Relaxation dispersion parameter constraints

To understand this section, please see Section 14.5 on page [*]. For a dispersion analysis, linear constraints are the most useful type of constraint.

For most models, the linear constraints in the notation of (14.18) for the relaxation rates are

$\displaystyle \begin{pmatrix}
1 & 0 & 0 \\
-1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
0 & 0 & -1 \\
\end{pmatrix}$$\displaystyle \begin{pmatrix}
\mathrm{R}_2^0\\
\mathrm{R}_{\mathrm{2A}}^0\\
\mathrm{R}_{\mathrm{2B}}^0\\
\end{pmatrix}$ $\displaystyle \geqslant$ $\displaystyle \begin{pmatrix}
0 \\
-200 \\
0 \\
-200 \\
0 \\
-200 \\
\end{pmatrix}$, (11.93)

for the Φex and Δω parameters as

$\displaystyle \begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 &...
...& 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{pmatrix}$$\displaystyle \begin{pmatrix}
\Phi_{\textrm{ex}}\\
\Phi_{\textrm{ex,B}}\\
\Ph...
...}\\
\Delta\omega^{\scriptscriptstyle\mathrm{H}}_{\textrm{BC}}\\
\end{pmatrix}$ $\displaystyle \geqslant$ $\displaystyle \begin{pmatrix}
0 \\
0 \\
0 \\
0 \\
0 \\
0 \\
0 \\
0 \\
0 \\
0 \\
\end{pmatrix}$, (11.94)

for the population parameters as

$\displaystyle \begin{pmatrix}
-1 & 0 \\
1 & 0 \\
1 & 0 \\
-1 &-1 \\
1 & 2 \\
\end{pmatrix}$$\displaystyle \begin{pmatrix}
p_{\textrm{A}}\\
p_{\textrm{B}}\\
\end{pmatrix}$ $\displaystyle \geqslant$ $\displaystyle \begin{pmatrix}
-1 \\
0.5 \\
0.85 \\
-1 \\
1 \\
\end{pmatrix}$, (11.95)

and for the exchange rate and time parameters as

$\displaystyle \begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 &...
...& 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{pmatrix}$$\displaystyle \begin{pmatrix}
\textrm{k}_{\textrm{ex}}\\
\textrm{k}_{\textrm{e...
...{\textrm{C}}\\
\textrm{k}_{\textrm{AB}}\\
\tau_{\textrm{ex}}\\
\end{pmatrix}$ $\displaystyle \geqslant$ $\displaystyle \begin{pmatrix}
0 \\
-2e^6 \\
0 \\
-2e^6 \\
0 \\
-2e^6 \\
0 \\
-2e^6 \\
0 \\
-2e^6 \\
0 \\
-100 \\
0 \\
\end{pmatrix}$. (11.96)

Through the isolation of each individual element, the constraints can be seen to be equivalent to

\begin{subequations}\begin{gather}
0 \leqslant \mathrm{R}_2^0\leqslant 200, \\
...
...\leqslant 100, \\
\tau_{\textrm{ex}}\geqslant 0.
\end{gather}\end{subequations}    

Note that the Δω and ΔωH constraints are not used for any of the MMQ-type models as sign differentiation is possible. These constraints are also turned off for the `NS R1ρ 3-site linear' and `NS R1ρ 3-site' models. And that the pA $\geqslant$ 0.85 constraint is used instead of the pA $\geqslant$ 0.5 constraint for all models which require pA $\gg$ pB. When not using the auto-analysis, constraints can be modified or turned off.

The relax user manual (PDF), created 2019-03-08.