Introduction to Inelastic x-ray scattering

Michael KrischEuropean Synchrotron Radiation Facility

Grenoble, France

krisch@esrf.fr

Outline of lecture

Introductionshort 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

2q

ii Ek ,r

f

fE

k,r

Qr E,

photon

photon

• Energy transfer: Ef - Ei = DE = 1 meV – several keV

• Momentum transfer: = 1 – 180 nm-1 Qkk if

Introduction II - schematic IXS spectrum

quasielastic

phonon, magnons, orbitons

valence electron

excitations

plasmon Compton profile

core-electron excitation

S. Galombosi, PhD thesis, Helsinki 2007

Introduction III – overview 1

-30 -20 -10 0 10 20 300

200

400

600

800

cou

nts

in 8

0 s

ecs

energy transfer [meV]

PhononsLattice dynamics

- elasticity- thermodynamics- phase stability- e--ph coupling

Lecture today!

Spin dynamics- magnon dispersions- exchange interactions

Lecture on Friday by Marco Moretti Sala!

Magnons

Introduction IV – overview 2

Nuclear resonance

prompt scattering

delayed scattering

±3/2¯

nuclear level scheme 57Fe

Ee

0

= 4.85 neV = 141 ns

3/2¯

1/2¯ 1/2¯

Lecture by Sasha Chumakov on Tuesday!

Introduction V – IXS instrumentation

Kout

Kin

Q

p = Rcrystal·sinqB

Rcrys = 2·RRowl

Detector

Sample

Spherical crystal

pRRowland

Energy analysis of scattered X-rays- DE/E = 10-4 – 10-8

- some solid angle

Rowland circle crystal spectrometer

Introduction VI – IXS at the ESRF

ID20: Electronic and

magnetic excitations

ID18: Nuclear resonance

ID28: Phonons

ID32: soft X-ray IXS

Relevance of phonon studies

Superconductivity

Thermal Conductivity Sound velocitiesand elasticity

Phase stability

Vibrational spectroscopy – a short history

Infrared absorption - 1881W. Abney and E. Festing, R. Phil. Trans. Roy. Soc. 172, 887 (1881)

Brillouin light scattering - 1922L. Brillouin, Ann. Phys. (Paris) 17, 88 (1922)

Raman scattering – 1928C. V. Raman and K. S. Krishnan, Nature 121, 501 (1928)

TDS: Phonon dispersion in Al – 1948P. Olmer, Acta Cryst. 1 (1948) 57

INS: Phonon dispersion in Al – 1955B.N. Brockhouse and A.T. Stewart, Phys. Rev. 100, 756 (1955)

IXS: Phonon dispersion in Be – 1987B. Dorner, E. Burkel, Th. Illini and J. Peisl, Z. Phys. B – Cond. Matt. 69, 179 (1987)

NIS: Phonon DOS in Fe – 1995M. 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 processesof 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

Burkel, Dorner and Peisl (1987)

Hard X-rays:

Ei = 18 keV

ki = 91.2 nm-1

DE/E 1x10-7

Thermal neutrons:

Ei = 25 meV

ki = 38.5 nm-1

DE/E = 0.01 – 0.1

Brockhouse (1955)

Inelastic x-ray scattering from phonons

HASYLAB

DE = 55 meV0.083 Hz

B. Dorner, E. Burkel, Th. Illini, and J. Peisl; Z. Phys. B 69, 179 (1987)

IXS scattering kinematics

dW

2q

ii Ek ,r

f

fE

k,r

Qr E,

photon

photon

)sin(2 qikQrr

=

fi EEE -=

momentum transfer is defined only by scattering angle

IXS from phonons – the low Q regime

· Interplay between structure and dynamics on nm length scale· Relaxations on the picosecond time scale· Excess of the VDOS (Boson peak)· Nature of sound propagation and attenuation

Q = 4p/lsin(q)DE = Ei - Ef

10-1 100 101 10210-4

10-3

10-2

10-1

100

101

102E

(meV

)

Q ( nm-1)

