Inelastic X-Ray Scattering
These slide are for
distribution to the Cheiron
Summer School Students.
Alfred Q.R. Baron
Materials Dynamics Laboratory (RIKEN)
Research and Utilization Division (JASRI)
SPring-8
AQRB @ AOFSRR Cheiron School 2013

Scope & Outline
Huge & Complex Topic - Appropriate for a semester, not an hour…

Main Goal:
Introduce Capabilities & Put them in Context
What properties can be measured?
Why consider these techniques?
Outline:Introduction
Instrumentation
Non-Resonant
Resonant
Others (time permitting)

AQRB @ AOFSRR Cheiron School 2013

Some References
Shulke, W. (2007), Electron Dynamics by Inelastic X-Ray Scattering.!
New York: Oxford University Press.!
& References therein (RIXS, X-Ray Raman, NRIXS…)!
Squires, G. L. (1978). Introduction to the Theory of Thermal Neutron Scattering.!
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, Springer-Verlag.!

Cooper, M.J. (1985). Compton Scattering Rep. Prog. Phys. 48 415-481
Ament, L.J., et al, (2011). RIXS, Rev. Mod. Phys. 83 705-767

AQRB @ AOFSRR Cheiron School 2013

Calvin & Hobbes (Watterson)
AQRB @ AOFSRR Cheiron School 2013

Scientific Information(from IXS)

Atomic Dynamics <-> Motions of atoms in a solid (phonons) or liquid.

Phase transitions, thermal properties, fundamental science (Atomic binding)Electron-phonon coupling, Magneto-elastic coupling
Superconductors, Ferroelectrics, multiferroics, etc

Electronic Dynamics <-> Motions/transitions of electrons
Chemical Bonding (Valence, etc)
Electronic Energy Levels (atomic/molecular)
Delocalized Electronic Excitations
Generalized Dielectric Response
Fermi-Surface Topology
Magnetic structure
Complex Materials
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Table Of IXS Techniques/Applications
TechniqueComment

Energy Scale

Information

X-Ray Raman(E)XAFS in Special Cases

Ein~10 keV
ΔE~100-1000 eV

Edge Structure, Bonding

ComptonOldestNote: Resolution Limited

Ein ~ 150 keV
ΔE ~ keVElectron Momentum Density
Fermi Surface Shape

Magnetic Compton

Weak But Possible

Ein ~ 150 keV
ΔE ~ keV

Density of Unpaired Spins

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

Electronic Structure

Under Development
Now Exploding

0.1-1.5 keV
ΔE ~ 0.05 - 5 eVElectronic & Magnetic
Structure

Low Rate
SimplerEin~10 keV
ΔE ~ <1-50 eV

Electronic Structure

RIXS

Resonant IXSSIXSSoft (Resonant) IXS

NRIXS

Non-Resonant IXS

IXS

Ein~16-26 keV

Large Instrument
ΔE = Typical
Energy Transfer (NotResolution)
ΔE ~ 1-100
meV
High-ResolutionNote:
IXS

Phonon Dispersion

Note also: Limit to FAST dynamics (~10 ps or faster)

NIS

Nuclear IXSAQRB @ AOFSRR Cheiron School 2013

Atom SpecificVia Mossbauer Nuclei

Ein ~ 14-25 keV
ΔE ~ 1-100 meVElement SpecificPhonon Density of States(DOS)(arb. units)

Excitation Energy Scales
Elastic

Phonons

S(Q,omega)

ComptonCore

QuasiElastic
Valence

-4

100!

10

-3

-2

-1

0

Plasmon

1

2

10
10
10
10
10
Energy Transfer to Sample [eV]103

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Spectroscopy

Absorption vs. Scattering
Absorption
Spectroscopy

Measure absorption as you
scan the incident energy

Optical, IR, NMR

When energy hits a resonance,or exceeds a gap, or… get a change

Raman

Free Parameters: E1, e1, k1-> In principle, 3+ dimensions
but in practice mostly 1 (E1)

Scattering
Spectroscopy
IXS, INSE1 k1 e1

Optical Spect. NiONewman, PR 1959

E2 k2 e2

Free Parameters: E1, e1, k1, E2, e2, k2-> In principle, 6+ dimensionsin practice, mostly 4: E1-E2, Q = k2-k1

Scattering is more complex, but gives more information.
Energy scales PLUS spatial structure on scale of probe wavelength

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Where We Are Measuring
Between the Bragg Peaks…
Conventional Diffraction
Linear Scale

On Log Scale
Meas.

