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%TCIDATA{Created=Fri Aug 30 08:56:12 1996}
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\markright{{\sf Frequency dependence of dielectric functions}}
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\begin{document}


\section{7.5\quad Dispersion Characteristics of Dielectrics, Conductors and
Plasmas.}

\subsection{7.5.1}

\quad \ \textbf{a.} In a dispersive medium, $\varepsilon $ and $\mu $ are
functions of the angular frequency $\omega .$ Hence $n=\sqrt{\varepsilon \mu 
}$ is also a function of $\omega .$ For simplicity, we assume that $\mu =1$
to a good approximation and concentrate on $\varepsilon (\omega ),$ but much
of the discussion holds for $\mu (\omega )$ and $n(\omega )$ as well.

\smallskip\ 

\textbf{b. }The important results will be:

(i) $\varepsilon (\omega )$ is in general complex and is an analytic
function of $\omega $ for $\func{Im}$ $(\omega )>0$ at thermal equilibrium;

(ii) $\func{Im}$ $\varepsilon $ can be identified with $4\pi \sigma /\omega
, $ where $\sigma $ is the real, frequency dependent conductivity, which is
positive for a medium in thermal equilibrium. $\func{Im}$ $\varepsilon $ $%
\rightarrow $ $0$ for $\omega \rightarrow \infty .$

(iii) Re $\varepsilon \geq 1$ for $\omega =0$ and $\func{Re}$ $\varepsilon
\rightarrow 1$ for $\omega \rightarrow \infty ,$ but in general $\func{Re}$ $%
\varepsilon $ is not necessarily positive. The conditions $\func{Re}$ $%
\varepsilon <0$ (for $\omega $ large enough), $\func{Im}\varepsilon \ll
\left| \func{Re}\varepsilon \right| $ are characteristic of a plasma.

(iv) The fundamental solutions for the fields in a medium are proportional
to $\exp (i(\mathbf{k\cdot x-}\omega t)),$ same as in vacuum, but $\mathbf{k}
$ is in general complex and its modulus is given by $k=n\omega /c$ (or $%
k^{2}=\varepsilon \omega ^{2}/c^{2}$ for $\mu =1$).

\smallskip\ 

\textbf{c. }Two further generalizations are sometimes needed:

(i) If the medium is \textsl{anisotropic}, the relation between $\mathbf{D}$
and $\mathbf{E}$ is still linear, but $\mathbf{D}$ is not always in the same
direction as $\mathbf{E.}$ The general relation is $D_p=$ $\sum_q$ $%
\varepsilon _{pq}E_q,$ where $\varepsilon _{pq}$ is the dielectric tensor.
For an isotropic medium, $\varepsilon _{pq}=$ $\varepsilon \delta _{pq.}$

(ii) If the field varies rapidly compared to some characteristic length of
the medium, the relation between $\mathbf{D}$ and $\mathbf{E}$ can be
non-local: for an isotropic medium, 
\[
\mathbf{D(r)=}\int d^3r^{\prime }\varepsilon (\mathbf{r,r}^{\prime })\mathbf{%
E(r}^{\prime }). 
\]
When the relation is local, $\varepsilon (\mathbf{r,r}^{\prime
})=\varepsilon \delta (\mathbf{r-r}^{\prime }).$ More generally, for a
homogeneous medium, $\varepsilon (\mathbf{r,r}^{\prime })$ depends only on $%
\mathbf{r-r}^{\prime }.$

\newpage

\subsection{7.5.2}

\quad \ \textbf{a. }The polarizability of a molecule in a uniform $\mathbf{E}
$ field of frequency $\omega $ is defined by $e\left\langle
X_{p}\right\rangle $ $=\sum_{q}\chi _{pq}\mathbf{E}_{p},$ where $X_{p}$ (for 
$p=1,2,3$) is the sum of all coordinates $x_{np}$ of the electrons in the
molecule. The general quantum formula for $\chi _{pq}$ is 
\begin{equation}
\chi _{pq}=-\frac{e^{2}}{\hbar }\sum_{ij}\rho _{i}\left[ \frac{\left\langle
i\mid X_{p}\mid j\right\rangle \left\langle j\mid x_{q}\mid i\right\rangle }{%
\omega -\omega _{ij}+i\eta /2}-\frac{\left\langle i\mid X_{q}\mid
j\right\rangle \left\langle j\mid x_{p}\mid i\right\rangle }{\omega +\omega
_{ij}+i\eta /2}\right] ,  \label{1}
\end{equation}
where $\rho _{i}$ is the occupation probability of the initial state $\left|
i\right\rangle ,$ $\hbar \omega _{ij}=E_{j}\,\,-$ $E_{i}$ is a transition
energy and $\eta $ $\rightarrow 0^{+}.$

