School on Synchrotron and Free-Electron-Laser Sources and their Multidisciplinary Applications. 26 April - 7 May, 2010

Transcription

1 School on Synchrotron and Free-Electron-Laser Sources and their Multidisciplinary Applications 26 April - 7 May, 2010 Inelastic x-ray scattering: principles Filippo Bencivenga Elettra, Trieste Italy

2 Inelastic x-ray scattering: principles Filippo Bencivenga Trieste 2010

3 OUTLINE Introduction High resolution inelastic x-ray scattering (IXS) Collective atomic dynamics Neutrons vs. X-rays Basic theory and instrumentation Experimental highlights Inelastic x-ray Raman scattering (XRS) Experimental/theoretical aspects Scattering vs. absorption spectroscopy Experimental highlights Resonant inelastic x-ray scattering (RIXS) Trieste 2010

4 Introduction: inelastic X-ray spectrum IXS RIXS XRS Energy Transfer [ev]

5 IXS: collective atomic dynamics The simpler case δ n (x,t) a x Information: Interatomic Structure (a) Interaction Potential (β) U = -βx 2 δ n (x,t) = δ n,0 exp[i(kx-ωt)] Phonons ω(k) Eigenstates of vibrational field Dispersion relation 1 st Brillouin Zone ω = (4β/M) 1/2 sin(ka/2) -π/a π/a k

6 IXS: collective atomic dynamics One step forward: 3D lattice z δx n (r,t) ω π/a a x y a a ω = (c ij /ρ) 1/2 sin(ka*) U = -β r 2 r = (x,y,z) ω k (along 100) π/ 2a Information: Interatomic Structure (a) Interaction Potential (β) Anisotropy (elasticity: c ij ) k (along 110)

7 IXS: collective atomic dynamics z ω 3N-3 optic branches ω(k=0) 0 π/ 2a ω TO /ω LO ω LA a x ω TA1 ω TA2 TA y a a Information: k (along 110) Interatomic Structure (a) TO k // x Interaction Potential (β) Anisotropy (elasticity: c ij ) Intramolecular vibrations

8 IXS: collective atomic dynamics The most complex case: disordered systems z δx n (r,t) z δx n (r,t) a x x y a a y Translational periodicity (a) ω Brillouin Zones π/a 2π/a k Eingenstates exp[i(kx-ωt)]

9 IXS: collective atomic dynamics The most complex case: disordered systems z δx n (r,t) z ξ ~ a τ ~ ps a x x y a a y Information: Interatomic Structure (ξ) ω τω~1 Interaction potential Topological disorder ~π/ξ ~ 2π/ξ k ~π/ξ Relaxation processes Lineshape (anharmonicity, etc )

10 An example Diamond: fcc symmetry + 2 C atoms each lattice (¼,¼,¼) Experimets (dots) Calculation (lines) Brillouin zones for fcc symmetry PRB 48, 3156 (1993) Information: Structure and Elasticity (sound velocities) Interaction Potential and Anharmonicity Dynamical Istabilities (phonon softening) Phonon-Electron coupling Thermodynamics (c V, λ, Θ D, S D, etc )

11 How can we measure Atomic Dynamics? ω mev K X Γ L Probe wavelenght (2π/ k ) < 0.1 nm Probe energy (E) > mev Inelastic scattering: ω=0 Inelastic Light Scattering (Brillouin & Raman) Ultrasonics Transient Grating Etc k E = E out - E in k = k out - k in k=0 k~20 nm -1 k=0 k~20 nm -1 N i n E in, k in 2θ Sample E, k Cross section: 2 σ ~ Nout /N in Ω E

12 Neutrons vs. X-rays λ in = 1Å E in = 82 mev λ in = 1Å E in = 12.4 kev E > 4 mev E/E in = 0.05 E > 4 mev E/E in = Moderate energy resolution Very high energy resolution Inelastic scattering: 100 INS instruments E = E out - E in 4 IXS instruments k = k out - k in + Spin sensitive Better contrast E in, k in 2θ Why X-rays? Older technique Sample E, k

13 Neutrons vs. X-rays λ in = 1Å E in = 82 mev λ in = 1Å E in = 12.4 kev E out E in E = E out - E in & k = k out - k in E out E in k 2 2 k in 2= 1-E/E k = 2 k in sin(θ) in+cos(2θ)(1-2e/e in ) 1/2 90 Kinematic constraint 90 θ= E (mev) E (mev) k (nm -1 ) k (nm -1 )

