+\item \begin{itemize}
+ \item Convention:\\
+ Atom type 1: $M_1$, $u_s$ (elongation of atom $s$ of type 1)\\
+ Atom type 2: $M_2$, $v_s$ (elongation of atom $s$ of type 2)\\
+ Lattice constant: $a$, Spring constant: $C$
+ \item Equations of motion:\\
+ $M_1\ddot{u}_s=C(v_s+v_{s-1}-2u_s)$\\
+ $M_2\ddot{v}_s=C(u_{s+1}+u_s-2v_s)$
+ \item Ansatz:\\
+ $u_s=u\exp(i(ska-\omega t))$\\
+ $v_s=v\exp(i(ska-\omega t))$
+ \item Solution of the equation system:\\
+ $-\omega^2M_1u\exp(i(ska-\omega t))=
+ C\exp(-i\omega t)[v\exp(iska)+v\exp(i(s-1)ka)-2u\exp(iska)]$\\
+ $\Rightarrow -\omega^2M_1u=Cv(1+\exp(-ika))-2Cu$\\
+ $-\omega^2M_2v\exp(i(ska-\omega t))=
+ C\exp(-i\omega t)[u\exp(i(s+1)ka)+u\exp(iska)-2v\exp(iska)]$\\
+ $\Rightarrow -\omega^2M_2v=Cu[\exp(ika)+1]-2Cv$\\
+ Non trivial solution only if determinant of coefficients
+ $u$ and $v$ is zero.\\
+ $\Rightarrow
+ \left|
+ \begin{array}{cc}
+ 2C-M_1\omega^2 & -C[1+\exp(-ika)]\\
+ -C[1+\exp(ika)] & 2C-M_2\omega^2
+ \end{array}
+ \right|=0$\\
+ $\Rightarrow
+ 4C^2+M_1M_2\omega^4-2C\omega^2(M_2+M_1)-
+ \underbrace{C^2(1+\exp(ika))(1+\exp(-ika))}_{
+ C^2(\underbrace{1+1+\exp(ika)+\exp(-ika)}_{
+ 2+2\cos(ka)=2(1+\cos(ka))})}$\\
+ $\Rightarrow
+ M_1M_2\omega^4-2C(M_1+M_2)\omega^2+2C^2(1-\cos(ka))=0$
+ \end{itemize}
+\item \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 $ka\ll 1$:\\
+ $\rightarrow \cos(ka)\approx 1-\frac{1}{2}k^2a^2$ (Taylor)\\
+ Optical branch: $\omega^2\approx
+ 2C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)$\\
+ Acoustic branch: $\omega^2\approx
+ \frac{C/2}{M_1+M_2}k^2a^2$\\
+ \item $k=0$:\\
+ Optical branch: $u/v = - M_2/M_1$ (out of phase)\\
+ \item $k=\pm \pi/a$:\\
+ $\rightarrow \omega^2=2C/M_2,2C/M_1$
+ \end{itemize}