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$\newline$ ${\bf I.\; INTRODUCTION}$
In X-ray diffraction one analyzes quantities like the unit
scattering power of a sample of N scatterers in a volume $V$. If the
scatterers are distributed with a density $\rho(u),\;u \in V$, then
the total scattering power for the sample is given by
$$I_{N}(s)=\int_{V}P(u)exp(-2\pi is \cdot u)\; du $$ where
$P(u)=\int_{V}\rho(u'+u)\rho(u')du'$ is the autocorrelation of $\rho$;
$I_{N}(s)$ is the measured intensity in the direction $S$, of
radiation diffracted from a beam of wavelength $\lambda$ propagating
in the direction $S_{0}$, with $s=(S-S_{0})/\lambda$ (equation 2.11 in
Guinier).$^{1}$ $I_{N}(s)$ clearly depends on the configuration of the
given sample through $P(u)$, and thus on the separations of scatterers
and the relative frequencies of such separations. Ergodic theory
enables us to relate $I_{N}(s)$ to part of the spectrum of a certain
group of unitary operators acting on a Hilbert space. In this paper
we prove that $$I_{N}(s)ds \cong |V|\;d\!$$
with equality in the limit of an infinite sample, where $E_{s}$ is a
spectral family of projection operators of those unitary operators
acting on a certain Hilbert space, $\tilde{f}$ is an element of this
space, and $ds$ is a volume element of ${\bf R}^3$ (the radial direction
corresponding to the wavelength $\lambda$). In this way ergodic theory demonstrates a link between $I_{N}(s)$
and $E_{s}$.
$\newline$ ${\bf II.\; THE \; THEORY}$
Let $X$ be the set of all ``configurations with hard core
restriction'', that is, the set of all sets of countably many points
in ${\bf R}^{3}$ (points representing the positions of scatterers)
such that the distance between any two points is at least one. Define
a metric on $X$ as follows. Enumerate the countable set of all finite
collections of open balls in ${\bf R}^{3}$ such that for each finite
set: (1) the configuration of their centers satisfies the hard core
condition, (2) all centers have rational coordinates, and (3) all
radii are rational and less than 1/2. Let $B_{c}(\epsilon)$ be the
open ball with center c and radius $\epsilon$, and to each such ball
define the continuous function $f_{c,\epsilon}: {\bf R}^{3}
\rightarrow {\bf R}$ by
\[ f_{c,\epsilon}(u) = \left\{ \begin{array}{ll}
\epsilon - \|u-c\| & \mbox{ if $u \in \! B_{c}(\epsilon)$} \\ 0 &
\mbox{ otherwise} \end{array} \right. \]
where $\|v\|$ denotes the usual euclidean norm in ${\bf R}^{3}$. Let
$C_{n}$ denote the set of open balls in the ${\rm n^{th}}$ collection
in the enumeration above, and let $\delta_{j}(x)$ be the position of
the $j^{th}$ particle of $x$. Define the function $f_{n} : X
\rightarrow {\bf R}$ as follows: $f_{n}(x) = \prod_{B_{c}(\epsilon)
\in C_{n}}\sum_{j}f_{c,\epsilon}(\delta_{j}(x))$. Finally, define the
metric $d\!:\!X \times X \rightarrow {\bf R}$ by $d(x,y)=
sup_{n}|f_{n}(x)-f_{n}(y)|/n$. To prove $X$ is compact in the metric
topology one need only show it has the property that any sequence of
configurations has a subsequence which converges.$^{2}$ This is
straightforward using Cantor diagonalization and the
Bolanzo-Weierstrass theorem applied to bounded cubes, as follows.
$\pagebreak$
$\newline$ ${\bf Lemma:}$ The space $X$ is compact. $\newline$
$Proof:$ Given a configuration $x \in X$ we may enumerate the points
of $x$ as follows. Form a lattice in ${\bf R}^{3}$ from the vectors
$n_{1}(1/\sqrt{3},0,0)$, $n_{2}(0,1/\sqrt{3},0)$,
$n_{3}(0,0,1/\sqrt{3})$, where $n_{1},n_{2},n_{3} \in \bf{Z}$.
