Inelastic X-Ray Scattering
AOFSRR 2nd Cheiron School
9 October 2008
Alfred Q.R. Baron
Materials Dynamics Laboratory (RIKEN)
Facilities and Utilization Division (JASRI)
SPring-8

Acknowledgement:
Y. Sakurai (Compton)

AQRB @ AOFSRR Cheiron School 2008

Scope & Outline
Main Goal: Introduce Capabilities & Put them in Context
Introduction & Relation to Other SR Work
What relevant properties of samples can be measured?
When/why would someone consider these techniques?
Outline:Introduction
Brief comments about instrumentation.
Non-Resonant Techniques
Resonant Techniques (Briefly)
Note: limit to some aspects of photon-in -> photon out scattering
Note: Huge & Complex Topic - Appropriate for a semester, not an hour…

AQRB @ AOFSRR Cheiron School 2008

Table Of IXS Techniques/Applications
TechniqueComment

Energy Scale

Information

X-Ray Raman(E)XAFS in Special Cases

Ein~10 keV!E~100-1000 eV

Edge Structure, Bonding

ComptonOldestNote: Resolution Limited

Ein ~ 150 keV!E ~ keVElectron Momentum Density
Fermi Surface Shape

Magnetic Compton

Weak But Possible

Ein ~ 150 keV!E ~ keV

Density of Unpaired Spins

High Rate
Somewhat ComplicatedEin ~ 4-15 keV!E ~ 1-50 eV

Electronic Structure

Non-Resonant IXS

Low Rate
SimplerEin~10 keV!E ~ <1-50 eV

Electronic Structure

IXS

Large Instrument

Ein~16-26 keV!E ~ 0.001-100 meV

Phonon Dispersion

Atom SpecificVia Mossbauer Nuclei

Ein ~ 14-25 keV!E ~ 0.001-100 meVElement SpecificPhonon Density of States(DOS)RIXS
Resonant IXS

NRIXS

High-Resolution IXSNIS
Nuclear IXS

Note: !E = Typical Energy Transfer (Not Resolution)
Note also: Limit to FAST dynamics (~10 ps or faster)
AQRB @ AOFSRR Cheiron School 2008Some References
Shulke, W. (2007). Electron Dynamics by Inelastic X-Ray Scattering.
New York: Oxford University Press.
& References therein
Squires, G. L. (1978). Introduction to the Theory of Thermal NeutronScattering. New York: Dover Publications, Inc.
van!Hove, L. (1954). Phys. Rev. 95, 249-262.
Born, M. & Huang, K. (1954). Dynamical Theory of Crystal
Lattices. Oxford: Clarendon press.
Bruesch, P. (1982). Phonons: Theory and Experiments 1. Berlin: Springer-Verlag.

Cooper, M.J. (1985). Rep. Prog. Phys. 48 415-481

AQRB @ AOFSRR Cheiron School 2008Spectroscopy

Absorption vs. Scattering
Absorption
Spectroscopy
Optical, IR, NMR

Measure absorption as you
scan the incident energy
When energy hits a resonance,or exceeds a gap, or… get a change
Optical Spect. NiONewman, PR 1959

Free Parameters: E1, e1, k1-> In principle, 6 dimensions
but in practice mostly 1 (E1)

Scattering
Spectroscopy

E2 k2!e2

E1 k1!e1

IXS, Raman, INS

Free Parameters: E1, e1, k1, E2, e2, k2-> In principle, 12 dimensionsin practice, mostly 4, E1-E2, Q = k2-k1

Scattering is much more complex, but gives more information.
AQRB @ AOFSRR Cheiron School 2008

X-Ray Scattering Diagramk1!
"E1 k1!e1

Q =!k"- k1hc
E 1 = h"1 == hck 1#1
k 1 = k1 =!

E2 k2!e2

2"

Two Main Quantities:#1

hc = 12.398 keV• ÅMomentum Transfer

Energy Transfer!
!

Q " k 2 # k1

E or "E = E 1 # E 2 $ h%

Q" Q $4%&1

sin '2( )Periodicity Probed in Sample!

Note: For Resonant Scattering
E1 and E2 and Poln.
Are also important!

d=2"
Qor d =2"
q = Q#$AQRB @ AOFSRR Cheiron School 2008!

