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\markright{{\sf Scattering at Long Wavelengths}}
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\section{9.6\quad Scattering at Long Wavelengths.}

\subsection{9.6.1}

\quad \thinspace \textbf{a.} A scattering process can be viewed as a
two-step process, where currents are induced in the scatterer by the
incident fields and in turn these currents radiate the scattered fields.

\smallskip\ 

\textbf{b}. The exact solution of a scattering problem involves the
difficult task of finding the induced currents. A great simplification
occurs if these can be computed in a static or quasi-static approximation.
This is a valid procedure if the dimensions of the scatterer are small
compared to the wavelength of the incident radiation, because then the
scattered fields in the near zone are well approximated by the static
fields. Furthermore a multipolar expansion of the radiated fields will be
rapidly convergent, so that often only the induced dipoles must be computed.

\smallskip\ 

\textbf{c.} We will use the above approach in the rest of this section. Here
we define some general terms and concepts.

\smallskip\ 

\textbf{d. }The incident field is conveniently taken to be a monochromatic
plane wave $\mathbf{E}_{inc}=\mathbf{\epsilon }_0E_0\exp (ik\mathbf{n}%
_0\cdot \mathbf{r),}$ $k=\omega /c,$ $\mathbf{B=n}_0\mathbf{\times E}_{inc}.$
The intensity is given by $(c/8\pi )\left| \mathbf{E}\right| ^2,$ both for
the incident and for the scattered waves. The intensity scattered in the
area $r^2d\Omega $ with polarization $\mathbf{\epsilon }$ is 
\[
r^2d\Omega (c/8\pi )\left| \mathbf{E}_{sc}\cdot \mathbf{\epsilon }%
^{*}\right| ^2. 
\]
This intensity, divided by $d\Omega $ and by the incident intensity gives
the differential scattering cross section 
\begin{equation}
\frac{d\sigma }{d\Omega }(\mathbf{n,\epsilon ;n}_0\mathbf{,\epsilon }_0)=%
\frac{r^2\left| \mathbf{\epsilon }^{*}\cdot \mathbf{E}_{sc}\right| ^2}{%
\left| E_0\right| ^2}  \label{1}
\end{equation}

\smallskip\ 

\textbf{e. }Define the (normalized) scattering amplitude $\mathbf{f}$ by $%
(\exp (ikr)/r)\mathbf{f=E}_{sc}/E_{0.}$

\newpage

\subsection{9.6.2}

\quad \thinspace \textbf{a.} To leading order, in the long wave limit, only
the dipole terms are kept and the dipole moments are found by solving a
problem of electrostatics or magnetostatics for $\mathbf{p}$ or $\mathbf{m}$
respectively. The scattered fields (in the radiation zone) are, according to
9.2.1c and 9.3.1c 
\[
\mathbf{E}_{sc}=k^2\frac{e^{ikr}}r\left( \mathbf{p}_{\perp }-\mathbf{n\times
m}\right) \quad \quad \quad \quad \mathbf{B}_{sc}=\mathbf{n\times E}_{sc} 
\]

\smallskip\ 

\textbf{b. }Hence the \textsl{differential cross section} is (recall $%
\mathbf{p}_{\perp }\cdot \mathbf{\epsilon }^{*}=\mathbf{p}\cdot \mathbf{%
\epsilon }^{*})$%
\begin{equation}
\frac{d\sigma }{d\Omega }(\mathbf{n,\epsilon ;n}_0\mathbf{,\epsilon }_0)=%
\frac{k^4}{E_0^2}\left| \mathbf{\epsilon }^{*}\cdot \mathbf{p+(n\times
\epsilon }^{*})\cdot \mathbf{m}\right| ^2  \label{2}
\end{equation}
where $\mathbf{p}$ and $\mathbf{m}$ are proportional to $E_0$ with
proportionality coefficients that are determined by the electric and
magnetic polarizability. If these depend weakly on $\omega ,$ the long wave
scattering cross section is proportional to $k^4,$ or to $\omega ^4.$ This
result was obtained by Rayleigh.

\smallskip\ 

\textbf{c. }As an example, consider the scattering by a \textsl{small
dielectric sphere} of radius $a$, with $\mu =1$ and an isotropic dielectric
constant $\varepsilon (\omega ).$

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If $ka\ll 1,$ the induced dipole can be computed by regarding $\mathbf{E}%
_{inc}$ as a uniform field. The result, from electrostatics, is 
\begin{equation}
\mathbf{p}=\frac{(\varepsilon -1)}{(\varepsilon +2)}a^{3}\mathbf{E}%
_{inc}=\chi ^{el}\mathbf{E}_{inc}.  \label{3}
\end{equation}
The magnetic dipole is of higher order in $ka,$ if the d.c. conductivity of
the material vanishes (see next page).

