Preparing an article for publication in an Institute of Physics Publishing journal using \LaTeXe
Tsampikos Kottos
37073 G\"ottingen
Germany
We review quantum chaos on graphs. We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wavefunction statistics. This operator is the analogue of the classical evolution operator on the graph. It allow us to establish a connection between the corresponding periodic orbits and the statistical properties of eigenvalues and eigenfunctions. Specifically, for the energy-averaged spectral form factor we derived an exact combinatorial expression which illustrate the role of correlations between families of isometric orbits. We also show that enhanced wave function localization due to the presence of short unstable periodic orbits and strong scarring can rely on completely different mechanisms. \\[0.2cm] {\em PACS}: 05.45.Mt, 03.65.Sq\\[0.1cm] {\em Keywords}: quantum chaos; random-matrix theory; periodic-orbit theory\\ %\hrule
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\begin{document}
\renewcommand{\baselinestretch}{1.5}
\title{ Quantum Graphology}
\author{Tsampikos Kottos \\
Max-Planck-Institute for Dynamics and Self-organization \\
37073 G\"ottingen, Germany} 
\date{\today }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}

We review quantum chaos on graphs. We construct a unitary operator which 
represents the quantum evolution on the graph and study its spectral and
wavefunction statistics. This operator is the analogue of the classical 
evolution operator on the graph. It allow us to establish a connection 
between the corresponding periodic orbits and the statistical properties 
of eigenvalues and eigenfunctions. Specifically, for the energy-averaged 
spectral form factor we derived an exact combinatorial expression which 
illustrate the role of correlations between families of isometric orbits. 
We also show that enhanced wave function localization due to the presence
of short unstable periodic orbits and strong scarring can rely on completely
different mechanisms.
\\[0.2cm]
{\em PACS}: 05.45.Mt, 03.65.Sq\\[0.1cm]
{\em Keywords}: quantum chaos; random-matrix theory; periodic-orbit theory\\
%\hrule
\end{abstract}


\section{Introduction}
%
Quantum graphs have recently attracted a lot of interest \cite{KS97,SS99,BK99,BG00,
ACDMT00,SS00,KS00,T01,KS01,K01,BSW02,BDJ02,SK03,SPG03,GSW04,GA04,HBPSZS04,K04,PSKG05}. 
A special volume containing a number of contributions can be found in \cite{K04}. The 
attention is due to the fact that quantum graphs can be viewed as typical and simple 
examples for the large class of systems in which classically chaotic dynamics implies 
universal spectral correlations in the semiclassical limit \cite{BGS84,berry}. Up to 
now we have only a very limited understanding of the reasons for this universality. 
In a semiclassical approach to this problem the main stumbling block is the intricate 
interference between the contributions of (exponentially many) periodic orbits 
\cite{ADDKKSS93,CPS98}. Using quantum graphs as model systems it is possible to pinpoint 
and isolate this central problem. In graphs, an exact trace formula exists which is 
based on the periodic orbits of a mixing classical dynamical system \cite{KS97,R83}. 
Moreover the orbits can be specified by a finite symbolic code with Markovian grammar. 
Based on these simplifications it is possible to rewrite the spectral form factor or 
any other two-point correlation functions in terms of a combinatorial problem \cite{KS97,
SS99,SS00,KS01}. This combinatorial problem on graphs has been solved with promising 
results: It was shown that the form factor, ensemble averaged over graphs with a single 
non-trivial vertex and two attached bonds (2-Hydra) coincides exactly with the random-matrix
result for the $2\times 2$ CUE \cite{SS99}. A simple algorithm which can evaluate the 
resulting combinatorial sum for any graph was presented in \cite{KS01}. In \cite{BK99} 
the short-time expansion of the form factor for $v$-Hydra graphs (i.~e. one central 
node with $v$ bonds attached) was computed in the limit $N\to\infty$. In \cite{SS00} 
a periodic-orbit sum was used to prove Anderson localization in an infinite chain graph 
with randomized bond lengths. In \cite{T01} the form factor of binary graphs was shown 
to approach the random-matrix prediction when the number of vertices increases. In 
\cite{BSW02} the second order contribution $-2\tau^2$ to the form factor, was derived 
and was shown to be related to correlations within pairs of orbits differing in the 
orientation of one of the two loops resulting from a self-intersection of the orbit. 
Finally, in \cite{GA04} a field theoretical method was used to evaluate exactly the 
form factor of large graphs. Very recently, the spectral properties of quantum graphs
were studied experimentally by the Warsaw group \cite{HBPSZS04} who constructed a 
microwave graph network.

The transport properties of open quantum graphs were also investigated quite thoroughly. 
In \cite{KS00} compact graphs were connected with leads to infinity and was shown that 
they display all the features which characterize quantum chaotic scattering. In \cite{SPG03} 
the open quantum graphs were used to calculate shot-noise corrections while in \cite{PSKG05} 
the same system was employed in order to understand current relaxation phenomena from 
open chaotic systems.

Quite recently the interest on quantum graphs moved towards understanding statistical 
properties of wavefunctions. In \cite{GSW04} the statistics of the nodal points was 
analyzed, while in \cite{K01,SK03} quantum graphs were used in order to understand 
scaring of quantum eigenstates. A scar is a quantum eigenfunction with excess density 
near an unstable classical periodic orbit (PO). Such states are not expected within 
Random-Matrix Theory (RMT), which predicts that wavefunctions must be evenly distributed 
over phase space, up to quantum fluctuations \cite{M90}. Experimental evidence and
applications of scars come from systems as diverse as microwave resonators \cite{Sri91}, 
quantum wells in a magnetic field \cite{W+96}, Faraday waves in confined geometries 
\cite{KAG01}, open quantum dots \cite{BAF01} and semiconductor diode lasers \cite{G+02}.


This contribution, is structured in the following way. In the following Section 2, the 
main definitions and properties of quantum graphs are given. We concentrate on the unitary 
bond-scattering matrix $U$ which can be interpreted as a quantum evolution operator on 
the graph. Section 3 deals with the corresponding classical dynamical system. In Section 
4, the statistical properties of the eigenphase spectrum of the bond-scattering matrix $U$ 
are analyzed and related to the periodic orbits of the classical dynamics. Scaring phenomenon 
is discussed and analyzed in Section 5. Finally, our conclusions and outlook are summarized 
in the last Section 6.
%
%-------------------------------------------------------------------------
%
\section{Quantum Graphs: Basic Facts}
%
We start with a presentation and discussion of the Schr\"odinger operator for
graphs.  Graphs consist of $V$ {\it vertices} connected by $B$ {\it bonds}.
The {\it valency} $v_{i}$ of a vertex $i$ is the number of bonds meeting at
that vertex.  The graph is called $v${\it -regular} if all the vertices have
the same valency $v$.  When the vertices $i$ and $j$ are connected, we denote
the connecting bond by $b=(i,j)$.  The same bond can also be referred to as
$\vec{b} \equiv (Min(i,j),Max(i,j))$ or $\smash{\stackrel{\leftarrow}{b}}
\equiv (Max(i,j),Min(i,j))$ whenever we need to assign a direction to the
bond. A bond with coinciding endpoints is called a {\it loop}. Finally, a
graph is called {\em bipartite} if the vertices can be divided into two
disjoint groups such that any vertices belonging to the same group are not
connected.

Associated to every graph is its {\it connectivity (adjacency) matrix}
$C_{i,j}$.  It is a square matrix of size $V$ whose matrix elements $C_{i,j}$
are given in the following way
$$
C_{i,j}=C_{j,i}=\left\{
\begin{array}{l}
1\,{\rm if }\,i,j\,\,{\rm are \,\,connected} \\
0\,\,{\rm otherwise}
\end{array}
\right\}\,.  \label{cmat}
$$
For graphs without loops the diagonal elements of $C$ are zero. The
connectivity matrix of connected graphs cannot be written as a block diagonal
matrix. The valency of a vertex is given in terms of the connectivity matrix,
by $v_i= \sum_{j=1}^V C_{i,j}$ and the total number of undirected bonds is $B=
{1\over 2} \sum_{i,j=1}^VC_{i,j}$.

For the quantum description we assign to each bond $b=(i,j)$ a coordinate
$x_{i,j}$ which indicates the position along the bond. $x_{i,j}$ takes the
value $0$ at the vertex $i$ and the value $L_{i,j} \equiv L_{j,i}$ at the
vertex $j$ while $x_{j,i}$ is zero at $j$ and $L_{i,j}$ at $i$. We have thus
defined the {\it length matrix} $L_{i,j}$ with matrix elements different from
zero, whenever $C_{i,j}\neq 0$ and $L_{i,j}=L_{j,i}$ for $b=1,...,B$.  The
wave function $\Psi$ contains $B$ components 
$\Psi_{b_1}(x_{b_1}),\Psi_{b_2}(x_{b_2}),...,\Psi_{b_B}(x_{b_B})$
where the set $\{b_i\}_{i=1}^B$ consists of $B$ different undirected bonds. 

The Schr\"{o}dinger operator (with $\hbar = 2m = 1$) is defined on a graph in
the following way: On each bond $b$, the component $\Psi_b$ of the total wave
function $\Psi$ is a solution of the one-dimensional equation
\begin{equation}
\left( -{\rm i}{\frac{{\rm d\ \ }}{{\rm d}x}}-A_b\right) ^2\Psi _b(x)=k^2\Psi
_b(x)\,.\label{schrodinger}
\end{equation}
We included a ``magnetic vector potential" $A_b$ (with $\Re e(A_{b})\ne 0$
and $A_{\vec{b}}= -A_{_{\smash{\stackrel{\leftarrow}{b}}}}$) which breaks
the time reversal symmetry. In most applications we shall assume that all
the $A_{b}$'s are equal and the bond index will be dropped.
%
On each of the bonds, the general solution of (\ref{schrodinger}) is a
superposition of two counter propagating waves
\begin {equation}
\Psi_{b=(i,j)} = a_{i,j} {\rm e}^{{\rm i}(k+A_{i,j})x_{i,j}}
+a_{j,i} {\rm e}^{{\rm i}(k+A_{j,i})x_{j,i}}
\label{propa}
\end{equation}
The coefficients $a_{i,j}$ form a vector ${\bf a}\equiv (a_{\vec{b}_1}$,
$\cdots$, $ a_{\vec{b}_B},$
$a_{_{\smash{\stackrel{\leftarrow}{b}_1}}}, \cdots, 
a_{_{\smash{\stackrel{\leftarrow}{b}_B}}})^T$ of complex numbers which
uniquely determines an element in a $2B-$dimensional Hilbert space. This space
corresponds to "free wave" solutions since we did not yet impose any
conditions which the solutions of (\ref{schrodinger}) have to satisfy at the
vertices.

The most general boundary conditions at the vertices are given in terms of
unitary $v_j\times v_j$ {\it vertex-scattering matrices}
$\sigma^{(j)}_{l,m}(k)$, where $l$ and $m$ go over all the vertices which are
connected to $j$.  At each vertex $j$, incoming and outgoing components of the
wave function are related by
\begin{equation}
a_{j,l}=\sum_{m=1}^{v_j}\sigma _{l,m}^{(j)}(k)e^{{\rm i}kL_{jm}} a_{m,j} \,,
\label{outin1}
\end{equation}
which implies current conservation. The particular form
\begin{equation}\label{Neumann}
\sigma^{(j)}_{l,m}={2\over v_{j}}-\delta_{l,m}
\end{equation}
for the vertex-scattering matrices was shown in \cite{KS97} to be compatible
with continuity of the wave function and current conservation at the vertices.
(\ref{Neumann}) is referred to as {\em Neumann} boundary conditions. Bellow,
we will concentrate on this type of graphs. Moreover we will always assume fully
connected graphs i.e. the valency is $v_j=v=V-1, \quad \forall j=1,\cdots,V$.

Stationary states of the graph satisfy (\ref{outin1}) at each vertex.  These
conditions can be combined into
\begin{equation}
{\bf a}= U(k)\, {\bf a}\,,
\label{outin2}
\end{equation}
such that the secular equation determining the eigenenergies and the
corresponding eigenfunctions of the graph is of the form \cite{KS97}
\begin{equation}
\det \left[ I-U(k,A)\right] =0\,.  \label{secular}
\end{equation}
Here, the unitary {\em bond-scattering matrix}
\begin{equation}\label{sb-matrix}
U(k,A)=D(k;A)\,T 
\end{equation}
acting in the $2B$-dimensional space of directed bonds has been
introduced. The matrices $D$ and $T$ are given by
\begin{eqnarray}
\label{DandT}
D_{ij,i^{\prime }j^{\prime }}(k,A) &=&\delta _{i,i^{\prime }}\delta
_{j,j^{\prime }}{\rm e}^{{\rm i}kL_{ij}+{\rm i}A_{i,j}L_{ij}}\ ;\label{scaco} \\
\ T_{ji,nm} &=&\delta _{n,i}C_{j,i}C_{i,m}\sigma _{j,m}^{(i)}\,. \nonumber
\end{eqnarray}
$T$ contains the topology of
the graph and is equivalent to the complete set of vertex-scattering matrices,
while $D$ contains the metric information about the bonds. Hereafter, the 
bond lengths $L_{m}$ ($m=1,\dots, B$) will be chosen to be incommensurate in 
order to avoid non-generic degeneracies.

It is instructive to interpret the action of $U$ on an arbitrary graph state as its time 
evolution over an interval corresponding to the mean bond length of the graph such that
\begin{equation}
\label{qevol}
{\bf a}(t) = U^t\,{\bf a}(0),\,\,\,\,t=0,1,2,...\,.
\end{equation}
Clearly the solutions of (\ref{outin2}) are stationary with respect to this
time evolution. $n$ in (\ref{qevol}) represents a discrete (topological) time
counting the collisions of the particle with vertices of the graph. In this 
"picture" the diagonal matrix $D_{mn}(k)=\delta_{mn}\,\e^{\i kl_{m}}$ describes 
the free propagation along the bonds of the network while $T$ assigns a scattering
amplitude for transitions between connected directed bonds. As we will see in
the next section it specifies a Markovian random walk on the graph which is 
the classical analogue of Eq.~(\ref{qevol}).

