X-ray Intensity Fluctuation Spectroscopy. Mark Sutton McGill University

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1 X-ray Intensity Fluctuation Spectroscopy Mark Sutton McGill University

2 McGill University Collaborators J-F. Pelletier K. Laaziri K. Hassani A. Fluerasu E. Dufresne G. Brown M. Grant Yale/MIT S. Mochrie D. Lumma P. Falus L. Lurio A. Rhüm MSD at Argonne National Labs G.B. Stephenson A. Malik CRNS Grenoble F. Bley F. Livet E. Geissler ESRF, Grenoble G. Grübel

3 Pinholes in Tinfoil 2 (a) 2 (b) (c) 2 (d)

4 SAXS of Au particles in PS Qy [Å 1].1 (a). (b)..1.2 Qx.3 [Å 1] (dσ/dω).1 (c) (d) S(Q) Q [Å 1].3.1 Q [Å 1] Small angle scattering of 6nm gold particles in polystyrene.

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

6 SAXS of Au particles in PS 2 t2 [sec] t 1 [sec].2.15 g time [sec] Calculating correlations functions.

7 Langevin Dynamics Ψ( x, t) t = M 2 F Ψ + η( x, t) where thermal noise is: η( x, t) = and forcing a generalized Einstein-Stokes or fluctuation dissipation relation requires: η( x, t)η( x, t ) = 2M 2k b T δ( x x )δ(t t ) Which is linear using the free energy: F = κ ψ( x, t) rψ2 2 d x

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

9 Requirements of XIFS 1. Scattering Volume comparable to coherence volume (diffraction limited beam resolved by detector). 2. Broad scattering (i.e. disorder so there is interesting structure within beam) 3. Sufficient counts per correlation time (like about 1) 4. Sufficient number of correlations times measured (either many times at one speckle or many speckles and times with the same time constant).

10 Why Coherence? Coherence allows one to measure the dynamics of a material (X-ray Intensity Fluctuation Spectroscopy, XIFS). I( R, t)i( R+δ R, t+τ) = I(R) 2 + β( κ) k 8 (4πR) 4V 2 I 2 where the coherence function is defined as: β( κ) = 1 V 2 I 2 V V ei κ ( r 2 r 1 ) S( Q, t) 2 Γ(, r 2 r 1 Q, ( r 2 r 1 ) ) ω 2 d r 1 d r 2 A good estimate for β is: β( ) V coherence V scattering

11 Signal to Noise One can show that the signal to noise in an X-ray intensity fluctuation spectroscopy measurement is: s n β( ) N speckles B λ 2 E E NB f x f y λ 2 λ λ max(1, f x f y f z ) B λ 3dσ dω dσ dω f z λ2 λ dσ dω L 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, (2).

12 Detector Resolution Speckle size (width of β( κ)) is given by diffraction limit of beam: θ λ d coh Need to resolve this on detector, so want detector size to be close to d coh i.e. λ d coh d coh R det or R det = d2 coh λ Problem if horizontal and vertical lengths are too different. Also, we don t want the coherence length to be too large as this would not match detector size. Similarly, don t want too large a mismatch between speckle size and the diffraction width of particles contributing to peak under study. With perfect optics, one can trade coherence length for angular spread. (Note: We also need better optics.)

13 Relationship to Brightness The conventional brightness has the form: B( r, ŝ, ν) = 1 4π 2 I H(ν) ( x2 2σ e 2 + y2 h 2σ 2) v e ( s2 x + s2 y 2σs h 2σsv ) σ sh σ sv σ h σ v where ŝ = (s x, s y, 1) = (x/z, y/z, 1), the σ s are the beam sizes and angular spreads and H(ν) is frequency spectrum. Brightness is related to coherence B( r, ŝ, ν) = k 2 1 I( r)h(ν) (2π) 2 and we can show that W ( r 1, r 2, ν) = I H(ν) 2πσ h σ v e ( (x 1 +x 2 ) 2 8σ 2 h + (y 1 +y 2 )2 8σ 2 v µ( v)e ikŝ v d v ) e ( (x 2 x 1 )2 2ρ 2 h where ρ i are coherence lengths and ρ i = 1/(kσ si ). + (y 2 y 1 )2 2ρ 2 ) v

14 Undulator A Coherence Lengths Can propagate the partial coherence to experimental station and the coherence lengths are: i = ρ i 1 + z kσ i ρ i 2 Source 35x5µm 2 by 25x5 µrad 2 dcoh (µm FWHM) vertical horizontal R source (m) Coherence lengths of an undulator source versus distance. The coherence lengths for an incoherent sources are plotted for comparison. Blue line is shrinking horizontal size half.

15 Optics With perfect optics, one can trade coherence length for angular spread. Thus one could demagnify the vertical (by.25) and magnify the horizontal (by 2) to get 2 µmx2 µm coherence lengths as this matches the typical CCD pixel size. But optics need not be 1% transmissive and tends to add aberrations. Need to consider which undulator output best matches required optics. (Note: We also need better optics.).5 Qy (Å 1 ) Q x (Å 1 ) (1) peak of disordered Fe 3 Al taken using a zone plate with 1 µm focal spot. 1 µm Fresnel Zone Plate

16 Time Correlations in Fe 3 Al g2(t) qξ=.94 qξ= 2.48 qξ= 5.72 (a) time (sec) τ(q) (sec) (b)..1.2 q (Å 1 ) (a) Measured autocorrelation functions versus time delays for various Q s for sample at T=553.3C. The solid line corresponds to a single exponential fit. (b) Fitted time constants versus Q. 4 T=552.9C Measured autocorrelation functions versus Q for several temperature near T c. The solid line is a scaled S(Q) and shows S(Q) is proportional to the measured time constants in this system. τ( q) (sec) T=553.C T=553.3C T=554.C q (Å 1 )

17 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 1 )Ψ( q, t 2 ) 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 S( q, t) = S( q, t, t)

18 Scattering from Cu 3 Au (1, k, l) [1,, ] (h,, ) * B.E.Warren, X-ray Diffraction, Dover, NY, 1969,199

19 Tranverse direction Two-Time Correlation Functions q=.6632 A 1 (265x645) t t t t 1 t t 1 q= A 1 (265x75) t t 1 (375 o C) (37 o C) (36 o C)