theabdus salaminternational centre for theoretical physics

strada costiera, 11 - 34014 trieste italy - tel. +39 040 2240111 fax +39 040 224163 - sci_info@ictp.trieste.it - www.ictp.trieste.it

united nationseducational, scientific

and culturalorganization

international atomicenergy agency

SCHOOL ON SYNCHROTRON RADIATION AND APPLICATIONSIn memory of J.C. Fuggle & L. Fonda

19 April - 21 May 2004

Miramare - Trieste, Italy

1561/22____________________________________________________________

Inelastic X-ray Scattering from

Collective Atom Dynamics

M. Krisch

Inelastic X-ray Scattering fromCollective Atom Dynamics

Michael Krisch

Introduction and Motivation

Inelastic neutron and x-ray scattering

IXS Instrumentation

High frequency collective dynamics of liquid water

Sound velocities in hcp-iron to 110 GPa

Elastic moduli in hcp-cobalt to 25 Gpa

Phonon dispersion in superconductors

European Synchrotron Radiation FacilityGrenoble, France

Aim of IXS (INS) Experiments

Study of phonon(-like) excitations in condensed matter.

Breakdown of the static lattice model

dynamical lattice model

• specific heat• thermal expansion• melting

• T-dependence of DC conductivity• thermal conductivity of insulators• sound transmission• superconductivity

Information obtained from IXS (INS) Experiments

• inter-atomic force constants• sound velocities• thermodynamic properties• dynamical instabilities (phonon softening)• electron-phonon coupling• relaxation phenomena….etc., etc.

X-rays and phonon studies ?

“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 large compared with phonon 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)

∆E ≈ meV !!!Einc > 10 keV !!!Photon flux ???

• Energy transfer E = Eout – Ein (E<< Ein)

• Momentum transfer Q = kout - kin = 2k sin (ϑ/2) (kout≈kin≡k)

♦ Directional analysis of the scattered photons Q

♦ Energy analysis of the scattered photons E

Scattering kinematics

Ein, kin, εin

Eout , k

out , εout

E, Qϑ

X-ray inelastic cross section

( )∑=−j

jThX trArH ,21 2

0

Electron-photon interaction Hamiltonian

Double-differential cross section

( ) ( )∑ −−∑⋅=∂Ω∂

∂

FIif

j

rQiI EEEIeFP

kkr

Ej

,

2

212

2120

2δεεσ rr

rr

Thomson scattering cross section

(i) Adiabatic approximation:

(ii)

ne SSS =

nene FIFIII ==

The central approximations

For a mono-atomic system:

( ) ( ) ( )EQSQfkkr

E,2

212

2120

2 rrr εεσ⋅=

∂Ω∂∂

Dynamical structure factor

Atomic form factorThomson scattering cross section

The dynamical structure factor

Pair correlation function GP(r,t):GP(r,t) is the probability to find two different particles at positions Rl’(t=0) and Rl’(t), separated by the distance r and the time intervall t.

Dynamical structure factor S(Q,ω):Space and time Fourier transform of GP(r,t).

l l’

The dynamical structure factor

( ) ( ) ( ) ( )∫∫ −= trGrQirdtidtQS ,expexp21, 3 rrr

h

rω

πω

S(Q,ω) is the space and time Fourier transform of the pair correlation function G(r,t).

The pair correlation function

G(r,t) is the probability to find two different particles at positions Rl’(t=0) and Rl(t), and separated by the distance r and the time intervall t.

Q ≈ 2π/ξ ξ is a characteristic length

ω = E/h ≈ 2π/τ τ is a characteristic (relaxation) time

( ) ( )( ) ( )( )∑ −+−∫='.

'3 '0''1,

llll tRrrRrrd

NtrG

rrrrrrr δδ

IXS versus INS

• no correlation between momentum- and energy transfer• ∆E/E = 10-7 to 10-8

• Cross section ~ Z2 (for small Q) • Cross section is dominated by photoelectric absorption (~ λ3Z4)• no incoherent scattering• small beams: 100 µm or smaller

( )EQSkkb

E,

212

2 r=

Ω∂∂∂ σ

INS

• strong correlation between momentum- and energy transfer• ∆E/E = 10-1 to 10-2

• Cross section ~ b2

• Weak absorption => multiple scattering• incoherent scattering contributions• large beams: several cm

IXS ( ) ( ) ( )EQSQfkkr

E,2

21212

0

2 rrr εεσ⋅=

Ω∂∂∂

IXS versus INS: scattering kinematics

λ1 = 1 Å => E1 = 12398 eV E = some meV E1 ≈ E2

λ1 = 1 Å => E1 = 82 meV E = some meV E1 ≠ E2

Energy Transfer:

