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NMR Spectroscopy: Larmor Precession

Nuclear Spin

One of the noteworthy properties of a proton and a neutron is that each has intrinsic angular momentum (p) and thus an intrinsic magnetic moment (μ). That is, each particle behaves as little magnet with a north pole and an opposing south pole. For this reason, μ is a vector: the magnetic moment has a magnitude and a direction in space.

The magnitude of the angular momentum, and thus the magnitude of the magnetic moment, is characterized by the nuclear spin (I), and both the proton and the neutron have I = ½. The magnetic moment of an atomic nucleus is simply the sum of the magnetic moments of all the protons and neutrons in the nucleus. For 1H the nucleus is just a proton and thus I = ½.

nuclear spin

For 4He there are two protons and two neutrons. The two protons are paired in the same orbital, with their magnetic moments (north and south poles) opposing and canceling each other. The same is true for the neutrons, leaving 4He with no net spin (I = 0) and thus no magnetic moment. Similarly, all of the protons and neutrons are paired in 12C, and thus is also has no magnetic moment, but there is one unpaired proton in 13C, leading to I = ½. There is one unpaired proton and one unpaired neutron in 14N and thus I = 1. Note that protons and neutrons are distinguishable particles and thus there are separate sets of orbitals for protons and for neutrons.

The magnitude of the angular moment depends upon the particle's spin:

|p| = ℏ [I (I + 1)]½

The physical constant ℏ equals h/2π, where h = 6.626 x 10-34 J sec is the Planck constant.

The magnetic moment is directly portional to the angular moment

μ = γ p

The proportionality constant (γ) is called the magnetogyric ratio. Each nucleus has an unique value of γ, which has units of sec-1 T-1. T = Tesla, a unit of magnetic field strength.

 NucleusSpin10-7 γ (sec-1 T-1)
 1H½26.7522
 13C½6.72828
 14N11.93378
 15N½-2.71262
 31P½10.8394

Notice that the magnetic moment of 1H is four times larger than that of 13C, even though both are spin ½ nuclei. For this reason, proton NMR is inherently four times more sensitive than 13C NMR. There are also other factors that affect sensitivity. Almost all naturally occurring hydrogen is 1H. Only 1% of naturally occurring carbon is 13C.

two magnets

If you obtain a pair of bar magnets and explore how they interact, you will quickly observe that the north pole of one magnet is attracted to the south pole of the other.

Imagine the large blue magnet in the image at left has a powerful magnetic field (B) and is held in a fixed position. The smaller red magnet is free to move. The magnetic forces between the two magnets create a force that will cause the movable red magnet to swing up against the stationary blue magnet in a north-to-south orientation, as shown in the image at the right.

The orientations of the magnets in the image at the right is the state with lowest energy. Once the system reaches this configuration, it is at equilibrium and no additional change will occur.

two magnets

There is a profound difference, however, between the behavior of the macroscopic magnets illustrated above and the magnetic moment of a nucleus. If a nucleus is placed in a stationary magnetic field (B) that is aligned along the z axis, the nucleus will attempt to align its magnetic moment against that of the magnetic field, similar to the bar magnets depicted above. Unlike the bar magnets, however, the alignment with the magnetic field is constrained to discrete levels.

The component of the angular moment (and hence magnetic moment) that can be aligned along the z axis (that is, aligned along B) is given by the quantum number mI, which has possible values of -I, -I+1, ... I.

For a spin ½ nucleus like 1H, the permissable values of mI are ± ½.

μz = γ pz = ℏ γ mI

The consequence of this quantitization is that the nuclear magnetic moment can never be perfectly aligned with B in the way the bar magnets can become perfectly aligned. There is always a residual component of the magnetic moment in the xy plane and thus there is always a non-zero torque on the nucleus in the xy plane.

Joseph Larmor worked out the mathematical description of this persistent torque on the nuclear magnetic moment. The torque causes the nuclear magnetic moment to precess or rotate about the z axis at a constant rate, as illustrated in the animations shown below. This behavior is called Larmor precession.

For a spin ½ nucleus, mI = +½ or -½, with +½ being the lower energy state. Both possible states are shown below.

Start the animation and observe the behavior of the nuclear magnetic moment (μ), which is shown by the red and green arrows. In particular, observe the following behaviors.

The frequency of precession on NMR spectrometers is in the MHz range, with some instruments as high as 1 GHz. Thus the time required for one full rotation is a few nsec. Thus actual Larmor precession is approximately a billion times faster than the simulation shown below.

mz = ½ (low energy state)mz = -½ (high energy state)
                               


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© Copyright 2013-2014, David N. Blauch