\usepackage{pstricks}
\usepackage{pst-node}
\usepackage{rotating}
+\usepackage{eepic}
\setlength{\headheight}{0mm} \setlength{\headsep}{0mm}
\setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm}
$\Rightarrow
M_1M_2\omega^4-2C(M_1+M_2)\omega^2+2C^2(1-\cos(ka))=0$
\end{itemize}
+\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}-
&=&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\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}
+\color{black}
+\put(1499,1677){\makebox(0,0){}}
+\color{black}
+\put(1499,1676){\makebox(0,0){}}
+\color{black}
+\put(681,124){\makebox(0,0)[l]{}}
+\color{black}
+\color{red}
+\color{black}
+\put(2550,1300){\makebox(0,0)[r]{$\sqrt{\frac{2C}{M_2}}$}}
+\put(2550,800){\makebox(0,0)[r]{$\sqrt{\frac{2C}{M_1}}$}}
+\put(2350,-10){\makebox(0,0)[r]{$\frac{\pi}{a}$}}
+\put(1500,-30){\makebox(0,0)[r]{$k$}}
+\put(700,-10){\makebox(0,0)[r]{$0$}}
+\put(650,1500){\makebox(0,0)[r]{$\sqrt{2C(\frac{1}{M_1}+\frac{1}{M_2})}$}}
+\put(600,800){\makebox(0,0)[r]{$\omega$}}
+\put(1800,1000){\makebox(0,0)[r]{$M_1>M_2$}}
+\put(1989,1636){\makebox(0,0)[r]{optical branch}}
+\color{red}
+\thinlines \path(2030,1636)(2235,1636)
+\thinlines \path(681,1490)(681,1490)(698,1490)(714,1490)(731,1490)(747,1490)(764,1489)(780,1489)(797,1488)(813,1488)(830,1487)(846,1487)(863,1486)(879,1485)(896,1484)(912,1484)(929,1483)(945,1482)(962,1481)(978,1479)(995,1478)(1012,1477)(1028,1476)(1045,1474)(1061,1473)(1078,1471)(1094,1470)(1111,1468)(1127,1466)(1144,1465)(1160,1463)(1177,1461)(1193,1459)(1210,1457)(1226,1455)(1243,1453)(1259,1451)(1276,1448)(1292,1446)(1309,1444)(1325,1442)(1342,1439)(1359,1437)(1375,1434)(1392,1432)(1408,1429)(1425,1427)(1441,1424)(1458,1421)(1474,1418)(1491,1416)
+\thinlines \path(1491,1416)(1507,1413)(1524,1410)(1540,1407)(1557,1404)(1573,1401)(1590,1398)(1606,1395)(1623,1393)(1639,1390)(1656,1387)(1673,1383)(1689,1380)(1706,1377)(1722,1374)(1739,1371)(1755,1368)(1772,1365)(1788,1362)(1805,1359)(1821,1356)(1838,1353)(1854,1351)(1871,1348)(1887,1345)(1904,1342)(1920,1339)(1937,1337)(1953,1334)(1970,1331)(1986,1329)(2003,1326)(2020,1324)(2036,1322)(2053,1320)(2069,1318)(2086,1316)(2102,1314)(2119,1312)(2135,1310)(2152,1309)(2168,1307)(2185,1306)(2201,1305)(2218,1304)(2234,1303)(2251,1303)(2267,1302)(2284,1302)(2300,1301)(2317,1301)
+\color{blue}
+\color{black}
+\put(1989,1553){\makebox(0,0)[r]{acoustic branch}}
+\color{blue}
+\thinlines \path(2030,1553)(2235,1553)
+\thinlines \path(681,82)(681,82)(698,92)(714,101)(731,111)(747,121)(764,130)(780,140)(797,150)(813,159)(830,169)(846,179)(863,188)(879,198)(896,207)(912,217)(929,227)(945,236)(962,246)(978,255)(995,265)(1012,274)(1028,283)(1045,293)(1061,302)(1078,312)(1094,321)(1111,330)(1127,340)(1144,349)(1160,358)(1177,367)(1193,376)(1210,386)(1226,395)(1243,404)(1259,413)(1276,422)(1292,431)(1309,439)(1325,448)(1342,457)(1359,466)(1375,474)(1392,483)(1408,492)(1425,500)(1441,509)(1458,517)(1474,525)(1491,534)
+\thinlines \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 $ka\ll 1$:\\
$\rightarrow \cos(ka)\approx 1-\frac{1}{2}k^2a^2$ (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 $k=0$:\\
- 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}
projects / lectures / latex.git / blobdiff