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\def\Etotrel{{E^{\rm tot}_{\rm rel}}}
\def\Eunrel{{E^{\rm un}_{\rm rel}}}
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\twocolumn[
\vspace*{30mm}
%  TITLE
\begin{LARGE} 
\begin{center}
Combined energy -- diffraction data refinement of decagonal AlNiCo
\end{center}
\end{LARGE}

%  AUTHORS
\begin{large}
\begin{center} 
M. Mihalkovi\v{c}$^{1,3}$, C. L. Henley$^{2}$ and M. Widom$^{1}$
\end{center}
\end{large}

%  ADDRESS
\begin{footnotesize}
\begin{it}
\begin{center}
$^{1}$Physics Dept., Wean Hall, Carnegie-Mellon University, Pittsburgh PA 15213 USA\\
$^{2}$Physics Dept., Clark Hall, Cornell University, Ithaca NY 14853 USA\\
$^{3}$Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia\\
\end{center}
\end{it}
\end{footnotesize}
\begin{footnotesize}
\begin{center}
%
%  DATE
%
\date{\today}
%Received 22 June 1998; received in revised form 11 August 1998
%
%  END DATE
%
\end{center}
\end{footnotesize}

%  ABSTRACT
%\vspace{4ex}
\begin{small}
\hrule\vspace{3ex}
\begin{minipage}{\textwidth}
{\bf Abstract}\vspace{2ex}\\
\hp 
We incorporate realistic pair potential energies directly into a
non-linear least-square fit of diffraction data to quantitatively
compare structure models with experiment for the Ni-rich $d$(AlNiCo)
quasicrystal.  The initial structure models are derived from a few {\it a
priori} assumptions (gross features of the Patterson function) and the
pair potentials.  In place of the common hyperspace approach to the
structure refinement of quasicrystals, we use a real-space tile
decoration scheme, which does not rely on strict quasiperiodicity, and
makes it easy to enforce sensible local arrangements of the atoms.
Inclusion of the energies protects the fit procedure from converging
on spurious solutions.  The method pinpoints sites which are likely to
break the symmetry of their local environment.
\OMIT{It is found that moderate-sized crystal approximants are 
adequate for the calculation of diffraction amplitudes.
Our refinement of a high-quality data set measured by
Cervellino {\it et al} yielded a solution similar to
a refinement published by Takakura {\it et al}.}
\vspace{2.5ex}\\
{\it Keywords:}\/ 
quasicrystals; structure refinement; d-AlNiCo

\end{minipage}\vspace{3ex}
\hrule
\end{small}\vspace{6ex}
]


\def\chem{\alpha}
\def\chemsite{\chi}
\def\orb{j}
\def\bxy{{b^{xy}}}
\def\bz{{b^{z}}}
\def\Esite{{E^{\rm site}}}
\def\Eorb{{E^{\rm orb}}}


\section{Introduction}
\label{sec:intro}
\hp

Recently, two high-quality refinements have been published from
single-crystal X-ray data from the ``basic Ni-rich'' phase of
$d$(AlNiCo)~\cite{CHS,TYT}.  At the same time, a Monte Carlo method
was developed~\cite{alnico} to {\it predict} the same structure,
approximating the Hamiltonian with pair potentials, and using as
inputs only the crystallographic parameters (plus the fact that the
structure is strongly layered).  Here, we present the first trial of a
combined method which uses energy and diffraction information to fix
complementary features of the atomic structure.

The decagonal AlNiCo system exhibits a variety of metastable and
stable phases, out of which the ``basic Ni-rich''
phase~\cite{Ni-basic} has attracted many detailed studies due to its
high structural quality.  The refined structures of \cite{CHS} and
\cite{TYT} agree on basic features, but they differ in many important
details.  The refinement presented by CHS~\cite{CHS} exhibits 8\AA\
periodicity along the vertical axis, lacks 10-fold screw symmetry
property, and does not lend itself an easy interpretation in terms of
a tiling geometry.  The structure solution of \cite{TYT} has 4\AA\
periodicity, and can be readily interpreted in terms of the
Hexagon-Boat-Star (HBS) tiling with edge length of $\sim$6.5\AA\
tiling geometry.  The R-factors achieved are comparable in the two
cases.


Quasicrystals, as aperiodic structures, typically include
local patterns which are rare enough to be ill-determined
from diffraction, but common enough to have a significant
effect on physical properties (such as total energy
or conductivity). 
Thus the problem of structure prediction from energies becomes
entangled with structure fitting from diffraction:
we suspect that, in the future, the best 
quasicrystal structure fits 
will combine the two inputs in some fashion. 

Here, we present the first structure fit  in 
quasicrystals, in which energy and diffraction
information are combined in the same objective function.
Previously, powder diffraction data on $i$-TiZnNi \cite{hennig} was
systematically refined with energy inputs. 
In that work, 
each type of information was used separately
to fix the degrees of freedom which are most sensitive to it:
a least-squares refinement from that data
determined the species occupying each site type, and this
was alternated with ab-initio calculations to relax the positions. 

In the present study we calculate energetics from pair potentials
microscopically derived using ``generalized pseudopotential theory''
(GPT) by Moriarty and Widom~\cite{GPT}, a total of six functions
(e.g. $V_{\rm AlNi}(r)$) for all combinations of the species.
The prominent feature of these potentials is Friedel oscillations, 
so that a typical  potential has three minima before the
cutoff radius (which we chose to be $10$~\AA).
The atom sites are called $\{ \R \}$, with 
$\chemsite(\R)$ designating the chemistry (Al, Ni, or Co)
of that site.
A convenient diagnostic tool in the refinement
of a decoration structure~\cite{AlMnI} is the ``site energy'', 
the portion of total energy ascribed to interactions of site $\R$:
   \begin{equation}
     \Esite (\R) \equiv \sum _{\R'} 
    V_{\chemsite(\R)\chemsite(\R')}(|\R-\R'|)  .
   \end{equation}
\OMIT{In the context
of the combined refinement, high site energies may indicate
conflicting trend from the diffraction data.}
%%%%%%%%%%%
Then the  total energy is 
   \begin{equation}
    E_{\rm tot} \equiv \frac{1}{2} \sum _{\R\R'} \Esite (\R) .
   \label{eq:totalE}
   \end{equation}


Whereas previous refinements of quasicrystals
have used the hyperspace-cut representation of the structure, 
our results demonstrate the equal effectiveness of 
an alternative approach, the tile-decoration framework. 
The value of the tiling-decoration approach is that 
the structure models do not rely on periodicity or quasiperiodicity, 
%% models are not biased by global symmetry, 
and they are naturally represented in ``real'' space. 
%%MM: The ``global'' was meant here as an opposite to ``local''; local
%% symmetries are the same for approximants and quasicrystal. So I guess
%% the ``global'' really meant presence/absence of translational symmetry.
Consequently, decoration is an elegant way~\cite{hennig} to transfer/combine 
information between the different system sizes appropriate for
different calculations, {\em e.g.}:
(i) numerical diffraction calculations from a large approximant;
(ii) eventual Monte Carlo simulations using a tile reshuffling update 
and a tile-tile Hamiltonian, in a smaller approximant;
(iii) molecular dynamics using pair potentials;
(iv) atom relaxation using an ab-initio total energy code, 
which is typically feasible only in  the smallest approximants.


The decoration formulation might be attractive for dealing
with {\it randomness} (an issue we ignored 
in the present contribution, however).
A realistic structure  model is an 
{\em ensemble} of atomic configurations.  When the contents of 
different sites are independent random variables, then a
stochastic, mixed  occupation of each site is a fair
description.  If, however, the sites are highly correlated, 
a random tiling may be a more reasonable model; indeed, 
that is our present picture of the ``basic Ni'' phase of
$d$-AlNiCo~\cite{alnico}, 
in contrast to $d$-AlCuCo,
with indications of obeying matching rules~\cite{lightbulb}.
A tile-decoration is superior
to a  hyperspace description in the random-tiling case:
all the randomness may be ascribed to the tile
configurations, while the decoration of each tiling is
deterministic. 
Some of the technical questions of a 
random-tiling  structure refinement were
discussed in Sec.5 of Ref.~\cite{rtdiff}.