%-------------------------------------------------------------------------
\section{Periodic orbits and classical dynamics on graphs}

In this section we discuss the classical dynamics corresponding to the quantum
evolution (\ref{qevol}) implied by $U$. To introduce this dynamics we
employ a Liouvillian approach, where a classical evolution operator assigns
transition probabilities in a phase space of $2B$ directed bonds
\cite{KS97}. If $\rho _b(t)$ denotes the probability to occupy the (directed)
bond $b$ at the (discrete) topological time $t$, we can write down a Markovian
Master equation of the form
\begin{equation}
\rho _b(t+1)=\sum_{b^{\prime }}M_{b,b^{\prime }}\rho _{b^{\prime }}(t)\,.
\label{master}
\end{equation}
The classical (Frobenius-Perron) evolution operator $M$ has matrix elements
\begin{equation}
M_{ij,nm}=\delta_{j,n} P^{(j)}_{i\rightarrow m}
\label{cl3}
\end{equation}
with $P_{ji\rightarrow ij^{\prime }}^{(i)}$ denoting the transition
probability between the directed bonds $b=(j,i)$ and $b^{\prime }=(i,j^{\prime
})$. To make the connection with the quantum description, we adopt the quantum
transition probabilities, expressed as the absolute squares of matrix elements
of $M$
\begin{equation}
P_{j\rightarrow j^{\prime }}^{(i)}=\left| \sigma_{j,j^{\prime
}}^{(i)}(k)\right| ^2\,.  \label{cl1}
\end{equation}
Note that $P_{j\rightarrow j^{\prime }}^{(i)}$ and $M$ do not involve any metric information 
on the graph. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\epsfxsize.5\textwidth%
%\includegraphics[width=10cm,angle=270]{fig4.eps}
\includegraphics[width=10cm]{fig1.eps}
\caption{ 
Upper panel: The spectrum of the classical evolution operator $M$ for the case of $V=15$ 
fully connected graph. Lower panel: The scaling of the second maximum eigenvalue $|\mu_2|$ with 
respect to $B$. Solid line is the best linear fitting indicating that $|\mu_2| \propto B^{-0.5}$ . }
\end{center}
\label{fig:fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The unitarity of the bond-scattering matrix $U$ guarantees $\sum_{b=1}^{2B} M_{b,b'}=1$ 
and $0\leq M_{b,b'}\leq 1$, so that the total probability that the particle is 
on any bond remains conserved during the evolution. The spectrum of $M$, denoted as $\{\mu_b
\}$ with $b=1,\cdots 2B$, is restricted to the interior of the unit circle and $\mu_1 = 1$ 
is always an eigenvalue with the corresponding eigenvector $|1\rangle = \frac 1{2B} 
\left( 1,1,...,1\right) ^T$.  In most cases, the eigenvalue $1$ is the only eigenvalue 
on the unit circle. Then, the evolution is ergodic since any initial density will evolve 
to the eigenvector $|1\rangle$ which corresponds to a uniform distribution (equilibrium). 
The rate at which equilibrium is approached is determined by the gap to the next largest 
eigenvalue. If this gap exists, the dynamics is also mixing. 

It was shown recently \cite{GA04} that mixing dynamics alone does not suffice to guarantee 
universality of the spectral statistics of quantum graphs \footnote{For an example of a 
mixing graph with non-universal spectral statistics, see \cite{BK99}}. An additional condition
proven recently by Gutzmann and Altland \cite{GA04} states that in the limit of $B\rightarrow 
\infty$, the spectral gap has to be constant or at least vanish slowly enough as $\Delta_g 
\equiv (1-|\mu_2|)\propto B^{-\alpha}$ with $0\leq \alpha < 0.5$ and $\mu_2$ being the second 
maximum eigenvalue of $M$. In Fig.~1 we report our numerical results for Neumann fully 
connected graphs. We see that this type of graph satisfies the condition requested by \cite{GA04}.

Graphs are one dimensional and the motion on the bonds is simple and stable. Ergodic (mixing) 
dynamics is generated because at each vertex a (Markovian) choice of one out of $v$ directions 
is made. Thus, chaos on graphs originates from the multiple connectivity of the (otherwise 
linear) system \cite{KS97}.

Despite the probabilistic nature of the classical dynamics, the concept of a classical 
orbit can be introduced.  A classical orbit on a graph is an itinerary of successively 
connected directed bonds $(i_1,i_2), (i_2,i_3),\cdots $. An orbit is {\it periodic} with 
period $t_p$ if for all $k$, $(i_{t_p+k},i_{t_p+k+1}) = (i_k,i_{k+1})$. For graphs without 
loops or multiple bonds, the sequence of vertices $i_1,i_2, \cdots$ with $i_m \in [1,V]$ 
and $C_{i_m,i_{m+1}} =1$ for all $m$ represents a unique code for the orbit. This is a 
finite coding which is governed by a Markovian grammar provided by the connectivity matrix. 
In this sense, the symbolic dynamics on the graph is Bernoulli. This analogy is strengthened 
by further evidence: The number of $t_p-$PO's on the graph is ${\frac 1t_p}{\rm tr}C^{t_p}$, 
where $C$ is the connectivity matrix. Since its largest eigenvalue $\Gamma_c$ is bounded
between the minimum and the maximum valency i.e.  $\min v_i \leq \Gamma_c \leq \max v_i$, 
periodic orbits proliferate exponentially with topological entropy $\approx \log \Gamma_c$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\epsfxsize.5\textwidth%
\includegraphics[width=10cm]{fig2.eps}
\caption{ 
The topology of the shortest PO's of a fully connected graph with valency $v$ are shown 
together with the classical probabilities to remain, and their corresponding Lyapunov exponent.
}
\end{center}
\label{fig:fig2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

From the previous discussion it is clear that all periodic orbits on a graph are unstable. 
The classical probability to remain at a specific PO of period $t_p$ is $M_p=\prod_{t=1}^{t_p} 
(M^{t})_{j,j}$. As $M_p<1$, the probability to follow the PO decreases exponentially with 
time. Assuming regular graphs of valency $v_j=v$ we can evaluate the rate of instability as
\begin{equation}
\label{lyapexp}
M_p = \prod_{s=1}^{r_p} \left( 1-{2\over v}\right)^2 
\prod_{f=1}^{t_p-r_p}\left({2\over v}\right)^2 \equiv \e^{-\Lambda_p t_p}
\end{equation}
where $\Lambda_p$ plays the role of the Lyapunov exponent (LE) and $r_p$ is the number of 
vertices where back scattering occurs. For the graphs studied in this contribution, some 
PO's $p$ and LE $\Lambda_p$, are listed in Fig.~2. The shortest PO's have period $2$ and 
bounce back and forth between two vertices. For large graphs $v\to\infty$ these are by far 
the least unstable ones, as their LE approaches 0 while all others become increasingly 
unstable $\Lambda_{p}\sim\ln v$. 

%-------------------------------------------------------------------------

\section {The spectral statistics of $U$}
%
We consider the matrix $U(k,A)$ defined in Eqs.~(\ref{sb-matrix}),\ref{scaco}). The 
spectrum consist of $2B$ points $e^{{\rm i}\epsilon_l(k)}$ confined to the unit circle 
(eigenphases). Unitary matrices of this type are frequently studied since they are the 
quantum analogues of classical, area preserving maps. Their spectral fluctuations depend 
on the nature of the underlying classical dynamics \cite{S89}. The quantum analogues of 
classically integrable maps display Poissonian statistics while in the opposite case of
classically chaotic maps, the eigenphase statistics conform with the results
of RMT for Dyson's {\it circular ensembles}. To describe the spectral
fluctuations of $U$ we consider the form factor
\begin{equation}\label{ff} 
K(t,2B)={\frac 1{2B}}\left\langle |{\rm tr}U^t|^2\right\rangle \qquad(t>0)\,.
\end{equation} 
The average $\langle\dots\rangle$ will be specified below. RMT predicts that $K(t,2B)$ 
depends on the scaled time $\tau=\frac {t}{2B}$ only \cite{S89}, and explicit expressions 
for the orthogonal and the unitary circular ensembles are known \cite{M90}. 

Using (\ref{sb-matrix}), (\ref{DandT}) we expand the matrix products in ${\rm tr}U^{t}$ 
and obtain a sum of the form
\begin{equation}
{\rm tr}U^t(k)
=\sum_{p\in{\cal P}_t}{\cal A}_p{\rm e}^{{{\rm i} (kL_p+Al_p)}}\,.
\label{posum}
\end{equation}
In this sum $p$ runs over all closed trajectories on the graph which are compatible with 
the connectivity matrix and which have the topological length $t$, i.~e. they visit 
exactly $t$ vertices. For graphs, the concepts of closed trajectories and periodic orbits 
coincide, hence (\ref{posum}) can also be interpreted as a periodic-orbit sum. From 
(\ref{posum}) it is clear that $K(t/2B)=0$ as long as $t$ is smaller than the period 
of the shortest periodic orbit. The phase associated with an orbit is determined by 
its total (metric) length $L_p = \sum_{b \in p} L_{b}$ and by the ``magnetic flux" 
through the orbit. The latter is given in terms of its total {\em directed} length
$l_{p}$ if we assume for simplicity that the magnitude of the magnetic vector potential 
is constant $|A_{b}|\equiv A$. The amplitude of the contribution from a periodic orbit by 
the product of all the elements of vertex-scattering matrices encountered
\begin{equation}
{\cal A}_p=\prod_{j=1}^{n_p}\sigma^{(i_j)}_{i_{j-1},i_{j+1}}
\equiv \prod_{[r,s,t]} \left(\sigma^{(s)}_{r,t}\right)^{n_p(r,s,t)}\,,
\end{equation}
i.~e.\ for fixed boundary conditions at the vertices it is completely
specified by the frequencies $n_p(r,s,t)$ of all transitions $(r,s)\to(s,t)$ .
Inserting (\ref{posum}) into the definition of the form factor we obtain a
double sum over periodic orbits
\begin{equation}
\label{double-sum}
K(t/2B)=\frac 1{2B}\, \Big\langle\sum_{p,p'\in {\cal P}_n} {\cal A}_p{\cal
A}_{p\prime}^{*} \quad\exp \left\{{\rm i} k(L_{p}-L_{p'})+{\rm i}
A(l_p-l_{p\prime})\right\}\Big\rangle\,. 
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\epsfxsize.7\textwidth%
\includegraphics[width=10cm]{fig3.eps}
\caption { 
The family  ${\cal L}$ of isometric orbits ${\cal F}_6$ of period $t_p=6$ and length 
$L_p=2 l_{1,2}+l_{1,4}+l_{1,3}+l_{2,3}+l_{2,4}$ for the tetrahedron. The various
orbits ($6$ in total) are indicated with the sequence of letters associated with the
arrows.
}
\end{center}
\label{fig:fig3}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Now we have to specify our averaging procedure which has to respect the restrictions
imposed by the underlying classical dynamics. To this end we will use the wavenumber $k$ 
for averaging i.e. $\langle \cdots \rangle_{k}=\lim_{k\to\infty}\,k^{-1} \int_{0}^{k}dk'\,
(\cdots)$ (and, if present, also the magnetic vector potential $A$). Provided that 
the bond lengths of the graph are rationally independent and that a sufficiently large 
interval is used for averaging, we have 
\begin{equation}
\label{average}
\langle \e^{\i k(L_p-L_{p'})}\rangle_{k}=\delta_{L_p,L_{p'}}\, \quad {\rm and}\quad
\langle \e^{\i A(l_p-l_{p'})}\rangle_{A}=\delta_{l_p,l_{p'}}
\end{equation}
i.e. only terms with $L_{p}=L_{p'}$ and $l_{p}=l_{p'}$ survive. 

Note that $L_{p}=L_{p'}$ does {\em not} necessarily imply $p=p'$ or that
$p,p'$ are related by some symmetry because there exist families $\cal L$ of
distinct but isometric orbits which can be used to write the result of
(\ref{double-sum}) in the form \cite{KS97,SS99,BK99,KS01}
\begin{equation}\label{family-sum}
K(t/2B)=\sum_{{\cal L}\in{{\cal F}_{n}}}\left|\sum_{p\in{\cal L}}{\cal A}_{p}\right|^2\,.
\end{equation}
The outer sum is over the set ${\cal F}_{n}$ of families, while the inner one
is a {\it coherent} sum over the orbits belonging to a given family (= metric
length). An example of such family for the tetrahedron is shown in Fig.~3.
Eq.~(\ref{family-sum}) is exact, and it represents a combinatorial problem since it 
does not depend any more on metric information about the graph (the bond lengths).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\includegraphics[width=10cm,angle=270]{fig4.eps}
\caption{\label{fig:fig4}
Form factor of $U$ for regular graphs with $V=4$
vertices (top) and $V=5$ vertices (bottom). In the right panels an additional
magnetic field destroyed time-reversal symmetry. Circles: exact results
obtained from (\protect\ref{family-sum}). Stars: numerical average over 2,000
values of $k$. The solid line is to guide the eye. The prediction of the
appropriate random matrix ensemble are shown with dashed lines. }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In general, the combinatorial problem (\ref{family-sum}) is very hard and cannot
be solved in closed form. Nevertheless {\em exact} result for finite $t$ can 
always be obtained from (\ref{family-sum}) using a computer algebra system such 
as Maple \cite{Maple}. To this end, one has to represent ${\rm tr} U^{t}$ as 
a multivariate polynomial of degree $t$ in the variables $\e^{ikL_i}$, i.~e.\
\begin{equation}
{\rm tr} U^{t}=\sum_{{\cal P}_{t}}c_{\cal P}\,(\e^{ikL_{1}})^{p_{1}}\,
(\e^{ikL_{2}})^{p_{2}}\dots\,,
\end{equation}
where ${\cal P}_{t}$ runs over all partitions of $t$ into non-negative
integers $t=p_{1}+p_{2}+\dots$ \cite{KS01}. The form factor is then simply 
given as
\begin{equation}
\label{coeffs}
K(t/2B)=\sum_{{\cal P}_{t}}|c_{\cal P}|^{2}\,.
\end{equation}
The task of finding the coefficients $c_{\cal P}$ can be expressed in Maple
with standard functions. In Fig.~\ref{fig:fig4} we compare the results of (\ref{coeffs}) 
with direct numerical averages for fully connected graphs with $V=4$ and $V=5$ 
vertices with and without magnetic field breaking the time-reversal symmetry. 
The results agree indeed to a high
precision. Although this could be regarded merely as an additional
confirmation of the numerical procedures used in \cite{KS97}, we see the
main merit of (\ref{coeffs}) in being a very useful tool for trying to find
the solution of (\ref{family-sum}) in closed form.


