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\markright{{\sf Scattering by Inhomogeneities in a Medium}}
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\begin{document}


\section{9.7\quad Scattering by Inhomogeneities in a Medium.}

\subsection{9.7.1}

\quad \thinspace \textbf{a. }We start this section by considering wave
propagation in a continuous medium with $\varepsilon $ and $\mu $ varying
weakly from point to point; we then develop the Rayleigh and
Einstein-Smoluchowski theories of scattering by density fluctuations both
from the macroscopic (variable $\varepsilon $) and the microscopic
(distribution of scatterers) point of view. The results are simply related,
since the macroscopic equations are based on a microscopic model. The role
of the structure factor is emphasized and the more general Ornstein-Zernicke
theory is outlined. The connection between scattering theory and dielectric
(linear response) theory in wave propagation is taken up again in Sect. 9.14
with a discussion of the optical theorem.

\smallskip\ 

\textbf{b}. Proceeding macroscopically, we write Maxwell's equations in a
way suitable for an expansion in the quantities $\mathbf{D-}\varepsilon _{0}%
\mathbf{E}$ and $\mathbf{B-}\mu _{0}\mathbf{H,}$ which are supposed to be
small if the true $\varepsilon $ and $\mu $ deviate little from the uniform
values $\varepsilon _{0}$ and $\mu _{0}$. We have, away from sources (where $%
\mathbf{\nabla }\cdot \mathbf{D=0}$) 
\[
\mathbf{\nabla \times \nabla \times (D-}\varepsilon _{0}\mathbf{E)=-}\nabla
^{2}\mathbf{D-}\varepsilon _{0}\mathbf{\nabla \times \nabla \times E} 
\]
and using $\mathbf{\nabla \times E=-}\dfrac{1}{c}\dfrac{\partial \mathbf{B}}{%
\partial t}$ 
\[
\mathbf{\nabla \times \nabla \times (D-}\varepsilon _{0}\mathbf{E)=-}\nabla
^{2}\mathbf{D+}\frac{\varepsilon _{0}}{c}\mathbf{\nabla \times }\frac{%
\partial \mathbf{B}}{\partial t} 
\]
Similarly, using $\mathbf{\nabla \times H}=\dfrac{1}{c}\dfrac{\partial 
\mathbf{D}}{\partial t}$%
\[
\frac{\varepsilon _{0}}{c}\frac{\partial }{\partial t}\mathbf{\nabla \times
(B-}\mu _{0}\mathbf{H)}=\frac{\varepsilon _{0}}{c}\frac{\partial }{\partial t%
}\mathbf{\nabla \times B-}\frac{\varepsilon _{0}\mu _{0}}{c^{2}}\frac{%
\partial ^{2}\mathbf{D}}{\partial t^{2}} 
\]
Subtracting side by side and rearranging we obtain an equation that is still
exact (but not very useful without making approximations): 
\[
\nabla ^{2}\mathbf{D-}\frac{\varepsilon _{0}\mu _{0}}{c^{2}}\frac{\partial
^{2}\mathbf{D}}{\partial t^{2}}=-\mathbf{\nabla \times \nabla \times (D-}%
\varepsilon _{0}\mathbf{E)+}\frac{\varepsilon _{0}}{c}\frac{\partial }{%
\partial t}\mathbf{\nabla \times (B-}\mu _{0}\mathbf{H).} 
\]

\newpage

\subsection{9.7.2}

\quad \thinspace \textbf{a.} Assuming that $\varepsilon _{0}$ and $\mu _{0}$
are time-independent and Fourier-analyzing everything, the equation for $%
\mathbf{D}$ becomes 
\begin{equation}
(\nabla ^{2}+k^{2})\mathbf{D=-\nabla \times \nabla \times (D-}\varepsilon
_{0}\mathbf{E)-}\frac{i\varepsilon _{0}\omega }{c}\mathbf{\nabla \times (B-}%
\mu _{0}\mathbf{H),}  \label{1}
\end{equation}
with $k^{2}=\mu _{0}\varepsilon _{0}\omega ^{2}/c^{2}.$ This can be turned
into an integral equation in the standard way, using 
\[
(\nabla ^{2}+k^{2})\exp (ikr)/r=-4\pi \delta (\mathbf{x)} 
\]
(with $r=\left| \mathbf{x}\right| $). We obtain, in cartesian components,

