Inelastic X-Ray Scattering

Alfred Q.R. Baron
Materials Dynamics Laboratory (RIKEN)
Research and Utilization Division (JASRI)
SPring-8
AQRB
A
AQ
QR
RB
B @ AOFSRR
AOF
OFSR
OFSR
SRR Cheiron
SRR
Che
Ch
eiiro
ron
on School
Scho
Sc
hoo
oll 2
2012
0112
0

AQRB
A
QRB
Q B @ AOFSRR
AOF
OF
FSRR
SR Cheiron
SR
Cheiror n School
S hool 2012Sc
2

Some References

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?

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.

Outline:Introduction
Instrumentation
Non-Resonant Techniques
Resonant Techniques (Briefly)

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

AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

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
Chemical Bonding (Valence, etc)
Electronic Energy Levels (atomic/molecular)
Delocalized Electronic Excitations
Generalized Dielectric Response
Fermi-Surface Topology
Magnetic structure
Calvin & Hobbes (Watterson)
AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

Table Of IXS Techniques/Applications
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

RIXS
Resonant IXSHigh Rate
Somewhat ComplicatedEin ~ 4-15 keV
ΔE ~ 1-50 eV

Electronic Structure

SIXS

Under Development

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

Non-Resonant IXS

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

Electronic Structure

IXS

Large Instrument

Ein~16-26 keV
ΔE ~ 1-100 meV

Phonon Dispersion

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

Soft (Resonant) IXSNRIXS

High-Resolution IXSNIS
Nuclear IXSAtom SpecificVia Mossbauer Nuclei(arb. units)

Comment

S(Q,omega)

Technique

Energy Scale of Excitations
Elastic

Phonons
ComptonCore
QuasiElastic

-4
100

PlasmonValence10-3 10-2 10-1 100
101
102
Energy Transfer to Sample [eV]103

Note: ΔE = Typical Energy Transfer (Not Resolution)
Note also: Limit to FAST dynamics (~10 ps or faster)

AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

Spectroscopy

Where We Are Measuring

Absorption vs. Scattering

Between the Bragg Peaks…
Conventional Diffraction

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

Meas.

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

Optical Spect. NiONewman, PR 1959

Silicon
Scattering
Spectroscopy
IXS, Raman, INS

On Log Scale

Linear Scale

Calc

Precession PhotoE2 k2 e2E1 k 1 e 1

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

Ruby

For IXS we are usually measuring between the
Bragg peaks where the intensity is weaker.
A strong signal is down by 108, weak by 1012

Laue Photo

Scattering is more complex, but gives more information.

Bragg peaks

AQRB @ AOFSRR Cheiron School 2012

X-Ray Scattering Diagram

AQRB @ AOFSRR Cheiron School 2012

Resonant vs Non-Resonantk1E1 k 1 e 1
E 1 = ω1 =hc= ck 1
λ1

k 1 = k1 =Q =k- k1

2π

λ1

E2 k 2 e 2Resonant:
RIXS
SIXS

hc = 12.398 keV• Å

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

Two Main Quantities:
Energy Transfer

Momentum Transfer

E or ΔE = E 1 − E 2 ≡ ωQ ≡ k 2 − k1

NonResonant:

4π

IXS
NRIS

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

λ1

Periodicity

( )

sin Θ
2

d=

2π
Q

Far from any atomic transition.
Small cross-section
Interpretation directly in terms of electron density
Choose energy to match optics -> good Resolution
Slightly Different Experimental Setup
Nuclear Resonant -> Different entirely… later.

AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

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 )
dΩdω
ω1

(

(

σ T hom son = re e2 •e1
2*Thomson Scattering
Cross Section
A Scale Factor

)

2

)

S(Q,ω )

Different Views of S(Q, )2

S(Q,ω ) = ∑ pλ λ 'λ ,λ '∑ eiQ•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 )2π11Im{− χ (Q,ω)}π 1− e−ω /kBT

Dynamic Structure Factor

=

→

N∑
q

∑ ∑
Modes

d
Atoms
/Cell

1
1
1
Im{−ε −1(Q,ω)}π 1− e−ω /kBT v(Q)fd (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)

The Science

AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

Kinematics

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 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 defined2

Conservation of EnergyMk1E1 k 1 e 1

Q =k- k1
E2 k 2 e 2

Kinetic Energy Given to Sample:

E recoil =p2
 2Q 2=2M
k′2M
E′ eˆ '

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

Non-equilibrium dynamics -> Real space (X,t) can be better.

2

f-sum rule:

Non-periodic (disordered) materials -> Expand in plane waves. (oh well)Compton Form: λ 2 − λ 1 =AQRB @ AOFSRR Cheiron School 2012Q
∫ dω ω S(Q,ω )=2M∫ dω S(Q,ω )h(1 − cos Θ)
Mc

λc =

2

h
= 0.0243Å
m ec

AQRB @ AOFSRR Cheiron School 2012

The IXS Spectrometer
An Optics Problem
Main Components

Basic Optical Concept
k ′ E ′ eˆ'

Braggs Law : λ = 2d sin(Θ) =>

Δθ = tan⎜⎛⎝Θ ⎟⎞⎠ ΔE
B E

Monochromator:
Modestly Difficult
Accepts 15x40 µrad2

Working closer to ΘB~90 deg. maximizes the angular
acceptance for a given energy resolution…

Sample Stages
Straightforward
Only Need Space

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

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…
AQRB @ AOFSRR Cheiron School 2012

High Resolution Analyzer Crystals
The more difficult optic…

Detector

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

SampleAnalyzer Crystal



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

AQRB @ AOFSRR Cheiron School 2012

Analyzer
Crystal

Method: Bond many small crystallites to a curved substrate.
1. Cut2. Etch3. Bond to 
Substrate

4. Remove BackX-Rays104 IndependentPerfect Crystals

Note: For resolution >300 meV, bending can be OK.
AQRB @ AOFSRR Cheiron School 2012

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 AreaAQRB @ AOFSRR Cheiron School 2012

A High (meV) Resolution Spectrometer

Spectrometer Photo
Spectrometer Photo

12 Chan. CZT Detector

Medium Resolution Spectrometer:
Arm Radius: 1 to 3 m
Resolution: ~0.1 to 1 eV
Used for RIXS and NRIXS

SampleAnalyzer Array

BL12XU

10m Arm & 12 Analyzers

3m Arm at BL12XU
BL12XUB
L12XU (Early)(Early)Vacuum Flight Path

IncidentBeam
ϕ~100 μmϕ 20 μm Possible

BL11XU

BL43LXU

Shorter Possible(later, if time)10
ΔE (meV)

Note difference between RIXS and NRIXS
NRIXS: Choose the energy to match the optics
RIXS: Resonance chooses energy -> usually worse resolution

ΔE (meV)

Slit System

A Medium Resolution Spectrometer

1

0.1

10

Granite Base w/Airpads

15

20

25

30

35

40

E(keV)AQRB @ AOFSRR Cheiron School 2012

Other Spectrometers @ SPring-8

AQRB @ AOFSRR Cheiron School 2012

Other High Resolution Spectometers

NRS Optics BL09
ESRF (ID28)

APS (Sector 30)

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

Compton Spectrometer BL08(~eV Resolution)
Hayashi, et al
AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

Disordered Materials

Atomic Dynamics: Systems and Questions

Liquids & Glasses

Disordered Materials (Liquids & Glasses):

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

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

l-Mg(Kawakita et al)a-Se (Scopigno et al)

AQRB @ AOFSRR Cheiron School 2012

The IXS Advantage“Fast Sound” at the Metal-Non-Metal Transitionin Liquid Hg
INS
Diagram

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

Universal Phenomenon in Liquids:
Expand a liquid metal enough and it becomes an insulator.

