From: hackbard Date: Thu, 15 Nov 2007 22:37:09 +0000 (+0100) Subject: more solution X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=b0536278b959f12a42c5856b21cb5d0502ae74d3;p=lectures%2Flatex.git more solution --- diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex index 8eb5881..e782f74 100644 --- a/solid_state_physics/tutorial/1_02s.tex +++ b/solid_state_physics/tutorial/1_02s.tex @@ -108,9 +108,49 @@ \section{Phonons 2} \begin{enumerate} -\item Derive the dispersion relation for a linear chain with two different - alternating types of atoms. -\item Discuss the two solutions for $\omega^2$. +\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:\\ + $-\omega^2M_1u=Cv[1+\exp(-ika)]-2Cu$\\ + $-\omega^2M_2v=Cu[\exp(ika)+1]-2Cv$\\ + Non trivial solution only if determinant of coefficients + $u$ and $v$ is zero.\\ + $\Rightarrow + \left| + \begin{array}{cc} + 2C-M_1\omega^2 & -C[1+\exp(-ika)]\\ + -C[1+\exp(ika)] & 2C-M_2\omega^2 + \end{array} + \right|=0$\\ + $\Rightarrow + M_1M_2\omega^4-2C(M_1+M_2)\omega^2+2C^2(1-\cos(ka))=0$ + \end{itemize} +\item \[ + \omega^2=C\left(\frac{1}{M_1}+\frac{1}{M_2}\right)\pm + C\sqrt{\left(\frac{1}{M_1}+\frac{1}{M_2}\right)^2- + \frac{2(1-\cos(ka))}{M_1M_2}} + \] + \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: $\omega^2\approx + \frac{C/2}{M_1+M_2}k^2a^2$\\ + \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} \end{enumerate} \end{document}