% This file has been produced for use with REVTEX 3.0.
%
\documentstyle[aps,preprint,epsfig]{revtex}

\begin{document}

%%\input amssym.def
%%\input amssym.tex

\title{Tile Hamiltonian for Decagonal AlCoCu}

\author{Ibrahim Al-Lehyani$^{a,b}$ and Mike Widom$^{a}$}

\address{$^{a}$Department of Physics, Carnegie Mellon University,
Pittsburgh, Pennsylvania 15213}
\address{$^{b}$Department of Physics, King AbdulAziz University, Jeddah, Saudi Arabia}

\date{\today}

\maketitle

\begin{abstract}
A tile Hamiltonian (TH) replaces the actual atomic 
interactions in a quasicrystal with effective interactions 
between and within tiles. We studied AlCoCu decagonal 
quasicrystals described as decorated Hexagon-Boat-Star (HBS) 
tiles using ab-initio methods. The dominant 
term in the TH counts the number of H, B and S tiles. Phason 
flips that replace an HS pair with a BB pair lower the 
energy. In Penrose tilings, quasiperiodicity is forced by 
arrow matching rules on rhombus edges. The edge orientation 
in AlCoCu is due to Co/Cu chemical ordering. Tile edges 
meet in vertices with 72$^\circ$ or 144$^\circ$ angles. We 
find strong interactions between edge orientations at 
72$^\circ$ vertices that force a subset of the Penrose 
matching rules. Interactions at 144$^\circ$ vertices are 
somewhat weaker. The ``vertical'' edge orientation disorder 
interactions are intermediate in strength.
\end{abstract}
\newpage

\section{Introduction}
\label{sec:intro}
Both quasicrystals and ordinaty crystals are made of elementary building blocks. In quasicrystals these building blocks do not repeat periodically. The way they are tiled to cover the space in a quasicrystal fasion and the decoration of individual tiles determines the structure completely. But the task is not as straight forward as in periodic crystals.

Matching rules govern local tile configurations by allowing a subgroup of all possible ones. Globally, they enforce quasiperiodicity. Penrose proposed his famous matching rules before the discovery of quasicrystal materials. In 2D, Penrose tiles are a fat and a thin rhombi [Fig~\ref{fig:penrose}]. Edges are assigned arrow decorations and they must match for common edges in adjacent tiles. In perfect quasicrystals these rules are obayed everywhere. However in the construction process one might have freedom of choice at one point that will affect future configurations. Thus backward tracking and correction procedures are needed when a conflict is reached because it might be a result of a degenerate choice in an early stage. Other sets of rules were studied too. Some represents a subgroup of Penrose matching rules like in AlCoNi [\cite{corrections}].

In quasicrystals, a actual atomic interactions in system Hamiltonian can be replaced with effective interactions between and within the tiles. The resulting tile Hamiltonian is a rearranged original Hamiltonian. In simple interaction pictures (pair potentials for example) the relation between the two is straight forward. It might be difficult for more complicated potential descriptions but it is theoretically possible. The tile Hamiltonian includes terms which depends only on the number of tiles and other terms for different interactions. The tile Hamiltonian simplifies the picture greatly and it is a reasonable way to describe the tiles.

Quasicrystals can be described as surface cuts with irrational slopes of higher dimensional space periodic structures. The irrational slope can be approximated by rational numbers and the resulting structure will be close to the quasicrystal. The better the approximation the closer the new structure (the approximant) to the quasicrystal and the bigger it is. The approximants are periodic and they can be studied with ``conventional'' tools. Their properties are very close to their corresponding quasicrystals and some aspects can be identical.  

In a previous paper we studied teh matching rules in decagonal AlCoCu using a limited group of quasicrystal approximats. Some angle interactions details couldn't be extractd. Here we study more thoroghly the set of rules controlling these compounds using different techniques and a much bigger set of approximants.

