Subsections

The alternative extended model-free gradient

Because of the equation S2 = S2fS2s and the form of the extended spectral density function (15.63) a convolution of the model-free space occurs if the model-free parameters {S2f, S2s, τf, τs} are optimised rather than the parameters {S2, S2f, τf, τs}. This convolution increases the complexity of the gradient. For completeness the first partial derivatives are presented below.

$ \mathfrak{G}_j$ partial derivative

The partial derivative of (15.63) with respect to the geometric parameter $ \mathfrak{G}_j$ is

\begin{multline}
\frac{\partial J(\omega)}{\partial \mathfrak{G}_j} = \frac{2}{...
...{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}
\Bigg)
\Bigg).
\end{multline}

$ \mathfrak{O}_j$ partial derivative

The partial derivative of (15.63) with respect to the orientational parameter $ \mathfrak{O}_j$ is

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial \mathfrak{O}_j}}}$ = $\displaystyle {\frac{{2}}{{5}}}$$\displaystyle \sum_{{i=-k}}^{k}$$\displaystyle {\frac{{\partial c_i}}{{\partial \mathfrak{O}_j}}}$τi$\displaystyle \Bigg($$\displaystyle {\frac{{S^2_f \cdot S^2_s}}{{1 + (\omega \tau_i)^2}}}$ + $\displaystyle {\frac{{(1 - S^2_f)(\tau_f + \tau_i)\tau_f}}{{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}}}$ + $\displaystyle {\frac{{S^2_f(1 - S^2_s)(\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}}$$\displaystyle \Bigg)$. (15.91)

S2f partial derivative

The partial derivative of (15.63) with respect to the order parameter S2f is

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial S^2_f}}}$ = $\displaystyle {\frac{{2}}{{5}}}$$\displaystyle \sum_{{i=-k}}^{k}$ciτi$\displaystyle \Bigg($$\displaystyle {\frac{{S^2_s}}{{1 + (\omega \tau_i)^2}}}$ - $\displaystyle {\frac{{(\tau_f + \tau_i)\tau_f}}{{(\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2}}}$ + $\displaystyle {\frac{{(1 - S^2_s) (\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}}$$\displaystyle \Bigg)$. (15.92)

S2s partial derivative

The partial derivative of (15.63) with respect to the order parameter S2s is

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial S^2_s}}}$ = $\displaystyle {\frac{{2}}{{5}}}$S2f$\displaystyle \sum_{{i=-k}}^{k}$ciτi$\displaystyle \Bigg($$\displaystyle {\frac{{1}}{{1 + (\omega \tau_i)^2}}}$ - $\displaystyle {\frac{{(\tau_s + \tau_i)\tau_s}}{{(\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2}}}$$\displaystyle \Bigg)$. (15.93)

τf partial derivative

The partial derivative of (15.63) with respect to the correlation time τf is

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial \tau_f}}}$ = $\displaystyle {\frac{{2}}{{5}}}$(1 - S2f)$\displaystyle \sum_{{i=-k}}^{k}$ciτi2$\displaystyle {\frac{{(\tau_f + \tau_i)^2 - (\omega \tau_f \tau_i)^2}}{{\left((\tau_f + \tau_i)^2 + (\omega \tau_f \tau_i)^2 \right)^2}}}$. (15.94)

τs partial derivative

The partial derivative of (15.63) with respect to the correlation time τs is

$\displaystyle {\frac{{\partial J(\omega)}}{{\partial \tau_s}}}$ = $\displaystyle {\frac{{2}}{{5}}}$S2f(1 - S2s)$\displaystyle \sum_{{i=-k}}^{k}$ciτi2$\displaystyle {\frac{{(\tau_s + \tau_i)^2 - (\omega \tau_s \tau_i)^2}}{{\left((\tau_s + \tau_i)^2 + (\omega \tau_s \tau_i)^2 \right)^2}}}$. (15.95)

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