CR72

The Carver and Richards 1972 2-site relaxation dispersion model for SQ CPMG-type data for most time scales whereby the simplification R_2A⁰ = R_2B⁰ is assumed. This model is labelled as CR72 in relax.

Equation

Please see the summary of the model parameters here.

[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_-)) \\ \phantom{R_{2,\textrm{eff}}} = 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}}) \\ \phantom{\zeta} = - 2 \Delta \omega \, ( p_A k_{\textrm{EX}} - p_B k_{\textrm{EX}}) \\ \phantom{\zeta} = - 2 \Delta \omega \, ( k_{\textrm{BA}} - k_{\textrm{AB}}) \\ \phantom{\zeta} = - 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 \\ \phantom{\Psi} = ( p_A k_{\textrm{EX}} + p_B k_{\textrm{EX}} )^2 - \Delta \omega^2 \\ \phantom{\Psi} = 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]

k_ex is the chemical exchange rate constant, p_A and p_B are the populations of states A and B, and Δω is the chemical shift difference between the two states in ppm.

Parameters

The CR72 model has the parameters {R₂⁰, ..., p_A, Δω, k_ex}.

Code

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

Reference

The reference for the CR72 model is:

Carver, J. P. and Richards, R. E. (1972). General 2-site solution for chemical exchange produced dependence of T2 upon Carr-Purcell pulse separation. J. Magn. Reson., 6(1), 89-105. (DOI: 10.1016/0022-2364(72)90090-X)

Related models

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

Links

The implementation of the CR72 model in relax can be seen in the: