Bloch equations

Then the Bloch equations read: where γ is the gyromagnetic ratio and B(t) = (Bx(t), By(t), B0 + ΔBz(t)) is the magnetic field experienced by the nuclei.

Their solution is: Thus the transverse magnetization, Mxy, rotates around the z axis with angular frequency ω0 = γB0 in clockwise direction (this is due to the negative sign in the exponent).

Mxy(t) is translated in the following way into observable quantities of Mx(t) and My(t): Since then where Re(z) and Im(z) are functions that return the real and imaginary part of complex number z.

Specifically, if the observer were rotating around the same axis in clockwise direction with angular frequency ω0, the transverse magnetization Mxy would appear to her or him stationary.

Substitute from the Bloch equation in laboratory frame of reference: But by assumption in the previous section: Bz′(t) = Bz(t) = B0 + ΔBz(t) and Mz(t) = Mz′(t).

Assume that: Then in the rotating frame of reference, the equation of motion for the longitudinal nuclear magnetization, Mz(t) simplifies to: This is a linear ordinary differential equation and its solution is where Mz(0) is the longitudinal nuclear magnetization in the rotating frame at time t = 0.