## J

I+Sz

2IzSz

90°13C

S+Iz the second XH 90° pulse converts I+Sz into 2Iz Sz, so that Ap(I) = -1. If instead we used a simultaneous XH 90° pulse and 13C 90° pulse to convert I+Sz directly into S+Iz, we would have Ap(I) =-1 for purposes of phase cycling the XH 90° pulse (I+ ^ Iz) and Ap(S) = + 1 for purposes of phase cycling the 13C 90° pulse (Sz ^ S+). For calculating gradients, we "inflate" the XH coherence order by the ratio of magnetogyric ratios yH/yo, where yo is the magnetogyric ratio of the lowest frequency nucleus in the experiment, but for phase cycling we always consider the different nuclei separately because a pulse is delivered on a specific channel and affects only that nucleus. Thus, for I = *H,

has p = 1, Iz has p = 0, and I- has p = -1 regardless of the definition of S (S = *H, 13C, 15N, 31P, etc.). Although a pulse can only affect one type of nucleus (e.g., XH or 13C), we will see below that gradients affect all coherences, and each coherence is affected according to its type of coherence (SQC, DQC, ZQC, etc.) and its magnetogyric ratio.

10.6.3 Coherence Pathway Selection with Pulsed Field Gradients

A pulsed field gradient is a distortion of the normally homogeneous magnetic field in the region of the sample that leads to a linear gradient of Bo along a single direction, usually the z axis, which is along the length of the NMR tube. This distortion is very small relative to Bo and is typically turned on for a period of around 1 ms and then turned off, returning the field to its homogeneous state. During the gradient, the Bo field depends on the position of a molecule within the NMR sample tube:

Bg = Bo + Gzz where z is the position along the z axis. The effect of the gradient on an SQC is to produce a position-dependent phase shift or "twist" that scrambles the coherence so that its net value, averaged through the whole sample volume, is zero. This comes about because the precession frequency of an SQC is proportional to Bo:

During the gradient pulse v(gradient) = Vg = y (Bo + Gzz)/2n = Vo + yGzZ/2n = Vo + Av For a gradient of duration t, the phase of the coherence will have changed by

The first part, which is not position dependent, is just the chemical-shift evolution that would be expected for a delay of duration t, but the second part is the result of the gradient. This is the position-dependent twist that results from the gradient pulse and is proportional to the gradient strength Gz and the gradient duration t. Usually, the gradient duration is the same for all of the gradients applied in a pulse sequence, and the gradient strength Gz is varied.

To see the power of gradients in coherence pathway selection, consider the effect of a gradient on a homonuclear DQC I+I+. The DQC undergoes evolution during a delay according to the sum of the two precession frequencies for the two protons Ha and Hb:

During a gradient, Bo is changed by the gradient field Gz z, leading to the following doublequantum precession rate during a gradient vdq (gradient) = v'a + Vb = [Va + YHGzZ/2n] + [vb + >HGzZ/2n] = vdq + 2yh Gzz/2n

After a gradient pulse of duration t, the phase shift for this DQC will be a\$dq = 2nvDQT + 2yGzZT

The first term is the evolution that would have occurred in the absence of the gradient, and the second term is the position-dependent phase twist. Note that the twist is twice as much as that experienced by a SQC in the same gradient. Thus, the coherence order (in this case p = 2) is encoded in the twist, and this gives us a way to select the coherence at any point during the pulse sequence by simply applying a gradient pulse.

When the same reasoning is applied to a homonuclear ZQC, we see that the gradient has no effect:

vzq(gradient) = v'a - v'b = [Va + yGzZ/2n] - [Vb + yGzZ/2n] = VZQ

The effect of gradients is especially simple to understand if we consider spherical operators. A Cartesian operator, such as Ix or Iy, will precess during a gradient according to

This is the position-dependent phase shift expressed in terms of the x and y components of the magnetization. Here, we assume that the I spin chemical shift is on-resonance, so that ordinary chemical-shift evolution can be ignored during the gradient. Now we can look at the effect of a gradient on the spherical operators by expressing them in terms of the Cartesian operators:

I+ = Ix + i Iy ^ Ixcos(yGzzr) + Iysin(yGzzr) + i Iycos(yGzzt) - iIxsin(yGzzr)

= Ix[cos(yGzzr) — i sin(yGzzr)] + iIy [cos(yGzzr) — i sin(yGzzr)]

In general, the effect of a gradient pulse on any operator is to multiply it by a phase factor that represents the position-dependent phase twist:

where p is the coherence order of the operator and yo is the lowest magnetogyric ratio in the spin system. For a homonuclear (XH) DQC

