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270°

—cos(2nvr t)

—y' axis

where vr is the frequency of the pulse. In the rotating frame of reference, we compensate for the physical violation of using an accelerating (rotating) frame of reference by including a fictitious magnetic field, oriented along the -z axis with magnitude 2nvr/y, where vr is the rate of rotation (in hertz) of the x' and y' axes of the rotating frame of reference (Fig. 8.1(a)). If the spins are on-resonance, then the pulse frequency, vr, is equal to the Larmor frequency, vo. In this case, the fictitious field strength is 2nvo/y, which is equal to Bo (because vo = yBo/2n). The fictitious field, which is oriented along -z, exactly cancels the real field Bo, which is oriented along +z, and there is no field at all in the absence of a pulse (Fig. 8.1(b)). During the pulse, the only magnetic field experienced by the spins is the B1 field, which is in the x'-y' plane.

Thus for an on-resonance pulse, the Bo field does not exist in the rotating frame and the effective field experienced by the spins is just the B1 field. The magnitude of the effective

field is Beff = B1, and the Beff vector is oriented in the x'-y' plane at a position determined by the pulse phase. The net magnetization vector M has magnitude Mo and, starting from its equilibrium orientation along the +z axis, precesses counterclockwise (ccw) about the B1 vector at a rate v1 = yB1/2n. If the pulse duration, tp, is adjusted so that M precesses exactly one fourth of a complete rotation (90° pulse: v1tp = 1/4; tp = 1/(4v1)), then the M vector ends up in the x'-y' plane at the end of the pulse. This is the picture we have been using so far for all pulses.

### 8.2.2 Off-Resonance Pulses

What happens if a resonance peak is not exactly in the center of the spectrum? In this case, the pulse is off-resonance (vr = vo). As always, we choose axes x' and y' that rotate around the z axis at a rate equal to vr, the frequency of the pulse (corresponding to the radio frequency at the center of the spectral window). As before, the B1 field can be described by a vector that is stationary in the x'-y' plane, but now the fictitious field, which is required to correct for the accelerating frame of reference, no longer perfectly cancels the Bo field. If the resonance peak is downfield (higher frequency) of the center of the spectral window, then vo > vr and the fictitious field (2nvr/y) is lower in magnitude than the Bo field (2nvo/y). Because the fictitious field is oriented along the -z axis and the slightly stronger Bo field is oriented along the +z axis, the result is a small residual field (Bres) oriented along the +z axis (Fig. 8.1(c)).

During the pulse, the spins do not experience the B1 field alone, but rather an effective field Beff, which is the vector sum of the small residual field along the +z axis (Bres) and the B1 field in the x'-y' plane (Fig. 8.2, left). If the resonance is not far from the center of the spectral window, the Beff vector will "tilt" slightly out of the x'-y' plane and get slightly longer than B1.

If the resonance peak is upfield (lower frequency) of the center of the spectral window, then vo < vr and the fictitious field along —z "wins out" over the Bo field along +z, leaving a small residual field (Bres) along — z (Fig. 8.1(d)). Now the spins experience an effective field vector Beff that is tilted slightly below the x'-y' plane during the pulse (Fig. 8.2, right). The exact angle of tilt can be calculated using simple trigonometry (tan © = Bres/B1) and the magnitude of Beff comes from Pythagoras (B^ff = B^es + B\), but we are concerned here only with a qualitative understanding: the Beff vector tilts out of the x'-y' plane and gets slightly longer than B1, and this effect depends on the relative magnitudes of B1 and Bres, that is, on how far we are off-resonance and how powerful the pulse is. A very high

power pulse can "resist" and minimize the off-resonance tilting, but a weak RF pulse will be more sensitive to resonance offset.

If the B1 field is weak (short B1 vector) or the pulse is far off-resonance (long Bres vector), the effective field vector Beff will tilt significantly up or down, out of the x'-y' plane. In this case, a 90o rotation about Beff ("90o pulse") will not really put the net magnetization M into the x'-y' plane and a 180o rotation ("180o pulse") will not really put the net magnetization on the — z axis. We can easily compare the magnitudes of B1 and Bres if we think of B1 amplitude in units of hertz: v1 = yB1/2n. This is analogous to expressing the field strength of a magnet in terms of its proton resonant frequency: vo = yBo/2n. Thus, we talk about a "300-MHz magnet" rather than a magnet with Bo = 7.4 T, and we can talk about a "25-kHz B1 field" for an RF power setting, which gives a 10-^s 90o pulse. In this B1 field, the 1H net magnetization rotates one full rotation in 40 |xs, so we can say that the net magnetization vector rotates around the B1 vector at a rate of v1 = 1/(0.000040 s) = 25,000 Hz.

