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Diffstat (limited to 'source/know/concept/drude-model/index.md')
| -rw-r--r-- | source/know/concept/drude-model/index.md | 27 |
1 files changed, 12 insertions, 15 deletions
diff --git a/source/know/concept/drude-model/index.md b/source/know/concept/drude-model/index.md index 0026d90..8fcd7fb 100644 --- a/source/know/concept/drude-model/index.md +++ b/source/know/concept/drude-model/index.md @@ -11,7 +11,7 @@ layout: "concept" The **Drude model**, also known as the **Drude-Lorentz model** due to its analogy -to the [Lorentz oscillator model](/know/concept/lorentz-oscillator-model/) +to the [Lorentz oscillator model](/know/concept/lorentz-oscillator-model/), classically predicts the [dielectric function](/know/concept/dielectric-function/) and electric conductivity of a gas of free charges, as found in metals and doped semiconductors. @@ -33,17 +33,16 @@ $$\begin{aligned} \end{aligned}$$ Where $$m$$ and $$q < 0$$ are the mass and charge of the electron. -The first term is Newton's third law, +The first term is Newton's second law, and the last term represents a damping force slowing down the electrons at rate $$\gamma$$. Inserting the ansatz $$\vb{x}(t) = \vb{x}_0 e^{- i \omega t}$$ -and isolating for the displacement $$\vb{x}$$, we find: +and isolating for the amplitude $$\vb{x}_0$$, we find: $$\begin{aligned} - \vb{x}(t) - = \vb{x}_0 e^{- i \omega t} - = - \frac{q \vb{E}}{m (\omega^2 + i \gamma \omega)} + \vb{x}_0 + = - \frac{q \vb{E}_0}{m (\omega^2 + i \gamma \omega)} \end{aligned}$$ The polarization density $$\vb{P}(t)$$ is therefore as shown below. @@ -76,8 +75,10 @@ leading to so-called **plasma oscillations** of the electron density (see also [Langmuir waves](/know/concept/langmuir-waves/)): $$\begin{aligned} - \varepsilon_r(\omega) - = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} + \boxed{ + \varepsilon_r(\omega) + = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} + } \qquad\qquad \boxed{ \omega_p @@ -96,7 +97,7 @@ then we can identify three distinct scenarios for $$\varepsilon_r$$ here: allowing for self-sustained plasma oscillations. * $$\omega > \omega_p$$, so $$\varepsilon_r > 0$$, so the index $$\sqrt{\varepsilon}$$ is real and asymptotically goes to $$1$$, - leading to high transparency and low reflectivity from air. + leading to high transparency and low reflectivity (coming from air). For most metals $$\omega_p$$ is ultraviolet, which explains why they typically appear shiny to us. @@ -158,12 +159,8 @@ the dielectric function $$\varepsilon_r(\omega)$$ can be written as: $$\begin{aligned} \boxed{ - \begin{aligned} - \varepsilon_r(\omega) - &= 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} - \\ - &= 1 + \frac{i \sigma(\omega)}{\varepsilon_0 \omega} - \end{aligned} + \varepsilon_r(\omega) + = 1 + \frac{i \sigma(\omega)}{\varepsilon_0 \omega} } \end{aligned}$$ |