source: sasview/src/sas/sasgui/perspectives/corfunc/media/corfunc_help.rst @ 501712f

magnetic_scattrelease-4.2.2ticket-1009ticket-1094-headlessticket-1242-2d-resolutionticket-1243ticket-1249ticket885unittest-saveload
Last change on this file since 501712f was 501712f, checked in by Adam Washington <adam.washington@…>, 6 years ago

Remove redundant mathrm from Average Interface Thickness

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[fb7fcec]1.. corfunc_help.rst
2
[da456fb]3.. _Correlation_Function_Analysis:
4
[32c5983]5Correlation Function Analysis
6=============================
[fb7fcec]7
8Description
9-----------
10
[ad476d1]11This currently performs correlation function analysis on SAXS/SANS data,
12but in the the future is also planned to generate model-independent volume
13fraction profiles from the SANS from adsorbed polymer/surfactant layers.
14The two types of analyses differ in the mathematical transform that is
15applied to the data (Fourier vs Hilbert). However, both functions are
16returned in *real space*.
[fb7fcec]17
[d78b5cb]18A correlation function may be interpreted in terms of an imaginary rod moving
[ad476d1]19through the structure of the material. Γ(x) is the probability that a rod of
20length x has equal electron/neutron scattering length density at either end.
21Hence a frequently occurring spacing within a structure will manifest itself
22as a peak in Γ(x). *SasView* will return both the one-dimensional ( Γ\ :sub:`1`\ (x) )
23and three-dimensional ( Γ\ :sub:`3`\ (x) ) correlation functions, the difference
24being that the former is only averaged in the plane of the scattering vector.
25
26A volume fraction profile :math:`\Phi`\ (z) describes how the density of polymer
27segments/surfactant molecules varies with distance, z, normal to an (assumed
28locally flat) interface. The form of :math:`\Phi`\ (z) can provide information
29about the arrangement of polymer/surfactant molecules at the interface. The width
30of the profile provides measures of the layer thickness, and the area under
31the profile is related to the amount of material that is adsorbed.
32
33Both analyses are performed in 3 stages:
34
35*  Extrapolation of the scattering curve to :math:`Q = 0` and toward
[fb7fcec]36   :math:`Q = \infty`
[1404cce]37*  Smoothed merging of the two extrapolations into the original data
38*  Fourier / Hilbert Transform of the smoothed data to give the correlation
[ad476d1]39   function or volume fraction profile, respectively
40*  (Optional) Interpretation of Γ\ :sub:`1`\ (x) assuming the sample conforms
41   to an ideal lamellar morphology
[fb7fcec]42
43.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
44
[ad476d1]45
[fb7fcec]46Extrapolation
47-------------
48
49To :math:`Q = 0`
[1404cce]50................
[fb7fcec]51
[ad476d1]52The data are extrapolated to q = 0 by fitting a Guinier function to the data
53points in the low-q range.
[1404cce]54
[fb7fcec]55The equation used is:
56
57.. math::
[ad476d1]58    I(q) = A e^{Bq^2}
[fb7fcec]59
[ad476d1]60Where the parameter :math:`B` is related to the effective radius-of-gyration of
61a spherical object having the same small-angle scattering in this region.
62       
63Note that as q tends to zero this function tends to a limiting value and is
64therefore less appropriate for use in systems where the form factor does not
65do likewise. However, because of the transform, the correlation functions are
66most affected by the Guinier back-extrapolation at *large* values of x where
67the impact on any extrapolated parameters will be least significant.
[fb7fcec]68
69To :math:`Q = \infty`
[1404cce]70.....................
[fb7fcec]71
[ad476d1]72The data are extrapolated towards q = :math:`\infty` by fitting a Porod model to
73the data points in the high-q range and then computing the extrapolation to 100
74times the maximum q value in the experimental dataset. This should be more than
75sufficient to ensure that on transformation any truncation artefacts introduced
76are at such small values of x that they can be safely ignored.
[fb7fcec]77
78The equation used is:
79
80.. math::
[ad476d1]81    I(q) = K q^{-4}e^{-q^2\sigma^2} + Bg
[fb7fcec]82
[ad476d1]83Where :math:`Bg` is the background, :math:`K` is the Porod constant, and :math:`\sigma` (which
84must be > 0) describes the width of the electron/neutron scattering length density
85profile at the interface between the crystalline and amorphous regions as shown below.
