Phase Cycling

The importance of allowing particular coherences to exist at certain times within an NMR experiment was shown in the previous section, where the existence of bothp=±1 coherences duringt1was necessary to achieve absorptive line shapes in the 2D spectrum.

In more complex experiments, coherences are often selected to allow the system to evolve under certain interactions, which provide useful information. For example, the

1_{H double-quantum experiment, which will be described in section 3.3.2, uses a series}

of pulses to excite double-quantum coherence between dipolar coupled nuclei. The coherence transfer pathway is chosen such that only p = ±2 coherences exist during

t1 and that this double-quantum coherence is reconverted to a state of p= 0 aftert1.

This experiment will be used as an example in this section, as it is extensively used throughout the experimental work presented in later chapters and involves a number of

3.1. PULSED FOURIER TRANSFORM NMR 47 t2 t1 1H π 2 Recoupling

A

B

C

Figure 3.4: Pulse sequence and coherence transfer pathway diagram for the 1H DQ correlation experiment. Further experimental details, including the construction of the recoupling pulses will be discussed in later sections.

coherence order transitions. The pulse sequence for a generic version of this experiment is shown in figure 3.4.

Application of rf pulses to a system of coupled spins can result in the creation of a variety of coherence orders. The pulse, or series of pulses, may be designed to excite particular changes in coherence with higher efficiencies than others, however allowing only these desired coherences to evolve while removing the undesired coherences requires the use of phase cycling.

This technique involves performing several experiments, incrementing the phases of several of the pulses through 2π radians over the duration of the phase cycle. For example, a pulse undergoing a four step phase cycle will have phases in successive experiments incremented by 2π/4 =π/2 radians, such that the pulse phase in the fifth experiment has returned to the same value as in the first experiment. The phases of the pulses are chosen such that when the transients recorded over the phase cycle are summed, the signals due to any undesired coherence pathways interfere destructively and cancel out, while the signal due to the desired coherence pathway remains.

The phase cycle necessary to follow a given coherence transfer pathway may be found by the application of the following two rules:

ˆ If the phase of a pulse, or of a group of pulses, is shifted by ∆φ, then a coherence undergoing an change in coherence order of ∆p will experience a phase shift of −∆φ·∆p, as detected by the receiver.

ˆ If a phase cycle uses steps of 360◦/N then, along with the desired coherence order change of ∆p, pathways corresponding to a change of ∆p±nN wheren= 1,2,3. . .

Table 3.1: Selection of double-quantum coherence by phase cycling.

∆p= +2 ∆p=−1

Step φA φR −∆φA∆p=−2φA Difference −∆φA∆p=φA Difference

1 0 0 0 0 0 0

2 90 180 180 0 90 90

3 180 0 0 0 180 -180

4 270 180 180 0 270 -90

will also be selected. All other pathways are suppressed.

The first rule makes reference to the phase of the receiver. While phases of pulses produced by the spectrometer can take any value, the receiver phase is often limited to multiples, of π/2. A particular pathway can be selected by making the receiver phase, φR, synchronise with the phase of the desired coherence,φR=−φ∆p. In practice, this

is achieved by multiplying the signal by −i for φR = π/2 (i.e. switching the real and

imaginary components) and by −1 for φR=π (i.e. inverting the FID).

It is clear from the second of the above rules that is is never possible to completely eliminate all pathways except the desired one, however it is important to consider which coherences can be created for a given nuclear spin system. When considering systems of dipolar coupled nuclei, as is relevant for the1H DQ correlation experiment, double-quantum coherences of order|p|= 2 are created between pairs of coupled nuclei, however significantly higher order coherences may safely be ignored.

One other relevant consideration is the predetermined states at the beginning and end of the experiment. The density matrix at thermal equilibrium, as derived in chapter 2, has no terms corresponding to a coherence, hence p = 0. Similarly, at the end of the experiment, the coherence p=−1 does not need to be selected as this is the only observable coherence, therefore there is no value in cancelling out other coherences. The result of these limitations is that the phase(s) of one pulse (or set of pulses) in the sequence need not be cycled. In the example of the 1H DQ experiment, the first pulse will be phase cycled to select double-quantum coherence, and the third pulse will be phase cycled to create a coherence of p=−1 only from p= 0.

The 1H DQ sequence is a simplified version of that used to record data presented later in the thesis. Notably, homonuclear decoupling sequences have been omitted and the details of the recoupling sequences used are not important for understanding of the

3.1. PULSED FOURIER TRANSFORM NMR 49

Table 3.2: Full phase cycle for the DQ experiment shown in figure 3.4. Pulses A and C are each cycled through four steps to produce the desired coherences.

Step φA −∆φA∆p=−2φA φC −∆φC∆p=φC φR=−2φA+φC 1 0 0 0 0 0 2 90 180 0 0 180 3 180 0 0 0 0 4 270 180 0 0 180 5 0 0 90 90 90 6 90 180 90 90 270 7 180 0 90 90 90 8 270 180 90 90 270 9 0 0 180 180 180 10 90 180 180 180 0 11 180 0 180 180 180 12 270 180 180 180 0 13 0 0 270 270 270 14 90 180 270 270 90 15 180 0 270 270 270 16 270 180 270 270 90

phase cycle. In practice, these blocks are formed of a combination of pulses and delays, which have been replaced in this example by a single block. Where phase shifts between successive experiments are derived in this example, the phases of all components of the blocks would be incremented in a real experiment.

Considering firstly the pulse sequence element used to excite double-quantum co- herence (labelled “A” in figure 3.4), a change in coherence order of ∆p=±2 is required. Using a four step phase cycle to selectp= +2 (fromp= 0) will also result in coherences ∆p=−2,6,−6. . . being selected. This successfully selects the two desired coherences, as well as other coherences of sufficiently high orders that they may be safely neglected. Table 3.1 shows the effect of a four step phase cycle on the first pulse, for a desired coherence (∆p = +2) and an undesired coherence (∆p =−1). The data presented in the table shows that the required ∆p = 2 coherence is selected as the receiver phase follows the phase of the coherence throughout the pulse sequence. Conversely the ∆p =−1 coherence is out of phase with the receiver and hence will be cancelled out when all four transients recorded over the phase cycle are added.

It is similarly possible to design a phase cycle for the final pulse, such that a coher- ence change of ∆p =−1 is selected. Since only p =−1 coherence is observable, only signal from the desired coherence pathway will then be detected. This is again achieved

with a four step phase cycle, selecting coherences ∆p = −1,3,−5,7 etc. In practice, these two phase cycles for the first and third pulses are performed in a nested fashion, i.e., for each pulse phase value in the cycle for pulse C, the entire cycle for pulse A is performed. The pulse phases used in the experiment are shown in table 3.2.