Temperature-dependent ``phason'' elasticity in a random tiling quasicrystal
M. Mihalkovi\v{c}~\protect\cite{MM-addr} and C. L. Henley
Both ``phason'' elastic constants have been measured from Monte Carlo simulations of a random-tiling icosahedral quasicrystal model with a Hamiltonian. The low-temperature limit approximates the ``canonical-cell'' tiling used to describe several real quasicrystals. The elastic constant $K_2$ changes sign from positive to negative with decreasing temperature; %%% $K_2/K_1=+0.6$ %%% at $T=\infty$ (the pure rhombohedral tiling, studied before), but in the ``canonical-cell'' limit, $K_2/K_1$ appears to approach $-0.7$, about the critical value for a phason-mode modulation instability. %% between two icosahedral random-tiling quasicrystal models. %% SEEMS NOT SO IMPORTANT HOW %% The configurations are lifted to a surface in six dimensions and %% the structure factor of its fluctuations is measured at %% several wavevectors; from this, both %% of the ``phason'' elastic constants were fitted. %% The principal elastic constant $K_1$ %% clearly disagrees with recent measurements on $i$-AlPdMn.] We compare to the experiments on $i$-AlPdMn and $i$-AlCuFe.
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\begin{document} 
\title
{Temperature-dependent ``phason'' elasticity in a
random tiling quasicrystal}

\author{M. Mihalkovi\v{c}~\protect\cite{MM-addr} and C. L. Henley}
\affiliation{Dept. of Physics, Cornell University, Ithaca NY 14853-2501}


%%
\def\ss #1 {\vskip 0.3cm {\noindent \large \bf{#1}}\\}
\newcommand{\Wperp}{{W^\perp}}
\newcommand{\aR}{{a_R}}
\newcommand{\Corr}{{\Gamma}}
\newcommand{\CorrH}{{\Gamma^{\rm H}}}
\newcommand{\CorrBragg}[1]{{\Corr^{\rm Bragg}_{#1}}}
\newcommand{\Corrnode}{{\Gamma^{\rm node}}}
\newcommand{\CCK}{{\bf C}}
\newcommand{\CK}{{C}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\newcommand{\Gpar}{{{\bf G}^\parallel}}
\newcommand{\epar}{{{\bf e}^\parallel}}
\newcommand{\eperp}{{{\bf e}^\perp}}
\newcommand{\vsix} {{v_{\rm 6D}}}
\newcommand{\DD}{{\bf D}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\fico}{f}    %%% form factor of icosahedron
\newcommand{\cfit}{c_1}    %%% coeff in (1+ \cfit q^2) in fitting form 
\newcommand{\rhomagic}{{\rho^*}}
\newcommand{\zetamagic}{{\zeta^*}}
\newcommand{\rhoo}{{\rho_0}}
\newcommand{\trhoo}{{\tilde{\rho}_0}}

\newcommand{\hh}{{\bf h}}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\def\PRLsection #1 {\section{#1}}     %%%try PRB RC section headings
%% \def\PRLsection #1 {{\it #1 --  }}     %%% PRL section headings

\begin{abstract}
Both ``phason'' elastic constants have been measured 
from Monte Carlo simulations of a random-tiling icosahedral 
quasicrystal model with a Hamiltonian. 
The low-temperature limit approximates the ``canonical-cell''
tiling used to describe several real quasicrystals. 
The elastic constant $K_2$ changes sign from positive to negative 
with decreasing temperature; 
%%% $K_2/K_1=+0.6$
%%% at $T=\infty$ (the pure rhombohedral tiling, studied before), but
in the ``canonical-cell'' limit, $K_2/K_1$ appears to approach 
$-0.7$, about the critical value 
for a phason-mode modulation instability. 
%% between two icosahedral random-tiling quasicrystal models.
%% SEEMS NOT SO IMPORTANT HOW 
%% The configurations are lifted to a surface in six dimensions and
%% the structure factor of its fluctuations is measured at
%% several wavevectors; from this, both 
%% of the ``phason'' elastic constants were fitted.
%% The principal elastic constant $K_1$ 
%% clearly disagrees with recent measurements on $i$-AlPdMn.] 
We compare to the experiments on $i$-AlPdMn and $i$-AlCuFe.
\end{abstract}


\pacs{
61.44.Br, %% Quasicrystals
62.20.Dc,  %% Elasticity, elastic  constants
5.10.Ln, %% Monte Carlo methods in statistical physics
%%% 64.70.-p ??, %%  Specific phase transitions
64.60.Cn}   %% Order-disorder transformations and statistical
             %% mechanics of model systems 
%%% SHAW's PACS (1990): 61.50.Em, 02.70.+d, 61.40.+b, 62.20.Dc 

\maketitle


Quasicrystals possess translational long-range order, 
as manifested by 
resolution-limited Bragg peaks~\cite{niceqxtal}, 
%%% in the family i(AlCuFe)--i(AlCuRu).
yet possess symmetries (e.g. icosahedral) incompatible with periodicity. 
The best-understood quasicrystals can be represented as
rigid tilings (or cluster networks)
decorated by atoms in a uniform fashion~\cite{Hen91ART,Hen91CCT,Wind94,Mi96a}.
Their long-range order can then be investigated in a purely tiling framework, 
being determined by the 
``tile Hamiltonian''\cite {Hen91ART,Mi96a}),
which assigns to every tiling the corresponding structural energy. 

Despite years of study, there remain two
plausible, competing scenarios that predict quasicrystal 
long-range order (in the sense that {\it both} predict Bragg peaks in
dimension $d>2$.)
The ``ideal tiling'' scenario postulates 
that the tile Hamiltonian 
enforces an essentially unique, perfectly quasiperiodic 
ground state analogous to a perfect crystal
(just as matching rules or covering rules~\cite{Matching,Jeong94}
enforce a Penrose tiling).
%%%%%%%%%
Alternatively,  the (equilibrium) ``random tiling'' scenario supposes 
%%% the quasicrystal phase is 
an ensemble of many {\it nearly} degenerate packings of the structural units.
Then the true ground state is normally a coexistence of crystal phases, 
but at higher temperatures the random-tiling quasicrystal
becomes thermodynamically stable owing to its larger entropy
(See references cited in \onlinecite{Hen91ART}). 
%%% \cite{entropic}.
%% It has been shown theoretically that an (equilibrium) random tiling
%% phase dimension $d>2$ indeed has Bragg peaks. 


The two scenarios 
%%% for the quasicrystal phase 
may be distinguished experimentally
by their diffuse scattering.  
In the random-tiling case,  every Bragg peak is surrounded
by wings of diffuse scattering with distinctive shapes predicted by
the ``phason'' elastic theory~\cite{diffuse91}. 
(See (\ref{eq-diffuse}), below.)
Those shapes were observed in X-ray and neutron diffraction
of $i$-AlPdMn\cite{Bois95,Le01} and $i$-AlCuFe\cite{Le99}, 
and the fitted phason-elastic constants were temperature-dependent. 


In this Letter, we simulate random
tilings of rhombohedra with a tile Hamiltonian that favors configurations
approximating a ``canonical cell tiling''~\cite{Hen91CCT,Mi93} (CCT). 
Our motives are (1) to infer the phason elastic constants of the CCT ,
which is difficult to simulate directly, and (2) 
to observe the crossover with decreasing temperature $T$
as clusters of small (rhombohedral) tiles get bound together into 
larger (CCT) ``supertiles''
% CLIPPED ~\cite{supertiles}, 
in a toy model.
The same techniques will be essential in computing the phason elastic 
constants for models of specific real quasicrystals.