\smallskip\ 

\textbf{d. }From (\ref{3}) 
\begin{equation}
\frac{d\sigma }{d\Omega }=k^4a^6\left| \frac{\varepsilon -1}{\varepsilon +2}%
\right| ^2\left| \mathbf{\epsilon }^{*}\cdot \mathbf{\epsilon }_0\right| ^2.
\label{4}
\end{equation}
The angular dependence $\left| \mathbf{\epsilon }_0\cdot \mathbf{\epsilon }%
^{*}\right| ^2$ is characteristic of electric dipole scattering by an
isotropic scatterer.

\newpage

\subsection{9.6.3}

\quad \thinspace \textbf{a.} The geometry of the scattering process is
illustrated in the sketches at the end of 9.6.5. The vectors $\mathbf{n}_{0}$
and $\mathbf{n}$ define the scattering plane. If $\mathbf{\epsilon }_{0}$ is
in this plane, $\mathbf{E}_{sc}$ is too; this state of polarization is
denoted as $/\!/$. If $\mathbf{\epsilon }_{0}$ is perpendicular ($\perp )$
to the scattering plane, so is $\mathbf{E}_{sc}.$ ($\perp $ and $/\!/$
correspond respectively to $S$ and $P$ of Sect. 7.3).

\smallskip \ 

\textbf{b. }The cross sections for several polarizations are tabulated below
in terms of the \textsl{forward scattering amplitude} $f_{0}=\mathbf{f\cdot
\epsilon }_{0}^{*}$ at $\theta =0.$ For scattering from a sphere, $%
f_{0}=k^{2}a^{3}\left( \dfrac{\varepsilon -1}{\varepsilon +2}\right) ;$ for
general electric dipole scattering, $f_{0}=k^{2}\chi ^{el}.$

\smallskip \ 

\begin{tabular}{|ccccc|}
\hline
$\text{Incident polarization}$ &  & $\left( \dfrac{d\sigma }{d\Omega }%
\right) _{/\!/}$ &  & $\left( \dfrac{d\sigma }{d\Omega }\right) _{\perp }$
\\ 
&  &  &  &  \\ 
$\perp $ &  & $0$ &  & $\left| f_{0}\right| ^{2}$ \\ 
&  &  &  &  \\ 
$/\!/$ &  & $\left| f_{0}\right| ^{2}\cos ^{2}\theta $ &  & $0$ \\ 
&  &  &  &  \\ 
$\mathbf{\epsilon }_{0}=(\cos \varphi _{0},\sin \varphi _{0})$ &  & $\left|
f_{0}\right| ^{2}\cos ^{2}\theta \cos ^{2}\varphi _{0}$ &  & $\left|
f_{0}\right| ^{2}\sin ^{2}\varphi _{0}$ \\ 
&  &  &  &  \\ 
$\text{unpolarized}$ &  & $\dfrac{1}{2_{\,}}\left| f_{0}\right| ^{2}\cos
^{2}\theta $ &  & $\dfrac{1}{2}\left| f_{0}\right| ^{2}$ \\ \hline
\end{tabular}

\smallskip\ 

\textbf{c. }The \textsl{total} differential cross section is $\left( \dfrac{%
d\sigma }{d\Omega }\right) _{\perp }+\left( \dfrac{d\sigma }{d\Omega }%
\right) _{/\!/}.$ The \textsl{polarization ratio }is defined as 
\[
\Pi =\frac{\left( \dfrac{d\sigma }{d\Omega }\right) _{\perp }-\left( \dfrac{%
d\sigma }{d\Omega }\right) _{/\!/}}{\left( \dfrac{d\sigma }{d\Omega }\right)
_{\perp }+\left( \dfrac{d\sigma }{d\Omega }\right) _{/\!/}} 
\]

\smallskip\ 

\textbf{d. }For unpolarized incident light, 
\begin{equation}
\left( \frac{d\sigma }{d\Omega }\right) _{\perp }+\left( \frac{d\sigma }{%
d\Omega }\right) _{/\!/}=\frac{1}{2}\left| f_{0}\right| ^{2}\left( 1+\cos
^{2}\theta \right) .  \label{tot}
\end{equation}
and 
\begin{equation}
\Pi =\dfrac{\sin ^{2}\theta }{1+\cos ^{2}\theta }  \label{pol}
\end{equation}