First, we describe the model of decagonal AlCoCu in section~\ref{decagonalalcocu}. Section~\ref{sec:abinitiostudy} gives our detailed calculations using ab-initio methods. Similar calculations but using pair potentials to cmpare with are described in section~\ref{pairpotentials}.In section~\ref{othereffects}, we talk about different other effects to be considered in the model. We analyse our finding and study their implications on AlCoCu compounds in section~\ref{diss}
\newpage


\section{Decagonal Al-Co-Cu}
\label{sec:decagonalalcocu}
Decagonal Al-Co-Cu quasicrystals have been studied by many authors~\cite{alconiexp}. Cockayne and Widom~\cite{CWmodel} proposed a model based on tiling of space by Hexagon, Boat and star tiles (HBS) decorated deterministically with atoms. Tile vertices are occupied by two pentagons of atoms on top of each other rotated by 36$^\circ$. One layer contains only Al atoms and the other layer contains a mixture of TM and can contain Al atoms. The decagon center, Al atom, is always of same hight as the TM atom layer. Decagons meet with pairs of Co-Cu at about 2.6~\AA~ away. Tile edges are drown by these pairs and they are 6.4~\AA~ long in quasiperiodic plane and $\frac{1}{2}c$=2.08~\AA~ along the periodic direction. All TM atoms surrounding a vertices Al atom are parts of tile edges. It was shown in the original model that TM atoms prefer to alternate on tile edges. This was confirmed later using locally self-consistent multiple scattering (LSMS) method~\cite{ICQ7}. Our study emphasizes the same conclusion.

The presented structure is slightly different form the original model. Originally, Cu atoms break the boat and star symmetries and composition is not conserved under phasom the flip between boats and stars. As discussed above, in our model Cu atoms reside only on tile edges and symmetry is broken by the internal Al atoms.

Tile edges meet in vertices at 72$^\circ$ or 144$^\circ$ angles. We assign arrows to edges based on the TM atom decorations. An arrow points from Cu to Co atoms. Meeting edges are of type ``i'' if both point towards their meeting vertices. Types ``o'' and ``m'' are out and mixed pointing.

Hexagon, boat and star tiles are the building blocks in the model. Some of their characteristics are summarized in table~\ref{tab:tiles}. The tile Hamiltonian for the system is:
\begin{equation}
\label{eq:tileham}
E_{\rm tot}=\sum_{\alpha}n_{t}^{\alpha}E_{t}^{\alpha}+\lambda_{1}\sum_{\alpha}n_{72}^{\alpha}E_{72}^{\alpha}+\lambda_{2}\sum_{\alpha}n_{144}^{\alpha}E_{144}^{\alpha}
\end{equation}
Where $n_{t}^{\alpha}$ is the number of specific tile $\alpha$ which can be s, b or h. Similarly, $n_{72}^{\alpha}$ and $n_{144}^{\alpha}$ are the numbers of 72$^\circ$ and 144$^\circ$ angles and $\alpha$ is the angle orientation. The tile energies are much bigger than the other terms and it turns out that the 72$^\circ$ angle interaction are bigger than the 144$^\circ$'s. It is desirable to check how every term can alter the system energies. We set $\lambda_1$ and $\lambda_2$ equal 1 or 0 for the purpose of including and excluding the angle interactions.

A hexagon contains 25 atoms, three of them are internal (one Al and two Co atoms). Internal atoms are not parts of any vertcis decagons. The Co atoms occupy symmetric sites and the Al atom breaks the symmetry by residing on one of two geometrically equivalent positions between the two Co atoms. Its interactions with tile edge TM atoms determine the preferred position. Centering it between the two sites symmetrizes the hexagon and can lower the energy even further. We chose to keep the hexagon unsymmetric because it makes phason flips easier and produces a more consistent decoration specially if one decides to go to smaller tile descriptions by deflating the tiles by $\tau^2$~\cite{au8}. A boat has 41 atoms, eight of which are internal. An Al atom breaks the symmetry by residing in one of two equivalent sites. In a star, two of the internal twelve atoms break the symmetry by occupying any two of five equivalent sites. A star has 57 atoms. A phason flip can switch a specially arranged star and a hexagon to a pair of boats and lowers the energy by 0.14 eV.


