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some corrected typos in documentation

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Kohlbrecher committed Mar 28, 2018
1 parent 0db1565 commit 3846d30e62381289522df3e1ea52d75fee6ca5a0
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  1. BIN CHANGES.txt
  2. BIN doc/images/form_factor/Ellipsoid/Ellipsoide.png
  3. BIN doc/images/form_factor/Ellipsoid/OblateSpheroid.png
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  8. BIN doc/images/form_factor/anisotropic/Pcs_homogeneousXS_txt.png
  9. BIN doc/images/form_factor/anisotropic/SketchdiffXSWorm.png
  10. BIN doc/images/form_factor/anisotropic/ThinDisc.png
  11. BIN doc/images/form_factor/anisotropic/ThinHollowCylinder.png
  12. BIN doc/images/form_factor/anisotropic/ThinRod.png
  13. BIN doc/images/form_factor/anisotropic/localplanar.png
  14. BIN doc/images/form_factor/anisotropic/planar2centrosymm_txt.png
  15. BIN doc/images/form_factor/anisotropic/rectangularparallelepiped.png
  16. BIN doc/images/form_factor/cylindrical_obj/CylShell2IQ.png
  17. BIN doc/images/form_factor/cylindrical_obj/DiscIQ.png
  18. BIN doc/images/form_factor/cylindrical_obj/ExactCylinder.png
  19. BIN doc/images/form_factor/cylindrical_obj/FlatCylinder.png
  20. BIN doc/images/form_factor/cylindrical_obj/LongCylinder.png
  21. BIN doc/images/form_factor/cylindrical_obj/Pcs_ellCylShell.png
  22. BIN doc/images/form_factor/cylindrical_obj/PorodCylinder.png
  23. BIN doc/images/form_factor/cylindrical_obj/RodIQ.png
  24. BIN doc/images/form_factor/cylindrical_obj/Torus.png
  25. BIN doc/images/form_factor/cylindrical_obj/beads_helix_model.png
  26. BIN doc/images/form_factor/cylindrical_obj/beads_helix_model_1.png
  27. BIN doc/images/form_factor/cylindrical_obj/beads_helix_model_2.png
  28. BIN doc/images/form_factor/cylindrical_obj/cylinder.png
  29. BIN doc/images/form_factor/cylindrical_obj/cylshell2D.png
  30. BIN doc/images/form_factor/cylindrical_obj/disc.png
  31. BIN doc/images/form_factor/cylindrical_obj/ellCylShell.png
  32. BIN doc/images/form_factor/cylindrical_obj/ellCylShell1.png
  33. BIN doc/images/form_factor/cylindrical_obj/ellCylShell1_2.png
  34. BIN doc/images/form_factor/cylindrical_obj/ellCylShell_shape.png
  35. BIN doc/images/form_factor/cylindrical_obj/fanlike_helices3D.png
  36. BIN doc/images/form_factor/cylindrical_obj/fanlike_helicesXS.png
  37. BIN doc/images/form_factor/cylindrical_obj/flat_cylinder.png
  38. BIN doc/images/form_factor/cylindrical_obj/long_cylinder.png
  39. BIN doc/images/form_factor/cylindrical_obj/partly_aligned_CylShell.png
  40. BIN doc/images/form_factor/cylindrical_obj/partly_aligned_cylinders.png
  41. BIN doc/images/form_factor/cylindrical_obj/partly_aligned_discs.png
  42. BIN doc/images/form_factor/cylindrical_obj/rod.png
  43. BIN doc/images/form_factor/cylindrical_obj/round_helices3D.png
  44. BIN doc/images/form_factor/cylindrical_obj/round_helices3D_2nd.png
  45. BIN doc/images/form_factor/cylindrical_obj/roundhelices1.png
  46. BIN doc/images/form_factor/cylindrical_obj/straightSuperhelix.png
  47. BIN doc/images/form_factor/cylindrical_obj/superhelixcoiled1.png
