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Slide1

Introduction to Inelastic x-ray scattering

Michael Krisch

European Synchrotron Radiation Facility

Grenoble, France

krisch@esrf.frSlide2

Outline of lecture

Introduction short overview of IXS and related techniques

IXS from phonons

why X-rays?

complementarity X-rays <-> neutrons

instrumental concepts & ID28 at the ESRF

study of single crystal materials

study of polycrystalline materials

revival of thermal diffuse scattering

Example I: plutonium

Example II: supercritical fluids

Other applications

ConclusionsSlide3

Introduction I – scattering kinematics

d

W

2

q

i

i

E

k

,

r

f

f

E

k

,

r

Q

r

E

,

photon

p

h

o

t

o

n

•

Energy transfer:

E

f

-

E

i

=

D

E =

1 meV – several keV

•

Momentum transfer: =

1 – 180 nm

-1

Slide4

Introduction II - schematic IXS spectrum

quasielastic

phonon, magnons, orbitons

valence electron excitations

plasmon

Compton profile

core-electron excitation

S. Galombosi, PhD thesis, Helsinki 2007Slide5

Introduction III – overview 1

Phonons

Lattice dynamics

- elasticity

- thermodynamics

- phase stability

- e

-

-ph couplingLecture today!Spin dynamics - magnon dispersions - exchange interactions Lecture on Friday by Marco Moretti Sala!MagnonsSlide6

Introduction IV – overview 2

Nuclear resonance

prompt scattering

delayed scattering

±3/2¯

nuclear level scheme

57

Fe



E

e

0

 = 4.85 neV

 = 141 ns

3/2¯

1/2¯

1/2

¯

Lecture by Sasha Chumakov on Tuesday!Slide7

Introduction V – IXS instrumentation

K

out

K

in

Q

p = R

crystal

·sinqBRcrys = 2·RRowl DetectorSampleSpherical crystal pRRowlandEnergy analysis of scattered X-rays - DE/E = 10-4 – 10-8 - some solid angleRowland circle crystal spectrometerSlide8

Introduction VI – IXS at the ESRF

ID20:

Electronic and

magnetic excitations

ID18:

Nuclear resonance

ID28:

Phonons

ID32: soft X-ray IXSSlide9

Relevance of phonon studies

Superconductivity

Thermal Conductivity

Sound velocities

and elasticity

Phase stabilitySlide10

Vibrational spectroscopy – a short history

Infrared absorption - 1881

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

Brillouin light scattering - 1922

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

Raman scattering – 1928

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

TDS: Phonon dispersion in Al – 1948P. Olmer, Acta Cryst. 1 (1948) 57INS: 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) Slide11

X-rays and phonons?

“ When a crystal is irradiated with X-rays, the processes

of photoelectric absorption and fluorescence are no doubt

accompanied by absorption and emission of phonons.

The energy changes involved are however so small compared

with photon energies that

information about the phonon

spectrum of the crystal cannot be obtained in this way.”W. Cochran in Dynamics of atoms in crystals, (1973)“…In general the resolution of such minute photon frequency is so difficult that one can only measure the total scattered radiation of all frequencies, … As a result of these considerations x-ray scattering is a far less powerful probe of the phonon spectrum than neutron scattering. ”Ashcroft and Mermin in Solid State Physics, (1975)b – tin, J. Bouman et al., Physica 12, 353 (1946)Slide12

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.”Slide13

IXS versus INS

Burkel, Dorner and Peisl (1987)

Hard X-rays:

E

i

= 18 keV

k

i = 91.2 nm-1DE/E  1x10-7Thermal neutrons: Ei = 25 meVki = 38.5 nm-1DE/E = 0.01 – 0.1

Brockhouse (1955)Slide14

Inelastic x-ray scattering from phonons

HASYLAB

D

E = 55 meV

0.083 Hz

B. Dorner, E. Burkel, Th. Illini, and J. Peisl; Z. Phys. B 69, 179 (1987)Slide15

IXS scattering kinematics

d

W

2q

i

i

E

k

,

r

f

f

E

k

,

r

Q

r

E

,

photon

p

h

o

t

o

n

)

sin(

2

q

i

k

Q

r

r

=

f

i

E

E

E

-

=

momentum transfer is defined only by scattering angle

Slide16

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 = 4

p

/

lsin(q)DE = Ei - EfIXS

INSv = 500 m/sv = 7000 m/sDENo kinematic limitations: DE independent of QDisordered systems: Explore new Q-DE rangeSlide17

