Parameter Guide¶

The heterodyne model uses 14 physics parameters (shared across all azimuthal angles) and 2 scaling parameters per angle. This page documents every parameter, its units, typical range, and physical interpretation.

Physics Parameters¶

All lengths are in Angstroms (Å), consistent with standard synchrotron beamline conventions (APS, ESRF, PETRA III).

Reference Component (Diffusion)¶

Name	Symbol	Units	Typical Range	Meaning
`D0_ref`	\(D_{0,r}\)	Å²/s^alpha	1 – 1e6	Reference diffusion prefactor. Sets the overall magnitude of the reference component’s mean-squared displacement. Larger values indicate faster diffusion.
`alpha_ref`	\(\alpha_r\)	–	-2 to 2	Reference transport exponent. \(\alpha = 0\) corresponds to normal Brownian diffusion. Positive values indicate super-diffusion (e.g., ballistic motion); negative values indicate sub-diffusion (e.g., caging, crowding). Default: 0.0.
`D_offset_ref`	\(D_\text{off,r}\)	Å²	0 – \(D_0\)	Reference diffusion baseline. A constant contribution to the cumulative transport coefficient, representing a time-independent offset in the MSD.

Sample Component (Diffusion)¶

Name	Symbol	Units	Typical Range	Meaning
`D0_sample`	\(D_{0,s}\)	Å²/s^alpha	1 – 1e6	Sample diffusion prefactor. Analogous to `D0_ref` for the sample component.
`alpha_sample`	\(\alpha_s\)	–	-2 to 2	Sample transport exponent. Same interpretation as `alpha_ref`. Default: 0.0.
`D_offset_sample`	\(D_\text{off,s}\)	Å²	0 – \(D_0\)	Sample diffusion baseline.

Velocity¶

Name	Symbol	Units	Typical Range	Meaning
`v0`	\(v_0\)	Å/s^beta	0 – 1e6	Velocity prefactor. Controls the magnitude of the relative velocity between reference and sample components. Default: 1e3.
`beta`	\(\beta_v\)	–	-2 to 2	Velocity exponent. \(\beta_v = 0\) corresponds to constant velocity; non-zero values describe acceleration or deceleration over time.
`v_offset`	\(v_\text{off}\)	Å/s	-100 to 100	Velocity offset. A constant contribution to the velocity field. Can be negative, indicating flow reversal relative to the dominant direction. Default: 0.0.

Fraction Evolution¶

The fraction function \(f(t)\) describes how the relative scattering weight of the two components evolves during the measurement:

\[f(t) = f_0\, \exp(-f_1\, t)\, + f_3\]

where \(f_1\) is the exponential decay rate and \(f_2\) is a time shift applied to \(t\).

Name	Symbol	Units	Typical Range	Meaning
`f0`	\(f_0\)	–	0 – 1	Fraction amplitude. The initial excess weight of one component above the baseline \(f_3\).
`f1`	\(f_1\)	1/s	> 0	Fraction exponential rate. Controls how quickly the component fraction relaxes toward the baseline.
`f2`	\(f_2\)	s	–	Fraction time shift. Offsets the origin of the exponential decay; useful when the dynamics do not begin at \(t=0\).
`f3`	\(f_3\)	–	0 – 1	Fraction baseline. The long-time asymptotic fraction.

Flow Geometry¶

Name	Symbol	Units	Typical Range	Meaning
`phi0`	\(\phi_0\)	degrees	-180 to 180	Flow angle offset. The angle between the in-plane flow direction and the reference axis of the detector. The velocity phase in the cross-correlation term is proportional to \(\cos(\phi - \phi_0)\).

Scaling Parameters (Per Angle)¶

Each azimuthal angle \(\phi_i\) carries two additional parameters that account for angle-dependent optical effects.

Name	Symbol	Units	Typical Range	Meaning
`contrast`	\(\beta\)	–	0 – 1	Speckle visibility (optical coherence factor). Values near 1 indicate a highly coherent beam; lower values reflect partial coherence, detector integration effects, or multiple-scattering contributions.
`offset`	–	–	~1	Baseline correlation level. Ideally exactly 1.0 for normalised \(g_2\), but small deviations arise from background subtraction imperfections or detector non-uniformity.

Parameter Count Summary¶

For an analysis with \(N_\phi\) azimuthal angles:

Physics parameters: 14 (shared across all angles).
Scaling parameters: \(2 \times N_\phi\).
Total: \(14 + 2 N_\phi\).

For a typical 8-angle dataset this gives \(14 + 16 = 30\) free parameters.

Interpreting Fitted Values¶

Diffusion prefactors: Compare \(D_{0,r}\) and \(D_{0,s}\) to estimate relative mobilities. If \(D_{0,r} \ll D_{0,s}\), the reference is nearly static (e.g., a gel network) and the sample is mobile (e.g., embedded nanoparticles).
Transport exponents: \(\alpha \approx 0\) is normal diffusion. \(\alpha \approx 1\) suggests ballistic motion on the probed timescale. Persistent \(\alpha < 0\) indicates sub-diffusive dynamics such as caging in a glass.
Velocity parameters: \(v_0\) gives the magnitude of relative flow. The \(\phi\)-dependent phase means that fitting multiple angles simultaneously greatly constrains \(v_0\) and \(\phi_0\). A non-zero v_offset indicates a constant drift component independent of the power-law time dependence.
Fraction evolution: A decaying \(f(t)\) (positive \(f_0\), positive \(f_1\)) indicates that one component progressively dominates (e.g., sedimentation or gelation removing scatterers from the beam volume).
Contrast: Low contrast (\(\beta < 0.1\)) may indicate a setup problem (poor coherence, wrong slit settings). If contrast varies strongly with \(\phi\), check for anisotropic beam profiles or parasitic scattering.