1_02s final

diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex
index 6f08cae..ccb4a26 100644 (file)
--- a/solid_state_physics/tutorial/1_02s.tex
+++ b/solid_state_physics/tutorial/1_02s.tex
@@ -152,7 +152,8 @@
               &=&C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)\pm
                  \sqrt{C^2\frac{(M_1+M_2)^2}{M_1^2M_2^2}-
                       \frac{1}{M_1M_2}2C^2(1-cos(ka))} \nonumber \\
-              &=&C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)\pm
+              &=&C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)
+                \stackrel{{\color{red}+}}{{\color{blue}-}}
                  C\sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2-
                        \frac{2(1-\cos(ka))}{M_1M_2}} \nonumber
       \end{eqnarray}
@@ -217,8 +218,8 @@
             Acoustic branch: $\omega^2\approx
                               \frac{C/2}{M_1+M_2}k^2a^2$\\
        \item $k=0$:\\
-             Optical branch: $u/v = - M_2/M_1$ (out of phase)\\
-       \item $k=\pm \pi/a$:\\
+             $\rightarrow u/v = - M_2/M_1$ (out of phase)\\
+       \item $k=\pi/a$\\
             $\rightarrow \omega^2=2C/M_2,2C/M_1$
       \end{itemize}
 \end{enumerate}