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
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
dW
r
ki , Ei
2q
photon
r
Q, E
• Energy transfer: Ef - Ei = DE = 1 meV – several keV
r
r r
• Momentum transfer: k f  ki  Q = 1 – 180 nm-1
Introduction II - schematic IXS spectrum
quasielastic
phonon,
magnons,
orbitons
plasmon
valence
electron
excitations
core-electron
excitation
S. Galombosi, PhD thesis, Helsinki 2007
Compton
profile
Introduction III – overview 1
Phonons
Lattice dynamics
- elasticity
- thermodynamics
- phase stability
- e--ph coupling
counts in 80 secs
800
600
400
200
0
-30
-20
-10
0
10
energy transfer [meV]
20
30
Lecture today!
Magnons
Spin dynamics
- magnon dispersions
- exchange interactions
Lecture on Friday by Marco Moretti Sala!
Introduction IV – overview 2
Nuclear resonance
Ee
3/2¯
 = 4.85 neV
 = 141 ns
0
prompt scattering
delayed scattering
Lecture by Sasha Chumakov on Tuesday!
1/2¯
D
±3/2¯
1/2¯
nuclear level scheme 57Fe
Introduction V – IXS instrumentation
Energy analysis of scattered X-rays
- DE/E = 10-4 – 10-8
- some solid angle
Kin
Kout
Detector
Sample
Rowland circle
crystal spectrometer
p = Rcrystal·sinqB
Rcrys = 2·RRowl
RRowland
p
Spherical crystal
Q
Introduction VI – IXS at the ESRF
ID28:
Phonons
ID32:
soft X-ray IXS
ID20:
Electronic and
magnetic excitations
ID18:
Nuclear resonance
Relevance of phonon studies
Phase stability
Superconductivity
Thermal Conductivity
Sound velocities
and elasticity
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)
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)
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
Brockhouse (1955)
Thermal neutrons:
Ei = 25 meV
ki = 38.5 nm-1
DE/E = 0.01 – 0.1
Burkel, Dorner and Peisl (1987)
Hard X-rays:
Ei = 18 keV
ki = 91.2 nm-1
DE/E  1x10-7
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)
IXS scattering kinematics
dW
r
ki , Ei
2q
photon
r
Q, E
E  Ei  E f
r
r
Q  2 ki sin(q )
momentum transfer is defined only
by scattering angle
IXS from phonons – the low Q regime
No kinematic limitations: DE independent of Q
2
10
1
Q = 4p/lsin(q)
DE = Ei - Ef
IXS
10
0
E (meV)
DE
10
-1
10
INS
-2
10
-3
10
-4
10
-1
10
0
10
1
-1
Q ( nm )
10
2
10
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
IXS from phonons – very small samples
Small sample volumes:
10-4 – 10-5 mm3
bcc Mo single
crystal
Ø 45 m
t=20 m
ruby
helium
Diamond
anvil cell
• (New) materials in very small quantities
• Very high pressures > 1Mbar
• Study of surface phenomena
IXS – dynamical structure factor
kin
Scattering function:
E, Q
r
r
r
S (Q, E )   G (Q, j ) F ( E, T , Q, j )
kout
j
Thermal factor:
r
F ( E , T , Q, j ) 

r
r
1
1


(
E

E
(
q
))


(
E

E
(
q
))
r
j
j
 E  E j (q )
1  exp 

kT


Dynamical structure factor:

