From: hackbard Date: Mon, 23 Jun 2008 13:11:36 +0000 (+0200) Subject: fixed a minus sign X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=914534d19da7f7c60fdc1db586f794416df49285;p=lectures%2Flatex.git fixed a minus sign --- diff --git a/solid_state_physics/tutorial/2_03s.tex b/solid_state_physics/tutorial/2_03s.tex index 67ad8c1..e592c99 100644 --- a/solid_state_physics/tutorial/2_03s.tex +++ b/solid_state_physics/tutorial/2_03s.tex @@ -185,15 +185,17 @@ w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}. e^{-\beta\hbar\omega_s({\bf k})}(-\hbar\omega_s({\bf k}))} {(1-e^{-\beta\hbar\omega_s({\bf k})})^2}\nonumber\\ &=&-\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k}) - \frac{e^{-\beta\hbar\omega_s({\bf k})}- + \frac{{\color{red}-}e^{-\beta\hbar\omega_s({\bf k})}- \frac{1}{2}(1-e^{-\beta\hbar\omega_s({\bf k})})} {1-e^{-\beta\hbar\omega_s({\bf k})}}\nonumber\\ &=&-\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k}) - \frac{\frac{1}{2}e^{-\beta\hbar\omega_s({\bf k})}-\frac{1}{2}} + \frac{{\color{red}-}\frac{1}{2}e^{-\beta\hbar\omega_s({\bf k})}-\frac{1}{2}} {1-e^{-\beta\hbar\omega_s({\bf k})}} =\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})\frac{1}{2} - \frac{1+e^{\beta\hbar\omega_s({\bf k})}} - {e^{\beta\hbar\omega_s({\bf k})}-1}\nonumber\\ + \frac{e^{-\beta\hbar\omega_s({\bf k})}+1} + {1-e^{-\beta\hbar\omega_s({\bf k})}}\cdot + \frac{e^{\beta\hbar\omega_s({\bf k})}}{e^{\beta\hbar\omega_s({\bf k})}} + \nonumber\\ &=&\frac{1}{V}\sum_{{\bf k}s}\hbar\omega_s({\bf k})\frac{1}{2} \frac{1+e^{\beta\hbar\omega_s({\bf k})}} {e^{\beta\hbar\omega_s({\bf k})}-1}