\section{Wavefunction statistics}

Following the quantization outlined in section 2 a quantum wavefunction is defined as a 
set of $2B$ complex amplitudes $a_{d}$, normalized according to $\sum_{d} |a_{d}|^2=1$. 
Here we will care about stationary solution satisfying Eq.~(\ref{outin2}) (i.e. eigenstates 
of the graph with corresponding wavelength $k$). The standard localization measure is 
the Inverse Participation Ratio (IPR) which is defined as
\begin{equation}
{\cal I}=\sum_{d=1}^{2B} |a_d|^4\,.
\end{equation}
Ergodic states which occupy each directed bond with the same probability have 
${\cal I}= 1/2B$ and up to a constant factor depending on the presence of 
symmetries this is also the RMT prediction. In the other extreme ${\cal I} =0.5$ 
indicates a state which is restricted to a single bond only, i.~e.\ the greatest 
possible degree of localization. Some representative eigenstates are shown in
Fig.~\ref{fig:fig5}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\epsfxsize.7\textwidth%
\includegraphics[width=10cm]{fig5.eps}
\caption { Representative eigenstates for a fully connected graph with $V=10$.
The corresponding IPR's are (from left to right) ${\cal I}={1\over 104}\approx 0.0096\,;
\, {\cal I}={1\over 6}\approx 0.1516\,;{\cal I}={1\over 2}=0.5$}
\end{center}
\label{fig:fig5}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The key theoretical idea discussed and applied in several recent works \cite{Hel84,KH98,Kap01}
is that wavefunction intensities in a complex system can often be separated into a product 
of short-time and long-time parts, the latter being a random variable. On the other hand 
the short time part can be evaluated using information about classical dynamics. Specifically 
we have that the probability amplitude $A_d$ to return to the original state $|d\rangle$ is
\begin{equation}
\label{rampl}
A_d\equiv \langle d| U^t|d\rangle= 
\sum_{m} \left|\langle d|m\rangle\right|^2 \e^{-i\epsilon_m t}.
\end{equation}
The return probability is then
\begin{equation}
\label{rprob}
P_d(t)\equiv \left|A_d\right|^2 = \sum_{m,n} \left|\langle d|m\rangle\right|^2 \left|\langle d|n\rangle\right|^2 
\e^{i(\epsilon_m-\epsilon_n) t}
\end{equation}
Averaging over initial states and over time (typically larger than the Heisenberg time $t_H=2\pi/\Delta=2B$) 
we get
\begin{equation}
\label{avrprob}
\langle \overline{P_d(t)}^t\rangle_d \equiv \langle {1\over 2B} \sum_{t=1}^{2B} P_d(t)\rangle_d =
{1\over 2B} \sum_{t=1}^{2B} P(t)=
\langle \sum_m \left|\langle d|m\rangle\right|^4 \rangle_d \equiv \langle {\cal I}\rangle_d
\end{equation}
where $\overline {\cdots}^t$ indicates an average over time and $\langle \cdots\rangle_d$ 
over initial states. Above $P(t)$ indicates the averaged (over initial states) return 
probability. In the last equality we had used the fact that due to time-average the off
-diagonal terms averaged out to zero. Eq.~(\ref{avrprob}) expresses the mean IPR in terms 
of the quantum return probability (RP), averaged over time and initial states. The next 
step is to argue that the quantum short-time dynamics, can be described by the classical 
time evolution (see Fig.~6). The latter can be approximated semiclassically quite well based 
only on period-two PO's which correspond to trajectories which bounce back and forth between 
two vertices. These type of orbits have the lowest Lyapunov exponent (LE) and it is expected 
to have the largest influence on eigenfunction localization because classical trajectories 
can cycle in their vicinity for a relatively long time and increase the RP beyond the ergodic 
average. The resulting survival probability is
\begin{equation}
\label{Spo2}
P(t) = \left\{
\begin{array}{ll}
0; & t \quad{\rm odd}\\
\left(-1+{2\over v}\right)^4; & t \quad{\rm even}
\end{array}
\right.
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\epsfxsize.7\textwidth%
\includegraphics[width=10cm]{fig6.eps}
\caption { 
The quantum survival probability (solid line), the classical survival probability 
(dashed line) and the classical short time approximation ($\star$) based only in 
the period-two orbits as it is given by Eq.(\ref{Spo2}).}
\end{center}
\label{fig:fig6}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Indeed the period $2$ orbits totally dominate the classical and quantum RP at short times 
as can be seen in Fig.~6. Including the contribution of these orbits only, Kaplan obtained 
a mean IPR which is by a factor $\sim v$ larger than the RMT expectation, in agreement with 
numerics \cite{Kap01}. Moreover, following the same line of argumentation as in \cite{KH98} 
we get that the bulk of the IPR distribution scales as \cite{SK03}
\begin{equation}
\label{bulkI}
\tilde P({\cal I}/\langle {\cal I}\rangle )=\langle {\cal I}\rangle \,P({\cal I})
\end{equation}
indicating that
the whole bulk of ${\cal P}({\cal I})$ is effectively determined by the least unstable orbits.
This result can be nicely verified from the numerical data presented in Fig.~7.

With all this evidence for their prominent role in wavefunction localization, one clearly 
expects to see strong scarring on the period 2 orbits. Such states would essentially be 
concentrated on two directed bonds and give rise to ${\cal I}\sim 1/2$. However, in this 
region ${\cal P}({\cal I})$ is negligible (see Fig.~7). We conclude therefore that {\em 
the shortest and least unstable orbits of our system produce no visible scars.} Note that 
the same applies also to the value ${\cal I}=1/4$ expected from the V-shaped orbits of 
Fig.~2. In fact ${\cal P}({\cal I})$ has an appreciable value only for ${\cal I}\le 1/6$ 
(Fig.~7). The position of this cutoff precisely coincides with the IPR expected for states 
which are scarred by triangular orbits. They occupy six directed bonds since, due to time-
reversal symmetry, scarring on a PO and its reversed must coincide. Indeed a closer 
inspection shows that the vast majority of states at ${\cal I}\approx 1/6$ look like the 
example shown in Fig.~5. Of course the step at ${\cal I}=1/6$, which is present for any 
graph size $V$, is incompatible with the scaling of $P({\cal I})$ mentioned above and 
indeed this relation breaks down in the tails at the expected points (inset of Fig.~7). 

These results \cite{SK03} provide clear evidence for the fact that {\em enhanced wavefunction 
localization due to the presence of short unstable orbits and strong scarring can 
in principle rely on completely unrelated mechanisms} and can also leave distinct 
traces in statistical measures such as the distribution of inverse participation 
ratios (IPR). As a matter of fact in \cite{SK03} we were able to identify a necessary 
and sufficient condition for the energies of perfect scars
\begin{equation}
\label{quantization}
(kL_{d})\,\mbox{mod}\,\pi=0\qquad\forall\, d\in p
\end{equation}
where $d$ is a directed bond which belongs to the specific PO $p$. Eq.~(\ref{quantization})
is reminiscent of a simple Bohr-Sommerfeld quantization condition $kL_{p}=2n\pi$, as it 
applies, e.~g., to strong scars in billiards. However, there is an important difference: 
not only does Eq.~(\ref{quantization}) require quantization of the total action $kL_{p}$ 
of the scarred orbit, it also implies action quantization on all the visited bonds $d$.
This stronger condition can only be met if the lengths of all bonds on $p$ are rationally 
related. As in general the bond lengths are incommensurate {\em there are no perfect scars 
for generic graphs}. Nevertheless, for incommensurate bond lengths Eq.~(\ref{quantization}) 
can be approximated with any given precision and then visible scars are expected \cite{SK03}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
\epsfxsize.7\textwidth%
\includegraphics[width=10cm]{fig7.eps}
\caption { 
Probability distribution ${\cal P}({\cal I})$ of the inverse participation numbers, 
showing a steplike cutoff at ${\cal I} = 1/6$ that can be attributed to scarring on 
triangular orbits. In the inset we report the rescaled distribution $\tilde {\cal P}
({\cal I}/\langle I\rangle)$. A nice scaling is observed.
}
\end{center}
\label{fig:fig7}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Conclusions and Outlook}

We have reviewed some of our results on the statistical properties of eigenvalues and 
eigenfunctions of the unitary quantum time evolution operator derived from quantum graphs. 
We have consentrated on fully connected quantum graphs. For this familly of graphs,
the gap $\Delta_g$ between the two maximum eigenvalues of the classical evolution operator
approaches $1$ as the number of directed bonds increases, thus satisfying the (sufficient) 
condition \cite{GA04} for a graph in order to show universal spectral statistics. One 
possible approach in understanding how universality emerge is the use of combinatorial 
methods to perform the periodic-orbit sums related to spectral two-point correlations.

At the same time, we show that the existing scar theory does not explain the appearance 
of visible scars (super-scars). As a matter of fact our numerical data indicated that 
enhanced wavefunction localization due to short unstable orbits and strong scarring are 
not the same thing. 

Quantum graphs were proven throughout the years very useful models. They allowed us to 
gain a good understanding of the spectrum and eigenfunctions properties of quantum systems
with underlying classical chaotic dynamics. Semiclassics on graphs is exact, and various
quantum mechanical quantities can be written in terms of classical periodic orbits. 
These studies and their conclusions are by now well documented in the quantum chaos 
literature. But quantum chaology has various other challenges that wait to be addressed.
Among them is a quantum mechanical theory of dynamical evolution which is still a missing 
chapter. Quantum dissipation, dephasing and irreversibility (also used in the framework 
of “fidelity” studies in quantum computation) of quantum chaotic motion are notions, which 
are related with specific aspects of this evolution. It is our believe that quantum graphs 
can play a prominent role in this ultimate challenge: to develop a general theory for the 
time evolution of quantum systems with underlying classical chaotic behavior.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgments}

We would like to express our gratitude to Prof.~Uzy Smilansky to whom we owe
our interest in the subject, and to Dr. Holger Schanz who contributed a great 
deal to the results discussed in this contribution.

\vspace{5mm}

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\bibitem{KS01}T. Kottos and H. Schanz, Physica E, {\bf 9} 523, (2001). 

\bibitem{K01} L. Kaplan, Phys. Rev. E {\bf 64}, 036225 (2001).

\bibitem{BSW02} G.~Berkolaiko, H.~Schanz, R. S. Whitney, Phys. Rev. Lett. {\bf 88}, 104101 (2002).

\bibitem{BDJ02} R. Bl\"umel, Yu. Dabaghian, and R. V. Jensen, Phys. Rev. Lett. {\bf 88}, 044101 (2002);
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\bibitem{SK03} H. Schanz and T. Kottos, Phys. Rev. Lett. {\bf 90}, 234101 (2003); 
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\bibitem{GSW04} S. Gnutzmann, U. Smilansky, J. Weber, J.~Phys.~A {\bf 14}, S61 (2004). 

\bibitem{GA04} S. Gnutzmann and A. Altland, Phys. Rev. Lett. {\bf 93}, 194101 (2004).

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Phys. Rev. E {\bf 69}, 056205 (2004). 

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264}, 108-170, (1998).

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du Potentiel}, A. Dold and B. Eckmann, eds. (Springer--Verlag) 521-539.

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\end{thebibliography}




\end{document}

ioplau2e.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    INSTITUTE OF PHYSICS PUBLISHING                                   %
%                                                                      %
%   `Preparing an article for publication in an Institute of Physics   %
%    Publishing journal using LaTeX'                                   %
%                                                                      %
%    LaTeX source code `ioplau2e.tex' used to generate `author         %
%    guidelines', the documentation explaining and demonstrating use   %
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%    `iopart.cls, iopart12.clo and iopart10.clo'.                      %
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\documentclass[12pt]{iopart}
% Uncomment next line if AMS fonts required
%\usepackage{iopams}  
\begin{document}
% Journal identifier can be put here if required, e.g.
%\jl{14}

\title[Author guidelines for IOP Journals]{Preparing an article for 
publication in an Institute of Physics Publishing journal using \LaTeXe}

\author{A J Cox\dag, Adrian Corrigan\ddag\ and 
Al Troyano\ddag\footnote[3]{To
whom correspondence should be addressed.}}

\address{\dag\ Electronic Production Manager, Institute of Physics 
Publishing, Dirac
House, Temple Back, Bristol BS1 6BE, UK}

\address{\ddag\ Senior Production Editor, Institute of Physics Publishing, 
Dirac House, Temple Back, Bristol BS1 6BE, UK}


\begin{abstract}
This document describes the  preparation of an article using \LaTeXe\ and 
\verb"iopart.cls" (the IOP \LaTeXe\ preprint class file).
This class file is designed to help 
authors produce preprints in a form suitable for submission to any of the
journals published by Institute of Physics Publishing.