\OMIT{This approach is also highly suitable for
quantitative analysis of the disorder effects
that impede accurate structure refinements, 
but for simplicity
the present exploration was based only on (approximants of)
ideal quasiperiodic tilings.} 

\section{Structure modeling for combined fit}

To explain the decoration approach, we review nomenclature introduced
in \cite{AlMnI}: A {\it decoration} is a mapping which, given any
valid tile-configuration, produces a set of atom sites.  The sites are
grouped into {\it orbits}, each of which is {\it bound} to a
particular kind of tile, and lies in the same positions on every tile
of that kind (An orbit plays a role like an orbit of symmetry-related
Wyckoff positions in a crystal structure.  In the hyperspace-cut
representation of a quasiperiodic structure, each member of the
decoration orbit on a representative tile would correspond to one
subdomain of the acceptance domain in perpendicular space.).  

\OMIT{Identical structures may be produced by a variety of decoration rules.}

More generally, orbits are bound to {\it tiling objects} which may be not
only tiles, but other geometrical constituents of the tiling such as
tile vertices -- e.g. we may choose to surround every vertex by an
identical symmetric cluster.  Another kind of a tiling object is
defined to overlay a certain local pattern in the tiling. We can then
introduce context dependence by {\it rebinding} certain atoms to the
new tiling object. Other atoms remain bound to the old objects, so we
do not define too many positional parameters.  One decoration rule is
a {\it rebinding} of another if it has more orbits, and the family of
structures produced from the original decoration (by varying the
parameter values) is a subset of the family produced from the
rebinding decoration.

By default we require that atomic decorations inherit the symmetry of
the tiling objects so that placing an atom at general position will
generate all symmetry equivalent sites. Furthermore, preservation of
the site symmetry constrains the degrees of freedom for atomic
displacements. In reality, forcing atomic motifs to obey the point
symmetry of tiling object is not always energetically favourable.

In the tile-decoration machinery,
such situations are handled by the concept of {\it symmetry-breaking}: 
%% the tile object symmetry may be arbitrarily broken by associating
a {\it flavour} (like an arrow on a tile)
is associated with each
(broken) symmetry related variant.~\cite{AlMnI}, so that the decoration
remains formally deterministic and reflects the symmetries
of the flavoured tiles.
%% In the decoration process. 
The flavours may be initialized randomly, and in the simulation process
(for example Monte Carlo annealing) the symmetry-breaking may
be effectively used as an update move.

\OMIT{ MM-15dec: this is somewhat misleading here, as the Decagon never
emerged as an idependent tile in our former study.
An example of symmetry breaking is the 20~\AA~ decagonal columnar
cluster~\cite{abe}. While its outer shells obey the decagonal
symmetry, the atomic motif near the centre has broken symmetry. This
is why we did not adopt the 20~\AA~ decagon as a tiling object in its own
right, but rather decomposed it into a Boat tile and a pair of Hexagons.
}

The decoration rule for our combined fit is adapted from the
deterministic model of~\cite{alnico}, in which atoms decorate an HBS
tiling with edge length $a_q\sim$6.4\AA. 
(similar to Fig.~3a of \cite{alnico}, but without the
complication of the ``bow-tie'' defects shown there.)

\OMIT{(Also as in Fig3b, 
but without the Al-Co substitutions  on some tile edges.
MM says Fig 3c there is closest, but not exactly the right thing, either.}
\OMIT{ (The lattice constant in CHS data is actually
6.37~\AA; the pair potentials we are using are matched to 6.43 A.
Thus, we are using the structure factors of CHS with a slightly
changed lattice constant.)}
%%%
This model was
derived by successive elimination of the degrees of freedom in
Monte-Carlo simulations with the GPT pair potentials.  Throughout the
paper we use a simple convention, with HBS edge vectors are $a_q
\vec{e}_i$, where $\vec{e}_i\equiv(\cos(2\pi i/5),\sin(2\pi i/5))$.
Notice that our Hexagon tile has a mirror plane running down its long
axis, but not transverse: we should visualize it as if marked by an
arrow to distinguish the two ends.  In the same way, the Boat's mirror
symmetry and the Star tile's fivefold rotation symmetry are broken to
allow for our decoration.

Our ``basic'' rule uses as supplementary tiling objects the
Fat rhombi (which appear naturally when the HBS tiles are
subdivided into Penrose rhombi), as well as the interior 
vertices thereby created; this captures the similarity
of the local environments in all the 72$^\circ$ corners
of HBS tiles.
This rule is shown in Fig.\ref{fig:model}.
All Ni atoms are ascribed to the HBS
edges, Co atoms decorate the supplementary
interior vertices \OMIT{inside the HBS tiles}, as well
as a site on the long diagonal of every Fat rhombus.
Orbits 12 and 16 implement the broken symmetries of the Boat and Star.
In the end, this decoration rule associates 16 orbits with
nine tiling objects, and yields 38 positional parameters.

Refinement using the ``basic'' binding motivated a more complex 
``augmented'' binding (Table \ref{tab:orbits}), 
that proved to be more successful in fitting the data. 
Firstly, since the Ni(3) orbit alone contains more than 20\% of all
atoms, we split it among 3 objects, ``context-sensitive''
HBS Nodes. In both approximants we used, three classes of
HBS vertex stars occure: 5-fold, 3-fold and 2-fold (in random HBS,
also 4-fold and another 3-fold vertex type occurs). The symmetry
of these objects gives rise to 4 Ni orbits, labelled Ni(3$_a$-3$_d$).
The new orbits are listed in Table \ref{tab:orbits} below the
horizontal line.
Second, one of the two Al(15) atoms has been replaced by vacancy implementing
symmetry-breaking of the Hexagon mirror (Al(15+) in the 
Table \ref{tab:orbits}), and new atom Al(17) has been
placed near the center of the Hexagon.
%(using the ``ideal'' positions, it creates a short 
%2.2\AA\ bond with Al(9))

\OMIT{MM : Chris was proposing we need another separate figure
of the augmented binding. I agree - currently augmented binding is
summarized in the tab:orbits, but it doesnt have separate figure.
Asymmetrically, the basic binding has its own figure, but is only
implicit in the table above the horiz. line - the data are on
augmented binding... CLH+: we do not have space for another figure.}

The atomic density for the decoration rule (in either binding variant)
is 0.0705 atoms/\AA$^3$,
and the composition is Al$_{70}$Ni$_{20.7}$Co$_{9.3}$, slightly richer
in Co than the CHS sample (Al$_{70.6}$Ni$_{22.7}$Co$_{6.7}$). 
When the atoms are fixed at ``ideal'' sites, the minimal distance
occuring between Al-Al, Al-Co and Al-Ni pairs is 2.46\AA. The
structural energy of this initial model is $-0.297$ eV/atom for
a small approximant with 214 atoms per unit cell; using
positional degrees of freedom of the decoration rule it relaxes
to $-0.393$ eV/atom. Fully relaxed energy (atomic displacements
for atoms in the same orbit no longer equivalent) is $-0.422$ eV/atom,
and finally allowing atoms to relax their positions in double
period (8\AA) unit cell, energy drops to $-0.442$ eV/atom.

{\it Periodic approximants.}
A drawback of the tile-decoration approach is that the infinite
non-periodic quasicrystal must usually be handled indirectly, being
approximated by an  periodic ``approximant''.
%% approximated by a sequence of approximants scaling down the
%% ``background phason strain'' (This may be approximately viewed as an
This necessarily has a 
``background phason strain'' which is, roughly, the
average tilt of the cut through hyperspace away from the irrational
orientation with 10-fold symmetry.)  
Consequently, each orbit of
reciprocal lattice vectors $\Q$ of the quasicrystal gets split into
several inequivalent orbits of the approximant's reciprocal lattice
vectors $\Q_{\rm app}$.  In the fit, we selected only one representative
$\Q_{\rm app}$ for each decagonal $\Q$, since the number of datapoints is
already large, but we monitored systematically the errors thus
introduced, calculating $\sigma_{\rm app}$ as the r.m.s. deviation 
of the difraction amplitudes among
all $\Q_{\rm app}$ mapping on a given $\Q$. 
%%[ to MM: If you want, replace the idea of a succession
%%by saying back.phas.strain.  scales
%%with the inverse approximant cell size.]
For the larger approximant
discussed below, $\sigma_{\rm app}$ was similar to or less than the
experimentally observed uncertainty for all peaks.

A more systematic treatment of the approximant diffraction amplitudes
would be to symmetrize over all those deriving from a single orbit of
quasicrystal $\Q$. This would demand careful handling of the phase factors,
by properly centering the approximant structure both in real and in
perpendicular space~\cite{hennig}.

A further difficulty is that for smaller approximants, different
quasiperiodic $\Q$ vectors may map on the same approximant $\Q_{\rm app}$.
We found the smallest orthorhombic approximant avoiding such ambiguous
mappings for {\it all} reflections in the CHS data set has edge
lengths 61\AA\ $\times$ 32\AA\ in the decagonal plane.  In the
following, we label this approximant T$_{42}$ (for the number of HBS
vertices per unit cell).  For our decoration rule it contains 560
atoms per unit cell.  We found it satisfactory to use just one more
bigger approximant, with both sides in the decagonal plane larger by a
factor of $\tau\equiv(\sqrt{5}+1)/2$. We denote this tiling T$_{110}$.


\section{Procedure}

{\it Dataset}. We used the CHS dataset~\cite{CHS} \OMIT{, crosschecked
with an older dataset from the same group~\cite{anc-oldref}}.
It dataset contains 2767 unique reflections under the assumption
of 10/$m$ Laue group; by averaging over symmetry-related amplitudes,
to enforce 10/$m mm$ Laue symmetry,  
a set with 1544 unique reflections is obtained.
The two possibilities gave about equal internal $R$-factors.
\OMIT{MM-15dec: in fact, Cervellino didnt bother in his paper to
tell what the ``internal R-factor'' is, while refering  it. Here
we just quote their statements - so I dont think we need to define.
In particular, I'm not sure if they used ``weighted'' or ``plain''
R-factor, but I presume it was plain R-factor, our Eq.7.
[@x4a. You aren't literally averaging over amplitudes since
you don't know signs; averaging |F| or intensities?]
@y: averaging |F|'s
[@x4b Need to define ``internal R-factor''? Your
explanation still doesn't tell me its definition so it could
be calculated.]}
\OMIT{MM:``observed'' amplitudes are ALL (not just sym.-equiv.) measured
amplitudes, and ``calc'' amplitudes are averaged through Laue symmetry.}
%%%%%%%
Our structure model has $P10_5/mmc$ space
group, so we used the smaller 10/$mmm$ data set.
(Using the 10/$m$ set gave
practically the same results, except that typically both the R$_w$ and R factor
are increased by 1\%.)

The partial diffraction amplitude for tiling orbit $j$~is
\begin{equation}
\label{eq:Forb}
F^{\rm orb}_\orb(\Q)= \sum _\chem W_{\orb\chem}(\Q) f_\chem(\Q)
\sum_{\R\in \orb: \chemsite(\R)=\chem} e^{i\Q\cdot\R}.
\end{equation}
The inner summation in (\ref{eq:Forb})
runs over all atoms of species $\chem$ in orbit $\orb$, 
and $f_\chem(\Q)$ is the atomic form factor for that species.
The Debye-Waller (DW) coefficients
$\bxy_{\orb,\chem}$ and $\bz_{\orb,\chem}$ couple with
decagonal-plane and vertical $\Q$ components, and take
different values not only for different orbits, but
also for different species occupying the same orbit:
\begin{equation}
W_{\orb\chem}(\Q)=
    \exp[{-{1\over 4}(\bxy_{\orb\chem}\Q^2_{xy}+\bz_{\orb\chem}\Q^2_z}]
\label{eq:DW}
\end{equation}
The total calculated diffraction amplitude is then
%@@ SIMPLIFIED
  \begin{equation}
  \label{eq:Fcalc}
  F^{calc}(\Q)=e^{-{1 \over 4} b_\perp\Q^2_\perp}|
       \sum_\orb{{F^{\rm orb}}_\orb(\Q)}.
\end{equation}
where we introduced a single perp-space DW coefficient $b_\perp$.
%%%%%
\OMIT{
An additional global parameter $I_0$
for a background intensity subtraction
(same for all peaks),  
was necessary for the fit with the dataset of \cite{anc-oldref}, 
but not for the CHS dataset as reported here.}
%%%%%
Finally, we compute goodness/reliability factors
\begin{equation}
\label{eq:chi}
\chi^2 \equiv \sum_{\Q}{(F^{obs}(\Q)-F^{calc}(\Q))^2/\sigma(\Q)^2}
\end{equation}
% \begin{equation}
% \label{eq:wr}
% R_w=\sum_{\Q}({|F^{obs}(\Q)- F^{calc}(\Q)|}/\sigma^2)/\sum_{\Q}{(F^{obs}(\Q)/\sigma^(\Q)2)}
% \end{equation}
\begin{equation}
\label{eq:r}
R\equiv\sum_{\Q} \frac   {\big|F^{obs}(\Q)-|F^{calc}(\Q)|\big|}
                         {\sum_{\Q} F^{obs}(\Q)}.
\end{equation}
As usual, the weighted $R$-factor, called $R_w$, is
defined like (\ref{eq:r}) except that each term in the numerator
and denominator acquires a factor $1/\sigma(\Q)^2$.

{\it Implementation.}
We couple pair-potential energy with $\chi^2$ via parameter $\lambda$
in the objective function
for the non-linear least-squares minimization:
\begin{equation}
\label{eq:obj}
U = \chi^2 + (E-E^{\rm targ})^2/\lambda^2
\end{equation}
where $E$ is the energy (\ref{eq:totalE}), per atom.
Here $E^{\rm targ}$ is a ``target'' energy,~ \footnote
{When $\Delta E \equiv E^{\rm tot}-E^{\rm targ} \gg \lambda$, 
as is usually the case in the orbit-constrained relaxation, then 
$U= \chi^2 + 2 (\Delta E/\lambda^2) E^{\rm tot}$
would be practically equivalent to (\ref{eq:obj});
since the likelihood of parameters is supposed to 
behave as $\exp(-U)$,  $\lambda^2/2\Delta E$ is 
evidently playing  the role of a temperature.
\OMIT{Thus only a single parameter is actually
meaning ful, the ratio $\Delta E/\lambda^2$.}}
%%%%
and is treated in the fit 
as a dummy datapoint with an ``error bar'' $\lambda$.
$E^{\rm targ}$ may be set to the total structural
energy per atom after an (unconstrained) relaxation $\Eunrel$.
%%%%
\OMIT{Occasionally interesting results may also 
be obtained when the target energy
is chosen higher than the relaxed energy.}

Our fitting form 
(Eq.~(\ref{eq:DW}))
allows independent DW 
parameters $\bxy$ and $\bz$
for each orbit/species combination, 
but these are underdetermined by the data; to obtain sensible
results, the fit 
needs to be biased towards having similar DW factors
  \OMIT{(the most uniform final set of DW factors)}
for similar atoms, in the spirit of the maximum-entropy method.
%%%%%%%%
\OMIT {(In the context of quasicrystals, 
deBoissieu, Steurer-Haibach, and Kato-Yamamoto all
have implemented MEM in the context of getting electron
density maps directly from the data by constraining 
$\rho_{el}$ to be positive everywhere. 
They proceed by idealizing the hyperspace density maps to
a hyperatom model, and the next stage is to the
non-linear fit to this model.)}
%%%%%%%%
We implement this by adding terms for $\bxy$ and $\bz$
to the $\chi^2$ sum:
before each iteration, we calculate the average DW factor for
each species and the $xy$ and $z$ components of DW factor,
and set them as datapoints for the next iteration, with appropriately
chosen $\sigma_{DW}$. 
\OMIT{(A more proper implementation would include,
right in the objective function, 
a sum of terms such as $|\bxy_{j\alpha}-\bxy_{j'\alpha'}|^2$
(and similarly for the $z$ components);
the coefficients 
would control how strongly we are biasing the coefficients to be equal.)}

