The LM63 3-site fast exchange CPMG model

This is the original Luz and Meiboom (1963) model for 3-site fast exchange for CPMG-type experiments. It is selected by setting the model to `LM63 3-site'. Taking the original Equation 11.16, the equation for 3-site exchange is simply:

R_ex = $\displaystyle \sum_{{i=2}}^{3}$ $\displaystyle {\frac{{\Phi_{\textrm{ex,i}}}}{{\textrm{k}_{\textrm{i}}}}}$ ⋅ $\displaystyle \left(\vphantom{ 1 - \frac{4\nu_{\textrm{CPMG}}}{\textrm{k}_{\tex... ...nh \left( \frac{\textrm{k}_{\textrm{i}}}{4\nu_{\textrm{CPMG}}} \right) }\right.$ 1 - $\displaystyle {\frac{{4\nu_{\textrm{CPMG}}}}{{\textrm{k}_{\textrm{i}}}}}$ ⋅tanh $\displaystyle \left(\vphantom{ \frac{\textrm{k}_{\textrm{i}}}{4\nu_{\textrm{CPMG}}} }\right.$ $\displaystyle {\frac{{\textrm{k}_{\textrm{i}}}}{{4\nu_{\textrm{CPMG}}}}}$ $\displaystyle \left.\vphantom{ \frac{\textrm{k}_{\textrm{i}}}{4\nu_{\textrm{CPMG}}} }\right)$ $\displaystyle \left.\vphantom{ 1 - \frac{4\nu_{\textrm{CPMG}}}{\textrm{k}_{\tex... ...nh \left( \frac{\textrm{k}_{\textrm{i}}}{4\nu_{\textrm{CPMG}}} \right) }\right)$ ,

(11.18)

The reference for this equation is:

Luz and Meiboom (1963)

This model is only provided as a demonstration and should not be used for a normal analysis. Without data at multiple temperatures, a feature not currently supported within relax, that there are infinite lines of solutions and that the Φ_{ex, B}, Φ_{ex, C}, k_B and k_C parameters are all convoluted together.

This equation was made more practically relevant in the paper of O'Connell et al. (2009). This relies on the assumption that site 1 (or A) has a much larger equilibrium population than the other sites ( p_A $\gg$ p_B and p_A $\gg$ p_C). As stated, “if the different values of ji are well-separated (by a factor of 3-10), then Eq. 3 reduces approximately to the sum of n - 1 independent two-state processes for exchange between site 1 and the n - 1 other sites”. In this situation, the following relationships hold

$\begin{subequations}\begin{align} &\textrm{k}_{\textrm{B}}\approx \textrm{k}_{\t... ...textrm{k}_{\textrm{AC}}+ \textrm{k}_{\textrm{CA}}, \end{align}\end{subequations}$

and

$\begin{subequations}\begin{align} &\Phi_{\textrm{ex,B}}= \overline{\Phi_{\textrm... ...}- \textrm{k}_{\textrm{ex}}^{\textrm{AB}}\right)} ,\end{align}\end{subequations}$

with

$\begin{subequations}\begin{align} &\overline{\Phi_{\textrm{ex,B}}} \approx (p_{\... ...{B}}) p_{\textrm{C}}\Delta\omega_{\textrm{AC}}^2 . \end{align}\end{subequations}$

The parameter deconvolutions for this model can be performed after a relax analysis, if desired.

More information about the LM63 3-site model is available from:

the relax wiki at http://wiki.nmr-relax.com/LM63_3-site,
the API documentation at http://www.nmr-relax.com/api/3.2/lib.dispersion.lm63_3site-module.html,
the relaxation dispersion page of the relax website at http://www.nmr-relax.com/analyses/relaxation_dispersion.html#LM63_3-site.

The relax user manual (PDF), created 2020-08-26.