I+I+ I+e —i(YHGzzr)I+e-i(yhGzZT) _ I+I+e-i(2>HGzzr) Ia Ib ^ Ia e Ib e = Ia Ib e

Note that the phase factor fits the general pattern e-i(pYoGzzT), with coherence order p = +2 and yo = yH. For a homonuclear ZQC:

As expected, homonuclear ZQC is not affected by a gradient as it has coherence order p = 0. For heteronuclear MQC, we need to consider the difference in magnetogyric ratio y. For example, if I = XH and S = 13 C

I+S+ I+e-i(YHGzzr)g+e-i(yCGzzr) = I+S+e-:'((YH+YCGzzr)

For heteronuclear systems, it is convenient to redefine the coherence order p so that it includes the relative magnetogyric ratio y/yC (or y/yo in the general case):

Ph = Yh/Yc = 4 and pc = Yc/Yc = 1 Thus, I+ has p = 4, I- has p = -4, S+ has p = 1 and S- has p =-1. Using these definitions,

I+S+ I+ S+e-i((YH+Yc)GZZT^ _ I+S+e-:((Yh/Yc+Yc/Yc)YcGZZT) = I+S+e-i((4+1)YcGzzr^ I+S+e-i(5YcGzzr)

This is consistent with the rule if we define p = pH + pC = 4 + 1 = 5 for I+S+. Following this rule, we have p = 3 for I+S-, p = -5 for I-S-, and p =-3 for I" -S+ . For nuclei other than 13C, we factor out the lowest magnetogyric ratio of the two; for example, if I = 1H and S = 31P, we can define for this case, pH = 2.5 and pP = 1 so that p = 3.5 for I+S+ and 1.5 for I+S" ". This may seem sloppy because the coherence order of proton SQC depends on the context, but it is quite convenient and easy to use. In this way, we can define the effect of a gradient on any kind of coherence, and we see that the degree of twist or the tightness of the helix formed by the position-dependent phase shift is proportional to the coherence order p at the time of the gradient pulse.

For several gradients applied at different points during the pulse sequence, the position-dependent phase shift accumulates:

Phase factor = e-i(PiYoGizT)e-i(P2yoG2zT) e-i(P3YoG3zr)...

where p1, p2, p3,... are the coherence orders at the time of the gradients of strength G1, G2, G3,..., respectively, for the desired coherence pathway. At the end of the pulse sequence (beginning of the FID), we want the phase to be the same at all levels of the NMR sample so that the signals will combine and induce a coherent signal in the probe coil. This will be the case if the exponent is zero, so that the phase factor equals 1, regardless of the position (z) in the NMR tube:

This is the rule for selecting a coherence pathway: The gradient strengths Gi are adjusted so that the sum of coherence order times gradient strength over all of the gradients is equal to zero. In this case, the last gradient will have

PnGn = —(piGi + P2G2 + P3G3 + •••, +pw-iGw_i)

and will impart a helical twist exactly equal and opposite to that accumulated so far, effectively "unwinding" the position-dependent phase twist to yield an observable coherence in the FID. The artifacts will not unwind at this point because they did not follow the same coherence pathway and will not satisfy the magic formula

These coherences will have a helical phase twist in the NMR sample tube and will add to give a net signal of zero in the probe coil during the FID.

Let's apply the gradient coherence pathway selection technique to the DQF-COSY pulse sequence. Using a phase cycle to select the coherence pathway that passes through DQC between the second and third pulses, it is required that we acquire at least four transients for each FID in the 2D acquisition. If we have sufficient sample concentration to get a good signal-to-noise ratio in a single scan, the experiment takes four times as long as is necessary based purely on sensitivity considerations. Using pulsed field gradients to select the desired coherence pathway, the experiment time could be reduced by a factor of 4!

Instead of focusing on pulses as we did for phase cycling, we will focus on the delay periods between pulses where we can insert gradient pulses. First, a gradient during the

Figure 10.32

evolution (t 1) period "twists" the observable single-quantum magnetization into a helix oriented along the z axis. A second gradient is applied during the short period between the second and third pulses (Fig. 10.32) to untwist the coherence. Because the desired magnetization component during this period is DQC, the twist accumulates twice as fast as it did during the first gradient. This is due to the fact that the DQ evolution occurs at a rate that is the sum of the two resonance frequencies:

Each of these frequencies is shifted the same amount by the gradient, so the total position-dependent change in frequency is twice as large for the DQ coherence:

For the second gradient to undo the twisting caused by the first, it needs to be of opposite sign and half the gradient strength of the first: G2 = -1/2 G1. For example, we could use relative gradient strengths of G1 = 2 and G2 = — 1 (Fig. 10.32). The sensitivity to twisting by gradients is p = 1 during 11 (SQC) and p = 2 during the short delay between pulses 2 and 3 (DQC). To get zero twist at the end, we use gradients G1 and G2 such that

