\htmladdnormallink{D.\ {}C.\ {}Hatton}{http://www.bib.hatton.btinternet.co.uk/dan/}\thanks{\htmladdnormallink{Physics and Chemistry of Solids Group}{http://www-pcs.phy.cam.ac.uk/}
\htmladdnormallink{Cavendish Laboratory}{http://www.phy.cam.ac.uk/}
abstract.tex
\section*{Abstract}

Recently, there has been a revival \cite{Weber:1999:EAF} of interest
\cite{Darwin:1928:DME} in mechanisms for changing the spin
polarization of an electron beam on transmission through, or
reflection from, a magnetic surface.  An understanding of these
mechanisms would \cite{Weber:1999:EAF} allow the use of an electron
beam as a polarized radiation probe for magnetic characterization,
like light in MOKE and neutrons in PNR.  Here, a mechanism is
described which, unlike simultaneously occurring processes proposed
elsewhere \cite{Weber:1999:EAF}, polarizes an unpolarized incident
beam without recourse to inelastic processes.

A magnetic field leads to a Zeeman term in an electron's Hamiltonian,
which depends on the angle $\theta$ between the electron's spin vector
and the magnetic flux.  As a result, when an electron wave is incident
on the surface of a bulk magnetic material (figure \ref{dhatton1},)
the wave-number of the transmitted wave depends on $\theta$.  When the
conditions of continuity of the wave-function, and of its first
spatial derivative, at the surface, and conservation of particles, are
applied, an electron reflection coefficient is obtained which also
depends on $\theta$.  Therefore, some polarizations are preferentially
reflected, while others are preferentially transmitted.  The amplitude
reflection and transmission coefficients can readily be converted to
intensity coefficients, and averaged over an incoherent superposition
of electron waves of different $\theta$, e.g.\ {}an unpolarized
incident beam.  The reflected polarization is
\begin{equation}
P = -\frac{2e\mu_BVB}{3e^2V^2+\mu_B^2B^2}\pnc{,}
\end{equation}
which can take values
\begin{equation}
-\frac{1}{\sqrt{3}}\leq{}P\leq\frac{1}{\sqrt{3}}\pnc{,}
\end{equation}
depending on the balance between $V$ and $B$.

The analysis can be extended to multi-layers using the theory of
Fabry-Perot etalons.
\begin{figure}
\includegraphics[width=\textwidth, height=0.75\textwidth]{dhatton1}
\caption{Surface of a Bulk Magnetic Sample}
\label{dhatton1}
\end{figure}

acknowledgements.tex
\section*{Acknowledgements}

Good morning.  As you've just heard, I'm Dan Hatton, and I'm going to
present a classical-field theory of electron waves as a polarized
radiation probe of magnetic surfaces.

We'd like to thank the Engineering and Physical Sciences Research
Council, in the UK, for paying for this work, and also Thanos
Mitrelias, Klaus Peter Kopper, and Peter Bode, our present and past
collaborators, in setting up the experiments for which we're putting
forward a theoretical interpretation.

author.tex
\author{\htmladdnormallink{D.\ {}C.\
{}Hatton}{http://www.bib.hatton.btinternet.co.uk/dan/}\thanks{\htmladdnormallink{Physics
and Chemistry of Solids Group}{http://www-pcs.phy.cam.ac.uk/},
\htmladdnormallink{Cavendish Laboratory}{http://www.phy.cam.ac.uk/},
\htmladdnormallink{University of Cambridge}{http://www.cam.ac.uk/},
Madingley Road, Cambridge, UK.  CB3
0HE}\latex{\newcounter{pcs}\setcounter{pcs}{\value{footnote}}}
$^{\pnc{,}}$ \thanks{\htmladdnormallink{Girton
College}{http://www.girton.cam.ac.uk/}, \htmladdnormallink{University
of Cambridge}{http://www.cam.ac.uk/}, Huntingdon Road, Cambridge, UK.
CB3 0JG} $^{\pnc{,}}$ \thanks{\htmailto{dan.hatton@btinternet.com}} and
\htmladdnormallink{J.\ {}A.\ {}C.\
{}Bland}{http://homer.phy.cam.ac.uk/Members/Tony_Bland/Home.html}\latex{\footnotemark[\value{pcs}]}\html{\thanks{\htmladdnormallink{Physics
and Chemistry of Solids Group}{http://www-pcs.phy.cam.ac.uk/},
\htmladdnormallink{Cavendish Laboratory}{http://www.phy.cam.ac.uk/},
\htmladdnormallink{University of Cambridge}{http://www.cam.ac.uk/},
Madingley Road, Cambridge, UK.  CB3 0HE}} $^{\pnc{,}}$
\thanks{\htmladdnormallink{Selwyn College}{http://www.sel.cam.ac.uk/},
\htmladdnormallink{University of Cambridge}{http://www.cam.ac.uk/},
Grange Road, Cambridge, UK.  CB3 9DQ} $^{\pnc{,}}$
\thanks{\htmailto{jacb1@phy.cam.ac.uk}}}

copying.data.tex
% fdl.tex 
% This file is a chapter.  It must be included in a larger document to work
% properly.

\section{GNU Free Documentation License}
\label{copying.data}

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copyright.data.tex
\section*{Copyright Notice}

      Copyright \copyright\ 2002 D.\ {}C.\ {}Hatton and J.\ {}A.\ 
      {}C.\ {}Bland.  Permission is granted to copy, distribute and/or
      modify this document under the terms of the GNU Free
      Documentation License, Version 1.1 published by the Free
      Software Foundation; with no Invariant Sections, with no
      Front-Cover Texts, and with no Back-Cover Texts.  A copy of the
      license is included in the section entitled ``GNU Free
      Documentation License'' (section \ref{copying.data}.)
      