\section{Canonical-cell tiling and Hamiltonian}
%%%
Atomic structure models of $i$-AlZnMg~\cite{Wind94} [or $i$-AlMnSi~\cite{Mi96a}], 
and many related alloys, 
%%% REAL MODELS USING CCT
may be built by placing icosahedral ``Bergman'' [or ``MI''] clusters
on the nodes of networks with inter-node linkages of two lengths:
$b\equiv 2.75\aR$ along twofold symmetry directions or $c\equiv 2.38\aR$ along
threefold directions, which we shall call a ``$bc$-network.''
(The quasilattice constant $\aR$ is set to unity in this paper;
$\aR \approx 0.5$nm in real quasicrystals.)
A ``canonical-cell'' tiling~\cite{Hen91CCT} (CCT) is a special
$bc$-network built from
the four smallest polyhedral cells having $b$ or $c$ edges. 


A Monte Carlo simulation of the CCT is practically intractable, 
since {\it local} tile reshufflings do not exist:
spatially extended clusters must be rearranged to reach another
valid tiling~\cite{Hen91CCT,Ox9397}.  
To circumvent this problem, our model system is a $bc$ network in 
which the CCT constraints are not imposed,  but are favored by a Hamiltonian 
and thus satisfied in the limit $T\to 0$. 
We emphasize that unlike matching-rule Hamiltonians,~\cite{steinhardt}
eq.~(\ref{eq-Ham12})  does {\it not} force a unique
quasiperiodic $T=0$ state; the ground state ensemble is still
a random tiling, but a different one (with less entropy) than at $T=\infty$.
Our configuration space 
is the random ``three-dimensional Penrose tiling'' (3DPT), 
which includes all packings of the well-known oblate and prolate
rhombohedra having edges of length $\aR\equiv 1$ along 5-fold symmetry directions. 
Similar simulations of another imperfect approximation of the CCT by
a $bc$-network, namely the lattice-gas 
of Ref.~\onlinecite{Mi93}, will be reported elsewhere.~\cite{LG} 

%%%%
The 3DPT contains a special subset of vertices, called ``12-fold'' 
nodes, at which rhombohedron edges emanate
in all 12 of the five-fold symmetry directions~\cite{Hen86}.   
%%% forming a local cluster of icosahedral symmetry. 
%%% (These become icosahedral cluster centers in structure models.) 
This 12-fold network approximates the CCT, since
the separation of nearby nodes is usually a $b$ or $c$ linkage;
however a ``short'' separation of $\aR$ (along 5-fold axes) 
occurs rarely.  For our simulations, we adopted the Hamiltonian 
%%% 4/29/02. MM confirms we are still using this one. 
   \begin{equation}
   {\cal H}_ {12} = - N_ {12}+\asb N_{sb}, 
   \label{eq-Ham12}
   \end{equation}
where $N_ {12}$
is the total number of 12-fold nodes, 
and $N_{sb}$ is the number of ``short'' node pairs with
separation  $\aR$. 
%%% ``short linkage'' (one rhombohedron edge). 

The purpose of $\asb$ is to suppress ``short'' pairs;
$\asb\equiv 1$ suffices, which we adopted after testing other values.
We conjecture (and simulations support) that
the ground states of (\ref{eq-Ham12}) are a subset of CCT's, 
namely those with maximum node density, 
and not the maximum-entropy random CCT. 

\section {Perpendicular space and fluctuations}
%%%%
Our rhombohedron edges are $\pm \epar_\alpha$, where~\cite{JarNel88}
     $\epar_1= (\tau,0,1)/(1+\tau^2)^{1/2}$
and the other basis vectors are made by 
reflections in the $x$, $y$, or $z$ planes, or by
cyclic permutation of the components. 
Thus every vertex has coordinates of the form
%%%%%%%%%%%%%%%%%%
\begin{equation}
   \xpar = \sum_{\alpha=1}^6 
   n_\alpha \epar_\alpha,
   \label{eq-xpar}
\end{equation}
%%%%%%%%%%%%%%%%%%
In a standard trick, the integer coefficients may be visualized as 
coordinates $\xx = [n_1, \ldots, n_6]$ of a 6-dimensional hypercubic lattice,  
into which eq.~(\ref{eq-xpar}) 
embeds a 3-surface  \cite{El85}. 
This 3-surface is conveniently parametrized by a 
``perpendicular'' (``perp'')
%%%also known as ``phason'') 
coordinate, defined for each vertex by
%%%%%%%%%%%%%%%%%%
\begin{equation}
   \xperp= \sum_{\alpha=1}^6 n_\alpha \eperp_\alpha, 
   \label{eq-xperp}
\end{equation}
%%%%%%%%%%%%%%%%%%
where $\eperp_1=\eta (1,0,-\tau)/(1+\tau^2)^{1/2}$
and the other basis vectors 
$\eperp_\alpha$ are derived by the same reflections and
cyclic permutations used to produce 
$\epar_ \alpha$ from $\epar_1 $.
We write $\hh(\rr)$ for the 
smoothed (coarse-grained) version of $\xperp$. 

The gradients $\nabla \hh(\rr)$, forming the ``phason strain''
tensor, quantify the local deviation from icosahedral symmetry.
The random-tiling scenario~\cite{Hen91ART} predicts that
a quasicrystal's free energy density has
a {\it phason elastic} form, 
\cite{Elastic}
i.e. is quadratic in components of $\nabla \hh$.
This was confirmed numerically for several completely random
($T=\infty$) tilings \cite{Hen91CCT,Ox9397,2DTM,Tang90,Shaw91a}, 
% CLIPPED 2DMC
but not until now for the CCT. 
%%%%%%%%%%%
In contrast, the ``ideal tiling'' behavior implies
a free energy scaling as $|\nabla \hh|$.
% CLIPPED ~\cite{Worms} 
An icosahedral quasicrystal has two elastic constants, 
$K_1$ and $K_2$.
Very roughly speaking, $K_1$ parametrizes the mean strength
of the elastic stiffness, and $K_2$ parametrizes its 
anisotropy (degree of coupling between direction of the
gradient in real space and components of $\hh$). 


For fluctuations around a state with 
%%% (which we will set to zero without loss of generality)
zero mean phason strain, the dimensionless elastic free energy 
is~\cite{Elastic,JarNel88}
  \begin{equation}
      F/T  = {1\over 2} \sum _{\qq,ij}
           \CK_{ij}(\qq) \tilde{h}_i(\qq) \tilde{h}_j(-\qq)
  \label{eq-Ffourier}
  \end{equation}
wehere it is convenient to use 
$\thh(\qq) \equiv V^{-1/2} \int d^3 \rr e^{-i\qq\cdot\rr} \hh(\rr)$.  
Here
  \begin{eqnarray}
    \CK_{ij}(\qq)& = & K_1|\qq|^2\delta_{ij}-
         K_2\; \big[({1\over 3}|\qq|^2+2q_i^2 \nonumber \\
    && + q_{i+1}^2/\tau-\tau q_{i-1}^2)\; \delta_{ij}-2q_iq_j \big].
  \label{eq-kquad}
  \end{eqnarray}
Eq.~(\ref{eq-Ffourier}) implies 
%%% by equipartition 
that each $\thh(\qq)$ consists of Gaussian random variables, and
their correlations 
  \begin{equation}
         \Corr_{ij}(\qq) \equiv 
          \langle {\tilde h}_i(-\qq)  {\tilde h}_j(\qq)\rangle. 
    \label{eq-Corr}
  \end{equation}
are given by
  \begin{equation}
         \Corr_{ij}(\qq) = [\CK(\qq)^{-1}]_{ij} .
  \label{eq-flucts}
  \end{equation}
The diffuse diffraction intensity~\cite{diffuse91}
near a Bragg peak with reciprocal lattice
vector $\Gpar$ is 
    \begin{equation}
      I(\qq) \propto \sum_{ij} G^\perp_i \Corr_{ij}(\qq) G^\perp_j , 
    \label{eq-diffuse}
  \end{equation}
where $\Gperp$ is the perp-space partner of $\Gpar$. 
(An equilibrium  random tiling 
in $d>2$ has true Bragg peaks~\cite{El85,Hen91ART} as
well as diffuse scattering.)
Experiments measure (\ref{eq-diffuse});
simulations can measure  (\ref{eq-Corr}) directly. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section {Simulation, measurements, and corrections}
%%%%
Our data is taken from tilings using periodic boundary conditions
in a cubic cell of side
$L=\tau^6 (2-2/\sqrt5)^{1/2}\aR=18.868\aR$.
This corresponds to the ``8/5 Fibonacci approximant'' 
of the 3DPT~\cite{Shaw91a}, 
and contains $N_v =10 336$ rhombohedron vertices. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We use the very simple update move for the 3DPT 
which rearranges the four tiles surrounding a vertex
with just four edges.\cite{Tang90,Shaw91a}.