!

Kinematics:

Simplistic energy-momentum relations

Mk1!
"E1 k1!e1

Q =!k"- k1
E2 k2!e2

Kinetic Energy Given to Sample:

E recoilp2
h 2Q 2==2M
k"2M
E" eˆ 'Take: M=57 amu, Q/c = 7 Å-1 -> Er=2.3 meV!

!

Compton Form:" 2 # "1 =h(1 # cos $)Mc

h= 0.0243Å
m ec"c =AQRB @ AOFSRR Cheiron School 2008!

!

Dynamic Structure Factor
It is convenient, especially for non-resonant scattering, to separate theproperties of the material and the properties of the interaction of thephoton with the material (electron)d 2"$2*= re (e 2 •e1 ) 2 S(Q,$ )d#d$$1" T hom son = re (e 2 •e1 ) ##2*!Thomson Scattering

21S(Q," ) =12 #h

Note: For resonant#-> #2$ dt e " $ dr $ dr& e (%i tiQ• r% r & )' (r&,t)'(r,t = 0)

Dynamic Structure Factor

Cross SectionhQ# d" h" S(Q," )=2M# d" S(Q," )2!

f-sum rule:

2AQRB @ AOFSRR Cheiron School 2008!

AQRB @ AOFSRR Cheiron School 2008

The IXS Spectrometer
An Optics Problem
k ! E ! eˆ'Main Components
Monochromator:
Modestly Difficult
Only needs to accept 15x40 mrad2Detector

Analyzer

Fixed bandwidth at aselected Q transfer

Sample Stages
Straightforward
Only Need SpaceSampleMonochromator!E small, E tunable

Analyzer:
Large Solid Angle
Difficult

The Goal: Put it all together andKeep Good Resolution, Not Lose Flux
AQRB @ AOFSRR Cheiron School 2008

Basic Optical Concepts
Bragg’s Law : $ = 2d sin("%) =>"# = tan%'&$ (*) "E
B E!
Working closer to "B~90 deg.
maximizes the angular
acceptance for a given energy resolution…

Better energy resolution-> Closer to 90 degrees-> Large Spectrometer

AQRB @ AOFSRR Cheiron School 2008

Analyzer Crystals
Detector

The more difficult optic…
Require:Correct Shape (Spherically Curved, R=9.8 m)
Not Strained (!E/E~few 10-8-> !d/d<<few10-8)

SampleAnalyzer
Crystal

Method: Bond many small crystallites to a curved substrate.

1. Cut

2. Etch

3. Bond toSubstrate

4. Remove BackX-Rays

104 Independent
Perfect Crystals
AQRB @ AOFSRR Cheiron School 2008

Analyzer Crystal
Collaborative R&D with NEC Fundamental Research Laboratory, H. Kimura, F. Yamamoto

Present Parameters (9.8 m Radius, 10cm Diameter)
50 or 60 µm blade, 2.9 mm depth, 0.74 mm pitch
Channel width (after etch): ~ 0.15 mm
60 to 65% Active Area

AQRB @ AOFSRR Cheiron School 2008

High (meV) Resolution Spectrometer

High Res Spectrometer

Spectrometer Photo
12 Chan. CZT Detector

Slit SystemSampleAnalyzer Array

10m Arm & 12 Analyzers
Vacuum Flight Path

10mHorizon

Granite Base w/Airpads

tal A

rmIncident!Beam&'100 µm~ 20 um Possible- to55°in 2"
AQRB @ AOFSRR Cheiron School 2008

Medium Resolution
Medium Resolution Spectrometer:
Arm Radius: 1 to 3 m
Resolution: ~0.1 to 1 eV
Used for RIXS and NRIXS
BL12XU (Cai, et al)
BL11XU (Ishii, et al)
BL12XU (Early)

Also: BL39XU (Hayashi, et al)

10Note difference between RIXS and NRIXS

NRIXS: Choose the energy to match the optics
RIXS: Resonance chooses energy -> usually worse resolution!E (meV)!E (meV)10.1

10152025303540E(keV)

AQRB @ AOFSRR Cheiron School 2008

AQRB @ AOFSRR Cheiron School 2008

Atomic Dynamics: Systems and Questions
Disordered Materials (Liquids & Glasses):
Still a new field -> Nearly all new data is interesting.
How do dynamical modes survive the cross-over from thelong-wavelength continuum/hydrodynamic regime toatomic length scales?

