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| -rw-r--r-- | source/know/concept/bernoullis-theorem/index.md | 2 | ||||
| -rw-r--r-- | source/know/concept/drude-model/index.md | 27 | ||||
| -rw-r--r-- | source/know/concept/lorentz-oscillator-model/index.md | 134 |
3 files changed, 147 insertions, 16 deletions
diff --git a/source/know/concept/bernoullis-theorem/index.md b/source/know/concept/bernoullis-theorem/index.md index 6b933d2..867c443 100644 --- a/source/know/concept/bernoullis-theorem/index.md +++ b/source/know/concept/bernoullis-theorem/index.md @@ -9,7 +9,7 @@ categories: layout: "concept" --- -For inviscid fluids, **Bernuilli's theorem** states +For inviscid fluids, **Bernouilli's theorem** states that an increase in flow velocity $$\va{v}$$ is paired with a decrease in pressure $$p$$ and/or potential energy. For a qualitative argument, look no further than 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}$$ diff --git a/source/know/concept/lorentz-oscillator-model/index.md b/source/know/concept/lorentz-oscillator-model/index.md new file mode 100644 index 0000000..61bbf6b --- /dev/null +++ b/source/know/concept/lorentz-oscillator-model/index.md @@ -0,0 +1,134 @@ +--- +title: "Lorentz oscillator model" +sort_title: "Lorentz oscillator model" +date: 2024-04-09 +categories: +- Physics +- Optics +- Electromagnetism +layout: "concept" +--- + +The **Lorentz oscillator model** or **dipole oscillator model** +is a classical description of light-matter interaction, +which treats the charged particles inside a solid as forming dipoles +that get pushed around by the electric field of passing light waves. +Quantitatively, it is of limited use, as it ignores quantum mechanics, +but qualitatively it captures the essential features. +It is similar to the [Drude model](/know/concept/drude-model/), +but applies to insulators instead of conductors. + +In insulators, the valence electrons are bound +to an immobile nucleus at a certain equilibrium distance +(this is a classical model, so we treat the electron as a particle). +If an [electric field](/know/concept/electric-field/) $$\vb{E}$$ +moves the electron, a restoring force brings it back to equilibrium, +so we can pretend that it is connected to the nucleus by a spring. +In other words, we treat it as a [harmonic oscillator](/know/concept/harmonic-oscillator/), +whose spring constant $$K$$ should be chosen such that: + +$$\begin{aligned} + \omega_0 = \sqrt{\frac{K}{m}} +\end{aligned}$$ + +Where $$m$$ is the electron's mass, and the resonance $$\omega_0$$ +is an empirically determined transition frequency of the atom. +When an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/) +travels through the material, its electric field +$$\vb{E}(t) = \vb{E}_0 e^{-i \omega t}$$ displaces +the electron by an amount $$\vb{x}(t)$$ governed by: + +$$\begin{aligned} + m \dvn{2}{\vb{x}}{t} + &= q \vb{E} - m \gamma \dv{\vb{x}}{t} - K \vb{x} +\end{aligned}$$ + +Where $$q < 0$$ is the electron's charge, +and $$\gamma$$ represents a weak damping effect. +The four terms represent Newton's second law, +the [Lorentz force](/know/concept/lorentz-force/), +the spring's damping force, and the spring's restoring force, respectively. + +Inserting the ansatz $$\vb{x}(t) = \vb{x}_0 e^{- i \omega t}$$ +and isolating for the amplitude $$\vb{x}_0$$, we find: + +$$\begin{gathered} + \vb{x}_0 + = \frac{q \vb{E}_0}{m (\omega_0^2 - \omega^2 - i \gamma \omega)} +\end{gathered}$$ + +The polarization density $$\vb{P}(t)$$ is therefore as shown below, +where $$N$$ is the number of atoms per unit of volume. +Note that the dipole moment vector $$\vb{p}$$ is defined +as pointing from negative to positive, +whereas the electric field $$\vb{E}$$ goes from positive to negative: + +$$\begin{aligned} + \vb{P}(t) + = N \vb{p}(t) + = N q \vb{x}(t) + = \frac{N q^2}{m (\omega_0^2 - \omega^2 - i \gamma \omega)} \vb{E}(t) +\end{aligned}$$ + +From the definition of the electric displacement field +$$\vb{D} = \varepsilon_0 \vb{E} + \vb{P} = \varepsilon_0 \varepsilon_r \vb{E}$$, +we find that the material's +[dielectric function](/know/concept/dielectric-function/) +$$\varepsilon_r(\omega)$$ is given by: + +$$\begin{aligned} + \boxed{ + \varepsilon_r(\omega) + = 1 + \frac{N q^2}{\varepsilon_0 m (\omega_0^2 - \omega^2 - i \gamma \omega)} + } +\end{aligned}$$ + +You may recognize the Drude model's plasma frequency $$\omega_p$$ here, +but the concept of plasma oscillation does not apply +because there are no conduction electrons. + +When the light's driving frequency $$\omega$$ is far from the resonance $$\omega_0$$, +we see that the "background" permittivity is higher at lower frequencies: + +$$\begin{aligned} + \varepsilon_{\mathrm{low}} + &= \, \lim_{\omega \to 0} \, \varepsilon_r(\omega) = 1 + \frac{N q^2}{\varepsilon_0 m \omega_0^2} + \\ + \varepsilon_{\mathrm{high}} + &= \lim_{\omega \to \infty} \varepsilon_r(\omega) = 1 +\end{aligned}$$ + +Using these limits, we can rewrite our previous expression for $$\varepsilon_r$$ as follows: + +$$\begin{aligned} + \varepsilon_r(\omega) + = \varepsilon_{\mathrm{high}} + + (\varepsilon_{\mathrm{low}} - \varepsilon_{\mathrm{high}}) \frac{\omega_0^2}{\omega_0^2 - \omega^2 - i \gamma \omega} +\end{aligned}$$ + +In reality, atoms have multiple spectral lines, +so we should treat them as if they have multiple oscillators +with different resonances $$\omega_\nu$$. +In that case, the relative permittivity $$\varepsilon_r$$ becomes: + + +$$\begin{aligned} + \boxed{ + \varepsilon_r(\omega) + = 1 + \frac{N q^2}{\varepsilon_0 m} \sum_{\nu} \frac{1}{(\omega_\nu^2 - \omega^2 - i \gamma_\nu \omega)} + } +\end{aligned}$$ + +This gives $$\varepsilon_r$$ the shape of a staircase, +descending from low to high $$\omega$$ in clear steps at each $$\omega_\nu$$. +Around each such resonance there is a distinctive "squiggle" in $$\Real\{\varepsilon_r\}$$ +corresponding to a peak in the material's reflectivity, +and there is an absorption peak in $$\Imag\{\varepsilon_r\}$$. +The damping from $$\gamma_\nu$$ broadens those peaks and reduces their amplitude. + + + +## References +1. M. Fox, + *Optical properties of solids*, 2nd edition, + Oxford. |