Pulsed Field Gradients

Field gradients were developed for magnetic resonance imaging, which is an NMR technique that encodes spatial information (x, y, and z axes) rather than chemical shift information in the FID. The resulting pictures of "slices" of the human body have provided a revolutionary new tool for medicine. More recently, this technology has been applied to NMR spectroscopy, with dramatic results. Pulsed field gradients make it possible to automatically optimize all of the shims simultaneously in a few minutes (Section 12.3). More importantly, with adequate sample concentration the total time required for a 2D experiment can be reduced from many hours to 15-45 min. Gradients make possible water suppression in 90% H2O that is far superior to the old presaturation method. Finally, many artifacts in 2D spectra can be eliminated and sensitivity can be further improved by selecting only the signals that you are interested in and suppressing all others.

8.5.1 What is a Gradient?

Normally, we go to great efforts to assure that the magnetic field is homogeneous throughout the sample volume. This means that the strength of the magnetic field, or Bo, is exactly the same everywhere in the sample leading to sharp peaks for each resonance in the spectrum. The gradient intentionally destroys this homogeneity in a linear and predictable way. For example, a z-axis gradient alters the magnetic field so that the magnetic field strength is reduced in the lower part of the sample and increased in the upper part in a linear fashion. In other words, the magnetic field strength is now a function of the position of a molecule in the NMR tube along the z axis:

Bg(z) = Bo + Gz x z where Bg is the magnetic field strength with the gradient turned on, Gz is the strength of the field gradient (usually given in gauss per centimeter, where 1 G is 10-4 T) and z is the position of the molecule along the z axis. We choose the zero of the z axis to be at the center of the sample, so molecules above the center experience a slightly increased magnetic field and molecules below the center experience a slightly decreased magnetic field. The relative magnitude of this change is very small; for example, a maximum gradient strength of 50 G/cm in a Bo field of 117,440 G (11.744 T, 500 MHz 1H). The gradient can be turned on and off very rapidly, so that typically the gradient is "pulsed" on for a period of 1-2 ms and then turned off.

8.5.2 Effect of Gradients on NMR Signals

What happens to the sample magnetization during a pulsed field gradient? Because the resonance frequency of a nucleus is always proportional to the magnetic field (vo = yB/2n), if we have a net magnetization vector in the x'-y' plane (e.g., after a 90° pulse), the magnetization vector will rotate in the x'-y' plane at a different rate depending on the molecule's position in the NMR tube (we can assume that diffusion is slow, so the molecule's position does not change). Magnetization in the upper part of the tube will precess faster than normal, and magnetization in the lower part will precess slower than normal. The result is that a "twist" or helix of magnetization will exist in the sample, so that at the end of the gradient period the magnetization rotates as a function of z coordinate throughout the sample (Fig. 8.9).

Assuming an on-resonance peak, spins at the center of the tube are unaffected by the gradient and do not move (Bg = Bo). Spins above the center experience a stronger magnetic field during the gradient (Bg > Bo), so they rotate counterclockwise in the x'-y' plane. As we move further up in the tube, this rotation is faster and the total angle of rotation during the gradient pulse is proportionally greater. To show this "twisting," we use an

Figure 8.9

open square to indicate magnetization pointing back into the page (— x' axis) and a filled square to indicate magnetization pointing out toward us (+x' axis). Moving up from the center, the net magnetization at each level progresses from +y to —x, to —y, to +x, and back to +y. All of the spins underwent precession in the x'-y' plane for the same amount of time (t) but at different rates (yGzz/2n) depending on their vertical distance from the center. The rotation is counterclockwise and is more pronounced as we move up. Moving down from the center, the field during the gradient pulse is slightly weaker (Bg < Bo) and the on-resonance spins now begin to fall behind the rotating frame, precessing in the clockwise direction. At the end of the gradient pulse, we have net magnetization on +y, +x, —y, —x, and back to +y as we move down. The result is a "phase twist" or a helix of coherence. This is shown in cartoon fashion as a spring or spiral of coherence (Fig. 8.9, right).

