Author: bugman
Date: Wed Oct 16 15:19:17 2013
New Revision: 21139
URL: http://svn.gna.org/viewcvs/relax?rev=21139&view=rev
Log:
Improvements for the LaTeX maths commands used in the dispersion chapter of
the user manual.
Modified:
branches/relax_disp/docs/latex/dispersion.tex
branches/relax_disp/docs/latex/relax.tex
Modified: branches/relax_disp/docs/latex/dispersion.tex
URL:
http://svn.gna.org/viewcvs/relax/branches/relax_disp/docs/latex/dispersion.tex?rev=21139&r1=21138&r2=21139&view=diff
==============================================================================
--- branches/relax_disp/docs/latex/dispersion.tex (original)
+++ branches/relax_disp/docs/latex/dispersion.tex Wed Oct 16 15:19:17 2013
@@ -159,22 +159,22 @@
For the fixed relaxation time period CPMG-type experiments, the
$\Rtwoeff$/$\Ronerho$ values are determined by direct calculation using the
formula
\begin{equation}
- R_{2\textrm{eff}}(\nucpmg) = - \frac{1}{T_\textrm{relax}} \cdot \ln
\left( \frac{I_1(\nucpmg)}{I_0} \right) .
+ \Rtwoeff(\nucpmg) = - \frac{1}{T_\textrm{relax}} \cdot \ln \left(
\frac{I_1(\nucpmg)}{I_0} \right) .
\end{equation}
The values and errors are determined with a single call of the \uf{calc}
user function. The $\Ronerho$ version of the equation is essentially the
same:
\begin{equation}
- R_{1\rho}(\omega_1) = - \frac{1}{T_\textrm{relax}} \cdot \ln \left(
\frac{I_1(\omega_1)}{I_0} \right) .
+ \Ronerho(\omega_1) = - \frac{1}{T_\textrm{relax}} \cdot \ln \left(
\frac{I_1(\omega_1)}{I_0} \right) .
\end{equation}
Errors are calculated using the formula
\begin{equation} \label{eq: dispersion error}
- \sigma_{\textrm{R}_2} = \frac{1}{T_\textrm{relax}} \sqrt{ \left(
\frac{\sigma_{I_1}}{I_1(\omega_1)} \right)^2 + \left(
\frac{\sigma_{I_0}}{I_0} \right)^2 } .
+ \sigma_{\Rtwo} = \frac{1}{T_\textrm{relax}} \sqrt{ \left(
\frac{\sigma_{I_1}}{I_1(\omega_1)} \right)^2 + \left(
\frac{\sigma_{I_0}}{I_0} \right)^2 } .
\end{equation}
In a number of publications, the error formula from \citet{IshimaTorchia05}
has been used. This is the collapse of Equation~\ref{eq: dispersion error}
by setting $\sigma_{I_0}$ to zero:
\begin{equation} \label{eq: IT05 dispersion error}
- \sigma_{\textrm{R}_2} = \frac{\sigma_{I_1}}{T_\textrm{relax}
I_1(\omega_1)} .
+ \sigma_{\Rtwo} = \frac{\sigma_{I_1}}{T_\textrm{relax} I_1(\omega_1)} .
\end{equation}
This is not implemented in relax as it can be shown by simple simulation
that the formula is incorrect (see Figure~\ref{fig: dispersion error
comparison}). This formula significantly underestimates the real errors.
The use of the same $I_0$ value for all dispersion points does not cause a
decrease in the $\Rtwoeff$ error but rather a correlation in the errors.
@@ -228,7 +228,7 @@
or rearranged as:
\begin{equation} \label{eq: Luz-Meiboom}
- \Rex = \sum_{i=2}^n \frac{\Phi_\textrm{ex,i}}{\textrm{k}_\textrm{i}}
\cdot \left( 1 - \frac{4\nucpmg}{\textrm{k}_\textrm{i}} \cdot \tanh \left(
\frac{\textrm{k}_\textrm{i}}{4\nucpmg} \right) \right) ,
+ \Rex = \sum_{i=2}^n \frac{\Phiexi}{\textrm{k}_\textrm{i}} \cdot \left( 1
- \frac{4\nucpmg}{\textrm{k}_\textrm{i}} \cdot \tanh \left(
\frac{\textrm{k}_\textrm{i}}{4\nucpmg} \right) \right) ,
\end{equation}
@@ -254,7 +254,7 @@
It is selected by setting the model to `LM63 3-site'.
Taking the original Equation~\ref{eq: Luz-Meiboom}, the equation for 3-site
exchange is simply:
\begin{equation}
- \Rex = \sum_{i=2}^3 \frac{\Phi_\textrm{ex,i}}{\textrm{k}_\textrm{i}}
\cdot \left( 1 - \frac{4\nucpmg}{\textrm{k}_\textrm{i}} \cdot \tanh \left(
\frac{\textrm{k}_\textrm{i}}{4\nucpmg} \right) \right) ,
+ \Rex = \sum_{i=2}^3 \frac{\Phiexi}{\textrm{k}_\textrm{i}} \cdot \left( 1
- \frac{4\nucpmg}{\textrm{k}_\textrm{i}} \cdot \tanh \left(
\frac{\textrm{k}_\textrm{i}}{4\nucpmg} \right) \right) ,
\end{equation}
The reference for this equation is:
@@ -307,15 +307,15 @@
It is selected by setting the model to `CR72 full'.