Ultrasonic
Velocity

Glassy-Se

Water
Teixeira, et al
INS (1985)

Data of Foret et al

INS (1998)

l-Si (Hosokawa, et al)
AQRB @ AOFSRR Cheiron School 2012

Scopigno et al
IXS (2004)

Suggests a change in the microscopic
density fluctuations…
Sette et al
IXS (1996)+15 meV

Probably general phenomenon…
but no confirmation yet.

-15 meV(Next M-I transition under discussion)

Also:

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

But:<1 meV resolution is hard
Low Rates for Heavy Materials
AQRB @ AOFSRR Cheiron School 2012

Ishikawa, Inui, et al, PRL 93 (2004) 97801~2 months of beam time…
AQRB @ AOFSRR Cheiron School 2012

On Positive Dispersion

Dynamical Distinction

Very General feature:

Widom Line
Maximum in Cp

As Q increases the phase velocity of theacoustic mode becomes larger than the
Low-Q (e.g. ultrasonic) sound velocity.
Casual explanation

Supercritical
Argon

Sette et al

For smaller length scales (high Q) and higher
frequencies, a liquid, locally, resembles a solid
which has a faster sound velocity.

Simeoni et al NPhys 2010

Partial explanation in terms of a visco-elastic model…
Take the presence of Positive Dispersion
as the definition of liquid-like behavior

Scopigno & Ruocco RMP 2005
Ruocco & Sette CMP 2008
Bryk et al JCP 2010
Hosokawa, et al

Gorelli et al, PRL (2006)
Simeoni et al, NPhys (2010)
Also Bencivenga et al EPL (2006)

AQRB @ AOFSRR Cheiron School 2012

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

Liquid Excitations = Solid + Disorder?
Giordano & Monaco, PNAS (2010)

Shear Wave -> Harder...
S (Q,ω ) ≈AQRB @ AOFSRR Cheiron School 2012∫ dte ∫ dr ∫ dr′−iωte iQ•( r− r ′) ρ (r′,t) ρ(r,t = 0)

IXS from Na: Above & Below TM

0.4
liquid Ga
40°C-1

Q (nm ) =

?

2DHO
0.3
residual x2
10.6
1DHO

Black = Polycrystalline NaBlue = Liquid Na

0.2

Red = Polycrystal + Scaling by
Density, T, & Blurring…

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 -> ?
AQRB @ AOFSRR Cheiron School 2012

Not bad …
AQRB @ AOFSRR Cheiron School 2012

Phonons in a Crystal

MgB2 As An Example

Normal Modes of Atomic Motion = Basis set for small displacements

Layered Material
Hexagonal Structure

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

B Layer

B-B Bond is Short& Stronger

Mg Layer

Mg-Mg Bond isLonger & Weaker

Generally: Consider the unit cell periodicity separately
by introducing a continuous momentum variable, q.
-> 3R modes for any given q

3 Atoms/cell -> 9 modes / Q PointAQRB @ AOFSRR Cheiron School 2012

Acoustic and Optical Modes

AQRB @ AOFSRR Cheiron School 2012

Dispersion of an Optical Mode

Acoustic Modes are Continuum (Smooth) Modes(0 0 0)LA ModeCompression ModeTA ModeShear Mode

(0.25 0 0)

(0.5 0 0)

Optical Mode
Atoms in one unit cellmove against each-other

AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

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

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

AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

MgB2

Superconductors

High Tc (39K)

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

Phonon Structure

Nagamatsu, et al, Nature 410, (2001) 63.

Simple Structure…
straightforward calculation.