There may be many hundreds of revolutions of the vector in the full vertical distance of the sample volume. If we try to acquire a spectrum at this point, after the gradient is turned off, we will not have any observable signal because the vectors point in all possible directions in the x'-y' plane equally throughout the sample volume and the net magnetization vector is zero (Fig. 8.9, bottom). When viewed from the top, we see magnetization vectors pointing equally in all directions in the x'-y' plane, leading to a net magnetization of zero throughout the whole sample. Figure 8.10(a) shows a spectrum of sucrose in D2O at 500 MHz, and Figure 8.10(b) shows the same spectrum with alms gradient applied between the 90° excitation pulse and the start of the FID. So, a gradient can completely annihilate an NMR signal! Why would we want to do that? The signal might be an artifact, a solvent signal, or some other feature of the spectrum that we do not want to see. In this way, gradients can be used to "clean up" or remove unwanted NMR signals. Compared to the older method

Figure 8.10

of removing unwanted signals, subtraction using a phase cycle, the gradient technique is far superior because it accomplishes the cleanup in one scan. The receiver never sees the artifacts so we do not have to turn down the receiver gain, and we are not dependent on perfect stability to give good subtraction.

8.5.3 Refocusing with Gradients—The Gradient Echo

We can not only kill coherence but also bring it back from the dead! The twisted magnetization in the sample can be "untwisted" by applying another gradient pulse of the same magnitude and duration but of opposite sign. This gradient decreases the magnetic field strength above the center of the sample and increases it below the center. During the second gradient pulse, the magnetization vectors rotate in the x'-y' plane clockwise in the upper part of the sample and counterclockwise in the lower part. The vectors which rotated counterclockwise in the first gradient pulse are now rotating clockwise at the same rate, and vice versa, so that at the end of the second gradient pulse all of the magnetization vectors are lined up again throughout the sample (Fig. 8.11). If we start the acquisition of the FID at this point, we will get a normal NMR spectrum, except for the phase "twist" that results from chemical shift evolution during the gradients. This result is shown for sucrose in Figure 8.10(c). Another way of saying this is that the first gradient pulse encoded the position of each molecule into its magnetization, scrambling the net magnetization of the whole sample, and the second gradient pulse decoded this information, unscrambling the net magnetization. So we can destroy with gradients, but we can also reverse the process and regenerate signals that were completely destroyed! All of this assumes that the molecules do not change their "level" in the tube between the time of the first gradient and the time of the second gradient. To the extent that the molecules undergo diffusion, which is faster for smaller molecules, there will be a loss of some signal. In fact, the gradient echo can be used as a way

Figure 8.11

of measuring diffusion rates or to distinguish between small molecules and large molecules. In order to minimize "stirring" of the sample, we do not use spinning during gradient experiments.

If this sounds a lot like a spin echo, you are right. In the spin echo, various factors affect the precession rate of the spins in a sample: chemical shift differences for nonequivalent spins or differences in Larmor frequencies for identical spins in a nonhomogeneous magnetic field (bad shimming). In either case, a time delay leads to a "fanning out" of phases in the x'-y' plane as they do not precess at exactly the same rates. A 180° pulse "flips" all the spins to the opposite side of the x'-y' plane, and as long as they continue at the same frequency for the second half, they will all line up again at the end. Each spin "remembers" its precession frequency, either due to its position within a molecule (chemical shift) or due to its physical location in the NMR tube (inhomogeneous field), and by repeating this behavior exactly in the second half, it ends up back where it started. In the gradient echo, we create the differences in frequency by applying a gradient pulse. This is just an inhomogeneous magnetic field. The spins "fan out" in phase during the time of the gradient, depending on where they are physically located within the NMR tube. The second gradient actually reverses the inhomogeneity of the field, so that the accumulated error in phase during the first gradient is exactly reversed during the second for each spin in the sample.

If the spins are not on-resonance, they will still undergo chemical shift evolution in the gradient echo. For a resonance in the downfield half of the spectral window, the spins in the upper part of the sample will precess faster in the counterclockwise direction and the spins in the lower part will precess slower during the first gradient. During the second gradient, the spins in the upper part will precess slower and the spins in lower part will precess faster. At the end of the second gradient, these two perturbations will exactly cancel out for each spin at each level in the sample, and all the magnetization vectors will point in the same direction, as if the gradients had just been simple delays.