SiliconCalc

Precession Photo

Ruby

Laue Photo

Bragg peaks

Holt et al

For IXS we are usually measuring between the
Bragg peaks where the intensity is weaker.
A strong signal is down by 107, weak by 1011
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X-Ray Scattering Diagramk1
Q	E1 k1 e1hc
E 1 = ω1 == ck 1
λ1

k 1 = k1 =Q = k２- k1

2π

λ1

E2 k2 e2

hc = 12.398 keV• Å

Two Main Quantities:
Energy Transfer

Momentum Transfer

E or ΔE = E 1 − E 2 ≡ ωQ ≡ k 2 − k1
Q≡ Q ≈

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

4π

λ1

Periodicity

( )

sin Θ
2

d=

2π
Q

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Resonant vs Non-Resonant

Resonant:

Tune near an atomic transition energy
ie: K, L or M Edge of an atom
Generally High Rate
Complex interpretation
Energy fixed by resonance -> poorer resolution

NonResonant:Far from any atomic transition.
Small cross-section
Interpretation directly in terms of electron density
Choose energy to match optics -> good Resolution

RIXS
SIXS

IXS
NRIXS

Different Experimental Setups -> Modern Specialization
Nuclear Resonant & Compton Scattering -> Different
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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)
2 ω
d 2σ
2
*
2
I scattered (Q,ω ) ∝= re e2 •e1
S(Q,ω )dΩdωω1(

σ T hom son = re

2

(

)

2
*
e2 •e1

Thomson Scattering
Cross Section“A Scale Factor”)

S(Q,ω )
Dynamic Structure Factor“The Science”

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Different Views of S(Q,w)
2

S(Q,ω ) = ∑ pλ λ '
λ,λ '

∑ e

iQ•r jλ δ ( Eλ '−Eλ − ω )Transition between stateselectronsjFluctuations in electron density==1
dt d 3r d 3r' e−iQ•r ρ (r',t = 0)ρ + (r + r',t )
∫
2π

1
1
Im{− χ (Q,ω)}π 1− e−ω /kBT

=

→

N∑
q

∑ ∑
Modes

d
Atoms
/Cell

1
1
1
Im{−ε −1(Q,ω)}−ω /k B T
π 1− ev(Q)2fd (Q) −W d (Q)eQ •eqjd eiQ•x d δ(Q−q),τ Fqj (ω)
2M dGeneralized Response(e.g. Dielectric functions)

See Squires, Lovesy, Shulke, Sinha (JPCM 13 (2001) 7511)

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Why is it Better to Measurein Momentum/Energy Space?
For diffraction (and diffractive/coherent imaging), one goes togreat lengths to convert from momentum space to real space.
If possible, a direct real-space measurement would (sometimes) be preferred.

Equilibrium Dynamics: Q,E space is what you want.
Normal modes -> peaks in energy space -> clear and “easy”
Periodicity of crystals -> Excitations are plane waves-> Q is well definedNon-equilibrium dynamics -> Real space (x,t) can be better.
Non-periodic (disordered) materials -> Expand in plane waves. (oh well)

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Kinematics
Conservation of Energy

M
k1
Q	

E1 k1 e1

Q = k２- k1
E2 k2 e2Kinetic Energy Given to Sample:

E recoil

Take: M=57 amu, Q/c = 7 Å-1 -> Er=2.3 meV

p2
 2Q 2==2M
k′2M
E′ eˆ '
Q
∫ dω ω S(Q,ω )=2M∫ dω S(Q,ω )2

f-sum rule:
Compton Form: λ 2 − λ 1 =h(1 − cos Θ)
Mc

λc =

2

h
= 0.0243Å
m ec

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The IXS Spectrometer
An Optics Problem
Main Components

k ′ E ′ eˆ'

Monochromator:
Modestly Difficult
Accepts 15x40 μrad2

Sample Stages
Straightforward
Only Need Space

Analyzer:
Large Solid Angle
Difficult

The Goal: Put it all together and
Keep Good Resolution, Not Lose Flux

Note: small bandwidth means starting flux reduced by 2 to 3 orders of magnitude…
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Basic Optical Concept(Hard x-rays)

Bragg’s Law : λ = 2d sin(ΘB)	

 =>	

Δθ = tan⎛⎜⎝Θ ⎞⎟⎠ ΔE
B E

Working closer to ΘB~90 deg. maximizes the angular
acceptance for a given energy resolution…
Better energy resolution-> Closer to 90 degrees-> Large Spectrometer

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High Resolution Analyzer Crystals
The more difficult optic…

Detector

Require:	Correct Shape (Spherically Curved, R=9.8 m)	

Sample		

Not Strained (ΔE/E~few 10-8-> Δd/d<<few10-8)	

		

