NMR Spectroscopy

The NMR Spectrum

The free induction decay (FID) shows the time-domain behavior of the spin system as it evolves to the equilibrium state. The use of quadrature detection provides signals from both the x and y detection coils: S_x and S_y. These two signals are combined into a single complex number that preserves the phase relation between the two signals:

S   =   S_x   +   i S_y

The symbol i represents the square root of -1. S_x is the real component of the signal S, and S_y is the imaginary component. This technique provides an easy way to package the components of a two-dimensional vector into a single variable.

The time-domain signals S_x and S_y show oscillations for each of the different types of nuclei with magnetic moments. The frequency of the oscillation from a particular nucleus corresponds with the Larmor frequency for that nucleus. While the time-domain signal is measured experimentally, users generally wish to know the frequency of the oscillations.

The frequency-domain response is obtained by taking the Fourier transform of the FID using the Fast Fourier Transform algorithm (FFT) to obtain the NMR spectrum. Like the FID itself, the spectrum is complex. In the simplest case, where the bulk magnetization M starts perfectly aligned with the y axis at t = 0, the real component of the spectrum contains the absorption signal, which is the signal normally used for a spectrum. The imaginary component of the spectrum contains the dispersion signal, which is the derivative of the absorption signal.

The frequency range extends from -f_N to f_N, where f_N is the Nyquist frequency.

The Nyquist frequency is the highest frequency that can be correctly extracted from the time-domain data. All components with higher frequencies appear in the spectrum at the wrong frequencies. Δt is the time interval between data points in the FID. To use the FFT algorithm, the data set for the FID must contain equally spaced points and the number of points must be a power of 2. In ¹H NMR spectroscopy, the FID might, for example, be acquired over 4.00 sec and contain 2¹⁴ = 16384 points. In this example, Δt = 4.00 sec / 16384 = 0.00024414 sec . The Nyquist frequency is therefore f_N = 2048 Hz.

Recall that the measured frequency is in the rotating frame: f = ν - ν_o. A negative frequency f simply means a component is precessing at a frequency less than the spectrometer frequency ν_o . Positive frequencies are for components with frequencies greater than ν_o. In either event, one readily obtains the chemical shift from

TMS refers to tetramethylsilane, Si(CH₃)₄, which is a commonly used reference compound in ¹H NMR spectroscopy. The factor of 10⁶ creates units of ppm for δ .

Nuclei existing in different chemical environments experience different magnetic fields. For example, in ¹H NMR spectroscopy, all protons experience the same magnetic field from the instrument magnet, B. However, the electrons around a particular proton produce a magnetic field of their own that opposes the magnetic field B. Thus the net magnetic field a given proton experiences will be less than B. For this reason, the proton is said to be shielded. The greater the shielding, the lower the magnetic field the proton experiences and thus the lower the Larmor frequency. Protons with a lower degree of shielding are said to be deshielded and have a larger Larmor frequency.

The protons in TMS are highly shielded and thus have a relatively low Larmor frequency. For this reason, convention places the TMS chemical shift at δ = 0. Almost all other protons in the system have a larger Larmor frequency and thus a positive chemical shift. That is, most protons are deshielded compared with the TMS protons. The convention in NMR spectroscopy is to plot spectra with the TMS peak on the far right and graph increasingly positive chemical shifts to the left.

The spectrometer frequency is usually in the MHz range. In the experiment below, ν_o = 100 MHz. On the other hand, the Larmor frequencies of the various protons in the sample usually differ a few kHz or less. Because relative differences in Larmor frequency are so small, the differences are reported as parts-per-million (ppm).

Experiment

The following simulation experiment offers several different samples, each of which contains a single type of nucleus with an unique Larmor frequency.

1. Select a sample and run the single-pulse experiment to obtain the FID.
   Note that both x and y detectors are used (quadrature detection).
2. Perform the Fourier transform and observe the spectrum (absorption data).
3. Use the results of the experiment to answer the following questions.

• For each component, determine its frequency f and its chemical shift δ in ppm downfield from TMS.
  The instrument is a 100. MHz spectrometer.
• Carefully examine the curves for the S_x (red) and S_y (blue) detectors.
  How do the detector signals differ for positive and negative frequencies?
  In the animation, how does the precession differ for positive and negative frequencies?
• How does T₂ affect the shape and position of a peak?

Notes

All nuclei relax with T₂ = 0.50 sec, except the indicated version of Sample C, for which T₂ = 0.05 sec.

The time scales for the 90°_x pulse and the FID are different and do not correspond with real NMR experiments. The real time for the 90°_x pulse is about 10 μsec. The duration of the FID (4.00 sec) is realistic, but the precession frequency in the animation is 100 times slower than the frequency shown in the spectrum. The time scale of the animation is much slower to allow better visualization of the behavior of the bulk magnetization.

The spectrum shows the precession frequency in the rotating frame. For a 10 ppm spectral window on a 100. MHz NMR spectrometer with the spectrometer frequency set to the exact middle of the spectral window, the frequencies ranging from f = +500 Hz to -500 Hz. Recall that large positive frequencies (deshielded nuclei) are plotted to the left, and large negative frequencies (shielded nuclei) are plotted to the right. Signals with large frequencies, and thus large chemical shifts, are said to be downfield.

TMS has the most upfield signal, which means it has the most negative f.