Magnetic propagation vectors - ajsteele/mmcalc GitHub Wiki

This has been copied almost verbatim from a DokuWiki version which had support for LaTeX math notation and consequently needs some serious cleaning up. Sorry! Please check my thesis for a more readable version, or feel free to fix this!

The magnetic propagation vector formalism allows compact description of any periodic magnetic structure. It is used by MµCalc to describe magnetic structures. This description is taken from Andrew Steele’s thesis, in Appendix B, The magnetic propagation vector, and more information can be found in Magnetic structures and their determination using group theory, A. Wills (2001) (DOI: 10.1051/jp4:2001906)

Magnetic periodicity often has a lower spatial frequency than crystal translation: even a relatively simple antiferromagnetic structure might be two unit cells in each dimension, creating an eight-fold increase in the number of moments which have to be specified in the magnetic unit cell. The magnetic unit cell becomes infinitely large if the magnetic structure is incommensurate along any axis. This limitation can be circumvented by using magnetic propagation vectors, sometimes referred to as $\mathbf{k}$-vectors, which allow large unit cells to be represented as a small number of Fourier components.

The $\mathbf{k}$-vector formalism is described in the familiar crystallographic notation. A crystal comprises an infinite array of atoms $j$ at positions $\mathbf{R}{j}$ each comprising the addition of a lattice translation vector $\mathbf{T{l}}$ and a basis vector $\mathbf{r}_{p}$:

\begin{eqnarray} \mathbf{R}{j} & = & \mathbf{T}{\mathbf{l}}+\mathbf{r}{p}\nonumber \ & = & n{l_{\mathrm{a}}}\mathbf{a}+n_{l_{\mathrm{b}}}\mathbf{b}+n_{l_{\mathrm{c}}}\mathbf{c}+x_{p}\mathbf{a}+y_{p}\mathbf{b}+z_{p}\mathbf{c}, \end{eqnarray}

where $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are the three non-coplanar lattice vectors; $n_{l_{\mathrm{a,b,c}}}\in\mathbb{Z}$ index the infinite three-dimensional array of unit cells; and $0\leq x_{p},y_{p},z_{p}<1$ together describe the fractional coordinates of the atom $p$ in the primitive unit cell. Magnetic order is then superposed using the formalism of Bloch waves, which allows decomposition of any periodic property into a Fourier series. This Fourier series is most easily expressed in terms of reciprocal lattice vectors $\mathbf{k}$, whose components are multiples of the three reciprocal lattice vectors

\begin{equation} \mathbf{a}^{*}=\frac{2\pi}{\Delta}\mathbf{b}\times\mathbf{c} \end{equation}

//etc//., where $\Delta=\mathbf{a}\cdot\mathbf{b}\times\mathbf{c}$ is the volume of the unit cell. Since the crystal properties are periodic in real space, only wavevectors in the first Brillouin zone need to be considered.

In the case of an ordered magnetic system, the periodic property is the atoms' magnetic moments $\mathbf{\mu}_{j}$. These are computed by a Fourier summation

\begin{equation} {\displaystyle \mathbf{\mu}{j}=\sum{\left{ \mathbf{k}\right} }\mathbf{m}_{p,\mathbf{k}}e^{-i\mathbf{k\cdot}\mathbf{T}}},\label{eq:propvectordefn} \end{equation}

where $\mathbf{m}{p,\mathbf{k}}$ is the Fourier component with wavevector $\mathbf{k}$ corresponding to an atom $p$ in the unit cell. The sum is taken over the set of wavevectors $\left{ \mathbf{k}\right} $, as explained below; in many systems, this set consists of a single element. In general, $\mathbf{m}{p,\mathbf{k}}$ is a complex vector, but $\mathbf{k}$ must be real (though it may be incommensurate with the lattice). An arbitrarily complicated periodic magnetic system can be described with every atom in the primitive unit cell being assigned a unique set of indefinitely many $\mathbf{m}_{p,\mathbf{k}}$ and $\mathbf{k}$ vectors. Thankfully, most systems can be described with fewer parameters than this!

The resulting magnetic moment $\mathbf{\mu}_{j}\in\mathbb{R}^{3}$ because it represents a physical magnetic moment on the $j$\textsuperscript{th} atom. However, $\mathbf{m}$ and $\mathbf{k}$ can be complex vectors, meaning that the Fourier components are in general complex. In non-trivial cases these vectors must be chosen carefully in order to ensure that the resulting magnetic structure contains only real moments.