Authors submitting to any IOP journal, i.e.\ 
to double-column journals as well as the single-column  
ones,
should follow the guidelines set out here. 
On acceptance, their source code will be converted to 
the appropriate journal format at Institute of Physics Publishing. 
For the printed version, Times fonts (and Helvetica in double-column 
journals) will be used instead of the Computer Modern used in the preprint
form. 

\end{abstract}

\pacs{00.00, 20.00, 42.10}

% Uncomment for Submitted to journal title message
%\submitted

% Comment out if separate title page not required
\maketitle

\section{Introduction}
Institute of Physics Publishing 
(IOPP) publishes a wide range of research and review 
journals and magazines. It is wholly owned by The Institute of Physics, 
the United Kingdom professional body for physicists.  
Many of our authors use \TeX\ or \LaTeX\ to 
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Institute of Physics Publishing\\
Dirac House\\
Temple Back\\
BRISTOL\\
BS1 6BE\\
UK.
\end{itemize}
For submitted articles and those not yet accepted, the disks should be 
mailed to:
\begin{itemize}
\item[]Managing Editor\\
{\it Journal Name}\\
Institute of Physics Publishing\\
Dirac House\\
Temple Back\\
BRISTOL\\
BS1 6BE\\
UK.
\end{itemize}
where {\it Journal Name} is the name of the journal to which the article
has been submitted.

\section{Preparing your article}
Using \LaTeX\ with the \verb"iopart" class file 
provides a simple way of 
producing an article in a form suitable for publication in one 
of the IOPP journals. Authors may add their own macros 
at the start of an article 
provided they do not overwrite existing definitions and 
that they send copies of their new macros with their text file.
\verb"iopart" can be used with other package files such
as those loading the AMS extension fonts 
\verb"msam" and \verb"msbm" (these fonts provide the 
blackboard bold alphabet 
and various extra maths symbols as well as symbols useful in figure 
captions); an extra style file \verb"iopams.sty" is provide to load these
packages and provide extra definitions for bold Greek letters.

In preparing your article you are 
requested to follow these guidelines as closely as possible; this will 
minimize the amount of copy editing required and will hasten the 
production process. This is particularly important with regard to the 
reference list.

The file name can be up to eight characters long with the suffix 
\verb".tex". 
Please use files names that are likely to be unique, and include 
commented material to identify the journal, author and reference number if 
known. The first non-commented line should be 
\verb"\documentclass[12pt]{iopart}"  to load the preprint class 
file. Other standard predeclared option files can be included in 
square brackets;
copies of any non-standard options must be sent in with the source code.
Omitting \verb"[12pt]" produces an article with the normal journal
page and type sizes. Macros for the individual paper not included 
in a style file
should be inserted in the preamble to the paper with comments to
describe any complex or non-obvious ones.
Authors of very long articles may find it convenient to separate 
their article 
into a series of files each containing a section, each of which is called 
in turn by the primary file. 

The start of the article text is signalled by 
\verb"\begin{document}".
The 
journal to which the article is to be submitted may be selected with the 
command \verb"\jl{#1}", where \verb"#1" is the journal reference number 
given in 
table~\ref{jlns}. This information is used if the command \verb"\submitted" 
is used (see later) but is not essential to the running of the 
file.

\begin{table}
\caption{The reference numbers for all the journals for which the 
IOPP macros can be used.}
\label{jlns}
\lineup
\begin{indented}
\item[]\begin{tabular}{@{}ll}
\br
No&{\rm Journal}\\
\mr
\01&Journal of Physics A: Mathematical and General\\
\02&Journal of Physics B: Atomic, Molecular and Optical Physics\\
\03&Journal of Physics: Condensed Matter\\
\04&Journal Physics G: Nuclear and Particle Physics\\
\05&Inverse Problems\\
\06&Classical and Quantum Gravity\\
\07&Network\\
\08&Nonlinearity\\
\09&Quantum Optics\\
10&Waves in Random Media\\
11&Pure and Applied Optics\\
12&Physics in Medecine and Biology\\
13&Modelling and Simulation in Materials Science and Engineering\\
14&Plasma Physics and Controlled Fusion\\ 
15&Physiological Measurement\\ 
16&Combustion Theory and Modelling (new in 1997)\\
17&High Performance Polymers\\
18&Public Understanding of Science\\
19&Reports on Progress in Physics\\
20&Journal of Physics D: Applied Physics\\
21&Superconductor Science and Technology\\
22&Semiconductor Science and Technology\\
23&Nanotechnology\\
24&Measurement Science and Technology\\ 
25&Plasma Sources Science and Technology\\
26&Smart Materials Structure\\ 
27&Journal of Micromechanics and Microengineering\\ 
28&Distributed Systems Engineering\\
29&Bioimaging\\
30&Journal of Radiological Protection\\
31&European Journal of Physics\\
32&Journal of Optics\\
\br
\end{tabular}
\end{indented}
\end{table}


\section{The title and abstract page} 
The code for setting the title page information is slightly different from
the normal default in \LaTeX. 

\subsection{Titles and article types}
The title is set in bold unjustified type using the command
\verb"\title{#1}", where \verb"#1" is the title of the article. The
first letter 
of the title should be capitalized with the rest in lower case. 
Mathematical expressions within the title may be left in light-face type 
rather than bold because the Computer Modern bold maths and symbol
fonts may not be available at the size required for the title.  
The final 
printed version will have bold mathematical expressions 
in the title. If the title is unsuitable for use as a running head a short
form can be provided as an optional argument (in square brackets)
before the full title, i.e. \verb"\title[Short title]{full title}".
A short title is required 
when the title itself is too long to 
be used 
as the short title or when the title contains a footnote. 

For articles other than papers the IOPP class file, \verb"iopart.cls", 
allows 
a generic heading \verb"\article[Short title]{TYPE}{Full title}" 
and the specific 
definitions given in table~\ref{arttype}. In each case (apart from Letters
to the Editor) an 
optional argument can be used immediately after the control sequence name
to specify the short title; where no short title is given the full title
will be used as the running head at the top of each page apart from the
first. 
For Letters no short title is required as 
the running head is automatically defined to be {\it Letter to the Editor}.
The generic heading could be used for 
articles such as those presented at a conference or workshop, e.g.
\begin{verbatim}
\article[Short title]{WORKSHOP ON HIGH-ENERGY PHYSICS}{Title}
\end{verbatim}


\begin{table}
\caption{Types of article defined in the {\tt iopart.cls} 
class file.\label{arttype}}
\begin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus12pt}}l}}
\br
Command&Type&Heading on first page\\
\mr
\verb"\title{#1}"&Paper&---\\
\verb"\review{#1}"&Review&REVIEW\\
\verb"\topical{#1}"&Topical review&TOPICAL REVIEW\\
\verb"\comment{#1}"&Comment&COMMENT\\
\verb"\note{#1}"&Note&NOTE\\
\verb"\paper{#1}"&Paper&---\\
\verb"\prelim{#1}"&Preliminary communication&PRELIMINARY COMMUNICATION\\
\verb"\rapid{#1}"&Rapid communication&RAPID COMMUNICATION\\
\verb"\letter{#1}"&Letter&LETTER TO THE EDITOR\\
\verb"\article{#1}{#2}"&Other articles&Whatever is entered as {\tt 
\#1}\\
\br
\end{tabular*}
\end{table}

\subsection{Authors' names and addresses}
The next information required is the list of authors' names and 
their affiliations. For the authors' names type \verb"\author{#1}", 
where \verb"#1" is the 
list of all authors' names. The style for the names is initials then
surname, with a comma after all but the last 
two names, which are separated by `and'. Initials should {\it not} have 
full stops. Christian names may be used if 
desired. If the authors are at different addresses one of the symbols 
\dag, \ddag, \S, $\|$, \P, $^+$, *, $\sharp$ should be used after each 
surname to reference an author to his/her address. 
The symbols should be used in the order given.  If 
an author has additional information to appear as a footnote, such as 
a permanent address, and the
footnote symbols are not being used to identify an address,
the footnote should be entered after the surname
as a normal \LaTeX\ footnote, without
specifying a sign. Where footnote symbols are being used to indicate
which address the author is at, the 
symbol used for a footnote should be the next one from the list given 
above and has to be selected individually using the command
\verb"\footnote[<num>]{Text of footnote}", where \verb"<num>" is a
number representing the position of the desired symbol in the list above,
i.e.\ for 1 for \dag, 2 for \ddag, etc.

The addresses of the authors' affiliations follow the list of authors. 
Each address is set by using
\verb"\address{#1}" with the address as the single parameter in braces. 
If there is more 
than one address then the appropriate symbol should come at the start of
the address. 

\subsection{The abstract}
The abstract follows the addresses and
should give readers concise information about the content 
of the article and indicate the main results obtained and conclusions 
drawn. It should be complete in itself with no table numbers, figure 
numbers or references included and should not normally exceed 200 
words.
To indicate the start 
of the abstract type \verb"\begin{abstract}" followed by the text of the 
abstract (not in braces).  The abstract should normally be restricted 
to a single paragraph and is terminated by the command
\verb"\end{abstract}"

\subsection{Subject classification numbers}
Following the abstract come any
Physics and Astronomy Classification System (PACS) codes
or American Mathematical Society 
(AMS) classification scheme numbers.  
The command 
\verb"\pacs{#1}", with the subject classification numbers from the  
Physics and Astronomy Classification Scheme as the parameter, defines 
the subject area of the paper (or for a single number \verb"\pacno{#1}").  
If PACS numbers are not readily 
available, {\it Physics Abstracts\/} classification scheme numbers can be 
given instead. If this command  is omitted the 
classification numbers for indexing
will be allocated by IOPP staff. It is 
unnecessary to supply PACS numbers for {\it Inverse 
Problems} and {\it Physics in Medicine and Biology}.
AMS classification numbers may be given as well as, or instead of, PACS 
numbers for mathematical articles, they are specified using the 
command \verb"\ams{#1}".

After any classification numbers the command \verb"\submitted" can be
inserted to print out a line indicating the article has been submitted to
the appropriate Institute of Physics Journal. This command is optional.
 
\subsection{Making a separate title page}
The command \verb"\maketitle" forces a page break after the point where it
is inserted and so to keep the header material on a separate page from the
body of the text insert \verb"\maketitle" or \verb"\newpage" after the
classification codes and \verb"\submitted" commands (if present) 
or the end of the abstract. 
If \verb"\maketitle" is not included the text of the
article will start immediately after the abstract.  


\subsection{Sample coding for the start of an article}
The code for the start of a title page of a typical paper might read:
\begin{verbatim}
\documentclass[12pt]{iopart}
\begin{document}
\jl{4}
\title[The anomalous magnetic moment of the 
neutrino]{The anomalous magnetic moment of the 
neutrino and its relation to the solar neutrino problem}

\author{P J Smith\dag, T M Collins\ddag, 
R J Jones\ddag\footnote[3]{Present address:
Department of Physics, University of Bristol, Tyndalls Park Road, 
Bristol BS8 1TS, UK.} and Janet Williams\P}

\address{\dag\ Mathematics Faculty, Open University, 
Milton Keynes MK7~6AA, UK}
\address{\ddag\ Department of Mathematics, 
Imperial College, Prince Consort Road, London SW7~2BZ, UK}
\address{\P\ Department of Computer Science, 
University College London, Gower Street, London WC1E~6BT, UK}

\begin{abstract}
...
\end{abstract}

\pacs{1315, 9440T}
\submitted
\maketitle
\end{verbatim}

\section{The text}
\subsection{Sections, subsections and subsubsections}
The text of papers and reviews, but not comments or letters, should be 
divided into sections, subsections and, where necessary, 
subsubsections. To start a new section, end the previous paragraph and 
then include \verb"\section" followed by the section heading within braces. 
Numbering of sections is done {\it automatically} in the headings: 
sections will be numbered 1, 2, 3, etc, subsections will be numbered 
2.1, 2.2,  3.1, etc, and subsubsections will be numbered 2.3.1, 2.3.2, 
etc.  Cross references to other sections in the text should, where
possible, be made using 
labels (see section~\ref{xrefs}) but can also
be made manually. Subsections and subsubsections are similar to 
sections but 
the commands are \verb"\subsection" and \verb"\subsubsection" respectively. 
Sections have a bold heading, subsections an italic heading and 
subsubsections an italic heading with the text following on directly.
\begin{verbatim}
\section{This is the section title}
\subsection{This is the subsection title}
\end{verbatim}


The first section is normally an introduction,  which should state clearly 
the object of the work, its scope and the main advances reported, with 
brief references to relevant results by other workers. In long papers it is 
helpful to indicate the way in which the paper is arranged and the results 
presented.

For articles not divided into sections, precede the 
start of the text (without leaving a blank line)
with the command \verb"\nosections", which provides the 
appropriate space and causes the paragraph indentation to be cancelled 
for the first paragraph.

Footnotes should be avoided whenever possible. If required they should be 
used only for brief notes that do not fit conveniently into the text. The 
standard \LaTeX\ macro \verb"\footnote" should be used and will normally
give an appropriate symbol; if a footnote sign needs to be specified
directly \verb"\footnote[<num>]{Text}" can be used instead where
\verb"<num>" is the number of the appropriate symbol as discussed in 
relation to authors (1~=~\verb"\dag", 2~=~\verb"\ddag", 3~=~\verb"\S", 
etc).  


\subsection{Appendices}
Technical detail that it is necessary to include, but that interrupts 
the flow of the article, may be consigned to an appendix. If there are 
two or more appendices they will be called Appendix A, Appendix B, etc. 
Numbered equations will be in the form (A.1), (A.2), etc,
figures will appear as figure A1, figure B1, etc and tables as table A1,
table B1, etc.

