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sasview/src/sas/sasgui/perspectives/corfunc/media/corfunc_help.rst
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Correlation Function Perspective
Description
This perspective performs a correlation function analysis of one-dimensional SANS data, or generates a model-independent volume fraction profile from a one-dimensional SANS pattern of an adsorbed layer.
The correlation function analysis is performed in 3 stages:
- Extrapolation of the scattering curve to Q = 0 and Q = ∞
- Fourier/Hilbert Transform of the extrapolated data to give the correlation function/volume fraction profile
- Interpretation of the 1D correlation function based on an ideal lamellar morphology
Extrapolation
To Q = 0
The data are extrapolated to Q = 0 by fitting a Guinier model to the data points in the lower Q range. The equation used is:
The Guinier model assumes that the small angle scattering arises from particles and that parameter B is related to the radius of gyration of those particles. This has dubious applicability to polymer systems. However, the correlation function is affected by the Guinier back-extrapolation to the greatest extent at large values of R and so the back-extrapolation only has a small effect on the analysis.
To Q = ∞
The data are extrapolated to Q = ∞ by fitting a Porod model to the data points in the upper Q range.
The equation used is:
Where B is the Bonart thermal background, K is the Porod constant, and σ describes the electron (or neutron scattering length) density profile at the interface between crystalline and amorphous regions (see figure 1).
Figure 1 The value of σ is a measure of the electron density profile at the interface between crystalline and amorphous regions.
Smoothing
The extrapolated data set consists of the Guinier back-extrapolation up to the highest Q value of the lower Q range, the original scattering data up to the highest value in the upper Q range, and the Porod tail-fit beyond this. The joins between the original data and the Guinier/Porod fits are smoothed using the algorithm below, to avoid the formation of ripples in the transformd data.
Functions f(xi) and g(xi) where xi ∈ {x1, x2, ..., xn} , are smoothed over the range [a, b] to produce y(xi) , by the following equations:
where:
Transform
Fourier
If Fourier is selected for the transform type, the perspective will perform a discrete cosine transform to the extrapolated data in order to calculate the correlation function. The following algoritm is applied:
Hilbert
If Hilbert is selected for the transform type, the perspective will perform a Hilbert transform to the extraplated data in order to calculate the Volume Fraction Profile.
Interpretation
Once the correlation function has been calculated by transforming the extrapolated data, it may be interpreted by clicking the "Compute Parameters" button. The correlation function is interpreted in terms of an ideal lamellar morphology, and structural parameters are obtained as shown in Figure 2 below. It should be noted that a small beam size is assumed; no de-smearing is performed.
Figure 2 Interpretation of the correlation function.
The structural parameters obtained are:
- Long Period = Lp
- Average Hard Block Thickness = Lc
- Average Core Thickness = D0
- Average Interface Thickness = Dtr
- Polydispersity = Γmin ⁄ Γmax
- Local Crystallinity = Lc ⁄ Lp
