Abstract

An introduction to x-ray intensity fluctuation spectroscopy is given by describing its relationship to speckle from coherent sources. Its use to measure two-time correlation functions is demonstrated using the equilibrium fluctuations of gold colloids in polystyrene and for non-equilibrium fluctuations in the unmixing below the miscibility gap in AlLi.

© 2003 Optical Society of America

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References

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  1. Wyn Brown, editor, Dynamic Light Scattering: The method and some applications (Clarendon Press, Oxford 1993).
  2. B. J. Berne and R. Pecora, Dynamic Light Scattering (Academic Press, Orlando, FL, 1976).
  3. M. Sutton, S. E. Nagler, S. G. J. Mochrie, T. Greytak, L. E. Bermann, G.Held, G. B. Stephenson, �??The Observation of Speckle by Diffraction with Coherent X-rays,�?? Nature 352, 608-610 (1991).
    [CrossRef]
  4. S. Brauer, G. B. Stephenson, M. Sutton, R. Br¨uning, E. Dufresne, S. G. J. Mochrie, G. Gr¨ubel, J. Als-Nielsen, D. L. Abernathy, �??X-ray intensity fluctuation spectroscopy observations of critical dynamics in Fe3Al,�?? Phys. Rev. Lett. 74, 2010-2013 (1995).
    [CrossRef] [PubMed]
  5. S. Dierker, R. Pindak, R. M. Fleming, I. K. Robinson and L. E. Berman, �??X-Ray Photon Correlation Spectroscopy Study of Brownian Motion of Gold Colloids in Glycerol,�?? Phys. Rev. Lett. 75, 449-552 (1995).
    [CrossRef] [PubMed]
  6. T. Thurn-Albrecht, W. Steffen, A. Patkowski, G. Meier, E. W. Fischer, G. Gr¨ubel and D. L. Abernathy, �??Photon Correlation Spectroscopy of Colloidal Palladium Using a Coherent X-Ray Beam,�?? Phys. Rev. Lett. 77, 5437-5440 (1996).
    [CrossRef] [PubMed]
  7. M. Sutton, �??Coherent X-ray Diffraction,�?? in Third-Generation Hard X-ray Synchrotron Radiation Sources: Source Properties, Optics, and Experimental Techniques, Dennis M. Mills ed. (John Wiley and Sons, Inc, New York, 2002), Chap. 3.
  8. D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, �??Diffusing Wave Spectroscopy,�?? Phys. Rev. Lett. 60, 1134-1137 (1988).
    [CrossRef] [PubMed]
  9. A. R. Sandy, L. B. Lurio, S. G. J. Mochrie, A. Malik, G. B. Stephenson, J. F. Pelletier and M. Sutton, �??Design and Characterization of an Undulator Beamline Optimized for Small-Angle Coherent X-ray Scattering at the Advanced Photon Source,�?? J. Synch. Rad. 6, 1174-1184 (1999).
    [CrossRef]
  10. D. Lumma, L. B. Lurio, S. G. J. Mochrie and M. Sutton, �??Area detector based photon correlation in the regime of short data batches: data reduction for dynamic x-ray scattering,�?? Rev. Sci. Inst. 71, 3274-3289 (2000).
    [CrossRef]
  11. M. K. Corbierre, N. S. Cameron, M. Sutton, S. G. J. Mochrie, L. B. Lurio, A. Rh¨um and R. B. Lennox, �??Polymer-Stablized Gold Nanoparticles and Their Incorporation into Polymer Matrices,�?? J. Am. Chem. Soc. 123, 10411-10412 (2001).
    [CrossRef] [PubMed]
  12. M. K. Corbierre, N. S. Cameron, M. Sutton, K. Laaziri, �??Polymer-Stablized Gold Nanoparticles and Their Incorporation into Polymer Matrices,�?? in preparation.
  13. K. Scha¨tzel, �??Single photon correlation techniques,�?? in Dynamic Light Scattering: The method and some applications, Wyn Brown, ed. (Clarendon Press, Oxford 1993), Chap. 2.
  14. E. Geissler, �??Dynamic light scattering from polymer gels,�?? in Dynamic Light Scattering: The method and some applications, Wyn Brown, ed. (Clarendon Press, Oxford 1993), Chap. 11.
  15. B. Crosignani, P. Di Porto and M. Bertolotti, Statistical Properties of Scattered Light (Academic Press, New York 1975), Sect. V.5.
  16. A. Malik, A. R. Sandy, L. B. Lurio, G. B. Stephenson, S. G. J. Mochrie, I. McNulty and M. Sutton, �??Coherent X-ray Study of Fluctuations During Domain Coarsening,�?? Phys. Rev. Lett. 81, 5832-5835 (1998).
    [CrossRef]
  17. F. Livet, F. Bley, R. Caudron, D. Abernathy, C. Detlefs, G. Gr¨ubel and M. Sutton, �??Kinetic evolution of unmixing in an AlLi alloy using x-ray intensity fluctuations spectroscopy,�?? Phys. Rev. E 63, Article 036108 (2000).
  18. G. Brown, P. A. Rikvold, M. Sutton and M. Grant, �??Speckle from phase-ordering systems,�?? Phys. Rev. E 56, 6601-6612 (1997).
    [CrossRef]
  19. G. Brown, P. A. Rikvold, M. Sutton and M. Grant, �??Evolution of speckle during spinodal decomposition,�?? Phys. Rev. E 60, 5151-5162 (1999).
    [CrossRef]
  20. L. Cipelletti and D. A. Weitz, �??Ultralow-angle dynamic light scattering with a charge coupled device camera based multispeckle, multitau correlator,�?? Rev. Sci Inst. 70, 3214-3221 (1999).
    [CrossRef]
  21. E. Geissler, A-M. Hecht, C. Rochas, F. Bley, F. Livet and M. Sutton, �??Aging in a filled polymer: Coherent small angle x-ray and light scattering,�?? Phys. Rev. E 62, 8308-8313 (2000).
    [CrossRef]

