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 p_A > p_B is used to halve the optimisation space, as both sides of the limit are mirror image spaces. The equation for the exchange process is

$${\frac{{1}}{{T_{\textrm{relax}}}}}$$ (11.44)

where

$${\frac{{\textrm{k}_{\textrm{ex}}}}{{2}}} \big( \big)$$ λ₁ (11.45)

$${\frac{{1}}{{2}}} \left(\vphantom{ \pm1 + \frac{\Psi + 2\Delta\omega ^2}{\sqrt{\Psi^2 + \zeta^2}} }\right. {\frac{{\Psi + 2\Delta\omega ^2}}{{\sqrt{\Psi^2 + \zeta^2}}}} \left.\vphantom{ \pm1 + \frac{\Psi + 2\Delta\omega ^2}{\sqrt{\Psi^2 + \zeta^2}} }\right)$$ D_± (11.46)

$${\frac{{2}}{{3}}} {\frac{{1}}{{\nu_{\textrm{CPMG}}}}} \left(\vphantom{ \pm\Psi + \sqrt{\Psi^2 + \zeta^2} }\right. \sqrt{{\Psi^2 + \zeta^2}} \left.\vphantom{ \pm\Psi + \sqrt{\Psi^2 + \zeta^2} }\right)^{{\frac{1}{2}}}_{}$$ η_± (11.47)

$$\left(\vphantom{ \imath \Delta\omega^{\scriptscriptstyle\mathrm{H... ...trm{A}}\textrm{k}_{\textrm{ex}}- p_{\textrm{B}}\textrm{k}_{\textrm{ex}}}\right. \left.\vphantom{ \imath \Delta\omega^{\scriptscriptstyle\mathrm{H... ...\textrm{k}_{\textrm{ex}}- p_{\textrm{B}}\textrm{k}_{\textrm{ex}}}\right)^{2}_{}$$ Ψ (11.48)

$$\left(\vphantom{ \imath \Delta\omega^{\scriptscriptstyle\mathrm{H... ...trm{A}}\textrm{k}_{\textrm{ex}}- p_{\textrm{B}}\textrm{k}_{\textrm{ex}}}\right. \left.\vphantom{ \imath \Delta\omega^{\scriptscriptstyle\mathrm{H... ...trm{A}}\textrm{k}_{\textrm{ex}}- p_{\textrm{B}}\textrm{k}_{\textrm{ex}}}\right)$$ ζ (11.49)

and where

$$\left(\vphantom{ 1 - m_{D+}^2 + m_{D+} m_{Z+} - m_{Z+}^2 + \frac{m_{D+} + m_{Z+}}{2} \sqrt{\frac{p_{\textrm{B}}}{p_{\textrm{A}}}} }\right. {\frac{{m_{D+} + m_{Z+}}}{{2}}} \sqrt{{\frac{p_{\textrm{B}}}{p_{\textrm{A}}}}} \left.\vphantom{ 1 - m_{D+}^2 + m_{D+} m_{Z+} - m_{Z+}^2 + \frac{m_{D+} + m_{Z+}}{2} \sqrt{\frac{p_{\textrm{B}}}{p_{\textrm{A}}}} }\right)$$ (11.50)

and

$${\frac{{\imath\textrm{k}_{\textrm{ex}}\sqrt{p_{\textrm{A}}p_{\textrm{B}}}}}{{d_{\pm} z_\pm}}} \left(\vphantom{ z_{\pm} + 2\Delta\omega \frac{\sin(z_{\pm}\delta)}{\sin((d_{\pm} + z_{\pm})\delta)} }\right. {\frac{{\sin(z_{\pm}\delta)}}{{\sin((d_{\pm} + z_{\pm})\delta)}}} \left.\vphantom{ z_{\pm} + 2\Delta\omega \frac{\sin(z_{\pm}\delta)}{\sin((d_{\pm} + z_{\pm})\delta)} }\right)$$ m_D± (11.51)

$$\mp {\frac{{\imath\textrm{k}_{\textrm{ex}}\sqrt{p_{\textrm{A}}p_{\textrm{B}}}}}{{d_{\pm} z_\pm}}} \left(\vphantom{ d_{\pm} - 2\Delta\omega \frac{\sin(d_{\pm}\delta)}{\sin((d_{\pm} + z_{\pm})\delta)} }\right. {\frac{{\sin(d_{\pm}\delta)}}{{\sin((d_{\pm} + z_{\pm})\delta)}}} \left.\vphantom{ d_{\pm} - 2\Delta\omega \frac{\sin(d_{\pm}\delta)}{\sin((d_{\pm} + z_{\pm})\delta)} }\right)$$ (11.52)

and

$$\left(\vphantom{ \Delta\omega^{\scriptscriptstyle\mathrm{H}}+ \Delta\omega }\right. \left.\vphantom{ \Delta\omega^{\scriptscriptstyle\mathrm{H}}+ \Delta\omega }\right)$$ d_± (11.53)

$$\left(\vphantom{ \Delta\omega^{\scriptscriptstyle\mathrm{H}}- \Delta\omega }\right. \left.\vphantom{ \Delta\omega^{\scriptscriptstyle\mathrm{H}}- \Delta\omega }\right)$$ z_± (11.54)

The symbol δ is half of τ_CPMG or

$${\frac{{1}}{{4\nu_{\textrm{CPMG}}}}}$$ (11.55)

The references for this model are:

• Korzhnev et al. (2004a)
• Korzhnev et al. (2004b)
• Korzhnev et al. (2005a)

More information about the MMQ CR72 model is available from:

• the relax wiki at http://wiki.nmr-relax.com/MMQ_CR72,
• the API documentation at http://www.nmr-relax.com/api/3.2/lib.dispersion.mmq_cr72-module.html,
• the relaxation dispersion page of the relax website at http://www.nmr-relax.com/analyses/relaxation_dispersion.html#MMQ_CR72.