From e1baa24b5ae38fb8f0e1e09dc6aa30d99f1fedba Mon Sep 17 00:00:00 2001 From: hackbard Date: Tue, 11 Dec 2007 17:51:37 +0100 Subject: [PATCH] somethiung new .. --- solid_state_physics/tutorial/1_02s.tex | 18 ++++++++++++------ 1 file changed, 12 insertions(+), 6 deletions(-) diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex index 6f920e8..a3e9b93 100644 --- a/solid_state_physics/tutorial/1_02s.tex +++ b/solid_state_physics/tutorial/1_02s.tex @@ -143,14 +143,20 @@ $\Rightarrow M_1M_2\omega^4-2C(M_1+M_2)\omega^2+2C^2(1-\cos(ka))=0$ \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 -- 2.20.1