\smallskip\ 

\textbf{b.} If the initial distribution of atomic states $\left|
i\right\rangle $ is isotropic, as it usually is in the absence of static
external fields, $\chi _{pq}$ $=\chi \delta _{pq}$ and (\ref{2}) below holds
for any component of $\mathbf{x}$\textbf{.}

\smallskip\ 

\textbf{c.} More generally, $\chi _{pq}$ may be reduced to a diagonal form
by a change to new axes (see for instance 7.6.2b); if $\mathbf{x}$\textbf{\ }%
is in the direction of one such axis, the corresponding diagonal element of
the polarizability tensor is 
\begin{equation}
\chi =\frac{e^2}m\sum_{ij}\rho _i\frac{f_{ij}}{\omega _{ij}^2-\omega
^2-i\omega \eta },  \label{2}
\end{equation}
where $f_{ij}$ is the \textsl{oscillator strength}: $f_{ij}=\omega
_{ij}\left| \left\langle i\mid x\mid j\right\rangle \right| ^22m/\hbar .$

\smallskip\ 

\textbf{d.} The \textsl{dipole sum rule} holds: 
\[
\sum_jf_{ij}=Z, 
\]
where $Z$ is the number of electrons in the molecule. In the optical range,
the deep-lying electrons do not contribute appreciably to the
polarizability; then one has an approximate partial sum rule, where $Z$ is
the number of ``polarizable'' electrons (typically, these are the
``valence'' electrons).

\newpage

\subsection{7.5.3}

\quad \ \textbf{a.} The imaginary part of $\chi $ arises when $\omega $ is
resonant with some $\omega _{ij}.$ Explicitly 
\begin{equation}
\omega \func{Im}\chi =\frac{e^2}m\sum_{ij}\rho _if_{ij}\pi \left[ \delta
(\omega -\omega _{ij})-\delta (\omega +\omega _{ij})\right] .  \label{3}
\end{equation}
Note that $\func{Im}\chi (-\omega )=-\func{Im}\chi (\omega )$. In fact, $%
\chi (-\omega )=$ $\chi ^{*}(\omega )$.

\smallskip\ 

\textbf{b.} If the system is initially in the ground state, only the term $%
i=0$ remains, with $\rho _0=1.$ In this case the oscillator strengths $%
f_{0j} $ and the frequencies $\omega _{0j}$ are non-negative and $\omega 
\func{Im}\chi (\omega )$ $\geq 0.$

\smallskip\ 

\textbf{c.} It can be shown that $\omega \func{Im}\chi $ $\geq $ $0$ is also
true if $\rho _i$ is the thermal equilibrium distribution. However if the
system has been ``pumped'' to an excited state, as in a laser, $\omega \func{%
Im}\chi (\omega )$ can be negative. Generally, $\omega \func{Im}\chi $ $<0$
means that the system is unstable with respect to the perturbation caused by
an electric field of frequency $\omega .$

\smallskip\ 

\textbf{d.} If there are $N$ non-interacting molecules per unit volume, the
dielectric constant is given by (writing $\chi ^{mol}$ instead of just $\chi 
$): 
\begin{equation}
\varepsilon =1+4\pi N\chi ^{mol},  \label{4}
\end{equation}
provided that local field effects can be neglected.

\smallskip\ 

\textbf{e.} In truth, the interactions between molecules cause the states $%
\left| j\right\rangle $ to have finite lifetimes. Even if these interactions
are negligible, the excited states can decay back to $\left| 0\right\rangle $
or to some lower state by emitting a photon, and thus have a finite lifetime
and an associated width (natural width or radiative width).

