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) = ci⋅e-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 { , , , α, β, γ} the variable k is equal to two and therefore the index i∈{ -2, -1, 0, 1, 2}. The geometric parameters { , , } are defined as

and are constrained by

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

The five weights ci are defined as

where

 d = 3δx4 + δy4 + δz4 - 1, (7.12) e = (1 + 3 (7.13)

and where

 (7.14)

The five correlation times τi are

### Diffusion as a spheroid

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

and are constrained by

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

The five correlation times τi are

### Diffusion as a sphere

In the situation of a molecule diffusing as a sphere either described by the single parameter τm or , 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 (7.20)

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

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