The Rotating Frame Of Reference

In a real sample, there will be more than one kind of nucleus of a given type. For example, a sample of ethanol has two different kinds of 13C nuclei: the CH3 carbons and the CH2OH carbons. Due to differences in the amount of nuclear shielding, these two kinds of 13C will have slightly different Larmor frequencies. For example, on a 300-MHz (7.05-T) spectrometer we might have 75.000000 MHz as the resonant frequency (vo) of the CH3 carbon and 75.003375 MHz for the resonant frequency of the CH2OH carbon. Because these differences are small (on the order of parts per million of the Larmor frequency), it is convenient to view the NMR experiment from a rotating frame of reference, so that the fundamental resonant frequency (near which all of the sample nuclei precess) is removed from consideration. In our example, all 13C nuclei have a resonant frequency very near to 75 MHz, so we are not really interested in this; it is the differences in resonant frequency, which are thousands of hertz at most, that are important. For example, the reference axes in the rotating frame, referred to as the X and y' axes, could be rotated counterclockwise (ccw) about the z axis at a frequency of 75.000000 MHz. In this case, the 13C nuclei of the CH3 group of ethanol precess at exactly the frequency of rotation of the X and y' axes, and the net magnetization vector for these nuclei after a pulse will appear to stand still in the x'-y' plane of the rotating frame of reference. The other 13C nuclei, corresponding to the CH2OH group of ethanol, will give rise to a net magnetization vector that appears to rotate slowly counterclockwise in the X-y' plane at a rate corresponding to the difference between its Larmor frequency and that of the reference frequency (75.003375 - 75.000000 = 0.003375 MHz = 3375 Hz). This rotation will be counterclockwise in this case because the Larmor frequency of the CH2OH group is greater than that of the CH3 group, but in other cases it could be clockwise (cw). The rotation frequency we choose for the rotating frame of reference establishes a reference frequency in the spectrum, which is just the center of the spectral window. This is also the frequency of the radio frequency (RF) pulses we apply to the sample and of the reference signal used in the detector of the NMR receiver. If a certain type of spin (e.g., CH3 of ethanol) has a chemical shift at the exact center of the spectral window, it is said to be "on resonance" and its magnetization vector will stand still in the X-y' plane. All other nuclei will rotate at a frequency and direction determined by their "offset": the chemical shift (in hertz or radians per second) relative to zero at the center of the spectral window.

Moving from the laboratory frame of reference to the rotating frame of reference is exactly analogous to the detection step in the NMR hardware. If the reference frequency is 75.000000 MHz, the detection step subtracts this frequency from the free induction decay (FID) frequency, which for the CH2OH carbon of ethanol is 75.003375 MHz. The resulting analog signal, 3375 Hz, is the audio signal that we digitize and record as the FID. After Fourier transformation, this leads to a single peak positioned 3375 Hz to the left of the center of the spectral window in the NMR spectrum.

Figure 6.1

For those of you who have studied physics, you may recognize that the rotating frame of reference is an accelerating frame of reference, which is a major no-no if we want the laws of physics to be preserved. A similar situation arises on the earth where all of us are in a rotating frame of reference at the surface of the earth. The laws of physics will not work in this accelerating frame of reference, so we have to invent fictitious forces called Coriolis forces to correct for the discrepancies. For example, it is the Coriolis force that makes storm systems rotate on the earth's surface. In the NMR world, the sins of the accelerating frame can be atoned for by inventing a fictitious magnetic field (or "pseudo-field") that is opposed to the applied magnetic field Bo (Fig. 6.1). This fictitious field has a strength of 2nvr/y, where vr is the reference frequency. The faster we spin the x'and y' axes in the rotating frame, the larger is the pseudofield required to maintain the laws of physics. If the spin in question is exactly on-resonance (2nvr/y = 2nvo/y = Bo), the fictitious field precisely cancels the Bo field and there is no magnetic field; hence, any net sample magnetization in the x'-y' plane remains stationary. If the spin is not on-resonance, the Bo field is not perfectly canceled and there remains a small residual field (Bres = Bo — 2nvr/y = 2nvo/y — 2nvr/y = 2n(vo — vr)/y), along either the +z or the —z axis, which is proportional in strength to the difference between the Larmor frequency (chemical shift) and the center of the spectral window. This residual field makes the net magnetization vector rotate in the X-y' plane the same way the Bo field makes the net magnetization rotate in the x-y plane of the laboratory frame:

Vo = yBo/2n, Av = Vo - vr = yBres/2n = y[2n(Vo - vr)/y]/2n

Note that we use Av to refer to the rotating-frame frequency (sometimes called the resonance offset). This is the difference between the Larmor frequency and the reference frequency: vo — vr. The above equation shows that the same physical law expressed in the equation on the left-hand side (precession rate is proportional to y and to Bo) is operating in the equation on the right-hand side (resonance offset is proportional to y and to Bres) in the rotating frame of reference, as long as we introduce the pseudofield. In the NMR spectrum, Av is the distance from the center of the spectral window to the NMR peak (Fig. 6.2), also represented as ^ in units of radians per second. If the peak is in the downfield half (left half) of the spectrum, the Larmor frequency is greater than the reference frequency (vo > vr) and we have a positive resonance offset (Av > 0). This corresponds to the motion of the net magnetization

Resonance offset

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