      This document can be found at
      $<$\urn{http://www.bib.hatton.btinternet.co.uk/dan/Natural_Sciences/Classical-Field_Theory_of_Electron_Waves_as_a_Polarized_Radiation_Probe_of_Magnetic_Surfaces/}$>$,
      and in transparent form at
      $<$\urn{http://www.bib.hatton.btinternet.co.uk/dan/Natural_Sciences/Classical-Field_Theory_of_Electron_Waves_as_a_Polarized_Radiation_Probe_of_Magnetic_Surfaces.tar.gz}$>$.


endorsements.tex
\section*{Endorsements}

The ``Acknowledgements'' section, and sections \ref{intro}, \ref{arc},
\ref{bulk}, \ref{multi}, \ref{incomplete}, and \ref{conclusions}, of
this document, form a transcript of a talk, given in the ``Novel
Techniques in Magnetism'' meeting of the \htmladdnormallink{CMD19CMMP
  2002}{http://physics.iop.org/IOP/Confs/CMD19/} conference, organized
by the \htmladdnormallink{European Physical
  Society}{http://www.eps.org/} and the \htmladdnormallink{Institute
  of Physics}{http://www.iop.org/}.  The ``Abstract'' is that which
appeared in the abstracts book for this conference, and on the basis
of which the talk was accepted for inclusion in the conference.


history.tex
\section*{History}

\begin{itemize}
\item 2002, April 10th: Version 13.10, amended from version 13.4, by
  D.\ {}C.\ {}Hatton, by virtue of having arXiv file generation use
  bbl instead of bib bibliography files, and include html, hthtml,
  copyright.programs, and latex2html-copyright, and typographical
  error corrected, released.
\item 2002, April 9th: Version 13.4, amended from version 13.3, by D.\ 
  {}C.\ {}Hatton, by virtue of having typographical error corrected,
  released.
\item 2002, April 9th: Version 13.3, amended from version 12.2, by D.\ 
  {}C.\ {}Hatton, by virtue of having details of appearance of this
  document as a conference talk added, some stylistic changes which
  happened on the fly, during the talk, included, marginal notes,
  describing some things that happened during the talk, added,
  conditionality of use of url of hturl sorted out, and Makefile set
  up to produce file for arXiv, released.
\item 2002, April 4th: Version 12.2, amended, from version 11.6, by
  D.\ {}C.\ {}Hatton, by virtue of having some comments added to
  improve flow of talk, and availability of url package to \LaTeX{}
  restored, released.
\item 2002, March 29th: Version 11.6, amended, from version 11.5, by
  D.\ {}C.\ {}Hatton, by virtue of (mostly) using hthtml's hturl and
  htmailto commands instead of url's url.
\item 2002, March 28th: Version 11.5, amended, from version 11.4, by
  D.\ {}C.\ {}Hatton, by virtue of working around lack of urldef
  command in \latextohtml{}.
\item 2002, March 28th: Version 11.4, amended, from version 11.3, by
  D.\ {}C.\ {}Hatton, by virtue of having ``Conclusions'' slide added,
  ``On the Diffraction of the Magnetic Electron'' included in slide
  bibliography, clarity improved, concision improved, use of URL
  package made \LaTeX{}-only, and typographical errors corrected,
  released.
\item 2002, March 27th: Version 11.3, amended, from version 10.3, by
  D.\ {}C.\ {}Hatton, by virtue of having construction of slides
  completed, clarity improved, typographical errors corrected, and
  use of URL package introduced, released.
\item 2002, March 26th: Version 10.3, amended, from version 7.2, by
  D.\ {}C.\ {}Hatton, by virtue of having construction of slides
  begun, acknowledgements adjusted, reference to ``A Bayesian
  Perspective on Recent Epistemological Developments'' added, clarity
  improved, and typographical errors corrected, released.
\item 2002, March 25th: Version 9.1, amended, from version 7.2, by D.\ 
  {}C.\ {}Hatton, by virtue of having discussion of extension to
  multi-layers completed, philosophical commentary written, derivation
  of reflected polarization from bulk sample tidied up, and author
  file amended to work around further \latextohtml{} bugs, released.
\item 2002, March 22nd: Version 7.2, amended, from version 7.1, by D.\ 
  {}C.\ {}Hatton, by virtue of having title, author and date placed
  after $\backslash$begin\{document\}, to work around \latextohtml{}
  bug, released.
\item 2002, March 21st: Version 7.1, amended, from version 5.2, by D.\
  {}C.\ {}Hatton, by virtue of having derivation of reflected
  polarization from bulk sample completed, discussion of extension to
  multi-layers begun, Acknowledgements section added, author
  affiliations expanded and tidied up, Abstract hived off into
  separate file and formatted as section instead of as abstract
  environment, an arithmetical error corrected, and typographical
  errors corrected, released.
\item 2002, March 20th: Version 5.2, written by D.\ {}C.\ {}Hatton and
  J.\ {}A.\ {}C.\ {}Bland, released.
\end{itemize}
newcommands.tex
\newcommand{\td}[1]{\mathrm{d}#1}
\newcommand{\pd}[1]{\partial #1}
\newcommand{\cof}{\mathrm{cof}}

\newcommand{\vc}[1]{\mathbf{#1}}
\newcommand{\mt}[1]{\mathsf{#1}}
\newcommand{\ut}[1]{\,\textrm{#1}}

\newcommand{\strong}[1]{\textbf{#1}}

\newcommand{\pnc}[1]{\textrm{#1}}
\newcommand{\br}{\nonumber\protect\\}

\html{\newcommand{\urn}[1]{\hturl{#1}}}
\latex{\newcommand{\urn}[1]{\url{#1}}}

packages.tex
\usepackage{html}

\usepackage{hthtml}

\latex{\usepackage{url}}

\usepackage{graphicx}

supplement.tex
\section{Supplementary Mathematical Details}
\subsection{Amplitude Reflection Coefficient for an Electron Pure State,
  at a Single Step in Electric Potential and Magnetic Flux Density}

