\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'})
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}
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}