.. corfunc_help.rst Correlation Function Analysis ============================= Description ----------- This 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 :math:`Q = 0` and :math:`Q = \infty` * 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 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ Extrapolation ------------- To :math:`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: .. math:: I(Q) = e^{A+Bq^2} The Guinier model assumes that the small angle scattering arises from particles and that parameter :math:`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 :math:`Q = \infty` ^^^^^^^^^^^^^^^^^^^^^ The data are extrapolated to Q = :math:`\infty` by fitting a Porod model to the data points in the upper Q range. The equation used is: .. math:: I(Q) = Bg + KQ^{-4}e^{-Q^2\sigma^2} Where :math:`Bg` is the Bonart thermal background, :math:`K` is the Porod constant, and :math:`\sigma > 0` describes the electron (or neutron scattering length) density profile at the interface between crystalline and amorphous regions (see figure 1). .. figure:: fig1.gif :align: center **Figure 1** The value of :math:`\sigma` 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 transformed data. Functions :math:`f(x_i)` and :math:`g(x_i)` where :math:`x_i \in \left\{ {x_1, x_2, ..., x_n} \right\}`, are smoothed over the range :math:`[a, b]` to produce :math:`y(x_i)`, by the following equations: .. math:: y(x_i) = h_ig(x_i) + (1-h_i)f(x_i) where: .. math:: h_i = \frac{1}{1 + \frac{(x_i-b)^2}{(x_i-a)^2}} Transform --------- Fourier ^^^^^^^ If Fourier is selected for the transform type, the analysis will perform a discrete cosine transform on the extrapolated data in order to calculate the correlation function. The following algorithm is applied: .. math:: \Gamma(x_k) = 2 \sum_{n=0}^{N-1} x_n \cos{\left[ \frac{\pi}{N} \left(n + \frac{1}{2} \right) k \right] } \text{ for } k = 0, 1, \ldots, N-1, N Hilbert ^^^^^^^ If Hilbert is selected for the transform type, the analysis will perform a Hilbert transform on the extrapolated 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:: fig2.gif :align: center **Figure 2** Interpretation of the correlation function. The structural parameters obtained are: * Long Period :math:`= L_p` * Average Hard Block Thickness :math:`= L_c` * Average Core Thickness :math:`= D_0` * Average Interface Thickness :math:`\text{} = D_{tr}` * Polydispersity :math:`= \Gamma_{\text{min}}/\Gamma_{\text{max}}` * Local Crystallinity :math:`= L_c/L_p` .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ Usage ----- Upon sending data for correlation function analysis, it will be plotted (minus the background value), along with a red bar indicating the lower Q range (used for back-extrapolation), and 2 purple bars indicating the upper Q range (used for forward-extrapolation) [figure 3]. These bars may be moved my clicking and dragging, or by entering the appropriate values in the Q range input boxes. .. figure:: tutorial1.png :align: center **Figure 3** A plot of some data showing the Q range bars Once the Q ranges have been set, click the "Calculate" button next to the background input field to calculate the Bonart thermal background level. Alternatively, enter your own value into the field. Click the "Extrapolate" button to extrapolate the data and plot the extrapolation in the same figure. The values of the parameters used for the Guinier and Porod models will also be shown in the "Extrapolation Parameters" section [figure 4] .. figure:: tutorial2.png :align: center **Figure 4** A plot showing the extrapolated data and the original data Then, select which type of transform you would like to perform, using the radio buttons: * **Fourier** Perform a Fourier Transform to calculate the correlation function of the extrapolated data * **Hilbert** Perform a Hilbert Transform to calculate the volume fraction profile of the extrapolated data Clicking the transform button will then perform the selected transform and plot it in a new figure. If a Fourier Transform was performed, the "Compute Parameters" button can also be clicked to calculate values for the output parameters [figure 5] .. figure:: tutorial3.png :align: center **Figure 5** The Fourier Transform (correlation function) of the extrapolated data, and the parameters extracted from it.