XpiGi = p1 G1 + p2G2 = 1 x 2 + 2 x (-1) = 0 In terms of spherical operators, the DQF-COSY experiment looks like this

Ha magnetization starts on z (equilibrium, p = 0) and is excited to I+ (SQC, p = 1) and I-(p = -1) by the first 90° pulse. Following just the I+ term, during the t1 delay we generate a mixture of I+ and the antiphase term I+(still p = 1), which is the only term that can lead to coherence transfer. The second 90° pulse generates DQC (I+I+) with coherence order p = 2 (along with five other undesired coherences!). The final 90° pulse "knocks down" Ha from I+ (p = 1) to Io (p = 0) and Hb from I+ (p = +1) to I- (p = -1) to generate the crosspeak I-Io. Here, we use Io = Iz for consistency with our spherical operators. In each step, we drop the phase factors (we do not bother specifying them) that accumulate because of pulse phases and evolution during delays. We are only interested in the general form of the coherence and its coherence order. This frees us from a lot of tedious bookkeeping! For example, each time an I+ or I- term is hit with a 90o pulse, we get three coherences:

, Io, and I (Fig. 10.29). So a product like I+I+ can generate nine different product operators (nine different coherences) with p ranging from —2 to +2! As long as we know the precise coherence order we want at each step, we can design the gradient strengths needed to select that "pathway" (sequence of coherences desired). We use the term "gradient selected" for any experiment where the gradients actually select the desired coherence pathway as opposed to "gradient enhanced" if they just clean up artifacts. The common feature of gradient selected experiments is the need for a precise ratio of gradient strengths (twist, untwist). In "gradient enhanced" experiments (e.g., gradient enhanced NOESY, Fig. 10.16) any old gradient strength will do because we are just "spoiling" coherences we do not want (twist, forget), rather than "selecting" the coherences we do want.

In terms of spherical operators, the improved DQF-COSY experiment looks like this

Io a

I+io

Ia Ib

180o T_

Ia Ib

180o

Ia Ib

Figure 10.34

The 180° pulses in the center of the spin echoes invert the spherical operators, with i+ becoming I- and I- becoming i+. We have to take this into account when designing gradients: now p1 = —1 (I-Ib) and p2 = +2 (I+I+), so we need gradient strengths of G1 = 2 and G2 = 1 to select this coherence pathway.

We might also want to have a third gradient applied after the final 90° pulse, so that we would "cover" the SQC period (Ha), the DQC period (Ha — Hb), and the SQC period (Hb) just before the FID (Fig. 10.34). We will have a huge phase twist in F2 if we simply delay starting the acquisition of the 12 FID, so we insert a spin echo between the third 90° pulse and the FID acquisition. In this case, one solution for gradient selection is G1 = 1, G2 = 1, and G3 = 3, selecting p = —1, p = —2, and p = +1, respectively. The accumulated "twist" at the end of each gradient is —1 (after G1), —3 (after G2), and 0 (after G3) for the selected pathway. Note that the alternative pathway p = —1, p = 2, and p = 1 (Fig. 10.34, dotted line) will lead to a twisted coherence and will not be observed:

This pathway is permitted in the phase cycled experiment, but the data from this pathway is lost in the gradient version. In this sense, gradients are sometimes too selective because they generally allow only one coherence level even when two or more pathways are equivalent in terms of the signal we want to see.

We have been annotating the pulse sequence to indicate what kind of evolution is going on at each stage: vH, — vC, J, and so on. Another way to diagram what is happening during the pulse sequence is to show the spherical operators at each stage for the coherence we are selecting with the gradients. This is shown in Figure 10.35 for the DQF-COSY sequence with gradients and spin echoes. The best way to do this is to start at the end with the FID: We always detect negative coherence, and as we are talking about a crosspeak at F1 = Ha, F2 = Hb, we will be detecting Hb SQC in the FID. Because there is no refocusing in the DQF-COSY, we will start the FID with antiphase Hb coherence: I— io with p = —1. Moving backward, the 180° pulse will always convert I— to I+ and vice versa, so before the final 180° pulse we have I+ Io with p = +1. Moving farther back in time, we encounter a 90° pulse. Many things happen to SQC with a 90° pulse (Fig. 10.29): The coherence order splits into three parts: it stays the same, goes to zero, or reverses sign. It is up to the gradients (or phase cycle) to select which of these three pathways is selected. A product operator with two operators such as I+ Io can come from many different levels: I+ can come from I+, or I— by a 90° pulse, and Io can come from I+ or I—. Taken together,

Figure 10.35

the coherence level before the 90o pulse could be -2, -1,0, +1, or +2! Fortunately, we know what coherence level we want to select for a DQF-COSY experiment: the coherence pathway diagram shows p = +2 at this point, and any other level will not survive the gradients. So we know that before the 90o pulse, we have I+1+ with p = 2. From I+ the 90o pulse creates I+, I£, and I-, but the gradient ratio selects only I+. Likewise, from I+ we select Io after the 90o pulse. It is important to keep in mind the big picture: DQF-COSY accomplishes coherence transfer from Ha antiphase SQC to Hb antiphase SQC by going through the obligatory intermediate state of {Ha, Hb} DQC.

