Summary Of Twospin Operators

For a system of two kinds of spins (I and S for heteronuclear, or Ia and Ib for homonuclear systems), there are 16 product operators, formed from the 16 matrix elements of the density matrix. So far we have discussed 14 of them:

Iz, Sz z magnetization(population difference)

Ix, Iy, Sx, Sj, in-phase magnetization in the x' - y' plane

2IxSz, 2IySz, 2SxIz, 2SyIz antiphase magnetization in thex' -y' plane

2IxSy, 2IySx? 2IxS x, 2Iy Sy zero-and double-quantum coherences (not observable)

The remaining two are the longitudinal spin order, which results when the macroscopic z magnetization of one nucleus (e.g., *H) is opposite depending on the microscopic z magnetization (a or P) state of the other nucleus (e.g., 13C), and the identity (1) operator, which simply represents the vast majority of spins that cancel each other out and play no role in NMR experiments. Longitudinal spin order can be viewed as an intermediate state in coherence transfer: 2Ix Sz ^ 2Iz Sz ^ 2Iz Sx = 2Sx Iz .Like z magnetization, it is not affected by gradients. The identity operator is usually ignored because we are interested only in population differences.

2IzSz longitudinal spin order

1 the identity operator(equal populations in all four states)

Together, these 16 product operators describe the 16 matrix elements in the 4 x 4 density matrix representation of a two-spin system (Chapter 10). In the matrix, each element represents coherence between (or superposition of) two spin states. As there are four spin states for a two-spin system (aIaS, aIpS, pIaS, and pIpS), there are 16 possible pairs of states, which can be superimposed or share coherence. The product operators are closer to the visually and geometrically concrete vector model representations, so in most cases they are preferable to writing down the 16 elements of the density matrix, especially as only a few of the elements are nonzero in most of the examples we discuss.

These operators and the rules that govern chemical shift and /-coupling evolution in time can be used to describe any combination of RF pulses and delays, giving a prediction of the observable magnetization (and thus the spectrum) at the end of the sequence. This gives us the means of understanding all of the 1D and 2D NMR experiments. By comparison, the vector model can explain only a few of the 1D experiments.


The INEPT sequence is not very useful because we cannot apply 1H decoupling to the antiphase signals observed in the FID. For example, the antiphase 13C doublet of a CH group (1:-1) would, with 1H decoupling, collapse to a single frequency with the positive peak right on top of the negative peak. These would exactly cancel, and we would see no peak at all. The same is true for a CH2 antiphase triplet (1:0:-1) and a CH3 antiphase quartet (1:1: — 1: — 1); all of these have zero net signal if they are collapsed by 1H decoupling into a single frequency. In order to apply 1H decoupling, we need to add a refocusing period to allow antiphase magnetization to evolve back into in-phase magnetization. For the CH group, this is very simple: we add a 1/(2/) delay to go from antiphase to in-phase: 2SxIz —1/(2 J) ^ Sy. To prevent phase twisting by chemical shift evolution, we need to add simultaneous 1H and 13C pulses in the center of the 1/(2/) delay, just as we did in the first ("defocusing") 1/(2/) delay. Figure 7.29 shows the full sequence with a general refocusing time of 2t. For the CH group, we would set 2t = 1/(2/). In the first spin echo, we have /-coupling evolution only from in-phase to antiphase: Iy —1/(2 J) ^ — 2IxSy. We think of the 180° 1H pulse on the yf axis as occurring at the start of the first delay, where it has no effect on Iy. The simultaneous 90° pulses on 1H and 13 C then lead to transfer of coherence: — 2IxSz ^ —2 [—Iz] [Sx] = 4[2SxIz]. Finally, the refocusing delay 1/(2/) brings us from antiphase 13C coherence back to in-phase: 4[2SxIz] ^ 4Sy. The 180° 13C pulse on the X axis has no effect because we imagine it at the beginning of the refocusing period, where 13C coherence is on the X axis. For a general refocusing delay A = 2t, we have

Figure 7.30

With decoupling, the antiphase term is not observed, so we see only the in-phase signal with an intensity of sin ©, where the "angle" © (in radians) is equal to nJA or 2nJr. The maximum signal occurs with © = n/2 (90o) or A = 1/(2J). We will see that while this is ideal for the CH group, we would see no signal at all for CH2 or CH3 groups! This can be useful because it allows us to get a 13C spectrum with only the CH peaks, so we can distinguish between CH and CH3 peaks—something that the APT experiment cannot do.

Figure 7.30 shows the INEPT spectrum of neat benzene, C6H6, acquired with the sequence of Figure 7.16. On the left is the spectrum without refocusing or 1H decoupling: we see an antiphase doublet with complex long-range (2JCH and 3JCH) couplings making the components "ragged." With refocusing (sequence of Fig. 7.29, A = 1/(2J), where J is 1JCH) we see an in-phase doublet, still showing the long-range couplings. When 1H decoupling is applied during acquisition of the FID, we see a sharp singlet. This singlet peak is about four times the height of the 13 C singlet obtained in a simple 13 C spectrum with 1H decoupling (Fig. 7.30, right) because of the enhancement coming from coherence transfer from the 1H.

Now we need to look at the refocusing step in general—for all three types of13 C nuclei that are coupled to 1H. The defocusing step (first 1/(2J) period) was simple because any proton, whether it is a part of a CH, CH2, or CH3 group, is still connected to only one 13C, so it can be looked at as a doublet. In the refocusing step, however, we have 13C coherence and it behaves differently depending on whether it is a doublet (CH), triplet (CH2), or quartet (CH3). We will have to define two 1H product operators — I1 and — I2 for the CH2 group and three — I1 ,I2 and I3 — for the CH3 group. As soon as we have 13C magnetization in the X-/ plane, we can have an in-phase or antiphase relationship to any of the attached protons. Thus, for pure 13C SQC with a CH2 group, we can have product operators like

Sx 2SxII, 2SxI2 4SxIII2

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