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documentation for ellipsoids

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Kohlbrecher committed Mar 19, 2018
1 parent 55704a8 commit 0db15656e2fc0d10dfd3b47c5244fbed438aa610
@@ -56,3 +56,8 @@ saskit/saskit_*
/examples/dll/libsasfit_fuzzysphere.dll
/examples/dll/libsasfit.dll
/src/fftw/fftw-3.3.7.tar.gz
/src/plugins/magcylshell/include
/src/plugins/magcylshell/include/private.h
/src/plugins/sin2
/src/plugins/sin2/include/private.h
/src/plugins/sin2/include/sasfit_sin2.h
BIN +246 Bytes (100%) CHANGES.txt
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@@ -1,9 +1,14 @@
\appendix{History}
\begin{description}
\item[201x-xx-xx] \SASfit 0.94.10
\item[2018-xx-xx] \SASfit 0.94.10
\begin{itemize}
\item bug fix in the algorithm for calculating the resolution parameter in case of averaging neighbouring data points
\item plugin implementation of several variants of helices
\item The form factors \texttt{"EllipsoidalCoreShell"}, \texttt{"Ellipsoid i"}, \texttt{"Ellipsoid ii"}, and \texttt{"triaxEllShell1"} have been disabled and replaced by a series of plug-in functions. Those replacements make use of a faster routine for multidimensional integrals (pcubature). Because of this an optional size distribution became part of the supplied plug-in form factors.
\item bug fix of unit conversion via clipboard
\item added a plug-in for a random flight structure factor
\item bug fix in the algorithm for calculating the resolution parameter in case of averaging neighboring data points
\item plug-in implementation of several variants of helices
\item plug-in for rectangular parallelepipeds have been extended so that it can have a size distribution of one, two or all three axis. The multiple integration is done by using the pcubature code from Steven G. Johnson
\item multi dimensional integration package "cubature" from Steven G. Johnson becomes generally available also in plug-ins (\href{https://github.com/stevengj/cubature}{cubature})
\end{itemize}
\item[2017-08-16] \SASfit 0.94.9
\begin{itemize}
@@ -41,7 +46,7 @@
\begin{itemize}
\item implementation of another cumulant formula for DLS
\item bug-fix in the unit conversion routine
\item implementation of GMRES, Bi-CGStab, TFQMR and Andersen Accelarion
\item implementation of GMRES, Bi-CGStab, TFQMR and Andersen Acceleration
for solving efficiently the Ornstein Zernicke fixpoint problem
\item bug fix: since version 0.94.4 the new interrupt option suppressed
a proper error reporting to the GUI for undefined input values
@@ -89,8 +94,8 @@
\item[2014-07-02] \SASfit 0.94.3
\begin{itemize}
\item bug fix in the plugin for parallel epiped
\item spelling errors in the menue interface
\item in case of slow convergence the OZ solver can be interupted now
\item spelling errors in the menu interface
\item in case of slow convergence the OZ solver can be interrupted now
\end{itemize}
\item[2014-02-06] \SASfit 0.94.2
\begin{itemize}
@@ -206,8 +211,8 @@
\item optical (layout) GUI improvements:
\begin{itemize}
\item removed thick margin around text boxes for ISP/analyt results
\item added resizeable file list in \texttt{ISP} window
\item added resizeable 'merge files' list when loading data files
\item added resizable file list in \texttt{ISP} window
\item added resizable 'merge files' list when loading data files
\end{itemize}
\item added menu->tools->toggle console to show the console, it is hidden by default now
\item added \texttt{OPTIM} parameter to \texttt{src/CMakeLists.txt} for optimized binary generation on the underlying hardware, use: '\texttt{cmake -DOPTIM=TRUE}'
@@ -5,7 +5,7 @@ \section{Ellipsoidal Objects}
\begin{center}
\includegraphics[width=0.85\textwidth]{../images/form_factor/Ellipsoid/Ellipsoide.png}
\end{center}
\caption{Ellipse of revolution or spheroid and triaxial ellipse}
\caption{Ellipse of revolution (spheroid) and triaxial ellipse}
\label{fig:EllipsoidalObjects}
\end{figure}
@@ -147,21 +147,22 @@ \subsection{Ellipsoid with two equal equatorial semi-axis $R$ and volume $V$}
\clearpage
\subsection{Ellipsoidal core shell structure}
\label{sect:EllipsoidalCoreShell} ~\\
For these form factor plugins a size distribution parameter has been included directly in the form factor to avoid apply nested numerical integration routines. A routine for multi-dimensional integration routines (\texttt{pcubature} \cite{Johnson}) has been used instead.
