Rewrite "Lindhard function", split off "dielectric function"

1 files changed, 11 insertions, 8 deletions

diff --git a/content/know/concept/electric-field/index.pdc b/content/know/concept/electric-field/index.pdc
index 6162e0b..62ce1f5 100644
--- a/content/know/concept/electric-field/index.pdc
+++ b/content/know/concept/electric-field/index.pdc
@@ -31,7 +31,7 @@ since opposite charges attracts and like charges repel.
 If two opposite point charges with magnitude $q$
 are observed from far away,
 they can be treated as a single object called a **dipole**,
-which has an **electric dipole moment** $\vb{p}$ defined as follows,
+which has an **electric dipole moment** $\vb{p}$ defined like so,
 where $\vb{d}$ is the vector going from
 the negative to the positive charge (opposite direction of $\vb{E}$):
 
@@ -88,7 +88,7 @@ and that $\vb{M}$ has the opposite sign of $\vb{P}$.
 The polarization $\vb{P}$ is a function of $\vb{E}$.
 In addition to the inherent polarity
 of the material $\vb{P}_0$ (zero in most cases),
-there is a possibly nonlinear response
+there is a (possibly nonlinear) response
 to the applied $\vb{E}$-field:
 
 $$\begin{aligned}
@@ -101,10 +101,7 @@ $$\begin{aligned}
 Where the $\chi_e^{(n)}$ are the **electric susceptibilities** of the medium.
 For simplicity, we often assume that only the $n\!=\!1$ term is nonzero,
 which is the linear response to $\vb{E}$.
-In that case, we define
-the **relative permittivity** $\varepsilon_r \equiv 1 + \chi_e^{(1)}$
-and the **absolute permittivity** $\varepsilon \equiv \varepsilon_r \varepsilon_0$,
-so that:
+In that case, we define the **absolute permittivity** $\varepsilon$ so that:
 
 $$\begin{aligned}
     \vb{D}
@@ -114,14 +111,20 @@ $$\begin{aligned}
     = \varepsilon \vb{E}
 \end{aligned}$$
 
+I.e. $\varepsilon \equiv \varepsilon_r \varepsilon_0$,
+where $\varepsilon_r \equiv 1 + \chi_e^{(1)}$ is
+the [**dielectric function**](/know/concept/dielectric-function/)
+or **relative permittivity**,
+whose calculation is of great interest in physics.
+
 In reality, a material cannot respond instantly to $\vb{E}$,
 meaning that $\chi_e^{(1)}$ is a function of time,
 and that $\vb{P}$ is the convolution of $\chi_e^{(1)}(t)$ and $\vb{E}(t)$:
 
 $$\begin{aligned}
     \vb{P}(t)
-    = (\chi_e^{(1)} * \vb{E})(t)
-    = \int_{-\infty}^\infty \chi_e^{(1)}(t - \tau) \: \vb{E}(\tau) \:d\tau
+    = \varepsilon_0 \big(\chi_e^{(1)} * \vb{E}\big)(t)
+    = \varepsilon_0 \int_{-\infty}^\infty \chi_e^{(1)}(t - \tau) \: \vb{E}(\tau) \:d\tau
 \end{aligned}$$
 
 Note that this definition requires $\chi_e^{(1)}(t) = 0$ for $t < 0$