Excitation profiles shaped pulses

The integral describes the spatial amplitude modulation of the excited magnetization. It represents the excitation or slice profile, g(z), of the pulse in real space. As drops to zero for t outside the pulse, the integration limits can be extended to infinity whereupon it is seen that the excitation profile is the Fourier transfonn of the pulse shape envelope ... 

A sine-shape has side lobes which impair the excitation of a distinct slice. Other pulse envelopes are therefore more commonly used. Ideally, one would like a rectangular excitation profile which results from a sine-shaped pulse with an infinite number of side lobes. In practice, a finite pulse duration is required and therefore the pulse has to be truncated, which causes oscillations in the excitation profile. Another frequently used pulse envelope is a Gaussian frmction ... 

Gaussian pulses are frequently applied as soft pulses in modern ID, 2D, and 3D NMR experiments. The power in such pulses is adjusted in milliwatts. Hard" pulses, on the other hand, are short-duration pulses (duration in microseconds), with their power adjusted in the 1-100 W range. Figures 1.15 and 1.16 illustrate schematically the excitation profiles of hard and soft pulses, respectively. Readers wishing to know more about the use of shaped pulses for frequency-selective excitation in modern NMR experiments are referred to an excellent review on the subject (Kessler et ai, 1991). 

The shape of any rf pulse can be chosen in such a way that the excitation profile is a rectangular slice. In the light of experimental restrictions, which often require pulses as short as possible, the slice shape will never be perfect. For instance, the commonly used 900 pulse is still acceptable, while a 1800 pulse produces a good profile only if it is used as a refocusing pulse. Sometimes pulses of even smaller flip angles are used which provide a better slice selection (for a discussion of imaging with small flip angles, see Section 1.7). 

Soft-pulse multiple irradiation In this method, pre-saturation is done using shaped pulses having a broader excitation profile. Therefore, it is a more suitable method for the suppression of multiplets. This technique is very effective, easy to apply and easy to implement within most NMR experiments. In aqueous solutions, however, slowly exchanging protons would be detectable due to the occurrence of transfer of saturation. In addition, the spins with resonances close to the solvent frequency will also be saturated. 

Frequency-selective REDOR (fsREDOR) is a very powerful technique developed for the study of 13C and 15N uniformly labeled peptides or proteins [92]. The basic idea of this technique is to combine REDOR and soft n pulses to recouple a selected 13C-15N dipole-dipole interaction in a multiple-spin system. Usually one could use Gaussian shaped pulses to achieve the required selective n inversions. Other band selective shaped pulses have been developed for a more uniform excitation profile [93]. In its original implementation, fsREDOR was used to extract the intemuclear distances of several model crystalline compounds [92], In the past few years, this technique has proven to be very useful for the study of amyloid fibrils as well. For the Ure2p10 39 fibril samples containing 13C and 15N uniformly... 

Fig. 7. The excitation profiles (n= — 2 to 2) by a periodic pulse of / Jt sin(jr//7) /Jt, where the solid lines are computer simulated results, solid circles are calculated with the effective RF fields, and open circles are obtained from the Fourier transformation of the RF shape. In the computer simulation, the pulse is composed of 4001 steps and has a pulsewidth r = 5 ms. Reprinted from Ref. 27 with permission from Elsevier.

TOPHAT-shaped 90° pulses are used in other cases as the best compromise with respect to the excitation profile, the phase homogeneity and length. Depending on the type of the detected spin-spin interaction - being either scalar or dipolar coupling - each selected spin is initially perturbed only once (ID TOCSY, ID INADEQUATE, ID C/H COSY, 2D TOCSY-COSY and 2D HMBC), or for several times (ID NOE). With each of the selected spins initially perturbed only once the inherently smaller transient NOEs would be detected in the latter case, whereas with the multiple excitation of a selected spin within the NOE build-up period the stronger steady-state NOEs are more or less approximated. 

It is possible to shape an excitation profile with weak pulses in such a way as to have zero excitation at a certain frequency. This was first due to Redfield [9],... 

Figure 8.15(b) shows the spectrum of sucrose with a 90° Gaussian pulse applied to the triplet at 3.99 ppm, compared to a normal 1H spectrum (Fig. 8.15(a)). This is done by moving the reference frequency to place the 3.99 ppm triplet at the center of the spectral window (on-resonance), which is the center of the Gaussian-shaped excitation profile... 