J. Am. Chem. Soc. (1)

M. K. Corbierre, N. S. Cameron, M. Sutton, S. G. J. Mochrie, L. B. Lurio, A. Rh¨um and R. B. Lennox, �??Polymer-Stablized Gold Nanoparticles and Their Incorporation into Polymer Matrices,�?? J. Am. Chem. Soc. 123, 10411-10412 (2001).
[CrossRef] [PubMed]

J. Synch. Rad. (1)

A. R. Sandy, L. B. Lurio, S. G. J. Mochrie, A. Malik, G. B. Stephenson, J. F. Pelletier and M. Sutton, �??Design and Characterization of an Undulator Beamline Optimized for Small-Angle Coherent X-ray Scattering at the Advanced Photon Source,�?? J. Synch. Rad. 6, 1174-1184 (1999).
[CrossRef]

Nature (1)

M. Sutton, S. E. Nagler, S. G. J. Mochrie, T. Greytak, L. E. Bermann, G.Held, G. B. Stephenson, �??The Observation of Speckle by Diffraction with Coherent X-rays,�?? Nature 352, 608-610 (1991).
[CrossRef]

Phys. Rev. E (4)

F. Livet, F. Bley, R. Caudron, D. Abernathy, C. Detlefs, G. Gr¨ubel and M. Sutton, �??Kinetic evolution of unmixing in an AlLi alloy using x-ray intensity fluctuations spectroscopy,�?? Phys. Rev. E 63, Article 036108 (2000).

G. Brown, P. A. Rikvold, M. Sutton and M. Grant, �??Speckle from phase-ordering systems,�?? Phys. Rev. E 56, 6601-6612 (1997).
[CrossRef]

G. Brown, P. A. Rikvold, M. Sutton and M. Grant, �??Evolution of speckle during spinodal decomposition,�?? Phys. Rev. E 60, 5151-5162 (1999).
[CrossRef]

E. Geissler, A-M. Hecht, C. Rochas, F. Bley, F. Livet and M. Sutton, �??Aging in a filled polymer: Coherent small angle x-ray and light scattering,�?? Phys. Rev. E 62, 8308-8313 (2000).
[CrossRef]

Phys. Rev. Lett. (5)

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, �??Diffusing Wave Spectroscopy,�?? Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

A. Malik, A. R. Sandy, L. B. Lurio, G. B. Stephenson, S. G. J. Mochrie, I. McNulty and M. Sutton, �??Coherent X-ray Study of Fluctuations During Domain Coarsening,�?? Phys. Rev. Lett. 81, 5832-5835 (1998).
[CrossRef]

S. Brauer, G. B. Stephenson, M. Sutton, R. Br¨uning, E. Dufresne, S. G. J. Mochrie, G. Gr¨ubel, J. Als-Nielsen, D. L. Abernathy, �??X-ray intensity fluctuation spectroscopy observations of critical dynamics in Fe3Al,�?? Phys. Rev. Lett. 74, 2010-2013 (1995).
[CrossRef] [PubMed]

S. Dierker, R. Pindak, R. M. Fleming, I. K. Robinson and L. E. Berman, �??X-Ray Photon Correlation Spectroscopy Study of Brownian Motion of Gold Colloids in Glycerol,�?? Phys. Rev. Lett. 75, 449-552 (1995).
[CrossRef] [PubMed]

