Inelastic X-ray Scattering study of the high frequency dynamics in Liquids and Glasses. Giancarlo Ruocco

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1 Inelastic X-ray Scattering study of the high frequency dynamics in Liquids and Glasses Giancarlo Ruocco Dipartimento di Fisica, Universita di Roma La sapienza, Italy Common features in the collective dynamics of liquids, glasses and topologically disordered systems: Five years of (and MD) studies 1

2 The technique: What do we measure? E o k o S Photon / Neutron E f k f Photon / Neutron "Phonon" E Q E = E o - E f Q = k o - k f σ ω Ω = N (e /mc ) k f /k i (ε f. ε i ) f(q) S(Q, ω) X-rays vs Neutron No kinematic limitations (i.e. low Q accessible) E o >> E Const. Q scan No incoherent scattering No multiple scattering But ω / ω v max = ω o / k o θ = o 3 o 6 o 9 o Low flux Typical counting rate : 1-1 photons / sec Q / k o Energy resolution 6. BL1 ESRF. Intensity ( counts ) PMMA Q = 1 nm -1 Si ( ) Lorentzian Fit E = 1.4 mev Energy ( mev )

3 The technique: Recent Applications Crystals Quartz (dispersion relations) Glasses CdTe Ice Ih (structural phase transition at high pressure) (dispersion relations) Glycerol, LiCl:6H O (Phonon-like modes at high Q in glasses) Silica [SiO ] Boron Oxide [B O 3 ] Zinc Chloride [ZnCl ] Polibutadiene [PB] Characteristic of the o-terphenyl [otp] vibrational modes in m-tricresylphospate [mtcp] disordered systems Propilene Carbonate [PC] Calcium Potassium Nitrate [CKN] PMMA (glass-glass transformation at high pressure) Liquids Fluids Lithium Glycerol o-terphenyl Propilene Carbonate m-tcp Viscoelastic behaviour and glass transition Polibutadiene Water (The fast sound issue and the Viscoelastic behaviour of water) Neon (Transition from hydrodynamic to kinetic regime) Helium (Density dependence of zero point kinetic energy) 3

4 The technique: Examples EXCITATIONS IN LIQUIDS Liquid Lithium 4

5 Lithium S(Q) from sum rules:. 4 S(Q) M o [S(Q,ω)]=S(Q) M 1 [S(Q,ω)]=h Q /M T = 475 K Si (7 7 7) Si (9 9 9) Si ( ) Q (nm -1 ) MD (sum rules) 4 3 Positive dispersion Ω(Q) ( mev ) 1 v s (Q) ( km/s ) Isothermal sound speed Q ( nm -1 ) Toward the single Particle regime Structural bend-down (Quasi Brillouin zone) Ω(Q) (mev) T=475 K isothermal sound speed v = 443 m/s Free particle Q (nm -1 ) 5

6 The technique: Examples VISCOELASTICITY.5 Q = nm T = 66 K Q = 4 nm -1 Water Intnesity ( count / s ) T = 314 K 1.5 T=78 K T=373 K T=493 K M r T = 413 K Energy ( mev ).5 v (Q)-v o v -v o Q ( nm -1 ) 6

7 The technique: Examples GLASS TRANSITION o-terphenyl c L ( Km / s ) US BLS T g OTP Temperature ( K ).9.8 T g T c f Q (T) OTP Temperature ( K ) 7

8 The technique: How to represent the S(Q,ω) (in glasses)? Langevin equation for F(Q,t) (= FT ω {S(Q,ω)} ) : t t F(Q,t) + ω o F(Q,t) + dt m(q,t-t ) F(Q,t ) = o t In a glass (approx: no hopping, no relax, ) or at very high frequency (ω 1 THz) freezing of the relaxation processes + very fast process (to be discussed later ) t F(Q,t) + Γ Q m 1 (t) const = Q m (t) = Γ Q δ(t) t m(t) m (t) F(Q,t) + (ω o + Q ) F(Q,t) = Q F(Q,) ω o =c o Q Ω Q =c Q Q f Q = F(Q,t )/ F(Q,) = Q /(ω o + Q ) = 1 (c o / c ) m 1 (t) t S(Q,ω) = S(Q) (1-f Q )Γ Q Ω Q ( ω - Ω Q ) + Γ Q ω + f Q δ(ω) DHO (Damped Harmonic Oscillator) C(Q,ω) S(Q,ω) S(Q,ω) C(Q,ω) = ω / Q S(Q,ω) C(Q,ω) Ω Q Γ Q ω 8