Be careful to note that v1 is not the frequency of the pulse. We call that frequency vr (reference frequency); it is very close to vo, or 300 MHz on a "300-MHz" NMR spectrometer. Think of v1 as a measure of the pulse amplitude rather than frequency: it is the rate at which the sample magnetization rotates around the B1 vector, a measure of the effect of the pulse (length of the B1 vector).

Now we can directly compare B1 (the magnetic field of the RF pulse) to Bres (what is left of Bo after subtracting out the pseudofield correction for the rotating frame of reference). If a resonance in the 1H spectrum is at 10.0 ppm and the center of the spectral window is at 5.0 ppm on a 300-MHz instrument, we have vo — vr = (10 — 5) 300 = 1500 Hz. This is how far the pulse is off-resonance and it is proportional to Bres. If the 90o pulse width is 10 |xs, we can describe the B1 field strength in hertz (v1 = yB1/2n) as 1/(4 x tp) = 25,000 Hz. This is proportional to B1 in the same way that vo — vr is proportional to Bres. Thus, the B1 vector is 16.67 times longer (25,000/1500) than the Bres vector, and the tilt will be insignificant. We can say that this pulse is strong enough to "cover" a spectral window 10 ppm wide (0-10 ppm) without any significant loss of effectiveness at the edges. The exact amount of tilt is 3.4o and the Beff vector is 0.18% longer than B1. Clearly, we can use the same simple vector model for all resonances within the 10 ppm (3000 Hz) wide spectral

window. We will have problems when Bres becomes comparable to B1:

Thus, we can avoid off-resonance effects by making B1 as strong as possible (highest power pulse possible, shortest duration) and by making vo — vr as small as possible (avoid having resonances peaks very far from the center of the spectral window).

As an example of how bad it can get, consider a case where B1 = Bres (i.e., v1 = vo — vr). In this case, if we apply the pulse on the x' axis the resultant vector Beff is tilted 45o out of the x'-y' plane toward the z axis (Fig. 8.3, left). The magnitude of the Beff vector is 1.414 (square root of 2) times B1 so that the net magnetization vector M will rotate about Beff about 41% faster than it would for an on-resonance pulse. Suppose we want to apply a 180o pulse to this off-resonance peak. We could compensate for the larger magnitude of Beff by using a pulse that is shorter in duration by a factor of 1.414. This would rotate the net magnetization M exactly 180o around the Beff vector (Fig. 8.3, right). Because Beff is tilted 45o up in the x'-z plane, the M vector rotates around it maintaining the 45o angle to Beff at all times, tracing out a conical path, and landing after a 180o rotation right on the x' axis! Thus, our "180o pulse" delivered to an off-resonance peak is really only a 90o pulse in our simple vector model.

This can be a real problem for 13C pulses at high field strengths. Consider a 600-MHz spectrometer with a 13 C spectral window stretching from 0 to 220 ppm. The center of the spectral window is at 110 ppm and a ketone carbonyl resonance is at 200 ppm. The strongest 13 C pulse you can muster is a 16-^s 90o pulse, corresponding to a B1 field strength of 15.63 kHz (1/0.000064 s). The resonance is 90 ppm or 13.5 kHz (90 x 150 Hz) from the center of the spectral window. Bres (13.5 kHz) is nearly equal to B1 (15.6 kHz). You've got problems!

8.2.3 Composite (Sandwich) Pulses

There are many tricks to get around the problem, such as sandwich 180o pulses (e.g., 90x-180y-90x) and "broadband" shaped pulses. Figure 8.4 (top) shows the inversion profile for asimple 180o pulse at the highest available power (tp = 28.4 |xs, yB1/2n = 17.6 kHz). The profile is obtained using an inversion-recovery sequence (180ox — t — 90oy) with recovery time t = 0. The final 90o pulse frequency and the 13C peak (13CH3I) are both at the center of the spectral window, but the frequency of the 180o pulse is moved in 10 ppm (1500 Hz)

steps away from the center right and left, each time printing the spectrum to the left or right of the previous spectrum. On-resonance the peak is upside down and has maximum intensity

-Sx) but as we move off-resonance the intensity diminishes and reaches zero at about 70 ppm off-resonance. At this point, we are getting a 90o pulse rather than a 180° pulse. Beyond this we actually see positive peaks, indicating that the z component of net magnetization after the "180°" pulse is positive.

A more "robust" way to invert the sample magnetization is the sequence 90°-180° -90°, with no space in between the pulses. This is called a composite pulse or sandwich pulse because a number of pulses are lined up right next to each other, like slices of cheese and meat in a sandwich. Suppose that what we think is a 90° pulse is really an 85° pulse due to miscalibration. The first "90°" pulse on y' rotates the sample magnetization ccw by only 85°, leaving it in the x'-z plane just 5° short of the x' axis (Fig. 8.5).

