source: sasview/src/sas/sasgui/perspectives/corfunc/media/corfunc_help.rst @ 204f628

ESS_GUIESS_GUI_DocsESS_GUI_batch_fittingESS_GUI_bumps_abstractionESS_GUI_iss1116ESS_GUI_iss879ESS_GUI_iss959ESS_GUI_openclESS_GUI_orderingESS_GUI_sync_sascalccostrafo411magnetic_scattrelease-4.1.1release-4.1.2release-4.2.2ticket-1009ticket-1094-headlessticket-1242-2d-resolutionticket-1243ticket-1249ticket885unittest-saveload
Last change on this file since 204f628 was f56770ef, checked in by lewis, 8 years ago

Tweak corfunc help file

  • Property mode set to 100644
File size: 6.0 KB
RevLine 
[fb7fcec]1.. corfunc_help.rst
2
[32c5983]3Correlation Function Analysis
4=============================
[fb7fcec]5
6Description
7-----------
8
[32c5983]9This performs a correlation function analysis of one-dimensional
[fb7fcec]10SANS data, or generates a model-independent volume fraction profile from a
11one-dimensional SANS pattern of an adsorbed layer.
12
13The correlation function analysis is performed in 3 stages:
14
15*  Extrapolation of the scattering curve to :math:`Q = 0` and
16   :math:`Q = \infty`
[41345d7e]17*  Fourier/Hilbert Transform of the extrapolated data to give the correlation
18   function/volume fraction profile
[fb7fcec]19*  Interpretation of the 1D correlation function based on an ideal lamellar
20   morphology
21
22.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
23
24Extrapolation
25-------------
26
27To :math:`Q = 0`
28^^^^^^^^^^^^^^^^
29
30The data are extrapolated to Q = 0 by fitting a Guinier model to the data
31points in the lower Q range.
32The equation used is:
33
34.. math::
[ff11b21]35    I(Q) = e^{A+Bq^2}
[fb7fcec]36
37The Guinier model assumes that the small angle scattering arises from particles
38and that parameter :math:`B` is related to the radius of gyration of those
39particles. This has dubious applicability to polymer systems. However, the
40correlation function is affected by the Guinier back-extrapolation to the
41greatest extent at large values of R and so the back-extrapolation only has a
42small effect on the analysis.
43
44To :math:`Q = \infty`
45^^^^^^^^^^^^^^^^^^^^^
46
47The data are extrapolated to Q = :math:`\infty` by fitting a Porod model to
48the data points in the upper Q range.
49
50The equation used is:
51
52.. math::
[c97d76a]53    I(Q) = Bg + KQ^{-4}e^{-Q^2\sigma^2}
[fb7fcec]54
[c97d76a]55Where :math:`Bg` is the Bonart thermal background, :math:`K` is the Porod
56constant, and :math:`\sigma > 0` describes the electron (or neutron scattering
[fb7fcec]57length) density profile at the interface between crystalline and amorphous
58regions (see figure 1).
59
60.. figure:: fig1.gif
61   :align: center
62
[41345d7e]63   **Figure 1** The value of :math:`\sigma` is a measure of the electron
[fb7fcec]64   density profile at the interface between crystalline and amorphous regions.
65
66Smoothing
67^^^^^^^^^
68
69The extrapolated data set consists of the Guinier back-extrapolation up to the
70highest Q value of the lower Q range, the original scattering data up to the
71highest value in the upper Q range, and the Porod tail-fit beyond this. The
72joins between the original data and the Guinier/Porod fits are smoothed using
[2fe78566]73the algorithm below, to avoid the formation of ripples in the transformed data.
[fb7fcec]74
[2fe78566]75Functions :math:`f(x_i)` and :math:`g(x_i)` where :math:`x_i \in \left\{
76{x_1, x_2, ..., x_n} \right\}`, are smoothed over the range :math:`[a, b]`
77to produce :math:`y(x_i)`, by the following equations:
[fb7fcec]78
79.. math::
80    y(x_i) = h_ig(x_i) + (1-h_i)f(x_i)
81
82where:
83
84.. math::
85    h_i = \frac{1}{1 + \frac{(x_i-b)^2}{(x_i-a)^2}}
[41345d7e]86
87Transform
88---------
89
