From: hackbard Date: Wed, 4 Jun 2008 00:04:02 +0000 (+0200) Subject: tarting of exc 2 X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=6893272d5705b9dff8c01c3bfbf6410631caf49e;p=lectures%2Flatex.git tarting of exc 2 --- diff --git a/solid_state_physics/tutorial/2_03s.tex b/solid_state_physics/tutorial/2_03s.tex index 882df74..fb0e83b 100644 --- a/solid_state_physics/tutorial/2_03s.tex +++ b/solid_state_physics/tutorial/2_03s.tex @@ -150,6 +150,30 @@ w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}. \[ w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \sum_i \exp(-\beta E_i). \] + \item \begin{itemize} + \item Total energy contribution of a particular normal mode: + $(n_{{\bf k}s} + \frac{1}{2})\hbar\omega_s({\bf k})$ + with $n_{{\bf k}s}=0,1,2,\ldots$ + \item A state of the crystal is specified by the excitation numbers + of the 3N normal modes. + \item The total energy is the sum of the energies of the individual + normal modes:\\ + $E=\sum_{{\bf k}s}(n_{{\bf k}s}+ + \frac{1}{2})\hbar\omega_s({\bf k})$ + \end{itemize} + \begin{eqnarray} + \Rightarrow + w&=&-\frac{1}{V}\frac{\partial}{\partial \beta} ln\left( + \prod_{{\bf k}s}(\exp(-\beta\hbar\omega_s({\bf k})/2)+ + \exp(-3\beta\hbar\omega_s({\bf k})/2)+ + \exp(-5\beta\hbar\omega_s({\bf k})/2)+ + \ldots) + \right)\nonumber\\ + &=&-\frac{1}{V}\frac{\partial}{\partial \beta} ln \prod_{{\bf k}s} + \frac{\exp(-\beta\hbar\omega_s({\bf k})/2)} + {1-\exp(-\beta\hbar\omega_s({\bf k}))}\nonumber + \end{eqnarray} + \item Evaluate the expression of the energy density. {\bf Hint:} The energy levels of a harmonic crystal of N ions