Electronic Structure

From T. Fukuda

Kortus, et al, PRL 86 (2001)4656

Dark Blue Line: Conventional, Phonon-Mediated Superconductors
AQRB @ AOFSRR Cheiron School 2012

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

BCS (Eliashberg) superconductor with mode-specific electron-phonon coupling.
AQRB @ AOFSRR Cheiron School 2012

Carbon Doped Mg(CxB1-x)2Electron-Phonon Coupling in MgB2
SpectraDispersion
100(2.31, 0, 0.01)

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

Energy (meV)

60

300K
44K
16K

(2.23, 0, - 0.03)

A2u

40

Γ

A

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(2.29,0,-0.03)
100

0

50

100
(2.29,0,0.03)

050 (2.22,0,0.03)
100

0

50

100
(2.22,0,-0.03)

0

50

0

50

0

50

0

50

0

50

0

50

0

50

0

50

100

(2.30,0,0.03)

10

(2.19, 0, 0.03)

20

M

20

0

E1u

0

M

30

FWHM

B1g
80

LinewidthΓ-M (Å-1)

100(2.24,0,-0.03)

M(2.15, 0, 0.01)

100(2.13,0,-0.01)

100(2.13,0,0.01)

100(2.16,0,0.01)(2.11, 0, -0.01)

Clear correlation between
0
50
100
linewidth & softening.
Excellent agreement with LDA Pseudopotential calculation.

Γ

100

100

100

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

PRL 92(2004) 197004: Baron, Uchiyama, Tanaka, ... Tajima
AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

Phonons in the Cuprates…

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

Everyone has their favorite mode, or modes, usually focus on Cu-O planes(0 0)In-Plane Mode:(0.5 0)Stretching mode
Extrapolation to the High Tc Copper Oxide Materials….
1. Much More Complex
2. Calculations Fail so interpretation in difficult

Out of Plane Modes:
Buckling Mode
Apical ModeAt the level of phonon spectra, the anomaly of theBond Stetching Mode is very large
AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

Copper Oxide Superconductors
Remain Challenging…

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)C-axis modes

De-Twinned YBCO:YBa2Cu3O7-
Tc = 91 K

In-Plane Modes

Lines = Calculation
Points = IXSStretching mode
Out of Plane Modes:
Buckling Mode
Apical ModeAt the level of phonon spectra, the anomaly of theBond Stetching Mode is very large
AQRB @ AOFSRR Cheiron School 2012

Beautiful Agreement
Shows Bond Stretching Anomaly
Is Huge (>> Buckling Anomaly)

Problems

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

Bohnen, et al.

AQRB @ AOFSRR Cheiron School 2012

Iron-Pnictide Superconductors

La1.48Nd0.4Sr0.12CuO4

High-Tc demonstrated February 2008 (Hosono’s group)
Phonon anomaly (blurring) is highly localized in
momentum space…

(Tc saturated within months…)Expt done by a neutron scatterer because
he could not get good enoughresolution using neutrons
Forces a reinterpretation of some Neutron
data (Reznik, Nature, 2006)

Note:
D. Reznik, et al

IXS Q Resolution
Analyzer array
Count rate limited.

2.5 days
AQRB @ AOFSRR Cheiron School 2012

Several families: Fe with Tetrahedral As (or Se)
Proximity to Magnetic Order
AQRB @ AOFSRR Cheiron School 2012

Phonons in the Iron Pnictides

Some Examples of Measured Spectra:
In-Plane1111 Materials -> 8 Atoms/cell-> 24 Modes (6 mostly oxygen)

C-axisMagnetism -> 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

ParentEstimateof modemotions
based oncalc

Clear differences in measured spectra (with doping, temperature)-> interpretations requires modeling…
AQRB @ AOFSRR Cheiron School 2012

Basic DFT (GGA) for PrFeAsO

AQRB @ AOFSRR Cheiron School 2012

Different Models:(No Magnetism)
Original: Straight GGA for Tetragonal stoichiometric PrFeAsO

Some agreement, but details are poor

O7/8: Super cell 2x2x1 with one oxygen removed
and softened Fe-As NN Force constant(31 atoms/prim cell, Tetragonal, No Magnetism)

Also, fails to get correct As height
above the Fe planes.

