%<PLAINTEXT>
\question
{Kouki Yonaga}                  % Enter your name
{12/7/11}                        % Enter mm/dd/yy
{yona@cmpt.phys.tohoku.ac.jp}   % Enter e-mail address
{3-3}                          % Enter the number and question.
%question
{
 We consider N - 1 atoms attached between the two walls. When we consider $x_0$ and $x_n$ as the coordinates of the two walls, show that all the
 equations of motion are expressed by the same fomula. Then considering that the Bloch theorem applies, substitute $x_{l} = A\exp(iqla - i\omega t)$
 into the equations of motion and obtain the dispersion of the phonon frequency $\omega(q)$. 
 Plot dispersion of the phonon within the first Brillouin zone.
} 
%answer
{
The equations of motion for the displacement of the $l-$th atom is given by
\begin{equation}
 M_{i}\ddot{x_{l}} = -2K(x_{l} - x_{l-1} - x_{l+1}),
 \label{Q3-3eq:equation_of_motion}
\end{equation}
 where $M$ and $K$ are, respectively, the mass and the force constant. Using $x_{l}$, the Eq.\,(\ref{Q3-3eq:equation_of_motion}) can be written as
\begin{equation}
 -M\omega^2x_{l} = -2K\left(1 - \exp(-iqa)-\exp(iqa)\right)x_{l},
 \label{Q3-3eq:equation_omega} 
\end{equation}
where $q$ and $a$ are, respectively, the wave number and the lattice constant.
For the boundary conditions that the end aoms are fixed, Re$(x_{0})$ and Re$(x_{N})$ are zero. When we conseder $t = 0$, Re$(x_{0}) = 0$ is written as
\begin{equation}
 {\rm Re}(A) = 0.
 \label{Q3-3eq:A}
\end{equation}
From Eq.\,(\ref{Q3-3eq:A}), $A$ is purely imaginary number and satisfies Eq.\,(\ref{Q3-3eq:imaginary});
\begin{equation}
 A = iB,
 \label{Q3-3eq:imaginary}
\end{equation}
where B is real number. Simiraly, from Re$(x_{N}) = 0$, the Re$(x_{N})$ is written as 
\begin{eqnarray}
 {\rm Re} (x_{N}) & = & -iB\sin(qna)  \nonumber \\
                  & = & 0 
\label{Q3-3eq:qna}
\end{eqnarray}

  The wave number satisfy Eq.\,(\ref{Q3-3eq:q});
\begin{equation}
 q = \frac{m\pi}{Na} \ \ \ (m = 1, 2, ..., N-1).
 \label{Q3-3eq:q}
\end{equation}

The first Brillouin zone is given by
\begin{equation}
 -\frac{\pi}{a} < q < \frac{\pi}{a}.
 \label{Q3-3eq:B.Z.}
\end{equation}

 
 From Eq.\,(\ref{Q3-3eq:equation_omega}),  $\omega$ is obtaind by

\begin{equation}
 \omega = 2\sqrt{\frac{K}{M}}\mid\sin(\frac{qa}{2})\mid.
 \label{Q3-3eq:omega}
\end{equation} 

In Fig.\,(\ref{Q3-3fig:dispersion}), the Eq.\,(\ref{Q3-3eq:omega}) is plotted within the first Brillouin zone, where $\omega_{0}$ is defined by $2\sqrt{\displaystyle\frac{K}{M}}$.
\begin{figure}[htbp]
 \begin{center}
   \includegraphics[width=6cm, clip]{Q3-3.eps} \\
 \begin{minipage}{15cm}
  \caption{Dispersion of frequency} 
\label{Q3-3fig:dispersion}
\end{minipage} 
\end{center}
\end{figure}

}