The incident and transmitted electron waves are modelled (figure
\ref{single_step}) as plane waves, allowing the well-defined
wave-vector components $p$ in the plane of the interface, and $q_i$
perpendicular to the interface.  $p$ must be the same for all the
waves, in order to satisfy the boundary condition of continuity of the
wave-function at the interface.  The amplitude reflection coefficient
is \cite{Rae:1992:QM}
\begin{equation}
r_{01} = \frac{q_0-q_1}{q_0+q_1}\pnc{,}
\end{equation}
or, for a general interface,
\begin{equation}
r_{ij} = \frac{q_i-q_j}{q_i+q_j}\pnc{.}
\end{equation}

Next, we need to build an expression for the energy of the electrons.
There will be kinetic energy terms, which, in the non-relativistic
limit, are
\[
\frac{\hbar^2p^2}{2m_e}\pnc{,}
\]
and
\[
\frac{\hbar^2q_i^2}{2m_e}\pnc{,}
\]
along with an electrostatic potential energy
\[
-eV_i\pnc{,}
\]
and a term due to the torque, on the electron magnetic moment, in a
magnetic field \cite{Rae:1992:QM}
\[
\frac{e\hbar{}B_i\cos{}S_i}{2m_e}\pnc{,}
\]
where $S_i$ is the angle between the electron spin and the magnetic
flux density.  The form of this last term assumes a well-defined
energy for all values of $S_i$.  Strictly, only certain $S_i$ values
are eigen-states of a Hamiltonian which includes a magnetic field;
more about this later (section \ref{incomplete}.)  The total energy is
\begin{equation}
E =
\frac{\hbar^2p^2}{2m_e}+\frac{\hbar^2q_i^2}{2m_e}-eV_i+\frac{e\hbar{}B_i\cos{}S_i}{2m_e}\pnc{,}
\end{equation}
or, where $p$ is expressed as a fraction $\sin{}I$ of the total wave-number
in the absence of any potential, $I$ being an angle of incidence like
that in figure \ref{dhatton1},
\begin{equation}
E = E\sin^2I+\frac{\hbar^2q_i^2}{2m_e}-eV_i+\frac{e\hbar{}B_i\cos{}S_i}{2m_e}
\end{equation}
\begin{eqnarray}
\Rightarrow{}q_i & = &
\left(\frac{2m_eE\cos^2I}{\hbar^2}\right)^{1/2}\left(1+\frac{eV_i}{E\cos^2I}-\frac{e\hbar{}B_i\cos{}S_i}{2m_eE\cos^2I}\right)^{1/2}\br
& = & \left(\frac{2m_eE\cos^2I}{\hbar^2}\right)^{1/2}(1+x_i)^{1/2}\pnc{,}
\end{eqnarray}
where $x_i = y_i+z_i\cos{}S_i$, $y_i = \frac{eV_i}{E\cos^2I}$, and
$z_i = -\frac{e\hbar{}B_i}{2m_eE\cos^2I}$.

We now use a binomial expansion \cite{Gallagher::MFH} for the case
where the potential energy terms are much smaller than the total
electron energy, where the dimensionless numbers we've just devised
are small.  The magnetic term associated with the Weiss field in a
ferromagnet is \cite{Moruzzi:1978:CEP} a few tenths of an
electron-volt, and the electrostatic contact potentials in the metals
which we study will not be more than a few volts, whereas, in our
experimental set-up, the incident electron energies range from a few
hundred to a few thousand electron volts, so this approximation seems
reasonable.
\begin{eqnarray}
q_i & = & \left(\frac{2m_eE\cos^2I}{\hbar^2}\right)^{1/2}\left(1+\frac{1}{2}x_i-\frac{1}{8}x_i^2+O(x_i^3)\right)\pnc{.}
\end{eqnarray}

The amplitude reflection coefficient is, therefore,
\begin{eqnarray}
r_{ij} & = &
\frac{1}{2}\left(\frac{1}{2}x_i-\frac{1}{2}x_j+\frac{1}{8}x_j^2-\frac{1}{8}x_i^2+O(\{x_i,x_j\}^3)\right)\left(1+\frac{1}{4}x_i+\frac{1}{4}x_j-\frac{1}{16}x_i^2-\frac{1}{16}x_j^2+O(\{x_i,x_j\}^3)\right)^{-1}\br
& = &
\frac{1}{2}\left(\frac{1}{2}x_i-\frac{1}{2}x_j+\frac{1}{8}x_j^2-\frac{1}{8}x_i^2+O(\{x_i,x_j\}^3)\right)\left(1-\frac{1}{4}x_i-\frac{1}{4}x_j+\frac{1}{8}x_i^2+\frac{1}{8}x_ix_j+\frac{1}{8}x_j^2+O(\{x_i,x_j\}^3)\right)\br
& = &
\frac{1}{2}\left(\frac{1}{2}x_i-\frac{1}{2}x_j-\frac{1}{4}x_i^2-\frac{1}{4}x_ix_j+\frac{1}{4}x_j^2+O(\{x_i,x_j\}^3)\right)\br
& = & \frac{1}{4}x_i-\frac{1}{4}x_j-\frac{1}{8}x_i^2-\frac{1}{8}x_ix_j+\frac{1}{8}x_j^2+O(\{x_i,x_j\}^3)\pnc{.}
\end{eqnarray}

\subsection{Reflection of an Unpolarized Beam from the Surface of a Bulk
  Magnetic Sample}
\label{bulk-details}

An unpolarized incident electron beam is \cite{Kessler:1976:PE,
  Kessler:1985:PE} an incoherent superposition of pure states
  representing all directions of the incident spin.  Each such
  direction can be represented by its spherical polar angle
  co-ordinates $(\theta{},\phi{})$.  That is to say, the incident beam
  contains a flux of electrons
\begin{equation}
F_1\td{\theta}\td{\phi} = A\sin\theta\td{\theta}\td{\phi}
\end{equation}
with polarization direction between $\theta$ and
$\theta{}+\td{\theta}$, and between $\phi$ and $\phi{}+\td{\phi}$.
The flux of such electrons in the reflected beam will, therefore, be
\begin{equation}
F_2\td{\theta}\td{\phi} = |r_{ij}|^2F_1\td{\theta}\td{\phi}\pnc{.}
\end{equation}

The reflection from the surface of a bulk sample is to be modelled as
a single reflection, of amplitude reflection coefficient $r_{01}$, in
a situation where $V_0$, $B_0$, and therefore $x_0$, are all zero.  In
this case,
\begin{equation}
r_{01} = -\frac{1}{4}x_1+\frac{1}{8}x_1^2+O(x_1^3)\pnc{,}
\end{equation}
and
\begin{equation}
|r_{01}|^2 = \frac{1}{16}x_1^2-\frac{1}{16}x_1^3+O(x_1^4)\pnc{,}
\end{equation}
assuming that $r_{01}$ is real.

