Model description

This chapter describes the mathematical models available for scattering populations in the Modeling package. Any population can use one of the following types:

“Size distribution”

“Unified level”

“Surface fractal”

“Mass Fractal”

“Diffraction peak”

This flexibility allows modeling of complex small-angle scattering data. The Small-angle diffraction tool and the Fractals tool also use some of these models.

Size distribution

Size distribution parameters are described on the main Modeling package page. The size distribution can use any of the form factors and structure factors available in Irena. See: Form and Structure factors. For GUI details, assumptions, and distribution shapes, see: Size distribution.

Unified Fit

The Unified Fit uses the formula explained in the Unified Fit model page: Unified Fit.

Diffraction Peaks

Diffraction peak shapes are used in the Small-Angle Diffraction tool and in the Modeling package. The following formulas define the peak profile Ψ(x) for each available shape:

1. Gaussian Function

\[\Psi(x)=M \cdot \exp\!\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\]

where σ is the Gaussian width, μ is the peak center, and M is the scaling factor.

2. Modified Gaussian Function

\[\Psi(x)=M \cdot \exp\!\left(-\frac{(x-\mu)^d}{2\sigma^d}\right)\]

where d ≥ 1 is the exponent controlling the falloff rate.

3. Lorentz Function (Lorentz-squared is this function squared)

\[\Psi(x)=M \cdot \frac{a}{\pi(a^2+(x-\mu)^2)}\]

where a is the Lorentzian width.

4. Pseudo-Voigt Function

\[ \begin{align}\begin{aligned}\Psi(x)=M \cdot \left(\eta\frac{1}{1+x^2}+(1-\eta)\exp(-(ln2)x^2)\right)\\x= \frac{2(x-x_0)}{w}\end{aligned}\end{align} \]

where x₀ is the peak center, w is the FWHM, and 0 ≤ η ≤ 1 is the weight parameter. η = 0 gives a pure Gaussian; η = 1 gives a pure Lorentzian.

5. Pearson type VII Function

\[\Psi(x)=M \cdot \left[1+\frac{(x-\mu)^2}{ma^2}\right]^{-m}\]

where a is proportional to the FWHM and m controls the tail falloff rate.

6. Gumbel Function

\[\Psi(x)=\frac{1}{\beta}\exp\!\left(\frac{x-\mu}{\beta}\right)\exp\!\left(-\exp\!\left(\frac{x-\mu}{\beta}\right)\right)\]

where β is the width and μ is the peak center.

7. Skew normal function

8. Percus-Yevick S(q) and Percus-Yevick S(q) × Sphere F(q) — described in the Form factor and Structure factor PDF (accessible from the SAS menu in Igor Pro). The P-Y S(q) code is derived from the NIST SANS data analysis macros.

Surface and Mass Fractal

This model was developed for analysis of fractal systems in cement, see: https://www.nature.com/articles/nmat1871. For more details, see also: Fractal model. When possible, use the dedicated Fractals tool — it is simpler to use than configuring this population type in Modeling.

The model predicts Q^Dv scattering (between Q^-1 and Q^-3) for mass or volume fractals, and Q^6-Ds scattering (between Q^-3 and Q^-4) for surface fractals.

The full model for dΣ/dΩ as a function of Q contains four components:

dΣ/dΩ = {VOLUME FRACTAL + SINGLE GLOBULE} TERM + SURFACE FRACTAL + FLAT BACKGROUND

These components are incorporated into the full theoretical expression as follows:

The first volume-fractal term contains Φ_CSH, ξ_v, and the mean radius R_o and shape aspect ratio β of the building-block C-S-H gel globules in the volume-fractal phase (assumed to be spheroids). It also contains a local volume fraction η and the mean correlation-hole radius R_c — the mean nearest-neighbor separation of gel-globule centers. R_c, weighted over spheroid surface contacts, is given by:

The single-globule term arises because nearest-neighbor solid particles cannot overlap (centers cannot approach within R_c). This correlation-hole effect makes individual particles distinguishable even within an aggregated structure, at length scales of order R_o. For a spheroid of aspect ratio β, the single-globule form factor F:sup:`2`(Q) is given by:

where V_p = (4βπR_o/3), J:sub:`3/2`(x) is a Bessel function of order 3/2, and X is an orientational parameter integrated over all orientations of the spheroid with respect to Q. A mildly spheroidal shape avoids the pronounced Bessel function oscillations for perfect spheres (β = 1) that can perturb fits at high Q. Both mildly oblate (β = 0.5) and mildly prolate (β = 2) shapes give equivalent fits for cement systems.

The surface fractal term includes ξ_s, the upper limit of surface-fractal behavior, and S_o, the measured smooth surface area per unit sample volume. The term Γ(5 − D_s) is the mathematical gamma function.

The BACKGROUND term represents incoherent flat background scattering, usually subtracted from both data and fits for convenience.