\[
\mathbf{D(x})\mathbf{=D}^{0}(\mathbf{x})-\frac{1}{4\pi }\int d^{3}x^{\prime }%
\frac{e^{ik\left| \mathbf{x-x}^{\prime }\right| }}{\left| \mathbf{x-x}%
^{\prime }\right| }\text{\{R.H.S. of (\ref{1}) evaluated at }\mathbf{x}%
^{\prime }\text{\}}
\]
where $\mathbf{D}^{0}$ is the incident field, solution of $(\nabla
^{2}+k^{2})\mathbf{D}^{0}=0$. (We must use cartesian components because $%
\exp (ikr)/r$ is the Green function of the wave equation for a scalar and
also for each cartesian component of the wave equation for a vector.)

We can take 
\[
\mathbf{D}^{0}(\mathbf{x})=D^{0}\mathbf{\varepsilon }^{0}\exp (ik\mathbf{n}%
^{0}\cdot \mathbf{x)} 
\]
and then we also have $\mathbf{B}^{0}=\mathbf{n}^{0}\times \mathbf{E}^{0}$

\smallskip\ 

\textbf{b. }For $r\gg r^{\prime }$, we can set 
\[
\frac{e^{ik\left| \mathbf{x-x}^{\prime }\right| }}{\left| \mathbf{x-x}%
^{\prime }\right| }\simeq \frac{e^{ikr}}{r}e^{-ik\mathbf{n\cdot x}^{\prime
}} 
\]
in the usual way. Then, in the radiation region, $\mathbf{D=D}^{0}+\mathbf{f}%
D^{0}\dfrac{e^{ikr}}{r}$ where $\mathbf{f}$ is the scattering amplitude: 
\[
\mathbf{f}D^{0}=\frac{1}{4\pi }\int d^{3}x^{\prime }e^{-ik\mathbf{n\cdot x}%
^{\prime }}\left[ \mathbf{\nabla }^{\prime }\mathbf{\times \nabla }^{\prime }%
\mathbf{\times (D-}\varepsilon _{0}\mathbf{E)+}\frac{i\varepsilon _{0}\omega 
}{c}\mathbf{\nabla }^{\prime }\mathbf{\times (B-}\mu _{0}\mathbf{H)}\right]
. 
\]
We can use integration by parts to replace $\mathbf{\nabla }^{\prime }%
\mathbf{\times }$ by $ik\mathbf{n}.$%
\[
\mathbf{f}D^{0}=\frac{k^{2}}{4\pi }\int d^{3}x^{\prime }e^{-ik\mathbf{n\cdot
x}^{\prime }}\left\{ -\mathbf{n\times }\left[ \mathbf{n\times (D-}%
\varepsilon _{0}\mathbf{E)}\right] \mathbf{-}\frac{\varepsilon _{0}\omega }{%
ck}\,\mathbf{n\times (B-}\mu _{0}\mathbf{H)}\right\} . 
\]

\smallskip\ 

\textbf{c. }The differential scattering cross section, for final
polarization $\mathbf{\epsilon }$ is just 
\[
\frac{d\sigma }{d\Omega }=\left| \mathbf{\epsilon }^{*}\cdot \mathbf{f}%
\right| ^{2}. 
\]

