The MMQ CR72 model

This is the analytic CR72 model for 2-site exchange on most times scales (Section 11.3.4 on page [*]) extended for multiple types of multiple quantum data (MMQ) by Korzhnev et al. (2004a). It is selected by setting the model to `MMQ CR72'. The simple constraint pA > pB is used to halve the optimisation space, as both sides of the limit are mirror image spaces. The equation for the exchange process is

R2eff = ℜ(λ1) - $\displaystyle {\frac{{1}}{{T_{\textrm{relax}}}}}$ln(Q), (11.44)

where

λ1 = R2, MQ0 + $\displaystyle {\frac{{\textrm{k}_{\textrm{ex}}}}{{2}}}$ - νCPMGcosh-1$\displaystyle \big($D+cosh(η+) - D-cos(η-)$\displaystyle \big)$, (11.45)
D± = $\displaystyle {\frac{{1}}{{2}}}$$\displaystyle \left(\vphantom{ \pm1 + \frac{\Psi + 2\Delta\omega ^2}{\sqrt{\Psi^2 + \zeta^2}} }\right.$±1 + $\displaystyle {\frac{{\Psi + 2\Delta\omega ^2}}{{\sqrt{\Psi^2 + \zeta^2}}}}$$\displaystyle \left.\vphantom{ \pm1 + \frac{\Psi + 2\Delta\omega ^2}{\sqrt{\Psi^2 + \zeta^2}} }\right)$, (11.46)
η± = 2$\scriptstyle {\frac{{2}}{{3}}}$$\displaystyle {\frac{{1}}{{\nu_{\textrm{CPMG}}}}}$$\displaystyle \left(\vphantom{ \pm\Psi + \sqrt{\Psi^2 + \zeta^2} }\right.$±Ψ + $\displaystyle \sqrt{{\Psi^2 + \zeta^2}}$$\displaystyle \left.\vphantom{ \pm\Psi + \sqrt{\Psi^2 + \zeta^2} }\right)^{{\frac{1}{2}}}_{}$, (11.47)
Ψ = $\displaystyle \left(\vphantom{ \imath \Delta\omega^{\scriptscriptstyle\mathrm{H...
...trm{A}}\textrm{k}_{\textrm{ex}}- p_{\textrm{B}}\textrm{k}_{\textrm{ex}}}\right.$ıΔωH + pAkex - pBkex$\displaystyle \left.\vphantom{ \imath \Delta\omega^{\scriptscriptstyle\mathrm{H...
...\textrm{k}_{\textrm{ex}}- p_{\textrm{B}}\textrm{k}_{\textrm{ex}}}\right)^{2}_{}$ - Δω2 +4pApBkex2, (11.48)
ζ = - 2Δω$\displaystyle \left(\vphantom{ \imath \Delta\omega^{\scriptscriptstyle\mathrm{H...
...trm{A}}\textrm{k}_{\textrm{ex}}- p_{\textrm{B}}\textrm{k}_{\textrm{ex}}}\right.$ıΔωH + pAkex - pBkex$\displaystyle \left.\vphantom{ \imath \Delta\omega^{\scriptscriptstyle\mathrm{H...
...trm{A}}\textrm{k}_{\textrm{ex}}- p_{\textrm{B}}\textrm{k}_{\textrm{ex}}}\right)$, (11.49)

and where

Q = ℜ$\displaystyle \left(\vphantom{ 1 - m_{D+}^2 + m_{D+} m_{Z+} - m_{Z+}^2 + \frac{m_{D+} + m_{Z+}}{2} \sqrt{\frac{p_{\textrm{B}}}{p_{\textrm{A}}}} }\right.$1 - mD+2 + mD+mZ+ - mZ+2 + $\displaystyle {\frac{{m_{D+} + m_{Z+}}}{{2}}}$$\displaystyle \sqrt{{\frac{p_{\textrm{B}}}{p_{\textrm{A}}}}}$$\displaystyle \left.\vphantom{ 1 - m_{D+}^2 + m_{D+} m_{Z+} - m_{Z+}^2 + \frac{m_{D+} + m_{Z+}}{2} \sqrt{\frac{p_{\textrm{B}}}{p_{\textrm{A}}}} }\right)$, (11.50)

and

m = ±$\displaystyle {\frac{{\imath\textrm{k}_{\textrm{ex}}\sqrt{p_{\textrm{A}}p_{\textrm{B}}}}}{{d_{\pm} z_\pm}}}$$\displaystyle \left(\vphantom{ z_{\pm} + 2\Delta\omega \frac{\sin(z_{\pm}\delta)}{\sin((d_{\pm} + z_{\pm})\delta)} }\right.$z± +2Δω$\displaystyle {\frac{{\sin(z_{\pm}\delta)}}{{\sin((d_{\pm} + z_{\pm})\delta)}}}$$\displaystyle \left.\vphantom{ z_{\pm} + 2\Delta\omega \frac{\sin(z_{\pm}\delta)}{\sin((d_{\pm} + z_{\pm})\delta)} }\right)$, (11.51)
mZ$\scriptstyle \mp$ = ±$\displaystyle {\frac{{\imath\textrm{k}_{\textrm{ex}}\sqrt{p_{\textrm{A}}p_{\textrm{B}}}}}{{d_{\pm} z_\pm}}}$$\displaystyle \left(\vphantom{ d_{\pm} - 2\Delta\omega \frac{\sin(d_{\pm}\delta)}{\sin((d_{\pm} + z_{\pm})\delta)} }\right.$d± -2Δω$\displaystyle {\frac{{\sin(d_{\pm}\delta)}}{{\sin((d_{\pm} + z_{\pm})\delta)}}}$$\displaystyle \left.\vphantom{ d_{\pm} - 2\Delta\omega \frac{\sin(d_{\pm}\delta)}{\sin((d_{\pm} + z_{\pm})\delta)} }\right)$, (11.52)

and

d± = $\displaystyle \left(\vphantom{ \Delta\omega^{\scriptscriptstyle\mathrm{H}}+ \Delta\omega }\right.$ΔωH + Δω$\displaystyle \left.\vphantom{ \Delta\omega^{\scriptscriptstyle\mathrm{H}}+ \Delta\omega }\right)$±ıkex, (11.53)
z± = $\displaystyle \left(\vphantom{ \Delta\omega^{\scriptscriptstyle\mathrm{H}}- \Delta\omega }\right.$ΔωH - Δω$\displaystyle \left.\vphantom{ \Delta\omega^{\scriptscriptstyle\mathrm{H}}- \Delta\omega }\right)$±ıkex. (11.54)

The symbol δ is half of τCPMG or

δ = $\displaystyle {\frac{{1}}{{4\nu_{\textrm{CPMG}}}}}$. (11.55)

The references for this model are:

More information about the MMQ CR72 model is available from:

The relax user manual (PDF), created 2016-10-28.