View source for CR72

= Intro =

The Carver and Richards 1972 2-site relaxation dispersion model for [[SQ CPMG-type data]] for most time scales whereby the simplification $R_{2A}^0$ = $R_{2B}^0$ is assumed.  This model is labelled as '''CR72''' in [[Relaxation dispersion citation for relax|relax]].

== Equation ==
[http://www.nmr-relax.com/manual/Dispersion_model_summary.html Please see the summary of the model parameters here.]

* relax manual http://www.nmr-relax.com/manual/reduced_CR72_2_site_CPMG_model.html
* relaxation dispersion page of the relax website http://www.nmr-relax.com/analyses/relaxation_dispersion.html#CR72

<math>
R_{2,\textrm{eff}} = \frac{R_2^A+R_2^B+k_{\textrm{EX}}}{2} - \nu_{\textrm{cpmg}} \cosh^{-1} (D_+\cosh(\eta_+) - D_-\cos(\eta_-))
= R_2 + \frac{k_{\textrm{EX}}}{2} - \nu_{\textrm{cpmg}} \cosh^{-1} (D_+\cosh(\eta_+) - D_-\cos(\eta_-)) 
</math>

Which have the following definitions

<math>
\zeta = 2 \Delta \omega \, (R_2^A - R_2^B - p_A k_{\textrm{EX}} + p_B k_{\textrm{EX}}) 
= - 2 \Delta \omega \, ( p_A k_{\textrm{EX}} - p_B k_{\textrm{EX}}) 
= - 2 \Delta \omega \, ( k_{\textrm{BA}} - k_{\textrm{AB}}) 
= - 2 \Delta \omega \, k_{\textrm{EX}} ( 2p_A - 1)  \\
\Psi = (p_B k_{\textrm{EX}} - p_A k_{\textrm{EX}})^2 + 4 p_A p_B k_{\textrm{ex}}^2 - \Delta \omega^2
= ( p_A k_{\textrm{EX}} + p_B k_{\textrm{EX}} )^2 - \Delta \omega^2 
= k_{\textrm{ex}}^2 - \Delta \omega^2 \\
\eta_+ = \frac{1}{2\sqrt{2} \, \nu_{\textrm{cpmg}}}\sqrt{+\Psi + \sqrt{\Psi^2 + \zeta^2}} \\
\eta_- = \frac{1}{2\sqrt{2} \, \nu_{\textrm{cpmg}}}\sqrt{-\Psi + \sqrt{\Psi^2 + \zeta^2}} \\
D_+=\frac{1}{2}\left(1+\frac{\Psi+2\Delta \omega^2}{\sqrt{\Psi^2+\zeta^2}} \right) \\
D_-=\frac{1}{2}\left(-1+\frac{\Psi+2\Delta \omega^2}{\sqrt{\Psi^2+\zeta^2}} \right)
</math>

<math>k_{\textrm{EX}}</math> is the chemical exchange rate constant, <math>p_A</math> and <math>p_B</math> are the populations of states A and B, and <math>\Delta \omega</math> is the chemical shift difference between the two states in ppm.

== Parameters ==

The CR72 model has the parameters {$R_2^0$, $...$, $p_A$, $\Delta\omega$, $k_{ex}$}.

== Code ==

The library code.<br>
http://svn.gna.org/viewcvs/*checkout*/relax/trunk/lib/dispersion/cr72.py?revision=HEAD

== Reference ==

The reference for the CR72 model is:

* Carver, J. and Richards, R. (1972).  General 2-site solution for chemical exchange produced dependence of T2 upon Carr-Purcell pulse separation. ''J. Magn. Reson.'', '''6'''(1), 89-105. ([http://dx.doi.org/10.1016/0022-2364(72)90090-X 10.1016/0022-2364(72)90090-X]).

== Related models ==

The CR72 model is a parametric restriction of the [[CR72 full]] model.

== Links ==

The [[Relaxation dispersion citation for relax|implementation of the CR72 model in relax]] can be seen in the:
* [http://www.nmr-relax.com/manual/reduced_CR72_2_site_CPMG_model.html relax manual],
* [http://www.nmr-relax.com/api/3.1/lib.dispersion.cr72-module.html API documentation],
* [http://www.nmr-relax.com/analyses/relaxation_dispersion.html#CR72 relaxation dispersion page of the relax website].

== See also ==
[[Category:Relaxation_dispersion]]