%%%%%%%%%%%%%%%%%%
Simulation time is measured in Monte Carlo sweeps, 
where each sweep makes $N_v$ actual flips on
randomly chosen vertices.
% CLIPPED ~\cite{FN-timenote}
A simulation run consisted of $500$ thermal cycles;
%% YES 4/2002
each cycle included 3000 MCS at $T=\infty$  (to erase 
memory of the preceding cycle), 
%% AS OF 4/2002; it was 500 in 2000
followed by 34 cooling stages each at a constant temperature. 
%% As of 4/2002: confirmed, 34 stages.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% {LG The temperatures decrease from $8$ to $1$ in powers of 2, 
%% from 0.9 to 0.60 (dotted) in steps of 0.05, 0.57, 0.54, 0.51, 
%% and from 0.48 down to 0.10 in steps of 0.02.} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Each stage takes 1000 MCS, except the first six stages ($T>0.8$)
used 200 to 800 MCS; 10 measurements were made during the last 100 MCS of each stage.
A typical 8/5 cooling run took 10 days of CPU time on a 1.5 GHz 
Athlon processor. 
%%% Checked 4/2002

We constructed $\xpar$ and $\xperp$ for each vertex, using
(\ref{eq-xpar}) and (\ref{eq-xperp}).
For each $\bf q$ on a standard list of 70 wavevectors, we took
   \begin{equation}
   \thvert(\qq)\equiv 
        {\sqrt{V}\over {N_v}}
        \sum_{\xx \in {\rm 3DPT}}\xperp e^{-i\qq\cdot \xpar}
   \label{eq-thvert}
   \end{equation}
as in Ref.~\onlinecite{Shaw91a},
where $V$ is the simulation cell volume and the
sum runs over the $N_v$ actual 3DPT vertices. 
The normalization in (\ref{eq-thvert}) ensures that
$\thvert(\qq) \approx \thh(\qq)$ for small $\qq$.
Fluctuation correlations analogous to (\ref{eq-Corr}), 
$\CorrH_{ij}(\qq)\equiv \langle \thvert_i(-\qq)\thvert_j(\qq)\rangle$
were accumulated, and were symmetrized with respect to 
cyclic permutations $(xyz)$. 

%%% \section{Bragg peak corrections}
%%%%
Although $\langle \thh(\qq) \rangle =0$, 
Eq.~(\ref{eq-thvert}) has a nonzero 
expectation when $\qq$ is a Bragg vector, 
just like the structure factor, from which 
the sum (\ref{eq-thvert})
differs only by the extra factor $\xperp$.
%%%%%%%%%%%%%%%%%%%%%%
As in the structure factor, each Bragg peak of our $\CorrH(\qq)$ function
is surrounded by a tail of diffuse scattering, which must also be
subtracted in order to correct the nearby wavevectors. 
%%%%%%%%%%
%%% Short version  
To calculate this tail, we adopt the ``undulating cut approximation''
(Ref.~\onlinecite{Hen91ART}, Sec. 7),  
in which the occupation probability of
a physical-space vertex $\xpar$ is $\rhoo(\xperp-\hh(\xpar))\leq 1$.
Here $\xpar$ and $\xperp$ are related as in eqs. (\ref{eq-xpar})
and (\ref{eq-xperp}), 
$\hh(\rr)$ is smooth and fluctuating according to (\ref{eq-flucts}), 
and $\rhoo(\hh)$ defines an ideal reference structure 
when $\hh$ is flat. 
%%%%%%%%%%%%%
The measured fluctuation is predicted to be~\cite{LG}
   \begin{equation}
       \CorrH_{ij}(\qq) = 
       \Corr_{ij}(\qq) + \sum _{\Gpar} \CorrBragg{ij}(\qq,\Gpar).
   \label{eq-sumBragg}
   \end{equation}
where
    \begin{equation}
          \CorrBragg{ij} (\qq;\Gpar) 
          =  D_i D_j \left[ V \delta_{\qq, \Gperp} +
              (\Gperp, \Corr (\qq-\Gpar) \Gperp) \right]. 
    \label{eq-Braggtail}
    \end{equation}
Here $\DD = \DD(\Gperp)$ can be expressed~\cite{LG} in terms of 
the Fourier transform of $\rhoo(\xperp)$;
in light of the symmetry, $\DD(\Gperp) \parallel \Gperp$ 
along any symmetry axis. 

Our measurements include
one orbit of weak Bragg peaks on 3-fold symmetry axes, 
-- at $\Gpar={{2\pi}\over L}(333)$ and ${{2\pi}\over L}(520)$ 
in our cell --
where the Bragg intensity contributes far stronger than the 
long-wavelength fluctuations (\ref{eq-Corr}). 
We determined each Bragg term (first term  in (\ref{eq-Braggtail}), 
i.e. $D_l D_m$) from
$\langle \thh(\qq)\rangle$ averaged 
over each temperature stage~\cite{rtdiff}.
>From our estimate of $K_1$ and $K_2$, we know $\Corr(\qq)$
via (\ref{eq-flucts}), hence the
second term in (\ref{eq-Braggtail}), so we can subtract the
sum in (\ref{eq-sumBragg}) from the measured data. 
The resulting 
%% $3\times 3$
matrices $\Corr(\qq)$ are inverted to obtain (via (\ref{eq-flucts})) 
estimates of (\ref{eq-kquad}), which are fitted to $K_1$ and $K_2$
using linear least squares.  The process is iterated until
converged. 


\section{Results and Discussion}
%%% 
To clarify two other artifacts which depend on $|\qq|$, we fitted
effective constants 
``$K_1(n)$'' and ``$K_2(n)$'' separately for each shell $n$ defined by
$|\qq|^2 = n (2\pi/L)^2$; the results are shown in Fig.~\ref{fig-shells}, 
for selected temperatures.  
The continuum approximation must break down 
at the CCT linkage length scale ($\sim 2.5\aR$), which explains the
deviations visible for $n\ge20$, i.e. $2\pi/|\qq|\leq 4.2 \aR$.
(Above the supertile formation temperature, the tile length
scale is the rhombohedron edge $\aR$, and no such deviations 
are seen in this wavevector range.)