\newpage

\subsection{9.6.4}

\quad \thinspace \textbf{a.} The total scattering cross section, integrated
over angles, summed over scattered polarization, averaged over incident
polarization (which corresponds to unpolarized incident light) is 
\[
\sigma =\frac 12\left| f_0\right| ^2\int_0^{2\pi }d\varphi \int_{-1}^1\left(
1+\cos ^2\theta \right) d(\cos \theta )=\frac{8\pi }3\left| f_0\right| ^2=%
\frac{8\pi }3\left| \chi ^{el}\right| ^2k^4. 
\]

\smallskip\ 

\textbf{b. }The behavior of the total differential cross section (\ref{tot})
and of the polarization ratio (\ref{pol}) (dash-dotted line) are shown in
the graph as a function of $\theta ,$ for unpolarized incident radiation.
The solid line gives $d\sigma /d\omega $ in units of $\left| f_{0}\right|
^{2},$ the dash-dotted line gives $\Pi .$

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\smallskip\ 

\textbf{c. }The \textsl{blue} color \textsl{of the sky} is due to
dipole-scattered light: the blue end of the solar spectrum is scattered more
effectively because $f_{0}$ is proportional to $\omega ^{2}.$ The light
scattered at $90^{\circ }$ is totally of $\perp $ polarization; thus the sky
in this direction looks black when viewed with polaroid glasses worn
``normally''. Tilting the head restores the blue color.

\smallskip\ 

\textbf{d.} Because the blue end of the spectrum is scattered out
preferentially, the direct light from the sun is shifted to the red,
especially when the sun is low on the horizon.

\newpage

\subsection{9.6.5}

\quad \thinspace \textbf{a.} We consider next the scattering from a \textsl{%
small} perfectly \textsl{conducting sphere}. In this case there is an
induced electric dipole moment $\mathbf{p}=a^3\mathbf{E}_{inc}$ and also a
magnetic dipole moment $\mathbf{m=-(}1/2)a^3\mathbf{B}_{inc}.$

\smallskip\ 

\textbf{b. }While $\mathbf{p}$ is easy to understand and is consistent with
the dielectric sphere result (\ref{3}) for $\varepsilon \rightarrow \infty ,$
the origin of $\mathbf{m}$ is a little more subtle. We have to assume that $%
\delta \ll a\ll \lambda ,$ where $\delta $ is the skin depth and $\lambda $
the wavelength. Then $\mathbf{m}$ can be found by solving a magnetostatic
problem with the boundary condition $\mathbf{B}_{\perp }=0$ on the surface
of the sphere, which amounts to taking $\mu =0$ in the sphere. The formal
analogy between electro- and magneto-statics assures that the result can be
obtained from (\ref{3}) with $\mathbf{p\rightarrow m,}$ $\varepsilon
\rightarrow \mu (=0)$ and $\mathbf{E}_{inc}\rightarrow \mathbf{H}%
_{inc}(\equiv \mathbf{B}_{inc}).$

\smallskip\ 

\textbf{c. }We now have 
\[
\frac{d\sigma }{d\Omega }(\mathbf{n,\epsilon ;n}_{0}\mathbf{,\epsilon }%
_{0})=k^{4}a^{6}\left| \mathbf{\epsilon }^{*}\cdot \mathbf{\epsilon }_{0}-%
\frac{1}{2}(\mathbf{n\times \epsilon }^{*})\cdot (\mathbf{n}_{0}\mathbf{%
\times \epsilon }_{0})\right| ^{2}\,. 
\]

\smallskip \ 

\textbf{d. }It is still true that $/\!/$ or $\perp $ incident polarizations
give only $/\!/$ or $\perp $ scattered radiation. See the sketches on next
page.

