In the Guinier region (low q), the scattering intensity is approximated by:
- I(q) = I0 · exp(−q²·Rg²/3)
- Linear form: ln(I) = ln(I0) − (Rg²/3)·q²
A straight-line fit of ln(I) versus q² yields slope b and intercept a. Then Rg = √(−3b) and I0 = exp(a).
- Paste your SAXS data as q and I(q) columns.
- Set column indices if your file has extra columns.
- Select a low-q range using q max or row indices.
- Enable weighted fit if you have σI values.
- Click Calculate and review the q·Rg check.
- Export the table using CSV or PDF buttons.
| q | I(q) | σI |
|---|---|---|
| 0.020 | 12500 | 320 |
| 0.040 | 9800 | 260 |
| 0.060 | 7600 | 220 |
| 0.080 | 6000 | 200 |
| 0.100 | 4800 | 180 |
1. Guinier analysis in SAXS
Small-angle X-ray scattering (SAXS) links nanoscale size to the intensity curve I(q) as a function of scattering vector q. In the lowest-q regime, many systems follow I(q) = I0 * exp(-q^2 * Rg^2 / 3). A linear fit of ln(I) versus q^2 estimates radius of gyration Rg and forward intensity I(0).
2. Data requirements and units
The fit uses ln(I), so q and I(q) must be positive. Common q units are 1/Angstrom or 1/nm, and the returned Rg uses the matching length unit. Use background-corrected, masked, and normalized data when possible, and avoid points affected by the beamstop or parasitic scattering.
3. Selecting a Guinier range
Guinier behavior is confined to low q where ln(I) versus q^2 appears linear. A widely used validity check is q*Rg <= 1.3 (sometimes extended to about 1.5 for screening). Start with roughly 10 to 20 low-q points, then reduce q max until the line is stable and residuals look random.
4. What the fit is solving
The linear model is y = a + b x with y = ln(I) and x = q^2. Intercept a corresponds to ln(I0), while slope b corresponds to -(Rg^2)/3. If b is not negative, the selected region is not Guinier-like and the reported Rg is not physically meaningful.
5. Weighted fit with sigmaI
If sigmaI values are available, weighting can improve robustness. Because the fit is in ln(I), the propagated uncertainty is sigma_lnI = sigmaI / I, and the weight becomes 1/sigma_lnI^2 = (I/sigmaI)^2. Enable weighting only when uncertainties are consistently estimated across the range.
6. Interpreting Rg and I(0)
Rg summarizes how electron density is distributed around the center of the scatterer, so it reflects overall size rather than sharp edges. I(0) scales with concentration and contrast, and it is commonly used for sample-to-sample comparisons, monitoring aggregation, or supporting molecular-weight estimation after absolute calibration.
7. Diagnostics and common pitfalls
Curvature in the linearized plot often signals interparticle interactions, polydispersity, aggregation, or an overly wide q window. If R2 drops or q*Rg is high, tighten the range and re-fit. Also re-check subtraction, solvent matching, and masking of detector artifacts that bias low-q points.
8. Reporting and exporting
For traceable reporting, record the selected q limits, number of fitted points, and whether weighting was used. Include Rg and I(0) with uncertainties, plus the linearized plot and the fitted table. Use CSV export for analysis pipelines and PDF export for quick reviews and sharing.
1) What q units should I use?
Use the same q units used by your instrument or data reduction, such as 1/Angstrom or 1/nm. The computed Rg will be in the corresponding length unit (Angstrom or nm).
2) Why must I(q) be positive?
The Guinier fit is performed in ln(I). If background subtraction yields negative or zero intensities, the logarithm is undefined, so those points must be excluded or corrected.
3) How many points are enough for a fit?
At least two points are required, but practical fits usually use 10 to 20 low-q points. Use fewer if curvature appears, and prefer the most linear segment.
4) What does the q*Rg check mean?
It is a quick Guinier-validity indicator. Values at or below about 1.3 are commonly considered safely within the Guinier regime. Higher values suggest you should reduce q max.
5) When should I enable the weighted fit?
Enable it when you have reliable sigmaI uncertainties for each point. The calculator uses (I/sigmaI)^2 weights in ln(I), reducing the influence of noisy measurements.
6) Why is my fitted slope positive?
A positive slope usually means the selected range is not Guinier-like, or the data contain artifacts. Tighten the low-q window, check background subtraction, and remove outliers.
7) What should I report with Rg and I(0)?
Report the q range, number of points, weighting choice, Rg and I(0) with uncertainties, and a plot of ln(I) versus q^2 with the fit line. This supports reproducibility.