included S(Q) according to Henderson and Grundke

.gitignore

@@ -61,3 +61,4 @@ saskit/saskit_*
 /src/plugins/sin2
 /src/plugins/sin2/include/private.h
 /src/plugins/sin2/include/sasfit_sin2.h
+/src/fftw/fftw-3.3.8.tar.gz

doc/manual/SASfit_pluginSQ_ordered_particle_systems.tex

@@ -1,11 +1,224 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Analytical static structure factor for hard spheres} \hspace{1pt}
+The rational function approximation method which is wholly compatible
+with the equation of state used for the fluid has been used for expressions of the hard sphere structure factor. 
+In general, integral equation theories require a closure approximations, and they lead to different equations of state if one takes
+the virial or the compressibility routes (thermodynamic inconsistency problem). In contrast the rational function approximation method
+completely avoids the thermodynamic inconsistency problem, in that the compressibility factor is involved in the derivation of $S(Q)$ \cite{Yuste1996,Robles1997,Haro2004}.
+\begin{align}
+S(Q) &= 1-24\eta\Re\left(\left.\frac{t^2G(t)-1}{t^3}\right|_{t=\imath Q\sigma}\right) \label{eq:RFAstart}\\
+G(t) &= \frac{t}{12\eta}\frac{1}{1-\exp(t)\Phi(t)} \\
+\Phi(t) &= \frac{1+S_1 t + S_2 t^2 + S_3 t^3 + S_4 t^4}{1+L_1 t + L_2 t^2}
+\end{align}
+where the six coefficients $S_1$, $S_2$, $S_3$, $S_4$, $L_1$, and $L_2$ may be evaluated in an
+algebraic form
+\begin{align}
+L_1 &= \frac{1}{2} \frac{η + 12\eta L_2 + 2 - 24\eta S_4}{2\eta + 1} \\
+S_1 &= \frac{3}{2}\eta\frac{-1+4L_2-8S_4}{2\eta +1}\\
+S_2 &= -\frac{1}{2} \frac{-\eta+8\eta L_2+1-2L_2-24\eta S_4}{2\eta +1} \\
+S_3 &= \frac{1}{12} \frac{2\eta-\eta^2+12\eta^2L_2-12\eta L_2-1-72\eta^2 S_4}{\eta (2\eta+1)}
+\end{align}
+with
+\begin{align}
+L_2 &= -3\left(Z-1\right)S_4 \\
+S_4 &= \frac{1-\eta}{36\eta\left(Z-\frac{1}{3}\right)}\left[1-\sqrt{1+\frac{Z-\frac{1}{3}}{Z-Z_\mathrm{PY}}\left(\frac{\chi}{\chi_\mathrm{PY}}-1\right)}\right]
+\end{align}
+Here, $Z_\mathrm{PY} = \frac{1+2\eta+3\eta^2}{(1−\eta)^2}$ and $\chi_\mathrm{PY} = \frac{(1−\eta)^4}{(1+2\eta)^2}$ are the compressibility factor and isothermal susceptibility arising in the PY theory. The isothermal compressibility $\chi$ and the compressibility factor $Z$ are related by
+\begin{align}
+\chi &= \left(\frac{\mathrm{d}(\rho Z)}{\mathrm{d}\rho}\right)^{-1} = \left(Z+\eta\frac{\mathrm{d} Z}{\mathrm{d}\eta}\right)^{-1}
+\label{eq:RFAend}
+\end{align}
+where $\eta = \frac{\pi}{6}\rho \sigma^3$ is the packing fraction of the spheres ($\rho$ is the number density and $\sigma$ the
+hard-sphere diameter).
+The only quantity which is left to calculate the structure factor is an explicit expression for the compressibility factor $Z$.