90Fourier
91^^^^^^^
92
[32c5983]93If Fourier is selected for the transform type, the analysis will perform a
[0390040]94discrete cosine transform on the extrapolated data in order to calculate the
[2fe78566]95correlation function. The following algorithm is applied:
[41345d7e]96
97.. math::
98    \Gamma(x_k) = 2 \sum_{n=0}^{N-1} x_n \cos{\left[ \frac{\pi}{N}
99    \left(n + \frac{1}{2} \right) k \right] } \text{ for } k = 0, 1, \ldots,
100    N-1, N
101
102Hilbert
103^^^^^^^
[32c5983]104If Hilbert is selected for the transform type, the analysis will perform a
[2fe78566]105Hilbert transform on the extrapolated data in order to calculate the Volume
[41345d7e]106Fraction Profile.
107
108Interpretation
109--------------
110Once the correlation function has been calculated by transforming the
111extrapolated data, it may be interpreted by clicking the "Compute Parameters"
112button. The correlation function is interpreted in terms of an ideal lamellar
113morphology, and structural parameters are obtained as shown in Figure 2 below.
114It should be noted that a small beam size is assumed; no de-smearing is
115performed.
116
117.. figure:: fig2.gif
118   :align: center
119
120   **Figure 2** Interpretation of the correlation function.
121
122The structural parameters obtained are:
123
124*   Long Period :math:`= L_p`
125*   Average Hard Block Thickness :math:`= L_c`
126*   Average Core Thickness :math:`= D_0`
127*   Average Interface Thickness :math:`\text{} = D_{tr}`
128*   Polydispersity :math:`= \Gamma_{\text{min}}/\Gamma_{\text{max}}`
129*   Local Crystallinity :math:`= L_c/L_p`
[0390040]130
131.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
132
133Usage
134-----
[f56770ef]135Upon sending data for correlation function analysis, it will be plotted (minus
136the background value), along with a red bar indicating the lower Q range (used
137for back-extrapolation), and 2 purple bars indicating the upper Q range (used
138for forward-extrapolation) [figure 3]. These bars may be moved my clicking and
139dragging, or by entering the appropriate values in the Q range input boxes.
[0390040]140
141.. figure:: tutorial1.png
142   :align: center
143
144   **Figure 3** A plot of some data showing the Q range bars
145
146Once the Q ranges have been set, click the "Calculate" button next to the
147background input field to calculate the Bonart thermal background level.
148Alternatively, enter your own value into the field. Click the "Extrapolate"
[f56770ef]149button to extrapolate the data and plot the extrapolation in the same figure.
150The values of the parameters used for the Guinier and Porod models will also be
151shown in the "Extrapolation Parameters" section [figure 4]
[0390040]152
153.. figure:: tutorial2.png
154   :align: center
155
156   **Figure 4** A plot showing the extrapolated data and the original data
157
158Then, select which type of transform you would like to perform, using the radio
159buttons:
160
161*   **Fourier** Perform a Fourier Transform to calculate the correlation
162    function of the extrapolated data
163*   **Hilbert** Perform a Hilbert Transform to calculate the volume fraction
164    profile of the extrapolated data
165
166Clicking the transform button will then perform the selected transform and plot
167it in a new figure. If a Fourier Transform was performed, the "Compute
168Parameters" button can also be clicked to calculate values for the output
169parameters [figure 5]
170
171 .. figure:: tutorial3.png
172    :align: center
173
174    **Figure 5** The Fourier Transform (correlation function) of the
175    extrapolated data, and the parameters extracted from it.
Note: See TracBrowser for help on using the repository browser.