Introduction to Inelastic x-ray scattering Michael Krisch European Synchrotron Radiation Facility Grenoble, France

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

Introduction I – scattering kinematics d  2  ii Ek,  f f E k,  Q  E, photon p h o t o n Energy transfer: E f - E i =  E = 1 meV – several keV Momentum transfer: = 1 – 180 nm -1

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

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

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

Introduction V – IXS instrumentation K out K in Q p = R crystal ·sin  B R crys = 2·R Rowl Detector Sample Spherical crystal p R Rowland Energy analysis of scattered X-rays -  E/E = – some solid angle Rowland circle crystal spectrometer

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

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

Vibrational spectroscopy – a short history Infrared absorption W. Abney and E. Festing, R. Phil. Trans. Roy. Soc. 172, 887 (1881) Brillouin light scattering 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)

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)  – tin, J. Bouman et al., Physica 12, 353 (1946)

X-rays and magnons? 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.”

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

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

IXS scattering kinematics d   ii Ek,  f f E k,  Q  E, photon p h o t o n )sin(2  i kQ    fi EEE  momentum transfer is defined only by scattering angle

IXS from phonons – the low Q regime  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 Q = 4  / sin(  )  E = E i - E f IXS INS v = 500 m/s v = 7000 m/s  No kinematic limitations:  E independent of Q Disordered systems: Explore new Q-  E range

IXS from phonons – very small samples Small sample volumes: – mm 3 Diamond anvil cell (New) materials in very small quantities Very high pressures > 1Mbar Study of surface phenomena Ø 45  m t=20  m bcc Mo single crystal ruby helium

IXS – dynamical structure factor Scattering function: Thermal factor: Dynamical structure factor: E, Q k in k out

Comparison IXS - INS no correlation between momentum- and energy transfer  E/E = to Cross section ~ Z 2 (for small Q) Cross section is dominated by photoelectric absorption (~ 3 Z 4 ) no incoherent scattering small beams: 100  m or smaller strong correlation between momentum- and energy transfer  E/E = to Cross section ~ b 2 Weak absorption => multiple scattering incoherent scattering contributions large beams: several cm IXS INS

Efficiency of the IXS technique L = sample length/thickness,  = photoelectric absorption, Z = atomic number  D = Debye temperature, M = atomic mass

IXS resolution function today  E and Q-independent Lorentzian shape Visibility of modes. Contrast between modes.

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

Instrumentation for IXS Monochromator: Si(n,n,n),  B = 89.98º n= tunable Analyser: Si(n,n,n),  B = 89.98º n= constant IXS set-up on ID28 at ESRF EE TT 1/K at room temperature  EE TT

Beamline ESRF ReflectionE inc [keV]  E [meV] Q range [nm -1 ]Relative Count rate (8 8 8) (9 9 9) /3 ( ) /17 ( ) /35 Spot size on sample: 270 x 60  m 2 -> 14 x 8  m 2 (H x V, FWHM) 9- analyser crystal spectrometer KB optics or Multilayer Mirror

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

Phonon dispersion scheme E, Q k in k out Diamond Diamond (INS + theory): P. Pavone, PRB 1993

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

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

Phonon dispersion and  -point phonons Raman scatteringBrillouin light scattering

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

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

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

IXS from polycrystalline materials - III Stishovite (SiO 2 ) rutile structure N = 6 18 phonon branches 27 IXS spectra A. Bosak et al; Geophysical Research Letters 36, L19309 (2009)

IXS from polycrystalline materials - IV SiO 2 stishovite: validation of ab initio calculation single scaling factor of 1.05 is introduced

IXS from polycrystalline materials - V Single crystal phonon dispersion the same scaling factor of 1.05 is applied F. Jiang et al.; Phys. Earth Planet. Inter. 172, 235 (2009) Ref.C 11 [GPa] C 33 [GPa] C 12 [GPa] C 13 [GPa] C 44 [GPa] C 66 [GPa] B [GPa] V D [km/s] Jiang et al. 455(1)762(2)199(2)192(2)258(1)321(1)310(2)7.97(2) this work 441(4)779(2)166(3)195(1)256(1)319(1)300(3)7.98(4)

Revival of thermal diffuse scattering = Å  /  = 1x10 -4 Angular step 0.1° ID29 ESRF Pilatus 6M hybrid silicon pixel detector

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

Diffuse scattering in Fe 3 O 4 A. Bosak et al.; Physical Review X (2014)

Diffuse scattering in Fe 3 O 4 Fe 3 O 4 A. Bosak et al.; Physical Review X (2014)