If the spherical polar representation $(\theta_i,\phi_i)$ is used for
the direction of the magnetic flux density in region $i$, then
\begin{equation}
\cos{}S_i = \sin\theta_i\cos\phi_i\sin\theta\cos\phi{}+\sin\theta_i\sin\phi_i\sin\theta\sin\phi{}+\cos\theta_i\cos\theta\pnc{.}
\end{equation}
Therefore,
\begin{equation}
x_i = y_i+z_i(\sin\theta_i\cos\phi_i\sin\theta\cos\phi{}+\sin\theta_i\sin\phi_i\sin\theta\sin\phi{}+\cos\theta_i\cos\theta{})\pnc{.}
\end{equation}

The polarization of the pure state represented by $(\theta{},\phi{})$,
in the Cartesian co-ordinate system associated with this spherical
polar system, is
\begin{equation}
\vc{P}(\theta{},\phi{}) =
(\sin\theta\cos\phi{},\sin\theta\sin\phi{},\cos\theta{})\pnc{,}
\end{equation}
and the average polarization of the reflected beam is
\begin{eqnarray}
\vc{P} & = &
\frac{\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}\vc{P}(\theta{},\phi{})F_2\td{\theta}\td{\phi}}{\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}F_2\td{\theta}\td{\phi}}\br
& = &
\frac{\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(\sin\theta\cos\phi{},\sin\theta\sin\phi{},\cos\theta{})|r_{01}|^2F_1\td{\theta}\td{\phi}}{\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}|r_{01}|^2F_1\td{\theta}\td{\phi}}\br
& = &
\frac{\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(\sin\theta\cos\phi{},\sin\theta\sin\phi{},\cos\theta{})\left(\frac{1}{16}x_1^2-\frac{1}{16}x_1^3+O(x_1^4)\right)\sin\theta\td{\theta}\td{\phi}}{\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}\left(\frac{1}{16}x_1^2-\frac{1}{16}x_1^3+O(x_1^4)\right)\sin\theta\td{\theta}\td{\phi}}\br
& = & \frac{(I_2-I_6,I_3-I_7,I_4-I_8)+O(\{y_1,z_1\}^4)}{I_1-I_5+O(\{y_1,z_1\}^4)}\pnc{.}
\end{eqnarray}

