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\begin_body

\begin_layout Chapter
The magnetic propagation vector
\begin_inset CommandInset label
LatexCommand label
name "chap:magnetic-propagation-vector"

\end_inset


\end_layout

\begin_layout Standard
The magnetic propagation vector formalism allows compact description of
 any periodic magnetic structure
\begin_inset CommandInset citation
LatexCommand citep
key "Wills2001-magnetic-structures-group-theory"

\end_inset

.
 Magnetic periodicity often has a lower spatial frequency than crystal translati
on: 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 
\begin_inset Formula $\bm{k}$
\end_inset

-vectors, which allow large unit cells to be represented as a small number
 of Fourier components.
\end_layout

\begin_layout Standard
The 
\begin_inset Formula $\bm{k}$
\end_inset

-vector formalism is described in the familiar crystallographic notation.
 A crystal comprises an infinite array of atoms 
\begin_inset Formula $j$
\end_inset

 at positions 
\begin_inset Formula $\bm{R}_{j}$
\end_inset

 each comprising the addition of a lattice translation vector 
\begin_inset Formula $\bm{T_{l}}$
\end_inset

 and a basis vector 
\begin_inset Formula $\bm{r}_{p}$
\end_inset

:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{eqnarray}
\bm{R}_{j} & = & \bm{T}_{\bm{l}}+\bm{r}_{p}\nonumber \\
 & = & n_{l_{\mathrm{a}}}\bm{a}+n_{l_{\mathrm{b}}}\bm{b}+n_{l_{\mathrm{c}}}\bm{c}+x_{p}\bm{a}+y_{p}\bm{b}+z_{p}\bm{c},
\end{eqnarray}

\end_inset

where 
\begin_inset Formula $\bm{a}$
\end_inset

, 
\begin_inset Formula $\bm{b}$
\end_inset

 and 
\begin_inset Formula $\bm{c}$
\end_inset

 are the three non-coplanar lattice vectors; 
\begin_inset Formula $n_{l_{\mathrm{a,b,c}}}\in\mathbb{Z}$
\end_inset

 index the infinite three-dimensional array of unit cells; and 
\begin_inset Formula $0\leq x_{p},y_{p},z_{p}<1$
\end_inset

 together describe the fractional coordinates of the atom 
\begin_inset Formula $p$
\end_inset

 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 
\begin_inset Formula $\bm{k}$
\end_inset

, whose components are multiples of the three reciprocal lattice vectors
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\bm{a}^{*}=\frac{2\uppi}{\Delta}\bm{b}\times\bm{c}
\end{equation}

\end_inset


\emph on
etc.
\emph default
, where 
\begin_inset Formula $\Delta=\bm{a}\cdot\bm{b}\times\bm{c}$
\end_inset

 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.
\end_layout

\begin_layout Standard
In the case of an ordered magnetic system, the periodic property is the
 atoms' magnetic moments 
\begin_inset Formula $\bm{\mu}_{j}$
\end_inset

.
 These are computed by a Fourier summation
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
{\displaystyle \bm{\mu}_{j}=\sum_{\left\{ \bm{k}\right\} }\bm{m}_{p,\bm{k}}e^{-i\bm{k\cdot}\bm{T}}},\label{eq:propvectordefn}
\end{equation}

\end_inset

 where 
\begin_inset Formula $\bm{m}_{p,\bm{k}}$
\end_inset

 is the Fourier component with wavevector 
\begin_inset Formula $\bm{k}$
\end_inset

 corresponding to an atom 
\begin_inset Formula $p$
\end_inset

 in the unit cell.
 The sum is taken over the set of wavevectors 
\begin_inset Formula $\left\{ \bm{k}\right\} $
\end_inset

, as explained below; in many systems, this set consists of a single element.
 In general, 
\begin_inset Formula $\bm{m}_{p,\bm{k}}$
\end_inset

 is a complex vector, but 
\begin_inset Formula $\bm{k}$
\end_inset

 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 indefinite
ly many 
\begin_inset Formula $\bm{m}_{p,\bm{k}}$
\end_inset

 and 
\begin_inset Formula $\bm{k}$
\end_inset

 vectors.
 Thankfully, most systems can be described with fewer parameters than this!
\end_layout

\begin_layout Standard
The resulting magnetic moment 
\begin_inset Formula $\bm{\mu}_{j}\in\mathbb{R}^{3}$
\end_inset

 because it represents a physical magnetic moment on the 
\begin_inset Formula $j$
\end_inset


\begin_inset script superscript

\begin_layout Plain Layout
th
\end_layout

\end_inset

 atom.
 However, 
\begin_inset Formula $\bm{m}$
\end_inset

 and 
\begin_inset Formula $\bm{k}$
\end_inset

 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.
\end_layout

\begin_layout Standard
Elucidating 
\begin_inset Formula $\bm{m}$
\end_inset

s and 
\begin_inset Formula $\bm{k}$
\end_inset

s from experimental data, particularly from neutron diffraction, can be
 done by considering the exchange interactions and symmetry of a system
 using representational theory
\begin_inset CommandInset citation
LatexCommand citep
key "Wills2005-magnetic-representation"

