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}}\\  \Phi...
...}\\  \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{ex...
...{\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, \\ 0 ...
...}\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 2016-10-28.