The crucial integrals are
\begin{eqnarray}
I_1 & = & \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^2\sin\theta\td{\theta}\td{\phi}\br
& = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1+z_1(\sin\theta_1\cos\phi_1\sin\theta\cos\phi{}+\sin\theta_1\sin\phi_1\sin\theta\sin\phi{}+\cos\theta_1\cos\theta{}))^2\sin\theta\td{\theta}\td{\phi}\br
& = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^2\sin\theta{}\br
& & +2y_1z_1(\sin\theta_1\cos\phi_1\sin^2\theta\cos\phi{}+\sin\theta_1\sin\phi_1\sin^2\theta\sin\phi{}+\cos\theta_1\sin\theta\cos\theta{})\br
& &
+z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^3\theta\cos^2\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^3\theta\sin\phi{}\cos\phi{}\br
& &
+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^2\theta\cos\theta\cos\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^3\theta\sin^2\phi\br
& & +2\sin\theta_1\cos\theta_1\sin\phi_1\sin^2\theta\cos\theta\sin\phi+\cos^2\theta_1\sin\theta\cos^2\theta{}))\td{\theta}\td{\phi}\br
& = & 4\pi{}y_1^2+\frac{4\pi{}z_1^2}{3}\pnc{,}
\end{eqnarray}
\begin{eqnarray}
I_2 & = & \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^2\sin^2\theta\cos\phi\td{\theta}\td{\phi}\br
& = & \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1+z_1(\sin\theta_1\cos\phi_1\sin\theta\cos\phi{}+\sin\theta_1\sin\phi_1\sin\theta\sin\phi{}+\cos\theta_1\cos\theta{}))^2\sin^2\theta\cos\phi\td{\theta}\td{\phi}\br
& = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^2\sin^2\theta{}\cos\phi{}\br
& & +2y_1z_1(\sin\theta_1\cos\phi_1\sin^3\theta\cos^2\phi{}+\sin\theta_1\sin\phi_1\sin^3\theta\sin\phi{}\cos\phi{}+\cos\theta_1\sin^2\theta\cos\theta{}\cos\phi{})\br
& &
+z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^4\theta\cos^3\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^4\theta\sin\phi{}\cos^2\phi{}\br
& &
+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^3\theta\cos\theta\cos^2\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^4\theta\sin^2\phi\cos\phi\br
& &
+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^3\theta\cos\theta\sin\phi\cos\phi{}+\cos^2\theta_1\sin^2\theta\cos^2\theta\cos\phi))\td{\theta}\td{\phi}\br
& = & \frac{8\pi{}y_1z_1\sin\theta_1\cos\phi_1}{3}
\pnc{,}
\end{eqnarray}
\begin{eqnarray}
I_3 & = & \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^2\sin^2\theta\sin\phi\td{\theta}\td{\phi}\br
& = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1+z_1(\sin\theta_1\cos\phi_1\sin\theta\cos\phi{}+\sin\theta_1\sin\phi_1\sin\theta\sin\phi{}+\cos\theta_1\cos\theta{}))^2\sin^2\theta\sin\phi\td{\theta}\td{\phi}\br
& = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^2\sin^2\theta{}\sin\phi\br
& & +2y_1z_1(\sin\theta_1\cos\phi_1\sin^3\theta\sin\phi\cos\phi{}+\sin\theta_1\sin\phi_1\sin^3\theta\sin^2\phi{}+\cos\theta_1\sin^2\theta\cos\theta\sin\phi{})\br
& &
+z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^4\theta\sin\phi\cos^2\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^4\theta\sin^2\phi{}\cos\phi{}\br
& &
+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^3\theta\cos\theta\sin\phi\cos\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^4\theta\sin^3\phi\br
& &
+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^3\theta\cos\theta\sin^2\phi+\cos^2\theta_1\sin^2\theta\cos^2\theta\sin\phi{}))\td{\theta}\td{\phi}\br
& = & \frac{8\pi{}y_1z_1\sin\theta_1\sin\phi_1}{3}
\pnc{,}
\end{eqnarray}
\begin{eqnarray}
I_4 & = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^2\sin\theta\cos\theta\td{\theta}\td{\phi}\br
& = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1+z_1(\sin\theta_1\cos\phi_1\sin\theta\cos\phi{}+\sin\theta_1\sin\phi_1\sin\theta\sin\phi{}+\cos\theta_1\cos\theta{}))^2\sin\theta\cos\theta\td{\theta}\td{\phi}\br
& = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^2\sin\theta\cos\theta{}\br
& & +2y_1z_1(\sin\theta_1\cos\phi_1\sin^2\theta\cos\theta\cos\phi{}+\sin\theta_1\sin\phi_1\sin^2\theta\cos\theta\sin\phi{}+\cos\theta_1\sin\theta\cos^2\theta{})\br
& &
+z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^3\theta\cos\theta\cos^2\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^3\theta\cos\theta\sin\phi{}\cos\phi{}\br
& &
+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^2\theta\cos^2\theta\cos\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^3\theta\cos\theta\sin^2\phi\br
& &
+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^2\theta\cos^2\theta\sin\phi+\cos^2\theta_1\sin\theta\cos^3\theta{}))\td{\theta}\td{\phi}\br
& = & \frac{8\pi{}y_1z_1\cos\theta_1}{3}\pnc{,}
\end{eqnarray}
\begin{eqnarray}
I_5 & = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^3\sin\theta\td{\theta}\td{\phi}\br
& = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1+z_1(\sin\theta_1\cos\phi_1\sin\theta\cos\phi{}+\sin\theta_1\sin\phi_1\sin\theta\sin\phi{}+\cos\theta_1\cos\theta{}))^3\sin\theta\td{\theta}\td{\phi}\br
& = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^3\sin\theta\br
& & +3y_1^2z_1(\sin\theta_1\cos\phi_1\sin^2\theta\cos\phi{}+\sin\theta_1\sin\phi_1\sin^2\theta\sin\phi{}+\cos\theta_1\sin\theta\cos\theta{})\br
& & +3y_1z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^3\theta\cos^2\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^3\theta\sin\phi{}\cos\phi{}\br
& &
+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^2\theta\cos\theta\cos\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^3\theta\sin^2\phi\br
& &
+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^2\theta\cos\theta\sin\phi{}+\cos^2\theta_1\sin\theta\cos^2\theta{})\br
& &
+z_1^3(\sin^3\theta_1\cos^3\phi_1\sin^4\theta\cos^3\phi{}+3\sin^3\theta_1\sin\phi_1\cos^2\phi_1\sin^4\theta\sin\phi\cos^2\phi{}\br
& &
+3\sin^2\theta_1\cos\theta_1\cos^2\phi_1\sin^3\theta\cos\theta\cos^2\phi{}+3\sin^3\theta_1\sin^2\phi_1\cos\phi_1\sin^4\theta\sin^2\phi\cos\phi{}\br
& &
+6\sin^2\theta_1\cos\theta_1\sin\phi_1\cos\phi_1\sin^3\theta\cos\theta\sin\phi\cos\phi{}+3\sin\theta_1\cos^2\theta_1\cos\phi_1\sin^2\theta\cos^2\theta\cos\phi{}\br
& &
+\sin^3\theta_1\sin^3\phi_1\sin^4\theta\sin^3\phi{}+3\sin^2\theta_1\cos\theta_1\sin^2\phi_1\sin^3\theta\cos\theta\sin^2\phi{}\br
& &
+3\sin\theta_1\cos^2\theta_1\sin\phi_1\sin^2\theta\cos^2\theta\sin\phi{}+\cos^3\theta_1\sin\theta\cos^3\theta{}))\td{\theta}\td{\phi}\br
& = & 4\pi{}y_1z_1^2\pnc{,}
\end{eqnarray}
\begin{eqnarray}
I_6 & = & \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^3\sin^2\theta\cos\phi\td{\theta}\td{\phi}\br
& = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^3\sin^2\theta\cos\phi\br
& & +3y_1^2z_1(\sin\theta_1\cos\phi_1\sin^3\theta\cos^2\phi{}+\sin\theta_1\sin\phi_1\sin^3\theta\sin\phi\cos\phi{}+\cos\theta_1\sin^2\theta\cos\theta\cos\phi{})\br
& & +3y_1z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^4\theta\cos^3\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^4\theta\sin\phi{}\cos^2\phi{}\br
& &
+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^3\theta\cos\theta\cos^2\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^4\theta\sin^2\phi\cos\phi\br
& &
+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^3\theta\cos\theta\sin\phi\cos\phi{}+\cos^2\theta_1\sin^2\theta\cos^2\theta\cos\phi{})\br
& &
+z_1^3(\sin^3\theta_1\cos^3\phi_1\sin^5\theta\cos^4\phi{}+3\sin^3\theta_1\sin\phi_1\cos^2\phi_1\sin^5\theta\sin\phi\cos^3\phi{}\br
& &
+3\sin^2\theta_1\cos\theta_1\cos^2\phi_1\sin^4\theta\cos\theta\cos^3\phi{}+3\sin^3\theta_1\sin^2\phi_1\cos\phi_1\sin^5\theta\sin^2\phi\cos^2\phi{}\br
& &
+6\sin^2\theta_1\cos\theta_1\sin\phi_1\cos\phi_1\sin^4\theta\cos\theta\sin\phi\cos^2\phi{}+3\sin\theta_1\cos^2\theta_1\cos\phi_1\sin^3\theta\cos^2\theta\cos^2\phi{}\br
& &
+\sin^3\theta_1\sin^3\phi_1\sin^5\theta\sin^3\phi\cos\phi{}+3\sin^2\theta_1\cos\theta_1\sin^2\phi_1\sin^4\theta\cos\theta\sin^2\phi\cos\phi{}\br
& &
+3\sin\theta_1\cos^2\theta_1\sin\phi_1\sin^3\theta\cos^2\theta\sin\phi\cos\phi{}+\cos^3\theta_1\sin^2\theta\cos^3\theta\cos\phi{}))\td{\theta}\td{\phi}\br
& = & \left(4\pi{}y_1^2z_1+\frac{4\pi{}z_1^3}{5}\right)\sin\theta_1\cos\phi_1\pnc{,}
\end{eqnarray}
\begin{eqnarray}
I_7 & = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^3\sin^2\theta\sin\phi\td{\theta}\td{\phi}\br
& = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^3\sin^2\theta\sin\phi\br
& & +3y_1^2z_1(\sin\theta_1\cos\phi_1\sin^3\theta\sin\phi\cos\phi{}+\sin\theta_1\sin\phi_1\sin^3\theta\sin^2\phi{}+\cos\theta_1\sin^2\theta\cos\theta\sin\phi{})\br
& & +3y_1z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^4\theta\sin\phi\cos^2\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^4\theta\sin^2\phi{}\cos\phi{}\br
& &
+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^3\theta\cos\theta\sin\phi\cos\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^4\theta\sin^3\phi\br
& &
+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^3\theta\cos\theta\sin^2\phi{}+\cos^2\theta_1\sin^2\theta\cos^2\theta\sin\phi{})\br
& &
+z_1^3(\sin^3\theta_1\cos^3\phi_1\sin^5\theta\sin\phi\cos^3\phi{}+3\sin^3\theta_1\sin\phi_1\cos^2\phi_1\sin^5\theta\sin^2\phi\cos^2\phi{}\br
& &
+3\sin^2\theta_1\cos\theta_1\cos^2\phi_1\sin^4\theta\cos\theta\sin\phi\cos^2\phi{}+3\sin^3\theta_1\sin^2\phi_1\cos\phi_1\sin^5\theta\sin^3\phi\cos\phi{}\br
& &
+6\sin^2\theta_1\cos\theta_1\sin\phi_1\cos\phi_1\sin^4\theta\cos\theta\sin^2\phi\cos\phi{}+3\sin\theta_1\cos^2\theta_1\cos\phi_1\sin^3\theta\cos^2\theta\sin\phi\cos\phi{}\br
& &
+\sin^3\theta_1\sin^3\phi_1\sin^5\theta\sin^4\phi{}+3\sin^2\theta_1\cos\theta_1\sin^2\phi_1\sin^4\theta\cos\theta\sin^3\phi{}\br
& &
+3\sin\theta_1\cos^2\theta_1\sin\phi_1\sin^3\theta\cos^2\theta\sin^2\phi{}+\cos^3\theta_1\sin^2\theta\cos^3\theta\sin\phi{}))\td{\theta}\td{\phi}\br
& = & \left(4\pi{}y_1^2z_1+\frac{4\pi{}z_1^3}{5}\right)\sin\theta_1\sin\phi_1\pnc{,}
\end{eqnarray}
and
\begin{eqnarray}
I_8 & = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^3\sin\theta\cos\theta\td{\theta}\td{\phi}\br
& = &
\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^3\sin\theta\cos\theta\br
& & +3y_1^2z_1(\sin\theta_1\cos\phi_1\sin^2\theta\cos\theta\cos\phi{}+\sin\theta_1\sin\phi_1\sin^2\theta\cos\theta\sin\phi{}+\cos\theta_1\sin\theta\cos^2\theta{})\br
& & +3y_1z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^3\theta\cos\theta\cos^2\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^3\theta\cos\theta\sin\phi{}\cos\phi{}\br
& &
+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^2\theta\cos^2\theta\cos\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^3\theta\cos\theta\sin^2\phi\br
& &
+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^2\theta\cos^2\theta\sin\phi{}+\cos^2\theta_1\sin\theta\cos^3\theta{})\br
& &
+z_1^3(\sin^3\theta_1\cos^3\phi_1\sin^4\theta\cos\theta\cos^3\phi{}+3\sin^3\theta_1\sin\phi_1\cos^2\phi_1\sin^4\theta\cos\theta\sin\phi\cos^2\phi{}\br
& &
+3\sin^2\theta_1\cos\theta_1\cos^2\phi_1\sin^3\theta\cos^2\theta\cos^2\phi{}+3\sin^3\theta_1\sin^2\phi_1\cos\phi_1\sin^4\theta\cos\theta\sin^2\phi\cos\phi{}\br
& &
+6\sin^2\theta_1\cos\theta_1\sin\phi_1\cos\phi_1\sin^3\theta\cos^2\theta\sin\phi\cos\phi{}+3\sin\theta_1\cos^2\theta_1\cos\phi_1\sin^2\theta\cos^3\theta\cos\phi{}\br
& &
+\sin^3\theta_1\sin^3\phi_1\sin^4\theta\cos\theta\sin^3\phi{}+3\sin^2\theta_1\cos\theta_1\sin^2\phi_1\sin^3\theta\cos^2\theta\sin^2\phi{}\br
& &
+3\sin\theta_1\cos^2\theta_1\sin\phi_1\sin^2\theta\cos^3\theta\sin\phi{}+\cos^3\theta_1\sin\theta\cos^4\theta{}))\td{\theta}\td{\phi}\br
& = & \left(4\pi{}y_1^2z_1+\frac{4\pi{}z_1^3}{5}\right)\cos\theta_1\pnc{.}
\end{eqnarray}