\OMIT{In these fits, we optionally added
the $\sigma_{\rm app}$ measuring deviation
from decagonal symmetry for each datapoint, to the experimental
error bars $\sigma_{obs}$. This proved useful especially for
the smaller T$_{42}$ tiling.
It can be dealt with,  more systematically, by
averaging over the full orbit as in \cite{hennig}.}

\OMIT { The energy constraint incorporated into the
objective function might push the fit away from spurious local
minima, or at least avoid it falling down into some. Our refinement
was set up to achieve this systematically: in ``inner'' iteration
cycle, we optionally reinitialize fit imposing a noise on 
current DW and positional parameters. In the ``outer''
loop, we sample plausible $E_targ$ and $\lambda$ ranges.
In the end of each inner loop, we record the solution.
This randomization  was unnecessary when the DW coefficients
are biased to be similar.}

\section{Results}

%%[@MM wants to add here: the atomic displacements are crucial for
%%accurate fit. Compare R-factors for ideal positions, and relaxed
%%positions, to see how far the two are from the fit-optimized
%%positions. Note- NOT INTENDED FOR USE: 
%%when CHS do their ``comparative analyzis'', 
%%from which their model comes totally outstanding, I suspect
%%they dont allow the atoms in the competing models to adjust
%%the displacements. So in reality the numbers they mention there,
%%like 30\% R-factors for competing models, are certainly blown
%%up... and as we show here, it is NOT sufficient to move the atoms,
%%they need also the right degrees of freedom (our notion of 
%%``binding'').]
%%[@MM We need to say something about the Cervellinos model.
%%I guess we can just say that it had lower symmetry than
%%ours (i) breaking 5m to 5 AND 4A->8A AND screw axis; importantly,
%%their model has TM atoms also at places were we dont!
%%We did not study the possibility of such symmetry breakings,
%%forcing the 5/m and the screw and 4A for now.
%%The interesting point here might be to try to link CHS 8A superorder
%%to the stacking faults.
%%Today, I spent a lot of time studying their model, but its really
%%awkward how it's described: they divide hyperatoms, but there
%%is no explicit link between say Al and TM occupying the same place
%%- the pieces are numbered differently even if they are on the same 
%%place for Al and TM. So it is really difficult to see what is going on.
%%From their tables, I painfully tried to calculate some overall 
%%characteristics, like chemical composition of each hyperatom,
%%but I couldnt get the sums correct...
%%]

Fitting the ``basic'' binding to the CHS dataset
we obtained reasonable, but not satisfactory R-factors $R_w$=0.122 
and $R$=0.202. The large displacement of atom Al(15) (see Fig.
%We notice that the
%biggest displacement occured for orbit Al(15) (Fig. \ref{fig:fmodel},
%top panel), pointing towards the center of the Hexagon tile; at
%the same time, $E_{site}$ for orbits Al(13) and Al(15) shown
%highest increase out of all atoms ($\sim$0.3eV).
\ref{fig:fmodel}, top) and other considerations motivated our
``augmented'' binding; Table  \ref{tab:summary} summarizes our
results for the latter.
\OMIT{(This rebinding has already been described above
under ``Structure modelling''.)}
%MMNEW: define relaxed energy as symmetry-constrained relaxed...
The symmetry-constrained relaxed energy $\Erel$
of the 
@@ ``augmented''
%% new decoration
rule is tolerably higher than $\Erel$
of the ``basic'' rule (see Table \ref{tab:summary}). 
However, the combined fit improved dramatically, displacing the
Al(15$_a$) atom towards the long body diagonal of H tile.
We have selected two structure solutions by the combined fit,
one with lowest R$_w$ factor but rather high energy, and one
with slightly worse R$_w$ factor, but comparable in energy
with the basic decoration rule (Table \ref{tab:summary}). 
@@
The $F_{calc}(\Q)$ from our fit is compared to the
$F_{obs}(\Q)$  from the diffraction data in Fig. \ref{fig:fit}.
%
%, together with errors
%arising from averaging $F_{calc}$ through the sets of $\Q_{\rm app}$ 
%for each $\Q$,
%as a function of $\Q_\perp$. In general, the approximants
%deviate from decagonal symmetry most significantly at low $\Q_\perp$,
%ei, for strong intensities; in this region, $\sigma_{\rm app}>\sigma_{obs}$ 
%for T$_{42}$ approximant, while $\sigma_{obs} \gg \sigma_{\rm app}$ at large
%$\Q_\perp$. (And there seems to be a puzzling correlation
%between $\sigma_{\rm app}$ and $\sigma_{obs}$ roughely up to 
%$|\Q_\perp|=$0.75\AA$^{-1}$). Deviation from decagonal symmetry
%for the T$_{110}$ is clearly within the error bars.

The fitted DW factors, especially those of Al, 
exhibit high degree of anisotropy. 
(Table \ref{tab:summary}\ ).
This agrees with the 
TYT refinement~\cite{TYT} of a different sample of AlNiCo.
(The TYT sample was slightly richer 
in Al and Co than the sample for the CHS data~\cite{CHS} 
that we are fitting here.)

Most of the TYT hyperatom orbits correspond to our HBS tile
decoration orbits, 
and their individual orbit DW factors are mostly 
consistent with ours for the corresponding atoms.
The most pronounced disagreement is just the orbit 
Al(15+) for which $b_{z}/\bxy \approx 5$ in our refinement, 
but $\approx 1/5$ in the TYT refinement (see Table 2 in Ref.
\cite{TYT}, site label 9). This atom is at the same time
displaced farthest from its ideal position for both
TYT and our refinements, and the displacements have similar 
magnitude and direction for TYT and our ``{\em basic}'' binding
(see Fig.\ref{fig:fmodel}).
In the ``augmented'' binding, Al(15+) occurs in a completely
different environment, displacing towards long diagonal of H tile,
which explains the discrepancy in the DW factors.

\OMIT{MM-dec15: fixed/clarified above
This atom  was suffering largest displacement in our ``basic''
binding (see Fig.\ref{fig:fmodel}, top panel); and the displacement 
had similar magnitude and direction compared with TYT.
This discrepancy
can be easily understood, since our rebinding entirely
changed the environment of this atom (which also displaced significantly
parallel to the long axis of the hexagon tile).
[@x9xa ``WAS displaced on the mirror plane'' DID I MISUNDERSTAND?]
@MM: well - it displaced ON the long diagonal of the H tile (this is
clearly seen in the figure).
CLH: So you intended ``displaced ALONG''?
But no, orbit 15 does NOT lie on the long diagonal.
Do you mean it displaced  TOWARDS the long diagonal???]
MM: YES
}
%%%%%%%

\OMIT{MM-15dec: merged para below into the text above, and hopefully
clarified the sense.@@
Furthermore, in the TYT refinement it is precisely this atom which
is displaced the farthest from its ideal position;
this can be compared with the displacement
of Al(15) in our ``basic'' binding (see Fig.\ref{fig:fmodel}, top panel);
indeed, the displacements have similar magnitude and direction.
}
\OMIT{
[@x9c I didn't get the logic here: to explain why you get a DIFFERENT
anisotropy of DW than TYT, you argue that the atoms
have displaced in the SAME way. CAN YOU FIX TEXT AS YOU SAY?
@MM: I forgot to tell - the displacements were the same for the BASIC
binding, when there was pair of Al(15) per H tile; in the ``augmented''
binding, one of the two sites was left vacant(symm.-break) and an
EXTRA atom Al(17) has been placed near the center of H tile.
This Al(17) in just 2.3A distant from the Al(9); TYT model has the
same short pair of Als near the center of H tile. Another issue related
to this which I didnt discuss are partial occupancies in the TYT model.]
}

