From 557ae9de164e6994e728188149e93a7f8380cc34 Mon Sep 17 00:00:00 2001 From: hackbard Date: Thu, 13 Dec 2007 18:30:51 +0100 Subject: [PATCH] final .. --- solid_state_physics/tutorial/1_02s.tex | 32 ++++++++++++++++++++++---- 1 file changed, 27 insertions(+), 5 deletions(-) diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex index ccb4a26..240d84e 100644 --- a/solid_state_physics/tutorial/1_02s.tex +++ b/solid_state_physics/tutorial/1_02s.tex @@ -160,7 +160,7 @@ \begin{figure}[!h] % GNUPLOT: LaTeX picture using EEPIC macros -\setlength{\unitlength}{0.120450pt} +\setlength{\unitlength}{0.130450pt} \begin{picture}(3000,1800)(0,0) \footnotesize \color{black} @@ -213,10 +213,32 @@ \begin{itemize} \item $ka\ll 1$:\\ $\rightarrow \cos(ka)\approx 1-\frac{1}{2}k^2a^2$ (Taylor)\\ - Optical branch: $\omega^2\approx - 2C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)$\\ - Acoustic branch: $\omega^2\approx - \frac{C/2}{M_1+M_2}k^2a^2$\\ + $\Rightarrow$\\ + $\sqrt{(\frac{1}{M_1}+\frac{1}{M_2})^2- + \frac{k^2a^2}{M_1M_2}}=$ + $(\frac{1}{M_1}+\frac{1}{M_2}) + \sqrt{1-\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2}} + \stackrel{Taylor}{\approx} + (\frac{1}{M_1}+\frac{1}{M_2}) + (1-\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2})$\\ + $\omega \approx \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})} + \sqrt{1\pm (1-\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2})}$\\ + $\stackrel{{\color{red}+}}{\rightarrow} + \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})} + \sqrt{2-\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2}} + \stackrel{Taylor}{\approx} + \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})}\sqrt{2} + (1-\frac{1}{2}\frac{1}{4}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2})$\\ + $\stackrel{{\color{blue}-}}{\rightarrow} + \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})} + \sqrt{\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2}}= + \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})} + \sqrt{\frac{1}{2}\frac{1}{M_1M_2(1/M_1+1/M_2)^2}}ka$\\ + {\color{red}Optical branch}: $\omega\stackrel{ka\ll 1}{\approx} + \sqrt{2C\left(\frac{1}{M_1}+ + \frac{1}{M_2}\right)}$\\ + {\color{blue}Acoustic branch}: $\omega\stackrel{ka\ll 1}{\approx} + \sqrt{\frac{C/2}{M_1+M_2}}ka$\\ \item $k=0$:\\ $\rightarrow u/v = - M_2/M_1$ (out of phase)\\ \item $k=\pi/a$\\ -- 2.20.1