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Figure 8.38

magnetization vector is 1/256 of its original magnitude, and after nine it is 1/512. A "trim pulse" or "purge pulse" is a short (1-2 ms) spin lock with high power (~25 kHz, the same as a hard pulse) placed on the axis of the desired magnetization to destroy all magnetization that is not on that axis. A simple application would be the INEPT sequence, where we allow in-phase 1H magnetization to undergo J-coupling evolution into antiphase: Ix ^ 2IySz. This conversion is perfect only if the delay is exactly equal to 1/(2J). Because we just set the delay to our best guess, there will be residual in-phase signals at the end of the delay:

At this point, a purge spin lock on the y axis would preserve the antiphase term 2Iy Sz and destroy the in-phase term Ix. We then proceed to the coherence transfer step (simultaneous 90o pulses on 13C and 1H) with pure antiphase 1H coherence.

8.10.3 Effect of the Spin Lock on Locked Magnetization

The locked magnetization is parallel to the only magnetic field that is present in the rotating frame: the B1 field. This is analogous to z magnetization in the Bo field in the laboratory frame during a delay, when there is no B1 field. If we follow this analogy further, we see that the spin lock axis is like the z axis and the B1 field is like the Bo field, except very much weaker (e.g., 8 kHz vs. 600 MHz, or 75,000 times smaller!). Thus, we can think of the spin lock as a way of temporarily "turning down" the Bo field to a vastly lower value. This has two effects: First, the tumbling rates required to stimulate SQ and DQ relaxation are very low (on the order of v 1 and 2v 1 instead of vo and 2vo), and second, the "chemical shift" differences (Av 1 instead of Avo) are extremely small when compared to the J values. The first effect means that the NOE in the spin-lock world (the "rotating-frame" NOE) will always be dominated by DQ cross-relaxation, which leads to negative NOEs (NOE enhancement), regardless of the size of the molecule. In other words, "all molecules are small molecules" in the spin lock. This is because the DQ and ZQ frequencies are now accessible to even the slowly tumbling biological molecules through dipole-dipole interactions. This is the basis of the ROESY experiment, which transfers magnetization along the spin-lock axis via through-space (NOE) effects, while making all molecules behave like small molecules in terms of their NOE behavior. The second effect means that the chemical shift differences between different protons within a molecule are almost completely eliminated, leading to "strong coupling" and "virtual coupling" within a spin system. In other words, the protons within a spin system behave like each one is J coupled to every other one, even if there is no direct J coupling between them. This is the basis of the TOCSY experiment, which transfers magnetization along the spin-lock axis from one proton to all other protons in the same spin system.

8.10.4 Off-Resonance Effects

So far we have assumed that the spin-locked nucleus is on-resonance or at the center of the spectral window (vo = vr). If the nucleus is off-resonance, the effective field in the rotating frame, Beff, is the vector sum of the B1 field vector along the axis of the spin lock (e.g., y') and the residual field along the z axis (Bres = 2n (vo - vr)/y = Bo - 2nvr/y). This means that the spin-lock axis tilts out of the x'-y' plane by an angle that increases as vo moves farther away from the reference frequency, or as the B1 field strength is decreased. Each proton in the molecule thus has a different spin-lock axis and must be considered separately. The length of the Beff vector is greater than B1 due to the vector sum: Beff (the magnitude of the Beff vector) is equal to (Bj + Bjes)1/2. The rate of precession about the spin lock axis is v1 = yBeff/2n, so the rotation of any magnetization that is not on the spin-lock axis around it becomes faster as we move off-resonance.

What does this mean for the effect of the spin lock on sample magnetization? If the sample magnetization starts on the y' axis, for example, the tilted spin-lock axis will destroy the component that is perpendicular to the spin-lock axis and retain the component that is on the spin-lock axis. This preserved component is "locked" because it is on the axis of the effective field and has no "reason" to precess around the z axis. So even if the spin is off-resonance, its magnetization does not precess around the z axis during the spin-lock period. Instead, the component that is not on the tilted spin-lock axis precesses around the spin-lock axis until it is destroyed by B1 inhomogeneity, and the component that is on the spin-lock axis is retained.

