The Spin Echo

The spin echo is one of the fundamental building blocks of pulse sequences and is used in a variety of 1D and 2D experiments. Consider first a 13C spectrum with three peaks, each one representing a separate 13C position in the molecule (singlet). The first half of the spin-echo sequence is very much like the primitive sequence described in Figure 6.11: a 90°13C pulse puts the sample magnetization on the y' axis in the rotating frame and a delay of duration t follows. The magnetization vectors of the three 13C resonances in the sample will "fan out" in both directions away from the y' axis with angular frequencies Av depending on the resonance offset Av of each NMR peak relative to the center of the spectral window (Fig. 6.24). Then a 180° 13C pulse is applied along the y' axis of the rotating frame. This rotates all three of the magnetization vectors in the x-y plane to the opposite side of the y' axis. Essentially, if you view all of the magnetization vectors as lines radiating out from the center of a pancake, you are flipping the pancake along the y' axis. Now a delay of the exact same duration t follows. Precession continues, with each carbon resonance rotating in the same direction and velocity as before, depending on its resonance offset (position of the peak in the spectrum). Because the two delays are of the same length, all of the magnetization vectors will be exactly aligned along the y' axis at the end of the second

Figure 6.25

delay (Fig. 6.24, right). This is analogous to a foot race around a circular track. The starting gun is fired; the runners fan out according to each runner's characteristic speed. After a certain period of time t, the gun is fired again and all of the runners turn around and run in the opposite direction. As long as all runners maintain the same characteristic speeds in the second half of the race, all of them will reach the starting line at the same time, exactly 2t after the start of the race. It is this coalescence of all the runners (or magnetization vectors) at the starting point at the end of the 2t period that is referred to as the "echo," because it bounces back and reappears after a specific length of time. The spin echo is sometimes described as "time reversal": With the spin echo we have reversed the effects of chemical shift evolution, effectively making time stand still for a period of time 2t. The second half of the echo (second t delay) is sometimes called the "refocusing" period because the chemical shift effects are focused back to the starting axis.

Consider the example of Figure 6.13 once again using a simple 13C spin-echo sequence (Fig. 6.25) with the delay t set to 1/(87). We saw how the initial 90oy pulse on the 13C channel rotates the two 13C net magnetization vectors (H = a and H = ft) representing the two components of the 13C doublet from the +z axis (equilibrium) to the +X axis. After a delay of 1/(87), the H = a vector has rotated one-fourth turn ccw and lies on the +/ axis, whereas the "slower" H = ft vector has rotated only 45o and lies between the +X and +/ axes (Fig. 6.26(c)). If we apply the 13C 180o pulse now (t = 1/(87)) on the +X axis, the H = a vector is rotated to the -/ axis and the H = ft vector ends up between the +x' and -y' axes (Fig. 6.26(d)).

It is very important to note that a 180o pulse (sometimes called an "inversion" pulse) does not always move a vector exactly to the opposite side of the x'-y' plane! The H = a vector in this case sweeps out a plane (the y'-z plane) as it rotates up to the +z axis and down to the — y' axis, but the H = ft vector sweeps out a cone as it rotates up to halfway between the +X and z axes and down to halfway between the x' and — y' axes. To get this concept clear you might want to think of the B1 vector (the pulse) as a physical object, an "axle" or broomstick glued to the net magnetization vector, another broomstick. The pulse "twists" the B1 vector (the axle) on the x' axis by a counterclockwise rotation of 180o, whereas the net magnetization vector is dragged along because it is physically attached. If the B1 vector forms a 90o angle with the net magnetization, the twisting moves the net magnetization to the opposite side—in this case from the +y' axis to the —y' axis. But if the net magnetization is at a different angle to B1 —45o in this example—it rotates in a conical path, reversing the sign of its y' axis projection without changing the x' axis projection at all. In the extreme case where the net magnetization is on the

Figure 6.26

Figure 6.26

same axis as the B1 vector (in this case, on the +x' or — x' axis), it is not affected at all by the pulse, as we can visualize a broomstick colinear with the B1 broomstick being twisted but not changing its direction as the Bi "axle" is rotated. To make this distinction, we refer to a 180° pulse as an "inversion" pulse only when the net magnetization moves from +z to —z, and we use the term "refocusing" pulse when the net magnetization starts and ends in the x'-y' plane.