Magnetic Orthorhombic: LSDA for LaFeAsO withstripe structure of De la Cruz (16 atoms/prim. cell, 72 Ibam)

Fe-As Bond Length
Expt: 2.41 Å
GGA: 2.31 - 2.33 Å

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%

100

Clipped: Mag. Ortho. with cut force constant

B1g

Energy (meV)

80

60

300K
44K
16K

E2g

Soft IP: Original but soften FeAs NN In Plane components

A2u

40

E1u
20

0

M

Γ

A Better Model is NeededA

Symbol size: Measured Intensity
Line Thickness: Calculated Intensity
AQRB @ AOFSRR Cheiron School 2012

Original ab-initio Calculations: Nakamura & Machida

AQRB @ AOFSRR Cheiron School 2012Comments

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

C-axisSize:
Intensity

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.

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

Still some missing ingredient(s) in the calculation-> Interpretation Difficult

AQRB @ AOFSRR Cheiron School 2012

Towards A Better Model?

AQRB @ AOFSRR Cheiron School 2012

Fit Full Spectra

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 2012

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

Some improvement by
allowing parts of nearly all
NN bonds to change.

AQRB @ AOFSRR Cheiron School 2012

Phonons in a Quasicrystal

Ferroelectrics

Mostly like a solid but some glassy character.

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.

Building a Quasicrystal (Zn-Mg-Sc)

Zeroth Approximation -> Two types of transitions“Displacive” transition where there is a “continuous” below T“Soft Mode” transition Examples: BaTiO3, KTaO3, Gd (MoO4)3

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

Soft Mode
Nomenclature

Compare to crystalline approximant &Simulation (2000 atoms/cell)

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.

General Trend: Blurring out
past a cutoff energy
Pseudo-Brillouin zone size

Examples KH2PO4(KDP), NaNO2, OrganicsA

De Boissieu, et al.
Nature Materials, Dec 2007

B

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

Red: Fits, Blue: Simulation
AQRB @ AOFSRR Cheiron School 2012

Multiferroic EuTiO3Perovskite – Similar to SrTiO3

AQRB @ AOFSRR Cheiron School 2012

Dispersion, Shell Model, & Approaching TN
Katsufuji& Takagi ,PRB,2001

But with magnetism & coupling ofmagnetic & dielectric response

q=(0.05 0 0)

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 2012

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 2012

IXS under High
PressureHigh Pressure & Temperature for Geology

Specific: elastic properties in extreme (geological) conditions
based on IXS sound velocity measurements
MgO Single Crystal in Ambient Conditions
135GPa,2000-3500K

Often: Just want the sound velocity
Precision/Accuracy 0.2/0.8% using
Christoffels Eqn & 12 Analyzer Array

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

General Viewpoint: Just another thermodynamic variable.

Vp, Vs Known
Need T, ρ,
Composition

H. Fukui, et al., JSR~1 Order Improvement in Precision
Over Previous IXSICB, 330GPa,
5000~7000K

One Scan with 12-Analyzers

Needed: Lab measurements relating T, Density & Composition to V

AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

IXS Data for Iron at 167 GPa

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

Earth’s Center, 365 GPa
6000~8000 K

So far with IXS:
170 GPa or 1800K

Std Cell
Laser

Very clear iron peak, but significant backgrounds(Note diamond background can be tricky-> careful orientation is required)

Sine fit gives velocity (Vp)

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

Small samples, Signal low, Poor signal to noise
AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

Sound Velocity in Pure Iron

Novel Uses of The Phonon Intensity

Birch’s Law: Approximate Linear relation between density and velocity.
SIMPLE, in principle

Phonon Cross Section:

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 needed: Diamond? Sine fit? Other?

