Figure 7.11

t delay

2IxSz-^ 2IxSz cos(nJT) + Iy sin(nJT) (refocusing: Fig. 7.9, right)

2IySz 1 delay 2IySz cos(nJT) — Ix sin(nJT) (refocusing: Fig. 7.9, left)

As before, the cosine term multiplies the starting spin state of the rotation, and the sine term multiplies the "next stop" on the counterclockwise rotation of /-coupling evolution. For example, for Ix, we know that the next stop for simple chemical shift evolution is Iy (if you have any doubts, draw a small set of coordinate axes (x, y, and z) and trace the counterclockwise rotation by 90o from the starting spin state). We multiply this by 2 in front and Sz after the Iy term (evolution into antiphase) to get 2Iy Sz. Starting from — 2Iy Sz, we move counterclockwise from — Iy to Ix and remove the 2 and the Sz (refocusing) to get Ix. This representation breaks down the vector model of two vectors moving in opposite directions into two components: the in-phase component and the antiphase component (Fig. 7.11). Note that the cosine term always goes with the unchanged product operator and the sine term always goes with the new product operator it evolves into. This makes sense, as the cosine function is 1 and the sine function is 0 at time zero.

Finally, consider the effect of both chemical shift evolution and J coupling. This gets pretty complicated, but we can consider either one of them first and then apply the effect of the other. Let's consider the chemical shift evolution first:

delay t

Now substitute for Ix and Iy, considering the effect of J-coupling evolution:

delay t

Ix cos^It + Iy sin^It-^ [Ix cosnJT + 2IySz sinnJT]cos^It

+ [Iy cos nJT — 2IxSz sin nJT]sin^It = Ix cosnJTcos^It + 2IySz sinnJTcos^It

+ Iy cosnJTsin^It — 2IxSz sinnJTsin^It C D

In the square brackets we have the result of J-coupling evolution starting from Ix (first term in brackets) and from Iy (second term in brackets). The result is four separate terms, and we can think of them like this: (A) coherence that underwent neither J-coupling evolution nor chemical shift evolution; (B) coherence that underwent /-coupling evolution but not chemical shift evolution; (C) coherence that underwent chemical shift evolution but not /-coupling evolution; and (D) coherence that underwent both chemical shift evolution and /-coupling evolution. In each case, we use a cosine term if that type of evolution (Qt for chemical shift evolution and nJt for /-coupling evolution) did not occur, and a sine term if it did occur. The correct product operators can also be written directly using this type of reasoning. Starting with Ix, we can go directly to the four terms: first, write Ix with two cosine terms (i.e., no evolution at all); then write 2IySz (counterclockwise rotation Ix ^ Iy plus evolution from in-phase into antiphase) with a sinn/t term (/-coupling evolution did occur) and a cosQIt term (chemical shift evolution did not occur); then write Iy with a cosn/T term (no /-coupling evolution) and a sinQIt term (chemical shift evolution); finally, write — 2Ix Sz with two sine terms (both kinds of evolution, resulting in a 180° rotation of the I operator and evolution into antiphase). If we use s and c for sinn/T and cosn/T, respectively, and s' and c' for sinQIT and cosQIT, respectively, this can be written quickly and simply as follows:

This is pretty complicated, but the advantage is that we can keep track of everything of importance. Any pulse sequence can, in principle, be examined to see what effect it will have on the sample magnetization and what observable signals will remain at the end. Product operator formalism represents the full quantum-mechanical phenomenon of NMR, so that any type of experiment including mysterious things like multiple-quantum coherences (MQCs) can be represented correctly.

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