Exact solution to the Bloch equations and application to the Hahn echo

The exact symbolic solution of the Bloch equations is given in the Lagrange form and illustrated with R2 experiments using a Hahn echo. Two different methods are also applied to approximately solve the Bloch equations, we find that splittings with effective-field interpretations are very substantially better than other approximations by comparing the errors. Estimates of transverse relaxation, R2, from Hahn echos are effected by frequency offset and field inhomogeneity. We use exact solutions of the Bloch equations and simulations to quantify both effects, and find that even in the presence of expected B0 inhomogeneity, off-resonance effects can be removed from R2 measurements, when∥ω∥⩽0.5γB1, by fitting the exact solutions of the Bloch equations. Further, the experiments and simulations show that the fitting models with the exact solutions of the Bloch equations do not depend on the sampling density and delay times.