  48. BIN doc/images/form_factor/cylindrical_obj/superhelixcoiled2.png
  49. BIN doc/images/form_factor/cylindrical_obj/superhelixcoiled3.png
  50. BIN doc/images/form_factor/cylindrical_obj/thinsinglehelices.png
  51. BIN doc/images/form_factor/ferrofluid/ferrfluidparticle.png
  52. BIN doc/images/form_factor/reptating_chain/lamb2t_tau_0.png
  53. BIN doc/images/form_factor/reptating_chain/lamb2t_tau_0_1.png
  54. BIN doc/images/form_factor/reptating_chain/lamb2t_tau_0_3.png
  55. BIN doc/images/form_factor/reptating_chain/lamb2t_tau_1.png
  56. BIN doc/images/form_factor/reptating_chain/lamb2t_tau_10.png
  57. BIN doc/images/form_factor/reptating_chain/lamb2t_tau_100.png
  58. BIN doc/images/form_factor/reptating_chain/lamb2t_tau_3.png
  59. BIN doc/images/form_factor/reptating_chain/lambda_2_reptating_chains.png
  60. BIN doc/images/form_factor/reptating_chain/reptating_chain.cdr
  61. BIN doc/images/form_factor/reptating_chain/reptating_chain.png
  62. +2 −2 doc/manual/IO_formats.tex
  63. +2 −2 doc/manual/SASfit_absInt.tex
  64. +0 −2 doc/manual/SASfit_ch2.tex
  65. +1 −1 doc/manual/SASfit_ch2_Spheres_Shells.tex
  66. +3 −41 doc/manual/SASfit_ch2_cluster.tex
  67. +2 −2 doc/manual/SASfit_ch2_otherObj.tex
  68. +9 −9 doc/manual/SASfit_ch3.tex
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  71. +1 −1 doc/manual/SASfit_pluginSD.tex
  72. +2 −0 doc/manual/SASfit_plugins.tex
  73. +2 −2 doc/manual/SASfit_pluginsFF_anisotropic.tex
  74. +11 −0 doc/manual/SASfit_pluginsFF_azimuthal.tex
  75. +11 −11 doc/manual/SASfit_pluginsFF_cylindrical_obj.tex
  76. +186 −127 doc/manual/{SASfit_ch2_ShearedObj.tex → SASfit_pluginsFF_deformed_sheared_obj.tex}
  77. +1 −1 doc/manual/SASfit_pluginsFF_elliptical_obj.tex
  78. +6 −6 doc/manual/basics.tex
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  82. +1 −1 doc/manual/intro.tex
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  90. BIN doc/manual/sasfit.pdf
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  94. +2 −2 doc/manual/sasfit_OZsolver.tex
  95. BIN saskit/libgcc_s_dw2-1.dll
  96. BIN saskit/libstdc++-6.dll
  97. +4,357 −3,754 saskit/src/mk-2.4.9.2/unix/configure
  98. +20 −20 src/plugins/ellipsoidal_shell/sasfit_ff_ellipsoidal_shell_0.c
  99. +20 −20 src/plugins/ellipsoidal_shell/sasfit_ff_ellipsoidal_shell_0t.c
  100. +20 −20 src/plugins/ellipsoidal_shell/sasfit_ff_ellipsoidal_shell_2.c
  101. +20 −20 src/plugins/ellipsoidal_shell/sasfit_ff_ellipsoidal_shell_2t.c
  102. +20 −20 src/plugins/ellipsoidal_shell/sasfit_ff_ellipsoidal_shell_re.c
  103. +21 −21 src/plugins/ellipsoidal_shell/sasfit_ff_ellipsoidal_shell_re_t.c
  104. +20 −20 src/plugins/ellipsoidal_shell/sasfit_ff_ellipsoidal_shell_rp.c
  105. +20 −20 src/plugins/ellipsoidal_shell/sasfit_ff_ellipsoidal_shell_rp_t.c
  106. +20 −20 src/plugins/ellipsoidal_shell/sasfit_ff_spheroid_R.c
  107. +20 −20 src/plugins/ellipsoidal_shell/sasfit_ff_spheroid_V.c
  108. +20 −20 src/plugins/ellipsoidal_shell/sasfit_ff_spheroid_nu.c
  109. +9 −0 src/plugins/groups.form_fac.def
  110. +3 −3 src/plugins/reptating_chain/include/sasfit_reptating_chain.h
  111. +4 −1 src/plugins/reptating_chain/sasfit_ff_reptating_chain.c