+
+\vspace{5mm}
+
+\subsection{Carnahan and Starling} \cite{Carnahan1969}
+
+\noindent In this approximation eqs.\ \ref{eq:RFAstart}-\ref{eq:RFAend} are used with $Z=Z^\mathrm{CS}$, where
+\begin{align}
+Z^\mathrm{CS} &= \frac{1+\eta+\eta^2-\eta^3}{\left(1-\eta\right)^3}
+\end{align}
+
+\vspace{5mm}
+
+\hspace{1pt}\\
+\underline{Input parameters for \texttt{Hard Sphere (CS)}:}
+\begin{description}
+    \item[\texttt{R}]  radius $R$
+    \item[\texttt{eta}] volume fraction $\eta$
+\end{description}
+
+\noindent
+\underline{Note}
+\begin{itemize}
+\item The structure factor accepts volume fractions between $\eta \in [0,1]$.
+\item The model lead to physically meaningful structural properties in the whole definition range of volume fractions.
+\end{itemize}
+
+\begin{figure}[htb]
+\subfigure[Comparison between $S^\mathrm{PY}$ and $S^\mathrm{CS}$]{\includegraphics[width=0.46\textwidth]{../images/structure_factor/HardSphere/SQCS.png}}
+\hfill
+\subfigure[residual between $S^\mathrm{PY}$ and $S^\mathrm{CS}$]{\includegraphics[width=0.48\textwidth]{../images/structure_factor/HardSphere/ResCS.png}} \\
+\caption{Comparison between analytical PY solution of hard sphere static structure factor and rational function approximation with a compressibility factor of Carnahan and Starling}
+\label{fig:SQ:CS}
+\end{figure}
+
+\clearpage
+\subsection{Pad\'{e}(4,3) of van Rensburg and S\'{a}nchez} \cite{Rensburg1993,Sanchez1994}
+
+\noindent In this approximation eqs.\ \ref{eq:RFAstart}-\ref{eq:RFAend} are used with $Z=Z^{(4,3)}$, where
+\begin{align}
+Z^{(4,3)} &= \frac{1+1.024385\eta+1.104537\eta^2-0.4611472\eta^3-0.7430382\eta^4}{1-2.975615\eta+3.007000\eta^2-1.097758\eta^3}
+\end{align}
+
+\vspace{5mm}
+
+\hspace{1pt}\\
+\underline{Input parameters for \texttt{Hard Sphere (4,3)}:}
+\begin{description}
+    \item[\texttt{R}]  radius $R$
+    \item[\texttt{eta}] volume fraction $\eta$
+\end{description}
+
+\noindent
+\underline{Note}
+\begin{itemize}
+\item The structure factor accepts volume fractions between $\eta \in [0,1]$.
+\item The threshold packing fraction (packing
+fraction at which a glass transition in the hard-sphere fluid takes place) of this model is $\eta^{(4,3)}_0 = 0.5604$  beyond
+which no meaningful fluid structure can be derived \cite{Haro2004}.
+\end{itemize}
+
+\begin{figure}[htb]
+\subfigure[Comparison between $S^\mathrm{PY}$ and $S^{(4,3)}$]{\includegraphics[width=0.46\textwidth]{../images/structure_factor/HardSphere/SQ(4,3).png}}
+\hfill
+\subfigure[residual between $S^\mathrm{PY}$ and $S^{(4,3)}$]{\includegraphics[width=0.48\textwidth]{../images/structure_factor/HardSphere/Res(4,3).png}} \\
+\caption{Comparison between analytical PY solution of hard sphere static structure factor and rational function approximation with a compressibility factor of van Rensburg and S\'{a}nchez}
+\label{fig:SQ:43}
+\end{figure}
+
+\clearpage
+\subsection{Malijevsk\'{y} and Veverka} \cite{Malijevsky1999}
+
+\noindent In this approximation eqs.\ \ref{eq:RFAstart}-\ref{eq:RFAend} are used with $Z=Z^\mathrm{MV}$, where
+\begin{align}
+Z^\mathrm{MV} &= \frac{1 + 1.0560\eta + 1.6539\eta^2 + 0.3262\eta^3}{\left(1- 3.8464\eta + 4.9574\eta^2 -2.1639\eta^3\right)\left(1-\eta\right)^3}
+\end{align}
+
+\vspace{5mm}
+
+\hspace{1pt}\\
+\underline{Input parameters for \texttt{Hard Sphere (MV)}:}
+\begin{description}
+    \item[\texttt{R}]  radius $R$
+    \item[\texttt{eta}] volume fraction $\eta$
+\end{description}
+
+\noindent
+\underline{Note}
+\begin{itemize}
+\item The structure factor accepts volume fractions between $\eta \in [0,1]$.
