From: hackbard Date: Thu, 19 Jun 2008 11:47:09 +0000 (+0200) Subject: finished tut 3 solutions X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=e9cc45dc889b19b16a0eb699b18da1334d6f4c5e;p=lectures%2Flatex.git finished tut 3 solutions --- diff --git a/solid_state_physics/tutorial/2_03s.tex b/solid_state_physics/tutorial/2_03s.tex index a3a3f24..67ad8c1 100644 --- a/solid_state_physics/tutorial/2_03s.tex +++ b/solid_state_physics/tutorial/2_03s.tex @@ -235,12 +235,16 @@ 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: \[ @@ -260,7 +264,20 @@ w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}. dk=\frac{1}{\beta\hbar c} dx \] \[ - c_{\text{V}}= + c_{\text{V}}=\frac{3\hbar c}{2\pi^2}\int_0^{\Theta_D/T} + \frac{x^3e^xx}{T(\beta\hbar c)^3(e^x-1)^2}\frac{dx}{\beta\hbar c}= + \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}\int_0^{\Theta_D/T} + \frac{x^4e^x}{(e^x-1)^2}dx + \] + \[ + \frac{3}{2\pi^2T\beta^4\hbar^3 c^3}= + \frac{3k_{\text{B}}}{2\pi^2\beta^3\hbar^3 c^3}= + \frac{3k_{\text{B}}T^33n}{\Theta_{\text{D}}^3} + \] + \[ + \Rightarrow + c_{\text{V}}=9nk_{\text{B}}\left(\frac{T}{\Theta_{\text{D}}} + \right)^3\int_0^{\Theta_D/T}\frac{x^4e^x}{(e^x-1)^2}dx \] \end{enumerate}