Comparison IXS - INS
IXS
r
 2
k1 r r
2
2
1   2  f Q  S Q, E 
 r0
EW
k2
• 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 m or smaller
INS
 2
k1 r
2
b
S Q, E 
EW
k2
• 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
Efficiency of the IXS technique
L = sample length/thickness,  = photoelectric absorption, Z = atomic number
QD = Debye temperature, M = atomic mass
IXS resolution function today
Signal [arb. units]
• DE and Q-independent
• Lorentzian shape
0,1
0,01
1E-3
1E-4
-40
-20
0
20
Energy Transfer [meV]
40
• Visibility of modes.
• Contrast between modes.
IXS resolution function tomorrow
Sub-meV IXS with sharp resolution
APS
E = 9.1 keV
DE = 0.1 – 1 meV
DE = 0.89 (0.6) meV at Petra-III
DE = 0.62 meV at APS
Dedicated instrument at NSLS-II
Y.V. Shvydk’o et al, PRL 97, 235502 (2006), PRA 84, 053823 (2011)
Instrumentation for IXS
IXS set-up on ID28 at ESRF
Monochromator:
Si(n,n,n), qB = 89.98º
n=7-13
l 1 tunable
sample
detector
Ei
Ef
q
DE
DT
1/K at room temperature
Analyser:
Si(n,n,n), qB = 89.98º
n=7-13
l 2 constant
Beamline ID28 @ ESRF
9- analyser crystal spectrometer
KB optics
or
Multilayer
Mirror
Reflection
Einc [keV]
DE [meV]
Q range [nm-1]
Relative
Count rate
(8 8 8)
15.816
6
2 - 73
1
(9 9 9)
17.794
3.0
1.5 - 82
2/3
(11 11 11)
21.747
1.6
1.0 - 91
1/17
(12 12 12)
23.725
1.3
0.7 - 100
1/35
Spot size on sample: 270 x 60 m2 -> 14 x 8 m2 (H x V, FWHM)
An untypical IXS scan
Diamond; Q=(1.04,1.04,1.04)
relative temperature [K]
0.44 0.22 0 -0.22 -0.44
Anti-Stokes peak:
phonon annihilation
energy gain
counts in 80 secs
800
600
400
200
0
-30
-20
-10
0
10
20
energy transfer [meV]
30
dscan monot 0.66 –0.66 132 80
Stokes peak:
phonon creation
energy loss
Phonon dispersion scheme
kin
E, Q
kout
Diamond
Diamond (INS + theory): P. Pavone, PRB 1993
Single crystal selection rules
S(Q,)  (Q·e)
ˆ 2
well-defined momentum transfer for given scattering geometry
Single crystal selection rules
S(Q,)  (Q·e)
ˆ 2
well-defined momentum transfer for given scattering geometry
Phonon dispersion and -point phonons
Brillouin light scattering
Raman scattering
Intensity [arb. units]
100000
10000
1000
100
0
500
1000
1500
2000
2500
-1
wave numbers [cm ]
3000
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
At high Q (50–80 nm-1)
At low Q (1. BZ)
0.03
30
Intensity [arb. units]
Energy [meV]
40
VL~E/q
20
10
0
0
2
4
6
8
10
12
14
16
-1
q [nm ]
Orientation averaged
longitudinal sound velocity
0.02
0.01
0.00
0
50
100
150
Energy [meV]
(Generalised)
phonon density-of-states
How to get the full lattice dynamics?
IXS from polycrystalline materials - II
New methodology
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
I. Fischer, A. Bosak, and M. Krisch; Phys. Rev. B 79, 134302 (2009)
IXS from polycrystalline materials - III
Stishovite (SiO2)
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
SiO2 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
Ref.
C11
C33
C12
C13
C44
C66
B
VD
[GPa]
[GPa]
[GPa]
[GPa]
[GPa]
[GPa]
[GPa]
[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)
F. Jiang et al.; Phys. Earth Planet. Inter. 172, 235 (2009)
Revival of thermal diffuse scattering
l = 0.7293 Å
Dl/l = 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 Fe3O4
A. Bosak et al.; Physical Review X (2014)
Diffuse scattering in Fe3O4
Fe3O4
A. Bosak et al.; Physical Review X (2014)
ZrTe3: IXS and (thermal) diffuse scattering
5
T = 292 K
T = 158 K
T = 100 K
T = 83 K
T = 78 K
T = 73 K
T = 68 K
non-interacting
(h0l)-plane
T=295 K
T=80K (1.3 TCDW)
(300)
energy (meV)
4
3
2
(400)
1
(301)
(401)
0
-4.00
M. Hoesch et al.; Phys. Rev. Lett. 2009
-3.96
-3.92
-3.88
momentum along CDW (a* component)
Example I: phonon dispersion of fcc -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 m
foil thickness: 10 m
J. Wong et al. Science 301, 1078 (2003); Phys. Rev. B 72, 064115 (2005)
Plutonium: the IXS experiment
ID28 at ESRF
• Energy resolution: 1.8 meV at 21.747 keV
• Beam size: 20 x 60 m2 (FWHM)
• On-line diffraction analysis
TA (0.2;0.2;0.2)
LA (0.2;0.2;0.2)
60
Counts in 180 secs
Counts in 180 secs
200
40
20
0
-20
-10
0
10
Energy [meV]
20
100
0
-10 -5 0
5 10
Energy [meV]
Plutonium phonon dispersion
soft-mode behaviour of
T[111] branch
proximity of structural
phase transition
(to monoclinic a’ phase
at 163 K)
Expt
B-vK fit 4NN
• Born-von Karman force constant model fit
- good convergence, if fourth nearest neighbours are included
Plutonium: elasticity
Proximity of -point: E = Vq
VL[100] = (C11
/r)1/2
<001>
<111>
VT[100] = (C44/r)1/2
VL[110] = ([C11+C12+2C44]/r)1/2
VT1[110] = ([C11 - C12] /2r)1/2
<110>
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
Plutonium: density of states
Specific heat
Cv  3 Nk B
2
 E  exp( E / k BT ) g ( E )dE

 
k
T
exp( E / k BT )  12
0  B 
E max
3R 
g(E)
0.0 0.2 0.4 0.6
Density of states
(arb. units)
• Born-von Karman fit
- density of states calculated
Cv (cal mole-1 K-1)
6
4
2
0
0
50
100
150
200
250
Temperature (K)
qD(T0) = 115K
qD(T  ) = 119.2K
300
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
Gas
P
Fluid
Liquid
Pc
Pc
A
Liquid
T<Tc
B
Supercritical Fluid
Gas
Fluid
Tc
T
T>Tc
IXS from fluids: behavior of liquids (below Tc)