Continuing with the simple spin-echo sequence (Fig. 6.25), during a second 1/(87) delay the H = a vector will rotate another one-fourth turn ccw, ending on the x! axis, and the H = ft vector will rotate by 45° ccw, also landing on the x! axis (Fig. 6.26(e)). This is exactly where we started after the initial 13C 90° pulse, with both vectors on the x' axis: the simple spin echo refocuses both the J-coupling evolution (divergence of the a and ft components) and the chemical shift evolution (rotation of the center position—the dotted line representing the 13C chemical shift of the doublet). Basically, we have wasted a period of time equal to 1/(4J) and nothing has happened! This might seem like a pointless exercise, but later we will see that there might be things you need to do during that delay, and the spin echo is a way to get everything back to where you started.

Applying this to the APT experiment, we could solve the problem of chemical shift differences affecting the final phase by applying a 180° pulse in the middle of the 1/J delay period, effectively making a spin echo with delay t = 1/(2J). The chemical shift evolution that occurs during the first half is now refocused in the second half, and we do not have to require that the 13 C peak be on-resonance. But the desired information would also be lost, because the evolution of magnetization vectors under the influence of J coupling would also be canceled in the second half of the spin echo! Each line of a 13C multiplet (doublet, triplet, or quartet) would undergo the same evolution and "de-evolution" described above and end up on the same axis at the end of the spin echo. We can, however, turn J coupling on and off at will using the proton decoupler. By using a spin-echo delay of t = 1/J and turning the decoupler on during the second half of the spin echo only, we generate the desired phase encoding due to J coupling during the first half of the spin echo and refocus the chemical shift effects during the second half, when each 13 C resonance behaves as a single line centered at the chemical shift position (Fig. 6.27). Because the proton decoupler eliminates the J couplings for the second half of the spin echo, the divergence of multiplets that occurred during the first half does not get refocused during the second half. Note that the 1H decoupler is on all of the time except for the first 1/J delay of the spin echo: During the relaxation delay, the decoupler is "pumping up" the 13 C z magnetization above Mo due to the heteronuclear NOE, and during the acquisition of the FID the decoupler is collapsing the 13C multiplets into singlets so that we see only one line for each 13C resonance.

Often a second, short spin echo is added to this sequence so that pulses shorter than 90° (e.g., 30°) can be used for the first pulse. A starting pulse of less than 90° is desirable to

Figure 6.27

allow for shorter relaxation delays, but in the simple APT sequence (Fig. 6.27) the sample magnetization left on the z axis is inverted by the 180° pulse of the spin echo and a long relaxation delay would be required to bring it back from the — z axis to equilibrium by T1 relaxation. The combination of two spin echoes (two 180° pulses) brings this z magnetization back to the positive z axis and allows for a more rapid return to equilibrium.

6.9.1 Measurement of T2 Values Using Multiple Spin Echoes: CPMG

We saw in Chapter 5 that in a perfectly homogeneous magnetic field the FID decays to zero in an exponential fashion with characteristic time T2. In the real world, field inhomogeneity makes the FID decay faster (Fig. 6.28), with a characteristic time we can call T2*(T* < T2). The envelope of the FID drops to 63% (e—1) of its original size after this period of time T2*. If we look at decay rate rather than characteristic time, the intrinsic rate (R2 = 1/T2) of decay of coherence is added to the rate of decay due to "fanning out" of individual vectors from different physical locations in the sample (Ri=1 /T^, the inhomogeneity decay rate) to obtain the experimental rate of decay (R* = 1/T*):

The Fourier transform converts the FID into a Lorentzian peak with absorptive lineshape (after phase correction). The full width of this peak at one half of the peak's height (the "linewidth") is inversely related to the decay time constant of the FID, T2*:

So just by measuring the width of the NMR peak we can determine the time constant for decay of the FID, but this is not an interesting number because it depends on shimming. The interesting number, which is a fundamental physical measurement for that particular spin in a specific environment, is the T2 value. How can we extract the T2 value from the easily measured T2* value, which is a combination of T2 and the inhomogeneity decay constant T2? Recall that in a spin echo the differences in chemical shift evolution that occur during a delay are "refocused" or removed during the second half of the spin echo following the 180° pulse. These differences may be real differences due to different positions within a molecule, or differences in resonant frequency due to the different locations of identical

