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Dynamic Case

The obvious generalization of the static model for the polarization is to assume a damped linear response to a harmonic (plane wave) driving electric field. That is, every molecule will be viewed as a collection of damped, driven (charged) harmonic oscillators. Magnetic and non-linear effects will be neglected. This is valid for a variety of materials subjected to ``weak'' harmonic EM fields11.10 which in practice (with optical frequencies) means nearly everything but laser light.

The equation of motion11.11 for a single damped, driven harmonically bound charged electron is:

$\displaystyle m\left [ \ddot{\Vec{x}} + \gamma \dot{\Vec{x}} + \omega_0^2 \Vec{x} \right ] = -e E(\Vec{x},t)$ (11.59)

where $ \gamma$ is the damping constant (so $ -m \gamma \dot{\Vec{x}}$ is the velocity dependent damping force). If we assume that the electric field $ \Vec{E}$ and $ \Vec{x}$ are harmonic in time at frequency $ \omega$ (or fourier transform the equation and find its solution for a single fourier component) and neglect the transients we get: \Vec{p} = -e\Vec{x} = e^2m \Vec{E} _&omega#omega; (&omega#omega;_0^2 - &omega#omega;^2 - i &omega#omega;&gamma#gamma;) for each electron11.12.

Actually, we have $ N$ molecules/unit volume each with $ Z$ electrons where $ f_i$ of them have frequencies and damping constants $ \omega_i$ and $ \gamma_i$ , respectively (whew!) then (since we will stick in the definitions $ \Vec{P}_\omega = \epsilon_0 \chi_e \Vec{E}_\omega$ and $ \epsilon = 1 + \chi_e$ )

$\displaystyle \epsilon(\omega) = \epsilon_0\left (1 + \frac{N e^2}{m} \sum_i \frac{f_i}{(\omega_i^2 - \omega^2 - i \omega \gamma_i)}\right)$ (11.60)

where the oscillator strengths satisfy the sum rule:

$\displaystyle \sum_i f_i = Z .$ (11.61)

These equations (within suitable approximations) are valid for quantum theories, and indeed, since quantum oscillators have certain discrete frequencies, they seem to ``naturally'' be quantum mechanical.


next up previous contents
Next: Things to Note Up: Dispersion Previous: Static Case   Contents
Robert G. Brown 2017-07-11