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Hund's rules

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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 <ref name="sblundell">S. Blundell. Magnetism in Condensed Matter. Oxford University Press, 2003</ref>.

==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 <ref name="sblundell" />.

\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, \({\bf S}\).

<references/>

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