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zWQegLQ-DF3YIsm!;1DKdjxqmuv?hoSchq`>*{@}qmi-J)~mhI568+Om3-~osxK-cdWFeGX$ zyuq&Sb690b(wyMF4hf@b+4$obH+QD*^XElb^WyjC=B5KWWgD;4^G<#JFd5cz>r?Rg zp^68pB2VHwc+arr7Z;1alwOnnagjD^q(#?$D{nYD_5nIpS)~s|p{QHZjydi*wK59_ zJ12!?E)qt2#p*MxysMjK)TiJLT&w4-EYTQ9zd(h+yo)v6pe1xEeR68TYgZI8W6&!Q zP=c_zL`DkgM2Wup6z%gjix0!HL)>Ipnqxo>9oh86W88y;?tZ0bfr-@xpY|8LTZb}V Ly=Zo^muAQRLo>A7 literal 0 HcmV?d00001 diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/commutivity.svg b/src/sas/qtgui/Perspectives/Corfunc/media/commutivity.svg new file mode 100644 index 0000000000..0bff18d8d0 --- /dev/null +++ b/src/sas/qtgui/Perspectives/Corfunc/media/commutivity.svg @@ -0,0 +1,316 @@ + + + + + + + + + + + + + + + + + + + γ(r)CorrelationFunction + + F(q)Form/StructureFactor + I(q)ScatteringAmplitude + ρ(r)ScatteringLength Density + + + + + + Autocorrelation + Square Magnitude + FourierTransform + FourierTransform + Loss of PhaseInformation + Loss of PhaseInformation + + diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst new file mode 100644 index 0000000000..a60e1dbc08 --- /dev/null +++ b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst @@ -0,0 +1,91 @@ +.. _corfunc-how-to: + +How To Use Corfunc +================== + +Running a Calculation +--------------------- + +Upon sending data for correlation function analysis, it will be plotted (minus +the background value), along with a bar indicating the upper end of the +low-Q range (used for Guinier back-extrapolation) and 2 bars indicating +the range to be used for Porod forward-extrapolation. + +This information is also shown on the orange and green interactive slider on the left. +You can drag bars on this slider (not the plot), or enter values manually. + +.. figure:: tutorial_data_loaded.png + :align: center + +Once the Q ranges have been set, click the "Go" button to run the analysis. +This will run through the process described in the technical documentation using the +default options shown above the button. + +The parameters used along the way can be overriden by changing them in the appropriate text boxes, +and whether or not they are recalculated is controlled by the check boxes. + +Options +------- + +In addition to the checkboxes that control whether parts of the calculation are executed, +there are options for choosing how the lamellar parameters are extracted. +These are intended to give flexibility to users and allow for manual control in certain edge +cases, or where there is small errors in the transformed data. + +Automatic selection is the most robust choice, this will choose the leftmost option +as long as it is feasible. + + +The extra options are as follows: + +1) There is an option to choose how the tangent slope used for calculating lamellar parameters +such as `hard block` is calculated. Usually this is done by finding the inflection point of the +curve. However, in some datasets it is more appropriate to use the point half way between the minimum +and maximum values of :math:`\Gamma_1`; + +2) The second option chooses whether to use the first minimum or subsequent maximum to infer the periodicity. +The standard option is to use the maximum, but in some cases, such as where the period is +long compared to experimental data, +it might be necessary to use twice the minimum as a proxy. + +The automatic selection should be appropriate for general use. + +Output +------ + +When the calculation is complete, the extrapolated curve will be shown on the Q-space plot. + + .. figure:: tutorial_after_go.png + :align: center + +The `Real Space` tab shows plots of :math:`\Gamma_1` and :math:`\Gamma_3` + + .. figure:: tutorial_real_space.png + :align: center + +To check the extrapolation parameters, a diagram shows the geometric construction used to +derive them in the `Extraction Diagram` tab. + + .. figure:: tutorial_extraction.png + :align: center + +Finally, you can see the interface distribution function in the `IDF` tab + + .. figure:: tutorial_idf.png + :align: center + +The export buttons allow you to produce CSV files containing either the extrapolated +Q-space data, or the transformed data. + +The structure of the transformed data file is shown below. + + .. figure:: tutorial_export_data.png + :align: center + +.. note:: If *Export Extrapolated* is selected a dialog box will appear in which the + bounds and binning of the extrapolated data can be selected. This is pre-populated + with the values for the experimental data; i.e. using these values would *not* + actually include any extrapolated data! + +.. figure:: tutorial_extrapolate.png + :align: center diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-technical.rst b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-technical.rst new file mode 100644 index 0000000000..a14143499a --- /dev/null +++ b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-technical.rst @@ -0,0 +1,189 @@ +.. _corfunc-technical: + +Corfunc Technical Documentation +=============================== + +In Brief +-------- + +.. figure:: tutorial_data_loaded.png + :align: center + +The correlation function analysis is performed in **3 steps**. + +First, the scattering curve is **extrapolated** to :math:`Q = 0` (Guinier) and toward +:math:`Q = \infty` (Porod), the details of the extrapolation is controlled by +the parameters `Guinier End`, `Porod Start` and `Porod End`, which +are settable by entering text, or by using the `Adjust` slider. +There is a smooth transition from the data to the extrapolation, start and end refer to +the region over which this occurs (see below). +Parameters for the extrapolation can manually entered in `Extrapolation Parameters` and +SasView can be told whether to recalcuate the parameters using the `Fit Background`, +`Fit Guinier` and `Fit