Figure 6.28

spins within the sample volume in an inhomogeneous magnetic field. It does not matter: In either case the vectors that lag behind in precession during the first half of the spin echo have less distance to travel in the second half, and the vectors that pull ahead in the first half have farther to travel in the second half. All of these individual vectors ("isochromats" if you like fancy words) representing different locations within the sample come together at the end of the second delay, creating a maximum in the net magnetization measured throughout the sample (an "echo": Fig. 6.29). In the extreme case, where field homogeneity is very bad (T2^T2), we would see the transverse magnetization we are measuring in the FID (e.g., Mx) decay rapidly (e-t/T2) during the first half of the FID as the individual vectors "fan out" in the x-y plane and become evenly distributed in all directions. After the 180° pulse, these vectors begin to gather together and the transverse magnetization grows exponentially until it reaches a maximum when all the vectors cross the "finish line" (the echo). After they cross, the vectors fan out again just as rapidly due to field inhomogeneity and the coherence is lost (Fig. 6.29, top).

Figure 6.29

Figure 6.29

The interesting thing is that the maximum intensity of the FID at the "top" of the echo is still less that that at the start of the FID: not all of the coherence is recovered by refocusing in the second half of the spin echo. The part that is lost is the intrinsic decay, the loss of coherence due to pure T2 relaxation, a fundamental relaxation process. The spin echo simply gets back the losses due to inhomogeneity of the magnetic field (72 losses). This gives us a method to measure T2: We could repeat the spin-echo experiment a number of times with different echo delays (t values) and start the acquisition of the FID at the "top" of the echo:

The signal intensity at the start of the FID is proportional to the peak height after Fourier transformation, so we could make a plot of peak height versus t delay and fit the exponential decay to a theoretical curve to measure the T2 value. This is the T2 equivalent of the inversion-recovery experiment (Section 5.8) for measurement of T1.

There is still one problem with this method. As we make the t delay longer and longer, it is possible that some molecules will diffuse from one part of the sample to another so that the spins do not experience the same magnetic field (Bo) in the two delays (t) of the spin echo. The refocusing of the spin echo works only if the inhomogeneity experienced in the first half is identical to that experienced in the second half for each of the identical spins. This diffusion problem will lead to loss of signal at the top of the echo in addition to the intrinsic T2 loss and will be more pronounced for smaller molecules or in less viscous solvents. One way to avoid this is to keep the echo time (t) short and use a large number of repeated spin echoes (Fig. 6.29, bottom). Instead of increasing the t value to explore the T2 decay curve, we increase the number of repeats of the spin-echo unit (t-180o-t) while keeping t constant. As long as the t delay is quite short, the loss of signal due to diffusion is kept to a minimum. This method is called CPMG, or Carr-Purcell-Meiboom-Gill, after the four investigators who developed it.

6.10 THE HETERONUCLEAR SPIN ECHO: CONTROLLING /-COUPLING EVOLUTION AND CHEMICAL SHIFT EVOLUTION

As we move on to more complex and more powerful pulse sequences for heteronuclear (e.g., 1H-13C) experiments, we would like to use the spin echo to refocus only one of the two kinds of evolution, J-coupling evolution or chemical shift evolution. This can be done in a very simple and elegant way by adding another 180° pulse to the 1H channel, simultaneous with the 13C 180° pulse. This "heteronuclear" spin-echo sequence is shown in Figure 6.30. Notice that we now are using the 1H channel (lower line) for more than just decoupling: We are delivering high-power pulses of defined duration, calibrated for a specific rotation—in this case, 180°. In the early commercial FT spectrometers this was an advance in hardware capability, because power could not be rapidly and repeatedly switched from high power ("hard" pulses) to low power (waltz-16 decoupling), so two separate sources of 1H RF were required.

Again using the example of Figure 6.26, consider what happens if we include the 1H 180° pulse at the center of the spin echo. At this point in the sequence (time "C" in Fig. 6.30), at the end of the first delay, we have the H = a vector on the +/ axis and the H = 5 vector halfway between the +X and +/ axes (Fig. 6.31 C). As before, the 13C 180° pulse on the X axis rotates the H = a vector to the — yf axis and the H = 5 vector to a position halfway between the — yf axis and the X axis. We can consider the effect of the 1H 180 pulse after the 13C 180° pulse, even though they are simultaneous. The 1H 180° pulse does not rotate the vectors because they represent13 C net magnetization and are not affected by a pulse at the 1H frequency. It does, however, affect the 1H nuclei that are attached to (and /-coupled to) to the 13C nuclei. The effect is to change every 1H nucleus that was in the a state to the 5 state and every 1H nucleus that was in the 5 state to the a state. This means that our "H = a" vector, representing the net magnetization of all 13C nuclei whose attached 1H is in the a state, is now an "H = ¡" vector, because all of those protons are now in the 5