IXS

INSv = 500 m/sv = 7000 m/s

DE

No kinematic limitations: DE independent of Q

Disordered systems:

Explore new Q-DE range

IXS from phonons – very small samples

Small sample volumes: 10-4 – 10-5 mm3

Diamond anvil cell

• (New) materials in very small quantities

• Very high pressures > 1Mbar

• Study of surface phenomena

Ø 45 m t=20 m

bcc Mo single crystal

ruby

helium

IXS – dynamical structure factor

j

jQTEFjQGEQS ),,,(),(),(

))(())(()(

1

exp1

1),,,( qEEqEE

qEkTE

jQTEF jjj

Scattering function:

Thermal factor:

Dynamical structure factor:

E, Q

kin

kout

Comparison IXS - INS

• 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 mm or smaller

EQSk

kb

E,

2

122

• 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

EQSQfk

kr

E,2

212

120

2

IXS

INS

Efficiency of the IXS technique

L = sample length/thickness, m = photoelectric absorption, Z = atomic numberQD = Debye temperature, M = atomic mass

IXS resolution function today

-40 -20 0 20 401E-4

1E-3

0,01

0,1

Sig

nal [

arb.

uni

ts]

Energy Transfer [meV]

• DE and Q-independent• Lorentzian shape

• Visibility of modes.• Contrast between modes.

IXS resolution function tomorrow Sub-meV IXS with sharp resolution

Y.V. Shvydk’o et al, PRL 97, 235502 (2006), PRA 84, 053823 (2011)

E = 9.1 keVDE = 0.1 – 1 meV

DE = 0.89 (0.6) meV at Petra-IIIDE = 0.62 meV at APS

Dedicated instrument at NSLS-II

APS

Instrumentation for IXS

sa m p le

E i

E f

d e te c to r

Monochromator:Si(n,n,n), qB = 89.98ºn=7-13l 1 tunable

Analyser:Si(n,n,n), qB = 89.98ºn=7-13l 2 constant

IXS set-up on ID28 at ESRF

DE DT

1/K at room temperature

q

DE DT

1/K at room temperature

Beamline ID28 @ ESRF

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 m m2 -> 14 x 8 m m2 (H x V, FWHM)

9- analyser crystal spectrometerKB opticsor

Multilayer Mirror

An untypical IXS scan

-30 -20 -10 0 10 20 300

200

400

600

800

Temperature difference [K]

+/-0 -0.44+0.44 -0.22+0.22

Co

un

ts in

4

0 se

cs

Energy transfer [meV]

-30 -20 -10 0 10 20 300

200

400

600

800

relative temperature [K]0.44 0.22 -0.22 -0.440

coun

ts in

80

secs

energy transfer [meV]

dscan monot 0.66 –0.66 132 80

Diamond; Q=(1.04,1.04,1.04)Stokes peak:phonon creationenergy loss

Anti-Stokes peak:phonon annihilationenergy gain

Phonon dispersion scheme

E, Q

kin

kout

Diamond

Diamond (INS + theory): P. Pavone, PRB 1993

Single crystal selection rules

well-defined momentum transfer for given scattering geometry

S(Q,w) (Q·e)2ˆ

Single crystal selection rules S(Q,w) (Q·e)2ˆ

well-defined momentum transfer for given scattering geometry

Phonon dispersion and G-point phonons

0 500 1000 1500 2000 2500 3000

100

1000

10000

100000

Inte

nsity

[arb

. uni

ts]

wave numbers [cm-1]

Raman scatteringBrillouin light scattering

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

0 2 4 6 8 10 12 14 160

10

20

30

40

En

erg

y [m

eV

]

q [nm-1]

VL~E/q

At low Q (1. BZ)

Orientation averaged longitudinal sound velocity

(Generalised) phonon density-of-states

At high Q (50–80 nm-1)

0 50 100 1500.00

0.01

0.02

0.03

Inte

nsi

ty [

arb

. u

nits

]

Energy [meV]

How to get the full lattice dynamics?