Porod` checkboxes. + +Secondly, the data is **transformed** to obtain the projected correlation functions. + +Finally, the transformed data is **interpreted** in terms of an ideal lamellar morphology + + +Extrapolation +------------- + +Small Q +....... + +The scattering data is extrapolated to :math:`q = 0` by fitting a Guinier function, defined as + +.. math:: + I(q) = e^{A + Bq^2} + +to data up to :math:`q` value specified by `Guinier Start`. + +This a Gaussian centred at :math:`q=0` (we only ever see the positive half). +The natural logarithm of the parameter :math:`A` is a constant of proportionality +equal to the scattering intensity at :math:`q=0`. +The parameter :math:`B` describes the width of the function and is related to the +size of the scattering object. For example, in systems of dispersed spherical +particles it is related to the radius of gyration :math:`R_g` by :math:`B = R_g^2 / 3`. + +*Note:* The Guinier model makes assumptions that do not hold for all systems +and so this approximation might not always be accurate. +If errors from the Guinier model fit occur, they will manifest as a constant offset in the correlation function, +because low :math:`q` values correspond to a long periodicity in :math:`x`. +Empirically, however, inaccuracies in the Guinier region have a very low impact on the +final analysis, and only some of the lamellar parameters will be affected at all. + +Large Q +....... + +The data is extrapolated towards :math:`q = \infty` by fitting a Porod model, to the region +between `Porod Start` and `Porod End`. This model is defined by + +.. math:: + I(q) = K q^{-4} e^{-q^2\sigma^2} + I_{B} + +Where :math:`I_B` is the background intensity, :math:`K` is the Porod constant, and :math:`\sigma` is a +parameter which, in a two phase system, describes the sharpness of the scattering length density +profile at the interface between phases. + +This model comprises three components, a constant background intensity, the standard Porod law, so + +.. math:: + I(q) - I_B \propto q^{-4} + +and a contribution which is attibutable to the sharpness of the boundaries between regions. so + +.. math:: + I(q) - I_B \propto e^{-q^2\sigma^2} + +SasView will use this formula to extrapolate to very large :math:`q` (100 +times the maximum data value). This ensures that the transform used in the +next stage does not contain artefacts (i.e. from treating secular data as periodic) + + +Merging +....... + +In the final step before transformation, the experimental and model data are merged together. +For the Guinier model, this happens on the region between the start of the experimental data and +the value specified by `Guinier End`. +For the Porod model, the merging happens between `Porod Start` and `Porod End`. + ++----------------+-------------------+-------------------+ +| From | To | Data | ++================+===================+===================+ +|| 0 || Start of data || Guinier Model | +|| Start of data || *Guinier End* || Guinier/Data mix | +|| *Guinier End* || *Porod Start* || Data | +|| *Porod Start* || *Porod End* || Data/Porod mix | +|| *Porod End* || 100x end of data || Porod model | ++----------------+-------------------+-------------------+ + +A smooth transition is achieved with sigmoid weighting defined as follows. +We start with two input functions, :math:`f(x)` on the 'left' and :math:`g(x)` on the 'right', and these +are to be smoothed over the range :math:`[a, b]`. +We use :math:`y(x)` to represent the transition over :math:`[a,b]`. :math:`y(x)` is given by the following convex combination + +.. math:: + y(x) = h(x) g(x) + (1-h(x))f(x) + +where :math:`h(x)` is a weighting between the two, with a value of zero at :math:`a` and one at :math:`b`, defined as + +.. math:: + h(x) = \frac{1}{1 + \frac{(x-b)^2}{(x-a)^2}} + + +Transformation +-------------- + +Corfunc uses a discrete cosine transform on the extrapolated data in order to calculate the +1D correlation function as: + +.. math:: + \Gamma _{1}(x) = \frac{1}{Q^{*}} \int_{0}^{\infty }I(q) q^{2} cos(qx) dq + +where Q\ :sup:`*` is the Scattering (also called Porod) Invariant. + +The following algorithm is applied: + +.. math:: + \Gamma(x_k) = 2 \sum_{n=0}^{N-1} x_n \cos{\left[ \frac{\pi}{N} + \left(n + \frac{1}{2} \right) k \right] } \text{ for } k = 0, 1, \ldots, + N-1, N + +The 3D correlation function is calculated as: + +.. math:: + \Gamma _{3}(x) = \frac{1}{Q^{*}} \int_{0}^{\infty}I(q) q^{2} + \frac{sin(qx)}{qx} dq + +.. note:: It is always advisable to inspect Γ\ :sub:`1`\ (x) and Γ\ :sub:`3`\ (x) + for artefacts arising from the extrapolation and transformation processes: + + - do they tend to zero as x tends to :math:`\infty`? + - do they smoothly curve onto the ordinate at x = 0? (if not check the value + of :math:`\sigma` is sensible) + - are there ripples at x values corresponding to 2 :math:`\pi` over the two + q values at which the extrapolated and experimental data are merged? + - are there any artefacts at x values corresponding to 2 :math:`\pi` / q\ :sub:`max` in + the experimental data? + - and lastly, do the significant features/peaks in the correlation functions + actually correspond to anticpated spacings in the sample?!!! + +Finally, the program calculates the interface distribution function (IDF) g\ :sub:`1`\ (x) as +the discrete cosine transform of: + +.. math:: + q^{4} I(q) + +The IDF is proportional to the second derivative of Γ\ :sub:`1`\ (x). + + +Interpretation +-------------- + +Once the correlation functions have been calculated *SasView* will +try and interpret Γ\ :sub:`1`\ (x) in terms of an ideal lamellar morphology +as shown below. + +.. figure:: fig2.png + :align: center + +**It is for the user to decide if this interpretation has any relevance to their system!