Changes in / [4b001f3:48154abb] in sasview


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  • src/sas/sascalc/corfunc/corfunc_calculator.py

    r92eee84 ra859f99  
    124124 
    125125        params, s2 = self._fit_data(q, iq) 
    126         # Extrapolate to 100*Qmax in experimental data 
    127126        qs = np.arange(0, q[-1]*100, (q[1]-q[0])) 
    128127        iqs = s2(qs) 

  • src/sas/sasgui/perspectives/corfunc/media/corfunc_help.rst

    rad476d1 rd78b5cb  
    99----------- 
    1010 
    11 This currently performs correlation function analysis on SAXS/SANS data,  
    12 but in the the future is also planned to generate model-independent 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*. 
     11This performs a correlation function analysis of one-dimensional 
     12SAXS/SANS data, or generates a model-independent volume fraction 
     13profile from the SANS from an adsorbed polymer/surfactant layer. 
    1714 
    1815A correlation function may be interpreted in terms of an imaginary rod moving 
    19 through the structure of the material. Γ(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 Γ(x). *SasView* will return both the one-dimensional ( Γ\ :sub:`1`\ (x) )  
    23 and three-dimensional ( Γ\ :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  
     16through the structure of the material. Γ\ :sub:`1D`\ (R) is the probability that 
     17a rod of length R moving through the material has equal electron/neutron scattering 
     18length density at either end. Hence a frequently occurring spacing within a structure 
     19manifests itself as a peak. 
     20 
     21A 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 
     23Both functions are returned in *real space*. 
     24 
     25The analysis is performed in 3 stages: 
     26 
     27*  Extrapolation of the scattering curve to :math:`Q = 0` and 
    3628   :math:`Q = \infty` 
    3729*  Smoothed merging of the two extrapolations into the original data 
    3830*  Fourier / Hilbert Transform of the smoothed data to give the correlation 
    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 
     31   function / volume fraction profile, respectively 
     32*  (Optional) Interpretation of the 1D correlation function based on an ideal 
     33   lamellar morphology 
    4234 
    4335.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
    44  
    4536 
    4637Extrapolation 
    … …  
    5041................ 
    5142 
    52 The data are extrapolated to q = 0 by fitting a Guinier function to the data 
    53 points in the low-q range. 
     43The data are extrapolated to Q = 0 by fitting a Guinier model to the data 
     44points in the low-Q range. 
    5445 
    5546The equation used is: 
    5647 
    5748.. math:: 
    58     I(q) = A e^{Bq^2} 
    59  
    60 Where the parameter :math:`B` is related to the effective radius-of-gyration of  
    61 a spherical object having the same small-angle 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 back-extrapolation at *large* values of x where  
    67 the impact on any extrapolated parameters will be least significant. 
     49    I(Q) = Ae^{Bq^2} 
     50 
     51The Guinier model assumes that the small angle scattering arises from particles 
     52and that parameter :math:`B` is related to the radius of gyration of those 
     53particles. This has dubious applicability to polymer systems. However, the 
     54correlation function is affected by the Guinier back-extrapolation to the 
     55greatest extent at large values of R and so only has a 
     56small effect on the final analysis. 
    6857 
    6958To :math:`Q = \infty` 
    7059..................... 
    7160 
    72 The data are extrapolated towards q = :math:`\infty` by fitting a Porod model to 
    73 the data points in the high-q 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. 
     61The data are extrapolated to Q = :math:`\infty` by fitting a Porod model to 
     62the data points in the high-Q range. 
    7763 
    7864The equation used is: 
    7965 
    8066.. math:: 
    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. 
     67    I(Q) = K Q^{-4}e^{-Q^2\sigma^2} + Bg 
     68 
     69Where :math:`Bg` is the background, :math:`K` is the Porod 
     70constant, 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 
     71regions as shown below. 
    8672 
    8773.. figure:: fig1.png 
    … …  
    9278--------- 
    9379 
    94 The extrapolated data set consists of the Guinier back-extrapolation from q ~ 0 
    95 up to the lowest q value in the original data, then the original scattering data,  
    96 and then the Porod tail-fit 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: 
     80The extrapolated data set consists of the Guinier back-extrapolation from Q~0 
     81up to the lowest Q value in the original data, then the original scattering data, 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. 
    9982 
    10083Functions :math:`f(x_i)` and :math:`g(x_i)` where :math:`x_i \in \left\{ 
    … …  
    11194 
    11295 
    113 Transformation 
    114 -------------- 
     96Transform 
     97--------- 
    11598 
    11699Fourier 
    117100....... 
    118101 
    119 If "Fourier" is selected for the transform type, *SasView* will perform a 
     102If "Fourier" is selected for the transform type, the analysis will perform a 
    120103discrete cosine transform on the extrapolated data in order to calculate the 
    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. 
     1041D correlation function: 
     105 
     106.. math:: 
     107    \Gamma _{1D}(R) = \frac{1}{Q^{*}} \int_{0}^{\infty }I(q) q^{2} cos(qR) dq 