The TYT model and our decoration (in either variant) 
differ from the CHS model in having a larger
space group symmetry (10-fold screw axis vs. a 5-fold axis, 
4\AA\ translation vs.  8\AA\ translation), 
but also in other respects: higher atomic
density, and absence of transition metals (TM)
in the sites corresponding to hyperatom ``B''
in Ref.~\cite{CHS}.
Our  decorations are
more constrained than the other models, in that
@@ mixed Al-TM occupancy is disallowed, partial 
occupation appears only through symmetry-breaking, 
and unphysical close pairs never appear on account
of the energy term in Eq.(\ref{eq:obj})
\OMIT{
MM: I meant we dont have ``partial occupancies'' in the strict sense,
and dont allow Al/TM mixes. CHS model has MANY partially occupied sites,
with low occupancy. TYT has few sites with low occupancy, but another
few with large fractional weight with Al/TM mix.}
%% using geometrically rigid assignement of species, with
%% strictly controlled ``partial occupancies'' (for symmetry-broken
%% orbits; by ``controlled'' we mean that pair of atoms may never appear
%% on forbiddingly close sites, due to the energy constraint).
The CHS model achieved a lower weighted R-factor ($\sim$0.06) 
than the present refinement of the same data set,
at the price of including many more parameters. 
It remains to be  resolved whether 
the CHS model is the essentially unique minimum for
their data set.

\section{Discussion}

The rearrangement of the atomic decoration inside the Hexagon
tile --  @@driven, we think, by the diffraction term in (\ref{eq:obj})
-- is reminiscent of the atomic shifts observed in 
molecular dynamics (MD) simulations under the same 
potentials with an initial atomic configuration similar
to the present model~\cite{sendai}.
\OMIT{
CLH: is the shift in the present fit is driven by
energy or diffraction term in $U$?
@MM: good question. I think it's driven by the diffraction term;
at least if I do pure constrained relaxation, the Al(17) doesnt
get there (on the diagonal of the H tile).]}
%%%%%
In those simulations, using supercells 8\AA\ or 16\AA\
thick in the stacking direction, slow
cooling led to a more favourable arrangement of atoms, in which 
one Al was pushed from the Al(15) position towards the center
of the Hexagon, while the other moved @@ onto the long body diagonal
of the Hexagon. However, in contrast to our present model
in which two such atoms get squeezed into a column, in 
\cite{sendai}
only one atom per 8\AA\ period was displaced in this fashion, 
so that the column in the Hexagon center contained
three Al atoms per 8\AA\ period.
A recent MD study~\cite{gaehler}
confirms that the optimal Hexagon decoration has 8\AA\ period,
with atomic arrangement topologically equivalent to the
result of our combined fit, but
in \cite{gaehler} the atomic sites in the second 4\AA\ bilayer 
are {\em flipped} in each Hexagon 
across a mirror plane perpendicular to its body diagonal.
\OMIT{That is,  in \cite{gaehler} the
Hexagon has a glide plane perpendicular to the layers.}
%%%%%%%%%%
\OMIT{Independent MD simulations, using our structure  
and potentials suggest that Al atoms can cyclically permute
in troughs around a Co atom or around the entire decagon.}
%%%%%%%%%%
@@
These findings again call for
study of the so-called ``stacking--fault'' mechanism,
in which the decoration rule would -- at least in a zero-order
approximation -- remain essentially a 4\AA\ rule, and the variations
in the decoration from one 4\AA\ bilayer to the next would be 
described by ``flips'' of the tiles, occuring independently
in each bilayer.

% Thus we suspect the
%solution obtained here provides only a hint for discovering
%the true optimum for an 8\AA\ structure. 
%Indeed the CHS structure solution~\cite{CHS} does
%have 8\AA translational symmetry, although it appears difficult to
%infer from their solution plausible microscopic arrangements
%of atoms.

@@
The discussed variations in the atomic structure also
%% highly relevant for understanding the structure of object
%% appearing as a generally important structural block for
illuminate a motif  generally considered important in
decagonal structures: the 20\AA\ diameter decagon.
One such cluster in marked in Fig.~\ref{fig:fmodel} by the inscribed  dashed 
circle. near the inner ring of 10 TM 
and .
Note the inner ring containing $\sim$20 Al atoms (Al(17) and Al(9))``squeezed''
in between 10 TM atoms (typically Co,  sometimes Ni), 
a feature that emerges in many HREM/HAADF
images~\cite{abe,yan}.
%%and it would never appear as an optimal 4\AA\ configuration 
%% {\em in a 4 \AA structure}
This could  appear as an optimal configuration in a 4\AA\  bilayer
%% {\em in a 4 \AA structure}
{\em only} when energy and diffraction data are combined. 
On the other hand, as suggested by the MD
studies, it {\em should} emerge as a low energy configuration 
from simulations performed
for stacked bilayers. 
\OMIT{
@MM: next sentence was silently referring to CHS result:
they claim it is very important for low R-factor that certain
atoms are assigned z-displacements that have opposite signs
for atoms related by 4A translation along c axis. They have
added superstructure reflections with zero amplitudes and very
small sigmas to dataset to ensure that this ``pseudosymmetry''
indeed cancels the superstructure reflections. So again, I've
been silently referring to the CHS paper - but I'm not opposed
to clipping, if you think the remark is not worthy]]
%%%%
It is not clear
whether such 8\AA\ superorder would give rise to any new diffraction
peaks located on odd Laue zones, due to the symmetries relating
adjacent bilayer motifs, like the one described for Hexagon tile.}

In conclusion, we used tile-decoration machinery to set up a combined 
energy--diffraction data fit
of the d-AlNiCo structure. 
%% encouraging 
The resulting R-factors are
comparable to those from a previous study~\cite{CHS},
while using fewer parameters, and not allowing ad-hoc averaged/mixed
site occupancies. Apart from the implementation of energies into
the diffraction fit, at this stage our refinement is completely
analogous to the hyperspace refinements, since we used strictly
approximants of the ideal, unperturbed quasiperiodic tiling.
The possible disorder correlations hidden in the experimental
data are still lumped into global parameters of the fit, 
mainly the perp-space
DW factor. 

Clearly, the future prospect is to take full advantage
of the real-space formulation of the problem and recover the 
correlations in atomic occupancies directly from the ``grand
combined fit''. 
This would use Monte Carlo annealing to sample
all relevant degrees of freedom (swapping individual atoms,
flipping symmetry-broken tiling objects) in conjunction with
optimizing R-factors fitting the diffraction data.~\cite{rtdiff} 
If the ``realistic'' pair potentials prove indeed sufficiently realistic,
the fit parameters may be further constrained by DW factors
pre-calculated from zero-T phonon spectra~\cite{phonon}, 
or extracted from room temperature MD annealing runs.

@@
Finally, we expect our approach will prove optimal
for refinements of large quasicrystal approximant structures,
%%% that dont have enough symmetry (and have too much complexity)
-- several such ``approximant''
phases are known to exist in the AlNiCo system~\cite{approximants} --
that are too complex to be handled by routine
crystallographic refinement approaches.
\OMIT{In such cases, the tile-decoration
approach to structure refinement has a clear 
%% obvious 
technical advantage over the common hyperspace approach, 
since hyperatom division
into symmetrically related subdomains -- the essential ingredient of the 
hyperspace refinements -- and derivation of the displacive degrees of 
freedom for atoms would be tedious, if not arbitrary.} 

%Here mainly goes explanation that all we described here
%could be perhaps equally achieved by hyperspace approach.
%But we want to explain energetics microscopically, so
%the hyperspace averaging description is not sufficient.
%We want to generate ensembles of samples, and fit somehow
%these to the diffraction data - but how to incorporate
%energy into such refinements?
%
%Second important point: possibly there is no really 4\AA\ phase?
%What is clearly doable - taking say the 560-atom approximant,
%repeat say 8 times vertically, and refine this directly.
%If the stacking fault energy is low enough such that there
%are many, this will introduce disorder also into deca plane
%(because for Q$_z$=0 $z$ positions dont matter) in the most natural
%way - better than time or space averaging, I think.
%
%Dont forget to say: Estermann's AlNiCo approximant huge
%data collection - ideal candidate for our type of refinement.