[fb7fcec]86
[6aad2e8]87.. figure:: fig1.png
[fb7fcec]88   :align: center
89
[d78b5cb]90
[fb7fcec]91Smoothing
[1404cce]92---------
[fb7fcec]93
[ad476d1]94The extrapolated data set consists of the Guinier back-extrapolation from q ~ 0
95up to the lowest q value in the original data, then the original scattering data,
96and then the Porod tail-fit beyond this. The joins between the original data and
97the Guinier/Porod extrapolations are smoothed using the algorithm below to try
98and avoid the formation of truncation ripples in the transformed data:
[fb7fcec]99
[2fe78566]100Functions :math:`f(x_i)` and :math:`g(x_i)` where :math:`x_i \in \left\{
101{x_1, x_2, ..., x_n} \right\}`, are smoothed over the range :math:`[a, b]`
102to produce :math:`y(x_i)`, by the following equations:
[fb7fcec]103
104.. math::
105    y(x_i) = h_ig(x_i) + (1-h_i)f(x_i)
106
107where:
108
109.. math::
110    h_i = \frac{1}{1 + \frac{(x_i-b)^2}{(x_i-a)^2}}
[41345d7e]111
[d78b5cb]112
[ad476d1]113Transformation
114--------------
[41345d7e]115
116Fourier
[1404cce]117.......
[41345d7e]118
[ad476d1]119If "Fourier" is selected for the transform type, *SasView* will perform a
[0390040]120discrete cosine transform on the extrapolated data in order to calculate the
[ad476d1]1211D correlation function as:
[1404cce]122
123.. math::
[ad476d1]124    \Gamma _{1}(x) = \frac{1}{Q^{*}} \int_{0}^{\infty }I(q) q^{2} cos(qx) dq
[1404cce]125
[ad476d1]126where Q\ :sup:`*` is the Scattering (also called Porod) Invariant.
[1404cce]127
128The following algorithm is applied:
[41345d7e]129
130.. math::
131    \Gamma(x_k) = 2 \sum_{n=0}^{N-1} x_n \cos{\left[ \frac{\pi}{N}
[4b4b746]132    \left(n + \frac{1}{2} \right) k \right] } \text{ for } k = 0, 1, \ldots,
[41345d7e]133    N-1, N
134
[ad476d1]135The 3D correlation function is calculated as:
[d78b5cb]136
137.. math::
[ad476d1]138    \Gamma _{3}(x) = \frac{1}{Q^{*}} \int_{0}^{\infty}I(q) q^{2}
139    \frac{sin(qx)}{qx} dq
140
141.. note:: It is always advisable to inspect Γ\ :sub:`1`\ (x) and Γ\ :sub:`3`\ (x)
142    for artefacts arising from the extrapolation and transformation processes:
143       
144        - do they tend to zero as x tends to :math:`\infty`?
145        - do they smoothly curve onto the ordinate at x = 0? (if not check the value
146          of :math:`\sigma` is sensible)
147        - are there ripples at x values corresponding to (2 :math:`pi` over) the two
148          q values at which the extrapolated and experimental data are merged?
149        - are there any artefacts at x values corresponding to 2 :math:`pi` / q\ :sub:`max` in
150          the experimental data?
151        - and lastly, do the significant features/peaks in the correlation functions
152          actually correspond to anticpated spacings in the sample?!!!
153
154Finally, the program calculates the interface distribution function (IDF) g\ :sub:`1`\ (x) as
155the discrete cosine transform of:
156
157.. math::
158    -q^{4} I(q)
159
160The IDF is proportional to the second derivative of Γ\ :sub:`1`\ (x).
[d78b5cb]161
[41345d7e]162Hilbert
[1404cce]163.......
[ad476d1]164       
[1404cce]165If "Hilbert" is selected for the transform type, the analysis will perform a
[2fe78566]166Hilbert transform on the extrapolated data in order to calculate the Volume
[41345d7e]167Fraction Profile.
168
[ad476d1]169.. note:: The Hilbert transform functionality is not yet implemented in SasView.
[1404cce]170
171
[41345d7e]172Interpretation
173--------------
[1404cce]174
175Correlation Function
176....................
177
[ad476d1]178Once the correlation functions have been calculated *SasView* can be asked to
179try and interpret Γ\ :sub:`1`\ (x) in terms of an ideal lamellar morphology
180as shown below.
[41345d7e]181
[6aad2e8]182.. figure:: fig2.png
[41345d7e]183   :align: center
184
[ad476d1]185The structural parameters extracted are:
[41345d7e]186
187*   Long Period :math:`= L_p`
188*   Average Hard Block Thickness :math:`= L_c`
189*   Average Core Thickness :math:`= D_0`
[501712f]190*   Average Interface Thickness :math:`= D_{tr}`
[8cc9048]191*   Polydispersity :math:`= \Gamma_{\mathrm{min}}/\Gamma_{\mathrm{max}}`
[41345d7e]192*   Local Crystallinity :math:`= L_c/L_p`
[0390040]193
[1404cce]194Volume Fraction Profile
195.......................
