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\title{Periodic diffraction patterns for 1D quasicrystals}
\author{
\textsc{Pawe\l{} Buczek}
\footnote{Faculty of Physics and Nuclear Techniques, AGH-UST, al. Mickiewicza 30, 30-059 Krak\'ow, Poland} \\
\texttt{\small{pawelbuczek@poczta.onet.pl}} \\
\textsc{Lorenzo Sadun}
\footnote{Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX 78703, USA} \\
\texttt{\small{sadun@math.utexas.edu}} \\
\textsc{Janusz Wolny} \footnotemark[1] \\
\texttt{\small{wolny@novell.ftj.agh.edu.pl}}}
\date{}
%\input epsf.tex
\begin{document}
\maketitle
\begin{sloppy}
\begin{abstract}
A simple model of 1D structure based on a Fibonacci sequence with
variable atomic spacings is proposed. The model allows for
observation of the continuous transition between periodic and
non-periodic diffraction patterns. The diffraction patterns are
calculated analytically both using ``cut and project" and
``average unit cell'' methods, taking advantage of the physical
space properties of the structure.
PACS numbers: 61.44.Br, 61.43.-j, 61.10.Dp
\end{abstract}
\section{Introduction}
\label{sec:Introduction}
For nearly 100 years, the analysis of diffraction patterns of solids has been
an essential tool for studying solids, since the diffraction pattern
of a solid is essentially the (squared)
Fourier transform of the set of atomic positions.
Classical crystallography considered periodic structures, whose
diffraction patterns consist entirely of sharp Bragg peaks.
The Fourier transform of such a periodic set can be computed from the
relevant unit cell. The discovery of quasicrystals showed that
discrete diffraction patterns are associated not only with periodic
structures but also with a large family of solids that have no discrete
translation symmetry -- quasicrystals. This fact was incorporated into
a new definition of ``crystal" proposed in 1992 by the Commission on
Aperiodic Crystals established by the International Union of
Crystallography: a crystal is defined to be any solid with an
essentially discrete diffraction pattern.
Diffraction patterns for periodic and aperiodic crystals differ in
substantial ways -- for instance, the diffraction patterns of quasicrystals
may exhibit ``forbidden symmetry". It is therefore illuminating to consider
a model that interpolates between periodic and aperiodic structures,
and observe how the diffraction pattern changes.
We consider such a one-parameter family of structures in this paper.
Specifically, we consider a fixed
(Fibonacci) sequence of two types of ``atoms'',
and vary the amount of space around each type of atom, while keeping
the overall density fixed. The control parameter $\kappa$
is the ratio of the two allowed distances between nearest neighbors.
In all cases, the
diffraction pattern is discrete, and the locations of the Bragg peaks
are independent of $\kappa$. However, the {\it intensities} of the peaks
are $\kappa$-dependent. When $\kappa$ is rational, the intensities
form a periodic pattern, while when $\kappa$ is irrational, the
diffraction pattern is aperiodic. We compute this diffraction pattern
in two independent but
equivalent ways: a) by recovering periodicity going to higher
dimension (the ``cut and project method" deeply discussed in many papers:
\cite{deBruijn1981, Duneau1985, Elser1986, Hof1995, Hof1997,
Jagodzinski1991, Janssen1998, Jaric1986, Kalugin1985, Kramer1984,
Senechal1997}); b)using the concept of the reference lattice.
These results are in accordance with the ergodic theory of tiling spaces.
It is known that the Bragg peaks of a tiling $T$ occur
at eigenvalues of the generator of translations on the hull of $T$ (i.e.,
the space of all tilings in the same local isomorphism class as $T$)
\cite{Dworkin1993}. It is also known \cite{Radin2001} that the hulls of
modified Fibonacci chains with the same average spacing are topologically
conjugate, hence that their generators of translations have the same
spectral decomposition. The question of when and how such a modification
affects the dynamical spectrum was addressed for one dimensional patterns in
\cite{ClarkSadun2003a}, and for higher dimensional patterns in
\cite{ClarkSadun2003b}. (It should be noted that for a substitution tiling
whose substitution matrix has two of more eigenvalues greater than 1, a
generic change in tile length will destroy the Bragg peaks altogether,
in sharp contrast to the behavior of modified Fibonacci chains, other
Pisot substitutions, and other Sturmian sequences.)
Ergodic theory says nothing, however, about the intensities of
the Bragg peaks. Although the spectrum of the generator of translations is
complicated, for special values of the control parameter
some of the peaks may have intensity zero, resulting in a simpler
diffraction pattern.
The calculations in this paper demonstrate that this does in fact happen.
\section{The Modified Fibonacci Chain}
\label{sec:RelevantPropertiesOfFibonacciSequence}
The properties of Fibonacci sequences have already been thoroughly
studied (see e.g., \cite{Senechal1995}). They are sequences of two
elements $A$ and $B$ obtained from a substitution rule:
%
\begin{align}
A \longrightarrow AB; \quad B \longrightarrow A.