Neutrons:

X-rays:

=> moderate energy resolution: E/E1 = 0.05

=> extremely high energy resolution: E/E1 = 10-7

IXS versus INS: scattering kinematics

MomentumTransfer:

Neutrons:

X-rays:

=> strong coupling between E and Q inaccessible E-Q region

=> Q only controlled by scattering angle ϑ

( )ϑcos2 2122

21 kkkkQ −+=

( )2sin2 1 ϑkQ =

10-3 10-2 10-1 100 101 10210-4

10-3

10-2

10-1

100

101

102

ILL-BRISP

INS

IXS

IN5

FP (S

ande

rcoc

k) (1

990)

DM

DP

(198

5)H

IRES

UV

prot

otyp

e (2

000)

V=500 m

/sV=7000

m/s

BLS

E (

meV

)

Q ( nm-1)

The Q-E Space of phonon spectroscopies

IXS has advantages in disordered systems with a high speed of sound (vg = Ε/Q)

Where do X-rays complement neutrons?

• In disordered systems with a high speed of sound (vg = E/Q).

Kinematic limitations for INS

• Samples (only available) in small quantities.- novel (exotic) materials (MgB2, …..)- samples under very high pressures

Neutron beams: several cm => signal lossSR beams: 10 - 100 µm, tµ = 10 -100 µm

large high pressure cell => P < 100 kbarDiamond anvil cell => P > 1 Mbar

Cou

nts

per 2

min

utes

Experimental principle

Monochromator

Analyser

Sample

Detector

30 m 40 m 50 m 60 m 70 m

-0.05 m

0.00 m

0.05 m

0.10 m

0.15 m

0.20 m

A

Side View

BCA

BC

FED

0.0 m

0.4 m

0.8 m

1.2 m

Top viewED, F

Beamline Layout ID16 and ID28 at the ESRF

• The x-ray source• High heatload premonochromator (A)• Main monochromator (B)• Focusing mirror (C)• Sample (D)• Analyser (E)• Detector (F)

High Energy Resolution Monochromator

• Silicon (n,n,n) at θB = 89.98°• ωD=∆E/E tan(θB) > sz’, sx’

Reflection Energy [eV] DE [meV] DE/E(7 7 7) 13840 5.3 3.8·10-7

(8 8 8 ) 15817 4.4 2.8·10-7

(9 9 9) 17794 2.2 1.2·10-7

(11 11 11) 21747 1.02 4.7·10-8

(12 12 12) 23725 0.73 3.0·10-8

(13 13 13) 25703 0.5 2.0·10-8

Energy Analysis: The Crystal Analyser

Requirements:

• Perfect single crystal properties (as HR mono)• Collect some solid angle (∆Qapp; flux)

Solution:

• Spherical crystal in Rowland Geometry (∆θ = 0)- R = 2.5 (6.5; 11.5) m- elastic deformation => ∆E/E =

• Approximation to sphere by 10000 0.6x0.6x3 mm3 cubes- preservation single crystal properties- θB = 89.98° to minimise cube size contribution to ∆E/E

-6 -4 -2 0 2 4 60

100

200

300

400

500

600

2f)Si(13,13,13)E = 25702 eV∆E = 0.87 ± 0.04 meV∆E/E =3.4·10-8

Inte

nsity

(a.u

.)

Energy (meV)

The Analyser in real life

Best Resolution so far obtained:

HR Mono: Energy Scans via Temperature Scans

λ = 2·d(T) sinθB∆d/d = -α(T)·∆T (α=2.58·10-6 at RT)

• Temperature controlled by AC Thermometer bridge∆Tmin = 0.25 mK∆Ttyp = 3, 5, 10, 15 mK/minute

The IXS spectrometers (schematics)

Side view

θsTop view

6.5 m

Q = 4π/λ·sin(θs/2)

detector analysers

sample

3 mm

5-Analyser Assembly

Analyser crystal

T-control mount

Goniometry (ath,achi)

0

100

200

300

400fit.outQ = 2 nm-1

T = 171 K

Glycerol

1

BG = 5.74086 +/- 0.43727SL = 0.17173 +/- 0.04063ZR = (-0.05627) +/- 0.01126PC = 811.0458 +/- 17.41197GC = 0.05275 +/- 0.01182PL = 172.9469 +/- 11.99725OL = 4.03667 +/- 0.04203GL = 0.11458 +/- 0.09244