+\item The model lead to physically meaningful structural properties in the whole definition range of volume fractions.
+\end{itemize}
+
+\begin{figure}[htb]
+\subfigure[Comparison between $S^\mathrm{PY}$ and $S^\mathrm{MV}$]{\includegraphics[width=0.46\textwidth]{../images/structure_factor/HardSphere/SQMV.png}}
+\hfill
+\subfigure[residual between $S^\mathrm{PY}$ and $S^\mathrm{MV}$]{\includegraphics[width=0.48\textwidth]{../images/structure_factor/HardSphere/ResMV.png}} \\
+\caption{Comparison between analytical PY solution of hard sphere static structure factor and rational function approximation with a compressibility factor of Malijevsk\'{y} and Veverka}
+\label{fig:SQ:MV}
+\end{figure}
+
+\clearpage
+\subsection{L\'{o}pez de Haro and Robles} \cite{Robles2003}
+
+\noindent In this approximation eqs.\ \ref{eq:RFAstart}-\ref{eq:RFAend} are used with $Z=Z^\mathrm{LHR}$, where
+\begin{align}
+Z^\mathrm{LHR} &= \frac{1   + 0.153555\eta
+                            - 0.428376\eta^2
+                            - 2.7987\eta^3
+                            - 0.317417\eta^4
+                            - 0.105806\eta^5}{1-3.84644\eta + 4.9574\eta^2 - 2.16386\eta^3}
+\end{align}
+
+\vspace{5mm}
+
+\hspace{1pt}\\
+\underline{Input parameters for \texttt{Hard Sphere (LHR)}:}
+\begin{description}
+    \item[\texttt{R}]  radius $R$
+    \item[\texttt{eta}] volume fraction $\eta$
+\end{description}
+
+\noindent
+\underline{Note}
+\begin{itemize}
+\item The structure factor accepts volume fractions between $\eta \in [0,1]$.
+\item The threshold packing fraction (packing
+fraction at which a glass transition in the hard-sphere fluid takes place) of this model is $\eta^\mathrm{LHR}_0 = 0.5684$  beyond
+which no meaningful fluid structure can be derived \cite{Haro2004}.
