Changeset ad476d1 in sasview
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 Sep 26, 2017 7:33:32 AM (15 months ago)
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src/sas/sasgui/perspectives/corfunc/media/corfunc_help.rst
rf80b416e rad476d1 9 9  10 10 11 This performs a correlation function analysis of onedimensional 12 SAXS/SANS data, or generates a modelindependent volume fraction 13 profile from the SANS from an adsorbed polymer/surfactant layer. 11 This currently performs correlation function analysis on SAXS/SANS data, 12 but in the the future is also planned to generate modelindependent volume 13 fraction profiles from the SANS from adsorbed polymer/surfactant layers. 14 The two types of analyses differ in the mathematical transform that is 15 applied to the data (Fourier vs Hilbert). However, both functions are 16 returned in *real space*. 14 17 15 18 A correlation function may be interpreted in terms of an imaginary rod moving 16 through the structure of the material. Î\ :sub:`1D`\ (R) is the probability that 17 a rod of length R moving through the material has equal electron/neutron scattering 18 length density at either end. Hence a frequently occurring spacing within a structure 19 manifests itself as a peak. 20 21 A volume fraction profile :math:`\Phi`\ (z) describes how the density of polymer segments/surfactant molecules varies with distance from an (assumed locally flat) interface. 22 23 Both functions are returned in *real space*. 24 25 The analysis is performed in 3 stages: 26 27 * Extrapolation of the scattering curve to :math:`Q = 0` and 19 through the structure of the material. Î(x) is the probability that a rod of 20 length x has equal electron/neutron scattering length density at either end. 21 Hence a frequently occurring spacing within a structure will manifest itself 22 as a peak in Î(x). *SasView* will return both the onedimensional ( Î\ :sub:`1`\ (x) ) 23 and threedimensional ( Î\ :sub:`3`\ (x) ) correlation functions, the difference 24 being that the former is only averaged in the plane of the scattering vector. 25 26 A volume fraction profile :math:`\Phi`\ (z) describes how the density of polymer 27 segments/surfactant molecules varies with distance, z, normal to an (assumed 28 locally flat) interface. The form of :math:`\Phi`\ (z) can provide information 29 about the arrangement of polymer/surfactant molecules at the interface. The width 30 of the profile provides measures of the layer thickness, and the area under 31 the profile is related to the amount of material that is adsorbed. 32 33 Both analyses are performed in 3 stages: 34 35 * Extrapolation of the scattering curve to :math:`Q = 0` and toward 28 36 :math:`Q = \infty` 29 37 * Smoothed merging of the two extrapolations into the original data 30 38 * Fourier / Hilbert Transform of the smoothed data to give the correlation 31 function /volume fraction profile, respectively32 * (Optional) Interpretation of the 1D correlation function based on an ideal33 lamellar morphology39 function or volume fraction profile, respectively 40 * (Optional) Interpretation of Î\ :sub:`1`\ (x) assuming the sample conforms 41 to an ideal lamellar morphology 34 42 35 43 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 44 36 45 37 46 Extrapolation … … 41 50 ................ 42 51 43 The data are extrapolated to Q = 0 by fitting a Guinier modelto the data44 points in the low Qrange.52 The data are extrapolated to q = 0 by fitting a Guinier function to the data 53 points in the lowq range. 45 54 46 55 The equation used is: 47 56 48 57 .. math:: 49 I(Q) = Ae^{Bq^2} 50 51 The Guinier model assumes that the small angle scattering arises from particles 52 and that parameter :math:`B` is related to the radius of gyration of those 53 particles. This has dubious applicability to polymer systems. However, the 54 correlation function is affected by the Guinier backextrapolation to the 55 greatest extent at large values of R and so only has a 56 small effect on the final analysis. 58 I(q) = A e^{Bq^2} 59 60 Where the parameter :math:`B` is related to the effective radiusofgyration of 61 a spherical object having the same smallangle scattering in this region. 62 63 Note that as q tends to zero this function tends to a limiting value and is 64 therefore less appropriate for use in systems where the form factor does not 65 do likewise. However, because of the transform, the correlation functions are 66 most affected by the Guinier backextrapolation at *large* values of x where 67 the impact on any extrapolated parameters will be least significant. 57 68 58 69 To :math:`Q = \infty` 59 70 ..................... 