Inelastic X ray Scattering

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1 Inelastic X ray Scattering with mev energy resolution Tullio Scopigno University of Rome La Sapienza INFM - Center for Complex Dynamics in Structured Systems Theoretical background: the scattering cross section Technical aspect for the achievement of mev resolutions An example: the fast sound in water

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3 Neutrons vs X-rays: competition or complementarity?

4 Non-relativistic Relativistic

5 Second order First order

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11 IXS: Technical aspects

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17 Resolution: the geometric contribution

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19 Inelastic scattering mev APS ESRF SPring8

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21 Definition of kinematics T E = E o - E f Q = k o - k f E o k o Photon / Neutron θ S E f k f Photon / Neutron Phonon E Q T=T f -T T f Q( θ ) k E( T ) E 2sinθ 2 = [ n + m + l ] 1 2 α T γ B [E(γ B ),θ] In general: E Q For X-rays (small E/E ): T E θ Q

22 Sample aspects N ( t) I( t) IQE (, ) 2 σ = N E QLρ E Q (, ) Forw ard x σ L x di Q E Ne E QLρ e E Q 2 µ µ ( ) 2 µ σ (, ) Forward L IQE N E QLe ρ E Q Maximum scattering efficiency L 5µ m 1cm

23 Multiple scattering: Contribution arising from the composition of several events with arbitrary scattering angles ending with the selected wavevector M I ( t) I( t) (1) I ( Q, E) = N E QL 2 σ ρ E Q (1) (1) 2 N Q = I E de f S ( ) (, ) () () E π (2) 2 2 ( ) ( θ ) ( θ) θ θ θ N Q h f S d Q = h Suppressed using samples of small transverse dimension N N π 2 2 (2) h (1) 2 Q π ρr f ( θ) S( θ) dθ Z S() 2

24 Experimental strategies G Q scan (fixed Energy) Energy scan (fixed Q) E o E(Q) E Q Q Xtal G - Q G + Q G MP -Q G MP +Q Periodic lattice Non-periodic lattice Glass G MP 2π/a Q S(Q)

25 Atomic vibrations and positional disorder ORDER DISORDER A real glass

26 Existence of high frequency propoagating modes (1) Energy scan (fixed Q) E(Q) E Q Glass G MP -Q E-scan Intensity ( counts ) 6 G 4 MP +Q Q = 5 nm -1 2 Intensity ( counts / 18 s ) G MP 2π/a Intensity ( counts / 1 s ) q = 3.7 nm -1 q = 2.5 nm -1 q = 1.6 nm -1 q = 1. nm -1 q =.75 nm Q = 4 nm -1 Q = 3 nm -1 Q = 2 nm -1 Q = 1 nm -1 Energy ( mev ) Energy ( mev ) Q Q = 5 nm Q = 4 nm Q = 3 nm Q = 2 nm Q = 1 nm Energy ( mev ) GLYCEROL 12 S(Q) SILICA Ω(Q) ( mev ) Ω(Q) ( mev ) OTP Ω(Q) ( mev ) ,,5 1, 1,5 2, 2,5 3, 3,5 4, Q ( nm -1 ) T=175 K T=29 K T=156 K T=223 K T=328 K T=45 K V=679 m/s Boson peak energy Q ( nm -1 ) Boson peak energy Boson peak position Q (nm -1 )

27 Existence of high frequency propoagating modes (2) Q-scan Liquid lithium Vitreous silica Q scan (fixed Energy) E(Q) Q G MP -Q G MP +Q E Glass G MP 2π/a Q S(Q) O. Pilla et al., PRL 85, 2136 (2) T. Scopigno et al., PRB 64, 1231 (21)

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41 Brief history of the fragility concept Not related to mechanical properties! Fragility: born as a property of the liquid state: Temperature dependence of the viscosity of a melt upon cooling

42 Brief history of the fragility concept Theory Phoenicians, Egyptians Sweet glasses: Murano ~1 Long-Short: Nemilov 1964 Strong-Fragile: Angell 1976 Practice

43 Viscosity in liquids approaching the glass transition m o-terphenyl α-process A = dlog( η( T) η ) dt ( T) g T g τ = 1 s η = G η = τ ( T g ) 1 1 poise Angell plot STRONG T = Tg SiO 2 FRAGILE B2 O 3 CKN OTP propanol

44 Fragility and glass blowing m A = d log( η( T) η ) dt ( T) g T g STRONG Working interval for glass molding FRAGILE l o n g short