\end{align}
%
Let $\vec{p}_{m}=(p_{m}^{A},p_{m}^{B})$ be the population vector, where
$p_{m}^{X}$ tells how many elements of type $X$ are among the first
$m$ terms of the sequence. Of course,
$p_{m}^{A}+p_{m}^{B}=m$. There are an uncountably infinite number of Fibonacci
sequences, but all have the same local properties and the same diffraction
pattern. It is easy to see that every Fibonacci sequence has
\begin{align}
\lim_{m \to \infty}\frac{p_{m}^{A}}{p_{m}^{B}}=\tau, \quad
\textrm{where} \quad \tau=\frac{1+\sqrt{5}}{2}. \label{eq:conctr}
\end{align}
For definiteness, we will work with the sequence
(\cite{Senechal1995})
\begin{align}
\vec{p}_{m}=(\left\|\frac{m}{\tau}\right\|,m-\left\|\frac{m}{\tau}\right\|).
\label{eq:3}
\end{align}
Here $\left\|\cdot\right\|$ is the nearest integer function: If $m
\in \mathbb{Z}$ and $m \leq x < m+1$ then:
\begin{align}
\left\|x\right\|=
\left\{
\begin{array}{ll}
m & \textrm{if} \quad x \in [m,m+1/2), \\
m+1 & \textrm{if} \quad x \in [m+1/2,m+1).
\end{array}\right. \label{eq:nif}
\end{align}
Now pick two positive numbers (also called $A$ and $B$) that determine the
space between each ``$A$'' or ``$B$'' atom and its predecessor. That
is, the atomic positions are given by
\begin{equation}
x_m = \vec p_m \cdot (A,B). \label{eq:positions}
\end{equation}
We call the sequence $\{ x_m \}$ a {\it modified Fibonacci chain}.
If $A=B$, then the atoms are equally spaced, and this is simply
a periodic array.
If $A/B = \tau$, then the atomic positions are those obtained from the
canonical ``cut and project'' method. By varying the ratio $A/B$,
we interpolate between these two cases.
Note that the average atomic spacing is $\frac{\tau A + B}{1+\tau}$. We will
keep this average spacing fixed and consider the one-parameter family
\begin{equation}
A = \tau - \frac{\epsilon}{\tau}, \qquad B = 1 + \epsilon,
\label{eq:a}
\end{equation}
depending on the control parameter $\epsilon$.
\section{2D Analysis of the Modified Fibonacci Chain}
\label{sec:MFCh2D}
The modified Fibonacci chain can be obtained by a ``cut and project''
method with a nonstandard projection. From this construction we
can compute the diffraction pattern.
Let $\mathcal{L}_{2}^{\nu}$ be the 2 dimensional square lattice with
spacing $\nu$. The Vorono\"\i{} cell of each lattice point is a square.
Let $l_{0}$ be
the line $y=x/\tau$ through the origin, making an angle
$\beta_0 = \cot^{-1}(\tau)$ with the $x$-axis. Let $X$ be the
subset of $\mathcal{L}_{2}^{\nu}$ whose Vorono\"\i{} cells
are cut by $l_{0}$. It is well known (see e.g.,
\cite{Senechal1995}) that $X = \left \{ \vec{x}\mid \vec{x}=\nu
\vec{p}_{m}, m \in \mathbb{Z} \right\}$ where $\vec{p}_{m}$ are
population vectors given by (\ref{eq:3}).
\begin{figure}[hftb]
\begin{center}
\includegraphics[width=0.75\textwidth, origin=c
angle=-90]{Figure1.eps}
\end{center}
\caption{The 2D construction of the modified Fibonacci chain. Details in text.}
\label{fig:projection}
\end{figure}
Let $l_\alpha$ be the line through the origin making an angle
\begin{align}
\beta=\beta_{0}+\alpha \label{eq:beta}
\end{align}
with the $x$-axis, and
define $\Pi_{\alpha}$ to be the orthogonal projection onto
$l_{\alpha}$. Finally,
let $\Lambda$ be the projection of $X$ onto $l_{\alpha}$:
\begin{equation}
\Lambda=\Pi_{\alpha}(X).
\label{eq:4}
\end{equation}
%
The set $\Lambda$ is then a modified Fibonacci chain. The two distances are
\begin{equation}
A=\nu \cos \beta, \qquad B=\nu \sin \beta. \label{eq:ABang}
\end{equation}
Their sequence is fully determined by $X$ and does not depend
on $\alpha$. The average distance between nearest neighbors in
$\Lambda$ is
\begin{align}
\lim_{m \to \infty} \frac{\vec{p}_{m} \cdot (A,B)}{\vec{p}_{m} \cdot (1,1)} %\\
%=\lim_{m \to \infty}(\frac{p_{m}^{A}}{p_{m}^{B}}A+B)/(\frac{p_{m}^{A}}{p_{m}^{B}}+1) \\
%=\nu \frac{t \cos \beta + \sin \beta}{\tau^{2}} \\
=\nu \frac{\sqrt{\tau+2}}{\tau+1} \cos \alpha.
