X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=solid_state_physics%2Ftutorial%2F2_03s.tex;h=b7f0f7d8c097b5fce0230953336133d221d7f184;hp=67ad8c12274af0bac1723ed387b1cb18ecdc4d9c;hb=183bd2b78445842ee859e2f10f8ab9dc84cf2776;hpb=e9cc45dc889b19b16a0eb699b18da1334d6f4c5e

diff --git a/solid_state_physics/tutorial/2_03s.tex b/solid_state_physics/tutorial/2_03s.tex
index 67ad8c1..b7f0f7d 100644
--- a/solid_state_physics/tutorial/2_03s.tex
+++ b/solid_state_physics/tutorial/2_03s.tex
@@ -119,7 +119,7 @@
        \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
+       -\sum\frac{1}{2M}\bar{{\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'})
@@ -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}