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

An implicit exact algebraic solution of CPMG experiments is presented and applied to fit experiments. Approximate solutions are also employed to explore oscillations and effective decay rates of CPMG experiments. The simplest algebraic approximate solution has illustrated that measured intensities will oscillate in the conventional CPMG experiments and that using even echoes can suppress errors of measurements of R₂ due to the imperfection of high-power pulses. To deal with low-power pulses with finite width, we adapt the effective field to calculate oscillations. An optimization model with the effective field approximation and dimensionless variables is proposed to quantify oscillations of measured intensities of CPMG experiments of different phases of the π pulses. We show, as was known using other methods, that repeating one group of four pulses with different phases in CPMG experiments, which we call phase variation, but others call phase alternation or phase cycling, can significantly smooth the dependence of measured intensities on frequency offset in the range of ±½γB₁. In this paper, a second-order expression with respect to the ratio of frequency offset to π-pulse amplitude is developed to describe the effective R₂ of CPMG experiments when using a group phase variation scheme. Experiments demonstrate that (1) the exact calculation of CPMG experiments can remarkably eliminate systematic errors in measured R₂s due to the effects of frequency offset, even in the absence of phase variation; (2) CPMG experiments with group phase variation can substantially remove oscillations and effects of the field inhomogeneity; (3) the second-order expression of the effective decay rate with phase variation is able to provide reliable estimates of R₂ when offsets are roughly within ±½γB₁; and, most significantly, (4) the more sophisticated optimization model using an exact solution of the discretized CPMG experiment extends, to ±γB₁, the range of offsets for which reliable estimates of R₂ can be obtained when using the preferred phase variation scheme.