\newpage

\subsection{9.7.3}

\quad \thinspace \textbf{a.} If $\varepsilon =\varepsilon _{0}+\delta
\varepsilon ,$ with $\delta \varepsilon \ll \varepsilon _{0},$ we can
replace $\mathbf{D-}\varepsilon _{0}\mathbf{E}=\delta \varepsilon \,\mathbf{E%
}$ by $\delta \varepsilon \,\mathbf{E}^{0}=\left( \delta \varepsilon
/\varepsilon _{0}\right) \mathbf{D}^{0}$, to lowest order (Born
approximation). Similarly, $\mathbf{B-}\mu _{0}\mathbf{H}\simeq (\delta \mu
/\mu _{0})\mathbf{B}^{0}=(\delta \mu /\mu _{0})\,\mathbf{n}_{0}\mathbf{%
\times E}^{0}.$ The scalar scattering amplitude for incident polarization $%
\mathbf{\epsilon }_{0}$ and scattered polarization $\mathbf{\epsilon }$, for
inhomogeneities $\delta \varepsilon $ and $\delta \mu $ in the properties of
the medium, is then 
\begin{equation}
\mathbf{\epsilon }^{*}\cdot \mathbf{f}=\frac{k^{2}}{4\pi }\int
d^{3}x^{\prime }e^{i\mathbf{q\cdot x}^{\prime }}\left\{ \mathbf{\epsilon }%
^{*}\cdot \mathbf{\epsilon }_{0}\frac{\delta \varepsilon (\mathbf{x}^{\prime
})}{\varepsilon _{0}}+(\mathbf{n\times \epsilon }^{*})\cdot (\mathbf{n}_{0}%
\mathbf{\times \epsilon }_{0})\frac{\delta \mu (\mathbf{x}^{\prime })}{\mu
_{0}}\right\} .  \label{2}
\end{equation}
The differential cross section is $d\sigma /d\Omega =\left| \mathbf{\epsilon 
}^{*}\cdot \mathbf{f}\right| ^{2}.$ Here $\mathbf{q}=k(\mathbf{n-n}_{0})$ as
usual, with $\mathbf{D}^{0}(\mathbf{x}^{\prime })=\mathbf{\epsilon }%
_{0}D_{0}\exp \left( ik\mathbf{n}_{0}\cdot \mathbf{x}^{\prime }\right) .$

\smallskip\ 

\textbf{b. }As an example, we consider again a sphere of radius $a$ having
dielectric constant $\varepsilon $, surrounded by vacuum ($\varepsilon
_0=1). $ An elementary integration in (\ref{2}) gives 
\begin{equation}
\mathbf{\epsilon }^{*}\cdot \mathbf{f}=k^2(\varepsilon -1)(\mathbf{\epsilon }%
^{*}\cdot \mathbf{\epsilon }_0)\left[ \frac{\sin qa-qa\cos qa}{q^3}\right] .
\label{3}
\end{equation}

\smallskip\ 

\textbf{c. }In the limit $q\rightarrow 0,$ we recover $\mathbf{f}=k^{2}a^{3}%
\dfrac{(\varepsilon -1)}{3}\,\mathbf{\epsilon }_{0\perp },$ which
corresponds to electric dipole scattering with an induced dipole moment $%
\mathbf{p}$ given by $\mathbf{f}E_{0}=k^{2}\mathbf{p}_{\perp },$ according
to 9.6.7d. The induced moment corresponds to a polarizability $\left( \dfrac{%
\varepsilon -1}{3}\right) a^{3},$ in agreement with the exact low-frequency
result $\left( \dfrac{\varepsilon -1}{\varepsilon +2}\right) a^{3}$ of
9.6.2c, for $\left| \varepsilon -1\right| \ll 1.$

\smallskip\ 

\textbf{d.} The same result as in b, is obtained by considering the sphere,
from the outset, as a collection of scatterers with polarizability $\chi
^{el},$ induced dipole moment $\chi ^{el}\mathbf{\epsilon }_{0}E_{0},$ and
scattering amplitude $k^{2}\chi ^{el}\mathbf{\epsilon }^{*}\cdot \mathbf{%
\epsilon }_{0}$ for final polarization $\mathbf{\epsilon .}$ The scattering
amplitude of the entire collection is, according to 9.6.8c, 
\begin{equation}
(\mathbf{\epsilon }^{*}\cdot \mathbf{\epsilon }_{0})\chi
^{el}k^{2}\sum_{i}e^{-i\mathbf{q\cdot x}_{i}}.  \label{4}
\end{equation}
Replacing $\sum_{i}$ by $N\int d^{3}x,$ where $N$ is the density of
scatterers, gives the desired result with the identification 
\[
\varepsilon =1+4\pi N\chi ^{el}.
\]

\newpage

\subsection{9.7.4}

\quad \thinspace \textbf{a.} We shall come back again and again to the
relation between scattering theory and dielectric response and to the role
of the structure factor. In the previous example, the discrete nature of the
scattering centers was averaged out by replacing $\sum_i$ with $N\int d^3%
\mathbf{x}.$ Next, we emphasize, by contrast, that the discreteness gives
rise to observable effects even if the scatterers are randomly distributed.