The command \verb"\appendix" is used to signify the start of the
appendixes. Thereafter \verb"\section", \verb"\subsection", etc, will 
give headings appropriate for an appendix. To obtain a simple heading of 
`Appendix' use the code \verb"\section*{Appendix}". If it contains
numbered equations, figures or tables the command \verb"\appendix" should
precede it and \verb"\setcounter{section}{1}" must follow it. 
 

\subsection{Acknowledgments}
Authors wishing to acknowledge assistance or encouragement from 
colleagues, special work by technical staff or financial support from 
organizations should do so in an unnumbered Acknowledgments section 
immediately following the last numbered section of the paper. The 
command \verb"\ack" sets the acknowledgments heading as an unnumbered
section. For Letters 
\verb"\ack" does not set a heading but leaves a line space and does not 
indent the next paragraph. 


\subsection{Some matters of style}
It will help your readers if your article is written in a clear,
consistent and concise manner. Copy preparation at Institute of Physics
Publishing will try to make sure that your work is presented to its
readers in the best possible way without sacrificing the individuality of
your writing.

The main elements of consistency and style we look for 
are presented in the booklet 
{\it Notes for Authors} (available upon request from Institute of Physics 
Publishing, 
Dirac House, Temple Back, Bristol, BS1 6BE, UK or from our World Wide
Web site (\verb"http://www.iop.org/Journals/nfa/"). Some recommended 
points to note, however, are the following.
\begin{enumerate}
\item Authors are often inconsistent in the use of `ize' and `ise' endings.
We recommend using `-ize' spellings (diagonalize, 
renormalization, minimization, etc) but there are some common 
exceptions to this, for example: devise, 
promise and advise.

\item English spellings are preferred (colour, flavour, behaviour, 
tunnelling, artefact, focused, focusing, fibre, etc). We write of a 
computer program on disk; otherwise, we use `programme' and `disc'.

\item Compound words beginning `non-' or `self-' are easier to read and
understand when hyphenated (non-existent, self-consistent, etc).

\item The words table, figure, equation and reference should be written 
in full and {\bf not} contracted to Tab., fig., eq. and ref.
\end{enumerate}

It will help the copy preparation and avoid unnecessary errors if you
carefully check your article for accuracy, consistency and clarity before
submission. Remember that your article will probably be read by many
people whose native language is not English and who may not therefore 
be aware of many of the subtle meanings of words or idiomatic phases
present in the English language. It therefore helps if you try and keep
sentences as short and simple as possible.

\section{Mathematics}
\subsection{Two-line constructions}
The great advantage of \TeX\ and \LaTeX\ 
over other text processing systems is their 
ability to handle mathematics to almost any degree of complexity. However, 
in order to produce an article suitable for publication within a journal, 
authors should exercise some restraint on the constructions used.
For simple fractions in the text the solidus \verb"/", as in 
$\lambda/2\pi$, should be used instead of \verb"\frac" or \verb"\over", 
care being taken to use parentheses where necessary to avoid ambiguity, 
for example to distinguish between $1/(n-1)$ and $1/n-1$. Exceptions to 
this are the proper fractions $\frac12$, $\frac13$, $\frac34$, 
etc, which are better left in this form. In displayed equations 
horizontal lines are preferable to solidi provided the equation is 
kept within a height of two lines. A two-line solidus should be 
avoided where possible; the construction $(\ldots)^{-1}$ should be 
used instead; for example use:
\begin{equation*}
\frac{1}{M_{\rm a}}\left(\int^\infty_0{\rm d}
\omega\;\frac{|S_o|^2}{N}\right)^{-1}
\end{equation*}
instead of
\begin{equation*}
\frac{1}{M_{\rm a}}\biggl/\int^\infty_0{\rm d}
\omega\;\frac{|S_o|^2}{N}.
\end{equation*}


\subsection{Roman and italic in mathematics}
In mathematics mode \LaTeX\ automatically sets variables in an italic 
font. In most cases authors should accept this italicization. However, 
there are some cases where it is better to use a Roman font; for 
instance, IOPP journals use a Roman d for a differential d, a Roman e 
for an exponential e and a Roman i for the square root of $-1$. To 
accommodate this and to simplify the  typing of equations we have 
provided some extra definitions. \verb"\rmd", \verb"\rme" and \verb"\rmi" 
now gives Roman d, e and i respectively for use in equations, 
e.g.\ $\rmi x\rme^{2x}\rmd x/\rmd y$ 
is obtained by typing \verb"$\rmi x\rme^{2x}\rmd x/\rmd y$". 
 

Certain other common mathematical functions, such as cos, sin, det and 
ker, should appear in Roman type. \LaTeX\ provides macros for 
most of these functions 
(in the cases above, \verb"\cos", \verb"\sin", \verb"\det" and \verb"\ker" 
respectively), we have also provided 
additional definitions for $\Tr$, $\tr$ and 
$\Or$ (\verb"\Tr", \verb"\tr" and \verb"\Or", respectively). 

Subscripts and superscripts should be in Roman type if they are labels 
rather than variables or characters that take values. For example in the 
equation
\[
\epsilon_m=-g\mu_{\rm n}Bm
\]
$m$, the $z$ component of the nuclear spin, is italic because it can have 
different values whereas n is Roman because it 
is a label meaning nuclear ($\mu_{\rm n}$ 
is the nuclear magneton).

\subsection{Alignment of mathematics}
\subsubsection{Alignment on the secondary margin.}
IOPP style for displayed mathematics in single-column journals
is not to centre equations, 
as \LaTeX\ normally does, but to have each equation indented to a 
secondary margin a fixed 
distance from the left-hand margin of the text, except for long 
equations that will just fit on one line, or need to be continued on 
subsequent lines, which start full left.  
Any continuation lines are indented the fixed amount. 
The macros in the IOPP preprint style automatically 
line equations up on the 
secondary margin unless they are set within double dollar signs.
Thus the use of double dollar signs should be avoided and the alternative
\verb"\[ ... \]" should be used instead for unnumbered equations.
The equation environment should normally be used for numbered single-line
equations and the eqnarray array environment for multiline equations.
It is then only necessary to indicate which lines 
should start full left and this is done by including \verb"\fl" (full
left) at the start of the lines. 
Thus the equations:
\begin{eqnarray}
\phi_{k}(\vec{r})=(2\pi)^{-3/2} \exp(\rmi\vec{k}\cdot\vec{r}) \\
N^+=\exp(\case12\pi\nu)\Gamma(1-\rmi\nu).
\end{eqnarray} 
are set with the code
\begin{verbatim}
\begin{eqnarray}
\phi_{k}(\vec{r})=(2\pi)^{-3/2} \exp(\rmi\vec{k}\bdot\vec{r}) \\
N^+=\exp(\case12\pi\nu)\Gamma(1-\rmi\nu).
\end{eqnarray} 
\end{verbatim}
Where an equation will not fit on a line if indented but would if it 
were not, then the equation is started full left and this is achieved 
simply by adding \verb"\fl" to the start of the line. For example
the equation
\begin{equation}
\fl R_{\rho l m,\rho'l'm'}(E)=\frac{1}{r_0}\sum_{i,j}\;\langle\rho l m r_0
\mid\Phi_i\rangle\; [(H_{\Omega}+B)-ES_{\Omega}]^{-1}_{i,j}\;\langle\Phi_j
\mid\rho'l'm'r_0\rangle. 
\end{equation}
does not fit on the line if indented to the secondary margin but fits in 
comfortably when full left.


For equations which do not fit on one line, even if started full left,
the first line should be set full left with the turnover lines at the 
secondary margin. This is achieved by using the eqnarray
environment
and adding \verb"\fl" at the start 
of the first line and \verb"\\" at the end of each line (apart from
the final line of the equation). Equations should be split at 
mathematically sound points, often at =, + or $-$ signs or between 
terms multiplied together. The connecting signs are not repeated and 
appear only at the beginning of the turned-over line. A multiplication 
sign should be added to the start of turned-over lines where the break 
is between two multiplied terms. Where an equation is broken at an 
equals sign (or similar, i.e.\ $\equiv$, $\le$, $\sim$, etc) the sign 
is made more prominent by aligning it to the left of the secondary 
margin; where it is a +, $-$ or $\times$ the sign goes to the right. 
Alignment to the left of the secondary margin is achieved by adding 
\verb"\lo" in front of the sign (and enclosing the sign within braces if it 
consists of more than one character or control sequence, e.g.\ 
\verb"\lo{:=}"). An example demonstrating these features is: 
\begin{eqnarray} 
\fl\langle\cos(q\Omega_s)\rangle=\frac12\int^\infty_0 
\frac{k_s(b)}{k^{\rm tot}_s}\{\cos[q\Omega^{\rm o}_S(b_f,R^s_{x})]+
\cos[q\Omega^{\rm i}_S(b_f,R^s_{x})]\}
2\pi b\, \rmd b\nonumber\\ 
\lo=\sum_c{(\mu^s_c)^2/\vert\Delta V'_s(R^s_{\rm c})\vert \over \sum_n 
(R^s_n\mu^s_n)^2(1-V^s_n/E)^{1/2}/\vert\Delta'_s(R^s_n)\vert}\nonumber\\ 
\times \frac12\int^{b_{\rm max}}_0 
\{\cos[q\Omega^{\rm o}_S(b_f,R^s_{\rm c})]
+\cos[q\Omega^{\rm i}_S(b_f,R^s_{\rm c})]\} 
b\, \rmd b/v_s(b,R^s_{\rm c}). 
\end{eqnarray} 
where a simplified version of the 
code used is: 
\begin{verbatim} 
\begin{eqnarray} 
\fl    <first line>  \nonumber\\ 
\lo=   <second line> \nonumber\\
\times <third line>  
\end{eqnarray} 
\end{verbatim} 

Note that alignment at the secondary margin normally takes 
precedence over aligning equals signs so there is usually no need 
to include any ampersands within the 
\verb"eqnarray" environment.


\subsubsection{Secondary alignment.}
While the primary alignment either on the secondary 
margin or full left is adequate in most cases 
there are examples where additional 
alignment is desirable. Firstly, for repeated series of short 
equations, secondly for
equations with attached conditions and thirdly for connected 
series of equations with a short left-hand side which together
occupy more than a full line but where each individual 
part is short. In these cases the \verb"eqnarray" environment 
should be used; there will still be alignment at the 
secondary margin but ampersands should be positioned to 
provide the secondary alignment. For equations with conditions the
space separating the longest part from its condition is provided by 
\verb"\qquad". Examples of equations requiring secondary alignment are:
\begin{eqnarray}
A^{(3/2)}=A^{(+)}-A^{(-)}&(I=\case32)\\
A^{(1/2)}=A^{(+)}+2A^{(-)}\qquad&(I=\case12)\\            
A^{(0)}&({\rm otherwise}).
\end{eqnarray}
which is obtained with the code
\begin{verbatim}
\begin{eqnarray}
A^{(3/2)}=A^{(+)}-A^{(-)}&(I=\case32)\\
A^{(1/2)}=A^{(+)}+2A^{(-)}\qquad&(I=\case12)\\            
A^{(0)}&({\rm otherwise}).
\end{eqnarray}
\end{verbatim}
and
\begin{eqnarray}
C(12)&=[\vec\pi(x)\cdot\vec\phi(x+r)]\nonumber\\
&\simeq 1-{\rm const}{r^2\over L^2}\int^L_r{x\, \rmd x\over 
x^2}+\cdots\\
&\simeq 1-{\rm const}{r^2\over L^2}\ln\left({L\over r}\right)+\cdots.
\end{eqnarray}
for which the code is
\begin{verbatim}
\begin{eqnarray}
C(12)&=[\vec\pi(x)\cdot\vec\phi(x+r)]\nonumber\\
&\simeq 1-{\rm const}{r^2\over L^2}\int^L_r{x\, \rmd x\over 
x^2}+\cdots\\
&\simeq 1-{\rm const}{r^2\over L^2}\ln\left({L\over r}\right)+\cdots.
\end{eqnarray}
\end{verbatim}

\subsection{Displayed equations in double-column journals}
The way equations are displayed in the Institute of Physics Publishing
double-column journals differs from that in single-column journals. 
However
authors submitting to double-column journals can produce their equations
as described above for single-column journals
and conversion to the proper double-column format will
take place here as proofs are being prepared. 

\subsection{Special characters for mathematics}
Bold italic characters are used in our journals to signify vectors (rather
than using an upright bold or an over arrow). To obtain this effect
use \verb"\bi{#1}" within maths mode, e.g. $\bi{ABCdef}$. If upright 
bold characters are required in maths use \verb"\mathbf{#1}" within maths
mode, e.g. $\mathbf{XYZabc}$. The calligraphic (script) uppercase alphabet
is obtained with \verb"\mathcal{AB}" or \verb"\cal{CD}" 
($\mathcal{AB}\cal{CD}$).

The American Mathematical Society provides a series of extra symbol fonts
to use with \LaTeX\ and packages containing the character definitions to
use these fonts. Authors wishing to use Fraktur 
\ifiopams$\mathfrak{ABC}$ \fi
or Blackboard Bold \ifiopams$\mathbb{XYZ}$ \fi can include the appropriate
AMS package (e.g. amsgen, amsfonts, amsbsy, amssymb) with a 
\verb"\usepackage" command or add the command \verb"\usepackage{iopams}"
which loads the four AMS packages mentioned above and also provides
definitions for extra bold characters (all Greek letters and some
additional other symbols). 

