Zhenghua Nie

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.

Problems, artifacts and solutions in the INADEQUATE NMR experiment.

The INADEQUATE experiment can provide unequalled, detailed information about the carbon skeleton of an organic molecule. However, it also has the reputation of requiring unreasonable amounts of sample. Modern spectrometers and probes have mitigated this problem, and it is now possible to get good structural data on a few milligrams of a typical organic small molecule. In this paper, we analyze the experiment step by step in some detail, to show how each part of the sequence can both contribute to maximum overall sensitivity and can lead to artifacts. We illustrate these methods on three molecules: 1‐octanol, the steroid 17α‐ethynylestradiol and the isoquinoline alkaloid β‐hydrastine. In particular, we show that not only is the standard experiment powerful, but also a version tuned to small couplings can contribute vital structural information on long‐range connectivities. If the delay in the spin echo is long, pairs of carbons with small couplings can create significant double‐quantum coherence and show correlations in the spectrum. These are two‐ and three‐bond correlations in a carbon chain or through a heteroatom in the molecule. All these mean that INADEQUATE can play a viable and important role in routine organic structure determination. Copyright

Designing optimal universal pulses using second-order, large-scale, non-linear optimization.

Recently, RF pulse design using first-order and quasi-second-order pulses has been actively investigated. We present a full second-order design method capable of incorporating relaxation, inhomogeneity in B(0) and B(1). Our model is formulated as a generic optimization problem making it easy to incorporate diverse pulse sequence features. To tame the computational cost, we present a method of calculating second derivatives in at most a constant multiple of the first derivative calculation time, this is further accelerated by using symbolic solutions of the Bloch equations. We illustrate the relative merits and performance of quasi-Newton and full second-order optimization with a series of examples, showing that even a pulse already optimized using other methods can be visibly improved. To be useful in CPMG experiments, a universal refocusing pulse should be independent of the delay time and insensitive of the relaxation time and RF inhomogeneity. We design such a pulse and show that, using it, we can obtain reliable R(2) measurements for offsets within ±γB(1). Finally, we compare our optimal refocusing pulse with other published refocusing pulses by doing CPMG experiments.

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.