\smallskip\ 

\textbf{b. }The fact is that when we form the structure factor $S(\mathbf{q)}%
=\sum_{ii^{\prime }}e^{i\mathbf{q\cdot (x}_i-\mathbf{x}_{i^{\prime }})}$
(see 9.6.8d), the terms with $i=i^{\prime }$ always contribute $N_{tot},$
the total number of scatterers. All the other terms give a negligible
contribution, except near the forward direction, if the assembly is large
enough and the scatterers are distributed at random. This can be seen from
equation (\ref{3}) of 3b in the case of a spherical assembly: the
``coherent'' part of $S(\mathbf{q)}$ is 
\[
S_{coh}=(4\pi N)^2\left[ \frac{\sin qa-qa\cos qa}{q^3}\right] ^2 
\]
(compare also 3.d). This is exactly $(NV)^2,$ where $V=4\pi a^3/3$ is the
volume, for $q\rightarrow 0,$ but it becomes $\sim (4\pi Na/q^2)^2$ for
large $qa$: thus it is seen to be a surface effect, proportional to the
surface area $4\pi a^2.$ By contrast, the incoherent contribution is just $%
N_{tot}=NV$ at all angles and thus is dominant away from the forward
direction, for large $a.$

\smallskip\ 

\textbf{c. }The incoherent scattering is then 
\[
\frac{d\sigma }{d\Omega }=k^{4}\left| \chi ^{el}\right| ^{2}\left| \mathbf{%
\epsilon }^{*}\cdot \mathbf{\epsilon }_{0}\right| ^{2} 
\]
per molecule. Using $\varepsilon -1=4\pi N\chi ^{el},$ this can be expressed
in terms of the macroscopic quantity $\varepsilon =n^{2}$ ($n$ is the
refractive index). Integration over angles replaces $\left| \mathbf{\epsilon 
}^{*}\cdot \mathbf{\epsilon }_{0}\right| ^{2}$ by $(8\pi /3)$ for natural
(unpolarized) incident light and all scattered polarizations summed over
(see 9.6.4a). The total cross section per molecule is then 
\[
\sigma =k^{4}\frac{\left| \varepsilon -1\right| ^{2}}{6\pi N^{2}}. 
\]

\newpage

\subsection{9.7.5}

\quad \thinspace \textbf{a.} If $\alpha V$ is the total scattering cross
section of an assembly of scatterers in the volume $V,$ the flux lost by a
beam of intensity $I$ and area $A$ in traversing a thickness $dx$ is $\alpha
IAdx=-AdI.$ Therefore $dI/dx=-\alpha I,$ $I(x)=I_0\exp (-\alpha x)$ and $%
\alpha $ has the meaning of an attenuation coefficient (improperly called
absorption coefficient in a case like this when no absorption is taking
place; another good name is \textsl{extinction coefficient}).

\smallskip\ 

\textbf{b. }From the theory of 4.c, the extinction coefficient, or cross
section per unit volume, for natural light traversing a medium with $N$
identical scatterers per unit volume, is 
\begin{equation}
\alpha =N\sigma =\frac{k^4}{6\pi N}\left| \varepsilon -1\right| ^2\simeq 
\frac{2k^4}{3\pi N}\left| n-1\right| ^2  \label{5}
\end{equation}

\smallskip\ 

\textbf{c. }Formula (\ref{5}) is due to Rayleigh and explains a variety of
atmospheric phenomena (see Jackson). The salient features are:

\quad - the $k^4,$ or $\omega ^4,$ dependence, typical of dipole scattering.

\quad - the dependence on the atomic structure of matter through the factor $%
N^{-1}.$ In fact (\ref{5}) provided one of the first estimates of Avogadro's
number.

\quad - $\alpha ^{-1}$ is of order 100 km for visible (blue) light in air.

