\section*{Problems}

\begin{enumerate}
\renewcommand{\labelenumi}{[\arabic{chapter}-\arabic{enumi}] }

\item Obtain the density of graphite in kgm$^{-3}$. Here we can use
      the in-plane C-C distance which is 1.42\AA \,and the interlayer
      distance between two graphene layers which is 3.35\AA\,
 and note that about
      2\% of the carbon atoms in graphite are $^{13}$C and the
      remaining 98\% are $^{12}$C. For this problem we need at least
      three digits in numerical accuracy.

\item When the radius
      of a zigzag SWNT is modified by $\delta R$, determined by
      how much the C-C distance is modified along the
      circumferential direction.

\item The Young's modulus of graphene is $Y = 1060$ GPa. Obtain the
      sound velocity of the LA phonon mode by calculating the nanotube
      density.

\item The Young's modulus of SiC, Fe and diamond are, respectively, $Y =
      450$, 200, 1200 GPa.  Obtain sound velocities for these
      materials.

\item Obtain the sound velocities for the ideal air in units of m/s
and km/h.

\item Obtain the formula for the sound velocity for the TA phonon mode $C_t
= \sqrt{G/\rho}$, where $G$, the shear modulus, is given by $G = Y/(2(1 +
\nu))$, where $Y$ and $\nu$ are the Young's modulus and Poisson ratio,
respectively.

\item Evaluate $A$ in Eq.\,(\ref{eq-elasticity3}) in cm$^{-1}$ units.

\item Check that all terms in Eq.\,(\ref{eq1}) are dimensionless.

\item Obtain Eq.\,(\ref{eq2}) from Eq.\,(\ref{eq1}). Note that a
factor $1/2\pi c$ appears when we measure $\omega$ in units of cm$^{-1}$.
Using the known values for the various factors, get the value of 227 for
$\omega_{\rm RBM}$ in units of cm$^{-1}$ for $d_t$ = 1\,nm.

\item Calculate the shift of $\omega_{\rm RBM}$ in Eq.\,(\ref{eq3})
for $d_t$=1\,nm and 2nm.

\item Estimate $C_e$ in Eq.\,(\ref{ch-RBMenv-simple}) and the
diameter at which the correction term $C_ed_t^2$ becomes 0.21.

\item Explain that the anharmonic term in the vibrational Hamiltonian
gives a finite lifetime to the phonon which is responsible for the
Raman spectral width.

\item Using the uncertainty relation between energy and time,
obtain the phonon lifetime for Raman spectra with a spectral
width of 1\,cm$^{-1}$ and 10\,cm$^{-1}$.

\item Evaluate the electric field in V/m for a laser power of 1mW/$\mu$m$^2$.

\item For T=300K and 77K, what is the intensity ratio of the Stokes to
 anti-Stokes {\it non-resonant} Raman signals for a 173.6\,cm$^{-1}$ RBM phonon? How about
for  a 1590\,cm$^{-1}$ G band?

\item Measure the resonance window values from Fig.\,\ref{f-SAStunablefit}
and estimate the lifetime of
photo-excited carriers. Explain by giving some reasons for which lifetime is
shorter, the lifetime of the photo-excited carriers or
the life time of phonons.

\item For 785\,nm laser light, obtain the wavelength in nm for Stokes
and anti-Stokes scattered light for a 173.6\,cm $^{-1}$ RBM phonon.

\item For 785\,nm laser light, what is the $E_{ii}$ energy of the scattered
light resonance conditions for the Stokes and anti-Stokes
Raman spectra for a 173.6\,cm $^{-1}$ RBM phonon?

\item Give the expected intensity ratio $I_{S}/I_{AS}$ between the
Stokes and anti-Stokes RBM signals shown in
Fig.\,\ref{f-SAStunablefit} for $E_{\rm laser} = 1.63$\,eV, $E_{\rm
laser} = 1.65$\,eV and $E_{\rm laser} = 1.67$\,eV. Consider both $T =
0$\,K and $T = 300$\,K.

\item Explain why the intensity ratio of the Stokes to anti-Stokes
resonance Raman intensity for one laser energy does not give the
temperature of the sample. For the non-resonance Raman intensity, on
the other hand, we may get the information needed to determine the
temperature. Why?

\item Consider that the spectra in Fig.\,\ref{fig1-8.5} were obtained
from a SWNT with $\gamma_{\rm RBM} = 8$\,meV. Using Eq.\,(\ref{eq8.1}), find
the value of $E_{ii}$ which gives the observed $I_{S}/I_{AS}$.

\item Build your own Kataura plot using Eq.\,(\ref{eq1-f}).
Evaluate  the $E_{22}^{S}$ and $E_{33}^{S}$ energies
 for  the (6,5),\, (11,1),\, (10,5) SWNTs by using Eq.\,(\ref{eq1-f}).

\item There are two definitions for the type of semiconducting SWNTs;
one is Type I and Type II using (${\rm mod}(2n+m,3)=1$ and =2), and the
other is Mod 1 and 2 using ${\rm mod}(n-m,3) = 1, 2$. Show that Type I and
Mod 2 (or Type II and Mod 1) are equivalent to each other.

\item In the $(n,m)$ map of SWNTs, show that SWNTs with $2n+m=$const
have a similar diameter while SWNTs with $n-m=$const have a similar
chiral angle. Explain that the $2n+m=$const family is suitable for
studying the chiral angle dependence while that $n-m=$const family is
suitable for studying the diameter dependence.



\end{enumerate}
