# NMR Spectroscopy

## Bulk Magnetization

Nuclei possessing a non-zero spin and thus a magnetic moment display Larmor precession when placed in a static magnetic field ( B ). The direction of B is selected as the z axis. Therefore |B| = Bz. In an experiment, the sample contains a vastly large number of nuclei. The magnetic moment of the entire sample, called the bulk magnetization ( M ), is the vector sum of the individual nuclear magnetic moments.

Just as an individual nuclear magnetic moment, μ, precesses about B, the bulk magnetization also precesses about B.

Unlike μ, which can never be perfectly aligned along B, the bulk magnetization can become perfectly aligned along B. That is, the bulk magnetization can become perfectly aligned along the z axis. This situation occurs when the μx and μy components of all the various nuclei become randomly distributed and perfectly offset each other.

Here are some comparison between μ and M that might be helpful.

The magnitude of μ never changes. Because of precession, the direction of μ is constantly changing. As a result, μx and μy are always oscillating.

The magnitude of M can change. At equilibrium, the bulk magnetization is oriented along the positive z axis and | M | = Mz = Meq. When the system is not at equilibrium, M can point in any direction. If M is not aligned with the z axis, then the bulk magnetization will precess around the z axis, with the result that Mx and My oscillate. When the system is not at equilibrium, the magnitude and direction of M change as the bulk magnetization evolves to the equilibrium state.

## Larmor Frequency

The rate of precession is related to the intrinsic properties of the nuclei ( γ ) and the strength of the static magnetic field ( B ).

ν = γ B

f = γ B / 2π

The frequency of precession ( ν ) has units of rad/sec. Often it is more convenient to use frequencies in Hz (that is, cycles per sec or revolutions per second). f is the frequency in Hz and

ν = 2 π f

It is possible to detect the precession of the bulk magnetization by placing a detector in the xy plane. A detector, represented by the red or blue disk in the animation below, is simply a coil of wire. If one passes a current through a coiled wire, a magnetic field is induced. Conversely, if a coil is placed in a changing magnetic field, a current is induced in the coil. This is the mode of detection in NMR spectroscopy. Note that the detector signal is proportional to the rate of change of the bulk magnetization ( dM/dt ) and not the bulk magnetization itself.

## Simulation

The simulation below shows the precession of the bulk magnetization (represented by the green arrow) for a 1H. The bulk magnetization precesses around the z axis, which means Mx and My oscillate. In this case, the bulk magnetization lies completely in the xy plane.

The red disk on the x axis represents a detector that provides the x signal, which is graphed with a red line as a function of time.

Similarly, the blue disk on the y axis is the detector that provides the y signal, which is graphed with a blue line as a function of time.

Notice how the signals are 90° out of phase with each other, which allows one to identify the direction of precession.

Allow the animation to run for awhile and then stop the simulation. Measure the Larmor frequency from the graph. Pressing the left mouse button will display the cursor position. Measure the period for the signal. The frequency (in Hz) is the reciprocal of the period (in sec).

The time scale for the animation is much longer than the real time scale. The graph displays time in nanoseconds (1 nsec = 10-9 sec). The simulation has been slowed drastically to facilitate visualization.

The simulation is for 1H, for which γ = 2.675222 x 108 sec-1 T-1.
Use your measured frequency to calculated the magnitude of B.

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