state. Likewise, our "H = ¡3" vector can now be called an "H = a" vector because all of the protons attached to those 13C nuclei are now in the a state. In other words, the effect of the !H 180o pulse is to "swap the labels" on the 13C net magnetization vectors without moving them. Now we have the H = a vector halfway between the +X and — / axes and the H = ¡3 vector on the — / axis (Fig. 6.31 D). Next we have the second delay, of duration 1/(87), of the spin echo. The H = a vector will rotate one-fourth turn ccw as before, ending up halfway between the +X and +/ axes, and the H = ¡3 vector will rotate 45° ccw as before, ending up halfway between the +X and — yf axes (Fig. 6.31 E). Note that the only thing that is different in the second half of the spin echo is the behavior of the two vectors, because we know that the H = a component of the 13 C doublet has a rotating-frame frequency (Av) of 300 Hz and the H = ¡3 component has a rotating-frame frequency of 150 Hz. By flipping the 1H from the a state to the ¡3 state, we changed the effective magnetic field experienced by the 13C nucleus, decreasing its Larmor frequency by 150 Hz (J).

So how does the result compare to the simple spin echo (without the 1H 180° pulse: Fig. 6.26)? The two vectors diverged during the first half of the spin echo, ending up with a 45° angle between them: this is J-coupling evolution. The center position between the two vectors rotated 67.5° from the +X axis to a position three-fourths of the way from the +X axis toward the +/ axis: this is chemical shift evolution. In the simple spin echo, both of these types of evolution were reversed during the second half, as the two vectors converged toward each other and the center position moved back to the +X axis. But with the 1H 180° pulse included, the divergence of the two vectors (J-coupling evolution) continues in the second half, resulting in an angle of 90° (twice as large) between them at the end of the delay. J-coupling evolution is "active" throughout the pulse sequence, and we have a divergence of 90° that is the result of J-coupling evolution for a total period of 1/(4J), which is the sum of the two delays. Recall that the "magic time" for J-coupling evolution is 1/(2J), which takes us from in-phase to antiphase (0° angle between the two vectors of a doublet to 180°), so a delay of 1/(4J) should give a divergence of 90°.

We can say that the 180° 13C pulse reverses the J-coupling evolution (Fig. 6.26), but the 180°

1H pulse "reverses the reversal" so that the two vectors continue to diverge during the second half (Fig. 6.31). Another way to look at it is that 13C chemical shift evolution (a result of the

Bo field interacting with the 13C nucleus) is sensitive only to 13C pulses, because only these can rotate the 13C net magnetization vectors. But /-coupling evolution is a mutual interaction between the 13C nucleus and the1H nucleus, so both of the 180° pulses affect it, and effectively they cancel each other out, just as the product of two negative numbers is a positive number.

What about the chemical shift evolution? During the second half, the center position between the two vectors (dotted line in Fig. 6.31) rotates ccw from a position three fourths of the way between the +X and -/ axes to end up back on the +X axis. We have refocused the chemical shift evolution which occurred during the first delay. The chemical shift evolution of the 13C net magnetization "sees" only the 13C 180° pulse, so it is refocused just as it was in the simple spin echo. In Figure 6.30, we represent the overall effect of the spin echo by writing "+vC" (13C chemical shift evolution) and "+/" (/-coupling evolution) in the space of the first delay, and "—vC" (13C chemical shift refocusing) and "+/" (/-coupling evolution continuing in the same sense) in the space of the second delay. Overall, we have +vC for time 1/(8/) and — vC for time 1/(8/), for a net chemical shift evolution of zero. We also have / evolution for 1/(8/) and again / evolution in the same sense for 1/(8/) for a total / evolution of 1/(4/), which leads to a total divergence of the two vectors by / Hz times 1/(4/) seconds or one-fourth turn (90°). This way of getting a simple overview of the effects of a pulse sequence building block will be crucial to your understanding of more complex experiments: Once you are very comfortable with the details of vector rotation in the x'-y' plane and the effects of pulses, you no longer will need to consider these details—only the overall effect of the pulse sequence building block is important. You will eventually be able to break up complicated pulse sequences into well-understood building blocks and guess the effect of each building block on the net magnetization.

