X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=solid_state_physics%2Ftutorial%2F2_03s.tex;h=b7f0f7d8c097b5fce0230953336133d221d7f184;hp=f1b880dbc7054ef003b3d797845221011fbc108d;hb=183bd2b78445842ee859e2f10f8ab9dc84cf2776;hpb=cd77e3b6e7e35332d6fedd7093cc9bdb86f53277

diff --git a/solid_state_physics/tutorial/2_03s.tex b/solid_state_physics/tutorial/2_03s.tex
index f1b880d..b7f0f7d 100644
--- 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}
@@ -221,26 +223,63 @@ 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:\\
-	      $(2\pi)^3/V \Rightarrow (2\pi)^3N/V=4\pi k_{\text{D}}^3/3
-	       \Rightarrow n=\frac{N}{V}=\frac{k_{\text{D}}^3}{6\pi^2}$
+        \item {\color{red}3} branches with Debye dispersion relation
+	      $w={\color{green}ck}$
+	\item Volume of $k$-space per wave vector: $(2\pi)^3/V$\\
+	      $\Rightarrow (2\pi)^3N/V=4\pi k_{\text{D}}^3/3
+	       \Rightarrow n=\frac{N}{V}=\frac{k_{\text{D}}^3}{6\pi^2}$,
+	       $k_{\text{D}}^3=6\pi^2 n$
+	\item $dn={\color{blue}\frac{k^2}{2\pi^2}dk}$
 	\item Debye frequency: $\omega_{\text{D}}=k_{\text{D}}c$
 	\item Debye temperature:
-	      $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$
+	      $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$,
+	      $\Theta_{\text{D}}=\hbar ck_{\text{D}}/k_{\text{B}}$,
+	      $\Theta_{\text{D}}^3=\frac{\hbar^3c^3k_{\text{D}}^3}
+	      {k_{\text{B}}^3}=
+	      \frac{\hbar^3c^3}{k_{\text{B}}^3}6\pi^2n$
        \end{itemize}
        Integral:
        \[
-       c_{\text{V}}=\ldots
+       c_{\text{V}}=\frac{\partial}{\partial T}\, {\color{red}3}\int_0^{k_D}
+       {\color{blue}\frac{k^2}{2\pi^2}dk} \frac{\hbar {\color{green}ck}}
+       {e^{\beta\hbar {\color{green}ck}}-1}=
+       \frac{\partial}{\partial T}\frac{3\hbar c}{2\pi^2}\int_0^{k_D}
+       \frac{k^3}{e^{\beta\hbar ck}-1}dk=
+       \frac{3\hbar c}{2\pi^2}\int_0^{k_D}
+       \frac{k^3e^{\beta\hbar ck}\beta\hbar ck\frac{1}{T}}
+       {(e^{\beta\hbar ck}-1)^2}dk
+       \]
+       Change of variables: $\beta\hbar ck=x$
+       \[
+       \Rightarrow
+       k=\frac{x}{\beta\hbar c} \quad \textrm{, } \quad
+       dk=\frac{1}{\beta\hbar c} dx
+       \]
+       \[
+       c_{\text{V}}=\frac{3\hbar c}{2\pi^2}\int_0^{\Theta_D/T}
+       \frac{x^3e^xx}{T(\beta\hbar c)^3(e^x-1)^2}\frac{dx}{\beta\hbar c}=
+       \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}\int_0^{\Theta_D/T}
+       \frac{x^4e^x}{(e^x-1)^2}dx
+       \]
+       \[
+       \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}=
+       \frac{3k_{\text{B}}}{2\pi^2\beta^3\hbar^3 c^3}=
+       \frac{3k_{\text{B}}T^33n}{\Theta_{\text{D}}^3}
+       \]
+       \[
+       \Rightarrow
+       c_{\text{V}}=9nk_{\text{B}}\left(\frac{T}{\Theta_{\text{D}}}
+       \right)^3\int_0^{\Theta_D/T}\frac{x^4e^x}{(e^x-1)^2}dx
        \]
 \end{enumerate}