This gives a polarization
\begin{eqnarray}
\vc{P} & = &
\frac{(I_2-I_6,I_3-I_7,I_4-I_8)+O(\{y_1,z_1\}^4)}{I_1-I_5+O(\{y_1,z_1\}^4)}\br
& = &
\frac{(10y_1z_1-15y_1^2z_1-3z_1^3)(\sin\theta_1\cos\phi_1,\sin\theta_1\sin\phi_1,\cos\theta_1)+O(\{y_1,z_1\}^4)}{15y_1^2+5z_1^2-15y_1z_1^2+O(\{y_1,z_1\}^4)}\br
& = &
\frac{(10y_1z_1-15y_1^2z_1-3z_1^3)\vc{\hat{B_1}}+O(\{y_1,z_1\}^4)}{15y_1^2+5z_1^2}\left(1-\frac{5y_1z_1^2}{3y_1^2+z_1^2}+O(\{y_1,z_1\}^2)\right)^{-1}\br
& = &
\frac{2y_1z_1\vc{\hat{B_1}}}{3y_1^2+z_1^2}+\frac{(36y_1^2z_1^3-45y_1^4z_1-3z_1^5)\vc{\hat{B_1}}}{45y_1^4+30y_1^2z_1^2+5z_1^4}+O(\{y_1,z_1\}^2)\pnc{,}
\end{eqnarray}
where $\vc{\hat{B_1}}$ is a unit vector, in the direction of the
magnetic flux density in region 1.