For these form factor plugins a size distribution parameter has been included directly in the form factor to avoid applying nested numerical integration routines. A routine for multi-dimensional integration (\texttt{pcubature} \cite{Johnson}) has been used instead.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.5\textwidth,height=0.28855\textwidth]{../images/form_factor/Ellipsoid/ellipsoidalShell.png}
\end{center}
\caption{Ellipsoid of revolution with an outer shell of constant thickness $t$} \label{ellipsoidalShell}
\end{figure}
\begin{align}
I_\text{ECSh}(Q) = \left\langle F^2(Q,\mu) \right\rangle
& = \int_0^1 \left[F(Q,\mu)\right]^2 d\mu \\
\left\langle F(Q,\mu) \right\rangle^2 & = \left[\int_0^1 F(Q,\mu)
d\mu \right]^2
I_\text{ECSh}(Q,R_\mathrm{p},R_\mathrm{e},t) = \left\langle F^2(Q,R_\mathrm{p},R_\mathrm{e},t) \right\rangle
& = \int_0^1 \left[F(Q,R_\mathrm{p},R_\mathrm{e},t,\mu)\right]^2 d\mu \\
\left\langle F(Q,R_\mathrm{p},R_\mathrm{e},t) \right\rangle & = \int_0^1 F(Q,R_\mathrm{p},R_\mathrm{e},t,\mu)
d\mu
\end{align}
with
\begin{align}
F(Q,\mu) &= \left(\eta_\text{core}-\eta_\text{shell}\right) V_c\left[
F(Q,R_\mathrm{p},R_\mathrm{e},t,\mu) &= \left(\eta_\text{core}-\eta_\text{shell}\right) V_c\left[
\frac{3j_1(x_c)}{x_c}\right]
+\left(\eta_\text{shell}-\eta_\text{sol}\right) V_t\left[ \frac{3j_1(x_t)}{x_t}\right]
\nonumber \\
@@ -175,16 +176,45 @@ \subsection{Ellipsoidal core shell structure}
\eta_\text{core} &: \text{scattering length density of core} \nonumber \\
\eta_\text{shell} &: \text{scattering length density of shell} \nonumber \\
\eta_\text{sol} &: \text{scattering length density of solvent} \nonumber \\
R_\mathrm{p} &: \text{semi-principal axes of elliptical core} \nonumber \\
R_\mathrm{p} &: \text{polar semi-axes of elliptical core} \nonumber \\
R_\mathrm{e} &: \text{equatorial semi-axis of elliptical core} \nonumber \\
t &: \text{thickness of shell} \nonumber \\
V_c &: \text{volume of core} \nonumber \\
V_t &: \text{total volume of core along with shell} \nonumber
\end{align}
Several variants including a size distribution have been implemented which just differ if only none, one or both axis scale with the same size distribution and if additionally also scales with that distribution.
\begin{align*}
\text{\texttt{Ellipsoidal Shell}} &: \left\langle F^n(Q,R_\mathrm{p},R_\mathrm{e},t)\right\rangle\\
\text{\texttt{Ellipsoidal Shell (t)}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,R_\mathrm{p},R_\mathrm{e},\nu t)\right\rangle \mathrm{d}\nu\\
\text{\texttt{Ellipsoidal Shell (Rp)}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,\nu R_\mathrm{p},R_\mathrm{e}, t)\right\rangle \mathrm{d}\nu\\
\text{\texttt{Ellipsoidal Shell (Rp t)}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,\nu R_\mathrm{p},R_\mathrm{e},\nu t)\right\rangle \mathrm{d}\nu\\
\text{\texttt{Ellipsoidal Shell (Re)}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,R_\mathrm{p},\nu R_\mathrm{e},t)\right\rangle \mathrm{d}\nu\\
\text{\texttt{Ellipsoidal Shell (Re t)}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,R_\mathrm{p},\nu R_\mathrm{e},\nu t)\right\rangle \mathrm{d}\nu\\
\text{\texttt{Ellipsoidal Shell (Re Rp)}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,\nu R_\mathrm{p},\nu R_\mathrm{e},t)\right\rangle \mathrm{d}\nu\\
\text{\texttt{Ellipsoidal Shell (Re Rp t)}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,\nu R_\mathrm{p},\nu R_\mathrm{e},\nu t)\right\rangle \mathrm{d}\nu
\end{align*}
with $n$ being either 1 or 2 depending if the size averaged amplitude or the size averaged intensity needs to be calculated. As a size distribution a lognormal distribution is assumed where all parameters are scaled simultaneously. The used lognormal distribution is defined as
