%<PLAINTEXT>
\question
{Kouki Yonaga}                  % Enter your name
{12/7/11}                        % Enter mm/dd/yy
{yona@cmpt.phys.tohoku.ac.jp}   % Enter e-mail address
{3-8}                          % Enter the number and question.
%question
{
In Figure\,3.3, choose any unit cell vibrational mode and show that the motion of the atoms for $q = 0\,(\lambda \rightarrow \infty)$ and for $q = 2\pi /a\,(\lambda = a)$
is the same.
}
%answer
{Let us consider 1D crystal system that has a $2N$ linear chain of atoms in which two different atoms A and B in the unit cell. If we use the Broch theorem, motions of two different atoms are in $j-$th unit cell written as
\begin{equation}
   x_{j}^{i}(q) = A^{i}\exp(iqja), \ \ \ (i = {\rm A, B,\ and}\ j = 1, 2, ..., N ), 
 \label{Q3-8eq:motion}
\end{equation}
where $x_{j}^{i}(q)$, $a$, $q$ and $A^{i}$ are, respectively, the displacement of A\,(B) atom, lattice constant, wave numbers and amplitude of A\,(B) atom. The first Brillouin zone is given by
\begin{equation}
 -\frac{\pi}{a} < q < \frac{\pi}{a},
\end{equation}
which is defined by the smallest basic region in $k-$space. When we consider $q = 0$ and $q = 2\pi /a$, $x_{j}^{i}(q)$ is obtained by
\begin{eqnarray}
 x_{j}^{i}(0) &=& A^{i}\exp(0) = A^{i}, \nonumber \\
 x_{j}^{i}(2\pi /a) &=& A^{i}\exp(2\pi j) = A^{i}. 
\end{eqnarray}
 Therefore, at $q = 0$ and $q = 2\pi /a$, the motion of atoms is the same (Fig.(\ref{Q3-8fig:B.Z.})).

\begin{figure}[htbp]
 \begin{center}
   \includegraphics[width=10cm, clip]{Q3-8.eps} \\
 \begin{minipage}{15cm}
  \caption{Schematic phonon dispersion relation of two atoms in the unit cell in the first Brillouin zone\,(black) and the second Brillouin zone\,(red).} 
\label{Q3-8fig:B.Z.}
\end{minipage} 
\end{center}
\end{figure}
}