Many different quasicrystal approximants are exploited here to study differnt terms in tile Hamiltonian. All the approximants are listed in table~\ref{tab:structures} with some of their properties. The list does not -and not meant to- include all the arrangements of HBS tiles for the number of atoms stated. The choice is based on the information which can be extracted from any approximant. The smallest structure is the mono-clinic single hexagon approximant~\ref{fig:h2_mono}. It contains one horizontal and two inclined tile edges. Equation~\ref{eq:tileham} becomes 
\begin{equation}
\label{eq:tileham_h1}
E_{\rm tot}=E_{t}^{h}+\lambda_{1}(E_{72}^{i}+E_{72}^{o})+\lambda_{2}(2E_{144}^{i}+2E_{144}^{o})
\end{equation}


The next bigger approximant is a 41-atom single boat [fig~\ref{fig:b1}]. Single stars do not cover the space so single star unit cell approximants are not possible. Two hexagons approximants can be constructed in orthogonal cells either by genuine orthogonal structure [fig~\ref{fig:h2_ortho}] or by doubling a mono-clinic cell [fig~\ref{fig:h2_mono}]. The doubling process gives more freedom to in controlling swaps. Similarly two boat unit cell can be constructed either by doubling the single boat structure or by tiling the two boats like in b2 [fig~\ref{fig:b2}].

The small approximants are mostly used to calculate $E_{t}^{\alpha}$. To explore the angle orientations larger approximants are needed. The last three approximants have 132 atoms per unit cell with different configurations. Two of them are related to each other by a phason flip (s1h3 and b2h2-2). The phason flip is outlined in fig~\ref{fig:s1h3}. This phason flip lowers the energy which indicates that stars are disfavored in these compounds. Angle orientations are investigated by swapping a TM pair on tile edges. An example of a pair used to explore pure 72$^\circ$ angle interactions is outlined by a dashed rectangle in fig~\ref{fig:b2h2_1}. The bond is surrounded by a single 144$^\circ$ on each side. Both point towards the middle bond making o144 on one side and m144 on the other. One extra 72$^\circ$ on one side making o72. If the middle pair is swapped the 144$^\circ$ will be conserved. The swap merely switch their sides. The o72 becomes m72. The difference between the swapped and the basic structures is the difference between m72 and o72. Arranging the 72$^\circ$ outer bond further and then making the swap can give m72-i72. Other swaps give an independent  i72-o72 difference. Similarly, pure 144$^\circ$ differences can be calculated from other swaps. The pair surrounded by a dashed oval shape in fig~\ref{fig:b2h2_1} is one example.

Long range interaction and other smaller terms in the tile Hamiltonian can be estimated from other swaps. The bonds on the sides of the horizontal tile edge of h2 can be specially arranged to cancel the 144$^\circ$ interactions and leave only other interactions.
\newpage
\newpage

\section{Ab-initio Study}
\label{sec:abinitiostudy}
For our calculations we employ an ab-initio pseudopotential utilizing the Vienna Ab-initio Simulation Package (VASP) method~\cite{VASP}. It is widely used for the purpose of structural studies. It starts with a specially constructed elemental pseudopotential and does the calculations in k-space. The k-space mesh size determines the accuracy of the calculations. Bigger k-space grids are more accurate but more expensive in calculation times. Usually one probes for the best k-space mesh before the production calculations. One has to balance between  the size of the system and the size of the k-space grid to fit within the available computer resources. Our calculations are carried out on the T3E and on the newly installed TCS machines at the Pittsburgh Supercomputer Center. As explained before, we use many different approximants for our study. The k-space mesh is increased until a consistency of about 0.02 eV is reached in worst cases. But within resonable use of our allocatd times we are able to get better convergence for most structures. The convergence pilot calculations are outlined in table~\ref{tab:convergence1} for our b2h2-1 and in table~\ref{tab:convergence2} for our b1 structures. There are two slightly different structures per approximant. They differ by a single arrow decoration. The tables show that at the highest k-space grid used the accuracy in base energies is within 0.01 eV. The energy differences converge faster. The b2h2-1 has 132 atoms per unit cell which is the largest number we studied. The largest grid of 2x2x10 (20 indep. k-points) takes about 6 hours on the T3E machine (450 MHz alpha processors) using 64 nodes. The smallest 1x1x1 (single k-point in the middle of the k-space unit cell) took about 2 hours on 8 processors. We note that for the same number of processors calculation time scales linearly with the number of independent k-points. But for the same structure calculation time decreases linearly (almost) with increasing number of processors only up to 16 processors. Beyond that, the total charging time (no. of processors*elapsed time) increases notably.Large structures and big k-space grids must use large numbers of processors because they need more memory. Another thing can be noted from tables~\ref{tab:convergence1},~\ref{tab:convergence2} is that the smoother the distribution of the k-space grid points along kx,kx and kz the closer the result to convergence.  For b2h2-1, the grid 1x1x4 is closer to converging result than 2x2x2 which has more k-points. In all out structures we choose the most homogeneous distribution of k-space points possible.

