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# NMR Spectroscopy: The NMR Spectrum

The FID shows the time-domain behavior of the spin system. The use of quadrature detection provides signals from both the x and y detection coils: Sx and Sy. These two signals are combined into a single complex number that preserves the phase relation between the two signals:

S = Sx + i Sy

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

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 provides the peaks normally associated with 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 -fN to fN, where fN = (2 Δt)-1 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.

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

δ = 106TMS - ν)/νo = 106 (fTMS - f )/νo

## Exercise 1

The following simulations 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 the 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 40 MHz spectrometer.
• Carefully examine the curves for the Sx (red) and Sy (blue) detectors. Which leads which for a positive frequency? Which leads which for a negative frequency?
• How does T1 affect the shape and position of a peak?
• How does T2 affect the shape and position of a peak?
 Samples: A       B       C TMS (T1 = 0.2 sec, T2 = 0.02 sec) TMS (T1 = 0.2 sec, T2 = 0.20 sec) TMS (T1 = 2.0 sec, T2 = 0.20 sec)

## Exercise 2

This simulation contains multiple nuclei with different Larmor frequencies. The bulk magnetization attributable to each type of nuclei is shown in the animation at the left. The graph of the FID shows only Sx, which is the sum of the signals from all the individual nuclei.

The most upfield (highest frequency) peak is for TMS. The instrument is a 40 MHz spectrometer.

Run the simulation and answer the following questions.

• What are the chemical shifts of the various nuclei?
• In what ways does the FID look similar to those in Exercise 1? In what ways does it have different features? Explain the differences.
• Why do the various peaks have different widths?
• Why do the various peaks have different heights? (Identify two factors that affect the relative heights.)

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Phase Angle
Single-Pulse Experiment