Notices of Lattice Dynamics

Nuclear Spin and Magnetic Resonance

The nuclear spin - Most elements have at least one isotope with a non-zero spin angular momentum, I, and an associated magnetic moment, µ, which are related by the gyromagnetic ratio, g I is a quantized characteristic of the nucleus, and its value describes the symmetry of the nuclear charge distribution. In this course we will limit the discussion to spin=1/2 nuclei for which the nuclear charge distribution is spherically symmetric. The nucleus then has the properties of a magnetic dipole (essentially a bar-magnet), whose strength is given above. All of the interaction of spin=1/2 nuclei are purely magnetic.
Spin =1/2 nuclei are the most often studied by NMR since they generally have both higher resolution and higher sensitivity spectra (therefore, it is perhaps easier to extract chemical information from these nuclei).

Quantum mechanics tells us a few important points about nuclear spins,
1. the projection of the nuclear magnetic moment along any direction is quantized and for spin=1/2 nuclear is restricted to the to values of +/- .
2. the uncertainty principle applies and places a limit on the amount of information we can know about the orientation of a nuclear magnetic moment. At any time, we can only know the magnitude of the vector and its projection along one axis. The projections along the other two axis are indeterminate (they are in a superposition state).

The Zeeman interaction

In an NMR experiment we are interested in exploring the interaction of the nuclear magnetic moment and an external magnetic field. The energy of this magnetic dipole-dipole interaction is given classically as,
where Bo is the strength of the external magnetic field. This external field has a direction and so this provides a coordinate system for the NMR experiment. From here on, we will work in coordinate systems where the applied magnetic field is oriented along the z-axis.
We can now see that for a spin=1/2 nuclei the two values of the spin along the z-direction, Iz= +/- 1/2, correspond to the nuclear magnetic moment oriented along and against the magnetic field. Classically we may compare this to the two stable positions of a compass needle in the earth’s magnetic field, a low energy configuration with the needle aligned with the earth’s field, and a higher energy (unstable equilibrium point) with the needle aligned against the earth’s field. In the case of the nuclear spins, the nuclear moment can not be aligned exactly along the applied field, since this would violate the uncertainty principle, and so there are two states, both of which are represented by cones.

The nuclear spin is restricted to being on these two cones oriented along the z-axis.

Let us explore the torque first, in the presence of an applied magnetic field, then the Larmor precession states that the bulk magnetic moment will revolve about the applied field direction. If the bulk magnetization is along the field direction, as it is at equilibrium, then there is no torque and hence no motion. As we expect, at equilibrium the system is stationary. Note, this is true of the detectable bulk magnetization, but is not true at the microscopic level. The dynamics of single spins can not be discussed in the classical terms that we are using.
If the system is away from equilibrium, if the bulk magnetization vector is oriented other than along the z-axis, then the magnetization presesses (rotates) about the z-axis with a angular velocity given by the energy separation of the two states (g B0). Notice that this torque will not change the length of the magnetization vector, it only varries its orientation.
This rotation can not be the only motion, sine then the system would never return to equilibrium. So along with the rotation, there is a relaxation of the vector to bring it back along the z-axis. Therefore the x and y-components of the nuclear magnetization decay towards zero, and the z-component decays towards the equilibrium value (typically called M0).

The above is a “quick-time” movie that shows the motion of the bulk magnetization vector starting from a position along the x-axis and then evolving towards its equilibrium position along the z-axis. The movie was created in Mathematica (see appendix 1-3) and may be run by double clicking on the figure. The red bar progressing across the figure is meant to represent the flow of time.
Latter we will show how a pulse of radio frequency radiation will tilt the bulk magnetization vector away from the z-axis and creat this non-equilibrium magnetization. For now, we are only interested in the spin’s return to equilibrium as shown in the above figure.

REFERENCIAS BIBLIOGRAFICAS:

web.mit.edu/22.058/www/documents/Fall2002/.../NMR.doc