NMR Spectroscopy: The Rotating Frame

Trigonometry

Magnetic moments for atomic nuclei precess at a very high frequency, typically tens to thousands of MHz, depending upon the magnitude of the magnetic field used in the NMR experiment and the nucleus of interest. For the ¹H nucleus, however, the range of Larmor frequencies for ¹H atoms in different chemical environments only varies by a few thousand Hz. Although the detectors directly measure the real-world precession (hundreds or thousands 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 = 125.66 rad/sec This frequency is 20 Hz.
b = 94.25 rad/sec This frequency is 15 Hz.

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.

The Rotating Frame of Reference

For a single Larmor frequency, the signal from the x detector simply equals cos( ν t ). This signal is multiplied by cos( ν_o t ), where ν_o 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 reference signal cos( ν_o t ) produces 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 of the nuclear precession frequency from the spectrometer frequency. f 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 accurate.

Chemical Shift

The convention in NMR spectroscopy is to report frequencies as a shift (or difference) from a reference compound. 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 =

ν - ν_TMS ν_o

f - f_TMS 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 10⁶ to obtain δ in ppm. The reference frequency, 0 ppm, is usually the lowest frequency one encounters in ¹H NMR, and this value is plotted at the far right of a spectrum. The chemical shift increases as one moves from right to left, and the increasing chemical shift corresponds with increasing (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 above 15 ppm.

Simulation

In the simulation below, the signal from the x detector is graphed in the rotating frame.

There is a challenge with interpreting data in the rotating frame. Because the Larmor frequency ( ν ) could be larger or smaller than the spectrometer frequency ( ν_o ), the frequency in the rotating frame, f = ν - ν_o , can be positive or negative. The cosine function is symmetric, making it impossible to distinguish between positive and negative f. In early versions of FT-NMR instruments, the spectrometer frequency was set sufficiently low that f was always positive. Modern FT-NMR instruments use quadrature detections, which allows one to determine the sign of f.

In this simulation, ν is always larger than ν_o , so that f is always positive.

Select the reference compound TMS or one of the unknown samples (A, B, C, or D). Then start the animation and measure the frequency. (Position the cursor on the graph and left-click to show the position of the cursor.) Measure the period of the cosine wave. The frequency in Hz is the reciprocal of the period.

Answer the following questions:

The spectrometer frequency is 200.000000 MHz. What are the frequencies of the samples in the real 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 displays time in milliseconds. The simulation has been slowed to facilitate visualization.