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 2012

AQRB @ AOFSRR Cheiron School 2012

Using Thermal Diffuse Scattering(TDS)

Atomic -> Electronic Dynamics

Phonon Intensity ~ 1/ -> In simple materials can use intensity togain insight about phonon frequencies

AtomicDynamics

Long history… at least to Colella and Batterman PR 1970 (Va dispersion)More sensitive -> See Kohn anomalies when
phonons span the Fermi surface

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

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
Learn where to look…
AQRB @ AOFSRR Cheiron School 2012

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 2012

d-d Excitations in NiOOrientation DependenceFirst 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

Orbitals
Results of Wanneir function
analysis of LDA+U calcsof Larson et al PRL (2007)
Larson, et al., PRL 99 (2007) 026401

Cai, et al, BL12XU, Unpublished

Cluster calculations
Haverkort, et al PRL (2007)

ScatteredIntensity

AQRB @ AOFSRR Cheiron School 2012

First High Resolution Experiment

AQRB @ AOFSRR Cheiron School 2012

Larger Energy Range

7 meV resolution at 1800 meV energy transfer

Cleaner Optical Spectroscopy due to
1. Non-resonant interaction S(Q, )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

Hiraoka et al
AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012Momentum Resolved Optical
Spectroscopy

The Orbiton Story(One, mostly experimental, viewpoint)

Orbital order exists -> there should be an equivalent excitation

Conventional Optical
Spectroscopy:(Absorption, Reflectivity)

Information on electronic energy levels but without
information on inter-atomic correlations or
atomic structure

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

With 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?

But some dissent:
Two phonon peak?
Gruninger (n), Kruger (prl), Marin-Carron (prl)Resonant experiment vs non-resonant IXS experiment.

Non-resonant experiment is simpler and can have higher resolution… but badly flux limited

And also corroboration
Calculated Dispersion

AQRB @ AOFSRR Cheiron School 2012

Raman spectra from different materials
AQRB @ AOFSRR Cheiron School 2012

Resonant IXS (RIXS)

Calculated Orbiton
Dispersion

Soft x-ray RIXS (SRIXS)

K-Edge RIXS (d-d excitations)

Ulrich, Ament, et al (PRL 2009)At SLS/ADDRESS
L3 in YTiO3, 55 meV Resolution at 450 eV

Oles, Feiner, Zaanen (PRB 2000)

Saitoh et al, (N 2001)

LaMnO3 Inami, et al (prb 2003)

KCuF3 Ishii, et al (PRB 2011)

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 2012

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

2-orbiton signal at 250 meV...

2011: STILL NO DISPERSING EXCITATIONSAQRB @ AOFSRR Cheiron School 2012

Recent WorkSIXS -> Spin Waves Possible

May, 2012

High Energy Excitation in Sr2CuO3AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

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 2012

AQRB @ AOFSRR Cheiron School 2012

X-Ray Raman Scattering

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

Suppose you would like to measure the structure of the oxygen kedge (at 532 eV) of a sample inside of complex sample environment…

Mössbauer Resonances Exist in Different Nuclei…

Diamond:labs < 0.5 um 500 eVlabs ~ 2 mm 10 keV
Easier at 10 keV than 0.5 keVSupercritical
Water
Ishikawa, et al,
Submitted

Resonances have relatively long lifetimes so that if one has apulsed source, one can separate the nuclear scattering by using afast time resolving detector.
Tetrohydrofuran Clathrate
Conrad, et al, PRL (2009) 218301

Nuclear Scattering
AQRB @ AOFSRR Cheiron School 2012

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

ΔE ~ eV

ΔE ~ meV

NIS: Good and Bad
1. Ein = Eres
2. Ein + Ephonon = Eres
3. Ein    = Eres

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)

SampleForward Detector(NFS)

High Heat Load
Monochromator
High Resolution
Monochromator

AQRB @ AOFSRR Cheiron School 2012

Incoherent Detector(NIS)

Single Photon
Creation
Single Photon
Annihilation

Element- SpeciﬁcProjected
Phonon DOS

Two Photon
Annihilation

-40

-30

-20

PhononCreation +Annihilation

-10

0

10

Energy (meV)Two Photon
Creation

20

30

5

40

10
15
20
25
30
Time After Excitation [ns]35Time DomainMossbauer Spectroscopy
AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

Surfaces by NIS

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

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

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?