9 The technique: example of fitting procedure Intensity ( counts ) Glycerol (Glass) Q = nm -1 T = 171 K Energy ( mev ) Fitting function = ( DHO + Elastic Peak ) * ( Exp Resolution ) Ω Q Γ Q f Q 9

10 Common features in the collective dynamics topologically disordered systems ( at ω/ω D.5 & Q/Q BZ.5 ) I Existence of Longitudinal collective excitations at small Q: Excitation energy proportional to Q Excitation broadening proportional to Q at increasing Q: Positive dispersion of the sound velocity Umklapp processes II Existence of Transverse collctive excitations (only appearing in the spectra at high Q) 1

11 Existence of Propagating Modes A S(Q,ω) 15 3 Vitreous Silica - - T=11 K Q = 1 nm -1 Q = nm Intensity ( counts / 6 s ) Energy ( mev ) ❶ Energy ( mev ) Q = nm -1 Q = 1 nm -1 Q = nm -1 1 Q = 1 nm Energy ( mev ) ❶ F. Sette et al., Science 8, 155 (1998) 11

12 Fitting Model (DHO) S(Q,ω) = S(Q) (1-f Q )Γ Q Ω Q ( ω - Ω Q ) + Γ Q ω + f Q δ(ω) Free parameters Glycerol - - T=175 K 6 Intensity ( counts / 18 s ) Q = 5 nm -1 Q = 4 nm -1 Q = 3 nm -1 Q = nm Q = 1 nm Energy ( mev ) 1

13 otp - - T=156 K v-sio --T=1375 K 4 75 Intensity ( counts / 1 s ) Q = 5 nm -1 Q = 4 nm -1 Q = 3 nm -1 Q = nm -1 Intensity ( counts ) q = 3.7 nm -1 q =.5 nm -1 q = 1.6 nm -1 q = 1. nm Q = 1 nm -1 ❶ q =.75 nm ❷ Energy ( mev ) Energy ( mev ) Also. Propilene Carbonate ❸ Polibutadiene ❹ m-tcp ❺ LiCl:6H O ❻ CKN ❼....and others ❶ G. Monaco et al., Physical Review Letters 8, 161 (1998). ❷ P. Benassi et al., Physical Review Letters 77, 3835 (1996). ❸ R. Di Leonardo et al., preprint. ❹ D. Fioretto et al., Physical Review E59, 447 (1999). ❺ M. Soltwisch et al., preprint. ❻ C. Masciovecchio et al., Physical Review Letters 76, 3356 (96) ❼ A. Matic et al. Europhysics Letters 54, 77 (1). 13

14 Dispersion relations: Glycerol T=175 K otp T= K Ω(Q) ( mev ) Ω(Q) ( mev ) T=175 K T=9 K T=156 K T=3 K T=38 K T=45 K Boson peak position Q (nm -1 ) ❶ Q ( nm -1 ) Boson peak energy ❷ 14 1 V=679 m/s v-sio T=1375 K Ω(Q) ( mev ) Boson peak energy ❸,,5 1, 1,5,,5 3, 3,5 4, Q ( nm -1 ) ❶ F. Sette et al., Science 8, 155 (1998) ❷ G. Monaco et al., Physical Review Letters 8, 161 (1998). ❸ C. Masciovecchio et al., Philosophical Magazine B79, 13 (96). 14

15 Existence of Propagating Modes B constant energy cut Glycerol - - T=17 K S (Q,E) (arb. units) S(E= mev,q) S(E=8 mev,q) difference ❶ Q (nm -1 ) ❶ S(Q,E) (arb. units) 1 E = 4.7 mev E = 8 mev Q (nm -1 ) ❶ C. Masciovecchio, Physical Review Letters 85, 166 (). 15

16 Existence of Propagating Modes B constant energy cut v-sio - - T=11 K -1 E=8.5 mev 1.5 nm E= mev.1 nm -1 I(E,Q) (arb. units) E=5.3 mev E= mev 1 3 Q (nm -1 ) 16

17 Existence of Propagating Modes B constant energy cut ❶ v-sio - - T=11 K E=8.5 mev I anel (E,Q) (arb. units) 1 E=5.3 mev Q (nm -1 ). DHO Model (P. Benassi et al. PRL 77, 3835 (1996)) EMA Model (M. Foret et al. PRL 77, 3831 (1996)) ❶ O. Pilla et al., Physical Review Letters 85, 136 (). 17