This is a very powerful pulse sequence building block! We now have control over the two kinds of evolution. We will see how in many heteronuclear experiments we want to get the 13C (or the 1H) into the "magic" antiphase state, but we do not want to complicate things with the chemical shift evolution, which would be different for every peak in the spectrum leading to a great confusion of phases and peak shapes at the end of the experiment. Let's see if we can apply this technique to the APT sequence. Recall that the goal of the APT sequence is to have /-coupling evolution for a period of time 1// to introduce the editing effect (CH and CH3 phases reversed, Cq and CH2 phases unaffected). We do not want to mess up all the phases by having chemical shift evolution, so we used a simple spin echo of total duration 2// with the XH decoupler on for one of the 1// delays (Fig. 6.27). Chemical shift evolution that occurs during the first half (+vC) is reversed during the second half (—vC) because of the 13C 180° pulse. Whether the XH decoupler is on or off is irrelevant to 13C shift evolution. We have /-coupling evolution (+/) during the first half, when the decoupler is off, and no /-coupling evolution during the second half. So the net /-coupling evolution is +/ for a period of time 1//, leading to a divergence of the two vectors of a CH group by an angle of /Hz times 1// seconds or one cycle (360°). The two vectors are once again in-phase but on the opposite side of the x'-y' plane, each individual vector having traveled by an angle of J/2 times 1// or one half cycle (180°): Each component of the doublet considered alone is //2 Hz away from the center or chemical shift position. Now let's try to achieve the same effect but with an "advanced" heteronuclear spin echo including a XH 180° pulse at the center along with the 13C 180° pulse. Using the sequence of Figure 6.30, we can set the delays to 1/(2/) each so that we get a total /-coupling evolution period of 1// without any chemical shift evolution. That is it! We do not need to play around with the XH decoupler, except to turn it on during the acquisition of the FID. This "improved" APT

Figure 6.32

sequence is shown in Figure 6.32. In what way is it superior to the original APT sequence? The total time that we have 13 C net magnetization in the x'-y' plane is reduced from 2// to 1//, cutting in half the loss of NMR signal due to T2 relaxation. Thus, the loss of sensitivity that we see in the APT compared to a simple 13C spectrum could be cut in half! Hardly anyone uses this sequence, however, because old methods die hard. We will also see in the next chapter that an even better alternative to the APT experiment exists, one that is actually quite a bit more sensitive than the simple 13 C experiment.

As a final illustration of the power and versatility of the heteronuclear spin echo, let's see if we can design a spin-echo sequence that allows chemical shift evolution but refocuses /-coupling evolution. We start with the simple spin echo, and we remove the 13C 180° pulse because it is responsible for the reversal of chemical shift evolution (changing +vC to — vC in our shortcut notation). But now we do not have any refocusing, just two delays of equal duration. How can we reverse the /-coupling evolution without rotating the 13C magnetization vectors? Remember that because the /-coupling evolution is due to a mutual interaction of the 1H and the 13C nucleus, it can be reversed with a 180° pulse on either the 1H channel or the 13 C channel, and that 180° pulses on both channels cancel each other in this effect. So we can put our 180° pulse on the 1H channel only, where it will reverse the /-coupling evolution without affecting the 13C chemical shift evolution. The pulse sequence is shown in Figure 6.33, with the notations "+vC" and "+/' in the first delay and "+vC" and "—J" in the second delay. The overall effect of this pulse sequence is 13 C

Figure 6.34

Figure 6.34

chemical shift evolution for a total time of 2t, leading to a total rotation of Av Hz times 2t seconds, where Av is the rotating-frame frequency at the center of the CH 13 C doublet. The /-coupling evolution is +J for time t and —J for time t, indicating that /-coupling evolution is refocused in the second half and the vectors will still be in-phase, without diverging at all. Figure 6.34 shows the effect of this pulse sequence using vector diagrams for our example. As before, at the end of the first 1/(8J) delay we have the H = a vector on the axis and the H = p vector halfway between the +X and +/ axes. As we saw before, the only effect of the 180° 1H pulse is to reverse the labels on the 13C net magnetization vectors: now the H = p vector is on the +/ axis and the H = a vector is halfway between the +X and +/ axes. The chemical shift position is in the same place between the two vectors. During the second delay, the chemical shift position continues to rotate counterclockwise. The H = p vector rotates 45° ccw to end up halfway between the +/ and — X axes, and the faster H = a vector rotates 90° ccw to end up at exactly the same place. The two vectors are in-phase again, and there has been no net J-coupling evolution. But the chemical shift evolution has continued, as if the pair of vectors never diverged, and moved a total of 135° (67.5° times 2) in a ccw direction (225 Hz x 0.833 ms x 2 = 0.375 rotations, 0.375 x 360° = 135°).

