Author: bugman
Date: Sun Nov 17 17:56:41 2013
New Revision: 21490
URL: http://svn.gna.org/viewcvs/relax?rev=21490&view=rev
Log:
Improvements to all of the R1rho model descriptions in the dispersion chapter
of the manual.
Modified:
branches/relax_disp/docs/latex/dispersion.tex
Modified: branches/relax_disp/docs/latex/dispersion.tex
URL:
http://svn.gna.org/viewcvs/relax/branches/relax_disp/docs/latex/dispersion.tex?rev=21490&r1=21489&r2=21490&view=diff
==============================================================================
--- branches/relax_disp/docs/latex/dispersion.tex (original)
+++ branches/relax_disp/docs/latex/dispersion.tex Sun Nov 17 17:56:41 2013
@@ -579,9 +579,11 @@
\label{sect: dispersion: M61 model}
\index{relaxation dispersion!M61 model|textbf}
-This is the model for 2-site fast exchange for on-resonance $\Ronerho$-type
experiments. It is selected by setting the model to `M61', here named after
\citet{Meiboom61}. The equation for the exchange process is:
-\begin{equation}
- \Ronerho = \Ronerhoprime + \frac{\Phiex\kex}{\kex^2 + \omegae^2} .
+This is the model for 2-site fast exchange for on-resonance $\Ronerho$-type
data.
+It is selected by setting the model to `M61', here named after
\citet{Meiboom61}.
+The equation for the exchange process is
+\begin{equation}
+ \Ronerho = \Ronerhoprime + \frac{\Phiex\kex}{\kex^2 + \omegae^2}.
\end{equation}
The reference for this equation is:
@@ -597,12 +599,15 @@
\label{sect: dispersion: M61 skew model}
\index{relaxation dispersion!M61 skew model|textbf}
-This is the second model for 2-site fast exchange for on-resonance
$\Ronerho$-type experiments from \citet{Meiboom61}. It is selected by
setting the model to `M61 skew'. The equation for the exchange process is:
-\begin{equation}
- \Ronerho = \Ronerhoprime + \frac{\pA^2\pB\dw^2\kex}{\kex^2 + \pA^2\dw^2
+ \omegaone^2} .
-\end{equation}
-
-Care must be taken as this model appears to have infinite lines of solutions
-- $\pA$ and $\dw$ are convoluted. Hence this model is disabled in the
dispersion auto-analysis.
+This is the second model for 2-site fast exchange for on-resonance
$\Ronerho$-type data from \citet{Meiboom61}.
+It is selected by setting the model to `M61 skew'.
+The equation for the exchange process is
+\begin{equation}
+ \Ronerho = \Ronerhoprime + \frac{\pA^2\pB\dw^2\kex}{\kex^2 + \pA^2\dw^2
+ \omegaone^2}.
+\end{equation}
+
+Care must be taken as this model appears to have infinite lines of solutions
-- $\pA$ and $\dw$ are convoluted.
+Hence this model is disabled in the dispersion auto-analysis.
% DPL94 model.
@@ -612,9 +617,13 @@
\label{sect: dispersion: DPL94 model}
\index{relaxation dispersion!DPL94 model|textbf}
-This is the model for 2-site fast exchange for $\Ronerho$-type experiments.
It is selected by setting the model to `DPL94', here named after
\citet{Davis94}. It extends the \citet{Meiboom61} model to off-resonance
data. The model collapses to the M61 model for on-resonance data. The
equation for the exchange process is:
-\begin{equation}
- \Ronerho = \Rone \cos^2\theta + \left( \Ronerhoprime +
\frac{\Phiex\kex}{\kex^2 + \omegae^2} \right) \sin^2\theta ,
+This is the model for 2-site fast exchange for $\Ronerho$-type data.
+It is selected by setting the model to `DPL94', here named after
\citet{Davis94}.
+It extends the \citet{Meiboom61} model to off-resonance data.
+The model collapses to the M61 model for on-resonance data.
+The equation for the exchange process is
+\begin{equation}
+ \Ronerho = \Rone \cos^2\theta + \left( \Ronerhoprime +
\frac{\Phiex\kex}{\kex^2 + \omegae^2} \right) \sin^2\theta,
\end{equation}
where $\theta$ is the rotating frame tilt angle. The reference for this
equation is:
@@ -630,15 +639,30 @@
\label{sect: dispersion: TP02 model}
\index{relaxation dispersion!TP02 model|textbf}
-This is the model for 2-site exchange for off-resonance $\Ronerho$-type
experiments from \citet{TrottPalmer02}.
+This is the model for 2-site exchange for off-resonance $\Ronerho$-type data
from \citet{TrottPalmer02}.
It is selected by setting the model to `TP02'.