\end_inset

; this is beyond the scope of this thesis.
\end_layout

\begin_layout Section
Examples
\end_layout

\begin_layout Standard
Ferromagnetic and simple antiferro- or ferrimagnetic structures can be simply
 described in this formalism, with a single wavevector 
\begin_inset Formula $\bm{k}$
\end_inset

 and therefore a single 
\begin_inset Formula $\bm{m}_{p}\equiv\bm{m}_{p,\bm{k}}\in\mathbb{R}^{3}$
\end_inset

 for each atom 
\begin_inset Formula $q$
\end_inset

.
 In this case, 
\begin_inset Formula $\bm{m}_{p}\equiv\bm{\mu}_{p}$
\end_inset

; they represent the actual magnetic moments of the atoms in the zeroth
 unit cell.
 The wavevector 
\begin_inset Formula $\bm{k}$
\end_inset

 describes how these moments transform between unit cells.
 In order that 
\begin_inset Formula $\bm{\mu}_{j}\in\mathbb{R}^{3}$
\end_inset

, the imaginary component of the complex exponential must be zero.
 This requires values of 
\begin_inset Formula $\bm{k}$
\end_inset

 such that 
\begin_inset Formula $\bm{k\cdot}\bm{T}=n\uppi$
\end_inset

, where 
\begin_inset Formula $n\in\mathbb{Z}$
\end_inset

.
 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:propvectordefn"

\end_inset

 then reduces to
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\bm{\mu}_{j}=\bm{m}_{p}\cos(\bm{k}\cdot\bm{T}).\label{eq:propvectorsimple}
\end{equation}

\end_inset

Since the sine component has vanished, the cosine is necessarily equal to
 
\begin_inset Formula $\pm1$
\end_inset

, 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 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:magnetic-prop-vector-examples"

\end_inset


\begin_inset space ~
\end_inset

(a)--(c).
\end_layout

\begin_layout Standard
The trivial case, 
\begin_inset Formula $\bm{k}=\bm{0}$
\end_inset

, can be used to describe any magnetic system where the spatial and magnetic
 unit cells are equivalent.
 
\begin_inset CommandInset ref
LatexCommand formatted
reference "eq:propvectorsimple"

\end_inset

 simplifies even further to
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{equation}
\bm{\mu}{}_{j}=\bm{m}_{p},\label{eq:propvectorsupersimple}
\end{equation}

\end_inset

 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.
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename ../chapters/figures/dipole/k-vector-afms/goodenough1.eps
	lyxscale 50
	scale 42.333300000000001

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
Four antiferromagnetic structures on a simple cubic lattice with one moment
 per unit cell
\begin_inset CommandInset citation
LatexCommand citep
key "blundell-magnetism-in-condensed-matter-book,goodenough1963-magnetism-book"

\end_inset

.
 (a), (b) and (c) have simple propagation vectors, perpendicular to planes
 of ferromagnetism within the material; 
\begin_inset Formula $\bm{m}_{\mathrm{a,b,c}}=\left(0,0,\mu\right)$
\end_inset

, 
\begin_inset Formula $\bm{k}_{\mathrm{a}}=\left(0,0,\frac{1}{2}\right)$
\end_inset

, 
\begin_inset Formula $\bm{k}_{\mathrm{b}}=\left(\frac{1}{2},\frac{1}{2},0\right)$
\end_inset

, 
\begin_inset Formula $\bm{k}_{\mathrm{c}}=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$
\end_inset

.
 (d) is a `zig-zag' structure, requiring two propagation vectors and basis
 vectors to describe: 
\begin_inset Formula $\bm{k}_{\mathrm{d}}^{1}=-\bm{k}_{\mathrm{d}}^{2}=\left(\frac{1}{4},\frac{1}{4},\frac{1}{2}\right)$
\end_inset

, 
\begin_inset Formula $\bm{m}_{\mathrm{d},\bm{k}_{1}}=\left(0,0,\frac{1+\mathrm{i}}{2}\mu\right)$
\end_inset

, 
\begin_inset Formula $\bm{m}_{\mathrm{d},\bm{k}_{2}}=\bm{m}_{\mathrm{d},\bm{k}_{1}}^{*}=\left(0,0,\frac{1-\mathrm{i}}{2}\mu\right)$
\end_inset

.
\begin_inset CommandInset label
LatexCommand label
name "fig:magnetic-prop-vector-examples"

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Standard
However, the propagation vector formalism can be turned to describing 
\emph on
any
\emph default
 periodic magnetic structure, and is consequently capable of representing
 significantly more complex examples than these.
 One slightly more subtle example is shown in 
\begin_inset CommandInset ref
LatexCommand formatted
reference "fig:magnetic-prop-vector-examples"

\end_inset


\begin_inset space ~
\end_inset

(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.
\begin_inset space ~
\end_inset

sine structures, helical structures 
\emph on
etc.
\emph default
) which vary either commensurately or incommensurately with the lattice
 by appropriate application of propagation vectors
\begin_inset CommandInset citation
LatexCommand citep
key "Wills2001-magnetic-structures-group-theory"

\end_inset

.
\end_layout

\end_body
\end_document
