# CR72 full

The full Carver and Richards 1972 2-site relaxation dispersion model for SQ CPMG-type data for most time scales. This model is labelled as CR72 full in relax.

## Equation

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

Which have the following definitions

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

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

## Parameters

The CR72 full model has the parameters {R2A0, R2B0, ..., pA, Δω, kex}.

## Reference

The reference for the CR72 full 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 this model.