\section{Problems}

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

\item Explain that the Coulomb interaction between positive and negative ions in an
ionic crystal changes the force constant for the LO phonon mode but not for
the iTO phonon mode.

\item When we consider the frequency dependent dielectric constant $\epsilon(\omega)$,
the ratio of the LO to the iTO phonon frequency in an
ionic crystals is given by $$
\frac{\omega_{\rm LO}}{\omega_{\rm iTO}} = \frac{\epsilon(0)}{\epsilon(\infty)}
$$
which was derived by Lyddane, Sachs, and Teller (LST theory).
Study the LST theory and obtain the above formula. Check that the formula works for some
ionic crystals such as NaCl.

\item The above LST relation is considered for ${\bf q} =0$. For a
general ${\bf q}$ vector, we can discuss how the phonon dispersion is
modified by the Coulomb interaction by considering the coupling of
Maxwell's equations to the equation of motions \ for the atoms:
$$
\ddot{\bf x} + \omega_{\rm iTO}^2{\bf x} - a {E_x} = 0,
$$
$$
{P_x} = a {\bf r} + ( \epsilon(\infty) -1 ) {E_x},
$$
where $a$ is given by
$$
a = \omega_{\rm iTO}[\epsilon(0) - \epsilon(\infty)]^{1/2}.
$$
Combining these two equations with  Maxwell's equations
for $D_x = E_x + P_x$ and for $H_x$, and considering
 their wave-vector $q$, obtain and plot $\omega(q)$.

\item When we pull a hexagonal net in one direction, how much
percent of the length of the net can be expanded
compared with the length of an undistorted hexagonal
net. When we rotate the pulling direction relative
to the C-C bond direction, how does the expansion
ratio change? Compare these results with the case
of a square or triangular net.

\item Consider two carbon atoms which are connected to each other by a spring.
Evaluate the force constant in units of
eV/\AA$^2$ (use the 1580\,cm$^{-1}$ LO phonon mode
frequency for sp$^2$ carbons).

\item Consider a $2\pi/3$ rotation around a carbon atom in the plane
of graphene.  Make a 3x3 rotation matrix $C(2\pi/3)$ for the $2\pi/3$
rotation around the $z$ axis.

\item Let us consider a function $f = ax + by + cz + d$. When a vector
$(x,y,z)$ is transformed by this $2\pi/3$ rotation into $(x',y',z')=
C(2\pi/3)(x,y,z)$ and suppose that $f$ is invariant for the $2\pi/3$
rotation, then show $a=b=c=0$.

\item Let us consider a function $g = ax^2 + by^2 + cz^2 + dyz + ezx +
 fxy$.  When a vector $(x,y,z)$ is transformed by the $2\pi/3$
 rotation as $(x',y',z')= C(2\pi/3)(x,y,z)$ and suppose that $g$ is
 invariant for the $2\pi/3$ rotation, what are the conditions imposed
 for the constants $a, b, c, d, e, f$.

\item A second rank tensor can be defined as a 3x3 matrix $a_{ij}$
in which $a$ is transformed by
any invariant operation $(x',y',z')= C(x,y,z)$, where
$C$ is a matrix that transforms into $a' = C^{-1}aC$. Because the
determinant of $a$ should not be changed by $C$,
show that det($C$)=1 and that $a_{ij}=a_{ji}$.
Thus a symmetric second rank tensor has 6 independent
components.

\item When we consider a fourth-rank tensor $K_{iklm}$ in three dimension
($i,k,l,m = 1, 2, 3$), 81 possible variables
exist. However, because of the relationship of
Eq.\,\ref{eq:Kiklm}, $K_{iklm}$ has only 21
independent variables even though there is no
symmetry for the system. Explain this statement.

\item What is the relationship between the LO and iTO phonon mode
 frequencies with the Young modulus and other force constants?

\item In the $C_N$ symmetry, show that only $A$, $E_1$, and $E_2$
phonon modes are Raman active.  If we plot the one-dimensional
Brillouin zone (cutting lines), which lines in the 2D Brillouin zone
correspond to the Raman active modes? Answer by constructing a figure.

\item Plot the $X$, $Y$, $Z$ axes and show the scattering geometries
with the corresponding resonance conditions for carbon nanotubes when
we put a carbon nanotube with its axis along the $Z$ axis and when the
light is coming from the $Y$ (or $Z$) direction.

\item The curvature of a nanotube is considered to have a constant
strain in one direction.  Then estimate the change in the iTO and LO
phonon frequencies.

\item Demonstrate equations \ref{eq-gruneisner2} and \ref{eq-beta2}.

\item Apply the theory developed in \S\ref{s-7yoyo} for carbon nanotubes,
and obtain the effect of strain on the transverse and longitudinal modes.
You can use Ref.\cite{reich00} as a guide.

\item Study the difference between Eqs.\,\ref{f-Adt} and \ref{eq-Gbandyoyo}.
Can these two equations be consistent?

\item Make a study of the Poison ratio and the Gr\"uneisen parameter for
graphene, graphite and carbon nanotubes.

\end{enumerate}