Analyzer
Crystal

Method: Bond many small crystallites to a curved substrate.
1. Cut	2. Etch	3. Bond to 	

Substrate	

4. Remove Back	X-Rays	104 Independent	Perfect Crystals	

Note: For resolution >300 meV, bending can be OK.
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Analyzer Crystal

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 2013

A High (meV) Resolution Spectrometer

Spectrometer Photo
Spectrometer Photo
Slit System	

12 Chan. CZT Detector	

Sample	Analyzer Array	

10m Arm & 12 Analyzers
Vacuum Flight Path	

Incident�Beam	

ϕ~100 µm	ϕ 20 µm Possible	

Granite Base w/Airpads	

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A Medium Resolution Spectrometer
Medium Resolution Spectrometer:
Arm Radius: 1 to 3 m
Resolution: ~0.1 to 1 eV
Used for RIXS and NRIXS
BL12XU
3m Arm at BL12XU
BL12XU (Early)

BL11XU

BL43LXU

Shorter Possible(later, if time)

10

Note difference between RIXS and NRIXS

NRIXS: Choose the energy to match the optics
RIXS: Resonance chooses energy -> usually worse resolution!E (meV)!E (meV)
1

0.1

10

15

20

25

30

35

40

E(keV)AQRB @ AOFSRR Cheiron School 2013

Other Spectrometers @ SPring-8
NRS Optics BL09

RIXS Spectrometer
2m Arm, BL11XU
Emission Spectrometerφ~1.5m Chamber

Compton Spectrometer BL08(~eV Resolution)
Hayashi, et al
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High Resolution Spectrometers
ESRF (ID28)

APS (Sector 30)

HRIXS~10m scale instruments
SPring-8 BL35XU

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Softer and Bigger…
SIXS @ NSLSII
W. Leonhardt, J. Dvorak, I. Jarrige

From ~ 2016
Soft inelastic x-ray scattering: 10 meV resolution at 1 keV via gratings
AQRB @ AOFSRR Cheiron School 2013

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 Conditions
Pairing mechanism in superconductors
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Disordered Materials
Liquids & Glasses

First Glance: Triplet response similar for most materials.
Dispersing Longitudinal Sound Mode
+ Quasi-Elastic peakl-Mg

(Kawakita et al)a-Se (Scopigno et al)l-Si (Hosokawa, et al)
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The IXS Advantage
INS
Diagram

IXS has no kinematic limitations (ΔE<<<Eγ)
Large energy transfer at small momentum transfer-> excellent access to mesoscopic length scales

Glassy-Se

Water
Teixeira, et al
INS (1985)

Data of Foret et al

INS (1998)

Scopigno et al
IXS (2004)

Sette et al
IXS (1996)+15 meV	-15 meV	

No Incoherent Background
Also:
Small Beam Size (φ<0.1mm)

But:<1 meV resolution is hard
Low Rates for Heavy Materials

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“Fast Sound” at the Metal-Non-Metal Transitionin Liquid Hg
Universal Phenomenon in Liquids:
Expand a liquid metal enough and it becomes an insulator.

Ultrasonic
Velocity

Suggests a change in the microscopic
density fluctuations…
Probably general phenomenon…
but no confirmation yet.
(Next M-I transition under discussion)

Ishikawa, Inui, et al, PRL 93 (2004) 97801~2 months of beam time…
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On Positive Dispersion
Very General feature:
As Q increases the phase velocity of theacoustic mode becomes larger than the
Low-Q (e.g. ultrasonic) sound velocity.
Casual explanation

Sette et al

For smaller length scales (high Q) and higher
frequencies, a liquid, locally, resembles a solid
which has a faster sound velocity.
Partial explanation in terms of a visco-elastic model…
Scopigno & Ruocco RMP 2005
Ruocco & Sette CMP 2008
Bryk et al JCP 2010
Hosokawa, et al

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Dynamical Distinction
Widom Line
Maximum in Cp

Supercritical
Argon

Simeoni et al NPhys 2010

Take the presence of Positive Dispersion
as the definition of liquid-like behavior
Gorelli et al, PRL (2006)
Simeoni et al, NPhys (2010)
Also Bencivenga et al EPL (2006)

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Shear Mode in a Simple Liquid
Pressure Wave in a Liquid:
Nearly Always Visible

Shear Wave -> Harder...
S (Q,ω ) ≈∫ dte ∫ dr ∫ dr′−iωte iQ•( r− r ′) ρ (r′,t) ρ(r,t = 0)0.4
liquid Ga
40°C-1

Q (nm ) =

?