+\end{itemize}
+
+\begin{figure}[htb]
+\subfigure[Comparison between $S^\mathrm{PY}$ and $S^\mathrm{LHR}$]{\includegraphics[width=0.46\textwidth]{../images/structure_factor/HardSphere/SQLHR.png}}
+\hfill
+\subfigure[residual between $S^\mathrm{PY}$ and $S^\mathrm{LHR}$]{\includegraphics[width=0.48\textwidth]{../images/structure_factor/HardSphere/ResLHR.png}} \\
+\caption{Comparison between analytical PY solution of hard sphere static structure factor and rational function approximation with a compressibility factor of L\'{o}pez de Haro and Robles}
+\label{fig:SQ:LHR}
+\end{figure}
+
+\clearpage
+\subsection{Grundke and Henderson} ~\\
+
+\noindent In this approximation the structure factor in the Percus Yevick approximation is corrected to have thermodynamic consistency \cite{Henderson1975} so that both routes leads to the expression of Carnahan and Starling 
+$pV/Nk_BT=\frac{1+\eta+\eta^2-\eta^3}{\left(1-\eta\right)^3}$
+\begin{align}
+\sigma &= 2R\\
+\left(\sigma_0/\sigma\right)^3 &= 1-\eta/16\\
+\eta_0 &= \eta (1-\eta/16) \\
+g_0(s,\eta_0) &\simeq \frac{1+\eta_0/2}{\left(1-\eta_0\right)^2} - \frac{9}{2}\eta_0\frac{1+\eta_0}{\left(1-\eta_0\right)^3}(s-1) \\
+\frac{C}{\sigma} &= \frac{2-\eta}{2(1-\eta)^3} - g_0(\sigma/\sigma_0,\eta_0) \\
+\frac{12\eta C}{m\sigma_0^2} &= \frac{(1-\eta)^4}{1+4\eta+4\eta^2-4\eta^3+\eta^4} - \frac{(1-\eta_0)^4}{(1+2\eta_0)^2} \nonumber \\
+&= 24\eta_0\int_0^{\sigma/\sigma_0} g_0(s,\eta_0)s^2 \mathrm{d}s \\
+S^\mathrm{GH}(Q,\sigma,\eta) &= S^\mathrm{PY}(Q,\sigma_0,\eta_0) + \frac{6\eta}{\pi\sigma^3}\tilde{h}_\mathrm{GH}(Q) \\
+\tilde{h}_\mathrm{GH}(Q)  &= -\frac{4\pi\sigma_0}{Q\sigma_0} \int_1^{\sigma/\sigma_0} sg_0(s,\eta_0)\sin(Q\sigma_0 s)\mathrm{d}s\\
+&+ \frac{2\pi\sigma^3}{Q\sigma}\frac{C}{\sigma} \left\{ \frac{\cos (Q\sigma)}{\sigma}\left[\frac{k+m}{m^2+(Q+m)^2}+\frac{Q-m}{m^2+(Q-m)^2}\right]\right.\nonumber \\
+& + \left. \frac{\sin (Q\sigma)}{\sigma}\left[\frac{m}{m^2+(Q+m)^2}+\frac{m}{m^2+(Q-m)^2}\right]\right\}\nonumber 
+\end{align}
+
+\vspace{5mm}
+
+\hspace{1pt}\\
+\underline{Input parameters for \texttt{Hard Sphere (GH)}:}
+\begin{description}
+    \item[\texttt{R}]  radius $R$
+    \item[\texttt{eta}] volume fraction $\eta$
+\end{description}
+
+\noindent
+\underline{Note}
+\begin{itemize}
+\item The structure factor accepts volume fractions between $\eta \in [0,1]$.
+\end{itemize}
+
+\begin{figure}[htb]
+\subfigure[comparison between $S^\mathrm{PY}$ and $S^\mathrm{GH}$]{\includegraphics[width=0.46\textwidth]{../images/structure_factor/HardSphere/SQGH.png}}
+\hfill
+\subfigure[residual between $S^\mathrm{PY}$ and $S^\mathrm{GH}$]{\includegraphics[width=0.48\textwidth]{../images/structure_factor/HardSphere/ResGH.png}} \\
+\caption{Comparison between analytical PY solution of hard sphere static structure factor and thermodynamically consistent correction as described by Henderson and Grundke}
+\label{fig:SQ:LHR}
+\end{figure}
+
+\clearpage
 \section{Structure factor for a two dimensional hard spheres/disks fluid} \hspace{1pt}
 The structure factor of hard disks or spheres with diameter/radius $\sigma=2R$ in two dimensions with an interaction potential
 \begin{align}
-U(r,\sigma) &= 
+U(r,\sigma) &=
 \begin{cases}
 \infty &\mathrm{for~}  r < \sigma \\
-0  &\mathrm{for~}  r \geq \sigma 
+0  &\mathrm{for~}  r \geq \sigma
 \end{cases}
 \end{align}
 have been implemented in several variants.