  112. +6 −2 src/plugins/triax_ellip_shell/include/private.h
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BIN +342 Bytes (100%) CHANGES.txt
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@@ -46,7 +46,7 @@ \subsection{Input Format} \hspace{1pt}\\
As the first lines start with a string, they will be automatically
ignored. To interpret the three columns as $Q$, $I(Q)$, $\Delta
I(Q)$ the format string should be simply {\tt xyz}. The BerSANS
format can also be read in by explicitly selecting
format can also be read in by explicitly selecting
the "BerSANS"-format button in the menu instead of the
"ASCII"-format. \\[1cm]
@@ -185,7 +185,7 @@ \subsection{Error bar} \hspace{1pt}\\
\begin{figure}[htb]
\begin{center}
\subfigure[simulated model data set with supplied error bars]{\label{fig:ModelDataWithError}\includegraphics[width=0.47\textwidth]{../images/ErrorBar/ModelDataWithError.png}}
\subfigure[Ratio between supplied error bar and the error bar guessed by \SASfit]{\label{fig:ErrorGUI}\includegraphics[width=0.47\textwidth]{../images/ErrorBar/RatioErrSuppGuess.png}}
\subfigure[Ratio between supplied error bar and the error bar guessed by \SASfit]{\label{fig:ErrorGUI2}\includegraphics[width=0.47\textwidth]{../images/ErrorBar/RatioErrSuppGuess.png}}
\end{center}
\caption{Comparison of a simulated model data set with supplied error bar and an error bar guessed by \SASfit.}
\label{fig:ErrBar}
@@ -143,7 +143,7 @@ \section{Contrast - Concentration - Forward Scattering - Particle Volume - Absol
so that
\begin{align}
\frac{d\sigma}{d\Omega}(Q) &= c\frac{N_A}{M_r M_u} P(Q)
\label{eq:xsabsolut1}
\label{eq:xsabsolut2}
\end{align}
$M_r$ is the relative molar mass of the particle
\footnote{
@@ -406,7 +406,7 @@ \section{Volume fractions}
f_p = \int_0^\infty n(L) V_\textrm{cyl}(R,L) \, dL
= \int_0^\infty N p(L) \pi R^2 L \, dL
= N R^2 \pi \langle L \rangle
\label{eq:fpMomentsCylR}
\label{eq:fpMomentsCylL}
\end{align}
depending if we have a distribution over the radius $R$ or the
length $L$. In both cases the volume fraction can be expressed in
@@ -14,8 +14,6 @@ \chapter{Form Factors}
\clearpage
\input{SASfit_ch2_planar.tex}
\clearpage
\input{SASfit_ch2_ShearedObj.tex}
\clearpage
\input{SASfit_ch2_magneticObj.tex}
\clearpage
\input{SASfit_ch2_MieSLS.tex}
@@ -10,7 +10,7 @@ \subsection{Sphere}
\end{center}
\caption{Sphere with diameter $2R$} \label{fig:Sketch_sphere}
\end{figure}
The scattering intensity and scattering amplitude of a homogeneous sphere with a small relative refraction index is given by \cite{Rayleigh1914}
\begin{subequations}
\begin{align}
I_\text{Sphere}(Q,R) = K^2(Q,R,\Delta\eta) \label{eq:I_sphere}
@@ -3,7 +3,7 @@ \section{Clustered Objects}
\label{sect:ClusteredObjects}
\subsection{Mass Fractal
\cite{Sorensen1999,Sorensen1992,Hurd1988,Lin1989,Lin1990,Lin1990a}}
\cite{Sorensen1999,Sorensen1992,Hurd1988,Lin1989,Lin1990,Lin1990a,Lin1990b}}
\label{sect:MassFractal}
\hspace{1pt} \\
@@ -84,49 +84,11 @@ \subsection{Mass Fractal
\label{fig:FFCluster}
\end{figure}
%REFERENCES: ~\\
%\cite{Sorensen1999}
%1. C. M. Sorensen and G. M. Wang,
%Size distribution effect on the power law regime of the structure factor of fractal aggregates,
