Coherence order pathway

This allows us to diagram the coherence order pathway of an NMR experiment in a very simple way. For example, in an INEPT experiment with intermediate DQC we have... 

Coherence Order Pathway Selection by Phase Cycling... 

Once we have diagramed the desired coherence order pathway, it is easy to add gradients to select that pathway. One simple solution is to use the 3,4,5 relationship of a right triangle 3x3 + 4x4 = 5x5. Put a gradient in the first half of t of relative amplitude G = 5, another in the second half with amplitude G2 = 3, and a third in the refocusing delay with amplitude G3 = 4. As the coherence order p is 5, —3, and —4, respectively, during these three periods, we have a total twist of ... 

Figure 1.45 Coherence transfer pathways in 2D NMR experiments. (A) Pathways in homonuclear 2D correlation spectroscopy. The first 90° pulse excites singlequantum coherence of order p= . The second mixing pulse of angle /3 converts the coherence into detectable magnetization (p= —1). (Bra) Coherence transfer pathways in NOESY/2D exchange spectroscopy (B b) relayed COSY (B c) doublequantum spectroscopy (B d) 2D COSY with double-quantum filter (t = 0). The pathways shown in (B a,b, and d) involve a fixed mixing interval (t ). (Reprinted from G. Bodenhausen et al, J. Magn. Resonance, 58, 370, copyright 1984, Rights and Permission Department, Academic Press Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887.)...

COHERENCE ORDER USING GRADIENTS TO SELECT A COHERENCE PATHWAY... 

The pathway S, S is unaffected by the first gradient (Fig. 8.26) because z magnetization does not precess, so the 13C SQC (Sx) is only twisted by the second gradient and arrives at the FID in a coherence helix that adds to zero over the whole sample. There is no need to subtract it out—it never reaches the receiver. We can add up the twists imparted by the two gradients using the fact that coherence order (p) equals zero for z magnetization ... 

Because the sum is not equal to zero, we end up with twisted coherence and no signal in the receiver. We call this a gradient-selected experiment because the gradients are being used to specifically refocus coherence in the desired coherence transfer pathway (XH SQC -> 13C SQC) and to reject all others. In Chapter 10, we will develop the idea of coherence order in a more precise manner, and we will see that coherence order can be either positive or negative. 

Below the pulse sequence we can show the desired coherence level, p, at each stage of the pulse sequence. This diagram defines the coherence pathway that is desired for a particular NMR experiment. Coherence order is mixed for Cartesian product operators... 

For a series of pulses (a pulse sequence), we can select the change in coherence order Ap resulting from each of the pulses if we phase cycle all of the pulses and then calculate the effect of the desired coherence pathway on the final phase. If we diagram the coherence pathway, we can note the change in coherence order Ap caused by each pulse and then calculate the receiver phase change necessary to make the desired combination of Ap s add together at the receiver while all other pathways cancel ... 

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 (A/-fold mask) and construct a phase cycle so that all of the selected pulses are independently stepped through their A-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... 

This is the rule for selecting a coherence pathway The gradient strengths G are adjusted so that the sum of coherence order times gradient strength over all of the gradients is equal to zero. In this case, the last gradient will have... 

To use the more formal analysis of phase cycling developed in Section 10.6, we first need to describe the coherence pathway in terms of spherical operators (I+, S, etc). Starting at the end and working backward and using the convention of positive coherence order during... 

Coherence Order. The coherence order, p, is zero for z magnetization and zero-quantum coherence, 1 or -1 for single-quantum coherence, and 2 or -2 for doublequantum coherence. The coherence order is useful for diagraming the coherence pathway in a pulse sequence and for predicting the effect of gradient pulses on the sample magnetization. 

FIGURE 11.2 Coherence pathways for a COSY experiment illustrated in a coherence order diagram, (a) All possible pathways resulting from the two 90° pulses. The width of the second pulse has been exaggerated to clarify the diagram. (b) The two pathways (A and B) selected from a) that lead to observable magnetization (p = —1). Adapted from Gunther.64... 

Pulsed field gradients are added to existing NMR pulse sequences in order to suppress artifacts and/or to select certain coherence transfer pathways. The application of gradients for these two purposes maintain different requirements and benefits. 

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