VL = 3075.943 +/- 32.0274Ianel / Iel = 0.21379

χ2 = 116.1882

µχ

2 = 96 σ

χ2 = 13.85641

Inte

nsity

( c

ount

s )

-20 -15 -10 -5 0 5 10 15 200

50

100

2

Energy ( meV )

-0.10 -0.05 0.00 0.05 0.10

-0.10

-0.05

0.00

0.05

0.10 3

∆ΓL

∆ΩL

-0.10 -0.05 0.00 0.05 0.10

-10

-5

0

5

10 4

∆P C

∆P L

∆ΩL

-0.10 -0.05 0.00 0.05 0.10-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

5

∆ΩL

∆ΓC

-0.10 -0.05 0.00 0.05 0.10

-15

-10

-5

0

5

10

15 6

∆ΩL

-20 -15 -10 -5 0 5 10 15 20

-2

0

27

Example of IXS spectrum and of data analysis

Si(11,11,11)∆E = 1.5 meV at 21747 eV8 hours accumulation

Data Analysis:Damped Harmonic Oscillator

0.36° -0.36°

The high-frequency dynamics of liquid water

1993 - 2004

Some key references:

J. Teixeira et al.; Phys. Rev. Lett. 54, 2681 (1985)G. Ruocco and F. Sette; J.Phys.: Condens.Matter 11, R259 (1999)M. Krisch et al.; Phys. Rev. Lett. 89, 125502 (2002)

INS and Molecular Dynamics Results(as of 1993)

Origin and nature of these two excitations?

0 2 4 6 8 10 120

10

20 MD Rahman et al. [9] " INS Bosi et al. [10] INS Teixeira et al. [11]

E (

meV

)

Q ( nm-1 )

IXS results with ∆E = 5 meV

Damped harmonic oscillator

( ) ( )[ ] ( ) ( ) ( )

( )[ ] ( ) 22222

21,

ωω

ωωωQQ

QQQInQFΓ+−Ω

ΩΓ+=

Dispersion relation for water at 5º C

• Confirmation of the existence of the “fast sound”.• Excitation involves the center of mass of the molecule.

No clear evidence for low energy excitations!

0 2 4 6 8 10 120

5

10

15

20

25

D2O INS Teixeira et al. [11] H

2O IXS Sette et al. [31]

E (m

eV)

Q (nm-1)

IXS results with ∆E = 1.5 meV

-10 0 10 20

E (meV)

E (meV)

Q=10.0 nm-1

Q=4.0 nm-1

Inte

nsity

( a

rb. u

nits

)

Energy (meV)

-5 0 5 10

-5 0 5 10

-5 0 5 10

0.5

0.5

0.5

0.0

0.0

0.0

0.5

0.0

Q=2.5 nm-1

2.0 nm-1

1.5 nm-1

1.0 nm-1

Inte

nsity

(ar

b. u

nits)

Energy (meV)

Complete dispersion relation for water at 5º C

• Existence of two sounds: v0 = 1500 m/s and v∞ = 3200 m/s• The “fast sound” is the continuation of the “hydrodynamic” sound. • The low energy mode only visible once the “fast sound” value is reached.

0 2 4 6 80

5

10

15

20

0 1 2 3 40

2

4

6

8

Ω (Q

) (m

eV)

Q (nm-1)

vinf=3200 m/s

v0 = 1500 m/s

Signature of a structural relaxation process (“fast” α-relaxation)

Low-frequency viscous regime (Ωτ << 1): liquid-like responsehigh-frequency elastic regime (Ωτ >> 1): solid-like response

Ωτ ≈ 1: Coupling of the phonon-like excitations with structural rearrangements.

E ≈ 3.3 meV τ = h/E = 1.3 psQ = 2 nm-1 ξ = 3.1 nm

Time and length scale can be associated withthe formation and breaking of hydrogen bond networks.

Q = 4 nm-1

-10 0 10

0.0

0.5

1.0

1.5

Intn

esit

y (

cou

nt /

s )

T = 314 K0.0

0.5

1.0

1.5

2.0

2.5Q = 2 nm-1

Energy ( meV )

T = 266 K

-10 -5 0 5 100.0

0.5

1.0

1.5

2.0 T = 413 K 0 2 4 6 8 10 12 140

5

10

15

20

25 T = 278 K T = 373 K T = 493 K

Ener

gy (

meV

)

Q ( nm-1 )

Fast sound transition as a function of T

Relaxation time τ(T) from: Ωtrans(T) τ=1

2,0 2,5 3,0 3,5 4,0

10-1

100

- Arrhenius behavior: ∆E = 2.3 + 0.2 kcal / mole

- Hydrogen bond in water : EHB

= 5 kcal / mole

∆E = 2.3 + 0.2 kcal / mole

IXS (const Q) IXS (const T)