Enumerate the corresponding lattice boxes $\{(u_{1},u_{2},u_{3}) \in
{\bf R}^{3}|\,1/\sqrt{3}\,n_{i} \le u_{i} < 1/\sqrt{3} \, (n_{i}+1),
\; i=1,2,3 \}$ so as to spiral outward from the origin. Then in each
copy (translate) of the half-open-half-closed box $[0,1/\sqrt{3})
\times [0,1/\sqrt{3}) \times [0,1/\sqrt{3})$ there exists at most one
point of the configuration $x$ (by hard core condition). Define
$x(j)$ as the position of the particle of $x$ in $j^{th}$ box, $B_{j}$
(let $x(j)=\{\eta\}$ if no such particle exists). Let
$\overline{B}_{j}$ be the closure of $B_{j}$ and assign ${\bf
B}_{j}=\overline{B}_{j}\cup\{\eta\}$ the usual product topology. Let
$x_{i}, \; i\ge 1$, be any sequence of configurations in $X$. Then
$x_{i}(1)$ is a sequence in ${\bf B}_{1}$. By the Bolanzo-Weierstrass
theorem$^{2}$ there exists a convergent subsequence, $x_{i}^{1}$, such
that $x_{i}^{1}(1)\rightarrow x^{1} \!
\in \! {\bf B}_{1} $ as $i \rightarrow \infty$. Again by the
Bolanzo-Weierstrass theorem we may take a subsequence of $x_{i}^{1}$,
call it $x_{i}^{2}$, such that $x_{i}^{2}(j) \rightarrow x^{j} \! \in
{\bf B}_{j}$ for $j=1,2$ as $i \rightarrow \infty$. Continuing in
this way we obtain subsequences $x_{i}^{m}$ such that $x_{i}^{m}(j)
\rightarrow x^{j} \! \in \! {\bf B}_{j}$ for $j=1,...,m$ as $i \rightarrow
\infty$. Taking the diagonal subsequence $x_{m}^{m}$ we have that
$x_{m}^{m}(j) \rightarrow x^{j} \! \in \! {\bf B}_{j}$ for all $j$ as $m \rightarrow \infty$. Note this gives rise to an
allowed configuration $x=\{x^{j}|\, j \ge 1 \}$ (that is, the hard core
condition is satisfied by the collection of points $x^{i},i \ge 1$).
The fact that $x_{m}^{m} \rightarrow x$ in the metric on $X$ follows
simply. Q.E.D.
$\pagebreak$
Consider the group of translations of ${\bf R}^{3}$ acting on $X$
in the usual way: that is, every particle of a given configuration is
translated by $t$; we denote this translation of $x$ by $t(x)$ for all
$ x \in X$, $t \in {\bf R}^{3}$. Since $X$ is compact and $\{t \,|\, t \in
{\bf R}^{3} \}$ is a group of commuting homeomorphisms on $X$, there exists a
Borel probability measure $\mu$ on $X$ invariant under $t$ for all $ t
\in {\bf R}^{3}$ (Markov-Kakutani theorem)$^{3}$. We assume the
action is ergodic, that is, $t(A)=A$ for all $ t \in {\bf R}^{3}$
implies $\mu(A)=0$ or $1$. Let $L=L^{2}(X,\mu )$ be the Hilbert space
of complex valued functions on $X$ which are square integrable with
respect to $\mu$. Define the group of unitary operators
$\{T^{t}\,|\,t \in {\bf R}^{3}
\}$ on $L$ by [$T^{t}(f)](x)=f[t(x)]$. Let $\delta (x) \in
{\bf R}^{3}$ be any point in the configuration $x$ closest to the
origin, and for each $f: {\bf R}^3 \rightarrow {\bf R}$ define
$\tilde{f}(x)=f[-\delta (x)]$. Take $f$ to be any positive, real
valued function on ${\bf R}^{3}$ such that: $supp(f) \subset
B_{0}(\epsilon)$ (that is, $f$=0 outside a ball of radius $\epsilon$
centered at the origin), and $\int_{B_{0}(\epsilon)} f^{2}(s)ds = 1$.
(For example, we may take $f$ to be the constant function
$1/vol[B_{0}(\epsilon)]$ on $B_{0}(\epsilon)$, 0 otherwise.)
Furthermore suppose $\epsilon > 0$ is sufficiently small (e.g.