2DHO
0.3
residual x2

10.6

1DHO

S(Q,ω)/S(Q) (meV-1)

0.2

2DHO
0.1
9.2
1DHO
0.0

Weak, but significant, signal.
-20

0
ω (meV)

20

Hosokawa, et al, PRL (2009)

Next experiment: l-Cu
2.5 Days -> ?
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Liquid Excitations = Solid + Disorder?
Giordano & Monaco, PNAS (2010)

IXS from Na: Above & Below TM

Black = Polycrystalline NaBlue = Liquid Na
Red = Polycrystal + Scaling by
Density, T, & Blurring…
Not bad …
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Phonons in a Crystal
Normal Modes of Atomic Motion = Basis set for small displacements
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

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MgB2 As An Example
Layered Material
Hexagonal Structure

B Layer

B-B Bond is Short& Stronger

Mg Layer

Mg-Mg Bond isLonger & Weaker3 Atoms/cell -> 9 modes / Q PointAQRB @ AOFSRR Cheiron School 2013

Acoustic and Optical Modes
Acoustic Modes are Continuum (Smooth) Modes

LA ModeCompression ModeTA ModeShear Mode

Optical Mode
Atoms in one unit cellmove against each-other

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Dispersion of an Optical Mode(0 0 0)(0.25 0 0)

(0.5 0 0)

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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
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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.

PhononMomentaQ<2kF

Large Momentum
Q>2kF
Can Not Couple to theElectronic system
Fermi Surface
Diameter = 2kf
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Superconductors
Systems Investigated include
MgB2, Doped MgB2, CaAlSi, B-Doped Diamond
Hg1201, LSCO, YBCO, LESCO, Tl2212, BKBO, NCCO,
Bi2201, Bi2212, Nickelates, Oxychlorides
Fe-As Systems: LaFeAsO, PrFeAsO, BaKFeAs

From T. Fukuda

Dark Blue Line: Conventional, Phonon-Mediated Superconductors
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MgB2
High Tc (39K)

Phonon Structure

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.
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Electron-Phonon Coupling in MgB2
SpectraDispersion
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

Linewidth

A2u

40

Γ	

(2.19, 0, 0.03)

20

0

!
Points: Measurements at BL35

10
0

E1u

M

20

A

!-M (Å-1)M	(2.15, 0, 0.01)(2.11, 0, -0.01)

Lines: Calculations by Bohnen et al.

Clear correlation between
0
50
100	

linewidth & softening.
Excellent agreement with LDA Pseudopotential calculation.
PRL 92(2004) 197004: Baron, Uchiyama, Tanaka, ... Tajima
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Carbon Doped Mg(CxB1-x)2MΓ

2%C, Tc=35.5K

12.5% C, Tc=2.5K

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

050(2.29,0,-0.03)
100

0

50

0

50(2.13,0,-0.01)
100

0

50

100
(2.22,0,0.03)

100

Phonon structure correlates nicely with Tc for charge doping.
(Electron doping fills the sigma Fermi surface)
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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

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Phonons in the Cuprates…
Everyone has their favorite mode, or modes, usually focus on Cu-O planes(0 0)In-Plane Mode:(0.5 0)Stretching mode
Out of Plane Modes:
Buckling Mode
Apical ModeAt the level of phonon spectra, the anomaly of theBond Stetching Mode is very large
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Copper Oxide Superconductors
Remain Challenging…
C-axis modes

De-Twinned YBCO:YBa2Cu3O7-!
Tc = 91 K

In-Plane Modes

Lines = Calculation
Points = IXSBeautiful Agreement
Shows Bond Stretching Anomaly
Is Huge (>> Buckling Anomaly)

Problems

Compare IXS to Calculation
At low T (~30K)

Bohnen, et al.

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Iron-Pnictide Superconductors
High-Tc demonstrated February 2008 (Hosono’s group)(Tc saturated within months…)

Several families: Fe with Tetrahedral As (or Se)
Proximity to Magnetic Order
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Phonons in the Iron Pnictides
1111 Materials -> 8 Atoms/cell-> 24 Modes (6 mostly oxygen)
Magnetism -> 16 Atoms / 48 Modes
No ab mirror plane-> Complex motions appear quickly
as one moves away from gamma.

Phonon response, in itself, is remarkably plain:
NO very large line-widths
NO obvious anomalies
NO asymmetric Raman lines

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Some Examples of Measured Spectra:
In-PlaneC-axisParentEstimateof modemotions
based oncalc

Clear differences in measured spectra (with doping, temperature)-> interpretations requires modeling…
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Basic DFT (GGA) for PrFeAsO(No Magnetism)

Some agreement, but details are poor
Also, fails to get correct As height
above the Fe planes.
Fe-As Bond Length
Expt: 2.41 Å
GGA: 2.31 - 2.33 Å
100

B1g

Energy (meV)

80

60

300K
44K
16K

E2g
A2u

40

E1u
20

0

M

Γ

A

Points: Measurements at BL35
Lines: Calculations by Bohnen et al.

