Derivation of the Bloch Equations

In a magnetic resonance experiment, we apply not only a static field B0 in the z-direction but an oscillating radiation field Bx in the xy-plane, so that the total field is  

Note that there are other possible ways to impose a time-dependent Bx. The one described in eqn (5.8) corresponds to a circularly polarized field initially aligned along the x-axis and rotating about the z-axis in a counterclockwise direction. The vector product of eqn (5.7) then becomes  

Inserting this expression in eqn (5.7) and separating it into components, we get  

This is equivalent to transformation into a coordinate system that rotates with the oscillating field u is that part of Mx which is in-phase with Bx and v is the part which is 90° out of phase. Differentiating eqn (5.11a) and substituting eqns (5.9a) and (5.9b), we get  

Equations (5.12a-c) are the Bloch equations in the rotating coordinate frame. 

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