$\epsilon < 1/2$) such that given any $x \in X$ there exists at most one
point of $x$ contained in $B_{0}(\epsilon)$. It then follows from the
hard core condition that $\tilde{f}(x) =
\sum_{j\ge 1} f[-\delta _{j}(x)]$, where $\delta _{j}(x)$ = position
of $j^{th}$ particle of $x$. Thus \begin{equation} T^{s}\tilde{f}(x)
= \sum_{j\ge 1} f[s-\delta _{j}(x)]. \end{equation} Consider the
quantity:
\begin{equation} = \int_{X} [T^{t}
\tilde{f}(x)][\tilde{f}(x)] d \mu (x) \end{equation}
which by Birkhoff's pointwise ergodic theorem$^{4}$ satisfies, for
$\mu$ - almost every $x \in X$,
\begin{equation}
=\lim_{V \rightarrow {\bf R}^{3}}
\frac{1}{|V|} \int_{V} [T^{t+u} \tilde{f}(x)][T^{u} \tilde{f}(x)] du
\end{equation}
and so,
\begin{equation} = \lim_{V
\rightarrow {\bf R}^{3}} \frac{1}{|V|} \int_{V} \sum_{j,k} f[t+u-\delta
_{j}(x)]f[u-\delta _{j}(x)] du \end{equation} Alternately, from
Naimark's generalization of Stone's theorem$^{5}$ we know that
\begin{equation} =
\int_{{\bf R}^{3}}\exp(i2\pi s\cdot t)\; d\!
\end{equation}
where $E_{s}: L \rightarrow L$ is a spectral family of
operators corresponding to the group of operators $\{T^{t}\, | \, t \in
{\bf R}^{3}\}$. Now if we have N identical scatterers, centered at N points
$\delta_{ j}(x)$ in the configuration $x$ of (4) which are in a volume
$V$, each scatterer represented, not by a delta-function mass (or
charge), but a mass (charge) distributed by a density $f[u-\delta
_{j}(x)]$, they produce a total intensity $I_{N}$ which satisfies
(taking the inverse Fourier transform of equation 2.11 in Guinier)$^{1}$,
\begin{equation} \int_{V} \sum_{j,k}
f[t+u-\delta_{j}(x)]f[u-\delta_{k}(x)] du = \int_{{\bf R}^{3}} \exp(i2\pi s
\cdot t)\; I_{N}(s) ds \end{equation}
Comparing (4), (5) and (6) we have that:
\begin{equation}\lim_{N \rightarrow \infty}
\frac{I_{N}(s)}{N} ds = |V_{0}|\; d\!
\end{equation}
where $|V_{0}| = V/N$ = average volume available to a
scatterer. Alternately we have approximately: \begin{equation}
I_{N}(s)ds \cong |V|\;d\! .\end{equation}
$\newline$ ${\bf III.\; CONCLUDING \; REMARKS}$
It is perhaps useful to mention the observation motivating the
application of ergodic theory to the study of the unit scattering
power. Consideration of the quantity $$ was
central to the above development and arose quite naturally in the
context of averaging. By (4), $$ is related
to the separation of scatterers- as can readily be seen in the case
where $f$ is uniformly supported over the ball of radius $\epsilon$
(that is, $f(u)=1/vol[B_{0}(\epsilon)]$ for $u \in B_{0}(\epsilon)$, 0
otherwise). In this case when $x \in X$ is a lattice (as in the case
of a simple crystal) and $t=\delta_{j}(x)-\delta_{k}(x)$ is a lattice
vector, then $$ is the density of pairs
separated by the vector $t$. If $t^{\prime} \cong t$ (say
$\|t^{\prime}-t\|<2\epsilon)$ then
$$ will be nonzero, so indeed this
quantity is related to the densities of pair separations.
It is hoped that the interplay between the two spectra will be of use to both
crystallographers and ergodic theorists.
$\newline$ ${\bf ACKNOWLEDGEMENT}$
I would like to thank C. Radin and M. Senechal for part of the above analysis.
\pagebreak
$\newline$ ${\bf BIBLIOGRAPHY}$
$\newline$ $^{1}$A. Guinier, {\it X-Ray Diffraction} (Freeman, San Francisco, 1963).
$\newline$ $^{2}$H. L. Royden, {\it Real Analysis} (Macmillan, New York, 1968), 2nd ed..
$\newline$ $^{3}$M. Reed and B. Simon, {\it Methods of Modern Mathematical Physics}
$\newline$ $\phantom{^{1}}$(Academic Press, New York, 1972).
$\newline$ $^{4}$N. Wiener, Duke Math. J., 5, 1 (1939).
$\newline$ $^{5}$F. Riesz and B. Sz.-Nagy, {\it Functional Analysis} (Frederick Ungar, New York,
$\newline$ $\phantom{^{1}}$1955), 2nd ed., section 140, p.392.
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