# NMR Spectroscopy

## Absorption and Dispersion Signals

The FID comprises signals from the detectors positioned on both the *x* and *y* axes. These two sets of signals are packaged as a complex number, with the real component being the signal from the *x* detector and the imaginary component being the signal from the *y* detector. The square root of -1 is represented by *i*.

*S* = *S _{x}* +

*i*

*S*

_{y}Applying the FFT to the FID produces a spectrum that is also complex. One component of the spectrum is the *absorption* spectrum and the other component is the *dispersion* spectrum. The absorption spectrum shows peaks at the Larmor frequencies for the various spins. The width of the peak is determined by the *effective* *T _{2}* value:

*T**.

_{2}*w*=

*T**

_{2}*T*

_{2}*w*

_{inst}The width of the peak ( *w* ) is the full width of the peak measured at a height half way between the baseline and the top of the peak (*w* = FWHM = full width at half maximum). The peak width is the sum of the natural linewidth ( determined by *T _{2}* ) and broadening from instrumental effects (characterized by

*w*).

_{inst}*w*accounts for instrumental factors that can cause the various nuclei of a given type to precess at slightly different frequencies. Variations in the strength of

_{inst}**B**across the sample is one source of instrumental broadening.

The dispersion spectrum is the derivative of the absorption spectrum, showing positive values on the left side of the peak and negative values on the right side. At the Larmor frequency, the absorption spectrum is at a maximum while the dispersion signal equals zero.

Both the absorption and dispersion spectra can be visualized in the graph below. Enter a value for *T _{2}** between 0.01 and 1 sec. Select the absorption and/or dispersion spectrum and observe the shapes of the peaks.

## The Phase Angle

The signal from the *S _{x}* detector is typically displayed as the FID. Mathematically, for a single Larmor frequency, the FID is a cosine function multiplied by an exponential decay. If the acquisition of the FID begins with the bulk magnetization perfectly aligned along the

*y*axis at

*t*= 0, then the

*S*signal begins at a maximum. The mathematical expression is

_{x}*S _{x}* = cos( 2 π

*f*

*t*) exp( -

*t*/

*T** )

_{2}The Fourier transform of this function produces a pure absorption spectrum as the real component of the NMR spectrum. The imaginary component of the spectrum is the pure dispersion spectrum.

Suppose the bulk magnetization does not begin perfectly aligned along the *y* axis at *t* = 0 when the FID is acquired. In this case, the signal from the *x* detector now includes a phase angle, θ.

*S _{x}* = cos( 2 π

*f*

*t*+ θ ) exp( -

*t*/

*T** )

_{2}For a nonzero phase angle, the Fourier transform produces a real component that is a mix of the absorption and dispersion spectra.

The experiment below illustrates this effect. The simulation shows the position of the bulk magnetization immediately after the 90°_{x} pulse, when **M** lies exclusively in the *xy* plane. The simulation shows the **M** vector as viewed along the *z* axis, which allows one to easily observe the phase angle. The phase angle θ is the angle between the **M** vector (represented by the green arrow) and the *y* axis.

There are several instrumental and experimental issues that can cause θ to differ from zero. For example, during the 90°_{x} pulse, as **M** developes nonzero *M _{x}* and

*M*components,

_{y}**M**begins to precess about the

*z*axis in addition to the precession about the

*x*axis caused y the 90°

_{x}pulse. This effect is one factor that leads to θ ≠ 0.

### Experimental Instructions

- Specify a phase angle and observe the initial position of the bulk magnetization.

Try different values for θ and observe the starting alignment of**M**and how the value of θ controls the position of the green arrow representing**M**. - Run the simulation and collect the FID. Observe where in the cosine cycle the FID signal begins and how that starting point depends upon the phase angle. (Only
*S*is shown in the FID.)_{x} - After the FID has been recorded, plot the spectrum, which displays only the real component of the spectrum. Observe how the absorption and dispersion spectra are mixed and how the extent of mixing depends upon θ.
- Vary the
*phase correction*until the displayed spectrum is the pure absorption spectrum (symmetric, positive peak).

This type of phase correction is a routine part of processing NMR data.

The phase correction consists of choosing an angle φ and multiplying the spectrum by exp( φ *i* ). This multiplication rotates the spectrum in the complex plane. By choosing φ = - θ , the absorption spectrum is rotated onto the real axis, which is the desired spectral representation.

In an actual NMR experiment, precession occurs on the millisecond or microsecond timescale. In order to facilitate visualization in this simulation, the Larmor frequency is 100 times slower than would actually be the case.

Answer the following questions by carefully observing the simulation and adjusting the phase correction.

- For what value of θ does one obtain the pure absorption spectrum?
- For what value of θ does one obtain the pure dispersion spectrum?
- For what value of θ does the spectrum contain a perfect, upside-down peak?

Why does this occur? - When the phase correction has been applied to obtain the absorption spectrum, how does the value of the phase correction φ compare with the value of θ?

*PhaseAngle.html version 2.0*

*© Copyright 2013, 2014, 2023 David N. Blauch*