$$
\mathrm{LogNorm}(x,\sigma) = \frac{1}{\sqrt{2\pi}x} \exp\left(-\frac{\ln^2(x)}{2\sigma^2}\right)
$$
\vspace{0.5cm}
\underline{Input Parameters for model \texttt{EllipsoidalCoreShell}:}
~\\
\underline{Input Parameters for model \texttt{Ellipsoidal Shell}:}
\begin{description}
\item[\texttt{R\_p}] semi-principal polar axes of elliptical core $R_\mathrm{p}$
\item[\texttt{R\_e}] equatorial semi-axis axes of elliptical core $R_\mathrm{e}$
\item[\texttt{dummy}] not used
\item[\texttt{t}] thickness of shell $t$
\item[\texttt{dummy}] not used
\item[\texttt{eta\_core}] scattering length density of core $\eta_\text{c}$
\item[\texttt{eta\_shell}] scattering length density of shell $\eta_\text{sh}$
\item[\texttt{eta\_sol}] scattering length density of solvent $\eta_\text{sol}$
\end{description}
~\\
\underline{Input Parameters for model \texttt{Ellipsoidal Shell (t)}, \texttt{Ellipsoidal Shell (Rp)},}
\underline{\texttt{Ellipsoidal Shell (Rp t)}, \texttt{Ellipsoidal Shell (Re)}, \texttt{Ellipsoidal Shell (Re t)},}
\underline{\texttt{Ellipsoidal Shell (Re Rp)}, \texttt{Ellipsoidal Shell (Re Rp t)}:}
\begin{description}
\item[\texttt{R\_p}] semi-principal polar axes of elliptical core $R_\mathrm{p}$
\item[\texttt{R\_e}] equatorial semi-axis axes of elliptical core $R_\mathrm{e}$
@@ -198,17 +228,17 @@ \subsection{Ellipsoidal core shell structure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.768\textwidth,height=0.588\textwidth]{../images/form_factor/Ellipsoid/ellipsoidal_core_shell.png}
\includegraphics[width=0.768\textwidth,height=0.588\textwidth]{../images/form_factor/Ellipsoid/spheroid_core_shell.png}
\end{center}
\caption{form factor of an ellipsoidal core shell $a$, $b$, $b$ and
$t$.} \label{fig:I_ellipsoidal_core_shell}
\caption{Form factor of an ellipsoidal core shell structure including a lognormal size distribution.} \label{fig:I_ellipsoidal_core_shell}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
\subsection{triaxial ellipsoidal core shell structure}
\label{sect:triaxEllShell1} ~\\
Also for these form factor plugins a size distribution parameter has been included directly in the form factor similar to the ellipsoidal shell of revolution to avoid applying nested numerical integration routines. As for the triaxial ellipsoid already a double integration for the orientational averaging is required it becomes even more important to have an optimized integration routine for including an additional size distribution, i.e an additional integration. The routine for multi-dimensional integration (\texttt{pcubature} \cite{Johnson}) has been used.
\begin{figure}[htb]
\begin{center}
@@ -220,7 +250,11 @@ \subsection{triaxial ellipsoidal core shell structure}
\end{figure}
\begin{align}
I_\text{triaxEllSh}(Q) &= \int^1_0 \int ^1_0 dx\,dy\, K_\text{sh}^2(Q,R,R_t)\\
I_\text{triaxEllSh}(Q) = \left\langle F^2(Q,a,b,c,t) \right\rangle &= \int^1_0 \int ^1_0 dx\,dy\, K_\text{sh}^2(Q,R,R_t)\\
\left\langle F(Q,a,b,c,t) \right\rangle & = \int^1_0 \int ^1_0 dx\,dy\, K_\text{sh}(Q,R,R_t)
\end{align}
with
\begin{align}
K(QR) &= 3 \frac{\sin QR - QR\cos QR}{(QR)^3} \\
K_\text{sh}(Q,R,R_t) &= \left(\eta_\text{c}-\eta_\text{sh}\right)K(QR)+\left(\eta_\text{sh}-\eta_\text{sol}\right)K(QR_t) \\
R^2 &= \left[a^2\cos^2\left(\pi x/2\right) + b^2\sin^2\left(\pi x/2\right)\right](1-y^2)+c^2y^2 \nonumber \\
@@ -241,23 +275,55 @@ \subsection{triaxial ellipsoidal core shell structure}
V_t &: \text{total volume of core along with shell} \nonumber
\end{align}
Several variants including a size distribution have been implemented which just differ if only none, one or both axis scale with the same size distribution and if additionally also scales with that distribution.
