somethiung new ..

diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex
index 6f920e8..a3e9b93 100644 (file)
--- a/solid_state_physics/tutorial/1_02s.tex
+++ b/solid_state_physics/tutorial/1_02s.tex
@@ -143,14 +143,20 @@
             $\Rightarrow
              M_1M_2\omega^4-2C(M_1+M_2)\omega^2+2C^2(1-\cos(ka))=0$
       \end{itemize}
-\item \[
-      \omega^2=C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)\pm
-               C\sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2-
-                     \frac{2(1-\cos(ka))}{M_1M_2}}
-      \]
+\item \begin{eqnarray}
+      \omega^2&=&C\left(\frac{2C(M_1+M_2)}{2M_1M_2}\right)\pm
+                 \sqrt{\frac{4C^2(M_1+M_2)^2}{4M_1^2M_2^2}-
+                      \frac{2C^2(1-cos(ka))}{M_1M_2}} \nonumber \\
+              &=&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\sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2-
+                       \frac{2(1-\cos(ka))}{M_1M_2}} \nonumber
+      \end{eqnarray}
       \begin{itemize}
        \item $ka\ll 1$:\\
-             $\rightarrow \cos(ka)\approx 1-\frac{1}{2}k^2a^2$\\
+             $\rightarrow \cos(ka)\approx 1-\frac{1}{2}k^2a^2$ (Taylor)\\
             Optical branch: $\omega^2\approx
                              2C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)$\\
             Acoustic branch: $\omega^2\approx