Correlation Functions in XPCS¶

This page derives the two-time intensity correlation function \(c_2(\mathbf{q}, t_1, t_2)\) from first principles, states the Siegert relation, and explains why the equilibrium projection \(g_2(q, \tau)\) is insufficient for the non-stationary samples that xpcsjax targets. The derivation follows the transport-coefficient framework of [He2024].

Position density and scattered field¶

For a sample of \(N\) scattering centres at positions \(\mathbf{r}_j(t)\), the position density in Fourier space is

(1)¶\[\rho(\mathbf{q}, t) \;=\; \sum_{j=1}^N f_j \exp\!\left(i\,\mathbf{q}\cdot\mathbf{r}_j(t)\right),\]

where \(f_j\) is the form factor of particle \(j\) and the magnitude of the momentum transfer is \(q = 4\pi \sin(\theta)/\lambda\).

The scattered electric field is

(2)¶\[E(\mathbf{q}, t) \;\propto\; \rho(\mathbf{q}, t),\]

and the measured intensity per pixel is

(3)¶\[I(\mathbf{q}, t) \;=\; |E(\mathbf{q}, t)|^2 \;=\; \left|\sum_j f_j\, e^{i\mathbf{q}\cdot \mathbf{r}_j(t)}\right|^2.\]

The fluctuations of \(I(\mathbf{q}, t)\) carry the information about the sample’s collective dynamics.

First-order correlation function c_1¶

The normalised first-order correlation function is

(4)¶\[c_1(\mathbf{q}, t_1, t_2) \;=\; \frac{\langle E^{*}(\mathbf{q}, t_1)\, E(\mathbf{q}, t_2)\rangle} {\sqrt{\langle I(\mathbf{q}, t_1)\rangle\, \langle I(\mathbf{q}, t_2)\rangle}}.\]

For a Gaussian process — valid for large \(N\) by the central limit theorem — \(c_1\) depends only on the transport coefficient \(J(t)\) and the mean particle velocity:

(5)¶\[c_1(\mathbf{q}, t_1, t_2) \;=\; \exp\!\left(-\tfrac{q^2}{2}\int_{t_1}^{t_2} J(t')\,dt'\right) \;\times\; \exp\!\left(i\,q\!\int_{t_1}^{t_2}\langle v(t')\rangle\,dt'\right).\]

The first factor is a generalised Debye–Waller decay; the second is a phase shift from the mean drift. This factorisation is exact for Gaussian displacement statistics. We split

(6)¶\[c_1 \;=\; c_1^{(\mathrm{in})} \;\times\; c_1^{(\mathrm{ex})},\]

with the internal (diffusive) piece

\[c_1^{(\mathrm{in})}(\mathbf{q}, t_1, t_2) \;=\; \exp\!\left(-\tfrac{q^2}{2}\,\mathcal{D}(t_1, t_2)\right), \qquad \mathcal{D}(t_1, t_2) \;=\; \int_{t_1}^{t_2} J(t')\,dt',\]

and the external (drift) piece

\[c_1^{(\mathrm{ex})}(\mathbf{q}, t_1, t_2) \;=\; \exp\!\left(i\,q\!\int_{t_1}^{t_2}\langle v(t')\rangle\,dt'\right).\]

The transport coefficient \(J(t)\) is the central physical quantity in xpcsjax; it is defined in Transport Coefficient J(t).

Second-order (intensity) correlation function c_2¶

The normalised second-order correlation function is

(7)¶\[c_2(\mathbf{q}, t_1, t_2) \;=\; \frac{\langle I(\mathbf{q}, t_1)\, I(\mathbf{q}, t_2)\rangle} {\langle I(\mathbf{q}, t_1)\rangle\, \langle I(\mathbf{q}, t_2)\rangle}.\]

This is the quantity measured directly in XPCS experiments. Each detector pixel contributes one time series \(I(\mathbf{q}, t)\) and the correlation is accumulated as a function of the absolute times \(t_1\) and \(t_2\).

Note

\(c_2\) is dimensionless and satisfies \(c_2 \geq 1\) for classical intensity fluctuations (Cauchy–Schwarz). For ergodic decorrelating dynamics, \(c_2(t_1, t_2) \to 1\) as \(|t_2 - t_1| \to \infty\). This is a statement about the true physical \(c_2\); xpcsjax’s fitted forward model \(\mathrm{offset} + \mathrm{contrast}\times g_1^2\) does not hard-enforce it – offset is bounded down to \(0.5\) (not \(1.0\)) to absorb baseline-calibration error in real data, and the model value is not clipped at runtime (clipping would kill gradients at the boundary). A fitted offset well below 1 is a data-quality signal worth checking, not something the solver is prevented from returning.

The Siegert relation¶

For a Gaussian field (single-mode thermal-light statistics), Wick’s theorem yields the Siegert relation connecting the measured intensity correlation to the unmeasurable field correlation [Sutton2008]:

(8)¶\[c_2(\mathbf{q}, t_1, t_2) \;=\; 1 + \beta(t_1, t_2)\, \bigl|c_1(\mathbf{q}, t_1, t_2)\bigr|^2,\]

where \(\beta \in (0, 1]\) is the speckle contrast (also called the optical coherence parameter). For a fully coherent beam with single-mode detection \(\beta = 1\); in realistic experiments \(\beta \approx 0.05\)–\(0.8\) depending on beam coherence, pixel geometry, and sample inhomogeneity.