\newpage

\subsection{7.5.4}

\quad \ \textbf{a.} The treatment of finite lifetime effects is rather
involved and depends on the system under consideration. A simple way out is
suggested by the formal identity of (\ref{2}) with the response of a set of
harmonic oscillators (classical or quantum mechanical) of frequency $\omega
_{ij}$ and infinitesimal damping $\eta $: finite lifetime effects can be
included by giving each effective harmonic oscillator a finite damping
constant $\gamma _{ij}>0.$

\smallskip\ 

\textbf{b.} Taking for simplicity a system in its ground state, so that $%
\rho _0=1$ and all other $\rho _i$'s vanish, we have then 
\begin{equation}
\varepsilon =1+4\pi N\frac{e^2}m\sum_j\frac{f_j}{\omega _j^2-\omega
^2-i\omega \gamma _j}  \label{5}
\end{equation}
with $f_j=$ $f_{0j}$ , $\omega _j=\omega _{0j}$ and $\gamma _j=\gamma _{0j}$
. It is still true that $\sum_jf_j=Z.$

\smallskip\ 

\textbf{c.} If the frequencies $\omega _j$ form a continuum, or if they are
spread apart by less than $\gamma _j,$ the precise value of $\gamma _j$ does
not matter as long as $\gamma _j$ $\ll $ $\omega .$ For $\omega $ near an
isolated frequency $\omega _j$ however, the behavior of $\func{Re}%
\varepsilon $ and $\func{Im}\varepsilon $ depends strongly on $\gamma _j$ as
shown in the figures, although the area under the peak of $\func{Im}%
\varepsilon ,$ or more precisely $\int \omega \func{Im}\varepsilon $ $%
d\omega ,$ does not depends on $\gamma _j.$ (See end of this section for
figures).

\smallskip\ 

\textbf{d.} For interacting molecules, as in a metal or a covalent solid, $%
\chi ^{mol}$ is meaningless. One writes then $\varepsilon =1+4\pi \chi
^{tot},$ where the (total) electric susceptibility $\chi ^{tot}$ is still
given by formulae (\ref{1}) and (\ref{2}) of 7.5.2. Now however $\left|
i\right\rangle $ are eigenstates of a unit volume of the material. For an
assembly of $N$ molecules, $\chi ^{tot}$ $=N\chi ^{mol}$ and (\ref{4}) is
recovered.

\newpage

\subsection{7.5.5}

\quad \ \textbf{a.} It is remarkable that the simple formula (\ref{5})
describes also a system containing ``free'' electrons, as are found in an
ionized plasma. It also describes ``itinerant'' electrons, as are found in
the conduction band of a metal or semiconductor. (Itinerant electrons behave
effectively as free electrons with an effective mass $m^{*}.$) In these
cases the lowest frequency $\omega _j$ in the sum is set equal to zero. It
is also customary to write the corresponding product $Nf/m$ in the form $%
N_c/m^{*}$, where $N_c$ is the number of conducting electrons per unit
volume and $m^{*}$ is their effective mass; one also puts $\gamma =1/\tau $,
where the \textsl{relaxation time} $\tau $ is related to the mean velocity $%
v $ and to the mean free path $l$ by $v=l/\tau .$ With these notations (\ref
{5}) becomes 
\begin{equation}
\varepsilon (\omega )=\varepsilon _b(\omega )+i\frac{4\pi N_ce^2}{%
m^{*}\omega }\frac \tau {1-i\omega \tau }\,\,,  \label{6}
\end{equation}
where $\varepsilon _b$ is the contribution of the ``localized'' or ``bound''
electrons.

\smallskip \ 

\textbf{b.} It is perhaps more natural to describe the conducting electrons
by a ``free current'' term in the Maxwell-Amp\`{e}re equation 
\begin{equation}
\mathbf{\nabla \times H}=\frac{4\pi }{c}\mathbf{J+}\frac{1}{c}\frac{\partial 
\mathbf{D}}{\partial t}  \label{7}
\end{equation}
with $\mathbf{J}$ related to $\mathbf{E}$ by Ohm's law, $\mathbf{J}=\sigma 
\mathbf{E,}$ while the ``bound'' electrons are described by a polarization $%
\mathbf{P}=\chi _{b}\mathbf{E,}$ leading to $\mathbf{D=(}1+4\pi \chi _{b})%
\mathbf{E}=\varepsilon _{b}\mathbf{E}$. Thus 
\begin{equation}
\mathbf{\nabla \times H}=-\frac{i\omega }{c}\left( \varepsilon _{b}+i\frac{%
4\pi \sigma }{\omega }\right) \mathbf{E}  \label{8}
\end{equation}

