more solution

diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex
index 8eb5881..e782f74 100644 (file)
--- a/solid_state_physics/tutorial/1_02s.tex
+++ b/solid_state_physics/tutorial/1_02s.tex
@@ -108,9 +108,49 @@
 
 \section{Phonons 2}
 \begin{enumerate}
-\item Derive the dispersion relation for a linear chain with two different
-      alternating types of atoms.
-\item Discuss the two solutions for $\omega^2$.
+\item \begin{itemize}
+       \item Convention:\\
+             Atom type 1: $M_1$, $u_s$ (elongation of atom $s$ of type 1)\\
+            Atom type 2: $M_2$, $v_s$ (elongation of atom $s$ of type 2)\\
+            Lattice constant: $a$, Spring constant: $C$
+       \item Equations of motion:\\
+             $M_1\ddot{u}_s=C(v_s+v_{s-1}-2u_s)$\\ 
+             $M_2\ddot{v}_s=C(u_{s+1}+u_s-2v_s)$
+       \item Ansatz:\\
+             $u_s=u\exp{i(ska-\omega t)}$\\
+            $v_s=v\exp{i(ska-\omega t)}$
+       \item Solution of the equation system:\\
+             $-\omega^2M_1u=Cv[1+\exp(-ika)]-2Cu$\\
+             $-\omega^2M_2v=Cu[\exp(ika)+1]-2Cv$\\
+            Non trivial solution only if determinant of coefficients
+            $u$ and $v$ is zero.\\
+            $\Rightarrow
+             \left|
+              \begin{array}{cc}
+              2C-M_1\omega^2 & -C[1+\exp(-ika)]\\
+             -C[1+\exp(ika)] & 2C-M_2\omega^2
+             \end{array}
+             \right|=0$\\
+            $\Rightarrow
+             M_1M_2\omega^4-2C(M_1+M_2)\omega^2+2C^2(1-\cos(ka))=0$
+      \end{itemize}
+\item \[
+      \omega^2=C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)\pm
+               C\sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2-
+                     \frac{2(1-\cos(ka))}{M_1M_2}}
+      \]
+      \begin{itemize}
+       \item $ka\ll 1$:\\
+             $\rightarrow \cos(ka)\approx 1-\frac{1}{2}k^2a^2$\\
+            Optical branch: $\omega^2\approx
+                             2C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)$\\
+            Acoustic branch: $\omega^2\approx
+                              \frac{C/2}{M_1+M_2}k^2a^2$\\
+       \item $k=0$:\\
+             Optical branch: $u/v = - M_2/M_1$ (out of phase)\\
+       \item $k=\pm \pi/a$:\\
+            $\rightarrow \omega^2=2C/M_2,2C/M_1$
+      \end{itemize}
 \end{enumerate}
 
 \end{document}