### Magnetic moment of atoms

Let an electric current $I$ flow around a closed loop $C$.

It gives rise to a magnetic dipole moment

$$\mathbf{\mu}=\frac{I}{2} \oint_C \mathbf{r}\times d\mathbf{s}$$

[The dipole moment is a source of a magnetic field.]

If we assume one electron (charge $e$, mass $m_e$) orbiting around a loop $C$, then

$$⁍$$

where

• $\gamma = g_e|e|/2m_e$ is the gyromagnetic ratio.
• *$\mathbf{L}$ *****is the angular momentum of the electron.

Electrons also have spin (internal angular momentum). Therefore, there is an additional magnetic moment due to the electron spin (in this case, $g_e=2$).

Fixed length vector.

An electron is orbiting around the atom in a fixed orbit (due to quantization) and this results in a vector $\mathbf{\mu}$ of fixed length.

[We will be primarily interested on the magnetic moments. Not on the magnetic field that they create.]

### A field with values on the sphere, $\mathbf{u}\in\mathbb{S}^2$

Assume a field $u = u(x)$ defined in the real space, $x \in \mathbb{R}$, and taking values on the unit sphere, $u \in \mathbb{S}^2$. Such a field is realised by a vector $\mathbf{u} \in \mathbb{R}^3$ with unit length $|\mathbf{u}| = 1$.