** + +The structural parameters extracted are: + +* Long Period :math:`= L_p` +* Average Hard Block Thickness :math:`= L_c` +* Average Core Thickness :math:`= D_0` +* Average Interface Thickness :math:`\text{} = D_{tr}` +* Eekhaut Polydispersity :math:`= \Gamma_{\text{min}}/\Gamma_{\text{max}}` +* Stribeck Polydispersity +* Local Crystallinity :math:`= L_c/L_p` + +which lead to: + +* Average Soft Block Thickness :math:`= L_p - L_c = L_a` +* Average Chord Length :math:`= ((1/L_c) + (1/L_a)) :sup:`-1`` +* Average Crystalline Chord Length :math:`= ((1/L_c) + (1/L_a)) :sup:`-1` / \Phi_{\text{c}}` +* Non-Ideality :math:`= (L_p – L_p*) :sup:`2` / L_p :sup:`2`` diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-theory.rst b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-theory.rst new file mode 100644 index 0000000000..4251ef081c --- /dev/null +++ b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-theory.rst @@ -0,0 +1,372 @@ +.. _corfunc-theory: + + + +Correlation Function Theory +=========================== + +Overview +-------- + +In small angle scattering we measure the tendency for probe particles (neutrons, photons, etc) +to transfer various amounts of momentum to a sample. The momentum is generally inferred from the scattering angle +of probe particles, along with other information about the probe particles (e.g. kinetic energy). +Small angle scattering is assumed to be elastic, which allows the momentum transfer to be directly related to a wavelength, and thus +a spatial distance. The correlation function represents the scattering intensity in terms of this spatial distance, +rather than in terms of momentum transfer. + +We can interpret the correlation function in terms of the sample structure by thinking about +pairs of points separated by a given displacement. When, on average over the sample, the pairs of points +have a high scattering length density, then the correlation function has a large value. Similarly, +when the pairs have a low scattering length density, the correlation function is low. +More concretely: the correlation function :math:`\Gamma(\vec{r})` for vector :math:`\vec{r} = (x,y,z)` is proportional to +the pairwise product of scattering length densities for all points separated by the vector :math:`(x,y,z)` +summed over all orientations and locations. + +Another way of thinking about the correlation function is as the scattering length +density but with phase information removed. As scattering experiments contain +no phase information, calculating the correlation function +is as close as one can get to calculating the scattering length density from +scattering data without incorporating additional information. + +The nature of small angle scattering further limits what spatial information +can be recovered. Whilst in its most general form the correlation +function takes a three dimensional vector input, +small angle scattering measurements are limited to one or two dimensions, +which in turn limits the amount of information about the correlation +function that can be obtained. For this reason, in the correlation function +analysis tool we consider various one dimensional projections of the full +correlation function, labelled :math:`\Gamma_1` and :math:`\Gamma_3` . + +The :math:`\Gamma_1` projection looks at changes in a single +direction perpendicular to the beam, with the other directions being averaged. +The direction is typically selected by hand from a 2D measurement prior to analysis. +Theoretically, the correlation function will be fully recoved as long as the system +being looked at is truly one dimensional and properly aligned. +However, one must remember the constraints of a small angle scattering experiment, +we only measure a small range of momentum transfer, and extrapolate the rest, +as such the extrapolation steps must be appropriate for the system. +This is in addition to the usual considerations of resolution and systematic +measurement error. + +The :math:`\Gamma_3` projection is motivated by a system of +monodisperse, randomly oriented particles in dilute suspension, +such that there is no spatial correlations between particles. +It is the kind of system described by the Debye equation. +Just as is the case with :math:`\Gamma_1`, as long as one truly +has this kind of system, and with caveats about extrapolation +and experimental constraints, one should be able to fully recover +the correlation function. + +Lorentz Correction +------------------ + +Lorentz corrections are often used in correlation function analysis. +Corfunc uses a Lorentz correction of + +.. math:: + I(q) = q^2 I_\text{measured}(q) + +In what follows, we assume that appropriate corrections have been made. +:math:`I(q)` here is what would be called :math:`I_1(q)` in :ref:`Stribeck`. + +Formal Description +------------------ + +More formally, the correlation function is a quantity that arises naturally from calculating the square magnitude +of the three dimensional Fourier transform, which is proportional to the scattering amplitude. + +.. math:: + \frac{d\sigma}{d\Omega} \propto F(\vec{q}) F^*(\vec{q}) + +where + +.. math:: + F(\vec{q}) = \int \rho(r) e^{i \vec{r}\cdot\vec{q}} \; dr^3 + +where :math:`dr^3` is the volume element (:math:`dx\;dy\;dz`). + + +.. figure:: commutivity.png + :align: center + + + +A couple of algebraic steps will bring us to the correlation function: first, +as :math:`\rho` is real, and the conjugate of :math:`e^{ix}` is :math:`e^{-ix}` we +know that the conjugate of :math:`F` is given by + +.. math:: + F^*(\vec{q}) = \int \rho(r) e^{-i \vec{r}\cdot\vec{q}} + +meaning that, with some renaming of variables, we have + +.. math:: + F(\vec{q}) F^*(\vec{q}) = \left(\int \rho(\vec{t}) e^{i \vec{t}\cdot\vec{q}} \; dt^3\right)\left( \int \rho(\vec{s}) e^{-i \vec{s}\cdot\vec{q}} \; ds^3 \right) + +With some rearrangement this becomes + +.. math:: + \int\int \rho(\vec{s}) \rho(\vec{t}) e^{i (t-s)\cdot\vec{q}} \; dr^3 \; ds^3 + + +and now letting :math:`\vec{r} = \vec{t} - \vec{s}` +and applying the Fourier translation theorem, we can rewrite the above as +(note this is not the same :math:`\vec{r}` as before, but a new variable): + +.. math:: + \int\int \rho(\vec{s}) \rho(\vec{s} + \vec{r}) e^{i \vec{r}\cdot\vec{q}} \; ds^3 \; dr^3 + +Some final reordering of the integration gives + +.. math:: + \int \left[ \int \rho(\vec{s}) \rho(\vec{s} + \vec{r}) \; ds^3 \right] \; e^{i \vec{r}\cdot\vec{q}} \; dr^3 + +The quantity in square brackets is what is called the correlation function, :math:`\gamma(\vec{r})`, so: + +.. math:: + \gamma(\vec{r}) = \int \rho(\vec{s}) \rho(\vec{s} + \vec{r}) \; ds^3 + +and it is the quantity that is Fourier transformed (with some appropriate scaling) +to get the magnitude of the scattering. + +Some useful properties of the Correlation Function +.................................................. + +As we have mentioned before, the correlation function contains no phase information, +mathematically this is the same as saying (1) that its Fourier transform is purely real, +or (2) that the correlation function is an even function. The consequence of this is +that we can write the Fourier transform of the correlation function using a cosine instead +of a complex exponential. + +Demonstrating the evenness of the correlation function is easily done by a change of +the variable of integration from :math:`\vec{s}` +to :math:`\vec{u} = \vec{s} + \vec{r}`. + +.. math:: + \gamma(\vec{r}) = \int \rho(\vec{s}) \rho(\vec{s} + \vec{r}) \; ds^3 = \int \rho(\vec{u}-\vec{r}) \rho(\vec{u}) \; du^3 = \gamma(-\vec{r}) + +and from this we can show that its Fourier transform is real by applying the following +to each dimension in turn (shown here in the 1D case for even :math:`f(x)`). + +First, we split the integral into negative and positive :math:`x` parts: + +.. math:: + \int_{-\infty}^\infty f(x) e^{i x \xi} dx = \int_{-\infty}^0 f(x) e^{i x \xi} dx + \int_{0}^\infty f(x) e^{i x \xi} dx + +Let :math:`u = -x` for the negative part, use the fact that :math:`f(-x)=f(x)` and +recalculate the bounds of integration + +.. math:: + = \int_0^\infty f(u) e^{-i u \xi} du + \int_{0}^\inf f(x) e^{i x \xi} dx + +Note that :math:`u` only appears within the integral, so we can rename it to :math:`x` +and recombine it with the positive part. We can also multiply the integral by two and +the integrand by two, giving + +.. math:: + = 2 \int_0^\infty f(x) \frac{e^{i x \xi} + e^{-i x \xi}}{2} dx + +The fractional part of which is the complex definition of cosine. +Applying this definition and using the fact that :math:`f(x)` is even +to restore the original bounds of integration we get + +.. math:: + = \int_{-\infty}^{\infty} f(x) cos(x \xi) dx + +which shows that the Fourier transform is purely real, reflecting the +fact that there is no phase information (which would be encoded in the imaginary part). + + +The :math:`\Gamma_1` Projection +............................... + +Consider the Fourier transform of the three dimensional correlation function, + +.. math:: + \int\int\int \gamma(\vec{r}) e^{i \vec{r} \cdot \vec{q}} \; dx \; dy \; dz + + +Now let :math:`q_z = q_y = 0`. +The motivation for this is (1) that during small angle scattering :math:`q_z` +is small enough to be neglected, and (2) that we are choosing to measure +in one direction of the :math:`q_x q_y` plane. +We assume, without loss of generality, this to be where :math:`q_y=0`. + +This gives us :math:`q \cdot r = x q_x`, and so the transform becomes + +.. math:: + I(q) = \int \gamma(\vec{r}) e^{i x q_x} \; dx \; dy \; dz + +which we can rewrite as + +.. math:: + \int\left( \int\int \gamma(\vec{r}) \; dy\;dz\right) e^{i x q_x} \; dx + +the quantity in the brackets is :math:`\Gamma_1(x)`. That is to say + +.. math:: + \Gamma_1(x) = \int\int \gamma(\vec{r}) \;dy\;dz + +If we now use the fact that :math:`\gamma(\vec{r})` is an even function, +we can use the result above to get + +.. math:: + I(q) = \int \Gamma_1(x) \cos(qx) dx + +The job of Corfunc is now to invert this. The following operation does the job: + +.. math:: + \Gamma_1(x) = \int I(q) \cos(qx) dx + +We can check this by showing that + +.. math:: + f(y) = \int \left( \int f(x) \cos(qx) dx \right) \cos(qy) dq + +Doing this formally requires a fair bit of algebraic legwork, +but there is an informal argument that will get us there. +First note that we can write it as (hand-waving away the convergence issues) + +.. math:: + f(y) = \int f(x) \int cos(qx) cos(qy) dq dx + +Then the equation corresponds to the identity function if the integral + +.. math:: + \int cos(qx) cos(qy) dq + +is the delta function. This the case, because cosine functions form an orthogonal basis. +When :math:`x=y` the integral is non-zero, being an +integral of the always positive :math:`cos^2(qx)`. +Conversely, when :math:`x \neq y` the integral is zero. + + + + +The :math:`\Gamma_3` Projection +............................... + +The :math:`\Gamma_3` projection is based on spherical symmetry. +It's derivation is essentially that of Debye's formula + +We begin with an expression for the scattered intensity as above + +.. math:: + I(\vec{q}) = \int_{\mathbb{R}^3} \gamma(\vec{r}) e^{i \vec{r} \cdot \vec{q}} dr^3 + +now, we want to average this over all angles, i.e. over all :math:`q`-vectors of a given length, and we do so in a coordinate +system relative to :math:`\vec{r}`. This is an unobvious choice of coordinate system, but it simplifies things greatly, +as in such a coordinate system, the dot product :math:`\vec{r}\cdot\vec{q}` becomes :math:`qr \cos\theta`. + +For our averaging there is a