## Spin Angular Momentum

Spin angular momentum is a specific type of angular momentum possessed by some nuclei. As such, it obeys all the relations given for angular momentum under the quantum mechanics of rotation, here. i.e. there are two quantum numbers associated with the spin angular momentum momentum that determine its properties.

All nuclei have a spin quantum number, I, which may be integral (including zero) or half-integral, but never negative. The value of I is characteristic of a given nucleus, and may vary between isotopes. Thus all ^{1}H hydrogen nuclei have I = ½ , but all ^{2}H deuterium nuclei have I = 1.

The magnitude of the spin angular momentum is determined by the quantum number I, and is given by:

Thus all nuclei with I > 0 have spin angular momentum.

Further, all nuclei with a spin angular momentum (i.e. all nuclei with nonzero I) have a magnetic moment with constant magnitude and an orientation determined by a second quantum number, m_{I}. (To say that a nucleus has a magnetic moment means that in some ways it behaves like a small bar magnet.)

The quantum number m_{I} properly determines the component of the spin angular momentum on an arbitrary axis, normally termed the z axis (The arbitrary axis may be defined, for example, by the field direction of an external electric or magnetic field.):

where m_{I} can take values from I to -I. (m_{I} = I, I – 1, I – 2 … -I)

This property indicates that for a given I, the spin (and thus the magnetic moment) of a nucleus can adopt 2I + 1 different orientations relative to a defined axis. A proton (^{1}H hydrogen nucleus) has I = ½ , and thus its spin may adopt 2 different orientations (m_{I} = -½ or m_{I} = +½).

A large number of the nuclei that are studied by nuclear magnetic resonance (NMR) spectroscopy have I = ½, as they give rise simpler spectra than nuclei with I > ½ , but **any nucleus with nonzero I may be studied by NMR**. Spin ½ nuclei that are commonly studied include ^{1}H (the most popular nucleus for NMR studies), ^{13}C, ^{19}F and ^{31}P.

The state with m_{I} = +½ is denoted α , while the state with m_{I} = -½ is denoted β.

Note that two very common isotopes, ^{12}C and ^{16}O, have I = 0, so have no spin angular momentum, no magnetic moment, and hence are invisible in NMR studies.