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 (file)
--- 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}