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documentation for random flight structure factor

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Kohlbrecher committed Dec 22, 2017
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Structure factor of a random flight model} \hspace{1pt}
\cite{Schweins2004,Burchard1970,Giehm2010}
The random flight model describes a discrete chain, where the positions of the $N$ units forming the discrete chains follow a 3D random walk. The distance between neighbouring units is constants.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.35\textwidth]{../images/structure_factor/randomflight3D.png}
\end{center}
\caption{Random flight of $N$ particles with constant distances $D$.}
\label{fig:randomflight3D}
\end{figure}
The structure factor $S_N(QD)$ describing such an arrangement of $N$ particles on a random flight
with a constant step size $D$ is given by \cite{Burchard1970} as
\begin{align}
S_N(QD)&=
\frac{2}{1-\frac{\sin⁡(QD)}{QD} )}-1-\frac{2\left[1-\left[\frac{\sin⁡(QD)}{QD}\right]^N \right]}{N\left[1-\frac{\sin⁡(QD)}{QD}\right]^2} \frac{\sin⁡(QD)}{QD}
\end{align}
The formula above is only defined for the whole real space for integer values of $N$. For non integer values $n$ a linear interpolation between $[n]$ and $[n]+1$ is taken, where $[n]$ is the largest integer smaller or equal to $n$. With $w=n-[n]$ we get
The formula above is only real for all values of $QD$ for integer values of $N$. For non integer values $n$ a linear interpolation between $[n]$ and $[n]+1$ is taken \cite{Giehm2010}, where $[n]$ is the largest integer smaller or equal to $n$. With $w=n-[n]$ we get
\begin{align}
S_N(QD)&= (1-w)S_{[n]} + wS_{[n]+1}
S_n(QD)&= (1-w)S_{[n]}(QD) + wS_{[n]+1}(QD)
\end{align}
\noindent \underline{Input Parameters for model \texttt{random flight}:}\\
\begin{description}
\item[\texttt{D}] step length $D$
\item[\texttt{n}] number of steps $n$
\end{description}
\noindent\underline{Note:}
\begin{itemize}
\item $N$ needs to be larger or equal to 1. For non-integer values of $N$ the curve is linear interpolated between the structure factor for $N$ and $N+1$.
\end{itemize}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.75\textwidth]{../images/structure_factor/randomflight.png}
\end{center}
\caption{Random flight structure factor with $N$ steps of length $D$.}
\label{fig:randomflight}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ordered particle systems} \hspace{1pt}
\label{sec:ops}
@@ -3,8 +3,8 @@
\section{Non spherical-symmetric randomly oriented particles}
\label{sec:anisotropic_randomlyoriented_particles}
\subsection{Rectangular parallelepiped}
In a rectangular parallelepiped, all angles are right angles, and opposite faces of a parallelepiped are equal. Also the terms rectangular cuboid or orthogonal parallelepiped are used to designate this polyhedron. Four variants of rectangular parallelepipeds have been implemented, where either none, one, two, or all three axis have a common size distribution.
\label{sec:reccuboid}
In a rectangular parallelepiped, all angles are right angles, and opposite faces of a parallelepiped are equal. Also the terms rectangular cuboid or orthogonal parallelepiped are used to designate this polyhedron. Four variants of rectangular parallelepipeds have been implemented, where either none, one, two, or all three axis have a common size distribution.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.75\textwidth]{../images/form_factor/anisotropic/rectangularparallelepiped.png}
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@@ -158,6 +158,13 @@ Ronald Bracewell.
\newblock {\em The Projection-Slice Theorem}, pages 493--504.
\newblock Springer US, Boston, MA, 2003.
\bibitem{Burchard1970}
W.~Burchard and K.~Kajiwara.
\newblock The statistics of stiff chain molecules i. the particle scattering
factor.
\newblock {\em Proceedings of the Royal Society of London A: Mathematical,
Physical and Engineering Sciences}, 316(1525):185--199, apr 1970.
\bibitem{Burchard1996}
W.~Burchard, E.~Michel, and V.~Trappe.
\newblock Conformational properties of multiply twisted ring systems and
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@@ -4781,6 +4781,7 @@ @Article{Burchard1970
url = {http://rspa.royalsocietypublishing.org/content/316/1525/185},
abstract = {Analytic expressions for the particle scattering factor of stiff chains have been derived, both for the wormlike and a discrete chain model with an axial symmetric potential of hindered rotation. The angular distribution functions agree well with the results of Monte Carlo calculations by Heine, Kratky \& Roeppert (1962), if the chains are longer than five persistence lengths. The particle scattering factor of short chains can be well represented by the simple Guinier approximation. A transition point from the behaviour of a coil to that of the rod-like short chain sections has been determined by graphical extrapolation and appears at Xa = 2.87 {\textpm} 0.05. The difference between the wormlike and the discrete chain models turned out to be smaller than 14\% even for an alkane type chain with free rotation of the chain elements and decreases with increasing chain stiffness. The influence of the cross-section has been taken into account by representing the chain by a pearl necklace. Comparison with X -ray small angle scattering measurements of a cellulosetricarbanilate reveals close similarities between calculated and experimental curves.},
publisher = {The Royal Society},
timestamp = {2017-12-21},
}
@Article{Schweins2004,
@@ -4794,6 +4795,7 @@ @Article{Schweins2004
pages = {25--42},
doi = {10.1002/masy.200450702},
publisher = {Wiley-Blackwell},
timestamp = {2017-12-21},
}
@Article{Giehm2010,
@@ -4807,6 +4809,24 @@ @Article{Giehm2010
pages = {115--133},
doi = {10.1016/j.jmb.2010.05.060},
publisher = {Elsevier {BV}},
timestamp = {2017-12-21},
}
@Article{Kuhn1934,
author = {Kuhn, Werner},
title = {{\"U}ber die Gestalt fadenf{\"o}rmiger Molek{\"u}le in L{\"o}sungen},
journal = {Kolloid-Zeitschrift},
year = {1934},
volume = {68},
number = {1},
month = {Feb},
pages = {2--15},
issn = {1435-1536},
doi = {10.1007/BF01451681},
url = {https://doi.org/10.1007/BF01451681},
abstract = {Es werden statistische Betrachtungen angestellt betreffend die Form, welche kettenf{\"o}rmig gebaute Molek{\"u}le, welche in einer L{\"o}sung (oder im Gasraum) suspendiert sind, infolge der Valenzwinkelung und der freien Drehbarkeit annehmen k{\"o}nnen.},
day = {01},
timestamp = {2017-12-22},
}
@Comment{jabref-meta: databaseType:biblatex;}
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