Mechanism for Relaxation Processes

Transitions between Zeeman energy levels, and hence nuclear magnetic relaxation, are caused by fluctuations (time variations) in the local interactions at the nucleus that can cause transitions. The transition rates between these energy levels that cause nuclear spin relaxation depend on two factors  

The fluctuations are often caused by atomic motion e.g. Brownian motion in liquids, ionic hopping, molecular rotations, librations and atomic vibrations. These motions are often complex and it is the range of frequencies that are present in the motions that determine relaxation. The spectral density function describes the relative intensities of different frequencies in the motions and can be used to calculate relaxation rates. 

Let y(t) be some function of time, such as the orientation of the intemuclear vector and some other function f (e.g. the dipolar interaction) that depends on y. It is possible to define a probability function p(y,t) which is the probability that at a time t, the inter-nuclear vector has some orientation y. Then 

Since G is the correlation of the function with itself it is termed the auto-correlation function of f(y). If the function y(t) is stationary then it is invariant to changes of the time origin and G depends only on the difference, t = t2 — t, so that 

The spectral density function is defined to be the temporal Fourier transform of G(t) so that 

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