The NS 3-site R1ρ model

This is the numerical model for 3-site exchange using 3D magnetisation vectors. It is selected by setting the model to `NS R1rho 3-site'. The constraints pA > pB and pA > pC is used to decrease the size of the optimisation space, as both sides of the limit are mirror image spaces.

For this model, as for the 2-site model above, the equations from Korzhnev et al. (2005b) have been used. These have been however rearranged to match the notation in Palmer and Massi (2006). The R1ρ value for state A magnetisation is defined as

R1ρ = - $\displaystyle {\frac{{1}}{{T_{\textrm{relax}}}}}$⋅ln$\displaystyle \left(\vphantom{ M_0^T \cdot e^{R \cdot T_{\textrm{relax}}} \cdot M_0 }\right.$M0TeR⋅TrelaxM0$\displaystyle \left.\vphantom{ M_0^T \cdot e^{R \cdot T_{\textrm{relax}}} \cdot M_0 }\right)$, (11.85)

where

M0 = $\displaystyle \begin{pmatrix}\sin{\theta} \\ 0 \\ \cos{\theta} \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix}$, (11.86)
θ = arctan$\displaystyle \left(\vphantom{ \frac{\omega_1 }{\Omega_{\textrm{A}}} }\right.$$\displaystyle {\frac{{\omega_1 }}{{\Omega_{\textrm{A}}}}}$$\displaystyle \left.\vphantom{ \frac{\omega_1 }{\Omega_{\textrm{A}}} }\right)$. (11.87)

This assumes that the starting magnetisation has an X and Z component only for the A state. The relaxation evolution matrix is defined as

R = $\displaystyle \begin{pmatrix}-\mathrm{R}_{1\rho A}'-\textrm{k}_{\textrm{AB}}-\t...
...{k}_{\textrm{AC}}& \cdots \\ \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix}$    
  + $\displaystyle \begin{pmatrix}\ddots & \vdots & \vdots & \vdots & \iddots \\ \cd...
...rm{BC}}& \cdots \\ \iddots & \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix}$    
  + $\displaystyle \begin{pmatrix}\ddots & \vdots & \vdots & \vdots \\ \cdots & -\ma...
...{\textrm{1C}}-\textrm{k}_{\textrm{CA}}-\textrm{k}_{\textrm{CB}}\\ \end{pmatrix}$    
  + $\displaystyle \begin{pmatrix}& & & \textrm{k}_{\textrm{BA}}& 0 & 0 & \cdots \\ ...
...{AB}}& & & & \\ \vdots & \vdots & \vdots & & \vdots & & \ddots \\ \end{pmatrix}$    
  + $\displaystyle \begin{pmatrix}& & & \cdots & \textrm{k}_{\textrm{CA}}& 0 & 0 \\ ...
...& & \ddots & \\ 0 & 0 & \textrm{k}_{\textrm{AC}}& \cdots & & & \\ \end{pmatrix}$    
  + $\displaystyle \begin{pmatrix}\ddots & & \vdots & & \vdots & \vdots & \vdots \\ ...
...& & \ddots & \\ \cdots & 0 & 0 & \textrm{k}_{\textrm{BC}}& & & \\ \end{pmatrix}$, (11.88)

where δA, B, C are defined as in Equations 11.78a and 11.78b. For the model, the assumptions R1ρA' = R1ρB' = R1ρC' = R1ρ' and R1A = R1B = R1C = R1 have been made.

More information about the NS R1rho 3-site model is available from:

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