A Better Model is Needed

Symbol size: Measured Intensity
Line Thickness: Calculated Intensity
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Different Models:
Original: Straight GGA for Tetragonal stoichiometric PrFeAsO
O7/8: Super cell 2x2x1 with one oxygen removed
and softened Fe-As NN Force constant(31 atoms/prim cell, Tetragonal, No Magnetism)
Magnetic Orthorhombic: LSDA for LaFeAsO withstripe structure of De la Cruz (16 atoms/prim. cell, 72 Ibam)
Magnetic Tetragonal: LSDA for LaFeAsO with stripes
Force a=b (to distinguish effects of structure vs magnetism)
Soft: As “Original” but soften the FeAs NN Force constant by 30%
Clipped: Mag. Ortho. with cut force constantSoft IP: “Original” but soften FeAs NN In Plane components

Original ab-initio Calculations: Nakamura & Machida

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Compare dispersion with various models
In-PlaneRed =Data (SC)Blue =
Calc.

C-axisSize:
Intensity

Over all: Better fit with magnetic calculations
And best fit with either “clipped” or “IP Soft” model

AQRB @ AOFSRR Cheiron School 2013

Comments
Of the straight ab-initio calculations, magnetic models do better thannon-magnetic due to softening of ferrmagnetically polarized modes
However, they get details wrong, including too high an energy for AF
polarized modes & predicting splitting that is not observed

Of the modified calculations, the in-plane soft generally seems best,
but still data-calc difference are larger than doping/T effects.
Many people have suggested some sort of fluctuating magnetism, especiallywhen magnetic calculations were seen to be better than non-magnetic
calcs for the (non-magnetic) superconducting materials.
However, phonon response of parent and SC are nearly the same, and it
seems unlikely that fluctuating magnetism is the answer
in the parent material which shows static magnetism.

Still some missing ingredient(s) in the calculation-> Interpretation Difficult
AQRB @ AOFSRR Cheiron School 2013

Towards A Better Model?
Fitting of full spectra: intensity vs energy transfer.
Zeroth Approximation: All Samples are the Same
Doping and Temperature Dependence are Weak
Differences between samples is generally much smaller
than between any calculation and the data-> Fit all spectra to a common modeland then fit subsets of the data to determine
effects of doping or phase transitions.

AQRB @ AOFSRR Cheiron School 2013

Fit Full Spectra

In-Plane Soft is
NOT bad but also
But also NOT great.

Some improvement by
allowing parts of nearly all
NN bonds to change.
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Phonons in a Quasicrystal

Mostly like a solid but some glassy character.
Building a Quasicrystal (Zn-Mg-Sc)

Periodic (BCC) -> Crystalline Approximant
Aperiodic -> Quasicrystal

Compare to crystalline approximant &Simulation (2000 atoms/cell)
General Trend: Blurring out
past a cutoff energy“Pseudo-Brillouin” zone size
De Boissieu, et al.
Nature Materials, Dec 2007

Red: Fits, Blue: Simulation
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Ferroelectrics
Develop spontaneous polarization over macroscopic (>~um) domains when T is below
the ferroelectric transition temperature (T0). The origin is a displacement (offcentering) of ions. This is switchable by an external (electric) field.
Zeroth Approximation -> Two types of transitions“Displacive” transition where there is a “continuous” below T“Soft Mode” transition Examples: BaTiO3, KTaO3, Gd (MoO4)3
Soft Mode
Nomenclature

Ferrodistortive transition involves softening of gamma point mode
(ferroelectric modes)
Antiferrodistortive involves softening of zone boundary mode
(unit cell size increases)

Lines & Glass“Order-Disorder” transition where displacements occur firstmetastably and then condense. No soft mode.
Examples KH2PO4(KDP), NaNO2, Organics

A

B

Perovskite structure (ABO3) popular as it is relatively simple
and the cubic structure is inherently unstable.. Why?
(3 atoms & one lattice constant)
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Multiferroic EuTiO3Perovskite – Similar to SrTiO3

Katsufuji& Takagi ,PRB,2001

But with magnetism & coupling ofmagnetic & dielectric response
Phase transition just below RT – putativerotation oxygen octahedra.
Calculations say disorder-order.
Bussman-Holder, PRB, 2011

1mm

IXS -> Phonon Softening-> Displacive
AQRB @ AOFSRR Cheiron School 2013

Dispersion, Shell Model, & Approaching TN
q=(0.05 0 0)

Shell model -> Good agreement
Suggests “soft” mode has Slater
character.