60 71 61 The data are extrapolated to Q = :math:`\infty` by fitting a Porod model to 62 the data points in the highQ range. 72 The data are extrapolated towards q = :math:`\infty` by fitting a Porod model to 73 the data points in the highq range and then computing the extrapolation to 100 74 times the maximum q value in the experimental dataset. This should be more than 75 sufficient to ensure that on transformation any truncation artefacts introduced 76 are at such small values of x that they can be safely ignored. 63 77 64 78 The equation used is: 65 79 66 80 .. math:: 67 I( Q) = K Q^{4}e^{Q^2\sigma^2} + Bg68 69 Where :math:`Bg` is the background, :math:`K` is the Porod 70 constant, and :math:`\sigma` (which must be > 0) describes the width of the electron or neutron scattering length density profile at the interface between the crystalline and amorphous 71 regions as shown below.81 I(q) = K q^{4}e^{q^2\sigma^2} + Bg 82 83 Where :math:`Bg` is the background, :math:`K` is the Porod constant, and :math:`\sigma` (which 84 must be > 0) describes the width of the electron/neutron scattering length density 85 profile at the interface between the crystalline and amorphous regions as shown below. 72 86 73 87 .. figure:: fig1.png … … 78 92  79 93 80 The extrapolated data set consists of the Guinier backextrapolation from Q~0 81 up to the lowest Q value in the original data, then the original scattering data, and the Porod tailfit 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. 94 The extrapolated data set consists of the Guinier backextrapolation from q ~ 0 95 up to the lowest q value in the original data, then the original scattering data, 96 and then the Porod tailfit beyond this. The joins between the original data and 97 the Guinier/Porod extrapolations are smoothed using the algorithm below to try 98 and avoid the formation of truncation ripples in the transformed data: 82 99 83 100 Functions :math:`f(x_i)` and :math:`g(x_i)` where :math:`x_i \in \left\{ … … 94 111 95 112 96 Transform 97  113 Transformation 114  98 115 99 116 Fourier 100 117 ....... 101 118 102 If "Fourier" is selected for the transform type, the analysiswill perform a119 If "Fourier" is selected for the transform type, *SasView* will perform a 103 120 discrete cosine transform on the extrapolated data in order to calculate the 104 1D correlation function :105 106 .. math:: 107 \Gamma _{1 D}(R) = \frac{1}{Q^{*}} \int_{0}^{\infty }I(q) q^{2} cos(qR) dq108 109 where Q\ :sup:`*` is the Scattering Invariant.121 1D correlation function as: 122 123 .. math:: 124 \Gamma _{1}(x) = \frac{1}{Q^{*}} \int_{0}^{\infty }I(q) q^{2} cos(qx) dq 125 126 where Q\ :sup:`*` is the Scattering (also called Porod) Invariant. 110 127 111 128 The following algorithm is applied: … … 116 133 N1, N 117 134 118 The 3D correlation function is also calculated: 119 120 .. math:: 121 \Gamma _{3D}(R) = \frac{1}{Q^{*}} \int_{0}^{\infty}I(q) q^{2} 122 \frac{sin(qR)}{qR} dq 135 The 3D correlation function is calculated as: 136 137 .. math:: 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 Î\ :sub:`1`\ (x) and Î\ :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?!!! 153 154 Finally, the program calculates the interface distribution function (IDF) g\ :sub:`1`\ (x) as 155 the discrete cosine transform of: 156 157 .. math:: 158 q^{4} I(q) 159 160 The IDF is proportional to the second derivative of Î\ :sub:`1`\ (x). 123 161 124 162 Hilbert 125 163 ....... 126 164 127 165 If "Hilbert" is selected for the transform type, the analysis will perform a 128 166 Hilbert transform on the extrapolated data in order to calculate the Volume 129 167 Fraction Profile. 130 168 131 .. note:: Th isfunctionality is not yet implemented in SasView.169 .. note:: The Hilbert transform functionality is not yet implemented in SasView. 132 170 133 171 … … 138 176 .................... 139 177 140 Once the correlation function has been calculated it may be interpreted by clicking the "Compute Parameters" button. 141 142 The correlation function is interpreted in terms of an ideal lamellar 143 morphology, and structural parameters are obtained from it as shown below. 144 It should be noted that a small beam size is assumed; ie, no desmearing is 145 performed. 178 Once the correlation functions have been calculated *SasView* can be asked to 179 try and interpret Î\ :sub:`1`\ (x) in terms of an ideal lamellar morphology 180 as shown below. 146 181 147 182 .. figure:: fig2.png 148 183 :align: center 149 184 150 The structural parameters obtained are:185 The structural parameters extracted are: 151 186 152 187 * Long Period :math:`= L_p` … … 160 195 ....................... 