X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=solid_state_physics%2Ftutorial%2F1_02s.tex;h=6f920e8bcfc9f7f56e909f52db0231d4658f45e4;hb=79c30f13ba31a3d5988133db6e6992235b64ca8a;hp=e782f746964382446eed2d49ff7a762e4c0ef3bb;hpb=b0536278b959f12a42c5856b21cb5d0502ae74d3;p=lectures%2Flatex.git

diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex
index e782f74..6f920e8 100644
--- 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}
@@ -80,7 +80,7 @@
 	\item $\sigma_{\perp} = \alpha \rho_0$, $\alpha \ll 1$\\
 	      $\sqrt{\rho_0^2+\sigma_{\perp}^2}=
 	       \sqrt{\rho_0^2+\alpha^2\rho_0^2}=
-	       \rho_0\sqrt{1+\alpha^2}=
+	       \rho_0\sqrt{1+\alpha^2}\stackrel{Taylor}{=}
 	       \rho_0(1+\frac{\alpha^2}{2}-\frac{\alpha^4}{8}+\ldots)$\\
 	      $\Rightarrow \Phi-\Phi_0=
 	       \frac{D}{2}\left[\rho_0^2\left(2+\alpha^2-
@@ -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}