Crystalline Materials:
Basic phonon model does very well ->Specific questions needed.

Phonon softening & Phase transitions (e.g. CDW Transition)
Thermal Properties:
Thermoelectricity & Clathrates
Sound Velocity in Geological ConditionsPairing mechanism in superconductors

AQRB @ AOFSRR Cheiron School 2008

Phonons in a Crystal
Normal Modes of Atomic Motion
Must have enough modes so that each atom in a crystal can be moved ineither x,y or z directions by a suitable superposition of modes.
If a crystal has N unit cells and R atoms/Cell then it has
3NR Normal Modes
Generally: Consider the unit cell periodicity separately
by introducing a continuous momentum variable, q.
-> 3R modes for any given q

AQRB @ AOFSRR Cheiron School 2008

Phonons in a Superconductor
Conventional superconductivity is driven by lattice motion.
“Phonon Mediated” - lattice “breathing” allows electron pairs to move without resistance.

Original Picture: Limited interest in specific phonons…
Now: Lots of interest as this makes a huge difference.
Particular phonons can couple very strongly to theelectronic system.

How does this coupling appear in the phonon spectra?
Softening:

Screening lowers the energy of the mode
(abrupt change <=> Kohn Anomaly)

Broadening:

Additional decay channel (phonon->e-h pair)reduces the phonon lifetime
AQRB @ AOFSRR Cheiron School 2008

Electron Phonon Coupling& Kohn Anomalies

On the scale of electron energies, a phonon has nearly no energy.
A phonon only has momentum.
So a phonon can move electrons from one part of the Fermi
surface to another, but NOT off the Fermi surface.

PhononMomenta
Q<2kF

Large Momentum
Q>2kF
Can Not Couple to theElectronic system
Fermi Surface
Diameter = 2kf
AQRB @ AOFSRR Cheiron School 2008

MgB2Phonon StructureHigh Tc (39K)
Nagamatsu, et al, Nature 410, (2001) 63.

Simple Structure…
straightforward calculation.

Electronic Structure

Kortus, et al, PRL 86 (2001)4656

Bohnen, et al. PRL. 86, (2001) 5771.

BCS (Eliashberg) superconductor with mode-specific electron-phonon coupling.
AQRB @ AOFSRR Cheiron School 2008

Electron-Phonon Coupling in MgB2Spectra

Dispersion
100(2.31, 0, 0.01)

E2g(2.27, 0, - 0.01)(meV)

Energy (meV)

60

300K
44K
16K

30

(2.23, 0, - 0.03)

FWHM

B1g
80

LinewidthA2u40((2.19, 0, 0.03)

200!
Points: Measurements at BL35

100E1uM

20A!-M (Å-1)

M(2.15, 0, 0.01)(2.11, 0, -0.01)Lines: Calculations by Bohnen et al.

Clear correlation between
050
100
linewidth & softening.
Excellent agreement with LDA Pseudopotential calculation.
PRL 92(2004) 197004: Baron, Uchiyama, Tanaka, ... Tajima
AQRB @ AOFSRR Cheiron School 2008

Carbon Doped Mg(CxB1-x)2M(2%C, Tc=35.5K

12.5% C, Tc=2.5K(2.37,0,0.01)

AlB2 (Not SC)(2.37,0,-0.01)(2.37,0,0.01)

050

050

50(2.13,0,0.01)
100

05050

050

050(2.29,0,-0.03)100

050 (2.29,0,0.03)
100050 (2.22,0,0.03)
100

050

050(2.13,0,-0.01)
10000500100

100(2.22,0,-0.03)

100

100
(2.30,0,0.03)

100(2.24,0,-0.03)

100(2.16,0,0.01)

100

Phonon structure correlates nicely with Tc for charge doping.
(Electron doping fills the sigma Fermi surface)
AQRB @ AOFSRR Cheiron School 2008

More Superconductors
Similar types of results for
Mn Doped MgB2
CaAlSi
Boron Doped Diamond

Extrapolation to the High Tc Copper Oxide Materials….
1. Much More Complex
2. Calculations Fail so interpretation in difficult

