Bloch equations revisited: New analytical solutions for the generalized Bloch equations

Abstract

The generalized Bloch equations in the rotating frame are solved in Cartesian space by an approach that is different from the earlier Torrey solutions. The solutions are cast into a compact and convenient matrix notation, which paves the way for a direct physical insight and comprehension of the evolution of various magnetization components. The solutions are expressed as a sum of two terms: One describes the decay of the initial state; the other describes the growth of the steady state. The representative trajectories of each component of the above terms plotted separately describe the complete time evolution of each magnetization component. © 1997 John Wiley & Sons, Inc.

Solution of the Bloch Equations including Relaxation

“…Case 4: General Solution. Equation (34) presents in compact form the solutions originally given by Torrey [1] and by Morris and Chilvers [6] as Laplace expressions and the tabulations of Madhu and Kumar [4,5] for a spin ensemble initially at equilibrium. ey are valid provided…”

A Complete Solution of the Bloch Equation

“…Equation (D5) is simply adj (s i 1 1 + Γ) from Eqs. (6)(7)(8). One therefore happens upon the modest result, apparently unrecognized, that an eigenvector υ corresponding to a distinct eigenvalue υ of operator Υ can be obtained as…”

Exact solution of the CPMG pulse sequence with phase variation down the echo train: Application to R2 measurements

“…The Bloch equations have been solved in different ways [38][39][40][41][42][43][44]. Recently, we presented an elegant exact solution of the Bloch equations and applied it to the Hahn echo [11].…”