Derivation of the Equations Related to NOE

A necessary step to derive Eq. (7.2) and other equations in Chapter 7 is the obtainment of a suitable expression for the time dependence of (/z) in a dipole-coupled two spin system. By referring to Fig. 7.1, (/z) and (Jz) are given by 

The rate of variation of (7Z) with time, d(/z(r))/dr, is proportional to the rate of population increase of levels + + and H— minus the rate of population increase of levels —I- and------  

Among the various transitions shown in Fig. 7.1, we can neglect those involving only changes in spin state of J. For instance, the first of the four terms in the right-hand side of Eq. (VII.2) is  

Analogous equations can be written for the other three terms in Eq. (VII.2). By grouping all the terms together, Eq. (VII.2) becomes 

The factor 2 derives from the fact that a single transition decreases the population of the starting state and at the same time increases the population of the target state, thereby changing the population difference by twice as much. 

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