Subsections

Brownian rotational diffusion

In equations (7.7) and (7.8) the generic Brownian diffusion NMR correlation function presented in d'Auvergne (2006) has been used. This function is

C(τ) = $\displaystyle {\frac{{1}}{{5}}}$$\displaystyle \sum_{{i=-k}}^{k}$cie-τ/τi, (7.8)

where the summation index i ranges over the number of exponential terms within the correlation function. This equation is generic in that it can describe the diffusion of an ellipsoid, a spheroid, or a sphere.


Diffusion as an ellipsoid

For the ellipsoid defined by the parameter set { $ \mathfrak{D}_{iso}$, $ \mathfrak{D}_a$, $ \mathfrak{D}_r$, α, β, γ} the variable k is equal to two and therefore the index i∈{ -2, -1, 0, 1, 2}. The geometric parameters { $ \mathfrak{D}_{iso}$, $ \mathfrak{D}_a$, $ \mathfrak{D}_r$} are defined as

\begin{subequations}\begin{align}& \mathfrak{D}_{iso} = \tfrac{1}{3} (\mathfrak{...
...{\mathfrak{D}_y - \mathfrak{D}_x}{2\mathfrak{D}_a},\end{align}\end{subequations}

and are constrained by

\begin{subequations}\begin{align}0 & < \mathfrak{D}_{iso} < \infty,\\ 0 & \le \m...
...\mathfrak{D}_{iso},\\ 0 & \le \mathfrak{D}_r \le 1.\end{align}\end{subequations}

The orientational parameters {α, β, γ} are the Euler angles using the z-y-z rotation notation.

The five weights ci are defined as

\begin{subequations}\begin{align}c_{-2} &= \tfrac{1}{4}(d - e),\\ c_{-1} &= 3\de...
...elta_x^2\delta_y^2,\\ c_{2} &= \tfrac{1}{4}(d + e),\end{align}\end{subequations}

where

d = 3$\displaystyle \left(\vphantom{ \delta_x^4 + \delta_y^4 + \delta_z^4 }\right.$δx4 + δy4 + δz4$\displaystyle \left.\vphantom{ \delta_x^4 + \delta_y^4 + \delta_z^4 }\right)$ - 1, (7.12)
e = $\displaystyle {\frac{{1}}{{\mathfrak{R}}}}$$\displaystyle \bigg[$(1 + 3$\displaystyle \mathfrak{D}_r) \left(\delta_x^4 + 2\delta_y^2\delta_z^2\right) +...
...^2\delta_z^2\right) - 2 \left(\delta_z^4 + 2\delta_x^2\delta_y^2\right) \bigg],$ (7.13)

and where

$\displaystyle \mathfrak{R} = \sqrt{1 + 3\mathfrak{D}_r^2}.$ (7.14)

The five correlation times τi are

\begin{subequations}\begin{align}1/\tau_{-2} &= 6 \mathfrak{D}_{iso} - 2\mathfra...
...6 \mathfrak{D}_{iso} + 2\mathfrak{D}_a\mathfrak{R}.\end{align}\end{subequations}


Diffusion as a spheroid

The variable k is equal to one in the case of the spheroid defined by the parameter set { $ \mathfrak{D}_{iso}$, $ \mathfrak{D}_a$, θ, φ}, hence i∈{ -1, 0, 1}. The geometric parameters { $ \mathfrak{D}_{iso}$, $ \mathfrak{D}_a$} are defined as

\begin{subequations}\begin{align}& \mathfrak{D}_{iso} = \tfrac{1}{3} (\mathfrak{...
...parallel}- \mathfrak{D}_{\scriptscriptstyle \perp}.\end{align}\end{subequations}

and are constrained by

\begin{subequations}\begin{gather}0 < \mathfrak{D}_{iso} < \infty, \\ -\tfrac{3}...
...{D}_{iso} < \mathfrak{D}_a < 3\mathfrak{D}_{iso}. \end{gather}\end{subequations}    

The orientational parameters {θ, φ} are the spherical angles defining the orientation of the major axis of the diffusion frame within the lab frame.

The three weights ci are

\begin{subequations}\begin{align}c_{-1} &= \tfrac{1}{4}(3\delta_z^2 - 1)^2,\\ c_...
...ta_z^2),\\ c_{1} &= \tfrac{3}{4}(\delta_z^2 - 1)^2.\end{align}\end{subequations}

The five correlation times τi are

\begin{subequations}\begin{align}1/\tau_{-1} &= 6\mathfrak{D}_{iso} - 2\mathfrak...
.../\tau_{1} &= 6\mathfrak{D}_{iso} + 2\mathfrak{D}_a.\end{align}\end{subequations}


Diffusion as a sphere

In the situation of a molecule diffusing as a sphere either described by the single parameter τm or $ \mathfrak{D}_{iso}$, the variable k is equal to zero. Therefore i∈{0}. The single weight c0 is equal to one and the single correlation time τ0 is equivalent to the global tumbling time τm given by

1/τm = 6$\displaystyle \mathfrak{D}_{iso}.$ (7.20)

This is diffusion equation presented in Bloembergen et al. (1948).

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