One technical question remains: What is the effect of the phase of the 13C 180° pulse in these spin-echo sequences? If we place the B1 field on the — X axis, the result is exactly the same, because rotation by 180° in one direction is the same as rotation by 180° in the opposite direction. But if we place the 13C 180° pulse on the / axis, the details of the vector motions will change, and the final result will be different, but the question of evolution remains the same. Do this as an exercise, going through our example for three types of spin echoes: the simple spin echo (13C 180°y only), the spin echo with 180° pulses on both channels, and the spin echo with 180° pulse on the 1H channel only. You will see that the simple spin echo refocuses J-coupling evolution and chemical shift evolution, but the vectors end up on the — xX axis instead of on the +X axis. You might have to try a different chemical shift (e.g., Av = 150 Hz) to convince yourself that the vectors always land on the —X axis, regardless of the resonance offset of the center of the doublet. The reason is that not only did the 13C 180° pulse refocus the chemical shift and J-coupling evolution, but it also flipped the vectors from the +X axis to the — X axis. We can say that the effect of the sequence is the same as it would be if the delay times were set to zero (90°y-180°y-): the two vectors are rotated to the +X axis by the 90°y pulse, and then to the — X axis by the 180° pulse on the y' axis. In the original example, the 13C 180° pulse was on the X axis, so it had no effect overall on the position of the vectors: If we imagine the delays being zero, we have a 90°y pulse rotating the vectors to the +X axis, and a 180°X pulse having no effect on the two vectors because they are colinear with the B1 field. Thus, the two vectors end up on the +X axis. But be careful about saying that the 180° 13C pulse on the X axis has "no effect": It has the effect of refocusing the chemical shift and J-coupling evolution, and if we leave it out the result will be entirely different. It just has no effect beyond the refocusing, whereas a 13 C 180° pulse on the y' axis has the additional effect of rotating the vectors from their original position on the +X axis to the —X axis.

The same reasoning can be applied to the sequence of Figure 6.30: Overall we have /-coupling evolution only, as if the13 C doublet were on-resonance. The two vectors diverge from each other but the center position does not move; the more downfield H = a line (7/2 Hz downfield of the center of the doublet) leads to a ccw rotation at a rate of 7/2, and the more upfield H = j line (—7/2 relative to the center of the doublet) leads to a cw rotation at a rate of 7/2. The effect of the phase of the 180° pulse is correctly accounted for if we imagine it happening at the beginning of the first delay: The two vectors rotate from the +X axis to the —X axis as a result of the 180° pulse on the y' axis, and then they diverge by an total angle of 90° (each one moving 7/2 Hz times 1/(47) or 45°) without changing the position of the center, which remains on the — X axis. The phase of the 180° *H pulse in either spin-echo sequence (Fig. 6.30 or Fig. 6.33) is irrelevant because it serves only to convert every XH from the a state to the j state and vice versa. It does not rotate the 13 C magnetization vector, so we do not care which axis the B\ field is on. We refer to this pulse as an inversion pulse, and like an inversion of net magnetization from +z to —z, it does not matter which axis the B1 field is on.

The overall lesson from this exercise is that we can ignore the details of vector rotation during a spin echo if we understand its overall effect: Which types of evolution are allowed and which are refocused? This is easily determined by looking at the 180° pulses at the center. A 180° pulse on the nucleus that has the net magnetization in the X-y1 plane (i.e., the coherence) will lead to refocusing of chemical shift evolution. A 180° pulse on one of the two nuclei involved in the 7 coupling will lead to refocusing of 7-coupling evolution. The 180° pulses on neither or both of the two nuclei will allow 7-coupling evolution to occur. The phase of the 180° pulse on the channel where we have coherence can be accounted for by imagining that the pulse occurs at the beginning of the first delay and accounting for the allowed evolution for the full delay time (2t) of the spin echo. The phase of any 180° pulse on the other channel, corresponding to the nucleus that is not evolving, is irrelevant. By looking at the spin echo in this overall view we can avoid the tedious analysis of considering the effect of each pulse and delay individually.

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