%%% A thought
%on the Steurer's refinement, the puzzling symmetry-related
%z-displacements (an atom displaced by $dz$ simultaneously with
%the corresponding atom in the NEXT 4A slab i.e. with $z$=$c$, 
%displaces by $-dz$! 
%I have an explanation based on the stacking-fault assumption:
%for the 4A vertical periodicity, there is no interaction between
%10-fold rotation related objects along $z$ direction, of course,
%because everyone is neighbor with himself.
%But when there is a stacking fault (take the ``half-moon'',
%Boat+Hexagon, for example) 

\vskip 0.1cm
{\bf Acknowledgments.}
We thank to W. Steurer for kindly  providing us his experimental
d-AlNiCo dataset, without which this work would not be possible.

\begin{footnotesize}
\begin{frenchspacing}
\begin{thebibliography}{99}
\vspace{3ex}
\itemsep 0.2ex                          

%
%  REFERENCES
%
\def\refer#1#2#3#4#5{{\frenchspacing\rm#1 #2\ #3 (#5) \rm#4}}
 

\OMIT{\bibitem{steinhardt}
\refer{P. J. Steinhardt, H.-C.~Jeong, K.~Saitoh, M.~Tanaka, E.~Abe, A.~P.~Tsai,}
{Nature}{396}{55}{1998};
%% ``Experimental verification of the quasi-unit-cell model of 
%% quasicrystal structure''
%% 55-57 NOV 5 1998 
\refer{}{Nature}{403}{267}{2000};
%% Alloys - Atomic structure of the quasicrystal Al72Ni20Co8 - Reply
%% {reply to a critique by Yan & Pennycook, with a new model}
}

\bibitem {abe}
\refer{E.~Abe, K.~Saitoh, H.~Takakura, A.~P.~Tsai, P.~J.~Steinhardt, H.-C.~Jeong,} 
{Phys. Rev. Lett.}
{84}{2000}{4609}. 
%% 4609-12. 
%% ``Quasi-Unit-Cell Model for an Al-Ni-Co Ideal Quasicrystal 
%% based on Clusters with Broken Tenfold Symmetry''


%MM: this is the first AlNiCo refinement
\refer{W. Steurer, T. Haibach, B. Zhang, S. Kek, R. L\"uck,}{Acta Cryst. B}{49}{661}{1993}.

\bibitem{Ni-basic}
%@MM added: IMPORTANT: BASIC-Ni phase discovery
%B. Zhang, Thesis No. 11439: Characterisation of decagonal Al-Co-Ni and
% its crystalline approximants. Ph.D. Thesis, ETH Swiss Federal Institute
% of Techonolgy - Zurich
%%%% and
\refer{B. Zhang, M. Estermann, W. Steurer,} {Z. Kristallogr. Suppl.} {10} {120} {1995};
%%%%%%%%%%% but this does not seem to be generally accepted
%%%%%% citation - Takakura instead cites:
\refer{S. Ritsch, C. Beeli, H. U. Nissen, T. G\"odecke, M. Scheffer,  R. L\"uck,} {Phil. Mag. Lett} {74} {99} {1996};
\refer{A.~P.~Tsai, A.~Fujiwara, A.~Inoue, T.~Masumoto,} {Phil. Mag. Lett.} {74} {223} {1996}


\bibitem{CHS}
\refer{A. Cervellino, T.~Haibach, W. Steurer,}
{Acta Crystallogr. B}{58}{8}{2002}. 
%%% 58, 8-33
%%``Structure solution of the basic decagonal Al-Co-Ni
%% phase by the atomic surfaces modelling method''

\bibitem{TYT}
\refer{H. Takakura, A. Yamamoto, A.-P. Tsai,}
{Acta Crystallogr. A}{57}{576}{2001}.
%% 576-585. 
%% ``The structure of a decagonal Al72Ni20Co8 quasicrystal''



\bibitem{hennig}
\refer{R.~G.~Hennig, E.~H.~Majzoub, A.~E.~Carlsson, K.~F.~Kelton, 
C.~L.~Henley, W.~B.~Yelon, S.~Misture, G.~Kresse, J.~Hafner,}
{Mater. Sci.~A}{294}{361}{2000};
%%% 361-365 (Dec 15 2000 
%%% ``Structural modelling of the Ti-Zr-Ni quasicrystal''
\refer{R.~G.~Hennig,   K.~F.~Kelton, C.~L.~Henley}
{preprint}{}{www.arxiv.org}{cond-mat/020217}.
%% ``Structure of the icosahedral Ti-Zr-Ni quasicrystal''
%%% Hennig & Henley energetic refinement


\bibitem{alnico}
\refer{M.~Mihalkovi\v{c}, I. Al-Lehyani, E.~Cockayne, 
C.~L.~Henley, N.~Moghadam, J.~A.~Moriarty, Y.~Wang, M.~Widom,}
{Phys. Rev.~B}{65}{104205}{2002}.
%%  (Mar. 1, 2002).
%% ``Total-energy-based prediction of a quasicrystal structure''
%%% for decagonal Al-Ni-Co, 
%%% see www.arXiv.org, cond-mat/0102085. 


\bibitem{sendai}
%% ``Total-energy-based structure prediction for d(AlNiCo)'', 
\refer{C.L. Henley, M.~Mihalkovi\v{c}, M.~Widom,}
{J.~Alloys. Compounds}{342}{221}{2002}. 
%%% {221-7 (2002). 


\bibitem{GPT} 
\refer{J.~A.~Moriarty, M.~Widom,}
{Phys. Rev. B}{56}{7905}{1997};
%%% (1997) 7905-17;
\refer{I.~Al-Lehyani, M.~Widom, Y.~Wang, N.~Moghadam, G.~M.~Stocks, %and
J.~A.~Moriarty,}{Phys.  Rev. B}{64}{075109}{2001}.

\bibitem{AlMnI}
\refer{M. Mihalkovi\v{c}, W.-J.~Zhu, C.~L.~Henley, M.~Oxborrow,}
{Phys. Rev. B}{53}{9002}{1996}. 
%%% 9002-20 
%%% ``Al-Mn Decoration Models: I. Geometrical Principles''


\bibitem{rtdiff}
%%% ``Structure determinations for random-tiling quasicrystals'',
\refer{C.~L.~Henley, V.~Elser, M.~Mihalkovi\v{c},}
{Z.~Kristallogr.}{215}{553}{2000}.
%% 553-568.

\bibitem{gaehler}
\refer{F~.G\"ahler, S.~Hocker,}
%% Franz G\"ahler and Silvia Hocker.
%% ``Atomic Dynamics in Decagonal Al-Ni-Co Quasicrystals,''
%% private http://www.itap.physik.uni-stuttgart.de/~gaehler/papers/icq8.pdf.
{J.~Non-Cryst. Solids}{}{this proceeding}{200x}.

\bibitem{yan}
\refer{Y.~Yan,  S.~J.~Pennycook,  A.~P.~Tsai,}
{Phys. Rev.  Lett.} {81}{1998}{5145}
%% 5145-8.