“Softening” (or weight shift) as T is
reduced toward TN consistent with
gradual change in dielectric response

AQRB @ AOFSRR Cheiron School 2013

Multiferroics
Sr1-xBaxMnO3

TbMnO3
Cycloidal spin structure coupled todevelopment of polarization below 28K

Classical ferroelectric softmode
Sakai, et al. PRL 2011 (Tokura-lab)

No observed phonon softening may suggest
a different (magnetic) driving force
Kajimoto, et al. PRL 2009 (Tokura-lab)
AQRB @ AOFSRR Cheiron School 2013

IXS under High
Pressure

General Viewpoint: Just another thermodynamic variable.
Specific: elastic properties in extreme (geological) conditions
based on IXS sound velocity measurements
MgO Single Crystal in Ambient Conditions

Often: Just want the sound velocity
Precision/Accuracy 0.2/0.8% using
Christoffel’s Eqn & 12 Analyzer Array
H. Fukui, et al., JSR~1 Order Improvement in Precision
Over Previous IXSOne Scan with 12-Analyzers
AQRB @ AOFSRR Cheiron School 2013

135GPa,
2000-3500K

ICB, 330GPa,
5000~7000K

Vp, Vs (km/sec), r (g/cm3)	

High Pressure & Temperature for Geology

Vp, Vs Known
Need T, ρ,
Composition

Earth’s Center, 365 GPa
6000~8000 K

Needed: Lab measurements relating T, Density & Composition to V
AQRB @ AOFSRR Cheiron School 2013

Diamond Anvil Cells
P > 200 GPa
T > 2000K (Laser Heating)

So far with IXS:
170 GPa or 1800K

Diamonds: 2 x 1.5mm Thk
Sample: ~ Φ20 μm x 5μm Thk
Also Gasket & Pressure Medium
P increases -> Smaller Sample & Gasket Hole

Std Cell
Laser

Cell with Internal Heating

Small samples, Signal low, Poor signal to noise
AQRB @ AOFSRR Cheiron School 2013

IXS Data for Iron at 167 GPa
Very clear iron peak, but significant backgrounds(Note diamond background can be tricky-> careful orientation is required)

Sine fit gives velocity (Vp)

AQRB @ AOFSRR Cheiron School 2013

Sound Velocity in Pure Iron
Birch’s Law: Linear relation between density and velocity.
SIMPLE, in principle
But 3 Facilities ->mostly different results
SP8 is faster than ESRF
and similar to APS
ESRF recently became
faster than before
Pure Iron to 170 GPa
Ohtani et al, submitted

T-Dependence:
APS is sensitive.
SP8 and ESRF are not.
Discussion is needed -> workshop
AQRB @ AOFSRR Cheiron School 2013

Novel Uses of The Phonon Intensity
Phonon Cross Section:

In principle, the phonon polarization is complex, but in some cases, it can be
simple, or smooth, letting one get information aboute.g.
the form factor from frequency resolved measurements or
sharp frequency changes from integrated measurements
AQRB @ AOFSRR Cheiron School 2013

Using Thermal Diffuse Scattering(TDS)
Phonon Intensity ~ 1/w -> In simple materials can use intensity togain insight about phonon frequencies
Long history… at least to Colella and Batterman PR 1970 (Va dispersion)More sensitive -> See Kohn anomalies when
phonons span the Fermi surface

Zn,
TDS from SiliconHolt, et al, PRL 1999

Bosak et al, PRL 2009

Detailed Phonon/FS behavior in SIMPLE materials
More generally very useful, but not so detailed… use it to learn where to look…
AQRB @ AOFSRR Cheiron School 2013

Atomic -> Electronic Dynamics
AtomicDynamics
Correlated atomic motions (phonons) play a role in many phenomena(e.g. superconductivity, CDWs, phase transitions, thermoelectricity, magneto-elastic phenomena etc)

Electronic excitations similar: Orbitons...?

