Author: tlinnet
Date: Wed May 7 15:54:28 2014
New Revision: 23036
URL: http://svn.gna.org/viewcvs/relax?rev=23036&view=rev
Log:
Changed manual, to the recipe at Appendix 1.
This was changed after the wish of the author.
Discussed in: http://thread.gmane.org/gmane.science.nmr.relax.devel/5632.
sr #3154: (https://gna.org/support/?3154) Implementation of Baldwin (2014)
B14 model - 2-site exact solution model for all time scales.
This follows the tutorial for adding relaxation dispersion models at:
http://wiki.nmr-relax.com/Tutorial_for_adding_relaxation_dispersion_models_to_relax#The_relax_manual
Modified:
trunk/docs/latex/dispersion.tex
Modified: trunk/docs/latex/dispersion.tex
URL:
http://svn.gna.org/viewcvs/relax/trunk/docs/latex/dispersion.tex?rev=23036&r1=23035&r2=23036&view=diff
==============================================================================
--- trunk/docs/latex/dispersion.tex (original)
+++ trunk/docs/latex/dispersion.tex Wed May 7 15:54:28 2014
@@ -574,15 +574,62 @@
\end{subequations}
-where
+where Appendix 1 in \citet{Baldwin2014} list the recipe for exact
calculation of $\Rtwoeff$.
+
+Establish the complex free precession Eigenfrequency.
\begin{subequations}
\begin{align}
- v_{1c} & =
F_0\cosh{\left(\tau_{\textrm{CP}}E_0\right)}-F_2\cosh{\left(\tau_{\textrm{CP}}E_2\right)}
\\
- v_{1s} & =
F_0\sinh{\left(\tau_{\textrm{CP}}E_0\right)}-F_2\sinh{\left(\tau_{\textrm{CP}}E_2\right)}
\\
- v_{2}N & = v_{1s}\left(O_B-O_A\right)+4O_B F_1^a
\sinh{\left(\tau_{\textrm{CP}}E_1\right)} \\
- p_D N & = v_{1s} +
\left(F_1^a+F_1^b\right)\sinh{\left(\tau_{\textrm{CP}}E_1\right)} \\
- v_3 & = \left( v_2^2 + 4 \kBA \kAB p_D^2 \right)^{1/2} \\
- y & = \left( \frac{v_{1c}-v_3}{v_{1c}+v_3} \right)^{N_{\textrm{CYC}}}
+ \Delta \Rtwozero & = \RtwozeroA - \RtwozeroB \\
+ \alpha_- & = \Delta \Rtwozero + \kAB - \kBA \\
+ \zeta & = 2 \dw \alpha_- \\
+ \Psi & = \alpha_-^2 + 4 \kAB \kBA - \dw^2 \\
+ h_3 &= \frac{1}{\sqrt{2}}\sqrt{ \Psi + \sqrt{\zeta^2 + \Psi^2} } \\
+ h_4 &= \frac{1}{\sqrt{2}}\sqrt{ -\Psi + \sqrt{\zeta^2 + \Psi^2} } \\
+\end{align}
+\end{subequations}
+
+The ground state ensemble evolution frequency $f_{00}$ expressed in
separated real and imaginary components, in terms
+of definitions $\zeta , \Psi , h_3 , h_4$.
+\begin{subequations}
+\begin{align}
+ f_{00} & = \frac{1}{2}(\RtwozeroA + \RtwozeroB + \kex) +
\frac{1}{2}(\dw - h_4) i
+\end{align}
+\end{subequations}
+
+Define substutions for 'stay' and 'swap' factors.
+\begin{subequations}
+\begin{align}
+ N & = h_3 + h_4 i \\
+ NN^* & = h_3^2 + h_42 \\
+ F_0 & = (\dw^2 + h_3^2) / NN^* \\
+ F_2 & = (\dw^2 - h_4^2) / NN^* \\
+ F_1^b & = (\dw + h_4) (\dw - h_3 i) / NN^* \\
+ F_1^{a+b} & = (2\dw^2 + \zeta i) / NN^* \\
+\end{align}
+\end{subequations}
+
+Weighting factors for frequencies ($E_{0-2}$) emerging from a single CPMG
block, ($F_{0-2}$).
+$\taucpmg = 1 / \nucpmg $.
+\begin{subequations}
+\begin{align}
+ E_0 & = 2 \taucpmg \cdot h_3 \\
+ E2 & = 2 \taucpmg \cdot h_4 \\
+ E1 & = (h_3 - h_4 i) \cdot \taucpmg
+\end{align}
+\end{subequations}
+
+Final result, with identities to assist efficient matrix exponentiation
optimised for numerical calculation.
+\begin{subequations}
+\begin{align}
+ \nu_{1c} = F_0 \cosh(E_0) - F_2 \cos(E_2)
+ \nu_{1s} & = F_0 \sinh(E_0) - F_2 \sin(E_2)i \\
+ \nu_{3} & = \sqrt({\nu_{1c}^2 - 1} \\
+ \nu_{4} & = F_1^b (-\alpha_- - h_3 ) + F_1^b (\dw - h_4) i \\
+ \nu_{5} & =(-\Delta \Rtwozero + \kex + \dw i) \nu_{1s} + 2 (\nu_{4} +
\kAB F_1^{a+b}) \sinh(E_1) \\
+ y & = \left( \frac{\nu_{1c} - \nu_{3}}{\nu_{1c} + \nu_{3}} \right) ^
{N_{\textrm{CYC}}} \\
+ T & = \frac{1}{2}(1 + y) + \frac{(1 - y)\nu_{5}}{2 \nu_{3}N} \\
+ \Rtwoeff & = \frac{(\RtwozeroA + \RtwozeroB + \kex)}{2} \nonumber \\
+ & \qquad - \frac{1}{\taucpmg} \left( N_{\textrm{CYC}} \cdot
\textrm{arcosh}(\operatorname{Re}(\nu_{1c})) - \log(\operatorname{Re}(T))
\right)
\end{align}
\end{subequations}
r23036 - /trunk/docs/latex/dispersion.tex