Calculator Inputs
Formula Used
Many systems show an approximately exponential decay of correlations: C(r) = A · exp(-r/ξ).
- Two-point decay: ξ = (r2 − r1) / ln(C(r1)/C(r2))
- Single-point with reference: ξ = −r / ln(C(r)/C0)
- Spectral width: For S(k) ∝ 1/(1+(kξ)^2), ξ = 1/HWHM and ξ = 2/FWHM
Use consistent normalization. Ensure values inside logarithms are positive.
How to Use This Calculator
- Select a method that matches your data type.
- Pick the length unit for r and ξ.
- Enter correlation values or a spectral peak width.
- Press Calculate to show results above the form.
- Use CSV or PDF buttons for exporting the output.
Example Data Table
Example for the two-point method with exponential decay. Values are dimensionless correlations measured at two distances.
| r1 (m) | r2 (m) | C(r1) | C(r2) | Computed ξ (m) |
|---|---|---|---|---|
| 2.5 | 7.5 | 0.62 | 0.18 | 4.1409 |
| 1.0 | 4.0 | 0.80 | 0.30 | 2.7816 |
| 3.0 | 9.0 | 0.55 | 0.12 | 4.0576 |
If your data include offsets, subtract backgrounds before applying the exponential form.
Professional notes for interpreting correlation length outputs.
1) What correlation length represents
Correlation length, ξ, is the characteristic distance over which fluctuations remain related. In magnets it describes how spin orientations stay aligned; in fluids it quantifies how density variations persist; in porous media it can describe structural similarity across space. Larger ξ means broader spatial memory, while smaller ξ indicates rapid decorrelation.
2) Common decay law in real space
Many experiments approximate the two-point correlation as C(r) ≈ A·exp(−r/ξ) once short-range structure is averaged out. If your data are noisy, selecting two well-separated distances can still yield a useful estimate. For best stability, choose points where C(r) is clearly above the noise floor.
3) Two-point estimate and practical data choices
The two-point method uses ξ = (r2−r1)/ln(C(r1)/C(r2)). A larger ratio C(r1)/C(r2) produces a stronger logarithmic signal and reduces sensitivity. If C(r2) is close to zero or negative after background subtraction, avoid the log step and consider fitting a full curve instead.
4) Single-point reference when C0 is known
When you have a reliable reference C0 near r≈0, the relation ξ = −r/ln(C(r)/C0) converts one measurement into a scale length. This approach is common for normalized autocorrelations where C0≈1. It is fastest, but it inherits any bias in C0 directly into ξ.
5) Reciprocal-space link to peak widths
Scattering and spectra often follow an Ornstein–Zernike form S(k) ∝ 1/(1+(kξ)^2), producing a Lorentzian peak. In that case, ξ is inversely proportional to the half-width at half-maximum (HWHM). This calculator supports ξ = 1/HWHM and ξ = 2/FWHM to match common reporting conventions.
6) Units, scaling, and reporting
Always treat r and ξ in the same length unit. For spectral widths, Δk must be in inverse length, so the output naturally has length units. When publishing results, include the method used, the chosen points or peak definition, and the unit system so the reported ξ is reproducible.
7) Uncertainty and quality checks
If you provide ΔC(r1) and ΔC(r2), the tool estimates first-order uncertainty for the two-point method by propagating errors through the logarithm. A quick sanity check is monotonic decay: typically C(r) decreases with r. Non-decaying trends, negative outputs, or identical correlations suggest re-normalization, filtering, or improved sampling.
8) Where correlation length is used
Correlation length appears in critical phenomena (diverging near phase transitions), polymer physics (coil statistics), turbulence (integral length scales), image texture analysis, and materials characterization from diffraction peaks. Even a simple estimate helps compare samples, quantify processing changes, and track how disorder or temperature reshapes spatial organization.
FAQs
1) Can correlation length be negative?
In most physical interpretations, ξ is positive. A negative value usually means the correlation did not decay between your chosen points or the inputs violate the assumed exponential model.
2) What if my correlation crosses zero?
Log-based formulas require positive values. If your correlation oscillates or crosses zero, use a different model, fit the full correlation function, or analyze an envelope or absolute correlation if justified.
3) How should I pick r1 and r2 for the two-point method?
Choose distances in a region that looks approximately exponential and above noise. Avoid extremely small separations dominated by instrument resolution and very large separations where the signal is near the background.
4) Does the spectral-width method assume a specific line shape?
Yes. It assumes a Lorentzian peak consistent with an Ornstein–Zernike form. If your peak is Gaussian or has multiple components, extract the appropriate width from a fit and use a model-consistent conversion to ξ.
5) Why is C0 needed for the single-point method?
The single-point formula compares a measured C(r) to a reference amplitude at r≈0. If C0 is uncertain, the derived ξ can be biased, so prefer multi-point fitting when possible.
6) How accurate is the uncertainty estimate Δξ?
It is a first-order propagation that captures how measurement errors in C(r1) and C(r2) affect the log ratio. It does not include systematic effects like offsets, finite-size bias, or model mismatch.
7) What is a typical range of correlation lengths?
It depends on the system and scale: nanometers in liquids and crystals, micrometers in soft matter, centimeters in porous media, and larger in geophysical patterns. Report ξ with units and context.