The package iopams uses the definition \verb"\boldsymbol" in amsbsy
which allows individual non-alphabetical symbols and Greek letters to be 
made bold within equations.
The bold Greek lowercase letters \ifiopams$\balpha \ldots\bomega$,\fi 
are obtained with the commands 
\verb"\balpha" \dots\ \verb"\bomega" (but note that
bold eta\ifiopams, $\bfeta$,\fi\ is \verb"\bfeta" rather than \verb"\beta")
and the capitals\ifiopams, $\bGamma\ldots\bOmega$,\fi\ with commands 
\verb"\bGamma" \dots\
\verb"\bOmega". Bold versions of the following symbols are
predefined in iopams: 
bold partial\ifiopams, $\bpartial$,\fi\ \verb"\bpartial",
bold `ell'\ifiopams, $\bell$,\fi\  \verb"\bell", 
bold imath\ifiopams, $\bimath$,\fi\  \verb"\bimath", 
bold jmath\ifiopams, $\bjmath$,\fi\  \verb"\bjmath", 
bold infinity\ifiopams, $\binfty$,\fi\ \verb"\binfty", 
bold nabla\ifiopams, $\bnabla$,\fi\ \verb"\bnabla", 
bold centred dot\ifiopams, $\bdot$,\fi\  \verb"\bdot", other 
characters are made bold using 
\verb"\boldsymbol{\symbolname}".

Table~\ref{math-tab2} lists some other macros for use in 
mathematics with a brief description of their purpose.
Both \verb"\ms" (medium space) and \verb"\bs" (big space) can be used to
provide extra spacing between lines of a displayed equation or table.
This space may be necessary when several separate equations are within the
same equation environment. 

\begin{table}
\caption{Other macros defined in IOPP macros for use in maths.
\label{math-tab2}}
\begin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus
12pt}}l}}
\br
Macro&Result&Description\\
\mr
Spaces\\
\verb"\fl"&&Start line of equation full left\\
\verb"\ms"&&Spread out lines in displayed equations slightly ($\sim$3pt)\\
\verb"\bs"&&Bigger space ($\sim$6pt) to separate lines in displays\\
\verb"\ns"&&Small negative space between lines in displays\\
\bs
\multispan{3}{For symbols to left of 5 pica indent\hfill}&\\
\verb"\lo{#1}"&$\#1$&Any symbol overhanging to left\\
\verb"\eql"&$=$&Left overhanging equals sign\\
\verb"\lsim"&$\sim$&Left overhanging tilde\\ 
\verb"\lsime"q&$\simeq$&Left overhanging approximately equals\\
\verb"\lequiv"&$\equiv$&Left overhanging equivalent sign\\
\bs
Miscellaneous\\
\verb"\case{#1}{#2}"&$\case{\#1}{\#2}$&Text style fraction in display\\
\verb"\Tr"&$\Tr$&Roman Tr (Trace)\\
\verb"\tr"&$\tr$&Roman tr (trace)\\
\verb"\Or"&$\Or$&Roman O (of order of)\\
\verb"\tdot{#1}"&$\tdot{x}$&Triple dot over character\\
\verb"\lshad"&$\lshad$&Text size left shadow bracket\\
\verb"\rshad"&$\rshad$&Text size right shadow bracket\\
\br
\end{tabular*}
\end{table}

\subsection{Miscellaneous points}
Exponential expressions, especially those containing subscripts or 
superscripts, are clearer if the notation $\exp(\ldots)$ is used except for 
simple examples. For instance $\exp[\rmi(kx-\omega t)]$ and $\exp(z^2)$ are 
preferred to $\e^{\rmi(kx-\omega t)}$ and $\e^{z^2}$, but 
$\e^2$ 
is acceptable. Similarly the square root sign $\sqrt{\phantom{b}}$ should 
only be used with relatively
simple expressions, e.g.\ $\sqrt2$ and $\sqrt{a^2+b^2}$;
in other cases the 
power $1/2$ should be used.

It is important to distinguish between $\ln = \log_\e$ and $\lg 
=\log_{10}$. Braces, brackets and parentheses should be used in the 
following order: $\{[(\;)]\}$. The same ordering of brackets should be 
used within each size. However, this ordering can be ignored if the
brackets have a 
special meaning (e.g.\ if they denote an average or a function).  Decimal 
fractions should 
always be preceded by a zero: for example 0.123 {\bf not} .123. For long 
numbers commas are not inserted but instead a thin space is added after 
every third character away from the position of the decimal point unless 
this leaves a single separated character: e.g.\ $60\,000$, $0.123\,456\,78$ 
but 4321 and 0.7325.

Equations that are referred to in the text should be numbered with 
the number on the right-hand side. 


\subsection{Equation numbering\label{eqnum}}
\LaTeX\ provides facilities for automatically numbering equations 
and these should be used where possible. Sequential numbering (1), (2), 
etc, is the default numbering system although, if the command
\verb"\eqnobysec" is included in the preamble, equation numbering
by section is obtained, e.g.\ 
(2.1), (2.2), etc. In articles with several appendixes equation numbering
by section is useful in the appendixes even when sequential numbering has
been used throughout the main body of the text and is switched on by the 
\verb"\appendix" command. Equation numbering by section {\it must} 
be used for {\it Reports on Progress in 
Physics}. When referring to an equation in the text, either put 
the equation number, in brackets, e.g.\ `as in (2)', or spell out the 
word equation in full, e.g.\ `if equation (2) is factorized'; do not 
use abbreviations such as eqn or eq.
When cross-referencing is used, \verb"\ref{<label>}"
 will produce `(\verb"<eqnum>")',
\verb"\eref{<label>}" produces `equation (\verb"<eqnum>")' and
\verb"Eref{<label>}" produces `Equation (\verb"<eqnum>")', 
where \verb"<label>" 
is the
label to produce equation number \verb"<eqnum>".


If an equation number is centred between lines then the 
command \verb"\eqalign{...}" 
can be used within the `equation' environment.
After \verb"\begin{equation}" enclose the lines over 
which the number is
to be centred 
within \verb"\eqalign{...}"  with 
\verb"\\" or \verb"\cr"
at the end of each line. Ampersands are unnecessary within the
\verb"\eqalign" but can be used for secondary alignment if necessary.
The code 
\begin{verbatim}
\begin{equation}
\eqalign{T_{11}&=(1+P_\e)I_{\uparrow\uparrow}-(1-P_\e)
I_{\uparrow\downarrow}\\
T_{-1-1}&=(1+P_\e)I_{\downarrow\downarrow}-(1-P_\e)
I_{\uparrow\downarrow}\\
S_{11}&=(3+P_\e)I_{\downarrow\uparrow}-(3-P_e)I_{\uparrow\uparrow}\\
S_{-1-1}&=(3+P_\e)I_{\uparrow\downarrow}-(3-P_\e)
I_{\downarrow\downarrow}}
\end{equation}
\end{verbatim}
gives four equations with a centred 
number:
\begin{equation}
\eqalign{T_{11}&=(1+P_\e)I_{\uparrow\uparrow}-(1-P_\e)
I_{\uparrow\downarrow}\\
T_{-1-1}&=(1+P_\e)I_{\downarrow\downarrow}-(1-P_\e)I_{\uparrow\downarrow}\\
S_{11}&=(3+P_\e)I_{\downarrow\uparrow}-(3-P_e)I_{\uparrow\uparrow}\\
S_{-1-1}&=(3+P_\e)I_{\uparrow\downarrow}-(3-P_\e)
I_{\downarrow\downarrow}}
\end{equation}
Note that the secondary alignment at the equals signs would not normally
be necessary but is included here for demonstration purposes.


Sometimes it is useful to number equations as parts of the same
basic equation. This can be accomplished by inserting the 
commands \verb"\numparts" before the equations concerned and 
\verb"\endnumparts" when reverting to the normal sequential numbering.
The equations below show the previous equations numbered as separate parts
using \verb"\numparts ... \endnumparts" and the \verb"eqnarray"
environment
\numparts
\begin{eqnarray}
T_{11}&=(1+P_\e)I_{\uparrow\uparrow}-(1-P_\e)
I_{\uparrow\downarrow}\\
T_{-1-1}&=(1+P_\e)I_{\downarrow\downarrow}-(1-P_\e)I_{\uparrow\downarrow}\\
S_{11}&=(3+P_\e)I_{\downarrow\uparrow}-(3-P_e)I_{\uparrow\uparrow}\\
S_{-1-1}&=(3+P_\e)I_{\uparrow\downarrow}-(3-P_\e)
I_{\downarrow\downarrow}
\end{eqnarray}
\endnumparts


\subsection{Miscellaneous extra commands for displayed equations}
The \verb"\cases" command of Plain \TeX\ is available
for use with \LaTeX\ but has been amended slightly to 
increase the space between the equation and the condition. 
\Eref{cases} 
demonstrates simply the output from the \verb"\cases" command
\begin{equation}
\label{cases}
X=\cases{1&for $x \ge 0$\\
-1&for $x<0$\\}
\end{equation}
The code used was:
\begin{verbatim}
\begin{equation}
\label{cases}
X=\cases{1&for $x \ge 0$\\
-1&for $x<0$\\}
\end{equation}
\end{verbatim}

To obtain text style fractions within displayed maths the command 
\verb"\case{#1}{#2}" can be used (see equations (2) and (5)) instead
of the usual \verb"\frac{#1}{#2}" command or \verb"{#1 \over #2}".

When two or more short equations are on the same line they should be 
separated by a `qquad space' (\verb"\qquad"), rather than
\verb"\quad" or any combination of \verb"\,", \verb"\>", \verb"\;" 
and \verb"\ ".
 


\section{Referencing}
Two different styles of referencing are in common use: the Harvard 
alphabetical system and the Vancouver
numerical system. All the IOPP journals allow 
the use of the Harvard system but the numerical system should {\bf not} be 
used in {\it Physics in Medicine and Biology}.
Brief descriptions of the use of the two 
referencing systems are given below.

\subsection{Harvard system}
In the Harvard system the name of the author appears in the text together 
with the year of publication. As appropriate, either the date or the name 
and date are included within parentheses. Where there are only two authors 
both names should be given in the text; if there are more than two 
authors only the first name should appear followed by `{\it et al}' 
(which can be obtained by 
typing \verb"\etal"). When two or 
more references to work by one author or group of authors occur for the 
same year they should be identified by including a, b, etc after the date 
(e.g.\ 1986a). If several references to different pages of the same article 
occur the appropriate page number may be given in the text, e.g.\ Kitchen 
(1982, p 39).

The reference list at the end of an article consists of an 
unnumbered section containing an
alphabetical listing by authors' names and in date order for each 
author or group of identical authors. The reference list in the 
preprint style is started by including the command
\verb"\section*{References}" and then
\verb"\begin{harvard}".
There will be two basic types of 
entries within the reference list: (i) those to journal articles and 
(ii) those to books, conference proceedings and reports. For both of 
these types of references \verb"\item[]"
is required before the start of an individual reference.
The reference list is completed with \verb"\end{harvard}".
There is also a shortened form of the coding; \verb"\section*{References}"
and \verb"\begin{harvard}" can be replaced by the single command
\verb"\References" and \verb"\end{harvard}" can be shortened to
\verb"\endrefs".


\subsubsection{References to journal articles.}
A normal reference to a journal article contains three changes of 
font:
the authors and date appear in Roman type, the journal title in 
italic, the volume number in bold and the page numbers in Roman again. 
A typical journal entry would be:

\smallskip
\begin{harvard}
\item[] Cisneros A 1971 {\it Astrophys.\ Space Sci.} {\bf 10} 87
\end{harvard}
\smallskip

\noindent which would be obtained by typing, within the references
environment 
\begin{verbatim}
\item[] Cisneros A 1971 {\it Astrophys. Space Sci.} {\bf 10} 87
\end{verbatim}

Features to note are the following.

(i) The authors should be in the form surname (with only the first 
letter capitalized) {\bf followed} by the initials with {\bf no} 
periods after the initials. Authors should be separated by a comma 
except for the last two which should be separated by `and' with no 
comma preceding it. For journals that accept titles of articles in the 
reference list,  the title should be in Roman (upright)
lower case letters, except for an initial 
capital, and should follow the date.

(ii) The journal is in italic and is abbreviated. \ref{jlabs} 
gives a list of 
macros that will give the correct abbreviation for 
many of the common journals. If a journal has several parts denoted by 
different letters the part letter
should be inserted after the journal in Roman type, e.g.\ 
{\it Phys.\ Rev.\ \rm A}. An exception to this is 
{\it Physics Letters} where 
the part letter is included in the volume number.

(iii) The volume number is bold; the page number is Roman.
 Both the initial and final page 
numbers should be given where possible. The final page number should be in 
the shortest possible form and separated from the initial page number by an 
en rule (\verb"--"), e.g.\ 1203--14.

(iv) Where there are two or more references with identical authors, 
the authors' names should not be repeated but should be replaced by 
\verb"\dash" on the second and following occasions. Thus
\begin{verbatim}
\item[]Davis R, Mann A K and Wolfenstein L 1989  {\it Ann. Rev. Nucl.
Part. Sci.} {\bf 39} 467
\item[]\dash 1990 Private communication 
\end{verbatim}



\subsubsection{References to books, conference proceedings and reports.}
References to books, proceedings and reports are similar, but have only two
changes of font. The authors and date of publication are in Roman, the 
title of the book is in italic, and the editors, publisher, 
town of publication 
and page number are in Roman. A typical reference to a book and a
conference paper might be

\smallskip
\begin{harvard}
\item[] Dorman L I 1975 {\it Variations of Galactic Cosmic Rays} 
(Moscow: Moscow State University Press) p~103
\item[] Caplar R and Kulisic P 1973 {\it Proc.\
Int.\ Conf.\ on Nuclear Physics (Munich)} vol~1 (Amsterdam:  
North-Holland/American Elsevier) p~517
\end{harvard}
\smallskip

\noindent which would be obtained by typing
\begin{verbatim}
\item[] Dorman L I 1975 {\it Variations of Galactic Cosmic Rays} 
(Moscow: Moscow State University Press) p~103
\item[] Caplar R and Kulisic P 1973 {\it Proc. Int. Conf. on Nuclear 
Physics (Munich)} vol~1 (Amsterdam: North-Holland/American 
Elsevier) p~517
\end{verbatim}
\noindent respectively.


