Information
来自 Wolfgang Kettler 的 AMO 课中提到
Model
原子由一个固定不动正电荷, 和一个质量为 \(m\) , 带电为 \(q\) 的负电荷组成, 二者由一个劲度系数 \(C\) 的弹簧连接. 原子处于外加驱动电场 \(E(t) = e^{-\mathrm{i}\omega t}\) 中, 运动时受到与速度成正比的阻尼 \(m\gamma \dot{x}\), 可列出牛顿第二定律的运动方程
\begin{align} qE(t) - Cx(t) - m\gamma \dot{x}(t) = m \ddot{x}(t) \end{align}记
\begin{align} \omega_0 = \sqrt{\frac{C}{m}} \end{align}上式解得
\begin{align} x(t) = \frac{q E(t)}{m} \frac{1}{\omega_0^2 - \omega^2 - \mathrm{i}\omega\gamma} \end{align}是一个 Lorentz shape 的形式.
Dipole moment
\begin{align}
p = q x = \frac{q^{2} E(t)}{m} \frac{1}{\omega_0^2 - \omega^2 - \mathrm{i}\omega\gamma}
\end{align}
Polarization
n is the electrons per volume
\begin{align} P = np = \frac{n q^{2} E(t)}{m} \frac{1}{\omega_0^2 - \omega^2 - \mathrm{i}\omega\gamma} \end{align}Susceptibility
\begin{align}
\chi = \frac{P}{\epsilon_0E} = \frac{n q^{2} E}{\epsilon_0 m} \frac{1}{\omega_0^2 - \omega^2 - \mathrm{i}\omega\gamma}
\end{align}
define plasma frequency
\begin{align} \omega_p = \sqrt{\frac{n q^2}{m\epsilon_0}} \end{align}so
\begin{align} \chi = \frac{\omega_p^2}{\omega_0^2 - \omega^2 - \mathrm{i}\omega\gamma} \end{align}Permittivity
\begin{align}
\epsilon_r = 1 + \chi = 1 + \frac{\omega_p^2}{\omega_0^2 - \omega^2 - \mathrm{i}\omega\gamma}
\end{align}