Changeset ad476d1 in sasview


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Timestamp:
Sep 26, 2017 5:33:32 AM (8 weeks ago)
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smk78
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master, ESS_GUI, ESS_GUI_Pr, ESS_GUI_corf, ESS_GUI_py2_OLD, SVCC-1, docs, ticket-1032-sldi, ticket-776-orientation, ticket-873, ticket1007, true, unittest-saveload
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Updated corfunc_help.rst

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

    rf80b416e rad476d1  
    99----------- 
    1010 
    11 This performs a correlation function analysis of one-dimensional 
    12 SAXS/SANS data, or generates a model-independent volume fraction 
    13 profile from the SANS from an adsorbed polymer/surfactant layer. 
     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*. 
    1417 
    1518A correlation function may be interpreted in terms of an imaginary rod moving 
    16 through the structure of the material. Γ\ :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 
     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  
    2836   :math:`Q = \infty` 
    2937*  Smoothed merging of the two extrapolations into the original data 
    3038*  Fourier / Hilbert Transform of the smoothed data to give the correlation 
    31    function / volume fraction profile, respectively 
    32 *  (Optional) Interpretation of the 1D correlation function based on an ideal 
    33    lamellar morphology 
     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 
    3442 
    3543.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
     44 
    3645 
    3746Extrapolation 
     
    4150................ 
    4251 
    43 The data are extrapolated to Q = 0 by fitting a Guinier model to the data 
    44 points in the low-Q range. 
     52The data are extrapolated to q = 0 by fitting a Guinier function to the data 
     53points in the low-q range. 
    4554 
    4655The equation used is: 
    4756 
    4857.. 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 back-extrapolation 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 
     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. 
    5768 
    5869To :math:`Q = \infty` 
    5970..................... 
    6071 
    61 The data are extrapolated to Q = :math:`\infty` by fitting a Porod model to 
    62 the data points in the high-Q range. 
     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. 
    6377 
    6478The equation used is: 
    6579 
    6680.. math:: 
    67     I(Q) = K Q^{-4}e^{-Q^2\sigma^2} + Bg 
    68  
    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 
     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. 
    7286 
    7387.. figure:: fig1.png 
     
    7892--------- 
    7993 
    80 The extrapolated data set consists of the Guinier back-extrapolation from Q~0 
    81 up 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. 
     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: 
    8299 
    83100Functions :math:`f(x_i)` and :math:`g(x_i)` where :math:`x_i \in \left\{ 
     
    94111 
    95112 
    96 Transform 
    97 --------- 
     113Transformation 
     114-------------- 
    98115 
    99116Fourier 
    100117....... 
    101118 
    102 If "Fourier" is selected for the transform type, the analysis will perform a 
     119If "Fourier" is selected for the transform type, *SasView* will perform a 
    103120discrete cosine transform on the extrapolated data in order to calculate the 
    104 1D correlation function: 
    105  
    106 .. math:: 
    107     \Gamma _{1D}(R) = \frac{1}{Q^{*}} \int_{0}^{\infty }I(q) q^{2} cos(qR) dq 
    108  
    109 where Q\ :sup:`*` is the Scattering Invariant. 
     1211D correlation function as: 
     122 
     123.. math:: 
     124    \Gamma _{1}(x) = \frac{1}{Q^{*}} \int_{0}^{\infty }I(q) q^{2} cos(qx) dq 
     125 
     126where Q\ :sup:`*` is the Scattering (also called Porod) Invariant. 
    110127 
    111128The following algorithm is applied: 
     
    116133    N-1, N 
    117134 
    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 
     135The 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 
     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). 
    123161 
    124162Hilbert 
    125163....... 
    126  
     164         
    127165If "Hilbert" is selected for the transform type, the analysis will perform a 
    128166Hilbert transform on the extrapolated data in order to calculate the Volume 
    129167Fraction Profile. 
    130168 
    131 .. note:: This functionality is not yet implemented in SasView. 
     169.. note:: The Hilbert transform functionality is not yet implemented in SasView. 
    132170 
    133171 
     
    138176.................... 
    139177 
    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 de-smearing is 
    145 performed. 
     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. 
    146181 
    147182.. figure:: fig2.png 
    148183   :align: center 
    149184 
    150 The structural parameters obtained are: 
     185The structural parameters extracted are: 
    151186 
    152187*   Long Period :math:`= L_p` 
     
    160195....................... 
    161196 
    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: 
     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: 
    163200 
    164201*   Surface Coverage :math:`=\theta` 
     
    175212   :align: center 
    176213 
     214The reader is directed to the references for information on these parameters. 
    177215 
    178216References 
    179217---------- 
    180218 
     219Correlation Function 
     220.................... 
     221 
    181222Strobl, G. R.; Schneider, M. *J. Polym. Sci.* (1980), 18, 1343-1359 
    182223 
     
    189230Baltá Calleja, F. J.; Vonk, C. G. *X-ray Scattering of Synthetic Poylmers*, Elsevier. Amsterdam (1989), 260-270 
    190231 
     232Göschel, U.; Urban, G. *Polymer* (1995), 36, 3633-3639 
     233 
     234Stribeck, N. *X-Ray Scattering of Soft Matter*, Springer. Berlin (2007), 138-161 
     235 
    191236:ref:`FDR` (PDF format) 
     237 
     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 
    192248 
    193249.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ 
     
    198254Upon sending data for correlation function analysis, it will be plotted (minus 
    199255the background value), along with a *red* bar indicating the *upper end of the 
    200 low-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 
    201 dragging, or by entering appropriate values in the Q range input boxes. 
     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. 
    202259 
    203260.. figure:: tutorial1.png 
    204261   :align: center 
    205262 
    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. 
     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. 
     266 
     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. 
    209271 
    210272.. figure:: tutorial2.png 
     
    214276buttons: 
    215277 
    216 *   **Fourier** Perform a Fourier Transform to calculate the correlation 
    217     function 
    218 *   **Hilbert** Perform a Hilbert Transform to calculate the volume fraction 
     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 
    219281    profile 
    220282 
    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. 
     283and click the "Transform" button to perform the selected transform and plot 
     284the results. 
    225285 
    226286 .. figure:: tutorial3.png 
    227287    :align: center 
    228288 
     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 
    229296 
    230297.. note:: 
    231     This help document was last changed by Steve King, 08Oct2016 
     298    This help document was last changed by Steve King, 26Sep2017 
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