At small wavevectors, equilibration time becomes problematic.
As shown in Ref.~\onlinecite{Shaw91a}, eqs. (20ff), 
the relaxation rates of $\thh(\qq)$ are $\gamma \CK_i(\qq)$, 
where $\gamma(T)$ is a relaxation coefficient and 
$\{\CK_1, \CK_2, \CK_3 \}$ are the eigenvalues of $\CCK(\qq)$. 
This becomes problematic when temperature 
is low (since $\gamma(T) \to 0$ as $T\to 0$)
and $|\qq|$ is small (as $\CCK \propto |\qq|^2$, 
eq.~(\ref{eq-kquad}). The consequence is that, at the
lowest temperatures in Fig.~\ref{fig-shells}, 
the smallest wavevectors 
%%% ($n<3$, and $n<10$ at $T=0.1$) 
are governed by the elastic constants 
of the higher temperature at which 
these modes fell out of equilibrium. 

The plots in Fig.~\ref{fig-shells} suggest two regimes 
as a function of wavevector and temperature.~\cite{FN-n20anomaly}
At high $T$ or large wavevector, 
a linear dependence on $n\propto |\qq|^2$ is visible, 
with the slope increasing as $T\to 0$. 
We attribute this to our use of the 3DPT vertices
in (\ref{eq-thvert}): the $\xperp$ differences 
between adjacent vertices (separated by $\aR$)
%%% exceed those between 12-fold nodes, and 
certainly exceed the fluctuations of the smooth $\hh(\rr)$ 
surface, extrapolated to this short scale.  Consequently, the 
inferred elastic constant is spuriously small. 

On the other hand, a second regime is visible 
at low $T$ and $|\qq|< \kappa(T)$, 
where $\kappa(T)$ defines a crossover. 
In this regime, $K_i(n)$ has a clear linear dependence, 
which we tentatively attribute to higher-order-gradient
terms of $O(q^4)$ 
in the harmonic phason-elastic coefficients (\ref{eq-kquad}), 
which are permitted by symmetry. 
Although it certainly affects the $n=1,2$
wavevector shells at the lowest temperatures,
lack of equilibration cannot explain all of the low $T$/small $|\qq|$ 
regime, as the data (in this regime) agree very well with 
our lattice-gas simulations~\cite{LG} which have a different dynamics.
% CLIPPED ~\cite{FN-instab}
We used the empirical fitting form 
  \begin{eqnarray}
   K_i(n) = a_i + b_i\sqrt{n^2+\nstar^2}+ c_i
        \frac{n}{\sqrt{n^2+\nstar^2}}
  \label{eq-fitform}
  \end{eqnarray}
to extrapolate $K_1(n)$ and $K_2(n)$ to $\qq=0$.

Our principal result is the 
temperature dependence of the elastic constants indicated on 
Fig.~\ref{fig-K1K2}.
At infinite temperature, i.e. for the maximally random tiling, 
we fit $K_1= 0.84$ and $K_2=0.52$, so 
$K_2/K_1=0.62$ 
%%% 4/29/02.  MM: Yes, for the 4/25 draft. 
in agreement with previous measurements 
on the same ensemble~\cite{Tang90,Shaw91a}. 
As temperature decreases, $K_1$ increases, 
as expected since the Hamiltonian (\ref{eq-Ham12}) favors 
formation of ``supertiles'' %
% CLIPPED ~\cite{supertiles}
%%% (canonical cells in this case) 
which have small perp-space differences
among their corner nodes.
Meanwhile, $K_2/K_1$ turns negative with decreasing temperature, 
consistent with our knowledge that 
the random CCT\cite{New95b}  has $K_2<0$. 
It appears that $K_1 \to 2.26$ and $K_2 \to -1.44$ as $T\to 0$, 
thus $K_2/K_1 \to -0.64$, with errors of order 10\%. 
Our lattice-gas simulations\cite{LG}
gave similar results.

Approaching $T=0$, the fit of our data to the angular dependence 
of elastic theory may be worsening.  Furthermore, 
$K_2/K_1$ seems to be approaching the critical 
value $-0.75$, at which the system goes unstable to a 
modulation with $\qq$ and $\thh(\qq)$ along a 3-fold axis.
This suggests that the true canonical-cell tiling -- more precisely
the maximum-density subensemble of the CCT -- might have a more
complicated dependence than elastic theory. This is not 
unprecedented: exactly that is known to happen in those 
10-fold and 12-fold symmetric two-dimensional tilings~\cite{Ox9397}
that have extra constraints like the CCT. 

%%% \section{Discussion}
%%% 
Neutron and X-ray diffraction measurements of diffuse wings around
Bragg peaks yield $K_2/K_1=-0.5$ for $i$-AlPdMn~\cite{Bois95,Le01,Le99}, 
agreeing with our result, so $i$-AlPdMn might
be modeled by a CCT-like network of microscopic clusters. 
On the other hand, $i$-AlCuFe has $K_2/K_1>0$~\cite{Le99}.
Thus, despite their similar face-centered icosahedral
atomic structures, $i$-AlPdMn and $i$-AlCuFe must be modeled by
different tile Hamiltonians.

The recent experimental calibration of $i$-AlPdMn data~\cite{Le01}
yielded $K_1/T=0.002 \Ao^{-3}$ in absolute physical units.
Multiplying by $\aR^3$ (with $\aR=4.5 \Ao$) converts this to 
$K_1 \approx 0.2$ in the units of this paper, which is puzzlingly small. 
%%% This suggests, surprisingly, that the real 
%%% system is right on the edge of a phason-elastic instability.
This is hard to explain by a misjudgment in our model formulation.
If our tiles are invalid, one expects that 
(since this alloy is so well ordered), 
they should be replaced by  large supertiles, 
so that $K_1$ becomes {\it larger}, or that matching rules
are satisfied implying $K_1 \to \infty$. 
Instead, the data suggest that, implausibly, atoms are free to fluctuate 
in groups even {\it smaller} than a rhombohedron.

The temperature dependence of $K_i(T)$ was
used for extrapolation  only;
%% technical purpose 
the Hamiltonian (\ref{eq-Ham12}) 
simply implements the CCT and is unlikely to model 
a realistic quasicrystal.
A physically relevant model would add to (\ref{eq-Ham12})
an additional perturbation $\alpha'$ that breaks the degeneracy 
among canonical-cell tilings, 
so that large $\alpha'$ corresponds to physical low temperature. 
%% so that a periodic packing
%% is stabilized, but only when $\alpha'/T$ is sufficiently large.
Such a model can be investigated using the techniques of this paper. 

In conclusion, we evaluated the phason elastic constants of 
the canonical-cell tiling for the first time, by a 
%%% Monte Carlo 
simulation in which the mean-square fluctuations 
of the abstract surface representing the tile configuration
were measured at many wavevectors.
%%
%% REMARK: Tang-Jaric, Dressel-Strandburg, Dotera-Steinhardt
%% and Jeong-Steinhardt 1993, evaluated some temperature 
%% dependences, but not of el. constants.
%% The Steinhardt projects evaluated elastic constants, but
%% not as a continuous function of $T$.
%%
%% A technique and formula (\ref{eq-sumBragg}) were also found for 
%% subtracting Bragg peaks from this data.  
This is the first theoretical measurement in any
quasicrystal model of the temperature dependence 
of the phason elastic constants. 


\begin{acknowledgments}
This work is supported by DOE grant DE-FG02-89ER45405;
computer facilities were provided by the Cornell Center for Materials
Research under NSF grant DMR-9632275. 
We thank M. de Boissieu for discussions. 
\end{acknowledgments}

\begin{thebibliography}{99}

\bibitem [*]{MM-addr} 
Materials Research and Liquids, Institute of Physics,
TU Chemnitz, D-09107 Chemnitz, Germany;
Current and permanent address, 
Institute of Physics, Slovak Academy of Sciences, 84228 Bratislava, Slovakia.

