What Is XPCS?

X-ray Photon Correlation Spectroscopy (XPCS) is a technique that uses the temporal correlations of coherent X-ray speckle patterns to probe nanoscale dynamics in condensed-matter systems. This page provides the conceptual foundation needed to understand the rest of the user guide.

Coherent X-rays and Speckle Patterns

When a partially coherent X-ray beam (wavelength on the order of 1Å) illuminates a disordered sample, interference among the scattered waves produces a granular intensity distribution on the detector known as a speckle pattern. Each speckle encodes the instantaneous spatial arrangement of scatterers at the probed wavevector q.

Key properties:

  • The speckle size is set by the coherence area of the beam, not by the sample structure.

  • As particles move, the speckle pattern evolves – fast dynamics produce rapid intensity fluctuations, slow dynamics produce slowly drifting speckles.

  • Recording a time series of 2-D detector frames gives direct access to the dynamics at every q pixel simultaneously.

From Speckles to Dynamics

The fundamental observable is the intensity–intensity autocorrelation function:

\[g_2(\mathbf{q}, \tau) = \frac{\langle I(\mathbf{q}, t)\, I(\mathbf{q}, t+\tau) \rangle_t} {\langle I(\mathbf{q}, t) \rangle_t^2}\]

For ergodic, stationary systems this depends only on the lag time \(\tau\). In non-stationary samples the full two-time correlation function must be retained:

\[C_2(\mathbf{q};\, t_1, t_2) = \frac{\langle I(\mathbf{q}, t_1)\, I(\mathbf{q}, t_2) \rangle} {\langle I(\mathbf{q}, t_1) \rangle \langle I(\mathbf{q}, t_2) \rangle}\]

The heterodyne package works with \(C_2(t_1, t_2)\) matrices directly, making it suitable for systems that age, flow, or otherwise evolve during the measurement.

The Siegert Relation

Under Gaussian statistics the measured intensity correlation is related to the field (amplitude) autocorrelation \(g_1\) through the Siegert relation:

\[g_2(\mathbf{q}, \tau) = 1 + \beta\, |g_1(\mathbf{q}, \tau)|^2\]

where \(\beta\) is the speckle contrast (optical coherence factor, \(0 < \beta \le 1\)). The physics enters through \(g_1\), which in turn depends on the transport coefficient and velocity fields of the scatterers.

Transport Coefficient

The field correlation for a single-component diffusive system is:

\[g_1(q, \tau) = \exp\!\bigl[-q^2\, J(\tau)\bigr]\]

where \(J(\tau)\) is the cumulative transport coefficient (integrated mean-squared displacement):

\[J(\tau) = D_0\, \frac{\tau^{1+\alpha}}{1+\alpha} + D_\text{offset}\, \tau\]

  • \(D_0\) (Å2/salpha) – diffusion prefactor.

  • \(\alpha\) (dimensionless) – transport exponent. \(\alpha = 0\) is normal Brownian diffusion; \(\alpha > 0\) is super-diffusive; \(\alpha < 0\) is sub-diffusive.

  • \(D_\text{offset}\) (Å2) – constant baseline.

Homodyne vs. Heterodyne Scattering

Homodyne scattering arises when a single scattering component dominates the signal. The standard Siegert relation applies directly, and \(g_2 - 1 \propto |g_1|^2\) gives the dynamics of that one component.

Heterodyne scattering occurs when two distinct scattering populations contribute coherently to the same detector pixel – for example a static reference component and a flowing sample component. The measured correlation then contains three terms:

  1. Reference self-correlation.

  2. Sample self-correlation.

  3. A cross-correlation that carries a velocity-dependent phase.

The heterodyne cross-term is sensitive to directed motion (flow velocity) in addition to diffusion, and its phase depends on the angle between the scattering vector and the flow direction. Extracting all three contributions from the measured \(C_2\) is the central inverse problem solved by this package.

Why XPCS?

Compared to alternative probes of nanoscale dynamics (DLS, DDM, particle tracking), XPCS offers several distinct advantages:

  • No dilution required – measurements are made on concentrated, opaque, or otherwise optically inaccessible samples.

  • Spatial selectivity – the wavevector q selects a specific length scale (from sub-nm to hundreds of nm).

  • Azimuthal resolution – anisotropic dynamics (e.g., flow) are resolved as a function of the in-plane angle \(\phi\).

  • Sub-|AA| wavelength – hard X-rays penetrate thick or absorbing materials that block visible light.

  • Non-invasive – the sample is probed in situ under realistic conditions (temperature, pressure, confinement).

These properties make XPCS the method of choice for studying dynamics in colloidal gels, metallic glasses, cement hydration, biological membranes, and other complex systems.