\smallskip \ 

\smallskip \ \textbf{e. }For unpolarized incident radiation (compare 3b for
the factor $1/2)$%
\[
\frac{d\sigma _{/\!/}}{d\Omega }=\frac{k^{4}a^{6}}{2}\left| \cos \theta -%
\frac{1}{2}\right| ^{2}\quad \quad \quad \frac{d\sigma _{\perp }}{d\Omega }=%
\frac{k^{4}a^{6}}{2}\left| 1-\frac{1}{2}\cos \theta \right| ^{2} 
\]
The sum of the two gives the total differential cross section 
\[
\frac{d\sigma }{d\Omega }=k^{4}a^{6}\left[ \dfrac{5}{8}(1+\cos ^{2}\theta
)-\cos \theta \right] 
\]

\newpage

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\smallskip\ 

\newpage

\subsection{9.6.6}

\quad \thinspace \textbf{a.} The polarization ratio is 
\[
\Pi =\frac{3\sin ^{2}\theta }{5(1+\cos ^{2}\theta )-8\cos \theta }.
\]
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\smallskip\ 

\textbf{b. }It is worth stressing that in this case the electric and
magnetic dipole terms are comparable. This should be contrasted with the
case of atoms, where the magnetic dipole term (for equally allowed
transitions) is down by a factor $(e^2/\hbar c)^2$ from the electric dipole
term. This factor is of the same order of magnitude as $(kd)^2$ for $%
d=a_0=\hbar ^2/me^2$ and makes the magnetic dipole term comparable in
magnitude to the electric quadrupole. A hydrogen atom, for instance, has the
same electric polarizability as a perfectly conducting sphere of radius $%
a_{0,}$ but negligible induced magnetic dipole moment.

\smallskip\ 

\textbf{c. }In nuclei, where the protons are the charge carriers, the
situation is somewhat more complex: here the magnetic dipole terms should be
half-way (on a logarithmic scale) between electric dipole and quadrupole,
based on size estimates; in fact the electric dipole is often sharply
suppressed (Jackson, p. 762).

\newpage

\subsection{9.6.7}

\quad \thinspace \textbf{a. }The scattered field (in the radiation region)
is related to the incident field strength $E_{0},$ quite generally, by 
\[
\mathbf{E}_{sc}(\mathbf{x)=f\,}\frac{e^{ik\left| \mathbf{x-x}^{\prime
}\right| }}{\left| \mathbf{x-x}^{\prime }\right| }E_{0}(\mathbf{x}^{\prime
}). 
\]
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\smallskip\ 

\textbf{b. }The vector \textsl{scattering amplitude} $\mathbf{f}$ depends on
the scattering angles and on the incident polarization $\mathbf{\epsilon }%
_{0}.$ $E_{0}(\mathbf{x}^{\prime })\mathbf{\epsilon }_{0}$ is the incident
field at the position of the scatterer; for a plane incident wave $E_{0}(%
\mathbf{x}^{\prime })=E_{0}\exp (ik\mathbf{n}_{0}\cdot \mathbf{x}^{\prime
}). $

\smallskip\ 

\textbf{c. }For electric dipole scattering $E_{0}(\mathbf{x}^{\prime })%
\mathbf{f}=k^{2}\mathbf{p}_{\perp }$ (9.2.1 c) where $\mathbf{p}$ is the
induced dipole. If the scatterer has isotropic polarizability $\chi ^{el},$ $%
\mathbf{p}=\chi ^{el}\mathbf{\epsilon }_{0}E_{0}(\mathbf{x}^{\prime })$ and $%
\mathbf{f}=\chi ^{el}\mathbf{\epsilon }_{0\perp }k^{2}.$ $\left( \mathbf{%
\epsilon }_{0\perp }=\mathbf{\epsilon }_{0}-(\mathbf{n\cdot \epsilon }_{0})%
\mathbf{n}\right) \mathbf{.}$ See 9.6.2 for an example.

\smallskip\ 

\textbf{d.} For magnetic dipole scattering, $E_{0}(\mathbf{x}^{\prime })%
\mathbf{f=-}k^{2}\mathbf{n\times m}$ (9.3.2a). If the scatterer has
isotropic susceptibility, $\mathbf{m}=\chi ^{mag}\mathbf{B}_{inc}=\chi ^{mag}%
\mathbf{n}_{0}\times \mathbf{\epsilon }_{0}E_{0}(\mathbf{x}^{\prime })%
\mathbf{\ }$and $\mathbf{f=}-k^{2}\chi ^{mag}\mathbf{n\times (n}_{0}\times 
\mathbf{\epsilon }_{0}).$

\smallskip\ 

\textbf{e.} The electric and magnetic scattering amplitudes are added
vectorially as in (9.6.2a).