Many different rearrangements of arrows, by swapping TM atom pairs, are performed and the different structure energies are calculated. Most of these are done for the large approximants because it gives more freedom. In general, pure 72$^\circ$ can be deduced if the middle bond has an equal number of 144$^\circ$ connections on each side (for 144$^\circ$ interaction to cancel) and total of one 72$^\circ$ connection which can be attached to any side or two from the same side. For pure 144$^\circ$ interactions there is only one configuration. That is a middle bond surrounded by two 144$^\circ$ and one 72 connection from one side and a single 72 and a single 144 connections from the other side. We compute an over complete set of structures and use a least square fit. First, we fit with $\lambda_1$ and $\lambda_2$ in equation ~\ref{equ:tileham} set to zero. The fitting is shown in fig.~\ref{fig:vasp3sep}. The five figures correspond to the different energy ranges our approximants span. The large approximants spread over a large range of energies because of the freedom we have in different swaps. That's why the range of the first graph is six times bigger than the range of the rest. The first graph shows that a pair of boats is favored over a star and a hexagon. Both have the same number of atoms but the last has one more 72$^\circ$ interaction. The graphs also show clearly that the three-parameter fit is not adequate.

The next set of graphs~\ref{fig:vasp9sep} shows when we set $\lambda_1$ and $\lambda_2$ back to one. The graphs are on different scales and the deviation from the y=x line is due to other longer-range effects. The root mean square of the deviations is (VALUE). Out of that, 0.013 eV is due to the central Al atom interactions with the tile edges. The rest can be due to collection of weaker longer range forces. In the first graph in figure~\ref{fig:vasp9sep} we intentionally control the swaps for all approximants to spread over the whole range.

The different parameters are listed in table~\ref{tab:energycosts}. The different energy constitutes of the model are as expected, tile energies are the biggest contribution. That is the first term in equation~\ref{equ:tileham}. The 72$^\circ$ angle orientation comes next. Then by an order of  magnitude from it follows the 144$^\circ$ angle contributions. The inward configurations are highly disfavored and outwards are lowest in energy by the same magnitude. Mixed arrowing are almost zero in energy. In previous paper we assumed that the Co-Cu interaction is the average of Co-Co and Cu-Cu interactions. This shows that such an assumption is very close to reality.

The HBS tiles are composed of Penrose tiles with double arrows always satisfied inside the tiles. The space can be constructed then by only watching out for the single arrows.

Arrowin 72 vs. 144 and matching rules (here or discu.)

\newpage



\section{Pair Potentials}
\label{sec:pairpotentials}
Using pair potentials in calculations simplifies it greatly. Many different pair potentials have been used to study this and other quasicrystals. Cockayne and Widom used mok-ternary potentials extracted from AlCo pair potentials. In Al-Co-Cu, Cu substitutes for an equal combination of Al and Co. So they approximated Cu interactions by the average interactions of Al and Co. They adopted AlCo pair potentials calculated by Phillips and others~\cite{phillips}. Alternation of TM atoms on tile edges and the effect of the edge decoration on the central Al are all consistent with our VASP results.

Later, more rigorous pair potentials in the context of Generalized Pseudopotential Theory (GPT) were used for Al-Co-Ni~\cite{corrections}. The original GPT pair potentials suffer from TM overbinding which is a very unphysical attraction between TM atoms at small separations. The strongest overbinding appears in Co-Co pair potentials. We modified the TM pair potentials at short distances using VASP~\cite{corrections}. Resulting potentials behave really well in the simulation~\cite{au8}. 