IXS from polycrystalline materials - II

Polycrystalline IXS dataQ = 2 – 80 nm-1

Lattice dynamics model + Orientation averaging

least-squares refinementor

direct comparison

Validated full lattice dynamicsSingle crystal dispersion

Elastic propertiesThermodynamic properties

New methodology

I. Fischer, A. Bosak, and M. Krisch; Phys. Rev. B 79, 134302 (2009)

IXS from polycrystalline materials - III Stishovite (SiO2)

rutile structureN = 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

F. Jiang et al.; Phys. Earth Planet. Inter. 172, 235 (2009)

Ref. C11

[GPa]

C33

[GPa]

C12

[GPa]

C13

[GPa]

C44

[GPa]

C66

[GPa]

B

[GPa]

VD

[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)

Revival of thermal diffuse scattering

l = 0.7293 ÅDl/ 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

M. Hoesch et al.; Phys. Rev. Lett. 2009

5

4

3

2

1

0e

ne

rgy

(me

V)

-4.00 -3.96 -3.92 -3.88

momentum along CDW (a* component)

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

(300) (400)

(301) (401)

T=295 KT=80K (1.3 TCDW)

Example I: phonon dispersion of fcc d-Plutonium

J. Wong et al. Science 301, 1078 (2003); Phys. Rev. B 72, 064115 (2005)

Pu is one of the most fascinating and exotic element known • Multitude of unusual properties • Central role of 5f electrons • Radioactive and highly toxic

typical grain size: 90 mmfoil thickness: 10 mm

strain enhanced recrystallisation of fcc Pu-Ga (0.6 wt%) alloy

Plutonium: the IXS experiment

ID28 at ESRF• Energy resolution: 1.8 meV at 21.747 keV• Beam size: 20 x 60 mm2 (FWHM)• On-line diffraction analysis

-2 0 -1 0 0 1 0 2 00

2 0

4 0

6 0

L A (0 .2 ;0 .2 ;0 .2 )

Cou

nts

in 1

80 s

ecs

E n e rg y [m e V ]-1 0 -5 0 5 1 0

0

1 0 0

2 0 0

TA (0 .2 ;0 .2 ;0 .2 )

Cou

nts

in 1

80 s

ecs

E n e rg y [m e V ]

Plutonium phonon dispersion

• Born-von Karman force constant model fit- good convergence, if fourth nearest neighbours are included

Expt

B-vK fit 4NN

Expt

B-vK fit 4NN

soft-mode behaviour of T[111] branch

proximity of structural phase transition (to monoclinic a’ phase at 163 K)

Plutonium: elasticity

Proximity of G-point: E = Vq

VL[100] = (C11/r)1/2

VT[100] = (C44/r)1/2

VL[110] = ([C11+C12+2C44]/r)1/2

VT1[110] = ([C11 - C12] /2r)1/2

VT2[110] = (C44/r)1/2

VL[111] = [C11+2C12+4C44]/3r)1/2

VT[111] = ([C11-C12+C44]/3r)1/2

C11 = 35.31.4 GPa

C12 = 25.51.5 GPa

C44 = 30.51.1 GPa

< 0 0 1 >

< 11 0 >

< 111 >

highest elastic anisotropy of all known fcc metals

Plutonium: density of states

0 .0 0 .2 0 .4 0 .6

(a rb . u n its )

0 .0 0 .2 0 .4 0 .6

D e n s ity o f s ta te s

• Born-von Karman fit- density of states calculated

22

0 1)/exp(

)()/exp(3

max

TkE

dEEgTkE

Tk

ENkC

B

BE

BBv

Specific heat

g(E)

Temperature (K)

0 50 100 150 200 250 300

Cv

(cal

mo

le-1

K-1

)

0

2

4

6 3R

qD(T0) = 115K

qD(T ) = 119.2K

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

T<Tc

Gas

Liquid

Supercritical Fluid

T>Tc

Fluid

Pc

P

T

Liquid

Gas

Fluid

Pc

Tc

A

B

IXS from fluids: behavior of liquids (below Tc)

Elastic"solidlike"dynamicsa>>1

Q

Viscous"liquidlike"dynamicsa<<1

Visco-elastictransition

a

w=CS*Q

w=C*Q

THz

nm-1

w=CL*Q

w = 1/ta: positive dispersion of the sound speed: cL > cS

Structural relaxation process ta interacting with the dynamics of the microscopic density fluctuations.