-The equation for the exchange process is:
-\begin{equation}
- \Ronerho = \Rone\cos^2\theta + \left( \Ronerhoprime +
\frac{\pA\pB\dw^2\kex}{\omega_\textrm{Aeff}^2\omega_\textrm{Beff}^2/\omega_\textrm{eff}^2
+ \kex^2} \right) \sin^2\theta.
-\end{equation}
+The equation for the exchange process is
+\begin{equation}
+ \Ronerho = \Rone\cos^2\theta + \Ronerhoprime\sin^2\theta +
\frac{\sin^2\theta\pA\pB\dw^2\kex}{\omega_\textrm{Aeff}^2\omega_\textrm{Beff}^2/\omega_\textrm{eff}^2
+ \kex^2},
+\end{equation}
+
+in which
+\begin{subequations}
+\begin{align}
+ \delta_A &= \omegaA - \omegarf, \\
+ \delta_B &= \omegaB - \omegarf, \\
+ \aveomega &= \pA\omegaA + \pB\omegaB, \\
+ \aveoffset &= \aveomega - \omegarf, \\
+ \omega_\textrm{Aeff}^2 &= \omegaone^2 + \delta_A^2, \\
+ \omega_\textrm{Beff}^2 &= \omegaone^2 + \delta_B^2, \\
+ \omega_\textrm{eff}^2 &= \omegaone^2 + \aveoffset^2, \\
+ \theta &= \arctan \left( \frac{\omegaone}{\aveoffset} \right).
+\end{align}
+\end{subequations}
The equation is accurate only when populations are highly skewed with $\pA
\gg \pB$.
And it is valid only for exchange processes which are not fast.
+Note that this model has been superseded by the `TAP03' and `MP05' models.
The reference for this equation is:
\begin{itemize}
\item \bibentry{TrottPalmer02}
@@ -652,7 +676,7 @@
\label{sect: dispersion: TAP03 model}
\index{relaxation dispersion!TAP03 model|textbf}
-This is the model for 2-site exchange for off-resonance $\Ronerho$-type
experiments from \citet{Trott03}.
+This is the model for 2-site exchange for off-resonance $\Ronerho$-type data
from \citet{Trott03}.
It is selected by setting the model to `TAP03'.
The equation for the exchange process is
\begin{align}
@@ -661,11 +685,9 @@
\frac{\sin^2\hat\theta\pA\pB\dw^2\kex}{\hat\omega_\textrm{Aeff}^2\hat\omega_\textrm{Beff}^2/\hat\omega_\textrm{eff}^2
+ \kex^2 - 2\sin^2\hat\theta\pA\pB\dw^2 + (1 - \gamma)\omegaone^2},
\end{align}
-in which
+in which, in addition to those parameters defined above for the `TP02' model,
\begin{subequations}
\begin{align}
- \delta_A &= \omegaA - \omegarf, \\
- \delta_B &= \omegaB - \omegarf, \\
\sigma &= \pB\delta_A + \pA\delta_B, \\
\gamma &= 1 - \pA\pB\dw^2 \frac{\sigma^2 - \kex^2 + \omegaone^2}{\left(
\sigma^2 + \kex^2 + \omegaone^2 \right)^2}, \\
\hat\omega_\textrm{Aeff}^2 &= \gamma\omegaone^2 + \delta_A^2, \\
@@ -690,14 +712,18 @@
\label{sect: dispersion: MP05 model}
\index{relaxation dispersion!MP05 model|textbf}
-This is the model for 2-site exchange for off-resonance $\Ronerho$-type
experiments from \citet{MiloushevPalmer05}. It is selected by setting the
model to `MP05'. The equation for the exchange process is:
-\begin{align}
- \Ronerho = & \Rone\cos^2\theta + \Ronerhoprime\sin^2\theta \\
+This is the model for 2-site exchange for off-resonance $\Ronerho$-type data
for all time scales from \citet{MiloushevPalmer05}.
+It is selected by setting the model to `MP05'.
+The equation for the exchange process is
+\begin{align}
+ \Ronerho = & \Rone\cos^2\theta + \Ronerhoprime\sin^2\theta \nonumber \\
& +
\frac{\sin^2\theta\pA\pB\dw^2\kex}{\omega_\textrm{Aeff}^2\omega_\textrm{Beff}^2/\omega_\textrm{eff}^2
+ \kex^2 - \sin^2\theta\pA\pB\dw^2 \left(
1 + \frac{2\kex^2 \left( \pA\omega_\textrm{Aeff}^2 +
\pB\omega_\textrm{Beff}^2
\right)}{\omega_\textrm{Aeff}^2\omega_\textrm{Beff}^2 +
\omega_\textrm{eff}^2\kex^2}
- \right)}
-\end{align}
-
+ \right)},
+\end{align}
+
+in which the parameters are defined as in the `TP02' model above.
+This model supersedes both the `TP02' and `TAP03' models.
The reference for this equation is:
\begin{itemize}
\item \bibentry{MiloushevPalmer05}
r21490 - /branches/relax_disp/docs/latex/dispersion.tex