%Physical Review E, Vol 60, No. 6, (1999) 7143-7148 \\
%\cite{Sorensen1992}
%2. C. M. Sorensen, J. Cai, and N. Lu,
%Test of Static Structure Factors for Describing Light Scattering from Fractal Soot Aggregates,
%Langmuir 1992,8, 2064-2069\\
%\cite{Hurd1988}
%3. A.J. Hurd and W.L. Flower,
%In Situ Growth and Structure of Fractal Silica Aggregates in a Flame,
%Journal of Colloid and Interface Science, Vol. 122, No 1, (1988) 178--192 \\
%\cite{Lin1989}
%4. M.Y. Lin, H.M. Lindsay, D.A. Weitz, R.C. Ball, R. Klein, P. Meakin,
%Universality of Fractal Aggregates as Probed by Light Scattering,
%Proc. R. Soc. Lond. A \textbf{423}, 71-87 (1989) \\
%\cite{Lin1990}
%5. M.Y. Lin, H.M. Lindsay, D.A. Weitz, R.C. Ball, R. Klein, P. Meakin,
%Universal reaction-limited colloidal aggregation,
%Physical Review A, Vol 41, No. 4, 2005--2020 (1990) \\
%\cite{Lin1990a}
%5. M.Y. Lin, H.M. Lindsay, D.A. Weitz, R.C. Ball, R. Klein, P. Meakin,
%Universal diffusion-limited colloid aggregation
%J. Phys.: Condens. Matter 2 (1990) 3093-3113. \\
%\begin{align}
%I_\text{MassFract}(Q,\xi,D,R) = \left(1 + \cfrac{D \Gamma(D-1)
%\sin\left([D-1]\arctan(Q \xi)\right)}{\left( Q R\right)^D
%\left[1+\cfrac{1}{Q^2\xi^2}\right]^{(D-1)/2}}
% \right) K^2(Q,R,\Delta\eta)
%\end{align}
%where $D$ is the fractal dimension, $\xi$ is a cut-off length for
%the fractal correlations, $\Gamma(x)$ is the gamma function, and
%$K^2(Q,R,\Delta\eta)$ the form factor of a sphere out of which the
%fractal consists.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
\subsection{Stacked Discs \cite{Kratky194935,Hanley2003}}
\subsection{Stacked Discs \cite{Kratky1949,Hanley2003}}
\label{sect:StackedDiscs}
~\\
@@ -148,7 +110,7 @@ \subsection{Stacked Discs \cite{Kratky194935,Hanley2003}}
Here it is assume that the nearest neighbor distance between the
platelets obeys a Gaussian distribution and consider an internal
structure factor, $S(Q,\Theta)$, first proposed by Kratky and Porod
in 1949 \cite{Kratky194935}
in 1949 \cite{Kratky1949}
\begin{align}
S(Q,\Theta) &= 1+\frac{2}{n} \sum_{k=1}^{n-1} (n-k)
\cos(kDQ\cos(\Theta))
@@ -129,7 +129,7 @@ \subsection{SuperParStroboPsi} ~\\
with $\alpha=BM_pV_p/(k_BT)$ being the Langevin parameter. For this orientation probability distribution
the first order parameters can be calculates as
\begin{subequations}
\label{eq:S_l_Boltzmann}
\label{eq:S_i_Boltzmann}
\begin{align}
S_0 & = 1 \\
S_1 & = L(\alpha) = \coth\alpha - \frac{1}{\alpha} \\
@@ -219,7 +219,7 @@ \subsection{SuperParStroboPsi} ~\\
In the case of a Boltzmann orientation distribution
$f(\theta)=\exp\left(\frac{\BM{B\mu}}{k_BT}\right)=\exp\left(\frac{B\mu\cos\theta}{k_BT}\right)$
the order parameter $S_l$ already have been given in eq.\ \ref{eq:S_l_Boltzmann}
the order parameter $S_l$ already have been given in eq.\ \ref{eq:S_i_Boltzmann}
and the spin dependent intensities can be written as
\begin{subequations}
\begin{align}
View
@@ -243,7 +243,7 @@ \subsection{Hard Sphere \cite{Percus1958,Vrij1979}
\subsection{Sticky Hard Sphere} ~\\
In Baxter's model \cite{Baxter1968,Robertus1989,Kruif1989,Barboy1974,Menon1991,Menon1991a}
of adhesive hard spheres the pair interaction potential $U(r)$ is replaces by