IXS from phonons – very small samples

Small sample volumes: 10-4 – 10

-5

mm

3

Diamond

anvil cell

• (

New) materials in very small quantities• Very high pressures > 1Mbar• Study of surface phenomenaØ 45 m t=20 m

bcc Mo single crystalrubyheliumSlide18

IXS – dynamical structure factor

Scattering function:

Thermal factor:

Dynamical structure factor:

E, Q

k

in

k

outSlide19

Comparison IXS - INS

• no correlation between momentum- and energy transfer •

D

E/E = 10

-7

to 10

-8

• Cross section ~ Z

2 (for small Q) • Cross section is dominated by photoelectric absorption (~ l3Z4)• no incoherent scattering• small beams: 100 mm or smaller• 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 IXSINSSlide20

Efficiency of the IXS technique

L = sample length/thickness,

m

= photoelectric absorption, Z = atomic number

Q

D

= Debye temperature, M = atomic massSlide21

IXS resolution function today

•

D

E and Q-independent

• Lorentzian shape

• Visibility of modes.

•

Contrast between modes.Slide22

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 keV

D

E = 0.1 – 1 meV

D

E = 0.89 (0.6) meV at Petra-IIIDE = 0.62 meV at APSDedicated instrument at NSLS-IIAPSSlide23

Instrumentation for IXS

Monochromator

:

Si(n,n,n

),

q

B

= 89.98

º

n=7

-

13

l

1

tunable

Analyser

:

Si(n,n,n

),

q

B

= 89.98

º

n=7

-

13

l

2

constant

IXS set-up on ID28 at ESRF

D

E

D

T

1/K at room temperature

q

D

E

D

T

1/K at room temperatureSlide24

Beamline ID28 @ ESRF

Reflection

E

inc

[keV]

D

E [meV]

Q range [nm

-1]RelativeCount rate(8 8 8)15.81662 - 731(9 9 9)17.7943.01.5 - 822/3(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

m

2

-> 14 x 8

m

m

2

(H x V, FWHM)

9- analyser crystal spectrometer

KB optics

or

Multilayer

MirrorSlide25

An untypical IXS scan

dscan monot 0.66 –0.66 132 80

Diamond; Q=(1.04,1.04,1.04)

Stokes peak:

phonon creation

energy loss

Anti-Stokes peak:

phonon annihilation

energy gainSlide26

Phonon dispersion scheme

E, Q

k

in

k

out

Diamond

Diamond (INS + theory): P. Pavone, PRB 1993Slide27

Single crystal selection rules

well-defined momentum transfer for given scattering geometry

S(Q,

w

)

 (Q·e)

2

ˆSlide28

Single crystal selection rules

S(Q,

w

)

 (Q·e)

2

ˆ

well-defined momentum transfer for given scattering geometrySlide29

Phonon dispersion and G-point phonons

Raman scattering

Brillouin light scatteringSlide30

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 contentSlide31

IXS from polycrystalline materials - I

V

L

~E/q

At low Q (1. BZ)

Orientation averaged

longitudinal sound velocity

(Generalised)

phonon density-of-statesAt high Q (50–80 nm-1)How to get the full lattice dynamics? Slide32

IXS from polycrystalline materials - II

Polycrystalline IXS dataQ = 2 – 80 nm-1

Lattice dynamics model

+

Orientation averaging

least-squares refinement

or

direct comparisonValidated full lattice dynamicsSingle crystal dispersionElastic propertiesThermodynamic propertiesNew methodologyI. Fischer, A. Bosak, and M. Krisch; Phys. Rev. B 79, 134302 (2009) Slide33