The 180° pulse rotates the M vector around the X axis in a very sharp cone, landing at a point 5° below the X axis. The final pulse (85°) rotates M precisely down to the —z axis. So we have a perfect 180° pulse even though it was miscalibrated by over 5%. This is not entirely true because the 180° pulse in the middle of the sandwich is really a 170° pulse, but this introduces a very small error as the sample magnetization is so close to the X axis and rotates in a very narrow cone. The effects of off-resonance pulses are more complex, but one can see in Figure 8.4 (middle) that this sandwich pulse does abetter job of inversion than the simple 180° pulse. The effective "bandwidth" or coverage of the pulse is 150 ppm compared to about 80 for the 180° pulse alone. This is still not wide enough for the entire range of 13 C shifts, which extends from 5 to around 220 ppm, requiring a 215 ppm bandwidth.

We will see that the major application of shaped pulses is to select a narrow region of the spectrum, thus displaying a narrow bandwidth. But there are also shaped pulses designed to do just the opposite — to give even excitation over a very wide range of frequencies. These "broadband" shaped pulses are specialized for inversion (Sz ^ —Sz) or refocusing (Sx ^ —Sx). Figure 8.4 (bottom) shows the inversion profile of an "adiabatic" inversion shaped pulse with maximum B1 field strength of 13.7 kHz (yB1l2n), average B1 field strength of 8 kHz and total duration 546 ^s. No discernible "droop" in inversion efficiency is seen over a range of 200 ppm, and even over a bandwidth of 260 ppm, only a 15% loss in efficiency is observed at the edges. This is accomplished with an average B1 field strength of less than half of that used for the simple 180° pulse. We will discuss how this works later in this chapter after we gain an understanding of shaped pulses and spin locks.

### 8.2.4 Precession in the Rotating Frame

What happens after the pulse for an off-resonance spin? Suppose that right after the pulse the net magnetization M of the sample is on the X axis. The Bi field is now turned off, so that in the rotating frame of reference the only field experienced by the spins is the residual field along the z axis (Bres = Bo — Bfict). If the resonance is downfield of the center of the spectral window (vo > vr), then Bo > Bf and the residual field is a small field oriented along the positive z axis of magnitude 2n(vo — vr)ly. The net magnetization vector M will precess about the effective field, which is now Bres, in the counterclockwise direction (viewed from the +z axis) at a rate equal to yBrtJ2n, which is simply vo — vr (Fig. 8.1(c), bottom). We can think of the Bres vector just like the B1 vector during the pulse or the Bo vector in the laboratory frame: Any magnetic field rotates the sample magnetization counterclockwise about the field axis. Precession is like a z-axis pulse!

This result is not surprising as the rotating frame of reference simply subtracts vr, the rate of rotation of the x and y axes in the rotating frame, from the Larmor frequency vo. The same thing goes on electronically in the receiver of the NMR instrument, where the FID (decaying signal of frequency vo) is mixed (mathematically multiplied at each time point using an analog device) with a reference frequency vr, which is the same frequency as the RF pulse. The result of this mixing consists of two signals added together, one with frequency vo + vr and another with frequency vo — vr. The first signal is a radio frequency (close to twice the Larmor frequency) and the second is an audio frequency (in the range 0-10 kHz), so it is easy to block the high frequency and pass the audio frequency with an analog filter. The result is the audio FID containing the frequency vo - vr, which is the precession frequency in the rotating frame of reference.

If the resonance is upfield of the center of the spectral window (vo < vr), then the fictitious field is stronger than the Bo field (Bo < Bfic) and the residual field Bres is oriented along the negative z axis. Under the influence of this effective field, the sample magnetization M rotates clockwise (viewed from above) at a frequency vr - vo (Fig. 8.1(d), bottom). In the rotating frame of reference, we consider this a negative frequency as the rotating-frame frequency is always defined as vo - vr, corresponding to precession in the counterclockwise direction. Some books use the uppercase omega to represent the rotating-frame angular velocity: ^ = mo - &r. Angular velocity is just frequency times 2n: m = 2nv. This angular velocity (in radians per second) is often referred to as the chemical shift, even though it has no relation to the 8 scale in parts per million. The rotating-frame frequencies (vo - vr in Hz) or the rotating-frame angular velocities (^ in radians per second) depend on the spectrometer field strength Bo and have zero value at the center of the spectral window. The chemical shift (8 in parts per million) is independent of Bo and is zero at the resonance position of tetramethylsilane (TMS).

If the peak is on-resonance (vo = vr), there is no residual field and the magnetization vector stands still in the x'-y' plane after the pulse (Fig. 8.1(b), bottom). This is to be expected as in the laboratory frame the magnetization vector is precessing counterclockwise at frequency vo, and in the rotating frame we are rotating the axes at exactly that rate (vr = vo), so relative to the rotating x' and y' axes the vector is not moving.

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