14 Neutrons vs. X-rays λ in = 1Å E in = 82 mev λ in = 1Å E in = 12.4 kev E out E in E = E out - E in & k = k out - k in E out E in k 2 2 k in 2= 1-E/E k = 2 k in sin(θ) in+cos(2θ)(1-2e/e in ) 1/2 E (mev) E (mev) θ= k (nm -1 ) k (nm -1 )

15 Neutrons vs. X-rays Inelastic excitations in disordered systems Neutrons X-rays λ in = 1Å E in = 82 mev λ in = 1Å E in = 12.4 kev E (mev V) !? E (mev V) θ= k (nm -1 ) k (nm -1 )

16 Neutrons vs. X-rays λ in = 1Å E in = 82 mev λ in = 1Å E in = 12.4 kev E > 4 mev E/E in = 0.05 E > 4 mev E/E in = Moderate energy resolution Very high energy resolution 100 INS instruments No kinematical + constraints (Disordered Spin sensitive systems) 3 IXS instruments Why Better contrast Older technique Small beams (small samples: high pressure, exotic materials, etc ) X-rays? No incoherent cross section

17 Basic theoretical aspects H int =(e/m e c) j [(e/2c)a j A j +A j p j +magnetic] A is the vector potential of electromagnetic field p is the momentum operator of the electrons j is the summation over all electrons of the system 1 st order perturbation theory A A term one photon (non-resonant) scattering 2 σ = r02 (ε in ε out ) 2 (k in /k out ) I P I <I exp{ik r j } F> 2 δ(e-e out +E in ) Ω E

18 Basic theoretical aspects 2 σ = r02 (ε in ε out ) 2 (k in /k out ) i P i <I exp{ik r j } F> 2 δ(e-e F +E I ) Ω E The key assumption: Adiabatic approximation I>= I n > I e > and F>= F n > F e > 2 σ = r02 (ε in ε out ) 2 (k in /k out ) F( k ) 2 S(k,E) Ω EE Thomson scattering cross section Molecular form factor ( I e >, F e >) Dynamic structure factor

19 The dynamic structure factor S(k,E) is the SPACE and TIME Fourier transform of G(r,t) y z x G(r,t) is the probability to find two distinct particles at positions r 1 (t 1 ) and r 2 (t 2 ), separated by the distance r=r 2 -r 1 and the time interval t=t 2 -t 1.

20 Basic IXS instrumentation Spot size: 30 x 60 µm 2 (H x V) θ B sample 2θ detector k = k out - k out Monochromator k = 4πsin(θ) Si(n,n,n) θb = n=7-13 λ in = 2hc/E in = 2d n sin(θ B ) θ B λ in (tunable) λ out (constant) E = E in - E out λ out = 2hc/E out = 2d n sin(θ B ) E/E = λ in /λ in ~ cot(θ B ) Analyser Si(n,n,n) θ B = n=7-13

21 Basic IXS instrumentation θ B sample 2θ detector k = k out - k out Monochromator k = 4πsin(θ) Si(n,n,n) θb = n=7-13 λ in = 2hc/E in = 2d n sin(θ B ) θ B d n /d n = -α(t) T α ~ ( E/E) MAX ~ T ~ mk < 1 27 kev Analyser Si(n,n,n) θ B = n=7-13

22 Basic IXS instrumentation θ B Monochromator Si(n,n,n) θb = n=7-13 sample 2θ θ detector θ ~ mrad ( k ~ 0.2 nm -1 ) >> θ D θ D θ B Analyser Si(n,n,n) θ B = n=7-13

23 Basic IXS instrumentation relative tem perature [K] θ B sample detector 2θ counts in 80 secs Diamond Monochromator Si(n,n,n) θb = n=7-13 θ energy transfer [m ev] θ B θ ~ mrad ( k ~ 0.2 nm -1 ) >> θ D θ B θ B flat Si perfect single crystals (0.6x0.6 mm 2 ) that approximate a spherical surface NIM 111, 181 (1996) Analyser Si(n,n,n) θ B = n=7-13

24 Experimental highlights (1) Collective dynamics in water Inelastic Neutron Scattering (D 2 O): 2 experiments, 2 results: why? 18 Bosi et al. - Nuovo Cimento Lett. (1978) Texeira et al. - Phys. Rev. Lett. (1985) A possible interpretation: High frequency mode D Low frequency mode O (mev) E 12 6 Ω HF /Ω LF ~ (m O /m D ) 1/2 ~ k (nm -1 )

