From 6b51a2bf7d43f9f83e668d0b97d24640da79e44d Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 9 Apr 2024 16:49:41 +0200
Subject: Expand knowledge base

---
 source/know/concept/bernoullis-theorem/index.md    |   2 +-
 source/know/concept/drude-model/index.md           |  27 ++---
 .../know/concept/lorentz-oscillator-model/index.md | 134 +++++++++++++++++++++
 3 files changed, 147 insertions(+), 16 deletions(-)
 create mode 100644 source/know/concept/lorentz-oscillator-model/index.md

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.
-- 
cgit v1.2.3