of adhesive hard spheres the pair interaction potential $U(r)$ is replaced by
\begin{equation}
\frac{U(r)}{k_BT} =
\begin{cases}
@@ -293,13 +293,13 @@ \subsection{Sticky Hard Sphere} ~\\
\subsection{Sticky Hard Sphere ($2^\text{nd}$ version \cite{Regnaut1989,Regnaut1990})} ~\\
In Baxter's model of adhesive hard spheres the pair interaction
potential $U(r)$ is replaces by
potential $U(r)$ is replaced by
\begin{equation}
\frac{U(r)}{k_BT} =
\begin{cases}
\infty & \text{for} \quad 0<r<\sigma \\
\ln\frac{12\tau\Delta}{\sigma+\Delta} & \text{for} \quad \sigma<r<\sigma+\Delta \\
0 & \text{for} \quad r>\sigma+\Delta
\infty & \text{for} \quad 0<r<\sigma-\Delta \\
\ln\frac{12\tau\Delta}{\sigma+\Delta} & \text{for} \quad \sigma-\Delta<r<\sigma \\
0 & \text{for} \quad r>\sigma
\end{cases}
\end{equation}
@@ -340,7 +340,7 @@ \subsection{Sticky Hard Sphere ($2^\text{nd}$ version \cite{Regnaut1989,Regnaut1
\end{center}
\caption{Structure factor $S(q)$ for a sticky hard sphere interaction potential for the different
stickiness parameters $\tau$.}
\label{fig:SQStickyHardSphere1}
\label{fig:SQStickyHardSphere2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -359,7 +359,7 @@ \subsection{Square Well Potential \cite{Sharma1977}} ~\\
\end{equation}
where $\lambda$ and $\epsilon$ correspond to the breadth and the
depth of the square well potential. The structure factor $S(Q)$ is
than given by the following relations:
then given by the following relations:
\begin{subequations}
\begin{align}
S(Q) = & \frac{1}{1-C(Q)} \\
@@ -407,7 +407,7 @@ \subsection{Square Well Potential 2} ~\\
\end{equation}
where $\Delta$ and $\epsilon$ correspond to the width and the
depth of the square well potential. The structure factor $S(Q)$ is
than given by the following relations:
then given by the following relations:
\begin{equation}
S(Q) = 1
-4\pi\rho\sigma^3\frac{\sin(Q\sigma)-Q\sigma\cos(Q\sigma)}{Q^3\sigma^3}
@@ -969,7 +969,7 @@ \subsection{Mass Fractal (OverlapSph Cut-Off)}
\end{center}
\caption{Structure factor of a mass fractal with a
cut-off function $h_\text{OverlapSph}(r,\xi) = \left(1+\frac{r}{4\xi}\right)\left(1-\frac{r}{2\xi}\right)^2$ for $r\leq 2\xi$.}
\label{fig:SQGaussCutOff}
\label{fig:SQOverlappSphCutOff}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -86,7 +86,7 @@ \section{Log-Normal distribution}
\caption{LogNormal distribution function ($R_0=1$ and $p=1$ has
been been set both to one here). Valid parameter ranges: $R \in
(0,\infty)$, $R_0 \in (0,\infty)$, $\sigma\geq 0$, $p \in
(-\infty,\infty)$} \label{NogNormal}
(-\infty,\infty)$} \label{fig:LogNormal}
\end{figure}
The \texttt{LogNorm} distribution is a continuous distribution in
@@ -134,7 +134,7 @@ \section{Schulz-Zimm (Flory) distribution}
\end{center}
\caption{The $\text{SZ}(X,N,X_a,k)$ distribution function. Valid
parameter ranges: $X \in [0,\infty)$, $X_a \in (0,\infty)$,
$k=X_a^2/\sigma^2 > 0$} \label{NogNormal}
$k=X_a^2/\sigma^2 > 0$} \label{fig:SchulzZimm}
\end{figure}
A function commonly used to present polymer molecular weight distributions is the Schulz-Zimm function
View
@@ -569,7 +569,7 @@ \subsection{F-Variance (Amplitude)} ~\\
