SASfit/SASfit

peak plugin

 @@ -364,7 +364,7 @@ \subsection{Star polymer with Gaussian statistic according to %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Polydisperse star polymer with Gaussian statistics \cite{burchard74}} +\subsection{Polydisperse star polymer with Gaussian statistics \cite{Burchard74}} \label{sect:PolydisperseStar} ~\\ \begin{figure}[htb] @@ -375,7 +375,7 @@ \subsection{Polydisperse star polymer with Gaussian statistics \cite{burchard74} \end{figure} For a SchulzFlory (most probable) distribution (SchulzZimm distribution -with $z = 1$) for the mass distribution of the arms, Burchard \cite{burchard74} has given the form factor: +with $z = 1$) for the mass distribution of the arms, Burchard \cite{Burchard74} has given the form factor: \begin{align} I_\text{PolydisperseStar}(Q) &= I_0 \frac{1+\frac{u^2}{3 f}}{\left(1+\frac{u^2(f +1)}{6 f }\right)^2}
 @@ -126,6 +126,7 @@ \section{Log-Normal distribution} \clearpage \section{Schulz-Zimm (Flory) distribution} +\label{sec:SchulzZimm} \begin{figure}[htb] \begin{center} @@ -207,6 +208,7 @@ \section{Gamma distribution} When $k$ is large, the gamma distribution closely approximates a normal distribution with the advantage that the gamma distribution has density only for positive real numbers. For small values of $k$ the distribution becomes a right tailed distribution. +The gamm distribution is equivalent to the Schulz-Zimm distribution (sec.\ \ref{sec:SchulzZimm}), which is using the mean value instead of $\theta=x_\mathrm{mean}/k$ for the parametrization. Also the Pearson III \ref{sec:PearsonIIIdist} is just another name for this distribution. The $m^\text{th}$ moment $\langle X^m\rangle$ of the size distribution is given by \begin{align} @@ -256,6 +258,7 @@ \section{Gamma distribution} \clearpage \section{PearsonIII distribution} +\label{sec:PearsonIIIdist} The Pearson distribution is a family of probability distributions that are a generalisation of the normal distribution. The Pearson @@ -317,23 +320,40 @@ \section{Gauss distribution} \text{erf}\left(\frac{R_0}{\sqrt{2}\sigma}\right) \right) \end{align} \end{subequations} -$c_\text{Gauss}$ is choosen so that $\int_0^\infty\! +$c_\text{Gauss}$is chosen so that$\int_0^\infty\! \text{Gauss}(R,\sigma,R_0)\,dR = N%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \clearpage \section{Generalized exponential distribution (GEX)} -\begin{subequations} +The generalised exponential distribution is a three parameter distribution. \begin{align} -\text{GEX}(R,\beta,\lambda,\gamma))&= N -\frac{\beta}{\gamma}\left(\frac{x}{\gamma}\right)^{\lambda+1} -\frac{e^{-(x/\gamma)^{\beta}}}{\Gamma \left( {\frac -{\lambda+2}{\beta}} \right)} +\text{GEX} (x,\beta,\lambda,\gamma) &= N \frac{\abs{\beta}}{\gamma} \left(\frac{x}{\gamma}\right)^{\lambda+1} +\frac{e^{-(x/\gamma)^\beta}}{\Gamma\left( \frac{\lambda+2}{\beta} \right)} \\ +\text{GEX}(x,\beta,p,\gamma) &= N +\frac{\abs{\beta}}{\gamma}\left(\frac{x}{\gamma}\right)^{p-1} +\frac{e^{-(x/\gamma)^{\beta}}}{\Gamma \left( {\frac{p}{\beta}} \right)} +\end{align} +For\lambda+2=p=\beta$it will transform in the Weibull distribution (eq.\ \ref{eq:Weibull}). For$\beta=1$one yields the Schulz-Zimm distribution (eq.\ \ref{eq:SZn(M)}) or the equivalent gamma distribution (eq.\ \ref{eq:GammaDistr}). In The limit$\beta \rightarrow 0$the GEX distribution transforms into a Lognormal distribution. +The mode$x_\mathrm{mode}$, mean$x_\mathrm{mean}$, and variance$\sigma^2of this distribution are given by +\begin{align} +x_\mathrm{mode} &= \gamma \left( \frac{\lambda+1}{\beta}\right)^\frac{1}{\beta} \\ +x_\mathrm{mean} &= +\begin{cases} +\gamma \Gamma\left(\frac{\lambda+3}{\beta}\right)/\Gamma\left(\frac{\lambda+2}{\beta}\right) & \mbox{for } \beta > 0 \wedge \lambda > -3 \\ +\mbox{undefined} & \mbox{otherwise} +\end{cases} \\ +\sigma^2 &= +\begin{cases} +\gamma^2 \left( +\frac{\Gamma\left(\frac{\lambda+4}{\beta}\right)}{\Gamma\left(\frac{\lambda+2}{\beta}\right)} - +\left(\frac{\Gamma\left(\frac{\lambda+3}{\beta}\right)}{\Gamma\left(\frac{\lambda+2}{\beta}\right)}\right)^2 +\right)& \mbox{for } \beta > 0 \wedge \lambda > -3 \\ +\mbox{undefined} & \mbox{otherwise} +\end{cases} \end{align} -\end{subequations} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \clearpage @@ -458,6 +478,7 @@ \section{Weibull distribution} \label{fig:Weibull} \end{figure} \begin{align} +\label{eq:Weibull} \text{Weibull}(R,\alpha,\lambda,\mu) = \frac{N \lambda}{\alpha} \left(\frac{R-\mu}{\alpha}\right)^{\lambda-1}  @@ -13,7 +13,7 @@ \item a first version of a plugin for ordered mesoscopic and nano structures. The plugin is providing a part of the structure factors available in the software package "scatter" from S. Förster. - \item plugin of a radial profile for a sphere resulting in a Porod law both below and above Q^-4 + \item plugin of a radial profile for a sphere resulting in a Porod law both below and aboveQ^{-4}(\texttt{Boucher Sphere}) \end{itemize} \item[2014-12-14] \SASfit 0.94.6 \begin{itemize}  @@ -764,8 +764,7 @@ \subsection{BoucherSphere, radial profile for a sphere resulting in a Porod law \end{align} By this normalisation the excess scattering length is for all\alpha$equal to the one of a homogeneous sphere with a scattering contrast$\Delta\eta$, i.e.$\beta_\mathrm{sp}=\Delta\eta \frac{4}{3}\pi R^3. -To get feeling which part of the spherical contributes most to the scattering ones can define -a fractional relative excess scattering length according to \cite{Boucher1983} which reads as +To get a feeling which part of the spherical scattering length density profile contributes most to the scattering it is convenient to define a fractional relative excess scattering length according to \cite{Boucher1983} which reads as \begin{align} Z(x) = & \frac{ \int_{0}^{x}\! \left( 1-{\frac {{r}^{2}}{{R}^{2}}} \right) ^{\frac{\alpha}{2}-2} 4\pi\,{r}^{2}\,{\rm d}r @@ -798,7 +797,7 @@ \subsection{BoucherSphere, radial profile for a sphere resulting in a Porod law &= \beta \left(\frac{2}{QR}\right)^{\frac{\alpha-1}{2}} J_{\frac{\alpha-1}{2}}(QR) \, \Gamma\left(\frac{\alpha+1}{2}\right) \nonumber \\ &= \beta \;_0F_1\left(\frac{\alpha+1}{2};-\tfrac{(QR)^2}{4}\right) \end{align} -making use of the fact that the bessel function can be expressed in terms of generalised hypergeometric functionsJ_\alpha(x)=\frac{(\frac{x}{2})^\alpha}{\Gamma(\alpha+1)} \;_0F_1 (\alpha+1; -\tfrac{x^2}{4})$. The scattering amplitude simplifies in case of$\alpha=4$exactly the one of a homogenous spherical shell and in case of$\alpha=2$to an infinitesimal thin spherical shell. For the scattering intensity$I(Q) = F^2(Q)$we get in the Porod limit$Q\rightarrow\infty$+making use of the fact that the bessel function can be expressed in terms of generalised hypergeometric functions$J_\alpha(x)=\frac{(\frac{x}{2})^\alpha}{\Gamma(\alpha+1)} \;_0F_1 (\alpha+1; -\tfrac{x^2}{4})$. The scattering amplitude simplifies in case of$\alpha=4$exactly the one of a homogenous sphere and in case of$\alpha=2$to an infinitesimal thin spherical shell. For the scattering intensity$I(Q) = F^2(Q)$we get in the Porod limit$Q\rightarrow\infty$\BE \lim_{Q\rightarrow\infty} I(Q)= \lim_{Q\rightarrow\infty} F^2(Q) = \beta^2 \frac{2^{\alpha-1}}{\pi} \Gamma^2\left(\frac{\alpha+1}{2}\right) \frac{1}{(QR)^\alpha} @@ -825,7 +824,7 @@ \subsection{BoucherSphere, radial profile for a sphere resulting in a Porod law \begin{description} \item[\texttt{R}] radius$R$\item[\texttt{alpha}] shape parameter of the profile$\alpha$, which is also equal to the potential law at large$Q$-values -\item[\texttt{Delta\_eta}] average scattering length density so that the excess scattering length corresponds to that one of a homogeneous sphere with contrast$\Delta\eta$+\item[\texttt{Delta\_eta}] is the average scattering length density so that the excess scattering length corresponds to the one of a homogeneous sphere with contrast$\Delta\eta$\end{description} \noindent\underline{Note for model \texttt{BoucherSphere2} and \texttt{Boucher profile}:}  @@ -127,12 +127,6 @@ B~Boucher, P~Chieux, P~Convert, and M~Tournarie. order in the amorphous metallic alloy tbcu 3.54. \newblock {\em Journal of Physics F: Metal Physics}, 13(7):1339, 1983. -\bibitem{burchard74} -W.