## Chemical Shift Evolution

We can consider the detection coil as lying along the y' axis of the rotating frame and recording a voltage proportional to My, the y' component of net magnetization. Of course, the coil is not rotating at hundreds of megahertz, but the electronics involved in detecting the FID signal and converting it to an audio signal are equivalent to placing the coil at a stationary position in the rotating frame of reference. If we start the experiment with a 90° RF pulse that places the B\ field along the y' axis, the net magnetization will be rotated to the x! axis and will start to precess in a counterclockwise direction at a rate corresponding to its resonance offset Av (or ^ in radians per second) relative to the center of the spectral window (Fig. 6.10). This motion induces an RF signal in the probe coil, which corresponds to a cosine function (Mx = +Mo, 0, —Mo, 0, etc., as the vector rotates) for the real audio signal (Mx) and a sine function (My = 0, +Mo, 0, —Mo, etc.) for the imaginary audio signal (My ). Fourier transformation of this signal leads to a peak in the spectrum with the normal absorptive lineshape.

Things get interesting if we insert a delay between the end of the 90° pulse and the beginning of the FID. Because of the delay, the net magnetization recorded in the FID will start at a different place in the x-y' plane due to precession during the delay. This motion is called chemical shift evolution. If the delay is just long enough to allow the net magnetization to precess from the x axis to the y' axis, the real FID signal will be a y

negative sine (-sin) function (Mx = 0, —Mo, 0, +Mo, etc.) and the peak in the resulting spectrum will have a dispersive lineshape in the absence of phase correction. A delay that is twice as long will allow the magnetization to precess to the — X axis, which will give a real FID that is a negative cosine (—cos) function (Mx = —Mo, 0, Mo, 0, etc.), leading to a negative or upside-down absorptive line. In this case, we can say that the reference axis is +x, so that magnetization starting on the +x axis at the beginning of the FID will give a positive absorptive peak, and if we start with the spin state — Ix we will get an upside-down peak.

The exact amount of rotation that occurs during the delay can be calculated in degrees as 360(vo — vr)t, where vo is the resonant frequency in hertz (Larmor frequency) in the laboratory frame, vr is the reference frequency in hertz (frequency of rotation of the axes in the rotating frame, frequency at the center of the spectral window), and t is the length of time of the delay in seconds. If the peak is in the upfield half of the spectral window, vo < vr and the rotating-frame frequency (vo — vr) will be negative. The total rotation will be negative, indicating a clockwise rotation looking down from the +z axis. If the peak is in the downfield half, vo > vr and the rotating-frame frequency (or resonance offset) will be positive. Total rotation in the above equation will be positive and the rotation will be counterclockwise. For example, if the NMR peak is 75 Hz downfield of the center of the spectral window (Av = 75 Hz), to get a rotation of 90° we would have to insert a delay t such that

360 (75 Hz)t = 90; t = (90/360)/75 = 0.25/75 Hz = 0.003333 s = 3.333 ms

Note that as Hz = s—1, when we divide by hertz we get seconds. With this delay, the net magnetization will rotate from the x axis to the yf axis, and the peak in the spectrum will be dispersive (reference axis = x').

6.6 SCALAR (J) COUPLING EVOLUTION

Now consider a two-spin system which is scalar (J) coupled, such as the 1H-13C pair in chloroform (CHQ3). The 13 C nuclei have two different resonant frequencies depending on whether the attached 1H nucleus is in the a or the 5 spin state. In the absence of proton

decoupling, the 13C spectrum will show a doublet with two peaks separated by the coupling constant J. The population of 13C nuclei in the sample can be divided into two parts. One half of the 13C nuclei are attached to a 1H nucleus in the a state, and the magnetization of these 13 C spins can be summed to give one net magnetization vector with a precession rate Av + J 12 (in hertz). The remaining half of the 13C nuclei are attached to a 1H nucleus in the p state, and they add up to form another net magnetization vector that precesses at a rate Av — J/2. Of course, there are slightly more 13C nuclei in the H = a group, but the difference is so small as to be insignificant in this analysis. To describe the motion of these two net magnetization vectors, it is convenient to choose the rotation rate of the rotating frame of reference to be vo, the chemical shift position of the 13C. This is the equivalent of placing the center of the spectral window exactly between the two components of the 13 C doublet in the spectrum, so that Av = 0 ("on resonance"). In this case, rotating-frame frequencies for the two components of the 13 C doublet are +J/2 for the H = a peak and —J/2 for the H = p peak.