\includegraphics[width=0.768\textwidth,height=0.588\textwidth]{FvarianceAmplitude.png}
\end{center}
\caption{Plot of \texttt{F-variance (Amplitude)} distribution.}
\label{fig:GammaAmplitude}
\label{fig:FvarianceAmplitude}
\end{figure}
\vspace{5mm}
@@ -1243,7 +1243,7 @@ \subsection{Giddings (Area)} ~\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
\section{Haarhoff - Van der Linde (Area)} ~\\
\label{sec:HaarhoffVanderLindeArea}
\label{sec:HaarhoffVanderLindeArea}
\begin{align}
\textrm{HVL}(x) & =
@@ -1261,7 +1261,7 @@ \section{Haarhoff - Van der Linde (Area)} ~\\
\frac{A}{\sigma\sqrt{2\pi}}
\exp\left[-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right]
\end{align}
as well as for $\mu\rightarrow 0$
as well as for $\mu\rightarrow 0$
\begin{align}
\lim_{\mu \rightarrow 0} \textrm{HVL}(x) & =
\frac{A}{\sigma\sqrt{2\pi}}
@@ -2026,7 +2026,7 @@ \subsection{Lorentzian (Area)} ~\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
\section{Maxwell-Boltzmann distribution}
\label{sec:Lorentzian}
\label{sec:MaxwellBoltzmann}
The Maxwell-Boltzmann distribution describes particle speeds in
gases, where the particles do not constantly interact with each
@@ -2493,10 +2493,10 @@ \subsection{Voigt (Amplitude)} ~\\[5mm]
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.768\textwidth,height=0.588\textwidth]{VoigtArea.png}
\includegraphics[width=0.768\textwidth,height=0.588\textwidth]{VoigtAmplitude.png}
\end{center}
\caption{Plot of \texttt{Voigt (Area)} distribution.}
\label{fig:VoigtArea}
\caption{Plot of \texttt{Voigt (Amplitude)} distribution.}
\label{fig:VoigtAmplitude}
\end{figure}
@@ -51,7 +51,7 @@ \section{LogNorm\_fp} \hspace{1pt}
cylinder length $L$ has a size distribution the volume fraction $f_p$ is calculated differently namely
in case for a radius distribution by
\begin{align}
f_p &= 10^{21} \int_0^\infty \mathrm{LogNorm}(R) V_\text{cyl}(R,L) \, dR \label{eq:fpMomentsV} \\
f_p &= 10^{21} \int_0^\infty \mathrm{LogNorm}(R) V_\text{cyl}(R,L) \, dR \label{eq:fpMomentsA} \\
&= 10^{21} \int_0^\infty \mathrm{LogNorm}(R) \pi R^2L \, dR = 10^{21} N \pi L \langle X^2 \rangle
\end{align}
and in case of a length distribution by
@@ -6,8 +6,10 @@ \chapter{Plugin functions for form factors}
\input{SASfit_pluginsFF_spheres_shells.tex}
\input{SASfit_pluginsFF_elliptical_obj.tex}
\input{SASfit_pluginsFF_cylindrical_obj.tex}
\input{SASfit_pluginsFF_deformed_sheared_obj.tex}
\input{SASfit_pluginsFF_magnetic_structures.tex}
\input{SASfit_pluginsFF_nonparticular.tex}
\input{SASfit_pluginsFF_azimuthal.tex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
@@ -260,7 +260,7 @@ \subsubsection{Pcs(Q) for a layered centro symmetric cross-section structure} ~\
\caption{Two layered centro symmetric structure with a core thickness of $L_\textrm{c}$ and an outer layer thickness
$L_{L_\textrm{sh}}$. The corresponding scattering length densities of the core, the shell layer and the solvent are
$L_{L_\textrm{c}}$, $L_{L_\textrm{sh}}$, and $L_{L_\textrm{solv}}$.}
\label{fig:Pcs:TwoInfinitelyThinLayers}
\label{fig:Pcs:LayeredCentroSymmetricCrossSection}
\end{figure}
This cross-section form factor describes the scattering of a layered centro symmetric cross-section structure.