25 Experimental highlights (1) Collective dynamics in water (mev) Inelastic Neutron Scattering vs. Inelastic X-ray Scattering (H 2 O vs. D 2 O) 24 Bosi et al. - Nuovo Cimento Lett. (1978) Texeira et al. - Phys. Rev. Lett. (1985) IXS: Sette et al. - Nature (1996) H 2 O 2 2 D 2 O A possible interpretation: High frequency mode D Low frequency mode O Ω HF /Ω LF ~ (m O /m D ) 1/2 ~ 2 E k (nm -1 ) High frequency mode: Expected for H 2 O Ω IXS /Ω INS ~ (m H /m D ) 1/2 ~ 1.4

26 Experimental highlights (1) Collective dynamics in water (mev) Inelastic Neutron Scattering vs. Inelastic X-ray Scattering (H 2 O vs. D 2 O) 24 Bosi et al. - Nuovo Cimento Lett. (1978) Texeira et al. - Phys. Rev. Lett. (1985) IXS: Sette et al. - Nature (1996) H 2 O 2 2 D 2 O A possible interpretation: High frequency mode D Low frequency mode O Ω HF /Ω LF ~ (m O /m D ) 1/2 ~ 2 E k (nm -1 ) High frequency mode: but Ω IXS =Ω INS

27 Experimental highlights (1) Collective dynamics in water IXS: no kinematic constraints low-k data can be obtained 9 24 Bosi et al. - Nuovo Cimento Lett. (1978) Texeira et al. - Phys. Rev. Lett. (1985) IXS: Sette et al. - Nature (1996) (mev) E k (nm -1 )

28 Experimental highlights (1) Collective dynamics in water 24 Bosi et al. - Nuovo Cimento Lett. (1978) Texeira et al. - Phys. Rev. Lett. (1985) IXS: Sette et al. - Nature (1996) 18 Viscoelasticity: The sound velocity (c=e/ħk) is not constant but depends on Eτ/ħ Solid-like (~ice) (mev) 12 c = c E 6 c c = c s k (nm -1 ) Liquid-like Eτ/ħ

29 Experimental highlights (1) Collective dynamics in water 18 Low frequency mode: Transverse-like sound propagation in the elastic (solid-like) limit: Eτ/ħ >> 1 24 Bosi et al. - Nuovo Cimento Lett. (1978) Texeira et al. - Phys. Rev. Lett. (1985) IXS water: Sette et al. - Nature (1996) IXS ice : Cimatoribus et al. - New J. Phys. (2010) Viscoelasticity: The sound velocity (c=e/ħk) is not constant but depends on Eτ/ħ Q = 8 nm -1 Water Q = 11 nm -1 TA LA Water TA LA (mev) E 12 6 Inelastic Signal (Arb. Units) Ice Ice k (nm -1 ) E (mev) E (mev) PRL 79, 1678 (1997) PRE 71, (2005)

30 Experimental highlights (2) Phonon dispersions in plutonium Putonium is one of the most fascinating and exotic element: Multitude of unusual properties Central role of 5f electrons ID28 at ESRF Energy resolution: 1.8 mev Beam size: 20 x 60 µmm 2 (HxV) Grain size: ~ 80 µm 2 On-line diffraction analysis LA (0.2;0.2;0.2) TA (0.2;0.2;0.2) Counts in 180 secs Counts in 180 se ecs 100 Science 301, 1078 (2003) Energy [mev] Energy [mev]

31 Experimental highlights (2) Phonon dispersions in plutonium soft-mode behavior of T[111] branch Expt B-vK fit 4NN Onset of phase transitions (here to monoclinic α phase at 163 K) Born-von Karman force constant model fit (fourth nearest neighbors) Science 301, 1078 (2003)

32 Experimental highlights (2) Phonon dispersions in plutonium Close to Γ-point: E = Vq/ħ <001> <111> VL[100] = (C11/ρ)1/2 VT[100] = (C44/ρ)1/2 VL[110] = ([C11+C12+2C44]/ρ)1/2 <110> VT1[110] = ([C11 - C12] /2ρ)1/2 VT2[110] = (C44/ρ)1/2 VL[111] = [C11+2C12+4C44]/3ρ)1/2 VT[111] = ([C11-C12+C44]/3ρ)1/2 C11 = 35.3±1.4 GPa C12 = 25.5±1.5 GPa C44 = 30.5±1.1 GPa highest elastic anisotropy of all known fcc metals Science 301, 1078 (2003)

33 Experimental highlights (2) Phonon dispersions in plutonium Specific heat: C v = 3Nk E 2 max E exp( / ) ( ) E kbt g E B 0 k BT B 1 de ( exp( E / k T ) ) 2 6 3R Born-von Karman fit g(e) Density of states (arb. units) Cv (cal mole -1 K -1 ) Temperature (K) Science 301, 1078 (2003)