total of :math:`4\pi` stradians in a sphere, giving a leading factor of :math:`1/4\pi`. + +.. math:: + I(\vec{q}) = \frac{1}{4\pi}\int_{\phi=0}{\phi=2\pi}\int_{\theta=0}^{\theta=\pi}\int_{\vec{r}\in\mathbb{R}} \gamma(\vec{r}) e^{i qr \cos\theta} d\vec{r}^3 \sin(\theta) d\theta d\phi + +The integral is constant with with respect to :math:`\phi`, so drops out as a factor of :math:`2\pi`. + +.. math:: + = \frac{1}{2}\int_{\theta=0}^{\theta=\pi}\int_{\vec{r}\in\mathbb{R}^3} \gamma(\vec{r}) e^{i \vec{r} \cdot \vec{q}} d\vec{r}^3 \sin(\theta) d\theta + +and we can adjust the order of integration, noting that because of our choice of coordinate system, :math:`\gamma(\vec{r})` is +independent of :math:`\theta`. + +.. math:: + = \frac{1}{2}\int_{\vec{r}\in\mathbb{R}^3} \gamma(\vec{r}) \int_{\theta=0}^{\theta=\pi} e^{i \vec{r} \cdot \vec{q}} \sin(\theta) d\theta d\vec{r}^3 + +Now, we can consider the inner integral specifically, firstly by doing a substitution of :math:`u = -\cos\theta`. This +means that :math:`du = \sin\theta d\theta`, the interval :math:`\theta\in[0,\pi]` becomes :math:`u\in[1, -1]`. + +.. math:: + \int_{\theta=0}^{\theta=\pi} e^{i \vec{r} \cdot \vec{q}} \sin(\theta) d\theta = \int_{u=-1}^{u=1} e^{i q r u} du + +which is just an exponential and easily integrated + +.. math:: + = \left[ \frac{-i e^{i q r u}}{qr} \right]_{-1}^{1} = \frac{i \left(e^{-i q r} - e^{i q r} \right) }{qr} = 2 \frac{i \sinh(-iqr)}{qr} + +which by the relationship between complex trigonometric and hyperbolic functions becomes + +.. math:: + = 2 \frac{\sin qr}{qr} + +The leading :math:`2` will cancel the leading :math:`1/2` and the value of :math:`I(q)` can be seen to be + +.. math:: + I(q) = \int_{\vec{r}\in\mathbb{R}^3} \gamma(\vec{r}) \frac{\sin qr}{qr} d\vec{r}^3 + +Note that this object is not dependent on the angular components of :math:`\vec{r}`, so the integral +over :math:`\mathbb{R}^3` can be written as + +.. math:: + = \int_0^\infty \int_\Omega\gamma(\vec{r})d\Omega \frac{\sin qr}{qr} dr + +Where :math:`\Omega` is a solid angle element. Letting + +.. math:: + `\Gamma_3(r) = \int_\Omega \gamma(\vec{r}) d\Omega` we have, + +finally, + +.. math:: + I(q) = \int_0^\infty \Gamma_3(r) \frac{\sin qr}{qr} dr + +In corfunc we don't invert this directly, but do so via :math:`\Gamma_1` + +Relationship between :math:`\Gamma_1` and :math:`\Gamma_3` +.......................................................... + +Internally, Corfunc calculates :math:`\Gamma_3` from :math:`\Gamma_1`. +Let's now look at how we can get one from the other, starting with :math:`\Gamma_3`. + +.. math:: + \Gamma_3 = \int I(q) \frac{\sin(q x)}{q x} dq + +First, multiply by :math:`x` + +.. math:: + x \Gamma_3 = x \int I(q) \frac{\sin(q x)}{q x} dq = \int I(q) \frac{\sin(q x)}{q} dq + +Now take the derivative with respect to :math:`x` + +.. math:: + \frac{d}{dx} x \Gamma_3 = \frac{d}{dx} \int I(q) \frac{\sin(q x)}{q} dq = \int I(q) \cos (q x) dq = \Gamma_1 + +Which, after expressing in terms of :math:`\Gamma_1` gives us the relation we use in corfunc, for +calculating :math:`\Gamma_3` + +.. math:: + \Gamma_3(x) = \int_0^x \frac{\Gamma_1(r)}{r} dr + + + + +References +---------- + +.. [Rutland] + Ruland, W. *Coll. Polym. Sci.* (1977), 255, 417-427 + +.. [Strobl] + Strobl, G. R.; Schneider, M. *J. Polym. Sci.* (1980), 18, 1343-1359 + +.. [Koberstein] + Koberstein, J.; Stein R. *J. Polym. Sci. Phys. Ed.* (1983), 21, 2181-2200 + +.. [Calleja1] + Baltá Calleja, F. J.; Vonk, C. G. *X-ray Scattering of Synthetic Poylmers*, Elsevier. Amsterdam (1989), 247-251 + +.. [Calleja2] + Baltá Calleja, F. J.; Vonk, C. G. *X-ray Scattering of Synthetic Poylmers*, Elsevier. Amsterdam (1989), 257-261 + +.. [Calleja3] + Baltá Calleja, F. J.; Vonk, C. G. *X-ray Scattering of Synthetic Poylmers*, Elsevier. Amsterdam (1989), 260-270 + +.. [Goschel] + Göschel, U.; Urban, G. *Polymer* (1995), 36, 3633-3639 + +.. [Stribeck] + Stribeck, N. *X-Ray Scattering of Soft Matter*, Springer. Berlin (2007), 138-161 + +:ref:`FDR` (PDF format) diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc_help.rst b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc_help.rst old mode 100755 new mode 100644 index e1f6fc8e44..812d106e5d --- a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc_help.rst +++ b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc_help.rst @@ -1,311 +1,21 @@ -.. corfunc_help.rst +.. _corfunc_help: -.. _Correlation_Function_Analysis: Correlation Function Analysis ============================= -Description ------------ +The Correlation Function Analysis documentation is split into three sections. +One on the general principles of correlation function analysis. +Another containing details on how the analysis is performed in SasView, and +another showing how to use the tool from a users perspective. -This currently performs correlation function analysis on SAXS/SANS data, -but in the the future is also planned to generate model-independent volume -fraction profiles from the SANS from adsorbed polymer/surfactant layers. -The two types of analyses differ in the mathematical transform that is -applied to the data (Fourier vs Hilbert). However, both functions are -returned in *real space*. +.. toctree:: + :maxdepth: 2 -A correlation function may be interpreted in terms of an imaginary rod moving -through the structure of the material. Γ(x) is the probability that a rod of -length x has equal electron/neutron scattering length density at either end. -Hence a frequently occurring spacing within a structure will manifest itself -as a peak in Γ(x). *SasView* will return both the one-dimensional ( Γ\ :sub:`1`\ (x) ) -and three-dimensional ( Γ\ :sub:`3`\ (x) ) correlation functions, the difference -being that the former is only averaged in the plane of the scattering