T. Thurn-Albrecht, W. Steffen, A. Patkowski, G. Meier, E. W. Fischer, G. Gr¨ubel and D. L. Abernathy, �??Photon Correlation Spectroscopy of Colloidal Palladium Using a Coherent X-Ray Beam,�?? Phys. Rev. Lett. 77, 5437-5440 (1996).
[CrossRef] [PubMed]

Rev. Sci Inst. (1)

L. Cipelletti and D. A. Weitz, �??Ultralow-angle dynamic light scattering with a charge coupled device camera based multispeckle, multitau correlator,�?? Rev. Sci Inst. 70, 3214-3221 (1999).
[CrossRef]

Rev. Sci. Inst. (1)

D. Lumma, L. B. Lurio, S. G. J. Mochrie and M. Sutton, �??Area detector based photon correlation in the regime of short data batches: data reduction for dynamic x-ray scattering,�?? Rev. Sci. Inst. 71, 3274-3289 (2000).
[CrossRef]

Other (7)

Wyn Brown, editor, Dynamic Light Scattering: The method and some applications (Clarendon Press, Oxford 1993).

B. J. Berne and R. Pecora, Dynamic Light Scattering (Academic Press, Orlando, FL, 1976).

M. Sutton, �??Coherent X-ray Diffraction,�?? in Third-Generation Hard X-ray Synchrotron Radiation Sources: Source Properties, Optics, and Experimental Techniques, Dennis M. Mills ed. (John Wiley and Sons, Inc, New York, 2002), Chap. 3.

M. K. Corbierre, N. S. Cameron, M. Sutton, K. Laaziri, �??Polymer-Stablized Gold Nanoparticles and Their Incorporation into Polymer Matrices,�?? in preparation.

K. Scha¨tzel, �??Single photon correlation techniques,�?? in Dynamic Light Scattering: The method and some applications, Wyn Brown, ed. (Clarendon Press, Oxford 1993), Chap. 2.

E. Geissler, �??Dynamic light scattering from polymer gels,�?? in Dynamic Light Scattering: The method and some applications, Wyn Brown, ed. (Clarendon Press, Oxford 1993), Chap. 11.

B. Crosignani, P. Di Porto and M. Bertolotti, Statistical Properties of Scattered Light (Academic Press, New York 1975), Sect. V.5.

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Figures (6)

Fig. 1.
Fig. 1.

(a) Scattering of Au particles in polystyrene (top). Black region is due to the shadow of a beam stop and the black arc consists of the wave-vectors for which fluctuations are shown (q=.028Å-1) in Fig. 2. The two directions of the wave-vector (horizontal and vertical) are labelled by qx and qy and q = q x 2 + q x 2 . (b) Circular average of (1a) (bottom). The 20 pixel wide stripe of the black arc contribute to the single point in this circular average for q=.028(±.0003)Å-1.

Fig. 2.
Fig. 2.

The intensity fluctuations versus time for selected individual pixels from the 20 pixel wide arc of scattering at wave-vectors for which | q |=.028(+-.0003)Å-1. Increasing pixel number corresponds to increasing angle around the arc. Lighter colours are higher intensities.

Fig. 3.
Fig. 3.

(a) Two-time correlation functions, using averages over the arc in Fig. 1(a) (top). (b) Time correlation functions (bottom). The noisy signal (blue) is the single slice corresponding to the diagonal line in 3(a) (a function of t 2 -t 1). Since the function is symmetric in t 1 and t 2 only the portion for t 2>t 1 is shown. The less noisy signal (red) is a simple average over all slices. The black curve on top is g 2 calculated using the multitau algorithm which obtains correlations for longer times after averaging intensities over successively longer times [13].

Fig. 4.
Fig. 4.

Time evolution of scattered intensity. Scattering was measured with a wavelength of 1.51 Å.

Fig. 5.
Fig. 5.

Contour plots of two-time correlation functions for | q |=.1055. Contour levels are 0.2, 0.4, 0.6 and 0.8. The diagonal t 1=t 2 has been normalized to 1.0 after suppressing the effects of Poisson noise. The black dot indicates the time at which the peak maximum sweeps by this particular wave-vector.

Fig. 6.
Fig. 6.

Correlation functions for q=.0155 Å-1value and t 1=2163 s; 7920 s; 16230 s. Solid lines are the fits. For comparison, the dash-dot line is a Gaussian and dashed line is a Lorentzian with the same height and width as given by the fit.

Equations (2)

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g 2 = I ( t 1 ) I ( t 2 ) ( I ( t 1 ) I ( t 2 ) ) ,
D ( q , t ) = I ( q , t ) I ( q , t ) I ( q , t ) .

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