The LM63 2-site fast exchange CPMG model

This is the original model for 2-site fast exchange for CPMG-type experiments. It is selected by setting the model to `LM63', here named after Luz and Meiboom (1963). The original n-site Equation (7) from their paper can be written as

Rex = $\displaystyle \left[\vphantom{ 1 - 2\tau_{\textrm{ex}}g \cdot \tanh \left( 2\tau_{\textrm{ex}}g \right)^{-1} }\right.$1 - 2τexg⋅tanh$\displaystyle \left(\vphantom{ 2\tau_{\textrm{ex}}g }\right.$2τexg$\displaystyle \left.\vphantom{ 2\tau_{\textrm{ex}}g }\right)^{{-1}}_{}$$\displaystyle \left.\vphantom{ 1 - 2\tau_{\textrm{ex}}g \cdot \tanh \left( 2\tau_{\textrm{ex}}g \right)^{-1} }\right]$τex$\displaystyle \sum_{{i=2}}^{n}$piΔωi, (11.14)

where g is the pulse repetition rate defined as

g = 2νCPMG. (11.15)

It can be rearranged as

Rex = $\displaystyle \sum_{{i=2}}^{n}$$\displaystyle {\frac{{\Phi_{\textrm{ex,i}}}}{{\textrm{k}_{\textrm{i}}}}}$$\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)$. (11.16)

The equation for the 2-site exchange process can be expressed as

R2eff = R20 + $\displaystyle {\frac{{\Phi_{\textrm{ex}}}}{{\textrm{k}_{\textrm{ex}}}}}$$\displaystyle \left(\vphantom{ 1 - \frac{4\nu_{\textrm{CPMG}}}{\textrm{k}_{\tex...
...h \left( \frac{\textrm{k}_{\textrm{ex}}}{4\nu_{\textrm{CPMG}}} \right) }\right.$1 - $\displaystyle {\frac{{4\nu_{\textrm{CPMG}}}}{{\textrm{k}_{\textrm{ex}}}}}$⋅tanh$\displaystyle \left(\vphantom{ \frac{\textrm{k}_{\textrm{ex}}}{4\nu_{\textrm{CPMG}}} }\right.$$\displaystyle {\frac{{\textrm{k}_{\textrm{ex}}}}{{4\nu_{\textrm{CPMG}}}}}$$\displaystyle \left.\vphantom{ \frac{\textrm{k}_{\textrm{ex}}}{4\nu_{\textrm{CPMG}}} }\right)$$\displaystyle \left.\vphantom{ 1 - \frac{4\nu_{\textrm{CPMG}}}{\textrm{k}_{\tex...
...h \left( \frac{\textrm{k}_{\textrm{ex}}}{4\nu_{\textrm{CPMG}}} \right) }\right)$. (11.17)

The reference for this equation is:

More information about the LM63 model is available from:

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