AQRB @ AOFSRR Cheiron School 2008

AQRB @ AOFSRR Cheiron School 2008

Disordered Materials
Hydrodynamic/Continuum (Q->0) Limit ->Brillouin Triplet: Quasi-Elastic Peak + Sound Mode
How does this evolve at larger Q?
“DHO model” = Lorentzian+ Damped Harmonic Oscillatorl-Mg(Kawakita et al)a-Se (Scopigno et al)l-Si (Hosokawa, et al)
AQRB @ AOFSRR Cheiron School 2008

Metal to Insulator Transition in Liquid Mercury
Universal Phenomenon in Liquids:
Expand a liquid metal enough and it becomes an insulator.

For Hg~1500 C & ~ 1.5 kbar

15 mm Be, 200 m He (STP), 0.15 mm Sapphire~ 20 microns Hg
D. Ishikawa, M. Inui, K. Tamura, et al.
AQRB @ AOFSRR Cheiron School 2008

“Fast Sound” at the Metal-Non-Metal Transition

Ultrasonic
Velocity

Suggests a change in the microscopic
density fluctuations, possibly due toa modification of the pair potential…

D. Ishikawa, et al, PRL 93 (2004) 97801
AQRB @ AOFSRR Cheiron School 2008

Elastic Inhomogeneity in a Glass
Ichitsubo, et al.

Pd42.5Ni7.5Cu30P20(More detailed analysis: possible failure of DHO Model)

AQRB @ AOFSRR Cheiron School 2008

AQRB @ AOFSRR Cheiron School 2008

NRIXSAQRB @ AOFSRR Cheiron School 2008

High Resolution Measurement
8 meV resolution at 15.8 keV, Si(888)Temperature scan of mono over 2 eV or 60K

Baron, et al, April 2007 @ BL35XU
Detwinned Sample

Havercourt, Sawatzky, et al, May 07
NiO6 cluster calculation
Note splitting of 1700 meV excitation.

AQRB @ AOFSRR Cheiron School 2008

NRIXS at Smaller (~eV) Energy Transfers“Momentum Resolved Optical Spectroscopy”
Conventional Optical
Spectroscopy:(Absorption, Reflectivity)

Information on electronic energy levels but without
information on inter-atomic correlations or
atomic structureWith x-rays, the short wavelength allows direct probe at atomic scale:

Is an excitation collective or local (does it disperse)?
What is the atomic symmetry of an excitation?
How does it interact with the surrounding environment?
Resonant experiment -> non-resonant IXS experiment.
Non-Resonant is simpler and can have higher resolution

But badly flux limited

AQRB @ AOFSRR Cheiron School 2008

d-d Excitations in NiOFirst something simple…

There exist well-defined excitations in the chargetransfer gap of NiO
Antiferromagnet (TN 523K), (111) Spin orderLong and Distinguished History

First (resonant) IXS experiments (Kao, et al)

Non-Resonant IXS, !E~300 meV

Larson, et al., PRL 99 (2007) 026401

Cai, Hiraoka, et al, BL12XU
UnpublishedAQRB @ AOFSRR Cheiron School 2008

Orientation Dependence

Orbitals
Results of Wanneir function
analysis of LDA+U calcsof Larson et al PRL (2007)

ScatteredIntensity

Cluster calculations
Haverkort, et al PRL (2007)

AQRB @ AOFSRR Cheiron School 2008

High Resolution Measurement
8 meV resolution at 15.8 keV, Si(888)Temperature scan of mono over 2 eV or 60K

Baron, et al, April 2007 @ BL35XU
Detwinned Sample

Havercourt, Sawatzky, et al, May 07
NiO6 cluster calculation
Note splitting of 1700 meV excitation.

AQRB @ AOFSRR Cheiron School 2008

Towards a Dispersive Excitation?