IXS from polycrystalline materials - III

Stishovite (SiO2

)

rutile structure

N = 6

18 phonon branches

27 IXS spectra

A. Bosak et al; Geophysical Research Letters 36, L19309 (2009)Slide34

IXS from polycrystalline materials - IV

SiO2 stishovite: validation of

ab initio

calculation

single scaling factor of

1.05

is introducedSlide35

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)Slide36

Revival of thermal diffuse scattering

= 0.7293 Å

Dl

/

l

= 1x10

-4Angular step 0.1°ID29 ESRFPilatus 6M hybrid silicon pixel detectorSlide37

TDS: theoretical formalism

with eigenfrequencies , temperature

and scattering factor

with eigenvectors Debye Waller factor ,

atomic scattering factor and mass .Slide38

Diffuse scattering in Fe3O4

A. Bosak et al.; Physical Review X (2014)Slide39

Diffuse scattering in Fe3O

4

Fe

3

O

4

A. Bosak et al.; Physical Review X (2014)Slide40

ZrTe3: IXS and (thermal) diffuse scattering

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

(h0l)-plane

(300)

(400)

(301)

(401)

T=295 K

T=80K (1.3 T

CDW)Slide41

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 toxictypical grain size: 90 mmfoil thickness: 10 mmstrain enhanced recrystallisation of fcc Pu-Ga (0.6 wt%) alloySlide42

Plutonium: the IXS experiment

ID28 at ESRF•

Energy resolution: 1.8 meV at 21.747 keV

•

Beam size: 20 x 60

m

m

2

(FWHM)• On-line diffraction analysisSlide43

Plutonium phonon dispersion

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

soft-mode behaviour of T[111] branch

proximity of structural phase transition

(to monoclinic

a

’ phase at 163 K)Slide44

Plutonium: elasticity

Proximity of

G

-point:

E

=

Vq

V

L[100] = (C11/r)1/2VT[100] = (C44/r)1/2VL[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)1/2VT[111] = ([C11-C12+C44]/3r)1/2C11 = 35.31.4 GPaC12 = 25.51.5 GPaC44 = 30.51.1 GPa highest elastic anisotropy of all known fcc metalsSlide45

Plutonium: density of states

• Born-von Karman fit- density of states calculated

Specific heat

g(E)

q

D

(T

0)

= 115KqD(T  ) = 119.2KSlide46

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)Slide47

Example II: IXS from fluids

Liquid–Gas Coexistence

T<T

c

Gas

Liquid

Supercritical Fluid

T>T

cFluid

PcPTLiquidGasFluidPcTcABSlide48

IXS from fluids: behavior of liquids (below Tc)

w

=C

S

*Q

w

=C

*QTHznm-1w=CL*Q

w = 1/ta: positive dispersion of the sound speed: cL > cSStructural relaxation process ta interacting with the dynamics of the microscopic density fluctuations.Slide49

IXS from fluids: oxygen at room T in a DAC

P/P

c

>> 1

DAC: diamond anvil cell; 80

m

m thick O

2

sampleT/Tc = 2Slide50

IXS from fluids: pressure-dependent dispersion

Positive dispersion is present in deep fluid oxygen!

C

L

/C

S

 1.2 typical of simple liquidsSlide51

IXS from fluids: reduced phase diagram

F. Gorelli et al; Phys. Rev. Lett. 97, 245702 (2006)Slide52

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?Slide53

IXS from fluids: Argon at high P and T

IXS and MD simulations

G.G. Simeoni et al; Nature Physics 6, 503 (2010)Slide54

IXS from fluids: reduced phase diagram (bis)

G.G. Simeoni et al; Nature Physics 6, 503 (2010)Slide55

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 C

P

maximum (Widom's line) in supercritical fluid ArSlide56

Applications: Strongly correlated electrons

Doping dependence in

SmFeAsO

1-x

F

y

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-US. Raymond et al.; PRL 107, (2011)Slide57

Applications: Functional materials

Piezoelectrics PbZr

1-x

Ti

x

O

3

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

SkutteruditesM.M. Koza et al.; PRB 84, 014306 InN thin film lattice dynamicsJ. Serrano et al.; PRL 106, 205501 Lecture by Benedict Klobes on Friday!Slide58

Applications: Earth & Planetary science

Elastic anisotropy in Mg

83

Fe

0.17

O

D. Antonangeli et al.; Science 331, 64

Sound velocities in Earth’s core

J. Badro et al.; Earth Plan. Science Lett. 98, 085501Lecture by Daniele Antonangeli on Friday!Slide59

Applications: Liquids & glasses

Nature of the Boson peak in glasses

A. Chumakov et al.; PRL 106, 225501

Liquid-like dynamical behaviour

in the supercritical region

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

Lecture by Sasha Chumakov on Tuesday!Slide60

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