Consider the sequence shown in Figure 6.11, which consists of a 90o 13 C pulse followed by a variable time delay before the start of acquisition of the (13C) FID. The vector diagram shows the X-y' plane, viewed from the +z axis (i.e., from above). At equilibrium, both net magnetization vectors will lie along the +z axis. A 90o pulse on the — X axis of the rotating frame will rotate both vectors to the +yr axis. Both magnetization vectors will precess, but the "P" vector will rotate with a velocity of —J/2 Hz (in a clockwise direction toward the X axis) and the "a" vector will rotate with a velocity of +J/2 Hz (in a counterclockwise direction toward the — X axis). After a delay of time 1/(2J) from the end of the 90o pulse, the two vectors will be opposite to each other on the +X and — X axes. This state means that the first half of the 13 C nuclei, which are attached to protons in the a state, give rise to a net magnetization vector that is 180o out of phase with the vector resulting from the other half, which are attached to protons in the p state. This condition is called antiphase magnetization, and it is crucial to many NMR experiments. We will see in the next chapter that this very special antiphase state is a prerequisite for magnetization transfer, the process of making the net magnetization "jump" from the 13C nucleus to the attached 1H. Collecting an FID beginning with this antiphase state would yield a spectrum in which both of the components of the 13 C doublet are dispersive, but one component peak is opposite in phase relative to the other (up-down vs. down-up). A 90o phase correction of this spectrum (i.e., changing the phase reference from yf to — X) would yield a spectrum in which the H = a component is positive absorptive ("up") and the H = j component is negative absorptive ("down") (Fig. 6.11, center spectrum). A further delay of 1/(27), for a total delay of 1/J, causes the two vectors to meet again along the —y' axis. This state is called "in-phase" because the two components have the same phase. Starting the FID at this point would yield an upside-down (negative) absorptive doublet using the original phase reference (y' axis). A total delay time of 2/J would allow both magnetization vectors to precess in opposite directions a full rotation, meeting back on the y' axis. This in-phase state would lead to a normal positive absorptive doublet in the spectrum.

It is important to recognize the difference between the terms absorptive and dispersive on the one hand, and in-phase and antiphase on the other hand. An antiphase doublet is sometimes confused with a dispersive peak because both have "up" and "down" components. Absorptive and dispersive lineshapes are characteristic of a single resonant frequency or "line" in a spectrum, and they can be interchanged by a 90° zero-order phase correction (i.e., changing the reference axis by 90°). In the vector model, using they' axis as a phase reference, absorptive and dispersive lineshapes correspond to net magnetization on the y and X axes, respectively, at the start of the FID. Thus, they differ only in the phase of the NMR signal resulting from a single magnetization vector. In-phase and antiphase states refer to the relative phase of the two components of a J-coupled doublet system (Fig. 6.12). The antiphase state is one in which the two magnetization vectors of a doublet, which correspond to the two lines in the spectrum (in the above 13 C example H = a and H = j), are directly opposite to each other in the X-y' plane. The in-phase state is one in which the two component vectors are aligned. In the spectrum, an antiphase doublet cannot be phase corrected to look like a normal doublet; if positive absorptive peak shape is achieved for one component, the other will be negative absorptive (upside down). It is quite possible to have a doublet that is in-phase absorptive, in-phase dispersive, antiphase absorptive, or antiphase dispersive (Fig. 6.12).

Evolution (rotation of net magnetization in the x'-y' plane) occurs during delays, and the direction and speed of motion in the x-y plane depend on the resonant frequency of the NMR line relative to the reference frequency (vo — ^r). In general, when the NMR peak is not on-resonance, there are two kinds of evolution. We think of the chemical shift as the frequency of the whole resonance or peak due to a nucleus or group of equivalent nuclei,

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