     108 
     109where Q\ :sup:`*` is the Scattering Invariant. 
    127110 
    128111The following algorithm is applied: 
    … …  
    133116    N-1, N 
    134117 
    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 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  
    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 Γ\ :sub:`1`\ (x). 
     118The 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 
    161123 
    162124Hilbert 
    163125....... 
    164          
     126 
    165127If "Hilbert" is selected for the transform type, the analysis will perform a 
    166128Hilbert transform on the extrapolated data in order to calculate the Volume 
    167129Fraction Profile. 
    168130 
    169 .. note:: The Hilbert transform functionality is not yet implemented in SasView. 
     131.. note:: This functionality is not yet implemented in SasView. 
    170132 
    171133 
    … …  
    176138.................... 
    177139 
    178 Once the correlation functions have been calculated *SasView* can be asked to  
    179 try and interpret Γ\ :sub:`1`\ (x) in terms of an ideal lamellar morphology  
    180 as shown below. 
     140Once the correlation function has been calculated it may be interpreted by clicking the "Compute Parameters" button. 
     141 
     142The correlation function is interpreted in terms of an ideal lamellar 
     143morphology, and structural parameters are obtained from it as shown below. 
     144It should be noted that a small beam size is assumed; ie, no de-smearing is 
     145performed. 
    181146 
    182147.. figure:: fig2.png 
    183148   :align: center 
    184149 
    185 The structural parameters extracted are: 
     150The structural parameters obtained are: 
    186151 
    187152*   Long Period :math:`= L_p` 
    … …  
    195160....................... 
    196161 
    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: 
     162SasView 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: 
    200163 
    201164*   Surface Coverage :math:`=\theta` 
    … …  
    212175   :align: center 
    213176 
    214 The reader is directed to the references for information on these parameters. 
    215177 
    216178References 
    217179---------- 
    218180 
    219 Correlation Function 
    220 .................... 
    221  
    222181Strobl, G. R.; Schneider, M. *J. Polym. Sci.* (1980), 18, 1343-1359 
    223182 
    … …  
    230189Baltá Calleja, F. J.; Vonk, C. G. *X-ray Scattering of Synthetic Poylmers*, Elsevier. Amsterdam (1989), 260-270 
    231190 
    232 Göschel, U.; Urban, G. *Polymer* (1995), 36, 3633-3639 
    233  
    234 Stribeck, N. *X-Ray Scattering of Soft Matter*, Springer. Berlin (2007), 138-161 
    235  
    236191:ref:`FDR` (PDF format) 
    237  
    238 Volume Fraction Profile 
    239 ....................... 
    240  
    241 Washington, C.; King, S. M. *J. Phys. Chem.*, (1996), 100, 7603-7609 
    242  
    243 Cosgrove, T.; King, S. M.; Griffiths, P. C. *Colloid-Polymer Interactions: From Fundamentals to Practice*, Wiley. New York (1999), 193-204 
    244  
    245 King, S. M.; Griffiths, P. C.; Cosgrove, T. *Applications of Neutron Scattering to Soft Condensed Matter*, Gordon & Breach. Amsterdam (2000), 77-105 
    246  
    247 King, S.; Griffiths, P.; Hone, J.; Cosgrove, T. *Macromol. Symp.* (2002), 190, 33-42 
    248192 
    249193.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
    … …  
    254198Upon sending data for correlation function analysis, it will be plotted (minus 
    255199the background value), along with a *red* bar indicating the *upper end of the 
    256 low-Q range* (used for Guinier back-extrapolation), and 2 *purple* bars indicating  
    257 the range to be used for Porod forward-extrapolation. These bars may be moved by  
    258 grabbing and dragging, or by entering appropriate values in the Q range input boxes. 
     200low-Q range* (used for back-extrapolation), and 2 *purple* bars indicating the range to be used for forward-extrapolation. These bars may be moved my clicking and 
     201dragging, or by entering appropriate values in the Q range input boxes. 
    259202 
    260203.. figure:: tutorial1.png 
    261204   :align: center 
    262205 
    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. 
     206Once 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 
     208Click 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. 
    271209 
    272210.. figure:: tutorial2.png 
    … …  
    276214buttons: 
    277215 
    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 
     216*   **Fourier** Perform a Fourier Transform to calculate the correlation 
     217    function 
     218*   **Hilbert** Perform a Hilbert Transform to calculate the volume fraction 
    281219    profile 
    282220 
    283 and click the "Transform" button to perform the selected transform and plot 
    284 the results. 
     221Click the "Transform" button to perform the selected transform and plot 
     222the result in a new graph window. 
     223 
     224If a Fourier Transform was performed, the "Compute Parameters" button can now be clicked to interpret the correlation function as described earlier. 
    285225 
    286226 .. figure:: tutorial3.png 
    287227    :align: center 
    288228 
    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  
    296229 
    297230.. note:: 
    298     This help document was last changed by Steve King, 26Sep2017 
     231    This help document was last changed by Steve King, 08Oct2016 
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