@@ -789,7 +789,7 @@ \subsubsection{P'(Q): Kholodenko's worm} ~\\
\includegraphics[width=0.617\textwidth,height=0.762\textwidth]{SemiflexiblePolymerTxt.png}
\end{center}
\caption{}
\label{fig:KholodenkoWorm}
\label{fig:Pprime4KholodenkoWorm}
\end{figure}
By using the analogy between Dirac's fermions
@@ -0,0 +1,11 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
\section{Functions for analysing azimuthal averaged data}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage
\subsection{$\sin^2$-$\sin^4$ azimuthal analysis} ~\\
\subsection{Maier-Saupe azimuthal analysis} ~\\
\subsection{Ellipsoidal azimuthal analysis} ~\\ \cite{Summerfield1983,Mildner1983,Reynolds1984,Hammouda1986,Hammouda1986a,Saraf1989,Svetogorsky1990,Gu2016,Gu2018}
@@ -16,11 +16,11 @@ \subsubsection{Fanlike helix} ~\\
Double helices with a fanlike cross section in the plane perpendicular to the helix axis have been described by by Schmidt and Pringle \cite{Schmidt1970,Pringle1971}. The model has been generalized by Fakuda et al.\ \cite{Fukuda2002} for an arbitrarily shaped cross section and by Teixeira et al.\ \cite{Teixeira2010} to a multi radial shell with fanlike cross section.
\begin{figure}[htb]
\begin{center}
\subfigure[top on view]{\label{fig:beadshelixside}\includegraphics[width=0.45\textwidth]{../images/form_factor/cylindrical_obj/fanlike_helicesXS.png}}
\subfigure[top on view]{\label{fig:fanhelixside1}\includegraphics[width=0.45\textwidth]{../images/form_factor/cylindrical_obj/fanlike_helicesXS.png}}
\hfill
\subfigure[side view]{\label{fig:beadshelixside}\includegraphics[width=0.45\textwidth]{../images/form_factor/cylindrical_obj/fanlike_helices3D.png}}
\subfigure[side view]{\label{fig:fanhelixside2}\includegraphics[width=0.45\textwidth]{../images/form_factor/cylindrical_obj/fanlike_helices3D.png}}
\end{center}
\caption{Double helix with strands of round cross sections.} \label{fig:roundhelix}
\caption{Double helix with strands of round cross sections.} \label{fig:fanhelix}
\end{figure}
\begin{align}
@@ -38,7 +38,7 @@ \subsubsection{Fanlike helix} ~\\
\end{cases} \\
P_\text{rod}(Q,H) &= H^2 \left(2\frac{\mathrm{Si}(QH)}{(QH)}-\left(\frac{\sin(QH/2)}{QH/2}\right)^2\right)
\end{align}
with $\epsilon_n=1$ for $n=0$ and $\epsilon_n=2$ for $n\geq 1$.
with $\epsilon_n=1$ for $n=0$ and $\epsilon_n=2$ for $n\geq 1$.
The sum converges very fast and for small $Q$-values the first few terms are already sufficient. However, {\tt SASfit} is continuing the sum until the relative change of the sum is less than $10^{-10}$.