\smallskip \ 

\textbf{c.} Comparing (\ref{8}) and (\ref{6}) we find that the conductivity $%
\sigma $ is given by \textsl{Drude's formula}

\begin{equation}
\sigma =\frac{N_ce^2}{m^{*}}\frac \tau {1-i\omega \tau }.  \label{9}
\end{equation}
There can be several groups of electrons (or holes), each with its $m^{*}$
and $\tau .$

\newpage

\subsection{7.5.6}

\quad \ \textbf{a.} Drude's formula is obtained more directly by considering
the effect of $\mathbf{E}$ on $N_c$ particles of charge $e$ and mass $m^{*}$
in the presence of a frictional force: the velocity $\mathbf{v}$ of each
particle satisfies $m^{*}\left[ (d\mathbf{v}/dt)+\mathbf{(v/}\tau )\right] =e%
\mathbf{E}$ and the current is $\mathbf{J}=N_ce\mathbf{v.}$ The most refined
quantum statistical treatments yield a formula of the same type and in
addition relate $\tau $ (which may depend, usually weakly, on $\omega $) to
the microscopic properties of the medium and to the temperature.

\smallskip \ 

\textbf{b.} The separation between ``free'' and ``bound'' electrons is
neither physically clearcut in general, nor mathematically convenient. Since
a polarization $\mathbf{P}$ gives rise to a current $\mathbf{J=}$ $d\mathbf{P%
}/dt=-i\omega \mathbf{P,}$ a single \textsl{complex} function, either $\chi
^{tot}(\omega )$ or $\sigma (\omega ),$ describes the response of the
system. Quite simply, $\sigma =-i\omega \chi $. (From here on $\chi $ means $%
\chi ^{tot}).$

\smallskip \ 

\textbf{c.} If the response of the system is purely resistive (current in
phase with the field, leading to power dissipation) then $\sigma $ is real
and is the more natural function to use; if the response is purely reactive, 
$\chi $ is real and $\varepsilon =1+4\pi \chi $ is the natural choice.

\smallskip \ 

\textbf{d.} If the d.c. conductivity $N_ce^2\tau /m^{*}$ (see(\ref{9})) is
finite, $\chi $ and $\varepsilon $ diverge as $\omega ^{-1}$ as $\omega $ $%
\rightarrow 0$ (see(\ref{6})). Clearly, $\sigma $ is more convenient to use
in this limit. It is also true that $\sigma $ $\rightarrow 0$ for $\omega
\rightarrow \infty ,$ while $\varepsilon $ $\rightarrow 1.$ These properties
of $\sigma $ make it easier to write dispersion relations (see 7.10). Note
also that the stability condition $\omega \func{Im}\chi >0,$ or $\omega 
\func{Im}\varepsilon >0,$ translates into $\func{Re}\sigma >0.$ The meaning
of this is clear since the power loss is given by $\frac 12$\thinspace $%
\func{Re}\sigma \left| \mathbf{E}\right| ^2.$

\newpage

\subsection{7.5.7}

\quad \ \textbf{a.} For $\omega $ larger than all the frequencies $\omega _j$
corresponding to significant oscillator strengths $f_j$, the dielectric
constant (\ref{5}) takes the form 
\begin{equation}
\varepsilon =1-\frac{\omega _p^2}{\omega ^2},  \label{10}
\end{equation}
where $\omega _p$ is the \textsl{plasma frequency}: $\omega _p^2=4\pi
NZe^2/m.$

\smallskip \ 

\textbf{b.} In an ionized plasma, the formula 
\begin{equation}
\varepsilon =1-\frac{4\pi N_ce^2}{m\omega }\frac 1{\omega -i/\tau }
\label{10a}
\end{equation}
holds down to frequencies of order $\tau ^{-1}$ . In the laboratory, $N_c$
is $10^{12}$ $-$ $10^{16}$ $/$cm$^3$ and $\omega _p=6\times 10^{10}$ $-$ $%
6\times 10^{12}$ sec$^{-1},$ while $\tau ^{-1}$ $\simeq $ $10^8$ $-10^{10}$
sec$^{-1}$\thinspace

