- \item Evaluate the expression of the energy density.
- {\bf Hint:}
- The energy levels of a harmonic crystal of N ions
- can be regarded as 3N independent oscillators,
- whose frequencies are those of the 3N classical normal modes.
- The contribution to the total energy of a particular normal mode
- with angular frequency $\omega_s({\bf k})$
- ($s$: branch, ${\bf k}$: wave vector) is given by
- $(n_{{\bf k}s} + \frac{1}{2})\hbar\omega_s({\bf k})$ with the
- excitation number $n_{{\bf k}s}$ being restricted to integers greater
- or equal zero.
- The total energy is given by the sum over the energies of the individual
- normal modes.
- Use the totals formula of the geometric series to expcitly calculate
- the sum of the exponential functions.
- \item Separate the above result into a term vanishing as $T$ goes to zero and
- a second term giving the energy of the zero-point vibrations of the
- normal modes.
- \item Write down an expression for the specific heat.
- Consider a large crystal and thus replace the sum over the discrete
- wave vectors with an integral.
- \item Debye replaced all branches of the vibrational spectrum with three
- branches, each of them obeying the dispersion relation
- $w=ck$.
- Additionally the integral is cut-off at a radius $k_{\text{D}}$
- to have a total amount of N allowed wave vectors.
- Determine $k_{\text{D}}$.
- Evaluate the simplified integral and introduce the
- Debye frequency $\omega_{\text{D}}=k_{\text{D}}c$
- and the Debye temperature $\Theta_{\text{D}}$ which is given by
- $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$.
- Write down the resulting expression for the specific heat.
+ \item \[
+ w=w_{\text{eq}}+
+ \frac{1}{V}\sum_{{\bf k}s}\frac{1}{2}\hbar\omega_s({\bf k})+
+ \frac{1}{V}\sum_{{\bf k}s}
+ \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
+ \]
+ \item \[
+ c_{\text{V}}=\frac{1}{V}\sum_{{\bf k}s}\frac{\partial}{\partial T}
+ \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
+ \]
+ Large crystal:
+ \[
+ \lim_{v\rightarrow\infty}c_{\text{V}}=\frac{1}{V}\sum_{{\bf k}s}
+ \frac{\partial}{\partial T}
+ \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
+ =\frac{\partial}{\partial T}
+ \sum_s\int\frac{d{\bf k}}{(2\pi)^3}
+ \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
+ \]
+ \item \begin{itemize}
+ \item {\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:
+ $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$,
+ $\Theta_{\text{D}}=\hbar ck_{\text{D}}/k_{\text{B}}$,
+ $\Theta_{\text{D}}^3=\frac{\hbar^3c^3k_{\text{D}}^3}
+ {k_{\text{B}}^3}=
+ \frac{\hbar^3c^3}{k_{\text{B}}^3}6\pi^2n$
+ \end{itemize}
+ Integral:
+ \[
+ c_{\text{V}}=\frac{\partial}{\partial T}\, {\color{red}3}\int_0^{k_D}
+ {\color{blue}\frac{k^2}{2\pi^2}dk} \frac{\hbar {\color{green}ck}}
+ {e^{\beta\hbar {\color{green}ck}}-1}=
+ \frac{\partial}{\partial T}\frac{3\hbar c}{2\pi^2}\int_0^{k_D}
+ \frac{k^3}{e^{\beta\hbar ck}-1}dk=
+ \frac{3\hbar c}{2\pi^2}\int_0^{k_D}
+ \frac{k^3e^{\beta\hbar ck}\beta\hbar ck\frac{1}{T}}
+ {(e^{\beta\hbar ck}-1)^2}dk
+ \]
+ Change of variables: $\beta\hbar ck=x$
+ \[
+ \Rightarrow
+ k=\frac{x}{\beta\hbar c} \quad \textrm{, } \quad
+ dk=\frac{1}{\beta\hbar c} dx
+ \]
+ \[
+ c_{\text{V}}=\frac{3\hbar c}{2\pi^2}\int_0^{\Theta_D/T}
+ \frac{x^3e^xx}{T(\beta\hbar c)^3(e^x-1)^2}\frac{dx}{\beta\hbar c}=
+ \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}\int_0^{\Theta_D/T}
+ \frac{x^4e^x}{(e^x-1)^2}dx
+ \]
+ \[
+ \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}=
+ \frac{3k_{\text{B}}}{2\pi^2\beta^3\hbar^3 c^3}=
+ \frac{3k_{\text{B}}T^33n}{\Theta_{\text{D}}^3}
+ \]
+ \[
+ \Rightarrow
+ c_{\text{V}}=9nk_{\text{B}}\left(\frac{T}{\Theta_{\text{D}}}
+ \right)^3\int_0^{\Theta_D/T}\frac{x^4e^x}{(e^x-1)^2}dx
+ \]