Chemical Exchange - The Modified Bloch Equations

Suppose we have a system in which a spin can exist in either of two different sites, A or B, and that these are distinguished by different resonant frequencies, coa and coB, and/or by different relaxation times, T2a and T2b If there is no exchange between sites, site A spins and site B spins can be described separately and independently by sets of Bloch equations. When exchange takes place, however, additional rate terms - completely analogous to terms in chemical rate equations - must be added to the Bloch equations. 

The algebra we are about to get into can be compacted somewhat by introducing the complex magnetization, G = u + iv, so that eqns (5.12a) and (5.12b) can be combined to obtain  

In eqn (5.16), we have ignored the difference between M0 and Mz and so have assumed that B is small. 

Spins at site A will have magnetization GA and those at site B will have magnetization Gb- We now assume that A and B are interconverted by first-order kinetics with an A - B rate constant xA 1 and a B — A rate constant xB 1. The site A magnetization thus decreases with a rate term -GA/xA and increases with rate GB/xb- Combining these terms with eqn (5.16) for site A  

The rather fearsome algebra can be somewhat simplified by defining  

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