G 1 2 S G 1 2 S

To summarize, the strategy for phase cycling is to select pulses at crucial points in the pulse sequence where the coherence order change Ap is different for the desired and undesired coherence pathways. Then decide how selective the mask must be for each pulse (N-fold mask) and construct a phase cycle so that all of the selected pulses are independently stepped through their N-fold phase progressions. Finally, the receiver phase is calculated by adding the phase shifts for the desired coherence pathway that would result from the phase cycle you have constructed for the pulses, using the rule

The effect of setting the receiver phase cycle is to position each N-fold mask so that it passes the desired coherence order change Ap resulting from that pulse.

As an example, consider the 2D DQF-COSY experiment. Except for the length of the delay between pulse 2 and pulse 3, the NOESY pulse sequence is identical to the DQF-COSY pulse sequence, so that in this case it is only the coherence pathway selection that makes it a DQF-COSY experiment! The coherence order diagram is shown in Figure 10.30, with the undesired NOESY pathway diagramed with dotted lines. The convention for 2D experiments is to always have the observed coherence in F2 (observed in the FID) be negative and the observed coherence in F1 (during the 11 delay) be positive. This is called the "echo" pathway because we have opposite sign of coherence in the two dimensions. The desired pathway has Ap = 1 for the first pulse, 1 or —3 for the second and —3 or 1 for the third, whereas the NOESY path has Ap = 1, —1, and —1 for the same pulses. We saw in detail how phase cycling the final pulse of the DQF-COSY experiment (N = 4) kills everything except DQC (p = 2 or —2) existing between pulse 2 and pulse 3. In that case we used AOp = + 90o and N = 4 (x, y, —x, —y). As we want to select Ap = —3 or Ap = + 1 for that pulse, we need to cycle the receiver as follows:

We see that these are the same, corresponding to a receiver phase increment of —1 or +3 in the 0, 1, 2, 3 system:

Pulse 3: 0123 = x, y, —x, —y Receiver: 0 3 21 = x, —y, —x, y

Moving backward from 0 you go to 3 (270° = — 90o). Every fourth coherence change is let through: Ap = —7, —3, 1, 5, and so on. The NOESY pathway (—I^cc' term from the homonuclear "front end") goes from p = 0top = —1, with Ap = — 1 in the final pulse. This is blocked by the fourfold mask of the phase cycle (Ap = —3 or +1). The ZQC part of the ZQ/DQ term is also blocked because it has the same Ap as the NOESY pathway (p = 0 to p = —1) in the final pulse. The COSY crosspeak and diagonal terms (+ixsC — 2IbI^ss') are SQC (p = 1 or —1) and would have to undergo a change of Ap = 0 or Ap = —2 (p = — 1 to — 1 or p = + 1to — 1) in order to be observed in the FID. These are also blocked by the phase cycle, which allows only Ap = —3 and Ap = + 1.

To select the NOESY pathway, we use exactly the same pulse sequence but change the ra Tm 90°

phase cycle of the final pulse. We want to select the pathway I^ — I+ — — I^ —m Ibb — I— , which has Ap = — 1 for the final pulse (Fig. 10.31). Using the same phase cycle x, y, —x, —y (AO = 90°) for the final pulse, we can calculate the receiver phase increment:

So instead of retarding the receiver phase by 90° with each scan, we will advance it by 90°: x, y, —x, —y. This will allow Ap of —5, —1, +3, and so on, and reject the DQF-COSY pathway (Ap = —3 or +1).

Better coherence pathway selection is achieved by cycling more than one of the pulses in the sequence. For a DQF-COSY sequence, we could set N to 2, 2, and 4 for the three 90° pulses, thus making the last pulse the most selective so that we can allow both Ap = —3 and Ap = 1 while blocking the NOESY Ap = —1. So we need to cycle the first two pulses with 180° phase shifts (360°/2) and the final pulse through a 90° phase shift (360°/4). To do all of these phase shifts independently will require 16 scans (2 x 2 x 4) because it requires two steps to sample all possible 180° phase shifts and four steps to sample all possible 90° phase shifts. Because these must be sampled independently, the number of scans required

(DQF-COSY)

is the product of the Ns for each pulse. Here is one way to do it:

Pulse 1: 0000222200002222 (N = 2) Pulse 2: 0000000022222222 (N = 2) Pulse 3: 0123012301230123 (N = 4)

where 0, 1, 2, and 3 correspond to phase shifts of 0°, 90°, 180°, and 270°, placing the B\ vector of the pulse on the x', y', -X, and —y' axes of the rotating frame of reference, respectively. Note that the goal of quadrature image elimination is also accomplished as the final pulse is cycled through all four axes in the rotating frame. If we acquire 16 scans for each FID in the 2D experiment, we will accomplish the full selectivity of the phase cycle. If we have a high concentration of sample so that we do not need 16 scans for signal averaging, we could use eight or 4 scans per FID. If we use only eight scans, the selectivity of pulse 2 is eliminated; with only four scans per FID, we lose the selectivity of the pulse 1 and have only the "mask" created by pulse 3. Clearly, the decision to cycle pulse 3 more rapidly (0123...) than the others means that its selection is more important than the others because it will be present even for the minimum number of scans (4).

The next step is to calculate the receiver phases. Each time we change the phase of a pulse, the receiver phase must be advanced by the appropriate amount so that the desired coherence (resulting from the selected coherence level change Ap) is summed in the receiver and not canceled. For more than one pulse, we can add together the phase shifts required at the receiver according to

The easiest way to do this is to calculate the receiver phase $r for each pulse independently and then add these receiver phase shifts together. For the first pulse, Ap is +1 and for the second pulse it is either +1 or -3 (Fig. 10.30), whereas in both cases A$p is 180° (N = 2). We calculate A$r = -(1) x 180° = 180° for Ap = 1 and A$r = -(-3) x 180° = 540°, which is the same as 180° for Ap = -3. Thus, every change of 180° in pulse phase results in a change of 180° in the desired receiver phase. For the third pulse, Ap is -3 (or + 1) and A$p is 90° (0 1 2 3), so A$r = -(-3) x 90° = 270° = -90°. So every time we advance the phase of the third pulse by 90°, we must also retard the phase of the receiver by 90°. Considering each pulse individually, the receiver phases are as follows:

Pulse phase Receiver phase

Pulse 1:0000 22220000 2222 ^ 0000222200002222

Pulse 2: 0000000022222222 ^ 0000000022222222

Pulse 3: 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 ^ 0321032103210321

$r (total): 0321254325434765

$r (corrected): 0321210321030321

The receiver phase is obtained simply by adding the phase shifts for the three pulses in each column. The total receiver phase can be simplified as we can subtract 360° from any phase greater than 360° to obtain a value between 0° and 360°. Thus, a phase of "4" is really 0, "5" is really 1, "6" is really 2, and "7" is really 3. If the receiver phases are calculated wrong, all or part of the desired signal will be eliminated by cancelation in the sum-to-memory. I have more than once experienced the unhappy result when a long 2D experiment results in a perfect 2D spectrum of noise! In this case, the elimination of artifacts is so efficient that even the desired signals are removed.

Now consider the 2D NOESY experiment. If we use the same selectivity for the three pulses, we could use the same pulse phases as we used for the DQF-COSY experiment, and only the receiver phases will be different:

Pulse phase Receiver phase

Pulse 1:0000 22220000 2222 ^ 0000222200002222

Pulse 2: 0000000022222222 ^ 0000000022222222

Pulse 3: 0123012301230123 ^ 0123012301230123

$r (total): 0123234523454567

$r (corrected): 0123230123010123

The first two pulses give the same result: pulse phase of 0 2 and receiver phase of 0 2, selecting Ap = +1, -1, -3, and so on. For the third pulse, Ap is -1 and A$p is 90°, so A$r = -(-1) x 90° = 90°. So every time we advance the phase of the third pulse by 90°, we must also advance the phase of the receiver by 90°. Note that we have done exactly the same experiment as the DQF-COSY, but we have changed the receiver phase cycle slightly from 0321210321030321 to 012323012301012 3, thus letting the NOESY signals accumulate in the sum-to-memory whereas the DQF-COSY signals cancel out.

For heteronuclear experiments, changing the phase of a pulse will only affect that part of the coherence that is sensitive to the pulse. For example, in the stepwise INEPT with 2Iz Sz as in intermediate step in coherence transfer

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