The Effect Of Rf Pulses On Product Operators

The effect of pulses is very simple: each individual operator is acted on by the pulse, and replaced by the result of that rotation about the B1 vector. For example, consider the effect of a 90° 1H pulse on the x axis:

These are exactly the same as the vector rotations, and you should draw a small set of coordinate axes in the margin of your paper to figure out these rotations as you work with product operators (Fig. 7.12). Note that 13C net magnetization is not affected by a 1H pulse (Sy ^ Sy). The effect of a 90° 13C pulse on the y axis is likewise the same as the vector model predicts:

180° pulses lead to inversion (z ^ —z and vice versa) or refocusing (x ^ —x, y ^ —y) as long as they are applied on an axis 90° from the axis of the starting magnetization. Any pulse applied on the same axis as the net magnetization has no effect. A 180° 13C pulse on the y axis brings about the following rotations (or "non-rotations"):

Figure 7.12

For two-spin operators, just figure out the effect of the pulse on each of the operators and replace the each starting operator with the result of the rotation. For example, for a 180° 13 C pulse on the x axis

2IySz ^ 2Iy(-Sz) = -2IySz (only Sz is affected by the13C pulse)

For a 90o !H pulse on the y axis

2IySz i 2IySz 2IxSx i 2(-Iz)Sx = -2SxIz 2SVIz i 2SVI

Dy Iz

V Ix

2IxSz i-2IzSz

Note that the observable operator (the operator representing coherence or net magnetization in the x-y plane) is always written first in the product. Also, we see above some examples where both operators are in the x-y plane, or both operators are on the z axis! These products represent nonobservable states which are nonetheless very important in NMR experiments. The only observable product operators are those with only one operator in the x-y plane ("single-quantum transitions").

For a homonuclear system (Ha and Hb with coupling constant J) we can do the same kinds of tricks:

Coupled protons also evolve into antiphase and refocus during delays:

2ix Ia

For example, — 2Ibis read as "proton Hb net magnetization on the —x axis, antiphase with respect to its coupling partner Ha." Again, we can think of the part as a multiplier, equal to +1/2 or —1/2 depending on the spin state (a or ft) of each individual Ha nucleus. The observable magnetization is on Hb, and as it rotates around the axes (y ^ —x ^ —y ^ x ^ y) it alternates between in-phase and antiphase with respect to its coupling partner Ha. The effect of pulses is similar to what we saw for a 1H-13C pair, except that with hard (high power, short duration, nonselective) 1H pulses we cannot deliver a rotation to one of the spins and not the other: all hard pulses affect both Ha and Hb. For example, for a 180° 1H pulse on the y axis:

ix ^ - ix - ib ^ ib 21»ib ^ 2iy(-ib) = -2iyib 2ibiz ^ 2(-ix)(-iz)=2ibiz 2ibiz ^ 2(ibv)(-iz)=-2ibiz

Likewise, a 90° 1H pulse on the x axis is viewed as a simultaneous pulse on Ha and on Hb:

iy ^ iz ib ^ - ib 2iyib ^ 2(iz)(-iy) = -2iyiz 2ib iz ^ 2(iX )(-iy) = -2iX iy - 2ib iz ^ 2iyib

In the two examples above on the right, observable magnetization (antiphase coherence) is transferred by the 90° 1H pulse from Hb to Ha (top) and from Ha to Hb (bottom). This is a key process in all advanced NMR experiments that depend on J couplings. The role of the two operators is reversed as the operator in the x-y plane (the observable net magnetization) rotates to the z axis and the operator on the z axis (the multiplier that represents microscopic z magnetization) rotates to the x-y plane. After the rotations, we reverse the order of the two operators because we always write the observable operator first in the product.


Now that we have the precise tools of product operator notation, we can look at coherence transfer in detail. In the NOE difference experiment (Chapter 5), we saw an example of transfer of z magnetization from one nucleus to another via the through-space interaction of cross-relaxation. It is also possible to transfer magnetization in the x-y plane (observable magnetization or coherence) via the through-bond J-coupling interaction. The simplest form of the INEPT pulse sequence is shown in Figure 7.13 for transfer of 1H coherence to 13C coherence. Consider a simple case of a single proton bonded to a 13C (e.g., benzene) and assume for the sake of simplicity that both the 1H and the 13 C frequencies are exactly on-resonance. The 90° 1H pulse (B1 vector on -x) rotates the proton z magnetization onto the yf axis of the rotating frame of reference (Fig. 7.14). The downfield component of the proton doublet, which arises from protons attached to 13 C nuclei in the a state, begins to rotate counterclockwise in the x!-y' plane toward the -x! axis with angular frequency J/2 Hz (or nJ

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