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) = $\displaystyle {\frac{{1}}{{5}}}$ $\displaystyle \sum_{{i=-k}}^{k}$ c_i⋅e^-τ/τ_i,

(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

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

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

The five weights c_i are defined as

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

where

d	= 3 $\displaystyle \left(\vphantom{ \delta_x^4 + \delta_y^4 + \delta_z^4 }\right.$ δ_x⁴ + δ_y⁴ + δ_z⁴ $\displaystyle \left.\vphantom{ \delta_x^4 + \delta_y^4 + \delta_z^4 }\right)$ - 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 τ_i are

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

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

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

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 c_i are

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

The five correlation times τ_i are

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

Diffusion as a sphere

In the situation of a molecule diffusing as a sphere either described by the single parameter τ_m 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 τ_m given by

1/τ_m = 6 $\displaystyle \mathfrak{D}_{iso}.$

(7.20)

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

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