How can x-ray intensity fluctuation spectroscopy push the frontiers of Materials Science. Mark Sutton McGill University

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1 How can x-ray intensity fluctuation spectroscopy push the frontiers of Materials Science Mark Sutton McGill University

2 Coherent diffraction (001) Cu 3 Au peak Sutton et al., The Observation of Speckle by Diffraction with Coherent X-rays, Nature, 352, (1991).

3 Why Coherence? Coherence allows one to measure the dynamics of a material (X-ray Intensity Fluctuation Spectroscopy, XIFS). I( Q,t)I( Q+δ κ,t + τ) = I(Q) 2 k 8 + β( κ) (4πR) 4V 2 I0 2 S( Q,t) where the coherence part is: β( κ) = 1 V 2 I 2 0 V V e i κ ( r 2 r 1 ) Γ( 0, r 2 r 1 Q, ( r 2 r 1 ) ) ω 0 and β( 0) V coherence V scattering with widths λ/v 1 3 Reference: M. Sutton, Coherent X-ray Diffraction, in Third-Generation Hard X-ray Synchrotron Radiation Sources: Source Properties, Optics, and Experimental Techniques, edited by. Dennis M. Mills, John Wiley and Sons, Inc, New York, (2002). 2 2 d r 1 d r 2

4 SAXS of Au particles in PS pixel time [sec] Define correlation function: g (2) ( Q,τ) 1 = I2 ( Q,t) I( Q,t) 2 I( Q,t) 2 = β g (1) ( Q,τ) = β e 2τ/τ Q 2 Time fluctuations in coherent scattering.

5 Au in polystyrene Two-time correlation functions Cu 3 Au 2000 t2 [sec] t 1 [sec] g contrast time [sec] t2 t1(10³sec)

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7 Real Equation of Everything Langevin dynamics (Models A through J): Ψ µ ( x,t) t = = {F,Ψ µ ( x,t)} PB M µν F Ψ ν + η µ ( x,t) {Ψ µ ( x,t),ψ ν ( x,t )} PB F Ψ ν d x M µν F Ψ ν + η µ ( x,t) F = V µ ( x,t) M µν + η µ ( x,t) Ψ ν where η µ ( x,t) = 0 and (generalized Einstein-Stokes/fluctuation-dissipation) η µ ( x,t)η ν ( x,t ) = 2M µν k b T δ( x x )δ(t t ) Reference: Section Principles of condensed matter physics,chaikin and Lubensky(1995).

8 Equilibrium Time Dependence In equilibrium F has no linear term in Ψ so equation of motion becomes: ( Ψ( x,t) 2 ) F = M Ψ+η( x,t) t Ψ 2 eq or Ψ( Q,t) = Mk bt t S( Q) Ψ+η( Q,t) Where the structure factor is: S( Q,t) = Ψ ( Q,t)Ψ( Q,t) ( 2 ) F = k b T/ Ψ 2 eq

9 Phase Separation of AlZn 1 S(q,t) 2Mq 2 t = ( κq f c 2 ( 4 ) f + k b T c 4 ) c 0 S(q,t) c 0 S 4 (q,t) 1) Equation of motion from J. S. Langer, M. Bar-on, and H. D. Miller, Phys. Rev. A, 11, 1417 (1975). 2) Data from: Mainville et al, Phys. Rev. Lett., 78, 2787 (1997).

10 Signal to Noise Signal is g 2 1 = β and variance of is var(g 2 ) 1/( n 2 N). So: s n = β n N = βiτ ttotal τ N speckles = βi τt total N pixels Note 1: This is linear in number of photons (as opposed to n). Note 2: For fixed s/n αi τ/α 2. Thus an α-fold increase in intensity is an α 2 -fold increase in time resolution. Need very fast detectors. Reference: Area detector based photon correlation in the regime of short data batches: data reduction for dynamic x-ray scattering, D. Lumma, L.B. Lurio, S.G.J. Mochrie, and M. Sutton, Rev. Sci. Instr. 71, (2000).

11 Signal to Noise More explicitly: s n β B 0dxdx dydy E 1 dσ E V dω L N sp βb 0 f x f y λ 2 E 1 dσ E V dω f λ 2 z Nsp λ 1 max(1, f i ) 3B 0 f x f y f z λ 2 λ 1 dσ λ 2 λ V dω λ B 0 λ 3 1 dσ Nsp V dω f B 0 λ 3 1 dσ Nsp (i f any f i < 1). V dω Note: should be a λ 3 /8 as normally use λ/2. Nsp

12 Detector Resolution Speckle size (width of β( κ)) is given by diffraction limit of beam: θ to resolve on detector. Thus λ d coh d coh R det R det = d2 coh λ Problem if horizontal and vertical lengths are too different or if any coherence length is too long. Similarly, don t want too large a mismatch between speckle size and the diffraction width of sample peak. Focus for a virtual source, for 1.5Å, a 100µm source at 10m gives d coh = 15µm. Ideally optics could tune coherence lengths.