Features to note are the following.

(i) Book titles are in italic and should be spelt out in full with 
initial capital letters for all except minor words. Words such as 
Proceedings, Symposium, International, Conference, Second, etc should 
be abbreviated to Proc., Symp., Int., Conf., 2nd, 
respectively, but the rest of the title should be given in full, 
followed by the date of the conference and the 
town or city where the conference was held. For 
Laboratory Reports the Laboratory should be spelt out wherever 
possible, e.g.\ {\it Argonne National Laboratory Report}.

(ii) The volume number as, for example, vol~2, should be followed by 
the editors, if any, in a form such as ed~A~J~Smith and P~R~Jones. Use 
\etal if there are more than two editors. Next comes the town of 
publication and publisher, within brackets and separated by a colon, 
and finally the page numbers preceded by p if only one number is given 
or pp if both the initial and final numbers are given.

Cross referencing between the text and the 
reference list is not necessary for alphabetic referencing 
in the Harvard system as adding or deleting a reference 
does not normally change any of the other references.

\subsection{Numerical system}
In the numerical system references are numbered sequentially 
throughout the text. The numbers occur within square brackets and one 
number can be used to designate several references. A numerical 
reference list in the preprint style is started by including the 
command \verb"\section*{References}" and then
\verb"\begin{thebibliography}{<num>}", where \verb"<num>" is the largest
number in the reference list (or any other number with the same number
of digits).  The 
reference list gives the references in 
numerical, not alphabetical, order and is completed by
\verb"\end{thebibliography}". Short forms of the commands are again
available: \verb"\Bibliography{<num>}" can be used at the start of the
references section and \verb"\endbib" at the end.
(Note that footnotes should not be 
part of a numerical reference system, but should be included in the 
text using the symbols \dag, \ddag, etc.)

References to journals and books are similar to those in the Harvard 
system, except that two or more references with identical authors are 
spelt out in full, i.e.\ they are {\bf not} replaced with \verb"\dash". 
When one number covers two or more separate references \verb"\nonum"
or \verb"\par\item[]" should be included at
the start of each reference in a group after the first.
A typical numerical reference list might begin

\smallskip

\numrefs{1}
\item Dorman L I 1975 {\it Variations of Galactic Cosmic Rays} (Moscow: 
Moscow State University Press) p~103
\item Caplar R and Kulisic P 1973 {\it Proc.\ Int.\ Conf.\ 
on Nuclear Physics (Munich)} vol~1 (Amsterdam:  
North-Holland/American Elsevier) p~517
\item Cisneros A 1971 {\it Astrophys. Space Sci.} {\bf 10} 87
\endnumrefs
\smallskip

\noindent which would be obtained by typing

\begin{verbatim}
\item Dorman L I 1975 {\it Variations of Galactic Cosmic Rays} 
(Moscow: Moscow State University Press) p~103
\item Caplar R and Kulisic P 1973 {\it Proc. Int. Conf. 
on Nuclear Physics (Munich)} vol~1 (Amsterdam:  
North-Holland/American Elsevier) p~517
\item Cisneros A 1971 {\it Astrophys. Space Sci.} {\bf 10} 87
\end{verbatim}

The point to note is that this is identical to the entries in the 
Harvard system except that square brackets following
\verb"\item" are no longer required.

\subsection{Reference lists}
A complete reference should provide the reader with enough information to 
locate the article concerned and should consist of: name(s) and initials, 
date published, title of journal or book, volume number, editors, if any, 
and town of publication and publisher in parentheses for books, 
and finally the 
page numbers. Titles of journal articles may also be included.
Up to twenty authors may be given in a particular reference; where 
there are more than twenty only the first should be given followed by 
`{\it et al}'. Abbreviations of the names of periodicals used by Institute
of Physics Publishing are usually the same as those 
given in British Standard 
BS 4148: 1985. If an author is unsure of an abbreviation and the 
journal is not given in \ref{jlabs}, 
it is best to leave the title in 
full. The terms {\it loc.\ cit.\ }and {\it ibid.\ }should not be used. 
Unpublished conferences and reports should generally not be included 
in the reference list and articles in the course of publication should 
be entered only if the journal of publication is known. References to 
preprints should give the title of the preprint and/or preprint number 
(if relevant). A thesis submitted for a higher degree may be included 
in the reference list if it has not been superseded by a published 
paper and is available through a library; sufficient information 
should be given for it to be traced readily.

\section{Cross referencing\label{xrefs}}
The facility to cross reference items in the text is very useful when 
composing articles the precise form of which is uncertain at the start 
and where revisions and amendments may subsequently be made. When using 
cross referencing labels are given to elements in the text, for 
instance sections, figures, tables or equations and the elements may 
be referred to elsewhere
in the text by using the label. When the article is 
first processed the labels are read in and assigned, e.g.\ 2.1 for a 
subsection or (4) for an equation number. When the article is 
processed a second or subsequent time the label assignments are read 
in at the start of the file and the correct values given in the text. 
\LaTeX\ provides excellent facilities for doing cross-referencing
and these can be very useful in preparing articles.

\subsection{References}
Cross referencing is useful for numeric reference lists because, if it 
is used, adding 
another reference to the list does not then involve renumbering all 
subsequent references. It is not necessary for referencing 
in the Harvard system where the final reference list is alphabetical 
and normally no other changes are necessary when a reference is added or
deleted.
Two passes are necessary initially to get the cross references right 
but once they are correct a single run is usually sufficient provided an 
\verb".aux" file is available and the file 
is run to the end each time.
\verb"\cite" and \verb"\bibitem" are used to link citations in the text
with their entry in the reference list;
if the 
reference list contains an entry \verb"\bibitem{label}", 
then \verb"\bibitem{label}" 
will produce the correct number in the reference list and 
\verb"\cite{label}" will produce the number within square brackets in the 
text. \verb"label" may contain alphabetic letters, 
or punctuation characters but must not contain spaces or commas. It is also
recommended that the underscore character \_{} is not used in cross
referencing. 
Thus labels for the form 
\verb"eq:partial", \verb"fig:run1", \verb"eq:dy'", 
etc, may be used. When several 
references occur together in the text \verb"\cite" may be used with 
multiple labels with commas but no spaces separating them; 
the output will be the 
numbers within a single pair of square brackets with a comma and a 
thin space separating the numbers. Thus \verb"\cite{label1,label2,label4}"
would give [1,\,2,\,4]. Note that no attempt is made to sort the 
labels and no shortening of groups of consecutive numbers is done.
Authors should therefore try to use multiple labels in the correct 
order.

The numbers for the cross referencing are generated in the order the 
references appear in the reference list, so that if the entries in the 
list are not in the order in which the references appear in the text 
then the 
numbering within the text will not be sequential. To correct this 
change the ordering of the entries in the reference list and then 
rerun {\it twice}.

\subsection{Equation numbers, sections, subsections, figures and 
tables}
Cross references can be made to equation numbers, sections, 
subsections, figures and tables or
any numbered environment
and this is a very useful feature when 
writing a document as its final structure is often not fully defined 
at the start. Thus a later section can be referred to by a label 
before its precise number is known and when it is defined there is no 
need to search back through the document to insert the correct value 
manually. For this reason the use of cross referencing 
can save considerable time. 

Labels for equation numbers, sections, subsections, figures and tables 
are all defined with the \verb"\label{label}" command and cross references 
to them are made with the \verb"\ref{label}" command. The \verb"\label" 
macro 
identifies the type of environment it is used in and converts \verb"label" 
into the correct form for that type of environment, thus \verb"\ref{label}" 
might give (2.3) for an equation number but 3.1 for a subsection and 2 
for a figure or table number. 

Any section, subsection, subsubsection, appendix or subappendix 
command defines a section type label, e.g. 1, 2.2, A2, A1.2 depending 
on context. A typical article might have in the code of its introduction 
`The results are discussed in section\verb"~\ref{disc}".' and
the heading for the discussion section would be:
\begin{verbatim}
\section{Results\label{disc}}
\end{verbatim}
Labels to sections, etc, may occur anywhere within that section except
within another numbered environment. 
Within a maths environment labels can be used to tag equations which are 
referred to within the text. 
An example of an equation with a label and a reference to it 
is:
\begin{verbatim}
\begin{equation}
X=a\cos\theta+ b\sin\phi. \label{cossin}
\end{equation}
Equation (\ref{cossin}) ...
\end{verbatim}
which produces
\begin{equation}
X=a\cos\theta+ b\sin\phi. \label{cossin}
\end{equation}
Equation (\ref{cossin}) ...

In addition to the standard \verb"\ref{<label>}" the abbreviated
forms given in the \tref{abrefs}
are available for reference to standard parts of the text

\Table{Alternatives to the normal references $\backslash${\tt ref} 
and the text generated by
them. Note it is not normally necessary to include the word equation
before an equation number except where the number starts a sentence. The
versions producing an initial capital should only be used at the start of
sentences.\label{abrefs}} 
\br
Reference&Text produced\\
\mr
\verb"\eref{<label>}"&(\verb"<num>")\\
\verb"\Eref{<label>}"&Equation (\verb"<num>")\\
\verb"\fref{<label>}"&figure \verb"<num>"\\
\verb"\Fref{<label>}"&Figure \verb"<num>"\\
\verb"\sref{<label>}"&section \verb"<num>"\\
\verb"\Sref{<label>}"&Section \verb"<num>"\\
\verb"\tref{<label>}"&table \verb"<num>"\\
\verb"\Tref{<label>}"&Table \verb"<num>"\\
\br
\endTable

\section{Tables and table captions}
Tables are numbered serially and referred to in the text 
by number (table 1, etc, {\bf not} tab. 1). Each table should have an 
explanatory caption which should be as concise as possible. If a table 
is divided into parts these should be labelled \pt(a), \pt(b), 
\pt(c), etc but there should be only one caption for the whole 
table, not separate ones for each part.

In the preprint style the tables may be included in the text 
or listed separately after the reference list 
starting on a new page. 

\subsection{The basic table format}
The standard form for a table is:
\begin{verbatim}
\begin{table}
\caption{Table caption.}
\begin{indented}
\item[]\begin{tabular}{@{}llll}
\br
Head 1&Head 2&Head 3&Head 4\\
\mr
1.1&1.2&1.3&1.4\\
2.1&2.2&2.3&2.4\\
\br
\end{tabular}
\end{indented}
\end{table}
\end{verbatim}

Points to note are:
\begin{enumerate}
\item The caption comes before the table. It should have a full stop at
the end.

\item Tables are normally set in a smaller type than the text.
The normal style is for tables to be indented in the same way as
equations. This is accomplished
by using \verb"\begin{indented}" \dots\ \verb"\end{indented}"
and putting \verb"\item[]" before the start of the tabular environment.
Omit these
commands for any tables which will not fit on the page when indented.

\item The default alignment of columns should be aligned left and 
adding \verb"@{}" omits the extra space before the first column.

\item Tables have only horizontal rules and no vertical ones. The rules at
the top and bottom are thicker than internal rules and are set with
\verb"\br" (bold rule). 
The rule separating the headings from the entries is set with
\verb"\mr" (medium rule).

\item Numbers in columns should be aligned on the decimal point;
to help do this a control sequence \verb"\lineup" has been defined 
which sets \verb"\0" equal to a space the size of a digit, \verb"\m"
to be a space the width of a minus sign, and \verb"\-" to be a left
overlapping minus sign. \verb"\-" is for use in text mode while the other
two commands may be used in maths or text.
(\verb"\lineup" should only be used within a table
environment after the caption so that \verb"\-" has its normal meaning
elsewhere.) See table~\ref{tabone} for an example of a table where
\verb"\lineup" has been used.
\end{enumerate}

\begin{table}
\caption{A simple example produced using the standard table commands 
and $\backslash${\tt lineup} to assist in aligning columns on the 
decimal point. The width of the 
table and rules is set automatically by the 
preamble.\label{tabone}} 

\begin{indented}
\lineup
\item[]\begin{tabular}{@{}*{7}{l}}
\br                              
$\0\0A$&$B$&$C$&\m$D$&\m$E$&$F$&$\0G$\cr 
\mr
\0\023.5&60  &0.53&$-20.2$&$-0.22$ &\01.7&\014.5\cr
\0\039.7&\-60&0.74&$-51.9$&$-0.208$&47.2 &146\cr 
\0123.7 &\00 &0.75&$-57.2$&\m---   &---  &---\cr 
3241.56 &60  &0.60&$-48.1$&$-0.29$ &41   &\015\cr 
\br
\end{tabular}
\end{indented}
\end{table}

\subsection{Simplified coding and extra features for tables}
The basic coding format can be simplified using extra commands provided in
the \verb"iopart" class file. The commands up to and including 
the start of the tabular environment
can be replaced by
\begin{verbatim}
\Table{Table caption}
\end{verbatim}
this also activates the definitions within \verb"\lineup".
The final three lines can also be reduced to \verb"\endTable" or
\verb"\endtab". Similarly for a table which does not fit in when indented
\verb"\fulltable{caption}" \dots\ \verb"\endfulltable" or \verb"\endtab"
can be used. \LaTeX\ optional positional parameters can, if desired, be added after 
\verb"\Table{caption}" and \verb"\fulltable{caption}".


\verb"\centre{#1}{#2}" can be used to centre a heading 
\verb"#2" over \verb"#1" 
columns and \verb"\crule{#1}" puts a rule across 
\verb"#1" columns. A negative 
space \verb"\ns" is usually useful to reduce the space between a centred 
heading and a centred rule. \verb"\ns" should occur immediately after the 
\verb"\\" of the row containing the centred heading (see code for
\tref{tabl3}). A small space can be 
inserted between rows of the table 
with \verb"\ms" and a half line space with \verb"\bs" 
(both must follow a \verb"\\" but should not have a 
\verb"\\" following them).
   