@@ -33,10 +246,10 @@ \section{Structure factor for a two dimensional hard spheres/disks fluid} \hspac
 with
 \begin{align}
 \begin{split}
-c(r',\eta) &= 
+c(r',\eta) &=
 \Theta(1-r') \left[-\frac{1-p\eta^2}{\left(1-2\eta+p\eta^2\right)^2}\right]  \\
 &\left\{1-a^2\eta-a^2\eta\frac{2}{\pi}\left[\arccos\left(\frac{r'}{a}\right)-\frac{r'}{a}\sqrt{1-\frac{r'^2}{a^2}}\right]\right\}
-\end{split}  
+\end{split}
 \end{align}
 with $p=\left(4\sqrt{3}\pi-12\right)/\pi^2$ and $\Theta()$ being the Heaviside function. The parameter $a$ is the root of a non-linear equation, which has been solved numerically and then approximated via a polynomial fitting by
 \begin{align}

doc/manual/SASfit_pluginsFF_nonparticular.tex

@@ -323,7 +323,7 @@ \subsection{Broad-Peak}
 \begin{center}
 \includegraphics[width=0.85\textwidth,height=0.6\textwidth]{BroadPeak.png}
 \end{center}
-\caption{Emical form factor of Broad-Peak} \label{fig:BroadPeakIq}
+\caption{Empirical form factor of Broad-Peak} \label{fig:BroadPeakIq}
 \end{figure}
 
 \clearpage

doc/manual/SASfit_pluginsFF_thin_objects.tex

@@ -1122,7 +1122,7 @@ \subsubsection{P'(Q): wormlike PS2} ~\\
 S_\mathrm{PS2}(Q,L,l_B) &=
 \begin{cases}
     S_\mathrm{SB}(Q,L,l_B) f_1 + S_\mathrm{loc}(Q,L,1) (1-f_1) & \mbox{ for~}  n_b > 2 \\
-    S_\mathrm{Debye}(Q,Q^2\langle R_g^2\rangle_0) f_2 + S_\mathrm{loc}(Q,L,a_1) (1-f_2) & \mbox{ for~}  n_b \leq 2
+    S_\mathrm{Debye}(Q,Q^2\langle R_g^2\rangle_0) f_2 + S_\mathrm{loc}(Q,L,l_B,a_1) (1-f_2) & \mbox{ for~}  n_b \leq 2
 \end{cases}
 \end{align}
 with $\langle R_g^2\rangle_0 = Ll_B/6\left(1-\frac{3}{2 n_b}
@@ -1147,7 +1147,7 @@ \subsubsection{P'(Q): wormlike PS2} ~\\
 \end{align}
 and in the large $Q$ range
 \begin{align}
-S_\mathrm{loc}(Q,L,a) &= \frac{a}{LbQ^2} + \frac{\pi}{LQ}
+S_\mathrm{loc}(Q,L,l_B,a_1) &= \frac{a_1}{Ll_BQ^2} + \frac{\pi}{LQ}
 \end{align}
 The optimized values of the parameters $q_1,p_1, a_1, a_2, q_2$, and $ p_2$ are
 $q_1 = 5.53$, $p_1 = 5.33$, $a_1 = 0.0625$, $a_2 = 11.7$, $p_2 = 3.95$ and
@@ -1175,27 +1175,27 @@ \subsubsection{P'(Q): wormlike PS2} ~\\
 \subsubsection{P'(Q): wormlike PS3} ~\\
 \label{plugin:Pprime4wormPS3}
 
-This version of the wormlike structure model originally from \cite{Pedersen96Macrom} is implemented following the suggestions for corrections given in \cite{Chen2006}. 
