From: hackbard Date: Wed, 18 Jun 2008 16:56:18 +0000 (+0200) Subject: more debye stuff X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=ba06829e4a37c48b3a578c4d684d94d48c25277e;p=lectures%2Flatex.git more debye stuff --- diff --git a/solid_state_physics/tutorial/2_03s.tex b/solid_state_physics/tutorial/2_03s.tex index f1b880d..a3a3f24 100644 --- a/solid_state_physics/tutorial/2_03s.tex +++ b/solid_state_physics/tutorial/2_03s.tex @@ -221,26 +221,46 @@ 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 + \item {\color{red}3} branches with Debye dispersion relation + $w={\color{green}ck}$ + \item Volume of $k$-space per wave vector: $(2\pi)^3/V$\\ + $\Rightarrow (2\pi)^3N/V=4\pi k_{\text{D}}^3/3 \Rightarrow n=\frac{N}{V}=\frac{k_{\text{D}}^3}{6\pi^2}$ + and $dn={\color{blue}\frac{k^2}{2\pi^2}dk}$ \item Debye frequency: $\omega_{\text{D}}=k_{\text{D}}c$ \item Debye temperature: - $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$ + $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$, + $\Theta_{\text{D}}=\hbar ck_{\text{D}}/k_{\text{B}}$ \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: $\beta\hbar ck=x$ + \[ + \Rightarrow + k=\frac{x}{\beta\hbar c} \quad \textrm{, } \quad + dk=\frac{1}{\beta\hbar c} dx + \] + \[ + c_{\text{V}}= \] \end{enumerate}