
\section*{Problems}

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

\item The carbon-carbon distance of graphene (see Fig.\,\ref{fig-carbon})
is 1.42\AA. How much area is occupied by a single
carbon atom in the graphene plane?

\item The inter-layer distance in multi-layer graphene or graphite
 (see Fig.\,\ref{fig-allsp2} (b)) is 3.35\AA. How much volume is
 occupied by a single carbon atom in graphite? From this information,
 estimate the density of graphite in g/cm$^3$. Compare your estimate
 with the literature value of 2.25 g/cm$^3$.

\item Figure\,\ref{fig-allsp2}(b) shows the AB stacking of
graphene layers in forming graphite.
Explain how two graphene layers are stacked in the
AB stacking sequence.

\item There are several ways to stack graphene layers. When we put a
third layer on the two graphene layers, there are two possible ways of
doing this stacking, which we call ABAB and ABC stacking. Show a
graphic picture of both ABAB and ABC stacking and explain your answer
in words, including the relation between the location of carbon
atoms in each relevant plane.

\item C$_{60}$ molecules form  face-centered cubic structures.
The density of the C$_{60}$ crystal is
1.72\,g/cm$^3$. From this value, estimate the C$_{60}$-C$_{60}$ distance
and the fcc lattice constant.

\item Each carbon atom in a C$_{60}$ molecule has one pentagonal
and two hexagonal rings. Calculate the angles (a)
between the two hexagonal rings and (b) between the
hexagonal ring and the pentagonal ring.

\item Diamond crystallizes in a cubic diamond structure.
The C-C distance in diamond is 1.544\AA. Estimate the
cube edge length and density in g/cm$^3$.

\item Diamond has four ($sp^3$)  chemical bonds in which all bond angles
 for any pair of chemical bonds are
 identical. Calculate the bond angle between two
 chemical bonds by using an analytical solution and
also give the numerical value in degrees.

\item Raman spectroscopy involves the inelastic light scattering
 process.  Part of the energy of the incident light is lost or gained,
 respectively, in materials in which some elemental excitation
 such as an atomic vibration (phonon) absorbs or releases the energy
 from or to the light. We call these two Raman processes
Stokes and anti-Stokes processes, respectively.
When light with the wavelength 632.8nm is incident on the
 sample and loses energy by creating a phonon with an energy of
 0.2eV, what is the scattered wavelength? Also give the scattered
 wavelength for the anti-Stokes Raman signal.

\item In spectroscopy, a wavevector is defined by $1/\lambda$
(where $\lambda$ is the wavelength) while in solid state physics, the
definition of a wavevector is $2\pi/\lambda$. Show that a 1eV photon
corresponds to  8065\,cm$^{-1}$ (wavenumbers).  In Raman spectroscopy,
the difference between the wavevectors for the incident and
scattered light is called the Raman shift whose units are generally given in
cm$^{-1}$.

\item Consider the optical electric field of the incident light with
an angular frequency $\omega_0 = 2\pi \nu_0$ and amplitude $E_0$,
$$
E = E_0 \cos \omega_0 t .
$$
Then the dipole moment $P$ of a diatomic molecule is proportional to $E$
such that $P = \alpha E$, in which $\alpha$ is called the polarizability.
When the molecule is vibrating with a frequency $\omega$, then $\alpha$
is also vibrating with the frequency $\omega$,
$$
\alpha = \alpha_0 + \alpha_1 \cos \omega t .
$$ When substituting $\alpha$ into the formula $P
= \alpha E$, show that there are three different
frequencies for the scattered light (or $P$),
$\omega_0$ (elastic, Rayleigh scattering) and
$\omega_0 \pm \omega$ (inelastic, Stokes (-) and
anti-Stokes (+) Raman scattering).

\item Let us consider a resonance effect. Here we consider a particle
 with a mass $m$ which is connected to a system by a spring
 with spring constant K. When we apply an
 oscillatory force $f\exp(i\omega t)$, the equation
 of motion for the amplitude $u$ of the vibration is
$$
m \ddot{u} + K u = f\exp(i\omega t) .
$$ Solve this differential equation and plot $u$ after a sufficiently
long time as a function of $\omega$. Show that a singularity occurs
when $\omega = \omega_0 \equiv \sqrt{K/m}$ and discuss the
significance of this singularity.

\item In a more realistic model than the previous model,
 we can consider the friction term $\gamma\dot{u}$ in
the vibration
$$
m \ddot{u} + \gamma\dot{u} + K u = f\exp(i\omega t).
$$
Plot $u$ as a function of $\omega$ for this case. Show that now a singularity no longer occurs.
How does $\gamma$ appear in the plot of $\omega(u)$? Consider the limits of weak
damping and strong damping and find what determines the transition between these limits.

\end{enumerate}
