From: hackbard Date: Tue, 3 Jun 2008 22:45:50 +0000 (+0200) Subject: finished exc 1 X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=9c0ce235d3e9a46383df0d81c43f48057846c6b0;p=lectures%2Flatex.git finished exc 1 --- diff --git a/solid_state_physics/tutorial/2_03s.tex b/solid_state_physics/tutorial/2_03s.tex index a8eab9d..882df74 100644 --- a/solid_state_physics/tutorial/2_03s.tex +++ b/solid_state_physics/tutorial/2_03s.tex @@ -109,30 +109,44 @@ \[ H_{\text{kin}}=\frac{{\bf P}({\bf R})^2}{2M} \] - Integral: - \[ - \int d\Gamma \exp(-\beta H)= + Integral (using change of variables): + \begin{eqnarray} + \int d\Gamma \exp(-\beta H)&=& \int d\Gamma \exp\left[-\beta\left(\sum \frac{{\bf P}({\bf R})^2}{2M}+ - U_{\text{eq}} + U_{\text{harm}}\right)\right] + U_{\text{eq}} + U_{\text{harm}}\right)\right]\nonumber\\ + &=& + \exp(-\beta U_{\text{eq}})\beta^{-3N} + \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 + -\frac{1}{4}\sum + [\bar{u}_{\mu}({\bf R})-\bar{u}_{\mu}({\bf R'})] + \Phi_{\mu v}({\bf R}-{\bf R'}) + [\bar{u}_v({\bf R})-\bar{u}_v({\bf R'})] + \right]\LARGE)\nonumber + \end{eqnarray} + \[ + \Rightarrow w=-\frac{1}{V}\frac{\partial}{\partial \beta} + ln\left((\exp(-\beta U_{\text{eq}})\beta^{-3N} \times \text{const} + \right) + =\frac{U_{\text{eq}}}{V}+3\frac{N}{V}k_{\text{B}}T + =u_{\text{eq}}+3nk_{\text{B}}T + \] + \[ + \Rightarrow + c_{\text{V}}=\frac{\partial w}{\partial T}=3nk_{\text{B}} \] - \end{enumerate} \section{Specific heat in the quantum theory of the harmonic crystal -\\ The Debye model} -As found in exercise 1, the specific heat of a classical harmonic crystal -is not depending on temeprature. -However, as temperature drops below room temperature -the specific heat of all solids is decreasing as $T^3$ in insulators -and $AT+BT^3$ in metals. -This can be explained in a quantum theory of the specific heat of -a harmonic crystal, in which the energy density $w$ is given by \[ w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}. \] \begin{enumerate} - \item Show that the energy density can be rewritten to read: + \item Energy: $\rightarrow$ 1(a) \[ w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \sum_i \exp(-\beta E_i). \]