We can look at this more precisely using the product operator formalism, even though it is more important to focus on the conceptual picture rather than the math. For a resonance with Larmor frequency vo, we have during the first gradient v'o = Y (Bo + zGz)/2n = yB0/2n + yzGz/2n = Vo + Vg where vg is the change in precession frequency due to the gradient. If the peak is on-resonance (vo = vr), we have a rotating-frame precession frequency of v'o — vr = vo + vg -vr = vg. Starting with magnetization on the y' axis, we have at the end of the first gradient

As vg is proportional to the z coordinate of the spin within the sample tube (vg = yzGz/2n), we see that this is a helical coherence spinning around in the x'-y' plane as we move up or down the z axis. The net magnetization throughout the whole sample, summed over a large range of values of z, is zero. The second gradient will modify the Larmor frequency to v'g = —yzGz/2n = —vg, and precession at this frequency for a time t will convert the pure Iy and Ix operators to

Iy ^ Iycos (—2nvgT) — Ixsin (—2nvgT) = Iycos (2nvgT) + Ixsin (2nvgT)

Ix ^ Ixcos (—2nvgT) + Iysin (—2nvgT) = Ixcos (2nvgT) — Iy sin (2nvgT)

Plugging these expressions for Ix and Iy into the product operator representation for the spin state at the end of the first gradient, we get

Iycos (2nvgT) — Ixsin (2nvgT) ^ [Iycos (2nvgT) + Ixsin (2nvgT)]cos (2nvgT) — [Ixcos (2nvgT) — Iysin (2nvgT )]sin (2nvgT)

= Iycos2 © + Ixsin©cos© — Ix cos©sin© + Iy sin2© = Iy(cos2© + sin2 ©) = Iy where © = 2nvgT and we make use of the trigonometric identity cos2© + sin2© = 1. We see that the spin state starts as Iy and ends as Iy regardless of the position (z) of the spin in the sample. The math is considerably more complicated if we do not assume that the peak is on-resonance, but the conclusion is that we would have the same result as if the gradients were just simple delays: chemical shift evolution for a period of time 2t (Fig. 8.10(c)).

8.5.4 The Pulsed Field Gradient Spin Echo (PFGSE)

We saw in Figure 8.10(c) the phase "twist" that results from chemical shift evolution, during the relatively long (ms) time of the two gradients. To refocus this chemical shift evolution, we place a 180° pulse at the center of the sequence, between the two gradients. This makes the gradient echo into a spin echo (r-180°-t) with gradients during the two delay times. But we have to consider what the 180° refocusing pulse does to the helix (the "twist") of coherence created by the first gradient. Figure 8.12 shows the effect that a 180° pulse on the x' axis has on the coherence helix: The sense of the twist is reversed, giving the mirror image of the original helix. Before the 180° pulse, we have coherence that moves from x to —j to —x to j (clockwise viewed from above) as we move down in the tube, but after the pulse the coherence moves the from x to j to —x to — j (counterclockwise viewed from above) as we move down. What kind of gradient do we need to untwist this coherence? We need a gradient identical to the first one, which will rotate coherence ccw in the upper part of the tube and cw in the lower part, exactly canceling the twist imparted by the first

Figure 8.14

gradient (Fig. 8.13). Without the 180° pulse, this second identical gradient would simply reinforce the effect of the first, twisting the helix twice as tightly (Fig. 8.14). This sequence (Gz-180°-Gz) is called PFGSE, and we will see that it forms the basis of many selective excitation experiments. The key to its utility is in the 180° pulse: If it truly flips the sample magnetization to the opposite side of the x'-y' plane, reversing the sense of the helix twist, the sample magnetization is lined up at the end (Fig. 8.10(d)). If the pulse does not give a 180° rotation, the sample magnetization is completely destroyed. We will see how this reinforces the selectivity of a shaped pulse in the next section.


We saw that the PFGSE acts as a spin echo if the central pulse is a 180° pulse, and as a gradient-based coherence annihilator if the central 180° pulse is absent. What happens if we put a 180° shaped pulse at the center of the PFGSE (Gz-180°(sel.)-Gz)? This pulse should deliver a 180° pulse to the selected resonance (peak) in the spectrum and have no effect (0° pulse) on all the other peaks. At the end of this sequence, we expect to have aligned coherence for the selected spins (Fig. 8.13) and completely scrambled coherence for all of the other spins (Fig. 8.14). As we started with a 90° pulse, we have no z magnetization and for the nonselected spins we end up with no net coherence either. So overall we have excited the selected resonance with a 90° pulse and we have destroyed all net magnetization on all other resonances in the spectrum. This is a very radical kind of selectivity, as nothing is left at all but the net magnetization of the desired spins in the xX-y' plane. A 90° shaped pulse will rotate the selected spins into the X-y' plane, but the other spins will still have their full equilibrium net magnetization on the +z axis. This strategy was developed by A.J. Shaka, who dubbed it "excitation sculpting" because we start by exciting all the spins equally and then we cut away all the magnetization we do not want using the gradients, just as a sculptor reveals the desired shape by cutting away marble from a formless block.