\smallskip\ 

\textbf{f. }The scalar product $\mathbf{\epsilon }^{*}\cdot \mathbf{f}$%
\textbf{\ }gives the \textsl{scalar} scattering amplitude for a process
where the final polarization is $\mathbf{\epsilon }.$ The differential cross
section for this process is $\left| \mathbf{\epsilon }^{*}\cdot \mathbf{f}%
\right| ^{2}.$

\newpage

\subsection{9.6.8}

\quad \thinspace \textbf{a.} The scattering of a collection of small
scatterers can be computed simply by adding the fields of each scatterer. As
long as the total scattering is weak (no shadowing of one scatterer by
another) we have simply, according to 2a, 
\[
\mathbf{E}_{sc}(\mathbf{x)}=\sum_{i}\frac{e^{ik\left| \mathbf{x-x}%
_{i}\right| }}{\left| \mathbf{x-x}_{i}\right| }\mathbf{f}_{i}E_{0}e^{ik%
\mathbf{n}_{0}\cdot \mathbf{x}_{i}}.
\]

\smallskip\ 

\textbf{b. }This formula holds as long as the distance from each scatterer
is large compared to $\lambda ,$ or $k\left| \mathbf{x-x}_{i}\right| \gg 1.$
At distances $r$ larger than the dimensions of the region where the
scattering occurs ($r\gg D,$ see sketch), we can set 
\[
\frac{e^{ik\left| \mathbf{x-x}_{i}\right| }}{\left| \mathbf{x-x}_{i}\right| }%
\simeq \frac{e^{ikr}}{r}e^{-ik\mathbf{n\cdot x}_{i}}. 
\]

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\smallskip\ 

\textbf{c.} Using this, one finds for the cross section of the entire system 
\[
\frac{d\sigma }{d\Omega }=\frac{r^2\left| \mathbf{E}_{sc}\cdot \mathbf{%
\epsilon }^{*}\right| ^2}{E_0^2}=\left| \sum_i(\mathbf{f}_i\cdot \mathbf{%
\epsilon }^{*})e^{-i\mathbf{q\cdot x}_i}\right| ^2. 
\]
where $\mathbf{q}=k\mathbf{n-}k\mathbf{n}_0$ is the change in wave vector ($%
\hbar \mathbf{q}$ is the momentum transfer for the photons involved).

\newpage

\subsection{9.6.9}

\quad \thinspace \textbf{a. }If all the scatterers are identical the result
is simply expressible in terms of the cross section for a single scatterer: 
\[
\frac{d\sigma }{d\Omega }=\left| \mathbf{f}\cdot \mathbf{\epsilon }%
^{*}\right| ^{2}S(\mathbf{q),} 
\]
where $S(\mathbf{q)}$ is the \textsl{structure factor} 
\[
S(\mathbf{q)}=\left| \sum_{i}e^{-i\mathbf{q\cdot x}_{i}}\right|
^{2}=\sum_{ii^{\prime }}e^{i\mathbf{q\cdot (x}_{i}-\mathbf{x}_{i^{^{\prime
}}})}. 
\]
\smallskip

\smallskip \ 

\textbf{b.} This separation of $d\sigma /d\Omega $ into a single-scatterer
cross section and a structure factor is not limited to the case when the
single scattering is described in the dipole approximation with isotropic
polarizabilities, nor is it limited to the scattering of electromagnetic
waves. Rather, it is entirely general (as long as multiple scattering can be
neglected: one does not really have to assume that the total scattering is
weak, because the attenuation of the incoming beam can be accounted for in a
rather simple way).

\smallskip\ 

\textbf{c. }Two limiting cases help to understand the great variety of
possible physical situations

\quad - if the scatterers form a large, regular array, such as a crystalline
solid, the structure factor consists of discrete spots having angular width $%
\lambda /L,$ where $\lambda $ is the wavelength and $L$ the size of the
array.

\quad - if the scatterers are arranged at random, only the terms with $%
i=i^{\prime }$ survive in the sum defining $S(\mathbf{q),}$ so that $%
S=N_{tot}$ (the number of scatterers). In this case the scattering is said
to be \textsl{incoherent} (totally so). As always, an incoherent
superposition of amplitudes results in a sum of amplitudes square, or
probabilities (if the radiation is regarded as a stream of photons).

\smallskip\ 

\textbf{d. }We take up the subject of incoherent and partially coherent
scattering in the next section. Here we remark that the superposition of
fields in the forward direction is always coherent, or in other words $S(%
\mathbf{q)}$ always equals $N_{tot}^{2}$ for $\mathbf{q}=0.$ Of course, in
the forward direction one has also the incident field interfering with the
scattered fields. We return to this important point in Sect. 9.14

\end{document}