34 Experimental highlights (3) Elasticity at high pressure Elasticity of hcp-metals under very high pressure (up to 1 Mbar): Geophysical interest (Earth core) DAC sample environment + IXS ga ske t sam ple X ra y diam ond hcp-co single crystal 1 m m helium PRL 93, (2004) Ø 45 µm 20 µm thickness ruby hcp-structure: 5 independent elastic moduli V L [001] = (C 33 /ρ) 1/2 V L [100] = (C 11 /ρ) 1/2 V T1 [110] = ([C 11 - C 12 ] /2ρ) 1/2 V T2 [110] = (C 44 /ρ) 1/2 V QL [101] = f(c ij,ρ) C 13

35 Experimental highlights (3) Elasticity at high pressure counts/120 s LA (1.05,0,0) 200 P=15 GPa Energy [mev] Energy [mev V] Energy [mev] V L [100]=(6140 +/-60) m/s P=9.5 GPa q [nm -1 ] V L [001]=(7550 +/-50) m/s P=31.5 GPa PRL 93, (2004) q [nm -1 ]

36 Experimental highlights (3) Elasticity at high pressure li (GPa) Elastic Modul C 33 C 11 C 13 Energy [mev V] V L [100]=(6140 +/-60) m/s P=9.5 GPa Elastic Modu uli (GPa) C 66 C 44 Energy [mev] q [nm -1 ] V L [001]=(7550 +/-50) m/s P=31.5 GPa PRL 93, (2004) Pressure (GPa) q [nm -1 ]

37 Experimental highlights (4) Liquids and supercritical fluids Counts / 65 s Counts / 150 s ID28 and ID16 at ESRF Energy resolution: 1.5 mev Moderate pressure (< 500 bar) Various temperatures ( K) bar 32 K 400 bar 87 K Q = 8 nm -1 Ne E (mev) N 2 PRL 98, (2007) 220 bar 293 K 400 bar 293 K NH 3 H 2 O E (mev) Counts / 150 s Counts / 65 s τ C / < τ > Viscoelasic data analysis: H 2 O NH 3 N 2 Ne T c / T Leading dynamics in fluids: T>Tc collisions T<Tc bond s lifetime

38 Basic instrumentation Monochromator Si(n,n,n) n=7-13 IXS XRS/RIXS sample detector 2θ Energy Transfer [ev] E ~ ev k = k out - k out E = E in - E out = 2hc(1/λ in -1/λ out ) λ = 2d n sin(θ B ) θ B Si(n,n,n) Analyser n=7-13 backscattering + high order reflections = E ~ mev

39 Basic instrumentation XRS / RIXS Monochromator Si(1,1,1); (2,2,0); λ in (tunable) E in, k in sample detector 2θ XRS/RIXS Energy Transfer [ev] Rowland circle spectrometer (1 m) k = k out - k out θ B E = 2hc(1/λ in -1/λ out ) λ = 2d n sin(θ B ); θ B <90 Si(h,k,l) Analyser λ out (tunable)

40 Basic instrumentation XRS / RIXS E in, k in sample detector 2θ Rowland circle spectrometer XRS/RIXS k = k out - k out E = 2hc(1/λ in -1/λ out ) θ B λ = 2d n sin(θ B ); θ B <90 Energy Transfer [ev] Scanning strategy 1. E out fixed, scanning E in IXS, XRS, RIXS 2. E in fixed, scanning E out RIXS (rotating crystal and follow with the detector) 3. Scanning E in and E out keeping E constant RIXS

41 Basic theoretical aspects Inelastic scattering: H int =(e/m e c) j [(e/2c)a j A j +A j p j ] E = E out - E in k = k out - k in A: vector potential of electromagnetic field P: momentum operator of the electrons j: summation over all electrons of the system E in, k in Sample 2θ E, k A A non-resonant scattering (example: IXS) A p resonant scattering, absorption followed by emission

42 Basic theoretical aspects Non resonant IXS cross section: 2 σ = r02 (ε in ε out ) 2 (k in /k out ) I P I <I exp{ik r j } F> 2 δ(e-e F +E I ) Ω E e k e k E F O 1s = 543 ev E F O 1s = 543 ev E in E out E in O K-shell O K-shell E 1s +e k =E in -E out =E E 1s +e k =E in X-ray absorption cross section (dipolar approximation): σ = 4π2 αe in I P I <I ε in r j F> 2 δ(e in -E F +E I ) E in

