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Static Case

Recall, (from sections 4.5 and 4.6 in Jackson) that when the electric field penetrates a medium made of bound charges, it polarizes those charges. The charges themselves then produce a field that opposes, and hence by superposition reduces, the applied field. The key assumption in these sections was that the polarization of the medium was a linear function of the total field in the vicinity of the atoms.

Linearity response was easily modelled by assuming a harmonic (linear) restoring force:

\begin{displaymath}
\mbox{\boldmath$F$} = -m\omega_0^2 \mbox{\boldmath$x$}
\end{displaymath} (9.100)

acting to pull a charge $e$ into a new neutral equilibrium in the presence of an electric field $vE$ acting on a presumed charge $e$. The field exerts a force $\mbox{\boldmath$F$}_e = e\mbox{\boldmath$E$}$, so:
\begin{displaymath}
e\mbox{\boldmath$E$} - m\omega_0^2\mbox{\boldmath$x$} = 0
\end{displaymath} (9.101)

is the condition for equilibrium. The dipole moment of this (presumed) molecular system is
\begin{displaymath}
\mbox{\boldmath$p$}_{\rm mol} = e\mbox{\boldmath$x$} = \fra...
...boldmath$E$}=
\gamma_{\rm mol} \epsilon_0 \mbox{\boldmath$E$}
\end{displaymath} (9.102)

where $\gamma_{\rm mol}$ is the ``molecular polarizability'' in suitable units.

Real molecules, of course, have many bound charges, each of which at equilibrium has an approximately linear restoring force with its own natural frequency, so a more general model of molecular polarizability is:

\begin{displaymath}
\gamma_{\rm mol} = \frac{1}{\epsilon_0} \sum_i \frac{e_i^2}{m_i\omega_i^2}.
\end{displaymath} (9.103)

This is for a single molecule. An actual medium consists of $N$ molecules per unit volume. From the linear approximation you obtained an equation for the total polarization (dipole moment per unit volume) of the material:

\begin{displaymath}
\mbox{\boldmath$P$} = N \gamma_{\rm mol} \left(\epsilon_0 \mbox{\boldmath$E$} + \frac{1}{3}\mbox{\boldmath$P$}
\right)
\end{displaymath} (9.104)

(equation 4.68) where the factor of 1/3 comes from averaging the linear response over a ``spherical'' molecule.

This can be put in many forms. For example, using the definition of the (dimensionless) electric susceptibility:

\begin{displaymath}
\mbox{\boldmath$P$} = \epsilon_0 \chi_e \mbox{\boldmath$E$}
\end{displaymath} (9.105)

we find that:
\begin{displaymath}
\chi_e = \frac{N \gamma_{\rm mol}}{1 - \frac{N \gamma_{\rm mol}}{3} }.
\end{displaymath} (9.106)

The susceptibility is one of the most often measured or discussed quantities of physical media in many contexts of physics.

However, as we've just seen, in the context of waves we will most often have occasion to use polarizability in terms of the permittivity of the medium, $\epsilon $. Recall that:

\begin{displaymath}
\mbox{\boldmath$D$}= \epsilon \mbox{\boldmath$E$}= \epsilon...
...box{\boldmath$P$}= \epsilon_0 (1 + \chi_e)\mbox{\boldmath$E$}
\end{displaymath} (9.107)

From this we can easily find $\epsilon $ in term of $\chi_e$:

\begin{displaymath}
\epsilon = \epsilon_0(1 + \chi_e)
\end{displaymath} (9.108)

From a knowledge of $\epsilon $ (in the regime of optical frequencies where $\mu \approx \mu_0$ for many materials of interest) we can easily obtain, e. g. the index of refraction:

\begin{displaymath}
n = \frac{c}{v} = \frac{\sqrt{\mu\epsilon}}{\sqrt{\mu_0\epsi...
... \sqrt{\frac{\epsilon}{\epsilon_0}} \approx \sqrt{1 + \chi_e}
\end{displaymath} (9.109)

or
\begin{displaymath}
n = \sqrt{\frac{1+\frac{2N\gamma_{\rm mol}}{3}}{1-\frac{N
\gamma_{\rm mol}}{3} }}
\end{displaymath} (9.110)

if $N$ and $\gamma_{\rm mol}$ are known or at least approximately computable using the (surprisingly accurate) expression above.

So much for static polarizability of insulators - it is readily understandable in terms of real physics of pushes and pulls, and the semi-quantitative models one uses to understand it work quite well. However, real fields aren't static, and real materials aren't all insulators. So we gotta

  1. Modify the model to make it dynamic.
  2. Evaluate the model (more or less as above, but we'll have to work harder).
  3. Understand what's going on.

Let's get started.


next up previous contents
Next: Dynamic Case Up: Dispersion Previous: Dispersion   Contents
Robert G. Brown 2013-01-04