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

57Fe on W(110)Toxicity of Nitric Oxide (NO)-> Reaction products previously believed to bemononuclear are dinuclear

57Fe with 56Fe

Stankov et al PRL (2007)

Slezak et al PRL 2007

Also: Multilayers - Cuenya et al, PRB 2008

AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

Three-Dimensional Momentum Density
Reconstruction

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

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

J(p z ) = ∫∫ n(p)dp x dp y

Typical: Q~100Å-1
E>100 keV

Reconstruction:
Direct Fourier Method
Fourier-Bessel Method
Cormack Method
Maximum Entropy Method

Ie: Compton scattering projects
out the electron momentum
density.

Momentum density, n(p)Note: a bulk probe that is tolerant of sample imperfections.

Typical of incoherent scattering…
From Y. Sakurai
AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

Hole Locations in La2-xSrxCuO4

Fermi surfaces of Cu and Cu alloys

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
Cu-15.8at%Al

Cu

& Some density that is not yet understood

Cu-27.5at%Pd

Band Structure Calculations

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

AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

Reducing the Two-Theta Arm Size
Dispersion Compensation: Houtari, et al JSR (2005)dAnimation
D. Ishikawad =AQRB @ AOFSRR Cheiron School 2012

4R 2 ΔE
p E

5 meV at 16 keV
R=2m, p=0.1 -> d=50 mm
AQRB @ AOFSRR Cheiron School 2012

Temperature Gradient Analyzer

A Nano-Volt Spectrometer(Ishikawa & Baron, JSR 2010)

Rayleigh Scattering of Synchrotron Mossbauer Radiation (RSSMR)

Longer Arm: T~0.1C

l=150 to 200mm

Short Arm: T: 1 to 10C~ meV resolution at 3m

Masuda, Mitsui, Seto, et al, JJAP (2008, 2009)~5 meV at 1m
AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012

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

S (Q,ω ) =

1
2π

x e

∫ dt eiQ•( r− r ′)−iωt

∫ dr ∫ dr′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 @ AOFSRRAOFSRR Cheiron
C
School 2012

AQRB @ AOFSRR Cheiron School 2012

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

For meV Resolution at 20 keV

A Next Generation Beamline
Dramatic Improvement to Source and Spectrometerallows new science…
New Field: Electronic excitations

NSLS-II(Not Proven)New
Beamline

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

?
ERL
(~$1B,
Not Funded)

Present
Status
Source & BL
Improvements

Machine Barrier

Expected
Upgrades

ESRF

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

AQRB @ AOFSRR Cheiron School 2012

Quantum NanoDynamics Beamline(BL43LXU)

Medium resolution: 10-100 meVHigh resolution spectrometer: <1 to 6 meV
2m Arm, Large Q Accepta
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 2012

AQRB @ AOFSRR Cheiron School 2012

Medium Resolution Spectrometer
Based on a 2m Arm & 3x3 Array Of Analzyers:
Energy resolution: ~10 to 100 meV (mono dependent)
Analyzers at Si(888) at 15.816 keV (reduced tails compared to lower order)
Dispersion compensation with Temperature Gradientkeeps high resolution with large space near sample.
Maximum momentum transfer ~15 Å-1 (phase plate needed @ 90 Deg.)
Solid Angle Gain: x25/Analyzer compared to high res spectrometer
Commissioning to begin late in 2012

AQRB @ AOFSRR Cheiron School 2012

Thanks for Your Attention!

AQRB @ AOFSRR Cheiron School 2012

AQRB @ AOFSRR Cheiron School 2012