X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=solid_state_physics%2Ftutorial%2F1_02s.tex;h=240d84e2e08560697271ac469ace514145e182db;hp=6f920e8bcfc9f7f56e909f52db0231d4658f45e4;hb=557ae9de164e6994e728188149e93a7f8380cc34;hpb=79c30f13ba31a3d5988133db6e6992235b64ca8a diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex index 6f920e8..240d84e 100644 --- a/solid_state_physics/tutorial/1_02s.tex +++ b/solid_state_physics/tutorial/1_02s.tex @@ -16,6 +16,7 @@ \usepackage{pstricks} \usepackage{pst-node} \usepackage{rotating} +\usepackage{eepic} \setlength{\headheight}{0mm} \setlength{\headsep}{0mm} \setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm} @@ -143,21 +144,104 @@ $\Rightarrow M_1M_2\omega^4-2C(M_1+M_2)\omega^2+2C^2(1-\cos(ka))=0$ \end{itemize} -\item \[ - \omega^2=C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)\pm - C\sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2- - \frac{2(1-\cos(ka))}{M_1M_2}} - \] +\newpage +\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) + \stackrel{{\color{red}+}}{{\color{blue}-}} + 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{figure}[!h] + +% GNUPLOT: LaTeX picture using EEPIC macros +\setlength{\unitlength}{0.130450pt} +\begin{picture}(3000,1800)(0,0) +\footnotesize +\color{black} +\color{black} +\thicklines \path(681,1718)(681,82)(2317,82)(2317,1718)(681,1718) +\color{black} +\put(681,900){\makebox(0,0)[l]{\shortstack{}}} +\color{black} +\color{black} +\put(2398,900){\makebox(0,0)[l]{\shortstack{}}} +\color{black} +\color{black} +\put(1499,41){\makebox(0,0){}} +\color{black} 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\path(1491,534)(1507,542)(1524,550)(1540,558)(1557,566)(1573,574)(1590,582)(1606,589)(1623,597)(1639,604)(1656,612)(1673,619)(1689,627)(1706,634)(1722,641)(1739,648)(1755,655)(1772,661)(1788,668)(1805,674)(1821,681)(1838,687)(1854,693)(1871,699)(1887,705)(1904,710)(1920,716)(1937,721)(1953,726)(1970,731)(1986,736)(2003,741)(2020,745)(2036,749)(2053,753)(2069,757)(2086,761)(2102,764)(2119,767)(2135,770)(2152,773)(2168,775)(2185,778)(2201,780)(2218,781)(2234,783)(2251,784)(2267,785)(2284,785)(2300,786)(2317,786) +\color{black} +\thicklines \path(681,1718)(681,82)(2317,82)(2317,1718)(681,1718) +\color{black} +\end{picture} + + \end{figure} \begin{itemize} \item $ka\ll 1$:\\ - $\rightarrow \cos(ka)\approx 1-\frac{1}{2}k^2a^2$\\ - Optical branch: $\omega^2\approx - 2C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)$\\ - Acoustic branch: $\omega^2\approx - \frac{C/2}{M_1+M_2}k^2a^2$\\ + $\rightarrow \cos(ka)\approx 1-\frac{1}{2}k^2a^2$ (Taylor)\\ + $\Rightarrow$\\ + $\sqrt{(\frac{1}{M_1}+\frac{1}{M_2})^2- + \frac{k^2a^2}{M_1M_2}}=$ + $(\frac{1}{M_1}+\frac{1}{M_2}) + \sqrt{1-\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2}} + \stackrel{Taylor}{\approx} + (\frac{1}{M_1}+\frac{1}{M_2}) + (1-\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2})$\\ + $\omega \approx \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})} + \sqrt{1\pm (1-\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2})}$\\ + $\stackrel{{\color{red}+}}{\rightarrow} + \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})} + \sqrt{2-\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2}} + \stackrel{Taylor}{\approx} + \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})}\sqrt{2} + (1-\frac{1}{2}\frac{1}{4}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2})$\\ + $\stackrel{{\color{blue}-}}{\rightarrow} + \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})} + \sqrt{\frac{1}{2}\frac{k^2a^2}{M_1M_2(1/M_1+1/M_2)^2}}= + \sqrt{C(\frac{1}{M_1}+\frac{1}{M_2})} + \sqrt{\frac{1}{2}\frac{1}{M_1M_2(1/M_1+1/M_2)^2}}ka$\\ + {\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 $k=0$:\\ - Optical branch: $u/v = - M_2/M_1$ (out of phase)\\ - \item $k=\pm \pi/a$:\\ + $\rightarrow u/v = - M_2/M_1$ (out of phase)\\ + \item $k=\pi/a$\\ $\rightarrow \omega^2=2C/M_2,2C/M_1$ \end{itemize} \end{enumerate}