= [—sin(^b?2)


i cos(^bi2)] x sin(^a11)

Now we have exactly the same kind of data we have with States mode acquisition: cosine modulation in t1 for the real FID and sine modulation in t1 for the imaginary FID. In 12 we can equate Mx with the real part and My with the imaginary part, so that in each case we have Mx = -sin(^bt2) and My = cos(^bt2). This represents a vector starting on the y' axis at t2 = 0 and rotating counter clockwise (positive offset at the rate of rad s-1. Once these rearrangements have been made in the computer, the data is processed just like States data, with a complex Fourier transform in t1.

10.6.5 Disadvantages of Phase Cycling Compared to Gradient Selection

Phase cycling operates by acquiring both the desired signals and the artifact signals in the FID. Then by accumulating a number of FIDs in the sum-to-memory of the spectrometer, the desired signals add together and the artifact signals subtract and cancel. Gradients destroy the artifacts by twisting their coherences spatially in the NMR sample. These artifact coherences never reach the receiver because their vector sum is zero at the probe coil. This accounts for the fundamental advantage of gradient pathway selection: it does not depend on mathematical subtraction of signals that have already been digitized; the subtraction occurs in the NMR sample. Thus phase cycling as a method suffers from a number of drawbacks:

1. Many Scans are Required for Each FID. To apply an N-fold mask at each crucial pulse (P1, P2, P3,...) in the sequence requires a phase cycle of N1 x N2 x N3 x ■■■ scans to completely cancel the blocked pathways. If the sample concentration is low, you might need 32 or 64 scans per FID in a 2D experiment just to get enough signal-to-noise ratio. In that case, a long phase cycle is not a disadvantage. But if you have a high sample concentration, as is often the case in small-molecule NMR, a single scan per FID might give you sufficient signal-to-noise in a 2D experiment. In that case, every additional scan required by the phase cycle will greatly increase the experiment time. For example, a 16 scan phase cycle will multiply the experiment time by 16 for a concentrated sample, making an 8 min experiment (with gradient selection) into a more than 2 h acquisition. This is the area where gradients give the most dramatic advantages: in time savings for 2D experiments on concentrated samples (10 mg or more of an organic molecule).

2. The Receiver Gain Must be Reduced to Avoid ADC Overflow From Artifact Signals. In many cases, the artifact signals are many times more intense than the desired signals. The large artifact signal would overflow the ADC if we set the receiver gain to be appropriate for the desired signal, so we have to reduce the gain by a large factor to make the FID "fit" in the input of the ADC. Since some of the noise observed in the digitized FID originates in the later stages of amplification, this means we are adding a smaller (less amplified) signal to this noise and therefore our signal-to-noise ratio is reduced as we decrease the receiver gain. Amplification is good! We do not want to turn it down, but we have to because of these large artifact signals. In a gradient-selected experiment, the artifacts never appear in the FID signal and the receiver gain can be dramatically increased, leading to higher sensitivity.

3. Dynamic Range Problems. Dynamic range is the complete range of signal intensities that can be observed in an NMR spectrum. The largest signal will almost fill the ADC if the receiver gain is adjusted properly, and the smallest detectable signal is then limited by the accuracy (number of bits) of the ADC, since no signal smaller than a single bit can be digitized. If the artifact signal is 100 times larger than the desired signal, then we are wasting 6.6 (26-6 = 100) bits of our 16-bit ADC digitizing a signal that will simply be canceled out in the sum-to-memory by the phase cycle. Now the smallest signal that can be digitized is only 328 times smaller than the largest desired signal, rather than 32,768 times smaller. In a gradient-selected experiment, the entire 16 bits of the digitizer are used to measure the desired signal, since the artifact never appears in the receiver.

4. Subtraction Artifacts. Any time you cancel an artifact signal by subtraction, you are assuming that the signal is exactly the same with each scan so that perfect cancellation will occur. If the frequency (chemical-shift position) of the signal changes very slightly, subtraction will lead to a large "dispersive" artifact because the negative (subtracted) peak is offset slightly to the left or right of the positive peak. If the intensity is not exactly equal, there will be a residual positive or negative peak. Many things can change slightly in a spectrometer from one scan to the next: temperature may be varying slightly; pulse frequency, intensity, or phase may not be perfectly reproducible; and vibration from air flow, building equipment, trains, and so on may reach the probe. Spinning of the sample is also a source of irreproducibility—for this reason, most experiments with a critical phase cycle are run without sample spinning. In a gradient experiment there is no subtraction of two or more separate measurements so none of these sources of irreproducibility can lead to artifacts.

10.6.6 Disadvantages of Gradient Selection

You might think that gradients are an endlessly beneficial technology, but in fact there are a few minor disadvantages of gradient coherence pathway selection:

1. Cost of Gradient Controllers, Amplifiers and Probes. Gradient technology is not inexpensive, and is simply not available on older NMR spectrometers. The hardware required consists of a gradient controller (digital timing control plus signal generation), a gradient amplifier (very stable source of large currents that can be accurately controlled) and a gradient probe that has gradient coils surrounding the active volume of the sample. The currents produced by the gradient amplifier run through these coils and produce the magnetic fields that add a gradient to the Bo field. In spite of this additional cost, the advantages of gradients are so powerful even for routine work that nearly all new NMR spectrometers are now purchased with gradient capability.

2. Diffusion. The primary assumption of the gradient approach to pathway selection is that a spin will be in the same physical location in the sample tube throughout the pulse sequence. Each spin receives a position-dependent phase shift with each gradient pulse, and if the spin changes its position it will not receive the correct "unwinding" Bo field in the last gradient. The distance between positive and negative phase in the "twisted coherence" can be on the order of microns (1 ^m = 10-6 m, smaller than a typical eukaryotic cell) so that diffusion will occur, especially for small molecules, and will lead to a reduced signal. This can be put to use to distinguish small molecule signals from large molecule signals, but in general it means that there will be some loss of sensitivity. If you look closely at gradient-selected pulse sequences, you will see that there is an attempt to place gradients as close to each other in time as possible to minimize the distance traveled as a result of molecular diffusion in solution.

3. Sensitivity Loss Due to T2 Relaxation. A typical gradient requires about 1 ms of time, usually followed by 200 ^s of recovery time to allow the field homogeneity to be reestablished. Unless the gradient can be placed in a fixed delay period (e.g., 1/(4/) in an INEPT), a spin echo will be required to refocus any evolution that occurs during the gradient, doubling the total time required for a gradient. For small molecules an additional delay of 2.5 ms per gradient does not lead to a large signal loss since T2 is relatively long. For large biological molecules (proteins and nucleic acids), however, or for paramagnetic molecules, the T2 values can be quite short and the extra delays may be intolerable because of the drastic loss of signal. In these special cases it may be necessary to use the old phase-cycled experiments or to use shorter, stronger gradient pulses without refocusing delays.

4. Loss of Sensitivity Due to Overselectivity. Gradient selection means that only a single coherence level can be present at the time of each gradient. With phase cycling we apply a mask with "holes" at regular intervals, so that more than one coherence level can be permitted. For example, in the DQF-COSY experiment with phase cycling we can have double-quantum coherence between the second and third pulses with coherence order +2 or -2. Both pathways are preserved and add together in the sum-to-memory since the mask (N = 4) used in the final pulse allows both Ap = -3 and Ap = +1. With gradient selection, we can allow p = 2 during the DQ filter or we can allow p = -2, but we cannot permit both pathways. Thus there is a loss of sensitivity in the gradient-selected experiment corresponding to a factor of V2.

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