\begin{align*}
\text{\texttt{triax ellip shell}} &: \left\langle F^n(Q, a, b, c, t)\right\rangle\\
\text{\texttt{triax ellip shell t}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q, a, b, c,\nu t)\right\rangle \mathrm{d}\nu\\
\text{\texttt{triax ellip shell 1}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,\nu a, b, c, t)\right\rangle \mathrm{d}\nu\\
\text{\texttt{triax ellip shell 1t}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,\nu a, b, c,\nu t)\right\rangle \mathrm{d}\nu\\
\text{\texttt{triax ellip shell 2}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,\nu a,\nu b, c, t)\right\rangle \mathrm{d}\nu\\
\text{\texttt{triax ellip shell 2t}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,\nu a,\nu b, c,\nu t)\right\rangle \mathrm{d}\nu\\
\text{\texttt{triax ellip shell 3}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,\nu a,\nu b,\nu c, t)\right\rangle \mathrm{d}\nu\\
\text{\texttt{triax ellip shell 3t}} &: \int_0^\infty \mathrm{LogNorm}(\nu,\sigma) \left\langle F^n(Q,\nu a,\nu b,\nu c,\nu t)\right\rangle \mathrm{d}\nu
\end{align*}
with $n$ being either 1 or 2 depending if the size averaged amplitude or the size averaged intensity needs to be calculated. As a size distribution a lognormal distribution is assumed where all parameters are scaled simultaneously. The used lognormal distribution is defined as
$$
\mathrm{LogNorm}(x,\sigma) = \frac{1}{\sqrt{2\pi}x} \exp\left(-\frac{\ln^2(x)}{2\sigma^2}\right)
$$
\vspace{3cm}
\underline{Input Parameters for model \texttt{triaxEllShell1}:}
\noindent \underline{Input Parameters for model \texttt{triax ellip shell}:}
\begin{description}
\item[\texttt{a}] semi-axes of elliptical core $a$
\item[\texttt{b}] semi-axes of elliptical core $b$
\item[\texttt{c}] semi-axes of elliptical core $c$
\item[\texttt{t}] thickness of shell $t$
\item[\texttt{dummy}] not used
\item[\texttt{eta\_c}] scattering length density of core $\eta_\text{c}$
\item[\texttt{eta\_sh}] scattering length density of shell $\eta_\text{sh}$
\item[\texttt{eta\_sol}] scattering length density of solvent $\eta_\text{sol}$
\end{description}
~\\
\underline{Input Parameters for model \texttt{triax ellip shell t}, \texttt{triax ellip shell 1},}
\underline{\texttt{triax ellip shell 1t}, \texttt{triax ellip shell 2}, \texttt{triax ellip shell 2t},}
\underline{\texttt{triax ellip shell 3}, \texttt{triax ellip shell 3t}}:
\begin{description}
\item[\texttt{a}] semi-axes of elliptical core $a$
\item[\texttt{b}] semi-axes of elliptical core $b$
\item[\texttt{c}] semi-axes of elliptical core $c$
\item[\texttt{t}] thickness of shell $t$
\item[\texttt{sigma}] width of the LogNorm size distribution $\sigma$
\item[\texttt{eta\_c}] scattering length density of core $\eta_\text{c}$
\item[\texttt{eta\_sh}] scattering length density of shell $\eta_\text{sh}$
\item[\texttt{eta\_sol}] scattering length density of solvent $\eta_\text{sol}$
\end{description}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.768\textwidth,height=0.588\textwidth]{../images/form_factor/Ellipsoid/triax_ellipsoidal_core_shell.png}
\includegraphics[width=0.48\textwidth]{../images/form_factor/Ellipsoid/triax_ellipsoidal_core_shell_1.png}
\includegraphics[width=0.48\textwidth]{../images/form_factor/Ellipsoid/triax_ellipsoidal_core_shell_2.png}
\end{center}
\caption{Form factor of a triaxial ellipsoidal core shell with semi
axis $a$, $b$ and $c$ and a shell thickness $t$.}
axis $a$, $b$ and $c$, with and without a shell thickness $t$ plus an additional size distribution.}
\label{fig:I_triax_ellipsoidal_core_shell}
\end{figure}
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