\smallskip\ 

\textbf{d. }If the scatterers are not distributed quite at random, as is the
case in a liquid, one can still obtain a general result in the limit $%
q\rightarrow 0$ (to be defined more fully in next sentence). Following
Einstein and Smoluchowski, the scattering in this limit can be attributed to
fluctuations in the dielectric constant of the medium on a scale $\xi $
small compared to the wavelength of light (hence $q\xi \rightarrow 0),$ but
still large compared to interatomic spacings. A brief account of this theory
is given next.

\newpage

\subsection{9.7.6}

\quad \thinspace \textbf{a.} If there are variations $\delta N$ in the
density of the medium, the dielectric properties can be described by a
locally varying function $\varepsilon +\delta \varepsilon (\mathbf{x),}$
with $\delta \varepsilon =(\partial \varepsilon /\partial N)\delta N(\mathbf{%
x)}$. Application of (\ref{2}) gives a scattering amplitude 
\[
\mathbf{\epsilon }^{*}\cdot \mathbf{f}=\frac{k^2}{4\pi }\int d^3\mathbf{x}%
e^{i\mathbf{q\cdot x}}\left( \mathbf{\epsilon }^{*}\cdot \mathbf{\epsilon }%
_0\right) \frac 1\varepsilon \frac{\partial \varepsilon }{\partial N}\delta
N(\mathbf{x)} 
\]
and a differential scattering cross section 
\begin{equation}
\frac{d\sigma }{d\Omega }=\frac{k^4}{16\pi ^2}\left| \frac 1\varepsilon 
\frac{\partial \varepsilon }{\partial N}\right| ^2\left| \mathbf{\epsilon }%
^{*}\cdot \mathbf{\epsilon }_0\right| ^2\int \delta N(\mathbf{x)}\delta N(%
\mathbf{x}^{\prime }\mathbf{)}e^{i\mathbf{q\cdot (x-x}^{\prime
})}d^3xd^3x^{\prime }.  \label{6}
\end{equation}

\smallskip\ 

\textbf{b. }Before proceeding to evaluate the integral in (\ref{6}), we
remark that it is closely related to the structure factor $S(\mathbf{q)}$ of
9.6.8d in the limit $\mathbf{q}\rightarrow 0$ (but not quite $=0),$ if one
is only interested in the part proportional to $V$ (see 9.7.4b). For small $%
q $ one can replace sums over particles by integrals over particle
densities, so that 
\[
S(\mathbf{q)=}\sum_{ii^{\prime }}e^{i\mathbf{q\cdot (x}_i\mathbf{-x}%
_{i^{^{\prime }}})}\rightarrow \int \left( N+\delta N(\mathbf{x)}\right)
\left( N+\delta N(\mathbf{x}^{\prime }\mathbf{)}\right) e^{i\mathbf{q\cdot
(x-x}^{\prime })}d^3xd^3x^{\prime }. 
\]
Now the term $N^2$ contributes only to the forward scattering ($\mathbf{q}%
=0),$ or else gives a surface contribution; the terms $N\delta N$ average to
zero over a sufficiently large volume $V,$ or as a function of time; the
remaining term gives the integral in (\ref{6}), which must also be
statistically averaged.

\smallskip\ 

\textbf{c. }Einstein pointed out that the average can be found by
thermodynamics. One must suppose that the correlation of density
fluctuations extends over a length $\xi $ with $\xi ^3=v\ll V$ and $\xi q\ll
1.$ Then the integral in (\ref{6}) reduces to $Vv\left\langle \delta
N^2\right\rangle ,$ where $\delta N$ is the fluctuation of particle density
in volume $v.$

\newpage

\subsection{9.7.7}

\quad \thinspace \textbf{a. }We shall derive, for completeness, the result $%
\left\langle \delta N^2\right\rangle =(N^2/v)kT\beta _T$ where $\beta _T$ is
the isothermal compressibility.