333 250500

τ (

ps )

1000 / T ( K-1 )

-15 -10 -5 0 5 10 150

50

100

150

T=277 KP=3.8 kBar

2 nm -1

Energy ( meV )

0

50

100

150

3 nm -1

0

50

100

150

4 nm -1

0

50

100

150

5 nm -1

0

100

200

300In

tens

ity

( nor

mal

ized

uni

ts )

Q = 6 nm -10 2 4 6 8

0

5

10

15

20v=3320 (40) m/s Ω(Q) (meV)

Q (nm-1)

Large volume cellV = 20 mm3

∆E=1.5 meV, Si (11,11,11)

-25 -20 -15 -10 -5 0 5 10 15 20 250

20

40

60

T=297 KP=7.9 kBar

3 nm -1

Energy ( meV )

010203040

4 nm -1

0

20

40

60

5 nm -1

0

20

40

60

6 nm -1

0

50

100

150

Inte

nsity

( n

orm

aliz

ed u

nits

)

Q = 8 nm -10 2 4 6 805101520

25v=3540 (60) m/s Ω(Q) (meV)

Q (nm-1)

Diamond anvil cellV = 0.04 mm3

∆E=3.0 meV, Si (9,9,9)

Water under pressure

Density dependence of sound velocity

• cIXS and c0 approach each other.• cIXS shows anomalous behavior at density, ρ~1.12 gcm-3

1,01,0 1,2 1,4

2

3

4

cIXS

T=277 K c

IXS T=297 K

cIXS

T=410 K c

0 T=277 K

c0 T=297 K

c0 T=410 K

Soun

d ve

loci

ty (k

m/s

)

Density (g/cm3)

• LVC

• DAC

1,0 1,1 1,2 1,3 1,4 1,5 1,61

2

3

4

5

(vix

s/v0)2

ρ [g/cm3]

• Decreasing role of H-bond network for water dynamics.• Water becomes a simple liquid.• ρ = 1.12 g/cm3 seems to mark transition region.

Pressure dependence of vIXS/v0

Geophysical applications of IXS

Jephcoat/RefsonNature 413,28 (2001)

• Earth’s shells• Earth’s composition• Earth’s dynamics

PHASE DIAGRAM OF IRONhcp –iron: the main constituent of Earth’s inner core

Determination of sound velocities• Comparison with seismic results• constraining compositional models • texturing mechanisms

Gasket

Sample

RubyDiamond anvil cell (DAC) techniques

1200 1250 1300 1350 1400 1450 1500 1550 16000

5000

10000

15000

20000

25000

30000

R2

R1

D

Inte

nsità

(u.

a.)

∆ν (cm-1)

Pressure measurement by frequency shift of ruby fluorescence

0 100 200 300 4000

100

200

300

400

500

Righa R1

∆ν

Hydrostatic - - -

Non Hydrostatic ........

dP/dν=1.3177 Kbar/cm-1

dν/dP=0.7589 cm-1/Kbarν(P=0)=14405 cm-1

∆P (K

bar)

polycrystalline ε (hcp)-iron

10 20 30 40 50

Inte

nsity

[a.u

.]

Energy [meV]

hcp iron at 28 GPa

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14En

ergy

[meV

]Q [nm-1]

G. Fiquet, J. Badro, F. Guyot, H. Requardt and M. Krisch; Science 291, 468 (2001)

( )2/1

maxcos1 ⎥

⎦

⎤⎢⎣

⎡∑ ⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅=

n QQnIQE π

(C*)

• ∆E = 5.5 meV at 15816 eV• focal size: 250 x 70 µm2 (FWHM)

ID 28

VP and VS as a function of density

9000 10000 11000 12000 13000

4000

6000

8000

10000

12000 IXS PREM

VS

VP

S

ound

Vel

ocity

[m/s

]

Density [Kg/m3]

ρ

ρ

/

/34

2

2

Gv

GKv

s

p

=

⎟⎠⎞

⎜⎝⎛ +=

D. Antonangeli et al; submitted to Earth and Planetary Science Letters (2004)

9000 10000 11000 12000 13000

4000

6000

8000

10000

12000

PREM NRIXS XRD Shock US PREM XRD

VS

VP

Soun

d Ve

loci

ty [m

/s]

Density [Kg/m3]

Comparison with other experimental techniques

• Good agreement between very different experimental techniques.- stringent test for theoretical approaches

• Shock results yield systematically lower VP’s.- VP(ρ,T)- Evidence that ε-iron VP too low w/r to PREM

different iron phase?presence of light elements?