+This version of the wormlike structure model originally from \cite{Pedersen96Macrom} is implemented together with the suggestions for corrections given in \cite{Chen2006}.
 \begin{align}
 q_0 &=
 \begin{cases}
 3.1 & \mbox{for~} L>4l_B \\
-\max\left\{a_3 l_B/R_G,4\right\} & \mbox{for~} L\leq 4l_B \wedge \mbox{without excluded volume}\\
-\max\left\{a_3 l_B/R_G,3\right\} & \mbox{for~} L\leq 4l_B \wedge \mbox{with excluded volume}\\
-\end{cases} \\
+\max\left\{a_3 l_B/R_G,4\right\} & \mbox{for~} L\leq 4l_B \wedge \mbox{without excl. vol.}\\
+\max\left\{a_3 l_B/R_G,3\right\} & \mbox{for~} L\leq 4l_B \wedge \mbox{with excl. vol.}\\
+\end{cases}\\
 R_G^2 &=
 \begin{cases}
-\frac{Ll_B}{6} & \mbox{for~} L > 4l_B \wedge \mbox{without excluded volume} \\
-\frac{Ll_B}{6} f\left(\frac{L}{l_B}\right) & \mbox{for~} L > 4l_B \wedge \mbox{with excluded volume} \\
-\frac{Ll_B}{6} f\left(\frac{L}{l_B}\right)& \mbox{for~} L\leq 4l_B \wedge \mbox{without excluded volume}\\
-\frac{Ll_B}{6} f\left(\frac{L}{l_B}\right)\alpha^2\left(\frac{L}{l_B}\right) & \mbox{for~} L\leq 4l_B \wedge \mbox{with excluded volume}\\
+\frac{Ll_B}{6} & \mbox{for~} L > 4l_B \wedge \mbox{without excl. vol.} \\
+\frac{Ll_B}{6} f\left(\frac{L}{l_B}\right) & \mbox{for~} L > 4l_B \wedge \mbox{with excl. vol.} \\
+\frac{Ll_B}{6} f\left(\frac{L}{l_B}\right)& \mbox{for~} L\leq 4l_B \wedge \mbox{without excl. vol.}\\
+\frac{Ll_B}{6} f\left(\frac{L}{l_B}\right)\alpha^2\left(\frac{L}{l_B}\right) & \mbox{for~} L\leq 4l_B \wedge \mbox{with excl. vol.}\\
 \end{cases} \\
 f(x) &= 1-\frac{3}{2x}+\frac{3}{2\left(x\right)^2}-\frac{3}{4\left(x\right)^3}\left[1-\exp\left(-2x\right)\right]\\
 \alpha(x) &= \left(1+(x/3.12)^2+(x/8.67)^3\right)^{\epsilon/3} \\
 \epsilon&=0.170 \\
 C\left(\frac{L}{l_B}\right) &=
 \begin{cases}
-a_4/\left(\frac{L}{l_B}\right)^{p_3} & \mbox{for~} L > 10l_B \wedge \mbox{with excluded volume} \\
+a_4/\left(\frac{L}{l_B}\right)^{p_3} & \mbox{for~} L > 10l_B \wedge \mbox{with excl. vol.} \\
 1 & \mbox{ otherwise}\\
 \end{cases}
 \end{align}
@@ -1204,31 +1204,31 @@ \subsubsection{P'(Q): wormlike PS3} ~\\
 S(Q,L,l_B) &=
 \begin{cases}
 L^2 S_{\textrm{small~} Q}(Q,L,l_B)& \mbox{for~}  Ql_B<q_0 \\
-L^2 \left(\frac{a_1}{(Ql_B)^{p_1}}+\frac{a_2}{(Ql_B)^{p_2}}+\frac{\pi}{QL}\right))& \mbox{for~}  Ql_B\geq q_0  \\
+L^2 \left(\frac{a_1}{(Ql_B)^{p_1}}+\frac{a_2}{(Ql_B)^{p_2}}+\frac{\pi}{QL}\right)& \mbox{for~}  Ql_B\geq q_0  \\
 \end{cases}\\
 p_1 &=
 \begin{cases}
-4.95 & \mbox{for~} L > 4l_B \wedge \mbox{without excluded volume} \\
