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X-Ray Photon Correlation Spectroscopy for the Characterization of Soft and Hard Condensed Matter

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Notes

  1. 1.

    The derivation of Eqs. (3.2) and (3.3) from the scattering of a set of N particles is very adequate to describe, for example, the scattering by a colloidal system. For other systems, such as surfaces, it may be more adequate to consider a continuous electron density and replace the summation in Eq. (3.2) by an integral. In that case, the instantaneous total electric field is

    $$ E\left(\overrightarrow{q},t\right)={\displaystyle \int \rho \left({\overrightarrow{r}}^{\prime },t\right) \exp \left[-\mathrm{i}\overrightarrow{q}\cdot {\overrightarrow{r}}^{\prime }(t)\right]d{\overrightarrow{r}}^{\prime }} $$
  2. 2.

    This is also referred to as ‘second-order statistical properties of the fields’ in the literature.

  3. 3.

    The q th moments of the distribution (3.4) are given by \( \overline{I^q}={\overline{I}}^qq! \). Thus, the second moment is \( \overline{I^2}=2{\overline{I}}^2 \). For more details, see [74].

  4. 4.

    This separation of the sample properties and the coherence properties of the beam is only valid under certain conditions: the coherence length of the light must be larger than the correlation length of the sample, and the scattering volume has to be larger than the correlation length of the sample fluctuations too. Especially in grazing incidence geometry, these conditions may not be fulfilled and a more elaborate treatment is needed. Gutt et al. have developed a rigorous treatment of the effects of partial coherence and detector resolution on the intensity autocorrelation function measured by XPCS [86].

  5. 5.

    In general, the specific prerequisites for the degree of coherence and wavefront curvature depend on the application. For example, other techniques such as phase contrast imaging typically require a planar wavefront [161], whereas curved wavefronts can even be advantageous in some cases for coherent diffraction imaging [249], and focused beams have been used to study nanoparticles [52, 204].

  6. 6.

    The number of photons that passes through an aperture with dimensions ξ v,h t per unit time is related to the brightness B of the source as I coh ∝ λ 2 B/4. The units of B are ph/photons/s/mrad2/mm2/0.1 % bandwidth [78].

  7. 7.

    Recent experiments on colloids in films or at surfaces are reviewed in Sect. 5.2.1.

  8. 8.

    As shown in the example of Sect. 3.3.1 for noninteracting spherical particles, the relaxation time obtained from the autocorrelation function is 1/D 0 q 2 (see Eq. (3.19)), and from it, one can obtain the hydrodynamic radius of the particles using the Stokes-Einstein equation (Eq. (3.20)).

  9. 9.

    Hexemer and Müller-Buschbaum [93] reviews in detail the application of neutron and X-ray grazing incidence techniques to the study of soft matter systems.

  10. 10.

    A recent review on erosion can be found in [119]. Ion beam sputtering is the focus of a special review issue [44].

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Authors and Affiliations

  1. Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK

    Oier Bikondoa

  2. XMaS, The UK-CRG Beamline, ESRF – The European Synchrotron, Grenoble, 38043, France

    Oier Bikondoa

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  1. Integrated Mesoscale Architectures for Sustainable Catalysis (IMASC), Rowland Institute of Science, Harvard University, Cambridge, Massachusetts, USA

    Challa S.S.R. Kumar

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Bikondoa, O. (2016). X-Ray Photon Correlation Spectroscopy for the Characterization of Soft and Hard Condensed Matter. In: Kumar, C. (eds) X-ray and Neutron Techniques for Nanomaterials Characterization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48606-1_3

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