\smallskip \ 

\textbf{c.} In a metal, $N_c$ is about $10^{23}$ cm$^{-3}$, so that $4\pi
N_ce^2/m^{*}$ is about $10^{15}$ sec$^{-1}$, while $\tau ^{-1}$ is typically 
$10^{12}$\thinspace sec$^{-1}$\thinspace . Thus, over the optical range and
the u. v.

\begin{equation}
\varepsilon (\omega )\simeq \varepsilon _b(\omega )-\frac{4\pi N_ce^2}{%
m^{*}\omega ^2}.  \label{11}
\end{equation}
This holds also for a doped semiconductor, where $N_c$ is much lower (down
to $10^{16}$\thinspace cm$^{-3}$).

\smallskip \ 

\textbf{d.} The striking feature of (\ref{10}) and (\ref{10a}) is that $%
\varepsilon (\omega )$ becomes negative below a cutoff frequency, which for (%
\ref{10}) is $\omega _{p}$ and for (\ref{10a}) is $\left[ 4\pi
N_{c}e^{2}/m^{*}\varepsilon _{b}\right] ^{1/2}$ $=$ $\omega _{p}^{*}.$

\smallskip \ 

\textbf{e.} Negative $\varepsilon $ implies that $n=\sqrt{\varepsilon }$ is
purely imaginary, and so is $k=n\omega /c.$ Then waves do not propagate far
in the medium, but decay exponentially, according to 
\[
E=E_0\exp (-\omega _p\xi /c-i\omega t),\quad \text{for}\,\,\,\omega \ll
\omega _p. 
\]

\newpage

\subsection{7.5.8}

\quad \ \textbf{a.} In general, a complex $\varepsilon $ implies a complex
index of refraction $n=\sqrt{\varepsilon }.$ Following Jackson, we will use $%
n$ for the real part of $\sqrt{\varepsilon }$ and introduce the \textsl{%
attenuation constant }$\alpha $\textsl{\ }by 
\begin{equation}
k=\sqrt{\varepsilon }\,\frac \omega c=n\frac \omega c+i\frac \alpha 2,
\label{12}
\end{equation}
or 
\begin{equation}
n^2-\left( \alpha c/2\omega \right) ^2=\func{Re}\varepsilon ,  \label{12a}
\end{equation}
\begin{equation}
n\alpha =\left( \omega /c\right) \func{Im}\varepsilon .  \label{12b}
\end{equation}

\smallskip

\textbf{b.} Three limiting cases are of interest:

(i) $\varepsilon $ nearly real and positive: then $n\simeq \sqrt{\func{Re}%
\varepsilon }$ and $\alpha \simeq \left( \omega /nc\right) \func{Im}%
\varepsilon $; $k$ is nearly real;

(ii) $\varepsilon $ nearly real and negative: then $\alpha \simeq \left(
2\omega /c\right) \sqrt{-\func{Re}\varepsilon }$ and $n=\func{Im}\varepsilon 
$ $/2\sqrt{-\func{Re}\varepsilon };$ $k$ is nearly imaginary;

(iii) $\varepsilon $ nearly imaginary: then $n(\omega /c)\simeq $ $\alpha /2$
and $k$ is of the form $\left( \alpha /2\right) \left( 1+i\right) .$ One
often sets $\alpha /2=\delta ^{-1}.$\thinspace

\smallskip \ 

\textbf{c.} Case (i) corresponds to ``ordinary'' dielectrics and case (ii)
corresponds to plasmas and metals for $\omega <\omega _{p}$ and $\omega \tau
>1.$ Case (iii) corresponds to good conductors at low frequencies ($\omega
\tau <1$) and will be taken up again in 7.7.\thinspace

\smallskip \ 

\textbf{d.} (Contrary to Jackson) it is \textsl{not} correct in general to
say that $\alpha $ is an absorption coefficient. Absorption is proportional
to $\func{Im}\varepsilon ,$ not to the imaginary part of $\sqrt{\varepsilon }%
.$ In particular a negative $\varepsilon $ (case (ii), plasmas) leads to a
large $\alpha ,$ but no absorption. The waves decay exponentially because
the medium is unsuitable for their propagation, not because they are
absorbed. Only in case (i) can $\alpha $ be called properly an absorption
coefficient. See the figures at the end for plots of $n$ and $\alpha $ in a
simple case.