Figure 8.15(b) shows the spectrum of sucrose with a 90° Gaussian pulse applied to the triplet at 3.99 ppm, compared to a normal 1H spectrum (Fig. 8.15(a)). This is done by moving the reference frequency to place the 3.99 ppm triplet at the center of the spectral window (on-resonance), which is the center of the Gaussian-shaped excitation profile

Figure 8.15

resulting from the shaped pulse. We can see some distortion of the peak shape as well as some undesired excitation, particularly of the two strong singlets (CH2OH peaks) and the HOD peak at 4.73 ppm. In Figure 8.15(c), we see the same spectrum using a PFGSE with a 180° Gaussian pulse at the center. The peak shape is improved and we see absolutely none of the nonselective peak intensity. This can be improved even further by repeating the PFGSE with a different gradient strength (Gaz - 180(sel.) - Gaz - Ghz - 180(sel.) - Gb) for an overall double pulsed field gradient spin-echo or DPFGSE (Fig. 8.15(d)). We can select other peaks in the spectrum by simply moving the reference frequency (vr) to place the desired peak on-resonance. For example, with the double doublet at 3.51 ppm on-resonance we see only this peak in the DPFGSE spectrum (Fig. 8.15(e)). The normal spectrum (Fig. 8.15(a)) has narrower lines because the sample is spinning.

We can actually measure the excitation profile of the Gaussian pulse in a DPFGSE sequence by selecting a peak (putting it on-resonance) and then repeating the experiment with the peak moved off-resonance in equal steps both upfield and downfield. The spectra are superimposed to give a series of peaks that map out the shape of the excitation. This is shown in Figure 8.16 for the HOD peak of sucrose in D2O, changing the reference and pulse frequency (vr) by 6 Hz for each successive spectrum. We see that the profile from a Gaussian-shaped pulse is indeed Gaussian, with a bandwidth at half-height of about 36 Hz. The bandwidth is inversely proportional to the pulse width (duration of the shaped pulse), so if we used a 70-ms Gaussian pulse (with half the maximum B1 field strength to maintain a 180° rotation), we would see a Gaussian excitation profile with a bandwidth of 18 Hz at half-height. Stretching the pulse squeezes the excitation profile and vice versa.

8.6.1 Frequency-Shifted Laminar Pulses

It is rather tedious to move the spectral window every time we want to select a peak with a shaped pulse, but it is necessary as the center of the Gaussian excitation profile is at

Figure 8.16

the center of the spectral window. One way to get around this is to change the phase of the individual rectangular pulses that make up the shaped pulse. If we consider a 35-ms Gaussian pulse made up of 35 rectangular pulses of 1 ms each, we could increase the phase of each pulse relative to the last one by an angle of 10o. This is easy to do because the shaped pulse is created from a list of 35 lines, each line specifying a pulse amplitude and a pulse phase. The first pulse would be delivered with B1 on the x' axis, the 10th pulse with B1 on the j' axis, the 19th pulse with B1 on the —x' axis, the 28th pulse with B1 on the — y' axis, and so on (Fig. 8.17). We see that the B1 vector is no longer stationary in the rotating frame of reference—it is moving counterclockwise (in jerks) at a rate of one cycle every 36 ms, which is a frequency of 1/(0.036 s) or 27.78 Hz. The effective frequency of the pulse is 27.78 Hz higher than vr, its nominal frequency, so the center of the excitation profile is shifted downfield by 27.78 Hz from the center of the spectral window. Until now, the pulse frequency has always been the same as the reference frequency, at the center of the spectral window. We are using the phase "ramp" as a way of tricking the spins into seeing the excitation pulse at a different position within the spectral window. Now we can place the Gaussian excitation

DPFGSE using 35 ms Gaussian pulse: sucrose in D20

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