Overview

SANS mode is used for fitting small-angle scattering data including:

• Instrumental resolution smearing integration
• Polydispersity (size distribution) integration

It enables fitting of real experimental data where measured intensity differs from the ideal model.

How to Initiate SANS Mode

1. Open Fit.Curve(s)
2. Select a model function
3. Enable Instrument mode (SANS)
4. Select table (q, I, dI [,Sigma])

Required Data Format

• Initiate SANS Mode
  
  
  

• Data Format
  
  
  

Resolution Smearing

Definition

$$ I_{meas}(q) = \int I(q') \cdot R(q,q', \sigma(q)) \, dq' $$

Resolution Options [modes]

Resolution Options [numerical integration]

Speed/Quality of the numerical integration could be adjusted.

Available resolution functions:

Gaussian: $ R(q,q') = \exp\left(-\frac{(q-q')^2}{2\sigma^2}\right) $

Triangular: Linear within \(q' \pm 2\sigma\)

Bessel: $R(q,q') \propto \frac{q}{\sigma^2} \exp\left(-\frac{q^2 + q'^2}{2\sigma^2}\right) I_0\left(\frac{q q'}{\sigma^2}\right) $

• Activate of the Resolution Mode (On)
• Select “Data_Sigma” column as $\sigma(q)$
• Select numerical integral option of the resolution integral
  

• Integration Options

Polydispersity

In SANS mode, polydispersity is implemented via numerical integration over a distribution of a selected parameter (e.g. radius, thickness). After activation of the polidispersity, polydisperse parameter (f.e. P) could be selected.

The scattering intensity is:

$$ I(q) = \int f(P)\, I_{\mathrm{model}}(q, P)\, dP $$

where:

• $P$ — polydisperse parameter
• $P_0$ — mean value (stored parameter)
• $f(P)$ — distribution function

The model function is evaluated dynamically by replacing \(P_0 \rightarrow P\) during integration.

$\sigma$ in the parameter's table is relative. Absolute $\sigma_{abs}$ could be calculated as: $$ \sigma_{abs} = \sigma \cdot P_0 $$

Polydispersity Type: Gaussian

Relative Gaussian distribution.

Parameters:

• $P_0$ — mean
• $\sigma$ — relative width (in parameters)
• $\sigma_{\text{abs}} = \sigma \cdot P_0$ - absolute width(should be calculated manually)

Distribution function: $$ f(P) = \frac{1}{\sqrt{2\pi}\,P_0\sigma} \exp\left(-\frac{(P-P_0)^2}{2(P_0\sigma)^2}\right) $$

Cutoff (truncation correction):
Because the physical domain is $P \ge 0$, the Gaussian is truncated and renormalized:

$$ C_{\mathrm{cutoff}} = 1 - \frac{1}{2}\operatorname{erfc}\left(\frac{1}{\sigma\sqrt{2}}\right) $$

$$ \int_{0}^{\infty} f(P)\, dP = 1 \quad \Rightarrow \quad f(P) \rightarrow \frac{f(P)}{C_{\mathrm{cutoff}}} $$

Scattering intensity:

$$ I(Q) = \int_{0}^{\infty} f(P)\, I_{\mathrm{model}}(Q,P)\, dP $$

Final implemented form:

$$ I(Q) = \int_{0}^{\infty} \frac{1}{\sqrt{2\pi}\,P_0\sigma\,C_{\mathrm{cutoff}}} \exp\left(-\frac{(P-P_0)^2}{2(P_0\sigma)^2}\right) I_{\mathrm{model}}(Q,P)\, dP$$

Polydispersity Type: Schultz–Zimm

Used for asymmetric size distributions.

• $\sigma$ — relative standard deviation
• $\sigma_{\text{abs}} = \sigma \cdot P_0$ - astandard deviation(should be calculated manually)
• $z=1/\sigma-1$ — shape parameter (should be calculated manually)

Internal variables: \[ k = \frac{1}{\sigma^2}, \quad t = \frac{kP}{P_0} \]

Weight: \[ f(P) = \frac{k}{P_0} \, t^{k-1} e^{-t} \frac{1}{\Gamma(k)} \]

Polydispersity Type: Gamma

Equivalent to Schultz–Zimm but parameterized differently.

Internal variables: $$ \theta = \sigma^2 P_0, \quad k = \frac{P_0}{\theta}, \quad t = \frac{P}{\theta} $$

Weight: $$ f(P) = \frac{1}{\theta} t^{k-1} e^{-t} \frac{1}{\Gamma(k)} $$

Polydispersity Type: Log-Normal

Logarithmic distribution.

Weight: $$ f(P) = \frac{1}{\sqrt{2\pi}\sigma P_0} \exp\left(-\frac{(\ln(P/P_0))^2}{2\sigma^2}\right) $$

Polydispersity Type: Uniform

Flat distribution.

Range: $$ P \in [P_0(1-\sigma),\; P_0(1+\sigma)] $$

Weight: $$ f(P) = \text{const} $$

Final result normalized by dividing by \((P_{\max}-P_{\min})\).

Polydispersity Type: Triangular

Piecewise linear distribution.

Parameters:

• \(A\) — minimum
• \(B\) — maximum

Weight: $$ f(P) = \begin{cases} \frac{2(P-A)}{(B-A)(P_0-A)}, & P < P_0 \\ \frac{2(B-P)}{(B-A)(B-P_0)}, & P \ge P_0 \end{cases} $$

Implemented in `polyFunctionTriangular()`.

Integration Range

General case: \[ P_{\min} = \max\left(0,\; P_0(1 - \sigma \cdot N_\sigma)\right) \] \[ P_{\max} = P_0(1 + \sigma \cdot N_\sigma) \]

Triangular: \[ P_{\min} = A, \quad P_{\max} = B \]