\newpage

\subsection{7.5.9}

\quad \ \textbf{a.} Jackson discusses in detail, as an example, the values
of $n$ and $\alpha $ for liquid water over the astounding frequency range $%
10^{2}$ $-$ $10^{22}$ sec$^{-1}.$ The most remarkable feature of the
spectrum is a sharp transparency window in the visible region: below it are
absorption lines due to vibrational and rotational transitions, above it are
absorption lines due to electronic transitions, which are broadened into
bands by intermolecular interactions and by admixture with rotational and
vibrational excitations.\thinspace

\smallskip \ 

\textbf{b.} In the low frequency region the absorption is dominated by a
phenomenon that we have neglected so far: \textsl{ionic conductivity}. It is
described by a formula identical to (\ref{9}), where $N_c$ and $m^{*}$
should be replaced by the ion concentration $N_i$ and the effective ion mass 
$M.$ Typically $N_i\ll $ $N_c$ and $M$ is $10^4$ times $m^{*},$ so that the
ionic term in the conductivity (and in $\varepsilon $) is important only
when all the electrons are effectively bound\textbf{.}\thinspace

\smallskip \ 

\textbf{c.} At wavelengths less than a few \AA ngstroms (frequencies greater
than $10^{18}$ sec$^{-1}$) the concepts of dielectric constant and index of
refraction do not make sense, because they are based on the assumption that
the spatial variation of the field is negligible on the scale of molecular
dimensions. It is possible to generalize the theory by introducing a
non-local response function as hinted in 7.5.1 c(ii). Physically, one can
think of the phenomenon of x-ray diffraction in crystals, where the
wavelength (or $\mathbf{k}$) is changed by interaction with the atoms of the
lattice. One can still define a total absorption coefficient $\alpha $ by
measuring the decrease in the total intensity along the path of a beam. For $%
\hbar \omega >1$ keV, intraatomic transitions determine $\alpha .$

\newpage

Here are various plots for the simple one-resonance dielectric constant 
\[
\varepsilon =1+\frac{\omega _p^2}{\omega _{res}^2-\omega ^2-i\omega \gamma }%
=1+\frac{z^2}{1-x^2-ixy} 
\]
For convenience, we measure frequencies in units of $\omega _{res}$ and put $%
z=\omega _p/\omega _{res}$ and $y=\gamma /\omega _{res}.$ The solid line is $%
\func{Re}\varepsilon $ and the dotted line is $\func{Im}\varepsilon ,$ for $%
z=1$ and $y=0.1:$

\FRAME{dtbpF}{9.6212cm}{5.039cm}{0pt}{}{}{Plot }{\special{language
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\TEXUX{$x$};valid_file "T";tempfilename
'D:/SWP25/DOCS/DWVW3G50.wmf';tempfile-properties "XP";}}Here are the
corresponding plots of $n=\func{Re}\sqrt{\varepsilon }$ and $\func{Im}\sqrt{%
\varepsilon }$ \FRAME{dtbpF}{9.6212cm}{5.0391cm}{0pt}{}{}{Plot }{%
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\TEXUX{$x$};valid_file "T";tempfilename
'D:/SWP25/DOCS/DWVW54XH.wmf';tempfile-properties "XP";}}and the real and
imaginary parts of $kc/\omega _{res}=\left( n\omega +i\alpha c/2\right)
/\omega _{res}=x\sqrt{\varepsilon }$\FRAME{dtbpF}{9.6212cm}{5.0391cm}{0pt}{}{%
}{Plot }{\special{language "Scientific Word";type "MAPLEPLOT";width
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TRUE;function \TEXUX{$\func{Re}\left( x\sqrt{\left(
1+\frac{1^{2}}{1-x^{2}-ix/10}\right) }\right) $};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";function \TEXUX{$\func{Im}\left(
x\sqrt{\left( 1+\frac{1^{2}}{1-x^{2}-ix/10}\right) }\right) $};linecolor
"black";linestyle 2;linethickness 1;pointstyle "point";xmin "0";xmax
"2.5";xviewmin "0";xviewmax "2.5";yviewmin "-0.05301";yviewmax
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"boxed";xis \TEXUX{x};var1name \TEXUX{$x$};valid_file "T";tempfilename
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\end{document}
