%
\question
{Riichiro Saito} % Enter your name
{02/14/11} % Enter mm/dd/yy
{rsaito@flex.phys.tohoku.ac.jp} % Enter e-mail address
{1-1} % Enter the number and question.
%question
{The carbon-carbon distance of graphene (see Fig.\,1.1)
is 1.42\AA. How much area is occupied by a single
carbon atom in the graphene plane?}
%answer
{
% if you want to include figure.
Graphene is a two-dimensional hexagonal lattice.
Each hexagonal ring consists of six carbon atoms
and each carbon atom is shared by three hexagons.
Thus there are two carbon atoms per a hexagonal
ring as shown in Fig.\,(\ref{fig:Q1-1-1}).
\begin{figure}[h]
\begin{center}
\includegraphics[width=6cm,clip]{Q1-1.eps}\\
\begin{minipage}{10cm}
\caption{Each hexagon has 6 carbon atoms and each carbon atom is shared by three hexagon. Thus there are 2 carbon atoms in the hexagon.
\label{fig:Q1-1-1}
}
\end{minipage}
\end{center}
\end{figure}
The area of the hexagon corresponds to six times of the regular triangle whose edge length is $a_{\rm C-C}$. Thus the area occupied by a single carbon atom isthree times of the triangle which is given by
\begin{equation}
3 \times \frac{\sqrt{3}}{4} a_{\rm C-C}^2 = 2.62 {\rm \AA}^2 = 2.62 \times 10^{-20} {\rm m}^2
\label{eq:Q1-1-1}
\end{equation}
Here we used the value $a_{\rm C-C}$=1.42\AA\, in Eq.\,(\ref{eq:Q1-1-1}).
}
% Do not forget this.