18 Existence of Propagating Modes B constant energy cut NH 3 T= K (liquid) S(Q,E) (arb. units) E=5 mev E=1 mev E=15 mev 3 1 Q=.8 nm Q ( nm -1 ) Intensity ( counts/18 s ) 4 Q=5.8 nm -1 5 Q=8.8 nm Q=11.8 nm Q=14.8 nm Energy shift ( mev ) F. Sette et al., Physical Review Letters 84, 4136 (). 18

19 Existence of Propagating Modes B constant energy cut NH 3 T= K (liquid) Q f ( nm -1 ) Q m S(Q, E) ( arb. units ) E ( mev ) Q ( nm -1 ) E (mev) 19

20 Existence of Propagating Modes C propagation threshold NH 3 T= K (liquid) Ω ( mev ) v=6 +/- 5 m/s v=8 m/s Γ ( mev ) Q ( nm -1 ) Q m

21 3 Q broadening A S(Q;ω) Experiments 1 v-sio T=11 K Linewidth (mev) BLS POT ❶ 1-4,1,1 1 1 Q ( nm -1 ) Glycerol T=145 K Γ (Q) (mev) Γ (Q) (mev) LiCl:6H O 6 4 T=8 K ❷ Q (nm - ) ❶ P. Benassi et al. Physical Review Letters 77, 3835 (1996) ❷ C. Masciovecchio et al. Physical Review Letters 76, 3356 (1996) 1

22 3 Q broadening B S(Q,ω) Simulations MD 3 MD T= Harmonic Model) T = 15 K v-sio MD (T=) (T=11 K) Ω(Q) (mev) 1 v = 73 m/sec v = 61 m/sec Also S(Q, ω) v-sio (S.N.Taraskin, S.R.Elliot) ZnCl (M.Ribeiro, M.Wilson, P.A.Madden) a-silicon (J.L.Feldman, P.B.Allen, S.R.Bickham),1,1,8,6,4, Q ( nm -1 ) ❶ Γ(Q) ( cm -1 ) Γ(Q) (mev) 4 6, v-ar MD T= ❶ G. Ruocco et al., preprint, cond-mat/13 ❷ R. Dell Anna et al. Physical Review Letters 8, 136 (1998). 3 1 ω ( cm -1 ) 1 slope = Q ( nm -1 ) Q (nm -1 ) ❷

23 3 Q broadening C Temperature dependence FRAGILE +BLS D=Γ Q /Q Polibutadiene + BLS 1-1 BLS (Q=.35 nm -1 ) Γ Q /Q ( cm /s ) 1 - (Q= nm -1 ) Temperature ( K ) D. Fioretto et al., Physical Review E59, 447 (1999). 3

24 otp +BLS Γ(Q)/Q ( cm s -1 ) ❶ 1-1 (Q= - 6 nm -1 ) 1 - BLS (Q=.45 nm -1 ) T g Temperature ( K ) Propilene Carbonate + BLS Γ(Q)/Q (1 - cm s -1 ) 1,1 ❷ BLS (Q=.45 nm -1 ) T g ❷ (Q= - 6 nm -1 ) Temperature (K) ❶ G. Monaco et al., Physical Review Letters 8, 1776 (1999) ❷ R. Di Leonardo et al., preprint 4

25 3 Q broadening C Temperature dependence STRONG +BLS v-silica Γ Q / Q ( mev/nm - ) BLS MD SiO Scopign o e t al. Vacher et al. Pine et al. Dell'Anna et al. Γ Q / Q ( mev/nm - ) , 5 1,,5, 1 Temperature ( K ) T ( K ) ❶ 1 1 BLS (.3 nm -1 ) Courtens e t al. Nelson et al. T g Glycerol Γ Q / Q ( mev/nm - ) 1 (-4 nm -1 ) ( nm -1 ) Temperature ( K ) ❶ G. Ruocco et al.,physical Review Letters, 83, 5583, (1999) ❶ 5

26 Glycerol v-silica Γ Q ( mev ) Γ Q ( mev ) BLS Glycerol MD POT BLS γ = γ =.6 γ = T=16 K (This w ork) T=167 K (This w ork) T=175 K [1] T= K [9] T=17 K [1] γ =.5 Γ Q ( mev ) T= K Dell'Anna et al. T=3 K Zhu et al. T=11 K T=5 K Vacher et al. T=3 K Q (nm -1 ) 1 1 Q ( nm -1 ) Γ Q ( mev ) ❶ MD T = K T=3 K IX S T=1 1 K T =137 5 K Q ( nm -1 ) Q ( nm-1 ) ❶ ❶ G. Ruocco et al., Physical Review Letters, 83, 5583, (1999) 6