Viscous
"liquidlike"
dynamics
a<<1
a
Visco-elastic
transition
=C*Q
=CL*Q
THz
=CS*Q
nm-1
Elastic
"solidlike"
dynamics
a>>1
Q
 = 1/a: positive dispersion of the sound speed: cL > cS
Structural relaxation process a interacting with the dynamics
of the microscopic density fluctuations.
IXS from fluids: oxygen at room T in a DAC
P=0.88 GPa
P=5.35 GPa
Q=10.2 nm
-1
Q=5.4 nm
-1
Q=7.8 nm
P/Pc>> 1
Q=3.0 nm
-1
T/Tc = 2
Intensity (a.u.)
-1
Q=12.6 nm
-1
P=2.88 GPa
-40
-20
0
20
40 -40
-20
0
20
40 -40
Energy (meV)
DAC: diamond anvil cell; 80 m thick O2 sample
-20
0
20
40
IXS from fluids: pressure-dependent dispersion
25
P=5.35 GPa
P=2.88 GPa
P=0.88 GPa
Energy (meV)
20
15
10
5
cISTS(m/s)
cIXS(m/s)
1920
2980
3680
2340
3600
4440
0
0
5
10
15
20
25
-1
Q (nm )
Positive dispersion is present in deep fluid oxygen!
CL/CS  1.2 typical of simple liquids
IXS from fluids: reduced phase diagram
F. Gorelli et al; Phys. Rev. Lett. 97, 245702 (2006)
IXS from fluids
Cross-over at the Widom line?
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”
IXS from fluids: Argon at high P and T
IXS and MD simulations
1.14
1.12
Positive sound dispersion
1.10
1.08
1.06
1.04
1.02
Widom line
1.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Pressure (GPa)
G.G. Simeoni et al; Nature Physics 6, 503 (2010)
3.5
4.0
IXS from fluids: reduced phase diagram (bis)
1000
100
P/Pc
10
1
Neon
Oxygen
Argon
Nitrogen
Mercury
Lithium
Sodium
Potassium
Rubidium
Cesium
Water
WIDOM
LINE FOR Ar
LIQUID-LIKE
GAS-LIKE
LIQUID
0.1
0.01
CRITICAL
ISOCHORE
FOR Ar
Pc
CP(J/mol K)
10000
40
T/Tc=1.46
1.79
2.45
3.12
30
GAS
20
0
1E-3
0.1
1
T/Tc
G.G. Simeoni et al; Nature Physics 6, 503 (2010)
5
10
15
P/Pc
20
10
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 supercritical fluid region
 Evidence of fluid-fluid phase transition-like behavior on the
locus of CP maximum (Widom's line) in supercritical fluid Ar
Applications: Strongly correlated electrons
Doping dependence in SmFeAsO1-xFy
Kohn anomaly in ZrTe3
e-ph coupling in a-U
M. Le Tacon et al.; Phys. Rev. B 80, (2009)
M. Hoesch et al.; PRL 102, (2009)
S. Raymond et al.; PRL 107, (2011)
Applications: Functional materials
Piezoelectrics PbZr1-xTixO3
Skutterudites
M.M. Koza et al.; PRB 84, 014306
InN thin film lattice dynamics
J. Hlinka et al.; PRB 83, 040101(R)
J. Serrano et al.; PRL 106, 205501
Lecture by Benedict Klobes on Friday!
Applications: Earth & Planetary science
Sound velocities in Earth’s core
Elastic anisotropy in Mg83Fe0.17O
J. Badro et al.; Earth Plan. Science Lett. 98, 085501
D. Antonangeli et al.; Science 331, 64
Lecture by Daniele Antonangeli on Friday!
Applications: Liquids & glasses
Liquid-like dynamical behaviour
in the supercritical region
0
100
200
300
P/Pc
400
500
600
700
800
1.14
Positive sound dispersion
1.12
1.10
1.08
Nature of the Boson peak in glasses
1.06
T= 573 K
T/Tc=3.80
1.04
Widom's line
1.02
1.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Pressure (GPa)
G. Simeoni et al.; Nature Phys. 6, 503
A. Chumakov et al.; PRL 106, 225501
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: 978-3-527-40812-2