-5.13 & \mbox{for~} L \leq 4l_B \wedge \mbox{without excluded volume} \\
-4.12 & \mbox{for~} L > 4l_B \wedge \mbox{with excluded volume} \\
-5.36 & \mbox{for~} L \leq 4l_B \wedge \mbox{with excluded volume} \\
+4.95 & \mbox{for~} L > 4l_B \wedge \mbox{without excl. vol.} \\
+5.13 & \mbox{for~} L \leq 4l_B \wedge \mbox{without excl. vol.} \\
+4.12 & \mbox{for~} L > 4l_B \wedge \mbox{with excl. vol.} \\
+5.36 & \mbox{for~} L \leq 4l_B \wedge \mbox{with excl. vol.} \\
 \end{cases}\\
 p_2 &=
 \begin{cases}
-5.29 & \mbox{for~} L > 4l_B \wedge \mbox{without excluded volume} \\
-7.47 & \mbox{for~} L \leq 4l_B \wedge \mbox{without excluded volume} \\
-4.42 & \mbox{for~} L > 4l_B \wedge \mbox{with excluded volume} \\
-5.62 & \mbox{for~} L \leq 4l_B \wedge \mbox{with excluded volume} \\
+5.29 & \mbox{for~} L > 4l_B \wedge \mbox{without excl. vol.} \\
+7.47 & \mbox{for~} L \leq 4l_B \wedge \mbox{without excl. vol.} \\
+4.42 & \mbox{for~} L > 4l_B \wedge \mbox{with excl. vol.} \\
+5.62 & \mbox{for~} L \leq 4l_B \wedge \mbox{with excl. vol.} \\
 \end{cases}
 \end{align}
 The parameters $a_1$ and $a_2$ are defined by the condition that for $Ql_B=q_0$ the small and large $Q$ region are continuous and smooth, i.e. that also the first derivative of $\mathrm{d}S(q,L,l_B)/\mathrm{d}q|_{q=q_0/l_B}$ is continuous.
 \begin{align}
 S_{\textrm{small~} Q}(Q,L,l_B) &=
 \begin{cases}
  S_\textrm{SB}(Q,R_G)    & \mbox{for~} L >    4l_B   \\
- S_\textrm{Debye}(Q,R_G) & \mbox{for~} L \leq 4l_B  \wedge \mbox{without excluded volume}\\
- S_\textrm{cexv}(Q,R_G)  & \mbox{for~} L \leq 4l_B  \wedge \mbox{with excluded volume} \\
-\end{cases}\\
+ S_\textrm{Debye}(Q,R_G) & \mbox{for~} L \leq 4l_B  \wedge \mbox{without excl. vol.}\\
+ S_\textrm{cexv}(Q,R_G)  & \mbox{for~} L \leq 4l_B  \wedge \mbox{with excl. vol.} \\
+\end{cases}
 \end{align}
 where
 \begin{align}
@@ -1240,7 +1240,7 @@ \subsubsection{P'(Q): wormlike PS3} ~\\
 x &= QR_G \\
 S_\textrm{EXV}(Q,R_G) &= (1-w)S_\textrm{Debye}(Q,R_G) \\
 &+ w f_\textrm{corr}(Q) \left(C_1x^{-1/\nu}+C_2x^{-2/\nu}+C_3x^{-3/\nu}\right) \nonumber \\
-w &= \frac12\left(1+\tanh((x-1.523)/0.1477)\right) \\
+w &= \frac12\left(1+\tanh((x-1.523)/0.1477)\right) 
 \end{align}
 with the optimized parameters $C_1 = 1.2220$, $C_2 = 0.4288$, and $C_3 = -1.651$.
 As $S_\textrm{EXV}$ should be a monotonic decreasing function also at very small $Q$-values the correction factor $f_\textrm{corr}(Q,R_G)$ has been introduced by \cite{Chen2006}