\bibitem{niceqxtal} 
%%% A. Tsai, A. Inoue, and T. Masumoto, 
%%% Jpn. J. Appl. Phys. {\bf 26}, L1505 (1987);
%%% AlCuFe
%%%
%%% C. A. Guryan, A. I. Goldman, P. W. Stephens, K. Hiraga,
%%%    A. Tsai, A. Inoue, and T. Masumoto, 
%%%   Phys. Rev. Lett. {\bf 62}, 2409 (1989).
%%% AlPdRu
See {\it e.g.} S.~W.~Kycia {\it et al}, 
Phys. Rev. B 48, 3544 (1993). 

\bibitem{Hen91ART}
C.~L.~Henley,
``Random tiling models,'' p. 429 in {\it Quasicrystals: The
State of the Art}, ed. P.~J.~Steinhardt and D.~P.~DiVincenzo,
(World Scientific, 1991).
% CLIPPED  a revised edition is in press.

\bibitem{Hen91CCT}
C. L. Henley, Phys. Rev. {\bf B43}, 993 (1991).
%% Canonical-cell tiling

\bibitem{Wind94}
M.~Windisch, J.~Hafner, M.~Kraj\v{c}i and 
M.~Mihalkovi\v{c}, Phys. Rev. B 49, 8701 (1994).

\bibitem{Mi96a}
M. Mihalkovi\v{c} {\it et al}, 
%%%% W.-J. Zhu, C. L. Henley, and M. Oxborrow,
Phys. Rev. B 53, 9002 (1996).
%% deco PRB I
%%% ``Al-Mn Decoration Models: I. Geometrical Principles''
%%% IS NOT CITED \bibitem{ElHen85} 
%%% V. Elser and C.~L.~Henley, Phys. Rev. Lett. {\bf 55}, 2883 (1985).
%% C. L. Henley and V. Elser,
%% Phil. Mag. B53, L59 (1986). 

\bibitem{Matching}
K.~Ingersent in {\it Quasicrystals: The State of the Art}, 
eds. D. P. DiVincenzo and 
P.~J.~Steinhardt (World Scientific, 1991). 
%%% see also J.~E.~S.~Socolar, in
%%% {\it Quasicrystals}, eds. M.~V.~Jari\'c and S.~Lundqvist
%%%(World Scientific, 1989). 
%% reviews about matching rules 

\bibitem{Jeong94} 
H.-C. Jeong and P.J. Steinhardt, {Phys. Rev. Lett.} {73}, 1943 (1994).
%%% (b) H.-C.~Jeong and P.~J.~Steinhardt, Phys. Rev. B55, 3520 (1997). 

\bibitem{diffuse91}
%%% a= \bibitem{Wid91}
M. Widom, Phil. Mag. Lett. 64, 297 (1991);
%%% b=\bibitem{I92}
 Y. Ishii, Phys. Rev. B45, 5228 (1992).

%% \bibitem{entropic} 
%% M. Widom, 
%% in {\it Quasicrystals}, 
%% ed. by M.~V.~Jari\'c and S.~Lundqvist
%% (World Scientific, 1989);
%% C.~L.~Henley, in 
%% {\it Quasicrystals and Incommensurate Structures in Condensed Matter}, 
%% edited by M. J. Yacam\'an, D. Romeu, V. Casta\~no, and A. G\'omez
%% (World Scientific, 1990). 

\bibitem{Bois95}
M.  de Boissieu, {\it et al},
%%% M. Boudard, B. Hennion, R. Bellissent,
%%% S.~Kycia, A.~Goldman, C.~Janot, and M.~Audier.
Phys. Rev. Lett. 75, 89 (1995).
%%%
%%% M. Boudard {\it et al},
%%%  M. de Boissieu, A.~Letoublon, B.~Hennion,
%%% R.~Bellissent, and C.~Janot,
%%% Europhys. Lett. 33, 199 (1996).
%%% Diffuse scatt. and phason elasticity in the AlPdMn Icosahedral phase
%%%
%%% \bibitem{Bou95}
%%% DROP THIS. M. Boudard, M. de Boissieu, M. Audier, S. Kycia, A.~I.~Goldman,
%%% B.~Hennion, R.~Bellissent, M.~Quilichini, and C.~Janot, p. xxx in 
%%% {\it Proc. 5th Int'l Conf. on Quasicrystals,}
%%% (Ed. C. Janot, and R.~Mosseri, World Scientific, 1995)
 %%% Modulated phase and diffuse scattering in the AlPdMn icosahedral phase

\bibitem{Le01}
%%% "Phason elastic constants of the icosahedral Al-Pd-Mn phase derived from 
%%% diffuse scattering measurements
A. L\'etoublon {\it et al}, 
%%% A. L\'etoublon, M.~de~Boissieu, M.~Boudard, L.~Mancini, J.~Gastaldi, 
%%% B.~Hennion, R.~Caudron, and R.~Bellissent, 
Phil. Mag. Lett.  81, 273 (2001). 
%%%  81 (4): 273-283 APR 2001


\bibitem{Le99}
A.~L\'etoublon, thesis (Grenoble, France), 1998.
%% ``Phase Transition and Diffuse Scattering studies in the
%% AlCuFe ternary system''. 
%% A.~Letoublon and M.~de~Boissieu, personal communications;
%%  It seems AlCuFe IS NOT YET PUBLISHED (2001,2002). 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% M. Boudard, A. Letoublon, M. de Boissieu, T. Ishimasa,
%% M. Mori, E. Elkaim, and J.P.Lauriat,
%% Proc. ICQ7 (1999).
%% {\sl Letoublon's own contributed talk at ICQ7, OA2, is
%% more about use of speckle to see phason domains.}
%%%
%%% A. Letoublon, F.~Yakhou, F.~Livet, F.~Bley, M.~de~Boissieu, L.~Mancini, 
%%% R.~Caudron, C.~Vettier, and J.~Gastaldi, 
%%% Europhys. Lett. 54, 753 (2001)
%%% 54 (6): 753-759 JUN 2001
%%% "Coherent X-ray diffraction and phason fluctuations in quasicrystals"
%%% This is very interesting: speckle patterns, q^2 dynamics -- but AlPdMn

\bibitem{Mi93} 
M. Mihalkovi\v{c} and P. Mrafko, Europhys. Lett. 21, 463 (1993).
%%% Tiling of canonical cells: large Pa3 approximants


%% CLIPPED \bibitem{supertiles}
%% ``Frank-Kasper decagonal quasicrystals: simulations and ideal models''
% The formation of supertiles is well-known in both toy and realistic
% decagonal models, such as Ref.~\onlinecite{Jeong94}; 
% J. Roth and C. L. Henley,
% Philos. Mag. A, Vol 75 (No 3), 861 (1997); and
%%% 861-887 (1997).
% M.~Mihalkovi\v{c} {\it et al}, 
%% M.~Mihalkovi\v{c}, I. Al-Lehyani, E.~Cockayne,
%% C.~L.~Henley, N.~Moghadam, J.~A.~Moriarty, Y.~Wang, and M.~Widom,
% Phys. Rev. B 65, 104205 (2002).

