Author: tlinnet
Date: Tue May 6 17:35:22 2014
New Revision: 23026
URL: http://svn.gna.org/viewcvs/relax?rev=23026&view=rev
Log:
Updated the math to be nicely ordered in array.
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=23026&r1=23025&r2=23026&view=diff
==============================================================================
--- trunk/docs/latex/dispersion.tex (original)
+++ trunk/docs/latex/dispersion.tex Tue May 6 17:35:22 2014
@@ -565,24 +565,25 @@
This is the model for 2-site exchange exact analytical derivation on all
time scales (with the constraint that $\pA > \pB$), named after
\citet{Baldwin2014}.
It is selected by setting the model to `B14 full'.
The equation is
-\begin{equation}
- R_{2,\textrm{eff}} =
\frac{R_2^A+R_2^B+k_{\textrm{EX}}}{2}-\frac{N_{\textrm{CYC}}}{T_{\textrm{rel}}}\cosh{}^{-1}(v_{1c})
- \frac{1}{T_{\textrm{rel}}}\ln{\left( \frac{1+y}{2} +
\frac{1-y}{2\sqrt{v_{1c}^2-1}}(v_2 + 2k_{\textrm{AB}}p_D )\right)} \\
- = R_{2,\textrm{eff}}^{\textrm{CR72}} -
\frac{1}{T_{\textrm{rel}}}\ln{\left( \frac{1+y}{2} +
\frac{1-y}{2\sqrt{v_{1c}^2-1}}(v_2 + 2k_{\textrm{AB}}p_D )\right)} ,
-\end{equation}
+\begin{eqnarray}
+ R_{2,\textrm{eff}} & = &
\frac{R_2^A+R_2^B+k_{\textrm{EX}}}{2}-\frac{N_{\textrm{CYC}}}{T_{\textrm{rel}}}\cosh{}^{-1}(v_{1c})
\nonumber \\
+ & - & \frac{1}{T_{\textrm{rel}}}\ln{\left(
\frac{1+y}{2} + \frac{1-y}{2\sqrt{v_{1c}^2-1}}(v_2 + 2k_{\textrm{AB}}p_D
)\right)} \nonumber \\
+ & = & R_{2,\textrm{eff}}^{\textrm{CR72}} -
\frac{1}{T_{\textrm{rel}}}\ln{\left( \frac{1+y}{2} +
\frac{1-y}{2\sqrt{v_{1c}^2-1}}(v_2 + 2k_{\textrm{AB}}p_D )\right)} ,
+\end{eqnarray}
where
-\begin{equation}
- 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 k_{\textrm{BA}} k_{\textrm{AB}} p_D^2
\right)^{1/2} \\
- y = \left( \frac{v_{1c}-v_3}{v_{1c}+v_3} \right)^{N_{\textrm{CYC}}}
-\end{equation}
+\begin{eqnarray}
+ v_{1c} & = &
F_0\cosh{\left(\tau_{\textrm{CP}}E_0\right)}-F_2\cosh{\left(\tau_{\textrm{CP}}E_2\right)}
\nonumber \\
+ v_{1s} & = &
F_0\sinh{\left(\tau_{\textrm{CP}}E_0\right)}-F_2\sinh{\left(\tau_{\textrm{CP}}E_2\right)}
\nonumber \\
+ v_{2}N & = & v_{1s}\left(O_B-O_A\right)+4O_B F_1^a
\sinh{\left(\tau_{\textrm{CP}}E_1\right)} \nonumber \\
+ p_D N & = & v_{1s} +
\left(F_1^a+F_1^b\right)\sinh{\left(\tau_{\textrm{CP}}E_1\right)} \nonumber \\
+ v_3 & = & \left( v_2^2 + 4 k_{\textrm{BA}} k_{\textrm{AB}} p_D^2
\right)^{1/2} \nonumber \\
+ y & = & \left( \frac{v_{1c}-v_3}{v_{1c}+v_3} \right)^{N_{\textrm{CYC}}}
+\end{eqnarray}
The advantage of this code will be that you will always get the right answer
provided you got 2-site exchange, in-phase magnetisation and on-resonance
pulses.
-The term $$p_D$$ is based on product of the off diagonal elements in the
CPMG propagator (Supplementary Section 3, \citet{Baldwin2014}).
+The term $p_D$ is based on product of the off diagonal elements in the CPMG
propagator, see supplementary Section 3, \citet{Baldwin2014}.
It is interesting to consider the region of validity of the Carver Richards
result. The two results are equal when the correction is zero, which is true
when
@@ -590,10 +591,10 @@
\sqrt{v_{1c}^2-1} \approx v_2 + 2k_{\textrm{AB}}p_D
\end{equation}
-This occurs when $$k_{\textrm{AB}}p_D$$ tends to zero, and so $$v_2=v_3$$.
-Setting $$k_{\textrm{AB}}p_D$$ to zero, amounts to neglecting magnetisation
that starts on the ground state ensemble and end on the excited state
ensemble and vice versa.
-This will be a good approximation when $$p_A \gg p_B$$.
-In practise, significant deviations from the Carver Richards equation can be
incurred if $$p_B > 1\%$$.
+This occurs when $k_{\textrm{AB}}p_D$ tends to zero, and so $v_2=v_3$.
+Setting $k_{\textrm{AB}}p_D$ to zero, amounts to neglecting magnetisation
that starts on the ground state ensemble and end on the excited state
ensemble and vice versa.
+This will be a good approximation when $p_A \gg p_B$.
+In practise, significant deviations from the Carver Richards equation can be
incurred if $p_B > 1$\%.
Incorporation of the correction term, results in an improved description of
the CPMG experiment over the Carver Richards equation
\citet{CarverRichards72}.
The reference for this equation is:
r23026 - /trunk/docs/latex/dispersion.tex