\label{eq:6}
\end{align}
To keep this average spacing constant we take
\begin{align}
\nu = \frac{\sqrt{\tau +2}}{\cos \alpha}. \label{eq:nu}
\end{align}
The angle $\alpha$, the displacement parameter $\epsilon$ of (\ref{eq:a})
and the ratio $\kappa=A/B$ are related by
\begin{align}
\epsilon = & \frac{\tau - \kappa}{\kappa + \tau -1} =
%\frac{(\tau + 2)\tan(\alpha) + 1-\tau}{2 \tau - 1},
\tau \tan (\alpha), &\\
\kappa = \cot(\beta) = & \frac{\tau - \tan(\alpha)}{1 + \tau \tan(\alpha)} =
\frac{\tau + \epsilon (1-\tau)}{1 + \epsilon}, & \\
\tan(\alpha) = %& \frac{(2 \tau -1)\epsilon + \tau - 1}{\tau + 2} =
%\frac{\tau - \kappa}{\kappa \tau + 1}.&
& \frac{\epsilon}{\tau} = \frac{\tau - \kappa}{\kappa \tau + 1}.&
\end{align}
We assume
that every point of the set $\Lambda$ is an atom with scattering power
equal to unity. Our aim is to calculate the diffraction pattern of
such a structure. We begin by calculating the 2-dimensional diffraction
pattern of $X$.
The diffraction pattern of
$\Lambda$ is then a section of the diffraction pattern of $X$ along the
direction $l_{\alpha}$.
To get the diffraction pattern of $X$ we note that $X$ is
$\mathcal{L}_{2}^{\nu}$ times the characteristic function of a strip of
width
\begin{align}
h_{\alpha} = \nu \frac{\tau + 1}{\sqrt{\tau + 2}}
=\frac{\tau + 1}{\cos \alpha} \label{eq:ha}
\end{align}
around $l_0$. The Fourier transform of a product is the convolution of
the Fourier transforms, and the Fourier transform of a lattice is the
reciprocal lattice. The
diffraction pattern of $X$ in 2D has normalized intensity
\begin{equation}
I(\vec{k})=\sum_{m_{x}}\sum_{m_{y}}\Big(\frac{\sin((h_{\alpha}| \vec{k}-\vec{k}_{m_{x}m_{y}} |)/2)}{(h_{\alpha}| \vec{k}-\vec{k}_{m_{x}m_{y}} |)/2)}\Big)^{2}
\delta\big((\vec{k}-\vec{k}_{m_{x}m_{y}}) \cdot (\tau,1)\big),
\label{eq:9}
\end{equation}
where $\vec{k}_{m_{x}m_{y}}=\frac{2\pi}{\nu}(m_{x},m_{y})$ and
$m_{x},\:m_{y}\in\mathbb{Z}$ label points of the
reciprocal lattice to $\mathcal{L}_{2}^{\nu}$.
Along the direction $l_\alpha$, the peaks can be observed at positions
\begin{equation}
k^{phys}=\frac{\vec{k}_{m_{x}m_{y}} \cdot (\tau,1)}{\cos \alpha \sqrt{\tau +2}}
= 2 \pi \frac{\tau m_x + m_y}{\tau + 2}. \label{eq:10}
\end{equation}
Note that these positions are independent of $\alpha$
(or $\epsilon$ or $\kappa$).
Their intensities are
\begin{equation}
I_{m_{x}m_{y}}=\Big(\frac{\sin w}{w}\Big)^{2}, \qquad w=
\frac{h_\alpha
\vec{k}_{m_{x}m_{y}} \cdot (-1-\tau \tan \alpha, \tau - \tan \alpha)}
{2 \sqrt{\tau + 2}}.
