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
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.
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 c_{i} are defined as
where
and where
(7.14) |
The five correlation times τ_{i} are
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 c_{i} are
The five correlation times τ_{i} are
In the situation of a molecule diffusing as a sphere either described by the single parameter τ_{m} or , the variable k is equal to zero. Therefore i∈{0}. The single weight c_{0} is equal to one and the single correlation time τ_{0} is equivalent to the global tumbling time τ_{m} given by
This is diffusion equation presented in Bloembergen et al. (1948).
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