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As was observed in the previous exercise, nuclei precess at a very high frequency, typically tens or hundreds of MHz.
For the ^{1}H 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.

- Select the individual cos(
*a**t*) and cos(*b**t*) functions and observe the pure signals. - 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. - Overlap the cos(
*a**t*) cos(*b**t*) and cos[(*a*+*b*)*t*] functions and observe the high frequency component of the product. - Overlap the cos(
*a**t*) cos(*b**t*) and cos[(*a*-*b*)*t*] functions and observe the low frequency component of the product.

For a single Larmor frequency, the signal from the detector simply equals cos(ν *t*). This signal is multiplied by cos(ν_{o} *t*),
where ν_{0} 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(ν_{o} *t*) products a signal with frequencies ν + ν_{o} and
ν - ν_{o}. A low-pass filter removes the high frequency component, leaving only the low frequency signal *f* = ν - ν_{o}.
Whereas ν_{o} 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 ν_{o} 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.

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 ^{1}H 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 = (ν_{TMS} - ν)/ν_{o} = (*f*_{TMS} - *f*)/ν_{o}

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 10^{6} 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, ^{1}H chemical shifts lie between 0 and 10 ppm. Highly *deshielded* protons can lie above 10 ppm, but rarely about 15 ppm.

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.)

Answer the following questions:

- If ν < ν
_{o}, 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 ν
_{o}= 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.

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Bloch Equations

Bulk Magnetization

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