Moving back further, we come to another 180o pulse at the center of the second spin echo. This is easy: I+ comes from I- and I+ comes from I-, so we have I-Ia- before the 180o pulse and p = -2. Before this we have another 90o pulse, the one that converts Ha antiphase SQC into {Ha, Hb} DQC. I- has to come from I£ because the starting Ha SQC is antiphase with respect to Hb, but I- can come from either I- or I+, both of which are Ha SQC. To decide which one, we look at the coherence pathway diagram (p = -1) or we look further back to the evolution delay t1, where we have to have positive coherence order for Ha SQC. This is the rule: negative coherence order during t2 and positive coherence order during t1 for the "echo" pathway. Because there is a 180o pulse between our state and the t1 period, we have to have negative coherence order here: I-I£. We can complete the sequence by moving back before the 180o pulse: I+Ib (Ib but we drop the "phase factor" of -1) and then we know that during t1 we have to get from in-phase Ha SQC to antiphase Ha SQC (i.e., selecting the sin(nJt1) term) so that we can move back to I+ just after the first (preparation) pulse. The whole concept here is very different from our analysis using Cartesian (Ix, Iy, etc.) operators: We ignore the phase factors that result from evolution and we focus on the coherence order only, letting the gradients do the work of choosing the pathway we desire. As an added bonus, we can just add up the superscripts of the spherical operators at each step to obtain the coherence order: for example, I-Ib has p = -1 + 0 = -1. We are definitely focusing on the big picture here, and we have come a long way from the laborious and meticulous analysis of vector rotations.

In Chapter 9, we saw that phase sensitive 2D NMR requires that we change the phase of the coherence observed in 11 (Fig. 9.47) using either the States technique (analogous

Figure 10.36

to simultaneous sampling of real and imaginary channels in t2) or the TPPI method (analogous to alternate sampling in t2). In either case, the phase of the ti FID is changed by changing the phase of the preparation pulse, the pulse just before the t1 delay, for each new FID in the 2D acquisition. With the advent of coherence pathway selection by gradients, it was discovered that the same phase shifting can be obtained by merely changing the sign of the gradient that selects coherence order during t1. If we select positive coherence order during t1 (Fig. 10.36, top, and solid lines), we have the so-called "echo" pathway: positive coherence order during t1 and negative coherence order during t2. The term "echo" refers to the reversal of sign of coherence order that occurs in the second half of a spin echo. If we select negative coherence order during t1 (Fig. 10.36, center, and dotted lines), we have the "antiecho" pathway: negative coherence order during both t1 and 12. The method ("echo-antiecho") is similar to the States method—for each t1 value, two FIDs are acquired: one selecting the echo pathway (G1 = 1) and one selecting the antiecho pathway (G1 = -1). The t1 delay is then incremented by 2 At1 = 1lsw(F1) and the process is repeated, acquiring two more FIDs.

The data are processed in a different way, combining the echo and antiecho FIDs for each t1 value to regenerate the real and imaginary (cosine modulated and sine modulated) FIDs. Then the data is processed just like States data. How this is done can be seen if we examine the phase factors that result from evolution during t1

I+ ^ I+Ibe-iQat1 sin(nJt1) echo I- ^ I-ioe!^at1 sin(nJt1) antiecho

The same phase factors attach themselves to the final spin state observed at the start of the FID:

I-I°e"!'"at1sin(nJt1) echo I-ioe!^at1sin(nJt1) antiecho

Now consider the signal recorded during t2, ignoring the /-coupling evolution for simplicity:

E(t1 ,t2) = e~iQat1 eiQbt2 echo FID A(t1 ,t2) = eiQat1 eiQbt2 antiecho FID

The chemical-shift evolution during the FID is taken care of by the exponential term in ^b t2, with a positive exponential because it is the I- that is evolving. In this complex arithmetic, the real part corresponds to the real FID in t2 (Mx component in the rotating frame) and the imaginary part is the imaginary FID in t2 (My component). We can substitute sines and cosines for the imaginary exponentials as

E(t1; t2) = [cos(^at1) - i sin(^atO][cos(^bt2) + i sin(^bt2)] echo FID A(t1; t2) = [cos(^at1) + i sin(^at1)][cos(^bt2) + i sin(^bt2)] antiecho FID

0 0