## I

7.5 TWO-SPIN OPERATORS: /-COUPLING EVOLUTION AND ANTIPHASE COHERENCE

For a system with two kinds of nuclei, the symbols I and S are used to represent the two nuclei. For example, in a 1H-13C system, the is usually represented as I and the 13C as S. The six simple spin states Ix, Iy, Iz and Sx, Sy, Sz can be represented, but we can also form products of these spin states to represent situations which cannot be described by the vector model. This is where the "product" of product operator comes from. Consider first the product of one nucleus in the X-y' plane with another nucleus on the z axis:

2IX Sz, 2Iy Sz: I (1H) magnetization in the x'-y' plane, antiphase with respect to the z orientation of the /-coupled spin S (13C)

2SXIz, 2SyIz: S (13C) magnetization in the xX-y' plane, antiphase with respect to the z orientation of the /-coupled spin I (1H)

The 2 is a normalization factor that will be explained in Chapter 10—it is needed any time we multiply two operators together. In the vector model, these product operators can each be represented by two vectors in the X-y' plane, 180° apart. For example, 2IX Sz represents a spin state where the half of the I nuclei that are coupled to an S nucleus in the a state add up to form a vector along the +X axis, whereas the other half of the I nuclei that are coupled to an S nucleus in the ¡5 state add up to form a vector along the — X axis (Fig. 7.7). We always put the a vector on the axis represented by the first part of the product: for example, for — 2IX Sz we put the 1H net magnetization (13C = a) vector on the —x axis and the 1H net magnetization (13C = ¡5) vector on the +x axis.

The chemical shift evolution of these product operators is obtained simply by plugging in the time evolution of the component single-nucleus operators. For example

2IxSz Tdelay 2{Ix cos^It + Iy sin^IT}{Sz} = 2IxSz cos^IT + 2IySz sin^IT.

This sequence of events can be represented schematically in a circle (Fig. 7.8) with the rotation rate (^I) in the center. This is simply the rotation of the two opposed (antiphase) vectors in the x'-y' plane at the frequency determined by the chemical shift of nucleus I. The Sz part just "goes along for the ride" because the antiphase relationship is retained throughout. Stand up with your arms outstretched at your sides: your right arm represents the 1H net magnetization vector (13C = a) on the +x axis, and your left arm represents the 1H net magnetization vector (13C = ¡¡) on the —x axis. The +y axis is in front of you and the —y axis is behind you. Now slowly turn your body counterclockwise (to your left), holding

your arms apart opposite each other. The "13C = a" vector (your right arm, corresponding to the downfield line of the 1H doublet) moves from the +x axis to the y axis, then the —x axis and then the —y axis, whereas the "13C = ¡" vector (your left arm, corresponding to the upfield line of the 1H doublet) moves from the —x axis to the —y axis, then the +x axis and then the +y axis, always opposite the "13C = a" vector. You are turning at a rate corresponding to QI, the chemical shift position of the proton resonance relative to the center of the spectral window. Of course, it requires some special tricks (decoupling, spin echo, etc.) to have a coupled nucleus affected only by its chemical shift and not by the J coupling, but it is easiest to understand these two effects separately.

Evolution under the influence of J coupling alone results in refocusing of antiphase magnetization, whereas in-phase magnetization evolves into antiphase:

Ix ^ 2Iy Sz ^ —Ix ^ — 2Iy Sz ^ Ix in-phase antiphase in-phase antiphase in-phase

Note that the operator in the x-y plane (1Hor I in this case) evolves, just like chemical shift evolution (Ix ^ Iy ^ —Ix ^ —Iy), a simple counterclockwise rotation in the x-y plane, but with each 90o rotation it alternates between in-phase (omitting the 2 and the Sz) and antiphase (including them). You can do some NMR calisthenics by first putting both arms forward in front of you (Ix , in-phase) and then moving them apart until they are at your sides sticking out (2Iy Sz, antiphase—your left arm is the 13 C = a component on the +y axis and your right arm is the 13 C = 5 component on the —y axis) and then moving them further around to meet in the back (—Ix, in-phase). You cannot go further without hurting yourself, but if you could move further your arms would cross and your right arm would point left and your left arm would point right (—2Iy Sz, antiphase in the opposite sense, with your left arm, 13C = a, on the right, the —y axis). Further rotation would bring your (broken) arms to the front (Ix, in-phase). This sequence can be represented in a circle (Fig. 7.9, left) with the rotation rate (nJ rad/s or J/2 Hz) in the center. If we start with Iy instead of Ix, we see the same progression of axes for the I spin (1H) going counterclockwise from Iy to -Ix to — Iy to Ix, but we start with in-phase on the y axis (Iy) and alternate in-phase and antiphase as we go around: Iy ^ — 2IxSz ^ — Iy ^ 2IxSz ^ Iy (Fig. 7.9, right).

We can also think about the spectrum that would be observed at each stage of this evolution ("J coupling evolution") if we started recording the FID at that point in time. For this purpose, we have to decide on a phase reference (receiver phase): let's use the +x axis as representing a positive absorptive peak in the spectrum. In other words, if a vector is on the +x axis at the start of the FID, it will give a peak in the spectrum that is positive and absorptive. Ix will give a nice positive absorptive peak for both components (Fig. 7.10) of

the doublet (1H-13C system with J coupling 1JCH ~ 150 Hz). 2Iy Sz will give a dispersive (up/down) peak for the 13 C = a component on the +y axis and an opposite dispersive peak (down/up) for the 13C = 5 component of the doublet on the —y axis. — Ix gives an upside-down (negative absorptive) doublet and —2Iy Sz gives the dispersive antiphase doublet in the opposite sense (down/up, up/down) to 2Iy Sz. If we use the +y axis as the phase reference, any vector on +y will give a positive absorptive peak, and any vector that leads +y by 90o (i.e., any vector on —x, which is 90o counterclockwise from +y) will be dispersive up/down. This gives absorptive phase for 2IySz and —2IySz (Fig. 7.10, bottom). A real life example of this evolution was shown for 13C labeled methyl iodide (13CH3I) in Chapter 6, Fig. 6.14. It is easier to think about the spectrum if we jump back and forth between the +x axis phase reference (for Ix and —Ix) and the +y axis reference (for 2Iy Sz and —2Iy Sz), avoiding the dispersive lineshape (Fig. 7.10, following the arrows).

We can also use the sine and cosine functions to describe any general rotation, not confined to the four "points of the compass":

t delay

> Ix cos(nJT) + 2IySz sin(nJT) (evolution into antiphase: Fig. 7.9, left)

t delay

Iy cos(nJT) — 2IxSz sin(nJT) (evolution into antiphase: Fig. 7.9, right)

t delay

Iy cos(nJT) — 2IxSz sin(nJT) (evolution into antiphase: Fig. 7.9, right)

## Post a comment