We calculated the energies of the approximants using both potentials to check how the pair potentials will do compared to VASP. Structures are calculated at fixed volume and composition which enables us to ignore the volume term. using the modified GPT, TM atoms favour alternation on the tile edges by  (VALUE) and the central position for the Al atom is preferred by at least (VALUE). 

Graphs in figures~\ref{fig:mgpt9sep} and~\ref{fig:mok9sep} show result fitting. Pair potentials give a very good agreement between the model and the calculations even for the mok-ternary pair potentials.

Results of the fitting are summarized in table~\ref{tab:energycosts} for all the methods used. Aside from a constant factor between the modified GPT and the mok-ternary pair potential calculations, they are very close to each other. The order of 144$^\circ$ angle interaction is reversed compared to VASP. They are very weak and spread symmetrically around zero. They exert no restriction on system freedom after satisfying only the 72$^\circ$ interactions.

\newpage

\section{Other Effects}
\label{sec:othereffects}
\begin{itemize}
\item Chemical ordering
The source of matching rules in the model is the chemical ordering of TM metal on tile edges. In fig~\ref{fig:h2_mono}, the heavey dashed unit cell is our h2-1 approximant. It contains two horizontal tile edges. We take a Cu atom from one of them and swap it with the Co atom on the other. Resulting edges contain pairs of similar spcies. The process raises the energy by 0.681 eV. The same swap were studied before with LSMS and gave same conclusion with slightly different value. TM atoms favor the alternations on tile edges greately.
\item Multi-layering
\item Centeral Al
\item Hidden 144
\end{itemize}

\section{Discussion}
\label{sec:dis}
\begin{itemize}
\item Optimal tile decoration - Does 144 matter?
\item What structure is favored?
\item What is 1000K structure?
\end{itemize}


\newpage

\acknowledgments
%
IA wishes to thank King Abdul Aziz University (Saudi Arabia) for
supporting his study and MW acknowledges support by the National
Science Foundation under grant DMR-9732567. PSC XXX

\begin{references}

%% 
\bibitem{corrections}
The AlCoNi mGPT corrections

%% 
\bibitem{alconiexp}
examples or review of expt. AlCoCu

%% 
\bibitem{CWmodel}
Cockayne and Widom PRL model 

%% 
\bibitem{LSMS}
ICQ7 paper

%% VASP references
\bibitem{VASP}
G. Kresse and J. Hafner, Phys. Rev. B {\bf 47}, RC558 (1993);
G. Kresse and J. Furthm\"uller, Phys. Rev. B {\bf 54}, 11169 (1996).

%% AlCo potentials
\bibitem{phillips}
R. Phillips, J. Zou, A.~E. Carlsson, and M. Widom, Phys. Rev. B {\bf 49},
9322  (1994)

%% 
\bibitem{au8}
The 8-author paper in PRB

%% 
\bibitem{}

%% 
\bibitem{}

%% 
\bibitem{}

%% 
\bibitem{}

%% 
\bibitem{}

%% 
\bibitem{}

%% 
\bibitem{}

%% 
\bibitem{}

%% 
\bibitem{}

%% 
\bibitem{}

%% 
\bibitem{}

\end{references}


\begin{table}
\caption{Tile edges meet in vertices with 72$^\circ$ or 144$^\circ$ angles. Arrows are defined to point from Cu to Co atoms. Bonds pointing towards the meeting vertices, away  and mixed pointing are i,o and m respectively. Energies listed here are the costs of introducing single i, o or m to a structure. The calculations are discussed in the text.}
\vspace{10pt}
\begin{tabular}{|c|r|r|r|}
 Parameters & Energy (VASP) & Energy (mGPT) & Energy (mokT)\\
   &  (eV)  & (eV) & (eV)   \\ 
\hline
 h &-133.176  &-6.00476 &2.12318 \\
 b &-216.046  &-6.82004 &5.41501 \\
 s &-298.781  &-7.4293  &8.758 \\
\hline
 i72 &0.295398  &0.294657 &0.0638355 \\
 m72 &-0.0320559  &0.00881958 &-0.00647638 \\
 o72 &-0.263342  &-0.303477 &-0.0573591 \\
\hline
 i144 &0.0321616  &-0.0497776 &-0.00941531 \\
 m144 &0.00139802  &0.0100431  &0.0000315417 \\
 o144 &-0.0335596  &0.0397346  &0.00938377 \\
\end{tabular}
\label{tab:energycosts}
\end{table}