IXS from fluids: oxygen at room T in a DAC

Q

=10

.2 n

m-1

Inte

nsity

(a.

u.)

Q=

3.0

nm-1

Q=

5.4

nm-1

Q=

12.6

nm

-1Q

=7.

8 nm

-1

-40 -20 0 20 40

-40 -20 0 20 40

P=2.88 GPa P=5.35 GPa

Energy (meV)

P=0.88 GPa

-40 -20 0 20 40

P/Pc>> 1

DAC: diamond anvil cell; 80 mm thick O2 sample

T/Tc = 2

0 5 10 15 20 250

5

10

15

20

25

cISTS

(m/s) cIXS

(m/s)

1920 2340 2980 3600 3680 4440

P=5.35 GPaP=2.88 GPaP=0.88 GPa

Ene

rgy

(meV

)

Q (nm-1)

IXS from fluids: pressure-dependent dispersion

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

Widom line: theoretical continuation into the supercritical region of the liquid-vapour coexistence line, considered as “locus of the extrema of the thermodynamic response functions”

Cross-over at the Widom line?

IXS from fluids: Argon at high P and T

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.01.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

Pos

itiv

e so

und

disp

ersi

on

Pressure (GPa)

Widom line

IXS and MD simulations

G.G. Simeoni et al; Nature Physics 6, 503 (2010)

IXS from fluids: reduced phase diagram (bis)

0.1 1 101E-3

0.01

0.1

1

10

100

1000

10000

CRITICALISOCHOREFOR Ar

Pc

WIDOMLINE FOR Ar

0 5 10 15 2020

30

40

2.45

3.12

1.79

CP(J

/mo

l K)

P/Pc

T/Tc=1.46

LIQUID-LIKE

P/P c

T/Tc

NeonOxygenArgonNitrogenMercuryLithiumSodiumPotassiumRubidiumCesiumWater

LIQUID

GAS

GAS-LIKE

G.G. Simeoni et al; Nature Physics 6, 503 (2010)

IXS from fluids: Conclusions Revisiting the notion of phase diagram beyond the critical point:

The positive sound dispersion is a physical observable able to distinguish liquid-like from gas-like behavior in the super-critical fluid region

Evidence of fluid-fluid phase transition-like behavior on the locus of CP maximum (Widom's line) in supercritical fluid Ar

Applications: Strongly correlated electrons

Doping dependence in SmFeAsO1-xFy

M. Le Tacon et al.; Phys. Rev. B 80, (2009)

Kohn anomaly in ZrTe3

M. Hoesch et al.; PRL 102, (2009)

e-ph coupling in a-U

S. Raymond et al.; PRL 107, (2011)

Applications: Functional materials Piezoelectrics PbZr1-xTixO3

J. Hlinka et al.; PRB 83, 040101(R)

Skutterudites

M.M. Koza et al.; PRB 84, 014306

InN thin film lattice dynamics

J. Serrano et al.; PRL 106, 205501

Lecture by Benedict Klobes on Friday!

Applications: Earth & Planetary science

Elastic anisotropy in Mg83Fe0.17O

D. Antonangeli et al.; Science 331, 64

Sound velocities in Earth’s core

J. Badro et al.; Earth Plan. Science Lett. 98, 085501

Lecture by Daniele Antonangeli on Friday!

Applications: Liquids & glasses

Nature of the Boson peak in glasses

A. Chumakov et al.; PRL 106, 225501

Liquid-like dynamical behaviourin the supercritical region

G. Simeoni et al.; Nature Phys. 6, 503

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.01.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

Pos

itive

sou

nd d

ispe

rsio

n

Pressure (GPa)

Widom's line

T= 573 KT/T

c=3.80

0 100 200 300 400 500 600 700 800 P/P

c

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