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
$$\begin{align} \epsilon_r = 1 + \chi = 1 + \frac{\omega_p^2}{\omega_0^2 - \omega^2 - \mathrm{i}\omega\gamma} \end{align}$$Reference
- https://physics.byu.edu/faculty/colton/docs/phy442-winter20/lecture-11-Lorentz-oscillator-model.pdf