\smallskip\ 

\textbf{b. }If $F(v,T)$ is the free energy of a \textsl{fixed} number of
particles in the \textsl{variable} volume $v,$ the probability that the
actual volume be $v+\delta v$ is $\sim $ $\exp [-F(v+\delta v,T)/k_BT],$ at
temperature $T.$ Expanding $F(v+\delta v,T)=F(v)+(\partial F/\partial
v)_T\delta v+(1/2)\partial ^2F/\partial v^2\left( \delta v\right) ^2,$ the
probability distribution is seen to be Gaussian. Recalling that $%
P=-(\partial F/\partial v)_T$ and $\beta _T=-(1/v)(\partial v/\partial P)_T,$
we find at once $\left\langle \delta v^2\right\rangle =v\beta _Tk_BT.$

\smallskip\ 

\textbf{c.} Now we must find the fluctuation of the density of particles in
a \textsl{fixed} volume. We simply look at a fixed number of particles $Nv.$
Then $\delta (Nv)=N\delta v+v\delta N=0$ and $\left\langle \delta
N^2\right\rangle =(N^2/v^2)\left\langle \delta v^2\right\rangle =(N^2/v)$ $%
\beta _Tk_BT.$

\smallskip\ 

\textbf{d. }From this and 6.c it follows that (\ref{6}) gives (see bottom of
9.4) 
\begin{equation}
\alpha =\frac{k^4}{6\pi }\left| \frac 1\varepsilon \frac{\partial
\varepsilon }{\partial N}\right| ^2N^2k_BT\beta _T  \label{7}
\end{equation}

\smallskip\ 

\textbf{e.} For a dilute gas, we can write $\varepsilon =1+4\pi \alpha
^{el}N,$ so that $\partial \varepsilon /\partial N=4\pi \alpha
^{el}=(\varepsilon -1)/N.$ Further, $Pv=$ const., so that $v\partial
P/\partial v=-P=-Nk_BT.$ It follows that $Nk_BT\beta _T=1$ and $\alpha
=(k^4/64N)\left| \varepsilon -1\right| ^2,$ in agreement with the Rayleigh
result (\ref{5}).

\textbf{f.} For a dense gas or a liquid, not only $Nk_BT\beta _T\neq 1$ but
we must also include local-field effects. Using the Clausius-Mossotti
equation, 
\[
\varepsilon =\frac{1+4\pi N\chi ^{el}}{1-\dfrac{4\pi }3N\chi ^{el}}, 
\]
in the form 
\[
\frac{4\pi }3N\chi ^{el}=\frac{\varepsilon -1}{\varepsilon +2}, 
\]
we find 
\[
\frac{\partial \varepsilon }{\partial N}=\frac 1N\frac{\left( \varepsilon
-1\right) \left( \varepsilon +2\right) }3, 
\]
and plugging back in (\ref{7}) 
\begin{equation}
\alpha =\frac{(\omega /c)^4}{6\pi N}\left| \frac{\left( \varepsilon
-1\right) \left( \varepsilon +2\right) }3\right| ^2Nk_BT\beta _T.  \label{7'}
\end{equation}

\newpage

\subsection{9.7.8}

\quad \thinspace \textbf{a.} The essence of Einstein's argument is that one
can make the replacement $(1/V)S(\mathbf{q)}\rightarrow N^2k_BT\beta _T$ for 
$q\xi \rightarrow 0.$ The argument fails near a critical point or a
second-order phase transition, where $\beta _T$ and $\xi $ both diverge.

\smallskip\ 

\textbf{b. }Ornstein and Zernicke built a more general theory by keeping
also terms due to $\mathbf{\nabla }N(\mathbf{x)}$ in the expansion of the
free energy. The relevant distribution of density fluctuations in the entire
volume $V$ is given by 
\[
\exp \left[ -\frac 1{k_BT}\frac 12\frac{\partial ^2F}{\partial N^2}\int
d^3xd^3x^{\prime }\left\{ \delta N(\mathbf{x)}\delta N(\mathbf{x}^{\prime }%
\mathbf{)+}\xi ^2\mathbf{\nabla }N(\mathbf{x)\cdot \nabla }N(\mathbf{x}%
^{\prime }\mathbf{)}\right\} \right] 
\]
where $\xi $ is an unknown quantity that will be identified with the
correlation length (see \textbf{e}.). According to thermodynamics (or from
7.c), $\partial ^2F/\partial N^2=V/\left( N^2\beta _T\right) .$