Orbiton Movie
S. Maekawa

1 electron-> Very Weak

Calculated Orbiton Dispersion
Ishihara

Key is to see momentum dependence (dispersion).
First Attempt via IXS: NJP 2004

AQRB @ AOFSRR Cheiron School 2013

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, ΔE~300 meV

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

Cai, et al, BL12XU, UnpublishedAQRB @ AOFSRR Cheiron School 2013

Orientation Dependence

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

Cluster calculations
Haverkort, et al PRL (2007)

ScatteredIntensity
AQRB @ AOFSRR Cheiron School 2013

First High Resolution Experiment
7 meV resolution at 1800 meV energy transfer

Cleaner “Optical Spectroscopy” due to
1. Non-resonant interaction S(Q,w)
2. Large Q & Q dependence-> selects multipole order.
-> atomic correlations.
Linewidth -> information about environment
Spin fluctuations
Lattice interactions (Franck-Condon)
d-d Excitation in NiO
3 Days/Spectrum

Collective interaction <-> dispersion

Relevance to correlated materials…
Gaps (Mott, Charge Transfer, SC) and
Mid-IR band in high Tcs
f-electron transitions, etc
AQRB @ AOFSRR Cheiron School 2013

Larger Energy Range

Hiraoka et al
AQRB @ AOFSRR Cheiron School 2013

“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 vs non-resonant IXS experiment.

Non-resonant experiment is simpler and can have higher resolution… but badly flux limited
AQRB @ AOFSRR Cheiron School 2013

The Orbiton Story(One, mostly experimental, viewpoint)

Orbital order exists -> there should be an equivalent excitation
Essential picture is of a correlated d-d excitation – change in electronic
wave function on one atom is correlated with change at other atoms.

2001

LaMnO3

But some dissent:

Two phonon peak?
Gruninger (n), Kruger (prl), Marin-Carron (prl)And also corroboration
Calculated Dispersion

Raman spectra from different materials
AQRB @ AOFSRR Cheiron School 2013

Calculated Orbiton
Dispersion

Oles, Feiner, Zaanen (PRB 2000)

Saitoh et al, (N 2001)

Khaliullin & Okamoto (PRL 2002)

van den Brink (PRL 2001)

Still Some Debate:
Ishihara (PRB 2004)
Energy scale?
Coupling to phonons and/or spin?
Linewidth small or large?
AQRB @ AOFSRR Cheiron School 2013

Resonant IXS (RIXS)Soft x-ray RIXS (SRIXS)

K-Edge RIXS (d-d excitations)

LaMnO3 Inami, et al (prb 2003)

Ulrich, Ament, et al (PRL 2009)At SLS/ADDRESS

L3 in YTiO3, 55 meV Resolution at 450 eV

KCuF3 Ishii, et al (PRB 2011)

Resolution Improving:
1000 -> 250 meV -> 70 meV

2-orbiton signal at 250 meV...

2011: STILL NO DISPERSING EXCITATIONSAQRB @ AOFSRR Cheiron School 2013

Recent Work
May, 2012

High Energy Excitation in Sr2CuO3AQRB @ AOFSRR Cheiron School 2013

Resonant IXS(RIXS/SIXS)

Fig: Ament et al, RMP 2011

Fig: Ament et al, RMP 2011

Direct RIXS

InDirect RIXSK Lines: Stronger, easier to reach for transition metals
L Lines: Weaker, lower energy
Photons: Orbital angular momentum change, Δl=1, dominates: s-> p or p->d
Science: Most interest is in d-orbitals
AQRB @ AOFSRR Cheiron School 2013

Spin Waves Possible

AQRB @ AOFSRR Cheiron School 2013

X-Ray Raman Scattering

Nagasawa, et al, J. Phys. Soc. Jpn. 58 (1989) pp. 710-717
Tohji&Udagawa, PRB 39 (1989) 7590
AQRB @ AOFSRR Cheiron School 2013

X-Ray Raman Scattering
Suppose you would like to measure the structure of the oxygen kedge (at 532 eV) of a sample inside of complex sample environment…
Diamond:labs < 0.5 um 500 eVlabs ~ 2 mm 10 keV
Easier at 10 keV than 0.5 keVSupercritical
Water
Ishikawa, et al,
Submitted

Tetrohydrofuran Clathrate
Conrad, et al, PRL (2009) 218301

AQRB @ AOFSRR Cheiron School 2013

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 2013

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

ΔE ~ eVHigh Heat Load
Monochromator

ΔE ~ meVHigh Resolution
Monochromator

1. Ein = Eres
2. Ein + Ephonon = Eres
3. Ein - Ephonon = Eres

SampleForward Detector(NFS)
Incoherent Detector(NIS)

Single Photon
Creation

Element- Specific	

Projected	

Phonon DOS	

Two Photon
Annihilation

-40

-30

-20

Y

Single Photon
Annihilation
PhononCreation +Annihilation

-10

0

10

Energy (meV)Two Photon
Creation

20

30

5

40

10
15
20
25
30
Time After Excitation [ns]35Time Domain	Mossbauer Spectroscopy	

AQRB @ AOFSRR Cheiron School 2013

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)
3. Resolution given only by monochromator(analyzer is ~ueV)
Easier optics but setup not optimized(compensated by large cross section)

AQRB @ AOFSRR Cheiron School 2013

Surfaces by NIS
The large nuclear cross section allows sensitivity even tomonolayers with relatively low backgrounds

In-Situ Deposition@ ESRF
Stankov et al, JP 2010

57Fe on W(110)

Stankov et al PRL (2007)

Also: Multilayers - Cuenya et al, PRB 2008

57Fe with 56Fe
Slezak et al PRL 2007

AQRB @ AOFSRR Cheiron School 2013

Example (NRVS/NIS/NRIXS) In Biology
S. Cramer, et al, JACS

Measurement to determine the products of biological reactions via site-selective
vibrational spectroscopy and comparison against calcs and model compounds
A compound in the nitrogen cycle…
Is X present? How many irons?