\bibitem{Ox9397}
%%% \bibitem{Ox93}
%%% ``Random square-triangle tilings: A model for
%%% twelvefold-symmetric quasicrystals''
M. Oxborrow and C. L. Henley, 
Phys. Rev. B 48, 6966 (1993);
%    \bibitem{Ox97} 
M. Oxborrow and M. Mihalkovi\v{c},
%%% p. 178 in {\it Aperiodic '94}, ed. G. Chapuis and W. Paciorek
%%% (World Scientific, 1995);
p.~451 in
{\it Aperiodic '97}, 
eds. M.~de Boissieu, J.-L. Verger-Gaugry, and R.~Currat
(World Scientific, 1998).

\bibitem{steinhardt}
H.-C. Jeong and P.J. Steinhardt {Phys. Rev. B} {48}, 9394 (1993);
%% 9394-9403 (1993);
%%"Finite-temperature elasticity phase transition in decagonal 
%% quasicrystals"  stacked
%%
%% "ISING-LIKE TRANSITION AND PHASON UNLOCKING IN ICOSAHEDRAL 
%% QUASI-CRYSTALS"  Hamiltonian favoring alt. cond.
T. Dotera and P. J. Steinhardt, 
Phys. Rev. Lett.  72, 1670 (1994). 


\bibitem{LG}
M.~Mihalkovi\v{c} and C.~L.~Henley, in preparation.
%%% Lattice-gas (other) paper.

\bibitem{Hen86}
C. L. Henley, 
Phys. Rev. B 34, 797 (1986).
%%% sphere packings '86.


\bibitem{JarNel88} 
M.~V.~Jari\'c and D.~R.~Nelson, Phys. Rev. B 37, 4458 (1988).
%%% diffuse part

\bibitem{El85} 
V. Elser, Phys. Rev. B {\bf 32}, 4892 (1985).

\bibitem{Elastic}
%%% T. C. Lubensky, S. Ramaswamy, and J. Toner,
%%% Phys. Rev. B32, 7444 (1985);
T. C. Lubensky, in
{\it Aperiodic Crystals I: Introduction to Quasicrystals},
%%% = {\it Aperiodicity and order}, vol. 1,
M. V. Jari\'c (ed.), (Academic Press, London, 1988),
and references therein.

% CLIPPED \bibitem{Worms}
%  J. E. S. Socolar, T.~C.~Lubensky, and P.~J.~Steinhardt, 
%  Phys. Rev. B34, 3345 (1986). 
%%% 3345-3360
%%% Phonons, phasons, and dislocations in quasicrystals
%%% old Lubensky-Socolar-Steinhardt, linear gradient.


% CLIPPED \bibitem{2DMC} 
%% \bibitem{2DRT} 
%%% \bibitem{Tang87} 
%  K. J. Strandburg, L. Tang, and M. V. Jari\'c,
%      Phys. Rev. Lett.  {\bf 63}, 314 (1989).
%%%  K. J. Strandburg, Phys. Rev. B {\bf 40}, 6071 (1989);
%%% Method of measuring phason elasticity from
%%% finite-size dependence of mean-square <|h|^2>?

\bibitem{2DTM} 
%%%  (a) M. Widom, D. P. Deng, and C. L. Henley, 
%%%     Phys. Rev. Lett.  {\bf 63}, 310 (1989);
%%%      %%% ``Transfer matrix analysis of a Two-dimensional Quasicrystal''
%%%  (b) L. J. Shaw and C. L. Henley, J. Phys. A. 24, 4129 (1991).
%%%       %% ``A transfer-matrix Monte Carlo study of random Penrose tilings''
  W. Li, {\it et al}, J. Stat. Phys. {\bf 66}, 1 (1992), and
references therein.
%%  (c) W. Li, H. Park, and M. Widom, 

%%% \bibitem{rhombo90}  
\bibitem{Tang90}  
L.-H. Tang, Phys. Rev. Lett. {\bf 64}, 2390 (1990).
%%% Tang PRL ico random. 90

\bibitem{Shaw91a}
L. J. Shaw, V. Elser, and C. L. Henley, 
Phys. Rev. B43, 3423 (1991). 
%%% ``Long-range order in a three-dimensional random tiling quasicrystal'',

% CLIPPED \bibitem{FN-timenote}
%  Our sweep converts to about 5 of the usual sweeps 
%  (Ref.~\onlinecite{Shaw91a})  defined as one flip
%  per {\it flippable} vertex.  
%%% Tang PRL defined MCS as ``one trial per site''
%%% but never gave numerical data. 

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

\bibitem{FN-n20anomaly}
We do not know the origin of the anomalies in the range
$12 \leq n \leq 20$ (the irregularities in
Fig.~\ref{fig-shells} are not statistical, 
except perhaps in the small-$|\qq|$/low-$T$ limit.)

% CLIPPED 
%  \bibitem{FN-instab}
%  An alternative interpretation, which is difficult to rule out, would
%  accept the large-$|\qq|$ curves as belonging to the proper elasticity theory. 
%  The ratio $K_2(n)/K_1(n)$ becomes more negative as $n \to 0$ and, 
%  at low $T$, it would cross $-0.75$, the threshold of an elastic instability
%  (Ref.~\onlinecite{diffuse91}).   
%  In this scenario, $\kappa$ is roughly the wavevector shell 
%  at which $K_2(n)/K_1(n) =-0.75$, and the small-$|\qq|$ regime
%  is where the longer-wavelength modes develop modulations limited by
%  anharmonic terms in the elastic free energy.
%%% and dependent on the boundary conditions


\bibitem{New95b}
%% ``Phason elasticity of a three-dimensional quasicrystal:
%% a transfer-matrix method'',
M.~E.~J.~Newman and C.~L.~Henley,
Phys. Rev.  B52, 6386 (1995).



%%% \bibitem{Ritsch96a} 
%%% S. Ritsch, H.-U. Nissen, and C. Beeli, 
%%% Phys. Rev. Lett. 76, 2507 (1996).
%%% stacking disorder

%%% \bibitem{Ritsch98}
%%% S. Ritsch,  C. Beeli, H. U. Nissen, {\it et al}
%%%      The existence regions of structural modifications in decagonal Al-Co-Ni
%%% Philos. Mag. Lett. 78, 67 (1998)
%%% also S. Ritsch, thesis, 1996. 

\end {thebibliography}

%%% \vfill\eject

\begin{figure}
%\includegraphics[width=2.9in,angle=0]{FIG1_K1.ps}
%\includegraphics[width=2.9in,angle=0]{FIG1_K2.ps}
\includegraphics[width=3.1in,angle=0]{FIG1.ps}
%%% \includegraphics[width=1.0\linewidth]{FIGS/k2_radial.ps}
%%% OLD \epsfxsize=2.8truein \epsfbox{FIGS/k2_radial.ps}}
\caption{
Phason elastic constants $K_1$ (bottom) and $K_2/K_1$ (top), where
$K_1$ and $K_2$ are fitted from
``perp'' space fluctuation $\CorrH_{lm}(\qq)$ of rhombohedron vertices, 
Fourier-transformed and separately fitted in each shell
of wavevector $\qq = \sqrt n (2\pi/L)$.  
Bragg peaks and tails were subtracted using (\protect\ref{eq-sumBragg}). 
%% from text 4/25/02 A rough plateau occurs in the range $7\leq n \leq 23$;
%% the dashed lines show the mean from this interval. 
Only five of the 34 temperatures are shown (labeled by
$\beta\equiv 1/T$. %%% MAREK: LABEL BY T INSTEAD? [no]
The fits to Eq.~(\ref{eq-fitform}) are shown as solid lines.
%%% This admits extrapolation to $|\qq|=0$ if one disregards the
%%% small-$\qq$ unequilibrated points at low $T$. 
}
\label{fig-shells}
\end{figure}