\smallskip\ 

\textbf{c.} Next one introduces the fourier transform, $\delta N_{\mathbf{q}%
}=V^{-1}\int d^3x\exp (-i\mathbf{q\cdot x)}\delta N(\mathbf{x),}$ of the
fluctuation $\delta N(\mathbf{x)}$ and notes that $S(\mathbf{q)}%
\Longrightarrow \left\langle \left| \delta N_{\mathbf{q}}\right|
^2\right\rangle V^2.$ According to \textbf{b}., the distribution of $\left|
\delta N_{\mathbf{q}}\right| ^2$ is given by 
\[
\exp \left[ -\frac 12\frac V{N^2k_BT\beta _T}\left( 1+\xi ^2q^2\right)
\left| \delta N_{\mathbf{q}}\right| ^2\right] 
\]
and therefore 
\begin{equation}
\frac 1VS(\mathbf{q)=}V\left\langle \left| \delta N_{\mathbf{q}}\right|
^2\right\rangle =\frac{N^2k_BT\beta _T}{1+\xi ^2q^2}.  \label{8}
\end{equation}

\smallskip\ 

\textbf{d. }The net outcome is that Einstein's result for $d\sigma /d\Omega $
must be divided by $\left( 1+\xi ^2q^2\right) =\left[ 1+2k^2\xi ^2(1+\cos
\theta )\right] .$ Thus near a critical point ($\xi \rightarrow \infty )$
the frequency dependence of $\alpha $ is $\omega ^2$ (instead of $\omega
^4), $ i.e. the \textsl{critical opalescence }looks bluish-white.

\smallskip\ 

\textbf{e.} The physical meaning of $\xi $ is made clear by the
density-density correlation function 
\[
\left\langle \delta N(\mathbf{x)}\delta N(\mathbf{x}^{\prime }\mathbf{)}%
\right\rangle =V\int \frac{d^{3}q}{(2\pi )^{3}}e^{i\mathbf{q\cdot (x-x}%
^{\prime })}\left\langle \left| \delta N_{\mathbf{q}}\right|
^{2}\right\rangle =\frac{e^{-R/\xi }}{4\pi R\xi ^{2}}N^{2}k_{B}T\beta _{T}. 
\]
where $R=\left| \mathbf{x-x}^{\prime }\right| .$

\newpage

\subsection{9.7.9}

\quad \thinspace \textbf{a.} There is much more to light scattering in bulk
matter than can be covered here. Fluctuations in the concentration of
mixtures, in the orientation of anisotropic molecules, in temperature and in
any quantity one can think of, all will contribute to light scattering.

\smallskip\ 

\textbf{b. }All light scattering with unshifted frequency is often called
Rayleigh scattering. It is also possible for photons to lose (or gain)
energy in the scattering process, so that the light will be
frequency-shifted (Brillouin scattering, Raman scattering, etc.).

\smallskip\ 

\textbf{c. }All Rayleigh-type scattering is essentially described by the
Fourier transform of the appropriate correlation function. (This is the
structure factor for density fluctuations). If a frequency shift $\omega
^{\prime }-\omega $ occurs, one must take the time-fourier transform of the
time-dependent correlations. Thus for density fluctuations one introduces 
\[
S(\mathbf{q,\,\,}\omega ^{\prime }-\omega )=\int d^3xd^3x^{\prime
}dt\left\langle \delta N(\mathbf{x,}t\mathbf{)}\delta N(\mathbf{x}^{\prime
},0\mathbf{)}\right\rangle e^{-i\mathbf{q(x-x}^{\prime })}e^{i(\omega
^{\prime }-\omega )t}. 
\]

\smallskip\ 

\textbf{d. }This formalism applies not only to the scattering from gases and
condensed matter, but also to the scattering from nuclei (as collections of
nucleons and pions) and, tentatively, to the scattering from subnuclear
particles (as collections of partons, maybe quarks).

\smallskip\ 

\textbf{e.} The connection between density correlations and compressibility
generalizes to a connection between the thermal fluctuations of any quantity 
$x$ and the response of the system to an external stimulus that couples to $x
$, since both are related to $\partial ^{2}F/\partial x^{2},$ where $F$ is
the appropriate free energy.

\end{document}
