the mpi publication, finally!

[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 a3a3f24..b7f0f7d 100644 (file)
--- a/solid_state_physics/tutorial/2_03s.tex
+++ b/solid_state_physics/tutorial/2_03s.tex
@@ -119,7 +119,7 @@
        \LARGE(\int\prod_{{\bf R}}d\bar{{\bf u}}({\bf R})d\bar{{\bf P}}({\bf R})
        \nonumber\\
        &&\times \exp\left[
-       -\sum\frac{1}{2M}{\bf P}({\bf R})^2
+       -\sum\frac{1}{2M}\bar{{\bf P}}({\bf R})^2
        -\frac{1}{4}\sum
        [\bar{u}_{\mu}({\bf R})-\bar{u}_{\mu}({\bf R'})]
        \Phi_{\mu v}({\bf R}-{\bf R'})
@@ -185,15 +185,17 @@ w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}.
             e^{-\beta\hbar\omega_s({\bf k})}(-\hbar\omega_s({\bf k}))}
             {(1-e^{-\beta\hbar\omega_s({\bf k})})^2}\nonumber\\
        &=&-\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
-       \frac{e^{-\beta\hbar\omega_s({\bf k})}-
+       \frac{{\color{red}-}e^{-\beta\hbar\omega_s({\bf k})}-
              \frac{1}{2}(1-e^{-\beta\hbar\omega_s({\bf k})})}
            {1-e^{-\beta\hbar\omega_s({\bf k})}}\nonumber\\
        &=&-\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
-       \frac{\frac{1}{2}e^{-\beta\hbar\omega_s({\bf k})}-\frac{1}{2}}
+       \frac{{\color{red}-}\frac{1}{2}e^{-\beta\hbar\omega_s({\bf k})}-\frac{1}{2}}
            {1-e^{-\beta\hbar\omega_s({\bf k})}}
        =\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})\frac{1}{2}
-       \frac{1+e^{\beta\hbar\omega_s({\bf k})}}
-            {e^{\beta\hbar\omega_s({\bf k})}-1}\nonumber\\
+       \frac{e^{-\beta\hbar\omega_s({\bf k})}+1}
+            {1-e^{-\beta\hbar\omega_s({\bf k})}}\cdot
+       \frac{e^{\beta\hbar\omega_s({\bf k})}}{e^{\beta\hbar\omega_s({\bf k})}}
+       \nonumber\\
        &=&\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})\frac{1}{2}
        \frac{1+e^{\beta\hbar\omega_s({\bf k})}}
             {e^{\beta\hbar\omega_s({\bf k})}-1}
@@ -235,12 +237,16 @@ 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:
        \[
@@ -260,7 +266,20 @@ w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}.
        dk=\frac{1}{\beta\hbar c} dx
        \]
        \[
-       c_{\text{V}}=
+       c_{\text{V}}=\frac{3\hbar c}{2\pi^2}\int_0^{\Theta_D/T}
+       \frac{x^3e^xx}{T(\beta\hbar c)^3(e^x-1)^2}\frac{dx}{\beta\hbar c}=
+       \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}\int_0^{\Theta_D/T}
+       \frac{x^4e^x}{(e^x-1)^2}dx
+       \]
+       \[
+       \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}=
+       \frac{3k_{\text{B}}}{2\pi^2\beta^3\hbar^3 c^3}=
+       \frac{3k_{\text{B}}T^33n}{\Theta_{\text{D}}^3}
+       \]
+       \[
+       \Rightarrow
+       c_{\text{V}}=9nk_{\text{B}}\left(\frac{T}{\Theta_{\text{D}}}
+       \right)^3\int_0^{\Theta_D/T}\frac{x^4e^x}{(e^x-1)^2}dx
        \]
 \end{enumerate}