fixes + tutorial 4

[lectures/latex.git] / solid_state_physics / tutorial / 1_02s.tex

diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex
index 2827bb7..6f920e8 100644 (file)
--- a/solid_state_physics/tutorial/1_02s.tex
+++ b/solid_state_physics/tutorial/1_02s.tex
@@ -34,7 +34,7 @@
  Prof. B. Stritzker\\
  WS 2007/08\\
  \vspace{8pt}
- {\Large\bf Tutorial 2}
+ {\Large\bf Tutorial 2 - proposed solutions}
 \end{center}
 
 \section{Phonons 1}
@@ -117,11 +117,15 @@
              $M_1\ddot{u}_s=C(v_s+v_{s-1}-2u_s)$\\ 
              $M_2\ddot{v}_s=C(u_{s+1}+u_s-2v_s)$
        \item Ansatz:\\
-             $u_s=u\exp{i(ska-\omega t)}$\\
-            $v_s=v\exp{i(ska-\omega t)}$
+             $u_s=u\exp(i(ska-\omega t))$\\
+            $v_s=v\exp(i(ska-\omega t))$
        \item Solution of the equation system:\\
-             $-\omega^2M_1u=Cv[1+\exp(-ika)]-2Cu$\\
-             $-\omega^2M_2v=Cu[\exp(ika)+1]-2Cv$\\
+             $-\omega^2M_1u\exp(i(ska-\omega t))=
+            C\exp(-i\omega t)[v\exp(iska)+v\exp(i(s-1)ka)-2u\exp(iska)]$\\
+            $\Rightarrow -\omega^2M_1u=Cv(1+\exp(-ika))-2Cu$\\
+            $-\omega^2M_2v\exp(i(ska-\omega t))=
+            C\exp(-i\omega t)[u\exp(i(s+1)ka)+u\exp(iska)-2v\exp(iska)]$\\
+             $\Rightarrow -\omega^2M_2v=Cu[\exp(ika)+1]-2Cv$\\
             Non trivial solution only if determinant of coefficients
             $u$ and $v$ is zero.\\
             $\Rightarrow
@@ -131,6 +135,11 @@
              -C[1+\exp(ika)] & 2C-M_2\omega^2
              \end{array}
              \right|=0$\\
+            $\Rightarrow
+             4C^2+M_1M_2\omega^4-2C\omega^2(M_2+M_1)-
+             \underbrace{C^2(1+\exp(ika))(1+\exp(-ika))}_{
+             C^2(\underbrace{1+1+\exp(ika)+\exp(-ika)}_{
+                             2+2\cos(ka)=2(1+\cos(ka))})}$\\
             $\Rightarrow
              M_1M_2\omega^4-2C(M_1+M_2)\omega^2+2C^2(1-\cos(ka))=0$
       \end{itemize}