%% This document created by Scientific Word (R) Version 2.0 \documentclass[12pt,thmsa]{report} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{sw20jrep} %TCIDATA{TCIstyle=Report/report.lat,jrep,report} %TCIDATA{Created=Wed Sep 25 17:10:55 1996} %TCIDATA{LastRevised=Mon Oct 14 15:02:54 1996} %TCIDATA{Language=American English} \input tcilatex \pagestyle{myheadings} \markright{{\sf Scattering at Long Wavelengths}} \addtolength{\topmargin}{-0.4in} \addtolength{\textheight}{1in} \setcounter{secnumdepth}{0} \setcounter{chapter}{6} \thispagestyle{empty} \begin{document} \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 $\varepsilon (\omega ).$ \FRAME{dtbpF}{1.689in}{1.42in}{0pt}{}{}{f09-6-2.eps}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width 1.689in;height 1.42in;depth 0pt;original-width 224.8125pt;original-height 188.6875pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename './F09-6-2.eps';file-properties "XNPEU";}} 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 $\Pi .$ \FRAME{dtbpF}{2.9992in}{1.9995in}{0pt}{}{}{Plot }{\special{language "Scientific Word";type "MAPLEPLOT";width 2.9992in;height 1.9995in;depth 0pt;display "FULL";plot_snapshots TRUE;function \TEXUX{$\dfrac{\sin ^{2}\theta }{1+\cos ^{2}\theta }$};linecolor "black";linestyle 5;linethickness 1;pointstyle "cross";function \TEXUX{$1/2(1+\cos ^{2}\theta )$};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";function \TEXUX{$1$};linecolor "black";linestyle 3;linethickness 1;pointstyle "cross";xmin "0.034333";xmax "3.176443";xviewmin "0.03433";xviewmax "3.176";yviewmin "-0.01959";yviewmax "1.02";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{v57683};var1name \TEXUX{$x$};valid_file "T";tempfilename './DZ9UC903.wmf';tempfile-properties "XP";}} \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 \FRAME{dbphF}{5.0548in}{2.776in}{0pt}{}{}{f09-6-51.eps}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 5.0548in;height 2.776in;depth 0pt;original-width 363.4375pt;original-height 198.75pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename './f09-6-51.eps';file-properties "XNPEU";}} \FRAME{dbphF}{5.0548in}{2.776in}{0pt}{}{}{f09-6-52.eps}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 5.0548in;height 2.776in;depth 0pt;original-width 363.4375pt;original-height 198.75pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename './f09-6-52.eps';file-properties "XNPEU";}} \smallskip\ \newpage \subsection{9.6.6} \quad \thinspace \textbf{a.} The polarization ratio is \[ \Pi =\frac{3\sin ^{2}\theta }{5(1+\cos ^{2}\theta )-8\cos \theta }. \] It is thus similar to that for a dielectric sphere (dashed line), even though the angular dependence of the differential cross section (solid line) is quite different and shows a marked peak in the backward direction, due to the interference between the electric and magnetic terms.\FRAME{dtbpF}{% 2.9992in}{1.9995in}{0pt}{}{}{Plot }{\special{language "Scientific Word";type "MAPLEPLOT";width 2.9992in;height 1.9995in;depth 0pt;display "FULL";plot_snapshots TRUE;function \TEXUX{$\frac{3\sin ^{2}\theta }{5(1+\cos ^{2}\theta )-8\cos \theta }$};linecolor "black";linestyle 5;linethickness 1;pointstyle "cross";function \TEXUX{$(5/8)(1+\cos ^{2}\theta )-\cos \theta $};linecolor "black";linestyle 1;linethickness 1;pointstyle "point";function \TEXUX{$1$};linecolor "black";linestyle 3;linethickness 1;pointstyle "cross";xmin "0.034333";xmax "3.176443";xviewmin "0.03433";xviewmax "3.176";yviewmin "-0.04485";yviewmax "2.295";rangeset"X";phi 45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis \TEXUX{v57683};var1name \TEXUX{$x$};valid_file "T";tempfilename './DZ9U7P01.wmf';tempfile-properties "XP";}} \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 \[ \mathbf{E}_{sc}(\mathbf{x)=f\,}\frac{e^{ik\left| \mathbf{x-x}^{\prime }\right| }}{\left| \mathbf{x-x}^{\prime }\right| }E_{0}(\mathbf{x}^{\prime }). \] \FRAME{dbphF}{5.0548in}{2.1923in}{0pt}{}{}{f09-6-7.eps}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 5.0548in;height 2.1923in;depth 0pt;original-width 363.4375pt;original-height 156.5625pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename './f09-6-7.eps';file-properties "XNPEU";}} \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, \[ \mathbf{E}_{sc}(\mathbf{x)}=\sum_i\frac{e^{ik\left| \mathbf{x-x}_i\right| }}{% \left| \mathbf{x-x}_i\right| }\mathbf{f}_iE_0e^{ik\mathbf{n}_0\times \mathbf{% x}_i}. \] \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 \[ \frac{e^{ik\left| \mathbf{x-x}_{i}\right| }}{\left| \mathbf{x-x}_{i}\right| }% \simeq \frac{e^{ikr}}{r}e^{-ik\mathbf{n\cdot x}_{i}}. \] \FRAME{dbphF}{5.0548in}{2.7899in}{0pt}{}{}{f09-6-8.eps}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 5.0548in;height 2.7899in;depth 0pt;original-width 363.4375pt;original-height 199.75pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename './F09-6-8.eps';file-properties "XNPEU";}} \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}