It is rather tedious to move the spectral window every time we want to select a peak with a shaped pulse, but it is necessary as the center of the Gaussian excitation profile is at... 

The amplitude modulated pulses may require special equipment such as a waveform generator which, however, has become a standard constituent of the modem commercial spectrometers. The amplitude modulated pulses are usually windowless and the sidebands produced by these pulses, in most cases, are very weak and can be neglected. The simplest amplitude modulated pulses are Gaussian pulse, sine pulse or sine-square pulse [1]. The main drawback of these simple shapes is that they produce a phase gradient over the excitation bandwidth and their excitation profiles are non-uniform over the bandwidth of interest. The amplitude modulated pulses can easily be shifted off-resonance by applying a phase ramp over the pulse according to equation (4). 

Hie selectivity of a rectangular RF pulse can be increased merely by lengthening the pulse and using a reduced power however, this approach is limited as it results in a sine function-shaped excitation profile. Also, to achieve the desired selectivity, the pulse duration will eventually become too long and solute and solvent relaxation effects may become significant during the pulse. Hence in attempts to keep the pulses short while retaining... 

Fig. 23. Excitation-sculpting TOCSY sequence. The shaped pulses have SEDUCE profiles.

Figure 9.10. Schematic excitation profiles for (a) a low-power rectangular pulse and (b) a smoothly truncated shaped pulse.

Figure 9.14. Simulated excitation profiles of selected shaped pulses (of 10 ms duration) see Table 9.3. The inversion profiles (lower trace) were simulated with a 180(soft)-90(hard) sequence.

The excitation profile of soft pulses is defined by the duration of the pulse, these two factors sharing an inverse proportionality. More precisely, pulse shapes have associated with them a dimensionless bandwidth factor which is the product of the pulse duration. At, and its effective excitation bandwidth, Af, for a correctly calibrated pulse. This is fixed for any given pulse envelope, and... 

A time domain function can be expressed as a Fourier series, an infinite series of sines and cosines. However in practise integrals related to the FOURIER series, rather than the series themselves are used to perform the Fourier transformation. Linear response theory shows that in addition to NMR time domain data and frequency domain data, pulse shape and its associated excitation profile are also a FOURIER pair. Although a more detailed study [3.5] has indicated that this is only a first order approximation, this approach can form the basis of an introductory discussion. 

A typical application for these commands is to display the excitation profile of a pulse or shaped pulse as a spectrum. The frequency of the pulse is held constant and the chemical shift of a spin is incremented. The signal intensity is then measured as a function of the frequency difference or offset between the spin s chemical shift and the pulse frequency. This method essentially corresponds to a sweep over the excitation range of the pulse and offers an alternative representation to the Bloch simulator approach. 

The calculation of the Excitation profile displays the effect of a shaped pulse on several magnetization vectors with different rf offsets. The result is a two dimensional graph of either one Cartesian or a combination of Cartesian coordinates as function of the relative offset. From the appearance of the graph the uniformity of the phase and the excitation of the magnetization may be determined within the frequency domain (frequency window of interest). 

The following Check its will use the Bloch simulator module of NMR-SIM to study and analyse a number of different shaped pulses. Time Evolution, the Excitation Profile and the Rf field profile simulation are illustrated using a 90° Gaussian pulse while an adiabatic CHIRP pulse is used for the Waveform analysis. 

The effect of a shaped pulse depends upon both the duration and the rf field intensity of the pulse. Normally the duration of a shaped pulse is adjusted to give the desired excitation range and then the rf field altered to obtain the desired tilt angle. As a rule of thumb the excitation range (selectivity) of shaped pulse in Hz is proportional to the reciprocal of the pulse duration. The RF field profile simulation can be used to study the effect of a shaped pulse and the rf field intensity of the pulse as function of the rf offset. Because of the correlation of pulse duration and tilt angle, the simulation does not accept normalized pulses instead the tilt angles of the individual magnetization vectors are calculated as a function of the rf field intensity of a specific rf offset. 

To be able to understand the effect of a shaped pulse, it would be useful to represent the excitation profile of a shaped pulse as a spectrum. This may be easily achieved in NMR-SIM by using a variable spin system. Since the application of a selective pulse creates a mixture of x- and y-transverse magnetization components over its excitation range, a phase distorted profile would be generated Check it 5.2.2.1(a)). Thus, a specific... 

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