\begin{figure}
%% OLD epsf \epsfxsize=3.4in {\epsfbox{FIGS00/fig1.eps}}
\includegraphics[width=2.8in,angle=-90]{FIG2.ps}
\caption{
%%% [K_1 in (a), K_2/K_1 in (b), vs. beta=1/T. 
Elastic constants as a function of inverse temperature $\beta$. 
%%% The three different symbols correspond to three different cooling programs. 
%%% Data are extracted from $\thnode(\qq)$, i.e. from the ``12-fold''
%%% subset of vertices, 
%%% except the [dotted line] is the high-temperature portion of the
%%% results from using all tile vertices $\thvert(\qq)$. 
%%% REVISED: Filled symbols are fitted from $\thvert(\qq)$
These are same fitted values indicated by solid lines
in Fig.~\ref{fig-shells},
extracted from the perp-coordinates of rhombohedron vertices
and extrapolated to $q\to0$.}
%%% and based on wavevectors in the 
%%% $5 \leq |\mm|^2 \leq 20$ CHECK in Fig.~\ref{fig-shells}; 
%%% The dashed curve is fitted from $\thvert(\qq)$ by a single
%%% fit combining the $\qq$ vectors from same range
%%% and including the ``$c_1$'' correction;
%%% the dotted curve is the same but using $\thnode(\qq)$, computed
%%% from the 12-fold vertices which (at low $T$) approximate
%%% the nodes of the canonical-cell tiling.} 
\label{fig-K1K2}
\end{figure}

\end{document}

cond-mat0205271.extracted_bib
\begin{thebibliography}{99}

\bibitem [*]{MM-addr} 
Materials Research and Liquids, Institute of Physics,
TU Chemnitz, D-09107 Chemnitz, Germany;
Current and permanent address, 
Institute of Physics, Slovak Academy of Sciences, 84228 Bratislava, Slovakia.

\bibitem{niceqxtal} 
%%% A. Tsai, A. Inoue, and T. Masumoto, 
%%% Jpn. J. Appl. Phys. {\bf 26}, L1505 (1987);
%%% AlCuFe
%%%
%%% C. A. Guryan, A. I. Goldman, P. W. Stephens, K. Hiraga,
%%%    A. Tsai, A. Inoue, and T. Masumoto, 
%%%   Phys. Rev. Lett. {\bf 62}, 2409 (1989).
%%% AlPdRu
See {\it e.g.} S.~W.~Kycia {\it et al}, 
Phys. Rev. B 48, 3544 (1993). 

\bibitem{Hen91ART}
C.~L.~Henley,
``Random tiling models,'' p. 429 in {\it Quasicrystals: The
State of the Art}, ed. P.~J.~Steinhardt and D.~P.~DiVincenzo,
(World Scientific, 1991).
% CLIPPED  a revised edition is in press.

\bibitem{Hen91CCT}
C. L. Henley, Phys. Rev. {\bf B43}, 993 (1991).
%% Canonical-cell tiling

\bibitem{Wind94}
M.~Windisch, J.~Hafner, M.~Kraj\v{c}i and 
M.~Mihalkovi\v{c}, Phys. Rev. B 49, 8701 (1994).

\bibitem{Mi96a}
M. Mihalkovi\v{c} {\it et al}, 
%%%% W.-J. Zhu, C. L. Henley, and M. Oxborrow,
Phys. Rev. B 53, 9002 (1996).
%% deco PRB I
%%% ``Al-Mn Decoration Models: I. Geometrical Principles''
%%% IS NOT CITED \bibitem{ElHen85} 
%%% V. Elser and C.~L.~Henley, Phys. Rev. Lett. {\bf 55}, 2883 (1985).
%% C. L. Henley and V. Elser,
%% Phil. Mag. B53, L59 (1986). 

\bibitem{Matching}
K.~Ingersent in {\it Quasicrystals: The State of the Art}, 
eds. D. P. DiVincenzo and 
P.~J.~Steinhardt (World Scientific, 1991). 
%%% see also J.~E.~S.~Socolar, in
%%% {\it Quasicrystals}, eds. M.~V.~Jari\'c and S.~Lundqvist
%%%(World Scientific, 1989). 
%% reviews about matching rules 

\bibitem{Jeong94} 
H.-C. Jeong and P.J. Steinhardt, {Phys. Rev. Lett.} {73}, 1943 (1994).
%%% (b) H.-C.~Jeong and P.~J.~Steinhardt, Phys. Rev. B55, 3520 (1997). 

\bibitem{diffuse91}
%%% a= \bibitem{Wid91}
M. Widom, Phil. Mag. Lett. 64, 297 (1991);
%%% b=\bibitem{I92}
 Y. Ishii, Phys. Rev. B45, 5228 (1992).

%% \bibitem{entropic} 
%% M. Widom, 
%% in {\it Quasicrystals}, 
%% ed. by M.~V.~Jari\'c and S.~Lundqvist
%% (World Scientific, 1989);
%% C.~L.~Henley, in 
%% {\it Quasicrystals and Incommensurate Structures in Condensed Matter}, 
%% edited by M. J. Yacam\'an, D. Romeu, V. Casta\~no, and A. G\'omez
%% (World Scientific, 1990). 

\bibitem{Bois95}
M.  de Boissieu, {\it et al},
%%% M. Boudard, B. Hennion, R. Bellissent,
%%% S.~Kycia, A.~Goldman, C.~Janot, and M.~Audier.
Phys. Rev. Lett. 75, 89 (1995).
%%%
%%% M. Boudard {\it et al},
%%%  M. de Boissieu, A.~Letoublon, B.~Hennion,
%%% R.~Bellissent, and C.~Janot,
%%% Europhys. Lett. 33, 199 (1996).
%%% Diffuse scatt. and phason elasticity in the AlPdMn Icosahedral phase
%%%
%%% \bibitem{Bou95}
%%% DROP THIS. M. Boudard, M. de Boissieu, M. Audier, S. Kycia, A.~I.~Goldman,
%%% B.~Hennion, R.~Bellissent, M.~Quilichini, and C.~Janot, p. xxx in 
%%% {\it Proc. 5th Int'l Conf. on Quasicrystals,}
%%% (Ed. C. Janot, and R.~Mosseri, World Scientific, 1995)
 %%% Modulated phase and diffuse scattering in the AlPdMn icosahedral phase

\bibitem{Le01}
%%% "Phason elastic constants of the icosahedral Al-Pd-Mn phase derived from 
%%% diffuse scattering measurements
A. L\'etoublon {\it et al}, 
%%% A. L\'etoublon, M.~de~Boissieu, M.~Boudard, L.~Mancini, J.~Gastaldi, 
%%% B.~Hennion, R.~Caudron, and R.~Bellissent, 
Phil. Mag. Lett.  81, 273 (2001). 
%%%  81 (4): 273-283 APR 2001


\bibitem{Le99}
A.~L\'etoublon, thesis (Grenoble, France), 1998.
%% ``Phase Transition and Diffuse Scattering studies in the
%% AlCuFe ternary system''. 
%% A.~Letoublon and M.~de~Boissieu, personal communications;
%%  It seems AlCuFe IS NOT YET PUBLISHED (2001,2002). 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% M. Boudard, A. Letoublon, M. de Boissieu, T. Ishimasa,
%% M. Mori, E. Elkaim, and J.P.Lauriat,
%% Proc. ICQ7 (1999).
%% {\sl Letoublon's own contributed talk at ICQ7, OA2, is
%% more about use of speckle to see phason domains.}
%%%
%%% A. Letoublon, F.~Yakhou, F.~Livet, F.~Bley, M.~de~Boissieu, L.~Mancini, 
%%% R.~Caudron, C.~Vettier, and J.~Gastaldi, 
%%% Europhys. Lett. 54, 753 (2001)
%%% 54 (6): 753-759 JUN 2001
%%% "Coherent X-ray diffraction and phason fluctuations in quasicrystals"
%%% This is very interesting: speckle patterns, q^2 dynamics -- but AlPdMn