\bibitem{phonon}
\refer{M.~Mihalkovi\v c, H.~Elhor, J.-B. Suck,}
{Phys. Rev. B}{63}{214301}{2001}

\bibitem{approximants}
\refer{M. Doblinger, R.~Wittmann, D.~Gerthsen, B.~Grushko,}
{Mater. Sci. Eng A}{294}{131}{2000}.
%% 131-134
%% "Structural relationship and mutual 
%% transformation of approximants of the decagonal
%% Al-Co-Ni phase."
\OMIT{Estermann has an unpublished collection of
data on AlNiCo  approximants, and MM has the data set.}

\bibitem{lightbulb}
\refer{M. Widom, I.~Al-Lehyani, M.~Mihalkovi\v c,}
{J.~Non-Cryst. Solids}{}{this proceeding}{200x}.
%% http://euler.phys.cmu.edu/widom/pubs/drafts/ICQ8/icq8sub.ps


%  END REFERENCES
%
\end{thebibliography}
\end{frenchspacing}
\end{footnotesize}

%%%%%%%%%%%%%%%%%%
\begin{table}
%\begin{tabular}{lldddd}
\begin{tabular}{llllll}
orbit & object &$x$ & $\bxy$ & $\bz$& $E_{site}$\\
\hline
Al(1) &  Node & 0.075 &  3.34 &  0.34 & 0.04\\ 
Co(2) &  Node$^r$ & 0.029 &  0.09 &  0.00 &$-2.21$\\ 
Al(4) &  Bond$^r$ & 0.104 &  0.92 &  2.79 &$-0.09$\\ 
Al(5) &   Fat & 0.064 &  0.00 &  4.73 &$-0.22$\\ 
Al(6) &   Fat & 0.128 &  2.15 &  2.45 & 0.24\\ 
Co(7) &   Fat$^r$ & 0.064 &  0.31 &  0.05 &$-2.28$\\ 
Al(8) &   Fat$^r$ & 0.128 &  2.91 &  3.25 & 0.49\\ 
Al(9) &   Fat$^r$ & 0.064 &  0.03 &  5.34 & 0.32\\ 
Al(10) &  Boat& 0.019 &  1.49 &  3.76 & 0.01\\ 
Al(11) &  Boat& 0.010 &  2.62 &  4.11 & 0.39\\ 
Al(12+) &  Boat& 0.010 &  0.00 &  0.21 & 0.65\\ 
Al(13) & Hex  & 0.030 &  1.23 &  3.63 &$-0.13$\\ 
Al(14) & Hex  & 0.030 &  0.00 &  3.82 & 0.06\\ 
Al(16+) & Star & 0.008 &  0.00 &  3.57 & 0.56\\  % lab 17
\hline
%% Ni(3$_a$)&Bond$_5$& 0.055 &  1.24 &  0.50 &$-1.10$\\ 
%% Ni(3$_b$)&Bond$_2$& 0.079 & 1.05  &  0.40 &-1.34\\   % lab 18
%% Ni(3$_c$)&Bond$_3$& 0.049 & 0.24  &  0.24 &-0.87\\   % lab 19
%% Ni(3$_d$)&Bond$_3$& 0.025 & 0.92  &  0.36 &-1.55\\   % lab 20
%% Al(9$_b$)& Hex & 0.015 &  2.22 &  4.83 & 0.82\\  % lab 16
%% Al(15$_a$) & Hex  & 0.015 &  0.82 &  5.37 & 0.61\\ 
Ni(3a)&Node$_2$& 0.079 & 1.05  &  0.40 &-1.34\\   % lab 18
Ni(3b)&Node$_3$& 0.049 & 0.24  &  0.24 &-0.87\\   % lab 19
Ni(3c)&Node$_3$& 0.025 & 0.92  &  0.36 &-1.55\\   % lab 20
Ni(3d)&Node$_5$& 0.055 &  1.24 &  0.50 &-1.10\\ 
Al(15+) & Hex  & 0.015 &  0.82 &  5.37 & 0.61\\ 
Al(17)& Hex & 0.015 &  2.22 &  4.83 & 0.82\\  % lab 16
\end{tabular}
\caption{
Atomic orbits in ``augmented'' binding. $x$ is fractional content, 
$\bxy$ and $\bz$ refined anisotropic DW coefficients, and $E_{site}$ 
are site energies in eV/atom, for
the approximant T$_{110}$. Orbits above the horizontal line were unchanged
from the ``basic'' binding.
Superscript $r$ denotes objects defined from the corners of the
vertices inside the HBS tiles (Fat$^r$ and Bond$^r$), see also
Fig. \ref{fig:model}
}
%%% Unit cell of the t123:
%% 60.87 x 83.79 x 4.08
%%ATOMS  1466    VACANT SITES  54
%$^a$ after Ref.X\\
%$^b$ extrapolated from the d-AlNiCo and $fcc$--Al, using Al--content.
\label{tab:orbits}
\end{table}


%%%%%%%MD: dw factors (18x2c approximant) MOLECULAR DYNAMICS
%%%%%%%% best data I think
%%%%%% 1    64     0.476     0.507     5.337
%%%%%% 2    24     0.254     0.247     0.281
%%%%%% 3   176     0.371     0.399     0.250
%%%%%% 4    88     0.675     0.673     0.669
%%%%%% 5    56     0.587     0.558     0.463
%%%%%% 6   112     1.716     1.717     0.595
%%%%%% 7    56     0.277     0.269     0.289
%%%%%% 8   112     1.114     1.136     0.481
%%%%%% 9    56     0.711     0.666     1.596
%%%%%%10    16     0.680     0.608     0.659
%%%%%%11     8     0.865     2.954     0.427
%%%%%%12    16     0.536     0.393     0.536
%%%%%%13    24     0.689     0.574     0.711
%%%%%%14    24     0.718     1.090     0.782
%%%%%%15    24     1.779     1.990     0.553
%%%%%%16    20     0.385     0.434     0.184
%%\begin{table}
%%\label{tab:dw}
%%\begin{tabular}{cc}
%%\end{tabular}
%%\begin{tabular}{cccccl}
%%& \multicolumn{2}{c}{MD}& \multicolumn{3}{l}{Refinement \protect\cite{TYT}}\\ 
%%orbit&$\bxy$ & $b_{z}$&$\bxy$& $b_{z}$&label\\
%% 1&  0.49& 5.46& 5.22&  3.31&11,12,13 \\
%% 2&  0.25& 0.47& 0.22&  0.83& 1,2 \\
%% 3&  0.40& 0.29& 0.16&  0.45& 3,6,4,7\\
%% 4&  0.66& 0.68& 0.14&  4.86& 13,14,15,16\\
%% 5&  0.61& 0.53& 0.21&  4.29& 15,17,21,22\\
%% 6&  1.66& 0.62& 1.19&  3.12& 20,21,22 \\
%% 7&  0.25& 0.34& 0.16&  0.64&  5     \\
%% 8&  0.98& 0.48& 1.65&  1.67&  8,9   \\
%% 9&  1.02& 6.43& 0.64&  3.70&  19,23 \\
%%10&  0.69& 1.22& 0.74&  7.55&  16,18 \\
%%11&  4.67& 0.46& 5.26&  0.29&  10    \\
%%12&  1.04& 5.55& 0.64&  3.70&  23   \\
%%13&  0.74& 0.84& 0.00&  4.54&  16    \\
%%14&  0.76& 1.69& 1.48& 10.55&  18    \\
%%15&  3.64& 0.53& 5.26&  0.29&  10    \\
%%16&   -- &  -- & 0.64&  3.70&  23    \\
%%\end{tabular}
%%\caption{
%%Debye-Waller parameters from our molecular
%%dynamics (MD) simulation  at 300K, using pair potentials (columns 2-3),
%%and from the diffraction refinement of Ref.~\cite{TYT}.
%%The first column shows our orbit labels for the
%%``basic'' decoration, and the
%%last column shows the corresponding hyperatom labels
%%of~\cite{TYT}. 
%%\OMIT{(This correspondence is not one-to-one.)}
%%Note that the small HBS tiling we used for the MD annealing has 
%%no Star tiles (hence no atoms from orbit 16).
%%}
%%\end{table}