8.10.5 Moving the Spin-Lock Axis

What would happen to the spin-locked sample magnetization if we moved the spin-lock axis? Would the spins follow, as the name "spin lock" implies, or would the magnetization vector be left behind, rotating around the spin-lock axis until the B1 field inhomogeneity "spins it out of existence"? The answer is: It depends on how fast we move the spin-lock axis. If we move it slowly enough (the "adiabatic" condition), the sample magnetization vector will get dragged along with it without any loss of intensity. How do we move the spin-lock axis? We can change the pulse frequency, moving it away from the resonance so that, in the rotating frame defined by the pulse frequency, the spins are off-resonance. This makes the Beff vector tilt out of the x'-y' plane. By a combination of adjusting the pulse amplitude (B1) and the resonance offset (vo - vr), we can put the Beff field vector anywhere

Figure 8.39

Figure 8.39

we want and give it any magnitude we want. Suppose, for example, that we start with the Beff field vector tilted 15° away from the +z axis in the y'-z plane and turn it on long enough to get a 180° rotation of the sample net magnetization, which starts on +z (a, Fig. 8.39). The magnetization will rotate to a point 30° away from the +z axis (b, Fig. 8.39). Now we move the Beff field vector to an angle of 45° with the +z axis and apply another 180° rotation. The sample magnetization will move around the Beff vector to a point 60° away from the +z axis (c). We could continue this process, placing the Beff vector at angles of 75°, 105°, 135°, and 165° from the +z axis until the last 180° rotation moves the sample magnetization down to the — z axis (g). We have inverted the sample magnetization by picking it up at +z and "shepherding" it around in a series of steps down to — z, always keeping the Beff vector close to the M vector. Imagine now that we decrease the increment of angle of the Beff vector and do the inversion in many more steps. Eventually, we would have a continuous RF irradiation with the Beff vector moving smoothly from +z to — z. What we have is a spin lock that "grabs" the sample net magnetization at +z and "drags" it down to — z physically, without doing any finite rotations. This can be accomplished by constructing a shaped pulse with a phase ramp that moves rapidly at first (large frequency shift in one direction) and slows down and stops at the center of the pulse, and then reversing the direction of the phase ramp, speeding it up continuously until the end of the pulse. The effective frequency of the pulse starts way downfield of the Larmor frequency, moves upfield until it equals the Larmor frequency at the center of the pulse, and then moves off upfield to a point far upfield of the Larmor frequency. At the same time, the magnitude of the B1 field is adjusted to be small at the start and end, to allow the Bres field (on +z or — z) to dominate and tilt the Beff vector up close to +z (start) or — z (end), and maximal at the center of the pulse to give a strong spin lock with no tilt out of the x-y plane. The classical adiabatic-inversion shaped pulsed is called "WURST" and there are many, many variations with equally cute names. Why go to all this trouble? If we consider a series of resonances widely spaced throughout a wide spectral window (like a 13C spectrum on a high-field spectrometer), as the effective frequency of the pulse "sweeps" from far downfield to far upfield, we will see the spins invert one at a time because the spin-lock axis depends on how far away the pulse effective frequency is from the Larmor frequency. As the pulse frequency passes through each peak in the spectrum, these spins are spin locked in the x-y plane halfway through their journey from +z to -z. It really does not matter where the spins are in the wide spectral window; they will be picked up on +z and dragged down to -z when their turn comes, like dominos falling in a row. If we sweep from downfield to upfield, the only difference will be that the spins on the downfield edge of the spectrum will be inverted a little earlier than the spins on the upfield edge of the spectrum. We saw an impressive example of this for inversion of the 13CH3113C signal using a "cawurst" adiabatic inversion pulse called "ad180" (Fig. 8.4, bottom). The bandwidth of this shaped pulse is far superior to hard pulses or hard pulse sandwiches. Decoupling schemes based on WURST are very effective, covering the wide bandwidth needed for 13 C decoupling with much lower power than rectangular pulse trains like waltz-16. The spin-lock field is just swept back and forth across the spectral window, inverting the spins over and over again as the Beff vector shuttles from +z to -z and back to +z. Rapid and continuous inversion gives good decoupling because the coupling partner (e.g., 1H) sees a spin (e.g., 13C) that is moving rapidly back and forth between the a state and the j state, blurring the difference in magnetic field experienced by the coupling partner into a single, constant field on the NMR time scale.

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