finished tut 3 solutions

diff --git a/solid_state_physics/tutorial/2_03s.tex b/solid_state_physics/tutorial/2_03s.tex
index a3a3f24..67ad8c1 100644 (file)
--- a/solid_state_physics/tutorial/2_03s.tex
+++ b/solid_state_physics/tutorial/2_03s.tex
@@ -235,12 +235,16 @@ w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}.
              $w={\color{green}ck}$
        \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 $dn={\color{blue}\frac{k^2}{2\pi^2}dk}$
+              \Rightarrow n=\frac{N}{V}=\frac{k_{\text{D}}^3}{6\pi^2}$,
+              $k_{\text{D}}^3=6\pi^2 n$
+       \item $dn={\color{blue}\frac{k^2}{2\pi^2}dk}$
        \item Debye frequency: $\omega_{\text{D}}=k_{\text{D}}c$
        \item Debye temperature:
              $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$,
-             $\Theta_{\text{D}}=\hbar ck_{\text{D}}/k_{\text{B}}$
+             $\Theta_{\text{D}}=\hbar ck_{\text{D}}/k_{\text{B}}$,
+             $\Theta_{\text{D}}^3=\frac{\hbar^3c^3k_{\text{D}}^3}
+             {k_{\text{B}}^3}=
+             \frac{\hbar^3c^3}{k_{\text{B}}^3}6\pi^2n$
        \end{itemize}
        Integral:
        \[
@@ -260,7 +264,20 @@ w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}.
        dk=\frac{1}{\beta\hbar c} dx
        \]
        \[
-       c_{\text{V}}=
+       c_{\text{V}}=\frac{3\hbar c}{2\pi^2}\int_0^{\Theta_D/T}
+       \frac{x^3e^xx}{T(\beta\hbar c)^3(e^x-1)^2}\frac{dx}{\beta\hbar c}=
+       \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}\int_0^{\Theta_D/T}
+       \frac{x^4e^x}{(e^x-1)^2}dx
+       \]
+       \[
+       \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}=
+       \frac{3k_{\text{B}}}{2\pi^2\beta^3\hbar^3 c^3}=
+       \frac{3k_{\text{B}}T^33n}{\Theta_{\text{D}}^3}
+       \]
+       \[
+       \Rightarrow
+       c_{\text{V}}=9nk_{\text{B}}\left(\frac{T}{\Theta_{\text{D}}}
+       \right)^3\int_0^{\Theta_D/T}\frac{x^4e^x}{(e^x-1)^2}dx
        \]
 \end{enumerate}