vector. + Theoretical basis of correlation function analysis -A volume fraction profile :math:`\Phi`\ (z) describes how the density of polymer -segments/surfactant molecules varies with distance, z, normal to an (assumed -locally flat) interface. The form of :math:`\Phi`\ (z) can provide information -about the arrangement of polymer/surfactant molecules at the interface. The width -of the profile provides measures of the layer thickness, and the area under -the profile is related to the amount of material that is adsorbed. + Details of how the Corfunc module in SasView works -Both analyses are performed in 3 stages: + How-to/tutorial on using Corfunc -* Extrapolation of the scattering curve to :math:`Q = 0` and toward - :math:`Q = \infty` -* Smoothed merging of the two extrapolations into the original data -* Fourier / Hilbert Transform of the smoothed data to give the correlation - function or volume fraction profile, respectively -* (Optional) Interpretation of Γ\ :sub:`1`\ (x) assuming the sample conforms - to an ideal lamellar morphology -.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ - - -Extrapolation -------------- - -To :math:`Q = 0` -................ - -The data are extrapolated to q = 0 by fitting a Guinier function to the data -points in the low-q range. - -The equation used is: - -.. math:: - I(q) = e^{A + Bq^2} - -Where the parameter :math:`B` is related to the effective radius-of-gyration of -a spherical object having the same small-angle scattering in this region. - -Note that as q tends to zero this function tends to a limiting value and is -therefore less appropriate for use in systems where the form factor does not -do likewise. However, because of the transform, the correlation functions are -most affected by the Guinier back-extrapolation at *large* values of x where -the impact on any extrapolated parameters will be least significant. - -To :math:`Q = \infty` -..................... - -The data are extrapolated towards q = :math:`\infty` by fitting a Porod model to -the data points in the high-q range and then computing the extrapolation to 100 -times the maximum q value in the experimental dataset. This should be more than -sufficient to ensure that on transformation any truncation artefacts introduced -are at such small values of x that they can be safely ignored. - -The equation used is: - -.. math:: - I(q) = K q^{-4}e^{-q^2\sigma^2} + Bg - -Where :math:`Bg` is the background, :math:`K` is the Porod constant, and :math:`\sigma` (which -must be > 0) describes the width of the electron/neutron scattering length density -profile at the interface between the crystalline and amorphous regions as shown below. - -.. figure:: fig1.png - :align: center - - -Smoothing ---------- - -The extrapolated data set consists of the Guinier back-extrapolation from q ~ 0 -up to the lowest q value in the original data, then the original scattering data, -and then the Porod tail-fit beyond this. The joins between the original data and -the Guinier/Porod extrapolations are smoothed using the algorithm below to try -and avoid the formation of truncation ripples in the transformed data: - -Functions :math:`f(x_i)` and :math:`g(x_i)` where :math:`x_i \in \left\{ -{x_1, x_2, ..., x_n} \right\}`, are smoothed over the range :math:`[a, b]` -to produce :math:`y(x_i)`, by the following equations: - -.. math:: - y(x_i) = h_ig(x_i) + (1-h_i)f(x_i) - -where: - -.. math:: - h_i = \frac{1}{1 + \frac{(x_i-b)^2}{(x_i-a)^2}} - - -Transformation --------------- - -Fourier -....... - -If "Fourier" is selected for the transform type, *SasView* will perform a -discrete cosine transform on the extrapolated data in order to calculate the -1D correlation function as: - -.. math:: - \Gamma _{1}(x) = \frac{1}{Q^{*}} \int_{0}^{\infty }I(q) q^{2} cos(qx) dq - -where Q\ :sup:`*` is the Scattering (also called Porod) Invariant. - -The following algorithm is applied: - -.. math:: - \Gamma(x_k) = 2 \sum_{n=0}^{N-1} x_n \cos{\left[ \frac{\pi}{N} - \left(n + \frac{1}{2} \right) k \right] } \text{ for } k = 0, 1, \ldots, - N-1, N - -The 3D correlation function is calculated as: - -.. math:: - \Gamma _{3}(x) = \frac{1}{Q^{*}} \int_{0}^{\infty}I(q) q^{2} - \frac{sin(qx)}{qx} dq - -.. note:: It is always advisable to inspect Γ\ :sub:`1`\ (x) and Γ\ :sub:`3`\ (x) - for artefacts arising from the extrapolation and transformation processes: - - - do they tend to zero as x tends to :math:`\infty`? - - do they smoothly curve onto the ordinate at x = 0? (if not check the value - of :math:`\sigma` is sensible) - - are there ripples at x values corresponding to (2 :math:`pi` over) the two - q values at which the extrapolated and experimental data are merged? - - are there any artefacts at x values corresponding to 2 :math:`pi` / q\ :sub:`max` in - the experimental data? - - and lastly, do the significant features/peaks in the correlation functions - actually correspond to anticpated spacings in the sample?!!! - -Finally, the program calculates the interface distribution function (IDF) g\ :sub:`1`\ (x) as -the discrete cosine transform of: - -.. math:: - -q^{4} I(q) - -The IDF is proportional to the second derivative of Γ\ :sub:`1`\ (x). - -Hilbert -....... - -If "Hilbert" is selected for the transform type, the analysis will perform a -Hilbert transform on the extrapolated data in order to calculate the Volume -Fraction Profile. - -.. note:: The Hilbert transform functionality is not yet implemented in SasView. - - -Interpretation --------------- - -Correlation Function -.................... - -Once the correlation functions have been calculated *SasView* can be asked to -try and interpret Γ\ :sub:`1`\ (x) in terms of an ideal lamellar morphology -as shown below. - -.. figure:: fig2.png - :align: center - -The structural parameters extracted are: - -* Long Period :math:`= L_p` -* Average Hard Block Thickness :math:`= L_c` -* Average Core Thickness :math:`= D_0` -* Average Interface Thickness :math:`\text{} = D_{tr}` -* Polydispersity :math:`= \Gamma_{\text{min}}/\Gamma_{\text{max}}` -* Local Crystallinity :math:`= L_c/L_p` - -Volume Fraction Profile -....................... - -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: - -* Surface Coverage :math:`=\theta` -* Anchor Separation :math:`= D` -* Bound Fraction :math:`=

