Excitation profiles binomial pulses

Fig. 9.1. (A) Gaussian (a) and sine (b) excitation profiles. (B) Composite (G3) Gaussian pulse. (C) Train of soft pulses modified after the DANTE sequence to achieve selective off-resonance excitation. (D) Redfield 21412 sequence. (E) Binomial 11, 121, 1331, 14641 sequences. (F) JR (a) and compensated JR (or 1111) (b) sequences. (G) Watergate sequence. (H) Weft (Superweft) sequence. (I) Modeft sequence. (J) MLEV16 sequence. (K) NOESY sequence with trim pulse. (L) MLEV17 sequence with trim pulses. (M) Clean-TOCSY sequence.

A somewhat better excitation profile can be obtained using so-called binomial pulses [8-10]. These pulses are similar to the DANTE pulses, except their number and length are given by binomial coefficients ... 

Special pulses containing one or several notches in their excitation profiles have been designed for purposes of solvent suppression. Obviously such pulses could be used also for suppression of parent lines in applications involving isotopi-cally diluted nuclei. One of the simplest experiments of this kind is the jump and return experiment [24] and corresponding higher order binomial pulses with alternating phases. Similarly to binomial excitation pulses these constant amplitude solvent suppression pulses create sidebands and excitation sidelobes. Corresponding amplitude modulated pulses [25,26] provide a better alternative. 

A very similar sequence is the ll sequence, which actually belongs to a family of binomial pulse sequences. Of these sequences, the 1331 (often referred to as 1—3) is the most popular due to its wider water suppression window (see Figure l).22 Due to the frequency-dependent sine modulation of the resonance amplitudes (the so-called excitation profile ), the resonances on either side of the water signal (carrier frequency) have opposite signs. In addition, the binomial sequences suffer from baseline distortions due to a strong linear phase gradient (see Figure 1). This baseline distortion can be particularly troublesome in multidimensional experiments and therefore the binomial sequences have not proved popular for multidimensional NMR studies. 

Figure 3 Excitation profiles of the Watergate sequence obtained using either a selective 180° pulse sandwiched between two hard 90° pulses (left panel) or a 3-9-19 binomial sequence in place of the S element of the gradient echo (right panel) (see also Figure 2).

The binomial sequences aim to improve 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 indieate the relative pulse widths, each separated by a delay x, and the overbar indicates phase inversion of the pulse. For off-resonance spins the pulse elements are additive at the exeitation maximum so for example, should one require 90° off-resonance excitation, 1-1 corresponds to the sequence 45x-t-45 x. Of this binomial series, it turns out that the 1-3-3-1 sequence [66] has good performance and is most tolerant of pulse imperfections by virtue of its symmetry [67]. The trajectory of spins with frequency offset l/2x from the transmitter for a net 90° pulse (1 = 11.25°) is shown in Fig. 9.26. During each... 

As apparent from the previous section, a binomial sequence has a suitably tailored profile for the element S, and the series 3a-9a-19a-19a-9a-3a (Fig. 9.28a, with 26a =180° and a delay t between pulses, here termed W3 [71]) has a desirable off-resonance inversion profile for this purpose. The WATERGATE excitation profile for this is shown in Fig. 9.29a. Once again characteristic nulls also occur at offsets of n/x Hz, but between these the excitation is quite uniform and does not suffer the phase inversion of the unaccompanied 90° binomials. More recently, extended binomial sequences have been shown to provide a narrower notch at the transmit-... 

Although the (3-9-19) sequence is the most commonly used binomial pulse sequence other sequences have been developed called W4 and W5 [5.19]. Only the W5 sequence will be considered here. The W5 sequence consists of a five hard pulse train, each pulse length being calculated to give the optimum excitation profile. In the first part of Check it 5.2.3.8 the spectrum with WATERGATE-3 (W5-binominal pulse sequence) is simulated and the results compared with Check it 5.2.3.7a. In the second part of this Check it the excitation profiles of three different WATERGATE sequences are calculated and compared. 

Symmetrically shifted pulses have been proposed as a means of solvent suppression. Symmetrically shifted pulses are symmetrically shifted laminar pulses that contain equal numbers of rectangular pulse components of the same phase at an offset frequency. The basis of the symmetrically shifted pulse family is the SS pulse which is conceptually equivalent to applying simultaneous ir/2 rectangular pulses with two separate, but in-phase, transmitters at offset frequency from the water. On a practical basis an SS pulse is obtained by a complete Itt cosine modulation of a single transmitter (see Fig. 15). An S pulse is half of an SS pulse (i.e. a half-cycle tt pulse) which results in a narrower null and a 180° phase inversion at the transmitter frequency. They are also the soft, continuous equivalent of binomial sequences. The SS and S pulses have broader excitation maxima than the sinusoidal profile of the JR sequence. The method has maximal excitation at an offeet frequency of second-order U-shaped water suppression. The exdtation profile is related to the maximum amplitude modulation and can be determined by numerical evaluation of the Bloch equations. Hence a new pulse shape must be used for each excitation window. The SS pulses give better water suppression than the JR sequence, but at the expense of poorer excitation of resonances closer to the water. Also, there is no phase inversion at zero frequency. The S pulse gives better excitation near the water frequency but with less water suppression. 

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