• Strong temperature effect on VS- partial melting of the core?- strong T-dependence of shear modulus G close to melting

(confirmed by calculations: Laio et al.; Science 287, 1027, 2000)

Conclusions

single crystals:

Limitations of polycrystals- orientationally averaged sound velocities

- no transverse modes

- all elastic moduli (constants)

hcp-Co

hcp-Co five independent elastic moduli: C11, C33, C44, C12, C13

3321

33 1 )001( C

CVLA →⎟⎟

⎠

⎞⎜⎜⎝

⎛=→

ρ

1121

11 1 )100( C

CVLA →⎟⎟

⎠

⎞⎜⎜⎝

⎛=→

ρ

( )12116621

66100 2

1 1 )110( CCCCV

LA −=→⎟⎟⎠

⎞⎜⎜⎝

⎛=→

ρ

4421

44001 1 )110( C

CVTA →⎟⎟

⎠

⎞⎜⎜⎝

⎛=→

ρ

( ) 131312443311 ,,,,f )101( CCCCCCLA →→

hcp-cobalt as an analogue to hcp-ironThesis work of Daniele Antonangeli

-5 0 5 10 15 20 25 3 0

300

350

400

450

500

550

600

elastic constan t: C33

δC33

/δP= 7.85 +/- 0.36 C33(P =0)= 339 +/-6 GPa

u ltrasonic measurements

Ela

stic

con

stan

ts (

GP

a)

P (GPa)

-5 0 5 10 15 2 0 25 30

70

80

90

100

110

120

130

elastic constant: C44

δC4 4/δP= 1.58 +/-0.18 C44(P=0)= 78 +/-2 GPa

ultra sonic mea sure ments

Ela

stic

cons

tant

(GPa

)

P (GPa)

-5 0 5 10 15 2 0 25 30

60

70

80

90

100

110

120

130

140

150

elastic constant: C 66

δC66

/δP= 2.6 +/- 0.2 C

66(P=0)= 71 +/-1 GPa

ultra sonic me asure men ts

Ela

stic

con

stan

t (G

Pa)

P (GPa)

-5 0 5 10 15 2 0 25 30260

280

300

320

340

360

380

400

420

440

460

480

elastic constant: C11

δC11/δP= 5.32 +/-0.3 7 C

11(P=0)= 29 4 + /-2 GPa

ultra sonic me asure men ts

Ela

stic

con

stan

t (G

Pa)

P (GPa)

0 10 20 30 40 50 60 70 80 9 0

6.2

6.4

6.6

6.8

7.0

7.2 P~ 14 GPa (V=7 0.0 Bohr3) IX S m easurem ents confid ence level (2 σ ) Steinle-Neum ann GGA Steinle-Neum ann LDA

ac

soun

d ve

loci

ty (K

m/s

)

direc tion respect c- axes (degree)

Ref.:Steinle-Neumann et al., Phys. Rev. B, 60, 791, (1999)

Comparison of IXS with theory

Phonon dispersion in Nd1.86Ce0.14CuO4+δ

M. D’Astuto et al.; Phys. Rev. Lett. 88, 167002 (2002)

Electron-phonon coupling in electron-doped high Tc superconducting cuprates?

Explanation of particle-hole asymmetry

Z.-X. Shen (2001)

Angle-resolved photoemission studies: Discontinuity in the quasi-particle dispersion.INS on hole-doped systems (La1.85Sr0.15CuO4): Anomalous softening of LO phonons.

Doping x

• Renormalisation of the highest LO branch with respect to undoped compound.• Anomalous softening of the highest LO branch.

=>Strong evidence for e-ph coupling as well in electron doped cuprates.

T = 15 KV = 5·10-3 mm3

Phonon dispersion in MgB2A. Shukla et al.; Phys. Rev. Lett. 90, 095506 (2003)

A selection of other investigated Systems

Liquids and glasses

- Teflon, Polyethylene (high presssure)- Na, Si, Ge, Sn (high temperature)- SiO2, aSi- H2, D2- glycerol, polybutadiene, salol,…….- water

Polycrystals- Ice IX, XII- Fe, FeS, FeO, FeSi (high pressure)

Single crystals

- Ice VI,VII; CH4-H2O (high pressure)- Co, Zn,Ta,Mo (high pressure)- SrTiO3 (high pressure)- SmS (high pressure) - C12,C13,GaN, SiC (ZB), ZnMgY- NbSe3, Pu- Nd1.86Ce0.14CuO4+δ, LaSrCuO4, HgBa2CuO4