161 196 162 SasView does not provide any automatic interpretation of volume fraction profiles in the same way that it does for correlation functions. However, a number of structural parameters are obtainable by other means: 197 SasView does not provide any automatic interpretation of volume fraction profiles 198 in the same way that it does for correlation functions. However, a number of 199 structural parameters are obtainable by other means: 163 200 164 201 * Surface Coverage :math:`=\theta` … … 175 212 :align: center 176 213 214 The reader is directed to the references for information on these parameters. 177 215 178 216 References 179 217  180 218 219 Correlation Function 220 .................... 221 181 222 Strobl, G. R.; Schneider, M. *J. Polym. Sci.* (1980), 18, 13431359 182 223 … … 189 230 BaltÃ¡ Calleja, F. J.; Vonk, C. G. *Xray Scattering of Synthetic Poylmers*, Elsevier. Amsterdam (1989), 260270 190 231 232 GÃ¶schel, U.; Urban, G. *Polymer* (1995), 36, 36333639 233 234 Stribeck, N. *XRay Scattering of Soft Matter*, Springer. Berlin (2007), 138161 235 191 236 :ref:`FDR` (PDF format) 237 238 Volume Fraction Profile 239 ....................... 240 241 Washington, C.; King, S. M. *J. Phys. Chem.*, (1996), 100, 76037609 242 243 Cosgrove, T.; King, S. M.; Griffiths, P. C. *ColloidPolymer Interactions: From Fundamentals to Practice*, Wiley. New York (1999), 193204 244 245 King, S. M.; Griffiths, P. C.; Cosgrove, T. *Applications of Neutron Scattering to Soft Condensed Matter*, Gordon & Breach. Amsterdam (2000), 77105 246 247 King, S.; Griffiths, P.; Hone, J.; Cosgrove, T. *Macromol. Symp.* (2002), 190, 3342 192 248 193 249 .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ … … 198 254 Upon sending data for correlation function analysis, it will be plotted (minus 199 255 the background value), along with a *red* bar indicating the *upper end of the 200 lowQ range* (used for backextrapolation), and 2 *purple* bars indicating the range to be used for forwardextrapolation. These bars may be moved my clicking and 201 dragging, or by entering appropriate values in the Q range input boxes. 256 lowQ range* (used for Guinier backextrapolation), and 2 *purple* bars indicating 257 the range to be used for Porod forwardextrapolation. These bars may be moved by 258 grabbing and dragging, or by entering appropriate values in the Q range input boxes. 202 259 203 260 .. figure:: tutorial1.png 204 261 :align: center 205 262 206 Once the Q ranges have been set, click the "Calculate" button to determine the background level. Alternatively, enter your own value into the field. If the box turns yellow this indicates that background subtraction has resulted in some negative intensities. 207 208 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 of the window. 263 Once the Q ranges have been set, click the "Calculate Bg" button to determine the 264 background level. Alternatively, enter your own value into the box. If the box turns 265 yellow this indicates that background subtraction has created some negative intensities. 266 267 Now click the "Extrapolate" button to extrapolate the data. The graph window will update 268 to show the extrapolated data, and the values of the parameters used for the Guinier and 269 Porod extrapolations will appear in the "Extrapolation Parameters" section of the SasView 270 GUI. 209 271 210 272 .. figure:: tutorial2.png … … 214 276 buttons: 215 277 216 * **Fourier** Perform a Fourier Transform to calculate the correlation217 function 218 * **Hilbert** Perform a Hilbert Transform to calculate the volume fraction278 * **Fourier**: to perform a Fourier Transform to calculate the correlation 279 functions 280 * **Hilbert**: to perform a Hilbert Transform to calculate the volume fraction 219 281 profile 220 282 221 Click the "Transform" button to perform the selected transform and plot 222 the result in a new graph window. 223 224 If a Fourier Transform was performed, the "Compute Parameters" button can now be clicked to interpret the correlation function as described earlier. 283 and click the "Transform" button to perform the selected transform and plot 284 the results. 225 285 226 286 .. figure:: tutorial3.png 227 287 :align: center 228 288 289 If a Fourier Transform was performed, the "Compute Parameters" button can now be 290 clicked to interpret the correlation function as described earlier. The parameters 291 will appear in the "Output Parameters" section of the SasView GUI. 292 293 .. figure:: tutorial4.png 294 :align: center 295 229 296 230 297 .. note:: 231 This help document was last changed by Steve King, 08Oct2016298 This help document was last changed by Steve King, 26Sep2017
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