%\begin{table}
%\label{tab:species}
%\begin{tabular}{cccccc}
%species &  $x$ &$\bxy$ &$\bz$& E\\
%\hline
%Al& 0.7208 &2.716& 3.717&  0.293$\pm$0.091\\
%Co& 0.0799 &0.574& 0.429& $-2.084\pm$0.011\\
%Ni& 0.1993 &1.166& 0.168& $-1.272\pm$0.055\\
%\end{tabular}
%\caption{
%Averages over species. Notation as in Table 1, $x$
%is fractional content. The atomic density of the model
%is 0.0704 atoms/\AA$^3$. 
%}
%\end{table}

\begin{table}
\label{tab:summary}
\begin{tabular}{cccccc}
binding & $N_{\rm obj}$ & $N_{\rm orb}$ & $N_{\rm pos}$ & $N_{\rm par}$ & $\protect\Erel$ \\
basic & 9 & 16 & 38 & 72 & $-0.400$\\
augmented& 13& 23 & 58 & 106 & $-0.395$\\
\end{tabular}
\begin{tabular}{cccccc}
\hline
model & $b_{perp}$& $R_w$ & $R$ & $E$\\
T$_{42}$-basic  & 3.65 & 0.122 & 0.202& $-0.372$\\
T$_{42}$-augmented& 4.15 & 0.089 & 0.169& $-0.344$\\
T$_{110}$-augmented& 3.88 & 0.082 & 0.152& $-0.330$\\
 (smaller $\lambda$) & 3.93 & 0.086 & 0.159& $-0.370$\\
\end{tabular}
\caption{
{\it Top:} Numbers of tiling objects $N_{\rm obj}$, orbits $N_{\rm orb}$, 
positional parameters $N_{\rm pos}$ and total number of fit parameters
$N_{\rm par}$ (including anisotropic DW factors and the global $b_\perp$
DW factor). The last column is the energy 
per atom after constrained relaxation, without use of 
diffraction data.
%% @MM: note I changed unconstr. to constrained relax.
{\it Bottom:} summary of the results for the combined fits, 
giving the perp-space Debye-Waller coefficient, 
the weighted ($R_w$) and unweighted $R$ factors.
The last row
reports an alternative refinement result for approximant T$_{110}$
with the ``augmented'' decoration rule, 
in which the $\lambda$ parameter in Eq. \ref{eq:obj}
was kept smaller to achieve lower energy.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%
%copa.final: 
%%%%%% apsig=1
%INIT: 1544  -0.330390 12.006 0.0841 0.1541
%%%%%% apsig=0
%INIT: 1544  -0.330390 52.203 0.0820 0.1520
%ocpa.final-lambda:
% apsig=1
%INIT: 1544  -0.369772 10.068 0.0777 0.1589 # ndat E chi wR R
% apsig=0:
%INIT: 1544  -0.369772 55.586 0.0863 0.1585 # ndat E chi wR R
% App 47-tt, apsig=0
%INIT: 1544  -0.347706 51.909 0.0885 0.1688 
% Det-basic, apsig=0
%INIT: 1544  -0.344036 8.858 0.1186 0.2171 # ndat E chi wR R
\end{table}


%%%%%%  FIGURES
%%%%%%%% FIGURE: model of atomic structure %%%%%%%%%%%%%
\begin{figure}
%\epsfxsize=240pt \epsfbox{T42.ps} 
\epsfxsize=240pt \epsfbox{model-basic.ps} 
\epsfxsize=240pt \epsfbox{model-aug.ps} 
\caption{
Atomic decoration of the HBS tiling for ``basic'' (top panel) and
``augmented'' (bottom panel) bindings. Dark circles are Ni atoms,
dark triangles Co, gray Al. Unoccupied sites are empty circles.
Tiling orbits are shown as framed labels (once for each tiling
object). Thick lines are HBS-tiling edges, thin lines edges
of the inscribed rhombus tiling. For orbits that are binded
to HBS tiles, the edges of rhombuses inside the tiles are
not drawn.
Each symbol occurs in two
sizes, for upper and lower layer in the 4\AA\ thick slice
cut perpendicularly to the periodic direction.
}
\label{fig:model}
\end{figure}

\begin{figure}
%%epsfxsize=220pt \epsfbox{cluster.eps}
\epsfxsize=240pt \epsfbox{fmodel-basic.ps} 
\epsfxsize=240pt \epsfbox{fmodel-aug.ps} 
\caption{
{\it Top:} ``basic'' decoration rule, refined model.
Arrows indicate displacements (empty symbols stay at 
the ``ideal'' starting position) that occured during combined fit.
Vacant sites are shown as small circles and marked by crosses.
{\it Bottom:} 
``augmented'' binding, refined model. % T$_{110}$ approximant tiling. 
One decagonal columnar cluster is marked by two dashed circles,
near the outer shell of atoms, and near the inner ring of 10 TM
atoms.
}
%\epsfxsize=200pt \epsfbox{100K.eps}
\label{fig:fmodel}
\end{figure}

\begin{figure}
%%epsfxsize=220pt \epsfbox{cluster.eps}
\epsfxsize=240pt \epsfbox{ii-final.ps} 
%\epsfxsize=200pt \epsfbox{100K.eps}
\caption{
%%{\it Bottom panel:} plot of the observed vs calculated diffraction amplitudes,
%%for the HBS approximant T$_{110}$.
%%The observed data are 1544 reflections of the CHS dataset
%%merged to Laue group 10/mmm. {\it Top:} comparison of the experimental
%%errors $\sigma_{obs}$ with $\sigma_{\rm app}$, obtained by averaging
%%$F_{calc}$ over the orbits of approximant $Q_{\rm app}$ mapping on the
%%same decagonal $Q$, as a function of Q$_{perp}$. 
%%Two approximants are shown:  T$_{42}$ and T$_{110}$. 
%%For clarity, signs of $\sigma_{\rm app}$ of T$_{110}$ and
%%$\sigma_{obs}$ are shifted along $y$ axis.
Observed vs calculated diffraction amplitudes for approximant T$_{110}$,
final refined model of the ``augmented'' binding.
%%MM: note binding name ``augmented''
}
\label{fig:fit}
\end{figure}



\end{document}

CLIPS (11/20/02)

In our decoration rule, we form a list of ``orbits'' (i.e. site
classes), each of which is ``bound'' to one kind of ``tiling object.''
A tiling object may be a tile, a tile edge or vertex, or any local
pattern that recurs in the tiling.  Every orbit has positional and
chemical occupation parameters.  Actual atomic structures are obtained
by applying a decoration rule to a particular tiling.

Though it is natural to seek an economical description of a structure,
the simplest decoration rule may not be the most sensible
energetically. For example, two atoms from the same orbit may feel
different forces due to differing tile environments. Associating
independent displacement to each may appreciably lower the energy.  We
define the orbit site energy $\Eorb_\orb$ to be the average site
energy in orbit $\orb$.  Orbits with high $E^{\rm orb}$ are candidates
for ``rebinding'', or differentiation, usually requiring splitting a
tile object into subclasses by increasing their context sensitivity.
The decoration rule is refined through rebinding, as a part of the
optimization process. The energetic refinement of the decoration rule
assigns a physical, not merely geometric, status to a particular
choice of tiling, and a set of tiling objects used in the decoration
rule.

\OMIT{In our decoration rules (unlike those of \cite{steinhardt}), 
each atom belongs to just one orbit.}

\OMIT{Thus, the set of decoration rules may be written 
as a sort of tree, where each branch is a rebinding
of a lower branch, and the  art consists of following
this tree out no further than is necessary to 
reasonably approximate the structure.}


{\sl THIS PARA FROM NEW ALTerNATE TEXT THAT CLH BEGAN.
We used two variants of the deterministic supertile decoration  
discovered in \cite{alnico}.
The difference is that one orbit in the first decoration
is subdivided into two or more orbits in the 
second decoration (according to the local context in the tiling), 
...}

...
The generic rule that would bind to the H, B, and S tiles
all atoms appearing in their interiors, with HBS nodes and
edges as the only other tiling objects, 
\OMIT{(to which are bound just the atoms on those HBS nodes and edges)}
would yield 92 positional parameters for 37 orbits 
\OMIT{(and the five tiling objects)}. In the rest of this
section, we consider only the ``basic'' decoration rule. 
as the simplest rule with moderate number of
positional degrees of freedom.

\OMIT{
The decoration rule includes also two symmetry-breaking orbits:
orbit Al(16) in the symmetry-broken Star has only mirror symmetry,
and there are five ``flavours'' of such star, each with different
Al-pair orientation. Similarly, orbit Al(12) breaks the mirror
plane of the Boat - only one of the two sites is occupied
simultaneously.}


\OMIT{
Ref.~\cite{widom-lightbulb}
exhibited two HBS tilings with neither zig-zag patterns
nor any Star tiles; here in approximant T$_{42}$
we have another example, which
(like the other two) is a pure Boat tiling.}