45 Why care so much- about fragility? T-dependence of the excess entropy (Martinez & Angell 21) Strength of the Boson peak at Tg (Sokolov 1993) Energy landscape statistics (Speedy 1999, Sastry 21, Ruocco et al. 24) T-dependence of the shear modulus-shoving model (Hall 1987, Dyre 22) Stretching exponent of the α-relaxation (Bohmer 2) Properties of the liquid on approaching T g We show here C ( a Tcorrelation T ) with: p g = + IMAX R = p( g ) I Connection between fragility and the MIN Tg temperature behaviour of the shear elastic modulus in the supercooled liquid (shoving I MAX model) I MIN F C T T + Non ergodicity factor f Q (T) Property of the glass in the T limit

46 f( Q, T) Φ( Q, t) S( Q, ω) How to measure f(q,t) in glasses: Φ && () t + ω ( Q) + Φ () t +Γ Φ& () t = Q Q Q Q Q Q 2 SQ (, ω) 1 ΩQ ΓQ = 1 fq f ( ) Q SQ ( ) + δ ω π ω 2 2 Ω Q + ω ΓQ VIBRATIONS 1. FROZEN DIFFUSION Non ergodicity factor Sound velocity Sound attenuation f Φ Q (t) c l ( ) Γ Q = Q.2 Q. I el = Ie l + I Ω Q 1/Ω Q in 1/Γ S(Q,ω)/S(Q) τ α log (t) f Q f Q ω f Q Ω Q Γ Q 2 Q 2 2 Q ω = + ( ) Q

47 IXS at work: Determination Measure S(Q,ω) of f Q (T) Glycerol Q=2 nm -1 f Intensity Q (T) ( counts / 18 s ) 1,,9,8, Q = 5 nm -1 f Q = 4 nm -1 Q = 3 nm -1 2 Low T Q How Q = 2 nm to parameterize the -1 Temperature dependence?,6 4,,2,4,6,8 1, 1,2 1,4 Q = 1 nm -1 = I el el T / T g I + I Energy ( mev ) in

48 Parameterizing f Q (T): harmonic + 1 phonon F ( Q, t) (1) ( S ( Q) + F ( Q, ) +...) ( ) W Q e is t Φ Q (1) Sis ( Q) + F ( Q, t) ( t) (1) S ( Q) + S ( Q) is f Q = 1+ 1 (1) S ( Q) S ( Q) is f Q ( T) 1 1 = = Qe i e + T iqx 2 i () K 1 ( ) BQ Ep Q 2 p KTQ B i m Sis ( Q) p ω p 2 m Sis ( Q) p ω p α / T g In general f Q ( T) 1 = 1 +αt T g

49 T= f Q = I el Iel + I in d ( f Q ) α = m gass gla l = dtt ( ) g T = m Angell Liquid = d(log η) dt ( T) g T g T g Glass T

50 IXS data analisys and related issues From I(θ, T) S(Q,ω) Checklist: Kinematics-Resolutions Sample aspects Measurement strategies Quantum aspects-normalization Background Empty cell Multiple scattering

51 Relaxation dynamics Q Q Constant T Q S(Q,ω) Constant Q T 1 S(Q,ω) Relaxation spectrum sound velocity ω ω B = v o Q 1 / τ R (T) ω B = v oo Q ω B << 1/τ R (T) v o ω B >> 1/τ R (T) v oo Q α = 1 / v τ(t) v oo Constant T v o S(Q,ω) S(Q,ω) ω B = v o Q 1 / τ R (T 2 ) 1 / τ R (T 1 ) T 2 T 3 Q 1 / τ R (T 3 ) ω B = v oo Q ω

52 Collective modes in glasses Act 3 The state of the art (selenium) Intensity (a.u.) 1 Q=9.5 3 Q= Q= Q=12.5 Q=7. Q=5. Q=3.5 Q=2.5 Q= Energy (mev) Q=4. Q= Ω(Q) (mev) Γ(Q) (mev) e 2W S(Q,ω) (ev -1 ) IXS vs INS (v=2 m/s) 5 Q (nm -1 ) 5 Q (nm -1 ) Q=7.5 nm -1 Q=5. nm -1 Q=3. nm -1 1 Q=2. nm -1 c (km/s) Γ (mev) Energy (mev) 5 1 Q (nm -1 ) 2 4 Ω 2 (mev 2 )

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