ZrTe 3 : IXS and (thermal) diffuse scattering M. Hoesch et al.; Phys. Rev. Lett (h0l)-plane (300) (400) (301) (401) T=295 K T=80K (1.3 T CDW )

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

Plutonium: the IXS experiment ID28 at ESRF Energy resolution: 1.8 meV at keV Beam size: 20 x 60  m 2 (FWHM) On-line diffraction analysis

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

Plutonium: elasticity Proximity of  -point: E = Vq V L [100] = (C 11 /  ) 1/2 V T [100] = (C 44 /  ) 1/2 V L [110] = ([C 11 +C 12 +2C 44 ]/  ) 1/2 V T1 [110] = ([C 11 - C 12 ] /2  ) 1/2 V T2 [110] = (C 44 /  ) 1/2 V L [111] = [C 11 +2C 12 +4C 44 ]/3  ) 1/2 V T [111] = ([C 11 -C 12 +C 44 ]/3  ) 1/2 C 11 = 35.3  1.4 GPa C 12 = 25.5  1.5 GPa C 44 = 30.5  1.1 GPa highest elastic anisotropy of all known fcc metals

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

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)

Example II: IXS from fluids Liquid–Gas Coexistence T<T c Gas Liquid Supercritical Fluid T>T c Fluid PcPc P T Liquid Gas Fluid PcPc TcTc A B

IXS from fluids: behavior of liquids (below T c )  =C S *Q  =C  *Q THz nm -1  =C L *Q  = 1/   : positive dispersion of the sound speed: c L > c S Structural relaxation process   interacting with the dynamics of the microscopic density fluctuations.

IXS from fluids: oxygen at room T in a DAC P/P c >> 1 DAC: diamond anvil cell; 80  m thick O 2 sample T/T c = 2

IXS from fluids: pressure-dependent dispersion Positive dispersion is present in deep fluid oxygen! C L /C S  1.2 typical of simple liquids

IXS from fluids: reduced phase diagram F. Gorelli et al; Phys. Rev. Lett. 97, (2006)

IXS from fluids Widom line: theoretical continuation into the supercritical region of the liquid-vapour coexistence line, considered as “locus of the extrema of the thermodynamic response functions” Cross-over at the Widom line?

IXS from fluids: Argon at high P and T IXS and MD simulations G.G. Simeoni et al; Nature Physics 6, 503 (2010)

IXS from fluids: reduced phase diagram (bis) G.G. Simeoni et al; Nature Physics 6, 503 (2010)

IXS from fluids: Conclusions Revisiting the notion of phase diagram beyond the critical point:  The positive sound dispersion is a physical observable able to distinguish liquid-like from gas-like behavior in the super- critical fluid region  Evidence of fluid-fluid phase transition-like behavior on the locus of C P maximum (Widom's line) in supercritical fluid Ar

Applications: Strongly correlated electrons Doping dependence in SmFeAsO 1-x F y M. Le Tacon et al.; Phys. Rev. B 80, (2009) Kohn anomaly in ZrTe 3 M. Hoesch et al.; PRL 102, (2009) e-ph coupling in  -U S. Raymond et al.; PRL 107, (2011)

Applications: Functional materials Piezoelectrics PbZr 1-x Ti x O 3 J. Hlinka et al.; PRB 83, (R) Skutterudites M.M. Koza et al.; PRB 84, InN thin film lattice dynamics J. Serrano et al.; PRL 106, Lecture by Benedict Klobes on Friday!

Applications: Earth & Planetary science Elastic anisotropy in Mg 83 Fe 0.17 O D. Antonangeli et al.; Science 331, 64 Sound velocities in Earth’s core J. Badro et al.; Earth Plan. Science Lett. 98, Lecture by Daniele Antonangeli on Friday!

Applications: Liquids & glasses Nature of the Boson peak in glasses A. Chumakov et al.; PRL 106, Liquid-like dynamical behaviour in the supercritical region G. Simeoni et al.; Nature Phys. 6, 503 Lecture by Sasha Chumakov on Tuesday!

Further reading  W. Schülke; Electron dynamics by inelastic x-ray scattering, Oxford University Press (2007)  M. Krisch and F. Sette; Inelastic x-ray scattering from Phonons, in Light Scattering in Solids, Novel Materials and Techniques, Topics in Applied Physics 108, Springer-Verlag (2007).  A. Bosak, I. Fischer, and M. Krisch, in Thermodynamic Properties of Solids. Experiment and Modeling, Eds. S.L. Chaplot, R. Mittal, N. Choudhury. Wiley-VCH Weinheim, Germany (2010) 342 p. ISBN: