The Excitation Profile For Rectangular Pulses

A simpler way to look at the effectiveness of an off-resonance pulse is to calculate the excitation profile, which is just a graph of the effective pulse rotation delivered by a pulse as a function of the resonance frequency in the rotating frame (vo - vr). This graph can be superimposed on the spectral window to view the effectiveness of the pulse with respect to each of the peaks in the spectrum (Fig. 8.6). It is actually quite simple to calculate this function, as the excitation profile is just the Fourier transform of the pulse. The pulse shape is rectangular as it is simply turned on at the beginning and held on for a time tp, which is the duration of the pulse (also called the pulse width), and then turned off. The height of this rectangular function is the pulse amplitude B1 (square root of the pulse power). We can also describe pulse amplitude using the effect of the pulse: the rotation rate of the sample magnetization around the B1 vector (v1 = yB1/2n). This is expressed in hertz and is equal to 1/(4 x tp(90o)). Using the B1 "field strength" in hertz as the height of the pulse, we see that the area of the pulse (height x width) is just the pulse rotation in cycles (0.25 for a 90o pulse). All three pulses in Figure 8.6 are 90o pulses, with the same area but different pulse widths tp.

The Fourier transform of a rectangular function turns out to be something called the "sinc" function: sin(x)/x. This function reaches a maximum at x = 0 and goes to zero on either side, with "wiggles" gradually dying out sort of like an FID in both the +x and -x directions (Fig. 8.6). Notice how the experimental result for inversion efficiency (Fig. 8.4, top) closely resembles this sinc function. The function first passes through zero at rotating-frame frequencies of -1/(2tp) and +1/(2tp), so we can say that the pulse excites a range

Figure 8.6

of frequencies corresponding to a "bandwidth" of 1/tp. Of course, if we superimpose this function on our spectral window with 1/tp = sw (spectral width), we will be getting very poor excitation (rotation much less than 90o for a 90o pulse) near the edge of our spectral window (Fig. 8.6, top), going down to zero at the edge (vo — vr = +sw/2 or — sw/2). We could take this even further by using a longer pulse with smaller B1 amplitude (Fig. 8.6, bottom) to get "selective excitation" of the peaks at the center of the spectral window. This is just the opposite strategy, using low-power ("soft") pulses to rotate only the spins corresponding to one peak in the spectrum.

To get nearly equal excitation all across our spectral window, we need to expand the excitation profile horizontally so that the zero points are far outside the spectral window and the function "droops" only slightly at the edges of the window (Fig. 8.6, center). This can be accomplished by using a shorter duration pulse, as the frequency domain has an inverse relationship to the time domain: squeezing the pulse in (shorter tp) has the effect of horizontally expanding the excitation profile (wider coverage or bandwidth 1/tp). We would like to have the "bandwidth" 1/tp much greater than the spectral width sw. Of course, if we use a shorter duration pulse, we must compensate by using a higher amplitude pulse so that the pulse rotation is not changed. For example, if we cut the duration tp of a 90° pulse in half, we will get a 45° rotation of the sample magnetization M, unless we also double the pulse amplitude (four times the pulse power) to compensate. To get nearly "flat" coverage of the entire spectral window, we want a very short duration pulse (on the order of tens of microseconds) with very high power (on the order of 50-300 W). This is a lot of radio frequency power to put into the small volume (about 300 ^l) of the NMR sample covered by the probe coil, so that we must be very careful to limit the pulse width to the microsecond range. Pulses of hundreds of milliseconds to seconds at this power level will boil the sample, fry the probe, and burn out the power amplifiers of the spectrometer. Pulse amplitude, which is the square root of pulse power, is limited not only by the maximum power output of the amplifiers, but also by the tendency of the probe coil to spark or "arc" at very high RF amplitudes. It is usually the arcing limit that sets a maximum on the B1 amplitude we can use.

A typical 90° pulse might have a duration of 10 ^s. Thus, 1/tp, the width of the main peak of the sinc function excitation profile, is 1/(10 ^s) = 100,000/(1 s) = 100 kHz. A typical spectral width for proton is 12 ppm or 12 x 300 = 3600 Hz on a 300-MHz (7.05-T) instrument. Thus, the "bandwidth" is 28 times (100,000/3600) wider than the spectral window, and we will have minimal "droop" of the excitation profile between the center and the edges of the spectral window. This pulse will deliver very close to a 90° pulse to all of the peaks in the spectrum. Problems arise with low-y nuclei because the rotation generated by the pulse is much slower (v1 = yB1/2n) and the 90° pulse width is, therefore, much longer even at the highest B1 amplitude (highest power) available. The longer pulse width corresponds to a narrower "coverage" (1/tp) in the frequency domain. This is compounded by the fact that many low-y nuclei have very wide ranges of chemical shifts, thus requiring very wide spectral windows. For example, 57Fe has a y that is 3.2% of yH (i.e., its nuclear magnet is only 3.2% of the strength of the proton nuclear magnet), and its range of chemical shifts is around 30,000 ppm. In terms of spectral width in hertz, this is about 1000 times wider than the typical proton spectral window. In these cases, it is often necessary to acquire several spectra with adjoining spectral windows in order to "cover" the entire range of chemical shifts.

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