@@ -85,9 +85,9 @@ \subsubsection{Helix with round cross-section} ~\\
Originally Franklin et al.\ \cite{Franklin1956} and Puigjaner et al.\ \cite{Puigjaner1974} have given the form factor of a cylinder with a groove of a double helix shape of round cross section. Fukada \cite{Fukuda2002} is discussing the form factor of the random oriented double helices alone, but with arbitrary cross-sections.
\begin{figure}[htb]
\begin{center}
\subfigure[top on view]{\label{fig:beadshelixside}\includegraphics[width=0.45\textwidth]{../images/form_factor/cylindrical_obj/roundhelices1.png}}
\subfigure[top on view]{\label{fig:roundhelix1}\includegraphics[width=0.45\textwidth]{../images/form_factor/cylindrical_obj/roundhelices1.png}}
\hfill
\subfigure[side view]{\label{fig:beadshelixside}\includegraphics[width=0.45\textwidth]{../images/form_factor/cylindrical_obj/round_helices3D.png}}
\subfigure[side view]{\label{fig:roundhelix2}\includegraphics[width=0.45\textwidth]{../images/form_factor/cylindrical_obj/round_helices3D.png}}
\end{center}
\caption{Double helix with strands of round cross sections. The cross-sections are round in the plane perpendicular to the helix axis.} \label{fig:roundhelix}
\end{figure}
@@ -150,9 +150,9 @@ \subsubsection{Beads model of a single helical strand} ~\\
\begin{figure}[htb]
\begin{center}
\subfigure[side view]{\label{fig:beadshelixside}\includegraphics[width=0.4\textwidth]{../images/form_factor/cylindrical_obj/beads_helix_model_1.png}}
\subfigure[side view]{\label{fig:beadshelixside1}\includegraphics[width=0.4\textwidth]{../images/form_factor/cylindrical_obj/beads_helix_model_1.png}}
\hfill
\subfigure[top on view]{\label{fig:beadshelixside}\includegraphics[width=0.4\textwidth]{../images/form_factor/cylindrical_obj/beads_helix_model_2.png}}
\subfigure[top on view]{\label{fig:beadshelixside2}\includegraphics[width=0.4\textwidth]{../images/form_factor/cylindrical_obj/beads_helix_model_2.png}}
%\includegraphics[width=0.4\textwidth,height=0.74\textwidth]{../images/form_factor/cylindrical_obj/beads_helix_model.png}
\end{center}
\caption{Bead model of a helix. In figure a) bead helices with different bead radius and different number of beads per turn are shown} \label{fig:beadshelix}
@@ -181,7 +181,7 @@ \subsubsection{Beads model of a single helical strand} ~\\
I(Q) &= P_\text{rod}(Q,H) \sum_{j=0}^{\infty} \epsilon_j \abs{\frac{nH}{P}\Psi_j\left(QD,\frac{2\pi j}{PQ}\right) \Phi(Q,R)}^2
\label{eq:beadshelix2}
\end{align}
with $\epsilon_j=1$ for $j=0$ and $\epsilon_j=2$ for $j\geq 1$.
with $\epsilon_j=1$ for $j=0$ and $\epsilon_j=2$ for $j\geq 1$.
The sum converges very fast and for small $Q$-values the first two term are already sufficient. However, {\tt SASfit} is continuing the sum until either the argument below the square root becomes negative or the relative change of the sum is less than $10^{-10}$.
@@ -239,7 +239,7 @@ \subsubsection{straight superhelix} ~\\
\mathcal{L} &= 2\pi \frac{H}{P} \sqrt{R_1^2+a^2} \\
a &= \frac{P}{2\pi}
\end{align}
$\mathcal{L}$ is the arclength of the helix, $R_1$ the distance to the helix axis and $P$ the pitch of the helix. The intensity is normalized to the squared arclength of the helix. i.e.
$\mathcal{L}$ is the arclength of the helix, $R_1$ the distance to the helix axis and $P$ the pitch of the helix. The intensity is normalized to the squared arclength of the helix. i.e.
\begin{align}
I(Q=0)&=\mathcal{L}^2
\end{align}
@@ -317,7 +317,7 @@ \subsubsection{coiled superhelix} ~\\
&+\frac{P^2}{(2\pi)^2} + 2R_2*R_1\cos\left(N\gamma\right) +2R_1^2N\cos(\alpha_2)
\end{split}
\end{align}
and
and
\begin{align}
\gamma_\mathrm{max} &= N_\mathrm{2nd}/(2\pi)
\end{align}
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