X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=solid_state_physics%2Ftutorial%2F1_02s.tex;h=a3e9b9352ec15efaafd8fe833953963bb8ce9fbb;hb=e1baa24b5ae38fb8f0e1e09dc6aa30d99f1fedba;hp=2827bb7d27dc620326c994cff48099e3ae72c6e4;hpb=32f786010b14f1197f69bddc12e6e35bff697fba;p=lectures%2Flatex.git

diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex
index 2827bb7..a3e9b93 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}
@@ -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.\\
 	     $\Rightarrow
@@ -131,17 +135,28 @@
 	      -C[1+\exp(ika)] & 2C-M_2\omega^2
 	      \end{array}
 	      \right|=0$\\
+	     $\Rightarrow
+	      4C^2+M_1M_2\omega^4-2C\omega^2(M_2+M_1)-
+	      \underbrace{C^2(1+\exp(ika))(1+\exp(-ika))}_{
+	      C^2(\underbrace{1+1+\exp(ika)+\exp(-ika)}_{
+	                      2+2\cos(ka)=2(1+\cos(ka))})}$\\
 	     $\Rightarrow
 	      M_1M_2\omega^4-2C(M_1+M_2)\omega^2+2C^2(1-\cos(ka))=0$
       \end{itemize}
-\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