%\frac{(-1,\tau)}{\sqrt{\tau+2}} -\vec{k}_{m_{x}m_{y}} \cdot \frac{(\tau,1)}{\sqrt{\tau+2}} \tan \alpha
\label{eq:11}
\end{equation}
After simplifying and rewriting in terms of $\epsilon$ we obtain
\begin{equation}
w= \frac{\pi(\tau + 1)}{\tau + 2}
(-m_{x}(1+\epsilon)+m_{y}(\tau - \epsilon / \tau)). \label{eq:intens_2D}
\end{equation}
\begin{figure}[hftb]
\begin{center}
\includegraphics[width=0.75\textwidth,
angle=0.0] {Figure2.eps}
\end{center}
\caption{Diffraction pattern of the modified Fibonacci structure is a section of the diffraction pattern of $X$ through direction $l_{\alpha}$. $k^{phys}=\overline{OP}$ is the position of the peak, $\overline{KP}$ determines the intensity.}
\label{fig:RecSpace}
\end{figure}
\section{Structure Factors and Average Unit Cells}
\label{sec:StructureFactor}
The concept of a reference lattice has previously been proposed in
\cite{Wolny1998PhM}. Suppose we have a 1 dimensional Delone set
$\left\lbrace r_{n} \right\rbrace$. Its points represent atoms, whose
scattering instensities are equal to unity. We get the following expression
for the structure factor:
\begin{align}
F(k) &= \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} \exp (i k r_{n}) %\nonumber \\
= \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} \exp (i k u_{n}) %\nonumber \\
= \int_{- \lambda /2}^{\lambda /2} P (u) \exp (i k u) \textrm{d} u,
\label{eq:1pFac}
\end{align}
where $P(u)$ is the probability distribution of distances $u_{n}$
from the atoms to reference lattice positions $m \lambda$, $\lambda=2 \pi /k$.
That is,
\begin{align}
u_n=r_n - \left\| \frac{r_n}{\lambda} \right\| \lambda.
\end{align}
We call the series $u_n$ the \textit{displacements sequence} of $r_n$ (induced
by the reference lattice with period $\lambda$).
Any series $u'_n$ such that
\begin{equation}
u_n = u'_n - \left\| \frac{u'_n}{\lambda} \right\| \lambda,
\end{equation}
will be called an \textit{unreduced displacements sequence} (of $r_{n}$).
\newtheorem{theo}{Theorem}
\begin{theo}
\label{theo:sumOfDis}
Let $r_{n}=\alpha_n+\beta_n$ be a sum of two real series.
%\begin{align}
%r_{n} = \alpha_{n} + \beta _{n}. \label{eq:rnDef}
%\end{align}
%
If $d_{n}$ is a displacements sequence of $\alpha _{n}$ induced by
a reference lattice,
%\begin{align}
%d_{n} &= \alpha _{n} - \left\| \frac{\alpha _{n}}{\lambda} \right\| \lambda ,
%\label{eq:dn}
%\end{align}
then
\begin{align}
u'_{n} &= d_{n} + \beta _{n} \label{eq:sumOfDis}
\end{align}
is an unreduced displacements sequence of $r_{n}$ induced by the same lattice.
\begin{proof}
Let $\lambda$ be the period of the reference lattice. We have to show that
\begin{align}
u_{n} = r_{n} - \left\| \frac{r_{n}}{\lambda} \right\| \lambda = u'_n - \left\| \frac{u'_n}{\lambda} \right\| \lambda \label{eq:toPr1}.
\end{align}
Note that for any real number $x,y$,
$\left \| x - \left\| y \: \right\| \right \|
= \left\| x \right\| - \left\| y \right\|$, since $\left\| y
\right\|$ is an integer. We can write the right hand side of
(\ref{eq:toPr1}) as
\begin{equation}
\beta_{n} + \alpha_{n} - \left\| \frac{\alpha _{n}}{\lambda}
\right\| \lambda - \left\| \frac{\beta_{n} + \alpha_{n} - \left\| \alpha _{n} /
\lambda \right\| \lambda}{\lambda} \right\| \lambda %\\
= r_{n} - \left\| \frac{r_{n}}{\lambda} \right\| \lambda,
\end{equation}
which is the left hand side.
\end{proof}
\end{theo}
The quantity $P(u)$ may be viewed as a probability distribution for an
{\em average unit cell}. The structure factor for the scattering vector
$k$ is just the first Fourier mode of this distribution.
Unfortunately, for each scattering vector we get, in principle, a different
average unit cell and a different distribution. However, the
structure factor for $mk, \: m \in \mathbb{Z}$ can be computed from the
reference lattice for $k$; it is the $m$-th Fourier mode of the
distribution $P(u)$. Thus, a single average unit cell is sufficient to analyze
structures whose scattering occurs at multiples of a fixed scattering vector $k_0$.
This situation includes, but is not limited to, the case where the
original point pattern was periodic with period $2 \pi / k_0$.