\bibitem{Mi93} 
M. Mihalkovi\v{c} and P. Mrafko, Europhys. Lett. 21, 463 (1993).
%%% Tiling of canonical cells: large Pa3 approximants


%% CLIPPED \bibitem{supertiles}
%% ``Frank-Kasper decagonal quasicrystals: simulations and ideal models''
% The formation of supertiles is well-known in both toy and realistic
% decagonal models, such as Ref.~\onlinecite{Jeong94}; 
% J. Roth and C. L. Henley,
% Philos. Mag. A, Vol 75 (No 3), 861 (1997); and
%%% 861-887 (1997).
% M.~Mihalkovi\v{c} {\it et al}, 
%% M.~Mihalkovi\v{c}, I. Al-Lehyani, E.~Cockayne,
%% C.~L.~Henley, N.~Moghadam, J.~A.~Moriarty, Y.~Wang, and M.~Widom,
% Phys. Rev. B 65, 104205 (2002).

\bibitem{Ox9397}
%%% \bibitem{Ox93}
%%% ``Random square-triangle tilings: A model for
%%% twelvefold-symmetric quasicrystals''
M. Oxborrow and C. L. Henley, 
Phys. Rev. B 48, 6966 (1993);
%    \bibitem{Ox97} 
M. Oxborrow and M. Mihalkovi\v{c},
%%% p. 178 in {\it Aperiodic '94}, ed. G. Chapuis and W. Paciorek
%%% (World Scientific, 1995);
p.~451 in
{\it Aperiodic '97}, 
eds. M.~de Boissieu, J.-L. Verger-Gaugry, and R.~Currat
(World Scientific, 1998).

\bibitem{steinhardt}
H.-C. Jeong and P.J. Steinhardt {Phys. Rev. B} {48}, 9394 (1993);
%% 9394-9403 (1993);
%%"Finite-temperature elasticity phase transition in decagonal 
%% quasicrystals"  stacked
%%
%% "ISING-LIKE TRANSITION AND PHASON UNLOCKING IN ICOSAHEDRAL 
%% QUASI-CRYSTALS"  Hamiltonian favoring alt. cond.
T. Dotera and P. J. Steinhardt, 
Phys. Rev. Lett.  72, 1670 (1994). 


\bibitem{LG}
M.~Mihalkovi\v{c} and C.~L.~Henley, in preparation.
%%% Lattice-gas (other) paper.

\bibitem{Hen86}
C. L. Henley, 
Phys. Rev. B 34, 797 (1986).
%%% sphere packings '86.


\bibitem{JarNel88} 
M.~V.~Jari\'c and D.~R.~Nelson, Phys. Rev. B 37, 4458 (1988).
%%% diffuse part

\bibitem{El85} 
V. Elser, Phys. Rev. B {\bf 32}, 4892 (1985).

\bibitem{Elastic}
%%% T. C. Lubensky, S. Ramaswamy, and J. Toner,
%%% Phys. Rev. B32, 7444 (1985);
T. C. Lubensky, in
{\it Aperiodic Crystals I: Introduction to Quasicrystals},
%%% = {\it Aperiodicity and order}, vol. 1,
M. V. Jari\'c (ed.), (Academic Press, London, 1988),
and references therein.

% CLIPPED \bibitem{Worms}
%  J. E. S. Socolar, T.~C.~Lubensky, and P.~J.~Steinhardt, 
%  Phys. Rev. B34, 3345 (1986). 
%%% 3345-3360
%%% Phonons, phasons, and dislocations in quasicrystals
%%% old Lubensky-Socolar-Steinhardt, linear gradient.


% CLIPPED \bibitem{2DMC} 
%% \bibitem{2DRT} 
%%% \bibitem{Tang87} 
%  K. J. Strandburg, L. Tang, and M. V. Jari\'c,
%      Phys. Rev. Lett.  {\bf 63}, 314 (1989).
%%%  K. J. Strandburg, Phys. Rev. B {\bf 40}, 6071 (1989);
%%% Method of measuring phason elasticity from
%%% finite-size dependence of mean-square <|h|^2>?

\bibitem{2DTM} 
%%%  (a) M. Widom, D. P. Deng, and C. L. Henley, 
%%%     Phys. Rev. Lett.  {\bf 63}, 310 (1989);
%%%      %%% ``Transfer matrix analysis of a Two-dimensional Quasicrystal''
%%%  (b) L. J. Shaw and C. L. Henley, J. Phys. A. 24, 4129 (1991).
%%%       %% ``A transfer-matrix Monte Carlo study of random Penrose tilings''
  W. Li, {\it et al}, J. Stat. Phys. {\bf 66}, 1 (1992), and
references therein.
%%  (c) W. Li, H. Park, and M. Widom, 

%%% \bibitem{rhombo90}  
\bibitem{Tang90}  
L.-H. Tang, Phys. Rev. Lett. {\bf 64}, 2390 (1990).
%%% Tang PRL ico random. 90

\bibitem{Shaw91a}
L. J. Shaw, V. Elser, and C. L. Henley, 
Phys. Rev. B43, 3423 (1991). 
%%% ``Long-range order in a three-dimensional random tiling quasicrystal'',

% CLIPPED \bibitem{FN-timenote}
%  Our sweep converts to about 5 of the usual sweeps 
%  (Ref.~\onlinecite{Shaw91a})  defined as one flip
%  per {\it flippable} vertex.  
%%% Tang PRL defined MCS as ``one trial per site''
%%% but never gave numerical data. 

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

\bibitem{FN-n20anomaly}
We do not know the origin of the anomalies in the range
$12 \leq n \leq 20$ (the irregularities in
Fig.~\ref{fig-shells} are not statistical, 
except perhaps in the small-$|\qq|$/low-$T$ limit.)

% CLIPPED 
%  \bibitem{FN-instab}
%  An alternative interpretation, which is difficult to rule out, would
%  accept the large-$|\qq|$ curves as belonging to the proper elasticity theory. 
%  The ratio $K_2(n)/K_1(n)$ becomes more negative as $n \to 0$ and, 
%  at low $T$, it would cross $-0.75$, the threshold of an elastic instability
%  (Ref.~\onlinecite{diffuse91}).   
%  In this scenario, $\kappa$ is roughly the wavevector shell 
%  at which $K_2(n)/K_1(n) =-0.75$, and the small-$|\qq|$ regime
%  is where the longer-wavelength modes develop modulations limited by
%  anharmonic terms in the elastic free energy.
%%% and dependent on the boundary conditions


\bibitem{New95b}
%% ``Phason elasticity of a three-dimensional quasicrystal:
%% a transfer-matrix method'',
M.~E.~J.~Newman and C.~L.~Henley,
Phys. Rev.  B52, 6386 (1995).



%%% \bibitem{Ritsch96a} 
%%% S. Ritsch, H.-U. Nissen, and C. Beeli, 
%%% Phys. Rev. Lett. 76, 2507 (1996).
%%% stacking disorder

%%% \bibitem{Ritsch98}
%%% S. Ritsch,  C. Beeli, H. U. Nissen, {\it et al}
%%%      The existence regions of structural modifications in decagonal Al-Co-Ni
%%% Philos. Mag. Lett. 78, 67 (1998)
%%% also S. Ritsch, thesis, 1996. 

\end {thebibliography}