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:

Rex = $\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:

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 Φex, B, Φex, C, kB and kC 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 ( pA $ \gg$ pB and pA $ \gg$ pC). 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}_{\te...
...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_{\t...
...{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 user manual (PDF), created 2016-10-28.