43 Basic theoretical aspects Non resonant IXS cross section: 2 σ = r02 (ε in ε out ) 2 (k in /k out ) I P I <I exp{ik r j } F> 2 δ(e-e F +E I ) Ω E k r j << 1 e ik r ~ 1 + ik r j k r j <<1 Dipolar regime: identical to photon absorption, where: i) The momentum transfer (k) plays the role of the photon polarization vector (ε in ) ii) The energy transfer (E) plays the role of the incident energy (E in ) X-ray absorption cross section (dipolar approximation): σ = 4π2 αe in I P I <I ε in r j F> 2 δ(e in -E F +E I ) E in

44 Basic theoretical aspects Non resonant IXS cross section: 2 σ = r02 (ε in ε out ) 2 (k in /k out ) F P F <I exp{ik r j } F> 2 δ(e-e F +E I ) Ω E X-ray Raman Scattering (XRS) e k E F E in E 1s +e k =E in -E out =E O 1s = 543 ev E out O K-shell Motivation: element-selective probe for local atomic structure XRS is alternative to: Neutron scattering (with isotopic substitution) X-ray (anomalous) scattering XANES and EXAFS

45 Experimental highlights (XRS) XRS from O K-edge in water and ice E in = 9686 ev ( E ~ 2 ev) k = 4.38 A -1 (k r ~ 0.29) PRB 62, R9223 (2000)

46 Experimental highlights (XRS) XRS from O K-edge in water and ice (EXAFS) Intensity (arb b. units) W ater 0.5 Ice Ih Energy [ev] PRB 62, R9223 (2000)

47 Experimental highlights (XRS) XRS from O K-edge in water and ice (EXAFS) XRS 10 fit parameters & constrained S(Q) Neutron diffraction XRS 15 fit parameters & unconstrained S(Q) XRS: O-O distance: 2.87 Å Coordination: 4-7 Neutrons: O-O distance: 2.85 Å Coordination: 4.4 PRB 62, R9223 (2000)

48 Experimental highlights (XRS) XRS from O K-edge in water and ice (XANES) XANES sensitive to the number and type of hydrogen bonds (HB) Pre-edge E ~ 1 ev k r ~ XRS agrees with XAS Pre-edge indicates a large number (~ 70%) of distorted or broken HB (supported by calculation) PRB 66, (2002)

49 Experimental highlights (XRS) XRS from O K-edge in ice under high pressure Eout = 9885 ev ( E ~ ev) k ~ 3 A-1 (k r ~ 0.2) DAC sample environment: hard x-ray needed d ia m o n d ga ske t sa m p le PRL 94, (2005) X ra y

50 Experimental highlights (XRS) XRS from O K-edge in ice under high pressure Slight increase of pre-edge with P (larger HB distortion) Increasing order of HB from liquid Ice III (tetrahedral) Ice II / IX New pre-edge low-t: new Ice phase? Observation of spectral changes: Need of much better statistics and theory to extract quantitative information PRL 94, (2005)

51 XRS in summary Soft x-ray spectroscopy in the hard x-ray regime Advantages simpler sample environment (high pressure/temperature, etc ) bulk sensitive indicated for studying (bulk) Oxygen and Carbon Drawbacks weak probe (practically limited to Z < 14) limited quality for structural analysis (EXAFS), reasonable quality in the XANES region Exploit information in the near-edge region

52 Basic theoretical aspects (RIXS) Inelastic scattering: H int =(e/m e c) j [(e/2c)a j A j +A j p j ] E = E out - E in k = k out - k in A: vector potential of electromagnetic field P: momentum operator of the electrons j: summation over all electrons of the system E in, k in Sample 2θ E, k A A non-resonant scattering (IXS - XRS) A p resonant scattering, absorption followed by emission

53 Basic theoretical aspects (RIXS) Absorption Resonant IXS cross section: Emission 2 σ <I C F k N> <N C k+ F> 2 δ(e-e Ω E E ~ F n F +E I ) E N -E I -E in -iγ N C k = j (ε p j )exp{ik r j } Resonant denominator e k E F E F 3 d 5/2 3 d 5/2 E in 2 p 3/2 2 p 3/2 E out E 2p +e k =E in E=E in -E out =E 2p -E 3d +e k

54 Basic theoretical aspects (RIXS) Final states for XAS are intermediate for RIXS (resolution ~Γ N ) E F e k E F 3 d 5/2 3 d 5/2 E in 2 p 3/2 2 p 3/2 E out E 2p +e k =E in Motivation: Final state core-hole lifetime < energy separation of the multiplet families Motivation: Keeping E fixed and tuning E in through edge (CFS) E=E in -E out =E 2p -E 3d +e k RIXS allows the separation of different excitation channels, which are obscured in XAS Resonant enhancement of intermediate states