` -* Second Moment :math:`= \sigma` -* Maximum Extent :math:`= \delta_{\text{h}}` -* Adsorbed Amount :math:`= \Gamma` - -.. figure:: profile1.png - :align: center - -.. figure:: profile2.png - :align: center - -The reader is directed to the references for information on these parameters. - -References ----------- - -Correlation Function -.................... - -Ruland, W. *Coll. Polym. Sci.* (1977), 255, 417-427 - -Strobl, G. R.; Schneider, M. *J. Polym. Sci.* (1980), 18, 1343-1359 - -Koberstein, J.; Stein R. *J. Polym. Sci. Phys. Ed.* (1983), 21, 2181-2200 - -Baltá Calleja, F. J.; Vonk, C. G. *X-ray Scattering of Synthetic Poylmers*, Elsevier. Amsterdam (1989), 247-251 - -Baltá Calleja, F. J.; Vonk, C. G. *X-ray Scattering of Synthetic Poylmers*, Elsevier. Amsterdam (1989), 257-261 - -Baltá Calleja, F. J.; Vonk, C. G. *X-ray Scattering of Synthetic Poylmers*, Elsevier. Amsterdam (1989), 260-270 - -Göschel, U.; Urban, G. *Polymer* (1995), 36, 3633-3639 - -Stribeck, N. *X-Ray Scattering of Soft Matter*, Springer. Berlin (2007), 138-161 - -:ref:`FDR` (PDF format) - -Volume Fraction Profile -....................... - -Washington, C.; King, S. M. *J. Phys. Chem.*, (1996), 100, 7603-7609 - -Cosgrove, T.; King, S. M.; Griffiths, P. C. *Colloid-Polymer Interactions: From Fundamentals to Practice*, Wiley. New York (1999), 193-204 - -King, S. M.; Griffiths, P. C.; Cosgrove, T. *Applications of Neutron Scattering to Soft Condensed Matter*, Gordon & Breach. Amsterdam (2000), 77-105 - -King, S.; Griffiths, P.; Hone, J.; Cosgrove, T. *Macromol. Symp.* (2002), 190, 33-42 - -.. ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ - - -Usage ------ -Upon sending data for correlation function analysis, it will be plotted (minus -the background value), along with a bar indicating the *upper end of the -low-Q range* (used for Guinier back-extrapolation), and 2 bars indicating -the range to be used for Porod forward-extrapolation. These bars may be moved by -entering appropriate values in the Q range input boxes or by clicking on them and -dragging them to the desired location.. - -.. figure:: tutorial1.png - :align: center - -Once the Q ranges have been set, click the "Calculate" button in the *Background* section -of the dialog to determine the background level. -Alternatively, enter your own value into the box. If the box turns -yellow this indicates that background subtraction has created some negative intensities. - -Now click the "Extrapolate" button to extrapolate the data. The graph window will update -to show the extrapolated data, and the values of the parameters used for the Guinier and -Porod extrapolations will appear in the "Extrapolation Parameters" section of the Corfunc -GUI. - -.. figure:: tutorial2.png - :align: center - -Now click the "Transform" button to perform the Fourier transform and plot -the results. The lower graph will display the 1D and 3D-averaged correlation functions. -The Interface Distribution Function (or IDF) is also computed, but is not displayed -for clarity. How to access the IDF, and the correlation functions themselves, is -explained shortly. - - .. figure:: tutorial3.png - :align: center - -*If* the sample morphology can be adequately described as an ideal lamellar morphology -the Corfunc GUI can attempt to derive morphological characterization parameters from the -1D correlation function. To do this, click the "Extract Parameters" button. - - .. figure:: tutorial4.png - :align: center - -Finally, it is possible to save the values of the real-space distance axis, the 1D and 3D -correlation functions, and the IDF to a simple ASCII text file by clicking on the "Save" -button. The file is given the unique file descriptor *.crf*. - - .. figure:: tutorial5.png - :align: center - -The structure of the file is shown below. - - .. figure:: tutorial6.png - :align: center - -.. note:: At the time of writing SasView will not load these *.crf* files, but they can - be easily loaded and displayed in most spreadsheet applications. - -.. note:: - This help document was last changed by Steve King, 21May2020 diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/fdr-pdfs.rst b/src/sas/qtgui/Perspectives/Corfunc/media/fdr-pdfs.rst old mode 100755 new mode 100644 diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/fig1.png b/src/sas/qtgui/Perspectives/Corfunc/media/fig1.png old mode 100755 new mode 100644 diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/fig2.png b/src/sas/qtgui/Perspectives/Corfunc/media/fig2.png old mode 100755 new mode 100644 diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/profile1.png b/src/sas/qtgui/Perspectives/Corfunc/media/profile1.png 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zBhnjEK?HO^Ln9OZC^qb0yvKuto$5PII||;ysNDjL05sKnbozEBA>Bk)w~1OFj5>b) zC46g7xHg`c@G^ft6u$UDJv=0lP|ir`KXKO`4~YIO64x7_><&-V#y@gl1oE@T4!`98 z!%w4+-wzq19=#8bx9S^tZ8sYd|HJtG;*I#beAT1k($Mq);9}89IBw9Qsjc^QC|}ei z^g*i1!&fBrTz268)yqF{Dek$^=o(l0ao^I1)!F`CevjL8wTDoj{@msY?lnpg!xQ~f z|Kbi1*$as7VZ$u*DS^zJxET zWh7^s0fuA#JM?