Toxicity of Nitric Oxide (NO)-> Reaction products previously believed to bemononuclear are dinuclear

AQRB @ AOFSRR Cheiron School 2013

Compton Scattering
For very large Q and ΔE<<E one can takem∫∫ dp x dp y ρ (p z = pQ )Q
m≡J(pQ )
Q

S(Q,ω ) =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 2013

Three-Dimensional Momentum Density
Reconstruction
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)Note: a bulk probe that is tolerant of sample imperfections.
From Y. Sakurai

AQRB @ AOFSRR Cheiron School 2013

Fermi surfaces of Cu and Cu alloys	

Cu-15.8at%Al	

Cu	

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 2013

Hole Locations in La2-xSrxCuO4
Sakurai, et al, Science 2011

Measured Results for Different Doping
Parent vs Optimal Doping:
Holes in ZR singlet state
Optimal vs Overdoped
Holes in Cu dz2 orbital
Cluster Calculations& Some density that is not yet understoodBand Structure Calculations

AQRB @ AOFSRR Cheiron School 2013

Reducing the Two-Theta Arm Size
Dispersion Compensation: Houtari, et al JSR (2005)dAnimation
D. Ishikawa

4R 2 ΔEd =p E

5 meV at 16 keV
R=2m, p=0.1 -> d=50 mm
AQRB @ AOFSRR Cheiron School 2013

Temperature Gradient Analyzer(Ishikawa & Baron, JSR 2010)

Longer Arm: DT~0.1C~ meV resolution at 3m

l=150 to 200mm

Short Arm: DT: 1 to 10C~5 meV at 1m

AQRB @ AOFSRR Cheiron School 2013

A Nano-Volt Spectrometer
Rayleigh Scattering of Synchrotron Mossbauer Radiation (RSSMR)

Masuda, Mitsui, Seto, et al, JJAP (2008, 2009)
AQRB @ AOFSRR Cheiron School 2013

Beyond Plane Waves
Usual Measurement is a
two-point correlation function:

S (Q,ω ) =

1
2π

∫

dt e−iωt∫ dr ∫ dr′

x e iQ•( r− r ′) ρ (r′,t) ρ(r,t = 0)Complete picture includes higher order correlation functions

I(Q,t)I(Q',t') ∝ ρ(r',t) ρ (r,0) ρ(s',t') ρ (s,0)
Ps Scales
XFEL
Or
XFELO
(A) Autocorrelation at One Q(B) Cross-Correlation, Different Q
Wochner et al, PNAS (2009)
AQRB @ AOFSRR Cheiron School 2013

IXS Beamline Evolution
Effective Flux [1014/s/0.1%BW]

For meV Resolution at 20 keV

NewBeamline

NSLS-II(Not Proven)?
Present
Status
Source & BL
Improvements

ESRF

ERL
(~$1B,
Not Funded)
Machine Barrier

Expected
UpgradesAQRB @ AOFSRR Cheiron School 2013

A Next Generation Beamline
Dramatic Improvement to Source and Spectrometerallows new science…
New Field: Electronic excitations
Also many expts now flux limited:

Phonons in complex materials
Extreme environments (HT, HP liquids)
High pressure DAC work (Geology)
Excitations in metal glasses
Super-cooled liquids
etc

Improvements

Flux On Sample: x10
Parallelization: x3
Small Spot Size: x5

AQRB @ AOFSRR Cheiron School 2013

Quantum NanoDynamics Beamline(BL43LXU)

Medium resolution: 10-100 meVHigh resolution spectrometer: <1 to 6 meV
2m Arm, Large Q Acceptance
10 m Arm, Good Q Resolution, to 12 Å-1
Good tails using (888)
Large (42 element) analyzer array.
10m ArmFirst Monochromatic Light: Sunday

AQRB @ AOFSRR Cheiron School 2013

AQRB @ AOFSRR Cheiron School 2013

Thanks for Your Attention!

AQRB @ AOFSRR Cheiron School 2013