BarOrbiton: Movie Maekawa, Dispersion Ishihara

AQRB @ AOFSRR Cheiron School 2008

Larger Energy Range

Hiraoka et al, Submitted
AQRB @ AOFSRR Cheiron School 2008

NRIXS

MgB2 Collective Excitation

Excitation repeats from one zone to the next…
AQRB @ AOFSRR Cheiron School 2008

AQRB @ AOFSRR Cheiron School 2008

X-Ray Raman Scattering

Nagasawa, et al, J. Phys. Soc. Jpn. 58 (1989) pp. 710-717
AQRB @ AOFSRR Cheiron School 2008

X-Ray Raman Scattering(Example of Ice Under Pressure)

Suppose you would like to measure the structure of the oxygen kedge (at 532 eV) of a sample inside of a high pressure cell with 1mm
thick diamond windows?
Diamond:labs < 0.5 um 500 eVlabs ~ 2 mm 10 keV
Easier at 10 keV than 0.5 keV
Note: need dipole approx. (Q.r<1) to be good tocompare with usual XAFS.

AQRB @ AOFSRR Cheiron School 2008

AQRB @ AOFSRR Cheiron School 2008

Compton Scattering
For very large Q and !E<<E one can takem## dp x dp y $ (p z = pQ )hQm%
J(pQ )hQ

S(Q," ) =!

From Shülke

Typical: Q~100Å-1
E>100 keV
Ie: Compton scattering projects
out the electron momentum
density.
Typical of incoherent scattering…
AQRB @ AOFSRR Cheiron School 2008

Three-Dimensional Momentum DensityReconstruction
Three-dimensional momentum density, n(p), can bereconstructed from ~10 Compton profiles.

J(p z ) = !! n(p)dp x dp y
Reconstruction:#Direct Fourier Method#Fourier-Bessel Method#Cormack Method#Maximum Entropy Method

Momentum density, n(p)AQRB @ AOFSRR Cheiron School 2008

Fermi surfaces of Cu and Cu alloys

Cu-15.8at%AlCu

Cu-27.5at%Pd

Determined by Compton scattering at KEK-AR
J. Kwiatkowska et al., Phys. Rev. B 70, 075106 (2005)

AQRB @ AOFSRR Cheiron School 2008

Fermi Surface of PdH0.84

Exp.
Experiment and theory
The Fermi surface is similar to that of Ag,
but the size of neck is about 2 times larger.

PdH

fcc-PdTheory

fcc-AgS. Mizusaki et al. Phys. Rev. B 73(2006)113101
AQRB @ AOFSRR Cheiron School 2008

AQRB @ AOFSRR Cheiron School 2008

Nuclear Inelastic ScatteringFirst Demonstrated (Clearly) by Seto et al 1995

Mössbauer Resonances Exist in Different Nuclei…

Resonances have relatively long lifetimes so that if one has apulsed source, one can separate the nuclear scattering by using afast time resolving detector.
Nuclear Scattering
AQRB @ AOFSRR Cheiron School 2008

NIS Setup
Use a narrow bandwidth monochormator
The nucelar resonance becomes the analyzer.

!E ~ eVHigh Heat LoadMonochromator!E ~ meVHigh Resolution
Monochromator

1. Ein = Eres
2. Ein + Ephonon = Eres
3. Ein -!$phonon = Eres

SampleForward Detector(NFS)
Incoherent Detector(NIS)Single Photon
Creation

Element- SpecificProjectedPhonon DOS

Two Photon
Annihilation-40-30-20Y

Single Photon
Annihilation
PhononCreation +Annihilation-10010Energy (meV)Two PhotonCreation

20305401015202530
Time After Excitation [ns]35Time Domain
Mossbauer Spectroscopy
AQRB @ AOFSRR Cheiron School 2008

NIS Gives the Partial Projected DOS
Example of the Fe-As Superconductors

Partial= Element SpecificProjected= Weakly Directional

I ~ " •k1!

Calculation: Boeri et al

Measurement: Higashitaniguchi et al

AQRB @ AOFSRR Cheiron School 2008

NIS: Good and Bad
Important things to note:
1. Element and isotope selective.
2. Gives Projected Density of states NOT Dispersion(But it does this nearly perfectly)

AQRB @ AOFSRR Cheiron School 2008

NIS Example: Surface DOS
Slezak et al PRL 99 (2007) 066103

57Fe monolayers near the surface of 56Fe

Note projection!
AQRB @ AOFSRR Cheiron School 2008

NIS Example: Biological Macro-Moleculese.g.

Where element specificity
can help a lot.

AQRB @ AOFSRR Cheiron School 2008