196
[ad476d1]197SasView does not provide any automatic interpretation of volume fraction profiles
198in the same way that it does for correlation functions. However, a number of
199structural parameters are obtainable by other means:
[1404cce]200
201*   Surface Coverage :math:`=\theta`
202*   Anchor Separation :math:`= D`
203*   Bound Fraction :math:`= <p>`
204*   Second Moment :math:`= \sigma`
[8cc9048]205*   Maximum Extent :math:`= \delta_{\mathrm{h}}`
[1404cce]206*   Adsorbed Amount :math:`= \Gamma`
207
208.. figure:: profile1.png
209   :align: center
[d78b5cb]210
[1404cce]211.. figure:: profile2.png
212   :align: center
[d78b5cb]213
[ad476d1]214The reader is directed to the references for information on these parameters.
[1404cce]215
216References
217----------
218
[ad476d1]219Correlation Function
220....................
221
[1404cce]222Strobl, G. R.; Schneider, M. *J. Polym. Sci.* (1980), 18, 1343-1359
223
224Koberstein, J.; Stein R. *J. Polym. Sci. Phys. Ed.* (1983), 21, 2181-2200
225
226Baltá Calleja, F. J.; Vonk, C. G. *X-ray Scattering of Synthetic Poylmers*, Elsevier. Amsterdam (1989), 247-251
227
228Baltá Calleja, F. J.; Vonk, C. G. *X-ray Scattering of Synthetic Poylmers*, Elsevier. Amsterdam (1989), 257-261
229
230Baltá Calleja, F. J.; Vonk, C. G. *X-ray Scattering of Synthetic Poylmers*, Elsevier. Amsterdam (1989), 260-270
231
[ad476d1]232Göschel, U.; Urban, G. *Polymer* (1995), 36, 3633-3639
233
234Stribeck, N. *X-Ray Scattering of Soft Matter*, Springer. Berlin (2007), 138-161
235
[1404cce]236:ref:`FDR` (PDF format)
237
[ad476d1]238Volume Fraction Profile
239.......................
240
241Washington, C.; King, S. M. *J. Phys. Chem.*, (1996), 100, 7603-7609
242
243Cosgrove, T.; King, S. M.; Griffiths, P. C. *Colloid-Polymer Interactions: From Fundamentals to Practice*, Wiley. New York (1999), 193-204
244
245King, S. M.; Griffiths, P. C.; Cosgrove, T. *Applications of Neutron Scattering to Soft Condensed Matter*, Gordon & Breach. Amsterdam (2000), 77-105
246
247King, S.; Griffiths, P.; Hone, J.; Cosgrove, T. *Macromol. Symp.* (2002), 190, 33-42
248
[0390040]249.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
250
[1404cce]251
[0390040]252Usage
253-----
[f56770ef]254Upon sending data for correlation function analysis, it will be plotted (minus
[d78b5cb]255the background value), along with a *red* bar indicating the *upper end of the
[ad476d1]256low-Q range* (used for Guinier back-extrapolation), and 2 *purple* bars indicating
257the range to be used for Porod forward-extrapolation. These bars may be moved by
258grabbing and dragging, or by entering appropriate values in the Q range input boxes.
[0390040]259
260.. figure:: tutorial1.png
261   :align: center
262
[ad476d1]263Once the Q ranges have been set, click the "Calculate Bg" button to determine the
264background level. Alternatively, enter your own value into the box. If the box turns
265yellow this indicates that background subtraction has created some negative intensities.
[0390040]266
[ad476d1]267Now click the "Extrapolate" button to extrapolate the data. The graph window will update
268to show the extrapolated data, and the values of the parameters used for the Guinier and
269Porod extrapolations will appear in the "Extrapolation Parameters" section of the SasView
270GUI.
[0390040]271
272.. figure:: tutorial2.png
273   :align: center
274
[1404cce]275Now select which type of transform you would like to perform, using the radio
[0390040]276buttons:
277
[ad476d1]278*   **Fourier**: to perform a Fourier Transform to calculate the correlation
279    functions
280*   **Hilbert**: to perform a Hilbert Transform to calculate the volume fraction
[1404cce]281    profile
282
[ad476d1]283and click the "Transform" button to perform the selected transform and plot
284the results.
[0390040]285
286 .. figure:: tutorial3.png
287    :align: center
288
[ad476d1]289If a Fourier Transform was performed, the "Compute Parameters" button can now be
290clicked to interpret the correlation function as described earlier. The parameters
291will appear in the "Output Parameters" section of the SasView GUI.
292
293 .. figure:: tutorial4.png
294    :align: center
295
[d78b5cb]296
[1404cce]297.. note::
[ad476d1]298    This help document was last changed by Steve King, 26Sep2017
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