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).

The relax user manual (PDF), created 2019-03-08.