Elucidating $\mathbf{m}$s and $\mathbf{k}$s from experimental data, particularly from neutron diffraction, can be done by considering the exchange interactions and symmetry of a system using representational theory\citep{Wills2005-magnetic-representation}; this is beyond the scope of this thesis.

===== Examples =====

Ferromagnetic and simple antiferro- or ferrimagnetic structures can be simply described in this formalism, with a single wavevector $\mathbf{k}$ and therefore a single $\mathbf{m}{p}\equiv\mathbf{m}{p,\mathbf{k}}\in\mathbb{R}^{3}$ for each atom $q$. In this case, $\mathbf{m}{p}\equiv\mathbf{\mu}{p}$; they represent the actual magnetic moments of the atoms in the zeroth unit cell. The wavevector $\mathbf{k}$ describes how these moments transform between unit cells. In order that $\mathbf{\mu}_{j}\in\mathbb{R}^{3}$, the imaginary component of the complex exponential must be zero. This requires values of $\mathbf{k}$ such that $\mathbf{k\cdot}\mathbf{T}=n\pi$, where $n\in\mathbb{Z}$. The equation for the propagation vector then reduces to

\begin{equation} \mathbf{\mu}{j}=\mathbf{m}{p}\cos(\mathbf{k}\cdot\mathbf{T}).\label{eq:propvectorsimple} \end{equation}

Since the sine component has vanished, the cosine is necessarily equal to $\pm1$, and so adjacent moments can either be identical or opposite, and equal in magnitude. In these cases, the propagation vector will inevitably be perpendicular to the planes of ferromagnetism in the material. Simple examples of structures of this type can be seen in the below figure, in examples a)–c).

The trivial case, $\mathbf{k}=\mathbf{0}$, can be used to describe any magnetic system where the spatial and magnetic unit cells are equivalent. The propagation vector equation then simplifies even further to

\begin{equation} \mathbf{\mu}{}{j}=\mathbf{m}{p},\label{eq:propvectorsupersimple} \end{equation} implying that any atom will have the same moment as its equivalent in the zeroth unit cell. Ferromagnets are a subset of this trivial description with the further precondition that mutiple moments within the crystallographic unit cell must be equal in magnitude and direction. It is also possible to envisage antiferro- and ferrimagnets with multiple moments per unit cell being described with this trivial propagation vector.

{{ :magnetic-propagation-vector-afms-goodenough.png?640 |}} Four antiferromagnetic structures on a simple cubic lattice with one moment per unit cell. a), b) and c) have simple propagation vectors, perpendicular to planes of ferromagnetism within the material; $\mathbf{m}{\mathrm{a,b,c}}=\left(0,0,\mu\right)$, $\mathbf{k}{\mathrm{a}}=\left(0,0,\frac{1}{2}\right)$, $\mathbf{k}{\mathrm{b}}=\left(\frac{1}{2},\frac{1}{2},0\right)$, $\mathbf{k}{\mathrm{c}}=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$. d) is a ‘zig-zag’ structure, requiring two propagation vectors and basis vectors to describe: $\mathbf{k}{\mathrm{d}}^{1}=-\mathbf{k}{\mathrm{d}}^{2}=\left(\frac{1}{4},\frac{1}{4},\frac{1}{2}\right)$, $\mathbf{m}{\mathrm{d},\mathbf{k}{1}}=\left(0,0,\frac{1+\mathrm{i}}{2}\mu\right)$, $\mathbf{m}{\mathrm{d},\mathbf{k}{2}}=\mathbf{m}{\mathrm{d},\mathbf{k}{1}}^{*}=\left(0,0,\frac{1-\mathrm{i}}{2}\mu\right)$.

However, the propagation vector formalism can be turned to describing //any// periodic magnetic structure, and is consequently capable of representing significantly more complex examples than these. One slightly more subtle example is shown in d), a zig-zag magnetic structure which does not contain obvious planes of ferromagnetism. This requires two propagation vectors and two corresponding complex basis vectors to specify. It is possible to specify structures with modulated moment size and/or varying moment orientation (e.g. sine structures, helical structures //etc//.) which vary either commensurately or incommensurately with the lattice by appropriate application of propagation vectors.