Magnetic ions
We begin with a short description of magnetic properties of materials. We will concentrate on materials where the magnetic moments are localised and interact via simple, so-called exchange interactions. We shall see how these interactions lead to a number of different magnetically ordered structures.
A number of textbooks are devoted to magnetic properties of materials. For a general introduction to the field, we recommend the one by S. Blundell [1].
Magnetic moments of electrons
The magnetic moment of atoms and ions stems from the angular moment of the electrons. The orbital angular moment, \({\bf l}\), generates a circular current, like a tiny coil. This produces a magnetic dipole moment of
\begin{equation} \label{dummy1253707594} {\boldsymbol\mu}_l = \mu_{\rm B} {\mathbf l} , \end{equation}
where the Bohr magneton is
\begin{equation} \label{dummy64749429} \mu_{\rm B} = \dfrac{\hbar e}{2 m_{\rm e}} = 9.274 \cdot 10^{-24} {\rm J/T} = 5.788 \cdot 10^{-5}\,{\rm eV/T}. \end{equation}
For similar reasons (enhanced by relativistic effects) the spin of the electron causes a magnetic dipole moment of
\begin{equation} \label{dummy633134625} {\boldsymbol\mu}_s = g \mu_{\rm B} {\mathbf s} , \end{equation}
where \(g=2.0023\) is the gyromagnetic ratio of the electron and \({\bf s}\) is the electron spin.
We have above taken \(\bf l\) and \(\bf s\) to be unitless (i.e. the orbital angular moment is actually \(\hbar {\bf l}\)). We will remain with this definition in all of these notes.
Hund's rules
We will now determine the total angular moments of a free atom or ion. In general, we use the \(z\)-axis as the quantization axis of angular momenta.
A general quantum mechanical result gives us the rather intuitive addition rule of angular momenta [1].
\begin{equation} \label{dummy521864502} {\mathbf L} = \displaystyle\sum_i {\mathbf l}_i , \qquad {\mathbf S} = \displaystyle\sum_i {\mathbf s}_i , \qquad {\mathbf J} = {\mathbf L} + {\mathbf S} , \end{equation}
where \({\bf J}\) is the total angular momentum. The quantum numbers, \(L\), \(S\), and \(J\) take integer or half-integer values. In general, due to the coupling between the magnetic field from the orbital motion and the spin magnetic moment (the spin-orbit coupling), \(J\) is the only constant of motion.
We immediately note that closed shells represent \(L=S=J=0\), since all positive and negative values of \(l_i^z\) and \(s_i^z\) are represented. Hence, we only need to consider partially filled shells.
Due to electrostatic repulsion between atoms, combined with quantum mechanics (the Pauli principle and the spin-orbit coupling), it is energetically favourable for the electrons to occupy the partially filled shells in a particular way. This is described by Hund's rules (in order of highest priority):
- Maximize \(S\).
- Maximize \(L\).
- For less-than-half-filled shells: Minimize \(J\). For more-than-half-filled shells: Maximize \(J\).
These rules are, however, only general rules of thumb that may be overruled by other effects, e.g. crystal electric fields as discussed below.
Quenching
In materials, the ions cannot be considered free, but instead they interact with their neighbouring ions with electrostatic forces. This implies a breaking of the rotational symmetry of the atomic orbitals. In many cases, \({\bf L}\) is then no longer a good quantum number, and the average contribution to the magnetic moment from \({\bf L}\) vanishes, whence \({\bf J} = {\bf S}\). This effect is denoted quenching.
Quenching is seen for most of the 3d-metals, i.e. the metals with a partially filled 3d shell (transition metals), which are some of the most prominent magnetic ions in solids. The other prominent group, the 4f-metals (the rare-earth metals), are less often prone to quenching due to the relatively smaller spatial extend of the 4f orbitals.
In much of the text to follow, we assume a complete quenching of the magnetic ions, so that the only magnetic degree of freedom is the spin quantum number, \({\bf S}\).