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# Nuclear magnetic resonance

**Nuclear magnetic resonance** is a consequence of a property possessed by the nucleus known as nuclear spin angular momentum. Some properties associated with nuclear spin angular momentum are similar to those of a spinning macroscopic body, however, nuclear spin angular momentum is a *fundamental property* that *cannot* be explained in terms of any other fundamental property such as mass, charge, etc.

Nuclear spin angular momentum—like the angular momentum of any other fundamental particle —is a *quantized vectorial* quantity^{[1]}. Its magnitude is restricted to certain fixed values *and* its direction is also restricted to certain directions in the presence of a magnetic field.
In the absence of a magnetic field, it is not possible to obtain *any* information regarding its direction.

Nuclei that have an even mass number and an even atomic number do not exhibit nuclear magnetic resonance, e.g., O-16, C-12. Some common nuclei that do exhibit nuclear magnetic resonance are: H-1, C-13, N-15, F-19. For a detailed list see the Catalog of magnetic nuclei.

The nuclear spin angular momentum is characterized by a quantum number *I* known as the nuclear spin angular momentum quantum number (often briefly referred to as "nuclear spin").
For example, the proton has a nuclear spin angular momentum quantum number of 1/2 and is known as a spin-1/2 particle. Similarly, the N-14 nucleus has a nuclear spin angular momentum quantum number of 1 and is known as a spin-1 nucleus.
The magnitude of the nuclear spin angular momentum with quantum number *I* is,

where is Planck's reduced constant.

In the presence of an external homogeneous magnetic field (i.e., a magnetic field that has same magnitude everywhere in the space of interest; the magnetic field has no transverse components since the *z*-direction is chosen to point along the direction of the magnetic field), the *z*-component of the nuclear spin angular momentum vector is restricted to certain values *mh*/(2π), where *m* is the spin magnetic quantum number, and can be any one of the values from +*I* to −*I*, that differ from each other in integral steps.

For example,

If *I*=1, then *m* = +1, 0 or −1

If *I*=1/2, then *m* = +1/2 or −1/2

(Note: Difference between different values of *m* should be integral; however, actual values of m may be integers or half-integers)

As a consequence of the restrictions on the magnitude of the *z*-component, the nuclear spin angular momentum vector can only point in certain (allowed) directions with reference to the external magnetic field. In the absence of other fields, there are no restrictions on the allowed directions in the *x*-*y* plane. The net result is that the spin angular momentum vectors of different nuclei point along the surface of cones that have a fixed angle with respect to the external magnetic field. The nuclear spin magnetic moment is proportional to the nuclear spin angular momentum and the constant of proportionality is known as the gyromagnetic ratio. The gyromagnetic ratio may be either positive or negative. Therefore, the nuclear spin magnetic moment vector is either parallel or antiparallel to the nuclear spin angular momentum vector.

The different allowed values of *m* define the allowed orientations of the nuclear spin angular momentum and each of these spin states is associated with a different energy. This is due to the fact that the energy of the spin states is proportional to the scalar product of the nuclear spin magnetic moment and the external magnetic field vectors. Electromagnetic radiation can efficiently cause transitions between the nuclear spin states if the frequency of the electromagnetic radiation, ν, is equal to the energy difference Δ*E* between the nuclear spin states divided by Planck's constant *h*.

## References

- ↑ W. Gerlach and O. Stern. (1922) Zeit. f. Physik 9, 349