[fb7fcec] | 1 | .. corfunc_help.rst |
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| 2 | |
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[da456fb] | 3 | .. _Correlation_Function_Analysis: |
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| 4 | |
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[32c5983] | 5 | Correlation Function Analysis |
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| 6 | ============================= |
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[fb7fcec] | 7 | |
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| 8 | Description |
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| 9 | ----------- |
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| 10 | |
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[32c5983] | 11 | This performs a correlation function analysis of one-dimensional |
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[fb7fcec] | 12 | SANS data, or generates a model-independent volume fraction profile from a |
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| 13 | one-dimensional SANS pattern of an adsorbed layer. |
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| 14 | |
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| 15 | The correlation function analysis is performed in 3 stages: |
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| 16 | |
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| 17 | * Extrapolation of the scattering curve to :math:`Q = 0` and |
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| 18 | :math:`Q = \infty` |
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[41345d7e] | 19 | * Fourier/Hilbert Transform of the extrapolated data to give the correlation |
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| 20 | function/volume fraction profile |
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[fb7fcec] | 21 | * Interpretation of the 1D correlation function based on an ideal lamellar |
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| 22 | morphology |
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| 23 | |
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| 24 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 25 | |
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| 26 | Extrapolation |
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| 27 | ------------- |
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| 28 | |
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| 29 | To :math:`Q = 0` |
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| 30 | ^^^^^^^^^^^^^^^^ |
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| 31 | |
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| 32 | The data are extrapolated to Q = 0 by fitting a Guinier model to the data |
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| 33 | points in the lower Q range. |
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| 34 | The equation used is: |
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| 35 | |
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| 36 | .. math:: |
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[ff11b21] | 37 | I(Q) = e^{A+Bq^2} |
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[fb7fcec] | 38 | |
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| 39 | The Guinier model assumes that the small angle scattering arises from particles |
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| 40 | and that parameter :math:`B` is related to the radius of gyration of those |
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| 41 | particles. This has dubious applicability to polymer systems. However, the |
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| 42 | correlation function is affected by the Guinier back-extrapolation to the |
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| 43 | greatest extent at large values of R and so the back-extrapolation only has a |
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| 44 | small effect on the analysis. |
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| 45 | |
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| 46 | To :math:`Q = \infty` |
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| 47 | ^^^^^^^^^^^^^^^^^^^^^ |
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| 48 | |
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| 49 | The data are extrapolated to Q = :math:`\infty` by fitting a Porod model to |
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| 50 | the data points in the upper Q range. |
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| 51 | |
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| 52 | The equation used is: |
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| 53 | |
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| 54 | .. math:: |
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[c97d76a] | 55 | I(Q) = Bg + KQ^{-4}e^{-Q^2\sigma^2} |
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[fb7fcec] | 56 | |
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[c97d76a] | 57 | Where :math:`Bg` is the Bonart thermal background, :math:`K` is the Porod |
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| 58 | constant, and :math:`\sigma > 0` describes the electron (or neutron scattering |
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[fb7fcec] | 59 | length) density profile at the interface between crystalline and amorphous |
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| 60 | regions (see figure 1). |
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| 61 | |
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| 62 | .. figure:: fig1.gif |
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| 63 | :align: center |
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| 64 | |
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[41345d7e] | 65 | **Figure 1** The value of :math:`\sigma` is a measure of the electron |
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[fb7fcec] | 66 | density profile at the interface between crystalline and amorphous regions. |
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| 67 | |
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| 68 | Smoothing |
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| 69 | ^^^^^^^^^ |
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| 70 | |
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| 71 | The extrapolated data set consists of the Guinier back-extrapolation up to the |
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| 72 | highest Q value of the lower Q range, the original scattering data up to the |
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| 73 | highest value in the upper Q range, and the Porod tail-fit beyond this. The |
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| 74 | joins between the original data and the Guinier/Porod fits are smoothed using |
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[2fe78566] | 75 | the algorithm below, to avoid the formation of ripples in the transformed data. |
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[fb7fcec] | 76 | |
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[2fe78566] | 77 | Functions :math:`f(x_i)` and :math:`g(x_i)` where :math:`x_i \in \left\{ |
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| 78 | {x_1, x_2, ..., x_n} \right\}`, are smoothed over the range :math:`[a, b]` |
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| 79 | to produce :math:`y(x_i)`, by the following equations: |
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[fb7fcec] | 80 | |
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| 81 | .. math:: |
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| 82 | y(x_i) = h_ig(x_i) + (1-h_i)f(x_i) |
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| 83 | |
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| 84 | where: |
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| 85 | |
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| 86 | .. math:: |
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| 87 | h_i = \frac{1}{1 + \frac{(x_i-b)^2}{(x_i-a)^2}} |
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[41345d7e] | 88 | |
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| 89 | Transform |
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| 90 | --------- |
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| 91 | |
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| 92 | Fourier |
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| 93 | ^^^^^^^ |
