more debye stuff

[lectures/latex.git] / solid_state_physics / tutorial / 2_03s.tex

diff --git a/solid_state_physics/tutorial/2_03s.tex b/solid_state_physics/tutorial/2_03s.tex
index f1b880d..a3a3f24 100644 (file)
--- a/solid_state_physics/tutorial/2_03s.tex
+++ b/solid_state_physics/tutorial/2_03s.tex
@@ -221,26 +221,46 @@ w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}.
        c_{\text{V}}=\frac{1}{V}\sum_{{\bf k}s}\frac{\partial}{\partial T}
        \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
        \]
-       Large crystal: ($\lim_{V\to\infty}\frac{1}{V}\sum_{{\bf k}}F({\bf k})
-                        =\int\frac{d{\bf k}}{(2\pi)^3}F({\bf k})$)
+       Large crystal:
        \[
-       \Rightarrow
-       c_{\text{V}}=\frac{\partial}{\partial T}
+       \lim_{v\rightarrow\infty}c_{\text{V}}=\frac{1}{V}\sum_{{\bf k}s}
+       \frac{\partial}{\partial T}
+       \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
+       =\frac{\partial}{\partial T}
        \sum_s\int\frac{d{\bf k}}{(2\pi)^3}
        \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
        \]
  \item \begin{itemize}
-        \item Debye dispersion relation: $w=ck$
-       \item Volume of $k$-space per wave vector:\\
-             $(2\pi)^3/V \Rightarrow (2\pi)^3N/V=4\pi k_{\text{D}}^3/3
+        \item {\color{red}3} branches with Debye dispersion relation
+             $w={\color{green}ck}$
+       \item Volume of $k$-space per wave vector: $(2\pi)^3/V$\\
+             $\Rightarrow (2\pi)^3N/V=4\pi k_{\text{D}}^3/3
               \Rightarrow n=\frac{N}{V}=\frac{k_{\text{D}}^3}{6\pi^2}$
+             and $dn={\color{blue}\frac{k^2}{2\pi^2}dk}$
        \item Debye frequency: $\omega_{\text{D}}=k_{\text{D}}c$
        \item Debye temperature:
-             $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$
+             $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$,
+             $\Theta_{\text{D}}=\hbar ck_{\text{D}}/k_{\text{B}}$
        \end{itemize}
        Integral:
        \[
-       c_{\text{V}}=\ldots
+       c_{\text{V}}=\frac{\partial}{\partial T}\, {\color{red}3}\int_0^{k_D}
+       {\color{blue}\frac{k^2}{2\pi^2}dk} \frac{\hbar {\color{green}ck}}
+       {e^{\beta\hbar {\color{green}ck}}-1}=
+       \frac{\partial}{\partial T}\frac{3\hbar c}{2\pi^2}\int_0^{k_D}
+       \frac{k^3}{e^{\beta\hbar ck}-1}dk=
+       \frac{3\hbar c}{2\pi^2}\int_0^{k_D}
+       \frac{k^3e^{\beta\hbar ck}\beta\hbar ck\frac{1}{T}}
+       {(e^{\beta\hbar ck}-1)^2}dk
+       \]
+       Change of variables: $\beta\hbar ck=x$
+       \[
+       \Rightarrow
+       k=\frac{x}{\beta\hbar c} \quad \textrm{, } \quad
+       dk=\frac{1}{\beta\hbar c} dx
+       \]
+       \[
+       c_{\text{V}}=
        \]
 \end{enumerate}