\Table{A table with headings spanning two columns and containing notes. 
To improve the 
visual effect a negative skip ($\backslash${\tt ns})
has been put in between the lines of the 
headings. Commands set-up by $\backslash${\tt lineup} are used to aid 
alignment in columns. $\backslash${\tt lineup} is defined within
the $\backslash${\tt Table} definition.\label{tabl3}}
\br
&&&\centre{2}{Separation energies}\\
\ns
&Thickness&&\crule{2}\\
Nucleus&(mg cm$^{-2}$)&Composition&$\gamma$, n (MeV)&$\gamma$, 2n (MeV)\\
\mr
$^{181}$Ta&$19.3\0\pm 0.1^{\rm a}$&Natural&7.6&14.2\\
$^{208}$Pb&$\03.8\0\pm 0.8^{\rm b}$&99\%\ enriched&7.4&14.1\\
$^{209}$Bi&$\02.86\pm 0.01^{\rm b}$&Natural&7.5&14.4\\
\br
\end{tabular}
\item[] $^{\rm a}$ Self-supporting.
\item[] $^{\rm b}$ Deposited over Al backing.
\end{indented}
\end{table}

Units should not normally be given within the body of a table but 
given in brackets following the column heading; however, they can be 
included in the caption for long column headings or complicated units. 
Where possible tables should not be broken over pages. 
If a table has related notes these should appear directly below the table
rather than at the bottom of the page. Notes can be designated with
footnote symbols (preferable when there are only a few notes) or
superscripted small roman letters. The notes are set to the same width as
the table and in normal tables follow after \verb"\end{tabular}", each
note preceded by \verb"\item[]". For a full width table \verb"\noindent"
should precede the note rather than \verb"\item[]". To simplify the coding 
\verb"\tabnotes" can, if desired, replace \verb"\end{tabular}" and 
\verb"\endtabnotes" replaces
\verb"\end{indented}\end{table}".

If all the tables are grouped at the end of a document
the command \verb"\Tables" is used to start a new page and 
set a heading `Tables and table captions'.
  
\section{Figures and figure captions}
Figures may be included in an article as encapsulated PostScript files or
using the \LaTeX\ picture environment. Alternatively authors may send in 
high quality printed versions of their figures (fair copies) and 
attach copies of the fair copies to each 
typescript. The fair copies should be in black 
Indian ink or printing on tracing paper, plastic or white card or 
paper, or glossy photographs.

Each figure should have a brief caption describing it and, if 
necessary, interpreting the various lines and symbols on the figure. 
As much lettering as possible should be removed from the figure itself and 
included in the caption. If a figure has parts, these should be 
labelled ($a$), ($b$), ($c$), etc. 
\Tref{blobs} gives the definitions for describing symbols and lines often
used within figure captions (more symbols are available
when using the optional packages loading the AMS extension fonts).

\begin{table}
\caption{Control sequences to describe lines and symbols in figure 
captions.\label{blobs}}
\begin{indented}
\item[]\begin{tabular}{@{}lllll}
\br
Control sequence&Output&&Control sequence&Output\\
\mr
\verb"\dotted"&\dotted        &&\verb"\opencircle"&\opencircle\\
\verb"\dashed"&\dashed        &&\verb"\opentriangle"&\opentriangle\\
\verb"\broken"&\broken&&\verb"\opentriangledown"&\opentriangledown\\
\verb"\longbroken"&\longbroken&&\verb"\fullsquare"&\fullsquare\\
\verb"\chain"&\chain          &&\verb"\opensquare"&\opensquare\\
\verb"\dashddot"&\dashddot    &&\verb"\fullcircle"&\fullcircle\\
\verb"\full"&\full            &&\verb"\opendiamond"&\opendiamond\\
\br
\end{tabular}
\end{indented}
\end{table}

Unless the figures files are incorporated into the text 
the captions should be listed at the end of the article.

\subsection{Figure captions where figure files are not available}
The command \verb"\Figures" starts a new page and an unnumbered section
with the heading `Figure captions'. 
The captions should then be set with the commands:
\begin{verbatim}
\begin{figure}
\caption{Figure caption.}
\end{figure}
\end{verbatim}
or more simply
\begin{verbatim}
\Figure{Figure caption.}                                         
\end{verbatim}
The caption should finish with a full stop and the printed version will be 
indented as in Institute of Physics Publishing single-column journals.

\subsection{Inclusion of graphics files}
If graphics files are available as encapsulated PostScript (EPS) files 
(or are created within the \LaTeX\ picture environment) they may
be included within the body of the text at an appropriate point using a
standard graphics inclusion package. Authors should ensure EPS files meet
the following criteria
\begin{itemize}
\item the Bounding Box should indicate the area of the figure 
with a minimum of white space around it and not the dimensions of the
page.
\item Any fonts used should be from the standard PostScript set (Times,
Helvetica, Courier and Symbol).
\item Scanned images should be of 600 dpi resolution for line art (black
and white) and 150 dpi resolution for grayscale or colour.
\item Captions and labels (e.g.\ Figure 1) should not be included in the
EPS file although part letters (e.g.\ ($a$)) are acceptable provided they
are placed close or within the boundary of the figure. 
\end{itemize}

The precise coding required will depend on the graphics package being used
and the printer driver. We use a printer driver compatible with DVIPS but
authors should avoid using special effects generated by including verbatim
PostScript code within the \LaTeX\ file with specials other than the
standard figure inclusion ones.

Using the epsf package figures can be included using code such as:
\begin{verbatim}
\begin{figure}
\begin{center}
\epsfbox{file.eps}
\end{center}
\caption{Figure caption}
\end{figure}
\end{verbatim}


\clearpage

\appendix
\section{List of macros for formatting text, figures and tables}


\begin{table}[hb]
\caption{Macros available for use in text. Parameters in square brackets 
are optional.}
\footnotesize\rm
\begin{tabular}{@{}*{7}{l}}
\br
Macro name&Purpose\\
\mr
\verb"\title[#1]{#2}"&Title of article and short title (optional)\\
\verb"\paper[#1]{#2}"&Title of paper and short title (optional)\\
\verb"\letter{#1}"&Title of Letter to the Editor\\
\verb"\comment[#1]{#2}"&Title of Comment and short title (optional)\\
\verb"\topical[#1]{#2}"&Title of Topical Review and short title 
(optional)\\
\verb"\review[#1]{#2}"&Title of review article and short title (optional)\\
\verb"\note[#1]{#2}"&Title of Note and short title (optional)\\
\verb"\prelim[#1]{#2}"&Title of Preliminary Communication \& short title\\
\verb"\author{#1}"&List of all authors\\
\verb"\article[#1]{#2}{#3}"&Type and title of other articles and 
short title (optional)\\
\verb"\address{#1}"&Address of author\\
\verb"\pacs{#1}"&PACS classification codes\\
\verb"\pacno{#1}"&Single PACS classification code\\
\verb"\ams{#1}"&American Mathematical Society classification code\\
\verb"\jl{#1}"&Number of journal article submitted to\\
\verb"\submitted"&`Submitted to' message\\
\verb"\maketitle"&Creates title page\\
\verb"\begin{abstract}"&Start of abstract\\
\verb"\end{abstract}"&End of abstract\\
\verb"\nosections"&Inserts space before text when no sections\\
\verb"\section{#1}"&Section heading\\
\verb"\subsection{#1}"&Subsection heading\\
\verb"\subsubsection{#1}"&Subsubsection heading\\
\verb"\appendix"&Start of appendixes\\
\verb"\ack"&Acknowledgments heading\\
\verb"\References"&Heading for reference list\\
\verb"\begin{harvard}"&Start of alphabetic reference list\\
\verb"\end{harvard}"&End of alphabetic reference list\\
\verb"\begin{thebibliography}{#1}"&Start of numeric reference list\\
\verb"\end{thebibliography}"&End of numeric reference list\\
\verb"\etal"&\etal for text and reference lists\\
\verb"\dash"&Rule for repeated authors in alphabetical reference list\\
\verb"\nonum"&Unnumbered entry in numerical reference list\\
\br
\end{tabular}
\end{table}


\begin{table}
\caption{Macros defined within {\tt iopart.cls}
for use with figures and tables.}
\begin{indented}
\item[]\begin{tabular}{@{}l*{15}{l}}
\br
Macro name&Purpose\\
\mr
\verb"\Figures"&Heading for list of figure captions\\
\verb"\Figure{#1}"&Figure caption\\
\verb"\Tables"&Heading for tables and table captions\\
\verb"\Table{#1}"&Table caption\\
\verb"\fulltable{#1}"&Table caption for full width table\\
\verb"\endTable"&End of table created with \verb"\Table"\\
\verb"\endfulltab"&End of table created with \verb"\fulltable"\\
\verb"\endtab"&End of table\\
\verb"\br"&Bold rule for tables\\
\verb"\mr"&Medium rule for tables\\
\verb"\ns"&Small negative space for use in table\\
\verb"\centre{#1}{#2}"&Centre heading over columns\\
\verb"\crule{#1}"&Centre rule over columns\\
\verb"\lineup"&Set macros for alignment in columns\\
\verb"\m"&Space equal to width of minus sign\\
\verb"\-"&Left overhanging minus sign\\
\verb"\0"&Space equal to width of a digit\\
\br
\end{tabular}
\end{indented}
\end{table}


\clearpage
\section{Control sequences for journal
abbreviations\label{jlabs}}



\begin{table}[hb]
\caption{Abbreviations for the IOPP journals.}
\begin{indented}
\item[]
\begin{tabular}{@{}lll}
\br
Macro name&{\rm Short form of journal title}&Years relevant\\
\mr
\verb"\CQG"&Class. Quantum Grav.\\
\verb"\CTM"&Combust. Theory Modelling\\
\verb"\EJP"&Eur. J. Phys.\\
\verb"\IP"&Inverse Problems\\
\verb"\JO"&J. Opt.\\
\verb"\JPA"&J. Phys. A: Math. Gen.\\
\verb"\JPB"&J. Phys. B: At. Mol. Phys.&1968--1987\\
\verb"\jpb"&J. Phys. B: At. Mol. Opt. Phys.&1988 and onwards\\
\verb"\JPC"&J. Phys. C: Solid State Phys.&1968--1988\\
\verb"\JPCM"&J. Phys: Condens. Matter&1989 and onwards\\
\verb"\JPD"&J. Phys. D: Appl. Phys.\\
\verb"\JPE"&J. Phys. E: Sci. Instrum.&1968--1989\\
\verb"\JPF"&J. Phys. F: Met. Phys.\\
\verb"\JPG"&J. Phys. G: Nucl. Phys.&1975--1988\\
\verb"\jpg"&J. Phys. G: Nucl. Part. Phys.&1989 and onwards\\
\verb"\MSMSE"&Modelling Simul. Mater. Sci. Eng.\\
\verb"\MST"&Meas. Sci. Technol.&1990 and onwards\\
\verb"\NET"&Network\\
\verb"\NL"&Nonlinearity\\
\verb"\NT"&Nanotechnology\\
\verb"\PAO"&Pure and Applied Optics\\
\verb"\PMB"&Phys. Med. Biol.\\
\verb"\PSST"&Plasma Sources Sci. Technol.\\
\verb"\PUS"&Public Understand. Sci.\\
\verb"\QO"&Quantum Opt.\\
\verb"\RPP"&Rep. Prog. Phys.\\
\verb"\SST"&Semicond. Sci. Technol.\\
\verb"\SUST"&Supercond. Sci. Technol.\\
\verb"\WRM"&Waves in Random Media\\
\br
\end{tabular}
\end{indented}
\end{table}



\begin{table}
\caption{Abbreviations for some more 
common non-IOPP journals.}
\begin{indented}
\item[]\begin{tabular}{@{}ll}
\br
Macro name&{\rm Short form of journal}\\
\mr
\verb"\AC"&Acta Crystallogr.\\
\verb"\AM"&Acta Metall.\\
\verb"\AP"&Ann. Phys., Lpz\\
\verb"\APNY"&Ann. Phys., NY\\
\verb"\APP"&Ann. Phys., Paris\\
\verb"\CJP"&Can. J. Phys.\\
\verb"\GRG"&Gen. Rel. Grav.\\
\verb"\JAP"&J. Appl. Phys.\\
\verb"\JCP"&J. Chem. Phys.\\
\verb"\JJAP"&Japan. J. Appl. Phys.\\
\verb"\JMMM"&J. Magn. Magn. Mater.\\
\verb"\JMP"&J. Math. Phys.\\
\verb"\JOSA"&J. Opt. Soc. Am.\\
\verb"\JP"&J. Physique\\
\verb"\JPhCh"&J. Phys. Chem.\\
\verb"\JPSJ"&J. Phys. Soc. Japan\\
\verb"\JQSRT"&J. Quant. Spectrosc. Radiat. Transfer\\
\verb"\NC"&Nuovo Cimento\\
\verb"\NIM"&Nucl. Instrum. Methods\\
\verb"\NP"&Nucl. Phys.\\
\verb"\PF"&Phys. Fluids\\
\verb"\PL"&Phys. Lett.\\
\verb"\PR"&Phys. Rev.\\
\verb"\PRL"&Phys. Rev. Lett.\\
\verb"\PRS"&Proc. R. Soc.\\
\verb"\PS"&Phys. Scr.\\
\verb"\PSS"&Phys. Status Solidi\\
\verb"\PTRS"&Phil. Trans. R. Soc.\\
\verb"\RMP"&Rev. Mod. Phys.\\
\verb"\RSI"&Rev. Sci. Instrum.\\
\verb"\SSC"&Solid State Commun.\\
\verb"\SPJ"&Sov. Phys.--JETP\\
\verb"\ZP"&Z. Phys.\\
\br
\end{tabular}
\end{indented}
\end{table}

\end{document}


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