Introduction to Inelastic x-ray scattering Michael Krisch
European Synchrotron Radiation Facility Grenoble, France
[email protected]
Slide 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
Slide 3
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
Slide 4
Introduction II - schematic IXS spectrum quasielastic phonon,
magnons, orbitons valence electron excitations plasmon Compton
profile core-electron excitation S. Galombosi, PhD thesis, Helsinki
2007
Slide 5
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
Slide 6
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!
Slide 7
Introduction V IXS instrumentation K out K in Q p = R crystal
sin B R crys = 2R Rowl Detector Sample Spherical crystal p R
Rowland Energy analysis of scattered X-rays - E/E = 10 -4 10 -8 -
some solid angle Rowland circle crystal spectrometer
Slide 8
Introduction VI IXS at the ESRF ID20: Electronic and magnetic
excitations ID18: Nuclear resonance ID28: Phonons ID32: soft X-ray
IXS
Slide 9
Relevance of phonon studies Superconductivity Thermal
Conductivity Sound velocities and elasticity Phase stability
Slide 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)
Slide 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) tin, J. Bouman et al., Physica 12, 353
(1946)
Slide 12
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.
Slide 13
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)
Slide 14
Inelastic x-ray scattering from phonons HASYLAB E = 55 meV
0.083 Hz B. Dorner, E. Burkel, Th. Illini, and J. Peisl; Z. Phys. B
69, 179 (1987)
Slide 15
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
Slide 16
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
Slide 17
IXS from phonons very small samples Small sample volumes: 10 -4
10 -5 mm 3 Diamond anvil cell (New) materials in very small
quantities Very high pressures > 1Mbar Study of surface
phenomena 45 m t=20 m bcc Mo single crystal ruby helium
Slide 18
IXS dynamical structure factor Scattering function: Thermal
factor: Dynamical structure factor: E, Q k in k out
Slide 19
Comparison IXS - INS no correlation between momentum- and
energy transfer E/E = 10 -7 to 10 -8 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 = 10 -1 to 10
-2 Cross section ~ b 2 Weak absorption => multiple scattering
incoherent scattering contributions large beams: several cm IXS
INS
Slide 20
Efficiency of the IXS technique L = sample length/thickness, =
photoelectric absorption, Z = atomic number D = Debye temperature,
M = atomic mass
Slide 21
IXS resolution function today E and Q-independent Lorentzian
shape Visibility of modes. Contrast between modes.
Slide 22
IXS resolution function tomorrow Sub-meV IXS with sharp
resolution Y.V. Shvydko et al, PRL 97, 235502 (2006), PRA 84,
053823 (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
Slide 23
Instrumentation for IXS Monochromator: Si(n,n,n), B = 89.98
n=7-13 1 tunable Analyser: Si(n,n,n), B = 89.98 n=7-13 2 constant
IXS set-up on ID28 at ESRF EE TT 1/K at room temperature EE TT
Slide 24
Beamline ID28 @ ESRF ReflectionE inc [keV] E [meV] Q range [nm
-1 ]Relative Count rate (8 8 8)15.81662 - 731 (9 9 9)17.7943.01.5 -
822/3 (11 11 11)21.7471.61.0 - 911/17 (12 12 12)23.7251.30.7 -
1001/35 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
Slide 25
An untypical IXS scan dscan monot 0.66 0.66 132 80 Diamond;
Q=(1.04,1.04,1.04) Stokes peak: phonon creation energy loss
Anti-Stokes peak: phonon annihilation energy gain
Slide 26
Phonon dispersion scheme E, Q k in k out Diamond Diamond (INS +
theory): P. Pavone, PRB 1993
Slide 27
Single crystal selection rules well-defined momentum transfer
for given scattering geometry S(Q, ) (Qe) 2
Slide 28
Single crystal selection rules S(Q, ) (Qe) 2 well-defined
momentum transfer for given scattering geometry
Slide 29
Phonon dispersion and -point phonons Raman scatteringBrillouin
light scattering
Slide 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
Slide 31
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 (5080 nm -1 ) How to get the
full lattice dynamics?
Slide 32
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, 134302 (2009)
Slide 33
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)
Slide 34
IXS from polycrystalline materials - IV SiO 2 stishovite:
validation of ab initio calculation single scaling factor of 1.05
is introduced
Slide 35
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)
TDS: theoretical formalism with eigenfrequencies, temperature
and scattering factor with eigenvectors Debye Waller factor, atomic
scattering factor and mass.
Slide 38
Diffuse scattering in Fe 3 O 4 A. Bosak et al.; Physical Review
X (2014)
Slide 39
Diffuse scattering in Fe 3 O 4 Fe 3 O 4 A. Bosak et al.;
Physical Review X (2014)
Slide 40
ZrTe 3 : IXS and (thermal) diffuse scattering M. Hoesch et al.;
Phys. Rev. Lett. 2009 (h0l)-plane (300) (400) (301) (401) T=295 K
T=80K (1.3 T CDW )
Slide 41
Example I: phonon dispersion of fcc -Plutonium J. Wong et al.
Science 301, 1078 (2003); Phys. Rev. B 72, 064115 (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
Slide 42
Plutonium: the IXS experiment ID28 at ESRF Energy resolution:
1.8 meV at 21.747 keV Beam size: 20 x 60 m 2 (FWHM) On-line
diffraction analysis
Slide 43
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)
Slide 44
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
Slide 45
Plutonium: density of states Born-von Karman fit - density of
states calculated Specific heat g(E) D (T 0) = 115K D (T ) =
119.2K
Slide 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)
Slide 47
Example II: IXS from fluids LiquidGas Coexistence TT c Fluid
PcPc P T Liquid Gas Fluid PcPc TcTc A B
Slide 48
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.
Slide 49
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
Slide 50
IXS from fluids: pressure-dependent dispersion Positive
dispersion is present in deep fluid oxygen! C L /C S 1.2 typical of
simple liquids
Slide 51
IXS from fluids: reduced phase diagram F. Gorelli et al; Phys.
Rev. Lett. 97, 245702 (2006)
Slide 52
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?
Slide 53
IXS from fluids: Argon at high P and T IXS and MD simulations
G.G. Simeoni et al; Nature Physics 6, 503 (2010)
Slide 54
IXS from fluids: reduced phase diagram (bis) G.G. Simeoni et
al; Nature Physics 6, 503 (2010)
Slide 55
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
Slide 56
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)
Slide 57
Applications: Functional materials Piezoelectrics PbZr 1-x Ti x
O 3 J. Hlinka et al.; PRB 83, 040101(R) Skutterudites M.M. Koza et
al.; PRB 84, 014306 InN thin film lattice dynamics J. Serrano et
al.; PRL 106, 205501 Lecture by Benedict Klobes on Friday!
Slide 58
Applications: Earth & Planetary science Elastic anisotropy
in Mg 83 Fe 0.17 O D. Antonangeli et al.; Science 331, 64 Sound
velocities in Earths core J. Badro et al.; Earth Plan. Science
Lett. 98, 085501 Lecture by Daniele Antonangeli on Friday!
Slide 59
Applications: Liquids & glasses Nature of the Boson peak in
glasses A. Chumakov et al.; PRL 106, 225501 Liquid-like dynamical
behaviour in the supercritical region G. Simeoni et al.; Nature
Phys. 6, 503 Lecture by Sasha Chumakov on Tuesday!
Slide 60
Further reading W. Schlke; 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: 978-3-527-40812-2