\begin{table}
\caption{Basic Tiles in HBS model and their compositions}
\vspace{10pt}
\begin{tabular}{|c|c|l|c|}
tile & No. of Atoms &Composition & Penrose tiles\\
\hline
H & 25 & Al$_{17}$Co$_{5}$Cu$_{3}$&2T+F\\
B & 41 & Al$_{29}$Co$_{8}$Cu$_{4}$&T+3F\\
S & 57 & Al$_{41}$Co$_{11}$Cu$_{5}$&5F\\
\end{tabular}
\label{tab:tiles}
\end{table}


\begin{table}
\caption{The different structures used and their characteristics and calculation parameters}
\vspace{10pt}
\begin{tabular}{|l|r|c|c|c|c|}
Structure & No of Stru.&unit cell & Composition & Indep. K-points &figure \\
\hline
 b1         & 1  &12.134,12.1342,108$^\circ$  &Al$_{29}$Co$_{8}$Cu$_{4}$    &48 &\ref{fig:b1} \\
 h2-1       & 5  &23.081, 7.5554              &Al$_{34}$Co$_{10}$Cu$_{6}$   &14 &\ref{fig:h2_mono}  \\
 h2-2       & 1  &14.265,12.1345              &Al$_{34}$Co$_{10}$Cu$_{6}$   &66 &\ref{fig:h2_ortho}  \\
 b2         & 3  &12.134,30.3025,134.5$^\circ$&Al$_{58}$Co$_{16}$Cu$_{8}$   &75 &\ref{fig:b2}  \\
 b2h2-1     & 20 &19.634,23.0812              &Al$_{92}$Co$_{26}$Cu$_{14}$  &20 &\ref{fig:b2h2_1}  \\
 b2h2-2     & 6  &19.634,23.0812              &Al$_{92}$Co$_{26}$Cu$_{14}$  &20 &\ref{fig:b2h2_2}  \\
 s1h3       & 12 &19.634,23.0812              &Al$_{92}$Co$_{26}$Cu$_{14}$  &20 &\ref{fig:s1h3}  \\
\end{tabular}
\label{tab:structures}
\end{table}




\begin{table}
\caption{}
\vspace{10pt}
\begin{tabular}{|l|r|c|c|c|}
K-point grid & indep. K-points& E$_{b1}$&E$_{b1-swapped}$& $\Delta$E \\
\hline
1x1x1  &  1 &-186.83238 &-186.78158 &0.05080 \\
1x1x2  &  1 &-217.02636 &-216.68522 &0.34114 \\
1x1x3  &  2 &-215.95009 &-215.63608 &0.31401 \\
2x2x2  &  4 &-218.18586 &-217.90238 &0.28348 \\
2x2x4  &  8 &-216.75311 &-216.51774 &0.23537 \\
2x2x6  & 12 &-216.74436 &-216.49251 &0.25185 \\
3x3x3  & 14 &-216.59210 &-216.32712 &0.26498 \\
3x3x9  & 41 &-216.74029 &-216.48619 &0.25410 \\
4x4x4  & 32 &-216.77625 &-216.53784 &0.23841 \\
4x4x12 & 96 &-216.74323 &-216.48616 &0.25707 \\
5x5x15 &188 &-216.74291 &-216.48615 &0.25676 \\
\end{tabular}
\label{tab:convergenceb1}
\end{table}