For modulated structures (including quasicrystals), there are usually two periods,
$a$ and $b$, which may be incommensurate. Using two reference
lattices, the first one having periodicity $a$ and the second having
periodicity $b$, the structure factor for the sum of two scattering
vectors $k_{0} \equiv 2 \pi/ a$ and $q_{0} \equiv 2 \pi/ b$ can be
expressed by:
\begin{align}
F(k_{0} + q_{0}) &= \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^{N} \exp(i(k_{0} + q_{0})x_{n}) %\nonumber \\
= \lim_{N \to \infty} \frac{1}{N} \sum _{n=1} ^{N} \exp(i(k_{0} u_{n} + q_{0} v_{n})) \nonumber \\
&= \int _{-a/2} ^{a/2} \int _{-b/2} ^{b/2} P(u,v) \exp(i(k_{0} u +
q_{0} v)) \textrm{d}u \textrm{d}v, \label{eq:2pFac}
\end{align}
where $u$ and $v$ are the shortest distances of the atomic position
from the appropriate points of two reference lattices and $P(u,v)$ is
the corresponding probability distribution, which thus describes a
{\em two dimensional} average unit cell. Likewise, the structure factor
for a linear combination $nk_{0} + mq_{0}, \:n,\:m \in \mathbb{Z}$ is given
by the $(n,m)$ Fourier mode of $P(u,v)$. This means that the
average unit cell, calculated for the wave vectors of the main
structure and its modulation, can be used to calculate the peak
intensities of any of the main reflections and its satellites of
arbitrary order. Using (\ref{eq:2pFac}) and its generalization,
it is possible to calculate
the intensities of all peaks observed in the diffraction patterns.
\section{1D Analysis of the Modified Fibonacci Chain}
\label{sec:MFCh1D}
Let $a=\frac{\tau + 2}{\tau + 1}$, the average spacing between atoms in a
modified Fibonacci chain, and let $b = \tau a$.
The positions of atoms in our chain are:
\begin{align}
x_{m} &= \vec{p}_{m} \cdot (A,B) \nonumber \\
&= \left\| \frac{m}{\tau}\right\|A+mB - \left\|
\frac{m}{\tau}\right\|B \nonumber \\
&= (A-B)\left\|\frac{m}{\tau}\right\| + mB %\nonumber \\
= (A-B)\Big[\frac{m}{\tau}+\frac{1}{2}\Big]+mB \nonumber \\
&= (A-B) \frac{m}{\tau} + mB + (A-B) \Big( \frac {1}{2} -
\left\{ \frac{m}{\tau}+\frac{1}{2}\right\}\Big) \nonumber \\
&= ma+{u_{\scriptscriptstyle{0}}}M(ma), \label{eq:xmFin}
\end{align}
where $M(x)=\frac{1}{2}-\{\frac{x}{b}+\frac{1}{2}\}$ is a periodic
function with period $b$ and
\begin{align}
u_{\scriptscriptstyle{0}}=\tau-1-\epsilon\tau. \label{eq:u0}
\end{align}
Here $[x] = \| x - \frac{1}{2} \|$ is the greatest integer function and
$\{x\} = x - [x]$ is the
fractional part of $x$.
%\begin{align}
%[x] &= \left\| x - \frac{1}{2} \right\| \\
%\{x\} &= x - [x].
%\end{align}
When $u_{\scriptscriptstyle{0}} \ne 0$, the modified Fibonacci chain is thus
an incommensurately modulated structure. A similar derivation for
$\epsilon=0$ can be found in \cite{Senechal1995}.
We are going to construct a two dimensional average unit cell based on
the two natural periodicities for $x_{m}$: $a$ and $b$.
For the scattering vector $k_0= 2 \pi / a$,
it is obvious that the series $u'_{m} = {u_{\scriptscriptstyle{0}}} M(ma)$
is an unreduced displacements sequence of $x_{m}$ induced by the reference
lattice with period $a$.
Next we consider the (one dimensional) average unit cell for
the scattering vector
$q_{0} = 2 \pi / b$. The series
\begin{align}
\mu_{m} = ma-\left\| \frac{ma}{b} \right\| b = -b M(ma) = -u'_{m} \frac{b}{{u_{\scriptscriptstyle{0}}}}
\label{eq:nuseq}
\end{align}
is an (unreduced) displacement sequence of $ma$
induced by a reference lattice with period $b$.
Using Theorem
\ref{theo:sumOfDis} we immediately get that the series
\begin{align}
v'_{m} = \mu _{m} + u'_{m} = (u_{\scriptscriptstyle{0}}-b) M(ma) = v_{\scriptscriptstyle{0}} M(ma) = \xi u'_{m} \label{eq:v_prim}
\end{align}
is an unreduced displacements sequence of $x_{m}$ induced by a lattice with
period $b$, where
\begin{equation}
{v_{\scriptscriptstyle{0}}}={u_{\scriptscriptstyle{0}}}-b.
\qquad
\xi = {v_{\scriptscriptstyle{0}}}/{u_{\scriptscriptstyle{0}}} =
\frac{-\tau^{2}(1+\epsilon)}{1-\epsilon\tau^{2}}. \label{eq:xi}
\end{equation}
By Kronecker's theorem (see \cite{Hardy1962}), the series $u'_{m}$ is
uniformly distributed in the interval
$[-\left|{u_{\scriptscriptstyle{0}}}\right|/2,
\left|{u_{\scriptscriptstyle{0}}}\right|/2]$.