~Burchard. -\newblock Statistics of star-shaped molecules. i. stars with polydisperse side - chains. -\newblock {\em Macromolecules}, 7:835--841, 1974. - \bibitem{Burchard1996} W.~Burchard, E.~Michel, and V.~Trappe. \newblock Conformational properties of multiply twisted ring systems and  @@ -4210,4 +4210,55 @@ @Article{Burchard1974 timestamp = {2016.04.20}, } +@Article{Jakes1991, + author = {Jaromír Jakeš}, + title = {A simple method of estimation of the polydispersity index of narrow molecular weight distributions by using quasielastic light scattering data}, + year = {1991}, + journal = {Collect. Czech. Chem. Commun.}, + volume = {56}, + issue = {8}, + pages = {1642-1652}, + doi = {http://dx.doi.org/10.1135/cccc19911642}, + owner = {kohlbrecher}, + timestamp = {2016.05.06}, +} + +@Article{Jakes1986, + author = {Jakeš, Jaromir and Saudek, Vladimir}, + title = {Empirical molecular weight distribution functions as determined from gel permeation chromatography data}, + year = {1986}, + journal = {Makromol. Chem.}, + volume = {187}, + number = {9}, + month = sep, + pages = {2223--2234}, + issn = {0025-116X}, + url = {http://dx.doi.org/10.1002/macp.1986.021870919}, + abstract = {Two- and three-parameter functions commonly used for the description + of the molecular weight distribution of polymers were determined + from gel permeation chromatography data for 80 water-soluble samples + of polymers derived from glutamic acid. Two methods were applied: + the moment method in which two or three different molecular weight + averages are used, and a least squares adjustment method using the + whole chromatogram. Convergence difficulties observed in the past + were overcome, and a fully automated computer procedure was developed. + The moment method, although virtually the only one still in use, + often gives inexact results: sometimes the type of two-parameter + distribution is wrongly assigned, or the description of experimental + data cannot be improved by adding yet another parameter. On the other + hand, in all cases under study the least squares method gave a function + which describes the real distribution within the limits of experimental + error. Although the majority of samples could be described by employing + one of the common two-parameter functions, the more general three-parameter + function, encompassing all two-parameter ones as special cases, has + the advantage of wider applicability. It is shown that the standard + gel permeation chromatography data often cannot provide reliable + values of higher molecular weight moment averages, including Mz. + Hence, it may be inferred that in the description of the experimental + molecular weight distribution curves the least squares method should + be preferred over the moment method.}, + owner = {kohlbrecher}, + timestamp = {2016.05.06}, +} + @Comment{jabref-meta: databaseType:biblatex;} Binary file not shown.  @@ -302,7 +302,7 @@ frame .openfile.layout2 frame .openfile.layout3 set format [tk_optionMenu .openfile.layout1.format tmpsasfit(actualdatatype) \ - Ascii BerSANS] + Ascii BerSANS ALV5000] .openfile.layout1.format configure -highlightthickness 0 label .openfile.layout1.label -text "File Format:" -highlightthickness 0 pack .openfile.layout1.label .openfile.layout1.format -side left -fill x @@ -785,13 +785,13 @@ set w .addfile.lay1 frame .addfile.layout3 set format [tk_optionMenu$w.layout1.format tmpsasfit(actualdatatype) \ - Ascii BerSANS] + Ascii BerSANS ALV5000] $w.layout1.format configure -highlightthickness 0 label$w.layout1.label -text "File Format:" \ -width 12 -highlightthickness 0 button $w.layout1.option -text "Options..." -command ReadOptionsCmd \ -highlightthickness 0 -pady 1m -button$w.layout1.read -text "Read File" \ +button $w.layout1.read -text "Read file" \ -command { global tmpsasfit set tmpfnlist$tmpsasfit(filename)