T ) = 119.2K • Born-von Karman fit - density of states calculated

  46. Example II: IXS from fluids High-frequency dynamics in fluids at high pressures and temperatures F. Gorelli, M. Santoro (LENS, Florence) G. Ruocco, T. Scopigno, G. Simeoni (University of Rome I) T. Bryk (National Polytechnic University Lviv) M. Krisch (ESRF)

  47. Liquid–Gas Coexistence Gas P Fluid Liquid Pc Pc T<Tc A Liquid B Supercritical Fluid Gas Fluid Tc T T>Tc Example II: IXS from fluids

  48. w=C*Q w=CL*Q THz w=CS*Q nm-1 IXS from fluids: behavior of liquids (below Tc) w = 1/ta: positive dispersion of the sound speed: cL > cS Structural relaxation process ta interacting with the dynamics of the microscopic density fluctuations.

  49. IXS from fluids: oxygen at room T in a DAC T/Tc = 2 P/Pc>> 1 DAC: diamond anvil cell; 80 mm thick O2 sample

  50. IXS from fluids: pressure-dependent dispersion Positive dispersion is present in deep fluid oxygen! CL/CS 1.2 typical of simple liquids

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