Author: bugman
Date: Tue Oct 22 15:14:38 2013
New Revision: 21212
URL: http://svn.gna.org/viewcvs/relax?rev=21212&view=rev
Log:
Added all of the equations for the 'NS CPMG 2-site expanded' dispersion model
to the relax manual.
These are essentially the source code modified to look good in LaTeX.
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=21212&r1=21211&r2=21212&view=diff
==============================================================================
--- branches/relax_disp/docs/latex/dispersion.tex (original)
+++ branches/relax_disp/docs/latex/dispersion.tex Tue Oct 22 15:14:38 2013
@@ -427,6 +427,81 @@
This is the numerical model for 2-site exchange expanded using Maple by
Nikolai Skrynnikov.
It is selected by setting the model to `NS CPMG 2-site expanded'.
The simple constraint $\pA > \pB$ is used to halve the optimisation space,
as both sides of the limit are mirror image spaces.
+
+This model will give the same results as the other numerical solutions
whereby $\RtwozeroA = \RtwozeroB$. The following is the set of equations of
the expansion used in relax. It has been modified from the original for
speed. See the \module{lib.dispersion.ns\_cpmg\_2site\_expanded} module for
more details including the original code. Further simplifications can be
found in the code.
+
+\begin{subequations}
+\begin{align}
+ t_{3} &= \imath, \\
+ t_{4} &= t_{3} \dw, \\
+ t_{5} &= \kBA^2, \\
+ t_{8} &= 2 t_{4} \kBA, \\
+ t_{10} &= 2 \kBA \kAB, \\
+ t_{11} &= \dw^2, \\
+ t_{14} &= 2 t_{4} \kAB, \\
+ t_{15} &= \kAB^2, \\
+ t_{17} &= \sqrt{t_{5} - t_{8} + t_{10} - t_{11} + t_{14} + t_{15}}, \\
+ t_{21} &= \exp \left(\frac{(-\kBA + t_{4} - \kAB + t_{17}) \taucpmg}{2}
\right), \\
+ t_{22} &= \frac{1}{t_{17}}, \\
+ t_{28} &= \exp \left(\frac{(-\kBA + t_{4} - \kAB - t_{17}) \taucpmg}{2}
\right), \\
+ t_{31} &= t_{22} \kAB (t_{21} - t_{28}), \\
+ t_{33} &= \sqrt{t_{5} + t_{8} + t_{10} - t_{11} - t_{14} + t_{15}}, \\
+ t_{34} &= \kBA + t_{4} - \kAB + t_{33}, \\
+ t_{37} &= \exp \left((-\kBA - t_{4} - \kAB + t_{33}) \taucpmg \right),
\\
+ t_{39} &= \frac{1}{t_{33}}, \\
+ t_{41} &= \kBA + t_{4} - \kAB - t_{33}, \\
+ t_{44} &= \exp \left((-\kBA - t_{4} - \kAB - t_{33}) \taucpmg \right),
\\
+ t_{47} &= \frac{t_{39}}{2} \left(t_{34} t_{37} - t_{41} t_{44} \right),
\\
+ t_{49} &= \kBA - t_{4} - \kAB - t_{17}, \\
+ t_{51} &= t_{21} t_{49} t_{22}, \\
+ t_{52} &= \kBA - t_{4} - \kAB + t_{17}, \\
+ t_{54} &= t_{28} t_{52} t_{22}, \\
+ t_{55} &= t_{54} - t_{51}, \\
+ t_{60} &= \frac{1}{2} t_{39} \kAB \left(t_{37} - t_{44} \right), \\
+ t_{62} &= t_{31} t_{47} + t_{55} t_{60}, \\
+ t_{63} &= \frac{1}{\kAB}, \\
+ t_{68} &= \frac{t_{63}}{2} \left(t_{49} t_{54} - t_{52} t_{51} \right),
\\
+ t_{69} &= \frac{t_{62} t_{68}}{2}, \\
+ t_{72} &= t_{37} t_{41} t_{39}, \\
+ t_{76} &= t_{44} t_{34} t_{39}, \\
+ t_{78} &= \frac{t_{63}}{2} \left(t_{41} t_{76} - t_{34} t_{72} \right),
\\
+ t_{80} &= \frac{1}{2} (t_{76} - t_{72}), \\
+ t_{82} &= \frac{1}{2} (t_{31} t_{78} + t_{55} t_{80}), \\
+ t_{83} &= \frac{t_{82} t_{55}}{2}, \\
+ t_{88} &= \frac{t_{22}}{2} \left(t_{52} t_{21} - t_{49} t_{28} \right),
\\
+ t_{91} &= t_{88} t_{47} + t_{68} t_{60}, \\
+ t_{92} &= t_{91} t_{88}, \\
+ t_{95} &= \frac{1}{2} (t_{88} t_{78} + t_{68} t_{80}), \\
+ t_{96} &= t_{95} t_{31}, \\
+ t_{97} &= t_{69} + t_{83}, \\
+ t_{98} &= t_{97}^2, \\
+ t_{99} &= t_{92} + t_{96}, \\
+ t_{102} &= t_{99}^2, \\
+ t_{108} &= t_{62} t_{88} + t_{82} t_{31}, \\
+ t_{112} &= \sqrt{t_{98} - 2 t_{99} t_{97} + t_{102} + 2 (t_{91} t_{68} +
t_{95} t_{55}) t_{108}}, \\
+ t_{113} &= t_{97} - t_{99} - t_{112}, \\
+ t_{115} &= n_\textrm{CPMG}, \\
+ t_{116} &= \left( \frac{t_{97} + t_{99} + t_{112}}{2} \right)^{t_{115}},
\\
+ t_{118} &= \frac{1}{t_{112}}, \\
+ t_{120} &= t_{97} - t_{99} + t_{112}, \\
+ t_{122} &= \left( \frac{t_{97} + t_{99} - t_{112}}{2} \right)^{t_{115}},
\\
+ t_{127} &= \frac{1}{2 t_{108}}, \\
+ t_{139} &= \frac{1}{2(\kAB + \kBA)} \Big[ (t_{120} t_{122} - t_{113}
t_{116}) t_{118} \kBA \nonumber \\
+ & \qquad \qquad + (t_{120} t_{122} - t_{116}
t_{120}) t_{113} t_{118} t_{127} \kAB \Big].
+\end{align}
+\end{subequations}
+
+The relative peak intensities, magnitisation, and effective $\Rtwo$
relaxation rate are calculated as
+\begin{subequations}
+\begin{align}
+ I_0 &= \pA, \\
+ I_1 &= \Re(t_{139}) \exp(-T_\textrm{relax} \Rtwozero), \\
+ M_x &= I_1 / I_0, \\
+ \Rtwoeff &= -\frac{1}{T_\textrm{relax}} \cdot \ln \left( M_x \right).
\label{eq: R2eff NS CPMG 2-site expanded}
+\end{align}
+\end{subequations}
+
+In these equations $\taucpmg$ and $n_\textrm{CPMG}$ are numpy arrays and
hence $t_{139}$ is also a numpy array. This avoids a Python loop over the
dispersion points until the very end of the calculation, required to populate
the $\Rtwoeff$ data structure, resulting in very fast calculations.
The reference for this model is:
\begin{itemize}
r21212 - /branches/relax_disp/docs/latex/dispersion.tex