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# NMR Spectroscopy: The Rotating Frame

## Trigonometry

As was observed in the previous exercise, nuclei precess at a very high frequency, typically tens or hundreds of MHz. For the 1H nucleus, however, that range of Larmor frequencies for nuclei in different chemical environments only varies by a few thousand Hz. Although the detectors directly measure the real-world precession (hundreds of MHz), the signal is transformed prior to digitization to better permit measurement of these small differences in frequency.

In order to understand this transformation, consider two pure cosine signals, one with frequency a and one with frequency b. If one multiplies these two signals, the product is a wave form that contains two frequency components, a + b and a - b. The relevant trigonometric relationship is

cos(a t) cos(b t) = ½ ( cos[(a + b) t] + cos[(a - b) t] )

This mathematical relationship is illustrated in the graph below, where a = 20 Hz = 125.66 rad/sec and b = 15 Hz = 94.25 rad/sec.

1. Select the individual cos(a t) and cos(b t) functions and observe the pure signals.
2. Select the cos(a t) cos(b t) product and observe the superposition of the signals. Note the beat pattern and the high and low frequency components.
3. Overlap the cos(a t) cos(b t) and cos[(a + b) t] functions and observe the high frequency component of the product.
4. Overlap the cos(a t) cos(b t) and cos[(a - b) t] functions and observe the low frequency component of the product.

cos(a t)               cos(b t)               cos(a t) cos(b t)               cos[(a+b) t]               cos[(a-b) t]

For a single Larmor frequency, the signal from the detector simply equals cos(ν t). This signal is multiplied by cos(νo t), where ν0 is the spectrometer frequency, which is usually set to the center of the spectral region of interest.

The product of the detector signal, cos(ν t), and cos(νo t) products a signal with frequencies ν + νo and ν - νo. A low-pass filter removes the high frequency component, leaving only the low frequency signal f = ν - νo. Whereas νo is typically hundreds of MHz, f is usually no more than a few kHz, but can be positive or negative.

The experimental frequency ν is in the real frame of reference, that is, it is the actual frequency of precession.

The processed frequency f is in the rotating frame of reference, that is, it is the frequency shift from the spectrometer frequency. It is the frequency one would expect to see if the detectors were rotating about the z axis at a frequency νo while the bulk magnetization is precessing at frequency ν.

The reason why the rotating frame is more useful than the real frame is that chemists are interested in changes or shifts in frequency rather than the absolute precession frequency. The rotating frame makes measuring small frequency shifts more precise and more accurate.

## Chemical Shift

The convention in NMR spectroscopy is to report frequencies as a shift (or difference) from a reference compound rather than f, ν or Δν. For 1H measurements, the most common reference compound is tetramethylsilane (TMS), which is defined to have a chemical shift of δ = 0. The chemical shift is defined as

δ = (νTMS - ν)/νo = (fTMS - f)/νo

According to this definition, δ is a tiny fraction that is usually reported in ppm. One ppm is one part per million or 10-6. Thus one multiplies the fraction obtained from the above equation by 106 to obtain δ in ppm. The reference frequency, 0 ppm, is usually the highest frequency one encounters, and this value is plotted at the far right of a spectrum. The chemical shift increases as one moves to the left, and the increasing chemical shift corresponds with decreasing (downfield) frequencies for the sample. For most organic compounds, 1H chemical shifts lie between 0 and 10 ppm. Highly deshielded protons can lie above 10 ppm, but rarely about 15 ppm.

## Exercise

The signal from the x detector coil is shown in the graph below. Select the reference compound TMS or one of the unknown samples (A, B, C, or D). Then start the animation and measure the frequency. (Hold down the left mouse button to show the position of the cursor on the graph. Measure the period of the signal; the frequency in Hz is f = 1/period.)

• If ν < νo, a negative frequency is obtained for f. How does precession of a negative f differ from that of a positive f?
• Can one tell if f is positive or negative solely on the basis of the detector signal as shown in the graph?
• The spectrometer frequency is νo = 200. MHz. What are the frequencies of the samples in the rotating frame?
• What are the chemical shifts (in ppm downfield from TMS) for each unknown?
• The audible frequency range for humans is 20 Hz to 20,000 Hz. The higher the frequency, the higher the pitch.
If the NMR signal in the rotating frame were played over speakers, would the signal be audible?

The time scale for the animation is not the same as the actual time scale. The graph display time is in milliseconds. The simulation has been slowed to facilitate visualization.

 Samples:     A     B     C     D     TMS

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Bloch Equations
Bulk Magnetization

RotatingFrame.html version 1.1