Substituting (5) into (8) gives the general homodyne result:

(9)¶\[c_2(\mathbf{q}, t_1, t_2) \;=\; 1 + \beta(t_1, t_2)\, \exp\!\left(-q^2\!\int_{t_1}^{t_2} J(t')\,dt'\right).\]

The drift phase drops out of \(|c_1|^2\) for homodyne detection. It is recovered in heterodyne detection (multi-component scattering) through cross terms between the populations, leading to the characteristic oscillations that xpcsjax fits in two_component mode (Heterodyne Model).

Note

In xpcsjax, \(\mathcal{D}(t_1, t_2) = \int_{t_1}^{t_2} J(t')\,dt'\) is evaluated numerically by cumulative trapezoidal integration on the experimental time grid — no closed-form antiderivative is ever substituted. A closed form (\(D_0 \tau^{\alpha+1}/(\alpha+1)\)) does exist for \(\alpha \neq -1\), but it is evaluated at the nominal grid spacing rather than the true (possibly irregular) sampling, and it drops the constant \(D_\mathrm{offset}\) term entirely – both introduce error relative to trapezoidal integration on the real time grid, which is why xpcsjax never substitutes it. See Transport Coefficient J(t) for the integration convention and the \(D_0 = 2 D_\mathrm{SE}\) factor.

Wiener–Khinchin and the equilibrium projection¶

For a wide-sense stationary process, the Wiener–Khinchin theorem relates the autocorrelation function to the power spectral density: the second-moment statistics of \(I(\mathbf{q}, t)\) are completely specified by the lag-only correlation. Equivalently, the two-time matrix collapses onto a single curve in \(\tau = t_2 - t_1\):

(10)¶\[g_2(q, \tau) \;=\; \frac{\langle I(q, t)\, I(q, t + \tau)\rangle}{\langle I(q, t)\rangle^2} \;=\; 1 + \beta\, e^{-2\Gamma \tau}.\]

For simple Brownian diffusion \(\Gamma = D q^2\) and the decay is a single exponential. The \(g_2(q, \tau)\) form is widely used in dynamic light scattering and equilibrium XPCS analysis.

Warning

Applying \(g_2(q, \tau)\) analysis to a non-stationary sample produces artefact-corrupted parameters: the effective \(D\) absorbs time-averaged heterogeneity, the apparent contrast \(\beta\) is depressed, and the functional form may stop being a single exponential even when the underlying physics is simple. For yielding, aging, or shear-banding samples, use the full two-time matrix.

The two-time correlation matrix¶

In practice \(c_2\) is represented as a matrix indexed by discrete frame times:

\[c_2^{ij} \;=\; c_2(q, t_i, t_j), \qquad i, j \in \{1, \dots, N_t\}.\]

This matrix is symmetric (\(c_2^{ij} = c_2^{ji}\)) and has \(c_2^{ii} = 1 + \beta\) on the diagonal (zero lag). The equilibrium projection \(g_2(\tau)\) corresponds to the mean along the anti-diagonal at lag \(\tau = (j - i)\,\Delta t\).

The xpcsjax data loader xpcsjax.data.xpcs_loader.load_xpcs_data() returns the array c2_exp of shape (n_phi, n_time, n_time) together with the two frame-time grids t1 / t2 (accessors .t1 / .t2) and the angle vector phi_angles_list (accessor .phi). The angle axis is preserved because the homodyne laminar-flow kernel and the heterodyne two-component kernel both depend explicitly on \(\phi\).

Multi-angle data and the per-angle scaling¶

A typical XPCS experiment captures the full \((q, \phi)\) plane on a 2D detector and resolves it into \(N_\phi\) azimuthal sectors at each \(q\). Each sector is fit simultaneously, but the speckle contrast \(\beta(\phi)\) and the baseline offset \(c_\mathrm{offset}(\phi)\) vary with angle for purely instrumental reasons (partial coherence, pixel geometry, gain variation, sample inhomogeneity).

The xpcsjax forward model factorises this as

(11)¶\[c_2^{\mathrm{model}}(\phi_k, t_1, t_2; \theta) \;=\; c_\mathrm{offset}(\phi_k) \;+\; \beta(\phi_k)\, \left|c_1(\mathbf{q}, t_1, t_2; \theta)\right|^2,\]

where \(\theta\) collects the physical parameters (\(D_0, \alpha, D_\mathrm{offset}\), plus shear parameters in laminar_flow) and \((\beta(\phi_k), c_\mathrm{offset}(\phi_k))\) are the per-angle scaling parameters. The naive choice of \(2 N_\phi\) free scaling parameters introduces a parameter absorption degeneracy; xpcsjax breaks the degeneracy through the strategies described in Anti-Degeneracy Defence.

Fitting the model to data¶

Given measured \(\{c_2^{kij,\,\mathrm{meas}}\}\) and the forward model in Equation (11), the NLSQ engine in xpcsjax minimises

\[\chi^2(\theta) \;=\; \sum_{k=1}^{N_\phi}\sum_{i, j} w_{kij}\!\left[c_2^{kij,\,\mathrm{meas}} - c_2^{kij,\,\mathrm{model}}(\theta)\right]^2,\]

through the trust-region Levenberg–Marquardt step in the upstream NLSQ library (xpcsjax.optimization.nlsq.fit_nlsq()). The weights \(w_{kij}\) default to uniform and can optionally encode Poisson photon statistics or shear-sensitivity reweighting. The shape parameters \((\beta(\phi_k), c_\mathrm{offset}(\phi_k))\) enter through the anti-degeneracy controller’s per-angle mode — constant (fixed from quantiles), averaged (averaged and optimised), or individual (per-angle); auto resolves to averaged or individual by angle count. See Anti-Degeneracy Defence.