%<PLAINTEXT>
\question
{Kouki Yonaga}                  % Enter your name
{12/7/11}                        % Enter mm/dd/yy
{yona@cmpt.phys.tohoku.ac.jp}   % Enter e-mail address
{3-15}                          % Enter the number and question.
%question
{How many normal modes exist for the ${\rm CH_{4}}$, ${\rm C_{2}H_{2}}$ and ${\rm C_{60}}$ molecules which, respectively, have the shapes of a regular tetrahedron, a linear
chain, and a truncated icosahedron?}
%answer
{
The molecular vibration $M$ is defined by
\begin{equation}
 M = 3N - d,
 \label{Q3-15eq:mode}
\end{equation}
where $N$ and $d$ are, respectively, the number of atoms in the molucle and the degree of freedom that the molecule has.
When we consider nonlinear molecules, $M$ is written as 
\begin{equation}
 M = 3N - 6
 \label{Q3-15eq:nonlinear},
\end{equation}
since nonlinear molecules have $3$ translations and $3$ rotations. When we consider linear molecules, $M$ is written as
\begin{equation}
 M = 3N - 5,
 \label{Q3-15eq:linear}
\end{equation}
 since linear molecules have $3$ translations and only $2$ rotations. Because, when we consider linear molecule along $z$ axis, the rotation around the $z$ axis dose not exist. 
Since ${\rm CH_{4}}$ is a nonlinear molecule, from Eq.\,(\ref{Q3-15eq:nonlinear}), the number of the modes of ${\rm CH_{4}}$ are given by
\begin{equation}
 M = 3 \times 5 - 6 = 9.
\end{equation}
Similarly, from Eq.\,(\ref{Q3-15eq:nonlinear}), the modes of ${\rm C_{60}}$ are given by
\begin{equation}
 M = 3 \times 60 - 6 = 174.
\end{equation}
Since ${\rm C_{2}H_{2}}$ is a linear molecule, from Eq.\,(\ref{Q3-15eq:linear}), the number of the modes of ${\rm C_{2}H_{2}}$ are obtained by
\begin{equation}
 M = 3 \times 4 - 5 = 7.
\end{equation}


}