27 4 Umklapp Ω(Q) Ω(Q) E Q Q Xtal G G - Q G + Q G FSDP -Q G FSDP +Q E Gl ass G FSDP S(Q) π/a Q 7

28 4 Umklapp A MD Lennard-Jones 4 E=.5 mev 4 E=.5 mev 4 E=1 mev ω S(Q, ω ) / Q 4 4 E= mev E=4 mev 4 E=6 mev 4 E=7.5 mev 4 E=1 mev Q ( nm -1 ) 8

29 B Liquid Lithium 4 Umklapp E = 15 mev ω S(Q,ω) / Q ( arb. units ) E = mev E = 5 mev E = 3 mev E = 35 mev E = 4 mev Q ( nm -1 ) 9

30 5 L/T mixing A Existence of L/T mixing in water +MD E ( mev ) 3 INS Teixeira et al. INS Bosi et al. MD Rahman et al. " MD Balucani et al. v = 33 m/s v = 15 m/s Q ( nm -1 ) Zone edge of the crystal (c-axis): Q=6.5 nm -1 - In the crystal, the transverse phonons can appear in the spectra only in Brillouin zones larger than the first (Q > 6.5 nm -1 ). - In the liquid, the transverse mode can appear also at smaller Q, but with intensity decreasing with Q. The similarity of the energies of the modes and the (relaxed) symmetry selection rules, suggest that the peak observed in liquid water at E 5 mev reflects the transverse dynamic. ❶ F. Sette et al., Physical Review Letters 77, 83 (1996). 3

31 HO - MD - T=3 K The transverse dynamic can be accessed by MD simulation (M. Sampoli et al. PRL79, 1678, 1997) 4 SPC/E water molecules Longitudinal current Transverse current 1..5 C L (Q,ω) Q=16. nm C T (Q,ω) Q=16. nm -1 C(Q,ω) ( arb. units ) Q=13.1 nm -1 Q=9.8 nm -1 Q=7.8 nm Q=13.1 nm -1 Q=9.8 nm -1 Q=7.8 nm -1. Q=5.8 nm -1.5 ❶ Energy ( mev )..5 Q=5.8 nm -1 ❶ Energy ( mev ) High Q : Longitudinal and Transverse peaks in both C L (Q,ω) and C T (Q,ω) Low Q : Longitudinal peak in C L (Q,ω) ❶ M. Sampoli et al., Physical Review Letters 79, 1678 (1997). 31

32 5 L/T mixing B Existence of L/T mixing in vitreous silica MD v-sio - MD - T= Q = 13.1 nm -1 Q = 1.1 nm -1 C L (Q,E)) Q = 11.7 nm -1 Q = 1.6 nm -1 Q = 9.9 nm -1 Q = 8.31 nm -1 ❶ Energy ( mev ) ❶ O. Pilla et al. Preprint () 3

33 9 Ω L Ω T Energy (mev) Q (nm -1 ) INS ❶ O. Pilla et al. Preprint () 33

34 5 L/T mixing C Evidence of the Transverse dynamics in GLYCEROL T=8 K ❶ T. Scopigno et al. Preprint () 34

35 GLYCEROL T=8 K Co-existence of two peaks L + T? ❶ T. Scopigno et al. Preprint () 35

36 GLYCEROL Q=17 nm -1 Evidence for the transverse nature of the second peak ❶ T. Scopigno et al. Preprint () 36

37 CONCLUSIONS In topologically disordered systems, there exist collective excitations in the mesoscopic energy-momentum region (Q=1-1 nm -1, E=-1 mev) (quasi-)longitudinal at small Q: Excitation energy proportional to Q Excitation broadening proportional to Q at increasing Q: Positive dispersion of the sound velocity Umklapp processes (quasi-)transverse only appearing in the spectra at high Q due to mixing 9 Ω L Ω T Energy (mev) Q (nm -1 ) 37

38 The work I presented wouldn t be possible without the joint efforts of: The ESRF (F) F. Sette, R. Angelici, A. Cunsolo, M. Krisch, C. Masciovecchio, G. Monaco, R. Verbeni D. Fioretto, L. Perugia (I) M. Firenze (I) O. Pilla, G. Trento (I) L. Angelani, R. Di Leonardo, E. Pontecorvo, B. Ruzcika, T. Scopigno, G. Roma (I) 38