Critical Exponent Fit Calculator

Formula used

Near a continuous transition, many observables follow a scaling law:

y(x) = y0 + a · |x − xc|^p

• xc: critical point (e.g., critical temperature).
• p: critical exponent to be fitted.
• a: amplitude (scale factor).
• y0: background offset (non-critical contribution).

The grid method searches over (p, xc) and solves the best a and y0 by least squares for each grid point.

How to use this calculator

1. Paste your measurements as x,y pairs, one per line.
2. Choose a fit method:
   • Grid search: best when xc is unknown.
   • Log-linear: faster, but requires fixed xc and y0.
3. Set ranges and grid sizes (or enable auto ranges).
4. Press Calculate to view results above the form.
5. Use Download CSV or Download PDF to save outputs.

Example data table

Sample dataset showing a smooth decrease toward a critical point.

Critical exponent fitting guide

1) Why critical exponents matter

Critical exponents summarize how an observable changes as a control variable approaches a continuous transition. They support comparisons across models and experiments, because different materials can share the same exponent when they belong to the same universality class.

2) Model used by this calculator

This tool fits a power-law with background, y = y0 + a·|x − xc|^p. Here xc is the critical point, p is the exponent, a sets the scale, and y0 captures non-critical offsets. The absolute value lets you use data from either side of the transition.

3) Preparing reliable input data

Use consistent units and record measurement uncertainty separately. A practical range is 10–50 points spanning the scaling window. Avoid including far-from-critical points where corrections-to-scaling dominate. If the observable can change sign, fit the magnitude or shift the baseline before applying the log method.

4) Choosing between grid and log fitting

The grid approach searches over p and xc and solves a and y0 by least squares at each grid point. It is slower but robust when xc is not known. The log-linear method is fast, but requires fixed xc and y0 and also needs y − y0 > 0.

5) Interpreting fit quality metrics

R² close to 1 indicates that the model explains most variance, but it should not be the only criterion. Check the residual column for patterns: curved residuals often mean the scaling window is too wide, while alternating residuals can indicate systematic measurement drift or an incorrect background term.

6) Uncertainty and confidence ranges

For grid fits, the calculator reports a practical envelope around the best solution by scanning points within a 10% SSE increase. This is not a strict statistical confidence interval, but it helps you see sensitivity to the chosen ranges and grid resolution. Increase grid density to refine the envelope.

7) Common physics use cases

Typical applications include magnetization near a Curie point, order parameters in lattice models, conductivity or susceptibility scaling, and finite-size studies where you compare effective exponents across system sizes. If you have multiple datasets, fit each one consistently and compare the extracted p and xc.

8) Reporting results clearly

When publishing, state the fitting window, method, and whether y0 was free or fixed. Include the extracted parameters, SSE, and a residual table. Use the built-in CSV export to archive the point-by-point fit and the print-to-PDF option to preserve a formatted snapshot for lab notes or reports.

FAQs

1) What does the exponent p represent?

It describes how quickly the fitted observable changes as x approaches the critical point xc. Larger p means a steeper rise or decay near xc for the same amplitude a and background y0.

2) When should I prefer the grid method?

Use grid fitting when xc is uncertain, the background y0 is not known, or your data is noisy. It explores many candidate (p, xc) pairs and chooses the combination that minimizes SSE.

3) Why does the log method reject some points?

Log-linear fitting requires x ≠ xc and y − y0 > 0, because it takes logarithms of both |x − xc| and y − y0. Points that violate these conditions are excluded automatically.

4) What ranges should I start with for p and xc?

Begin with a broad xc range spanning the transition and a p range such as 0.01–3. If you know the universality class, narrow p. The auto-range option is useful for a quick first pass.

5) How can I improve stability of the fitted xc?

Increase the number of points close to the transition, narrow the xc search window based on physics expectations, and raise grid resolution. Also remove far-from-critical data where scaling corrections can bias xc.

6) What does a negative amplitude a mean?

A negative a means the observable decreases as |x − xc| increases, after accounting for y0. This can be physical for some definitions or plotting conventions. If the sign is unexpected, verify that x and y columns are not swapped.

7) Can I fit data on only one side of xc?

Yes. The model uses |x − xc|, so it can fit points from one side if there is enough spread in x. However, xc may become less constrained, so tighter ranges and more near-critical points help.