As pointed out by Elser (\cite{Elser1986}, for a more precise discussion
see also \cite{Senechal1995}) the uniformity of this
distribution is crucial for our deliberations. Likewise, the
series $v'_{m}$ is uniformly distributed in the interval
$[-\left|{v_{\scriptscriptstyle{0}}}\right|/2,\left|{v_{\scriptscriptstyle{0}}}\right|/2]$.
The structure factor (see (\ref{eq:2pFac})) is
\begin{equation}
F(n_{1}k_{0} + n_{2}q_{0}) = \int _{-a/2} ^{a/2} \int _{-b/2} ^{b/2} P(u,v) \exp(i(n_{1}k_{0}u + n_{2}q_{0}v)) \textrm{d} u \textrm{d} v. \label{eq:factor}
\end{equation}
The unreduced displacements sequences $u'_{m}$ and $v'_{m}$ can be used to
calculate $P(u,v)$. However, this cannot be done directly because their
terms may
lie outside the average unit cell (i.e.:
$\left|u_{\scriptscriptstyle{0}}\right|>a$ or
$\left|v_{\scriptscriptstyle{0}}\right|>b$). Such a situation is
shown in figure \ref{fig:AvCell1}. We have to reduce the series to the
interior of the cell. The probability function $P(u,v)$ is nonzero
only along segments with slope $\xi$ (as a result of the strong
correlation between $u'_{m}$ and $v'_{m}$ given by (\ref{eq:v_prim}))
and has constant value. This last fact follows from the uniformity of the
marginal distributions.
The formula (\ref{eq:factor}) is invariant under the changes
\begin{equation}
u \to u+\gamma_{1}a, \qquad
v \to v+\gamma_{2}b, \label{eq:inv}
\end{equation}
where $\gamma_{1,2}$ are arbitrary integers.
Likewise, the formula does not change if we use
$P'(u,v)=C\delta(v-\xi u)$ instead of
$P(u,v)$ and if we change the region of integration from
$[-a/2,a/2]$,$[-b/2,b/2]$ to
$[-\left|{u_{\scriptscriptstyle{0}}}\right|/2,\left|{u_{\scriptscriptstyle{0}}}\right|/2]$,
$[-\left|{v_{\scriptscriptstyle{0}}}\right|/2,\left|{v_{\scriptscriptstyle{0}}}\right|/2]$.
That is, we are free to integrate a part of distribution in the
neighboring unit cells.
\begin{figure}[hftb]
\begin{center}
\includegraphics[width=0.75\textwidth,
angle=0.0]{Figure3.eps}
\end{center}
\caption{(a) shows two parameter average unit cell.
The distribution $P(u,v)$ is non zero only along the thick
lines and has constant value. Projections of it onto
directions $u$ and $v$ determine the probability
distributions for scattering vectors $k_{0}$ and $q_{0}$
((b) and (c) respectively). (d) presents set of vectors
$\left\lbrace (u'_{m},v'_{m}) \mid m \in \mathbb{Z}
\right\rbrace$. It may happen its elements lie outside the
average unit cell and have to be reduced to its interior
(like for the presented example with $\epsilon = -0.7$).
Invariance under the substitution (\ref{eq:inv}) assures that
integration of functions on (a) and (d) gives the same
results. Our parameter space has discrete translational
symmetry like a periodic crystal.}
\label{fig:AvCell1}
\end{figure}
For integers $n_1, n_2$ we compute the location $K_{n_1,n_2}$, the
structure factor $F(K_{n_1,n_2})$ and the normalized
intensity $I$ of the corresponding
peak:
\begin{equation}
K_{n_{1}n_{2}} =(n_{1}k_{0}+n_{2}q_{0}) = \frac{2 \pi (\tau n_{1} + n_2)}
{\tau a} \qquad
F(K_{n_1,n_2}) = \frac{\sin(w)}{w}, \qquad I = \left| F \right|^2,
\label{eq:FinK}
\end{equation}
where
\begin{equation}
w = (n_{1}k_{0}+n_{2}q_{0}\xi){u_{\scriptscriptstyle{0}}}/2 =
(K_{n_1,n_2}-n_{2}q_{1}){u_{\scriptscriptstyle{0}}}/2, \qquad
q_{1}=q_{0} (1 - \xi) = \frac{2 \pi \tau}{1 - \epsilon \tau^2}.
\label{eq:Fin_w}
\end{equation}
%= \frac{2\pi}{b} (1+\tau^{2}\frac{1+\epsilon}{1-\epsilon\tau^{2}})
%\end{align}
%
%\begin{align}
The integers $n_{1}$ and $n_{2}$ label the main reflection and its
satellites, respectively. Equations (\ref{eq:FinK}, \ref{eq:Fin_w})
can be used to calculate
the positions and intensities of all peaks.
