# Energies of Nuclei in Magnetic Fields

The nuclear magnetic moment of a nucleus is denoted μ, and the component of this magnetic moment on the z axis, μ_{z}, is proportional to the component of the nuclear magnetic moment along this axis, m_{I}. We thus write:

where γ is called the magnetogyric ratio of the particular nucleus under consideration, an empirically determined quantity.

Each value of m_{I} indicates a different orientation of the nuclear spin and thus of the nuclear magnetic moment. In a magnetic field of strength B, the direction of the field defining the direction of the nuclear z-axis, all 2I + 1 orientations of the nucleus have different energies, given by:

It is common to express these energies in terms of the Larmor frequency, ν_{L}:

Note that the Larmor frequency depends upon the field strength. The energy separation of the two states of a spin ½ nucleus is equal to hν_{L}.

For most nuclei γ is positive; in such cases the β state lies higher in energy than the α state meaning that there are slightly more α than β spins in a sample of material in a magnetic field. (The population imbalance is only small, as the energy separation between the two states is very small compared to kT at most temperatures.)

When the sample is irradiated with radiation of the correct frequency, the spins are induced into making the transition α → β until the populations of the two states have equalised, and absorption of the incident radiation can be observed.

For this to happen, the energy of the incoming photons must match the energy difference between the two states; the following resonance condition must be fulfilled:

i.e. the frequency of the incident radiation must be equal to the Larmor frequency of the nucleus to induce a transition between the states of different m_{I}.

At typical magnetic field strengths, Larmor frequencies normally lie in the radiofrequency portion of the electromagnetic spectrum, so NMR normally uses radio waves to cause transitions. At its most basic, NMR involves applying a magnetic field to a sample of material and then observing the frequencies required to bring the different magnetic nuclei in the sample into resonance.