index.bbl
\begin{thebibliography}{10}

\bibitem{Anderson:1972:MID}
P.~W. Anderson.
\newblock More is different: Broken symmetry and the nature of the hierarchical
  structure of science.
\newblock {\em Science}, 177(4047):393--396, Aug. 1972.

\bibitem{Blundell:1992:PNR}
S.~J. Blundell and J.~A.~C. Bland.
\newblock Polarized neutron reflection as a probe of magnetic-films and
  multilayers.
\newblock {\em Phys. Rev. B}, 46(6):3391--3400, Aug. 1992.

\bibitem{Blundell:1993:PNR}
S.~J. Blundell and J.~A.~C. Bland.
\newblock Polarised neutron reflection as a probe of in-plane magnetisation
  vector rotation in magnetic multilayers.
\newblock {\em J. Magn. Magn. Mater.}, 121:185--188, 1993.

\bibitem{Darwin:1928:DME}
C.~G. Darwin.
\newblock On the diffraction of the magnetic electron.
\newblock {\em P. Roy. Soc. Lond. A Mat.}, 120:631--642, 1928.

\bibitem{Dunning:1994:MEP}
F.~B. Dunning.
\newblock Mott electron polarimetry.
\newblock {\em Nucl. Instrum. Meth. A}, 347(1--3):152--160, Aug. 1994.

\bibitem{Gallagher::MFH}
F.~G. Gallagher and J.~R. Shakeshaft.
\newblock Mathematical formulae: A handbook with table of physical constants.
\newblock On sale at the {C}avendish Stores, Cavendish Laboratory, University
  of Cambridge, Madingley Road, Cambridge, UK. CB3 0HE.
\newblock Version 1.4.

\bibitem{Gay:1992:MEP}
T.~J. Gay and F.~B. Dunning.
\newblock {M}ott electron polarimetry.
\newblock {\em Rev. Sci. Instrum.}, 63(2):1635--1651, Feb. 1992.

\bibitem{Hatton:2002:BPR}
D.~C. Hatton.
\newblock A {B}ayesian perspective on recent epistemological developments.
\newblock World-Wide Web page, 2002.
\newblock In preparation. URL:
  $<$\urn{http://www.bib.hatton.btinternet.co.uk/dan/Mathematics/A_Bayesian_Pe%
rspective_on_Recent_Epistemological_Developments/}$>$.

\bibitem{Hodge:1979:MES}
L.~A. Hodge, T.~J. Moravec, F.~B. Dunning, and G.~K. Walters.
\newblock {M}ott electron spin polarization analysis system of novel design.
\newblock {\em Rev. Sci. Instrum.}, 50:5--8, 1979.

\bibitem{Kessler:1976:PE}
J.~Kessler.
\newblock {\em Polarized Electrons}.
\newblock Texts and Monographs in Physics. Springer-Verlag, Berlin, 1976.

\bibitem{Kessler:1985:PE}
J.~Kessler.
\newblock {\em Polarized Electrons}.
\newblock Springer Series on Atoms and Plasmas. Springer-Verlag, Berlin, 2nd
  edition, 1985.

\bibitem{Lind:1987:SSA}
D.~M. Lind, F.~B. Dunning, G.~K. Walters, and H.~L. Davis.
\newblock Surface-structural analysis by use of spin-polarized low-energy
  electron diffraction: An investigation of the {$Cu(100)$} surface.
\newblock {\em Phys. Rev. B}, 35(17):9037--9044, June 1987.

\bibitem{Lind:1986:ETC}
D.~M. Lind, K.~D. Jamison, F.~B. Dunning, G.~K. Walters, and H.~L. Davis.
\newblock An experimental/theoretical comparison of spin-polarized low energy
  electron diffraction from the {$Cu(100)$} surface.
\newblock {\em J. Vac. Sci. Technol. A}, 4(3):1271--1273, May--June 1986.

\bibitem{MacKay:1992:BMA}
D.~J.~C. MacKay.
\newblock {\em {B}ayesian Methods for Adaptive Models}.
\newblock PhD thesis, California Institute of Technology, Computation and
  Neural Systems, 1992.
\newblock URL: $<$\urn{http://www.inference.phy.cam.ac.uk/mackay/PhD.html}$>$.