The correspondence with the previous 2 dimensional calculation is
given by
\begin{equation}
n_{1}=m_{y}, \qquad
n_{2}=m_{x} - m_{y}. \label{eq:m_to_n}
\end{equation}
By equations (\ref{eq:10}) and
(\ref{eq:FinK}), the peaks are located at
\begin{equation}
K_{n_{1},n_{2}} = K_{m_{y}, m_{x}-m_{y}} = 2 \pi \frac{m_y \tau + m_x - m_y}
{\tau} \frac{\tau + 1}{\tau + 2} = 2 \pi \frac{m_x \tau + m_y}{\tau + 2}
= k^{phys}.
\end{equation}
Likewise, substituting (\ref{eq:m_to_n}) into (\ref{eq:Fin_w})
and simplifying yields (\ref{eq:intens_2D}).
%Their intensities are given by equation (\ref{eq:FinRes}) with (\ref{eq:Fin_w}) where:
%\begin{align}
%w_{n_{1}n_{2}} &= w_{m_{y}(m_{x}-m_{y})} = (m_{y}k_{0}+(m_{x}-m_{y})q_{0}\xi)\left|{u_{\scriptscriptstyle{0}}}\right|/2 \nonumber
%\end{align}
%Using equations (\ref{eq:nu0}), (\ref{eq:a}), (\ref{eq:ha}), (\ref{eq:b}), (\ref{eq:u0}), (\ref{eq:k0}) and (\ref{eq:q0}) we also gets:
%\begin{align}
% w_{m_{y}(m_{x}-m_{y})} &= \frac{1 - \epsilon \tau ^2}{2 \tau} \frac{h_{0}}{\tau^{2}} (m_{y} \frac{2\pi}{a} + (m_{x} - m_{y}) \frac{-\tau^2(1+\epsilon)}{1-\epsilon \tau ^2} \frac{2\pi}{a \tau}) \\
% &= \frac{h_{0}}{\tau^{2}} \frac{\pi}{a}(-m_{x}(1+\epsilon) + m_{y}(1 + \frac{1}{\tau} + \epsilon - \epsilon \tau )) \\
% &= \frac{(-m_{x}(1+\epsilon)+m_{y}(\tau - \epsilon /
% \tau))}{\sqrt{1+\tau^{2}}} \frac{h_{0}}{2} \frac{2\pi}{\nu_{0}}
%\end{align}
%The last expression is identical with (\ref{eq:intens_2D}), what was to be proved.
\section{Discussion of the Results}
\label{sec:Discussion}
It has been shown that the deformation rule in physical space changes
only the amplitude of modulation (equation (\ref{eq:xmFin})).
Positions of peaks do not depend on the
parameter $\epsilon$; only their
intensities vary. Using equations (\ref{eq:FinK}) and
(\ref{eq:Fin_w}) we can easily build envelope functions, which go
through the satellite reflections of the same order (indexed by
$n_{2}$). The shift of the envelope functions is $q_1$, as given by
(\ref{eq:Fin_w}).
The set of positions of Bragg peaks is always periodic, since the spectrum of a
one-dimensional dynamical system is an Abelian group.
By a \textit{commensurate} diffraction pattern we mean a pattern in which
the amplitudes are periodic as well. However, aside from the special case
$A=B$, the Bragg peaks are described by two incommensurate periods, and
should not be confused with the diffraction of a periodic crystal.
For our diffraction patterns,
one period (of length $q_1$) is
connected with envelope functions, while the
second, with period $k_0$, is associated with peaks ascribed to each
envelope function. This behavior is characteristic of modulated crystals and
was discussed in \cite{Wolny1995}. Only the first periodicity could
assure equality of intensities.
It is convenient to describe our results in terms of the ratio $\kappa=A/B$.
When $\kappa$ is rational, say equal to $p/q$, then every atomic location
$x_m$ is a multiple of $A/p=B/q$. Plane waves whose frequencies are multiples
of $2 \pi p/A$ have value 1 at each atomic position, and the entire diffraction
pattern is periodic with period $2 \pi p/A$.
Conversely, if the diffraction pattern is periodic, then the underlying
periods $k_0$ and $q_1$ must be commensurate. A simple algebraic calculation
then shows that $\kappa$ must be rational.
Thus, the pattern is commensurate if and only if
$\kappa$ is rational, which corresponds to projecting the
set $X$ onto a rational direction ($\kappa=\cot \beta$; see
(\ref{eq:ABang})).
