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

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

“…The difficult task is to extract the unknown parameters we want, for example, T 2 . We have solved the Hahn echo problem, and this is a continuation to deal with more complicated experiments using similar methods and notation of [11].…”

“…Recently, we presented an elegant exact solution of the Bloch equations and applied it to the Hahn echo [11]. In this paper, we will continue to use the exact solution of the Bloch equations presented in [11].…”

“…Recently, we presented an elegant exact solution of the Bloch equations and applied it to the Hahn echo [11]. In this paper, we will continue to use the exact solution of the Bloch equations presented in [11]. Without loss of generality, the excitation pulse can be supposed to be perfect and it is along the y axis, thus, after the excitation pulse, the magnetization will be along with the x axis.…”

“…It is a simple experiment and works well with a perfect pulse which can make the echo intensities to be a smooth exponential decay, but it lacks a mechanism to suppress oscillations of measured intensities which depend on the effects of imperfect pulses [9] or offsets [11]. However, there is an exact, and several approximate, solutions of the Hahn echo which provide some sophisticated analysis tools to account for oscillations due to pulse imperfections [11]. The second method for R 2 measurement, which uses a train of echoes, is called the CPMG pulse sequence [9,10].…”

“…Previously [11], we have found that an exact symbolic solution of the Hahn echo experiment incorporating offset and relaxation effects during pulses, delays and acquisition can be used to fit experimental data and obtain accurate R 2 estimates for a wider range of offsets than was previously possible. The goal of this paper is to similarly develop exact and approximate solutions of the Bloch equations for CPMG experiments (including soft p pulses) and, by fitting them to measurements, to obtain better R 2 estimates.…”