\bibitem{Moruzzi:1978:CEP}
V.~L. Moruzzi, J.~F. Janak, and A.~R. Williams.
\newblock {\em Calculated Electronic Properties of Metals}.
\newblock Pergamon, New York, 1978.
\newblock Cited in \cite{Weber:1999:EAF}.

\bibitem{Rae:1992:QM}
A.~I.~M. Rae.
\newblock {\em Quantum Mechanics}.
\newblock Institute of Physics Publishing, Bristol, 3rd edition, 1992.

\bibitem{Weber:1999:EAF}
W.~Weber, D.~Oberli, S.~Riesen, and H.~C. Siegmann.
\newblock The electron analogue to the {F}araday rotation.
\newblock {\em New J. Phys.}, 1:9, May 1999.
\newblock URL: $<$\urn{http://stacks.iop.org/NJP/1/9/}$>$.

\end{thebibliography}

index.ltx.bbl
\begin{thebibliography}{10}

\bibitem{Anderson:1972:MID}
P.~W. Anderson.
\newblock More is different: Broken symmetry and the nature of the hierarchical
  structure of science.
\newblock {\em Science}, 177(4047):393--396, Aug. 1972.

\bibitem{Blundell:1992:PNR}
S.~J. Blundell and J.~A.~C. Bland.
\newblock Polarized neutron reflection as a probe of magnetic-films and
  multilayers.
\newblock {\em Phys. Rev. B}, 46(6):3391--3400, Aug. 1992.

\bibitem{Blundell:1993:PNR}
S.~J. Blundell and J.~A.~C. Bland.
\newblock Polarised neutron reflection as a probe of in-plane magnetisation
  vector rotation in magnetic multilayers.
\newblock {\em J. Magn. Magn. Mater.}, 121:185--188, 1993.

\bibitem{Darwin:1928:DME}
C.~G. Darwin.
\newblock On the diffraction of the magnetic electron.
\newblock {\em P. Roy. Soc. Lond. A Mat.}, 120:631--642, 1928.

\bibitem{Dunning:1994:MEP}
F.~B. Dunning.
\newblock Mott electron polarimetry.
\newblock {\em Nucl. Instrum. Meth. A}, 347(1--3):152--160, Aug. 1994.

\bibitem{Gallagher::MFH}
F.~G. Gallagher and J.~R. Shakeshaft.
\newblock Mathematical formulae: A handbook with table of physical constants.
\newblock On sale at the {C}avendish Stores, Cavendish Laboratory, University
  of Cambridge, Madingley Road, Cambridge, UK. CB3 0HE.
\newblock Version 1.4.

\bibitem{Gay:1992:MEP}
T.~J. Gay and F.~B. Dunning.
\newblock {M}ott electron polarimetry.
\newblock {\em Rev. Sci. Instrum.}, 63(2):1635--1651, Feb. 1992.

\bibitem{Hatton:2002:BPR}
D.~C. Hatton.
\newblock A {B}ayesian perspective on recent epistemological developments.
\newblock World-Wide Web page, 2002.
\newblock In preparation. URL:
  $<$\urn{http://www.bib.hatton.btinternet.co.uk/dan/Mathematics/A_Bayesian_Pe%
rspective_on_Recent_Epistemological_Developments/}$>$.

\bibitem{Hodge:1979:MES}
L.~A. Hodge, T.~J. Moravec, F.~B. Dunning, and G.~K. Walters.
\newblock {M}ott electron spin polarization analysis system of novel design.
\newblock {\em Rev. Sci. Instrum.}, 50:5--8, 1979.

\bibitem{Kessler:1976:PE}
J.~Kessler.
\newblock {\em Polarized Electrons}.
\newblock Texts and Monographs in Physics. Springer-Verlag, Berlin, 1976.

\bibitem{Kessler:1985:PE}
J.~Kessler.
\newblock {\em Polarized Electrons}.
\newblock Springer Series on Atoms and Plasmas. Springer-Verlag, Berlin, 2nd
  edition, 1985.

\bibitem{Lind:1987:SSA}
D.~M. Lind, F.~B. Dunning, G.~K. Walters, and H.~L. Davis.
\newblock Surface-structural analysis by use of spin-polarized low-energy
  electron diffraction: An investigation of the {$Cu(100)$} surface.
\newblock {\em Phys. Rev. B}, 35(17):9037--9044, June 1987.

\bibitem{Lind:1986:ETC}
D.~M. Lind, K.~D. Jamison, F.~B. Dunning, G.~K. Walters, and H.~L. Davis.
\newblock An experimental/theoretical comparison of spin-polarized low energy
  electron diffraction from the {$Cu(100)$} surface.
\newblock {\em J. Vac. Sci. Technol. A}, 4(3):1271--1273, May--June 1986.

\bibitem{MacKay:1992:BMA}
D.~J.~C. MacKay.
\newblock {\em {B}ayesian Methods for Adaptive Models}.
\newblock PhD thesis, California Institute of Technology, Computation and
  Neural Systems, 1992.
\newblock URL: $<$\urn{http://www.inference.phy.cam.ac.uk/mackay/PhD.html}$>$.

\bibitem{Moruzzi:1978:CEP}
V.~L. Moruzzi, J.~F. Janak, and A.~R. Williams.
\newblock {\em Calculated Electronic Properties of Metals}.
\newblock Pergamon, New York, 1978.
\newblock Cited in \cite{Weber:1999:EAF}.

\bibitem{Rae:1992:QM}
A.~I.~M. Rae.
\newblock {\em Quantum Mechanics}.
\newblock Institute of Physics Publishing, Bristol, 3rd edition, 1992.

\bibitem{Weber:1999:EAF}
W.~Weber, D.~Oberli, S.~Riesen, and H.~C. Siegmann.
\newblock The electron analogue to the {F}araday rotation.
\newblock {\em New J. Phys.}, 1:9, May 1999.
\newblock URL: $<$\urn{http://stacks.iop.org/NJP/1/9/}$>$.

\end{thebibliography}