%\begin{align}
% \kappa = A / B = \frac{\tau - \epsilon / \tau}{1 + \epsilon},
% \label{eq:kappa}
%\end{align}
%which will be used to state the nature of the pattern. It will be also convenient to express $\epsilon$ and $q_{1}$ in terms of $\kappa$:
%\begin{align}
%\epsilon &= \frac{\tau - \kappa}{\kappa + 1 / \tau} \\
%q_{1} &= \frac{2 \pi}{b} \frac{\tau \kappa + 1}{\tau (\kappa - \tau) + 1}
%\end{align}
%Period of diagram have to be $nq_{1}, \: n \in \mathbb{Z}$. $\Delta k$
%is position of the given peak $\mathcal{P}$ in relation to the maximum
%of the appropriate envelope function. Peak in distance $nq_{1}$ must be
%a satellite of an order $n_{2} + n$, where $n_{2}$ is the order of the
%peak $\mathcal{P}$. It is expressed by the following condition:
%\begin{align}
%\left\{
%\begin{array}{ll}
%n_{1} k_{0} + n_{2} q_{0} = n_{2} q_{1} + \Delta k \nonumber \\
%(n_{1} + m) k_{0} + (n_{2} + n) q_{0} = (n_{2} + n) q_{1} + \Delta k \nonumber \\
%\end{array}\right.
%\end{align}
%Using (\ref{eq:q0}), (\ref{eq:k0}), (\ref{eq:xi}) and (\ref{eq:kappa}), we get
%\begin{align}
%&m k_{0} + n q_{0} = n q_{1} \\
%&m k_{0} = n (q_{1} - q_{0}) = - \xi q_{0} n \nonumber \\
%&\frac{n}{m} = - \frac{q_{0}}{k_{0}} \xi = \frac{1}{\kappa - 1} \label{eq:comm} \\
%&n_{1}, n_{2}, n, m \in \mathbb{Z}. \nonumber
%\end{align}
Figure \ref{fig:ModFib0}a shows the diffraction pattern of an unmodified
Fibonacci chain. The pattern is clearly non-periodic, as $A/B$ equals to
$\tau$. For $\epsilon=0$ our approach is identical with that presented
in \cite{Wolny1998ACr}. Figure \ref{fig:ModFib0}b shows the pattern for
$\epsilon=-0.7$ (corresponding to the average unit cell presented in
figure \ref{fig:AvCell1}). As we can see, analytical calculations of
envelope functions are in full agreement with numerical calculations of
the diffraction pattern.
\begin{figure}[hftb]
\begin{center}
\includegraphics[width=0.75\textwidth,
angle=0.0]{Figure4a.eps}
\includegraphics[width=0.75\textwidth,
angle=0.0]{Figure4b.eps}
\end{center}
\caption{Examples of diffraction patterns: (a) unmodified Fibonacci chain ($\kappa = \tau$ and $\epsilon = 0$); (b) $\kappa \approx 6.836$ ($\epsilon = -0.7$). Broken lines present envelope functions. Given envelope function goes through satellite peaks of the same order. All envelopes have the same shape; their shift is $q_{1}$. As we can see analytical results are in full compatibility with numerical calculations.}
\label{fig:ModFib0}
\end{figure}
Figures \ref{fig:ModFib5} and \ref{fig:ModFib6} show diffractions
patterns for $\kappa$ equal to $2/3$ and $3/2$ respectively. The
regular series of peaks are clearly visible, but the diffraction patterns
are still quasicrystaline. It is significant that for any value of
$\epsilon$ except $1-1/\tau$ (discussed below) the structure is not
periodic in physical space, but may have periodic diffraction
patterns.
\begin{figure}[hftb]
\begin{center}
\includegraphics[width=0.8\textwidth,
angle=0.0]{Figure5.eps}
\end{center}
\caption{Modified Fibonacci chain for $\kappa=2/3$ ($\epsilon \approx 0.741$, $q_{1} \approx 10.816$).}
\label{fig:ModFib5}
\end{figure}
\begin{figure}[hftb]
\begin{center}
\includegraphics[width=0.8\textwidth,
angle=0.0]{Figure6.eps}
\end{center}
\caption{Modified Fibonacci chain for $\kappa=3/2$ ($\epsilon \approx 0.056$, $q_{1} \approx 11.913$).}
\label{fig:ModFib6}
\end{figure}
For $\epsilon=1 - 1 / \tau$ one gets fully periodic structure
with $\kappa=1$, hence $A=B$. Our deformed Fibonacci chain is then simply a
lattice, and its diffraction pattern is the reciprocal lattice, with
period $k_{0}$. At this special value of $\kappa$, all the other Bragg
peaks have intensity zero.
Finally, it must be noted that the amplitude of
each peak is a continuous
function of $\kappa$. In fact, it is infinitely differentiable.
As $\kappa$ is varied, there is no
phase transition between commensurate and incommensurate diffraction patterns;
the evolution is smooth. As such, with measurement apparatus of fixed accuracy,
it is impossible to determine whether a given pattern is precisely
periodic.
%From the experimental point of view for a given measurement accuracy
%and for a limited range of scattering vector one cannot distinguish
%between periodic and aperiodic diffraction pattern. One has still to
%remember that periodicity in this case does not need to mean that the
%diagram is the lattice of diffraction peaks.
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\end{sloppy}
\end{document}