\begin{table}
\caption{}
\vspace{10pt}
\begin{tabular}{|l|r|c|c|c|}
K-point grid & indep. K-points& E$_{b2h2-1}$&E$_{b2h2-1-swapped}$& $\Delta$E \\
\hline
 1x1x1&  1 &-606.30421 &-606.22773 &.07648\\
 1x1x2&  1 &-704.05048 &-703.77620 &.27428\\
 1x1x4&  2 &-700.48922 &-700.24548 &.24374\\
 2x2x2&  4 &-704.02670 &-703.77286 &.25384\\
 1x2x8&  8 &-700.38011 &-700.12094 &.25917\\
 2x2x4&  8 &-700.73700 &-700.49654 &.24046\\
 1x2x10& 10&-700.42044 &-700.15409 &.26635\\
 2x2x8&  16&-700.44533 &-700.18672 &.25861\\
 2x2x10& 20&-700.45159 &-700.19033 &.26126\\
\end{tabular}
\label{tab:convergenceb2h2-1}
\end{table}


\newpage

\begin{figure}
\epsfig{figure=rules.eps}
%%\vspace{100pt}
\caption{}
\label{fig:penrose}
\end{figure}

%% start trial

\newpage
\begin{figure}[t]
\centerline{\epsfig{file=h2_mono.eps}}
\centerline{\epsfig{file=b1.eps}}
\label{fig:str1}
\end{figure}
%% end trial




\newpage
\begin{figure}
\epsfig{figure=h2_mono.eps}
%%\vspace{100pt}
\caption{Filling space with hexagons. This approximant has one hexagon per monoclinic cell (fine dashing, h1 strcutre in the text). The unit cell has 25 atoms. The cell can be doubled to get an ortho-rhombic unit cell (coarsed dashing, h2-1 structure in the text).}
\label{fig:h2_mono}
\end{figure}

\begin{figure}
\epsfig{figure=b1.eps}
%%\vspace{100pt}
\caption{Single boat (b1) approximant. It contains 41 atoms per monoclinic cell. Swapping the pair in the small rectangle changes the distribution of angle decorations. The two 144$^\circ$'s surrounding the middle bond are images of each other. So total cancellation of 144$^\circ$ interactions is not possible in this unit cell. Enlarging the unit cell gives the freedom of assigning different decorations to the outer bonds to cancel that.}
\label{fig:b1}
\end{figure}


\newpage
\begin{figure}
\epsfig{figure=h2_ortho.eps}
%%\vspace{100pt}
\caption{Ortho-rhombic two hexagon approximants (h2-2). Different arrangements of hexagons from fig~\ref{h2_mono}. Even with a single tile, space can be covered in different ways.}
\label{fig:h2_ortho}
\end{figure}


\begin{figure}
\epsfig{figure=b2.eps}
%%\vspace{100pt}
\caption{Two boat approximant (b2). One of the heel bonds in the boats particpates only in 144$^\circ$ angles.}
\label{fig:b2}
\end{figure}

\begin{figure}
\epsfig{figure=b2h2_1.eps}
%%\vspace{100pt}
\caption{Space can be tile in many ways using HBS tiles even with the same number of atoms or tiles. All these approximants (b2h2-1, b2h2-2 and h3s1) have 132 atoms per unit cell. Structures in (b) and (c) are related to each othere by a phason flip outlined in (c) with a small dashed line. Bonds in dashed shapes in (a) are bonds that can give information about pure angle interactions (after slight rearrangements)}
\label{fig:b2h2_1}
\end{figure}

\begin{figure}
\epsfig{figure=b2h2_2.eps}
%%\vspace{100pt}
\caption{}
\label{fig:b2h2_2}
\end{figure}

\begin{figure}
\epsfig{figure=s1h3.eps}
%%\vspace{100pt}
\caption{}
\label{fig:s1h3}
\end{figure}

\begin{figure}
\epsfig{figure=vasp3sep.eps}
%%\vspace{100pt}
\caption{}
\label{fig:vasp3}
\end{figure}

\begin{figure}
\epsfig{figure=vasp9sep.eps}
%%\vspace{100pt}
\caption{}
\label{fig:vasp9}
\end{figure}

\begin{figure}
\epsfig{figure=mok9sep.eps}
%%\vspace{100pt}
\caption{}
\label{fig:mok9}
\end{figure}

\begin{figure}
\epsfig{figure=mgpt9sep.eps}
%%\vspace{100pt}
\caption{}
\label{fig:mgpt9}
\end{figure}

\end{document}        
