From 9abf096e836103e3e28d961cf2c1022d916927af Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 14 Apr 2024 16:55:58 +0200
Subject: Expand knowledge base

---
 .../know/concept/lorentz-oscillator-model/index.md | 27 ++++++++++++++--------
 1 file changed, 17 insertions(+), 10 deletions(-)

(limited to 'source/know/concept/lorentz-oscillator-model')

diff --git a/source/know/concept/lorentz-oscillator-model/index.md b/source/know/concept/lorentz-oscillator-model/index.md
index 61bbf6b..580ba99 100644
--- a/source/know/concept/lorentz-oscillator-model/index.md
+++ b/source/know/concept/lorentz-oscillator-model/index.md
@@ -61,20 +61,22 @@ 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:
+whereas the electric field $$\vb{E}$$ points from positive to negative:
 
 $$\begin{aligned}
     \vb{P}(t)
-    = N \vb{p}(t)
+    \approx N \vb{p}(t)
     = N q \vb{x}(t)
     = \frac{N q^2}{m (\omega_0^2 - \omega^2 - i \gamma \omega)} \vb{E}(t)
 \end{aligned}$$
 
+Also note that $$\vb{P}$$ is not equal to $$N \vb{p}$$;
+this will be clarified later.
 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
+we see that the material's
 [dielectric function](/know/concept/dielectric-function/)
-$$\varepsilon_r(\omega)$$ is given by:
+$$\varepsilon_r(\omega)$$ must be given by:
 
 $$\begin{aligned}
     \boxed{
@@ -84,7 +86,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 You may recognize the Drude model's plasma frequency $$\omega_p$$ here,
-but the concept of plasma oscillation does not apply
+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 $$\omega_0$$,
@@ -108,23 +110,28 @@ $$\begin{aligned}
 
 In reality, atoms have multiple spectral lines,
 so we should treat them as if they have multiple oscillators
-with different resonances $$\omega_\nu$$.
+with different resonances $$\omega_n$$.
 In that case, the relative permittivity $$\varepsilon_r$$ becomes:
 
-
 $$\begin{aligned}
     \boxed{
         \varepsilon_r(\omega)
-        = 1 + \frac{N q^2}{\varepsilon_0 m} \sum_{\nu} \frac{1}{(\omega_\nu^2 - \omega^2 - i \gamma_\nu \omega)}
+        = 1 + \frac{N q^2}{\varepsilon_0 m} \sum_{n} \frac{1}{(\omega_n^2 - \omega^2 - i \gamma_n \omega)}
     }
 \end{aligned}$$
 
 This gives $$\varepsilon_r$$ the shape of a staircase,
-descending from low to high $$\omega$$ in clear steps at each $$\omega_\nu$$.
+descending from low to high $$\omega$$ in clear steps at each $$\omega_n$$.
 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.
+The damping from $$\gamma_n$$ broadens those peaks and reduces their amplitude.
+
+Finally, recall that $$\vb{P}$$ was not exactly equal to $$N \vb{p}$$.
+This is because each atomic dipole generates its own electric field,
+affecting the response of its neighbors.
+There exists a formula to correct for this effect:
+the [Clausius-Mossotti relation](/know/concept/clausius-mossotti-relation/).
 
 
 
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