Imperfect inverting pulse

It turns out that, in the case of an imperfect inverting pulse, the factor of 2 in equation (3) must be substituted by an unknown factor K (< 2) it is thus recommended to turn to a non-linear fit of M fx) = Mo[l—ifexp(—x/Tj)], where the three quantities Mq, K, and Ti have to be refined, starting for example from values deduced from (4). It must be stressed that the measurement, as described above, does not require the system to have returned to equilibrium between two experiments with different x values, or between two consecutive scans if accumulation is necessary for improving the S/N ratio (4). If the repetition time T is smaller than 5Ti, the factor K of the above equation depends on the ratio T/Ti here and, just as before, has to be adjusted for each resonance. 

Imperfections in pulses also may be corrected by using composite pulses instead of single pulses. The 180° pulse that inverts longitudinal magnetization for the measurement of T or other purposes may be replaced by the series 90, 180°, 90°, which results in the same net 180° pulse angle, but reduces the error from as much as 20% to as little as 1%. As Figure 5-27 shows, the 180° pulse compensates for whatever imperfection existed in the 90° pulse. (Normally, 180° is taken as double the optimized 90° pulse, so errors in one are present in the other). The three components of the WALTZ-16 method (90°, 180°, 270°) also constitute a composite pulse for 180°. 

The binomial sequences aim at improving the zero excitation profile and provide schemes that are less sensitive to spectrometer imperfections. The series may be written 1-1,1-2-1,1-3-3-1 and so on, where the numbers indicate the relative pulse widths, each separated by a delay r, and the overbar indicates phase inversion of the pulse. For off-resonance spins, the pulse elements are additive at the excitation maximum, so, for example, should one require 90° off-resonance excitation, 1-1 corresponds to the sequence A5x-t-A5-x- Of this binomial series, it turns out that the 1-3-3-1 sequence [75] has good performance and is most tolerant of pulse imperfections by virtue of its symmetry [76]. The trajectory of spins with frequency offset l/2r from the transmitter for a net 90° pulse (1 = 11.25°) is shown in Fig. 10.33. During each r period, the spins precess in the rotating frame by half a revolution, so the effect of the phase-inverted pulses is additive and the magnetisation vector is driven stepwise into the transverse plane. As before, the on-resonance solvent vector shows no precession, so it is simply tipped back and forth, finally terminating at the North Pole. 

Where there is a combination of different sites, each of which produces a significant number of sidebands, simplification of the spectrum by removal of the spinning sidebands would be extremely helpful. Dixon (1982) introduced the concept of breaking up the rotation period into intervals separated by 180° pulses. The phase of each spin is inverted by the pulses and the total phase accumulated by each spin can be calculated. Then for all the sidebands to be removed, the sum of the phase for spins contributing to the spinning sidebands needs to vanish, independent of the orientation of the particular crystallite. The basic TOSS sequence has been developed to compensate for imperfections by extension of the applied phase cycles and by paying careful... 

A perfect inversion pulse simply inverts z-magnetization or, more generally, all z-operators Iz — -Iz. If the pulse is imperfect, it will generate transverse magnetization or other coherences. Inversion pulses are used extensively in heteronuclear experiments to control the evolution of heteronuclear couplings. 

Composite pulse A sequence of RF pulses designed to generate a desired spectral response. The individual pulses within a composite pulse may vary in time, frequency, and orientation to achieve the desired response. Composite pulses are commonly used to invert spins over a broad frequency range where a single pulse gives imperfect inversion far from resonance. Composite pulses are also used for excitation of selected frequencies. 

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