From 88537b82784f71104c3a2771330c9e492f57fb03 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 30 Jan 2022 11:27:00 +0100
Subject: Expand knowledge base, remove comments
---
content/know/concept/jellium/index.pdc | 7 +------
1 file changed, 1 insertion(+), 6 deletions(-)
(limited to 'content/know/concept/jellium')
diff --git a/content/know/concept/jellium/index.pdc b/content/know/concept/jellium/index.pdc
index 15c3308..9743514 100644
--- a/content/know/concept/jellium/index.pdc
+++ b/content/know/concept/jellium/index.pdc
@@ -5,6 +5,7 @@ publishDate: 2021-11-23
categories:
- Physics
- Quantum mechanics
+- Perturbation
date: 2021-10-15T20:31:12+02:00
draft: false
@@ -363,17 +364,11 @@ $$\begin{aligned}
&= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \int_0^{2 k_F}
\bigg[ \frac{k_F^3}{3} x + \frac{|\vb{q}|^3}{48 \xi^2} \bigg]_{|\vb{q}| / (2 k_F)}^1 \dd{|\vb{q}|}
\\
- %&= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \int_0^{2 k_F}
- %\bigg( \frac{k_F^3}{3} + \frac{|\vb{q}|^3}{48} - \frac{k_F^3}{3} \frac{|\vb{q}|}{2 k_F} - \frac{4 k_F^2 |\vb{q}|^3}{48 |\vb{q}|^2} \bigg)
- %\dd{|\vb{q}|}
- %\\
&= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \int_0^{2 k_F}
\bigg( \frac{k_F^3}{3} + \frac{|\vb{q}|^3}{48} - \frac{k_F^2 |\vb{q}|}{4} \bigg) \dd{|\vb{q}|}
\\
&= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \bigg[ \frac{k_F^3 |\vb{q}|}{3} + \frac{|\vb{q}|^4}{192} - \frac{k_F^2 |\vb{q}|^2}{8} \bigg]_0^{2 k_F}
\\
- %&= \frac{- e^2 V}{4 \pi^4 \varepsilon_0} \bigg( \frac{k_F^3}{3} 2 k_F + \frac{16 k_F^4}{192} - \frac{k_F^2 4 k_F^2}{8} \bigg)
- %\\
&= \frac{- e^2 V}{16 \pi^4 \varepsilon_0} k_F^4
= \frac{- e^2 N}{16 \pi^4 \varepsilon_0 n} k_F^4
= -\frac{3 e^2 N}{16 \pi^2 \varepsilon_0} k_F
--
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