*v!TPx# literal 0 HcmV?d00001 From e375e2a22d801e3f7c5fdce8bdc8570643558329 Mon Sep 17 00:00:00 2001 From: lucas-wilkins Date: Mon, 25 Mar 2024 17:45:03 +0000 Subject: [PATCH 2/5] Typos in corfunc-theory --- .../Perspectives/Corfunc/media/corfunc-theory.rst | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-theory.rst b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-theory.rst index 4251ef081c..df8121c4c7 100644 --- a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-theory.rst +++ b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-theory.rst @@ -51,7 +51,7 @@ measurement error. The :math:`\Gamma_3` projection is motivated by a system of monodisperse, randomly oriented particles in dilute suspension, -such that there is no spatial correlations between particles. +such that there are no spatial correlations between particles. It is the kind of system described by the Debye equation. Just as is the case with :math:`\Gamma_1`, as long as one truly has this kind of system, and with caveats about extrapolation @@ -99,7 +99,7 @@ know that the conjugate of :math:`F` is given by .. math:: F^*(\vec{q}) = \int \rho(r) e^{-i \vec{r}\cdot\vec{q}} -meaning that, with some renaming of variables, we have +meaning that, with some renaming of variables (from :math:`r` to :math:`s` and :math:`t`), we have .. math:: F(\vec{q}) F^*(\vec{q}) = \left(\int \rho(\vec{t}) e^{i \vec{t}\cdot\vec{q}} \; dt^3\right)\left( \int \rho(\vec{s}) e^{-i \vec{s}\cdot\vec{q}} \; ds^3 \right) @@ -259,7 +259,7 @@ now, we want to average this over all angles, i.e. over all :math:`q`-vectors of system relative to :math:`\vec{r}`. This is an unobvious choice of coordinate system, but it simplifies things greatly, as in such a coordinate system, the dot product :math:`\vec{r}\cdot\vec{q}` becomes :math:`qr \cos\theta`. -For our averaging there is a total of :math:`4\pi` stradians in a sphere, giving a leading factor of :math:`1/4\pi`. +For our averaging there is a total of :math:`4\pi` steradians in a sphere, giving a leading factor of :math:`1/4\pi`. .. math:: I(\vec{q}) = \frac{1}{4\pi}\int_{\phi=0}{\phi=2\pi}\int_{\theta=0}^{\theta=\pi}\int_{\vec{r}\in\mathbb{R}} \gamma(\vec{r}) e^{i qr \cos\theta} d\vec{r}^3 \sin(\theta) d\theta d\phi @@ -305,9 +305,9 @@ over :math:`\mathbb{R}^3` can be written as Where :math:`\Omega` is a solid angle element. Letting .. math:: - `\Gamma_3(r) = \int_\Omega \gamma(\vec{r}) d\Omega` we have, + \Gamma_3(r) = \int_\Omega \gamma(\vec{r}) d\Omega -finally, +we have, finally, .. math:: I(q) = \int_0^\infty \Gamma_3(r) \frac{\sin qr}{qr} dr From fc0dcec3810303958666b2ffee88820512d0e767 Mon Sep 17 00:00:00 2001 From: lucas-wilkins Date: Mon, 25 Mar 2024 17:56:26 +0000 Subject: [PATCH 3/5] Changes to corfunc-how-to --- src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst index a60e1dbc08..91bda98ac8 100644 --- a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst +++ b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst @@ -39,7 +39,7 @@ as long as it is feasible. The extra options are as follows: 1) There is an option to choose how the tangent slope used for calculating lamellar parameters -such as `hard block` is calculated. Usually this is done by finding the inflection point of the +such as `hard block` are calculated. Usually this is done by finding the inflection point of the curve. However, in some datasets it is more appropriate to use the point half way between the minimum and maximum values of :math:`\Gamma_1`; @@ -77,7 +77,9 @@ Finally, you can see the interface distribution function in the `IDF` tab The export buttons allow you to produce CSV files containing either the extrapolated Q-space data, or the transformed data. -The structure of the transformed data file is shown below. +The structure of the transformed data file is shown below (note that as this is a 4 column .csv file, +SasView will interpret this as a :math:`I(q)` vs :math:`q` curve if you try to load it from the main +SasView window). .. figure:: tutorial_export_data.png :align: center From 30c7579bc3b51da6eec45d82d949c7afd86d5dd3 Mon Sep 17 00:00:00 2001 From: Steve K Date: Tue, 26 Mar 2024 13:42:14 +0000 Subject: [PATCH 4/5] Update corfunc-theory.rst Correct grammar --- src/sas/qtgui/Perspectives/Corfunc/media/corfunc-theory.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-theory.rst b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-theory.rst index df8121c4c7..8cecb2de94 100644 --- a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-theory.rst +++ b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-theory.rst @@ -236,7 +236,7 @@ Then the equation corresponds to the identity function if the integral .. math:: \int cos(qx) cos(qy) dq -is the delta function. This the case, because cosine functions form an orthogonal basis. +is the delta function. This is the case, because cosine functions form an orthogonal basis. When :math:`x=y` the integral is non-zero, being an integral of the always positive :math:`cos^2(qx)`. Conversely, when :math:`x \neq y` the integral is zero. From e221911ab1daa88de998964ddb79f4da16b21823 Mon Sep 17 00:00:00 2001 From: Steve K Date: Tue, 26 Mar 2024 13:45:53 +0000 Subject: [PATCH 5/5] Update corfunc-how-to.rst Correct grammar --- src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst index 91bda98ac8..697fca63bc 100644 --- a/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst +++ b/src/sas/qtgui/Perspectives/Corfunc/media/corfunc-how-to.rst @@ -30,7 +30,7 @@ Options In addition to the checkboxes that control whether parts of the calculation are executed, there are options for choosing how the lamellar parameters are extracted. These are intended to give flexibility to users and allow for manual control in certain edge -cases, or where there is small errors in the transformed data. +cases, or where there are small errors in the transformed data. Automatic selection is the most robust choice, this will choose the leftmost option as long as it is feasible.

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