final ..

diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex
index ccb4a260ba3c51c592e7ddf4e039c415a0594b8b..240d84e2e08560697271ac469ace514145e182db 100644 (file)
--- a/solid_state_physics/tutorial/1_02s.tex
+++ b/solid_state_physics/tutorial/1_02s.tex
@@ -160,7 +160,7 @@
       \begin{figure}[!h]
 
 % GNUPLOT: LaTeX picture using EEPIC macros
-\setlength{\unitlength}{0.120450pt}
+\setlength{\unitlength}{0.130450pt}
 \begin{picture}(3000,1800)(0,0)
 \footnotesize
 \color{black}
@@ -213,10 +213,32 @@
       \begin{itemize}
        \item $ka\ll 1$:\\
              $\rightarrow \cos(ka)\approx 1-\frac{1}{2}k^2a^2$ (Taylor)\\
-            Optical branch: $\omega^2\approx
-                             2C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)$\\
-            Acoustic branch: $\omega^2\approx
-                              \frac{C/2}{M_1+M_2}k^2a^2$\\
+            $\Rightarrow$\\
+            $\sqrt{(\frac{1}{M_1}+\frac{1}{M_2})^2-
+             \frac{k^2a^2}{M_1M_2}}=$
+            $(\frac{1}{M_1}+\frac{1}{M_2})
+             \sqrt{1-\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2}}
+             \stackrel{Taylor}{\approx}
+             (\frac{1}{M_1}+\frac{1}{M_2})
+             (1-\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2})$\\
+            $\omega \approx \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})}
+             \sqrt{1\pm (1-\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2})}$\\
+            $\stackrel{{\color{red}+}}{\rightarrow}
+             \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})}
+             \sqrt{2-\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2}}
+             \stackrel{Taylor}{\approx}
+             \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})}\sqrt{2}
+             (1-\frac{1}{2}\frac{1}{4}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2})$\\
+            $\stackrel{{\color{blue}-}}{\rightarrow}
+             \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})}
+             \sqrt{\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2}}=
+             \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})}
+             \sqrt{\frac{1}{2}\frac{1}{M_1M_2(1/M_1+1/M_2)^2}}ka$\\
+            {\color{red}Optical branch}: $\omega\stackrel{ka\ll 1}{\approx}
+                             \sqrt{2C\left(\frac{1}{M_1}+
+                                           \frac{1}{M_2}\right)}$\\
+            {\color{blue}Acoustic branch}: $\omega\stackrel{ka\ll 1}{\approx}
+                              \sqrt{\frac{C/2}{M_1+M_2}}ka$\\
        \item $k=0$:\\
              $\rightarrow u/v = - M_2/M_1$ (out of phase)\\
        \item $k=\pi/a$\\