13 Order-disorder phase transitions in Cu 3 Au Disorder: Order: f =0.75 f Cu f Au f Cu, f Au

14 Scattering from Cu 3 Au (1, k, l) [1, 0, 0] (h, 0, 0) S(q,t) Movie B.E.Warren, X-ray Diffraction, Dover, NY, 1969,1990

15 Scattering from Cu3Au

16 Two-Time Correlation Functions Non-stationary so autocorrelate I(q,t 1) I(q,t 1 ) I(q,t 1 )

17 Upgrades to the beamline allow us to obtain better data, especially important for the early time region contrast t2 t1(10³sec) Cross-section of several slices

18 New Data horizontal 15 yh(yfit1) 0.05 Time(10³ sec) xh vertical 5 yv(yfit2) Q(Å 1 ) xv

19 ~ two-time Two-Q Quench from 425 C to 383 C t2(10³sec) t1(10³sec) 15

20 Particle Size Effects 1.5 S(Q) Q(Å 1 ) Q and φ dependence

21 Non-Gaussian Nature Intensity related to Fourier transform of density-density correlation function. For non-zero correlations, must have pair of points within a correlation length ξ. δρ( r 1,0)δρ( r 2,0)δρ( r 3,τ)δρ( r 4,τ) δρ( r 1,0)δρ( r 2,0) δρ( r 3,τ)δρ( r 4,τ) + δρ( r 1,0)δρ( r 3,0) δρ( r 2,τ)δρ( r 4,τ) + δρ( r 1,0)δρ( r 4,0) δρ( r 3,τ)δρ( r 2,τ) This overcounts the volume when all four r i are within the same correlation length. When Fourier transformed, this correction is of order (ξ 3 /V) 2.

22 Gaussian Decoupling I( q,t 1 )I( q,t 2 ) T = Ψ ( q,t 1 )Ψ( q,t 1 )Ψ ( q,t 2 )Ψ( q,t 2 ) T = Ψ ( q,t 1 )Ψ( q,t 1 ) T Ψ ( q,t 2 )Ψ( q,t 2 ) T + Ψ ( q,t 1 )Ψ( q,t 2 ) T Ψ ( q,t 2 )Ψ( q,t 1 ) T + Ψ ( q,t 1 )Ψ ( q,t 2 ) T Ψ( q,t 1 )Ψ( q,t 2 ) T = [1+δ( q)]s 2 ( q,t 1,t 2 )+ I( q,t 1 ) T I( q,t 2 ) T Where: S( q,t 1,t 2 ) = Ψ ( q,t 1 )Ψ( q,t 2 ) T and I( q,t) = S( q,t,t)

23 Metrology (high resolution from broad scattering) Quasistatic X-Ray Speckle Metrology of Microscopic Magnetic Return-Point Memory, M.S. Pierce, R.G. Moore, L.B. Sorensen, S.D. Kevan, O. Hellwig, E.E. Fullerton, and J.B. Kortright, Phys. Rev. Lett., 90, , (2003).

24 Experimental Setup Scattering geometry

25 Rubber Scattering of model rubber (EPR with carbon black)

26 Heterodyne G 2 ( q,t) = I 2 r + I s (t) 2 t (1+β g 1 (t) 2 )+ 2I r I s (t) t + 2I r I s (t) t βre(g 1 (t)) Moving at constant velocity gives phase factor e i q vt = e iωt So correlation becomes (x = I s /(I s + I r )) g 2 (q,φ,t) = 1+β(1 x) 2 +x 2 βγ 2 (t/τ)+2x(1 x)βcos(ωt)γ(t/τ)

27 Rubber Q and φ dependence

28 qvcos(ω) Azimuth dependence Wavevector dependence

29 Velocity

30 Wave vector dependence

31 Hydrodynamics For instance given density ρ( x,t) and momentum density ρ( x,t) v( x,t): {ρ( x,t),ρ( x,t )v i ( x,t )} PB = i (ρ( x,t)δ( x x )). Thus ρ t = (ρ v) and (linearizing for simplicity) [( ) ] (ρ v) = t p δρ( x) + η, ρ S (p is pressure and S is entropy). These are the linearized hydrodynamic equations. Reference: Section Principles of condensed matter physics,chaikin and Lubensky(1995).

32 Conclusions 1. Understanding the time evolution of the microstructure is key to understanding Materials Science. 2. XIFS is a powerful and direct way to measure this dynamics, both in and out of equilibrium. 3. This dynamics is controlled by the thermodynamic fluctuations of the system and XIFS directly measures these. 4. Did not show many examples, but has been done in SAXS, Diffuse Reflectivity, Short-range quasi-bragg peaks. Essentially all forms of diffuse scattering 5. Can heterodyne. This will give access to understanding how microscoptic structure controls visco-elastic properties (rubber). Can separate advection from disspation effects in the equation of motion.

33 6. Can use speckle to obtain high resolution structural information from very disordered materials. (Thermal expansion of a glass, reversible and irreversible hysterisis effects in magnetic domains). 7. Two-Q Two-time correlations give unique information about microstructure. non-guassian fluctuations.