The equation is
\begin{equation}
- \Rtwoeff = \frac{1}{2} \Big( \textrm{R}_\textrm{2A}^0 +
\textrm{R}_\textrm{2B}^0 + \kex - 2\nucpmg\cosh^{-1} \big( D_+\cosh(\eta_+) -
D_-\cos(\eta_-) \big) \Big) ,
+ \Rtwoeff = \frac{1}{2} \Big( \RtwozeroA + \RtwozeroB + \kex -
2\nucpmg\cosh^{-1} \big( D_+\cosh(\eta_+) - D_-\cos(\eta_-) \big) \Big) ,
\end{equation}
where
\begin{align}
D_\pm &= \frac{1}{2} \left( \pm1 + \frac{\Psi + 2\dw^2}{\sqrt{\Psi^2
+ \zeta^2}} \right) , \\
\eta_\pm &= 2^{\frac{2}{3}}\frac{1}{\nucpmg} \left( \pm\Psi +
\sqrt{\Psi^2 + \zeta^2} \right)^{\frac{1}{2}} , \\
- \Psi &= \left( \textrm{R}_\textrm{2A}^0 - \textrm{R}_\textrm{2B}^0 -
\pA\kex + \pB\kex \right)^2 - \dw^2 + 4\pA\pB\kex^2 , \\
- \zeta &= 2\dw \left( \textrm{R}_\textrm{2A}^0 -
\textrm{R}_\textrm{2B}^0 - \pA\kex + \pB\kex \right).
+ \Psi &= \left( \RtwozeroA - \RtwozeroB - \pA\kex + \pB\kex \right)^2
- \dw^2 + 4\pA\pB\kex^2 , \\
+ \zeta &= 2\dw \left( \RtwozeroA - \RtwozeroB - \pA\kex + \pB\kex
\right).
\end{align}
The reference for this equation is:
Modified: branches/relax_disp/docs/latex/relax.tex
URL:
http://svn.gna.org/viewcvs/relax/branches/relax_disp/docs/latex/relax.tex?rev=21139&r1=21138&r2=21139&view=diff
==============================================================================
--- branches/relax_disp/docs/latex/relax.tex (original)
+++ branches/relax_disp/docs/latex/relax.tex Wed Oct 16 15:19:17 2013
@@ -158,17 +158,19 @@
\newcommand{\Phiex}{\Phi_\textrm{ex}}
\newcommand{\PhiexB}{\Phi_\textrm{ex,B}}
\newcommand{\PhiexC}{\Phi_\textrm{ex,C}}
+\newcommand{\Phiexi}{\Phi_\textrm{ex,i}}
\newcommand{\Rex}{\mathrm{R}_\textrm{ex}}
\newcommand{\Ronerhoprime}{\mathrm{R}_{1\rho}'}
-\newcommand{\Rtwoeff}{\mathrm{R}_{2\textrm{eff}}}
+\newcommand{\Rtwoeff}{\mathrm{R}_\textrm{2eff}}
\newcommand{\Rtwozero}{\mathrm{R}_2^0}
-\newcommand{\RtwozeroA}{\mathrm{R}_{2A}^0}
-\newcommand{\RtwozeroB}{\mathrm{R}_{2B}^0}
-\newcommand{\RtwozeroC}{\mathrm{R}_{2C}^0}
-\newcommand{\RtwoDQA}{\mathrm{R}_{2,DQ}^A}
-\newcommand{\RtwoDQB}{\mathrm{R}_{2,DQ}^B}
-\newcommand{\RtwoZQA}{\mathrm{R}_{2,ZQ}^A}
-\newcommand{\RtwoZQB}{\mathrm{R}_{2,ZQ}^B}
+\newcommand{\RtwozeroA}{\mathrm{R}_\mathrm{2A}^0}
+\newcommand{\RtwozeroB}{\mathrm{R}_\mathrm{2B}^0}
+\newcommand{\RtwozeroC}{\mathrm{R}_\mathrm{2C}^0}
+\newcommand{\RtwozeroMQ}{\mathrm{R}_\mathrm{2,MQ}^0}
+\newcommand{\RtwoDQA}{\mathrm{R}_\mathrm{2,DQ}^\mathrm{A}}
+\newcommand{\RtwoDQB}{\mathrm{R}_\mathrm{2,DQ}^\mathrm{B}}
+\newcommand{\RtwoZQA}{\mathrm{R}_\mathrm{2,ZQ}^\mathrm{A}}
+\newcommand{\RtwoZQB}{\mathrm{R}_\mathrm{2,ZQ}^\mathrm{B}}
\newcommand{\tex}{\tau_\textrm{ex}}
\newcommand{\taucpmg}{\tau_\textrm{CPMG}}
r21139 - in /branches/relax_disp/docs/latex: dispersion.tex relax.tex