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| 94 | |
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[32c5983] | 95 | If Fourier is selected for the transform type, the analysis will perform a |
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[0390040] | 96 | discrete cosine transform on the extrapolated data in order to calculate the |
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[2fe78566] | 97 | correlation function. The following algorithm is applied: |
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[41345d7e] | 98 | |
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| 99 | .. math:: |
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| 100 | \Gamma(x_k) = 2 \sum_{n=0}^{N-1} x_n \cos{\left[ \frac{\pi}{N} |
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| 101 | \left(n + \frac{1}{2} \right) k \right] } \text{ for } k = 0, 1, \ldots, |
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| 102 | N-1, N |
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| 103 | |
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| 104 | Hilbert |
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| 105 | ^^^^^^^ |
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[32c5983] | 106 | If Hilbert is selected for the transform type, the analysis will perform a |
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[2fe78566] | 107 | Hilbert transform on the extrapolated data in order to calculate the Volume |
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[41345d7e] | 108 | Fraction Profile. |
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| 109 | |
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| 110 | Interpretation |
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| 111 | -------------- |
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| 112 | Once the correlation function has been calculated by transforming the |
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| 113 | extrapolated data, it may be interpreted by clicking the "Compute Parameters" |
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| 114 | button. The correlation function is interpreted in terms of an ideal lamellar |
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| 115 | morphology, and structural parameters are obtained as shown in Figure 2 below. |
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| 116 | It should be noted that a small beam size is assumed; no de-smearing is |
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| 117 | performed. |
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| 118 | |
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| 119 | .. figure:: fig2.gif |
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| 120 | :align: center |
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| 121 | |
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| 122 | **Figure 2** Interpretation of the correlation function. |
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| 123 | |
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| 124 | The structural parameters obtained are: |
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| 125 | |
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| 126 | * Long Period :math:`= L_p` |
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| 127 | * Average Hard Block Thickness :math:`= L_c` |
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| 128 | * Average Core Thickness :math:`= D_0` |
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| 129 | * Average Interface Thickness :math:`\text{} = D_{tr}` |
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| 130 | * Polydispersity :math:`= \Gamma_{\text{min}}/\Gamma_{\text{max}}` |
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| 131 | * Local Crystallinity :math:`= L_c/L_p` |
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[0390040] | 132 | |
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| 133 | .. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ |
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| 134 | |
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| 135 | Usage |
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| 136 | ----- |
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[f56770ef] | 137 | Upon sending data for correlation function analysis, it will be plotted (minus |
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| 138 | the background value), along with a red bar indicating the lower Q range (used |
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| 139 | for back-extrapolation), and 2 purple bars indicating the upper Q range (used |
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| 140 | for forward-extrapolation) [figure 3]. These bars may be moved my clicking and |
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| 141 | dragging, or by entering the appropriate values in the Q range input boxes. |
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[0390040] | 142 | |
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| 143 | .. figure:: tutorial1.png |
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| 144 | :align: center |
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| 145 | |
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| 146 | **Figure 3** A plot of some data showing the Q range bars |
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| 147 | |
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| 148 | Once the Q ranges have been set, click the "Calculate" button next to the |
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| 149 | background input field to calculate the Bonart thermal background level. |
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| 150 | Alternatively, enter your own value into the field. Click the "Extrapolate" |
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[f56770ef] | 151 | button to extrapolate the data and plot the extrapolation in the same figure. |
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| 152 | The values of the parameters used for the Guinier and Porod models will also be |
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| 153 | shown in the "Extrapolation Parameters" section [figure 4] |
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[0390040] | 154 | |
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| 155 | .. figure:: tutorial2.png |
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| 156 | :align: center |
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| 157 | |
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| 158 | **Figure 4** A plot showing the extrapolated data and the original data |
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| 159 | |
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| 160 | Then, select which type of transform you would like to perform, using the radio |
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| 161 | buttons: |
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| 162 | |
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| 163 | * **Fourier** Perform a Fourier Transform to calculate the correlation |
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| 164 | function of the extrapolated data |
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| 165 | * **Hilbert** Perform a Hilbert Transform to calculate the volume fraction |
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| 166 | profile of the extrapolated data |
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| 167 | |
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| 168 | Clicking the transform button will then perform the selected transform and plot |
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| 169 | it in a new figure. If a Fourier Transform was performed, the "Compute |
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| 170 | Parameters" button can also be clicked to calculate values for the output |
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| 171 | parameters [figure 5] |
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| 172 | |
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| 173 | .. figure:: tutorial3.png |
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| 174 | :align: center |
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| 175 | |
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| 176 | **Figure 5** The Fourier Transform (correlation function) of the |
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| 177 | extrapolated data, and the parameters extracted from it. |
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