Gaussian function (general form e-x ) has the useful property that its Fourier transform is also a Gaussian function (Fig. 8.8, top). The Gaussian is symmetrical and goes smoothly to zero quickly without wiggles, so it is an ideal shape for a selective pulse. We can adjust the selectivity of the Gaussian pulse by adjusting its pulse duration tp, just as we do with rectangular pulses. A long, low-power Gaussian pulse corresponds to a narrow (highly selective) excitation profile and a faster, higher-power Gaussian pulse leads to a wider Gaussian excitation profile. We can even create a rectangular excitation profile, exciting a precise region of the spectrum with flat response throughout the region, by using a sinc function for our pulse shape (Fig. 8.8, bottom). Just as the Fourier transform of a rectangular pulse is a sinc function excitation profile, the Fourier transform of a sinc-shaped pulse is a rectangular excitation profile.

The ability to produce pulses with amplitude variation during the pulse according to a precise mathematical function became commonly available in the 1990s as a result of new hardware technology, called waveform generators (Varian) or amplitude setting units (Bruker). These nonrectangular pulses are called shaped pulses, and they are put together not with a continuous function but rather as a "sandwich" of short rectangular pulses. For example, a 35-ms Gaussian pulse might be put together by executing a long string of 350 rectangular pulses, each one 0.1 ms (100 ^s) in duration. The amplitudes are set from a list of amplitudes calculated from the mathematical Gaussian function. This list can also contain RF phases (0-360°) that also vary in a precise predetermined fashion during the course of the long pulse.

Because nonselective pulses use high power, they are sometimes called "hard" pulses, whereas the low-power selective pulses are called "soft" pulses. Thus, for selectivity we use low-power (soft), long-duration shaped pulses and when we want to excite all of the signals in the spectral window equally, we use high-power (hard), short-duration rectangular pulses. Pulse power can vary over an enormous range, so we use a logarithmic scale to measure it. In the decibel scale, the pulse power in decibels is ten times the logarithm (base 10) of the pulse power:

dB = 10 log (power) = 10 log [(amplitude)2] = 20 log (amplitude)

To double the pulse power, simply increase the power by 3 dB, as log(2) = 0.3. Because pulse power is the square of pulse amplitude B1, to double the amplitude we need to multiply pulse power by a factor of 4, which corresponds to increasing power by 6 dB, as log (4) = 0.6. This leads to a simple rule of thumb: Every time you increase the pulse power by 6 dB, you will cut the 90° pulse (tp) in half (because Bi is doubled). Likewise, each 6 dB decrease in pulse power will double the 90° pulse width. This is a good rule of thumb, but as the actual power settings are not precise, you will normally have to calibrate the 90° pulse at the new power setting to be sure. To make matters worse, Bruker uses the dB scale to describe power attenuation rather than power itself, so that the higher the dB value the lower the power. This is the opposite of Varian's system. Be careful whenever you are setting power levels! If you get it wrong, you can burn up the probe, the amplifiers, and your sample!

As good as shaped pulses sound, they have some unpleasant features. The phase properties of the pulse are often less than ideal, leading to phase distortions of the resonance peak being excited. These problems can be eliminated by using pulsed field gradients (PFGs) to "clean up" the selective excitation. PFGs can effectively scramble any undesired excitation, leaving only the absolutely clean pure-phase excitation at the desired resonance in the spectrum. When these two new technologies, shaped pulses and pulsed field gradients, pair up, we get a truly powerful new way to pick apart the spectrum and establish connectivities within a molecule through space and through bonds.

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